macro prelim solutions

7/27/2019 Macro Prelim Solutions

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Macro Prelim Solutions

Joseph Steinberg

May 25, 2009

Contents

1 Spring 2009, 8107 final exam, question 1 (Perri, Ayagari) 2

2 Fall 2008, 8106 final exam, question 1 (Chari, Asset allocation) 6

3 Fall 2008, 8106 final exam, question 2 (Chari, cash-credit/Friedman rule) 9

4 Fall 2008, I.1 (Larry, labor-augmenting tech. change) 13

5 Fall 2008, I.2 (Chari, Search and R&D) 16

6 Fall 2008, I.4 (Perri, income fluctuation problem) 20

7 Fall 2008, II.1 (Larry, TDCE with government spending in utility function) 22

8 Fall 2008. II.2 (Victor, externality) 26

9 Fall 2008, II.3 (Chari, cash-credit) 30

10 Spring 2008, I.1 (Larry, DP) 33

11 Spring 2008, I.2 (Chari, optimal asset allocation) 34

12 Spring 2008, I.3 (Victor, bargaining/monopolistic competition) 36

13 Spring 2008, I.4 (Perri, IFP) 38

14 Spring 2008, II.1 (Larry, DP) 41

15 Spring 2008, II.2 (Victor, recursive competitive equilibrium) 43

16 Spring 2008, II.3 (Chari, search and human capital) 46

17 Spring 2008, II.4 (Perri, Ayagari) 50

18 Fall 2007, I.1 (Larry, TDCE) 53

19 Fall 2007, I.2 (Perri, Ayagari) 55

20 Fall 2007, I.3 (Chari, optimal asset allocation) 57

21 Fall 2007, II.1 (Larry, durable goods) 57

22 Fall 2007. II.2 (Perri, IFP in small open economy) 63

23 Spring 2007, I.1 (Larry, different discount factors) 64

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24 Spring 2007, I.2 (Victor, OLG) 66

25 Spring 2007, I.4 (Chari, on-the-job search) 69

26 Spring 2007 II.1 (T. Kehoe, DP) 72

27 Spring 2007 II.2 (Larry, TDCE with government spending in the utility function) 76

28 Fall 2006 I.1 (Larry, TDCE) 76

29 Fall 2006 I.4 (Chari, on the job search) 79

30 Fall 2006 II.1 (T. Kehoe, DP/guess & verify) 79

31 Fall 2006 II.2 (Larry, TDCE with government spending in the production function) 83

32 Spring 2006, I.1 (Larry, TDCE) 87

33 Spring 2006, I.3 (Chari, DP) 89

34 Fall 2005, II.4 (Chari, cash-credit) 91

35 UPenn prelim, fall 2007, industry equilibria 93

1 Spring 2009, 8107 final exam, question 1 (Perri, Ayagari)

Thanks to Tayyar and Jan for their solution.

Part (a)

Let A = [a, ) and let E = [y, ). Several things to note: both sets are compact, A is the state spacefor type 1, and A E is the state space for type 2. Let AE be a -algebra for A E. Let (A E,AE)be a measurable space. Finally, let be the set of probability measures on (A E,A E).

There are two ways to approach this problem. The first way is to try to come up with a way to have astationary measure that includes both types of agents and have a Huggett economy where aggregate demandfor saving is zero. The other way is to have a stationary measure for just the type 2 agents and consider thetype 1 agents as the demand for borrowing (like the exogenous K(r) in Ayagaris paper). In order to avoidtrouble defining a measure over all the agents (we would have to have some way of differentiating type 1agents from type 2 agents that just happen to draw a shock of zero), I will use the second approach.

A stationary equilibrium in this economy is: Value functions V1 : A R and V2 : A E R; policyfunctions g1a : A A, g

1c : A R+, g

2a : A E A, and g

2c : A E R+; a stationary measure

;and an interest rate r such that

(i) Given r, V1, g1a, and g1c solve type 1s problem:

V1(a) = max(a,c)1(a)

{u(c) + V1(a)}

where 1(a) = {(a, c) : c + a = (1 + r)a + y, c 0, a A}.

(ii) Given r, V2, g2a, g2c solve type 2s problem:

V2(a, ) = max(a,c)2(a,)

u(c) +

E

V2(a, ) dF()

where 2(a, ) = {(a, c) : c + a = (1 + r)a + y + , c 0, a A}.

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(iii) The market for saving clears:

g1a(a) + (1 )

AE

g2a(a, ) d(a, ) = 0.

Note that Walras law implies that the market for consumption goods will clear if the above equationholds so we dont need to explicitly include a market-clearing condition for the goods market.

(iv) is invariant, i.e., for all A E A E,

(A E) =

AE

Q[(a, ), (A E)] d(a, ),

where

Q[(a, ), (A E)] =

E

A[g2a(a, )] dF(

).

In other words, Q gives the probability that a type 2 agent starting with (a, ) will choose g2a A andreceive a shock E. If the shocks were not independently distributed, we would replace dF() withsomething like (, d).

Part (b)Autarky is not a stationary equilibrium. To see this, first consider the type 1 agents Euler equation:

u(c1t ) (1 + r)u(c1t+1).

Suppose that (1 + r) > 1. Define Mt = u(c1t )[(1 + r)]

t. Then the Euler equation implies that

Mt Mt+1 > 0.

Thus Mt is a bounded sequence. Since [(1 + r)]t , it must be that u(c1t ) 0. Inada conditions then

imply that c1t . By iterating forward on the budget constraint from time t, we can express the type 1agents budget constraint as

a1t (1 + r) +

j=0y

(1 + r)j

j=0c1t+j

(1 + r)j

.

Since c1t is unbounded, the sum on the RHS is also unbounded. But

j=0y

(1+r)j is clearly bounded, so it

must be that a1t . Thus autarky cannot be a stationary equilibrium when (1 + r) > 1.Now suppose that (1 + r) < 1. For simplicity, consider the simple case in which a = 0. We will use

x (1 + r)a + y to represent the type 1 agents cash in hand. Then his problem is

V1(x) = maxc,a

{u(c) + V1(x)}

s.t. x = (1 + r)(x c) + y

a = x c

a 0

The envelope condition for this problem isV1x (x) = uc(g

1c (x)).

Assuming that V1 is twice-differentiable, we get

V1xx(x) = ucc(g1c (x))

dg1c (x)

dx

which impliesdg1c (x)

dx=

Vxx(x)

ucc(g1c (x))> 0

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since both u and V1 are strictly concave. Thus type 1s consumption is increasing in cash-in-hand. If theborrowing constraint is not binding, the FOC holds with equality. Using the envelope condition in the FOC,we get

Vx(x) = (1 + r)Vx(x) < Vx(x

).

Thus x < x, so cash-in-hand is decreasing over time when the borrowing constraint is not binding. Sinceincome is constant, this implies that a1

t

is decreasing over time. Further, a1

t

will hit the borrowing constraintin finite time. Suppose not. Then cash-in-hand is greater than y for all t, i.e., xt > y, t. Then we have

0 < u(c1t ) = limj

[(1 + r)]j u(g1c (xt+j )) limj

[(1 + r)]j u(g1c (y)) = 0.

Contradiction. Therefore it must be that cash-in-hand converges to y in finite time, i.e., T such that for allt T, a1t = 0 = a. This analysis yields a similar result when a < 0, so autarky cannot be a stationaryequilibrium when (1 + r) > 1.

So far we have shown that just considering the type 1 agents, the only viable candidate for parametervalues that are consistent with autarky is (1 + r) = 1. However, if (1 + r) = 1, type 2 agents willaccumulate assets. This is because the utility function is strictly concave and marginal utility is strictlyconvex. Consider the Euler equation for type 2 agents when (1 + r) = 1:

u(c) E[u(c)].

Since marginal utility is strictly convex, Jensens inequality implies that

E[u(c)] > u (E[c]).

Thusu(c) > u(E[c])

so type 2s consumption will tend to increase over time. This requires that type 2 agents have precautionarysaving. Thus when (1 + r) = 1, type 2 agents will tend to acucmulate assets over time, so (1 + r) = 1 isnot consistent with autarky either. Therefore there is no value of (1 + r) that is consistent with autarky,so autarky cannot be a stationary equilibrium.

Part (c)In part (b), we saw that (1 + r) = 1 cannot be not a stationary equilibrium because type 1 agents will holdzero assets while type 2 agents will accumulate assets, so aggregate asset holdings would be greater thanzero, i.e., the market for saving would not clear. Further, if (1 + r) > 1, both types will accumulate assets,so in a stationary equilibrium it must be that (1 + r) < 1.

In this case, type 1 agents will hit the borrowing constraint in finite time, so in stationary equilibrium,all type 1 agents will have a1t = a. Thus average assets holdings for type 1 agents is a and the varianceis zero.

Part (d)

Since g1a(a) = a in stationary equilibrium, market clearing requires that

a + (1 )

AE

g2(a, ) d(a, ) = 0

or

E[a2t ] =

A

g2(a, ) d(a, ) =

1 a.

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Part (e)

Since type 1 agents all have assets equal to a, their budget constraint implies that

c1t = y ra

for all type 1 agents. Thus average consumption for type 1 agents is y ra and the variance is zero.

For type 2 agents, market clearing for goods implies that

(y ra) +

AE

g2c (a, ) d(a, ) = y.

Thus we get

E[c2t ] =

AE

g2c (a, ) d(a, ) = y +

1 ra.

Then average type 2 consumption is higher than average type 1 consumption if and only if

y +

1 ra > y ra.

Simplifying this expression, we getra > 0.

ThusE[c2t ] > c

1t ra > 0.

If idiosyncratic risk of type 2 agents increases, they will want to save more. Since type 1 agents will stillbe at the borrowing constraint (i.e., their assets will remain unchanged), the interest rate must go down inorder to clear the market for saving (since a lower interest rate will decrease the amount of saving done bytype 2 agents). Since

E[c2t ] c1t = y +

1 ra y + ra

we have(E[c2t ] c

1t )

r=

1 a + a > 0.

Since r must decrease to clear the market for saving, the difference in consumption of the two types will go

down. The graph below shows how the increase in idiosyncratic risk causes type 2 agents demand for savingto increase and causes equilibrium interest rate to drop.

Include graph!

Part (f)

To answer this problem, we have to consider what happens during the transition from time zero to thestationary equilibrium as well as the stationary equilibrium itself. Note that since has mean zero, theexpected present value of lifetime endowments is the same for both types: y1 . Further, the law of largenumbers implies that there is no aggregate uncertainty; the aggregate endowment in every period and stateis y.

If we had complete markets, the second welfare theorem implies that we could solve the planners problemand use the Negiishi algorithm to calculate the planners problem weights that would give us the competitive

equilibrium allocations. Since both types have the same expected present value of lifetime endowments andthe same initial asset holdings, the correct weights would be the same for both types. This implies that thecomplete markets CE would have each type consuming y in every period and state.

Since the utility function is strictly concave, the type 1 agents problem has a unique solution, so settingct = y is strictly better than every other affordable allocation for type 1 in the complete markets CE. Notethat consuming y in every period is still feasible for type 1 agents in our Ayagari economy with non-contingentbonds only. Since the type 1 agents choose a different consumption stream in this economy, strict concavityof the utility function implies that they are strictly better off in the Ayagari economy than in the completemarkets economy. Therefore the type 2 agents must be strictly worse off, so ex ante it would be better tobe a type 1 agent in this economy.

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Part (g)

The market clearing condition is now

g 1a(a) + (1 )

AE

g2a(a, ) d(a, ) = 0.

Since type 1 agents will still be at the borrowing constraint and g2(a, ) has not changed, the increase in thefraction of type 1 agents causes aggregate supply of lending to decrease (the curve (1 )A2 shifts left to(1 )A2 in the graph below) and causes aggregate demand for borrowing to increase (from a to a in thegraph). In order for the market to clear, type 2 agents will have to save more on average so the equilibriuminterest rate will have to rise. See the graph below.

Part (h)

As 1, aggregate supply of lending will go to zero and demand for borrowing will go to a for all r suchthat (1 + r) < 1. In order to clear the market for assets, the equilibrium interest rate will go to r = 1 1at the limit, at which point the type 1 agents have no incentive to borrow or lend.

2 Fall 2008, 8106 final exam, question 1 (Chari, Asset allocation)Part (a)

I assume that the entrepreneur can consume his wealth as well as allocate it to capital and bonds. Thismeans that the problem is similar to Charis other asset allocation problem in which an agent allocates wealthbetween consumption, a risky asset, and a safe asset. Here, the risky asset is the profit from entrepreneurialactivity. I also assume that the production function is CRS and the agent gets to choose labor input afterhi productivity and wage shocks are drawn.

To write the entrepreneurs problem as a dynamic program, first note that in the final period, theentrepreneur will consume all of his wealth since he receives no benefit from allocating any of it to capitalor bonds. Thus VT(W) = u(W). The entrepreneurs problem for t < T can be written as follows:

Vt(W) = maxc,k,b u(c) + A maxn {Vt+1(W)} dF(A) dG()s.t. c + k + b W

W = AF(k, n) n + Rb

Substituting the second constraint into the functional equation gives us a more compact version:

Vt(W) = maxc,k,b

u(c) +

A

maxn

{Vt+1[AF(k, n) n + Rb]} dF(A) dG()

s.t. c + k + b W

If we let k = (W c) and b = (1 )(W c), we can write the problem in a different way:

Vt(W) = maxc0[0,1]

u(c) + A

maxn {Vt+1[AF((W c), n) n + R(1 )(W c)]} dF(A) dG() .

(1)

Part (b)

Given my interpretation of the question, I assume that we want to show that the entrepreneur allocates afixed fraction of his after-consumption wealth W c to capital.

Proposition. For all t T, Vt(W) = atu(W) for some constant at.

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Proof. By induction. For period T, we have VT(W) = u(W), so the condition holds for period T withaT = 1. Now suppose that Vt+1(W) = at+1u(W). We want to show that this implies that Vt(W) = atu(W).Given the assumption about Vt+1, we can write Vt as

Vt(W) = maxc0

[0,1]

u(c) +

A

maxn

{at+1u[AF((W c), n) n + R(1 )(W c)]} dF(A) dG()

.

(2)Take arbitrary k, b and fix A, . Consider the general problem of the form

maxn

{u[AF(k, n) n + Rb]}.

Since the utility function is strictly increasing, the agent will choose n to maximize profits. Thus the maximalutility from this problem is equivalent to

u

maxn

{AF(k, n) n} + Rb

.

Let n(A ,k,) denote the optimal choice ofn given A ,k,. Then we have

u maxn {AF(k, n) n} + Rb = u [AF(k, n(A ,k,)) n(A ,k,) + Rb] .Since F is CRS, the optimal choice n(A ,k,) will be a linear function of k, i.e., the ratio n

(A,k,)k will be

equal to some constant function of the parameters A, . Suppose we multiply k and b by the same numberQ. Then this implies that

u [AF(Qk,n(A,Qk,)) n(A,Qk,) + RQb] = u [AF(Qk,Qn(A ,k,)) Qn(A ,k,) + RQb]

= u [AQF(k, n(A ,k,)) Qn(A ,k,) + RQb]

= u [Q(AF(k, n(A ,k,)) n(A ,k,) + Rb)]

Note that the specification of u implies that for any a, b, u(ab) = (1 )u(a)u(b). Given the above result,

this implies that

u [AF(Qk,n(A,Qk,)) n(A,Qk,) + RQb] = (1 )u(Q)u(AF(k, n(A ,k,)) n(A ,k,) + Rb).

Since n is the maximizing value, we have

maxn

{u [AF(Qk,n) n + RQb]} = (1 )u(Q)u

maxn

{AF(k, n)} + Rb

. (3)

Plugging this result into (2), we get

Vt(W) = maxc0

[0,1]

u(c) +

A

maxn

{at+1u[AF((W c), n) n + R(1 )(W c)]} dF(A) dG()

= maxc0[0,1]

u(c) + at+1(1 )u(W c) A

u maxn {AF(, n)} + R(1 ) dF(A) dG()

Since the term A

u

maxn

{AF(, n)} + R(1 )

dF(A) dG()

does not depend on W or c, we can write our functional equation as

Vt(W) = maxc0

u(c) + at+1(1 )u(W c) max

[0,1]

A

u

maxn

{AF(, n)} + R(1 )

dF(A) dG()

.

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Since u is strictly concave,

argmax[0,1]

A

u

maxn

{AF(, n)} + R(1 )

dF(A) dG()

is unique. Let denote the maximizing value, and note that does not depend on c, W, or t. This

implies that if our induction proof goes through, the entrepreneur will always allocate the same fraction ofafter-consumption income W c to capital.Now we have

Vt(W) = maxc0

u(c) + u(W c)at+1(1 )

A

u

maxn

{AF(, n)} + R(1 )

dF(A) dG()

.

(4)Let Jt denote

at+1(1 )

A

u

maxn

{AF(, n)} + R(1 )

dF(A) dG()

Then we can write this as simply

Vt(W) = maxc0

{u(c) + Jtu(W c)} . (5)

Since u is strictly increasing and strictly concave, the FOC is sufficient to guarantee a maximum. The FOCis

u(c) = Jtu(W c).

Lets use the specification of u now. The FOC is

c = Jt(W c).

Solving for c, we get

c =J

1/t W

1 + J1/

t

=W

1 + J1/t

.

Plugging this into (5), we have

Vt(W) = u

W

1 + J1/t

+ Jtu

W

W

1 + J1/t

.

Using the fact that u(ab) = (1 )u(a)u(b) again, we have

Vt(W) = u(W)(1 )

u

1

1 + J1/t

+ Jtu

1

1

1 + J1/t

.

Let

at = (1 )

u

1

1 + J1/t

+ Jtu

1

1

1 + J1/t

.

So now we haveVt(W) = atu(W).

This completes the induction.

With our proposition verified, we can see that the entrepreneur indeed allocates a fixed fraction of hisafter-consumption wealth to capital.

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3 Fall 2008, 8106 final exam, question 2 (Chari, cash-credit/Friedmanrule)

Part (a): Constant money supply

A CE in this environment is

An allocation (z1, z2), where zi = {ci1t, ci2t,

it, M

it , B

it}t=0

A price system P = {pt, wt}t=0

A policy = {Bt, M, Rt}t=0 (Mt = M, t)

such that1) Given (P, ),

zi argmax{ci1t,c

i2t,

it,M

it ,B

it}

t=0

t=0

tui(ci1t, ci2t,

it)

s.t. Mit + Bit (M

it1 pt1c

it1) + wt1

it1 pt1c

i2t1 + Rt1B

it1

ptci

1t Mi

t

0 it 1

non-negativity, no Ponzi, Ai0 given

2) Given (P, ), (z1, z2) solves the firms problem

max{c11t,c

21t,c

12t,c

22t,

1t ,

2t}

t=0

t=0

[pt(c11t + c

21t + c

12t + c

22t) wt(

1t +

2t )]

s.t. c11t + c21t + c

12t + c

22t =

1t +

2t

non-negativity

3) The governments budget balances: Bt+1 = RtBt (since money supply is fixed, money cancels out on

both sides)4) The market clears for all t: B1t + B

2t = Bt, M

1t + M

2t = M,

2i=1[c

i1t + c

i2t] =

1t +

2t .

3.1 Part (b): Constant interest rate

A CE in this environment is


it, M

it , B

it}t=0

A price system P = {pt, wt}t=0

A policy = {Bt, Mt, R}t=0 (Rt = R, t)

such that

1) Given (P, ),

zi argmax{ci1t,c

i2t,

it,M

it ,B

it}

t=0

t=0

tui(ci1t, ci2t,

it)

s.t. Mit + Bit (M

it1 pt1c

it1) + wt1

it1 pt1c

i2t1 + RB

it1

ptci1t M

it

0 it 1

non-negativity, no Ponzi, Ai0 given

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2) Given (P, ), (z1, z2) solves the firms problem

max{c11t,c

21t,c

12t,c

22t,

1t ,

2t}

t=0

t=0

[pt(c11t + c

21t + c

12t + c

22t) wt(

1t +

2t )]

s.t. c11t + c21t + c

12t + c

22t =

1t +

2t

non-negativity

3) The governments budget balances: (Mt+1 Mt) + Bt+1 = RBt.4) The market clears for all t: B1t + B

2t = Bt, M

1t + M

2t = M,

2i=1[c

i1t + c

i2t] =

1t +

2t .

Part (c): Friedman rule

I assume that utility for both agents is given by

u1(c1, c2, 1 ) = u2(c1, c2, 1 ) =

c111

+c121

+ v(1 ).

Let = {Bt, Mt, Rt}t=0 denote a policy. Let be the set of all possible policies. To find the optimalpolicy, we solve the Ramsey problem:

max

t=0

2i=1

iui(ci1t(), ci2t(),

it())

,

where i is the planner weight for consumer i, subject to

{(ci1t(), ci2t(),

it(), M

it (), B

it ())

2i=1}

t=0

is a competitive equilibrium allocation.A competitive equilibrium in this environment (remember that labor is supplied inelastically)


it, M

it , B

it}t=0

A price system P = {pt, wt}t

=0

A policy = {Bt, Mt, Rt}t=0

such that1) Given (P, ),

zi argmax{ci1t,c

i2t,

it,M

it ,B

it}

t=0

t=0

tui(ci1t, ci2t, 1

it)

s.t. Mit + Bit (M

it1 pt1c

it1) + wt1

it1 pt1c

i2t1 + Rt1B

it1

ptci1t M

it

0 it 1

non-negativity, no Ponzi, A

i

0 given2) Given (P, ), (z1, z2) solves the firms problem

max{c11t,c

21t,c

12t,c

22t,

1t ,

2t}

t=0

t=0

[pt(c11t + c

21t + c

12t + c

22t) wt(

1t +

2t )]

s.t. c11t + c21t + c

12t + c

22t =

1t +

2t

non-negativity

3) The governments budget balances: (Mt+1 Mt) + Bt+1 = RtBt.

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4) The market clears for all t: B1t + B2t = Bt, M

1t + M

2t = M,

2i=1[c

i1t + c

i2t] =

2i=1

it.

The firms problem implies that wt = pt, t. Given our assumption, both consumers utilities arestrictly increasing and strictly concave. Then their securities market constraints will hold with equality.Their constraint sets are convex, so FOCs are sufficient to guarantee a maximum. The FOCs for consumeri are:

c

i

1t :

t

u

i

1(t) = pt

i

t+1 +ptt (1)ci2t :

tui2(t) = ptit+1 (2)

it : tui(t) = wt

it+1 (3)

Mit : it =

it+1 +

it (4)

Bit : it = Rt

it+1 (5)

it : Mit + B

it = (M

it1 pt1c

it1) + wt1

it1 pt1c

i2t1 + Rt1B

it1 (6)

it : ptci1t M

it (7)

where it and it are the lagrange multipliers on the securities market and cash-in-advance constraints re-

spectively. Note that by using (1), (2), (4), and (5), we get

ui

1

(t)

ui2(t) = Rt. (8)

We can use these FOCs to get an implementability constraint for each consumer which we will use to solvethe Ramsey problem.

Start with (5), the securities market constraint:

Mit + Bit = (M

it1 pt1c

it1) + wt1

it1 pt1c

i2t1 + Rt1B

it1.

Assume the CIA constraint holds with equality. Then ptci1t = Mit , t.

Mit + Bit = wt1

it1 pt1c

i2t1 + Rt1B

it1.

Multiply by it:itM

it +

itB

it =

itwt1

it1

itpt1c

i2t1 +

itRt1B

it1.

Use (2), (3), and (5):

itMit +

itB

it =

t1ui(t 1)it1

t1ui2(t 1)ci2t1 +

it1B

it1.

Sum from period t = 1 to T:

Tt=1

[itMit +

itB

it] =

Tt=1

t1[ui(t 1)it1 + u

i2(t 1)c

i2t1] +

Tt=1

it1Bit1.

Reindex on the RHS:

T

t=1[itM

it +

itB

it] =

T1

t=0t[ui(t)

it + u

i2(t)c

i2t] +

T1

t=0itB

it.

Cancel out some of the bond terms:

Tt=1

itMit +

iTB

iT =

T1t=0

t[ui(t)it + u

i2(t)c

i2t] +

i0B

i0.

Add i0Mi0 to both sides:

T1t=0

itMit +

iT[M

iT + B

iT] =

T1t=0

t[ui(t)it + u

i2(t)c

i2t] +

i0[M

i0 + B

i0].

11


12/94

Use the fact that the CIA holds with equality again:

T1t=0

itptci1t +

iT[M

iT + B

iT] =

T1t=0

t[ui(t)it + u

i2(t)c

i2t] +

i0[M

i0 + B

i0].

Use (1) and (4) on itptci1t and rearrange some terms:

T1t=0

t[ui1(t)ci1t + u

i2(t)c

i2t + u

i(t)

it] =

i0[M

i0 + B

i0]

iT[M

iT + B

iT].

Take limits as T and apply the transversality condition limT[MiT + Bit] = 0:

t=0

t[ui1(t)ci1t + u

i2(t)c

i2t + u

i(t)

it] =

i0[M

i0 + B

i0].

Finally, set p0 = 1 and use (2):

t=0

t[ui1(t)ci1t + u

i2(t)c

i2t + u

i(t)

it] = u

i2(0)[M

i0 + B

i0].

WLOG, set Mi0 + Bi0 = 0. Thus our implementability constraint for consumer i is

t=0

t[ui1(t)ci1t + u

i2(t)c

i2t + u

i(t)

it] = 0. (9)

The competitive equilibrium allocation is thus fully characterized by (8), (9), and the resource constraint.Note that for the problem to be well-posed, we require that Rt 1, t. Therefore we can write the

Ramsey problem as

max{(ci1t,c

i2t,

it)2i=1}

t=0

t=0

2i=1

iui(ci1t, ci2t,

it)

s.t.

t=0 t[ui1(t)ci1t + ui2(t)ci2t + ui(t)it] = 0, i = 1, 2ui1(t)

ui2(t) 1, i = 1, 2, t

2i=1

[ci1t + ci2t] =

2i=1

it, t

Consider the Ramsey problem with the second constraint dropped:

max{(ci1t,c

i2t,

it)2i=1}

t=0

t=0

2i=1

iui(ci1t, ci2t,

it)

s.t.

t=0 t[ui1(t)ci1t + ui2(t)ci2t + ui(t)it] = 0, i = 1, 22

i=1

[ci1t + ci2t] =

2i=1

it, t

Let i be the multiplier on consumer is implementability constraint and let t be the multiplier on theresource constraint. Given the assumption about the form of the utility function, the implentability constraintfor consumer i is

t=0

t[(ci1t)1 + (ci2t)

1 + v(1 it)it] = 0.

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13/94

Then the FOCs for ci1t and ci2t are

ci1t : it(ci1t)

= ti(1 )(ci1t) + t

ci2t : it(ci2t)

= ti(1 )(ci2t) + t

Combining the two conditions, we get

(ci1t)t[i i(1 )] = (ci2t)

t[i i(1 )].

This implies thatui1(t)

ui2(t)=

(ci1t)

(ci2t)

= 1.

This satisfies the constraint we dropped, so it must be that

ui1(t)

ui2(t)= 1

is optimal. Looking at (8), we see that this implies that Rt = 1, t, i.e., the Friedman rule is optimal.

4 Fall 2008, I.1 (Larry, labor-augmenting tech. change)

Part (a)

Proposition. Assume the following conditions hold:

(i) F is continuous.

(ii) F has CRS.

(iii) F(0, ) = 0.

(iv) k such that for all k [k, ) and all n, F(k, n) + (1 )k k.

(v) u is a continuous function of the form

u(c, ) =c1

1 v().

(vi) 0 < 1 < 1.

Then the planners problem for the single-sector growth model with labor augmenting technological changecan be obtained by solving a stationary dynamic program.

Proof. Define the following variables:

ct =ctAt

xt = xtAt

ky =ktAt

13


14/94

Using the assumption about the form of u, the planners problem becomes

max{ct,t,nt,xt,kt}t=0

t=0

t(tct)

1

1 v(t)

s.t. tct + txt F(tkt, tA0nt)

nt + t 1t+1kt+1 + (1 )

tkt + txt

k0 =k0A0

given

non-negativity

Since F is CRS, this simplifies to

max{ct,t,nt,xt,kt}t=0

t=0

tc1t1

v(t)

s.t. ct + xt F(kt, A0nt)

nt + t 1kt+1 + (1 )kt + xt

k0 =k0A0

given

non-negativity

where = 1, 1 = 1 , and =1. Since F and u are bounded on [0, k] and 0 < < 1, this problem

satisfies assumptions of the principle of optimality. Thus the supremum v(k0) from the problem above isthe unique solution to the following stationary dynamic program:

v(k) = maxc,,n,x,k

c1

1 v() + v(k)

s.t. c + x F(k, A0n)

n + 1

k (1 )k + x

c, x, k, n , non-negative

Part (b)

Proposition. Suppose that in addition to (i) - (vi) above, the following conditions hold:

(vii) F is strictly increasing.

(viii) F is strictly concave.

(xi) F is continously differentiable.

(x) F satisfies Inada conditions.

(xi) v() is constant (labor is supplied inelastically).

Then for any k0,kt+1

kt .

14


15/94

Proof. Since labor is supplied inelastically and the utility function is strictly increasing, n = 1 and allconstraints hold with equality. Define the function f as f(k) = F(k, A0) + (1 )k. WLOG, assume thatv() = 1. Then we can write the Bellman equation above in a simpler form:

v(k) = maxk(k)

1

1 [f(k) k]1 + v(k)

where

(k) = [0, f(k)].

We already showed that assumptions 4.3 and 4.4 hold, so theorem 4.6 from SLP implies that v is continuousand bounded. Since u and F are strictly increasing, the return function of the above Bellman equationis strictly increasing and is monotone. Thus assumptions 4.5 and 4.6 are also satisfied, so theorem 4.7implies that v is strictly increasing. Since u and F are strictly concave, f is also strictly concave, so the returnfunction is strictly concave as well, i.e., assumption 4.7 holds. Our conditions on u and F also guaranteethat is convex in the sense of assumption 4.8, so theorem 4.8 implies that v is strictly concave and thepolicy correspondence is a single-valued function g. Finally, since u and F are continuously differentiable,assumption 4.9 holds, so theorem 4.11 implies that v is differentiable for any k > 0 such that g(k)

(k).

Thus the FOC and envelope condition fully characterize the solution to our dynamic programming problem.

We want to show that

kt+1

kt . Since kt = t

A0kt, it is equivalent to show that

kt+1

kt 1. By theprinciple of optimality, the optimal sequence {kt}t=0 in the sequence problem from part (a) is generatedby iteratively applying the policy function g to the initial value k0. Thus it suffices to show that g has apositive, globally stable fixed point.

The FOC for the Bellman equation is

[f(k) g(k)] = v (g(k)).

The envelope condition isv(k) = [f(k) g(k)]f(k).

Combining the two gives the Euler equation:

[f(k) g(k)] = [f(g(k)) g(g(k))]f(g(k)).

Thus any fixed point k of g must solve

[f(k) k] = [f(k) k]f(k)

which simplifies to = f(k).

Our conditions on F imply that f is continuously differentiable, strictly concave, and satisfies Inada condi-tions. Therefore there exists a unique positive solution to this equation, i.e., there exists a unique positivefixed point k of g.

First, we show that g is strictly increasing. Suppose not. Then there exists k1 < k2 such that g(k1) g(k2). Since v is strictly concave, we have v(g(k1)) v(g(k2)). Then the FOC implies that

[f(k1) g (k1)] [f(k2) g(k2)]

.

This in turn implies thatf(k1) g (k1) f(k2) g (k2).

Since g(k1) g(k2), we havef(k1) f(k2).

But since f is strictly increasing by our conditions on F, this means that k1 k2. Contradiction. Thereforeg is strictly increasing.

15


16/94

Since v is concave, the following inequality must hold for all k, with equality if and only if k = k:

[v(g(k)) v(k)][g(k) k] 0.

Using the FOC and envelope condition to substitute for v(g(k)) and v(k) respectively, we get

[f(k) g(k)] [f(k) g (k)]f(k) [g(k) k] 0.Simplifying and using the fact that f(k) =

we have

(f(k) g (k))

f(k) f(k)

[g(k) k] 0.

Since the utility function is strictly increasing, marginal utility is always strictly positive. Therefore we cansimplify even further to get the following inequality, which again holds only for k = k:

f(k) f(k)

[g(k) k] 0. (*)

Suppose k < k. Since f is strictly concave, f(k) < f(k). Since (*) holds with equality only fork =

k

, g(k) >

k. Since g is strictly increasing, we know that g(

k) < g (

k

) =k

. Thus fork k. Again, since f is strictly increasing, f(k) > f(k). Since (*) holds with

equality only for k = k, g(k) < k. Since g is strictly increasing, we know that g(k) > g(k) = k. Thus for

k > k,k > g(k) > k.

Take k0 > k

and define the sequence {kt}t=0 as above. Now we have

k0 > k1 > k2 > . . . > kt > kt+1 > k, t.

Thus kt k.So we have shown that for all possible values ofk0, the sequence {kt}t=0 defined by iteratively applying

the policy function converges to k. Therefore k, the unique positive fixed point of g, is globally stable. Asexplained above, this implies that

kt+1kt

.

5 Fall 2008, I.2 (Chari, Search and R&D)

Part (a)

The typical inventors problem can be written as a functional equation as:

V(z) = max{VM(z), VI} (2.1)

where vM(z), the value of managing an invention, is

VM(z) = z + (1 p)VM(z) + pVI (2.2)

and vI, the value of inventing, is

VI = b +

V(z)dF(z). (2.3)

16


17/94

Part (b)

I assume that there exists a maximum invention quality z, the lowest invention quality is 0 (the inventionfails to meet government standards or some other reason for failure) and that 0 < < 1. Note that becauseVI is constant, VM(z) is continuous and strictly increasing. We can write VM(z) as

VM

(z) =

z + pVI

1 (1 p) .

Then we have

V(z) = max

z + pVI

1 (1 p), VI

so V is nondecreasing.

I assume that b 0 and 0 < < 1. Now suppose that the inventor chooses to sell a worthless productthan to invent a new one, i.e.,

V(0) = VM(0) VI.

Thenp

1 (1 p)VI VI.

This implies that p

1 (1 p) 1.

Rewriting, this is 1

1 (1 p) 0.

Since < 1 and (1 p) < 1, this is a contradiction, so it must be that the inventor chooses to invent, i.e.,

V(0) = VI > VM(0).

Now suppose that given an invention of the maximum quality, the inventor stil chooses to invent. Then

V(z) = VI VM(z).

Since V is nondecreasing, V(z) V(z) for all z [0, z]. Then

V(z) b + V(z).

Since b 0 and < 1, this is a contradiction. Then it must be that the worker chooses to manage:

V(z) = VM(z) > VI.

Thus we have shown that

VM(0) =pVI

1 (1 p)< VI

and

VM(z) =z + pVI

1 (1 p)> VI.

Since VM is continuous and strictly increasing, the Intermediate Value Theorem implies that that thereexists a unique z such that

VM(z) =z + pVI

1 (1 p)= VI. (2.4)

Therefore this dynamic program exhibits the reservation wage property, i.e.,

V(z) =

VM(z) z z

VI z z.

17


18/94

We will now solve for the reservation invention quality in terms of the given parameters. We can rewrite(2.4) as

z + pVI = [1 (1 p)]VI.

Rearranging gives usz = (1 )VI. (2.5)

We know that for all z z

, we have

V(z) = z + [(1 p)V(z) +pVI]

=z + pVI

1 (1 p).

=z

1 (1 p)+

pz

[1 (1 p)](1 ), z z (2.6)

Similarly, since V(z) = VI for all z < z, we can use (2.5) to get

V(z) =z

1 , z z (2.7)

Then by (2.5)-(2.7) we have

z = b(1 ) + (1 )

z0

V(z)dF(z)

= b(1 ) + (1 )

z0

V(z)dF(z) +

zz

V(z)dF(z)

= b(1 ) +

z0

zdF(z) +

zz

(1 )z

1 (1 p)+

pz

1 (1 p)

dF(z)

= b(1 ) +

z0

zdF(z) +

zz

(1 )z

1 (1 p)+

[1 (1 p) (1 )]z

1 (1 p)

dF(z)

= b(1 ) + z

0

zdF(z) + z

z (1 )z

1 (1 p)

(1 )z

1 (1 p)+ z dF(z)

= b(1 ) +

z0

zdF(z) +

zz

(1 )(z z)

1 (1 p)

dF(z) +

zz

zdF(z)

= b(1 ) +

z0

zdF(z) +

zz

(1 )(z z)

1 (1 p)

dF(z)

= b(1 ) + z +

zz

(1 )(z z)

1 (1 p)

dF(z)

Then we have

z(1 ) = b(1 ) +

zz

(1 )(z z)

1 (1 p)

dF(z)

which can be rewritten as

z = b + 1 (1 p)

zz

(z z)dF(z). (2.8)

Let Mt denote the fraction of inventors engaged in management in period t, and let It denote the fractionengaged in invention. Since this economy has a large number of inventors, we can express Mt+1 as a functionof Mt and It:

Mt+1 = (1 p)Mt + (1 F(z))It.

Since Mt + It = 1 t, we can rewrite this as a first-order difference equation in Mt:

Mt+1 = (1 p)Mt + (1 F(z))(1 Mt).

18


19/94

Similarly, the fraction of inventors engaged in invention in period t + 1 is

It+1 = p(1 It) + F(z)It.

A stationary equilibrium in this economy is (z, I) such that (2.8) holds and

I = p(1 I) + F(z)I. (2.9)

Rearranging (2.9) we get I as a function of p and F(z):

I =p

1 +p F(z). (2.10)

Part (c)

Consider the following rearrangement of (2.8):

z

1 (1 p)

zz

(z z)dF(z) = b. (2.11)

If b falls, the RHS rises, so the LHS must also rise. Note thatzz

(z z)dF(z)

is a decreasing function ofz. This implies that

z

1 (1 p)

zz

(z z)dF(z)

is a strictly increasing function of z. So if the LHS of (2.11) rises, then z also rises. Thus z rises when bfalls. This means that F(z) will go up. Examining (2.10), we can see that this will cause I, the fractionof agents engaged in invention, to rise.

Part (d)

We can rewrite (2.8) as

z[1 (1 p)] = b[1 (1 p)] +

zz

(z z)dF(z) +

z0

(z z)dF(z)

z0

(z z)dF(z)

= b[1 (1 p)] +

z0

(z z)dF(z)

z0

(z z)dF(z)

= b[1 (1 p)] +

E[z] z

z0

(z z)dF(z)

.

Rearranging the last equation and applying integration by parts gives us

z = 1 (1 p)

1 pb +

1 pE[z] +

1 p

z0

F(z)dz. (2.12)

Suppose that G is a mean-preserving spread ofF. Thenz0

G(z) dz

z0

F(z) dz, z [0, z].

19


20/94

For clarity, I will use zF and zG to denote the reservation invention qualities for distributions F and G.

Define the functions hF and hG as

hF(z) = 1 (1 p)

1 pb +

1 pE[z] +

1 p

z0

F(z)dz

hG(z) = 1 (1 p)

1 pb +

1 pE[z] +

1 p z

0

G(z)dz

We can see from (2.12) that zF = hF(z) and zG = hG(z

). Since G is a mean-preserving spread of F, wecan see that hG(z) hF(z), z [0, z]. This means zF z

G. In other words, a mean-preserving increase

in risk will cause the reservation invention quality to rise. This may seem counterintuitive at first, but itactually makes sense. We can think of inventing as an option which will be exercised only when the drawis above the reservation invention quality. Thus the higher incidence of very good draws increases the valueof inventing, while the higher incidence of very bad draws has little effect since the option will never beexercised upon receiving such draws. So a mean-preserving increase in risk actually increases the value ofinventing, so the reservation invention quality has to rise.

However, the fact that zG zF does not necessarily imply that G(z

G) F(z

F) nor does it necessarily

imply that G(zG) F(zF). Thus the effect of a MPS on the fraction of agents engaged in invention is

ambiguous.

6 Fall 2008, I.4 (Perri, income fluctuation problem)

Part (a)

The utility function is strictly concave and the constraint set is convex so FOCs are sufficient for maximiza-tion. Let t be the lagrange multiplier on the budget constraint. The FOCs are:

ct : t(b1 2b2ct) = t

at+1 : t = Et[(1 + r)t+1]

This implies that(b1 2b2ct) = Et[(1 + r)(b1 2b2ct+1)]

which becomesct = Et[ct+1].

since (1 + r) = 1. Note that this implies that ct is a Martingale, i.e.,

ct = Et[ct+j ], j 0. (1)

Starting from period t, multiply all the budget constraints by

11+r

j, j 0. Add them up,

take expectations (keep in mind the law of iterated expectations!), and use the transversality condition

E0

limt

1

1+r

tat

= 0, we get

j=01

1 + rj

Et[ct+j ] =

j=01

1 + rj

Et[yt+j ] + (1 + r)at.

Using (1), we get

j=0

1

1 + r

jct =

j=0

1

1 + r

jEt[yt+j ] + (1 + r)at.

This simplifies to

ct =r

1 + r

j=0

1

1 + r

jEt[yt+j]

+ rat. (2)20


21/94

By (1), ct = ct+1 Et[ct+1], which we can write as

ct = ct+1 Et[ct+1] =r

1 + r

j=0

1

1 + r

j Et+1[yt+1+j] Et[yt+1+j ]

+ rat+1 Et[rat+1].Since a

t+1is chosen at time t, the last part drops out and we simply have

ct =r

1 + r

j=0

1

1 + r

j Et+1[yt+1+j ] Et[yt+1+j]

. (3)Notice that

Et+1[yt+1+j ] Et[yt+1+j] = Et+1[zt+1+j + t+1+j ] Et[zt+1+j + t+1+j ]

= zt+1 zt

= zt + t+1 zt

= t+1

And for j = 0, we have

Et+1[yt+1] Et[yt+1] = yt+1 Et[yt+1]

= zt+1 + t+1 zt

= zt + t+1 + t+1 zt

= t+1 + t+1

Thus

ct =r

1 + r

j=0

1

1 + r

jt+1 + t+1

.This simplifies to

ct = t+1 +

r

1 + r t+1. (4)

This is an intuitive result; it says that the agent will consume the entirety of the permanent shock t+1 butwill only consume the annuity value of the temporary shock t+1.

The BC implies thatat+1 = yt + at(1 + r) ct.

Thusat = at+1 at = yt + at(1 + r) = ct at = yt + rat ct.

Using (2), we have

at = yt r

1 + r

j=0

1

1 + r

jEt[yt+j ]

. (5)

Since yt = zt + t and Et[yt+j ] = zt for all j > 0, we have

at = zt + t r

1 + r(zt + t)

r

1 + r

j=1

1

1 + r

jzt

.which simplifies to

at =1

1 + rt. (6)

This is reasonable in light of (4).

21


22/94

Part (b)

The specification of the income process implies that

yt = zt+1 + t+1 zt t = zt + t+1 + t+1 zt t = t+1 + t+1 t.

Then

E[yt] = 0.Note that if x and y are independently distributed (like and are), E(xy) = E(x)E(y). Then sinceE(t+1) = E(t+1) = E(t) = 0, for the variance of yt we get

var(yt) = E[(yt E[yt])2]

= E[(yt)2]

= E[(t+1 + t+1 t)2]

= E[2t+1 + 2t+1 +

2t ]

= 2 + 22

Note the formula for covariance:

cov(x, y) = E[(x E(x))(y E(y))]= E[xy + E(x)E(y) E(x)y xE(y)]

= E(xy) + E(x)E(y) 2E(x)E(y)

= E(xy) E(x)E(y)

Then for the covariance of yt and ct we get

cov(yt, ct) = E[ytct] E(yt)E(ct)

= E

(t+1 + t+1 t)

t+1 +

r

1 + rt+1

E[t+1 + t+1 t]E

t+1 +

r

1 + rt+1

= E(2t+1) +

r

1 + rE(2t+1)

= 2 + r1 + r2

Thus we have

0.2 =cov(yt, ct)

var(yt)=

2 +r

1+r 2

2 + 22

.

Rearranging stuff a few times, we have

0.2[2 + 22 ] =

2 +

r

1 + r2

then0.82 = [0.4

r

1 + r]2

so

=0.4 r1+r

0.8.

7 Fall 2008, II.1 (Larry, TDCE with government spending in util-ity function)

Part (a)

Given a fixed sequence {gt, kt, nt}t=0, a TDCE in this economy is:

22


23/94

household allocations xt = {ct, xt, kt+1, nt, lt}t=0;

firm allocations yt = {cft , x

ft , k

ft+1, n

ft , g

ft }

t=0; and

prices {pt, rt, wt}t=0

such that given prices,

1. xt solves the households problem:

max{ct,xt,kt+1,nt,lt}t=0

t=0

tu(ct, lt, gt)

s.t.

t=0

pt(ct + xt)

t=0

[wt(1 nt)nt + rt(1 kt)kt] +

kt+1 (1 )kt + xt

nt + lt n

k0 given, all quantities non-negative

2. yt solves the firms problem:

max{cft ,x

ft ,k

ft+1,n

ft ,g

ft }

t=0

=

t=0

[pt(cft + xft + gft ) wtnft rtkft ]

s.t. cft + xft + g

ft F(k

ft , n

ft )

all quantities non-negative

3. The market clears: cft = ct, xft = xt, k

ft = kt, n

ft = nt, and g

ft = gt for all t.

4. The governments budget balances:

t=0

ptgt =

t=0

[rtktkt + wtntnt]

Part (b)

Without loss of generality, let n = 1. I assume that the utility function is strictly increasing and F has CRS.Then the first three of the households contraints hold with equality and = 0. I can then eliminate theconstraint on the law of motion for capital by replacing xt with kt+1 (1 )kt in the budget constraint:

t=0

pt(ct + kt+1) =

t=0

[wt(1 nt)nt + [rt(1 kt)kt +pt(1 )]kt]

I can also replace lt with 1 nt throughout. For simplicity, I will use the shorthand uc(t) to refer touc(ct, 1 nt, gt), etc. Let be the multiplier on the budget constraint. Then the FOCs for the householdare:

ct : tuc(t) = pt, t

kt+1 : pt = rt+1(1 kt+1) +pt+1(1 ), t

nt : tul(t) = wt(1 nt), t

Note that there is no FOC for gt. This is because the consumer does not choose gt; he takes it as given.Since the market clears, we can replace cft with ct (and the same for the other quantities) in the firms

FOCs. Since the firm maximizes profits, it will not waste resources, i.e., ct + xt + gt = F(kt, nt). Thereforewe can eliminate the resource constraint in the firms problem by replacing ct + xt + gt with F(kt, nt) in thefirms objective function. The firms FOCs are:

kt : ptFk(t) = rt, t

nt : ptFn(t) = wt, t

23


24/94

WLOG, assume that p0 = 1. Using the households FOC for ct, we get

pt = t uc(t)

uc(0).

Using the firms FOCs to substitute into the households FOCs, we obtain the following equations thatcharacterize the equilibrium allocations:

uc(t)

uc(t + 1)= [(1 kt+1)Fk(t + 1) + 1 ], t

ul(t)

uc(t)= (1 nt)Fn(t), t

Since F has CRS, the firms profits are zero, so

t=0

pt(ct + xt + gt) =

t=0

[rtkt + wtnt].

Subtracting the households budget constraint from this equation, we see that

t=0pt(ct + xt + gt)

t=0pt(ct xt) =

t=0[rtkt + wtnt]

t=0[rt(1 kt)kt + wt(1 nt)nt].This implies that

t=0

ptgt =

t=0

[rtktkt + wtntnt]

so the governments budget constraint is satisifes. This shows that the governments budget constraint isredundant.

With the above results, we have shown that given taxes and government spending, a TDCE in thiseconomy is a sequence of prices {pt, rt, wt}t=0 and a sequence of allocations allocations {ct, kt+1, nt}

t=0

characterized by the following conditions:

pt = t uc(t)

uc(0), t (1)

rt = ptFk(t), t (2)

wt = ptFn(t), t (3)

uc(t)

uc(t + 1)= [(1 kt+1)Fk(t + 1) + 1 ], t (4)

ul(t)

uc(t)= (1 nt)Fn(t), t (5)

ct + kt+1 + gt = F(kt, nt) + (1 )kt, t (6)

t=0

pt(ct + kt+1) =

t=0

[wt(1 nt)nt + [rt(1 kt)kt +pt(1 )]kt] (7)

Part (c)Rearrange (7) as follows:

t=0

[ptct wt(1 nt)nt] =

t=0

[rt(1 kt)kt +pt(1 )kt ptkt+1].

Substituting (3), we get

t=0

[ptct ptFn(t)(1 nt)nt] =

t=0

[rt+1(1 kt)kt +pt(1 )kt ptkt+1].

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Pulling out some terms for period 0, we get

t=0

[ptct ptFn(t)(1 nt)nt] = Fk(0)(1 k0)k0 + (1 )k0

+

t=0[rt+1(1 kt+1)kt+1 +pt+1(1 )kt+1 ptkt+1] lim

TpTkT+1.

From the households FOCs, we know that the second term on the RHS iz zero. I assume the transversalitycondition holds. Then the third term is zero as well. Thus we have

t=0

[ptct ptFn(t)(1 nt)nt] = Fk(0)(1 k0)k0 + (1 )k0.

Substituting (1), we have

t=0t

uc(0)

[uc(t)ct uc(t)Fn(t)(1 nt)nt] = Fk(0)(1 k0)k0 + (1 )k0.

Substituting (5) and moving uc(0) to the RHS, we get

t=0

t[uc(t)ct ul(t)nt] = uc(0)k0[Fk(0)(1 k0) + (1 )]. (8)

This is the implementability constraint. Together with (6), it completely characterizes the set of TDCEallocations. Given a sequence of allocations that satisfies (6) and (8), we can find the prices and taxes withwhich the sequence is a TDCE.

Define the set

A1 =

{ct, nt}

t=0 : {(gt, kt, nt), (pt, rt, wt), (ct, kt, nt)}

t=0 is a TDCE

.

Thus the Ramsey Problem is:

max{ct,nt}t=0

t=0

u(ct, nt, g0) s.t. {ct, nt}t=0 A

1.

A1 can also be described as

A1 =

{ct, nt}t=0 : {ct, nt}

t=0 satisfies (6) and (8)

.

So we can also write the Ramsey problem as

max{ct,kt,nt}

t=0,

t=0 tu(ct, nt, gt)

s.t. ct + kt+1 + gt = F(kt, nt) + (1 )kt

t=0

t[uc(t)ct ul(t)nt] = uc(0)k0[Fk(0)(1 k0) + (1 )]


Let be the multiplier on the implementability constraint. Define W0(c0, n0, g0; ) as

W(c0, n0, g0; ) = u(ct, nt, g0) + [uc(0)k0(Fk(0)(1 k0) + (1 )) (uc(t)ct ul(t)nt)], t = 0

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and V(ct, nt, gt; ) as

V(ct, nt, gt; ) = u(ct, nt, gt) [uc(t)ct ul(t)nt], t > 0.

Thus we can rewrite the Ramsey Problem as

max{ct,kt,nt}t=0,W0(c0, n0, g0; ) +

t=1

tV(ct, nt, gt; )

s.t. ct + kt+1 + gt = F(kt, nt) + (1 )kt


Let t be the multiplier in the resource constraint.As before, I will use the shorthand Vc(t) to refer toVc(ct, nt, gt; ). The FOCs for this problem for all t > 0 are:

ct : tVc(t) = t

nt : tVl(t) = tFn(t)

kt+1 : t = t+1[Fk(t + 1) + (1 )]

Using the first and last FOCs, we obtain the following Euler equation that characterizes the Ramsey alloca-tion:Vc(t)

Vc(t + 1)= [Fk(t + 1) + (1 )], t > 0.

Assume that the system converges to a steady state. Then ct c, kt k, etc. The Euler equationthen implies that

Vc()

Vc()= [Fk() + (1 )]

or1 = [Fk() + (1 )]. (9)

Recall the Euler equation used to characterize all TDCE allocations:

uc(t)

uc(t + 1) = [(1 kt+1)Fk(t + 1) + 1 ], t.

Since the Ramsey allocation is a TDCE, it must also satisfy this equation. Steady state convergence impliesthat

uc()

uc()= [(1 )Fk() + 1 ]

or1 = [(1 k)Fk() + 1 ]. (10)

So both (9) and (10) must hold at the steady state level of capital. This is only possible if k = 0. Thuskt 0, so the Chamley-Judd result holds.

8 Fall 2008. II.2 (Victor, externality)Part (a)

TO BE COMPLETED.

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Part (b)

Let ht(z0, . . . , zt) denote the history of shocks up to period t. Let Ht be the set of all possible histories. Thecommodity space is

L =

(1, 2, 3) : i = {lit(ht)}tN,htHt , lit(ht) R i,t ,ht, sup

t,ht

|lit(ht)| < i = 1, 2, 3.

The consumption sets are

Xi =

xi L : {cit(ht), k

it(ht1)}tN,htHt such that :

cit(ht), kit(ht1) 0, t, ht

cit(ht) + kit+1(ht) x

i1t(ht) + (1 )k

it(ht1)

kit(ht1) xi2t(ht) 0

1 xi3t(ht) 0

ki0 = ki0

, i = s, n

The production set is

Y =

y L : 0 y1t(ht) zt(ht)F(y2t(ht), y3t(ht)), t, ht

.

An Arrow-Debreu equilibrium in this economy is (xs, xn, y) Xs Xn Y, Ns , and acontinuous linear functional : L R such that

(i) For i = s, n, xi argmaxxiXi, (xi)0 Ui(xi)

(ii) y argmaxyY (y)

(iii) sxs + (1 s)xn = y

(iv) Ns = xs3

where

Us = E

t=0

t

(cst (ht))

1

1 +

1 xs3t(ht)

1 Nst (ht)

1/2and

Us = E

t=0

t

(cnt (ht))1

1

.

Part (c)

The first welfare theorem will hold when s = 0, in which case there is no externality and all agents have

locally nonsatiated preferences. When s > 0, the sociable agents choices of xs3 (their labor) has anexternality effect on the other sociable agents, but these agents do not take this effect into account whenchoosing xs3 . Thus the first welfare theorem will not hold when

s > 0.

Part (d)

We need to know how much capital each type holds in the aggregate to know prices in the future. Thereforethe aggregate state is (z, Ks, Kn). I will let the agents trade contingent claims on capital goods. This allows

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28/94

us to use a cash-in-hand individual state variable rather than one each for capital and bonds (in the case ofcontingent claims on consumption goods). The problem of an agent of type i is

vi(z, Ks, Kn, a; G, H) = maxc,n,y,b(z)

ui(c,n,Ns) +

zZ

zzvi(z, Ks, Kn, y + b(z); G, H)

s.t. c + y + zZ

q(z, Ks

, Kn

, z

)b(z

) = [1 + r(z, Ks

, Kn

, N)]a + w(z, Ks

, Kn

, N)n

Ki(z) = Gi(z, Ks, Kn, z)

Ns = Hs(z, Ks, Kn)

N = HA(z, Ks, Kn)

c 0, y 0, b(z) b(z), n [0, 1]

where ui is the period utility utility given in the problem.A rational expectations RCE is: (vi, giy , g

ib , h

i)i=s,n, r, w, q, G, H such that

(i) For i = s, n, (vi, giy , gib , h

i) solves type is problem shown above given r, w, q, G, H.

(ii) Prices are set by a competitive, profit-maximizing firm:

r(z, Ks, Kn, N) = zFk(sKs + (1 s)Kn, N)

w(z, Ks, Kn, N) = zFn(sKs + (1 s)Kn, N)

(iii) Representative agent conditions hold:

Gi(z, Ks, Kn, z) = giy (z, Ks, Kn, Ki) + gib (z, K

s, Kn, Ki, z), i = s, n

Hs(z, Ks, Kn) = hs(z, Ks, Kn, Ks)

HA(z, Ks, Kn) = shs(z, Ks, Kn, Ks) + (1 s)

(iv) The market for contigent claims clears:

s

g

s

b (z, K

s

, K

n

, K

s

, z

) + (1

s

)g

n

b (z, K

s

, K

n

, K

n

, z

) = 0, z

Z.

(v) No arbitrage condition holds: zZ

q(z, Ks, Kn, z) = 1, (z, Kn, Ks).

Part (e)

Without contingent claims, the agents no longer choose b(z). Therefore we no longer need the claim pricesq nor conditions (iv) and (v). This means that the aggregate capital stocks tomorrow are known today withcertainty. The agents problem simplifies to

vi

(z, Ks

, Kn

, a; G, H) = maxc,n,y ui(c,n,Ns) + zZ

zzvi

(z

, Ks

, Kn

, y; G, H)s.t. c + y = [1 + r(z, Ks, Kn, N)]a + w(z, Ks, Kn, N)n

Ki = Gi(z, Ks, Kn)

Ns = Hs(z, Ks, Kn)

N = HA(z, Ks, Kn)

c 0, y 0, n [0, 1]

A rational expectations RCE is now: (vi, giy , hi)i=s,n, r, w, G, H such that

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(i) For i = s, n, (vi, giy , hi) solves type is problem shown above given r, w, G, H.

(ii) Prices are set by a competitive, profit-maximizing firm:

r(z, Ks, Kn, N) = zFk(sKs + (1 s)Kn, N)

w(z, Ks, Kn, N) = zFn(sKs + (1 s)Kn, N)


Gi(z, Ks, Kn) = giy (z, Ks, Kn, Ki), i = s, n

Hs(z, Ks, Kn) = hs(z, Ks, Kn, Ks)

HA(z, Ks, Kn) = shs(z, Ks, Kn, Ks) + (1 s)

Part (f)

In this scenario, the firm now has a dynamic problem:

(z , K , k; G, H) = maxk,n zF(k, n) k

w(z , K , N )n + q(z, K) zZzz(z, K, k; G, H)

s.t. K = G(z, K)

N = HA(z, K)

Let gf(z , K , k) denote the policy function for capital for this problem. Let a denote the value of an agentsshares in firms. His problem is now

vi(z , K , a; G, H) = maxc,n,a,b(z)

ui(c,n,Ns) +

zZ

zzvi(z, K, a + b(z); G, H)

s.t. c + q(z, K)a +

zZq(z , K , z)b(z) = a + w(z , K , N )n

K = G(z, K)

Ns = Hs(z, K)

N = HA(z, K)

c 0, y 0, n [0, 1]

Let gia and gib denote the policy functions shares of firms and contingent claims respectively.

NOT SURE WHETHER WE NEED TO KEEP TRACK OF BOTH TYPES SHARES. WHAT IS THEAGGREGATE STATE?

Part (g)

TO BE COMPLETED.

Part (h)

TO BE COMPLETED.

Part (i)

TO BE COMPLETED.

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9 Fall 2008, II.3 (Chari, cash-credit)

Part (a)

A competitive equilibrium in this economy is:

Allocations {c1t, c2t, Mt, Bt}t=0;

Prices {pt}t=0; and

Policy {Mt, Bt, Rt, Tt}t=0

such that:1. Given prices and policy, the allocation solves the households problem:

max{c1t,c2t,Mt,Bt}t=0

t=0

tu(c1t, c2t)

s.t. ptc1t Mt

Mt + Bt = (Mt1 pt1c1t1) pt1c2t1 +pt1y + Rt1Bt1 + Tt1

initial nominal holdings given, all quantities non-negative, no ponzi schemes

2. The governments budget constraint is satisfied:

Mt+1 Mt + Bt+1 = Tt + RtBt, t.

3. Markets clear for all t: c1t + c2t = y, Mt = Mt, Bt = Bt.

Part (b)

Since u is strictly increasing and strictly concave, FOCs are sufficient for maximization. Let t be themultiplier on the cash-in-advance constraint and let t be the multiplier on the securities market constraint.I will use the shorthand u1(t) to refer to u1(c1t, c2t) (and the same for the second variable). The householdsFOCs are:

c1t : tu1(t) = tpt + t+1pt (1)

c2t : tu2t = t+1pt (2)

Mt : t = t + t+1 (3)

Bt : t = Rt+1 (4)

Note that since R > 1, the CIA constraint will bind in every period. A sufficient condition for the constraintto bind is t > 0. Looking at (4), R > 1 implies that

t+1t

> 1 for all t. Rearranging gives us 1 t+1t > 0.Multiplying both sides by t, we have t t+1 > 0. Looking at (3), this means that t > 0 for all t, so theCIA constraint binds in every period.

Combining (1) and (2), we see that

u1

(t)

u2(t) =

t+

t+1t+1 , t.

Using (3), we getu1(t)

u2(t)=

tt+1

, t

and using (4), we getu1(t)

u2(t)=

Rt+1t+1

, t.

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Thus we get the following relationship between c1t and c2t:

u1(t)

u2(t)= R, t. (5)

The resource constraint isc1t + c2t = y, t. (6)

Looking at the resource constraint, we can write the equilibrium value of c2t in terms of c1t:

c2t = y c1t.

Using this substituting, we can rewrite (b.5) as a function of c1t only:

u1(c1t, y c1t)

u2(c1t, y c1t)= R, t. (7)

This single equation fully characterizes the set of equilibrium real allocations. Call this set AR. Since Rand y are constant, the values of c1t that solve these equations are the same for every period. Then we canexpress AR as

AR = c1 R+ : u1(c1, y c1)u2(c1, y c1) = R .Claim: AR is a singleton.Proof: Suppose not. Then there exists c1, c1 AR such that c1 = c

1. Since (7) is a FOC and both c1

and c1 both satisfy it, it must be that both c1 and c1 maximize u(x, y x). In other words, u(c1, y c1) =

u(c1, y c1). But since u is strictly concave, any linear combination of (c1, y c1) and (c

1, y c

1) will give

strictly higher utility than (c1, y c1). This contradicts the fact that c1 AR, so it must be that AR is asingleton.

Now we want to characterize the equilibrium values of the nominal variables. We know that c1t =c1t+1 = c1 and c2t = c2t+1 = y c1 for all t. Equality in the CIA constraint implies that

Mt+1Mt

=pt+1pt

c1c1

=pt+1pt

. (8)

Divide (2) for period t + 1 by (2) for period t:

u2(t + 1)

u2(t)=

t+2t+1

pt+1pt

, t.

Using (4), we getu2(t + 1)

u2(t)=

t+2Rt+2

pt+1pt

, t.

Thenpt+1

pt= R

u2(t + 1)

u2(t), t. (9)

Since equilibrium values of the consumption goods are constant, I will use u1 and u2 to refer to u1(c1, y c1)

and u2(c1, y c1) respectively. Then u2(t) = u2(t + 1) = u2 for all t, so (9) becomespt+1

pt= R

u2u2

= R, t.

With (8) and the market clearing condition, this implies that

Mt+1

Mt=

Mt+1Mt

=pt+1pt

= R, t. (10)

So the growth rates of both money and prices are constant in equilibrium.

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Given the results above, we can write the securities market constraint as

ptc1 + Bt = pt1y pt1(y c1) + Rt1Bt1 + Tt1.

This simplifies toptc1 + Bt = pt1c1 + Rt1Bt1 + Tt1. (11)

We can use (11) along with the government budget constraint to calculate the other nominal variables interms of p0.

Suppose {Mt, Bt, pt, Tt}t=0 form an equilibrium with the unique equilibrium real allocation c1. Then{Mt, Bt, pt, Tt}t=0 satisfy (10), (11) and the GBC. Looking at these equations, we can see that for any > 0,{Mt, Bt, pt, Tt}t=0 also satisfy (10), (11) and the GBC. Therefore > 0, {Mt, Bt, pt, Tt}

t=0 also

form an equilibrium with c1, so the equilibrium is not unique.

Part (c1)

The same FOCs as in part (b) are valid here. They are rewritten below using the specified utility function:

c1t :t

c1t= tpt + t+1pt (12)

c2t :

t

c2t = t+1pt (13)

Mt : t = t + t+1 (14)

Bt : t = Rtt+1 (15)

Note that in (15), the interest rate is no longer constant. By all 4 of the FOCs in part (c1), we know that

u1(t)

u2(t)=

c2tc1t

= Rt+1. (16)

Divide (13) for period t + 1 by period t:

u2(t + 1)

u2(t)=

c2tc2t+1

=t+2t+1

pt+1pt

, t.

Using (15), we get u2(t + 1)

u2(t)=

c2tc2t+1

=t+2

Rt+1t+2

pt+1pt

, t.

Rearranging, we get

Rt+1 =1

u2(t)

u2(t + 1)

pt+1pt

=1

c2t+1c2t

pt+1pt

.

Assume the CIA constraint holds with equality. Using (16) in (17), we get

Rt+1 =1

c1t+1c1t

M c1tMc1t+1

which simplifies toRt+1 =

1. (17)

The resource constraint isc1t + c2t = y. (18)

So (16) - (19) characterize the equilibrium real allocations and interest rate given prices.Since money supply is constant and the markets clear, Mt = M for all t. Then the securities market

constraint becomesBt = pt1c1t1 pt1c2t1 +pt1y + Rt1Bt1 + Tt1 (19)

and the GBC isBt+1 = Tt + RtBt. (20)

Equations (16) - (20) characterize the equilibrium.

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Part (c2)

By (12),t1u2(t 1)

pt1= t.

By (13),

t

u1(t)pt

= t+1 + t.

Using (14), we getu1(t)

pt=

u2(t 1)

pt1.

Multiply both sides by M:u1(t)M

pt=

u2(t 1)M

pt1.

I assume the cash-in-advance constraint is binding for all t. Then ptc1t = M, so the previous equationbecomes

u1(t)c1t = u2(t 1)c1t1.

As in the previous section, c2t = y c1t. Thus we have

u1(c1t, y c1t)c1t = u2(c1t1, y c1t1)c1t1.

The problem states that F(c1) = c1u2(c1, y c1) and G(c1) = c1u1(c1, y c1). So we can rewrite the lastequation as

F(c1t) = G(c1t1). (21)

Suppose the economy converges to a steady-state level c1. Then c1 is characterized by

F(c1) = G(c1).

Take a first-order Taylor expansion of (21) around c1:

F(c1)(c1t c1) = G(c1)(c1t1 c1).

This can be rearranged as

c1t c1 =

G(c1)

F(c1)(c1t1 c

1). (22)

The problem states thatF(c1)

G(c1)<

soc1t c

1 < c1t1 c

1. (23)

Therefore c1t c1. This implies that the economy has a continuum of equilibria, each converging to c1,

indexed by c10. For each equilibrium, the sequence {c1t}t=0 is given by (22).

10 Spring 2008, I.1 (Larry, DP)

Since log ct is strictly increasing, the resource contraint will hold with equality. Since nt doesnt enter theobjective function, we know that nt = 1for all t. Then we can rewrite the set of 5 contraints as

ct = Akt h

1t kt+1 ht+1 + kt+1(1 k) + ht+1(1 h) 0 (1)

(h0, k0) = (h, k) given (2)

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Define the feasibility correspondence : R2+ R2+ by

(ht, kt) = {(ht+1, kt+1) R2+ : s.t. (1) holds}.

Define (h, k) as the set of feasible sequences given (h, k), i.e., (h

, k

) (h, k) if (h

0, k

0) = (h, k) and

(h

t+1, k

t+1) (h

t, k

t) for all t > 0. Then we can rewrite V(h, k) as

V(h, k) = sup(h,k)(h,k)

t=0

t log(ct)

where ct is given by (1) for all t.

Proposition. Suppose that (h, k) attains V(h, k). Then for > 0, (h, k) attains V(h, k).Proof. First note that

A(kt)(ht)

1 = Akt h1t .

Then the resource constraint is homogeneous of degree one in (h, k), so is also homogeneous of degree onein (h, k). This implies that (h, k) (h, k).Suppose for contradiction that (h

, k

) does not attain V(h,k). Then (h

, k

) (h, k) such

that

t=0t log(ct) >

t=0

t log(ct).

Since (h, k) (h, k), homogeneity of implies that

h , k (h, k). Note that log(ct) = log() +log(ct). This implies that log(ct) is homothetic, i.e., log(x) = log(y) log(x) = log(y). Then

t=0

t log

ct

>

t=0

t log(ct ).

But this is a contradiction since (h, k) attains V(h, k). Therefore it must be that (h, k) attainsV(h, k).The above proposition implies that we can express V(h, k) as


t=0

t log(ct)

where ct is given by (1). Since log(ct) = log() + log(ct), this can be rewritten as


log()

1 +

t=0

t log(ct)

=log()

1 + sup

(h

,k

)(h,k)

t=0

t log(ct)

=log()

1 + V(h, k)

= B() + V(h, k)

11 Spring 2008, I.2 (Chari, optimal asset allocation)

Part (a)

Since the agents lifetime ends after period T, we know that he will consume all of his wealth in that period,i.e.,

VT(WT) = u(WT).

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For all t < T, the dynamic program representing the agents decision can be written as

Vt(Wt) = maxct+xt+ytWt

u(ct) + E[Vt+1(Rxt + Rfyt)]

Part (b)

We can express xt and yt as tWt and (1 t)Wt respectively, where t [0, 1]. Then we can rewrite theDP as

Vt(Wt) = maxctWt, t[0,1]

u(ct) + E[Vt+1((Wt ct)(tR + (1 t)Rf))]

Proposition. Vt(Wt) = atu(Wt) for allt T, and the optimal choice t does not depend ont, i.e.,

t =

for all t.

Proof. We know that VT(WT) = u(WT), so aT = 1. Now suppose that Vt+1 = at+1u(Wt+1) for some uniqueat+1 that is independent of Wt+1. Then


u(ct) + E[at+1u((Wt ct)(tR + (1 t)Rf))]

Note that given the specification of the utility function, u(ab) = (1 )u(a)u(b). Then


u(ct) + u(Wt ct)(1 )at+1E[u(tR + (1 t)Rf)]

Since E[u(tR + (1 t)Rf)] does not depend on Wt, we can rewrite this as

Vt(Wt) = maxctWt

u(ct) + u(Wt ct)(1 )at+1 max

t[0,1]

E[u(tR + (1 t)Rf)]

Let

Qt = (1 )at+1 maxt[0,1]

E[u(tR + (1 t)Rf)]

ThenVt(Wt) = max

ctWt

u(ct) + u(Wt ct)Qt

The utility function is strictly increasing and strictly concave and the constraint set is compact and convex,so first-order conditions are sufficient for a maximum. The FOC for ct is

u(ct) = u(Wt ct)Qt

Plugging in the specification of the utility function, this is

ct = (Wt ct)Qt

or

ct = (Wt ct)Q1/

t

Solving for ct, we get

ct =Q

1/t Wt

1 + Q1/t

=Wt

1 + Q1/t

Plugging this solution into the last expression of the dynamic program, we have

Vt(Wt) = u

Wt

1 + Q1/t

+ u

Wt

Wt

1 + Q1/t

Qt

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Using homotheticity of the utility function again, we can rewrite this as

Vt(Wt) = u(Wt)

u

1

1 + Q1/t

+ u

1

1

1 + Q1/t

Qt

Let

at = u 11 + Q1/t + u1 1

1 + Q1/t QtThen we have

Vt(Wt) = atu(Wt)

Recall thatQt = (1 )at+1 max

t[0,1]

E[u(tR + (1 t)Rf)]

Since R is i.i.d., the value of max t[0,1]

E[u(tR + (1 t)Rf)]

is the same for all t, and the solution

will be the same for all t as well. Then we can write

Qt = (1 )at+1 max[0,1]

E[u(R + (1 )Rf)]

Since u is strictly concave, there is a unique such that

argmax[0,1] E[u(R + (1 )Rf)]Then

Qt = (1 )at+1E[u(R + (1 )Rf)]

Therefore the agent will always allocate the same portion of his disposable wealth (Wt ct) to the riskyasset. In other words, his portfolio allocation is constant for all t < T.

12 Spring 2008, I.3 (Victor, bargaining/monopolistic competi-tion)

Part (a)

I dont really see what he is asking. I will assume that there are only ten periods, so once those ten periodsis up, everyone dies. The worker either takes the offer and works for 10 periods earning w, or rejects theoffer and gets 0.1 per period plus 0.5 in each of the first 5 periods. I assume that a zero interest rate impliesa discount rate of 1. Then the value of taking the offer is

VE(w) = 10w

and the value of rejecting the offer is

VU = 0.1(10) + 0.5(5) = 3.5.

The minimum wage w that the worker would take solves

VE(w) = Vu 10w = 3.5 w = 0.35.

Part (b)The value of the firm is

(w) = 10 10w.

The firm gets nothing if it doesnt hire the worker, so the firms threat point is 0. The workers threat pointis VU. Thus the Nash bargaining solution with the worker having twice the weight of the firm is

w = argmaxw

(10 10w)1/3(VE(w) VU)2/3

= argmaxw

(10 10w)1/3(10w 3.5)2/3

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Part (c)

Let W denote the representative consumers wealth. His utility functions strictly increasing in all goods, sothe budget constraint will hold with equality. Then his problem is

max{c(i)}i[0,A]

A

0

c(i) di1/

s.t.

A0

p(i)c(i) di = W.

The FOCs for this problem are

c(i) : U1/1c(i)1 = p(i)

:

A0

p(i)c(i) di = W

where

U =

A

0

c(i) di.

This implies thatp(i)

p(j)=

c(i)1

c(j)1, i, j [0, A].

or

p(i)c(i) =p(j)c(i)

c(j)1, i, j [0, A].

Integrating over i, we get

M =

A0

p(i)c(i) di =p(j)

c(j)1

A0

c(i) = Up(j)

c(j)1, j [0, A].

Thus

p(j) =W

U c(j)1, j [0, A]. (1)

Each firm solves(i) = max

p(i),c(i)p(i)c(i) c(i).

Plugging in the result we got above about the consumers demand, this problem becomes

(i) = maxc(i)

W

Uc(i) c(i).

The FOC of this problem is

W

Uc(i)1 = .

Plugging in our formula for p(i) in (1) above, we get

p(i) =

.

Rearranging the FOC a bit, we haveU

Wc(i) = c(i).

If we integrate, we getU

Wc =

A0

c(i) di =

A0

c(i) = U.

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Then we have an expression for industry output:

c =W

.

If the firms are owned by the workers, the representative consumers problem is now

max{c(i)}i[0,A]

A0

c(i) di1/

s.t.

A0

p(i)c(i) di = W +

A0

(i) di.

The consumer takes profits as given, so the analysis above.. . TO BE COMPLETED.

13 Spring 2008, I.4 (Perri, IFP)

Part (a)

Both utility functions are strictly concave and the constraint sets are convex, so FOCs are sufficient for

maximization. The FOCs for consumer 1 are

c1t : t(b1 2b2c

1t ) =

1t

a1t+1 : 1t = Et[(1 + r)

1t+1]

This implies thatb1 2b2c

1t = (1 + r)Et[b1 2b2c

1t+1

or simplyc1t = (1 + r)Et[c

1t+1].

Since (1 + r) = 1, we havec1t = Et[c

1t+1].

This implies that

c1t = Et[c1t+j ], j 0. (1)

In other words, c1t is a martingale. Iterating on consumer 1s budget constraint starting with period t, takingexpectations, and using the transversaility condition

E0

lim

t

1

1 + r

ta1t

= 0,

we have

j=0

1

1 + r

jEt[c

1t+j ] =

j=0

1

1 + r

jEt[y

1t+j ] + (1 + r)a

1t .

Using (1), we have

j=0 11 + r

j

c1t =

j=0

11 + rj

Et[y1t+j ] + (1 + r)a1t .

This simplifies to

c1t =r

1 + r

j=0

1

1 + r

jEt[y

1t+j]

+ ra1t . (2)We know that for all j > 0, Et[y

1t+j ] = y. Thus

c1t = y +r

1 + r1t + ra

1t . (3)

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For consumer 2, the form of the utility function will implies that c2t will not be a martingale. Insteadwe will use guess and verify method on the dynamic program

V(a, y) = maxc,a

u(c) + E[V(a, y)]

s.t. c 0

c + a

= (1 + r)a + y

I assume V is strictly concave and differentiable at x. The FOCs are

c : u(c) =

a : E[V1(a, y)] =

This implies the first order conditionu(c) = E[V1(a

, y)]. (4)

The envelope condition for a isV1(a, y) = (1 + r)

or

V1(a, y) = (1 + r)E[V1(a

, y

).Since (1 + r) = 1, this becomes

V1(a, y) = E[V1(a, y)]. (5)

Combining (4) and (5) givesu(c) = (1 + r)V1(a, y). (6)

Guess that

V(a, y) = 1

rer(a+By+D)

ThenV1(a, y) = e

r (a+By+D).

Using this in (6) gives

ec

= er(a+By+D)

.Taking logs, we have

c = r(a + By + D) log(1 + r).

Thus our candidate policy function for consumption is

c = r

a + By + D +

1

rlog(1 + r)

. (7)

Looking back at (5), if we plug in the guess and do some substitutions, we get

er (a+By+D) = E

er (a+By+D)

= Eer ((1+r)a+yc+By+B+D)= er((1+r)a+yc+By+D)E

erB

Then we haveerc = er(ra+(1B)y+By)E

erB

.

We can solve this for c:

c = r

a +

1 B

ry +

B

ry

1

rlog

E

erB

. (8)

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Comparing with (7), we see that B = 11+r and

D =B

ry

1

rlog

E

erB

1

rlog(1 + r).

This verifies the guess. Thus our consumption policy function for type 2 is

c = r

a +1

1 + ry +

1r(1 + r)

y 1

rlog

E

er1+r

.

Note that

E[er1+r

] = e2r22

2(1+r)2 .

Then type 2s consumption is

c = r

a +

1

1 + ry +

1

r(1 + r)y

r2

2(1 + r)2

. (9)

Lets rewrite (9) a little bit:

c = r1 + r y + 11 + r y + ra r

2

2(1 + r)2 .

Since y = y + , we have

c =r

1 + r(y + ) +

1

1 + ry + ra

r2

2(1 + r)2.

This simplifies to

c = y +r

1 + r + ra

r2

2(1 + r)2.

Adding time subscripts and denoting consumer 2s consumption by c2t , we have

c2t = y +r

1 + r2t + ra

2t

r2

2(1 + r)2. (10)

Comparing (10) to (3), we can see that if we assume that a1t = a2t and

1t =

2t , consumer 2s consumption is

lower by exactly r2

2(1+r)2 .

Part (b)

Note that if consumer 1 starts with zero assets and receives a shock of zero in every period, (3) implies that

c1t = y, t. Let denoter2

2(1+r)2 . If consumer 2 also starts with zero assets and receives a shock of zero in

every period, (10) implies that his consumption and asset paths are as follows:

t c2t a2t

0 y 0

1 y + r 2 y + 2r 2...

......

t y + tr t

and so on. Thus c2t starts out below c1t , but once t is large enough so that tr > , c

2t will be larger than

c1t . This is because (1 + r) = 1 implies that both consumers value consumption in all periods equally (sothey have no incentive to borrow), but the uncertainty in type 2s income causes precuationary saving sothat type 2s assets go to infinity.

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14 Spring 2008, II.1 (Larry, DP)

Part (a)

Let C(X) be the set of continuous, bounded, real-valued functions defined on X. We know that C(X) isa Banach space. Since C(X) is a closed subset of C(X), C(X) can be viewed as a Banach space itself.In order to show that T

has a fixed point, we need to show that T

and T

satisfy Blackwells sufficiency

conditions. Let f, g C(X) such that f(k) g(k) for all k X. Then for any k X,

maxk(k)

g(k) maxk(k)

f(k).

Thus Tg(k) Tf(k) and T g(k) Tf(k). Therefore T and T both satisfy monotonicity. I assume that

, (0, 1). Then T satisfies discounting since

T(f + a)(k) = maxk(k)

{u(F(k) k) + [f(k) + a]}

= maxk(k)

{u(F(k) k) + f(k) + a}

= maxk(k)

{u(F(k) k) + f(k)} + a

= T(f)(k) + a

Similarly, T (f+ a) = T(f)+ a, so T satisfies discounting as well. Thus both operators satisfy Blackwells

sufficiency conditions, so both are contractions. Since C(X) is a Banach space, the Contraction MappingTheorem implies that T and T each have a unique fixed point in C

(X).

Part (b)

Since v(; ) is the fixed point for T,

v(k; ) = maxk(k)

{u(F(k) k) + v(k; )}.

g(; ) is the policy function associated with v(; ), so

g(k; ) argmaxk(k)

{u(F(k) k) + v(k; )}.

Thus g(k; ) must satisfy the FOC:

u(F(k) g(k; )) = v (g(k; ); ).. (1)

Part (c)

Given the initial guess v(k, ), we have

v1(k; ) = T(v(k; )) = max

k

(k)

{u(F(k) k) + v(k; )}.

g1(; ) is the policy function associated with v1(; ), so

g1(k; ) argmaxk(k)

{u(F(k) k) + v(k; )}.

Thus g1(k; ) must satisfy the FOC:

u(F(k) g1(k; )) = v(g1(k; ); ).. (2)

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Part (d)

I assume that u is strictly concave. Suppose for contradiction that k such that g1(k; ) g(k; ). Thenstrict concavity of u implies

u(F(k) g(k; )) u(F(k) g1(k; )).

By (1) and (2), we have

v (g(k; ); ) v (g1(k; ); ).

Since v(; ) is weakly concave,v(g(k; ); ) v(g1(k; ); ).

Since < ,v (g(k; ); ) < v (g1(k; ); ).

This is a contradiction, so it just be that g1(k; ) > g(k; ) for all k X.

Part (e)

Since v(; ) and v1(; ) are differentiable and weakly concave, we obtain the following envelope conditions(see Th 4.10 in SLP; weak concavity is sufficient for the envelope condition to hold):

v(k; ) = u(F(k) g(k; ))F(k) (3)

v1(k; ) = u(F(k) g1(k; ))F

(k) (4)

By part (d), g1(k; ) > g(k; ). Then by strict concavity of u,

u(F(k) g(k; )) < u(F(k) g1(k; ))

so (3) and (4) imply that v(k; ) < v 1(k; ) for all k X.

Part (f)

Proposition. gn(k; ) > g(k; ) and vn(k; ) > v(k; ), k X, n N.

Proof. By induction. By part (d) we have g1(k; ) > g(k; ), k X. By part (e), v(k; ) < v 1(k; ), k X. Thus the conditions hold for n = 1. Now suppose that gn1(k; ) > g(k; ) and vn1(k; ) > v

(k; )

for all k X. We have the following FOC and envelope condition for vn(; ) and gn(; ):

u(F(k) gn(k; )) = vn1(gn(k; ); ) (5)

vn(k; ) = u(F(k) gn(k; ))F

(k) (6)

Suppose for contradiction that k X such that gn(k; ) g(k; ). By strict concavity ofu,

u(F(k) gn(k; )) u(F(k) g(k; )).

Then by (1) and (5),

v n1(gn(k; ); ) v (g(k; ); ).

Since > , this implies thatvn1(gn(k; ); ) < v

(g(k; ); ).

Since gn(k; ) g(k; ) and v is weakly concave,

v(g(k; ); ) v(gn(k; ); ).

But then we havevn1(gn(k; ); ) < v

(gn(k; ); )

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which contradicts the assumption of the induction. Therefore it must be that gn(k; ) > g(k; ) for all

k X. Then by strict concavity of u,

u(F(k) g(k; )) < u(F(k) gn(k; )), k X.

By (3) and (6), we have

v

(k; ) < v

n(k;), k X.

Since the conditions hold for n = 1 and arbitrary n given that they hold for n 1, induction impliesthat gn(k; ) > g

(k; ) and vn(k; ) > v(k; ), k X, n N.

15 Spring 2008, II.2 (Victor, recursive competitive equilibrium)

Part (a)

The utility function is locally nonsatiated so the first welfare theorem will hold. Therefore we can just usethe FOCs from the planners problem to get the following three equations that characterize the steady stateequilibrium:

AFK(K, NA) = 1

0.5(1 NA NB)0.5 = AFN(K, NA)

0.5(1 NA NB)0.5 = (C)

We can use the second two equations to get the following condition:

AFN(K, NA) = 1.

So if we want to have NA = NB = 0.5, we just have to solve the system

AFK(K, 0.5) = 1

AFN(K, 0.5) = 1

Since we have two equations and two unknowns we will be able to get the answer.

Part (b)

The commodity space is

L =

(1, 2, 3) : i = {lit(ht)}tN,htHt , lit(ht) R i,t,ht, sup

t,ht

|lit(ht)| < i = 1, 2, 3

.

The consumption sets are

X =

x L : {(cit(ht), k

it(ht1), n

it(ht))i=A,B}tN,htHt such that :

ci

t

(ht), ki

t

(ht1), ni

t

(ht) 0, t, ht, i

cAt (ht) + kAt+1(ht) x1t(ht)

cBt (ht) + kBt+1(ht) = n

Bt (ht)

kAt (ht1) kBt (ht1) x2t(ht) 0

nAt (ht) = xi3t(ht)

0 nAt (ht) + nBt (ht) 1

ki0 = ki0

, i = s, n

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The production set is

Y =

y L : 0 y1t(ht) zt(ht)AF(y2t(ht), y3t(ht)), t, ht

.

An Arrow-Debreu equilibrium without fireworks is (x, y) X Y and a continuous linear functional such that

(i) x argmaxxX,(x)0 U(x)

(ii) y argmaxyY (y)

(iii) x = y

where U(x) is the consumers lifetime utility.

Part (c)

We can write the consumers problem in recursive form as

V(z , K , a; G, HA) = maxc,nA,nB,au(c, nA + nB , 0) + zZzzV(z, Ka; G, HA)

s.t. c + a = r(z , K , N A)a + w(z , K , N A)nA + nB

nA + nB 1

K = G(z, K)

NA = HA(z, K)

non-negativity

where u(c,n,P) is the consumers period utility.A rational expectations RCE is: (V, g, hA, hB), r, w, G, HA such that:

(i) (V, g, hA, hB) solves the consumers problem shown above given r, w, G, HA

(ii) r and w are marginal products of a profit-maximizing firm:

r(z , K , N A) = zAFK(K, NA)

w(z , K , N A) = zAFN(K, NA)


G(z, K) = g(z , K , K )

HA(z, K) = h(z , K , K )

Part (d)

TO BE COMPLETED.

Part (e)

I assume home production is not counted in GDP. Then we want

P = 0.1AF(K, H(1, K))

where all starred variables are steady-state values. We are going to tax all income, so we have

P = [r(1, K, HA(1, K))K + w(1, K, HA(1, K))HA(1, K)].

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I assume that F is homogeneous of degree one in (K, NA). Then by Eulers theorem,

AFK(K, NA)K+ AFN(K, N

A)NA = AF(K, NA).

Since r and w are marginal products, this implies that

P = AF(K, H(1, K)).

Thus = 0.1.

Part (f)

Now we are just taxing labor income. Then

P = w(1, K, HA(1, K))HA(1, K) = AFN(K, H(1, K)).

This means that

= 0.1AF(K, H(1, K))

AFN(K, H(1, K)).

Part (g)If sector B becomes a market activity, then the equilibrium wage must be the same in both sectors. LetHB(z, K) denote hB(z , K , K ). Assuming that we still are not counting sector B in GDP, we have

P = 0.1AF(K, H(1, K))

andP = w(1, K, HA(1, K))[HA(1, K) + HB(1, K)].

Thus

= 0.1AF(K, H(1, K))

w(1, K, HA(1, K))[HA(1, K) + HB(1, K)].

If we count sector B in GDP, then this changes to

P = 0.1[AF(K, H(1, K)) + HB(1, K)]

P = w(1, K, HA(1, K))[HA(1, K) + HB(1, K)]

= 0.1AF(K, H(1, K)) + HB(1, K)

w(1, K, HA(1, K))[HA(1, K) + HB(1, K)]

Part (h)

Lump-sum taxes are nondistortionary so I would choose to fully finance the fireworks with a lump-sum taxof 10% of GDP, i.e.,

T = 0.1AF(K, HA(1, K))

orT = 0.1[AF(K, HA(1, K)) + HB(1, K)]

depending on whether we could sector be as part of GDP.

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Part (i)

Since sector B is now intermediated by firms, we can simplify the consumers labor choice so that he justchooses his total amount of labor. Again, the wages in both sectors have to be the same in equilibrium. Theconsumers problem in recursive form as

V(z,K,B,a,b; G, HA

, HB

, , P , t) = maxc,nA,nB,y,bu(c,n,P) + zZ

zzV(z

, K

a

, b

; G, HA

, HB

, , P , t)s.t. c + y + b = (1 )[r(z , K , B)(a + b) + w(z , K , B)n]

n 1

K = G(z , K , B)

P = P(z , K , B)

= (z , K , B)

non-negativity

where u(c,n,P) is the consumers period utility.A rational expectations RCE is: (V, ga, g

b , h

), r, w, G, HA, HB, , P, , such that:

(i) (V

, g

a, g

b , h

) solves the consumers problem shown above given r

, w

, G

, HA

, HB

,

,

P

,

(ii) r and w are marginal products of a profit-maximizing firm:

r(z , K , B) = zAFK(K, HA(z , K , B))

w(z , K , B) = zAFN(K, HA(z , K , B))


G(z , K , B) = ga(z,K,B,K,B)

HA(z , K , B) + HB(z , K , B) = h(z , K , B , K , B)

(z , K , B) = gb (z , K , B)

(iv) The governments budget constraint holds:

r(z , K , B)B+P(z , K , B) = (z , K , B)[r(z , K , B)(K+B)+w(z , K , B)(HA(z , K , B)+HB(z , K , B))]+(z , K

where

P(z , K , B) =

0.5

(1 )(z , K , B)[r(z , K , B)(K+ B) + w(z , K , B)(HA(z , K , B) + HB(z , K , B))] G(z , K , B) (z , K

16 Spring 2008, II.3 (Chari, search and human capital)

Part (a)

Given human capital h and an offer of z, the value of the choice faced by the wor

macro prelim solutions

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