3. Consumption Problem. Suppose period utility is of the constant relative risk

aversion type:

u(c) --0-1,0>0.

Consumer's objective is to maximise expected lifetime utility:

=

U = E[Σ_Bu(c)], βε(0,1),

Eolt

t-0

with 3 = p > 0. Assume the real interest rate, r, is constant but not necessary

equal to the time preference, p. Eo is the expectation conditional on information at

time 0. The budget constraint is:

c+at+1 = yr + (1+r)at,

where do is given, yt is a stochastic random variable and at time t the consumer

knows the realisation of y, but does not know its future values.

a. What is the Euler equation relating the marginal utility of ce to expectations

concerning the marginal utility of C++1? Explain.

b. Suppose that the (natural) logarithm of income is normally distributed with

zero mean, and that as a consequence the logarithm of ce+1 is also normally

distributed, with mean E₂(lnc+1). Let o² denote the variance of the log of con-

sumption conditional on information available at time t. Rewrite the expression

in part (a) in terms of In(c₂), E₂(ln(+1), ² and the parameters r, p, and 0.

(Hint: If a variable z is distributed normally with mean and variance V, then

E(e²) = c+V).

c. Show that if rand o are constant over time, the result in part (b) implies that the

log of consumption follows a random walk with drift: In(+1) = a+ln(c₂)+u+1,

where u is a white noise.

d. How do changes in each of r and o² affect expected consumption growth,

E₂(ln(+1)-In())? Interpret the effect of o² on expected consumption growth

in light of the discussion on precautionary savings in Romer.

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