Economics 306 Doyle Fall 2022 Practice Problems for Chapter 3: Solutions Coverage: Chapter 3, Static

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Economics 306 Doyle Fall 2022

Practice Problems for Chapter 3: Solutions Coverage: Chapter 3, Static General Equilibrium with Production: Diagrams

Problem 1 Answer this question using diagrams.

Consider an economy in which the representative household has preferences given by U (c, l) = max{2 · c, l}.

The representative rm's production function, F (K̄, N d ) is a standard, concave production function. Suppose that the capital input is xed.

There is no government in this question. (a) Formally de ne a competitive equilibrium in this economy.

Answer: A competitive equilibrium is:

An allocation (i.e. quantities): {C, N s , l, Y, N d , π}, and Prices: {w} such that:

i) the household chooses C , N s , and l to maximize its utility given its budget constraint, while taking prices as given, ii) the rm chooses Y and N d to maximize pro ts given its production technology, while taking prices as given, and iii) both the goods and labour markets clear. That is:

N d = N s, and

Y = C. 1

(b) Write down the household's optimization problem. Depict the solution to the consumer's

problem in a diagram.

Answer: The household's optimization problem can be written as:

max max{2 · c, l} subject to: C = w · (h − l) + π, C,l

c ≥ 0, and 0 ≤ l ≤ h

Note that since the preferences the+|Thousehold will choose a corner soluu(C)¢`are not S-,t convex, Q-I_wlvi-t`

tion. There are two possible solutions, depending on which corner is optimal. Figures 1 c2o oS+,,(±\ and 2 depict the two possibilities.

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wV\+IT

Figure 1: Corner solution: zero leisure (c) Depict a case where no competitive equilibrium exists. Explain carefully why no equilib-

rium exists in your example.

Answer: Figure 3 depicts a case where there is no equilibrium.

The fundamental

problem is that there is no wage that induces both the household and the rm to choose the same optimal point.

To get the rm to produce at point a, the wage must be w0 . In this case, however, the household is better o at the other end of the budget line. To get the rm to produce at point b, the wage must be w1 . In this case, however, the household is better o at 1 the other end of the budget line.

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1 Figure 2: Corner solution: zero work Therefore, no price will get the household and rm to choose the same point. Therefore, no equilibrium exists. (d) Depict a case where a competitive equilibrium does exist.

Answer: Figure 4 depicts a case where an equilibrium does exist. The key di erence

is that, given how the production function is drawn, there is a value of the real wage such that both the rm and household optimally choose point E . This allows for both decision makers to optimize and for both markets to clear.

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(e) Show that your diagram satis es the de nition of a competitive equilibrium.

Answer: To show this, we need to go through each of the components of the equilibrium

de nition and show/argue that it is met by the allocation in Figure 4. This is indeed the case: All of the equilibrium prices and quantities are solved. So we have solved for everything included in the equilibrium allocation and prices.

Point E is on the highest indi erence curve that is a ordable. Hence the equilibrium maximizes the household's utility, given prices.

At point E, the wage bill is tangent to the PPF. Hence the equilibrium solves the rm's optimization problem.

Both markets clear at point E. (f) Show whether the equilibrium in part d) is or is not e cient. Explain.

Answer: It is. Point E is on the highest indi erence curve on the PPF. This is the

characteristic of a point that solves the social planner's problem. Hence the equilibrium allocation implements the solution to the planner's problem, which means that the equilibrium is e cient.

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Problem 2 1. Answer this question with diagrams.

Consider a variant of the static G.E. model in which the representative consumer has preferences given by U (C, l), where U (·) is a standard, quasi-concave utility function. The representative rm's production function, F (K̄, N d ) is a standard, concave production function. Suppose that the capital input is xed.

The government di ers from the model in the notes. Like the notes, suppose that the government has to nance an exogenous level of government spending given by G. For simplicity, assume that these government expenditures are useless. Unlike the notes, we suppose that the government nances its expenditures by taxing rms. The government does this by levying a proportional tax, where the tax rate is denote by τ , on employment. That is, the rm pays τ units of output to the government for every unit of labour the rm hires. (a) Formally de ne a competitive equilibrium in this economy.

Make sure to include an expression for the government's budget constraint in your de nitions.

Answer:

A competitive equilibrium is: An allocation (i.e. quantities): {C, N s , l, Y, N d , π, τ }, and Prices: {w} such that:

i) the household chooses C , N s , and l to maximize its utility given its budget constraint, while taking prices as given, ii) the rm chooses Y and N d to maximize pro ts given its production technology, while taking prices and taxes as given, iii) the government chooses τ to balance its budget given G, where the government's budget constraint is τ · N = G, and iv) both the goods and labour markets clear. That is:

N d = N s, and

Y = C + G. 6

(b) Depict the competitive equilibrium in a diagram.

Note: Make sure that the key part of your diagram is large enough that I can evaluate whether you have or have not depicted an equilibrium correctly.

Answer:

The key di erence between this model and the one from the notes is that here the households pay no taxes, while the rm pays a tax for every unit of labour it employs.

This means that the household's budget constraint is:

c = w(h − l) + π, while the rm's pro t function is:

π = zF (K̄, N ) − (w + τ )N

a Ap

Figure 5: Equilibrium (c) Show that your diagram in part (b) satis es your de nition in part (a).

Don't worry about depicting the fact that the government balances its budget for this question.

Answer:

Our equilibrium de nition tells us that we have to solve for the variables {C, N s , l, Y, N d , π, T w}.

At a solution, i) the household's optimization problem must be solved, ii) the rm's optimization problem must be solved, iii) the government's budget must be balanced, and 7

iv) both markets must clear. We need to show that the point labeled as an equilibrium

in the above diagram satis es all of these condition.

The household makes an optimal an optimal choice when the indi erence curve is tangent to the budget line. The red lines in the gure correspond to the indi erence curve and budget line. We can see that a tangency occurs at point E in the diagram. Hence, at point E the household is making an optimal choice..

The black lines on the diagram represent the key objects in the rm's problem. The rm makes an optimal choice when the marginal product of labour equals the marginal cost of labour, which in this case is just w. This also occurs at point E in the diagram.

Note that tN = G, though you are not required to show this.

It should be clear that both markets clear at point E . The goods market clearing condition is simply Y ∗ = C ∗ + G∗ , which is clearly satis ed at point E . That labour supply equals labour demand is also clearly apparent from the diagram.

Finally, we can see that all of the equilibrium variables have been solved for at point E .

Hence, point E is a competitive equilibrium of the new model. (d) Show whether the equilibrium in part (b) is or is not e cient. Explain.

Answer:

The equilibrium is e cient if it maximizes the household's utility subject only to the constraint of the production function. We can characterize the e cient allocation using the social planner's problem. Since the planner's problem doesn't have prices or taxes in it, the planner's problem for the model in this question is identical to the planner's problem we saw in section 7.1 of the notes. This is given by gure 6.

We can see immediately that the equilibrium point depicted in part b) does not correspond to the solution to the planner's problem. In particular, the shaded lens in gure 5 corresponds to those allocations which are both feasible (i.e. the economy can produce those combinations of c and l) and preferred by the household (i.e. on a higher indi erent curve) to point E .

All households would be made better o by the implementation of an allocation within the shaded lens area, all of which are technologically feasible. However, the distortionary tax makes it impossible for the market to clear at any such point. In order to get the household to supply more labour, the wage paid to the household would have to increase.

However, given the existence of the tax, this would require the after-tax wage paid by the rm to increase as well. If this were to happen, rms would not be willing to 8

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1

Figure 6: The planner's problem hire additional labour instead they would want to reduce hiring. Excess supply in the labour market would then push the wage back to its equilibrium level. Essentially, the economy gets stuck at point E , even though that is not the most desirable of the feasible allocations.

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Problem 3 Answer this question with diagrams.

Consider a variant of the static G.E. model in which the representative consumer has preferences given by U (C, l), where U (·) is a standard, quasi-concave utility function. The representative rm's production function, F (K̄, N d ) is a standard, concave production function. Suppose that the capital input is xed.

The government di ers from the model in the notes. First o , let G = T =. Now suppose that the government introduces a policy of subsidizing consumption. It does this by paying households a subsidy of rate s for each unit of consumption that the household purchases.

The government nances this subsidy by taxing rms using a lump sum tax. The government balances its budget. The subsidy rate is s > 0, and the lump sum tax is given by τ . Treat the subsidy rate as an exogenous variable and the tax rate as endogenous. (a) Formally de ne a competitive equilibrium in this economy.

The only change to the model is the government's policy. This means that most of the equilibrium de nition remains as in the notes. We simply have to modify those aspects of the model that pertain to the government (including the description of the household and rm problems, the government's balanced budget constraint, and the goods market clearing condition).

A competitive equilibrium is: An allocation (i.e. quantities): {C, N s , l, Y, N d , π, τ }, and Prices: {w} such that:

i) the household chooses C , N s , and l to maximize its utility given its budget constraint, while taking prices and the subsidy rate as given, ii) the rm chooses Y and N d to maximize pro ts given its production technology, while taking prices and the level of the tax as given, iii) the government chooses τ to balance its budget given s (the government's balanced budget constraint becomes sC = τ ), and

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iv) both the goods and labour markets clear. That is:

N d = N s, and

Y = C. (b) Depict the competitive equilibrium in a diagram.

Note: Make sure that the key part of your diagram is large enough that I can evaluate whether you have or have not depicted an equilibrium correctly.

First note that while the tax reduces the rm's pro ts, it doesn't a ect either the production function or the rm's decision to produce where the marginal product of labour equals the wage. Hence the diagram illustrating the solution to the rm's problem is the same as gure 9 in the notes.

The household's problem is changed by this policy. In particular, the household's budget constraint becomes:

(1 − s)C = w · (h − l) + π.

This can be rearranged as follows:

C=

π w · (h − l) + .

1−s 1−s

It should be clear that the diagram illustrating the solution to the household's looks like gure 4 in the notes. The key di erence is that the slope of the budget line now equals

−w/(1 − s), because of the subsidy. Since s > 0 this means that the budget line is steeper (leisure has become more expensive relative to consumption). Incidentally, there π∗ is a typo in the gure as the right endpoint of the budget line should be labeled 1−s rather than π ∗ (it's quicker for me to just write this, rather than re-label and re-scan the gure).

Essentially, the subsidy means that the rm and the household face di erent prices, as was the case in section 7.3 of the notes. The equilibrium is depicted in gure 7

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N>= Nd

Figure 7: The Equilibrium (c) Show that your diagram in part (b) satis es your de nition in part (a).

about the government variables for this question.

Don't worry

We simply check that all of the conditions of the equilibrium de nition have been satis ed: i) All of the endogenous variables have been solved for on the diagram (with the exception of τ , but you are not asked to show this). ii) The blue lines depict a solution to the household's problem. That is, the indi erence curve is tangent to the budget line at point E. iii) The black lines depict a solution to the rm's problem. That is, the marginal product of labour equals the wage at point E. iv) Both markets clear at point E:

C * = y*

and N s* = N d*

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(d) Show whether the equilibrium in part (b) is or is not e cient. Explain.

It isn't.

The solution to the planner's problem is unchanged by the introduction of the subsidy.

It is reproduced in gure 6 below.

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1

Figure 8: The planner's problem At an e cient allocation, the indi erence curve is tangent to the PPF. The equilibrium allocation fails to satisfy this property (the shaded lens in gure 7 represents feasible allocations that are preferred to the equilibrium allocation). Even though the subsidy appears like it would bene t households, note that i) the households pay all taxes in the economy, as the households own the rms, and ii) the subsidy itself distorts the households' labour leisure choice.

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