Macroeconomics Qualifying Exam-303 Module
Claremont Graduate University
September, 2007
Lamar
There are 100 points on this part of the Qualifying Exam. All parts are equally
weighted. Show all your work.
Part I. Answer all questions. (50 points).
1. Consider the following endowment OLG model (there is no production and
capital accumulation). There is a large (finite) number of individuals
denoted by i = 1,2,3...N who live for two periods. Individuals are
heterogeneous in their endowments of the single good of the economy.
They are endowed with (e1i , e2i ) of the single good of the economy when
young (1) and old (2) respectively. Their endowments are perishable and
no storage technology is available. That is, if endowments are not
consumed within the period they vanish. Preferences of individuals are
represented by a utility function u (c1 , c2 ) that is continuous, differentiable,
strictly concave, and strictly increasing in both arguments.
a) Suppose that a market is opened up in which individuals can borrow and
lend at the interest rate r. Argue that all trade will occur between young
individuals. Argue that all individuals are better-off by the introduction of
this market. A graph can help your argument.
b) Define a competitive equilibrium for this economy.
c) Suppose now that u (c1 , c2 ) = c1c2β and N = 1000. Furthermore the
endowment pattern are (e1i , e2i ) = (0,1) for i = 1,2,…500, and (e1i , e2i ) = (1,0) for
i = 501,502,…1000. What is the equilibrium interest rate.
2. The US Congress decides to change immigration laws, allowing for a period,
and only for one period, the entry of Nf young people from abroad at the
beginning of period t0. This implies that the labor force in the economy changes
at t0 from N to N’ = N + Nf forever. Assume for simplicity that population growth
and growth in the efficiency of labor are both zero. Using an OLG model with
productive capital analyze the following questions:
a) Describe the dynamic evolution of the capital to labor ratio, output and
output per capita from t0 on.
b) Is the steady state of the economy affected?
c) Would old people at time t0 favor the change in immigration law for one
period?
d) What about the young natives at t0. Explain
Part II. Answer 1 of the 2 questions. (50 points).
3. Assume an OLG model with productive capital and no population growth.
Individuals live for two periods, working and saving in the first and living off
capital in the second period. Assume that capital is the only asset and it is paid
its marginal product. Also assume that there is no technological progress. This
problem wants you to analyze the impact of different forms of government
taxation on the steady state and dynamics of the capital stock.
The individual’s utility function is U = log(c1,t ) + log(c2,t ) (no discount factor), and
the production function in per capita terms is yt = ktα .
a) Assume that there exist a government who imposes a lump-sum tax of an
amount Tt on each time to young individuals, and that each individual
maximizes utility subject to his/her budget constraint.
i)
Determine the intertemporal budget constraint of each individual
ii)
Use the budget constraint to solve for the first period
consumption c1,t and the first period saving.
b) Using the saving function derived on a) and the production function:
i)
Determine the relationship between kt and kt +1 , and show it in a
graph.
ii)
Write down the expression that implicitly defines the equilibrium
capital stock k * , assuming that the tax is constant: Tt = T
iii)
How many equilibria does the economy exhibit? Which one are
stable?
c) The government now decides to use a proportional tax on labor income
instead of a lump-sum tax. Therefore the after-tax income of each worker
is wt (1 − τ wt ) , where τ wt is the labor income tax.
i)
Determine the relationship between kt and kt +1 , and show it in a
graph. Hint: you can use the expressions you derived above,
and just replace wt (1 − τ wt ) for wt,a nd remove the effect of a
lump sum tax.
ii)
Write down the expression that implicitly defines the equilibrium
capital stock k * , assuming τ wt is constant.
iii)
Assuming that the government sets τ wt such that in equilibrium,
the revenue is equal than with the lump-sum tax, that is,
w*τ w = T , how does the new level of equilibrium capital per
capita compare with the one obtained in part b).
d) Provide an explanation for the comparison of the capital per capita of b)
and c)
4. Consider a standard optimal growth model with population growth.
a) Write down and solve the planning problem.
b) Write down the market problem and show that is equivalent to the
planning problem you solved in a). Under what assumptions the
two problems are equivalent.
c) Derive the phase portrait, showing vector fields, carefully
determining the relative positions of the KK and CC curves.
d) Assume you are the central planner, given an initial ko, what would
you like to do and why?