1. Consider a two period model. An agent gains utility from consumption today (c1) and in the future (c2) according to U(c1, c2), that satisfied the usual assumptions. The budget constraints are c1 = w1 + s and c2 = w2 + (1+r)s, where wt is labor income in period t, s is savings, and r is the interest rate.
A. Set up the decision problem and find the first order conditions.
B. Suppose U = log(c1) + blog(c2), for 0 < b < 1. Find the consumption function for c1 and the savings function.
2. Suppose the person lives for two periods, U = u(c1) + bu(c2), and can acquire an asset at price q, with c1 = w1 - qa and c2 = (d + q*)a + w2, where d = dividend and q* = selling price.
A. Find the first order conditions.
B. Find the asset demand function for a when utility takes the log form, as in 1B above.