Reference: Sargent, Recursive macroeconomic theorem

(b) Use the "guess-and-verify" method to find the value function and the corresponding decision rules. (Hint: The value function takes the form v(a) = A + BIn(a + D), where A, B, D are coefficients to be solved for.) Characterize the dynamics of the consumer's asset holdings. Do they converge or diverge? If they converge, to what value do they converge? (d) Consider two consumers with the same initial asset holdings, but with different preferences. In particular, suppose that one of the consumers is more patient than the other one (i.e. the two con- sumers have different values of B). How does the relative wealth of the two consumers evolve over time?Question 2. Consider a consumer with the following optimization [35] problem: s.t.: c::+at+1=Rat-lw, t=1,2,..., where 6 (0,1). ct is the consumer's level of consumption in period t, at is the consumer's asset holdings at the beginning of period t, R is the gross interest rate. and w is the consumer's labour earnings in each penod. (a) Formulate the consumer's problem as a recursive