Question 1

Consider the following utility maximization problem:

{ct ,kt+1>0}0 s.t.

E0  ,  pt (ln ct + y ln kt+1)┐

kt+1 + ct  = A t(1) -a kt(a)

where a e (0, 1), p e (0, 1), y | 0, ct  denotes consumption in period t , kt+1  is the amount of capital stock held at the   end of period t (and thus at the beginning of period t + 1), and At  is the productivity of capital stock in period t .

Assume that

ln At+1 = p ln At + et+1

for all t, where p e (0, 1) and et+1  is an independent white noise.

You can guess and verify that the value function in the       Bellman equation for this problem takes the following form:

V (At , kt) = F + G ln At + H ln kt

where F , G , and H are constants.

Suppose that a = 0.6, p = 0.9, y = 0.2, and p = 0.5. Given these parameter values, derive the values of F , G , and H with 2 decimal places (i.e., if the value of F is 1.6875, only answer 1.68). (Note: you do not need to de-trend the model, because there is no trend in At .)

Question 2                             

Consider the following utility maximization problem for a household:

{ct ,at+1 ,bt1a,dt(x)+1>0}0    pt  /  + y \

s.t. at+1 + bt+1 + dt+1 + ct  = (1 + i)at + (1 + ib)bt + (1 + id)dt

where f | 0, p e (0, 1), y | 0, ct  denotes consumption in    period t, and at+1 , bt+1 , dt+1  is the amounts of capital stock, government bonds, and bank deposits held at the end of         period t (and thus at the beginning of period t + 1). i , ib, and id  denote the constant gross rates of return on at , bt, and dt , respectively, at period t . Thus, the household faces no            uncertainty.

In the utility maximization problem, qt+1  measures the         “liquidity aggregate” that consists of government bonds and bank deposits. The value of this variable represents the         amount of convenience provided by the two types of financial assets, which are easily convertible into cash when the           household needs to make payments. The utility of this          convenience is assumed to be determined by the following     function form:

qt  三 [(1 - 入)bt(p) + 入dt(p)]

where 入 e (0, 1) and p ) 1.