asset price in static economies

8/14/2019 Asset Price In Static Economies

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Chapter B

Solutions to Exercises

1 Complete Contingent Claims

Question 1 Define the setA = {a RI : ai = ui(xi) ui(c

i), x } andA = {a RI+: a = 0}.

We first note that A

A is empty by the Pareto optimality of the allocation(c1, . . . , cI). Furthermore, A is a convex set by the concavity of the utilityfunctions. Thus, the Separating Hyperplane Theorem implies that there existsa nonzero vector y z for each y Aand each z A. Since 0 A (byPareto optimality), 0.

Question 2 In a perfect foresight competitive equilibrium for this economy(which is just a special case of the complete contingent claims equilibrium)both consumers solve the problem:

maxci1,c

i2

Ui(ci1) + Ui(ci2) s.t. c

i1+pc

i2 1+p2.

a) For individual a, construct the Lagrangian function as follows:

L=(ca1)

1 1

1 +

(ca2)1 1

1 + a[1+p2 c

a1 pc

a2].

The first-order conditions are as follows:

L

ca1= 0 (ca1)

=a

L

ca2= 0 (ca2)

=ap.

From the equations above, we obtain ca2 in terms ofca1 as:

ca2 = p 1ca1. (1.1)

1


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2 Solutions to Exercises

The remaining first-order condition,L/a = 0, gives us back the consumersbudget constraint:

1+p2= ca1+pc

a2.

If we substitute the valueca2 obtained from the equation (1.1) to the conditionabove and then use the resulting expression in equation (1.1), we find thevalues ofca1 and c

a2 as follows:

ca1 = 1+p21 +p(1)/

, (1.2)

ca2 =

p1/(1+p2)

1 +p(1)/ . (1.3)

For individual b, construct the Lagrangian function as follows:

L= log(cb1) + log(cb2) +

b[1+p2 cb1 pc

b2].

The first-order conditions are as follows:

L

cb1= 0

1

cb1=b

L

cb2= 0

1

cb2=bp.

From the equations above, we obtain cb1 in terms ofcb2 as:

cb1= pcb2. (1.4)

The remaining first-order condition L/b = 0 yields the budget constraintfor individualb as:

1+p2= cb1+pc

b2.

If we substitute the valuecb1 obtained from the equation (1.4) to the conditionabove and use the resulting expression in equation (1.4), we find the values of

cb

1 and cb

2 as follows:

cb1 = 1+p2

2 (1.5)

cb2 = 1+p2

2p . (1.6)


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3

(b) Notice that if both individuals have logarithmic utility functions,then the equilibrium price is given by p = 1/2 with r = 1/p 1. For themore general case with > 1, p < 1/2. To show this, we can use themarket-clearing conditions for periods 1 and 2, which are given by:

1+p21 +p(1)/

+1+p2

2 = 21

p1/(1+p2)

1 +p(1)/ +

1+p22p

= 22.

Expressing these conditions in terms of a common denominator and takingtheir ratios yields:

p(3 +p(1)/)

3p(1)/+ 1 =

12

.

As 1, the left side of this equation goes to p and

p= 12

To show that p < 1/2 when > 1, assume the contrary that p = 1/2.Then the condition describing the equilibrium price (or real interest rate)simplifies as

3 +p(1)/

= 3p(1)/

+ 1 p(1)/

= 1.

For > 1, the only solution is p = 1, contradicting the assumption thatp = 1/2 < 1. Henceforth, we assume that p < 1/2. Since p < 1,p(1)/


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c) When we look at the consumption inequality,

ca1 cb1 =

(1+p2)

2(1 +p(1)/)(1 p(1)/)

ca2 cb2 =

(1+p2)

2(1 +p(1)/)

p(1)/ 1

p

,

and since 0< p < 1 and >1,

|1 p(1)/| 0 for each j and s, the collateral constraint

jxj,s+ fjws 0

can be equivalently written as:

j fjwsxj,s

.


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Under short sales,j


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11

Simplifying first-order conditions imply the following:

ci2(s) =(s)ci1

p(s) , for i = 1, 2, s= 1, 2 (2.12)

Define the world output as

y11+ y21 =y

w1 and y

12(s) + y

22(s) =y

w2(s)

The market clearing conditions are:

c11+ c21 = y

w1 = 4

c12(s) + c22(s) = y

w2(s) = 4

Substituting the FOC in Eq. (2.12) into the market clearing condition yields:

c12(s) + c22(s) = (c

11+ c

21)

(s)

p(s)

yw2(s)

yw1=

(s)

p(s) = 1.

Then, the state price is given by:

p(s) =(s)

Taking the ratio of the first-order conditions in the different states yields:

ci2(2)

ci2

(1)=

p(1)

p(2)

(2)

(1) =

(1)

(2)

(2)

(1) ci2(1) =c

i2(2) for i = 1, 2.

Notice that the first-order conditions in (2.12) together with the condition thatp(s) =(s) fors = 1, 2 imply that:

ci2(s)

ci1=

yw2(s)

yw1= 1 ci1= c

i2(1) =c

i2(2) for i = 1, 2

Since countries are symmetric in their endowments and have identical pref-erences, the optimal consumption allocations and Arrow-Debreu security po-sitions are given by:

c1 = (2, 2, 2), c2 = (2, 2, 2)

z1 = (2, 2, 0), z2 = (2, 2, 0)

and equilibrium prices satisfy:

p(1) =p(2) =3

8, p(3) =

3

4.

Thus, by trading in the contingent securities in the absence of any shortsales constraint, each consumer is able to achieve a perfectly smooth consump-tion stream. Notice that the equilibrium holdings of the third security for bothconsumers is zero. Hence, as argued before, the price of the third security inequilibrium is equal to the sum of the other two securities.


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b) Under the short sales constraint, since the optimal security holdingswill be on the boundary, the multiplier associated with the constraint will bepositive. Then, the first-order conditions imply that:

p(1)

ci1

(1)

ci2(1) > 0

p(2)

ci1

(2)

ci2(2) > 0

p(3)

ci1

(1)

ci2(1)+

(2)

ci2(2)

= 0 for i = 1, 2.

The last inequality follows from the fact that none of the countries need asecurity that will pay in both states, so that the third security is not subjectto the short sales constraint. An equilibrium for this world economy is acorner solution, where countries share the risk to the extent that the shortsales constraint allows:

c1 = (2, 3, 1), c2= (2, 1, 3)

z1 = (1, 1, 0), z2 = (1, 1, 0).

Notice that the short sales constraint affects the consumption of consumer 1 in

state 2 and of consumer 2 in state 1. Substituting these equilibrium allocationsand the values for and given in the question into the first-order conditionsfor the state-contingent securities yields:

p(2) =p(1)> (3/4)(1/2)2

1 =

3

4.

Using the third first-order condition, one can easily check that price of thethird asset is not equal to sum ofp(1) and p(2). Since,

p(s)

ci1>

(s)

ci2(s) for s = 1, 2

then,p(1)

ci1+

p(2)

ci1>

(1)

ci2(1)+

(2)

ci2(2)

=

p(3)

ci1,

which implies

p(1) +p(2)> p(3)

so that the law of one price does not hold.


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13

Question 5 We examine the value of the two portfolios at date T. Let STdenote the price of the stock at date T.

Value ofA =

ST X+ X=ST ifST > X

X ifST X

and

Value ofB =

X ST+ ST =X ifST < X

ST ifST X

Thus, both portfolios are worth max(ST, X) at the expiration date of the

options. Thus, if there is no arbitrage, then two portfolios must have identicalvalues today. This implies

c + Xr(Tt) =p + St, (2.13)

This relation is known asput-call parity. It shows that the value of a Europeancall with a certain exercise price and date can be deduced from the value ofa European put option with the same exercise price and date, and vice versa.If this relationship does not hold, then there exists an arbitrage opportunity.Let us illustrate this with an example. Suppose this equality does not holdsuch that:

p + St> c + Xr(Tt)

The inequality above says that Portfolio B is overpriced relative to PortfolioA. Thus, the correct arbitrage strategy is to buy the relatively underpricedsecurity (which is the call in this case) and to short sell the one which isrelatively overpriced (the put and the stock). Such a strategy will generate acurrent cash flow ofp+St c, which will grow to r

Tt(p+St c) in T tperiods if invested at the risk free return. At the expiration date, there aretwo possible scenarios for this strategy:

ifST > X , then only the call will be exercised. The investor will buythe stock at a price ofXat time Tand will close out the short positionwith it.

ifST < X, then only the put will be exercised. The investor is obligedto buy the stock at a price of X, at time T and will use this stock toclose his short position.

Therefore for both scenarios, rTt(p + St c)X is the net profitat time T from this strategy. Since, we assumed that rTt(p+ St) >rTtc+ X , net profits from this strategy will be positive such thatrTt(p+St c) X > 0 . The table shows the cash flows associatedwith such an investment strategy.


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ST > X ST < X

Call c ST X 0P ut/stock p + St ST ST (X ST)Bond p + St c (p + St c)r

Tt (p + St c)rTt

Total 0 (p + St c)rTt X (p + St c)r

Tt X

Using the same steps described above, one can easily show that an ar-bitrage strategy also exists when Portfolio A is overpriced relative toPortfolio B .


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15

3 Expected Utility

Question 1 Let L1 = ((x11, . . . , x

1m),

1) and L2 = ((x21, . . . , x

2n),

2) be twolotteries and suppose that rewards are ordered such that xi1 . . . x

im.

Choose xh such that it is preferred to both x11 and x21 and choose x

l suchthat x1m and x

2n are preferred to it. For each x

ij compute q

ij such that x

ij

((xh, xl), (qij, 1 qij)). This can be done the continuity axiom. By the inde-

pendence axiom, Li ((xh, xl), (Qi, 1 Qi)), where Qi =

qijij. By the

dominance axiom, L1 L2 if and only ifQ1> Q2.Notice thatqijis a valid utility measure for x

ijbecause by Axiom 3.6x

ij x

ik

if and only ifqij > qik. This implies thatq is increasing and by Axiom 3.5, it

is continuous.

Question 2

a) The first-order condition with respect tox consists of

E[U(W1)]

x = E

U(W1)

x

= E

U(W1)

W1

W1x

= EU(W1)(r rf)W0= 0.b) The first-order condition is a nonlinear function of x, which enters

W1. To linearize it, we expand U(W1)(r r

f)W0 around x = 0 as

E

U(W1)(r rf)W0

U(W0(1 + r

f))(r rf)W0

+E

x(r rf)2W20 U(W0(1 + r

f))

.

Solving forx yields

x = UW0UW20

E(r) rfE(r rf)2

=

E(r) rf

V ar(r) + [E(r) rf]2

1

CRRA

(3.1)

where E(r rf)2 =V ar(r) + [E(r) rf]2.


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c) The fraction of wealth invested in the risky asset,x, varies positively with E(r) rf. The higher the expected return on the

risky asset relative to the risk-free, the larger is the fraction of wealthheld in the risky asset. If the risky asset is a fair gamblewithE(r) =rf,then x = 0.

varies negatively with V ar(r).

varies negatively with CRRA, which shows the investors attitude to-wards risk. As CRRA increases, the fraction of wealth held in the riskyasset decreases.

Question 3

a) Here, is practically useful for two things. First, it is defined inunits of consumption. Second, it summarizes behavior towards risk directlysuch that a more risk-averse person is expected to have a higher certaintyequivalent of a particular set of state-contingent consumptions compared to aless risk-averse person. In this case, the appropriate measure of risk is simplythe difference between the certainty equivalent and expected consumption.Since

E(u(c)) =U({c(s)}) =u(({c(s)})),

it follows thatSs=1

p(s)u(c(s)) =Ss=1

p(s)u(({c(s)})) =u(({c(s)})).

Because u(.) is strictly increasing, it has an inverse so that we can solve forthe certainty equivalent:

({c(s)}) =u1 Ss=1

p(s)u(c(s))

= u1[E(u(c))].

b) Substituting the information we are given into the expected utility

yields:2s=1

p(s)u(c(s)) =1

4.0 +

3

4.2 =

3

2.

The inverse of the CRRA utility function is:

u1(x) = [1 + (1 )x]1

1 = [1 + 0.5x]2,

where x = u(c). Then,

u1

3

2

=

1 +

3

4

2=

49

16 3


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Question 4

a) i. The certainty eqivalent function can be found from the relation:

=

sp(s)c(s)

1

+

1

1

.

Solving for yields:

=s

p(s)c(s).

ii. The certainty equivalent function satisfies the relation:

E(u(c)) =U({c}) =u(({c(s)})).Since the person is risk-averse,

u(({c(s)}))< u[E(c(s))] ({c(s)})< E(c(s)),

or equivalently,

=Ss=1

p(s)M[c(s), ]< E(c(s)).

We can write the right side of this equation as an expectation:

E[M(c(s), )]< E(c(s)).

We claim that M[E(c(s)), ] =E(c(s)). To show this, consider:

=Ss=1

p(s)M[E(c(s)), ] =

sp(s)E(c(s))

1

+

1

1

,

which implies that:

=E(c(s)), or = E(c(s)).

It follows thatE[M(c(s)), ]< E(c(s)) =M[E(c(s)), ]. But if this inequalityholds, then by Jensens inequality, the function M(.) is concave.

b) Substituting the risk aggregatorMinto the definition for the certaintyequivalent function yields:(i) Weighted utility

=s

p(s)

c(s)+1

+

1

c(s)

= 1

s

p(s)c(s)+ + 1

s

p(s)c(s)

=

sp(s)c(s)

+

zp(z)c(z)


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Notice that the probabilities placed on each outcome are weightedby the factorzp(z)c(z)

.

(ii) Disappointment aversion

=s

p(s)

c(s)1

+

1

1

+ I[c(s) ]

(c(s)1 )

= 1

s

p(s)c(s) +

1

1

+s

p(s)I[c(s) ](c(s)1 )

Solving for yields:

=s

p(s)c(s) + s

p(s)I[c(s) ](c(s) )

where I is an indicator function that equals 1 ifc(s) and zero otherwise.Notice that disappointment aversion places greater weight on outcomes thatare worse than the certainty equivalent.

Question 5

a) According to the max-min utility function, the expected utility fromowning security 1 is given by

min

Ss=1

scs = min 1

4 14

1

2

=

1

4

and the expected utility from owning security 2 is given by

min

Ss=1

scs = min 14

14

1

2+

=

1

4.

b) By contrast, when the probabilities of the 2 states are known to be12 ,

12 , then each security is worth 1/2. Thus, we find that ambiguityabout the

probability of a given state leads agents to value payoffs less than in a situationwhere the probabilities are known for sure.


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19

4 CAPM and APT

Question 1

a) The problem is to minimize V(rp) by choice of x. The first-ordercondition is

V(rp)

x = 2x21 2(1 x)

22+ 2(1 2x)12= 0,

which has the solution

x = 22 12

21 212+

22

=Cov(r2, r2 r1)

V(r2 r1)

. (4.1)

We can calculate the minimum value of the portfolio variance by substitutingfor the optimal value ofx into the expression for V(rp) as

V(rp) =x2V(r1) + (1 x)

2V(r2) + 2x(1 x)12

= 22 2x(22 12) + x

2(21+ 22 212)

= 22 2 (22 12)

2

21 212+ 22

+(22 12)

2(21+ 22 212)

(21 212+ 22)

2

= 21

22

212

21+ 22 212. (4.2)

b) We now consider the optimal solution under alternative covariancestructures for the risky assets. Figure A.1 shows the possible configurations.

- Perfectly positively correlated returns (12 = 1). In this case, 12= 12and

x=2(2 1)

(2 1)2 =

22 1

. (4.3)

Recall that we did not impose any restrictions on the value ofx as part

of the optimization problem. Thus, x is not necessarily restricted tobe between zero and one. The above expression shows that when theunderlying risky assets are perfectly positively correlated, the optimalstrategy involves selling one asset (shorting it) and buying the otherasset. But this implies the presence of arbitrage opportunities. Supposethat the variance on the first asset is smaller than the variance of thereturn for the second asset, i.e., 1 < 2, then the minimum varianceportfolio has the share of asset 1 given by

x= 2

2 1>1.


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px y

E(R)

E(Ry)

E(Rx)

= 1

= 1

Figure B.1: Efficiency Locus with Different Values of

Thus, the minimum variance portfolio involves short selling the secondasset and buying the first asset. Since a security that has a higher

variance will also typically have a higher mean, if the investor followsthis strategy, s/he can obtain a riskless return at zero outlay. Note thatfor any x satisfying (4.3), the variance ofrp is:

V(rp) = 21

22

212

21+ 22 212

= 21

22

21

22

21+ 22 212

= 0,

To evaluate its mean, supposeE(r1) = 6,E(r2) = 10,1= 4 and2= 8.Then,

x= 8

8 4

= 2 or 200%,

and

E(rp) = 2(6) 10 = 2.

Since such arbitrage is inconsistent with individual optimization andmarket equilibrium, we typically rule out the case of perfectly positivelycorrelated assets.


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- Perfectly negatively correlated returns (12 = 1). In this case, 12 =12 and

x=2(2+ 1)

(2+ 1)2 =

22+ 1

, (4.4)

and V(rp) = 0. In this case, the investor holds positive amounts of bothassets. Perfect negative correlation implies that when one asset has areturn above average, the other asset has a return below average. Thus,by holding both assets in the proportions described above, the investorcan obtain a positive expected return with no risk.

- Uncorrelated returns (12= 0). In this case,

x= 21

22+ 21

, (4.5)

and

V(rp) = 21

22

22+ 21

. (4.6)

Question 2

a) Define l to be the n 1 vector of ones, i.e., l = (1, 1, . . . , 1). Hence,we can write the investors end-of-period wealth as

Wt+1= Wt

xt(l+ rt+1) + (1 xtl)(1 + r

ft+1)

= Wt

xtl+ x

trt+1+ 1 x

tl+ r

ft+1 x

tlrft+1

= Wt

1 + xtrt+1 lr

ft+1+ r

ft+1

= Wt(1 + r

pt+1).

b) We assume that the consumer maximizes the expected utility of nextperiods wealth Et[U(Wt+1)]. As before, we expand Et[U(Wt+1)] aroundEt(r

pt+1). It follows that

Et[U(Wt+1)] =EtU

Wt(1 + rpt+1)

Et

UWt(1 + Etr

pt+1)

+ Wt(r

pt+1 Etr

pt+1)U

+

1

2W2t(r

pt+1 Et(r

pt+1))

2U


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= UWt(1 + Etrpt+1) +12 W2tVt(rpt+1)U= F

Et(r

pt+1), Vt(r

pt+1)

, (4.7)

where F >0, F1 =F/Et(rpt+1)> 0, F2 =F/Vt(r

pt+1) 0, and F2/F1 =

(WtU)/(2U). Thus, we have that maximizing expected utility is equivalent

to maximizing a function of expected return and the variance of returns, wherethe two are traded off.

Using the definition ofrpt+1, we have that

Et(rpt+1) =x

tEt(rt+1) + (1 x

tl)r

ft+1 (4.8)

and

Vt(rpt+1) =x

tEt

(rt+1 Et(rt+1))(rt+1 Et(rt+1))

xt= xtVt(rt+1)xt. (4.9)

Hence, the investor can choose to invest nothing in the risky assets (by settingxt = 0) and obtain a zero variance on the portfolio Vt(r

pt+1) = 0 and the sure

portfolio return rpt+1 = rft+1, or s/he can invest in both the risk-free and the

risky assets and obtain a portfolio return that is greater than the risk-freereturn Et(r

pt+1)> r

ft+1 and a positive portfolio variance Vt(r

pt+1)> 0.

c) Consider the problem

maxxt

F

Et(rpt+1), Vt(r

pt+1)

subject to

rpt+1= xt(rt+1 lr

ft+1) + r

ft+1 and Vt(r

pt+1) =x

tVt(rt+1)xt.

The first-order conditions with respect to xt are

F

xt= F1

Et(r

pt+1)

xt

+ F2

Vt(r

pt+1)

xt

= F1

Et(rt+1) lrft+1

+ F22Vt(rt+1)xt = 0.

d) We can manipulate the first-order conditions to show that:

Et(rt+1) lrft+1 =

2

F2F1

Vt(rt+1)xt

= tVt(rt+1)xt, (4.10)


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23

where t = WtU/U = 2F2/F1 = CRRA. Pre-multiply both sides ofthe expression in (4.10) by xt. Thus, at the consumers optimum, the excessreturn on the optimal portfolio relative to the risk-free return is given by

Et(rpt+1) r

ft+1 = x

t(Et(rt+1) lr

ft+1)

= txtVt(rt+1)xt

= tVt(rpt+1). (4.11)

We can use this expression to solve for t as

t=Et(r

pt+1) r

ft+1

Vt(rpt+1)

. (4.12)

Eliminating t from (4.11) using the expression in (4.12) yields

Et(rt+1) lrft+1= [Etr

pt+1 r

ft+1]

Vt(rt+1)xtVt(r

pt+1)

.

e) Notice that this relation holds at each consumers optimum, regardlessof her/his attitude towards risk. If all market investors have identical beliefs,then in equilibrium the return on the portfolio rp will be the market returnrm, too. Thus, in equilibrium the return on each risky asset relative to therisk-free asset is given by

Et(rt+1) lr

f

t+1= [Etr

m

t+1 r

f

t+1]

Vt(rt+1)xt

Vt(rmt+1) ,or

Et(rpt+1) r

ft+1 = [Etr

mt+1 r

ft+1]

xtVt(rt+1)xtVt(rmt+1)

= [Etrmt+1 r

ft+1]

Vt(rpt+1)

Vt(rmt+1). (4.13)

The relationship in (4.13) is an equilibrium relation which provides a linearrepresentation for the risk and return on efficient portfolios. It represents themean-variance portfolio frontier in the presence of a risk-free asset. This isknown as theCapital Market Line (CML). The CML is graphed in Figure A.2below.

The picture depicts investment opportunity (or feasible) set with finitelymany risky assets. The point on the left side of the set, denoted by + signis the minimum variance portfolio, which is the portfolio with the smallestrisk and maximum expected return. The point, where the line is tangent tothe mean-variance frontier denotes the market (tangency) portfolio, where theCapital Market Line (CML) is tangent to the frontier. The portfolio providesthe largest slope for the line drawn from a given expected risk-free return.In other words, the market portfolio, when combined with a risk-free assetprovides highest return per risk.


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MarketPortfolio

MinimumVariancePortfolio

CML

E(Rf)

E(R)

Figure B.2: Capital Market Line (CML)

Question 3

a) This follows directly from part (e) of Exercise 2 above by noting that

V(rt+1)xt= C ovt(rt+1, rpt+1).

Substituting for rpt+1 = rmt+1 and evaluating Covt(rt+1, r

mt+1) for a single asset

yields the result in the text.

b) The market beta shows the covariance of the return on asset i with

the market return. The quantity [Et(rmt+1) r

ft+1] is referred to as the price of

risk, and thequantity of riskis the beta for asseti,it. The linear relationship

between the excess expected return on asset i relative to the risk-free asset,on the one hand, and the price and quantity of risk, on the other, is referredto as the Capital Asset Pricing Model.

c) The beta of a risk-free asset is 0:

Covt(rft+1, r

mt+1)

V art(rmt+1) = 0.


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25

d) The beta of the market return is 1:Covt(r

mt+1, r

mt+1)

V art(rmt+1) = 1.

Question 4 Let us begin with writing the first-order condition given belowonce more:

ql = E

U Nn=1

inxn

xl

.

Since, this first-order condition is valid for all securities, we can write it forthe riskless security, too:

qf =E

U

Nn=1

inxn

xf

Equating the s and arranging the resulting equation we obtain:

E

U

Nn=1

inxn

xlql

xfqf

= 0

Since, Rn= xn/qn n= 1, 2,...,N, we can write this equation as:

EUN

n=1 inxn (Rl rf)= 0

or

E

UNn=1

Wi

(Rl rf)

= 0

where we make use of the definition of next periods wealth: Wis =Nn=1

inxn.

Let us apply the covariance decomposition in order to rewrite the above equa-tion as:

Cov

U Nn=1

Wi

, (Rl rf)

= E(Rl rf)E

U

Nn=1

Wi

Now, we can use the lemma given in the question on the left side of thisequation:

E[U(Wi)]Cov

Wi, (Rl rf)

= E(Rl rf)E

U

Nn=1

Wi

Substituting the definitions of win and Rn,s, we can write the next periodswealth in terms ofw in, Rn,s and W

i0 as follows:

Wi =Wi0

Nn=1

winRn


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Substituting this definition into the equation yields:

E(Rl rf)E[U(Wi)] = E[U(Wi)]Cov

Wi0

Nn=1

winRn, (Rl rf)

= E[U(Wi)]Wi0Cov

Nn=1

winRn, (Rl rf)

Rearranging yields:

E(Rl rf)E[U

(Wi)]

E[U(Wi)]Wi

0

= Cov N

n=1winRn, (Rl rf)

E(Rl rf)

A(Wi)Wi0= Cov

Nn=1

winRn, (Rl rf)

whereA(Wi) is the coefficient of absolute risk aversion evaluated at Wi. Now,we can sum the left and right sides of this equation over individuals:

E(Rl rf)Ii=1

1

A(Wi)Wi0

=

Ii=1

Cov

Nn=1

winRn, (Rl rf)

= Cov Ii=1

Nn=1

wi

nRn, (Rl rf)SinceRWs

Ii=1

Nn=1 w

inRn,s and the return to the market portfolio and to

the riskless asset are uncorrelated, this reduces to

E(Rl rf)Ii=1

1

A(Wi)Wi0

= Cov(Rl, RW) (4.14)

Since Eq.(4.14) is valid for all assets, we can write it for the market portfolio,too:

E(RW rf)

Ii=1

1A(Wi)Wi0

= Cov(RW, RW) =V ar(RW) (4.15)Dividing Eq.(4.14) to Eq.(4.15) yields the CAPM representation:

E(Rl rf) =E(RW rf)Cov(Rl, RW)

V ar(RW) .


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27

5 Consumption and Saving

Question 1 Under certainty, we solve the following problem:

max {U(c1) + U(c2)}

with respect to

c1+ S y1

c2 y2+ (1 + r)S.

If we substitute the values ofc1 and c2, then we obtain

max {U(y1 S) + U[y2+ (1 + r)S]} .

The F.O.C is given by:

U(c1) =(1 + r)U(c2). (5.1)

Since our utility function is quadratic, then U(c1) = a bc1 and U(c2) =

a bc2. If we substitute these values into the equation (5.1), and solve for c2,we obtain the value ofc2 under certainty as following;

c2= a[(1 + r) 1] + bc1

b(1 + r) . (5.2)

On the other hand, when we look at the solution under uncertainty, wesolve the following problem:

max E{U(c1) + U(c2)}

with respect to

c1+ S y1

c2 y2+ (1 + r)S.

If we also substitute the values ofc1 and c2, then we obtain

max E{U(y1 S) + U[y2+ (1 + r)S]} .

The F.O.C is given by:

EU(c1)= E(1 + r)U(c2) . (5.3)If we substitute values ofc1 and c2 associated with the quadratic utility func-tion into the equation (5.3), and solve for c2, we obtain the value ofc2 underuncertainty as follows:

E[c2] = a[(1 + r) 1] + bE(c1)

b(1 + r) . (5.4)

Thus, we find that the solution under uncertainty is equal to the expectationof the solution under certainty. This is known as the certainty equivalentproperty. It is worth noting that the quadratic utility function is the onlyutility function for which this property holds.


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Question 2 To solve this problem, we construct the Lagrange function asfollows;

L=i=0

i log(ct+i) +

t [yt+ St(1 + r) ct St+1]

.

The F.O.Cs are

L

ct= 0

t

ct=t

L

ct+1= 0

t+1

ct+1=t+1

L

St+1= 0

tt+1

= 1 + r.

a) The stochastic discount factor is

mt+1= U(ct+1)

U(ct) =

ctct+1

.

b) The price of the wealth portfolio is calculated as:

PWt = Et

j=1

jU(ct+j)

U(ct) ct+j

= Et

j=1

jctct+j

ct+j =Et

j=1

jct,

which implies

PWt = Et

j=1

jct= Etct(+ 2 + 3 + ...)

= Etct

1 =ct

1 .

c) Finally the return on the wealth portfolio is defined as:

RWt+1=PWt+1+ ct+1

PWt=

ct+1

1+ ct+1

ct

1

=ct+1

ct,

which implies that

1

RWt+1=

ctct+1

=mt+1.


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29

Hence, the inverse of the return on the wealth portfolio is equal to the stochas-tic discount factor, which is strictly positive. Thus, the log utility CAPMprovides a way to obtain a atrictly positive discount factor that is based on(the inverse of) the return on the wealth portfolio.

Question 3 This is an example of a dynamic optimization problem with aquadratic utility function and a present-value form of the budget constraint.Let the single-period utility function be given by U(ct) = a (b/2)c

2t . The

problem is:

max{ct,bt+1}E0 t=0

U(ct)subject to

bt+1+ yt= ct+ it+ gt+ (1 + r)bt, t 0,

givenb0. It is straightforward to show that the intertemporal Euler equationis:

(1 + r)Et[ct+s] = a[(1 + r) 1]

b + Et[ct+s1]

= B+ Et[ct+s1], s 1, (5.5)

where B = a[(1 + r) 1]/b.

a) To express the solution for consumption in terms of national cash flowsnft, solve the budget constraint forward at time t, imposing the transversalitycondition that

limh

Et

bt+h+1

(1 + r)t+h

0.

This yields:

s=0

ct+s(1 + r)s =

s=0

yt+s it+s gt+s(1 + r)s (1 + r)bt

=s=0

nft+s(1 + r)s

(1 + r)bt.

Now take expectations of both sides of the present-value budget constraintand substitute recursively forEt[ct+s] using the first-order condition:

ct+s=1

cts(1 + r)2s

+ B

(1 + r)2

1

1 1(1+r)

2 =

s=0

Et

nft+s(1 + r)s

(1 + r)bt.


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Ignoring the constant term and simplifying yields:

ct= r

1

1 + rEt

s=0

nft+s(1 + r)s

bt

,

where

= (1 + r)r

(1 + r)2 1.

This result shows that predicted consumption is proportional (r/) to netproductive wealth, which is defined as the present value of expected nationalcash flows, net of the stock of foreign debt today.

The parameter is known as the consumption-tiltingparameter. Consider3 cases:

= 1. This corresponds to the permanent income theory. In this case,the consumption-smoothing level of consumption is equal to permanentnet cash flow or the countrys wealth, and the country is neither accu-mulating nor decumulating foreign assets.

(1 + r)1, which means that the subjective discountfactor is greater than the discount factor offered by world capital mar-kets. Equivalently, the rate of return from deferring consumption is lowerthan the rate of return offered by world capital markets.

>1. In this case,


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31

and substitute for the optimal consumption evaluated at = 1, ct :CAt =nft c

t rbt

= nft r

1

1 + rEt

s=0

nft+s(1 + r)s

bt

rbt

= nft r

1 + rEt

s=0

nft+s(1 + r)s

= nft

1 + r

r

1 + rEt

s=1nft+s

(1 + r)s

= Et

nft+11 + r

+ nft+1(1 + r)2

r

1 + rEt

s=2

nft+s(1 + r)s

= Et

nft+11 + r

nft+2(1 + r)2

+ nft+2(1 + r)3

r

1 + rEt

s=3

nft+s(1 + r)s

= Et

s=1

nft+s(1 + r)s

.

The last relation shows that the consumption-smoothing component of thecurrent account is the negative of the present discounted value of expected

changes in national cash flow. The current account is in deficit when thepresent discounted value of expected changes in national cash flow is positive.The current account is in surplus in the opposite case.

c) By this argument, permanent increases in real output (yt+s for all s)have no effect on the current account since optimal consumption also increasesby the same amount, leaving the current account unchanged. For the same rea-son, permanent decreases in investment or government expenditures result inequal increase in consumption; hence, the current account remains unchanged.Note, however, that a permanent positive shock to investment may worsen thecurrent account, if the investment shock is expected to increase output in thefuture.

On the other hand, any temporary shock will have an effect on the cur-rent account. If there is a temporary increase in investment or governmentspending, a temporary decrease in national cash flow occurs. Therefore, thecurrent account deficit increases and the economy increases its foreign liabil-ities. For the opposite case, any temporary positive shock to real output ora temporary decrease in investment improves the current account, hence theeconomy increases its foreign assets. For these reasons, we can say that thecurrent account plays a role of buffer to smooth consumption in the presenceof temporary shocks.


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Question 4

a) Eliminate lt using the time constraint. The problem is

max{ct,nt,St+1}

t=0

t {U(ct, 1 nt) + t[wtnt+ (1 + r)St ct St+1]} .

The first-order conditions are

U1(ct, 1 nt) = t, (5.6)

U2(ct, 1 nt) = twt, (5.7)

t = t+1(1 + r). (5.8)

Eliminate t in the first two equations to obtain

U2(ct, 1 nt) U1(ct, 1 nt)wt= 0. (5.9)

Differentiate with respect to nt to show that the left side is monotonicallyincreasing in nt, assuming both consumption and leisure are normal goods.We can expressnt as a function of (ct, wt), using the inverse function theorem.Hence,

nt= H(ct, wt).

Totally differentiate (5.9) with respect to ct after substituting in the functionH,

H

ct=

U21 U11wtU22 U12wt

0.

Hence the higher wages, the more agents work (no surprise). Next substitutethe solution into the first-order condition (5.8). This yields the intertemporalEuler equation:

1 =U1(ct+1, 1 H(ct+1, wt+1))(1 + r)

U1(ct, 1 H(ct, wt)) . (5.10)

Using this equation along with the budget constraint results in a second-ordernonlinear difference equation in savings (specifically solve the budget con-straint for ct and substitute into (5.10)).


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33

b) Under this assumption,

U1(ct, 1 H(ct, wt)) =U1(ct+1, 1 H(ct+1, wt+1)), (5.11)

so that the marginal utility of consumption is constant over time. If utilitywere a function of consumption only, then this implies that consumption isconstant since the utility function is concave. Let K denote the constantmarginal utility of consumption. We know then that the pair (ct, nt) thatsolve the first-order conditions is

U1(ct, 1 H(ct, wt)) =K.

Totally differentiate with respect to ct,

U11 U12Hct

= U11 U12U21 U11wt

U22 U12wt

= U11U22 U

212

U22 U12wt>0

Since the left side is increasing in ct, there exists a unique solution, given K,

ct= g(ct, K).

The problem now is to find Ksuch that the lifetime budget constraint holds.That step is explained later.

c) The first-order conditions imply that

21 nt

= 1wt

ct

ct+1ct

= (1 + r)

Under the assumption (1 + r) = 1, the system can be rewritten as

nt = 1 2ct1wt

,

ct = ct+1.

Under this preference assumption, specifically that the utility function is sepa-rable inct, lt, and the assumption that(1+r) = 1, consumption is a constant;call it c. Substitute this into the first-order condition,

nt = 1 c 21wt

. (5.12)

Notice that labor supply does fluctuate in response to fluctuations in the wagerate. We need to find the lifetime budget constraint. Solve the initial budgetconstraint for S0,

S0= 1

1 + r[c0+ S1 w0n0]


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Now solve this forward to obtain

S0+t=1

1

(1 + r)

twtnt= c

t=1

1

(1 + r)

t, (5.13)

assumingc is constant. Substitute for nt using (5.12) and simplify to obtain

S0+t=1

1

(1 + r)

t wt c

21

=

c

r

or

S0

+

t=1

1(1 + r)t

wt=

c

r 1 +2

1 .GivenS0and the wage sequence, ccan be determined from the equation above.

d) If we assume the logarithmic utility function above, then consumptionwill be a constant. When wages are high, labor will be high and labor is lowwhen wages are low. This can be verified from the labor function above.Specifically,

nl = 1 c 21wl

< nh 1 c 21wh

.


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35

6 Dynamic Programming

Question 1

a) Write the Bellman equation as follows:

V(y) = maxc,k

U(c) + V(f(k))

subject to c+k = y = k. Since the technology shock is deterministic, wecan define ()1=.1 Now try to make a guess about the form of the valuefunction. For instance:

V(y) =G + Fy1

1

where F and G are undetermined constants. Use the first-order condition:

U(c) =V(y)

k

and substitute our guess into this condition to obtain:

c = (y k)= (k k)

= F(y) = F(k).

Then, the values of next periods capital stock and consumption are given by:

k = k1 + (F)1/

,

c = k[(F)1/]

1 + (F)1/.

Now use the envelope condition V (k) =U(c)f(k)together with the expres-sion for the value function to obtain:

F1/= [(F)1/]

1 + (F)1/,

which implies that

(1F)1/= ()1/ 1.

Substituting this result back into the decision rules for capital and consumptionyields:

c = g(y) =y y

()1/,

k = h(y) = y

()1/.

1We note that the solution to the problem with {t}

t=0 i.i.d. will be the same providedwe define = E()1 and take expectations of the future value function.


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b) We follow the approach in the text to generate a solution.t=T:

The value function at the terminal date again has the form:

maxkT+1

V(kT) = c1T1

+ V(kT+1)

If we are at time t = T, next periods utility will be 0, so that V(kT+1) = 0.The Bellman equation at time t = Treduces to:

maxkT+1

V(kT) = c1T1

=[kT kT+1]

1

1 .

Obviously, solution to this maximization problem is kT+1 = 0. The valuefunction at time T is:

V(kT) = [kT]

1

1 ,

and the optimal consumption level is cT =kT.

t=T-1:The social planner solves

maxkT

V(kT1) =c1T11

+ V(kT).

Substituting for V(kT) and feasibility constraints yields:

maxkT

V(kT1) =[kT1 kT]

1

1 +

(kT)1

1 .

The first-order condition with respect to kT is:

(kT1 kT)=(kT)

.

LettingFT = (1)1/, the solution for kT is:

kT =

1 + FTkT1.

Using the law of motion for capital and the feasibility constraint, one can showthat the optimal consumption is:

cT1= FT1 + FT

kT1.

Now we need to re-write the value function at time T 1 in terms ofkT1:

V(kT1) =

FT1+FT

1+

2

1+FT

11

k1T1.


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37

t=T-2:Define

RT1=

FT

1 + FT

1+

FT(1 + FT)1

.

The Bellman equation for time T 2 is:

maxkT1

V(kT2) =[kT2 kT1]

1

1 +

RT1(kT1)1

1 .

Taking the derivative with respect to kT1 yields:

(kT2 kT1)

=1

RT1kT1.

Solving forkT1 and cT2 yields:

kT1 =

1 + (1RT1)1/kT2 (6.14)

cT2 = (1RT1)

1/

1 + (1RT1)1/kT2. (6.15)

Let

FT1= (1RT1)

1/.

As before, we can re-write the value function at time T 2 in terms ofkT2 as:

V(kT2) =

FT11+FT1

1+ RT1

2

1+FT1

11

k1T2.

Let

RT2 =

FT1

1 + FT1

1+

FT1(1 + FT1)1

.

t=T-3: We can now observe a pattern for the optimal policy functions.Using the same steps, one can show that

kT2 =

1 + (1RT2)1/kT3

cT3 = (1RT2)

1/

1 + (1RT2)1/kT3.

Let

FT2= (1RT2)

1/.


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These results yield a pattern for the evolution of optimal capital and consump-tion for t = 0, . . . , T as

kt =

1 + Ftkt1 (6.16)

ct1 = Ft1 + Ft

kt1, (6.17)

where

Ft = (1)1/

Ft+11 + Ft+1

, given FT = (1)1/.

c) We will write the solution to finite horizon problem in terms of theterminal period T:

cFt = Ft1 + Ft

kt =(1)1/ Ft+11+Ft+1

1 + (1)1/ Ft+11+Ft+1

kt

= (1)1/Ft+1

1 + Ft+1+ (1)1/Ft+1kt=

(1)1/

1 + (1)1/+ 1Ft+1kt

If we substitute for Ft+1 recursively, we see that the consumption can be

written as:

cFt = (1)(Tt)/Tti=0(

1)i/+ F1Tkt

= (1)(Tt)/

1(1)(Tt+1)/

1(1)1/ + 1()1/

kt

In order to take the limit of this expression, we can use LHospital Rule andtake the derivative of the numerator and the denominator with respect to Tto obtain:

limT cFt =limT (

1

)

(Tt)/

ln(

1

)(

1

)limT

(1)(Tt+1)/ ln(1)(1)1(1)1/

kt

= (1)1/[1 (1)1/]kt

=yt (1)1/yt= c

It .

So, in the limit, the optimal consumption we found for the finite horizoncase will converge to the deterministic version of the solution for the infinitehorizon.


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39

Question 2

a) Let Prob(t = h) =G(h) for h = 1, . . . , k. Then,

li,j = Prob(Xt+1= Xj|Xt = Xi, ut= ul)

=kh=1

G(h)f(Xt= Xi, ut = ul, t= h).

b) Since the shockt is i.i.d. and the law of motion forXt depends onlyon its past value, the state variable for the problem consists ofXt. Thus, the

value function may be expressed as:

V(Xt) = maxut

v(Xt, ut) +

rh=1

V(Xt+1)G(h)

.

However, using our answer to part a) and the notation defined in the problem,we can simplify this expression as

Vi = maxl

vi,l+

rj=1

li,jVj

(T V)i,

for i = 1, . . . , r.

c) Using the notation in the problem, we can stack the values ofVi eval-uated at the optimal solution Ui as:

V1V2...

Vr

=

S1S2...

Ss

+

11 12 . . . 1r21 22 . . . 2r

... ... . . .

...r1 r2 . . . rr

V1V2...

Vr

,

which can be re-written as:

V =S+ V V = (1 )1S.

This shows that the solution for the value function at every possible currentstate Xi is equal to the current evaluated at the optimal policy for that state,v(Xi, Ui), plus expected discounted value of the future value, conditional onthe current state, Xi, and the control variable that is optimal conditional onthat state, Ui.


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d) To do value iteration, guess that the initial value function for anystate Xj is 0, V

0j = 0. Then

V1i = (T V0)i = max

lvi,l, i= 1, . . . , n

which says that the value function at the next iteration is just equal to thevalue of the current return v(Xi, ul) maximized with respect to the controlvariable ul for l= 1, . . . , m. Now proceed in this way and use V

1 in the nextiteration as:

V2i = (T V1)i = (T

2V0)i

= maxl

vi,l+ nj=1

li,jV1j

,for i = 1, . . . , m. We continue in this way until maxh |V

n+1h V

nh| , where

> 0 is some small number.

e) To understand how policy function iteration is implemented, startwith V0 = 0. Then

U1i = UV0i = max

lvi,l, i= 1, . . . , r

S1i = v(Xi, U1i), i= 1, . . . , r .

To find 1i,j, choose the value ofli,j that corresponds to the optimal control

at stage 1, namely, U1i. Once 1 is determined, the value function at the next

stage can be determined as

V1 = (1 1)1S1,

where we have stacked all the relevant variables in vectors and matrices. Wecontinue to iterate in this way until maxl |U

n+1l U

nl | , where >0 is some

small number.The difference between value iteration and policy function iteration is that

we seek convergence in terms of the optimal policy function instead of thevalue function. However, at each stage of the policy function iteration, we alsoupdate the value function according to the new policy function and the newprobabilities of the future state, conditional on the new control variable.

Question 3 Notice that

Vn V Vn Vn+1 + Vn+1 V.


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41

But

Vn+1 V = T Vn T V V

n V

sinceV satisfies T V= V and Tis a contraction. Combining these resultsyields

Vn V Vn+1 Vn + Vn V,

which implies that

Vn V 1

1 Vn+1 Vn.

b) We can use this result to bound the error in approximating the true valuefunction by Vn since the result in part a) says that for any n, this error is(1 )1 of the distance between the value function at the nth and (n + 1)th

iterations.


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7 Intertemporal Risk Sharing

Question 1

a) LetR+denote the positive real numbers. The commodity space is thespace of sequences {ci} RTS+ .

b) To answer this question, you need to solve for the competitive equi-librium. Agent i solves

maxT

t=1 stSt tU(ci(st))t(s

t)

+iTt=1

st

pt(st)[wi(st) c

i(st)].

Denote

p(st) pt(s

t)

tt(st).

Suppose we are looking for a stationary solution. This means that for anystate st, agent i has the consumption c

i(st), regardless of the time period.The first-order conditions are

U(ci(st))

i = p(st). (7.18)

The right side is the same for all agents i= 1, . . . , I . Define the inverse functionfor marginal utility as G= (U)1, or ifU(c) =x then c = G(x). It followsthat

ci(st) =G(p(st)i).

These functions are known as theFrisch demands, which express consumptionallocations in terms of the marginal utility-weighted prices.

The feasibility condition is

Ii=1

wi(st) =Ii=1

ci(st) =Ii=1

G(p(st)i). (7.19)

For each agent, his lifetime budget constraint must hold, or

Tt=1

st

tt(st)p(st)[w

i(st) G(p(st)i)] (7.20)

where the only unknown isi, given prices. Hence there areIequations in theunknowni. For each sj Sthere are Sequations of the form (7.19). Hencethe solution is a system ofI+ Sequations.


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43

Now to answer the original question: observe that the consumption allo-cation for an agent depends on his i and prices, where i is invariant withrespect to st or t. The budget constraint does not require, for a given timepath of realizations st, that

Tt=1

pt(st)[wi(st) c

i(st)] = 0.

Hence an agent may expire with a surplus or a deficit along a particular samplepath. What the budget constraint does require is that the weighted averageequals zero over all possible time paths, where the weights are the prices.

c) Observe that feasibility requires the sum of all endowments at a pointin time equal the sum of consumption for any st. If we take the equationabove, the budget constraint along a sample path, and sum over all agents,

Ii=1

Tt=1

pt(st)

Ii=1

[wi(st) ci(st)]

= 0 (7.21)

where the equality follows because

Tt=1

pt(st)

Ii=1

[wi(st) ci(st)]

= 0 (7.22)

where the term in brackets must equal 0 by feasibility. Hence if some agent idies in debt then there is some agent j that dies in a surplus.

d) LetM(st+1) U(ci(st+1))/U(ci(st)). Observe thatMis not a func-tion ofi since the intertemporal marginal rate of substitution is equal acrossagents. To see this, observe for agent i

tU(ci(st))t(st)

pt(st) =i =

t+1U(ci(st+1))t+1(st+1)

pt+1(st+1)

so that

(st+1, st)U

(ci

(st+1

)U(ci(st))

= pt+1(st+1

)pt(st)

.

which can be rewritten as

pt+1(st+1) =M(st+1)(st+1, st)pt(s

t).

Since units of the price are arbitrary set p0(s0) = 1. Now the discountedexpected present value of total wealth is

Tt=1

st

pt(st)w(st)


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which is equal to

Tt=0

tst

tj=0

M(sj)

t(st)w(st). (7.23)

Observe thatM(sj) converges to a constant andt is decreasing ast increaseswhile w(st) is positive and finite for all st. Hence the expected discountedpresent value of total resources is finite. This isnt always true for infinitehorizon economies.

Question 2 The feasibility condition for the economy at any point in time

and for any history is

Ii=1

wi(st) Ii=1

ci(st).

The central planner solves

Ii=1

Tt=1

st

iU(ci(st))t(s

t) +Tt=1

st

t(st)

Ii=1

[wi(st) ci(st)

.

The first-order conditions are

iU

(ci

(st

))t(st

) =t(st

). (7.24)

Observe for each i,

ci(st) =G

t(s

t)

t(st)i

, G= (U)1. (7.25)

Recall that the i are known so that feasibility requires

Ii=1

wi(st) G

t(s

t)

t(st)i

= 0. (7.26)

The solution is a value oft(st

) solving the equation above. In both the contin-gent claims equilibrium and social planning problem, the optimal consumptionallocations are given by a function of marginal utilities.


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45

Question 3

a) For the time-zero trading model, the intertemporal marginal rate ofsubstitution is

U(ci(st+1)

U(ci(st)) =

pt+1(st+1)

pt(st)(st+1, st).

Hence we require

qt(st+1, st) =

pt+1(st+1)

pt(st)(st+1, st),

which can be rewritten as

pt+1(st+1) =qt(st+1, s

t)pt(st)(st+1, st).

If the prices for the two versions of the model are related as above, then thetwo solutions are equivalent.

b) We will just use the contingent claims solution above. Solve theagents budget constraint with respect to zt1(s

t):

zt1(st) =ci(st) wi(st) +

t+1q(st+1, st)zt(s

t+1)

In period 1, assume that z0(s1) = 0. Starting from time period 1, for whichthere are S possible realizations of the first period state, solve the equationabove forward in time forzt(s

t+1) to obtain

0 =s1

q(s1, s0)[ci(s1) w

i(s1) +s2

q(s2, s1)[ci(s2) w

i(s2)

+s3

q(s3, s2)[ci(s3) w

i(s3) + . . .

=T

t=1stt

j=1 q(sj, sj1)[ci(st) w

i(st)],

so that the expected discounted present value of lifetime consumption is justequal to the expected discounted present value of lifetime wealth. Observethat the discount factor is the product of the one-period ahead contingentclaims price.


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c) For the sum to converge in (7.27), we require that the producttj=1

q(sj , sj1)

tend to zero when averaged over all states, as t . The price qis just theintertemporal marginal rate of substitution (IMRS). Hence the IMRS mustbe less than one on average for the sum to converge - this is equivalent tothe requirement that the one-period interest rate be positive on average. Theno-Ponzi scheme condition is that debt at some future date t+Nmust tendto zero in expected present value as N .

Question 4 This problem can be set up as in the previous question. Wewill write down the sequential equilibrium but realize that the problem canbe correctly solved in other ways. We will use notation developed earlier forconvenience (so there are other correct ways of setting this up.) Let st bean exogenous Markov process such that yt = y(st) and xt = x(st), so thatthe state of the system can be easily described. Let (st+1 | st) denote theprobability of moving to statest+1 from state st in one period. Let q(st+1, st)denote the period t price of one unit of consumption delivered at time t+ 1.Letza(st, st1) denote the contingent claims purchased by type A at time t 1and zb(st, st1) denote the claims purchased by type B .

a) The typeA consumer solves

V(st, za(st)) = maxcat ,za(st+1,st)

[U(cat ) + EtV(st+1, za(st+1))]

subject to

y(st) + za(st, st1) cat +

st+1

q(st+1, st)za(st+1, st).

Letat denote the Lagrange multiplier. The first-order conditions are

(st+1, st)Va(st+1, za(st+1)) =

at q(st+1, st) (7.27)

at = U(cat ). (7.28)

A similar expression can be derived for the type B agent. The first-orderconditions are

(st+1, st)Vb (st+1, zb(st+1)) =

btq(st+1, st) (7.29)

bt = U(cbt). (7.30)

For each state, in equilibrium

za(st+1, st) + zb(st+1, st) = 0.


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47

Market clearing requires that

y(st) + x(st) =cat + c

bt .

Let y(st) =y(st) + x(st). Solve the first-order conditions forqfor each agent,equate and rewrite to obtain

q(st+1, st)

(st+1, st) =

Va(st+1, za(st+1))

U(cat ) =

Vb (st+1, zb(st+1))

U(cbt) . (7.31)

A property of complete markets is that, for every state, the ratio of marginalutilities for different agents is equal to a constant, or

U(cat )

U(cbt) =

or

U(y(st) cbt)

U(cbt) =.

Let cbt = g(y(st), ) The result for permanent income becomes clearer if alltrading is done at time 0. Let st = (s0, s1, . . . , st) denote the history of thesystem up to time t and let t(s

t) denote the probability of st. Finally, letqt(s

t) denote the time 0 price of a unit of consumption delivered at time t instate st. The type B s lifetime budget constraint is

t

st

qt(st)[x(st) g(y(st), )] = 0.

A similar expression holds for type A. Substitute the solution to obtaint

st

qt(st)[g(y(st), ) x(st)] = 0.

There exists a unique solving this equation.

b) Under complete markets, the ratio of the marginal utilities is equalto a constant, so that although the aggregate endowment is random and con-sumption of each type of agent is fluctuating, the aggregate risk is shared. If

the two endowments are negatively correlated, then large transfers would takeplace from the high endowment agent to the low endowment agent. If thereis no aggregate and only idiosyncratic risk, then the consumption of the twoagents would be constant and there would be a large volume of trade providingcomplete insurance against idiosyncratic risk.

c) Suppose in an extreme case, that the endowments were always equal.Then there would be no trade because the agents are identical. If the endow-ments are positively but imperfectly correlated, there may be some scope fortrade, but there is little insurance value in the trade.


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d) This question is trickier than you might at first guess. The reasonis that we need to look at what happens to the price qt(s

t). Suppose thataggregate output is low, so that the marginal utility of consumption is highfor each type of agent. Then the contingent claims price is high. If oneagent has most of the endowment during that period, then the value of theendowment will be high. Hence we can compute the unconditional mean of thetwo processesx, y, and determine which one is higher. But permanent incomedepends on the expected value of the endowment and so it is possible for theagent with the lower average endowment to have higher permanent income.


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49

8 Consumption and Asset Pricing

Question 1 The asset pricing function satisfies the relation:

U(ct)qet =Et

U(ct+1)(q

et+1+ yt+1)

.

Since the consumer is risk neutral, marginal utility of consumption is a con-stant. Furthermore, output equals consumption in equilibrium. Hence, weobtain the relation:

qet =Etqet+1+ yt+1

. (8.1)

Since yt follows a first-order autoregressive process, the state variable is the

current level of output,yt. Furthermore, {yt} is a stationary stochastic processsince || < 1. Therefore, for any bounded continuous function qe(y), noticethat the right-hand side of the above equation maps the space of bounded,continuous functions into itself. We can define an operator from the right-sideof this equations as:

(Tqe)(y) =Ey(qe) + y

.

Since 0 < < 1, T is a contraction and its fixed point can be found byiterating on equation (8.1) as follows:

qet = Et qet+1+ yt+1

= Et

yt+1+ Et+1qet+2+ yt+2

=

...

=s=0

sEt[yt+s].

But

Et[yt+1] =Et[yt+ t+1] =yt since {t} is i.i.d.

Likewise,

Et[yt+2] =Et{Et+1[yt+1+ t+2]} =Et[yt+1] = 2yt.

Iterating in this way, we find that

Et[yt+s] =syt.

Substituting this result into the solution for qet yields:

qet =s=1

ssyt =

1 yt.

To interpret these results, we note:


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- Under risk neutrality, the equity price is just equal to the discountedvalue of expected dividends, the discounting being done with the con-stant rate of subjective time preference r = 1/ 1, which also equalsthe real interest rate for this economy.

- The autoregressive parameter shows the impact of future dividendson the current stock price. If is small, then current output has littleeffect in predicting future dividends. Hence, the response of the currentequity price to current output is small. By contrast, if is large, thenthe impact of current output on future dividends decays slowly. In thiscase, the equity price shows a large response to changes in current output

because changes in currentyt are a good predictor of changes in yt+s fors 0.

Question 2

a) The price-dividend ratio satisfies the relation

qetyt

=Et

1t+1

1 +

qet+1yt+1

.

In the text, we showed that the equation describing qet /ytsatisfies a contractionand has unique fixed which can be found by iterating on this equation as

qetyt

= Et

1t+1 +

1t+1

1t+2

1 +

qet+2yt+2

= Et

i=1

i

ij=1

1t+j

.

b) If dividend growth satisfies

t = exp( + t), t i.i.d, N(0, 2),

then

Et

ij=1

1t+j

= i

j=1

Et(1t+j)

=ij=1

exp[(1 ) + (1 )2(2/2)]

= exp

i((1 ) + (1 )2(2/2))

.


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51

Substituting into the expression for the price-dividend ratio yields

qetyt

=i=1

i exp

i((1 ) + (1 )2(2/2))

= i=0

i

=

1 ,

where

= exp[(1 ) + (1 )2(2/2)].

c) In the version of the model with a growing endowment, a sufficient con-dition for a recursive competitive equilibrium to exist is that Es[(s

)1]< 1.Evaluating this condition under the distributional assumptions for t yieldsthe condition < 1.

Question 3

a) Let and 2

be a function of an underlying parameter . To keepE(t) constant while increasing V ar(t), we require that:

()/= (2()/2)/.

b) The price-dividend ratio has the form:

qetyt

=

1 ,

where = exp[(1 )+ (1 )2(2/2)]. Consider a change in qe/y as afunction of:

(qe/y)

=

1 +

(1 )2

=

(1 )2

(1 )

+ (1 )2

(2/2)

=

(1 )2[(1 )(1 1 + )]

=

(1 )2[(1 )]

.


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Hence, we see that the price-dividend ratio decreases with a mean-preservingspread in the distribution for t if > 1 and increases otherwise. Since denotes the coefficient of relative risk aversion, we note that if consumers arerelatively more risk averse an increase in risk reduces the price-dividend ratioand increases it otherwise.

Question 4 Consider a period coupon bond that is issued at date t. Atdate t the consumers budget constraint satisfies

ct+ bt+1Qc,t yt.

Consider the strategy of purchasing the bond att + s 1 that is speriodsfrom maturity for 0 s 1, and holding it for one period. At t+ s,the consumer receives the coupon and also the price at date t+s of a bond.Hence, its budget constraint satisfies:

ct+s+ bt+s+1Qsc,t+s yt+s+ bt+s(c + Q

rc,t+s), 0 s 1, r= s + 1.

The condition characterizing the optimal choice ofbs+t+1 is:

U(ct)Qsc,t =Et

U(ct+1)(c + Q

s1c,t+1 )

, 0 s 1,

where Q0c,t= 1.Define mt+s = U

(ct+s+1)/U(ct+s) for s 1. Substituting into the

above condition and solving this equation forward subject to the conditionthat Q0t+s = 1, we obtain:

Qc,t = Et

mt+1(c + Q1c,t+1)

= Et

mt+1c + mt+1

mt+2(c + Q

2c,t+2)

= Et

1i=1

ij=1

mt+j

c +

j=1

mt+j

=

1i=1

Et(mt,i)c + Et(mt,)

=1i=1

Qitc + Qt ,

where mt,i = iU(yt+i)/U

(yt) and Qit is the price of a pure discount bond

with maturity i for i = 1, . . . , .


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53

Question 5 The delivery price must satisfy

0 =Etmt,n(q

e(st+n) S)

,

where mt,n = nU(yt+n)/U

(yy). Otherwise, there would exist an arbitrageopportunity. Solving for Sequals

S = 1

Et(mt,n)Et[mt,nq

e(st+n)]

= rntEt[mt,nqe(st+n)].

Question 6 We will compute the price of a-period bond for each model.

Consider the first time series model and suppose that = 1. The price of aone-period bond satisfies

Q1t =Et

yt+1

yt

.

Substituting the expression for output using the representation yields

Q1t = Et {exp[(1+ t+1 t)]}

= exp(1)Et{exp[((2 1)t+ et+1)]}

= exp(1+

2

2

e/2) exp[(1 2)t]= A1exp(a1t).

Now let = 2. Then

Q2t = 2Et {exp[(21+ t+2 t)]}

= 2Et{exp[21 (2t+1+ et+2 t)]}

= 2 exp((1 22)t)Et[exp(21 (2et+1+ et+2)]

= A1exp[1+ (a1 )22e/2] exp(a1(1 + 2)t).

More generally, for any , we have

Qt =Aexp(at),

where

a1= (1 2), a =a1

1i=0

i2,

A1= exp[1+ 22e/2],

A =A1exp[1+ (a1 )2(2e/2)].


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Next, we derive the price of a -period bond for the second time seriesmodel. In this case, we use the version of the model with a growing endowment.The expression for a one-period bond is given by:

Q1t =Et(t+1).

Using the representation for log(t), we have

Q1t = Et{exp[(0+ 1log(t) + ut)]}

= exp[0+ 22u/2]exp(1log(t))

= B1b1t .

For = 2, we have

Q2t = 2Et(t+2t+1)

= 2Et{exp[(20+ 1(log(t+1) + log(t)) + ut+2+ ut+1)]}

= 2Et[exp((20+ 10+ 1(1 + 1)log(t) + (1 + 1)ut+1+ ut+2)]

= 2 exp((0+ 0(1 + 1))] exp[22u/2 + (1 +

21)

22u/2]1(1+1)t .

More generally,

Qt =Bbt ,

where

b1= 1, b =b1

i=1

i11 ,

B1= exp[0+ 22u/2],

B =B1exp[(b1 )0+ (b21+

2)2u/2].

Question 7 Recall that the price of a put option satisfies

Ppt =Et[mt,1. max(0,q qet+1)].

Assuming that preferences are CRRA, we can rewrite the price as follows

Ppt =Et

ct+1

ct

max(0,q qet+1)

(8.2)

Using the assumptions that ln(qet+1) and ln(ct+1/ct) are lognormally distributedwith covariance matrix given in the text, we can rewrite the equation in (8.2)as

Ppt =qe

ln(q/qe)

(q/qe exp(z)) exp(y)f(z, y)dzdy,


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55

where f(z, y) is the joint density function for z= ln[(qe)/qe] and y = ln[(c/c)].This last equation can be expressed as the sum of two integrals:

Ppt = q

ln(q/qe)

exp(y)f(z, y)dzdy qe

ln(q/qe)

exp(z+ y)f(z, y)dzdy.

We can make use of Rubinsteins double integration formula to show that

q

ln(q/qe)

exp(y)f(z, y)dzdy= qexp(c+ 2c/2)

ln(q/qe) q

q c

and

qe

ln(q/qe)

exp(z+ y)f(z, y)dzdy

=qe exp

q+ c+

2q+2qc+

2c

2

ln(q/qe)q

q c q

Notice that we made use of the relation 1 (x) = (x). Under the pricingequations of C-CAPM and the distributional assumptions, one can easily showthat

Ppt = q(rf)1(A) qe(A q).


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9 Nonseparable Preferences

Question 1

a) Substituting the budget constraint into the problem yields:

maxWs

U

W0

Ss=1

psWs

+

Ss=1

sU(Ws)

Now, substitute the exponential utility function given in the question:

maxWs A expSs=1psWs W0

A ASs=1

sexpWsA The first-order condition is given by:

sexp

Ws

A

psexp

Ss=1psWs W0

A

= 0

Simplifying this equation yields:

log(s/ps) =

Ss=1psWs W0+ Ws

A

Then,

Ws = A log(s/ps) + W0 Ss=1

psWs

= A log(s/ps) + c0. (9.1)

Multiplying both sides of the last relation by ps and summing over s yields:

Ss=1

psWs = ASs=1

pslog(s/ps) + c0

Ss=1

ps= W0 c0

where the second equality is derived using the budget constraint. Solving for

c0 yields:

c0=W0 A

Ss=1pslog(s/ps)

1 +Ss=1ps

(9.2)

One can derive next periods state contingent wealth using Eq. (9.1):

Ws = A log(s/ps) +W0 A

Ss=1pslog(s/ps)

1 +Ss=1ps

(9.3)


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57

Dividing and multiplying the first term in Eq. (9.3) bySs=1pslog(s/ps)and the second term by

Ss=1ps yields:

Ws = 1Ss=1ps

Ss=1ps{W0 A

Ss=1pslog(s/ps)}

1 +Ss=1ps

+ log(s/ps)Ss=1pslog(s/ps)

ASs=1

pslog(s/ps)

Define = 1 +Ss=1ps and 1 +r

f (Ss=1ps)

1 and substitute into theequation above:

Ws = (1 + rf)( 1){W0 ASs=1pslog(s/ps)}

+ log(s/ps)Ss=1pslog(s/ps)

ASs=1

pslog(s/ps)

One can argue that buying state-contingent wealth is like investing into a risk-free and a risky asset. The terms in braces show how much of current wealth isinvested into each asset and the terms before the braces give the gross returnon each asset.

b) Notice that the first term in braces using Eq. (9.2) equals:Ss=1ps{W0 A

Ss=1pslog(s/ps)}

1 +Ss=1ps

=c0

Ss=1

ps

In part (a) in Eq. (9.2), we showed that:

ASs=1

pslog(s/ps) + c0

Ss=1

ps = W0 c0

So, we showed that terms in braces sum to the amount of current wealth net

of current consumption, invested into a risk free and a risky asset, where thegross return to the risk free asset is 1/Ss=1ps and the return to the risky

asset is log(s/ps)/Ss=1pslog(s/ps). Notice that the risk-free asset is

given by the ratio of one sure amount of next periods wealth divided by thesum of all contingent prices. Hence, the consumer divides her wealth net ofconsumption between the risk-free asset and a risky asset. This is knownas the portfolio separation property, and it characterizes the class of HARApreferences described in Chapter 3.


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Question 2

a) The state variables for the consumers problem are given byWt andct1. The value function is given by:

V(Wt, ct1) = maxct

{U(ct ) + V(Wt+1, ct)}

subject to

ct = ct hct1, 0< h


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59

Notice the value function can be written in the form:(AWt+ F ct1)

1

1 = max

ct

(ct hct1)

1

1 +

(AWt+1+ F ct)1

1

subject to

Wt+1= (Wt ct)R.

Substituting forXt= AWt+ F ct1, the value function can be written:

X1t = maxct

(ct hct1)

1+ X1t+1

.

Substituting forXt+1 into the value function, the following should hold:

X1t = (ct hct1)1+ [(R)1/Xt]1

= (ct hct1)1+ (R)(1)/X1t ,

X1t [1 (R)(1)/] = (ct hct1)

1.

Solving forXt :

Xt = (ct hct1)

[1 (R)(1)/]/(1)

Now, equate this to Eq. (9.6) to obtain:

h

F

=h(R h)

hRA

= [1 (R)(1)/]/(1).

Therefore,

A = R1(R h)[1 (R)(1)/]/(1),

F = h[1 (R)(1)/]/(1).

Note that since h[1 (R)(1)/]/(1) is a common term for F and A,we can write the value function as follows:

V(Wt, ct1) = K

1

Wt

hR

(R h)ct1

1.

With some further algebra, we can show that

K = h[1 (R)(1)/]

= (R h)1R21

R (R)1/

.

Substituting for these coefficients in the expression for the value functionand the first-order condition for consumption implies the solution:

ct =

R (R)1/

(R h)R2Wt+ (R)1/hR1ct1,

V(Wt, ct1) = K

1

Wt

hRct1R h

1.


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c) We can solve for the time path of consumption from the first-ordercondition directly. Iterating the envelope conditions one period forward andsubstituting in the first-order condition yields:

U(ct ) (R)2VW(Wt+2, ct+1) hU

(ct+1) = 0.

By the first-order condition for period t + 1, we have that

RVW(Wt+2, ct+1) =U(ct+1) hU

(ct+2).

Define xt+i (ct+i hct+i1). Therefore, the first-order condition can be

written as:

xt hxt+1= R [xt+1 hxt+2] .

Collecting terms, we can re-write this as:

2Rhxt+2 (R+ h)xt+1+ xt= 0.

Notice that this is a second-order homogeneous difference equation in xt+2.Dividing through by 2Rh and using lag operator notation, we have:

1 (R1)L

1 (h)1L

xt+2= 0.

Recall that 1/h >1. Hence, we solve the first root backward and the second

root forward as:(h)1L

1 (R)1L

1 hL1

xt+2= 0,

or

xt+1= (R)1xt.

Substituting for the definition ofxt+1 and xt yields:

(ct+1 hct)= (R)1(ct hct1)

.

Inverting this equation and simplifying yields:

ct+1

h + (R)1/

ct+ h(R)1/ct1= 0. (9.7)

To solve this equation and find the steady state consumption, define zt+1 ct+1 hct. Then we can write Eq. (9.7) as:

1 (R)1/L

zt+1= 0,

which has the solution:

zt+1= (R)(t+1)/z0.


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61

Substituting for the definition ofzt+1 and z0, we obtain:ct+1 hct= (R)

(t+1)/(c0 hc1).

Iterating on this condition, we obtain:

ct+1 = ht+1c0+ (R)

(t+1)/(c0 hc1)

+

h(R)t/) + h2(R)(t1) . . . + ht(R)1/

(c0 hc1)

= ht+1c0+ (R)(t+1)/(c0 hc1)

+h(R)t/1 + h

(R)1/

+ . . . + ht1

(R)(t1)/ (c0 hc1)

= ht+1c0+ (R)(t+1)/(c0 hc1)

+h(R)t/

(R)1/

(R)1/ h

ht

(R)t/(R)1/

(R)1/ h

(c0 hc1)

= 1

(R)1/ h

ht+2c0+ h

t+2(R)1/c1+ (R)(t+2)/(c0 hc1)

.

Simplifying the last expression and using the solution for period t implies:

ct= c0 hc1

(R)1/

h (R)(t+1)/+

(R)1/c1 c0

(R)1/

h ht+1.

Since (R)1/ >1 and 0 < h < 1, the first term dominates the second term.Hence, ct/ct1 (R)

1/ and ct1/Wt (R)1/ R1 as t .

d) The coefficient of relative risk aversion (CRRA) is defined as:

CRRA = WtVWW

VW

=

1 hRct1/(R h)Wt

=

1 h[R(R)1/ 1]/(R h) (in the steady state)

Since 1< (R)1/< R,

CRRA

1 h(R 1)/(R h).

Thus we find that the coefficient of relative risk aversion differs from to theextent that the habit persistence parameter differs from zero. However, evenfor large values of h, the upper bound on the CRRA is close to . Hence,in approximate terms, we can take CRRA = and this approximation is notsensitive to the value ofh.


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e) The elasticity of consumption with respect to the interest rate is givenby

1

ct

ctln(r)

=

(R)1/ct1/ct h(R)1/ct2/ct1

/

=

1 h(R)1/

/ (in the steady state).

In contrast to the CRRA, the consumption elasticity is sensitive to the valueofh. If R)1/ 1, then (1 h)/, which varies significantly with h.

f ) The product of the coefficient of relative risk aversion and consumptionelasticity is given by

CRRA= 1 hR1.

This product equals one if there is no habit persistence; in the presence ofhabit persistence, this product may be substantially below one. Thus, we findthat habit persistence drives a wedge between CRRA and the intertemporalelasticity of substitution in consumption.

Question 3

a) In order to solve this problem with the Weil specification, we will usethe transformation of Epstein and Zin. Notice that the Weil utility functioncan be rewritten as:

ut= [1 + (1 )(1 )Ut]1/(1)

So, ut is a monotonic transformation of the original utility function Ut. Thusthe consumption stream that solves the transformed problem will also be thesolution to the original maximization problem. Remember the Epstein-Zinspecification in the text:

ut=

(1 )ct + (Etut+1)

/1/

,

where= 1 1/and = 1 . The consumers problem can be formulatedas a dynamic programming problem as follows:

V(At, Rt) maxct

(1 )ct + [EtV(At+1, Rt+1)

]

1

subject to

At+1 = Rt(At ct).


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Since Rt is distributed i.i.d, we can define V(At, Rt) = V(At) At andct= At. We can rewrite the Bellman equation as follows:

At maxct

(1 )ct + [Et(At+1)

]

1

subject to

At+1= Rt(At ct).

The first-order condition with respect to ct is:

(1 )c1t [(ERt)

1/](At ct)1 = 0.

Let Rt= (ERt)

1/. Substituting for ct= At yields:

(1 )

(Rt) =

1

1

1

(1 )

1 1

1Rt

=. (9.8)

We know that the value function calculated at the optimal consumptionlevel should satisfy:

At =

(1 )ct + [Et(At+1)]

1

(At) = (1 )ct + [Et(At+1)

]

= (1 )ct + (1 )c1t (At ct)

= (1 )c1t At = (1 )1At

= (1 )1 (9.9)

Equating this expression to the expression that we found for in Eq. (9.8)yields:

(1 )1 =Rt = 1 (Rt)

/(1).

Substituting for Rt and using the definition for and gives the expressionfor in the text:

= 1

(E(R1t )1/(1)

(11/).

To solve for , notice that

= (1 )1 =

(1 )11/(11/)

,


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or

= [(1 )1/]/(1).

Finally, to derive the form of the value function given in the question, noticethat

ut= [1 + (1 )(1 )Ut]1/(1),

which implies that

Ut= u1t 1

(1 )(1 )=

(At)1 1

(1 )(1 )

as claimed.

b) The coefficient of relative risk aversion can be obtained as:

W V

(W)

V(W) =

(W)12W

(W)

= .

Thus, the coefficient of relative risk aversion for this class of preferences differsfrom the elasticity of intertemporal substitution in consumption, .

Question 4

a) The state variables for the consumers problem consist of

ht (ct1, . . . , ctm, kt1, b0t, . . . , bN1,t).

The value function can be expressed as

V(ht) = maxct,dt,{bj,t+1}Nj=0

{U(ct , dt ) + V(ht+1)}

subject to (9.71), (9.72), (9.73), and (9.75).

b) The first-order conditions are given by:

t = MU(ct), (9.10)

tpd,t = MU(dt), (9.11)

tQjt = Et(t+1Q

j1t+1), j = 1, . . . , N . (9.12)

where MU(ct) and MU(dt) denote the marginal utility from nondurable anddurable consumption goods.


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c) The holding return on the three investment strategies are

h1t,1 = 1

Q1t,

h2t,3 = 1

Q3tor h2t,3 =

Q3t+3Q6t

,

h3t,3 =

2i=0

1

Q1t+i

.

Using Equations (9.10) and (9.12), these holding returns satisfy the relation:

nEt

MU(ct+n)MU(ct)

hkt,n

= 1, k= 1, , K. (9.13)

Equations (9.10) and (9.11) yield an expression for the price of durable goodpd,t as:

pd,t = MU(dt)

MU(ct). (9.14)

d) To evaluate expression (9.13) and (9.14), we need expressions forMU(ct) and MU(dt). Notice that MU(dt) depends on the expected, infinite

sum of marginal utility from current durable goods acquisitions. To see this,substitute sequentially in the expression for dt to obtain in the expression fordt yields:

dt =kt1+ dt = j=0

(1 )jdtj.

Hence,

M U(dt) =j=0

(1 )jM U(dt ).

However, if consumers can trade in consumption services, then the price ofservices from durable consumption goods expressed in units of the numerairegood must equal the ratio of the marginal utility of services from durableconsumption goods and the marginal utility of nondurable consumption goodsacquisitions:

MU(dt ) =pd,tMU(ct), (9.15)

where MU(dt ) denotes the marginal utility with respect to dt .


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Using the utility function defined above, notice that:

MU(ct) = Et

mj=0

jt+jc1t+j d

(1)t+j

,

MU(dt ) = (1 )tct d

(1)1t .

In order to evaluate Equation (9.15), we need an observable counterpart forpd,t. But,

pd,t=1

pd,t (1 )EtMU(ct+1)

MU(ct)

pd,t+1 .Using these results and scaling the resulting expressions byc1t d

(1)t , we

obtain the following relations:

Et

mj=0

jj(c1t+j d

(1)t+j )

n

mj=0

jj(c1t+j+nd

(1)t+j+n )

hkt,n

/[c1t d(1)t ]

= 0,

for k = 1, . . . , K . Likewise,

Et

(1 (1 )L1)[pd,t m

j=0

jjc1t+j d

(1)t+j ]

(1 )(ct d(1)1)t )

/[c1t d

(1)t ]

= 0.

These conditions are used to generate orthogonality condition estimatorsin Dunn and Singletons application.


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67

10 Economies with Production

Question 1

a) The household takes as given the wage and rental rate. The factorprices are themselves functions of aggregate capital and aggregate laborNt, or w(t, Nt, t) and r(, Nt, t). Define the vector of state variables asSt= (kt, t, t, Nt, gt)

. The household solves

V(St) = maxct,it,nt

[U(ct+ gt, 1 nt) + EtV(St+1)]

subject to

ct+ it rtkt+ ntwt+ t wtnt (rt )kt,

wherekt+1= (1 )kt+ it. Alternatively, you can write the Bellman equationtaking the wage and rental rate sequences as given. Rewrite the constraint as

ct+ kt+1= rt(1 )kt+ ntwt(1 ) + t+ (1 (1 ))kt. (10.1)

Let t denote the Lagrange multiplier for the budget constraint. The first-order conditions and envelope condition are

U1 = t, (10.2)

U2 = twt(1 ), (10.3)

t = Et[V1(St+1)], (10.4)

V1(St) = t[rt(1 ) + (1 (1 ))]. (10.5)

We can use the definition of the utility function to re-write these conditionsas:

(ct+ gt) = (1 )wt,

1

ct+ gt

= Et 1

ct+1+ gt+1

(rt(1 ) + (1 (1 )) .Notice that the income tax rate reduces the after-tax wage and the capitaltax reduces the after-tax rental rate. Since the capital tax is assessed on therental rate of capital net of depreciation, the effective depeciation rate alsofalls. Since utility is linear in hours of work, the income tax rate affects laborsupply only through an income effect on consumption. Finally, with > 0,an increase in government consumption reduces private consumption througha negative wealth effect.


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b) The firm solves

t = maxkt,nt

[tktn

(1)t rtkt wtnt]. (10.6)

The first-order conditions are:

tk1t n

1t = rt, (10.7)

t(1 )ktn

t = wt. (10.8)

c) The aggregate state vector ist, Nt, t, gt. The first two variables arestate variables because they help to determine the factor prices taken as given

by the representative household and firm. The last two variables are statevariables if they help predict future t+1, gt+1.

Feasibility requires that gt+ct+it =tktn

(1)t + (1 )kt. This is also

the goods market-clearing condition. The capital market and the labor marketmust also clear. What this means is thatnt =Nt and kt = t. The reason isthat individual firms and households determine their supply and demand foreach factor at the given wage and rental rate and then markets are requiredto clear.

d) First of all, as specified, no closed-form solution can be derived. Todo so requires the additional assumption that = 1. Under the assumptionsfor the utility function and parameter values above, the first-order conditionssimplify to

1

ct= t, (10.9)

= twt, (10.10)

t = Et[V1(St+1)], (10.11)

V1(St) = trt. (10.12)

Whenever you see a logarithmic structure like this, think make a clever

guess. Try this: suppose that

kt+1= Atktn

1t ,

where 0< A < 1. From the resource constraint ct = (1 A)tktn

1t . Then

the first-order condition for (ct, nt) simplify to:

t(1 )ktn

t

(1 A)tktn1t

=, nt= 1

(1 A).


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Next, use the envelope condition in the first-order condition for capital toobtain

1

(1 A)tktn1t

=Et

t+1k

1t+1 n

1t+1

(1 A)t+1kt+1n1t+1

. (10.13)

Notice that this simplifies to kt+1= tktn

1t so thatA = .

e) If the taxes are equal, then in the agents budget constraint

ct+ kt+1 = (1 )[rtkt+ ntwt] + t+ (1 (1 ))kt]

= (1 )tkt n

1t + t+ (1 (1 ))kt].

The first-order conditions and envelope condition are

U1 = t, (10.14)

U2 = twt(1 ), (10.15)

t = Et[V1(St+1)], (10.16)

V1(St) = t[rt(1 ) + (1 (1 ))]. (10.17)

We can derive the intratemporal and intertemporal conditions characterizingthe optimal choice ofct, nt and kt+1 as:

ct = (1 )wt,

1

ct= Et

1

ct+1(rt(1 ) + (1 (1 ))

.

In contrast to the version of the model in which > 0 and = , the taxrate affects both the return to labor and the return to capital. Furthermore,unlike the model in part a), there is no exogenous government consumption,implying that labor varies only in response to a shock to productivity.

Question 2

a) Lets start with constructing the problem under the assumption thatthe firm does not issue any equities:

max{ct,bt+1}t=0

E0

t=0

tU(ct)

subject to

ct+ bt+1 wtlt+ (1 + rt)bt.


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b) This time, we will take the taxes into account:

max{ct,bt+1}t=0

E0

t=0

tU(ct)

subject to

ct+ bt+1 (1 y)[wtlt+ rtbt] + bt

The first-order conditions are:

t = U(ct), (10.20)

t = Et[t+1(1 + rt+1(1 y))]. (10.21)

The firms gross profits are distributed as:

(1 )t= REt+ (1 + rt)bt prtbt

and the net investment is financed such that the equation holds:

kt+1 (1 )kt= bt+1+ REt.

Using the definition of net cash flows and equations above we can re-write itas follows:

Nt = (1 + rt)bt bt+1 prtbt

The only source of financing for the firm is still through debt, so that Wet =

bt+1. Using the (10.21), we obtain:

Wet =Et {mt+1[1 + rt+1(1 y)]bt+1} .

Again, by adding and subtractingbt+2from the equation above and simplifyingwe obtain:

Wet =Et

m

asset price in static economies

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