Question

1. Consider the following two period consumption savings problem. A consumer cares about consumption (c and future consumption c according to Assume that U(c) is given by for some constant y. In the present the consumer chooses how much to consume and how much to save out of her income y>0 This decision is made in the knowledge that in the future she will be retired, have no income, and thus future consumption will be entirely out of savings: c)a, where r is the real interest rate. The consumer chooses c, c/, and a to maximize her well being, which is given by (1) (a) What is the consumers's present value budget constraint? (Hint: substitute for a in (3) using (4)) (b) The consumer's marginal utility from current consumption is given by MUc c and marginal utility of future consumption is given by MUd (cB(c). What condition is required for the consumer to be behaving optimally when making her consumption and savings decisions (Hint Remember the two good problem.) (c) Derive an expression for the agent's desired consumption that depends on β, γ, r and y (d) Show that when γ-1, we recover the expression derived in class when T-yf Tf a-0 (e) If y < 1 will the consumer's consumption be increasing or decreasing in r? Is the substitution or income effect dominating? (f) If γ > 1 will the consumer's consumption be increasing or decreasing in r? Is the substitution or income effect dominating?

Answer 1

Given that rightarrow A consumer cares about consumption and future consumption (c^f) according to u(c)_+ f u (c^t). the budget constraint is given by c + a^f - y now, c^f = (1 + r) a^f a^f = c^f/1 + r therefore [[c + c^f/1 + r] = y] downarrow pv of budget constraint. At the optimal choice mu u c/mu u c^f = 1/1/1 + r = 1 + r or c^-r/beta(c^f)-r = 1 + r or c^-r/(c^f)^-r = beta (1 + r) \r\n or c/c^f = [beta (1 + r)]^-1/r c = [beta(1 + r)]^-1/r c^f from the budget constraint. C + c^f/1 + r = 4 or [beta(1 + r)]^yr c^f + c^f/1 + r = 4 or [beta(1 + r)]^-l/r c^f + c^f/1 + r = 4 or [1/beta(1 + r)]^1/r + 1/1 + r] e^f = 4 therefore c^f = 4 [beta^1/r (1 + r)^1/r + 1/(1 + r) + {beta(1 + r)}^1/r