Econ 241 A
Joshua Aizenman
Problem set #1
Textbook, Chapter 1, Exercise 2 and 7, and
Question # 3:
Consider a two periods, small open economy. There are two goods: the home good, h, and the
foreign good, m. The price of the domestic good is normalized to 1, the international relative
price of the two goods is one. The consumer starts with endowment W0 of the home good in
period zero. The real interest rate in terms of the home good is r (= the world real interest rate).
V  (h0 ) ( m0 )1   (h1 ) ( m1 )1 ;   1. Trade liberalization
Consumer’s utility is (1)
eliminates the tariff on good h at the beginning of period zero. Hence, the period-0 tariff is zero.
The trade reform is not credible - it will be reversed in period one with probability .
Consequently, the expected future relative price of the foreign good in the second period is
t Pr . 
(2)
T1   1
;
t1  1 . The tariff’s income is reimbursed to the consumer in
1 Pr . 1  
a lump sum manner.
A.
Find the impact of the anticipated policy reversal on the patterns of consumption and
saving.
B.
Denote by C t the real consumption expenditure, associated with the dual price index p t
[it has the from of Ct  (ht  Tt m) / pt ]. Characterize p t .
C.
D.
Find the expected value of C1 , C1  E C1  (E is the expectation operator).
Consider now the case where the consumer’s preferences are modified to


1
(3) V~  C0 1    C1, 1  1  where C1,  E C1 1 1 , and   0 . Explain the economic
interpretation of  and .
E. Characterize the impact of higher  on consumer’s saving for the case outlined in part a.
1
Econ 241 A
Joshua Aizenman
Problem set #2; October 8, due date: October 17.
Chapter 2, question 2, and the following three questions:
1. Consider a two periods, small open economy, under free trade. There are two goods: the home
good, h, and the foreign good, m. The price of the domestic good is normalized to 1, the
international relative price of the two goods is one in period zero. The consumer starts
with endowment W0 of the home good in period zero. The real interest rate in terms of
the home good is r (= the world real interest rate). The utility in period t is a CobbDouglass one
( ht ) ( mt )1
(1)
The future terms of trade are random: the future relative price of the foreign good in the second
period is
1 /(1   ) Pr . 0.5
(2)
.
T1  
1 /(1   ) Pr . 0.5
Denote by C t the real consumption expenditure, associated with the dual price index p t [it has
h Tm
the from of Ct  t 1t ]. The consumer’s preferences are
(Tt )
(3)

1 
~
1 
V  C0    C1, 

1
1 

where C1,  E C1 
1

1
1
, and   0 .
Characterize the impact of greater terms of trade volatility (higher ) on consumer’s saving.
Explain.
Consider a two period economy. The first period output is Y0  1 . The second period
output is random:
(1)
Y1  1   ;  is the second productivity,       for 0   .
Let f ( ) be the density function of productivity shocks. Collecting taxes is costly: a tax at rate t
yields a net tax revenue of
(2)
Ti  Yi [t  (t )]; '  0; "  0. The tax revenue is used to purchase public goods G (like
defense, etc.), and to service the outstanding debt.
The demand for fiscal services is inelastic:
2.
(3)
G0  G1  G .
The country may borrow internationally at a risk free real interest rate of r. Let B stand for
borrowing in period zero, and assume that the initial debt is zero. The government does
all the international borrowing. All agents are risk-neutral:
(4)
a.
b.
V  C0  C1
Characterize the patterns of optimal taxes and borrowing for the case where the
government is maximizing the expected utility of the representative agent, subject to the
constraints provided by (1)-(4).
Redo part a. for the case where the private sector does the borrowing.
c.
Redo part a. for the case where agents are risk averse [i.e.,
V  u(C0 )  u(C1 ); u' ; u"  0 ].
3.
Loss aversion is the tendency of agents to be more sensitive to reductions in their
consumption than to increases, relative to some reference point. It is modeled using a
generalized expected utility framework that attaches bigger weights to ‘bad’ states of
nature and smaller ones to ‘good’ states than in the conventional expected utility set up.
Consider a two period economy. The first period output is Y0  1 . The second period output is
random:
(1)
Y1  1   ;
There is an equal chance of the productivity shock being good or bad:
with probabilit y 0.5
 
(2)
.
 
with probabilit y 0.5
 
Consider the case where agents’ initial income is allocated across saving (S), and consumption
(C). Hence, the intertemporal budget constraints are
C0  1  S ; C1  S (1  r )  1  
(1)
Private agents choose a level of saving to maximize the utility (V) of loss-averse agents:
(3)
(4)
MAX
S
V
; where
V  u0  0.5 [(1   )u 1,H (1   )u 1,L ] ;
u0  u(1  S ); u1,H  u( S (1  r )  1   ); u1,L  u( S (1  r )  1   ) ;
and u is a neo-classical utility function [u’>0, u” < 0]. The extent of loss aversion is captured by
the extra weight (  ) attached to the bad state of nature in the utility function ( V ).1
Characterize the optimal saving, S. What factors determine the size of precautionary saving?
1
The loss aversion ratio is the marginal utility of a loss relative to the marginal utility of a gain. It is equal to
(1   ) /(1   ) . The ratio measures the tendency of agents to be more sensitive to reductions in their utility than to
increases. [See Kahneman, Knetsch and Thaler (JPE, 1990)]. The ratio has a value of one in the conventional utility
framework where agents assign no extra weight to bad outcomes, but it exceeds one for agents exhibiting first-order loss
aversion. Empirical estimates of the loss-aversion ratio are typically in the neighborhood of 2 (corresponding to a weight
of   1 / 3 ).
Econ 241 A
Problem set # 3. Due Date: November 4.
Joshua Aizenman
1.
The Classical Transfer Problem:
Evaluate the welfare effects of increasing the transfer TB in a D-F-S model with non-traded
goods. What factors determine the size of the welfare effect? Explain.
2.
Chapter #3, question 2.
3.
Consider a two period economy. Suppose  is a productivity shock that occurs only in
period one. Then GDP in period i (i = 0, 1) is
(1)
1  
Y0  1; Y1  
1  
with probabilit y
0.5
with probabilit y
0.5
.
Collecting taxes is costly: a tax at rate t yields a net tax revenue of
Ti  Yi [t  (t )]; '  0; "  0. The tax revenue is used to purchase public goods G (like
(2)
defense, etc.), and to service the outstanding debt.
The demand for fiscal services is inelastic:
(3)
G0  G1  G .
All agents are risk-neutral:
The emerging market can borrow in international capital markets. It borrows B in period 0 at a
contractual rate r and owes (1 r)B in period 2 [assume that the initial debt is zero]. If it
faces a bad enough productivity shock in the second period, it defaults. Default is not
without penalty, however. International creditors can confiscate some of the emerging
market’s export revenues or other resources equal to a share  of its output. The more
open the economy, the greater  is likely to be. We assume that the defaulting country’s
international reserve holdings are beyond the reach of creditors.2
In period 1, the country repays its international obligations if repayment is less costly than the
default penalty. The country ends up transferring S1 real resources to international
creditors in the second period, where:
(3)
S1  MIN (1  r) B; Y1  ,
0  1
Suppose the risk-free interest rate is rf . The interest rate attached to the country’s acquired debt,
r , is determined by the condition that the expected return on the debt is equal to the riskfree return:
This is a realistic assumption. For example, on January 5, 2002, The Economist reported “[President Duhalde]
confirmed that Argentina will formally default on its debt, an overdue admission of an inescapable reality. The
government has not had access to international credit (except from the IMF) since July. It had already repatriated nearly
all of its liquid foreign assets to avoid their seizure by creditors.” (The Economist, p. 29)
2
(4)
E[ S1 ](1  rf ) B
a.
Find the supply of funds facing the economy.
b.
Characterize the optimal borrowing and the demand for international reserves for the case
where (1  rf )   1 .
Evaluate the impact of lowering  on the optimal demand for B and R. Explain.
Problem set # 4. Due Date: Questions 1-2 November 19. Questions 3-4 November 26.
1.
Assume a Cagan demand for currency:
(1)
m  exp( 0   ) .
The banking system is competitive, subject to a binding reserve requirement . The
real loan interest rate is zero. The nominal interest rate on reserves is zero. The
demand for deposits is
(2)
d  exp(  0   [  id ]) , where id is the deposit nominal interest rate.
a.
Find the equilibrium id .
b.
Find the inflation tax collected for given inflation and reserve requirement.
c.
Find the inflation and the reserve requirement that would maximize the inflation tax.
d.
Suppose now that the lending interest rate is il   (1   ) ,   0 . What is the economic
significance of  . How this will modify your answer to part c? Who is paying the
inflation tax? Explain.
2.
Find the first order condition characterizing the optimal  CB in the model discussed in
section 9.5.2.1.
3.
Agents in the emerging market are also risk neutral, having preferences represented by
C*2
*
*
V  C1 
* . The only source of macro uncertainty is the shock,  , to second period output in
1 
*
*
the emerging-market Y2  Y (1  ) where  is governed by the probability density function f ( ) ,
    ,   0 . All private agents are price takers.
*
In period 2, the borrowing country is required to repay its aggregate loan, B1 , but because of
the stochastic output it may default partially. Let S2 denote the debt repayment to foreign creditors in
period 2. In the event of default, creditors are assumed to be able to penalize the borrowing country,
*
reducing its net output by Y2 . The parameter  reflects the bargaining power of foreign lenders,
indicating that up to a fraction  of output can "confiscated" due to the threat of embargoes, etc.
*
*
Consequently, the effective ceiling on net resource transfers to creditors is: S2  min [(1  r)B1 , Y2 ] .
The international credit market is risk neutral, characterized by competition among banks that are
fully informed regarding the debt exposure of the country. The risk free interest rate is r*. Partial
default by the emerging markets requires that banks should spend real resources  in order to verify
the productivity shock and to enforce the repayment. We assume that the partial default decision is
centralized, being made and enforced by the policy maker (like the Central Bank or the Treasury),
*
*
who follows the default rule described by S 2  min[( 1  r ) B1 , Y2 ] . If default would occur, the debt
*
would be serviced by the treasury, imposing the lump sum taxes need to collect Y 2 .
a. Find the equilibrium financial spread [i.e. r –r*].
b. Characterize the optimal borrowing from the point of view of a small consumer (recall that
each borrower is small, and price taker).
c. Extra credit: Consider the case where f( ) is a uniform distribution. Characterize the
optimal borrowing from the point of view of the central bank (note that the central bank is
not a price taker). Describe policies that may be needed to implement the optimal
borrowing.
4.
Chapter 6, question #4.