Consider the basic asset pricing model of Chapter 8. In this basic (BASIC) model, the household maximizes lifetime discounted utility:

Sum (infinity , s=0) Beta^s u(c_t+s),

subject to each period's nominal budget constraint of:

P_tc_t + S_ta_t = S_ta_t-1 + D_ta_t-1 + Y_t,

where Y_t is the current given labor income. Assume the utility function is u(c_t) = c_t^{1 - gamma}/ 1 - gamma.

(a) Form a Lagrangian for the BASIC model and nd the FOCs with respect to c_t , a_t , and lambda_t .

(b) Based on your BASIC FOCs, find an optimality equation that relates the lost utility from purchasing one more asset share to the discounted future benet of the purchase. Explain the economic reasoning behind this equation.

(c) Construct an expression for the real stock price, s_t = S_t/P_t , as a function of future real values (Finance view).

(d) Assuming the utility function is linear (Gamma = 0), find a numerical value for the steady-state real stock price s-bar given the steady-state values d-bar = 1, a-bar = 1, and y-bar = 0 and parameter value Beta = .95.

(e) Suppose now that dividends future dividends are uncertain but follow a Markov chain: d ⁽¹⁾ = :50 or d ⁽²⁾ = 1:50 with probability transition matrix:

II = [ .80 .20

.40 .60 ].

Based on the Markov chain, construct two expressions for the real stock price in each state, s⁽¹⁾ and s⁽²⁾, that relate to the expected future marginal benet of the stock purchase. Finally, show how to write these two equations in linear algebraform.