Problem Set 2 Macroeconomic Theory, Spring 2015 April 2, 2015 Do the following problems, some of which are in the textbook. The number in the parenthesis corresponds to the problem number in the textbook. Students are encouraged to work together, but need to submit their own homework. The problem set is due on April 9, 2015 in the class time. 1 Intertemporal Choice of Consumers 1. Consider a two-period model. Assume the following: • The utility of households is time separable: U (c1 , c2 ) = u(c1 ) + βu(c2 ), where β is a discount factor • Households have initial wealth, w, in period 1 and labor income, y1 and y2 , in both periods. • The interest rate between period 1 and period 2 is given by R. (a) Write down the dynamic budget constraint of households. (b) Define h ≡ y1 + R1 y2 . Economists sometimes call h as human wealth. Rewrite the budget constraint using h and discuss the reason why it’s called human wealth. (c) By solving the maximization problem, derive the Euler equation in this economy. The answer involves marginal utility, u0 (·). (d) Suppose that households deviate from optimal consumption by reducing consumption by the amount of ε in period 1, and save it for the consumption in period 2. Show that household cannot increase the utility by this action. (Hint: Apply the first-order Taylor approximation to the utility function, around the optimal consumption.) (e) Suppose that utility in each period is given by the CRRA utility function: u(c) = c1−θ . Compute the intertemporal elasticity of substitution, which is a measure of 1−θ responsiveness of the growth rate of consumption to the real interest rate. (Hint: more specifically, it is given by d lnd R (ln cc21 ).) 1 (f) Recall that the coefficient of relative risk aversion in CRRA utility function is given by θ. Can the intertemporal elasticity of substitution be independently identified from the coefficient of relative risk aversion? Explain. 2. Consider an individual who lives for two periods. The individual has no wealth and earns labor income of amount y1 and y2 in the two periods. y1 is known, but y2 is random. Assume that E[y2 ] = y1 . The government taxes income at rate t1 in period 1 and t2 in period 2. The individual can borrow and lend at an interest rate of zero. Therefore, the second-period consumption is c2 = (1 − t1 )y1 − c1 + (1 − t2 )y2 . The individual chooses c1 to maximize expected lifetime utility, U = u(c1 ) + E[u(c2 )]. (a) Find the first-order condition for c1 . (b) Show that E[c2 ] = c1 , if y2 is not random. c − c)2 where c¯ is a bliss (c) Suppose that utility is in the quadratic form: u(c) = − 21 (¯ point, which a consumer always favors. Show that we also have E[c2 ] = c1 . (d) Show that if u000 (c) > 0 and y2 is random, then we have E[c2 ] > c1 . (Hint: show that marginal utility is a convex function and use the Jensen’s inequality to show that E[u0 (c)] > u(E[c]).) (e) Suppose that the government marginally lowers t1 and raises t2 by the same amount, so that its expected total revenue, t1 y1 + t2 E[y2 ] is unchanged. Implicitly differentiate the first-order condition in part (a) to find an expression for how c1 responds to this change. (f) Show that c1 is unaffected by this change if y2 is not random or if utility is quadratic. (g) Some economists argue that time path of taxation doesn’t affect the consumption, if taxation doesn’t change the life time income, which is called Ricardian equivalence. Discuss the relevance of Ricardian equivalence in the real world using the implication of the results you have derived. 2 Ramsey-Cass-Koopmans Model 1. (Problem2.7) Describe how each of the following affects the c˙ = 0 and k˙ = 0 curves, and thus how they affect the balanced-growth-path values of c and k: (a) A rise in θ. (b) A downward shift of the production function. (c) A change in the rate of depreciation from the value of zero assumed in the text to some positive level. 2