Advanced Macro I, Part 2, Goethe University Frankfurt, WS 2010-2011

Final Examination, March 10, 2011

Instructor: Ctirad Slavk

Instructions: The exam has 3 parts. Answer only 1 question in part 2. The duration of the exam is 60 minutes.The total amount of points for this exam is 60. The number of points needed to pass the exam is 30. You are notallowed to use books, notes, calculators etc. You can keep the question sheet. Please, mark your student ID numberclearly on the answer sheets.

1. Competitive equilibria, Pareto efficient allocations and the First Welfare Theorem. (15 points)

Consider the following pure exchange economy. There are 2 consumers and 2 goods, apples and oranges. Bothconsumers have the same preferences u(ca, co) with u strictly increasing in both arguments. Initial endowmentsare e1a, e

1o, e

2a, e

2o; they are all strictly greater than 0.

(a) (2 points) Define the competitive equilibrium for this economy. Denote the set of competitive equilibriumallocations CE.

(b) (2 points) Explain why we can set the price of one good to one, i.e. use it as numeraire.

(c) (2 points) What is Walras law? Does it apply to this economy?

(d) (2 points) Define the set of feasible allocations and denote it Z.

(e) (2 points) Define the set of Pareto efficient allocations and denote it PE.

(f) (5 points) State and prove the First Welfare Theorem for this economy. Clearly explain how you use theassumption on u. Can we relax this assumption?

2. This section is worth 15 points. Answer only one question in this section, either 2a or 2b, but NOT both.

2a. Solving the one sector growth model using the guess and verify method. (15 points)

Consider the following growth model with inelastic labor supply, full depreciation, log utility and CRS technology.The Bellman equation is:

v(k) = maxk

log(k k) + v(k)k 0k k 0

(a) (5 points) Guess that the value function has the form v(k) = a1 + a2 log k. Plug the guess into themaximization problem:

maxk

log(k k) + (a1 + a2 log k)

Solve this problem for k as a function of k and the other constants.

(b) (10 points) Plug back into v(k) = maxk log(k k) + v(k) for the optimal k(k) equation to get:

a1 + a2 log k = log [k k(k)] + [a1 + a2 log k(k)]

Find the a1 and a2 that satisfy the equation. This verifies the guess. Write down the value function v(k)and the policy function k(k) in terms of the parameters of the model.

2b. The stochastic Ak model. (15 points)

Consider the following version of the Ak model:

v(A0, k0) := max{ct,kt+1}t=0

t=0

tAt

P (At)ct(A

t)1

1 s.t.

kt+1(At) + ct(A

t) Atkt(At1)ct(A

t) 0, kt+1(At) 0,tA0, k0 given

Here, At is an i.i.d. stochastic process with mean 1, P (At) denotes the probability of a sequence (A0, A1, ..., At)

and we assume that > 0, (0, 1).

(a) (5 points) Derive the Euler equation.

(b) (10 points) Assume that the solution has the following property kt+1 = sAtkt (we have shown in class thatit is true). Derive the savings rate s in terms of the parameters of the model. Explain why s is constantover time.

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3. Workers vs. capitalists and the kt 0 proposition. (30 points)Consider an economy with 2 representative consumers with the following endowments of capital and time n thatcan be used as leisure or labor:

k10 = k0 > 0, k20 = 0,t : n1t = 0,t : n2t = 1

This means that agent 1 does not have any time endowment in all periods and thus cannot work at all, we callher the capitalist. Capital depreciates at rate (0, 1). Agent 2 has no initial endowment of capital and he isnot allowed to accumulate capital, i.e. t : x2t , k2t = 0. We thus call him the worker. Preferences are the samefor the two agents and are given by:

t=0

tu(ct, lt)

There is also one representative firm facing a time stationary CRS technology, and a government that financesa given stream of expenditures {gt}t=0 by proportional taxes on capital and labor income, i.e. the budgetconstraints for the two agents are:

t=0

pt(c1t + x

1t )

t=0

(1 kt)rtk1t . (1)

t=0

ptc2t

t=0

(1 nt)wtn2t . (2)

The government is NOT required to balance its budget in every period.

(a) (8 points) Define the Tax Distorted Competitive Equilibrium (TCDE) for this economy.

(b) (4 points) Derive the non-arbitrage condition for this economy. What is the intuition for this condition?

(c) (4 points) Derive the the 2 implementability conditions. Hint: for agent one, use the non-arbitrage conditionto rewrite the budget constraint as one in which the agents wealth only depends on initial capital and thestream of labor endowments. State clearly the transversality condition that you are using.

(d) (4 points) Suppose k0 1 is given. Set up the Ramsey problem in which the government chooses allocationsdirectly so as to maximize the utility of the agent with no capital subject to a feasibility constraint and thetwo implementability constraints.

(e) (10 points) Suppose there is a steady state and the economy converges to it. Show that the solution to theRamsey problem will have k 0. What is your intuition for this result?

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