SDP Macroeconomics Final exam, 2014 Professor Ricardo Reis

Transcription

1 SDP Macroeconomics Final exam, 2014 Professor Ricardo Reis 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). Consider a neoclassical growth model, where the felicity function of the representative agents is u(c t ) = ln(c t s) where s > 0 is a subsistence level of consumption that consumption must always be above. Is the model with these preferences consistent with the facts that US output has grown at a roughly constant rate g > 0 and the capital-output ratio and consumption-output ratios have been roughly constant? 2. Question 2 (10 points). During recessions, aggregate income goes down, but the individual income of each household moves in different ways, depending at least partly on luck: some barely see a change, while those that become unemployed have a dramatic fall in income. If there is full insurance leading to efficient risk-sharing within the households in the United States (and assuming identical preferences), then which of the following is true: (i) there is no welfare costs of business cycles, (ii) the welfare cost is positive but smaller than with no insurance, (iii) the welfare costs of business cycles is the same with or without insurance. 1

2 Sergio Rebelo 20 points Consider the following version of the neoclassical model in which preferences are not time separable: U = β t log(c t ψnt θ X t ), t=0 where C t denotes consumption, N t hours worked, and X t is a geometric average of past consumption: X t = C γ t X 1 γ t 1. Output is producing by combining capital (K t ) and labor according to a Cobb-Douglas production function: Y t = AK 1 α t N α t. Output can be consumed or invested: Y t = C t + I t. The capital stock evolves according to: K t+1 = I t + (1 δ)k t. a) (10 points) Derive the first-order conditions for the planner s problem of this economy (hint: write the problem as a Lagrangean with two constraints, the resource constraint and the equation for X t ). b) (5 points) Derive the system of equations that characterize the steady state of the economy. c) (5 points) Suppose that total factor productivity, A t, grows at a constant rate: A t+1 = (1 + g A )A t, Y t = A t Kt 1 α Nt α. Does the economy have a balanced growth path? 1

3 SZG Macro Final Exam Week Preference Shocks and Monetary Policy (70 points) The representative consumer maximizes E 0 f P 1 t=0 t U(C t ; N t ; Z t )g where the period utility is given by! N 1+' t U(C t ; N t ; Z t ) = log C t Z t 1 + ' where C t is a CES function of the quantities consumed of the di erent types of goods, N t is employment and z t log Z t is a preference shifter which follows and AR(1) process z t = z z t 1 + " t The budget constraint (conditional on optimal alocation of expenditures across goods) is P t C t + Q t B t = B t 1 + W t N t + D t where P t is the price level, W t is the nominal wage, B t represents purchases of one-period bonds (at a price Q t ), D t is a lump-sum component of income (including dividends). All output is consumed. The labor market is perfectly competitive. Technology is given by: Y t = N t where Y t and N t denote output and employment, respectively. Price-setting is staggered à la Calvo, which gives rise to an in ation equation: t = E t f t+1 g ( t ) where t p t p t 1 is the rate of in ation between t 1 and t and where t is the average (log) price markup. We assume that in the absence of constraints on price adjustment rms would set a price equal to a constant markup over marginal cost given by (in logs). The central bank follows the simple interest rate rule where i t log Q t and log. i t = + t a) Derive the consumer s intertemporal and intratemporal optimality conditions and express them in terms of logs. b) Determine the natural (i.e. exible price) level of (log) output y n t. Show that it is independent of the preference shock and explain why. 1

4 y t c) Derive the relationship between the markup gap t and (log) output d) Determine the equilibrium behavior of output, in ation and nominal and real interest rates (20 points). e) How does the response of the economy to a preference shock di er from the response to a monetary policy shock (i.e. an additive shock to the rule above). Explain. f) Derive an interest rate rule (with nite coe cients) that would fully stabilize in ation. Explain. 2

5 Final-Home Exam, Macro 3rd week, Gerzensee, Fall 2014 Fernando Alvarez Instructions: There is a total of 70 points You can consult your notes, but not with your classmates Good luck!

6 1 Cost of Inflation (12 points) Consider the Sidrausky s model (or money-in-the-utility-function model ) discussed in the notes about the cost of inflation. Let s continue with the assumption that the representative agent utility function is: for some positive parameters A and φ. U(c, z) = c + A z1 φ 1 φ Assume that the representative agent discount utility with discount rate 1/(1 + ρ). Assume also that the (exogenous) output (or labor income) of the representative agent is constant and equal to y t = y for all t 0. Inverse velocity m t is defined in the same way, i.e m t z t /y t. We assume that the nominal money supply Z t+1 = (1 + µ) Z t for all t 0, so that it grows at the constant (net) rate µ. Note that the representative agent s budget constraints in nominal terms and in real terms are exactly the same in the in the class notes. The first order conditions are also the same. Recall that we concentrate in a case where output is constant. real balances z t is constant the net inflation rate π t = π is constant through time the net nominal interest rate R t = R is constant through time the inverse velocity m t = z t /y t is constant through time 1. What should be the (gross) inflation rate be? (The solution is one formula) 2. Use the first order condition for b t+1, and also use the properties of the utility function to obtain an expression for the gross nominal interest rate (1 + R) (The solution is one formula) 3. Use the first order condition for b t+1 and z t+1, evaluated in a constant equilibrium output to obtain an equation relating the nominal interest rate R with the inverse velocity m. (The solution is one formula). 4. Use your previous answers to write an equation for the area under the money demand as a function of the growth rate of money µ, the discount rate ρ, and the constants A and φ. How does this area change as function of µ? Why? How does this area change as a function of A? What does a higher value of A represents? 1

7 2 Looking for a good deal (16 points) Consider as a version of the sequential search model as the one described in the partial equilibrium search lecture notes and discussed in class but applied to a consumer buying a durable good. Suppose that a buyer receives offers to purchase a good every period. If the good is bought the buyer enjoys a flow benefit equal to u where u is a strictly positive constant for all future periods. If it buys the good during the current periods it pays the price p, so during this period the net utility flow is u p. If the buyer purchase the good, it will enjoy the utility flow u every future periods and it will not purchase any more goods. Each period the buyer receives an offer to purchase the good a price p drawn form a distribution F. We assume that the support of F is finite given by [b, B] where B > u/(1 β) b 0. If the buyer does not purchase the good the period utility is zero and next period will get another offer. The buyer discounts utility at the rate β (0, 1) per period. 1. Suppose the buyer accepts an offer with price p, what is her expected discounted utility from now on? (the answer is a very simple expression) 2. Write the bellman equation equation for a buyer that has not bought the good yet but that it has an offer with price p at hand. Use v(p) for the value function. (The answer one function equation with v(p) in the left hand side and the expression with the maximum between the two relevant values on the right hand side). 3. Is the value function v increasing or decreasing in p? (the answer is one word) 4. What is the nature of the optimal decision rule? Is a threshold rule? Describe it in words. 5. Write the Bellman equation you derived above evaluated at the highest price at which the buyer will buy the good, denoting this price by p. 6. How does the price p change if the distribution of price becomes higher? (in the first order stochastic dominance sense) 7. How does the price p change if the distribution of price is more disperse? (in the second order stochastic dominance sense) 8. How does the price p change if the discount rate β increases? 2

8 3 Lucas-Prescott model of the islands (8 points) Consider the benchmark version with directed search model of the islands by Lucas and Prescott discussed in the lecture notes. To simplify fix distribution of the shocks z t with finitely many values. Suppose the discount rate β goes to 0, which is the limit as the agents become extremely patient -or equivalent, where the length of the period it takes to move from location to location becomes very large. Can a stationary (i.e. a steady state) equilibrium (solving the value functions v) involve agents leaving any location? (The solution is a yes or no answer.) 3

9 4 DMP seach-and-matching (18 points) In this question we analyze the DMP model with constant exogenous wages. We let z be the productivity of the match, γ be the cost of working for the agent, c the flow cost of a vacancy, r the discount rate of agents (and firms), x the destruction rate of the matches, µ(θ) the vacancy filling rate and f(θ) the job finding rate where both f and µ are functions defined in terms of the matching function m and where θ is the vacancy-unemployment ratio, i.e. f(θ) = m(1, θ) and µ(θ) = f(θ)/θ. To simplify the analysis we will use m(u, v) = A u α v 1 α for some constants A > 0 and α (0, 1). We will assume that wages w are fixed at a value γ < w < z. We let U be the value of unemployment, J the value of match for the firm, E the value for an employed worker, and V the value for a firm posting vacancies. Each of these values satisfy the equations ru = f(θ)(e U) re = w γ + x(u E) rv = c + µ(θ)(j V ) with the free entry condition: V = 0 rj = z w xj 1. Use the equations above to derive an equation relating θ, actually µ(θ), with (r + x), (z w), and c. 2. Use the equation you derived above to find an expression for the elasticity of θ with respect to z, i.e. (z/θ)(dθ/dz), and the elasticity of µ, i.e. (θ/µ(θ))µ (θ) and the ratio z/(z w). 3. Use the law of motion of unemployment u = x (1 u) m(u, v), the definition of steady state, i.e. u = 0, and the definition of market tightness θ = v/u to derive an equation for the steady state unemployment and x, x + f(θ). 4. Use the answer to the previous question to derive an expression for the elasticity of unemployment u with respect to the market tightness θ. 5. Use the answer to the previous questions to derive an expression for the elasticity of unemployment u with respect to productivity z. How does it depends on z w? What is the intuition of this? Is the effect larger or smaller than in the case where wages are bargaining between firms and workers? 4

10 5 Ramsey Taxation without capital (16 points) Consider the case of a model with linear production and no capital, so that c t + g t = A n t where g t is government expenditures, c t consumption, n t labor and where A is the marginal (and average) productivity. Use v(c, l) for the period-utility function where l = 1 n and β for the discount rate. Consider the case where there are (linear) taxes on labor income which finance the stream of purchases. 1. Write down the first order condition of the agent s problem for each period. 2. Write down the implementability constraint as defined in the notes for this case. 3. Use the f.o.c. for the agent s problem as well as feasibility to show that D v c c + v n n equals the government deficit measured in utils. 4. What is the effect on the utility of period t of changing labor taxes from τ = 0 to τ = ɛ where ɛ is small? i.e. what is the derivative of v(c(ɛ), l(ɛ)) at ɛ = 0? [Hint: this is a simple envelope argument] 5. What is the effect on the period t of deficit, in utils, of changing labor taxes from τ = 0 to τɛ where ɛ is small? i.e. what is the derivative of D(ɛ) at ɛ = 0? 6. Use the answer to your previous questions and assume g T > 0 for at least some period(s) T. Are taxes be strictly positive or zero in a period t for which g t = 0? 5