SDP Macroeconomics Midterm exam, 2017 Professor Ricardo Reis PART I: Answer each question in three or four sentences and perhaps one equation or graph. Remember that the explanation determines the grade. 1. Question 1 (10 points). Some authors have argued that during the process of development, as economies grow and capital becomes more abundant, then the politicians become more tempted to protect some industries at the expense of others, the financial sector more willing to support well-connected firms, and that capitalists become stronger to lobby for proteccionist measures that let them raise markup. The allocation of capital across sectors and firms therefore becomes worse as capital rises. How would adding this mechanism to the Solow model affect the uniqueness of a steady state? How would it affect the speed of convergence to that steady state? 2. Question 2 (10 points). A closed economy has one representative agent with constant relative risk aversion utility, income risk, but no net savings in equilibrium as the capital stock is fixed. There are three assets (claims on parts of the capital stock) in the economy. Asset A has the same return next period regardless of the state of the world. The return on asset B has positive variance and is uncorrelated with aggregate income. The return on asset C has positive variance and is negatively correlated with income. Do you, this period, expect the return on asset A to be higher / lower/ the same as that of asset B? Do you expect the return of asset B to be higher / lower/ the same as that of asset C? Explain. PART II: Longer question. Question 3 (40 points). expected discounted utility: A consumer starts with initial wealth w 0 and wishes to maximize his [ ( )] E β t c 1 γ t 1 γ t=0 where β (0, 1) and γ > 0 are preference parameters. His wealth evolves according to the budget constraint: w t+1 = R(w t c t ) + y t+1 1
where R is a fixed gross interest rate such that R 1 (Rβ) 1/γ < 1. He is subject to a natural debt limit. If he is able, the consumer faces no risk, since his income is equal to y t = 1. However, with probability p an able individual can become disabled. If, so then y t = 0 forever. That is, the effective income process is: y t = 1 if able, y t = 0 if disabled; if able there is a probability p of becoming disabled, 1 p of staying able; if disabled, the agents stays that way forever. a) What is the natural debt limit? Will this ever bind? Do you expect there to be precautionary savings in this economy? b) Write down the Bellman equation for a disabled agent. c) Derive the full set of sufficient and necessary optimality conditions for this problem. d) Guess that the consumption of a disabled agent c d t is a linear function of his cash on hand c d t = kw t. Verify that this guess satisfies all the optimality conditions. e) Does k rise or fall with β? Explain the intuition. f) Write down the Bellman equation for an able agent and derive the Euler equation. Explain the intuition behind the Euler equation. g) Write out the expectation in terms of the two states of the world and their probabilities, and solve for the consumption growth of an able agent c a t+1 /ca t in terms of the consumption wealth ratio c a t+1 /w t+1. h) Is the consumption growth higher than what it would be without uncertainty βr? (Hint: c a t+1 /cd t+1 > 1) Does the risk of disability cause precautionary savings? 2
Macro, Week II, Midter, Gerzensee Fernando Alvarez Instructions. Your answers should be very short. Often one line, and at most four lines per item.
1 Mc Call Search Model Consider the Mc Call Seach model with two values for the wage takes two values, w 1 < w 2 with probabilities π 1,π 2 so that π 1 +π 2 = 1. Assume that w 2 b. The discount factor is β and the benefit per period if the worker is unemployed is b. Let v 1,v 2 denotes the value function of an unemployed worker with wage offer i = 1,2 at hand. 1. Write an expression for the value functions v i for i = 1,2. In your expressions write explicitly the expectations using π 1,π 2 and v 1,v 2. Your answer should consist of two equations. It should also explicitly depends on β,w 1,w 2 and should use max{, }. 2. Assume that the optimal policy is to accept the offer w 2 and reject the offer w 1. Write explicitly the solution of the value functions v 1,v 2 (so v 1 should be only in the left hand side of one equation, and v 2 should be only on the left hand side of the other equations. No max should be involved. 3. Use the answer of the previous question to find an expression for the highest value of w 1 for which the policy assumed in the previous question is optimal. Your answer should be a weighed average between b and w 2 with weights depending on β and π 1. 2 Stigler s Search Model Consider the Stigler search model with cost c, discount factor β, benefit b = 0 and where wages are uniformly distributed between 0 and B > 0. In this case the CDF of wages is F(w) = w/b for w [0,B]. 1. Write the objective function of the Stigler s search model. Use w for wages and n for the number of offers. Your expression should be a function of B, β and c so do NOT use a general expression for the CDF, instead use the assumed functional form for the uniform distribution, but don t explicitly solve for the expectation. Hint: use the expression in the notes for the expected value the maximum of n identical, independent random variables. 2. Solve explicitly for the integral in the objective function. Your final expression should be a function of n,b,c and β. 3. Assume that n can take any real value, take the first order condition for the objective function with respect to n and solve for the optimal number of offers as a function of B,c and β. 3 Linear Taxes on the steady state of the Neoclassical Growth Model Consider a version of the neoclassical growth model where the period utility function over consumption c and leisure l is v(c,l) = log(c)+bl for a constant B > 0, where leisure and labor add up to one, i.e. l + n = 1, where the write discount rate β = 1/(1 + ρ) with ρ > 0, where 1
δ is the depreciation rate of capital, and where the production function is given by Ak α n 1 α. Use w,v for the before tax wage and before tax rental rate of capital. Use τ l and τ k for the tax rate on labor income and tax rate on (net of depreciation) capital income. Let g be the steady state government purchases, and assume that there is a lump sum tax τ which for any given {τ l,τ k,g} balanced the government budget constraint. Use x for investment. 1. Write a set of 7 equations in the 7 unknowns, the steady state values {c,x,k,n,v,w,n} whose solution give a steady state of the model, given {τ l,τ k,g}. Use the functional forms given above: 2. Let κ = k/n. Give an explicit function of n as a function of the endogenous variables κ,w and the parameters α,b,δ, and the policy parameters g,τ l. 3. Using the previous answer, what is the value of dc/dg? What is the reason? 4. Using the previous answer, what is the value of [(1 τ l )/n][dn/d(1 τ l )], if g = 0. How does this elasticity [dn/d(1 τ l )] changes if g > 0? 5. Find an expression for n in the special case where g = 0 and δ = 0. What is the effect of taxes τ k in this case? 2