Macroeconomics Qualifying Exam
Part 2
September 2010
Claremont Graduate University
Please answer question ONE and also either question two or three. Each
part is equally weighted. HINT: “Prove” means to show mathematically
using a formal derivation. “Describe” or “explain” means to provide the
behavioral intuition for a result.
1. Consider an OLG economy with productive capital, K, and government
debt B that accumulates over time and debt holders earn rate Rt+1 from t
to t+1 from purchasing debt. Population grows geometrically, Nt+1 =
(1+n)Nt, n > -1, N0=1, and young people pay tax τ >0 on labor income. A
person who is born at time t faces the decision problem
Maxc0, c1
s.t.
U(c0, t) + βU(c1, t+1)
c0,t = wt (1-τ)– st
c1,t+1 = Rt+1st,
where c0, c1, s, w, β∈ (0,1), and R have the standard definitions, and the
utility function has the standard properties.
a) Identify the choice and state variables for this model at time t.
b) Write down the government budget constraint (GBC) at time t in
aggregate terms including government debt. Identify each term. Now
state this in per worker terms where bt=Bt/Nt is debt per worker.
c) Take the FOC write down the term the equation is solving for and what
it depends upon from the agent's perspective.
d) Set up and solve the firm's optimization problem using a standard
neoclassical production function Yt=F(Kt, Nt), where Y is output.
e) There are two equations that determine equilibrium in this economy,
write them down in per worker terms.
f) In plain English, tell which each equation in (e) means. Maximum two
sentences per equation.
g) Define a competitive equilibrium for this model C3: clearly, completely,
and concisely.
h) For the next two questions, let τ=0.
economy.
Find all steady states for this
i) Derive the phase portrait for this economy. You will not get credit by
drawing a phase portrait from memory. Show all steady states and their
stability properties.
j) The US government has incurred substantial debt in the last 10 years.
What does this model predict is the effect of additional debt on
aggregate outcomes.
2. Consider a two period life pure exchange OLG economy. In this model,
there are N > 1 young people and there is no population growth.
Individuals have endowments {eo, e1} >> 0. Young people pay a tax, τ>0,
that is pure waste. Let co and c1 be young and old consumption, β∈ (0,1),
and R be the yield on savings, s. Then an individual's optimization problem
is to solve
Max
C0, C1
(1-β) ln c0 + β ln c1
c0 = eo – s - τ
c1 = e1 + R s
a. Solve for optimal savings relation.
b. Prove that there exists a value of τ such that savings is zero.
c. Prove that there exists a value of R such that savings is zero.
d. State the equilibrium condition in the loan market.
e. Prove or disprove: there is a unique equilibrium for this model.
f. Define a general equilibrium for this model completely and carefully.
g. Now suppose that instead of the tax being wasted, it is stored costlessly
by a benevolent tribal chief. Tax τ yields Rf τ > τ in the individual's second
period of life. Set up an individual's lifetime utility maximization problem.
Note that she can still save from her untaxed endowment at rate R.
h. Solve for optimal savings relation for the model in (g).
i. Prove that there exists a value of τ such that savings is zero.
j. Prove or disprove: The equilibrium value of R in the model with a
benevolent chief (second model) is higher than the model without him
(first model).
3. Consider an infinitely-lived representative agent (Cass) economy. In this
model, population is normalized to 2 and utility only comes from
consumption. The decision problem at time t for a type j=1,2 individual is
described by
Maxct
s.t.
βt U(cjt)
cjt = wjt + rt kjt - ijt
kjt+1 = (1-δ)kjt +ijt,
where c is consumption, w is wage, k is capital owned by the individual, i
is investment, r is the interest rate, δ∈ [0,1] is depreciation, β∈ (0,1) is the
discount rate, and the utility function has the standard properties
including the Inada conditions. Note that there is a single unified capital
market (i.e. all individuals earn the same interest rate r on their
investments).
a. Identify the state variable or variables for this model at time t for both
types of people.
b. Find the first order condition(s) (FOCs) for an optimal solution to this
problem for some j.
c. Identify the variable(s) the FOC(s) solve for in (a) with a star (*) and
show which other variables these depend on from the decision-maker’s
perspective.
d. Let w be the average wage in the economy. Assume that type 1
individuals have an earning advantage over type 2 people, w1 = w (1+ ε),
and w2 = w (1- ε), for ε∈ (0,1) determined exogenously. Set up and solve
the profit maximization problem for a representative firm in a perfectly
competitive input market with intensive production function y=f(Κ), where
Κ is per capita capital, i.e. Κ = K/2, y is output per capita, K is aggregate
capital, and f(Κ) has the standard properties including the Inada
conditions and constant returns to scale. Use the FOCs to find the return
on investment and the average wage.
e. Construct the capital market equilibrium for next period's aggregate
capital Kt+1.
f. State all the conditions needed to identify an optimal equilibrium path in
this economy.
g. Find all steady states.
h. Derive the phase portrait for this model using aggregate variables.
Show the dynamics in the phase space and identify the stability properties
of each steady state.
i. Prove or disprove: ε has no effect on the dynamics of this economy.
j. Prove or disprove: in a steady state, consumption is equal for both types
of people.