Macroeconomics Qualifying Exam, Part 2
September, 2011
Please answer BOTH questions completely. This exam does not have a
choice of questions. 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 model with production and a public good σ. In this
model, agents live two periods and have preferences over the public
good when they are youngsters. The public good is funded by a lumpsum tax τ on youngsters. Assume population is constant and normalized
to one. The model is given by
Max
C0, C1, σ
(1-β) ln c0t + β ln c1t+1 + γ ln σt
c0t = wt -τt – st
c1t+1 = Rt+1st
Notation:
γ >0 : preference weight for how much youngsters value the public good
w : wage
R: yield earned on savings
τ : tax on youngsters
c0: youngster’s consumption
c1: oldster’s consumption
β>0: patience parameter
k: capital stock.
The production function f(kt) has the standard properties and satisfies the
Inada conditions. The public good has no effect on the firm.
a. Identify the state and choice variables for this model.
b. State the government budget constraint (GBC) noting that there is a
deadweight loss 0<d<1 to funding the public good.
c. Using the GBC to substitute out σ in the consumer's problem and take
the first order conditions for s and τ. Solve for the optimal savings and tax
equations. [LONG]
d. Using the behavioral relationship you just derived for the desired tax,
explain using simple words how i) the preference for public goods (γ); ii)
patience (β), iii) the deadweight loss d, and iv) wages (w) affect the
amount of the public good that is provided. [Note: “explain” does NOT
mean to just sign of the derivative, but the behavioral reason for this
effect.]
e. Let f(k)= kα be the representative firm's production function for 0<α<1.
Set up and solve the firm’s optimization problem to show how wages and
the yield on savings depend on the state variable(s).
f. Construct the capital market clearing condition in terms of the state
variable(s).
g. Define a general equilibrium for this model clearly and completely.
h. Identify all steady states for this model.
i. Draw a phase portrait by deriving arrows of motion and graphing the
dynamics in the state space.
j. Identify stability properties of each steady state.
k. Redraw the phase portrait for γ=0 and γ>0 in the same graph. Describe
what a positive preference for the public good does to the dynamics of
this model in one paragraph.
l. Redraw the phase portrait for d=0 and d >0 in the same graph. Describe
what deadweight loss in funding the public good does to the dynamics of
this model in one paragraph.
m. Explain in simply language what happens to people's demand for the
public good at d approaches 1. Describe how government inefficiency
affects welfare without deriving the welfare implications formally.
2. Consider a two period life pure exchange OLG economy. In this model,
there are N identical agents. There is no population growth. Endowments
are {e0, e1} >>0. Let co and c1 be young and old consumption, and R be
the yield on savings with R= 1+r, where r is the interest rate, and s is savings.
Let e1 = e0/(1-β). Then a person in this world solves
Max
C0, C1
(1-β) ln c0 + β ln c1
c0 = e0 – st
c1 = e0/(1-β) + Rs
a. True or false and explain why: e0 is the state variable of this model.
b. Solve for optimal savings.
c. Prove or disprove: for all positive and bounded R, people have positive
savings.
d. Identify equilibrium in the loan market. What is the equilibrium interest
factor R*?
e. Prove or disprove: Because the equilibrium interest factor does not
depend on the endowments, individuals cannot be optimizing lifetime
utility.
f. Define a general equilibrium for this model completely and carefully.
g. Now assume there is a second set of M people in this economy. These
people have endowments {(1-β)e1/β, e1} for e1 > 0. Solve for the savings
function for the 2nd type of agents (call them type M, and the previous
type call them type N).
h). Let the ratio individuals M/N = g. Construct the equilibrium condition in
the loan market when there are both type N and type M agents active in
the economy. HINT: remember that e1 = e0/(1-β).
i) Now, let g=1. Solve for the new equilibrium interest factor R**.
j). Prove or disprove: R** is strictly decreasing in the patience parameter β.