Suggested Solutions to Homework #4Econ 511b (Part I), Spring 2004

1. Consider a neoclassical growth model with valued leisure. The (represen-tative) consumer values streams of consumption and leisure according toP∞

t=0 βtU (ct, lt), where lt is hours of leisure in period t. The felicity function

U is strictly concave and strictly increasing in both of its arguments. Out-put is produced according to yt = F (kt, nt), where nt = L− lt is hours of laborsupply in period t (F is the total number of hours available for either leisureor work in period t) and F exhibits constant returns to scale. Output inperiod t can be used for either consumption ct or investment it and capitalaccumulates according to kt+1 = (1− δ) kt + it.

(a) Display Bellman’s equation for the problem faced by the social plan-ner in this economy. Identify clearly the state and control (or choice)variable(s).The recursive formulation for the planning problem is

v¡k¢= maxn

c,l,k0oU ¡c, l¢+ βv

³k0´

s.t.

c+ k0= F (k, L− l) + (1− δ)k

orv¡k¢= max{l,k0}

U³F (k, L− l) + (1− δ)k − k

0, l´+ βv

³k0´

From the way we write it, we can see that the state variable is k, and controlvariables are l, k

0.

(b) Derive the first-order and envelope conditions for the planning problem.The F.O.C. is ©

lª: U1

¡c, l¢F2(k, L− l) = U2

¡c, l¢n

k0o

: U1¡c, l¢= βv

0³k0´

From Envelope Theorem, we have

v0 ¡k¢= U1

¡c, l¢ ¡

F1(k, L− l) + (1− δ)¢

Iterate forward for one period, it becomes

v0³k0´= U1

³c0, l0´³

F1(k0, L− l

0) + (1− δ)

´Plug it into F.O.C., we get the final optimality conditions:©

ltª: U1

¡ct, lt

¢F2(kt, L− lt) = U2

¡ct, lt

¢©kt+1

ª: U1

¡ct, lt

¢= βU1

¡ct+1, lt+1

¢ ¡F1(kt+1, L− lt+1) + (1− δ)

¢1

(c) Use the conditions from part (b) to determine the economy’s steadystate. Show how the steady state depends on primitives and compareyour results to those for a growth model without valued leisure.In steady state, the optimality condition becomesn

l∗o

: U1³c∗, l

∗´F2(k

∗, L− l

∗) = U2

³c∗, l

∗´nk∗o

: F1(k∗, L− l

∗) =

1

β− (1− δ)

where c∗ = F (k∗, L− l

∗)− δk

∗. We can see that k

∗and l

∗depends on both β, δ,

production techonology F (kt, nt), and utility function U (ct, lt).In a growth model without valued leisure, the steady state is determined by theequation

F1(k∗, L) =

1

β− (1− δ)

which does not depend on the utility function U (ct, lt).Now let’s compare two models. First, in the model with leisure choice we add anadditional equation which states that the marginal rate of substitution betweenconsumption and leisure must equal to the marginal rate of transformation. Sec-ond, the equation about k

∗is the same, except that the level of steady-state

leisure is different. Third, for the equation F1(k∗, L − l

∗) = 1

β− (1 − δ), due to

the difference in the steady-state leisure level, the steady-state capital stock isalso different. For example, if F12 > 0 as in the case of Cobb-Douglas produc-tion function, the capital stock in the model with leisure will be lower than thatwithout leisure choice (since L− l

∗< L).

(d) Let F (k, n) = kαn(1−α) and U (c, l) = λ log (c) + (1− λ) log (l), where 0 <α < 1 and 0 < λ < 1. Solve explicitly for the steady state in terms ofparameters.With F (k, n) = kαn(1−α) and U (c, l) = λ log (c)+(1− λ) log (l) , the steady-statecondition becomes

nl∗o

:

λ

k∗α ³

L− l∗´(1−α) − δk

∗

(1− α)

Ãk∗

L− l∗

!α

=1− λ

l∗

nk∗o

: α

Ãk∗

L− l∗

!α−1

=1

β− (1− δ)

which leads to

n∗ = 1

1−λλ

α1−α

Ã1α− δ

1β−(1−δ)

!+1

L

l∗= L− n∗

k∗=

µ1β−(1−δ)α

¶ 1α−1

n∗

c∗ = k∗α(n∗)(1−α) − δk

∗

2

2. (a) Carefully define a recursive competitive equilibrium for the neoclassicalgrowth model with valued leisure. (Hint: You need two functions todescribe the behavior of the aggregate economy.)A recursive competitive equilibrium for the neoclassical growth model with valuedleisure is a set of functions:

price function : r¡k¢, w¡k¢

policy function : k0= gk

¡k, k

¢, l = gl

¡k, k

¢value function : v

¡k, k

¢aggregate state : k

0= G

¡k¢, l = l(k)

such that:(1) Given k

0= G

¡k¢, k

0= gk

¡k, k

¢, l = gl

¡k, k

¢and v

¡k, k

¢solves consumer’s

problem:

v¡k, k

¢= max

{c,l,k0}U (c, l) + βv

³k0, k

0´s.t.

c+ k0= r

¡k¢k + (1− δ) k + w

¡k¢(L− l)

k0= G

¡k¢

(2) Price is competitively determined:

r¡k¢= F1

¡k, L− l(k)

¢w¡k¢= F2

¡k, L− l(k)

¢(3) Consistency:

G¡k¢= gk

¡k, k

¢l(k) = gl

¡k, k

¢(b) Find the (functional) first-order conditions of a typical consumer who

takes as given the economy’s aggregate laws of motion.Solve a typical consumer’s problem

v¡k, k

¢= max

{l,k0}U³r¡k¢k + (1− δ) k + w

¡k¢(L− l)− k

0, l´+ βv

³k0, k

0´s.t.

k0= G

¡k¢

We get the F.O.C. as

{l} : U1 (c, l)w¡k¢= U2 (c, l)n

k0o: U1 (c, l) = βv1

³k0, k

0´3

Use the envelope condition

v1¡k, k

¢= U1 (c, l)

¡r¡k¢+ (1− δ)

¢We get the optimality condition:

{lt} : U1 (ct, lt)w¡kt¢= U2 (ct, lt)

{kt+1} : U1 (ct, lt) = βU1 (ct+1, lt+1)¡r¡kt+1

¢+ (1− δ)

¢Correspondingly, the functional F.O.C. is

{lt} : U1¡ct, gl

¡kt, kt

¢¢w¡kt¢= U2

¡ct, gl

¡kt, kt

¢¢{kt+1} : U1

¡ct, gl

¡kt, kt

¢¢= βU1

¡ct+1, gl

¡kt+1, kt+1

¢¢ ¡r¡G¡kt¢¢+ (1− δ)

¢where ct = r

¡kt¢kt + (1− δ) kt + w

¡kt¢ ¡

L− gl¡kt, kt

¢¢ − gk¡kt, kt

¢, r¡k¢=

F1¡k, L− l

¢= F1

¡k, L− gl

¡k, k

¢¢, andw

¡k¢= F2

¡k, L− l

¢= F2

¡k, L− gl

¡k, k

¢¢.

(c) Impose equilibrium conditions on the first-order conditions from part(b) and verify that the resulting equations are identical to the first-order conditions associated with the planning problem for this economy.Impose the equilibrium conditions kt = kt, lt = lt, we have

{lt} : U1¡ct, lt

¢F2¡kt, L− lt

¢= U2

¡ct, lt

¢{kt+1} : U1

¡ct, lt

¢= βU1

¡ct+1, lt+1

¢ ¡F1¡kt, L− lt

¢+ (1− δ)

¢where

ct = F1¡kt, L− lt

¢kt + (1− δ) kt + F2

¡kt, L− lt

¢ ¡L− lt

¢− kt+1

= F¡kt, L− lt

¢+ (1− δ) kt − kt+1

which are identical to the first-order conditions associated with the planning prob-lem for this economy.

3. Consider a competitive equilibrium one-sector growthmodel without valuedleisure in which consumers own capital and labor and rent their services tofirms. There is no uncertainty and the felicity function of a typical consumerhas constant elasticity of intertemporal substitution σ−1.

(a) Show that the decision rule of a typical consumer takes the form:

k0= µ

¡k¢+ λ

¡k¢k

where the functions µ and λ satisfy the following pair of functionalequations (i.e., these two equations must hold for all values of k):

µ¡k¢+

w³k0´− µ

³k0´

r³k0´− λ

³k0´ = w

¡k¢− µ

¡k¢

r¡k¢− λ

¡k¢ λ ¡k¢

4

and

1

βr³k0´ =

³r³k0´− λ

³k0´´

λ¡k¢

r¡k¢− λ

¡k¢

−σ ,where k

0= µ

¡k¢+ λ

¡k¢k. In these equations, k is the individual’s

holdings of capital, k is aggregate capital, r¡k¢is the rental rate of

capital plus one minus the depreciation rate, and w¡k¢is the wage

rate. (Hint: Obtain the Euler equation for a typical consumer, guessthat the consumer’s decision rule takes the conjectured form, and thenfind restrictions that the “coefficients” µ

¡k¢and λ

¡k¢must satisfy in

order for the Euler equation to hold for all values of k and k.)A recursive competitive equilibrium for this economy is a set of functions:

price function : r¡k¢, w¡k¢

policy function : k0= g

¡k, k

¢value function : v

¡k, k

¢transition function : k

0= G

¡k¢

such that:(1) k

0= g

¡k, k

¢and v

¡k, k

¢solves consumer’s problem:

v¡k, k

¢= max

{c,k0}c1−σ

1− σ+ βv

³k0, k

0´s.t.

c+ k0= r

¡k¢k + w

¡k¢

k0= G

¡k¢

(2) Price is competitively determined:

r¡k¢= F1

¡k, 1¢+ 1− δ

w¡k¢= F2

¡k, 1¢

(3) Consistency:G¡k¢= g(k, k)

By taking F.O.C. and using envelope condition, we can get the Euler equation as

βu0(ct+1)

u0 (ct)r¡kt+1

¢= 1

⇒ β

µct+1ct

¶−σr¡kt+1

¢= 1

⇒ ct+1 =£βr¡kt+1

¢¤ 1σ ct

⇒ r¡kt+1

¢kt+1 + w

¡kt+1

¢− kt+2

=£βr¡kt+1

¢¤ 1σ · £r ¡kt¢ kt + w

¡kt¢− kt+1

¤5

Since this is a second-order linear difference equation in kt, we conjecture thesolution form as

k0= µ

¡k¢+ λ

¡k¢k

Plug back into Euler equation, we get

LHS = r¡kt+1

¢kt+1 + w

¡kt+1

¢− kt+2

= r¡kt+1

¢ ¡µ¡kt¢+ λ

¡kt¢kt¢+ w

¡kt+1

¢− ¡µ ¡kt+1¢+ λ

¡kt+1

¢ ¡µ¡kt¢+ λ

¡kt¢kt¢¢

=£¡r¡kt+1

¢− λ¡kt+1

¢¢λ¡kt¢¤kt +£

µ¡kt¢ ¡

r¡kt+1

¢− λ¡kt+1

¢¢+ w

¡kt+1

¢− µ¡kt+1

¢¤and

RHS =£βr¡kt+1

¢¤ 1σ£r¡kt¢kt + w

¡kt¢− ¡µ ¡kt¢+ λ

¡kt¢kt¢¤

=£βr¡kt+1

¢¤ 1σ£¡r¡kt¢− λ

¡kt¢¢

kt +¡w¡kt¢− µ

¡kt¢¢¤

To make LHS and RHS equate for every kt and kt+1, we must have

(1) :¡r¡kt+1

¢− λ¡kt+1

¢¢λ¡kt¢=£βr¡kt+1

¢¤ 1σ¡r¡kt¢− λ

¡kt¢¢

(2) : µ¡kt¢ ¡

r¡kt+1

¢− λ¡kt+1

¢¢+ w

¡kt+1

¢− µ¡kt+1

¢=

£βr¡kt+1

¢¤ 1σ¡w¡kt¢− µ

¡kt¢¢

i.e.

(1) ⇒ 1

βr³k0´ =

³r³k0´− λ

³k0´´

λ¡k¢

r¡k¢− λ

¡k¢

−σ

(2) ⇒ µ¡k¢+

w³k0´− µ

³k0´

r³k0´− λ

³k0´ = hβr ³k0´i 1σ w

¡k¢− µ

¡k¢

r³k0´− λ

³k0´

⇒ µ¡k¢+

w³k0´− µ

³k0´

r³k0´− λ

³k0´ =

³r³k0´− λ

³k0´´

λ¡k¢

r¡k¢− λ

¡k¢

w¡k¢− µ

¡k¢

r³k0´− λ

³k0´

⇒ µ¡k¢+

w³k0´− µ

³k0´

r³k0´− λ

³k0´ = w

¡k¢− µ

¡k¢

r¡k¢− λ

¡k¢ λ ¡k¢

This finishes the proof.

(b) Suppose now that there are two (types of) consumers in the economywho differ only in their initial capital holdings. Each consumer repre-sents half of the economy’s population. Use the result from part (a)to argue that a redistribution of capital (holding aggregate capital con-stant) across the two consumers at time 0 has no effect on equilibriuminterest rates and wages.

6

To prove the result, we need to see what is the difference between a representativeagent economy and heterogeneous agent economy. So we first clearly define arecursive competitive equilibrium for heterogeneous agent economy.A recursive competitive equilibrium for the economy with two types of agents isa set of functions:

price function : r¡k¢, w¡k¢

policy function : k0i = gi

¡ki, k1, k2

¢value function : vi

¡ki, k1, k2

¢transition function : k

0

i = Gi

¡k1, k2

¢where k = 1

2(k1 + k2), such that:

(1) Given G1

¡k1, k2

¢and G2

¡k1, k2

¢, k

0i = gi

¡ki, k1, k2

¢and vi

¡ki, k1, k2

¢solve

consumer’s problem:

vi¡ki, k1, k2

¢= max

{ci,k0i}ui (ci) + βvi

³k0i, k

0

1, k0

2

´s.t.

ci + k0i = r

¡k¢ki + w

¡k¢

k0

1 = G1

¡k1, k2

¢k0

2 = G2

¡k1, k2

¢(2) Price is competitively determined:

r¡k¢= F1

¡k, 1¢+ 1− δ

w¡k¢= F2

¡k, 1¢

(3) Consistency:Gi

¡k1, k2

¢= gi(ki, k1, k2)

Notice the difference between the definition of recursive competitive equilibriumwe met before: in heterogeneous agent economy, normally the whole distributionof state variable counts.Now we start to prove the result. Due to the special preference relation here,first we assume that only average capital level matters for the evolution of theeconomy, i.e. k

0= G

¡k¢. Then the result from part (a) tells us that in recursive

competitive equilibrium we will have

k0i = µ

¡k¢+ λ

¡k¢ki

Therefore, we have

k0=1

2

³k0

1 + k0

2

´= µ

¡k¢+ λ

¡k¢k

7

Therefore, indeed the evolution of average capital stock does not depend on thedistribution of capital. In other words, this verifies our guess that k

0= G

¡k¢.

Since equilibrium interest rates and wages depend only on the aggregate averagecapital stock (remember that r

¡k¢= F1

¡k, 1¢+ 1 − δ and w

¡k¢= F2

¡k, 1¢),

a redistribution of capital (holding aggregate capital constant) across the twoconsumers at time 0 has no effect on equilibrium interest rates and wages.

8