Question

Only need the answer to problem 3. Please show all steps.

In(a) + ? ln(a) where ?, satisfying 0 < ? < 1, determines how patient(? larger) or impatient (? smaller) the consumer is. (a) Set up the Lagraungian for this cousumers problen: choose co and c to imaxinize tihe utility function subject to the lifetime budget constraint. [10 pointsl (b) Usc the first-order conditions to obtain an expression that shows how the consiumers optimal choice of cunsumption growth depends un the discount factor ? and the interest rate r. [15 points (c) Rationalize the answer you found in part (b). In other words, interpret the rela- tionships between the variables on the right-hand side and the left-hand side. 10 points] 3. Stocks, Bonds, and Contingent Claims Consider an economic environment with risk, similar to one we studied in class, in which there are two periods, t 0 and t-1, and two possible states at t 1: a good state that occurs with probability 1/2 and a bad state that occurs with probability 1 -1/2. Suppose that investors trade two assets in this economy. The first is a stock, which sells for q 1.1 at t 0 and pays a large dividend of dG = 2 in the good state at t = 1 and a small dividend of dB-1 in the bad state at t- 1. The second is a bond that sells for 0.9 at t 0 and makes a payoff of one with certainty in both states, at t-1. (a) Just as we did in class, assume that investors can take long and short positions in both assets. Given the above payoffs for the stock and bond, find the combination of stock and bond holdings s and b that replicate the payoff provided by a contingent claim for the good state: one in good state and zero in the bad. [10 points] provided by the contingent claim for the bad state.[10 points the prices at which cach contingent claim should sell at t 0.110 points! (b) Then, find the combination of stock and bond holdings that replicate the payoff (c) Finally, given the answers you just derived and the prices of the stock and bond, find

Answer 1