Recall the basic model of consumption choice. There are two periods: present and future. Assume households have the following lifetime utility: u(c)+\beta u\left(c^{f}\right) where u(c) = Vc, the discount rate

3 <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. The interest rate, r, is equal to 4%. Assume that the discount rate 3 is such that: \beta(1+\tau)=\sqrt{1.02} \approx 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^{\prime}(c)=\beta(1+\tau) u^{\prime}\left(c^{f}\right) and substitute u'(·) with its actual value in this exercise. (4) Incomes today and tomorrow are such that y = yf = 50, 000. Using the Euler equation and the inter temporal budget constraint, solve for consumption today, c, and tomorrow, c (5) Assume now that today's income increases by 10%. Compute the new optimal consumption choices for today and tomorrow. (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 yf increased by 10%?