Macroeconomics Qualifying Exam May 2011
Part 2
Claremont Graduate University
Please answer BOTH questions. This qual does not have a choice
between questions. Each subpart 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 infinitely-lived representative agent (Cass) economy. In this
model, population is normalized to 2, where there is one individual of
each type j=1,2, and utility only comes from consumption. The decision
problem at time t for a type j individual is
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, K
is aggregate 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
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, and f(Κ) has the
standard properties including the Inada conditions and constant returns
to scale. Use the FOC(s) to find the return on investment and the average
wage.
e. Write down 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.
2. Consider an OLG economy with productive capital, K, and population
growth, Nt+1 = (1+n)Nt, n > -1, N0=1. In this economy, young people pay
tax τ>0, while old people pay no tax. Taxes are used to subsidize young
generation consumption. Let σ be the consumption subsidy. The agent’s
decision problem at time t is described by
Maxc0, c1
s.t.
U(c0, t, c1, t+1)
c0,t (1-σ) = wt -τ – st
c1,t+1 = Rt+1st,
where c0, c1, s, w and R have the standard definitions, and the utility
function has the standard properties.
a. Find the agent’s first order condition (FOC) for an optimal solution to this
problem.
b. Identify the state variable or variables for this model.
c. Set up and solve the profit maximization problem for a representative
firm in a perfectly competitive market with production function Y=F(K,N),
where Y is output, and F(K,N) has the standard properties.
d. Carefully and completely, define a general equilibrium for this model.
e. Let U(c0, c1)= (1-β)lnc0 + βlnc1 and find the FOC for consumers and solve
for the savings function.
f. Construct the capital market clearing condition using the utility function
in part (e). Show the dependence of each variable on the state
variable(s).
g. Find the steady state capital market equation and identify all steady
states.
h. Assuming the theorem in (g) holds, draw (don't derive) the phase
portrait for this model, showing all steady states and their stability
properties.
Now we modify the model above so that there is no consumption subsidy
(σ=0) and government spending is productive. Let Q be aggregate
government investment funded by collected tax revenue; Q may be used
for example, to build roads or other infrastructure. Further, assume that
the per worker production function is y= qγkα, where γ, α ∈ (0,1) and
q=Q/N.
i. Find the government budget constraint (GBC) in per worker terms.
j. Using the per worker GBC, replace government investment in the
production function in terms of the tax τ.
k. Draw a graph of net-of-tax output, y(τ)-τ. There is something unusual
about net-of-tax output, identify it. Offer a justification for this modeling
choice.
l. Assuming competitive markets, state a representative firm’s profit
maximization problem using y(τ). Assume that firms take government
investment q as given. Derive the equilibrium wage, w, and interest
factor, R.
m. Now using part 2l, and the utility function in part 2e, construct the
capital market clearing condition in terms of the state variable(s) and
parameters.
n. Derive a condition showing that if τ is “not too big”, then raising taxes
increases kt+1.
o. State the capital market equilibrium condition at a steady state. Prove
or disprove: there are two interior steady states. HINT: a geometric proof is
easiest and will earn full credit.
p. Draw the phase portrait for this economy, identifying all steady states
and their stability properties.
q. Describe in simple language what the tax and government investment
do in this model. For example, how does it affect the steady states(s),
growth in poor countries, growth in rich countries, etc. Limit your answer to
10 sentences.