Need to solve the exercise 17.5 from the Recursive macroeconomic theory by Lars Ljungqvist.

Exercise 17.5TradejQuery2240583310651680935_1616879480665

A consumer orders consumption streams according to

1

(1)E₀

^t ct,(0, 1)

t=01

where > 1 and E₀ is the mathematical expectation conditional on time 0 information. The consumer can borrow or lend a one-period risk-free security that bears a xed rate of return of R = ¹ . The consumer's budget constraint at time t is

(2)c_t + R¹b_t+1 = y_t + b_t

where b_t is the level of the asset that the consumer brings into period t . The household is subject to a"natural" borrowing limit. Thehousehold's initial asset level is b₀ = 0 and his endowment sequence y_t follows the process

(3)y_t+1 = y_t exp(_t+1 + )

where _t+1 is an i.i.d. Gaussian process with mean zero and variance 1 , =

.5² , and > 0 . The consumer chooses a process {c_t, b_t+1}

to maximize

(1) subject to (2), (3), and the natural borrowing limit.

t=0

a.Give a closed-form expression for the consumer's optimal consumption and asset accumulation plan.

Hint 1: If log x is N (, ²), then Ex = exp( + ²/2).

Hint 2: You could start by trying to verify the following guess: the optimal policy has b_t+1 = 0 for all t 0.

b.Discuss the solution that you obtained in part a in terms of Friedman's permanent income hypothesis.

c.Does the household engage in precautionary savings?