Ph.D. Macroeconomics
Exam Practice Problems
Fall 2015
Part 1: Economic Growth
These problems are to help you prepare for the exam covering economic growth. Some of them are
longer than others. The exam will have a mix of shorter and longer problems. Work some or all of
the problems and ask me if you have questions. The exam will include one or more problems that
relate to these practice problems or to the homework problems you have done.
1. A macroeconomist is found dead on the floor of his office from poisoned candy. The
following message is written on the white board above the body:
If U=
∫
T
0
e −ρt u ( t ) dt < 0 for all T > 0, then T= 0 maximizes U .
The campus police are baffled: is it murder or is it suicide? Assist them in solving the
case by explaining the message.
2. Suppose that the aggregate production function is Y ( t ) = BK ( t ) , where B is a positive
constant. Investment is given by K ( t ) = sY ( t ) , where the saving rate s is a positive
constant. There is no depreciation or technological progress and the labor force is
constant.
a. Does this model have constant, decreasing, or increasing returns to produced
inputs? Based on this, do you expect this model to exhibit endogenous growth?
Explain.
b. What are the growth rates of capital and output in this economy? Is there a
process of convergence through which the growth rate moves gradually to a
constant, steady-state value, or is this economy’s growth rate constant regardless
of initial conditions? Explain.
c. Do changes in the saving rate and the productivity parameter B have growth
effects or just level effects? Explain.
d. Suppose that the economies of the Czech Republic and Slovakia follow this
model and that they have identical values of s and B, but that the Czech Republic
has a higher initial level of per-capita income. Will per-capita income levels in the
two countries converge to the same path as t → ∞ ? Explain.
3. Suppose that Italy and Germany initially have totally separate economies governed by
the Ramsey growth model and that both are on steady-state growth paths. Everything in
the two economies is identical except that Germans are more patient than Italians:
ρG < ρI. Recall that the equations for the two curves in the Ramsey model phase diagram
are
c = 0 : r = f ′ ( k ) = ρ + θg
=
k 0 : =
c f ( k=
) ( n + g ) k.
a. Draw a phase diagram showing the relative positions of the c = 0 and k = 0
curves for the two countries.
(
)
*
*
b. How do the steady-state values of ( kG* , c G* ) and kI , c I compare? Why?
c. Now suppose that Germany and Italy join an economic union in which people in
each country are free to own capital in the other. Assume that individuals in both
countries choose to invest in the country with the higher rate of return. Where
will the return to capital be initially higher? Which country’s citizens will begin to
buy capital in the other country?
d. Once the economies reach a new steady-state equilibrium, what must be true of
the rates of return in the two countries: rG* and rI* ? What must be true of the
capital-to-effective-labor ratios kG* and kI* being used for production in the two
countries in the new steady-state equilibrium? Why? (Note: Do not try to do this
on a phase diagram.)
4. In the human-capital model presented in Chapter 4 of the Romer text and discussed in
class, explain intuitively how it is possible that an increase in education levels could lead
to lower steady-state per-capita GDP in an economy.
5. Use the Solow growth model (including a diagram) to explain why growth in France
and Germany was very high from 1950–1973 (after World War II destroyed much of
their countries) relative to other times and other countries.
6. We have worked with Romer’s R&D model under two conditions: β + θ = 1 with
n = 0, and β + θ < 1 and n > 0. We now consider the case where β + θ = 1 and n > 0.
Recall that the growth-rate equations for K and A at every moment in this model are:
gK ( t ) =
s (1 − aK ) (1 − aL )
α
1−α
 A (t ) L (t ) 


 K ( t ) 
g A ( t ) = BaKβ aLγ K ( t ) L ( t ) A ( t )
β
γ
θ−1
1−α
,
.
Recall also that the steady-state conditions are:
*
g=
0 : g=
g A* + n,
K
K
γn 1 − θ *
g A =
0 : g K* =
gA.
− +
β
β
a. Briefly describe how the steady-state conditions are derived from the growth-rate
equations.
b. Sketch the phase plane (with gK on the vertical axis and gA on the horizontal axis)
for the case of β + θ = 1 and n > 0. Show the directions of motion in each region.
c. If this model has a steady-state balanced growth equilibrium, show it and
demonstrate its stability. If it does not, explain the long-run dynamics of the
model.
7. Suppose that technological progress in both the United States and Canada occurs at
1% per year in the steady state. Canada’s population is growing at 2% per year and the
U.S. population is growing at 1%. The Canadian central bank expands the supply of
Canadian dollars at 8% while the U.S. money supply grows at 4%. The money-demand
functions of both countries are stable and the elasticity of money demand with respect to
income is one. What will be the steady-state inflation rates in Canada and the United
States? Explain how you arrived at your answers.
8. The Euler equation of the Ramsey model (in terms of consumption per person, not
C ( t ) r ( t ) − ρ
=
consumption per effective worker) is
. What does ρ measure? For a given
θ
C (t )
value of ρ, use this equation to explain the effect of a change in the rate of return r on
consumption/saving behavior.