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ECOS3010: Practice Midterm Exam
Q1. Consider an economy with a shrinking stock of
at money. Let Nt = N; a constant,
and Mt = zMt1 for every period t, where z is positive and less than 1. The government
taxes each old person goods in each period, payable in
at money. It destroys the money
it collects.
a. Find and explain the rate of return in a monetary equilibrium.
b. Prove that the monetary equilibrium does not maximize the utility of the future
generations.
c. Do the initial old prefer this policy to the policy that maintains a constant stock of
at money? Explain.
Q2. Assume that people face a lump-sum tax of goods when old and a rate of expansion
of the
at money supply of z > 1. The tax and the expansion of the
at money stock are
used to
nance government purchases of g goods per young person in every period. There
are N people in every generation. Assume that the utility function of people in the economy
is log(c1;t) + log(c2;t):
a. Find the real demand for money (q = vtmt) as a function of z and .
b. Find the government budget constraint in a stationary equilibrium. Solve it for as
a function of z. The expression will also involve y and g.
c. Substitue your expression for from the government budget constraint (part b) into
the demand for money (part a). Use this to repreent seigniorage as a function of z alone.
Graph seigniorage as a function of z. For the graph, use the following parameter values:
N = 1; 000; y = 100; and g = 10:
Q3. Let Nt = nNt1 and Mt = zMt1 for every perioid t, where z and n are both
greater than 1. The money created in each period is used to
nance government purchase
of g goods per young person. Prove that the monetary equilibrium does not maximize the
utility of future generations.
Q4. Consider the following version of the Lucas model. The number of young individuals
born on island i in period t; N it is random according to the following speci
cation:
N it =
3
4
N with probability 0:5
=
1
4
N with probability 0:5:
Assume that the money supply grows at the constant rate zt = z in all periods.
(a) Set up the budget constraints of the individuals when young and when old. Also
set up the government budget constraint and money market clearing condition. Find the
lifetime budget constraint (combine the budget constraints of the young and old).
(b) On which island would you prefer to be born? Explain with reference to the rate of
return to labor.
(c) Show how the rate of return to labor and the individuals labor supply depend on
the value of z.
For the following parts, assume that the growth rate of money supply zt is random
1
according to
zt = 1 with probability
= 3 with probability 1 :
The realization of zt is kept secret from the young until all purchases of goods have occurred
(i.e., individuals do not learnMt until period t is over). Given these changes in assumption,
answer the following questions:
(d) How many states of the world would individuals be able to observe if information
about every variable were perfectly available? Describe those possible states.
(e) How many states of the world are the individuals able to distinguish when there is
limited information (i.e., they do not know the value of zt)?
(f) Draw a graph of labor supply and the growth rate of money supply in each possible
state of the world when there is limited information. What is the correlation observed
between money creation and output?
(g) Suppose the government wanted to take advantage of the relation between money
creation and output. If it always inate ( = 0), will the graph you derived in part (f)
remain the same. Explain your answer.