Macroeconomics Qualifying Exam
Part 2
May, 2009
Claremont Graduate University
Please answer question ONE and EITHER questions two or three
completely. 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 an
economic "stimulus" package. Population grows geometrically, Nt+1 =
(1+n)Nt, n > -1, N0=1, and old people pay tax τ >0, while young people get
a transfer σ >0. The tax and transfer occur in the same period. An agent
who is born at time t faces the decision problem
Maxc0, c1
s.t.
(1-β)ln (c0, t) + βln(c1, t+1)
c0,t = wt– st + σt
c1,t+1 = Rt+1st - τt+1,
where c0, c1, s, w 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. Identify
each term. Now state this in per worker terms.
c) Use the per worker GBC to substitute σ out of the flow (2 period) budget
constraint in terms of τ. Assume τ is constant and derive a condition
showing that lifetime resources might be higher with the stimulus package
than without it if some condition is met. Assume the economy is at a
steady state.
d) Briefly describe what the condition you derived in (c) means in plain
English. Agree or disagree and explain your reasoning for: this is a
politically viable stimulus plan.
e) Take the FOC and solve for the savings relation.
f) Prove or disprove that the savings relation is increasing in R. Why might
this be important?
g) Prove or disprove this statement: Savings is higher for the economy
above when τ=σ= 0 than for τ,σ> 0.
h) In plain English, tell me why you obtained the result in (g).
i) Let y=kα where α ∈(0,1). Set up and solve the firm's profit maximization
problem.
j) Define a competitive equilibrium for this model C3: clearly, completely,
and concisely.
k) Find all steady states for this economy.
l) Phase portrait [THIS IS SOMEWHAT HARD]: Deriving the phase portrait is
difficult, but use the capital market clearing condition and your intuition to
draw one as well as you can. A full derivation is not required, but show
your work. You will not get credit by drawing a phase portrait from
memory.
2. Consider a two period life pure exchange OLG economy. In this model,
there are two types of agents, type 1 and type 2, with N1 and N2 young
members, respectively. There is no population growth. Agent types differ
by their endowments. Type 1 endowments are {e, ε}, and type 2
endowments are {ε,e}, where e,ε>0, with e large while ε is very close to
zero. Let co and c1 be young and old consumption, and R be the yield on
savings, s. Then type 1 agents solve
Max
C0, C1
(1-β) ln c10 + β ln c11
c10 = e – s1t
c11 = ε + Rt+1s1t
Type 2 agents solve
Max
C0, C1
(1-β) ln c20 + β ln c21
c20 = ε – s2t
c21 = e + Rt+1s2t
a. Solve for optimal savings by type 1 and type 2 agents.
b. Prove or disprove: for all positive and bounded values of R, type 2
agents have positive savings.
c. Define a general equilibrium for this model completely and carefully.
d. Let the number of youngsters in each group have the following relation
N1=γN2, for ∞ > γ > 1. Use this population relation and the loan market
equilibrium condition to solve for the equilibrium yield R* in per youngster
terms.
e. Using (d), prove that i) R*>0; ii) R* is decreasing in γ. Explain the reason
for result (ii) in plain language.
Now let’s add a tax on type 1 agents to subsidize their type 2 brethren.
Type 1 youngsters now pay a lump-sum tax τ<e that is immediately and
costlessly sent to type 2 youngsters as transfer σ. Type 1s now solve
Max
(1-β) ln c10 + β ln c11
C0, C1
c10 = e – s1t - τ
c11 = ε + Rt+1s1t
Type 2 agents solve
Max
(1-β) ln c20 + β ln c21
C0, C1
c20 = ε – s2t + σ
c21 = e + Rt+1s2t
f. Find the optimal savings relations for types 1 and 2.
g. State the government budget constraint (GBC). Write the GBC in per
youngster terms (i.e. using population relation given in part (d)).
h. Using the per youngster GBC, substitute out σ for τ in the savings
relations, and construct the loan market equilibrium using the population
relation given in part (d). Find the equilibrium yield R* for this version of
the model.
i. Prove or disprove: R* is increasing in the tax τ. In plain language, explain
why this relationship between the tax and the yield is occurring.
3. Consider an infinitely-lived representative agent (Cass) economy. In this
model, population is normalized to 1 and utility only comes from
consumption. The decision problem at time t is described by
Maxct
βt U(ct)
s.t.
ct = wt (1- τ)+ rt kt - it
kt+1 = (1-δ)kt +it,
where c is consumption, w is wage, k is capital, 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 tax τ ∈ (0,1) that funds government investment, γ.
a. Identify the state variable or variables for this model at time t. Prove or
disprove: This is a market problem.
b. Find the first order condition(s) (FOCs) for an optimal solution to this
problem.
c. Identify the variables the FOC(s) solve for in (a) with a star (*) and show
which other variables these depend on from the decision-maker’s
perspective. Now state the government budget constraint.
d. Set up and solve the profit maximization problem for a representative
firm in a perfectly competitive input market with production function
Y=F(γ, K), where Y is output, K is aggregate capital, γ is government
investment, and F(γ,K) has the standard properties including the Inada
conditions and constant returns to scale.
Assume that firms take
government investment as given and fixed, with F1(γ,K)>0, and F2(γ,K)<0.
e. Carefully and completely, define a general equilibrium for this model.
f. Identify all steady states.
g. Construct the phase portrait for this model by deriving arrows of motion
and the dynamics in the phase space. Identify the stability properties of
each steady state.
h. State the transversality condition (TVC). Provide a logical argument for
why the TVC identifies a single two-branch dynamical path for this
economy out of the infinite number of possible paths that satisfy the
equilibrium conditions. Now redraw your phase portrait and add second
phase portrait for τ=0 [assume when τ=0 F(γ, K)= F(K) which is a standard
production function].
i. Prove or disprove: For Kt less than the interior steady state value of K and
on the saddle path, it > δKt.