Question

Recall the basic model of consumption choice. There are two periods: present and future. Assume households have the following lifetime utility: u(c) + βu(c f ) where u(c) = √ c, the discount rate β < 1 (i.e., the rate at which households discount utility from future consumption). Assume also that households start with no initial wealth, a = 0, but receive income today, y, and income tomorrow, y f . The interest rate, r, is equal to 4%. Assume that the discount rate β is such that: β(1 + r) = √ 1.02 ≈ 1.01

(1) Write down the consumers’ intertemporal budget constraint and indicate which terms stand for the present value of lifetime income and lifetime consumption. State the full consumers’ maximization problem. [Do not solve yet]

(2) Substitute the budget constraint in the utility function to turn the consumers’ problem into an unconstrained maximization problem.

(3) Derive the first-order condition that characterizes the optimal consumption choice. Show that the condition you obtain is the standard consumer Euler equation: u 0 (c) = β(1 + r)u 0 (c f ) and substitute u 0 (·) with its actual value in this exercise.

(4) Incomes today and tomorrow are such that y = y f = 50, 000. Using the Euler equation and the intertemporal budget constraint, solve for consumption today, c, and tomorrow, c f

(5) Assume now that today’s income increases by 10%. Compute the new optimal consumption choices for today and tomorrow. 2

(6) Using your answers to parts (4) and (5), compute the percent increase in c after the income shock. Do the same for tomorrow’s consumption. How do these rates of increase compare to the income shock? Explain how this captures the idea of consumption smoothing [i.e., the idea that people tend to prefer a stable consumption path, so temporary shocks to income are spread over time]

(7) How would your answers to parts (4) and (5) change if the increase in income was in fact permanent, i.e. if y and y f increased by 10%?

Answer 1