Macroeconomics Qualifying Exam-302 Module
Claremont Graduate University
September, 2010
Lamar
There are 100 points on this part of the Qualifying Exam. All parts are equally weighted.
Show all your work.
Part I. Answer the following two questions. (50 points)
1. Consider the following production function,
, where represents
an externality. The parameter η is an indicator of the importance of the
externality. Initially consider κ = k, where k is capital per person (a la Lucas)
a) Is the production function neoclassical?
b) Express the production function in per capita terms.
c) Derive the Solow Fundamental equation.
d) Analyze how the time path of the economy depends on the externality.
e) Now analyze the case where κ = K (a la Romer), and no population growth,
and α + η = 1. Obtain an expression for the rate of growth of k. What the
model predicts?
2. Assume a closed economy with the following two-period utility function
and Cobb-Douglas technology,
, where technology follows an
AR1 process. Capital depreciation rate is equal to 1 every period.
a) Obtain expressions for wages and the rental return of capital.
b) Set up the utility maximization problem for the representative household and from
the First Order Conditions for the problem find expressions for relative
consumption and labor.
c) Interpret your findings.
d) Assuming that the economy responds optimally to technology shocks; explain
what happens if the economy is hit with a negative and temporal technology
shock.
e) Show the persistency effects of the technological shock on output.
Part II. Answer two of the following three questions. (50 points)
3. Modify the Solow model of growth by including government spending as follows.
The government purchases G units of consumption goods in the current period,
where G = gL, where L is population and g a positive constant. Population grows
at the rate n > 0. The government finances its purchases through lump-sum
taxes on consumers, and the government budget is balanced each period, so
that G = T, where T denotes total taxes. Consumers consume a constant fraction
of disposable income, that is, C = (1 – s)(Y-T), where s is the saving rate and
0<s<1. Assume a Cobb=Douglas production function.
a) Derive the law of motion of capital per worker in this economy and write the
equation describing the steady state level of capital per worker k*.
b) Show that in this economy there can be two steady states levels of capital per
capita, one with high k* , one with low k*.
c) Ignore the steady state with low k*. Determine the effects of an increase in g on
capital per worker and on output per worker in the steady state. What are the
effects on the growth rates of aggregate output, aggregate consumption and
aggregate investment?
d) Explain your results.
4. Consider the Lucas Island Model. Explain the validity of the following statements.
Use the corresponding equations and graphs when appropriate.
a) According to the Lucas imperfect information model, a 5% drop in the money
supply has a larger effect on output in an economy where monetary conditions in
general are more stable than it does in an economy with normally volatile
monetary conditions.
b) Models that assume a long-run natural rate, rational expectations, and
instantaneous market-clearing will generate a vertical short-run AS curve and
imply the policy irrelevant results of Sargent and Wallace.
5. Consider an individual who lives T periods and whose lifetime utility is given by,
, for simplicity assume a rate of time preference equal to zero.
The typical individual has initial wealth A0 and earns income Yt over his/her
lifetime. The individual can borrow/save at the interest rate zero.
a) Set and solve the individual maximization problem and from the first order
conditions obtain an expression for consumption at time t. Interpret the
expression you found for Ct.
b) Assume an unanticipated and transitory (say, one year) deviation in income.
Find the change in consumption. Is it high/low? How the change in
consumption depends on the remaining lifetime?
Now assume uncertainty and Rational expectations, where the consumer
solves the following maximization problem,
Utility is strictly concave and separable.
c) Find and interpret the Euler equation.
d) Show that under rational Expectations consumption follows a random walk.
What is the meaning of consumption follows a random walk?