Macroeconomics Qualifying Exam
Part 2
May, 2008
Please answer TWO of the following three questions completely. Each
question has the same number of subparts, and each subpart is equally
weighted. HINT: “Prove” means to show mathematically using a formal
derivation.
1. Consider two period OLG economy with productive capital, K, and
endogenous fertility. In this model, young agents choose how many
children b (births) to have (individuals do this alone, ignoring the marriage
choice). Each child born costs d>0 units of consumption to raise, and
provides utility to one’s parent, with η>0 the weight on utility from children.
All variables have the standard definitions, i.e. β ∈(0,1) is patience, co and
c1 are young and old consumption, w is wage, s is savings, R is the yield on
savings, K is capital stock, and N is the young population. The model is
given by
Max
C0, C1, b
(1-β) ln c0t + β ln c1t+1 + η ln bt
c0t = wt– dt bt – st
c1t+1 = Rt+1st
Note that the number of young people next period is Nt+1 = Nt bt.
a. Identify the state variable(s) for this model.
b. Take the first order conditions.
number of children.
Solve for the optimal savings and
c. Prove or disprove: Children are a normal good.
d. Let K be capital stock and the production function be
Yt = KtαNt1-α
for α∈(0, 1). Set up and solve the firm’s profit maximization problem.
e. Let the cost of children by proportional to the wage, dt=γwt, for γ∈ (0,1).
Construct the capital market equilibrium conditions in per youngster terms.
HINT: start with the aggregate equilibrium condition.
f. Define a general equilibrium for this model clearly and completely.
g. Find all steady states.
h. Prove or disprove: If k0 is large enough, this model produces
endogenous growth (growth forever without hitting a steady state).
i. Draw a phase portrait by deriving arrows of motion and graphing them
in the state space. Identify the stability properties of all steady states.
2. Consider an infinitely-lived representative agent (Cass) economy. In this
model, people (normalized to 1) pay tax τ>0 on their labor income, and
leisure is valued. Assume that taxes are unproductive and are dumped
into Lake Tahoe. The agent’s decision problem at time t is described by
Maxct, lt
s.t.
βt U(ct, lt)
ct = wt ht (1-τ)+ rt kt - it
kt+1 = (1-δ)kt +it,
lt = 1-ht,
where c is consumption, l is leisure, h is hours worked, 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.
a. Find the agent’s first order conditions (FOCs) for an optimal solution to
this problem.
b. Identify the variables the agent’s optima solve for in (a) with a star (*)
and show which other variables these depend on from the agent’s
perspective.
c. Identify the state variable or variables for this model.
d. Set up and solve the profit maximization problem for a representative
firm in a perfectly competitive market with production function Y=F(K,H),
where Y is output, K is aggregate capital, H is aggregate labor, and F(K,H)
has the standard properties.
d. State the capital markets equilibrium condition for this model.
e. Carefully and completely, define a general equilibrium for this model.
f. What additional condition is required to ensure that the equilibrium is an
optimal solution? State it. What does this rule out?
g. Let h* equal to a constant, call is d>0. Now, construct the phase portrait
for this model by deriving arrows of motion and showing them in the
phase space. Identify each steady state and its stability properties.
h. Draw another phase portrait showing what happens dynamically when
taxes increase from τ0>0 to τ1>τ0.
i. Prove or disprove: The equilibrium allocation is Pareto optimal.
3. Consider a growth model with human capital. In this model, agents live
3 periods. The first period is childhood where children make no choices
and are supported by their parents. In middle age (age 0), agents work,
save, and reproduce a single child. Note that this means that population
is constant and can be assumed to be 1. In old age (age 1), agents are
retired and consume from the principal and interest on savings. All terms
are written relative to time t human capital. The model is given by
Max
C0, C1, z
(1-β) ln c0t + β ln c1t+1 + γ ln zt+1
c0t = wtzt(1-τt)– vt – st
c1t+1 = Rt+1st
Notation:
γ : parent’s preference weight for how much they value their children’s’
human capital
zt : parent’s human capital
zt+1 : child’s human capital
θ : productivity of educational expenditures, θ ∈ (0, 1),
wtzt : labor income
vt: cost of raising a child’s human capital, i.e. education
τt : tax on labor income, assumed unproductive.
The production function for human capital is
zt+1 = ωztvtθ,
where ω relates kids’ human capital to the environment in which they live.
a. Identify the state variables for this model.
b. Take the first order conditions.
education spending.
Solve for the optimal savings and
c. Let K be capital stock and the production function be
Yt = Ktαzt1-α
for α ∈ (0, 1). Set up and solve the firm’s profit maximization problem.
d. Define a general equilibrium for this model clearly and completely.
e. Identify all steady states for this model.
f. Draw a phase portrait by deriving arrows of motion and graphing them
in the state space.
g. Identify stability properties of each steady state.
f. Describe in simple language the dynamics below and above the
inflection point. Use the model to show why each behavior occurs.
g. Derive a condition showing how taxes affect investment in a child’s
human capital.
h. Prove or disprove: If taxes are high enough, the economy will converge
toward the poverty trap.
i. Now assume that tax revenue, call it σt, is used to fund additional tutors
for kids. The human capital production function is now
zt+1 = ωσtztvtθ,
Prove or disprove: there is a tax level τ* that will guarantee in equilibrium
positive growth in human capital for any value of zt.