THE UNIVERSITY OF MANCHESTER MACROECONOMIC THEORY Semester 1 2022/23
Release Date: 18 January 2023, 2:00 pm Submission Deadline: 20 January 2023, 2:00 pm
INSTRUCTIONS SPECIFIC TO THIS EXAM:
• Answer ONE question from Section A and TWO questions from Section B. Each section is worth 50 points.
• You must submit typed responses. Hand-written responses are not acceptable. Any equations must be typed. However, hand-drawn diagrams are acceptable, as long as they are included within the main document of your submission.
• You must submit your answers either as a Word document or as a PDF.
• Ensure that any included diagrams are oriented correctly. Marks will be deducted
if your diagrams are rotated 90 degrees, upside down, etc.
• Do not submit an image of typeset answers. Do not include equations by inserting
pictures of equations.
• Students are not permitted to discuss their answers with other students before
submission.
• Candidates are expected to demonstrate to the examiners a competent knowledge
of all computations.
• Candidates are also advised that the examiners attach considerable importance to
the clarity with which answers are expressed.
⃝c The University of Manchester, 2022/23 Page 1 of 4
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SECTION A Answer ONE question
1. Efficiency of the decentralized equilibrium in a search economy
Consider the steady state of the Mortensen-Pissarides model. Suppose r > 0 and assume that firms are owned by households. Therefore, total welfare can be measured as the discounted sum of utility and profits per unit time:
∞ −rt
e [yE(t)−(E(t)+V(t))c+b(1−E(t))]dt.
0
Consider a social planner who chooses the path of V (t) to maximize W , subject to the constraint that E ̇ (t) = M (1 − E (t), V (t)) − λE (t). The solution to this problem is the efficient allocation. Letting MU and MV denote the partial derivatives of M (U, V ) with respect to U and V , respectively, the efficient allocation is then determined by the following first-order condition:
e−rt [y − c − b] = μ(t) [MU (1 − E(t), V (t)) + λ] − μ ̇ (t) where μ(t) ≡ ce−rt/MV (1 − E(t), V (t)).
(a) Using the first-order condition for the planner’s problem, and imposing the steady state where E ̇ = V ̇ = 0, show that the equilibrium allocation will be constrained efficient when γ = 1 − φ. Show analytically whether equilibrium employment is above or below the efficient level when (i) 1 − φ > γ and (ii) 1 − φ < γ. Explain your steps clearly and give economic intuition. (15 points)
(b) Suppose the government imposes a tax Tv (or subsidy if Tv < 0) on firms to post vacancies and rebates the revenue back to workers as a lump-sum T . That is, the firm’s profits are now y−w−c if the job is filled and −c−Tv if vacant. Workers receive w+T if employed and b+T if unemployed. Solve for the new equilibrium condition, rVV (E) = 0. Find an expression for the optimal Tv (i.e., the value of Tv which shifts the equilibrium allocation to the efficient one). Explain and give economic intuition for what you find. (10 points)
(c) Suppose there is an increase in y. How would the efficient level of employment respond? Can you tell what would happen to the optimal Tv from part (b)? Explain and give economic intuition for what you find. (10 points)
(d)Assumek=1,γ=1/2,r=0.05,b=0.2,c=0.5,λ=0.3,φ=0.25. Plot rVV (E) against E with and without the optimal Tv for two scenarios: (i) y = 1 and (ii) y = 2. Report the value of Tv and the equilibrium E in each case.
W =
Explain what you find.
(15 points)
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Consider the Barro tax-smoothing model. Suppose that output, Y , and the real interest rate, r > 0, are constant, and that the level of government debt outstanding at time 0 is D0 = 0. Suppose there are two possible values of government purchases: either GL or GH , where GL < GH . Assume distortion costs are quadratic.
(a) Suppose initially that there is no uncertainty in the path of government pur- chases. Specifically, assume Gt = GH when t is even and Gt = GL when t is odd. What are the optimal paths of taxes, Tt, the primary deficit, Gt − Tt, and government debt, Dt? Give economic intuition for your answer. (10 points)
Now, for the rest of this question, suppose there is uncertainty over the path of government purchases, Gt. Specifically, if Gt = GL, the probability that Gt+1 = GH is pL ∈ (0,1). If Gt = GH, the probability that Gt+1 = GL is pH ∈ (0,1).
(b) Solve for the optimal rule for taxes Tt as a function of existing debt Dt and gov- ernment purchases Gt. Give economic intuition for what you find. (20 points)
(c) AssumeGL =5,GH =10,r=0.04,pL =1/5andpH =1/10. Plotthepathof taxes Tt, the primary deficit Gt − Tt and government debt Dt over 60 periods, assuming that the realized path of Gt alternates with GH for 10 periods, then GL for 5 periods, starting with GH. In other words, Gt = GH for the first 10 periods, Gt = GL for the next 5 periods, Gt = GH for the next 10 periods, and so on. Give economic intuition for what you find. (10 points)
(d) Suppose the realized value of Gt was GH forever. What would happen to the path of government debt? Is the no-Ponzi game condition violated in this situation?
Why or why not?
(10 points)
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2. Barro tax smoothing model
SECTION B
Answer TWO questions
Word limit for each question: 500 words
1. Solve the log-linearized version of the RBC model using the method of undetermined coefficients, where your solution takes the following form:
C ̃ t = a C K K ̃ t + a C A A ̃ t + a C G G ̃ t
L ̃t =aLKK ̃t +aLAA ̃t +aLGG ̃t K ̃t+1 =bKKK ̃t +bKAA ̃t +bKGG ̃t
Assumeα=1/3,g=0.5%,n=0.25%,δ=2.5%,(G/Y)∗ =0.2,r∗ =1.5%, l∗ = 1/3, ρG = 0.75 and ρA = 0.75. Explain carefully your solution technique and report the coefficients (a,b) for your solution. Then, trace out the impulse responses (over 60 periods) for capital, labor, consumption, output, the wage and the interest rate to a 1% technology shock. Give economic intuition for how the RBC economy responds. How does the persistence of productivity (ρA) affect the dynamics? (25 points)
2. Suppose that output is determined by the Lucas supply curve, y = yn + b(π − πe). Moreover, suppose that social welfare is quadratic in both output and inflation. In other words, the social loss function is
L=12(y−y∗)2+12a(π−π∗)2, y∗>yn,a>0.
Assume the policymaker operates under discretion and chooses inflation π to minimize L subject to the Lucas supply curve. Give economic interpretation for the parameters of this model, and show what happens to equilibrium social welfare when a falls. Give economic intuition for your answer. (25 points)
3. The average income of farmers is less than the average income of non-farmers, but fluctuates more from year to year. Given this, how does the permanent-income hypothesis predict that estimated consumption-income functions for farmers and non- farmers differ? Give economic intuition. (25 points)
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