Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Intermediate Microeconomics UA10

Homework 7 Fall 2025

Upload to Brightspace by midnight (11:59 pm) Thursday October 30

1. Consider a two period choice model with perfect capital markets withd per period interest rate r > 0 and income m1 > 0 available in period 1 for consumption or saving and with m2 = 0. The utility function is,

(a) Suppose that . In this case show that optimal consumption path involves a decreasing path of consumption over time, .

(b) Now identify conditions on the discount rate and interest rate that are equivalent to an increasing path of consumption over time, .

(c) Specialize now to r = 0.2 with m1 = 90 and m2 =0. In this case write the formula for the budget line that identifies the maximum affordable level of c2 written as a function of c1.

(d) Using the parameter values and assumptions stated in part (c), solve for the optimal consumption in each period, (β) and (β), as functions of the discount factor β ∈ (0,1).

2. Suppose there are two periods, no discounting of future utility, no interest on savings, and overall utility function,

where v is strictly increasing, strictly concave, and differentiable. In period 1, the consumer is given a wealth of W, from which they can spend on period 1 consumption, or save for period 2 consumption: there is no other source of money for spending in period 2 than wealth retained from period 1. In the second period, the consumer faces a lossd L<W with probability pL ∈ (0,1)

(a) In class we explained the optimality condition determining as

Explain this condition in a few words.

(b) Setting v(x) = In x, draw a graph of v'(x) with on the horizontal axis. It is not critical to make this accurate but you must ensure it has the right shape properties.

(c) Set pL = 0.5, W = 4, and L = 2. Use the figure above to illustrate why equalizing expected consumption across periods does not satisfy the optimality condition in (a).

(d) Illustrate qualitatively in the figure why the optimal solution is to set less than the expected consumption in period 2. You do not need to solve the model quantitively.