Solutions Manual for A Course in Modern Macroeconomics Pablo Kuralt

SOLUTION MANUAL

This is a solutions manual for A Course in Modern Macroeconomics. Many exercises admit more than one correct answer; the model answers provided here are just suggestions. If you nd mistakes or have comments, let me know at pkurlat@gmail.com.

3

CHAPTER 1

GDP Exercises 1.1

Accounting How does GDP accounting record the following events?

For each of them, describe how they would

be computed in GDP accounts using the income method, the production method and the expenditure method.

(a) A car manufacturer buys components from Japan for $1 to be used in production later on and stores them at its warehouse.

Production

Income 0

Total

0

Expenditure 0

Total

0

Imports

-1

Investment

1

Total

0

(b) A car manufacturer buys components from Japan for $1 and uses half of those components in the production of a car, that it sells to Andy for $2. It stores the rest of the components.

Production Manufacturing

Income 1.5

Pro ts

Expenditure 1.5

Imports

-1

Investment

0.5

Consumption

Total

1.5

Total

1.5

Total

2

1.5

(c) An army battalion is deployed to the border to repel a threatened Canadian invasion. The soldiers earn wages of $10,000 and use ammunition that the government buys for $5,000. The ammunition is produced using $2,000 of imported steel and 100 hours of work, for which the workers were paid $1,000.

5

Production

Income

Expenditure

Manufacturing

3,000

Wages

11,000

Gov. services

10,000

Pro ts

2,000

Total

13,000

Total

13,000

Imports

-2,000

Gov. Spending

15,000

Total

13,000

(d) Walmart sells 1000 bottles of Coca-Cola for $1,500. It had previously paid $1,200 for them.

Production Retail

Income 300

Total

300

(e) A shipyard builds a cruise ship.

Pro ts

300

Total

300

Consumption

1,500

Investment

-1,200

Total

300

It pays wages of $200,000, interest on loans (from US residents)

of $100,000 and $300,000 for imported raw materials. company.

Expenditure

The ship is sold for $1,000,000 to a cruise

In the same year, the cruise company has revenue for $50,000 from operating cruises,

pays wages of $20,000 to its workers and has no other expenses. Half the cruise revenue comes from tourists who reside in the United States and half comes from tourists who reside abroad.

Production Manufacturing Tourism

Total

Income

Expenditure

700,000

Wages

220,000

Investment

50,000

Interest

100,000

Consumption

25,000

Pro ts

430,000

Exports

25,000

Imports

-300,000

Total

750,000

750,000

Total

750,000

1,000,000

(f ) The government collects $1000 in income taxes from Roger.

Production

Income

Expenditure

0

0

0

(g) Roger earns $4000 for working as a babysitter and pays $1000 in income taxes.

Production Services 1.2

Income 4,000

Wages

Expenditure 4,000

Consumption

4,000

Comparisons Across Time and Across Countries Suppose these are the prices (in US dollars) and quantities of goods A and B produced in the US in 2017 and 2018:

(a) What was nominal GDP in 2017? (b) What was nominal GDP in 2018?

6

pA

qA

pB

qB

2017

4

5

3

3

2018

1

10

4

2

(c) What was real GDP in 2018 at 2017 prices (computed using xed 2017 prices)? Using this measure, how much did GDP grow between 2017 and 2018? (d) What was real GDP in 2017 at 2018 prices (computed using xed 2018 prices)? Using this measure, how much did GDP grow between 2017 and 2018? What explains the di erence between the two measures? (e) How much did GDP grow between 2017 and 2018 using the chain-weighted method?

In Thailand, prices (in Thai baht) and quantities in 2017 were:

2017

pA

qA

pB

qB

30

1

10

2

(f ) What was nominal GDP in Thailand in 2017, expressed in baht. (g) Suppose the exchange rate in 2017 was 25 baht per dollar.

What was GDP in Thailand in 2017,

expressed in US dollars at market exchange rates? (h) What was GDP in Thailand in 2017 at PPP? What accounts for the di erence between the market exchange rate measure and the PPP measure? (i) What was the PPP exchange rate between the baht and the dollar?

Suggested Answer (a)

4 × 5 + 3 × 3 = 29 US dollars.

(b)

1 × 10 + 4 × 2 = 18 US dollars.

(c)

4 × 10 + 3 × 2 = 46 US dollars. GDP grew 46 29 − 1 ≈ 59%.

18 1 × 5 + 4 × 3 = 17 US dollars. GDP grew 17 − 1 ≈ 6%.

46 0.5 18 0.5 (e) GDP grew − 1 ≈ 30% 29 17

(d)

1.3

(f )

30 × 1 + 10 × 2 = 50 Thai baht.

(g)

50 25 = 2 US dollars.

(h)

4 × 1 + 3 × 2 = 10 US dollars.

(i)

50 10 = 5 baht per dollar.

Chained GDP The country of Fructus produces Apples, Bananas and Cherries. Its production statistics are given below:

7

Apples quantity price

Bananas quantity price

Cherries quantity price

2000

100

40

50

30

200

20

2001

103

39

47

35

206

20

2002

107

39

47

38

213

19

2003

109

39

45

38

215

18

2004

112

40

46

41

224

18

2005

112

39

46

39

227

17

2006

114

40

47

36

237

16

2007

115

40

46

42

249

16

2008

119

40

43

44

255

16

2009

118

42

40

46

262

15

2010

116

43

41

44

268

15

2011

118

42

40

50

280

15

2012

117

42

41

50

289

15

2013

119

43

40

53

290

14

2014

120

44

40

56

297

13

2015

125

45

41

59

308

13

2016

128

45

38

66

320

13

2017

129

46

38

61

330

13

2018

131

47

37

64

344

13

2019

136

47

37

61

353

13

2020

143

47

34

61

361

12

(you can download them as an Excel spreadsheet from the book website)

(a) Compute a real GDP series at year-2000 prices using base year prices and using the chained method.

(b) Plot both series and comment on any di erences.

(c) What was the average growth rate according to each method?

Suggested Answer

(a) The various measures of GDP are:

8

Nominal

Real (base 2000)

Growth (initial year)

Growth ( nal year)

Growth (chained)

Real (chained)

2000

9,500

9,500

2001

9,782

9,650

1.6%

1.4%

1.5%

9,640

2002

10,006

9,950

3.0%

3.0%

3.0%

9,929

2003

9,831

10,010

0.4%

0.4%

0.4%

9,968

2004

10,398

10,340

3.2%

3.2%

3.2%

10,289

2005

10,021

10,400

0.5%

0.5%

0.5%

10,342

2006

10,044

10,710

2.9%

2.8%

2.8%

10,636

2007

10,516

10,960

2.0%

1.8%

1.9%

10,838

2008

10,732

11,150

1.2%

1.2%

1.2%

10,968

2009

10,726

11,160

-0.6%

-0.7%

-0.6%

10,899

2010

10,812

11,230

0.5%

0.4%

0.5%

10,950

2011

11,156

11,520

2.1%

2.0%

2.0%

11,169

2012

11,299

11,690

1.3%

1.3%

1.3%

11,313

2013

11,297

11,760

0.4%

0.4%

0.4%

11,361

2014

11,381

11,940

1.2%

1.2%

1.2%

11,500

2015

12,048

12,390

3.7%

3.7%

3.7%

11,923

2016

12,428

12,660

0.9%

0.8%

0.9%

12,024

2017

12,542

12,900

1.4%

1.4%

1.4%

12,194

2018

12,997

13,230

1.7%

1.7%

1.7%

12,399

2019

13,238

13,610

2.7%

2.7%

2.7%

12,736

2020

13,127

13,960

1.9%

1.9%

1.9%

12,976

9,500

(b) Graphically:

9

The chained series shows lower growth than the base-2000 series. The goods whose output is growing more have prices growing the least. When everything is computed at base year prices, then these goods are weighted more and the resulting growth is higher. (c) The average growth rates are 1.9% per year for base-2000 real GDP and 1.6% for chained real GDP. 1.4

Drugs and Prostitution Read the following article:

http://www.economist.com/news/ nance-and-economics/21603073-italys-

inclusion-illicit-activities-its- gures-excites-much-interest-sex.

What do you think?

Should drug pro-

duction and prostitution be included in the calculation of GDP?

Suggested Answer Some possible arguments: (a) These are market transactions and we want to include all market transactions. (b) Both of these activities are valuable and should be included. (c) Both of these activities are harmful or reprehensible and should not be included. (d) What is included and not included in GDP is arbitrary anyway so it doesn't really matter what the rule is as long as it's clear. (e) Traditionally these have not been included so it's better to keep it that way to facilitate historical comparisons.

1.5

Changes in Relative Prices We saw that using an earlier year as the base year to compute real GDP results in a high rate of growth if the sectors that are expanding most are those whose relative price is falling. Can you think of reasons why that should be the case (i.e., economic forces that make the same types of goods become relatively cheaper and be produced in higher quantities)? Can you think of reasons why the opposite should be the case (i.e., economic forces that make the same types of goods become relatively expensive and be produced in higher quantities)?

Suggested Answer One possible reason for a sector to expand while its relative price falls is sectoral technological progress.

As technology in a sector improves, production costs falls, producers lower prices and the quantity rises.

In terms of supply and demand, this would be a technology-driven increase in supply. A reason for the opposite to happen is an increase in demand, driven by changes in tastes or incomes. If demand for a sector's output rises, prices rise, attracting more producers.

10

CHAPTER 2

Beyond GDP Exercises 2.1

Comparing the HDI and the Jones & Klenow Welfare Measure Download the UN data that goes into building the HDI from http://hdr.undp.org/en/data and the Jones & Klenow Data from

http://web.stanford.edu/~chadj/papers.html#rawls. For each country, construct:

(a) Relative GDP: GDP per capita GDP per capita in the US

(b) Welfare-to-GDP:

λ Relative GDP

(c) HDI-to-GDP: HDI Relative GDP and plot a scatterplot of Welfare-to-GDP against HDI-to-GDP. What do each of these ratios measure?

Is the impression we get from the Jones & Klenow measure very di erent from the one we get from the HDI? What countries look better under each measure and why?

Suggested Answer Here is the gure: 11

Each of the ratios measures whether a country compares more favorably with the US using GDP per capita or each of the alternative measures. Overall, the two ratios are negatively related. Poor countries tend to look better under HDI than under GDP per capita because, given the way the components of the index are scaled, they are not so low in life expectancy and education as they are in GDP per capita.

For instance, Ethiopia's GDP per capita is only 3.2% of the US level but its life expectancy is 66 years so it receives a relatively high score in this category. As a result, its HDI is 0.46. The Jones & Klenow measure, instead of saying its GDP per capita is low but its life expectancy is not so bad , says: its GDP per capita is low and on top of that, its life expectancy is lower than in the US , so the welfare measure is

0.6%. Among rich countries, some that look good under HDI also look good under the Jones & Klenow measure. For instance, Iceland an HDI of 0.93 (just above the US) and a welfare measure of 1.11 despite a relative GDP of 0.89.

2.2

Other Things that Matter The welfare measure λ takes into account data on consumption levels, inequality, leisure and mortality. (a) Name one other variable that might be an important determinant of welfare that is not included in standard GDP calculations. (b) What data would you need in order to gure out how much weight to give to this variable? Describe how you would use that data to come up with the correct way to include the variable in question in the welfare calculation. What choices that people make might reveal how much they care about this variable? Don't worry too much about the feasibility of the data-collection procedure, but think carefully about why the observed choices would be informative.

Suggested Answer (a) One example is local pollution.

12

(b) One way to measure how much people are willing to trade o consumption against local pollution is to compare the prices of real estate in more- and less-locally-polluted areas.

2.3

Healthcare When we calculate consumption, one of the (many) categories of consumption is healthcare services. (a) Look up how much healthcare is consumed in the United States per year for a recent year. State the total dollar amount and also what fraction of GDP and what fraction of consumption is accounted for by healthcare.

A good place to look for this data is the National Income and Product Accounts (NIPA) at

https:

//apps.bea.gov/iTable/index_nipa.cfm. Browse around a little to get a sense of how the NIPA data is presented, and nd the correct place to look up this particular fact. (b) Suppose that we are calculating welfare in the style of Jones and Klenow, taking into account the impact of life expectancy on utility. Should we therefore subtract consumption of healthcare services from our measure of consumption? What do you think?

Suggested Answer (a) According to the National Health Expenditure Accounts, healthcare spending in 2018 was 3.65 trillion dollars, equivalent to 17.7% of GDP, 26.1% of consumption or 20.7% of consumption-plus-governmentspending (note that not all health spending is included in private consumption C since part of it is included in government spending G). (b) One argument is that including both consumption of health care services and life expectancy in the welfare calculation is a form of double counting. If we measure the output (life expectancy) directly, why also measure the input (healthcare spending)? On the other hand, healthcare a ects the quality of life, not just its length, so Rawls might prefer a country with higher healthcare spending even if he has the same chances of being alive.

2.4

Construction and the Armed Forces Suppose we compared welfare in two otherwise identical countries. Country A dedicates 50% of GDP to maintaining a large army.

Country B dedicates 50% of GDP to building houses.

In which of the two

countries would welfare be higher, according to the measure used by Jones and Klenow. What do you think of this?

Suggested Answer In the Jones & Klenow measure, government spending (including the armed forces) is included in the consumption measure but investment is not. Therefore country A would be measured as having higher welfare.

If Rawls were to spend his entire lifetime in these countries and both countries were equally

peaceful, Rawls would prefer country B, where over time housing services will become plentiful. If Rawls will just spend one year, then he doesn't care about the houses that are being built.

In addition, it's

possible that country A is more peaceful thanks to its large army, which Rawls would enjoy during his year there.

13

2.5

Measuring the Value of Life Expectancy Would each of the following observations be useful in determining how much to weigh life expectancy in measured welfare? Explain. (a) Observing how life expectancy varies across people with di erent income levels within a country. (b) Observing how life expectancy varies across countries. (c) Observing how much people are willing to pay for funerals. (d) Observing the di erence in house prices in neighborhoods with di erent murder rates.

Suggested Answer (d). Living in a neighborhood with a higher murder rate implies a lower life expectancy. The price discount that will persuade someone to live there gives an indication of how much consumption compensates them for their lower life expectancy. (a) and (b) can give an indication of what determines life expectancy, but not necessarily how much people value it. (c) tells us something about how people deal with death, but not really about life expectancy.

2.6

The Value of Life Expectancy In the US, average consumption per capita is about $35,000. Life expectancy is 79 years, so in the Jones and Klenow experiment, Rawls would have a 79% chance of being alive. In Bolivia, average consumption per capita is $3,700 and life expectancy is 68 years. Assume that there is no inequality in either country so that everyone who is alive gets the average level of consumption. Assume the following utility function:

c1−σ a u(c, a) = ū + 1−σ with σ = 0.5 and ū = 48.1. (a) Suppose you made a US resident the following o er: you can buy a health plan that costs x dollars per year and will extend your lifetime by one year.

What is the price x that would make them

indi erent between getting the health plan or not? (b) How much would a Bolivian resident be willing to pay for such a plan? (c) Write down the the special case of equation (2.2.1) that applies to this example and solve for λ for Bolivia. Don't replace any numbers yet, leave it in terms of aU S , aBol , cU S , cBol , σ and ū. (d) Replace the values of aU S , aBol , cU S , cBol , σ and ū to nd a value for λ. How does it compare to

cBol cU S ? Why?

Suggested Answer (a) They would be indi erent if:

c1−σ (c − x)1−σ ū + a = ū + (a + 0.01) 1−σ 1−σ 14

1−σ 1 c a − 0.01(1 − σ)ū 1−σ x=c− a + 0.01

2 35, 0000.5 × 0.79 − 0.01 × 0.5 × 48.1 = 35, 000 − 0.8 ≈ 980 i.e. if the plan cost $980. (b) They would be indi erent if:

1−σ 1 c a − 0.01(1 − σ)ū 1−σ a + 0.01

2 3, 7000.5 × 0.68 − 0.01 × 0.5 × 48.1 = 3, 700 − 0.69

x=c−

≈ 148 i.e. if the plan cost $148. (c)

c1−σ (λcU S )1−σ ū + aU S = ū + Bol aBol 1−σ 1−σ

1

1 aBol 1−σ 1−σ aBol λ= − 1 ū + c (1 − σ) cU S aU S aU S Bol (d)

λ=

1 35, 000

2 0.68 0.68 0.5 − 1 48.1 + 3, 7000.5 0.79 0.79

≈ 0.069 whereas

3, 700 cBol = ≈ 0.11 cU S 35, 000

The relative welfare measure is lower than the relative consumption measure because, in addition to lower consumption, Bolivians have lower life expectancy.

2.7

Measuring Risk Aversion The students of Concave University are all identical in their ability and preferences, which are well described by the function:

u(c) = E Two employers recruit CU graduates.

c1−σ 1−σ

Stabilis o ers each graduate a xed salary of $50,000.

Lotteris

instead o ers them a base salary of $20,000 plus a bonus scheme that depends on their performance,

15

which is entirely driven by luck. The bonus is either $10,000 or $100,000, with equal probability. The job itself is the same in both rms. About half the CU graduates choose to work for Stabilis and half prefer Lotteris, and they all say they found it hard to choose because both o ers were similarly attractive. What value of σ is consistent with their decisions?

Suggested Answer

50, 0001−σ 1 30, 0001−σ 1 120, 0001−σ = + 1−σ 2 1−σ 2 1−σ

This does not have a closed-form solution but, solving numerically, σ ≈ 1.8.

2.8

Inequality and Risk Aversion Compare the following two countries.

Both have a population of 100.

Within each country, we label

individuals in order of increasing consumption. In country A, the consumption of individual j is:

cA (j) = 100 + 8j while in country B the consumption of individual j is:

cB (j) = 200 + 4j (a) Plot the consumption patterns of each country, with an individual's label j (which ranges from 1 to

100) on the horizontal axis and their consumption on the vertical axis.

(b) Compute per capita consumption in each country. [Note: if you want, you can approximate sums with integrals]

(c) Suppose the utility function in both countries is:

u (c) =

c1−σ 1−σ

For what values of σ is expected utility higher in country A? Interpret your results.

Suggested Answer 16

(a) (b) Average consumption is, respectively:

Z 100 1 100 + 8jdj 100 0 1 100 = 100j + 4j 2 0 100 = 500

cA =

and

Z 100 1 200 + 4jdj 100 0 1 100 200j + 2j 2 0 = 100 = 400

cB =

(c) Expected utility is higher in country A if

Z 100 0

1−σ

(100 + 8j) 1−σ

Z 100 dj > 0

100

1 1 2−σ (100 + 8j) 8 (2 − σ)(1 − σ) 0

(200 + 4j) 1−σ

1−σ

dj 100

>

1 1 2−σ (200 + 4j) 4 (2 − σ)(1 − σ) 0

This does not have a closed-form solution but, solving numerically, country A has higher utility if

σ < 1.

If risk aversion is su ciently low, expected utility is higher in the more-unequal, higher-

average-consumption country, and vice versa.

2.9

Risk Aversion and Welfare in Western Europe 17

If instead σ = 1, Jones and Klenow had set a higher value of σ , what would this do to measured welfare in Western European countries? Explain.

Suggested Answer A higher value of σ increases Rawls' dislike of risk and therefore inequality.

Since Western European

countries have less inequality than the point of comparison, which is the US, then a higher value of σ would make their measured welfare higher.

2.10

Risk Aversion and the Di erence Principle Suppose a individual is trying to evaluate a society from behind the veil of ignorance. He knows that he can either be rich or poor, with equal probability. Expected utility is

u (cRich ) + u (cP oor ) 2

E [u (c)] = where the utility function u is

u (c) =

c1−σ 1−σ

Let's now de ne an indi erence curve. For any utility level Ū , the indi erence curve plots all the combinations of cRich and cP oor such that expected utility is Ū . (a) Take the following equation that de nes an indi erence curve:

Ū =

1 c1−σ 1 c1−σ Rich P oor + 21−σ 21−σ

and solve for cRich . This will tell you what value of cRich corresponds to each value of cP oor in the indi erence curve with level Ū . (b) Plot (using Excel, Matlab or some other software that produces nice graphs) a map of indi erence curves for the following values of σ :

√

10 and 4 respectively

i.

σ = 0.5. Plot three curves, for Ū = 2,

ii.

σ = 2. Plot three curves, for Ū = −1, −0.4 and −0.25 respectively.

Note that when σ > 1 utility values will be negative, but that's OK. Also note that

Ū > is impossible, i.e.

even if cRich

1 cP oor 1−σ 2 1−σ

= ∞ expected utility will be lower than Ū , so when you are

1 2Ū (1 − σ) 1−σ you should

plotting this indi erence curve, whenever you set cP oor to the left of set cRich = ∞. iii.

σ = 5. Plot three curves, for Ū = −0.25, − 544 and −4−5 respectively.

iv.

4 4 σ = 11. Plot three curves, for Ū = −0.1, − 10 11 and − 10

10

−10

respectively.

In all cases, have cP oor on the horizontal axis and cRich on the vertical axis and let the range of the horizontal axis be [0, 10] and the range of the vertical axis be [0, 10].

18

(c) Notice that as σ becomes large, the indi erence curves start to look like right angles. Explain how this relates to Rawls's di erence principle.

Suggested Answer (a) Solving for cRich :

1 1−σ cRich = 2(1 − σ)Ū − c1−σ P oor

(b) Plotting:

(c) When the indi erence curves become right angles, this means that utility depends on min{cRich , cP oor }, i.e. on the worst-case scenario. This means that someone who is highly risk averse will evaluate their prospective welfare the way the philosopher John Rawls argued one should evaluate a society: by looking at the well-being of the least well-o person in that society.

19

CHAPTER 3

Basic Facts About Economic Growth Exercises

3.1

The Past is a Foreign Country Find the data compiled by Bolt et.

al.

(2018) at

https://www.rug.nl/ggdc/historicaldevelopment/

maddison/releases/maddison-project-database-2018. Starting from 1800, look up the US GDP per capita at intervals of one decade. For each of these points in time, nd the country that currently has the closest GDP per capita to the past US level. When did the US become richer than present-day India, present-day China, and present-day Portugal?

Suggested Answer

The exact comparison will depend on what year you are looking at. This is the table using 2016 data:

21

Year

US GDP p.c.

Closest

1800

1980

Gambia

1810

2120

Benin

1820

2080

Benin

1830

2364

Tanzania

1840

2583

Nepal

1850

2825

Cameroon

1860

3425

Djibouti

1870

3736

Sudan

1880

4866

Nicaragua

1890

5184

Pakistan

1900

6252

Laos

1910

7586

Morocco

1920

8485

Angola

1930

9490

Armenia

1940

11307

Albania

1950

15241

Azerbaijan

1960

18058

Bulgaria

1970

23958

Hungary

1980

29613

Trinidad & Tobago

1990

36982

Oman

2000

45887

Denmark

2010

49267

Netherlands

The US had the GDP per capita of these countries in the following years:

3.2

Country

GDP p.c.

Year

India

5,961

1898

China

12,320

1941

Portugal

27,726

1977

The Kaldor Facts in Other Countries Look up GDP accounts for some countries other than the US (a good source is https://www.rug.nl/ggdc/ productivity/pwt/). Reproduce Figures 3.2.1-3.2.3 using data from those countries (try a rich country, a middle-income country and a poor country). Do the Kaldor facts seem to hold for those countries as well?

Suggested Answer These are the gures for Japan:

22

For Brazil:

And for Chad:

None of them look quite like the Kaldor facts. Japan had fast growth at rst and then a deceleration, as well as an increase in the capital-output ratio and a fall in the labor share. Brazil is a bit closer to the Kaldor facts, although it also had a surge in growth and then a deceleration. Chad had large fall in GDP per capita, then stagnation and later a recovery, large changes in the capital-output ratio (in part due to changes in the denominator) and an increase in the labor share.

23

3.3

Within-Region Convergence Look up cross-country GDP data (a good source is

https://www.rug.nl/ggdc/productivity/pwt/). For

each country, compute average GDP-per-capita growth 1960-2014. Produce a scatterplot of growth against initial GDP per capita like Figure 3.3.1 but separately for countries in each of three regions of the world: Europe, Latin America and Africa. Is it the case in any of these regions that initially-poor countries have grown faster?

In Europe, initially-poor countries have grown faster than initially-rich countries. It's also true, though to a lesser extent, in Latin America, and to some extent in Africa, though with much more variation around the trend (note also the di erent axes in each of the graphs).

24

CHAPTER 4

The Solow Growth Model Exercises 4.1

An Earthquake Suppose an economy behaves according to the Solow growth model. It starts out at t = 0 at a steady state, with no technological progress and no population growth. Suppose an earthquake destroys half the capital stock. As a a consequence of this, what would happen in the short run (i.e. immediately) and in the long run (i.e. once the economy reaches steady state) to:

(a) GDP, (b) GDP per capita, (c) wages, (d) interest rates.

Suggested Answer In the short run, the capital stock is lower. Simply looking at the production function we can infer that GDP and GDP per capita will be lower. Since k is lower, equations (4.4.10) and (4.4.11) imply that the rental rate of capital will be higher higher and wages will be lower. By equation (4.4.12), a higher rental rate of capital implies higher interest rates.

The earthquake does not change the fundamental determinants of the steady state (s, δ , the production function, etc.). Therefore in the long run the economy will return to the same steady state that prevailed before the earthquake, with the same level of GDP, GDP per capita, wages and interest rates.

4.2

The Black Death In the middle of the 14th century, a plague killed about a third of the population of Europe. Assume that the economy of Europe was well described by the Solow model. How would the following variables change in response to the Black Death in the short run (i.e. immediately)?

(a) GDP,

25

(b) GDP per capita, (c) wages, (d) interest rates.

Suggested Answer In the short run, the population is lower. Simply looking at the production function we can infer that GDP will be lower. However, k is higher, so GDP per capita will be higher, and equations (4.4.10) and (4.4.11) imply that the rental rate of capital will be lower and wages will be higher. By equation (4.4.12), a lower rental rate of capital implies lower interest rates.

4.3

Korean Uni cation Suppose both North and South Korea have the same technology level (!?), which doesn't grow, and also have constant population. Their respective populations and capital stocks are:

LSouth = 10 K South = 2430 LN orth = 5 K N orth = 160 The production function is:

Y = K α L1−α with α = 0.4. The depreciation rate is δ = 0.08. (a) Compute GDP per capita in each country. (b) Compute wages and interest rates in each country. (c) Suppose North and South Korea unify. What is the new

K L in the uni ed country? What is GDP

per capita in the uni ed country? (d) Compute wages and interest rates in the uni ed country. Interpret (e) Will people and/or machines physically move between the North and the South? In what direction?

Suggested Answer (a) Plugging the numbers into the production function:

24300.4 × 100.6 =9 10 1600.4 × 50.6 yN = =4 10 yS =

(b) Using equations (4.4.8), (4.4.9) and (4.4.12):

wS = 0.6 × 24300.4 × 10−0.4 = 5.4 26

wN = 0.6 × 1600.4 × 5−0.4 = 2.4 rS = 0.4 × 2430−0.6 × 100.6 − 0.08 = −6.52% rN = 0.4 × 160−0.6 × 50.6 − 0.08 = −3% (c) Adding up the countries:

2430 + 160 K = = 172.667 L 10 + 5 (2430 + 160)0.4 × (10 + 5)0.6 y= = 7.85 10 + 5 (d) Using equations (4.4.8), (4.4.9) and (4.4.12):

w = 0.6 × (2430 + 160)0.4 × (10 + 5)−0.4 = 4.71 r = 0.4 × (2430 + 160)−0.6 × (10 + 5)0.6 − 0.08 = −6.18% (e) Before uni cation, the rental rate of capital is higher in the North and the wage is higher in the South. After uni cation, people and machines will move towards those places where the factor prices are higher, up to the point where k and therefore factor prices are equalized across North and South.

This means that machines will move North and/or people will move South (the absolute numbers that end up in each place are indeterminate)

4.4

An AK Model Suppose the production function takes the form:

F (K, L) = AK (a) Which of the assumptions that we made about F does this satisfy and which does it not satisfy? (b) Suppose every other assumption we used in the Solow model holds, and there is no technological progress. Find an expression for

kt+1 kt (it may be useful to follow the steps that lead to expression

(4.2.2)). (c) Will this economy grow in the long run? Explain.

Suggested Answer (a) It satis es 4.1 (constant returns to scale). It does not satisfy 4.2 because the marginal product of labor is zero. It does not satisfy 4.3 because FKK = FLL = 0. It does not satisfy 4.4 because FK = A for any K . (b) Using (4.2.2) and f (kt ) = Akt we have:

kt+1 − kt =

sA − δ − n kt 1+n 27

⇒

kt+1 1 + sA − δ = kt 1+n

(c) The growth rate will be:

Akt+1 kt+1 1 + sA − δ yt+1 = = = yt Akt kt 1+n so it will be constant over time. Therefore this economy will grow in the long run. The force that eventually makes growth stop in the Solow model is the diminishing marginal product of capital. This economy has a constant marginal product, so it can grow forever just from capital accumulation.

4.5

A Malthusian Model Look up Thomas Malthus and read a little about his work. We are going to formalize a simple version of his ideas in a little model. The production function is:

Yt = At Lα t

(4.1)

where At is technology, L is labor supply and α < 1.

There is no capital, no saving and no investment, so consumption is given by

Ct = Yt

(4.2)

The population evolves according to

Lt+1 = Lt 1 + γ

Ct −c Lt

(4.3)

where γ and c are parameters.

(a) Malthus had in mind an agricultural economy where the total amount of land is xed. What does that have to do with the way we wrote down the production function? What is the signi cance of assuming α < 1? (b) What does equation (4.3) mean? What is c? What is γ ? Why did Malthus think that something like equation (4.3) applied? (c) Assume

At = A is constant over time.

Using (4.1), (4.2) and (4.3), nd an expression for the Lt+1 percentage rate of population growth Lt as a function of the level of population Lt . Plot this function.

(d) Assume At = A is constant over time. Will GDP per capita grow in the long run? Will the population grow in the long run? (e) Suppose there is a one-time increase in At , from A to A

0

> A. What will happen to GDP per capita

and the size of the population in the short run and in the long run? Show your reasoning graphically and/or algebraically.

28

(f ) Suppose now that instead of being constant, technology improves at a constant rate, so that

At+1 = At (1 + g) Will GDP per capita grow in the long run? Will the population grow in the long run? Explain. (g) When we have a constant rate of technological progress, how does the long-run level of consumption per capita depend on γ ? Why?

Suggested Answer (a)

α < 1 means that if we add more workers to a xed supply of land, then they each add less output than the previous one. Even if agriculture is a constant-returns-to-scale technology, there is no way to accumulate additional land.

(b) The equation says population growth is increasing in the level of consumption per capita. c is the level of consumption per capita such that the population will remain constant. γ measures the sensitivity of population growth to the level of consumption per capita.

Malthus thought that population

does invariably increase when the means of subsistence increase because poverty and the associated mortality was the only thing preventing population growth. (c) Replacing:

Ct = At Ltα−1 Lt so:

Lt+1 = 1 + γ At Ltα−1 − c Lt

Graphically:

(d) In the long run, GDP per capita will be constant at c and the population will be constant too.

29

(e) At rst,

Ct Lt

t > c, so the population will grow. Over time, a higher population means that C Lt falls

back to c and the population stops growing.

(f ) Guess and verify that consumption per capita will be constant at a level css in the long run, and the population will grow at a constant rate gL . For this to be true, it must be that:

Ct Ct+1 = Lt Lt+1 At Lα−1 = At+1 Lα−1 t t+1 At Lα−1 = At (1 + g) (Lt (1 + gL )) t

α−1

1

1 + gL = (1 + g) 1−α Furthermore, we know that

1 + gL = (1 + γ (css − c)) so for our guess to be correct, it must be that 1

(1 + g) 1−α = (1 + γ (css − c)) 1

css = c +

(1 + g) 1−α γ

In the long run the economy will converge to a level of consumption per capita such that the increase in population exactly o sets technological progress. If consumption per capita is higher, the population will grow more than this, causing consumption per capita to fall, and vice versa. (g) From the expression above, we can see that css is decreasing in γ . This is because γ measures how strongly population growth responds to living standards. If γ is high, it only takes a small distance

30

between

Ct Lt and c for population growth to o set technological progress. If γ is small, a large increase

in consumption is needed to generate a given rate of population growth.

31

CHAPTER 5

Confronting Theory and Evidence 5.1

Quantifying the Solow Model Suppose the economy is described by the Solow growth model with the parameter values we used in Section 5.2.

(a) Suppose the economy starts out with a level of GDP per e ciency unit of labor that is only 10% of its steady state level, i.e.

ỹ0 = 0.1ỹss . What is the initial capital stock k̃0 ? Use a spreadsheet to

compute k̃t and ỹt for t = 1, 2, . . . , 100. How many years does it take for

ỹt ỹss > 0.95?

(b) Suppose the economy begins in steady state but there is a fall in the rate of population growth from

n = 0.01 to n = 0. How much higher will GDP per e ciency unit of labor be in the new steady state? Use a spreadsheet to compute k̃t and ỹt for t = 1, 2, . . . , 100. How many years does it take for

ỹt ỹss > 0.95?

Suggested Answer

(a) The initial capital stock is 1

k0 = 0.1 α kss Using the parameters from Section 5.2, we have kss = 5.64 and k0 = 0.0078. The rst 100 periods are:

33

year

kt

yt

yt yss |

year

kt

yt

yt yss |

year

kt

yt

yt yss

0

0.01

0.18

0.10

34

3.64

1.57

0.86

68

5.13

1.77

0.97

1

0.04

0.33

0.18

35

3.72

1.58

0.86

69

5.15

1.77

0.97

2

0.11

0.45

0.25

36

3.79

1.59

0.87

70

5.17

1.78

0.97

3

0.19

0.56

0.30

37

3.86

1.60

0.88

71

5.19

1.78

0.97

4

0.28

0.64

0.35

38

3.93

1.61

0.88

72

5.21

1.78

0.97

5

0.39

0.72

0.39

39

4.00

1.62

0.89

73

5.23

1.78

0.97

6

0.51

0.79

0.43

40

4.06

1.63

0.89

74

5.24

1.79

0.97

7

0.63

0.85

0.46

41

4.12

1.64

0.90

75

5.26

1.79

0.98

8

0.75

0.91

0.49

42

4.18

1.65

0.90

76

5.27

1.79

0.98

9

0.88

0.96

0.52

43

4.24

1.66

0.90

77

5.29

1.79

0.98

10

1.01

1.01

0.55

44

4.29

1.66

0.91

78

5.30

1.79

0.98

11

1.15

1.05

0.57

45

4.34

1.67

0.91

79

5.32

1.79

0.98

12

1.28

1.09

0.59

46

4.39

1.68

0.92

80

5.33

1.80

0.98

13

1.41

1.13

0.62

47

4.44

1.69

0.92

81

5.34

1.80

0.98

14

1.54

1.16

0.64

48

4.49

1.69

0.92

82

5.35

1.80

0.98

15

1.67

1.20

0.65

49

4.54

1.70

0.93

83

5.36

1.80

0.98

16

1.80

1.23

0.67

50

4.58

1.70

0.93

84

5.38

1.80

0.98

17

1.92

1.26

0.69

51

4.62

1.71

0.93

85

5.39

1.80

0.98

18

2.05

1.28

0.70

52

4.66

1.71

0.94

86

5.40

1.80

0.98

19

2.17

1.31

0.72

53

4.70

1.72

0.94

87

5.41

1.81

0.99

20

2.29

1.34

0.73

54

4.74

1.72

0.94

88

5.42

1.81

0.99

21

2.40

1.36

0.74

55

4.77

1.73

0.94

89

5.42

1.81

0.99

22

2.51

1.38

0.75

56

4.81

1.73

0.95

90

5.43

1.81

0.99

23

2.62

1.40

0.77

57

4.84

1.74

0.95

91

5.44

1.81

0.99

24

2.73

1.42

0.78

58

4.87

1.74

0.95

92

5.45

1.81

0.99

25

2.84

1.44

0.79

59

4.90

1.74

0.95

93

5.46

1.81

0.99

26

2.94

1.46

0.80

60

4.93

1.75

0.95

94

5.46

1.81

0.99

27

3.03

1.47

0.81

61

4.96

1.75

0.96

95

5.47

1.81

0.99

28

3.13

1.49

0.81

62

4.99

1.76

0.96

96

5.48

1.81

0.99

29

3.22

1.51

0.82

63

5.01

1.76

0.96

97

5.48

1.81

0.99

30

3.31

1.52

0.83

64

5.04

1.76

0.96

98

5.49

1.82

0.99

31

3.40

1.53

0.84

65

5.06

1.76

0.96

99

5.50

1.82

0.99

32

3.48

1.55

0.84

66

5.09

1.77

0.96

100

5.50

1.82

0.99

33

3.56

1.56

0.85

67

5.11

1.77

0.97

It takes 56 years for

(b)

34

ỹ to reach 0.95. y˜ ss

ỹss = 2.00, which is 9.4% higher than in the previous steady state. The rst 100 periods are:

year

kt

yt

yt yss |

year

kt

yt

yt yss |

year

kt

yt

yt yss

0

5.64

1.83

0.91

34

6.78

1.95

0.98

68

7.14

1.99

0.99

1

5.69

1.84

0.92

35

6.80

1.96

0.98

69

7.14

1.99

0.99

2

5.74

1.84

0.92

36

6.82

1.96

0.98

70

7.15

1.99

0.99

3

5.80

1.85

0.92

37

6.83

1.96

0.98

71

7.15

1.99

0.99

4

5.85

1.86

0.93

38

6.85

1.96

0.98

72

7.16

1.99

0.99

5

5.90

1.86

0.93

39

6.87

1.96

0.98

73

7.16

1.99

0.99

6

5.94

1.87

0.93

40

6.88

1.96

0.98

74

7.17

1.99

0.99

7

5.99

1.87

0.93

41

6.89

1.97

0.98

75

7.17

1.99

0.99

8

6.03

1.88

0.94

42

6.91

1.97

0.98

76

7.17

1.99

0.99

9

6.08

1.88

0.94

43

6.92

1.97

0.98

77

7.18

1.99

0.99

10

6.12

1.88

0.94

44

6.93

1.97

0.98

78

7.18

1.99

0.99

11

6.16

1.89

0.94

45

6.95

1.97

0.98

79

7.19

1.99

1.00

12

6.20

1.89

0.94

46

6.96

1.97

0.98

80

7.19

1.99

1.00

13

6.23

1.90

0.95

47

6.97

1.97

0.98

81

7.19

1.99

1.00

14

6.27

1.90

0.95

48

6.98

1.97

0.99

82

7.20

2.00

1.00

15

6.30

1.90

0.95

49

6.99

1.98

0.99

83

7.20

2.00

1.00

16

6.34

1.91

0.95

50

7.00

1.98

0.99

84

7.20

2.00

1.00

17

6.37

1.91

0.95

51

7.01

1.98

0.99

85

7.21

2.00

1.00

18

6.40

1.92

0.96

52

7.02

1.98

0.99

86

7.21

2.00

1.00

19

6.43

1.92

0.96

53

7.03

1.98

0.99

87

7.21

2.00

1.00

20

6.46

1.92

0.96

54

7.04

1.98

0.99

88

7.21

2.00

1.00

21

6.49

1.92

0.96

55

7.05

1.98

0.99

89

7.22

2.00

1.00

22

6.52

1.93

0.96

56

7.06

1.98

0.99

90

7.22

2.00

1.00

23

6.54

1.93

0.96

57

7.06

1.98

0.99

91

7.22

2.00

1.00

24

6.57

1.93

0.96

58

7.07

1.98

0.99

92

7.22

2.00

1.00

25

6.59

1.94

0.97

59

7.08

1.98

0.99

93

7.23

2.00

1.00

26

6.62

1.94

0.97

60

7.09

1.98

0.99

94

7.23

2.00

1.00

27

6.64

1.94

0.97

61

7.09

1.99

0.99

95

7.23

2.00

1.00

28

6.66

1.94

0.97

62

7.10

1.99

0.99

96

7.23

2.00

1.00

29

6.69

1.94

0.97

63

7.11

1.99

0.99

97

7.23

2.00

1.00

30

6.71

1.95

0.97

64

7.11

1.99

0.99

98

7.24

2.00

1.00

31

6.73

1.95

0.97

65

7.12

1.99

0.99

99

7.24

2.00

1.00

32

6.75

1.95

0.97

66

7.13

1.99

0.99

100

7.24

2.00

1.00

33

6.77

1.95

0.97

67

7.13

1.99

0.99

It takes 13 years for

5.2

ỹ to reach 0.95. y˜ ss

The Speed of Convergence Suppose the economy is described by the Solow growth model with a Cobb-Douglas production function, no population growth and no technological progress. The economy is not in steady state. GDP capita in

35

period t is yt = θyss , with θ ∈ (0, 1). (a) How far away from steady state is the capital stock? Find an expression for the level of

kt kss that

would be consistent with yt = θyss , in terms of θ and parameters. (b) Find an expression for kss , the steady state level of capital, in terms of parameters. (c) Use your results from parts (a) and (b) and equation (5.3.1) to nd an expression for gy , the rate of growth of GDP, as a function of θ and parameters. (d) Let α = 0.35 and δ = 0.04. Compute gy for an economy whose GDP is θ = 0.8 times its steady-state level? Compute

gy 1−θ . What fraction of the gap between yt and yss does the economy close in one

year? Repeat this calculation for θ = 0.9 and θ = 0.99.

Suggested Answer (a)

1 kt α kss = θ .

(b) Using the formula on page 60:

kss =

1 s 1−α

δ

(c) Using equation (5.3.1):

gy ≈ sf 0 (kt ) − δα = sαk α−1 − δα

α−1 α−1 = α skss θ α −δ !

1 α−1 α−1 s 1−α θ α −δ =α s δ α−1

= αδ θ α − 1 (d) Replacing:

`

gy 1−`

0.8

0.036

0.9

0.030

0.99

0.026

As the economy approaches the steady state, it closes around 2.6% of the remaining gap every year (a bit more than that if it's further away from steady state).

5.3

Causes of Growth and Growth Accounting Suppose an economy is well described by the steady state of the Solow Growth Model with constant technological progress and no population growth.

Imagine we take data from this economy and do a

growth-accounting exercise. How much growth will we attribute to capital accumulation and how much to technology? How does this relate to the result that says that without technological progress there would be no growth in the steady state?

36

Suggested Answer We know that in a steady state GDP and the capital stock will grow at the rate of technological progress

g . Letting α be the capital share and using equation (5.4.2) we'll have: g = α × g + (1 − α) × 0 + Solow Residual ⇒ Solow Residual = (1 − α) × g Therefore we will attribute α × g of growth to capital accumulation and (1 − α) × g to technological progress. Note that the Solow decomposition measures the proximate sources of growth. In an ultimate causal sense all the growth is due to technological progress. Part of what technological progress does is make it possible for the capital stock to grow in the steady state by counteracting diminishing returns.

5.4

Measurement of the Capital Stock The capital stock is not easy to measure. A common way to try to do this is to:

• guess a value for K0 at some date in the past (maybe when statistics were rst collected), • measure investment every year thereafter (which is also not that easy), and • use the equation Kt+1 = (1 − δ) Kt + It

(5.5.1)

to compute the capital stock for every year thereafter.

Consider an economy that is in a steady state without technological progress (and has been there for an in nitely long time). Every year the level of investment is:

It = 0.2 and the depreciation rate is:

δ = 0.1 (a) What is the capital stock? (b) The statistical o ce only started measuring investment in year 0. By then the economy was already in steady state. They guessed (incorrectly) that the capital stock in year 0 was K0 = 1. Use equation (5.5.1) to compute how the estimate of the capital stock changed over time after year 0 and plot your answer. (You don't need to provide a closed form solution, a nice graph made with a spreadsheet is enough). Compute the ratio:

KEST IM AT E KT RU E for the years 5, 10, and 50. (Again, you can just take the numbers from a spreadsheet). (c) Suppose now that the statistics o ce started collecting statistics a long time ago (an in nite time ago) so we don't have the problem of making the wrong initial guess. However, instead of using the

37

correct value of δ , the statistical o ce incorrectly believes that δ = 0.05. What is their estimate of the capital stock? Why does it di er from the truth? (d) The production function in this economy is

Y = AK α L1−α

(5.5.2)

and there are accurate data on GDP and on total hours of labor supply. Using the inaccurate estimate of K from part (c), plus accurate data on total labor and GDP, an economist is trying to measure the economy's productivity, i.e. to solve for A in equation (5.5.2). Find an expression for

AEST IM AT E AT RU E as a function of

KEST IM AT E KT RU E

Given the numbers from part (c) and using α = 0.35, how far would the economist's estimate be from the truth?

Suggested Answer (a) Using (5.5.1) and imposing Kt+1 = Kt because we are in a steady state:

K=

I =2 δ

(b) The measured capital stock will be:

For years 5, 10 and 50, the ratio

38

KEST IM AT E will be 0.705, 0.826 and 0.997 respectively.

KT RU E

(c) Using K =

I δ the statistics o ce will incorrectly think that K = 4.

(d) Solving for A:

AEST IM AT E =

Y α 1−α KEST IM AT E L

Y KTαRU E L1−α

α AEST IM AT E KT RU E ⇒ = AT RU E KEST IM AT E AT RU E =

= 0.785 By overestimating the capital stock, the economist will infer that the level of productivity is 21.5% lower than it actually is.

5.5

Sources of TFP Suppose that the true production function in Gotham is

Y = AK α L1−α

(5.1)

Unfortunately, crime is a huge problem in Gotham, so that for each worker doing actual work rms need to hire γ security guards just to protect their products from being stolen.

The security guards will of

course describe their activity as work even though they are not actually producing anything. Use the notation N to refer to the total labor force (including workers and guards) and denote the number of actual production workers by L. (a) Find an expression for total output as a function of A, K , N , γ and α. (b) Write down the problem of a rm that has to choose capital and labor to maximize pro ts. Notice that the rm will have to pay a wage to the security guards even though they will not produce anything. (c) If the representative rm hires all the workers and rents all the capital, what will be the wage and the rental rate of capital? Express it as a function of A, K , N , γ and α. (d) Suppose an economist studying Gotham is trying to estimate A using equation (5.1). The economist has accurate data on K , N and Y .

However, the economist doesn't really know whether workers

are involved in production or in security services: in national statistics they all look employed .

Therefore the economist will plug in the value of N instead of the value of L into the estimate of A. What will the economist's estimate of A be? How does it compare to the true value of A? (e) How does this relate to the ndings that link GDP levels to social and political institutions? (f ) Suppose that, in a economy that didn't have the crime problem of Gotham, the government attempted to create jobs by mandating that rms hire γ assistants for every production worker. The job of assistants is to look at production workers all day long and not do anything. Using the analysis above, what would be the e ects of such a policy?

39

Suggested Answer (a) If there are γ guards per production worker, then L =

Y = AK α

N 1+γ . Using the production function:

N 1+γ

1−α

We can treat this as the production function, which tells us what output will be as a function of the capital stock K and the total number of workers N . (b) The rm solves:

max AK α K,N

N 1+γ

1−α

− wN − rK K

(c) Using (4.4.9) and (4.4.8):

w = (1 − α)AK α (1 + γ)−(1−α) N −α rK = αAK α−1 (1 + γ)−(1−α) N 1−α (d) Solving for A:

AEST IM AT E =

=

Y K α N 1−α 1−α

N AK α 1+γ

K α N 1−α = A(1 + γ)−(1−α) −(1−α)

Since (1 + γ)

< 1, the economist will infer that this economy has a lower level of productivity

than A. (e) This is an example of how bad institutions can lead to low productivity. Whether due to a lack of social norms or e ective law enforcement, Gotham has a crime problem.

This leads to a fraction

of the labor force to be diverted away from production. This means that productivity is e ectively lower than it would be if it depended just on pure technological productivity. (f ) Assistants have the same impact on the economy as security guards. Since they are doing something other than production, mandating them lowers productivity.

5.6

Interest Rates Suppose we observe that Usuria (a closed economy) grows at approximately 6% a year, and we are trying to understand why. We know the labor force has been constant.

• Conjecture 1: The economy has been at a steady-state-with-technological-progress. There has been capital accumulation just to maintain

K AL constant but the cause of growth has been TFP growth.

• Conjecture 2: The economy started from a very low level of capital stock (below steady state) and has been growing because it is converging to the steady state, but TFP has been constant.

40

Ideally, if we wanted to distinguish between Conjecture 1 and Conjecture 2 we could do a growth-accounting exercise.

Unfortunately, Usuria does not collect reliable statistics on capital accumulation that would

enable us to do this. We do, however, have data on interest rates in Usuria. How would one use these data to distinguish between Conjecture 1 and Conjecture 2? Be as mathematically precise as possible.

Suggested Answer We know that under Conjecture 1, the Kaldor Facts would hold. This means the rental rate of capital and therefore the interest rate would be constant over time. Instead, under Conjecture 2, equation (4.4.8) implies that as the capital stock grows, the rental rate of capital and therefore the interest rate falls over time. Therefore, if we observe that interest rates are constant over time, this would indicate that Conjecture 1 is correct, whereas if they are falling, this would indicate that Conjecture 2 is correct.

5.7

GDP Accounting, Interest Rates and Growth Accounting Proletaria is a mythical country in Central Asia, where the currency is the ruble. The exchange rate is 10 rubles per euro.

Suppose the production function in Proletaria is given by:

Y = K α L1−α (a) Find expressions for the output-to-capital ratio

Y K and the marginal product of capital FK as functions

of K and L. During 2014, the following events took place in Proletaria.

• Kapitas, the main manufacturer in the country, imported an industrial welder made in Germany, for which it paid 1,000 euro.

• By the end of the year, the industrial welder was no longer new. Its estimated value in the resale market was 960 euro.

• 100 workers worked for Kapitas the entire year making screws and nails, using the new welder. Each of them received wages for 350 rubles.

• Each of the Kapitas workers paid 100 rubles in income taxes.

• The total output of Kapitas consisted of 1 million screws and 1 million nails. All of it was exported to Austria for a total of 4,000 euro.

• The government of Proletaria employed one of the 100 workers as Chief of the Secret Police (in addition to his factory job) to maintain law and order, and paid her a salary of 10,000 rubles.

• The workers ate beef imported from France, which cost a total of 2,500 euro. (b) Construct GDP accounts (in rubles) for Proletaria by production, income and expenditure. (c) What were the income shares of labor and capital? (d) Suppose that we know that the capital stock in Proletaria is 100,000 rubles and that the depreciation rate of the industrial welder is typical for this country's capital stock. What interest rate should we expect to observe?

41

(e) During 2015, additional investment in Proletaria has exceeded depreciation so that the capital stock now stands at 120,000 rubles. The labor force also grew thanks to immigration, and now consists of 105 workers instead of 100. GDP during 2015 was 55,000 rubles (prices were constant). How much of the growth in GDP between 2014 and 2015 can be attributed to growth in Total Factor Productivity?

Suggested Answer (a) The output-to-capital ratio is

αK

α−1

α

L

Y K

= K α−1 Lα .

The marginal product of capital is

FK (K, L) =

Y . = αK

(b) The events are recorded as follows:

Event

Production

Welder

Income

Expenditure

0

Depreciation

0

Wages Paid

0

Income Tax

0

Pro ts

-400

Depreciation

400

Pro ts

-35,000

Wages

35,000

0

Imports

-10,000

Investment

10,000 0

0

0

0

Screws

Manuf.

40,000

Pro ts

40,000

Exports

40,000

Police

Govt.

10,000

Wages

10,000

Govt.

10,000

Imports

-25,000

Consumption

25,000

Beef

Total (c) Labor Share =

0

0

0

10,000

50,000

50,000

50,000

35,000+10,000 = 0.9. Capital share = −400+400−35,000+40,000 = 0.1 50,000 50,000

(d) Compute:

r = rK − δ = FK (K, L) − δ Y −δ K 50, 000 400 = 0.1 − 100, 000 10, 000

=α

= 0.01 The rst step is equation (4.4.12). The second is equation (4.4.8). The third uses the results from part (a). The fourth uses that α is the capital share and that the depreciation rate can be determined by looking at how much the welder depreciated in the rst year. (e) Solow decomposition:

Solow Residual = GDP growth

42

− Capital share × Capital Growth − Labor Share × Labor Growth

= 0.1 − 0.1 × 0.2 − 0.9 × 0.05 = 0.035 Of the 10% growth that the economy experienced, 3.5% can be attributed to growth in Total Factor Productivity.

5.8

National Accounts and The Golden Rule In 2018, the following events happened in Aurum, which is a closed economy that uses the Denarius (plural: Denarii) as its currency.

• Grano, Inc. hired Cornelia to plant and harvest 500 tons of wheat on a large plot of land using a tractor. Both the land and the tractor are owned by Grano, Inc.

It paid Cornelia 200 Denarii for her work on the harvest.

It sold the wheat to Panem, Inc. for 1,000 Denarii.

• The tractor the Cornelia used:

Was new at the beginning of the year. It was worth 2,000 Denarii.

Was no longer new at the end of the year. By then it was worth 1,800 Denarii.

• Panem, Inc. hired Gaius to grind the wheat in its mill and produce a million loaves of bread.

It paid Gaius 1,400 Denarii for his work.

It sold the bread for 4,000 Denarii to the hungry citizens, who eat it.

• Fabrica, Inc. hired Livia to build a combine harvester.

It paid Livia 400 Denarii for her work.

It sold the combine harvester to Grano, Inc. for 1,000 Denarii. (a) Construct GDP accounts (in Denarii) for Aurum by production, income and expenditure. (b) What were the income shares of labor and capital? (c) What was the savings rate? (d) Assume the following:

• the savings rate is constant, • the production function has the form: Y = AK α L1−α and there is no technological progress,

• the depreciation rate for the tractor is typical for the capital stock as a whole, • the population is constant.

In the long run, would the level of consumption be higher if the economy slightly increased its saving rate?

43

Suggested Answer (a) The events are recorded as follows:

Event Grano

Production Agric

Income 1,000

Tractor

Panem

Fabrica

0

Manuf.

Manuf.

Total

3,000

1,000

Expenditure

Wages

200

Pro ts

800

Pro ts

-200

Depreciation

200

Investment

1,000

0

Wages

1,400

Investment

-1,000

Pro ts

1,600

Consumption

4,000

Wages

400

Investment

1,000

Pro ts

600

5,000

5,000

(b) Labor Share =

200+1,400+400 = 0.4. Capital share = 800−200+200+1,600+600 = 0.6.

5,000 5,000

(c) Savings rate =

Y −C = 5,000−4,000 = 0.2.

Y 5,000

5,000

(d) We know that for the Cobb-Douglas production function the Golden Rule savings rate is s = α. Here we have that s = 0.2 and α = 0.6. Since the savings rate is below the Golden Rule savings rate, we know that (locally) steady state consumption is increasing in the savings rate. Alternatively, we can start from equation (4.3.3) to get:

∂css = f 0 (kss ) − δ − n ∂kss Y =α −δ−n K δ+n =α −δ−n s 0.1 = 0.6 − 0.1 0.2 = 0.2 > 0 The rst step is taking a derivative. The second step is the same as in the previous question. The third step uses equation (4.3.2). The fourth step uses the results from parts (b) and (c), and that the depreciation rate can be determined by looking at how much the tractor depreciated in the rst year.

The reasoning shows that a higher steady state capital stock would be associated with higher steady state consumption, so raising the savings rate, which brings about a higher steady state capital stock, would result in higher steady state consumption.

5.9

Disease and TFP The production function in In uenzistan is:

1−α

Y = K α (AH)

44

(5.5.4)

where H is the total number of hours of work.

(a) Solve equation (5.5.4) for A, so that you end up with an expression for A in terms of all the other variables and parameters. (b) Suppose an economist wants to measure A. The economist has o cial data from the In uenzistan Statistics O ce on:

• total hours worked by all workers, • the total level of investment, which has been constant for many many years, • an estimate of the average rate of depreciation of the capital stock, • GDP and its subcomponents, measured by the income method.

How can he use this data to get an estimate of A?

Don't be vague: provide exact formulas that

describe each step precisely. (c) Suppose now that the economist has the same data as above except he doesn't know total hours worked. Instead, the In uenzistan Statistics O ce only keeps records on the total number of employed workers (call this number L), but not on how many hours each of them works. In order to get an estimate of A, the economist assumes that the average worker works N hours per year, which is the number of hours worked by the average worker in neighboring Healthistan. Write down an expression for the economist's estimate of A (call this number Â). (d) Unfortunately, the residents of In uenzistan keep getting sick, so they have to spend a signi cant portion of their time recovering at home instead of working. As a result, they each work xN hours

 A , i.e. for the ratio of the economists'

instead of N hours, where x < 1. Derive an expression for estimate of A to its true value. Interpret your nding.

Suggested Answer (a) Solving: 1

α

A = Y 1−α K − 1−α H −1 (b) First measure the capital stock as: K =

I δ . Then replace this value, measured GDP and total hours

into the expression above and recover A. (c) The estimate of A is: 1

α

−1

 = Y 1−α K − 1−α (N L) (d) Dividing  by A:

 H = A NL xN L = NL =x

45

The economist will infer that productivity is only a fraction x of its true value.

He will wrongly

attribute low output to low productivity per hour worked when it really is a matter of a low number of hours worked.

46

CHAPTER 6

Consumption and Saving 6.1

Two-Period Problem with Taxes and Initial Wealth Suppose a household solves the following two-period consumption-savings problem with taxes:

max u (c1 ) + βu (c2 )

c1 ,a,c2

s.t. a = a0 + y1 − τ1 − c1 c2 = y2 − τ2 + (1 + r) a with u (c) =

c1−σ 1−σ , where: c1 is consumption at time 1, c2 is consumption at time 2, y1 is household income

at time 1, y2 is household income at time 2, τ1 are taxes at time 1, τ2 are taxes at time 2, and a0 is initial wealth. (a) Solve for the household's choice of c1 , c2 and a in closed form. (b) How does

1

c1 y1 depend on y2 ? What would happen if households suddenly became optimistic about

the future?

c1 y1 depend on a0 ? Interpret. c (d) How does 1 depend on β ? Interpret. y1 (c) How does

(e) Suppose y2

1 = τ1 = τ2 = 0 and compute ∂c ∂r .

How does the answer depend on σ ?

Interpret the

answer. [Hint: this is a hard question, not the maths but the interpretation. Think about what σ means for the relative importance of income and substitution e ects]

Suggested Answer (a) The same steps that lead to equation (6.2.9) imply:

c1 =

2 −τ2 a0 + y1 − τ1 + y1+r 1

1

1 + β σ (1 + r) σ

−1

(6.1)

1 In closed form means that you have an explicit expression for something. Suppose I ask you to determine some variable x.

Then something like x = y 2 − z is a solution in closed form ; something like ex + x − b = 0 also implicitly tells what x should be but is not in closed form.

47

and using 1

c2 = c1 [β (1 + r)] σ we get 1

c2 =

y2 − τ2 a + y − τ + 0 1 1 1 1 −1 1+r 1 + β σ (1 + r) σ [β (1 + r)] σ

(6.2)

(b) Dividing:

y2 −τ2 a0 τ1 c1 y + 1 − y1 + y1 (1+r) = 1 1 1 −1 y1 1 + β σ (1 + r) σ so

(6.3)

c1 y1 is increasing in y2 . If households become optimistic about their future income, they will increase

consumption already in the present. (c)

c1 y1 is increasing in a0 . A household that starts with some wealth will consume a higher fraction of their income.

(d)

c1 y1 is decreasing in β . A household that places higher weight on the future will consume less and save more.

(e) If we set y2 = τ1 = τ2 = 0 then we have:

c1 =

a 0 + y1 1

1

1 + β σ (1 + r) σ

−1

Taking the derivative:

−2 1 1 1 1 ∂c1 1 σ −1 σ −2 σ σ = (a0 + y1 ) 1 + β (1 + r) 1− β (1 + r) ∂r σ so consumption is increasing in the interest rate if and only if σ > 1. With these preferences, σ governs the strength of income and substitution e ects. A high value of σ means a sharply decreasing marginal utility, so the household places great importance in smoothing consumption. Since the household has income and wealth only in the initial period, higher interest rates means that the budget set expands.

If σ is high, the household takes advantage of this by consuming more in both periods. Instead, if

σ is low, the household is more willing to substitute consumption in one period for consumption in the other.

Therefore it takes advantage of higher interest rates to reduce present consumption to

increase future consumption even more.

6.2

Savings Rates for Di erent People An economist has data on the occupation, this year's income (denoted y ) and this year's consumption (denoted c) of a sample of 26-year-olds. Within this sample, some are top professional athletes and others are medical doctors in their rst year of residence. [Make whatever assumptions you think are reasonable and, if you wish, refer back to Exercise 6.1] (a) Suppose one computed s = for athletes or doctors?

48

y−c y for each individual in the sample. Should we expect s to be higher

(b) Suppose interest rates go down. How should we expect the response of s to di er between the two groups?

Suggested Answer

(a) Doctors in their rst year of residence probably have higher income in the future than in the present.

Using (6.3), this implies a high level of

c1 y1 and therefore low s.

The opposite is the case for top

athletes, whose short typical careers mean higher income in the present than in the future.

(b) Lower interest rates have the same substitution e ect for everyone. However, they have a negative wealth e ect for those whose income is in the present and positive wealth e ects for those whose income is in the future.

Therefore we should expect the consumption of doctors to increase more

than that of athletes, or equivalently their savings to fall more.

6.3

Consumption and Interest Rates A household solves a special case of the problem in Exercise 6.1, with τ1 = τ2 = a0 = 0. Suppose that, by coincidence, the values of β, r, y1 and y2 are such that it is optimal for the household to consume:

c1 = y1 c2 = y2 (a) What will happen to c1 if interest rates increase? [A graph will be helpful. Make sure you draw it carefully, it's probably useful to make it large]

(b) Does the overall utility achieved by the household increase, decrease or stay the same?

(c) Suppose this household was the only household in the economy and Jones & Klenow, using this year's data, applied their measure of welfare to this economy. How would their measure of welfare change with the change in interest rates?

(d) Explain the relationship between the answer to part (b) and the answer to part (c).

Suggested Answer

(a)

c1 will fall. Graphically: 49

(b) As the graph shows, the household attains higher utility. Note that the household could have stayed at its original consumption choice buy prefers to substitute. (c) In Jones & Klenow's main thought experiment, Rawls spends just one year (the present) in the country. If during this year consumption is lower because interest rates have risen, Rawls's measured welfare will be lower. (d) The di erence comes from the fact that the actual residents of this country will spend the entire two years of their life in it, so they and will actually be better o thanks to the higher consumption they will enjoy once Rawls has left.

6.4

Consumption and Income Suppose there are two households in the economy.

Each of them solves a special case of Exercise 6.1,

where τ1 = τ2 = a0 = 0, β = 1 and r = 0. (a) Solve for c1 as a function of y1 and y2 . (b) Suppose the incomes of each of the households are:

y1

y2

Household A

2

4

Household B

6

4

Compute c1 and c2 for each of the households. (c) Suppose an economist is trying to decide what is a reasonable model for consumption behavior and only has data for period 1. Is the data supportive of the Keynesian view of the consumption function?

Explain.

50

(d) Suppose the same economist now looks at data for period 2 in addition to data for period 1. Is the data supportive of the Keynesian view of the consumption function? Explain.

Suggested Answer (a) Using (b):

c1 =

y1 + y2 2

(b) For household A, c1 = c2 = 3. For household B, c1 = c2 = 5. (c) The data for the rst period ts a Keynesian view relatively well. Household B has 4 higher income but only 2 more consumption, which looks like a marginal propensity to consume of 0.5. In proportional terms, household B has 3 times higher income but only 1.667 times higher consumption. (d) The data for the second period is less easy to reconcile with a pure Keynesian view:

2

the two

households have the same income but di erent consumption levels.

6.5

Credit Constraints and Ricardian Equivalence Suppose a household solves the following variant of the problem in Exercise 6.1:

max u (c1 ) + βu (c2 )

c1 ,a,c2

s.t. a = y1 − τ1 − c1 c2 = y2 − τ2 + (1 + r) a a ≥ −b

(6.4.1)

(a) What does equation (6.4.1) mean? What does b represent? (b) Plot the household's budget constraint. In the same graph, plot constraint (6.4.1). (c) Solve for c1 , c2 and a.

[Hint: notice that the constraint (6.4.1) is a weak inequality, not an equality, so it may or may not be binding. If it is not binding, then you can use the answer from Exercise 6.1. Then think about what happens if it is indeed binding. Then gure out whether or not it will be binding] (d) Show that, other things being equal, constraint (6.4.1) is more likely to be binding if i.

y2 − τ2 is high,

ii.

y1 − τ1 is low,

iii.

b is low.

Interpret each of these conditions.

2 Though one could conjecture that household B consumes more than household A despite identical income because they are di erent in ways other than income.

51

(e) Suppose that the government announces a stimulus package of size ∆. This involves lowering τ1 by ∆ and increasing τ2 by ∆ (1 + r) so that the present value of taxes is unchanged. How does c1 respond to the stimulus package if we start from a situation where constraint (6.4.1) is NOT binding?

How does c1 respond to the stimulus package if we start from a situation where constraint (6.4.1) is binding? Explain. (f ) Suppose that instead of announcing a stimulus package, the government announces that it will allow households to borrow ∆ from the government and repay it back (with interest) at t = 2. How do the e ects of this policy compare with the e ects of the stimulus package? Explain.

Suggested Answer (a) Equation (6.4.1) is a borrowing constraint. The household must have at least −b assets or, equivalently, can borrow at most b (b) Rearranging:

y1 − τ1 − c1 ≥ −b c1 ≤ y1 − τ1 + b Graphically:

(c) If the borrowing constraint does not bind, then we have the solution from Exercise 6.1:

c1 given by

(b), c2 given by (6.2) (replacing a0 = 0) and a = y1 − τ1 − c1 . If the borrowing constraint binds, then we have: c1 = y1 − τ1 + b, a = −b and c2 = y2 − τ2 − (1 + r)b. The constraint will bind whenever the unconstrained consumption level requires borrowing more than b, i.e. whenever: 2 −τ2 y1 − τ1 + y1+r 1

1

1 + β σ (1 + r) σ 52

−1

> y1 − τ1 + b

(6.4)

(d) Rearranging (6.4), the borrowing constraint will bind if: 1

1

β σ (1 + r) σ 1 σ

−1

1 + β (1 + r)

1 σ −1

(y1 − τ1 ) + b <

1 1 σ

1 + β (1 + r)

1 σ −1

y2 − τ2 1+r

This immediately implies that it will bind if i.

y2 − τ2 is high, because this makes the household want to consume by borrowing against future income.

ii.

y1 −τ1 is low, because this makes the household want to borrow to maintain consumption despite low current income.

iii.

b is low, because this limits how much the household can borrow.

(e) If the constraint is not binding, then we can rewrite the household's budget using equation (6.2.15):

c1 +

1 1 c2 = y1 − τ1 − ∆ + (y2 − τ2 + ∆(1 + r)) 1+r 1+r

and it's immediate that ∆ cancels out so the stimulus package has no e ect. Instead, if the constraint is binding, we have that

c1 = y1 − (τ1 − ∆) + b so consumption increases in response to the stimulus package. In this scenario Ricardian equivalence does not hold because what determines the level of consumption is how much the household can consume before hitting its borrowing constraint, and the stimulus package a ects this. (f ) This program has the same e ect as the stimulus package. If the household is not constrained, it is

3

irrelevant. If the household is constrained, it works by relaxing this constraint.

6.6

A Tax on Savings A household solves a special case of the problem from Exercise 6.1, with τ1 = τ2 = a0 = 0. Suppose now that the government introduces a tax on interest income, so that a household that saves a (and therefore earns interest ra) will have to pay τ ra in taxes. (If the household borrows instead of saving it pays no tax). (a) Plot the new budget constraint. (b) Show graphically: i. an example where the new policy persuades households to save more, ii. an example where the new policy persuades the household to save less, iii. an example where the household does not change its decision in response to the new policy.

Explain.

3 Note that for either the stimulus package or the lending program to work, we need to assume that the borrowing limit does not change in response to the government's policy. To judge whether this is reasonable, we would need a theory of what determines

b. One possibility is that lenders react by saying: this household will be burdened in the future by having to pay taxes / repay public loans; it is better not to lend to them now . If this reaction by lenders reduced b one-for-one with the size of the public program, then there would be no e ect even on constrained households.

53

Suggested Answer (a) The budget introduces a kink in the budget constraint. To the right of y1 , the slope is −(1 + r). To the left of y1 , the slope is −(1 + r(1 − τ )). Graphically:

(b) Examples: i. This household is saving in the original allocation and has a strong preference for smooth consumption. Therefore the tax on savings has a strong wealth e ect and it reduces consumption (i.e. increase saving)

54

ii. This household is saving in the original allocation and is willing to substitute consumption across periods. The savings tax makes future consumption expensive relative to present consumption so the household increases consumption (i.e. decreases savings)

iii. This household is borrowing in the original allocation so the tax has no e ect on its decisions.

6.7

One Rational Household Consider an economy that is well described by the Solow growth model. The production function is:

Y = K α L1−α

55

The population is constant an equal to 1 and there is no technological progress. The saving rate is s and the depreciation rate is δ . (a) What will the capital stock be in the long run? (b) What will be the real interest rate in the long run?

For the rest of the question, assume that s = 0.4, α = 0.35 and δ = 0.1 and the economy is initially at a steady state (c) If the savings rate increases from s = 0.4 to s = 0.5: i. Will GDP increase in the long run? ii. Will consumption increase in the long run? (d) Suppose a single household in the economy (the Friedmans) decides that, instead of just saving an exogenous fraction s of their income, they are going to start choosing consumption and saving to maximize the following standard preferences:

∞ X

β t u (ct )

t=0 The Friedmans can borrow or lend at the market interest rate. Since it's just the Friedmans who start acting this way, and they are small relative to the economy, we are going to assume that the aggregate economy (aggregate quantities, prices, etc.) remains unchanged. Will the Friedmans' consumption be high initially and then fall over time, be low initially and then rise over time, or remain constant?

Suggested Answer (a) We know from the calculations on page 60 that (setting n = 0):

kss =

1 s 1−α

δ

(b) Using (4.4.8) and (4.4.12):

rss = FK (Kss , 1) − δ α−1 = αkss −δ α

= −1 δ s (c) If the savings rate increases: i. Long run GDP is given by:

α yss = kss =

α s 1−α

δ

so long-run GDP increases with s, for any value of s

56

ii. Long-run consumption is:

css = yss − δkss Replacing the values, we have that when s rises from 0.4 to 0.5, steady-state consumption falls from 1.27 to 1.19. This means that the economy is on the wrong side of the Golden Rule. We know that with a Cobb-Douglas production function the Golden Rule requires s = α. In this example s > α. (d) Replacing the values (with s = 0.4), the steady-state interest rate is r = −0.0125. (Negative interest rates are a sign of being on the wrong side of the Golden Rule).

0

0

With a negative interest rate,

0

equation (6.3.9) implies that the u (ct ) < u (ct+1 ). Since u (c) is a decreasing function, this implies that ct > ct+1 . The Friedmans take advantage of the negative interest rates to have an initially-highbut-decreasing consumption plan.

6.8

The Marginal Propensity to Consume Suppose a household has preferences given by (6.3.1) over consumption across in nite periods. Its discount factor and the real interest rate satisfy:

1+r =

1 β

The household starts with no wealth, will receive income

y0 in period t = 0 and income ȳ in every

subsequent period, from t = 1 onwards. (a) Use the Euler equation (6.3.9) and the budget constraint (6.3.8) to nd an expression for initial-period consumption c0 . (b) Compute

∂c0 ∂y0

(c) Suppose r = 0.04. What is the marginal propensity to consume out of a purely temporary increase in income? Describe in words what a household does with a temporary increase in income.

Suggested Answer (a) With 1 + r =

1 β , the Euler equation (6.3.9) implies that consumption is constant over time: ct = c0

for all t. We can therefore rewrite the budget constraint as

c0

T X 1 ȳ = y + 0 t t t=1 (1 + r) t=0 (1 + r) 1+r 1 c0 = y0 + ȳ r r ry0 + ȳ c0 = 1+r

T X

(b) Taking the derivative:

∂c0 r = ∂y0 1+r (c) The marginal propensity to consume is

0.04 1.04 ≈ 0.0385. The household spreads the temporary increase

in income over time by saving it and only consuming the interest.

57

6.9

The Marginal Propensity to Consume in Proportional Terms A household solves a special case of the problem in Exercise 6.1, with τ1 = τ2 = a0 = 0 and y2 = my1 , where m is some number. (a) Suppose that both y1 and y2 increase by x%. By what percentage do c1 and c2 increase? (b) Suppose y1 increases by x% but y2 remains unchanged. By what percentage does c1 increase? How does this depend on m and r ? Explain.

Suggested Answer (a) We have:

m 1 + 1+r

c1 =

1

1

1 + β σ (1 + r) σ Therefore:

y −1 1

∂c1 y1 =1 ∂y1 c1

If both y1 and y2 increase x%, then c1 increases x% as well. (b) Now we have:

y2 y1 + 1+r

c1 =

1

1

1 + β σ (1 + r) σ

−1

Therefore:

∂c1 y1 1 = 1 1 −1 ∂y1 c1 σ 1 + β (1 + r) σ

y1 y2 y1 + 1+r 1

1

1+β σ (1+r) σ −1

y1 y2 y1 + 1+r y1 = 1 y1 + my 1+r 1 = m 1 + 1+r =

The proportional increase is larger when r is high or m is low. The reason is that consumption is proportional to the present value of income.

y1 represents a higher fraction of the present value of

income when r is high (because this increases the present value of present income relative to future income) or when m is low (because this means lower future income).

58

CHAPTER 7

Labor and Leisure Exercises 7.1

Labor Supply Suppose household preferences are given by:

u (c, l) = c + θ log (l) where c is consumption, l is leisure and θ is a parameter. Households have a total of one unit of time and can supply labor in a competitive labor market at a wage w . (a) Find an expression for the fraction of their time that households spend in market work. (b) If this was the right model and one looked at households in di erent countries, how would hours of work correlate with wage levels? How does this compare to the empirical evidence?

Suggested Answer (a) The FOC (7.2.3) is:

θ =w l so, using that L = 1 − l:

L=1−

θ w

(b) If this was the right model, then households in higher-wage countries would spend a higher fraction of their time in market work. The empirical evidence in Figures 7.3.1 and 7.3.2 shows that the opposite is true.

7.2

Military Service An economy is populated by a representative household, whose preferences are described by:

u (c) + v (l)

59

where c is consumption and l is leisure. The household has 1 unit of time so its budget is:

c = w (1 − l) − τ where w is the wage rate and τ is a tax that the government collects. Notice that τ is a lump-sum tax, not an income tax: the household must pay the same amount regardless of how much income it earns. (a) Set up the Lagrangian for the household's optimization program and nd rst-order conditions. (b) Use the budget constraint to replace c in the rst-order condition to obtain a single equation that relates the household's choice of l to w and τ . (c) Suppose that the government uses all the tax revenue to hire soldiers for the army, paying them the prevailing wage rate. How many units of soldiers' time can the government a ord? Denote this number by m. (d) Now suppose that instead of taxing citizens to hire soldiers, the government imposes compulsory military service: the representative household has to dedicate m units of time to serving in the army, unpaid.

The rest of their time they can use as they please, and they pay no taxes.

Set up the

household's optimization program under this policy. (e) Show that the representative household's level of consumption, leisure, army labor and non-army labor are the same under both policies. Explain.

Suggested Answer (a) The Lagrangian is:

L(c, l, λ) = u(c) + v(l) − λ (c − w(1 − l) + τ ) with rst order conditions:

u0 (c) − λ = 0 v 0 (l) − λw0 (b) Replacing:

v 0 (l) = u0 (w(1 − l) − τ ) w (c)

τ m= w

(d) The household solves:

max u(c) + v(l) c,l

s.t. c = w(1 − l − m) (e) The household's FOC is, as before:

v 0 (l) = u0 (c)w

60

and the budget constraint is:

c = w(1 − l − m) If m =

τ w , then this reduces to

c = w(1 − l) − τ which is the same budget constraint as before. Therefore the household will make the same choices.

It is equivalent for the government to take a xed amount of money from the household and then buy the household's time with it or to take the household's time directly.

7.3

Consumption Taxes and Labor Supply Suppose a household solves the following problem:

max u (c) + v (l) c,l

s.t.

(1 + τc ) c ≤ w (1 − τ ) (1 − l)

where c is consumption, l is leisure and τ is an income tax. τc is a consumption tax, so that if the household wishes to consume c it must pay (1 + τc ) c.

(a) Draw the household's budget constraint. What is the slope?

(b) Find the rst-order conditions for the household's consumption-leisure decision. How do the e ects of τ and τc relate to each other? Explain.

(c) If the household chooses c

∗

∗

and l , how much tax revenue does the government collect?

(d) Suppose the government had originally set τc = 0 and τ > 0 and now wants to enact a tax reform that uses consumption taxes instead of income taxes. What would be the level of τc that leaves the decisions of the household unchanged?

(e) How much revenue does the government collect under the new system? How does it compare to the old system? Explain.

Suggested Answer

(a) The slope is −

w(1−τ ) 1+τc

61

(b) The FOC is:

v 0 (l) =

w(1 − τ ) 0 u (c) 1 + τc

Both taxes work in the same way: if the household decides to sell a marginal unit of time in the market in order to buy consumption, then it will have to pay a tax, either on the earnings or on the spending. If the household instead dedicates that unit of time to leisure, there is no tax. (c) The government's total tax revenue is:

R = τ w(1 − l∗ ) + τc c∗ (d) In order to leave the household's decisions unchanged, setting:

τc =

w(1−τ ) 1+τ )c must remain the same. This means

τ 1−τ

The government will collect:

R = τc c∗ τ = c∗ 1−τ τ = w(1 − τ )(1 − l∗ ) 1−τ = τ w(1 − l∗ ) which is the same as under the old system.

Whether the government chooses to tax income or

consumption, as long as the overall tax on selling time to the market to buy consumption is the same, the government's revenue will be the same.

62

7.4

Puerto Rico Read the following article: island.

https://www.economist.com/united-states/2006/05/25/trouble-on-welfare-

Is what was going on in Puerto Rico consistent with the theories discussed in this chapter?

Discuss.

Suggested Answer There are many possible interpretations of what the article describes.

Here is one.

Puerto Ricans on

average receive positive net transfers from the government. This has an income e ect on labor supply, as illustrated in Figure 7.2.4. As a result, Puerto Ricans rationally choose higher leisure than they otherwise would.

7.5

Prescott's Calculation Suppose preferences for consumption and leisure are:

u (c, l) = log (c) + α log (l) and households solve:

max u (c, l) c,l

s.t.

c = w (1 − τ ) (1 − l) + T

(a) Find rst-order conditions for the consumption-leisure decision. (b) Use the budget constraint to solve for leisure l.

You should get an explicit expression for l as a

function of w , τ , T and α. (c) Suppose T = 0. How does l respond to the tax rate τ ? What does this mean?

Now suppose that in both Europe and the US we have:

α = 1.54 w=1 but in the US we have:

τ = 0.34 T = 0.102 while in Europe we have:

τ = 0.53 T = 0.124

63

(d) Compute the amount of leisure chosen in the US and Europe. If we interpret 1 as your entire adult lifetime, what fraction of their adult lives do people in Europe and the US work? Comment on the respective role of taxes and transfers in this analysis using your answers to parts (b) and (c). (e) The values for τ and T above are not arbitrary. If you did the calculations correctly, you should nd that both governments have balanced budgets (up to rounding error), i.e. they redistribute all the tax revenue back as transfers. Check that this is the case. (f ) Assuming the production function is Y

= L = 1 − l, how much lower is GDP per capita in Europe

compared to the US? (g) Compute the relative welfare of Europe by solving for λ in the following equation:

u (cEurope , lEurope ) = u (λcU S , lU S ) Interpret the number λ that you nd (h) How do the answers to questions (f) and (g) compare? Why? (i) Suppose a European policymaker sees Prescott's calculation and concludes that Europe could increase its welfare by a factor of

1 λ by reducing its tax rate and level of transfers to US levels. Do you think

they are right? Why? Don't answer this question mechanically: think about what this calculation does and what it leaves out.

In any calculation of this sort, an important parameter is the elasticity of labor supply. One de nition of elasticity that is often looked at by labor economists is known as the Frisch elasticity. It is based on the answer to the following question: suppose we increased wages but adjusted the household's income so that consumption remained constant: how would labor supply change?

1 Let's calculate the Frisch

elasticity in Prescott's model. (j) Use your answer to part (a) to nd an expression for labor supply (1 − l) in terms of w (1 − τ ), c and

α. Notice that now we are holding consumption constant, so the idea is that we don't replace c from the budget constraint like we did in part (b). (k) Use your answer to part (j) to nd an expression for

∂(1−l) w(1−τ ) ∂w(1−τ ) 1−l , i.e. the elasticity of labor supply

with respect to after-tax wages, holding consumption constant. (l) Plug in the values of α, τ , w , c and l that you found for the US case into the expression for elasticity.

What number do you get? Empirical estimates of this elasticity are usually in the range of 0.4 to 1.

How does that compare to the elasticity implied by Prescott's model? Why does that matter for our conclusions about tax policy?

Suggested Answer (a) The FOC is:

α 1 = w(1 − τ ) l c

1 Notice that it's not exactly the same as isolating the substitution e ect as in Figure 7.2.2, because that was holding utility constant instead of consumption; but it's related.

64

(b) Using the budget constraint:

α w(1 − τ ) = l w(1 − τ )(1 − l) + T αw(1 − τ ) + αT l (1 + α)w(1 − τ ) (c) If T = 0, then:

l= so it does not depend on τ .

α 1+α

Income and substitution e ects exactly cancel out.

The log utility

function is like having σ = 1, so we know from equation 7.2.4 that this is what happens. (d) For the US, we have l = 0.7 and for Europe l = 0.77. In the US, people spend 30% of their lifetime working while in Europe it's 23%. This is due to higher taxes and transfers in Europe. As we saw above, if it was just a di erence in taxes then Europeans would work the same as Americans. It's the combination of taxes and transfers which makes the di erence. (e) For the US, τ (1 − l) ≈ 0.102 and for Europe τ (1 − l) ≈ 0.124, so the budget constraints hold. (f ) GDP per capita is 1 −

0.23 0.3 ≈ 22% lower in Europe.

(g) Solving:

log(cE ) + α log(lE ) = log(λcU ) + α log(lU ) α cE lE α cU lU α (1 − lE )lE = α (1 − lU )lU

λ=

≈ 0.9 GDP per capita is 22% lower in Europe but it welfare is not equivalent to the US with 22% lower consumption. It is higher than this because the increased leisure partly compensates for the lower consumption. Welfare is equivalent to the US with 10% lower consumption (h) If we thought this was the right model, then the conclusion would be correct. Indeed, it would be even better to reduce taxes and transfers to zero since they serve no purpose in the model and only distort labor supply. One important feature of the world that is not in this model in inequality. Is some people have higher income than others, a tax-and-transfer system redistributes from the rich to the poor, which can be desirable from a behind-the-veil-of-ignorance perspective (or some other philosophical principle).

If we accept that, it is no longer clear that shifting from the European

tax levels to the US tax levels would be desirable. This would depend on the levels of underlying inequality and the degree of risk aversion. (i) Using the FOC and solving for 1 − l

1−l =1−

αc w(1 − τ ) 65

(j)

∂ (1 − l) w (1 − τ ) αc = ∂w (1 − τ ) 1 − l w(1 − τ ) − αc l = 1−l (k) Replacing l

= 0.7, we get an elasticity of 2.33, quite a bit higher than empirical exercises.

This

matters because if we overestimate the elasticity of labor supply, then we overestimate how much labor supply responds to di erences in tax policy, and therefore we overestimate how much of the di erences in labor supply between the US and Europe can be attributed to tax di erences and what the welfare e ects of tax policy are, even leaving inequality out of the calculation.

7.6

The Protestant Work Ethic Suppose that the representative household has preferences given by:

u(c, l) = log(c) + θ log(l) where c is consumption, l is leisure and θ is a parameter. The production function is given by:

F (K, L) = K α L1−α where K is the capital stock, which we take as given and equal to 1, L is labor and α is a parameter.

Firms hire workers in a competitive labor market where the wage is w .

(a) What will be the equilibrium wage w ? (b) How does w depend on θ ?

Suppose the household suddenly develops a so-called Protestant work

ethic and stops enjoying leisure so much, what will happen to real wages? Explain.

Suggested Answer (a) The worker's FOC is:

θ w = c l and the budget constraint is:

c = w(1 − l) Replacing, we get:

L=1−l =

1 1+θ

From the rm's side we have the FOC:

w = (1 − α)K α L−α

66

so

α

w = (1 − α)K α (1 + θ) (b) Wages are increasing in θ .

θ measures how much workers enjoy leisure. If they enjoy leisure more,

they will supply less labor and rms will bid up the wage as they compete for scarce workers.

If

workers develop a Protestant work ethic , which we can capture as a fall in θ , wages will fall as labor becomes abundant.

7.7

The Beveridge Curve Assume the labor market is well described by the search model presented in Section 7.5. (a) Suppose there is an improvement in recruiting technology, so that the matching function becomes

Am(V, U ), with A > 1. What happens to the number of vacancies? What happens to the after-hiring unemployment rate? Would this produce something that looks like a Beveridge Curve? (b) Suppose there is an increase in the initial (before-hiring) unemployment rate. What happens to the number of vacancies? What happens to the after-hiring unemployment rate? Would this produce something that looks like a Beveridge Curve?

Suggested Answer (a) In terms of Figure 7.5.3, this produces an upward shift in the free-entry curve, so in equilibrium there are more vacancies. More vacancies, combined with a better matching technology, results in lower after-hiring unemployment, a movement consistent with a Beveridge Curve. (b) In terms of Figure 7.5.3, this also results in an upward shift in the free-entry curve. The reason is that

m and therefore q is increasing in U : the chances of nding a suitable worker are higher when there is more unemployment, making vacancy-creation more attractive. Without making more assumptions about the matching function m we cannot say whether after-hiring unemployment will be higher or lower: there are more hires but pre-hiring unemployment is higher. Quite possibly after-hiring unemployment will be higher, so we would have unemployment and vacancies moving in the same direction, the opposite of what the Beveridge Curve postulates.

67

CHAPTER 8

Investment Exercises 8.1

Valuation Charlie's cheese factory has a very precise business plan for 2019-2028, shown below (you can download it as an Excel spreadsheet from the book website):

Year

Pro ts

Investment

Dividend

2019

80

50

30

2020

77

49

28

2021

86

49

37

2022

91

47

44

2023

98

47

51

2024

98

48

50

2025

109

200

-91

2026

120

57

63

2027

119

57

62

2028

125

61

64

(in 2025 the main storage facility will need to be replaced, hence the higher investment).

From 2029

onwards, dividends will increase at a rate of 2% a year forever. The interest rate is 6%. (a) Use the Gordon growth formula (8.1.2) to calculate the value that the factory will have in 2028 after paying dividends (i.e. the value not including the value of the dividends it will pay in 2028). (b) Compute the present value of the entire in nite stream of dividends that starts in 2019.

Suggested Answer (a)

V =

64 × 1.02 = 1, 632 0.06 − 0.02 69

(b) To compute the present value, just add 1, 632 to the 2028 dividend and compute the discounted sum.

If we assume that dividends are paid at the end of the year, so even the 2019 dividend is discounted once, this gives a present value of 1, 155. If we assume that dividends are paid at the beginning of the year (or that we are at the 2019 dividend date when we start the calculation), then the present value is 1, 225.

8.2

Bond Pricing Suppose the interest rate is 4% and is expected to stay at 4% forever. On January 1, 2018, the government issued a T -year bond, which pays coupons of 4 cents every year on January 1 (starting in 2019) and then pays 1 dollar (in addition to the 4-cent coupon) on January 1 of the year 2018 + T . (a) Use formula (8.1.1) to compute what the market price of the bond should be. How does it depend on T ? Explain. (b) Suppose that after the bond has been issued, market conditions change and interest rates fall to 3%, and are expected to remain at 3% forever. What is the market price of the bond now? How does it depend on T ? Explain.

Suggested Answer (a)

P =

T X 0.04

1.04t t=1

+

1 =1 1.04T

This price does not depend on T . No matter what the maturity of the bond is, if the bond's coupon rate is equal to the market interest rate, the bond will trade at a price of 1 (known as trading at par ). (b)

T X 0.04

1 1.03T

0.04 1 0.04 = + 1− 0.03 1.03T 0.03

P =

1.03)t t=1

+

This price is increasing in T . The bond's coupon rate is higher than the market rate. The longer the bond, the more attractive it is to obtain this rate, so the higher the bond will trade.

8.3

Looking at the Stock Market Speci c Mills is a company that has just started operations. Its business consists of buying wheat from farmers, grinding it to produce our and selling the our.

It is very small relative to both the wheat

market and the our market. Its assets consist of a single plant that cost $10 million to build. It issued 1 million shares, which currently trade in the stock market at a price of $10 per share. (a) What is Q for Speci c Mills?

70

(b) Speci c Mills' share price suddenly rises to $15 per share. What is Q now?

Management is considering expanding the plant. It has calculated that in order to expand the plant to λ times its current capacity, it is going to have to carry out additional investment that will cost

(λ − 1) + ψ(λ − 1)2 times its original investment, with ψ = 1. Assume that management trusts the stock market investors' opinion about the pro tability of the grain-processing industry. (c) By how much should it expand its capacity? (d) How much will it spend in total to do so? (e) Now suppose that instead of Speci c Mills, which is small relative to the industry it operates in, the same situation arose for a company that was large relative to its market. How does its reaction to its stock market price compare to that of Speci c Mills? Why?

Suggested Answer (a)

Q=1

(b)

Q = 1.5

(c) If management trusts the market's opinion, then they believe that each dollar of installed capacity is worth 1.5 dollars. To maximize the value of the company they solve:

max λλKQ − (λ − 1) + ψ(λ − 1)2 K with FOC:

KQ − (1 + 2ψ(λ − 1)) K0 λ=1+

Q−1 2ψ

Management will expand capacity by a factor of λ. Setting Q = 1.5, ψ = 1 and K = $10 million we get λ = 1.25%, so Speci c Mills will expand capacity by 25%. (d) It will spend

S = (λ − 1) + ψ(λ − 1)2 K

Q−1 Q−1 = +ψ K 2ψ 2ψ = $3.125 million (e) A large rm understands that if it expands capacity, it will signi cantly increase demand in the wheat market (driving up the price of inputs) and supply in the our market (driving down the price), so

71

it realizes that its market value per unit of installed capacity will fall if it expands. Therefore it will react less strongly to the stock market than a small rm would.

8.4

The Magnitude of Adjustment Costs The slope of the relationship illustrated in Figure 8.1.1 is approximately 0.01, so an increase in Q of 1 is associated with an increase in the investment-to-capital ratio

I K of 0.01.

(a) If adjustment costs are given by formula (8.1.4), what value of

ψ would be consistent with this

observation? (b) If a rm's investment is equal to 10% of its capital stock, how much does it have to spend in adjustment costs as a fraction of its total investment? How about if a rm's investment is 20% of its capital?

Suggested Answer (a) We know that

I Q−1 = K ψ so

I ∂K 1 = ∂Q ψ

If the left hand side is 0.01 this means that ψ = 100. (b) Adjustment costs are

ψI 2 2K . As a fraction of investment, they are:

µ=

ψI 2 2K

I ψI = 2K

Replacing ψ = 100 and

I K

= 0.1 we get µ = 5, so the rm will spend 5 times more in adjustment

costs than it does in investment (equivalently, adjustment costs are costs). If

I K

5 6 of investment-plus-adjustment-

= 0.2 we get µ = 10, so the rm spends 10 times more on adjustment costs than on

investment.

8.5

A Risky Asset The world lasts two periods. Danube.com is a large internet-based retailer. At t = 2, it will pay a dividend of $150 per share if the economy is doing well and $50 per share if the economy is doing poorly.

The

representative household will consume $40,000 if the economy is doing well and $30,000 if the economy is doing poorly. Its preferences are given by E

1−σ c 1−σ

. The interest rate is 10%. Use formula (8.2.5) to

determine at what price the representative household would be indi erent with respect to buying shares in Danube.com, for the following values of σ : 0.5,1,2 and 10. Explain.

Suggested Answer 72

σ

p

0.5

87.6

1

84.4

2

78.2

10

50.3

In all cases the price is in between the average discounted dividend (90.9) and the worst-case-scenario discounted dividend (45.5). This is because Danube's dividends are higher in the state of the world where the household has higher consumption and therefore lower marginal utility. Higher risk aversion means that weighting by marginal utility makes more of a di erence so the price is lower.

8.6

A Risky Investment Ingrid plans to live for two periods and only cares about consumption in the second period.

She is

considering getting her MBA in the rst period. If she does it, it's going to cost her $200,000 between tuition and foregone wages.

Once she graduates, there is a 50% chance that in the second period she

will get the job she wants, which will pay her $5,000,000 and an 50% chance she'll get a regular job that will pay her $80,000.

If she doesn't get her MBA, she'll put $200,000 in the bank, where it will earn

10% interest, and then get a regular job that pays her $80,000. In all cases, since she will only live for two periods, she will consume everything she has in period 2. Suppose Ingrid's preferences over period-2 consumption are given by E

1−σ c 1−σ

.

(a) For what values of σ is it a good idea to get an MBA? Explain. (b) Now suppose σ = 1.2. For what values of the interest rate is it a good idea to get an MBA? Explain.

Suggested Answer (a)

σ solves: 1 2

5, 000, 0001−σ 80, 0001−σ + 1−σ 1−σ

=

300, 0001−σ 1−σ

which gives σ ≈ 1.38. It is a good idea to get an MBA as long as σ < 1.38. Otherwise, an MBA is too risky. (b)

r solves: 1 2

5, 000, 0001−σ 5, 000, 0001−σ + 1−σ 1−σ

1−σ

=

(200, 000(1 + r) + 80, 000) 1−σ

which gives r ≈ 69%. It is a good idea to get an MBA as long as the interest rate is lower than 69%.

Otherwise, the opportunity cost of the interest Ingrid would earn on her tuition money is too high.

8.7

Credit Constraints Two world lasts two periods. A rm's production function is F (K, L) = At K

α

L1−α , where At can take

di erent values in each period. In each period, the rm can hire labor in a competitive labor market at the same wage w , which the rm takes as given.

However, there is no market for renting capital: the

73

rm can only use capital that it owns. If the rm owns K units of capital and decides to hire L workers, then it earns F (K, L) − wL. The rm starts o having K1 units of capital in the rst period. There is no depreciation. At the end of period 1, the rm can buy capital for period 2 by either using its earnings or by borrowing. Loans must be paid back in period 2. The interest rate is r . The rm can also use its earnings to pay dividends to its shareholders in period 1. The maximum amount that lenders are willing to lend to the rm is b.

(a) Set up the problem of the rm that needs to decide how many workers to hire in each period. Note that this problem can be solved period-by-period taking Kt as given.

Find an expression for the

period-t pro ts of a rm that takes as given its capital stock Kt and chooses how much labor to hire, i.e. for π(Kt ) ≡ maxL F (Kt , L) − wL. (b) Set up the problem of the rm that must decide how much to pay in dividends, how much to borrow and how much to invest. (c) Assume b = ∞. Show that the optimal amount of investment depends on A2 but not on A1 . Explain. (d) Assume b = 0. Show that if A1 is su ciently high, then the optimal amount of investment is the

∗

same as in part (c). Find the minimum level of A1 such that this is the case, and denote it A1 . Show

∗

that for A1 < A1 , investment depends on A1 . Explain.

∗

(e) Suppose A1 < A1 . How does the rm react to an increase in A2 ? How does the rm react to an increase in b? Explain.

Suggested Answer (a)

max F (Kt , L) − wL = At Ktα L1−α − wL L

FOC:

(1 − α)At Ktα L−α = w

L=K

(1 − α)A w

α1

so

(1 − α)A π(Kt ) = AK K w

α1 −1 1 1−α = αKA α w α

α1 !1−α

− wK

1

Unfortunately, this function is linear in K . This means that if αA α

(1 − α)A w

1−α w

α1

α1 −1

> 1 + r then the rms

wants in nite investment, and if the opposite is true the rms wants negative investment. In order to get an interior solution for investment we need to have a concave pro t function. One example of

74

α

γ

this is if the production function is At Kt L

with γ < 1 − α. This has decreasing returns to scale

and a concave pro t function. Below is a suggested solution for this case.

Suggested Answer for alternative version (a)

max F (Kt , L) − wL = At Ktα Lγ − wL L

FOC:

γAt Ktα Lγ−1 = w 1

γAK α 1−γ L= w so

π(Kt , At ) = AK

α

γAK α w

γ 1−γ

α

−w

1

= (1 − γ)K 1−γ A 1−γ

γAK α w

1 1−γ

γ γ 1−γ

w

(b)

max d1 +

d1 ,d2 ,b,I

d2 1+r

s.t.d1 + I = π(K1 , A1 ) + a d2 = π(K1 + I, A2 ) − a(1 + r) a≤b where a > 0 represents borrowing and a < 0 represents saving. (c) If b = ∞ then we can ignore the last constraint and, replacing the other constraints in the objective function, the problem reduces to:

max π(K1 , A1 ) − I + I

π(K1 + I, A2 ) 1+r

and it's immediate that the solution does not depend in A1 . (d) In the solution from part (c), setting d1 = 0, then

a = I − π(K1 , A1 ) Since π(K1 , A1 ) is increasing in A1 and limA1 →∞ π(K1 , A1 ) = ∞, then for su ciently high A1 , the

∗

rm does not need to borrow to invest I , so the constraint does not bind. To nd A1 we must rst

75

nd the level of investment from part (c). The FOC is:

∂π(K2 , A2 ) =1+r ∂K2 γ γ 1−γ α+γ−1 1 α K2 1−γ (1 − γ)A21−γ =1+r 1−γ w 1−γ

γ γ γ−1 α+γ−1 1 1−γ γ−1 K2 = (1 + r)(1 − γ)A2 α w 1−γ

γ

1 γ γ−1 α+γ−1 1−γ γ−1 (1 + r)(1 − γ)A2 − K1 I= α w Setting d1 = 0, the rm can invest I without borrowing if:

π(K1 , A1 ) ≥ I 1−γ

γ γ γ 1−γ γ γ−1 α+γ−1 1 1 α 1−γ 1−γ γ−1 1−γ (1 + r)(1 − γ)A2 (1 − γ)K A1 ≥ − K1 w α w A∗1 is the number such that this condition holds with equality. If A1 < A∗1 then this condition does not hold, so the borrowing constraint will bind and the rm will invest I = π(K1 , A1 ), which is increasing in A1 . If the rm's is unable to borrow, then it can only invest out of retained earnings. Therefore it will increase investment if it has has higher productivity and therefore higher earnings. If the rm were not borrowing constrained, desired investment would depend only on future productivity, not current productivity. (e) The rm does not increase I in response to an increase in A2 because it already is investing all its retained earnings, which is the maximum investment it can reach. It reacts to an increase in b (a relaxation of the borrowing limit) by borrowing in order to invest.

8.8

An Earthquake Suppose an earthquake destroys a large part of the capital stock at time t.

Assume interest rates and

future labor supply are not a ected by the earthquake, and there are no adjustment costs.

(a) What will happen to investment? (b) How does Kt+1 compare with and without the earthquake?

Suggested Answer If interest rates and future labor supply (and the production function) are unchanged, then the level of capital that equated the marginal product of capital to r + δ is the same as before the earthquake.

Therefore Kt+1 is the same as before the earthquake. This requires higher investment, since the starting point is lower.

8.9

76

Aggregate Investment with Risk

The world lasts two periods. The aggregate production technology for period 2 is given by:

F (K, L) = AK α L1−α K is the aggregate capital stock in period 2 and L is the labor force, which is exogenous and normalized to L = 1. Period 2 is the end of the world, so capital depreciates fully (δ = 1) and the representative household will consume F (K, L)

= AK α L1−α .

variable, which can take two possible values:

The utility function is u(c)

= log(c). A is a random

AH = 1 + ε or AL = 1 − ε, with equal probability. The

interest rate between periods 1 and 2 is r .

Now consider the following investment project:

investing an additional unit in period 1 to obtain an

additional unit of capital in period 2. (a) What is the dividend produced by the marginal unit of capital?

Express it as a function of the

aggregate capital stock K and realized productivity A. (b) Suppose ε = 0. For what value of K is the net present value of additional investment exactly zero? (c) Now suppose ε > 0. For what value of K is the net present value of additional investment exactly zero? (Use equation (8.2.5) to guide your answer) (d) How does K depend on ε? Explain.

Suggested Answer (a)

αAK α−1 L1−α

(b)

αAK α−1 L1−α = 1 + r

1 αA 1−α L K= 1+r (c)

1 1 1 1 α−1 1−α α−1 1−α L L 2 αAL K AL K α L1−α + 2 αAH K AH K α L1−α 1 1 1 1 2 AL K α L1−α + 2 AH K α L1−α

=1+r

αK α−1 L1−α 1 − 2 = 1 + r K=

(d)

αA 1 − 2 1+r

1 ! 1−α

L

K is decreasing in . Both the marginal product of capital and consumption depend on realized productivity. Therefore the dividend from an investment project co-moves negatively with the marginal utility of consumption. The higher the risk, the less attractive it is to invest, and therefore the lower the level of investment.

77

CHAPTER 9

General Equilibrium 9.1

The First Welfare Theorem Consider the following policies. In each case, explain whether, in your view, the policy is justi ed and why. (a) Setting a maximum price that electrical utilities may charge. (b) Mandatory vaccinations. (c) Workplace safety standards. (d) Rent control. (e) A 35-hours-per-week limit on working hours. (f ) Banning grocery stores from opening on Sundays. (g) Banning self-service at gas stations.

Suggested Answer All of these are debatable. Here are some possible arguments in favor of each of these policies, which you may nd more or less convincing. (a) Utilities are a natural monopoly, unregulated they would set price above marginal cost. (b) Getting vaccinated creates a positive externality since one does not spread diseases onto other people. (c) This one is tricky:

why can't workers and rms negotiate safety standards by themselves?

One

possibility is that checking a rm's safety standards is a public good / positive externality: it only needs to be done once and then the information is provided to all workers. A pass-fail test is a way to convey this information to workers in a simple way. Open question: should workers be allowed to voluntarily and knowingly work in a rm that does not meet safety standards? If not, why not? (d) This can be a way to redistribute wealth from landlords to renters (not necessarily the only or the best way to do this). Possibly it's also a way for interesting-but-poor people to live in certain neighborhoods, where they create positive externalities on other residents.

79

(e) One argument is that by restricting labor supply, this raises equilibrium wages, bene ting workers at the expense of capital owners (see Exercise 9.4). (f ) One argument is that the inability to shop on Sundays forces people to spend their time in positiveexternality activities like hanging out with friends and family. (g) This one is very hard to justify. Presumably it's designed to increase labor demand and equilibrium wages, but see Exercise 5.5.

9.2

Storage Suppose the production function is the following:

F (K, L) = K and the rate of depreciation is δ = 1. (a) Why is this production function called storage ? (b) What will be the real interest rate and the wage in this economy? (c) With standard preferences, will consumption increase over time, remain constant over time or decrease over time?

Suggested Answer (a) If you transform one good at time t into capital , this capital provides one good at time t + 1, so it's just like storing the good. (b) The real interest rate will be FK − δ = 0 and the wage will be FL = 0. (c) With standard preferences (with β

< 1) and a real interest rate of 0, the Euler equation implies

consumption decreasing over time.

9.3

Baumol's Cost Disease The representative household consumes two di erent goods:

x (clothes) and y (performances by string

1 quartets). Its preferences are given by:

1 −1 −1 1 −1 u (x, y) = α x + (1 − α) y

Each of the two goods is produced using only labor. If Lx units of labor work in producing clothes, output of clothes is x = Ax Lx ; if Ly units of labor play in string quartets, output is y = Ay Ly . The household supplies one unit of labor inelastically and is indi erent as to how much it supplies to each industry. The price of clothes is denoted px and the price of a quartet performance is py . The wage rate is w .

1 This is known as a constant elasticity of substitution utility function. The parameter represents the elasticity of substitution between goods x and y : a higher number means that that are close to perfect substitutes. preferences converge to the Cobb-Douglas case u(x, y) = xα y 1−α .

80

For reference, when → 1, these

(a) Use a software like Matlab or Excel to plot two sets of indi erence curves for this utility function.

Set α = 0.5 in both cases, = 0.5 in one of them and = 2 in the other. (b) Set up the problem of a household that obtains income w from supplying one unit of labor inelastically and has to decide how much to consume each of the two goods. Show that the household will consume:

x=

w α p

px p

−

w y = (1 − α) p where p ≡

py p

−

1 1− . [This takes a bit of work. If you want, you can look up the αp1− + (1 − α) p1− x y

derivation on the book website]. (c) Set of the problem of a rm in each of the two industries that needs to decide how much labor to hire to maximize pro ts. What values of

py px w and w are consistent with each industry hiring a positive

but not in nite amount of labor? (d) What will be the price of string quartet performances relative to clothes

py px ?

(e) Suppose that over time Ax rises and Ay stays the same. What does this mean? Do you nd this plausible? What will be the e ect on the relative price of string quartets? Explain. (f ) Use your answers to parts (b) and (d) to compute the relative quantities of each of the two goods that will be consumed, i.e. solve for the equilibrium level of

y x.

(g) Use your answer to part (f) to compute the relative allocation of labor across the two industries, i.e. solve for the equilibrium level of

Ly Lx .

(h) Suppose that over time Ax rises and Ay stays the same.

What happens to the quantity of string

quartet performances over time? How does the answer depend on ? Explain.

Suggested Answer

(a) (b) The household's preferences are:

1 −1 −1 1 −1 u (x, y) = α x + (1 − α) y

81

The household solves:

max u (x, y) x,y

(9.1)

s.t. px x + py y ≤ w The FOCs are: 1 −1 1 −1 −1 1 − 1 1 −1 α x + (1 − α) y x − λpx = 0 α −1

1 −1 1 1 − 1 −1 −1 1 −1 α x + (1 − α) y (1 − α) y − λpy = 0 −1

(9.2)

(9.3)

Divide (9.2) by (9.3) and rearrange:

x α = y 1−α

px py

− (9.4)

Replace (9.4) in the budget constraint (9.1):

px y

α 1−α

px py

− + py y = w

Rearrange:

αp1− + (1 − α) p1− = (1 − α) x y

w − p y y

De ne: 1 1− p ≡ αp1− + (1 − α) p1− x y

Therefore (9.5) can be rewritten as:

w − p y y

p1− = (1 − α) and, rearranging:

w y = (1 − α) p

py p

−

Similarly:

x=

w α p

px p

−

(c) In industry x the problem is:

max px Ax L − wL L

and similarly in industry y :

max py Ay L − wL L

82

(9.5)

If both industries have an interior solution, it must be that (d)

py px 1 1 w = Ax and w = Ay .

py Ax px = Ay

(e) Productivity increases in clothes manufacturing: each worker produces more clothes per hour as time goes by and technology improves. However, it still takes four musicians to perform a string quartet productivity is not improving. Therefore the price of string quartets relative to clothes will increase over time. (f )

− py w p (1 − α) p y = − x px w pα p − 1 − α py = α px − 1 − α Ax = α Ay (g)

y/Ay Ly = Lx x/Ax =

(h) If > 1, then

1−α α

Ax Ay

1−

Ly Lx falls over time, which implies that Ly , the amount of labor dedicated to string

quartets (and thus the number of string quartet performances) falls over time. Instead, if < 1, the opposite happens. If > 1, then the representative household is quite willing to substitute between di erent goods.

As clothes productivity improves and lowers their relative price, the household

strongly substitutes towards them, and demands fewer string quartets. Instead, if < 1, the household is less willing to substitute. As clothes productivity improves, the household reallocates workers away from the clothes sector in order to also increase its consumption of string quartets.

9.4

Pro-Worker Policies Consider the following one-period economy.

The production function is F (K, L).

household owns all the capital stock K , which is exogenously given.

The representative

The household's preferences are

u(c, l) where, as usual, l = 1 − L. The labor and capital markets are perfectly competitive.

Denote by L

∗

the amount of labor that households supply in a competitive equilibrium with no government

intervention.

Now suppose that the government dictates a new law that prohibits the household from

∗

working more than L

− ε, where ε is a small positive number.

(a) Show that the policy will increase wages and lower the rental rate of capital. (b) Show that the policy will make the representative household worse o .

83

(c) Now suppose that there are two representative households in the economy. Household A owns the capital stock and does not work.

Its preferences are given by u(c).

Household B has preferences

u(c, l) and does not own any capital. Suppose the government enacts the same policy as before (i.e. ∗

it prohibits household B from working more than L

− ε, where L∗ is the amount it works under a

no-intervention equilibrium). Show that this policy makes household A worse o and household B better o .

Suggested Answer (a) This follows immediately from equations (9.1.9) and (9.1.10) (b) This is an implication of the First Welfare Theorem. One way to see this is to de ne:

l (ε) ≡ l∗ + ε c (ε) ≡ F (K, L∗ − ε) U (ε) ≡ u (c (ε) , l (ε)) Now use a second order Taylor expansion:

1 U (ε) ≈ U (0) + U 0 (0) ε + U 00 (0) ε2 2 " #

2

2 1 ∂l ∂c ∂2c ∂2l 2 ∂l ∂c + ul ε+ ucc + uc 2 + ull + ul 2 ε = U (0) + uc ∂ε ∂ε 2 ∂ε ∂ε ∂ε ∂ε   i 1h 2 = U (0) + −uc FL + ul  ε + ucc (FL ) + uc FLL + ull + 0 ε2 | {z } 2 =0

i 1h 2 ucc (FL ) + uc FLL + ull ε2 < 0 ⇒ U (ε) − U (0) ≈ 2 This shows that for the utility of the household is higher at ε = 0 than for a positive value of ε (c) For household A we have:

cA (ε) = FK (K, L∗ − ε) K so

∂cA = −FKL K < 0 ∂ε

Since the household does not work it only cares about consumption, so it is worse o .

For household B we have:

l (ε) = l∗ + ε cB (ε) = FL (K, L∗ − ε) (L − ) so its utility is given by

U (ε) ≡ u cB (ε) , l (ε)

84

Now take the derivative:

∂l ∂cB + ul ∂ε ∂ε = −uc FLL L − uc FL + ul

U 0 (0) = uc

= −uc FLL L > 0 so household B's utility is increasing in ε, as long as ε is small.

9.5

Di erences in Preferences There are two countries: Industria and Lethargia. Both countries have a population of 1 and identical production functions:

Y = AL where L is the total amount of time spent in market labor.

Preferences are given by:

u (c, l) = log (c) + θ log (l)

(9.6)

where c is consumption and l is leisure. Within each country, everyone is identical but θ is di erent for residents of the two countries. Denote the two values by θ

I

and θ

L

and assume θ

L

> θI .

Both countries run free market economies, with perfectly competitive labor markets.

(a) What is the wage in each country? (b) Set up the problem of a representative household that has to decide how to divide its time (normalized to a total of 1) between market work and leisure. (c) What fraction of its time will the representative household spend on market work in each country? (d) Which country has higher GDP? (e) Suppose a researcher does the following: i. assumes preferences are the same across the world, with functional form (9.6), ii. estimates θ using data from Lethargia, iii. gathers data on consumption and leisure in each country, iv. computes λ: the relative welfare of residents of Industria, taking Lethargia as a benchmark with

λ = 1 the way Jones and Klenow took the US as a benchmark.

Find an expression for λ.

Which country would the researcher conclude has a higher standard of

living? Explain.

Suggested Answer (a) The wage in both countries is A, which is the marginal product of labor.

85

(b) The household solves:

max log(c) + θ log(l) c,l

s.t. c ≤ A(1 − l) (c) The rst order condition is

θ A = l c Solving for c, replacing in the budget constraint and solving for l:

l= The household will spend a fraction 1 − l =

θ 1+θ

1 1+θ of its time in market work. Consumption will be

A 1+θ (d) GDP is higher in Industria because its residents work more.

L

(e) The researcher estimates θ using data for Lethargia, obtaining θ . For Industria, he calculates that utility is:

u = log(cI ) + θL log(lI )

A 1 L = log + θ log 1 + θI 1 + θI Then the researcher computes λ by solving:

log

A 1 + θI

+ θL log

λA 1 L + θ log 1 + θL 1 + θL

1+θL 1 + θL ⇒λ= <1 1 + θI

1 1 + θI

= log

The researcher would conclude that Industria has a lower standard of living than Lethargia.

By

assuming that everyone's preferences are like those in Lethargia, the researcher would conclude that the extra consumption that the residents of Industria obtain is not worth the extra work they need to do to get it. In fact, neither of the two peoples would want to switch places.

9.6

Capital Income Taxes An economy has the production function

Y = K α L1−α It is populated by two types of households: workers and capitalists.

Workers supply a total of L units of labor inelastically and consume all their income in every period, so

86

they don't really make any decisions. Their consumption in any given period is given by:

ct = wt L + Tt where wt is the wage and Tt is a transfer they get from the government.

Capitalists have preferences given by:

∞ X

β t u (ct )

t=0 They do not work: they get their income from capital, and it is taxed at a rate τ . However, they can choose how much to consume and how much to save in the standard way. Their budget constraint is:

Kt+1 = −ct + 1 + (1 − τ ) rtK − δ Kt K

where Kt is the capital stock in period t, δ is the rate of depreciation, τ is the tax rate and rt

is the rental

rate of capital (which each capitalist household takes as given but depends on the total capital stock in the usual way).

What does this budget constraint mean? For each unit of capital the capitalist has, it obtains a rental.

The government taxes this rental (net of depreciation) at a rate τ . Assume that the government taxes interest income in the same way as it taxes income from renting capital, so that equation (9.1.11) holds. (a) Set up the maximization problem of a capitalist household. (b) From the rst order conditions, nd an after-tax version of the Euler equation. (You can skip steps if you want) (c) If in the long run this economy reaches a steady state in which consumption of the capitalists is constant, what are the pre-tax and after-tax interest rates in this steady state? (d) If the economy is in a steady state, what is the rental rate of capital? How does it depend on τ ? (e) If the economy is in a steady state, what is the capital stock? (f ) How much revenue does the government collect from the capital-income tax in steady state? (g) What is the level of wages in steady state? (h) Suppose that the government uses all the revenue from the capital-income tax to nance transfers to the workers. What level of consumption do workers attain in steady state? How does it depend on

τ ? Explain.

Suggested Answer (a) The capitalist household solves:

max

ct ,Kt+1

∞ X

β t u (ct )

t=0

s.t.

87

Kt+1 = −ct + 1 + (1 − τ ) rtK − δ Kt K0

given

(b) The Euler equation is:

K u0 (ct ) = β 1 + (1 − τ )(rt+1 − δ) u0 (ct+1 ) K

(c) By the usual argument, the pre-tax interest rate is rt+1 = rt+1 − δ and the after-tax interest rate is

K (1 − τ )rt+1 = (1 − τ )(rt+1 − δ) If consumption is constant in the steady state, then K 1 = β 1 + (1 − τ )(rss − δ)

so the after tax interest rate mus be:

(1 − τ )rss =

1 −1 β

and the before-tax interest rate must be

1 rss = 1−τ

1 −1 β

(d) The rental rate of capital must be:

K rss =

1 1−τ

1 −1 +δ β

which means it is increasing in τ . The after-tax interest rate has to be

1 β − 1, so higher taxes mean

the before-tax interest rate must be higher, which means the rental rate of capital must be higher. (e) Equating the marginal product of capital to the rental rate:

K FK (Kss , L) = rss

1 1 −1 +δ 1−τ β 1

− 1−α 1 1 1 ⇒ Kss = L −1 +δ α 1−τ β

α−1 1−α αKss L =

(f ) The government collects:

T = τ rK − δ K 1

− 1−α τ 1 1 1 1 = −1 L −1 +δ 1−τ β α 1−τ β (g) The wage is:

w = (1 − α)K α L−α

88

= (1 − α)

1 α

1 1−τ

α

− 1−α 1 −1 +δ β

(h) Consumption per worker is

cw = w +

T L

= (1 − α)

1 α

1 1−τ

α 1

− 1−α

− 1−α τ 1 1 1 1 1 + −1 +δ −1 −1 +δ β 1−τ β α 1−τ β

This is a decreasing function of τ , so capital income taxes for redistributing to workers is counterproductive in the steady state. No matter what the tax rate, in the steady state the capitalists will get an after-tax return of

1 β − 1 and the workers will get everything else, either in the form of wages

or transfers. So workers' total consumption will be:

cw = F (K, L) −

This is maximized when FK (K, L) = 9.7

1 −1+δ K β

1 β − 1 + δ , which is attained with τ = 0.

Getting Old An economy has two types of households: young and old. Let µ be the fraction of households that are young. We hold the total population constant and consider the e ects of higher or lower µ (which could be the result of di erences in fertility and mortality).

Each young household supplies x units of labor

Y

= µx; old households supply z units of labor, also

inelastically, so the total amount of young labor is L

O

inelastically, so the total amount of old labor is L

= (1 − µ) z . Assume x > z .

The production function is:

Y = K α L1−α Y

where K is the capital stock and L is the total amount of labor: L ≡ L

+ LO .

(a) Holding K constant, how do GDP, wages and and the rental rate of capital depend on µ? (b) How will the steady-state capital stock capital compare between two economies with di erent levels of µ? Explain. (c) Now imagine instead that the economy doesn't use capital and the production function is:

Y = LY

γ

LO

1−γ

How do the wage levels of young and old workers depend on µ?

Suggested Answer (a)

L = µx + (1 − µ)z which is increasing in µ. Therefore wages are decreasing and the rental rate of capital is increasing in µ.

89

(b) In steady state we must have:

1 − 1 = FK (K, L) − δ β Higher µ means higher L and therefore higher K in order for this to hold. (c) The wage of young workers is:

γ−1

wY = γ (µx)

((1 − µ)y)

1−γ

which is decreasing in µ, while

−γ

γ

wO = (1 − γ) (µx) ((1 − µ)y) which is increasing in µ.

9.8

A New Technology Consider a one-period economy where the representative worker has preferences:

log (c) + θ log (l) where c is consumption and l is leisure. The worker must choose how to divide its time between leisure l and market work L, so:

l+L=1 The labor market is perfectly competitive. The equilibrium wage is denoted w .

(a) Set up the problem of the representative worker, who must decide how to allocate his time. Assume the worker has no source of income other that its earnings from labor.

Solve for the household's

choice of L.

The production technology uses labor and capital to produce goods according to:

Y = K α L1−α where K is the capital stock, which is exogenously given.

The capital-rental market is also perfectly

competitive. The equilibrium rental rate of capital is denoted r

K

.

(b) Set up the problem of the representative rm, which must choose capital and labor inputs to maximize pro ts. Derive rst order conditions and nd the level of w and r (c) How do w and r

K

K

such that markets clear.

depend on K and θ ? Explain.

(d) Suppose a team of scientists develops a new technology for producing goods. Using this new technology, it's possible to produce goods according to:

Y = AK

90

where A is a parameter. The scientists tried out the technology at a very small scale, so anything they did was too small to a ect equilibrium prices. Prove that the new technology turned out to be pro table if and only if K > K̄ , where K̄ is some number. Find an expression for K̄ . How does K̄ depend on θ ? Explain.

Suggested Answer (a) The worker solves:

max log(c) + θ log(l) c,l

s.t. c ≤ w(1 − l) The FOC is

θ w = c l w θ = w(1 − l) l θ 1+θ 1 L= 1+θ l=

(b) Firm solves

max K α L1−α − rK K − wL K,L

with FOCs:

αK α−1 L1−α = rK (1 − α)K α L−α = w so, replacing:

1−α 1 αK = rK 1+θ −α

1 (1 − α)K α =w 1+θ α−1

(c)

w is increasing and rK decreasing in θ.

If workers have a higher preference for leisure, they will

supply less labor, and scarce labor drives up wages and drives down the rental rate of capital. is increasing and r

K

w

is decreasing in K . More abundant capital drives up the marginal product of

labor and drives down the marginal product of capital.

91

(d) If they tried the new technology at scale k , pro ts are:

π = Ak − rK k =

A − αK

α−1

1 1+θ

1−α ! k

which is positive if and only if:

K > K̄ =

1 α 1−α

A

1 1+θ

K̄ is decreasing in θ. The new technology uses capital but no labor. It will be pro table if capital is abundant and therefore cheap and labor is expensive.

Higher θ means lower labor supply and

therefore higher wages, so capital doesn't need to be so abundant before the technology is pro table.

9.9

An Oil-Producing Economy Suppose that the production function is given by F (K, L) = K

α

L1−α + e.

GDP is the sum of regu-

lar output, which is produced using capital and labor, and oil extraction e, which is exogenous. representative household has preferences:

∞ X

The

β t log (ct )

t=0 and supplies L = 1 units of labor inelastically. Capital depreciates at rate δ .

(a) On the same graph, draw two phase diagrams, one for e = 0 and one for e = ē > 0 and label the steady state in each case.

(b) Suppose that the economy starts at t = 0 at the ē steady state but everyone suddenly realizes that oil will run out at time T , so that:

( et =

ē

if t < T

0

if t ≥ T

Using the phase diagrams, plot the evolution of Kt and ct from t = 0 until t = T and from t = T onwards.

Describe in words why consumption and investment evolve the way they do, and what

happens to wages and the interest rate.

Suggested Answer

(a) The phase diagram is:

92

The constant-consumption line is the same before and after time T : it takes the same capital stock to get the interest rate to

1 β − 1, which keeps consumption constant. The constant-capital line shifts

down for the period after oil runs out, because without oil maintaining the capital stock requires lowering consumption. (b) The evolution of c and K is shown in the gure. Upon realizing that oil will run out, consumption falls from its initial level

css to c0 .

Since the dynamics are still governed by the

e = ē phase

diagram, consumption falls over time and capital grows over time. In economic terms, the economy is accumulating capital in order to smooth consumption between the present and the post-oil period.

Higher capital stock means a lower interest rate, which is the reason consumption falls over time. As capital accumulates, wages rise. At time T , the dynamics start to be determined by the e = 0 phase diagram. The economy has higher-than-steady-state capital, so it starts decumulating capital. Both

0

consumption and the capital stock fall over time, until they converge to Kss and css . Interest rates start rising again and wages start falling. Note that c0 is such that at time T the economy is exactly on the saddle path for the new phase diagram.

9.10

Wizards and Cows The Calevingians have two basic economic activities: gathering fruit and raising cows.

Wild fruit grows in the Western part of the kingdom.

The Calevingians have not mastered the art of

growing fruit trees, so they have no control over how much fruit is available. They just visit the fruit trees every week and pick whatever fruit is ripe. Luckily, the kingdom includes fruits with di erent seasonal patters so there is ripe fruit more or less evenly throughout the year. Not all years are the same, however.

Some years are warm and fruit is plentiful, while others are colder and result in less fruit. The Calevingians have complete trust in their chief wizard, who gives them a weather forecast (and thus a fruit forecast) with a horizon of a few years.

93

Cows graze in pastures in the Eastern part of the kingdom. The grass on which cows feed grows just as well in cold and warm weather. Each family owns a plot of land and keeps their own cows there. Cows are raised primarily for meat; the Calevingians have not learned to milk them. One of the main decisions that Calevingians need to make is how many of their cows to slaughter for meat each year and how many (including newborn calves) to keep fattening from one year to the next. They understand that the more cows they keep in their plot of land, the less grass each of them will have available for grazing.

(a) Write down mathematically the economic decision problem faced by a Calevingian household. Explain the meaning of each equation you write down. (b) Derive rst-order conditions (you may skip steps if you want). (c) Suppose the chief wizard announces that the next few years will be cold and therefore yield less fruit.

How do households react to this announcement? (d) Despite their rather simple economy, the Calevingians have a fairly sophisticated legal system; the concept of a sale, a loan and an interest rate are well-established and contracts are enforced very e ectively. It's possible, for instance, to borrow in order to buy more cows or, conversely, to sell one's cows and lend the proceeds of the sale to someone else. What happens to interest rates when the wizard announces the upcoming cold weather?

Suggested Answer (a) There are two periods.

Let K1 be the number of cows the household starts with and K2 be the

number of cows it keeps for the following period. Let xt be fruit available in period t. The household solves:

max u(c1 ) + βu(c2 ) s.t.c1 + K2 = K1 + x1 c2 = F (K2 ) + x2 Assume that F (K2 ) is concave because the cows will grow more when there is more land per cow. (b) The Euler equation will be:

u0 (c1 ) = βF 0 (K2 )u0 (c2 ) (c) The announcement means a fall in x2 . To maintain he Euler equation, both c1 and c2 would fall.

Households want to smooth consumption, keeping more cows to make up for the fact that there will be less fruit tomorrow. (d) The constraint c1 + K2 = K1 + x1 implies that a fall in c1 means an increase in K2 , and therefore

0

a fall in F (K2 ).

A fall in the marginal product of capital means that the interest rate will fall

too. Another way to interpret it is that since period-2 goods have become scarce, they become more expensive, which is what lower interest rates mean.

94

9.11

Patience and Investment Suppose the production function is F (K, L) = K

α

L1−α , and the labor supply is exogenous and equal to

L = 1. The representative household's preferences are given by: ∞ X

βt

t=0

c1−σ t 1−σ

and the depreciation rate is δ .

(a) Use equation (9.3.17) to nd an expression for the level of capital that this economy will have in the steady state. (b) Find an expression for the level of GDP that the economy will have in the steady state. (c) Find an expression for the steady-state investment-to-GDP ratio. How does it depend on β ? Explain.

Suggested Answer (a)

1 −1 β 1

− 1−α 1 1 K= −1+δ L α β

αK α−1 L1−α − δ =

(b)

Y = K α L1−α α

− 1−α 1 1 = −1+δ L α β (c)

δK I = Y Y 1 h i− 1−α δ α1 β1 − 1 + δ L = h α i− 1−α 1 1 L α β −1+δ

−1 1 1 =δ −1+δ α β Investment-to-GDP is increasing in β . If households are more patient, it takes a lower interest rate to persuade them to keep consumption constant. This in turn requires a lower marginal product of capital and therefore a higher capital stock. In order to maintain a higher capital stock, the economy dedicates a higher fraction of GDP to investment.

95

9.12

Optimal vs Fixed Savings Rates Suppose the economy is as in Exercise 9.11, with α = 0.4, β = 0.95, δ = 0.08 and σ = 2. The economy starts with an initial capital stock K0 = 2. We are going to compare how this economy behaves under a xed investment rate and under an optimally-chosen investment rate. (a) First suppose that, like we assumed in Chapter 4, the investment rate is exogenous and constant, and equal to the steady-state investment-to-GDP ratio that you found in Exercise 9.11. Compute GDP, consumption, investment and the capital stock for the rst 150 years of this economy. (b) Now suppose that consumption and investment are chosen in accordance with (9.3.13).

Find the

level of initial consumption that is consistent with the system of equations (9.3.15) - (9.3.13) and the economy converging to a steady state. (You can nd a Matlab code that will compute this at the book website). Compute GDP, consumption, investment and the capital stock for the rst 150 years of this economy. (c) Plot the consumption paths of the two economies over the rst 150 years on the same graph. (d) Replace the computed paths of consumption from parts (a) and (b) into the utility function to compute how much utility the representative household will obtain in the rst 150 years of each economy. (e) By what fraction would one have to reduce consumption in the optimal-savings economy for the household to get the same utility as in the exogenous-savings economy?

Suggested Solution (c) These are the respective paths:

(d) In the xed-savings economy, the discounted sum of utility is −15.01. In the optimal savings economy, it's −14.96.

96

(e) The utility of the xed-savings economy is the same as in the optimal-savings economy with consumption reduced by 0.36%.

9.13

The Enclosure Acts The Enclosure Acts were a series of laws that converted common property into private property in Great Britain starting around the XVII century.

This exercise asks you to think about their macroeconomic

e ects.

Consider the following one-period economy.

• There are two sectors:

Agriculture produces output according to: YA = N α L1−α A where YA is agricultural output,

N is natural resources (land) and LA is labor dedicated to

agriculture.

Industry produces output according to: YI = K α L1−α I where YI is industrial output, K is capital and LI is labor dedicated to industry.

For simplicity, assume that both sectors produce the same good (or that both goods are perfect substitutes), so GDP is Y

= YA + YI .

• The industrial sector operates as a competitive market, with a wage rate w and a rental rate of capital r

K

. The capital stock is exogenously given and is owned by capitalists, who do not work.

• Agriculture is organized as follows:

no one owns the land and anyone can use it, once all the agricultural output is produced, it gets shared equally among agricultural workers.

• The amount of land available and the capital stock are exogenously given.

• Each worker supplies one unit of labor. The total number of workers is L. • Each worker freely decides whether to work in industry or in agriculture:

if the worker chooses industry, he gets a wage w, if the worker chooses agriculture, he gets a share of total agricultural. output (a) Find an expression for the ratio

LA LI that would lead to maximum GDP.

(b) If there are LA workers in agriculture, what is the income of a worker who chooses agriculture? (c) If there are LI workers in industry, what is the income of a worker who chooses industry?

97

LA LI that would make workers indi erent between choosing agriculture and industry?

(d) What is the ratio

Argue that this is the allocation of labor that the economy will have. How does it compare to the allocation that would maximize GDP? Explain. (e) Suppose there is an Enclosure Act that does the following:

• It splits up the land equally among agricultural workers, giving each of them ownership of a speci c piece of land.

• It establishes a competitive market for renting land and hiring agricultural workers.

After the reform: i. Which of the sectors will increase the number of workers it employs? Explain. ii. What will happen to GDP? Explain. iii. What will happen to industrial wages? Explain. iv. What will happen to the rental rate on industrial capital? Explain. (f ) How would the answers to part (e) change if instead of being split up equally the land was given to only some of the agricultural workers? Why?

Suggested Answer (a) Let x =

LA x 1 LI so that LA = 1+x L and LI = 1+x L. Maximize GDP by solving:

max N α x

1−α

1−α 1 x L + Kα L 1+x 1+x

with FOC:

α

N (1 − α)

α

α x 1 1 1 α L L = K (1 − α) L L 2 1+x (1 + x) 1+x (1 + x)2 N x= K

(b) A worker in agriculture gets:

CA =

N α L1−α A LA

= N α L−α A (c) A worker in industry gets:

CI = (1 − α)K α L−α I 9.14 Equating CA and CI :

α −α N α L−α A = (1 − α)K LI 1

x = (1 − α)− α

98

N N > K K

Since workers can freely choose sectors, they must be indi erent between them. The fraction of workers that choose agriculture will be higher than the one that maximizes GDP. To maximize GDP the marginal product of labor has to be equated across sectors. But in agriculture workers get the average product of labor, which is higher than the marginal because (keeping N constant), the marginal product of labor is diminishing. Hence more workers are attracted to agriculture.

9.15 After the reform, being an agricultural worker gives the worker

N CA = (1 − α)N α L−α A +r N (i.e. a wage equal to the marginal product of labor plus a rent on the land they own). Instead, going to industry gives the worker

CI = (1 − α)K α L−α + rN N I (i.e.

they get a wage equal to the marginal product of labor plus they still get a rent on their land,

even if they have left agriculture). Indi erence between sectors implies equating wages (because they get the rent either way), therefore equating marginal product and therefore x =

N K , which maximizes GDP.

Industry expands and agriculture contracts their workforce. Industrial wages fall because they are equal to (1 − α)K

α

K L−α and LI increases. The rent on industrial capital is r = αK α−1 L1−α , so it rises because I I

LI rises.

9.16 Giving the land to only some agricultural workers does not change the answer. It means that the rent they earn will be high for some and zero for others, but the rent does not a ect which sector they choose.

Of course, those that get land will be better o than those that don't, but the sectoral allocation of labor will be the same.

99

CHAPTER 10

Money Exercises 10.1

Central Bank Instruments Suppose the Central Bank wants to reduce the M1 money supply but does not want to change the monetary base. In what direction should it change reserve requirements?

Suggested Answer As formula (10.3.1) shows, lowering reserve requirements raises the money multiplier, increasing the M1 money supply for a given monetary base.

10.2

Pickpockets Suppose there is an increase in the number of pickpockets. How would that change the fraction of their money that people want to have in cash as opposed to checking deposits? If the Central Bank keeps the monetary base constant, what will happen to the M1 money supply?

Suggested Answer Presumably, if there are more pickpockets, people will prefer to hold less of their money as cash.

In

terms of formula (10.3.1), this means a fall in χ and therefore an increase in the money multiplier. If the monetary base is held constant, this will result in an increase in M 1.

10.3

Interest on Reserves Consider an economy where households almost use no cash (so χ → 0). Furthermore, assume that the Central Bank makes lowers the reserve requirement to almost zero (so ρ → 0). (a) What is the value of the money multiplier as χ and ρ become small? (b) Suppose that the Central Bank decides to pay interest on reserves at the market rate i.

Does

equation (10.3.1) still hold? Why or why not? (c) Under this policy regime, what determines the quantity of money?

Suggested Answer 101

(a) As χ → 0 and ρ → 0, then the multiplier ω → ∞. (b) If the Central Bank pays interest on reserves, then there is no longer a reason to expect (10.3.1) to hold, since it's derived under the assumption that banks hold the minimum required level of reserves. If the Central Bank pays interest on reserves, this need not be the case. (c) There still is a money demand curve, so the quantity of money will be a function of the interest rate. This raises the question of what determines the interest under such a system. One possibility is for the Central Bank to set the interest rate by o ering to borrow or lend unlimited amounts at the interest rate it chooses.

10.4

ATMs Suppose one day, suddenly and unexpectedly, ATMs are invented, which make getting cash more convenient than before. What would be the e ect of this invention on the demand for money?

Suggested Answer It depends a little bit on what de nition of money we have in mind. If we are thinking about cash, then we can think about ATMs as reducing the cost of going to the bank . Then formula (10.4.4) and Figure 10.4.2 show that this leads to a fall in money demand. If we are thinking about a broader de nition of money like M1 or M2, it's less clear what ATMs do.

10.5

Going to the Bank The average person in the US has about $10,000 in M1 money and spends around $60,000 per year.

Suppose this person uses its checking account to pay for all its purchases, behaves according to the Baumol-Tobin model and only has two investment options: its checking account (which pays zero interest) and an investment account (which pays interest). De ne going to the bank as transferring a balance from the investment account to the checking account.

(a) How often does the average person go to the bank? (b) Suppose the interest rate is 2%. What must be the perceived cost of each visit to the bank for this type of behavior to be optimal? (c) How often would the average person go to the bank if the interest rate rose to 3%? (d) Now suppose the checking account does pay interest, but it only pays 0.5% instead of the full 2% one can get from the investment account. What is the opportunity cost of holding balances in the checking account? What do we infer now about the perceived cost of going to the bank?

Suggested Answer (a) Using formula (10.4.1):

pY 2M 60, 000 = 2 × 10, 000

N=

102

=3 The average person goes to the bank three times a year, takes out $20, 000 each time and therefore holds $10, 000 on average. (b) Using formula (10.4.3):

ipY 2N 2 0.02 × 60, 000 = 2 × 32

pF =

= 66.67 The perceived cost of going to the bank is $66.67. (c) Using formula (10.4.3):

ipY 2pF 0.5

0.03 × 60, 000 = 2 × 66.67

N=

= 3.67 The average person would go to the bank 3.67 times a year if the interest rate rose to 3%. (d) The opportunity cost of keeping money in the checking account is 1.5%. Using formula (10.4.3):

ipY 2N 2 0.015 × 60, 000 = 2 × 32

pF =

= 50 The perceived cost of going to the bank is $50.

103

CHAPTER 11

The Price Level and In ation Exercises 11.1

The Elasticity of Money Demand Recall the Baumol-Tobin model of Chapter 10. Compute η , the elasticity of money demand with respect to GDP, in the money-demand function that arises from this model.

Suggested Answer

∂mD Y ∂Y mD

−0.5 F YF = 0.5 Y 2iY 2iY

η≡

= 0.5 The elasticity with respect to GDP is 0.5.

11.2

The Quantity Theory The quantity theory of money assumes that money velocity is constant. Try to come up with a moneydemand function that would imply that velocity is indeed constant.

What does this money-demand

function say about how money holdings depend on interest rates? What does it say about how households manage their trips to the bank?

Suggested Answer D

One possibility is m

(Y, i) = Yv for some constant v . Velocity is then: V ≡

which is constant.

Y =v mD (Y, i)

With this demand function, money demand does not depend on the interest rate.

Households go to the bank with the same frequency no matter what the interest rate is.

105

11.3

Seignorage with Zero In ation and Growth Suppose an economy is growing at a constant rate g , money demand comes from the Baumol-Tobin model and the money multiplier is ω . Suppose the government wants to maintain zero in ation. How much seignorage revenue can it obtain as a fraction of GDP? In other words, nd an expression for the level of:

B MtB − Mt−1 pt Yt

that is consistent with zero in ation. How does this depend on g , F and ω ? Why? [Hint: use the result from Exercise 11.1.]

Suggested Answer From equation (11.2.3) we know that, for zero in ation we need

µ = ηg We then have:

B MtB − Mt−1 1 ∆M = pt Yt ω pY 1 µM = ω pY r 1 µ YF = ωY 2i r 1 F = µ ω 2Y r r F 1 = ηg ω 2Y r r 11 F g = ω2 2Y r

Seignorage is increasing in g : the faster the economy is growing, the more the money supply can increase without generating in ation, so the more seignorage the government can obtain.

Seignorage is decreasing in ω . If the money multiplier is higher, the monetary base is a lower fraction of GDP, other things being equal. Therefore the lower the seignorage the government gets for a given percentage increase in the monetary base.

Seignorage is increasing in F . If going to the bank is more costly, the money demand is higher so the government gets more seignorage for a given percentage increase in the monetary base.

11.4

Growth in the Money Supply There are two otherwise identical economies. In both of them GDP is constant, prices are exible and the real interest rate is constant. The rate of growth of the money supply in each of them is:

106

Year

Growth rate of the money supply Country A Country B

2009

3%

6%

2010

3%

5%

2011

3%

4%

2012

3%

3%

2013

3%

2%

2014

3%

1%

2015

3%

0%

How does in ation in the two countries in 2012 compare? Explain.

Suggested Answer In ation will be higher in country A than in country B. Both countries increased the money supply by

3%. In country A, where everything is constant, this will imply in ation of 3%. In country B, where in ation is coming down over time, the nominal interest will also be coming down over time. This implies that the demand for real money balances

M p is increasing over time, and therefore p is rising less than

M. 11.5

In ation Targeting (a) Suppose a Central Bank has decided it wants to keep in ation at exactly 2% every year.

The

economy is not growing and the real interest rate is 3%. Describe what the Central Bank must do to the money supply to achieve its objective. (b) Suppose the real interest rate suddenly and unexpectedly falls from 3% to 1%.

How should the

Central Bank respond? Explain in detail.

Suggested Answer (a) The Central Bank needs to set the quantity of money at M

= pmD (Y, 0.05) and then increase it

2% per year. (0.05 comes from r = 0.03 and π = 0.02). (b) The Central Bank needs to immediately increase the money supply to M = pm

D

(Y, 0.03), and then

continue to increase it 2% every year. If the Central Bank keeps in ation constant, a fall in the real interest rate translates into a fall of the nominal interest rate to 3%, and therefore an increase in the money demand. The Central Bank accommodates the increase in money demand with an immediate increase in the money supply and then carries on as usual.

11.6

Seignorage with High In ation Suppose that money demand is given by:

−µ

mD (Yt , it+1 ) = Yt · (1 + it+1 )

107

where µ is a parameter. This is sometimes known as the Cagan money-demand function. Equilibrium in the money market is given by:

MtS = Yt · (1 + it+1 )

−µ

pt

The real economy is in a steady state so that:

Yt = Yss rt+1 = rss The government obtains seignorage revenue St (in real terms) by expanding the monetary base.

St is

given by:

St = Let ω be the money multiplier, so that M

S

B MtB − Mt−1 Pt

(11.1)

= ωM B .

Suppose that the rate of growth of the money supply is constant at rate γ , i.e.:

S MtS = (1 + γ) Mt−1

(11.2)

(a) What will be the rate of in ation? (b) What will be nominal interest rate? (c) Find an expression for the level of real money balances. How does it depend on γ ? Why?

S

(d) Solve equation (11.2) for Mt−1 and replace this in (11.1) to obtain an expression for seignorage revenue St in terms of γ and (e) Replace the value of

MtS MtS Pt . How does St depend on γ ? How does it depend on Pt ? Why?

MtS Pt that you found in part (c) into the expression for St that you found in part

(d) to obtain an expression for seignorage revenues St in terms of γ , Yss , rss , ω and µ. (f ) Assume the following parameter values:

Yss = 1 µ=4 rss = 0.0025 ω=5 Note that µ = 4 is close to the value that has been empirically estimated in some high in ation countries when the time period is one month.

To be consistent,

rss should be interpreted as a

monthly real interest rate and Yss as monthly GDP. (g) Plot St against γ for values of γ between 0 and 0.8. (h) What is the monthly rate of growth of the money supply that maximizes seignorage revenue? Call

∗

this γ .

108

∗

(i) Why is seignorage revenue decreasing in γ for γ > γ ? (j) What monthly in ation rate does γ = γ

∗

imply?

(k) What monthly nominal interest rate does γ = γ (l) What yearly rate of in ation does γ = γ

∗

∗

imply?

imply?

Mt ∗ (m) What is Pt if γ = γ ? (n) What is the maximum revenue as a fraction of GDP that the government in this example can constantly collect from seignorage? (o) Look at the data from Sargent (1982) on the German hyperin ation of the 1920s (you can nd it at the book website). What was the maximum monthly in ation rate that Germany experienced?

∗

How does it compare to γ ?

∗

(p) Suppose that the government is increasing the money supply at a rate γ = γ , and in month t it

B

B

credibly announces that (i) Mt+1 will not be equal to Mt

(1 + γ ∗ ) the way it would have been if

the policy had continued as usual, but it will be some other level M̄ (maybe higher, maybe lower

B

than Mt

B = M̄ for every s ≥ 1 (i.e. the money supply (1 + γ ∗ )) (ii) from then on, it will set Mt+s

will be constant). (q) What will be the rate of in ation from period t + 1 onwards? (r) What will be the price level Pt+1 in period t + 1? (This will depend on the government's choice of

M̄ ) Suppose the government sets M̄ at the level that will ensure that Pt+1

= Pt (i.e.

the level that will

immediately stop in ation).

B

(s) What value of M̄ will it need to choose? How does it compare to Mt

(1 + γ ∗ )?

(t) How much seignorage revenue will the government obtain in period t + 1? How does this level of seignorage compare to what the government was getting every month before making this change? (u) Look carefully at the timing of the end of hyperin ation in Germany? When exactly did the money supply stop growing? When exactly did hyperin ation stop? How can we make sense of this?

Suggested Answer (a) In ation will be γ . (b) The nominal interest rate will be i = rss + γ . (c) Real money balances will be Yss (1 + rss + γ)

−µ

. They are decreasing in γ because higher money

growth leads to higher in ation and therefore higher nominal interest rates. (d)

St =

S 1 MtS − Mt−1 ω Pt

109

MS

S t 1 Mt − 1+γ = ω Pt 1 γ MtS = ω 1 + γ Pt t St is increasing in γ because (holding M Pt constant!), the faster the money supply increases, the

more resources the government obtains. It is increasing in

Mt Pt because higher money demand means

that a given percentage increase raises more seignorage. (e) Replacing:

1 γ MtS ω 1 + γ Pt 1 γ Yss (1 + rss + γ)−µ = ω1+γ

St =

(f ) (g) Plotting:

(h)

γ ∗ ≈ 0.25

(i) There are two e ects of γ on S . The direct e ect is to increase S : the government obtains more seignorage from creating more money. The indirect e ect is to lower the money demand and therefore

∗

decrease seignorage revenue. For γ > γ , the indirect e ect dominates. (j)

γ ∗ implies monthly in ation of 25%.

(k) The monthly nominal interest rate is i ≈ r + γ

110

∗

≈ 0.253, or 25.3%.

(l) Yearly in ation is (1 + γ

∗ 12

)

− 1 ≈ 1, 360%.

Mt (m) Replacing: Pt ≈ 0.41. (n) Replacing: S ≈ 0.016. The government can collect up to 1.6% of GDP in seignorage revenue. (o) Germany experienced a peak monthly in ation of around 29, 500% in October 1923, much higher

∗

than γ . (p) (q) From t + 1 onwards, in ation will be zero. (r) Equilibrium in the money market requires:

ω M̄ = Yss (1 + rss )−µ Pt+1 −1 Pt+1 = ω M̄ Yss (1 + rss )µ

≈ 5.05M̄ (s) The government needs to choose M̄ such that:

−1 Pt = ω M̄ Yss (1 + rss )µ so

M̄ =

Pt Yss (1 + rss )−µ ω

We know that at time t, equilibrium in the money market implies:

Mt = Yss (1 + rss + γ ∗ )−µ Pt ωMtB = Yss (1 + rss + γ ∗ )−µ Pt and therefore

MtB (1 + γ ∗ ) = (1 + γ ∗ ) B

Comparing, M̄ will be greater than Mt

Pt Yss (1 + rss + γ ∗ )−µ ω

(1 + γ ∗ ) if:

(1 + rss )−µ > (1 + γ ∗ )(1 + rss + γ ∗ )−µ 0.99 > 0.51 which is indeed the case. (t) The government will obtain:

St+1 =

M̄ − MtB Pt+1 Pt

= ω

Yss (1 + rss )−µ − Pωt Yss (1 + rss + γ ∗ )−µ Pt 111

Yss (1 + rss )−µ − (1 + rss + γ ∗ )−µ ω ≈ 0.117 =

The government can instantly obtain seignorage equal to 11.7% of monthly GDP by satisfying the increased money demand that is a consequence of stabilization.

This is about 7.2 times what it

∗

obtains every month by expanding the monetary base at rate γ . (u) In ation stopped quite suddenly in December 1923-January 1924.

However, the money supply

continued to increase, almost fourfold over the course of 1924. One possible interpretation of what happened is that as the public became convinced that in ation was actually over, they greatly increased their money demand, and the government could increase the money supply without causing in ation. What convinced the public that in ation was over? According to Sargent, a series of scal reforms that persuaded people that the government was no longer going to need a lot of seignorage to satisfy the budget.

11.7

Bank Seignorage = A−ηi where M is the quantity of M1 money, pY is nominal GDP, i is the nominal interest rate and A and η are parameters. Households want to hold

Suppose that the demand for M1 money is given by

M pY

a fraction χ of their M1 money in the form of cash and a fraction 1 − χ in the form of checking accounts, which earn no interest. Banks earn seignorage by taking checking deposits and investing in assets that earn the nominal interest rate.

(a) Find an expression for the ratio of total seignorage earned by banks to GDP. Call this ratio s. (b) Compute

∂s ∂i . Why does this number depend on η ?

(c) Look up data on M1 and its components in 2018. What is a reasonable value for χ? What was the value of

M pY ? If the average nominal interest was 2%, how much seignorage did banks earn as a

fraction of GDP? (d) If η = 0.2, how much seignorage would banks earn as a fraction of GDP if the nominal interest rate went up to 3%?

Suggested Answer (a)

s=

(1 − χ)M i pY

= (1 − χ)A−ηi i (b)

∂s = (1 − χ)A−ηi (1 − η log(A)i) ∂i 112

This depends on η because η governs the elasticity of money demand. Raising the price of any good increases or decreases total revenue depending on whether the elasticity of demand is smaller or greater than 1. The less elastic the money demand, the more bank seignorage increases with the interest rate. (c) During 2018, M1 averaged about 3.7 trillion dollars and cash averaged about 1.6 trillion, so a reasonable value for χ might be χ = 0.43.

GDP was around 20.5 trillion, so

M pY

= 0.18.

Bank

seignorage was s ≈ 0.002, so 0.2% of GDP. (d) Solving for A, we have:

=

ηi1

i(1 − χ) s

0.02(1 − 0.43) 0.002

A=

1 0.2×0.02

≈ 7.7 × 10185 Now if the interest rate goes up to 3%, then seignorage will be:

s = (1 − χ)A−ηi i = (1 − 0.43) × 7.7 × 10185

−0.2×0.03

× 0.03

= 0.0013 so bank seignorage would decrease to

0.13% of GDP. In this example, even though banks are

increasing the seignorage they earn per dollar of deposits, the elasticity of money demand is so high that the fall in deposit quantities more than o sets the increase in interest rates. Empirically, this is probably not the more realistic case, so a better example would use a lower value for η or, as the example below, a di erent functional form for the money demand function, with parameter values that imply a lower demand elasticity.

11.8

Suggested Answer - Alternative This question actually works out more nicely with a di erent functional form for the money demand.

Assume that

M pY

= Ai−η

(a)

s=

(1 − χ)M i pY

= (1 − χ)Ai1−η (b)

∂s = (1 − χ)A(1 − η)i−η ∂i 113

This depends on η because η is equal to the elasticity of money demand. Raising the price of any good increases or decreases total revenue depending on whether the elasticity of demand is smaller or greater than 1. The less elastic the money demand, the more bank seignorage increases with the interest rate. (c) During 2018, M1 averaged about 3.7 trillion dollars and cash averaged about 1.6 trillion, so a reasonable value for χ might be χ = 0.43.

GDP was around 20.5 trillion, so

M pY

= 0.18.

Bank

seignorage was s ≈ 0.002, so 0.2% of GDP. (d) Solving for A, we have:

s −(1−η) i 1−χ 0.002 = 0.02−(1−0.2) 1 − 0.43

A=

≈ 0.083 Now if the interest rate goes up to 3%, then seignorage will be:

s = (1 − χ)Ai1−η = (1 − 0.43) × 0.083 × 0.031−0.2 = 0.0028 so bank seignorage would increase to 0.28% of GDP. 11.9

Real Interest Rates De ne

realized real interest rates rt+1 as: rt+1 ≡ it+1 − πt+1

where it+1 is the nominal interest rate between period t and period t + 1 and πt+1 is the rate of in ation between period t and period t + 1. If there is uncertainty about what the rate of in ation is going to be, this implies that there is uncertainty about what realized real interest rates will be. De ne the

expected

real interest rate as:

E rt+1 ≡ it+1 − E (πt+1 ) where E (πt+1 ) is the expected rate of in ation.

Suppose there are two parties to a nominal lending contract: a borrower and a lender. Who bene ts when

realized real interest rates turn out to be higher than expected real interest rates? What monetary policy will each of them support?

Suggested Answer Lower-than-expected real interest rates bene t borrowers at the expense of lenders, so borrowers will support policies that generate in ation (for instance, rapid increases in the money supply) and lenders

114

will support the opposite policies.

11.10

Money among Prisoners of War Read The Economic Organisation of a P.O.W. Camp , which you can nd at:

https://www.jstor.org/

stable/2550133. Pick one passage out of the article and explain how it relates to the models of money supply, money demand and in ation from Chapters 10 and 11.

Suggested Answer My favorite passage is the following, which describes how the price level reacts to changes in the money supply: Consequently our economy was repeatedly subject to de ation and to periods of monetary stringency. While the Red Cross issue of 50 or 25 cigarettes per man per week came in regularly, and while there were fair stocks held, the cigarette currency suited its purpose admirably. But when the issue was interrupted, stocks soon ran out, prices fell, trading declined in volume and became increasingly a matter of barter. This de ationary tendency was periodically o set by the sudden injection of new currency. Private cigarette parcels arrived in a trickle throughout the year, but the big numbers came in quarterly when the Red Cross received its allocation of transport.

Several hundred thousand cigarettes might arrive in the space of a fortnight.

Prices soared, and then began to fall, slowly at rst but with increasing rapidity as stocks ran out, until the next big delivery.

Most of our economic troubles could be attributed to this

fundamental instability.

115

CHAPTER 12

Business Cycle Facts Exercises 12.1

Business Cycles in Other Countries (a) Look up GDP accounts for a country other than the US. Download quarterly data for GDP, consumption and investment. (b) Apply the HP lter to create trend and cyclical components. To do this, rst transform the data by taking logarithms, i.e. compute log(Y ), log(C) and log(I). Then apply formula (12.1.1) to compute a trend component for each

log series.

If you use Excel, you can nd an add-in to do this at

https://web-reg.de/webreg-hodrick-prescott- lter/; if you use Matlab, the command hp lter does it for you. Finally, compute the cyclical component by subtracting the trend from the un ltered log series. (c) Compute the standard deviation of the cyclical component of GDP, consumption and investment. (d) Compute the correlation between the cyclical components of consumption and investment with the cyclical component of GDP. (e) How do the patterns compare with the US? What is similar and what is di erent?

Suggested Answer Here are the gures for Australia 1960-2019:

Standard deviation

Relative standard deviation

Correlation with GDP

GDP

1.2%

1

1

Consumption

1.1%

0.86

0.42

Investment

3.9%

3.15

0.63

Variable

This gures are similar to those for the US: consumption is less volatile than GDP and investment is more volatile than GDP, and both are positively correlated with GDP, though the correlations are less strong than in the US. In terms of magnitudes, Australia has had milder business cycles than the US.

117

12.2

Stabilizing a Very Volatile Economy We are going to redo Lucas's calculation for an economy that keeps undergoing the Great Depression all the time. Look up the NIPA GDP data for 1929 (the peak before the Great Depression) and 1933 (the trough of the Great Depression). Imagine an economy where half the time consumption is at the level of 1929 and half the time consumption is at the level of 1933. What is the value for the representative household in this economy of complete consumption stabilization at the average level (which, incidentally, is pretty close to the level of consumption in 1931)? Try the following values of σ : 2, 5 and 10.

Suggested Answer Real GDP in 1933 was approximately 27% below its 1929 level, which means that the level in 1929 was a (1 + f ) times the average and the level in 1933 was (1 − f ) times the average, where f satis es:

Y1929 = (1 + f )

Y1929 + (1 − 0.27)Y1929 2

f ≈ 0.152 Replacing this value of f in formula (12.3.3), we get:

σ

λ

2

1.0235

5

1.0572

10

1.0989

In contrast to the tiny numbers in Table 12.2, a household that kept experiencing the Great Depression would value stabilization quite a bit. For σ = 5, the household feels that eliminating the business cycle is as attractive as increasing its average consumption by 5.7%.

12.3

Heterogeneity and the Value of Stabilization Imagine an economy that spends half the time in a recession and half the time in an expansion.

In

this economy there are two kinds of households: stable and volatile. Stable households consume c every period, so they are immune to the business cycle. Volatile households consume:

( c

V olatile

=

c (1 + v)

in an expansion

c (1 − v)

in a recession

Suppose that a fraction µ of the households in the economy is volatile and a fraction 1 − µ is stable. (a) What is total consumption in the economy, in an expansion and a recession respectively? (b) Suppose aggregate consumption is well described by equation (12.3.1).

Given a value of µ, what

value of v would make aggregate consumption have uctuations of amplitude equal to f ?

For the remaining questions, assume σ = 5, f = 0.012 and µ = 0.05. (c) What is the value for a stable household of eliminating the business cycle?

118

(d) What is the value for a volatile household of eliminating the business cycle? (e) What is the value of eliminating the business cycle for a household that, behind the veil of ignorance, is not sure whether they are going to be stable or volatile?

Suggested Answer (a) Respectively:

cE = c (µ(1 + v) + (1 − µ)) cR = c (µ(1 − v) + (1 − µ)) (b)

1 + f = µ(1 + v) + (1 − µ) v=

f µ

(c) For a stable household, there is no value in eliminating the business cycle. (d) For a volatile household, applying formula (12.3.3) but with

f µ = 0.24 instead of f , we get λ = 1.14.

The volatile household feels that eliminating the business cycle is as attractive as increasing its average consumption by 14%. (e)

λ satis es:

f W (c0 , g, 0) = µW λc0 , g, + (1 − µ)W (λc0 , g, 0) µ " #

1−σ 1−σ !

1−σ 1−σ (c0 ) (λc0 ) f f µ = 1+ + 1− + (1 − µ) 1−σ 1−σ 2 µ µ 1 # σ−1 "

1−σ 1−σ ! f µ f 1+ + (1 − µ) λ= + 1− 2 µ µ = 1.0088 For a household who doesn't know if they will be volatile or stable, eliminating the business cycle is as attractive as a 0.88% increase in average consumption. This is about 24 times more valuable that it would be for a household who knows they will bear the business cycle in the same proportion as everyone else.

12.4

The Value of Growth Lucas (1988) argued:

Is there some action a government of India could take that would lead the Indian economy to

119

grow like Indonesia's or Egypt's?

1 If so, what exactly? If not, what is it about the nature of

India that makes it so? The consequences for human welfare involved in questions like these are simply staggering: once one starts to think about them, it is hard to think about anything else.

In this exercise we'll try to make sense of what Lucas had in mind. (a) Solve for λ in the following equation:

W (λc0 , g, f ) = W (c0 , g 0 , f ) with g = 0.01, g

0

= 0.02, f = 0.01 and σ = 2.

(b) What does the number you found represent?

Suggested Answer (a) Setting β = 0.97 in addition to the given parameter values:

1−σ

1 (λc0 ) 2 1−σ

1−σ

(1 + f )

1−σ

+ (1 − f ) 1−σ

1 − β (1 + g)

1−σ

=

λ=

1−σ

1−σ

1 (c0 ) (1 + f ) + (1 − f ) 1−σ 2 1−σ 1 − β (1 + g 0 ) 1 ! 1−σ 1−σ 1 − β (1 + g) 1−σ

1 − β (1 + g 0 )

≈ 1.238 (b) The representative household feels that increasing the growth rate of consumption from 1% per year to 2% per year is as attractive as increasing the starting level of consumption by 23.8%.

1 This was back before the India started to grow fast. India's GDP per capita grew less than 1% per year in the 1970s, just over 3% per year in the 1980s and 1990s and over 5% per year since 2000.

120

CHAPTER 13

The RBC Model Exercises 13.1

Prices in the RBC Model Consider the two-period RBC model summarized by equations (13.1.2)-(13.1.5). (a) Suppose there is a temporary, positive, productivity shock in period 1 only. What happens to the wage w and the real interest rate r ? Explain in words why it is that each of the prices change. (b) Suppose there is temporary increase in households preference for leisure. What happens to the wage

w and the real interest rate r? Explain in words why it is that each of the prices change.

Suggested Answer Assume that the shock is like the one depicted in Figure 13.3.1, where the net e ect is to increase labor supply. We know from equation (13.2.1) that the wage is:

w=

v 0 (1 − L) u0 (c1 )

Both c1 and L increase, so the numerator increases and the denominator decreases, so wages go up. What makes the worker work more is a temporary increase in wages, made possible by the higher marginal product of labor.

0

(If the income e ect were to dominate and labor supply falls, then we have that w = F1 (L), and since L

0

goes down and the whole F (·) schedule shifts up, wages also increase).

Furthermore, we know that c1 increases, which by the Euler equation (panel 3 of Figure 13.3.1) implies

K increases, which by equation (13.2.3) implies that the interest rate falls, due to diminishing marginal product of capital.

13.2

Stabilizing a Real Business Cycle Suppose the economy is well described by the Real Business Cycle model. The economy has just had a positive shock to aggregate productivity.

121

(a) Suppose the government wants GDP to remain the same as it would have been without the productivity shock. Describe one possible policy instrument that could be used for this. Explain why it works. (b) Suppose instead that the government wants employment to remain the same as it would have been without the productivity shock. Does the policy need to be applied more intensely or less intensely?

Why?

Suggested Answer (a) One possibility is to impose a temporary tax on labor income in order to discourage labor supply, preventing GDP from rising, as illustrated in Figure 13.4.3. (b) The policy needs to be applied less intensely to stabilize employment than to stabilize GDP. In order to stabilize GDP when there is a positive productivity shock, employment has to fall rather than remain constant, which requires a bigger tax increase than to merely prevent employment from rising.

13.3

Good News Consider the two-period RBC model summarized by equations (13.1.2)-(13.1.5). Suppose that the representative household develops a very particular form of optimism about the future. It believes that the period-2 production function is:

Y2 = F2 (K) + A where A is some positive number.

(a) How does this type of optimism di er from the one we looked at in Section 13.4? (b) What will happen to employment, output, consumption, investment, the wage and the real interest rate? (c) Does this type of optimism produce something that looks like a business cycle? Why or why not?

Suggested Answer (a) Here there is no optimism about the marginal product of capital in period 2, just a belief that GDP will be higher. (b) The shock produces a leftward shift in the consumption-investment tradeo : since there will be extra output in period 2, the household is less interested in saving for any given level of consumption. For the same reason, there is a leftward shift in the market clearing condition

122

Output and employment fall (because of the income e ect on labor supply), consumption rises, and investment falls. Wages rise because of the lower labor supply and the interest rate rises because lower investment implies a higher marginal product of capital (and at the same time because future goods have become more abundant and thus cheaper). (c) This doesn't look like a business cycle because consumption moves in the opposite direction than output, employment and investment.

13.4

Government Spending Consider an economy that is well described by the RBC model with one modi cation: there is a government that spends G in period 1. It pays for this by collecting lump-sum taxes from the representative household (Ricardian equivalence holds, so we don't need to specify when the government collects these taxes). G does not enter the representative household's utility function.

(a) Where in the system of equations (13.1.2)-(13.1.5) would G show up? Explain. (b) Suppose there is an increase in G. What happens to output, consumption, employment and investment? What happens to wages and interest rates? Show this graphically and explain in words what is going on.

123

Suggested Answer (a)

G would show up in the market clearing condition. Start from: c1 + K + G = Y1 Solve for c1 :

c1 = Y1 − K − G 0 −1

Equate this to c1 = (u )

[βF20 (K) u0 (F2 (K))] to obtain: −1

Y1 = (u0 )

[βF20 (K) u0 (F2 (K))] + K + G

The market clearing condition describes how much output needs to be produced to meet the requirements of investment and consumption. If there is government spending, there needs to be more production to satisfy a given amount of investment-plus-consumption. (b) The market-clearing curve shifts upwards by the amount G. consumption and investment fall.

124

Output and employment increase,

Households feel poorer because the government is taxing them to pay for G. working more and consuming less.

They respond by

They smooth out the fall in consumption by also reducing

investment.

13.5

Fish In the town of New Oldport in Maine, the main economic activity is shing.

Crews sail out to New

Oldport Bay in the morning and return in the evening with their catch. It's a very unstable occupation because it depends on the migration patterns of cod and bass, which are erratic. Sometimes (when a large school of sh is passing by) sh are plentiful in the New Oldport Bay; otherwise they are scarcer.

Sonar tracking gives residents a reasonably reliable daily estimate of the number of sh in the bay. On average, large schools of sh come by the Bay a few days a year.

(a) Write down two production functions for the New Oldport economy: one for when sh are plentiful and one for when they are not. (b) Write down the problem of a household that resides in New Oldport and has to decide how much to consume and how much to work each day. Derive rst-order conditions (you may skip steps if you want). (c) Suppose you compare a day when sh are plentiful and one when they are not. On what day will workers choose to work longer hours? (d) Suppose in 2018 the migration patterns of cod and bass change such that now sh in the New Oldport Bay are always plentiful. How will the number of hours that households choose to work on a typical day in 2018 compare to the number of hours they worked on a plentiful- sh day in 2017?

How will they compare to the average daily number of hours in 2017?

Suggested Answer (a) For example:

Y = AH L1−α when sh are abundant and

Y = AL L1−α when sh are scarce, with AH > AL . (b) The usual FOCs:

u0 (ct ) = β(1 + r)u0 (ct+1 ) u (ct )At (1 − α)L−α = v 0 (1 − Lt ) t 0

(13.1) (13.2)

We have written a production function without capital, but we may assume that New Oldport just takes the national interest rate r as given (it is equal to the marginal product of capital elsewhere in the country).

125

(c) At a daily frequency, consumption is unlikely to depend strongly on how many sh there are, because

1 So we may assume that the term c in the consumption-labor t

the household smooths consumption.

equation (13.2) doesn't vary strongly with the realization of A, or in other words that the income e ect is small. Then (13.2) implies that the household works more when sh are abundant. (d) Now if sh are abundant every day, consumption will be higher. If At is the same in 2018 as in a plentiful- sh day in 2017 but ct is higher, then (13.2) implies that L will be lower. This is the income e ect on labor supply. Instead, the comparison between average hours in 2018 and average hours in 2017 is ambiguous: it depends on whether income or substitution e ects dominate. Some evidence suggests that the income e ect may dominate, but in general it could go either way.

1 One way to look at this rigorously is to specify the problem of a household with uncertainty, coming from the uncertainty about how many sh there will be, and arrive at an equation like (6.2.17)

126

CHAPTER 14

The New Keynesian Model Exercises 14.1

Too Many Customers Jackson & Jackson is a shampoo manufacturer. Its engineering department estimates that if it decides to produce q units of shampoo, the total production cost would be c(q) = aq +

b 2 2 q dollars. Its marketing

department estimates that if it sets a price of p dollars per unit of shampoo, it will sell q(p) = α − βp units, where

α β > a.

(a) What price should Jackson & Jackson set in order to maximize pro ts? (b) Suppose Jackson & Jackson has chosen the pro t-maximizing price and advertised it to all its clients, so that at this point it is impossible to change it. Soon after, it learns that the marketing department turned out to be too pessimistic, and actual demand for shampoo is q(p) = α

0

− βp, where α0 > α.

If Jackson & Jackson decides to satisfy every purchase order it receives, how many units will it end

0

up selling? How does this number depend on α and α ? Explain.

0

(c) For what values of α would it be advantageous for Jackson & Jackson to turn away some customers? (d) Now suppose that the marketing department had the correct prediction but the engineering depart-

0

ment underestimated production costs, which turn out to be c(q) = a q +

b 2 0 2 q , with a > a. For

0

what values of a would Jackson & Jackson want to turn away some customers?

Suggested Answer (a)

max p(α − βp) − a(α − betap) − p

with FOC:

p=

b 2 (α − βp) 2

α(1 + βb) + aβ 2β + β 2 b

(b)

q = α0 − βp

127

= α0 − This is increasing in α

0

α(1 + βb) + aβ 2 + βb

(more demand means more sales) and decreasing in α (higher expected

demand makes J&J raise prices and reduces sales). (c) J&J wants to turn away customers if marginal cost is higher than the price, so if:

c0 (q) > p α(1 + βb) + aβ a + bq > 2β + β 2 b

α(1 + βb) + aβ α(1 + βb) + aβ > a + b α0 − 2 + βb 2β + β 2 b

1 βb + 1 α0 > (α(1 + βb) + aβ) − a b 2β + β 2 b (d) J&J wants to turn away customers if marginal cost is higher than the price, so if:

c0 (q) > p α(1 + βb) + aβ 2β + β 2 b α − βa α(1 + βb) + aβ a0 + b > 2 + βb 2β + β 2 b α + aβ + β 2 ab a0 > 2β + β 2 b a0 + bq >

14.2

Stickiness and the Phillips Curve How does the slope of the Phillips Curve (14.7.4) depend on µ? What does this mean?

Suggested Answer Let

m(Y1 ) ≡

v 0 (1 − L(Y1 )) u0 (Y1 − K̄)F10 (L(Y1 ))

m(Y1 ) is the marginal cost of production, expressed as a function of GDP, which we know is an increasing function. Replace this in equation (14.7.5) to get:

p1 =

µpS1 η 1 − (1 − µ) m (Y1 ) η−1

and now compute the rst and cross-partial derivatives:

−2 ∂p1 η η S =µ (1 − µ) p1 1 − (1 − µ) m (Y1 ) m0 (Y1 ) > 0 ∂Y1 η−1 η−1

−2 ∂ 2 p1 η η = (1 − 2µ) pS1 1 − (1 − µ) m (Y1 ) m0 (Y1 ) ∂Y1 ∂µ η−1 η−1 128

− 2µ (1 − µ) pS1

1 − (1 − µ) m (Y1 )

η η−1

−3

η η−1

2

m0 (Y1 ) m (Y1 )

−3 η η 0 S = m (Y1 ) p1 1 − (1 − µ) m (Y1 ) η−1 η−1

η η (1 − 2µ) 1 − (1 − µ) m (Y1 ) − 2µ (1 − µ) m (Y1 ) η−1 η−1 −3

η η η m0 (Y1 ) pS1 1 − (1 − µ) m (Y1 ) 1 − (1 − µ) m (Y1 ) − 2µ = η−1 η−1 η−1 Now use that with exible prices we have that:

m (Y1 )

η =1 η−1

which implies that as long as we are not too far from exible prices we have that:

−3 η η ∂ 2 p1 ≈− m0 (Y1 ) pS1 1 − (1 − µ) m (Y1 ) µ<0 ∂Y1 ∂µ η−1 η−1 The positive rst derivative means that, as we know, if we write the Phillips Curve with GDP on the horizontal axis, the slope of the Phillips Curve is positive. The negative cross-partial derivative means that the curve becomes atter when µ is higher. Recall that µ is the fraction of rms whose prices are sticky. The more of these there are, the more that increased demand can raise GDP without generating in ation. In the limit where µ = 1 (all rms have sticky prices), the Phillips Curve becomes horizontal, and higher demand can raise GDP without causing any in ation. In the limit where µ = 0, the Phillips Curve becomes vertical, and GDP will be at its natural level for any level of in ation.

14.3

Reserve Requirements Suppose the government decides to lower banks' reserve requirements. (a) What will happen to the (M1) money multiplier? (b) What will happen to the (M1) money supply? (c) If prices are partially sticky, what will happen to GDP, interest rates and the price level?

Suggested Answer (a) The money multiplier will rise, as equation (10.3.1) says. (b) The money supply will rise (c) GDP will rise, the interest rate will fall and prices will rise.

14.4

Gold For a long time, gold was used as money.

This meant that the quantity of money depended on the

quantity of gold that had been dug up from mines. Suppose a new mine is discovered.

129

(a) What should we expect to happen to GDP, nominal interest rates and the price level? (b) How does the answer depend on how exible prices are?

Suggested Answer (a) A new mine increases the money supply. GDP will rise, nominal interest rates will fall and prices will rise. (b) The more exible prices are, the less GDP will rise and the more prices will rise.

14.5

Future In ation and Present In ation Suppose the economy is well described by a New Keynesian model with partially sticky prices. Let π1 refer to in ation between period 0 and period 1 and π2 refer to in ation between period 1 and period 2, which we have taken as exogenous. Suppose there is an exogenous increase in π2 , for instance because the government announces that it will increase the money supply in the future. What will happen to π1 ?

What are all the steps that lead to this conclusion?

Suggested Answer As in Figure 14.5.4, higher expected in ation causes an upward parallel shift in the IS curve. The overall e ect is like Figure 14.7.2. The nominal interest rate rises, GDP rise and prices rise. Higher in ation means, other things being equal, lower real interest rates and higher demand.

Higher demand means

that exible-price producers raise prices, so π1 rises.

14.6

The E ect of Investment Consider the simpli ed New Keynesian model with exogenous investment from Section 14.6, with a period-2 production function given by (14.6.1). Suppose the utility function is u(c) = log(c), the household's discount factor is β = 1 and in ation is π = 0. (a) Derive an IS equation for this special case. This should be a relationship between Y1 and i for given values of the exogenous parameters A and K̄ . (b) Holding the interest rate constant, by how much does output increase if K̄ increases by one unit?

Explain in words why the answer depends on A. Now consider a simpli ed version of the Old Keynesian model with exogenous investment. The consumption function is c(Y1 ) = a + bY1 and the period-2 production function is given by (14.6.1). (c) Use the market-clearing condition to solve for Y1 as a function of the exogenous parameters A, K̄ ,

a and b. Explain in words why it does not depend on i. (d) By how much does output increase if K̄ increases by one unit? Explain in words why the answer depends on b but not on A. Explain why the answer is di erent from the answer to part (b).

Suggested Answer 130

(a) Using equation (14.6.2):

i=

AK̄ −1+π β(Y1 − K̄)

(b) Rearranging:

Y1 = so

1 AK̄ + K̄ β 1+i−π

∂Y1 1 A = +1 β 1+i−π ∂ K̄

Higher K̄ raises GDP directly because investment is part of demand and indirectly because it raises expected future consumption and therefore present consumption due to consumption smoothing.

This second e ect depends on A, which governs the size of the increase in future consumption. (c) Market clearing says:

Y1 = c(Y1 ) + K = a + bY1 + K̄ =

a + K̄ 1−b

This does not depend on i because the assumption is that neither consumption nor investment are sensitive to interest rates. (d)

1 ∂Y1 = 1 − b ∂ K̄ This depends on b because b is the marginal propensity to consume. If investment goes up by one dollar, this raises GDP directly but it also means that someone gets the extra income, and they consume a fraction b of the extra income. This raises someone else's income, who goes on to consume

2

a fraction b of it, etc. This de nes a geometric series: 1 + b + b

1 + b3 . . . whose sum is 1−b .

This does not depend on A because consumption is not forward-looking, so the consumption response does not depend on the expected increase in future GDP that is governed by A.

14.7

Blowing Stu Up Imagine that the economy is well described by the New Keynesian model with perfectly sticky prices, with one twist. There is already some capital in place, which can be used to produce output in period 2. Denote this capital by K0 and, for simplicity, assume that it does not depreciate. The total capital stock that's used in production in period 2 is therefore:

K = K0 + I where I is investment. (a) Let I (K0 , r) be the level of investment as a function of K0 and r . How does it depend on K0 ?

131

(b) Write down the market-clearing condition for period-1 goods. (c) Derive an IS equation for this economy. (d) Suppose someone blows up part of the original capital stock, so that we start with K0 − X instead of K0 . How does the IS curve shift in response to X? What happens to GDP and interest rates?

Explain.

Suggested Answer (a)

I(K0 , r) is decreasing one-for-one in K0 : the more capital there is already in place, the less invest0

ment is needed to reach the point where F2 (K) = 1 + r . (b)

Y1 = c1 + I(K0 , r) (c) Let K(r) = K0 + I(K0 , r) be de ned by equation (14.4.2). The IS equation is:

u0 (Y1 − I (K0 , i − π)) = β (1 + i − π) u0 (F2 (K (i − π))) (d) The IS curve shifts to the right by the amount X .

This raises GDP and interest rates.

If there

is less capital to begin with, then there needs to be more investment to reach the target level of capital, so GDP rises to produce the additional investment.

14.8

A Recession Consider the following data:

2015

2016

Capital Stock

10, 000

10, 000

Employment

1

0.81

GDP

4, 000

3, 600

Labor Income

2, 000

1, 800

Capital Income

2, 000

1, 800

800

600

Investment

3, 200

3, 000

Price level

100

90

Nominal Interest Rate

5%

10%

Consumption

Suppose that 2015 was a totally normal year (almost identical to 2012, 2013 and 2014). This question asks you to think about what are the possible causes of recessions and how to use the data to determine what may have caused the 2016 recession. (a) Name two possible causes of recessions that don't seem to t the data. For each of them, explain: i. Why, according to some model, it's possible for this to cause a recession. You can make the argument using equations, graphs or words, as long as you are precise.

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ii. What features of the data indicate that this is not what happened. (b) Name one possible cause of recessions that does t the data. Explain: i. Why, according to some model, it's possible for this to cause a recession. You can make the argument using equations, graphs or words, as long as you are precise. ii. What features of the data indicate that this is what may be going on?

Suggested Answer (a) This recession doesn't seem to be caused by a fall in productivity, which could cause a recession in an RBC model. If we compute a Solow residual, using the fact that 2015 was identical to 2014, including the investment level, so the capital stock in 2016 is the same as in 2015) we get:

Solow Residual = gY − αgK − (1 − α)gL

= −0.1 − 0 − 0.5 × (−0.19)

= −0.005

Even this tiny fall in the measured Solow residual is a result of the local approximation used to compute a Solow residual. If we t a Cobb-Douglas production function Y

= AK α L1−α then we

get:

−α α−1 A2016 Y2016 K2016 L2016 = −α α−1 A2015 Y2015 K2015 L2015

=

3, 600 × 0.81−0.5 4, 000 × 1−0.5

=1 so productivity was exactly the same in both years.

The recession also doesn't seem to be caused by pessimism about the future, which could cause a recession in a New Keynesian model. As shown in Figure 14.5.2 (in the opposite direction), this would have the e ect of lowering nominal interest rates, not raising them . (b) This recession may have been caused by a tightening of monetary policy. This can be represented by a leftward shift in the LM curve. It has the e ect of raising interest rates, lowering GDP (both consumption and investment) and, if prices are partially sticky, creating de ation, all of which happened.

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CHAPTER 15

Fiscal and Monetary Policy Exercises 15.1

Interaction Between Fiscal and Monetary Policy The US government has decided to go to war in order to conquer Canada an incorporate it as the 51st US state. The Canadian government has politely agreed to carry out the war according to the following rules:

• Each country will build a large amount of tanks and set them on re.

• The country whose tanks make the most noise will be declared the winner of the war.

• If the US wins the war, Canada will become a US state; if Canada wins the war we will leave them alone.

• Either way, no policies will change in any of the two countries, nothing besides the tanks will be destroyed and no one will be hurt.

In preparation for the war , the US government orders a large amount of new tanks from its suppliers of military equipment.

Suppose throughout that the economy is well described by a New Keynesian model with partially sticky prices. (a) Suppose the Federal Reserve follows a constant-money-supply policy. i. What will happen to GDP in the US? ii. What will happen to nominal interest rates? iii. What will happen to the price level? (b) Suppose now that the Federal Reserve decides to adjust the money supply to keep nominal interest rates constant. i. What does the Federal Reserve need to do to the money supply? ii. How does the reaction of GDP compare to part (a)?

135

iii. How does the reaction of the price level compare to part (a)? (c) Suppose now that the Federal Reserve follows a strict in ation-targeting policy. i. What does the Federal Reserve need to do to the money supply? ii. How does the reaction of GDP compare to part (a)? iii. How does the reaction of nominal interest rates compare to part (a)?

Suggested Answer (a) This will de a rightward shift in the IS curve, so GDP, interest rates and the price level will all rise (b) To prevent interest rates from rising, the Federal Reserve has to increase the money supply. The LM curve shifts outwards, GDP and prices rise even more. (c) To prevent in ation, the Federal Reserve needs to prevent the increase in GDP by contracting the money supply / raising the interest more than in part (a).

15.2

Money Demand and Fiscal Policy Let's look at some special cases of money-demand functions:

• Case 1: mD = a · Y0 • Case 2: mD = b − x · i1 where a, b and x are positive constants.

In case 1 the money demand depends on GDP but is completely insensitive to interest rates. Case 2 is the opposite: money demand depends on the nominal interest rate but not on GDP. (a) Find an expression for money velocity in each of the two cases. Does the quantity theory of money hold in each case? (b) Draw the LM curve that results from each of these two assumptions. How does the LM curve shift in response to an increase in the money supply? (c) Suppose the government decides to undertake a scal expansion, i.e. increases G. i. How does that shift the IS curve? ii. What is the e ect on interest rates and on GDP in case 1 and case 2 respectively? iii. Suppose the scal expansion was undertaken with the objective of increasing GDP. You are trying to judge whether the policy is working, but you don't have the latest GDP gures yet.

You do, however, have very good data on interest rates. How could you use the IS-LM model together with data on interest rates to get a sense of the e ectiveness of the policy?

Suggested Answer 136

(a) By de nition, V

Y0 1 ≡ pY M , so in case 1: V = a and in case 2: V = b−xi1 . The quantity theory holds in

case 1 but not case 2. (b) Graphically:

(c) This shifts the IS curve to the right. In case 1, the interest rate rises; in case 2 it does not. One could use this to see whether the policy is working before seeing the GDP gures: if interest rates are not rising in response to the policy, it is more likely that we are closer to case 2, where scal policy would be e ective.

15.3

Taxes and Spending in the Old Keynesian Model Suppose household consumption behavior is well described by the Old Keynesian model, with a consumption function:

c1 = a + b(Y1 − τ1 ) where a and b are constants, b < 1 and τ1 is the level of taxes collected by the government, so that Y1 − τ1 is the household's after-tax income. The level of government spending is G. (a) Use equation (15.1.3) and the implicit function theorem to compute how much the IS curve shifts

137

to the right in response to an increase in government spending G and in response to a decrease in taxes τ1 , i.e. compute

∂Y1 ∂Y1 ∂G and ∂τ1 .

(b) If this model is correct, will period-1 GDP increase more in response to an increase in government spending of 100 million dollars or a tax cut of 100 million dollars? Explain.

Suggested Answer (a) Solving for Y1 :

Y1 = so

a − bτ1 + K + G 1−b

∂Y1 ∂Y1 1 b ∂G = 1−b and ∂τ1 = 1−b .

(b) GDP increases more in response to an increase in spending because it generates the same increase in after-tax income and therefore consumption but there is the public spending on top of that.

15.4

Patriotic Consumption The President has just tweeted: We need to help our economy. I'm going to ask that each of you go out and buy stu . If you all do that, that'll get the economy going.

Suppose the typical household reacts to the announcement as follows: I wasn't planning to do this, but the President seems like such a wise leader that I'm going to do my patriotic duty and buy myself a lawnmower. (a) Describe one model according to which the overall e ect of the appeal to patriotic consumption is detrimental to welfare. Describe what happens and why it's detrimental. (b) Describe one model according to which the overall e ect of the appeal to patriotic consumption has desirable consequences. Describe what happens and why its desirable.

You can make the argument using equations, graphs or words, as long as you are precise.

Suggested Answer (a) If the economy is well described by the RBC model, where the FWT holds, then persuading people to change plans that they thought were optimal can only make them worse o . (b) If the economy is well described by the New Keynesian model, then output is typically below the e cient level (due to market power), so it would be desirable to raise it. Furthermore, because of sticky prices, if people decide to consume more than they normally would, GDP will indeed rise, so this might turn out to be a good plan.

15.5

Taxes and In ation Suppose the economy is well described by a New Keynesian model with partially sticky prices.

The

government wants to bring down in ation. The Central Bank, for some reason, is unable or unwilling to

138

change monetary policy so the government decides to try to use scal policy. The rst idea it considers is to lower the level of government spending but it decides not to do it because this would cut into public services that are considered too important. Two other proposals are considered:

• Proposal 1: An immediate, temporary, increase in the level of taxes, done in a lump-sum way: everyone must pay an extra ∆ in taxes this year.

• Proposal 2: An immediate, temporary, increase in consumption taxes: everyone must pay extra taxes in proportion to this year's consumption Suppose that the size of the tax increase is such that the government will raise the same revenue from both plans. (a) If the present and future level of government spending is unchanged, what should households expect about future taxes? (b) Will either or the two policies be e ective in lowering in ation? i. If yes, what is the mechanism? ii. If no, why not? (c) Suppose a lot of households are borrowing-constrained, how does that a ect the answer?

Suggested Answer (a) Households should expect a tax cut in the future (b) Proposal 1 will not be e ective. Households will not change their behavior due to Ricardian equivalence. Proposal 2 will be e ective. Households pay taxes if they consume today but not if they consume tomorrow (this is equivalent to raising the interest rate consumers face: today's consumption has become expensive relative to tomorrow's). Consumption will fall (a leftward shift in the IS curve), so GDP will be lower and in ation will fall (a movement along the Phillips Curve). (c) If households are borrowing-constrained, then Ricardian equivalence does not hold.

Constrained

households will have to reduce present consumption in order to pay today's taxes even if they fully understand that tomorrow's taxes will be lower. Hence even proposal 1 would work.

15.6

The Timing of Fiscal Policy Equation (15.1.1) describes how to modify the New Keynesian IS equation when there is government spending in period 1. Now imagine that there is government spending in both periods: G1 and G2 . (a) Write down the market-clearing condition for goods in period 2. (b) Derive an IS equation that includes both G1 and G2 . (c) How does a change in G2 move the IS curve? Interpret what this means. (d) Suppose that the government wants to use increased spending (higher G1 ) to raise GDP. However, it chooses some spending projects that will take some time to get started, so it ends up increasing

G2 instead, leaving G1 unchanged. What will happen?

139

Suggested Answer (a)

F (K) = c2 + G2

(b)

u0 (Y1 − G1 − K (i − π)) = β (1 + i − π) u0 (F2 (K (i − π)) − G2 ) (c) A rise in G2 leads to a leftward/downward movement of the IS curve. Higher G2 means, other things being equal, lower consumption in period 2. Anticipating this, the household smooths consumption by reducing period-1 consumption, which lowers demand and therefore GDP. (d) The policy will be counterproductive in the short run - GDP will fall.

15.7

Government Spending in the Utility Function Suppose that government spending enters the utility function (because households care about public goods) and consider two possible utility functions:

u (ct , Gt , lt ) = u (ct ) + w (Gt ) + v (lt )

(15.1)

u (ct , Gt , lt ) = u (ct + Gt ) + v (lt )

(15.2)

(a) Explain in words what each of these two utility functions mean. (b) What would the Euler equation for intertemporal choice look like under each of these utility functions? (c) Derive an IS equation in each case. (d) Assume prices are sticky. How e ective are increases in Gt in increasing output in each of the two cases? Explain why.

Suggested Answer (a) In the rst case, the utility a household gets from private consumption does not depend on the level of public spending. In the other, private and public spending are perfect substitutes, which implies that the marginal utility of consumption falls as G increases. (b) In the rst case we have the usual Euler equation. In the second case,

u0 (ct + Gt ) = β(1 + rt+1 )u0 (ct+1 + Gt+1 ) (c) In the rst case we have, as in the previous question:

u0 (Y1 − G1 − K (i − π)) = β (1 + i − π) u0 (F2 (K (i − π)) − G2 ) In the second case, we have the usual IS equation (14.4.5) that doesn't have G in it. (d) In the rst case, increases in G work just like in Figure 15.1.1. Households enjoy the extra G but don't alter any decision in response to it. In the second case, G has no e ect on GDP because it replaces private consumption one-for-one.

140

15.8

ATMs Assume that the economy is well described by the New Keynesian model with partially sticky prices.

As we did in Exercise 10.4 suppose that one day, suddenly and unexpectedly, ATMs are invented, which make getting cash more convenient than before. (a) If the Central Bank follows a policy of keeping the money supply constant, what will happen to GDP, interest rates and prices? (b) Describe in detail what the Central Bank should do if it wanted to prevent GDP from changing in response to this change.

Suggested Answer (a) Having ATMs means that the money demand is lower. This means that it takes higher GDP (more transactions) or lower interest rates to persuade households to hold the entire stock of money. This is a rightward / downward shift in the LM curve, leading to higher GDP, lower interest rates and higher in ation. (b) To prevent GDP from changing, the Central Bank should contract to money supply to the point where the interest rate is the same as before.

15.9

The Bond Market Suppose that the Central Bank is known to use monetary policy to try to stabilize GDP. Suppose that a new employment report is released, which suggests that employment growth is lower than people thought it was going to be. What will happen to the price of government bonds? Explain.

Suggested Answer If monetary policy is used to try to stabilize GDP, then signs of a coming recession will lead the Central Bank to lower interest rates, relative to what it was planning to do before this news. The present value formula (8.1.1) implies that the price of government bonds will rise.

15.10

Reputation In this exercise we are going to think about how the government can build a reputation. The government and producers play the game we looked at in Section 15.2, with two di erences:

• Instead of just playing the game once, they are going to play the same game twice.

• Producers don't actually know the value of φ (i.e.

how much the government dislikes in ation).

They are going to try to gure it out by watching what the government does the rst time they play the game.

Let's start with the rst time they play the game. Producers don't know the value of φ but they have a guess, which we denote by φ̂. Assume that producers believe that the government's preferences are given by φ̂ and that the government will play the rst game just like the single-game case. (a) What level of in ation do they expect to see? Call this level of in ation π

E

.

141

(b) Now assume that (i) φ is not actually equal to φ̂ (it could be higher or lower) and (ii) the government does indeed behave in the rst game just like in the single-game example. What level of in ation does the government actually choose? (c) Now suppose that in preparation for the second game, producers try to gure out the true value of φ by looking at what the level of in ation turned out to be. Use your result from part (b) to

∗

derive an expression for how the new guess about about φ (let's call this φ ) depends on the level of in ation. Explain in words how φ

∗

depends on π and why.

∗

(d) Now let's consider the second game. If producers' guess about φ is φ , what level of in ation do they expect? (e) Find an expression for W (the level of welfare the government attains in the second game) as a

∗

function of φ and φ . Does the government want producers to think that φ is high or low? Why? (f ) Now go back to thinking about the rst game. Suppose that in the rst game, the government is not just trying to maximize W (as we assumed in part (b)), but is also trying to a ect how producers

∗

will form their beliefs φ . i. Will they choose higher or lower in ation than what you found in part (b)? Why? ii. Will GDP be higher or lower than what you found in part (b)?

Suggested Answer (a) Using equation (15.2.14):

πE =

1 φ̂a

Y1∗ − Y1N

(b) Using equation (15.2.12), the government will choose:

a 1 Y ∗ − Y1N + πE 1 + φa2 1 1 + φa2

1 + φ̂a2 1 Y1∗ − Y1N = 2 1 + φa φ̂a

π=

(15.3)

(c) Producers will solve (15.3) for φ to infer the government's preferences from the observation of π1 :

1 φ = 2 a ∗

!

1 + φ̂a2 1 ∗ N Y1 − Y1 − 1 π φ̂a

φ∗ is decreasing in π . If they see high in ation, producers will infer that the Central Bank doesn't actually care too much about in ation, and vice-versa. (d) In the second game, producers will expect:

πE =

142

1 Y1∗ − Y1N ∗ φ a

(e) Actual in ation in the second game will be:

π=

1 + φ∗ a2 1 Y ∗ − Y1N 1 + φa2 φ∗ a 1

Using the Phillips Curve (15.2.10), GDP will be:

Y1 = =

π − πE +YN a

1+φ∗ a2 1 ∗ N − φ1∗ a Y1∗ − Y1N 1+φa2 φ∗ a Y1 − Y1 a

=YN +

φ∗ − φ φ∗ (1 + φa2 )

Y∗−YN

+YN

Replacing in equation (15.2.9):

2

W = − (Y − Y ∗ ) − φπ 2

2 =− Y∗−YN 1− = − Y1∗ − Y1N

2

2 2 1 + φ∗ a2 1 Y1∗ − Y1N 2 ∗ 1 + φa φ a 2

2 !

∗ 1 + φ∗ a2 1 φ −φ +φ 1− ∗ φ (1 + φa2 ) 1 + φa2 φ∗ a φ∗ − φ ∗ φ (1 + φa2 )

2

−φ

∗

This function is increasing in φ . The government bene ts from producers believing that it dislikes in ation.

If producers believe the government dislikes in ation, in ation expectations are lower,

which improves the tradeo between actual in ation and GDP (graphically, the Phillips Curve in Figure 15.2.3 shifts down). The government takes advantage of this by generating lower in ation and higher GDP. (f ) If the government is trying to use the rst game to a ect φ additional reason to have low in ation.

Therefore (taking

∗

for the second game, this gives an

π

E

in the rst game as given), the

government will choose lower in ation and lower GDP.

143