This paper examines the argument that the fixed exchange rate regime should be preferred to the flexible rate regime because the former allows risk sharing across countries while the latter does not. The analysis is performed in a two-country overlapping generations model, where markets are incomplete under either exchange regime. In this second best world, it is demonstrated that the ability to share risk across countries in the fixed rate regime does not necessarily lead to higher welfare than the inability to share risk in the flexible rate regime.

David A. Hsieh

English

Research Papers in Economics

NBER WORKING PAPER SERIES
INTERNATIONAL RISKSHARING AND THE
CHOICEOF EXCHANGE RATE REGIME
DavidA. Hsieh
Working Paper No. 842
NATIONAL BuREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge MA 02138
January 1982
I would like to thank Rudiger Dornbusch and Michael Mussa for their
very helpful comments on earlier drafts of this paper.
The research reported here is part of the NBER's research program
in International Studies. Any opinions expressed are those of the
author and not those of the National Bureau of Economic Research.NBER Working Paper #842
January 1982
INTERNATIONAL RISK SHARING AND THE CHOICE OF EXCHANGE RATE REGIME
Abstract
This paper examines the argument that the fixed exchange rate regime
should be preferred to the flexible rate regime because the former allows
risk sharing across countries while the latter does not. The analysis
is performed in a two-country overlapping generations model, where markets
are incomplete under either exchange regime. In this second best world, it
is demonstrated that the ability to share risk across countries in the
fixed rate regime does not necessarily lead to higher welfare than the
inability to share risk in the flexible rate regime.
David A. Hsieh
Graduate School of Business
University of Chicago
1101 East 58th Street
Chicago, IL 60637
(312) 962—74181 •Introduction
One argument which has been advanced in favor of the fixed exchange rate
regime over the flexible rate regime is that the former allows risk sharing
across countries (See Mundell [1973]). This is not necessarily welfare
improving, however, when the set of markets are not complete (in an Arrow—
Debreu sense) in either regime.
This paper makes use of a two—country overlapping generation model similar
to that in Karekan and Wallace [1981], and outlined in Section 2. Individuals
in one country face an exogenous uncertainty in their endowments, while individ-'
uals in the other country receive a nonrandom endowment. The flexible rate
regime, discussed in Section 3, permits no risk sharing. The fixed rate regime,
on the other hand, forces agents in the country with nonrandom endowments to
share the risk of agents in the country with random endowments, as explained in
Section 4. Explicit welfare comparison in Section 5 shows that the choice of
the exchange rate regime rests on the degree of relative risk aversion. Some
concluding remarks are offered in the final section.
2. The de1 andNotation
There are two countries, denoted A and B, and one nonstorable con-
sumption good. All agents are identical,!' live for two periods, and receive
an endowment of the good only in the first period. Each agent is indexed by a
vector (i, t h), where i is the country of residency, t the current date,
and h his age. A young person is denoted by h =1,and an old person by
h =2.There are Na agents in each generation residing in country A,
and Nb in country B.
.iLThe case where preferences are different across countries is considered
later. It does not alter any results of the paper.2
When young, an agent of generation t in country i receives w1(t)
units of the consumption good, which depends on the state of nature z(t).
For simplicity, there are two states of nature, denoted by "one" and "two",
occuring with probability q and (1 -q)respectively. Each draw of z(t)
is stochastically independent of any other draw. An A—young gets w units in
state one, and w units in state two. Without loss of generality, assume
w > w. Each B—young gets the same endowment in both states, i.e.,
b b
Wi =W2.
He consumes c'it, 1) units when young, selling the rest for m1(t)
units of real balance. Money is the only store of value. Each country issues
its own currency, the nominal quantities of each are fixed at M1. An agent
is allowed to hold only money of his own country..Y When old, he purchases
c1(t +1,2) units of consumption, leaving no bequest.
The problem faced by each young person is to maximize his expected hf e—
time utility, which is separable in the two periods:21
(2.1) maximize U(c1(t, 1)) +E[V(c1(t+1,2))]
subject to c1(t, 1) =w1(t)—m1(t)
c1(t +1,2) =t)p]t)/pl(+1)
c'it, 1), ci(t +1,2), &(t) > 0
The agents know the current prices, Pa(t) and pb(t), and the distribution
of prices in the next period. E[ Iisthe expectations operator.
.YOther regimes, allowing agents to hold money from both countries are
considered in Karekan and Wallace. The equilibria under these regimes all
lead to a constant exchange rate, and can be identified with the "fixed
exchange rate regime" of this paper.
2/This is not essential, but simplifies the mathematics.3
We consider actions of agents only in stationary equilibria. A
stationary equilibrium is one in which the sante prices, a and b, occur
whenever the state of nature is z. It is completely characterized by the
aa a b bb4/ ratios of prices in the two states, =
p1/p2and =
Anindividual born in state one will solve the first-order condition:
(2.2) U'(w —m)
=qV'(m)
+(1—q)V(m1),ia, b
and it is straightforward to show that the optimal real balance, m(4), is
a decreasing function if R < 1, a constant function if R =1,and an
increasing function if R > 1, where R is the degree of relative risk
aversion of the second period utility:
(2.3) R(x) =—xV"(x)/V'(x)
as defined in Pratt [1964]. His expected lifetime welfare is:
(2.4) w(4) =U(w
—m())
+qV(m(4?))
+(1—q)V(m($1)4),
i =a,b
An individual born in state two will solve the first—order condition:
(2.5) U'(w —m)4
=qV'(m/)+(1—q)V'(m)4,
i =a,b
The optimal real balance, m(), is an increasing function if R < 1,a
constant function if R =1,and a decreasing function if R > 1. His
expected lifetime welfare is:
(2.6) W(1) =U(w
—m())
+qV(m(4)/cf)
+(1—q)V(m(4)),
i =a,b
!"isa measure of uncertainty of prices. A rise in 4increases
the mean and variance of prices.4
From (2.4) and (2.6), we can calculate the expected lifetime welfare of
an unborn individual:
(2.7) W1() =W(4)
+(1
W'(4) is a convex function when R < 1, a constant function when R =1,
and a concave function when R > 1. This fact is crucial in the comparison of
welfare in the two exchange rate regimes.
3.uilibriumunderthe FlexibleRate Regime
Each country has its own central bank. In the flexible rate regime, the
sole function of the central banks is to distribute money to the old at time
t =0.They do not intervene in the exchange market. The trade balance in
the two countries must therefore be zero, since agents are not allowed to hold
the other country's money. Hence prices are random in A, and nonrandom in B.
In country A, the goods market at time t clears when the money market
clears:
(3.1) Napa(t)ma(t) =NaPa(—1)ma(t-1)=Ma
The stationary equilibrium is a price pair (pt, p), which satisfies:
(3.2) =Ma/ENama],z =1,2
where m and m are the optimal real balances in the two states. We can
a a a a a a show that m2 > m1, and w —
m2> w1 —in2,i.e., A—young born in state
"two" will save and consume more. So the stationary equilibrium price ratio
aa aa a =p1/pm2/m1 is greater than unity.
In country B, there is no uncertainty in the stationary equilibrium.
Hence prices and real balances are constant:5
b b
(3.3) m1 =m
(3.4) p p =Mb/[Nbm]
The equilibrium price ratio is:
(35) =1
The equilibrium exchange rate in state z is given by the "law of one
price" :/
ab
(3.6) z"z' z =1,2
which is the B—currency price in terms of A—currency. The crucial result is
that the exchange rate is different in the two states:
(3.7) si/s2 =a>
4.ui1ibriuunder the Fixed Rate Regime
In the fixed rate regime, the central banks cooperatively fix the ex-
change rate at S forever, and are willing to trade any amount of currencies
at this rate. This fixes the world private stock of money at MMa +SMb,
whose composition is determined by demand conditions. The world goods market
clears when the money market clears:
(4.1) (t)[Nama(t) +Nbmb(t)]=ir(t-l)[N%a(t-1)+Nbmb(t-1)]=M
where ir(t) is the price of consumption in units of A—currency.
!/we assume that there are no barriers to trade, i.e., no transport
costs, tariffs, quotas, etc. Therefore goods arbitrage ensures the validity
of the "law of one price."6
The stationary equilibrium under this regime is a price pair it2)
whichsatisfies:
raa bbi
(4.2) =M/LN+ N z =1,2
where i4gp,and are the optimal real balances in the two
states. The equilibrium price ratio, =1t1/1t2 isthe fixed point of
[Nam() +Nbm($)J/[Nam()
+Nbm()].Itcan be shown that is between
=1and •a > 1.(The proof is outlined in Appendix 1.)
There are several interesting points about this equilibrium. One, the
real balances and the price ratio are independent of the money supplies and
the (fixed) exchange rate. Second, the larger the population of A relative
* a to that of B, the closer is to > 1; conversely, the smaller the
* b population of A relative to that of B, the closer 4isto c= 1.
Third, A—young born in state two desires to hold a higher real balance
than those born in state one. In fact, they hold a higher nominal balance:
> iru. (A proof is furnished in Appendix 1.)
Four, the nominal trade balance is zero on average, while the real trade
balance is positive (negative) for country A (B) on average. This comes from
the previous observation. There are three cases: (a) If the state of nature
was "one" last period and "two" this period, the current A-young will want to
accumulate a higher nominal balance than the current A—old. They sell some of
their endowment to B—old, running a trade surplus. (b) Conversely, if the
state of nature was "two" last period and "one" this period, A will run a
trade deficit of an equal nominal amount. (c) When the same state of nature
occurs consecutively, the trade balance in the second period is zero. Since
the events (a) and (b) occur with equal probability, the nominal trade balance
is zero on average. But the real trade balance is nonzero, since A tends to7
run a surplus when the price of consumption is low, and a deficit when it is
high, which implies that A expects to run a real trade surplus.
Intuitively, the fixed regime unifies the currency and goods markets,
forcing B—residents to share the endowment risks of A—residents. In a good
state, i.e., z =2,A "gives" B consumption by running a trade surplus.
In a bad state, i.e.,z =1,A "takes" consumption from B by running a
trade deficit. This risk sharing arrangement increases the mean consumption
in B, i.e., A expects to run a real trade surplus.
5.Comparisonof ExchangeRate Regimes
First,we compare the distribution of random variables under the two
regimes.From the point of view of country B, the fixed rate regime
increases the price variance, because the disturbance is external to country
B. On the other hand, the fixed rate regime reduces the price variance for
country A, since the disturbance is internal to country A. It is also easy
to show that the mean and variance of aggregate consumption are lower (higher)
in country A (B) under the fixed rate regime. This agrees with the usual
findings, as in Fischer [1977].
Second, we compare the expected lifetime welfare of an unborn agent,
W1(), between the two regimes. As noted in Section 2, W1() depends onO<R<l
Figure 1
Welfare Comparisons Between Exchange Regimes
8
R= 1 R> 1
1
1'1' c 1 q'a9
the degrees of relative risk aversion. The various cases are exhibited in
Figure 1, and summarized as follows:
(1)If R < 1, w1() is convex. Wa() is decreasing over the
interval (1, p),forsome4i between 1 and 4aand increasing
over p,co•wb() is decreasing over (0, 1), and increasing
over (1, co)
1 (2)If R =1,W ()isconstant.
(3) If R > 1, W1() is concave. Wa(,) is increasing over the
interval (1, iv),forsome
i4ibetween1 and ca and decreasing
over ,. isincreasing over (0, 1), and decreasing
over (1, ).
(4)If either R < 1 or R >1,Wa(l) >Wa()
(The proofs are in Appendix 2.)
Clearly, B—individuals prefer the fixed regime if R < 1. They are
indifferent if R =1,and they prefer the flexible rate regime if R > 1.
These results are simple to explain. For a given increase in both the mean
and variance of consumption, B—residents are better off if they are not very
risk averse, and worse off if they are very risk averse.
The cases for A—re5idents are not so clear. If they are highly risk
averse, i.e., R > 1, they prefer the fixed rate regime, because the reduction
in the variance of consumption more than compensates for the reduction in the
mean of consumption. If R =1,they are indifferent. But if they are not
very risk averse, i.e., R < 1, then the case is ambiguous. When A is large,
* a is close to c,andthey prefer the flexible rate regime. When A is
* b
small, is close to =1,and they prefer the fixed rate regime.
Thus far, the analysis has assumed that preferences are identical across
countries. All the results will obtain, when we allow preferences to differ10
Table 1
WelfareComparison Between Exchange Regimes
Country B
b b b R >1 R =1 R <1
Ra 1 (+,—) (+, 0) (+, +)
aR =1 (0, ) (0,0) (0, +)
Country A
Ra 1 (?,) (7,0) (7, +)
* (—,—) (—, 0) (—fl,+)
** (+,-) (+, 0) (+, +)
Thefirst entry of the ordered pair pertains to country A, the second entry
pertains to B. The following conventions are used:
+: prefer fixed regime
0: indifferent
-:preferflexible regime
* Note: A is large relative to B
**
Ais small relative to B.11
across countries (but remain identical within each country). Table 1 gives the
welfare comparisons. The polar cases are the most interesting ones. For example,
Ra >1and Rb < 1 imply that both countries prefer the fixed rate regime.
When Ra < 1 and Rb > 1, and when A is large, both prefer the flexible rate
regime. There are also examples where the countries disagree——when Ra > 1and
Rb > 1, or when Ra < 1 and Rb <1with A large compared to B.
5.Concluding iinrks
When there are no missing markets under either exchange regime, money plays
no essential role in the model, and so the allocation of real resources is
invariant to the exchange regime.2! In this case, there is no reason to prefer
either regime. On the other hand, when some markets are missing under both
regimes, it is not clear that the ability to share risk across countries in the
fixed rate regime improves welfare over the flexible rate regime which prevents
any risk sharing. This is a standard result in a second best world, and is
likely to be robust against all modifications of the model as long as markets
remain incomplete in both regimes.
By the same token, there are circumstances in which one exchange regime has
a complete set of markets while the other regime does not. We then opt for the
regime which attains the first best equilibrium. For example, we can introduce
a forward exchange market into our model, thus completing the set of markets in
the flexible regime but not in the fixed regime, making the former preferable.
Another modification isto include random endowments in country B, such that
w =w,w =w,and Na =Nb.There will be no aggregate uncertainty in the
fixed rate regime, which makes it preferable over the flexible rate regime.
2./See Lucas [1981].A-i
AppendIx 1
Lmm 1: ( m()/ if and only if 1.
Proof: Define h() =cfm'(4)
-m()/.
Dropping the superscripts to save
notation, we have:
h'={i +m/m]mi ÷ [im2/m2][n2] 0
itis easy to show that[i + 4rn/m1} > 0, and[1 —m/m2J> 0. Thus, h' > 0.
Now h(1) =0.Thus h() > 0 for 4>1, h(4) < 0 for < 1. Q.E.D.
Corollary 1: m() > m() when > 1.
Ta 2: zn() m(4)/ if and only if 4i,for some 4in(1, 4).
Proof:Define h() =m($)
-m($)/.
Dropping superscripts to save nota-
tion, we have:
h' =[1+cm;/m]m+[i—m2/m2J[m2/]
0
Clearly,h' > 0, for the same reasons as in Lemma 1. Note that h(1) < 0,
since m(1) < m(1).
Also h(4) >0,since aa(a) =a($a)> m()/. Therefore, there
exists a unique 4between1 and •a, such that h(4) = 0.
*a TflEOEI1: 1< <,.
Nam() + Nbm() Proof:Define =
aa b b
•Clearly,f(1) > 1, since
N m1() +N
m(1)> m(i), and m(1) =ui(i).Also, fct,a < 4aThis is shownasfollows:
m() =4m(4).Hencef(4a) < a if and only if m(4)<A- 2
a * whichis true by Corollary 1 (since > 1). By continuity, there exists
* * b * a such that $= f(),and =1<
Corollary 2: =a
> c)*m(4)*)=
* aabbaa bb b *b b a a*a Proof: c[Np,1 + N =N + N So N [
p1
—
1.12]
=N2 —
* *b b a*a Since 4,>1, we know 4,1i.i > U2, from Corollary 1.Hence 112 > 4,A- 3
Appendix 2
Lmm 3: Let f(x) =xV'(x),where V( )is increasing and concave.
Then ft(x) > 0 if R(x) < 1, f(x) =0if R(x) =1,and f'(x) < 0 if
R(x) > 1.
Proof: f'(x) =V'(x)+ xV"(x) =V'(x)[1+XV"(X)] =V'(x)[l—R(x)]. Q.E.D.
Now, define:
g.(q.) =E[w1()]=qw(4)
+ (1 —q)w(4)
=
qU(w
—m(4))
+ (1 —q)U(w
—m(4))
÷ q2V(m(4)) ÷ q(1 —q)V(m(4))
+ q(l —q)V(m()/4)
÷ (1 —q)2V(m(4))
,fori =a,b
Note that g(4) =q(1—q)m()V'(4n($)) —
Considerthe case for i =b.At •= 1,g,(1) =0.Suppose < 1. Then
< by Lemma 2. Suppose R < 1. Then
< [m(4)/4]V'(m($)/$), by Lemma 4.In other words, g,() < 0 for < 1.
Similarly, g() > 0 for c > 1.This means that Ewb(cf)) is increasing
over (0, 1), and decreasing over (1, °°), reaching a maximum at t= 1.
The other cases for B are shown analogously.
Now consider the case for i =a.By Lemma 2 we know there exists 4
between 1and •a such that = Hence g(4i) =0.Suppose
< .Thenm($) ( m(4)/, by Lemma 3. Suppose R > 1. Then
> [m(4)/4]V'(m($)/), by Lemma 4. So, g($) > 0 for
< .Similarly,g'($) < 0 for > 4. This means E[wa()] is
decreasing over (0, ), reaching a minimum at =,and increasing over
(p,). Theother cases for A are shown analogously.R- 1
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