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ECOS 3010 Monetary Economics
MID-SEMESTER EXAMINATION
Instructions:
1. Answer all 3 questions. Each question is worth 10 marks. The whole
exam is worth 30 marks.
2. Handwrite your answers using a dark pen on white paper, A4 size, then
scan your handwritten answer to create a single PDF and upload the PDF file.
Organise your answers section by section, in the same order as the questions
below, clearly labelling each part ‘(a)’, ‘(b)’, etc. Do not write the questions
again on your answer sheet. Show all working. Present your answers clearly and
concisely. Draw and label charts in a format that is accurate and comprehensive.
The way you organise your answers will be a factor in your overall mark.
3. Carefully explain your work.
1
Answer all 3 questions. Each question is worth 10 marks.
1. Consider the standard OLG model of money. Individuals are endowed with y units of
the endowment good when young and nothing when old. Assume that people face a lump-
sum tax of τ goods when old and a rate of expansion of the fiat money supply of z > 1. The
tax and the expansion of the fiat money stock are used to finance government purchases of
g goods per young person in every period. There are N people in every generation.
a. Find the individual’s budget constraints when young and when old. Combine them
to form the individual’s lifetime budget constraint and graph this constraint. (1 mark)
b. Find the government’s budget constraint. (1 mark)
c. Graph together the feasible set and the stationary monetary equilibrium. (2 marks)
d. Find the stationary monetary equilibrium when z = 1 and add it to the graph in
part c. (1 marks)
e. Use a ruler on your graph to compare the real balances of fiat money when z > 1 to
the values when z = 1.(1 mark)
Now assume that the utility function of people in the economy described in this economy
is
log(c1,t) + log(c2,t+1).
f. Find the real demand for money (vtmt) as a function of z and τ . (1 mark)
g. Find the government budget constraint in a stationary equilibrium. Solve it for τ as
a function of z . (The expression will also involve y and g.) (1 mark)
h. Substitute your expression for τ from the government budget constraint (part g) into
the demand for money (part f). Use this to represent seigniorage as a function of z alone.
Graph seigniorage as a function of z. For the graph, use the following parameter values:
N = 1000, y = 100, and g = 10. (2 marks)
2
2. Consider the following version of the Lucas model in which the money growth rate
is a random variable. Let the probability be 45 that zt = 1 and the probability be
1
5 that
zt = 2. The realization of monetary policy (the realized value of zt) is kept secret from the
young until all purchases have occurred – that is, people do not learn until period is over.
Prices are the only thing directly observable by the young. Let l(pit) = 5+ 0.2p
i
t. The total
population across the two islands is constant over time. Half of the old individuals in any
period live on each of the islands. The old are randomly distributed across the two islands,
independently of where they lived when young. The distribution of young individuals
are unknown. Use N1 and N2 to represent the size of young population on Island 1 and
Island 2, respectively.
a. Solve for the equilibrium price level on Island 1 and Island 2. (6 marks)
b. What does the price level tell the worker about the money supply change? What if
the distribution of young individuals are known, that is, the young are distributed unequally
across the islands and in any period each island has an equal chance of having the large
population of young? (4 marks)
3
3. Consider the following OLG economy: individuals are endowed with y units of the
consumption good when young and nothing when old. Preferences are such that individuals
would like to consume in both periods of life. Fiat money is supplied by the government and
is constant. The population grows at rate n, Nt = nNt−1. In each period, the government
taxes each young individual τ goods. The total revenue of the tax is then distributed among
the old who are alive in that period as lump-sum transfers.
(a) What is the amount of lump-sum transfer received by each old in period t? (1 mark)
(b) Write down the first- and second-period budget constraints facing a typical individual
in period t. Combine the constraints to find the lifetime budget constraint. (2 marks)
(c) Find the rate of return to money vt+1/vt in a stationary monetary equilibrium. (1
mark)
(d) Graph the stationary monetary equilibrium and indicate the levels of c1 and c2 that
would be chosen by an individual in this equilibrium. (2 marks)
(e) Write down the resource constraint facing the planner. On the graph you drew in
part (d), find the golden rule allocation. (2 marks)
(f) Suppose that the tax τ is not larger than the real value of money individuals would
choose to hold in the absence of the tax. Does a change in τ affect an individual’s utility
in our economy? (1 mark)
(g) Suppose that tax collection and redistribution are costly, so that for every unit of
tax collected from the young, only 0.5 unit is available to distribute to the old. How does
your answer in (f) change? (1 mark)