Download Macro monetaria y financiera Tarea 4.

Macro monetaria y financiera
Tarea 4.
April 23, 2016
Exercise 1
Consider an OLG economy where each generation lives for 2 periods. Preferences are given
by:
Uth [cht (t), cht (t + 1)] = cht (t) cht (t + 1)
β
Suppose that there exists two segmented markets and 2 types of agents: rich and poor.
In particular suppose that N/2 individuals of each generation are rich and receive an endowment ω R,h (t) = [ωtR,h (t), 0] and N/2 individuals of each generation are poor and receive an
endowment ω P,h (t) = [ωtP,h (t), 0], with ωtR,h (t) > ωtP,h (t). The population is constant.
Assume that there exists a capital technology that pays a return of x each period per unit
invested (linear technology). Moreover, suppose that the minimum level of capital investment is k ∗ , with ωtR,h (t) > k ∗ > ωtP,h (t).
Suppose that in the economy there is a government that issues money and that the quantity of money M is fixed. In conclusion, suppose that there is no possibility of financial
intermediation (i.e., there are no bank institutions that collect deposits).
1. Write down the budget constraints of rich and poor individuals in the first and second
period (recall to consider the restriction on the capital accumulation). Moreover, suppose that there exists a market for debt, where individuals can buy and sell lh (t), as
in the models we have seen in class.
2. Write down the inter temporal budget constraint (IBC). Explain.
3. Use the IBC to discuss the portfolio saving decision of the rich individuals. On what
does it depend? Use the no arbitrage condition to show it. Provide the economic
intuition.
4. Solve the problem of the agent h (both rich and poor), i.e. maximise the utility function
subject to the budget constraints. Write down the FOC and find the consumption and
saving of each agent (both rich and poor) in every period.
5. Show that the rich save only in capital if x > 1.
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Exercise 2
Suppose that: n > 1 (there is population growth) and µ > 1 (there is growth in the money
supply). What is the necessary and sufficient condition (if any) for the rich to save in capital?
1. Solve the previous exercise with the new assumptions
2. Find the maximum level of seignorage that can be financed under the assumption that
x > n/µ.
3. Find the maximum level of seignorage that can be financed under the assumption that
x < n/µ.
Exercise 3
Suppose that: n > 1 (there is population growth) and µ > 1 (there is growth in the money
supply). Suppose that the government issues public debt with minimum price of k ∗ .
1. What is the minimum return that the government has to promise for the government
debt to be hold in equilibrium?
2. Write down the budget constraint of rich and poor agents in the two periods, and the
inter temporal budget constraint.
3. What is the maximum amount (of goods) that the government can extract from the
private sector (by using seignorage and public debt) if x > n/µ
4. What is the maximum amount (of goods) that the government can extract from the
private sector (by using seignorage and public debt) if x < n/µ
Exercise 4
Recall the model of three-period-lived people, in which agents receive an endowment when
they are young and nothing else in future periods ωth = [y, 0, 0]. Keep standard assumptions related to preferences. Money supply is constant and equal to M and population grows
at rate n, N (t) = nN (t − 1), with n > 1. In this model, capital paid the rate of return X but
only after two periods. Assume that intermediation (or IOU issue) is costless but observable
by the government and that capital creation is not observable by the government. Assume
further that the government prevent financial intermediaries from offering negotiable notes
backed by interest-paying bonds.
1. Assume the bonds are given a two-period maturity and are made non-negotiable (i.e.,
they can’t be traded before maturity). Is the bond going to be a substitute of capital or
fiat money? Will the government be able to raise money through seignorage? Explain
[Max 6 lines].
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2. How would your answer change if the bonds are made negotiable before maturity?
Would the government be able to raise revenue through seignorage? Explain [Max 4
lines].
3. Assume the bonds are given a two-period maturity and are made non-negotiable (i.e.,
they can’t be traded before maturity) and that government can prevent financial intermediaries from offering negotiable notes backed by interest-paying bond.
• What is the minimum return on debt for which which agents are willing to hold
debt?
• For what values of X can the government roll over the debt?
• Under the previously computed values of X for which the government can roll
over the debt, are people going to hold debt in equilibrium?
Exercise 5
Consider an OLG economy where each generation lives for 2 periods, and where all the
individuals of the same generation are identical (symmetry assumption). The utility of an
individual who was born in t ≥ 1 is given by
U (ct (t), ct (t + 1)) = ln(ct (t)) + ln(ct (t + 1))
while the utility of the old at t = 1 is simply U (c0 (1)) = ln(c0 (1)).
The initial old are endowed with m(0) = 1 units of money, while the agents of the generations
t ≥ 1 have an endowment of ωt = [ωt (t), ωt (t + 1)] = [4, 2]. The government force agents
to invest φ = 1 units of goods in money (i.e., pm (t) ∗ m(t) = ψ). The individuals can also
invest in a technology k(t) (say, capital), which allows you to transform 1 unit of good in
t to X = 1.04 units of good in t + 1. Suppose that there N (0) = 100 initial old, and that
the population grows at the gross rate n = 1.02. Moreover, suppose that is not possible to
lend/borrow.
a Suppose that M (0) = M (1) = ... = M (t) = 100 ∀t. Find the stationary equilibrium in
this economy. Compute the GDP (gross domestic product) in the first 3 periods.
b Find the stationary equilibrium for the following values of ψ: ψ = 0.5 y ψ = 0.
Compute the GDP, compare it with the result in a) and explain.
c Suppose that M (0) = 100 and that the government increase the growth rate of money
µ = 1.03 to finance government expenditure g. Find the stationary equilibrium of this
economy. Quantify the seignorage. Graph g in function of µ. Comment the results.
d Compute the stationary equilibrium for ψ = 2 with the assumption in a) (Hint: remember that k(t) ≥ 0).
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Exercise 6
Consider the model of demand deposits described in class. Suppose N = 1000, y = 10,
v k − θ = 0.6, and X = 1.3. Let each person have a 13 chance of being a type 1 and a 23 chance
of being a type 2.
a What bank portfolio can guarantee the rate of return 1 to all type 1 people and the
rate of return 1.3 to all type 2 people? How many goods are placed in storage? In
capital?
b Now suppose the type 2 people pretend to be type 1 people and withdraw early. How
many people can be paid before the bank runs out of assets? Assuming that all type
one are repaid, what is the fraction of type II agents that are not refunded?
c Suppose now that because of a government reform, capital investors are required to
provide a more transparent documentation on the initial capital investment, which
reduces the effort cost of verifying the quality of capital investment, v k − θ = 1.
c.1 How this change would affect your answer to section b)?
c.2 Is the economy going to experience a bank run in equilibrium? Explain.
d Assume v k − θ = 0.99 and repeat point c.
Exercise 10
Consider an OLG economy, where capital offers a net interest rate of 25%. The population
grows at 10% per period. In the first period (period 1) there are 100 people and the stock of
money supplied is M0 = 1 million. In each period, the government expenditure exceed the
government revenues (without consider the seignorage) by 50 goods per young person. Each
young people wants to maintain stable the real money balances at 200, and avoid to import
the inflation rate.
1 Use the budget constraint of the government to find the growth rate of money which is
necessary to finance the deficit. Find the stock of money and level of prices in period
1 and 2.
2 Suppose now that in the initial period the monetary authority does not print new
money, forcing the government to issues new debt at the new interest rate in the
market. Of course, in the second period, we assume that the monetary policy prints
enough money to repay back all the debt and to finance the deficit in that period. Find
the stock of money in period 2 and compare it with the answer in the previous bullet
point. Explain the differences.
3 Suppose now that in the initial period you anticipate the monetary policy intervention
described in the previous bullet point. What is the expected inflation rate? If contrary
to our assumptions, a greater inflation in the future discourage the use of money, for
what reason the price level will increase in period 1, despite there is no issue of new
stock of money in that period?
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