Question

2. TIME PREFERENCE

In class, we solved a two-period savings model where a consumer allocates income across two

periods. We assumed the consumer’s intertemporal utility function was given by: U(c1,c2) =

log(c1) + δlog(c2) and that their intertemporal budget constraint was M1 + M2 = c1 + c2 . 1+r 1+r

Along the way to solving that problem, we found that consumers should select their consumption in each period so that:

u′(c1) = δ(1 + r)u′(c2),

where δ is the exponential discount rate and r is the interest rate.
In this problem, we will extend this problem from two to three periods. We will solve it with

exponential discounting and quasi-hyperbolic discounting.

1. 2.1. Assume the consumer’s “flow” utility is given by log(ct) in each period and that the consumer has an exponential discount rate δ. What is the intertemporal utility function with three periods (instead of two)? (5 points)

2. 2.2. Assume the consumer receives income M1 in period 1, M2 in period 2, and M3 in period 3 and that the interest rate is still r. What is the intertemporal budget constraint with three periods (instead of two)? (5 points)

3. 2.3. We normally solve utility maximization problems where the consumer only chooses two things. We use two equations to solve for these two unknowns: (1) the marginal rate of substitution equals the price ratio and (2) the budget constraint. When solving for more than two things, we can replace the MRS=price ratio equation with the requirement that the marginal utility per dollar for each good must be the same. If it is helpful, you can write for each good, g, MUg/pg = Λ where Λ is sometimes called the “Lagrange Multiplier.” With only two goods, this gives you exactly the same information as MRS=price ratio, but with three goods it gives you more equations.

   Use this to extend the condition that u′(c1) = δ(1 + r)u′(c2) to include utility in the third period. (5 points)

4. 2.4. Solve for consumption in each period assuming u(ct) = log(ct) in every period, δ = 0.95, r=0.05,andM1 =M2 =M3 =100. (15points)

5. 2.5. Now, assume the consumer has quasi-hyperbolic time preferences with additional parameter β = 0.9. Solve for consumption in each period assuming u(ct) = log(ct) in every period, δ=0.95,r=0.05,andM1 =M2 =M3 =100. (20points)

6. 2.6. Finally, assume the consumer still has quasi-hyperbolic time preferences with additional pa- rameter β = 0.9, but it is now the second period. Solve for consumption in each period assuming u(ct) = log(ct) in every period, δ = 0.95, r = 0.05, and M2 = M3 = 100. Ex- plain why the difference between this answer and the previous question is an example of a commitment problem. (20 points)

Answer 1