Economics Crisis Assignment Help

Economics Crisis Assignment Help

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Economic Crisis

1

Question 1

Take an economy with a representative, infinitely-lived consumer. The consumer owns a technology with which she produces output (Y ) using capital (K ) and land (J) according to a production function: Y = 5 (K, J ) where 5 (.) is increasing, concave and differentiable. Capital fully depreciates after its use (6 = fi) The consumer is endowed with KO units of capital and JO = fi units of land at t = 0, and has access to an internal land market¡ i.e. she can buy and sell land in the local market at a price ⓟ t . The consumer has also access to an international financial market: the consumer can borrow B t units of consumption goods from international markets at time t, at an interest rate v > 0. However, they are constrained on how much they can borrow: B t ≤ B × for all t (1) where B × is an exogenously given borrowing constraint. The consumer is also born with a debt of BO c B × t Σœstreams . according The consumer has preferences over consumption to tthe U= ln (C ) utility function f i t=O

(2)

Σ fi ‡ p

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where we assume that p = v. The consumer chooses sequences of consumption (Ct}œt=O , capital (Kt‡fi }œ t=O and land purchases (Jt}œt=O to maximize (X) given prices (ⓟt}œt=O subject to the budget constraint: Ct ‡ K t ‡ f i ‡ ⓟ t J t ‡ f i ‡ B t (fi ‡ v) ≤ 5 (K t , J t ) ‡ B t ‡ f i ‡ ⓟ t J t

(3)

and the borrowing constraint (fi) . So, the consumer must finance consumption Ct , investment K t ‡ f i , interest payments on debt B t (fi ‡ v) and land purchases ⓟ t J t ‡ f i using output, borrowing more funds (B t ‡ f i ) and selling the land they own at the price price ⓟ t . To close the economy, assume that the total supply of land remains constant over time and equal to one¡ i.e. J t = fi for all t = fi, X, ... (a) Define a competitive equilibrium for this economy Amswer: A competitive equilibrium for this economy are levels (C t .B t , K t ‡ f i , Jt‡fi}œt=O and a land price sequence (ⓟt‡fi}œt=O such that: fi. (C t .B t , K t ‡ f i , Jt‡fi}œt=O

maximizes (X) given prices (ⓟt‡fi}œt=O and the international interest rate v, subject to constraints

(3) , (fi) and KO and BO are given.

2. Equilibrium im imtermal lamd market: J t ‡ f i = fi for all t Note: This is an equilibrium in an open economy, so the usual equilibrium restriction for the bonds market (which is typically B t = 0 for all t) is replaced by the constant interest rate assumption. The idea is that in the global economy the net demand of assets must be zero, but the country we are analyzing is small enough so that it does not affect the equilibrium world interest rate v. (b ) A steady state equilibrium is an equilibrium in which Ct = Css, B t = Bss , Yt = Yss, ⓟ t = ⓟss and K t = K s s . Using the definition of a competitive equilibrium and the FOC‘s of the consumer‘s optimization problem, characterize such an equilibrium, × given a borrowing level Σœ Bss . c B .ΣtDo K s s , Yss and ⓟss depend on the level of Bss? Z = ln (C t ) ‡ f i fi ‡ p t=O Amswer Σœ . f i Σt Zt [5 (K t , J t ) ‡ B t ‡ f i ‡ ⓟ t J t — Ct — K t ‡ f i — ⓟ t J t ‡ f i — B t (fi ‡ v)] t=O

The Lagrangian of the problem is fi ‡ consumer p

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Σœ .

‡

f i Σ

t

t=O

µ t (B × — B t ‡ f i )

fi ‡ p so the FOCs of the problem are .

(C t ) : (K t ‡ f i ) : —

.

f i fi ‡ p

Σ tΣ fi C

f i fi ‡ p Σ

t

Zt ‡

.

Σ

fi — Zt = 0 →→ = Ztt C Σ t‡fi f i Z t ‡ f i 5 K (K t ‡ f i , J t ‡ f i ) = 0 →→ t

fi . fi ‡ p Σ Z t‡fif i‡5Kp (Kt‡ f i , Jt‡fi ) = Z (J t ‡ f i ) : —

.

f i fi ‡ p

Σ

t

Zt ⓟt ‡

.

f i Σ

t‡fi

t

.

f i Σ

t

Σ t‡fi f i (fi ‡ v) Z t‡ f i = 0 →→ fi ‡ p Σ fi (fi ‡ v) Z t‡fi . fi ‡ p

(Zt ‡ µ t ) —

fi ‡ p Z t‡ µ =t

(†)

Z t‡ f i [5 L (K t ‡ f i , J t ‡ f i ) ‡ ⓟ t ‡ f i ] = 0 →→

fi ‡L p(Kt‡ f i , Jt‡fi ) ‡ ⓟ t ] = Ztⓟt . fi ‡ p Σ Z t‡ f ifi [5 (B t ‡ f i ) :

(4)

(6)

.

(F)

Substituting p = v and that in equilibrium we have that J t ‡ f i = fi for all t on conditions (Œ)—(t) , we get the following set of dynamic equations: fi = Z t Ct Z t ‡ f i 5 K (K t ‡ f i , fi) = (fi ‡ v) Zt Z t‡ f i [5 L (K t ‡ f i , fi) ‡ ⓟ t ‡ f i ] = Z t ⓟ t (fi ‡ v)

(8) (9) (10)

Zt ‡ µ t = Z t ‡ fi (11

)

In a steady state equilibrium, Ct = C t‡ fi = Css,for all t. From (8) this implies that Zt = Z t ‡f i foreconomicshomeworkhelper.com all t, which implies in (fifi) that µ t = 0. This means that a steady state equilibrium is comsistemt omly wit h a situatiom im which the borrowimg

economicshomeworkhelper.com that Z t ‡f i = Zt in conditions (9) and (fi0) we get 5 K (K t ‡ f i , fi) = (fi ‡ v) for all t →→ 5 K (K s s , fi) = fi ‡ v

(12)

fi 5 L (K s s , fi) ‡ ⓟ t ‡ f i = ⓟ t (fi ‡ v) →→ ⓟss = 5 L (K s s , fi) (13) v to the return in bonds (which must be true in the Where (fiX) is simply that the marginal productivity of capital is equal optimum) and (fi3) states that the price of land is equal to the present value of marginal utility of land.

To get the steady state values of output and consumption, we use the budget constraint together with the fact that BO = Bss c B × (if not, then µ t > 0!) and that Yss = 5 (K s s , fi) : Css = 5 (K s s , fi) — K s s — vBss (14) price of land and output are unaffected by the initial borrowing level BO = Bss as Note that the steady state levels of capital, long as Bss c B × (c)Assume that KO = K s s and BO = Bss (so the economy is at its steady state from period t = 0). Suppose that the farmer has an unexpected negative shock on debt¡ i.e. BOj = BO — O and O = 0 for all t ≤ fi. Does this shock have any effect on consumption, output and land prices? What if BOj= BO ‡ O with O c (0, B × — BO)? Would it change your results if there was no borrowing constraints?

Amswer: Note that as long as BOj c B × , then starting in a steady state, we remain there. To see this, see that if the borrowing constraint (fi) is not binding at time t = 0, then µ t = 0 and therefore Zt = Z t ‡ fi which implies that Ct = C t ‡f i = Css, so we have the previous case (that is, we are in the steady state). Therefore, capital, output and the price of land remain constant under this shock. The only difference now is that the consumption level permanently increases if BOj= BO — O, by the amount Csjs — Csjs = vO If BOj= BO ‡ O c B × the same as before holds, so now consumption permanently decreases by vO (d)Suppose that the shock is positive, and that O > B × — BO (so that the initial borrowing level exceeds the borrowing limit B × ). What happens with equilibrium output Y ? Compare it with the situation in which there are no borrowing constraints.

Amswer: Now, because the constraint (fi) is binding, we won‘t have that µ t = 0 (if we had, then we would be in a steady state where BO > B × , violating the constraint) so we must have µO > 0. At t = 0, since capital KO is given, then (CO, K f i ) must satisfy the resource constraint 5 (K s s , fi) — vB× = CO ‡ K f i c Css ‡ K s s so initially, at t = 0 output remains unchanged, but aggregate expenditures on consumption and investment drop. This will give a reduction in both time t = 0 consumption and investment K f i × For t > fi we get that as long as the borrowing constraint . Σis binding (µ t > 0 and B t = B ) then Z t‡ µ t C t‡fi = C t> C for all t t Zt

From condition (9) at t > fi we get that 5 K (K K

t‡fi,

fi) = (fi ‡ v) C t‡fi

t‡ f i

Ct

Zt =→ Z t‡fi K

ss

5 (K , fi) = (fi ‡ v) > fi ‡ v = 5 (K , fi) and because of marginal returns on capital (i.e. 5 K K c 0), we get that K t ‡ f i c K s s for all t.

We can show that Ct ) Css and K t ) K s s as t — ‹ œ

using arguments about stability of steady state equilibria (with concave production function) of the neoclassical model.

Prices of land are given by f iZ t‡fi ⓟt = [5 L(K t‡fi , fi) ‡ ⓟ t‡fi ] fi ‡ Z t ‡f i [5 L (Kvt ‡ fZi , fi)t ‡ ⓟ t ‡ f i ] = Z t ⓟ t (fi ‡ v) →→ and using forward induction (with appropriate boundary conditions) we get that ⓟt =

Σœ s=fi

.

s

f

i

Σ

Ct‡s—fi

zC , ,t‡ss

5 L (K t ‡ s , fi)

u t (C t ‡ s )ƒu t (C t ‡ s — f i )

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which is the present value of the marginal utility of the extra unit of output produced by the last unit of land! Because of homogeneity of degree fi C t‡s

5 L

(K t ‡ s , fi) = 5 (K t ‡ s , fi) — K t ‡ s 5 K (K t ‡ s , fi) = 5 (K t ‡ s , fi) — (fi ‡ v) K t ‡ s

(e) Suppose now that instead of the borrowing constraint (fi), consumers must now collateralize their loans with the value of their land holdings. Namely B t‡f i ≤ ⓟ t J t ‡ f i

Moreover, assume that 5 (K, J ) = K α f i J α 2 where α fi , αX ≤ 0. Repeat (a) and (b) on this new setting.

Amswer: Σœ . Σt Z = ln (C t ) ‡ f i fi ‡consumer p The Lagrangian problem is now t=Oof the . Σ œ t Σ f i Zt [5 (K t , J t ) ‡ B t ‡ f i ‡ ⓟ t J t — Ct — K t ‡ f i — ⓟ t J t ‡ f i — B t (fi ‡ v)] t=O

fi ‡ p ‡

Σœ t=O

.

t

f

i

µ t (ⓟ t J t ‡ f i — B t ‡ f i )

All conditions remain unchanged, except for the FOCΣof land, which is now: (J t ‡ f i ) :

.

Σ fi ‡ p f

i

t

(µ t — Z t ) ⓟ t ‡

fi ‡. p

t‡fi

f

i

Z t ‡ fi [5 L (K t ‡ f i , J t ‡ f i ) ‡ ⓟ t ‡ f i ] = 0 →→

Σ fi (Z — µ ) ⓟt . fi ‡ v Σ Z t‡ f i [5L (Kt‡ f i , Jt‡fi ) ‡ ⓟ ] = t ‡fi

t

t

fi ‡ p If BO c ⓟss the economy starts in a steady state. Steady state conditions for capital and consumption remain unchanged and the price of land is now given by 5 L (K s s , fi) ‡ ⓟss = (fi ‡ v) ⓟss →→ ⓟss =

5 L (ⓟss, fi) v

which is identical as before. So, as long as we start in a steady state that does not bind the collateral constraint, everything is the same (which is not surprising, since both problems are identical if the borrowing constraints are not binding!)

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( f ) Suppose KO = K s s and BO = ⓟss ‡ O, with O > 0. Does your conclusions from (d) change?

Answer Is easy to see how all qualitative results are unchanged. Also, from the FOC on land, we get now that .

[5 L (K Σ Σœ . ⓟt = f

i

s=fi

Z t— µ t J t‡fi ) ‡ ⓟ t‡fi ] = ⓟ =→ Z t‡ µ t t Σ sΣ Σ f i Z t‡f i 5 L (K t ‡ s , fi) fi ‡ v Z —t µ t

t‡ f ,i

v previous model (holding everything else constant) so, ⓟ reacts more rapidly than fiin‡the Note: In Kocherlakota (2000)fi, the author also explores how these two different models create different amplification mechanisms of the credit constraint channel. Which one do you expect to be the one that generates more amplification?

Compare to Kiyotaki and Moore (fi99F)

2 Question 2 This problem analyzes the welfare effects of a ”capital injection™in a model with financial frictions. There are two periods, 0 and fi. Consumers and entrepreneurs have a linear utility function, sO‡ sfi. Consumers have a large endowment of consumption goods in each period and a unit endowment of labor in period fi, which they sell inelastically on a competitive labor market at the wage u f i .

Entrepreneurs have a given endowment of consumption goods EO and no capital. Then they borrow bfi and invest hfi. In period fi they hire workers at the wage u f i and produce consumption goods according to the Cobb-Douglas production function: (fi†) fi fi α fi—α 4fi = h 1

The entrepreneurs face the collateral constraint bfi ≤ Z (4fi — ufi1fi) where Z c (0, fi) is a given scalar (think that in period fi the entrepreneurs can run away with a fraction (fi — Z) of the firm‘s profits). Assume the consumers endowment is large enough that the gross interest rate is always fi in equilibrium.

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The entrepreneur‘s problem is then max

sO‡ sfi

(c0,cfi )≤O,hfi ,1fi,bfi

sO ‡ hfi ≤ EO ‡ bfi : (t = 0 budget constraint) sfi ‡ bfi ≤ 4fi —

,

‹.t :

.

ufi1fi : (t = fi budget constraint) bfi ≤ Z (4fi — ufi1fi) :

.,

(collateral constraint)

(a) Argue that the entrepreneurs will always choose 1fi to maximize profits in period fi and that then profits are a linear function of the capital stock hfi, that is: 4fi — ufi1fi = R (u f i ) hfi where R (u f i ) is some (potentially non-linear) function of u f i . Restate the entrepreneur‘s problem as a simpler linear problem.

Amswer: Is easy to see that the only way profits affect the consumer problem is by making both the t = fi budget constraint and the collateral constraint slacker when profits go up. Therefore, the entrepre- neur would want to make profits as big as possible.

Therefore, in any optimum the entrepreneur must maximize profits (relate this result to the result shown in fi4.fi2fi about the consumer that owns a technology: we showed there that the consumer would always maximize profits). α fi—α v × = max 4 — fi u 1fi fi= h 1 — fi fiu 1 fi fi 1 fi

The profit maximization problem (given a capital stock hfi) is FOCs: (1 fi) : (fi — α) h 1 fiα ×—α = u fi→→fi1 = fi

×

.

fi — α uf i

Σ

α

hfi

and plugging in into the profit maximization problem ×

α ×fi—α

v = hfi 1fi

×

— u 1= fih fi

. z

f i —α u fi

Σ

α

Σ. ,,

a . Σ Σ fi—a f i —α fi—α f i —α hfi — uf i u fi u fi Σ f i Σ fi—a fi — α — fi hfi = R (u f i ) hfi fi u s

α fi

.

α

hfi =

R(m ) fi

as we wanted to show. The maximization problem is now

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max

sO‡ sfi

(c0 ,c fi )≤O,h fi ,b fi

,

‹.t :

.

.,

sO ‡ hfi ≤ EO ‡ bfi : (t = 0 budget constraint) sfi ‡ bfi ≤ R (u f i ) hfi : (t = fi budget constraint) bfi ≤ Zhfi : (collateral constraint)

which is a linear program in sO, sfi, hfi and 1fi (b) Argue that if ZR(u fi ) c fi ≤ R(u f i ) then the entrepreneur‘s problem is well defined and the entrepreneur‘s demand for capital is finite. Derive it.

Amswer Let‘s study the different cases:

Case 1 R (u f i ) = fi fhen ‹.t :

, . .,

max

sO‡ sfi

(c0,cfi)≤O,hfi,1fi,bfi s O ‡ hfi ≤ EO ‡

bfi : (t = 0 budget gonstvasnt) sfi ‡ bfi ≤ hfi : (t = fi budget gonstvasnt) bfi ≤ Zhfi : (go11ateva1 gonstvasnt)

so the entvepveneuv ss sndsflevent betmeen snsestsng sn b ov h, as 1ong as bfi ≤

Zhfi

Case 2 R (u f i ) c fi In thss gase, snsestsng ss gapsta1 gsses negatsse vetuvn (begause R (u f i ) hfi c hfi ), and henge thss teghno1og4 ss domsnated b4 sassng sn bonds on14. fhevefove, sn the optsmum me must hase hfi×= 0 and sO‡ sfi = EO. Case 3 ZR (u f i ) ≤ fi In thss gase, me a1so hase that R (u f i ) > fi. We ms11 shom that the pvob1em does not hase a so1utson (s.e. the demand fov gapsta1 ss snflnst4). Fov that, flx a gapsta1 1ese1 A > 0 and me ms11 flnd a feassb1e p1an (sO, sfi, hfi, bfi) sugh that gapsta1 hfi = A and uts1st4 ss U = EO ‡ [R (u f i ) — fi] A

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As me tahe A — ‹ œ me see that the pvob1em does not hase so1utson. smp1ementsng hfi = A

A feassb1e p1an fov

(a1though st ss not the on14 one) ss hfi sfi

= bfi = A sO = EO

= [R (u f i ) — fi] A

Is eas4 to ghegh that st satssfles a11 the gonstvasnts. In pavtsgu1av, begause ZR (u f i ) > fi me hase that the go11ateva1 gonstvasnt ss: bfi ≤ ZR (u f i ) hfi →→ A ≤ ZR (u f i ) A mhsgh ss a1ma4s tvue, begause ZR (u f i ) > fi. If, on the gontvav4, me had that ZR (u f i ) c fi, then thss p1an mou1d nesev be feassb1e, ssnge A > ZR (u f i ) A, sso1atsng the go11ateva1 gonstvasnt.

Case 4 ZR (u f i ) c fi c R (u f i ) We ms11 shom that sn thss gase, the pvob1em ss me11 deflned, and has a posstsse so1utson fov gapsta1.

See that sn the optsmum me must hase that sO

= EO ‡ bfi — hfi ≤ 0

sfi

= R (u f i ) hfi — bfi ≤ 0

c0

cfi

max E, ‡ — h— O bsz fi h ,‡ fi R, (u )fisz fi b, fi= E ‡ O max [R (u fi) — fi] h so me gan vemvste the hfi,bfioptsmssatson pvob1em as hfi ,bfi sO ÷ EO ‡ bfi — hfi ≤ 0 sfi ÷

, ‹.t :

.

→→

fi

(fi) : hfi ≤ EO ‡ bfi

R (u f i ) hfi — bfi ≤ 0 bfi ≤ ZR (X) : bfi ≤ R (u f i ) hfi ., (u f i ) hfi (3)mahe : bfi ≤hZR (u f i ) hfi See that begause R (u f i ) > fi me must hase that the entvepveneuv mants to fi as bsg as posssb1e. A1so, see that vestvsgtson (3) smp1ses vestvsgtson (X), so then me on14 need to flnd the bsggest hfi that satssf4 the gonstvasnts hfi

≤ EO ‡ bfi

bfi

≤ ZR (u f i ) hfi

Begause bfi gsses vetuvns of fi c R (u f i ), the go11ateva1 gonstvasnt ms11 be bsndsng (s.e. bfi =

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ZR (u f i ) hfi). Ussng that sn the flvst gonstvasnt, me get E O fi — ZR (u )

× fi

h fi= E ‡O ZR (u fi) h fi→→ h =

fi

So, the entvepveneuv gan flnange msth hss omn funds EO of gapsta1, and bovvoms fvom outssde E0 fi—ZR(mfi)

— EO

^ if E > E, the entrepreneurs ^ (c) Show that there is a cutoff Ef isuch that can finance the first-best O × f i — a level of capital hfi = h = α , and the collateral constraint is not binding.

Amswer The first best level of capital solves

.

×

α—fi max hα — fi hfi →→ αh fi = fi →→ h = fiα h

—fiΣ

f i a —fi

fi

= α

f i fi—a

Also, remember that R (u f i ) = 5 K (h) = αhα—fi

fi

so, if we are at the first best level of capital, we have that R (u )fi = α α

.

Σ

f i fi—a

α —fi

= fi

So the collateral constraint is just satisfied (but may not bind), and the entrepreneur is indifferent Σ Σ ΣE 0 E 0Σ between all levels of investment h c 0, = 0, fi—Z fi fi—ZR Σ Σ 0 E fi—Z Then, h c fi× 0, →→ h fi×c

E O × →→ E > O (fi — Z) h = (fifi — Z) α fi — Z

f i fi—a

÷ E^

(d ) Show that if E c E there ^is an equilibrium where the entrepreneurs are constrained and the equilibrium value of hfi is an increasing function of EO. Amswer If E cO E =^ →fi h c h ×fi→ R (u ) > fifi = = R (u f i ) = αhα—fi fiso hfi is given by

→ optimal h to finance is h = fi fi

h=

fi

E 0 fi—ZR(m fi)

. In equilibrium

EO α —fi fi — Zαh fi

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Using the implicit function theorem on G (h , E) = dhfi dEO

=

—GE Gh

fi

= —

h fiα —XαZE

fi—Zαh O

f

E 0 fi—Zαha—fi

i

=

— h we fi get that

i

fi a —fi

α—f

fi

( h fia —fiαZ—fi)

2

— fi

fi—Zαh fi fi ‡ hα—X αZE

f

i

fi a —fi

O fi—α ( h fia —fiαZ—fi)

2

> 0

proving the result. Note that as EO — ‹ 0 =→ hfi (EO) — ‹ 0 as well. If EO — ‹ 0 then lim E 0 —‹O

dh dEO

fi

=

fi ‡ h

α—X fi αZ

f i fi—ZαOa—fi

m0m

fi—α

( h fia —fiαZ—fi)

2

= fi

(e) Suppose the consumers pay a lump-sum tax t in period 0. The receipts from the tax are transferred directly to the entrepreneurs. Derive an expression for the expected utility of consumers and entrepreneurs as a function of t ^ utility of entrepreneurs is maximal utility ‡ t , and for consumers is the utility If E >O E then of equilibrium —t If E Oc E, ^then equilibrium capital will change, from h (E fi) toOh (E ‡fi t )O > hfi (EO ‡ t ) .

Utility for consumers will then be Σ Σ —α f i UC (t, EO) = u f i (hfi (EO ‡ t ) ) — t = (fi — α) —t h fi(E ‡O t ) ( f ) Show, analytically or by numerical example, that there is a non-monotone relation between t and the expected utility of the consumers. In particular, if EO is sufficiently small, a small positive tax can increase the utility of both consumers and entrepreneurs.

Amswer Suppose EO = o is sufficiently small, and t is sufficiently small as well, so that UC (t, o) = (fi — α) hfi (o ‡ t ) α — t α —fi 6h fi U Cj(0, o) = α (fi — α) [h (o ‡ fit)] — fi c α (fi — α) [h (ofi ‡ t )] — fiα —fi 6E O At t = 0, the derivative of UC is and j lim U (0, !!

C o) = ‡ œ o—‹ O

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so, a small increase in taxes from really small endowment of entrepreneurs EO increase utility of consumers marginally more than one to one, so they are better off by paying the tax.

If, however, t is big enough so that o ‡ t > EO Then the tax won‘t affect the entrepreneur‘s investment in hfi, because in equilibrium R (u f i ) = fi and capital will be at its first best level. Therefore, wages won‘t change in equilibrium, and the tax will just be a lump-sum transfer from consumers to entrepreneurs with no effects on equilibrium.

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