Consider a standard Solow growth model in continuous time with no population or tech-

nological growth (i.e. the one from the lecture with n = g = 0). As seen in the lecture,

the equilibrium of that economy is characterised by the following law of motion of capital:

_K

(t) = sF(K(t); L;A) ? K(t), 8t 0;

and an initial condition K(0) = K0, where s 2 (0; 1) is an exogenous saving rate and

2 (0; 1) is the rate of capital depreciation. The production function of output Y (t) =

F(K(t); L(t);A) considered here is a general constant elasticity of substitution (CES)

function of the form

F(K; L;A) =

K

?1

+ (1 ? ) (AL)

?1

?1

where 2 (0; 1) [ (1;1) is the elasticity of substitution between capital K and labour

L (A is technology), and 2 (0; 1) denotes the relative intensity in capital. The CES

production function is very useful since it nests interesting cases. For ! 1 it be-

comes linear, F(K; L) = K + (1 ? )AL, for ! 1 is becomes a Cobb-Douglas

of the form F(K; L) = K(AL)1?, and for ! 0 it becomes a Leontie function

F(K; L) = minfK; (1?)ALg. In general, measures the degree of substitutability of

production factors, so that when < 1 production factors are said to be gross comple-

ments while for > 1 they are considered gross substitutes.

a) Show that F is strictly concave in K. Show that F has constant returns to scale.

What is the economic meaning of strict concavity? What is the economic meaning of

constant returns to scale?

b) Does F full any of the two Inada conditions with respect to K, and if yes under

what conditions? Distinguish between the cases 2 (0; 1) and 2 (1;1). Provide an

economic interpretation.

c) What is the rental rate of capital R(t)? What is therefore the capital share xK(t)

R(t)K(t)=F(K(t); L(t);A)?

d) We can safely x L(t) = L (because of labour market clearing). Does the capital

share grow or fall in K(t)? How does that depend on , and which value of ensures that

the capital share is always constant? Provide an intuition for these results by appealing

to the meaning of .

e) Find the steady state level of capital K? > 0, assuming that it exists.

f) Show that the growth rate of capital is

gK(t)

_K

(t)

K(t)

= s

+ (1 ? )

AL

K(t)

?1

!

?1

? :

g) Let us assume that

?

s

?1

> . For the case where > 1, show that the system con-

verges to the steady state K? by showing that the growth rate of capital gK(t) is strictly

positive if and only if K(t) < K?, and strictly negative if and only if K(t) > K?. Repeat

the same for the case where 2 (0; 1).

h) Now, assume that

?

s

?1

and suppose that > 1. Show that the growth rate of

capital gK(t) is always positive. What is the level of capital in the long run as t ! 1?

Does a steady state exist? Provide an intuition for that result.

i) Assume again that

?

s

?1

, but now suppose that 2 (0; 1). Show that the

growth rate of capital gK(t) is always negative. What is the level of capital in the long

run as t ! 1? Does a steady state exist? Provide an intuition for that result.

Exercise 2: Neoclassical model in discrete time

Consider a standard neoclassical growth model in discrete time, with no population or

technological growth. The household maximises present-value discounted utility where

time t 2 f0; 1; :::;1g is discounted at the rate 2 (0; 1), and the period utility is u(C(t))

with standard properties. The household owns (a xed amount of) labour L and claims

on the capital stock B(t). It rents these out to a representative rm operating a produc-

tion function F(K(t); L(t)) with standard properties. All markets are competitive, and

market clearing implies that wages w(t) and rental rates of capital R(t) are such that

L = L(t) and K(t) = B(t). The law of motion of capital is K(t+1) = (1?)K(t)+I(t),

the household's budget constraint is B(t + 1) = [1 + r(t)]B(t) + w(t)L ? C(t), and the

aggregate resource constraint is C(t)+I(t) = F(K(t); L(t)). Net and gross capital rental

rates are related according to r(t) = R(t) ? , where 2 [0; 1]. The household must

also respect a no-Ponzi condition. The market equilibrium, as seen in the lecture, can be

summarised by two dierence equations:

u0(C(t))

u0(C(t + 1))

= [1 + F0

K(K(t + 1); L) ? ]; 8t 0;

K(t + 1) = (1 ? )K(t) + F(K(t); L); 8t 0;

an initial condition K0 as well as a terminal (transversality) condition:

lim

t!1

"

K(t)

Yt?1

s=1

1

1 + F0K

(K(s); L) ?

#

= 0:

a) We suppose that the Social Planner maximises the household's welfare subject to fea-

sibility constraints. Write down the Social Planner's problem in terms of C and K only.

b) Set up the Planner's Lagrangian and characterise the equilibrium. Comparing it to

the results in the lecture, does it coincide with the market outcome in all time periods

t 2 f0; 1; :::;1g?

c) Suppose you are only interested in the equilibrium allocation of variables C and K

at period 0, i.e. C0 and K0. Can you write the associated characterising equation without

reference to any future variables? And if you were interested in only in the the combi-

nation of period 0 and 1 variables, could you then write a system of equations without

referencing future variables? Explain your answer.

d) Set up the Planner's Bellman equation, i.e. express the Planner's problem in re-

cursive form.

e) From now, suppose that = 1 (full depreciation). Suppose that the production func-

tion is Cobb-Douglas, F(K; L) = KL1?. Suppose that the period utility function is

u(C) = log(C). That very special case allows for an analytic solution of the value func-

tion. To simplify things (this is not necessary), also suppose that L = 1. Guess and verify

that the value function then takes the form V (K) =

0 +

1 log(K). Now show that the

exact expression for

1 is

1 =

1?.

f) Using these results, nd explicit expressions for the policy functions C(K) and K0(K)

(in terms of and only). Do you agree that in this special case the model delivers

an identical solution to that of a standard discrete-time Solow model with a particular

savings rate and = 1?