The welfare case for the European Monetary System

Jortmal of Intrmotional

,Wonry and Finance (1985).

4, 485-506

The Welfare Case for the European Monetary System JACQCES MELITZ* C’nitk de Recherche,

INSEE,

18 Blvd Adolphe

Pinard, Paris,

France

This paper argues that the European Monetary System can be interpreted as a cooperative game yielding benefits to its members. A simple model is set up in which it is shown that the EMS solution is superior to Nash. If the member countries differ, this superiority to Nash may require an occasional realignment. It will also then require that there be no sterilization of foreign exchange interventions, since otherwise the system would collapse. The EhIS is interpreted in this paper as an arrangement where all the members bear the burden of intervention in a symmetrical manner, and not a Bretton-Woods type of arrangement where there is a leader-follower relationship inside.

The

object

of this paper

can be interpreted

is to show

as a cooperative

that game

the European that

Monetary

yields

benefits

System

(EMS)

to its members.

the member nations happened to be perfectly identical in all respects joint decision entailed some bargaining costs, the EMS would even

while

If any

be the best cooperative game possible. When the nations differ, there may be a better cooperative game. But still, in principle, the EMS will always improve on the noncooperative solution (unless of course the latter coincides with the optimal cooperative one). It will do so, though, because of the possibility of a realignment. This possibility is therefore an integral part of the argument for the EMS. It follows, quite significantly, that the case for the EhIS does not imply a similar case for a currency union. All of the special issues in this paper arise in the case of differences between the members of the EMS. Such differences raise a fundamental problem of overdetermination, or one of too few endogenous variables. It is a problem that we shall only be able to resolve in case the members do not optimize to the hilt. A central message of the paper, therefore, is that the benefit and survival of the EMS requires some limitations which go beyond the formal rules of membership themselves. This essential limitation is that the members must abstain from any sterilization of foreign exchange interventions. This limitation applies to everyone. No member of the EMS retains complete monetary control; that is, the EMS is not a system where one member controls his money * I wish to thank Barry Eichengreen. Daniel Laskar, Jorge de Macedo, Marcus ~liiler. Gilles Oudiz, Jeffrey Sachs, and Charles

U’yplosz

for very

useful

comments.

Cl261-5606/85/04/048j_22503.00 c 1985 Butterworth & Co (Publishers) Ltd

486

The WeIJare Casefor the European

‘Llonetary System

supply while everybody else bears the responsibility for keeping his eschange rate fixed. Otherwise, there would be no issue of overdetermination. But then also the EMS would not be an eschange-rate union. In an exchange-rate union, all of the members share in the burden of intervention in order to keep the eschange rates fixed. Another basic feature of the paper is thus a sharp distinction between an exchange-rate union, like the EMS, and a BrettonWoods type of arrangement where one country-namely, the one that is supposed to issue the reserve currencv-stands in an asymmetric relation uis-2vir all of the rest.’ Costs of bargaining also play a crucial role in the argument. Not only do they lay down the basis for the possible superiority of the EMS to alternative cooperative schemes, but they limit the optimal composition and membership of the EXIS. Since shocks may require an occasional realignment and since, in addition, every realignment entails some bargaining costs, the indiscriminate or the indefinite admission of new members may totally undo the net advantage of the system. The order of the discussion will be as follows. First, I will develop a basic framework of analysis involving two nations, and I will esamine the noncooperative solution in this context. Next I will compare the noncooperative solution with the EMS one in two stages: first, if the nations are second, if they differ. Finally, I will examine the issue of perfectly identical; third countries in the EhIS. The comparisons of the EhlS with alternative cooperative schemes will not take place separately, but will be integrated within the rest of the discussion. The EMS can be understood simply as an eschangerate union in this paper. But I prefer the European interpretation and will comment explicitly upon this interpretation in the concluding section.

I. The

Model

Consider two nations which pursue two objectives, one about volume, the other about price. Each country has only one instrument, money, and therefore must compromise between the two objectives. Furthermore, each nation’s behavior affects the performance of the other. More precisely, each nation affects the other’s performance differently from its own, that is, its effects on the two performances are not co-linear. The only difference between the two nations that I will suppose, in order to simplify, derives from their possible pursuit of different targets for output and inflation. Otherwise they are perfectly identical (or symmetric) escept possibly in size since it makes no difference to the argument if everything is stated in per capita terms. In that case, simply interpret output as output per capita and money as money per capita throughout. The fundamental equations are as follows: (1)

q = v(p*+

e-p)

(I*)

(3)

m-p=/lg-i.i

p-p-,=+-

- v(p*+

e-p)

-w(i*-p:;+p*)

-“(i-p:,+p) (2)

q*=

<2*>

4/i)

(3*)

me-p*= p*-p-“,

/lq*-i.i* =qq*-

42)

487

J.ACQUES M~LITZ e"_,-e=i-i*

(4) (5)

I=‘I/(p-p-J+

(l-I(/)

x (p*-p-*,+ 0.5clj (6)

L=

(q-

e<

rj)‘f

I* =Il/@*-p+,)+

<5*>

e- ,)

(l-4)

x(p-p-,-e+e_,)

1

&(I-- I>’

(6*)

L* = (q*-

i*)Z+&(I*-

I>’

where q stands for quantity of output,p for price ofoutput, e for exchange rate, i price and 8+, for the anticipated for interest rate, p;, for the anticipated exchange rate in the next period, qIc for full-employment output, I for inflation, 4 and ffor the target levels of q and I, and L for loss of utility. Asterisks signify foreign variables. All variables except interest rates are in logarithms. Greek letters refer to parameters, all of which are positive. The exchange rate is the price of a unit of foreign currency in terms of home currency. Equations (1) and (l*) are the respective demands for home and foreign output, equations (2) and (2*) the respective demands for home and foreign money, and equations (3) and (3*) the respective output-price or supply of output equations. Equation (4) is a statement of interest-rate parity. It implies perfect capital mobility. Equations (5) and (5*) define the rate of inflation. In effect, they spell out the price indices that enter the official utility functions. The latter are written as quadratic loss functions of deviations from desired levels. It is assumed that the demand for money depends on p rather than the general price index Ii/p+ (1 -I(/)@*+ e) in order to simplify. Without this last and admittedly awkward simplification, the algebra would be monstrous though the qualitative results the same.2 This is a one-period model which includes no long-run relations. Hence perfect foresight has no meaning. Expectations about the next period must either be exogenous or else based, at least partly, upon current values. We shall assume endogenous expectations. Quite specifically, our assumptions will be: P”c,-P=P-P-i

(7)

p:;-p*=

(7*) (8)

et,+PZ-

p”t,=

p*-p-*, e-,-I-p?,-p_,

People are thus understood to form their expectations about inflation by projecting the current inflation into the future, while quite differently, they expect the terms of trade to go back where they were in the previous period, that is, to return to a reference level. Let us also treat the natural values of output prices and the exchange rate of the previous period, as well as the fullemployment output of both countries, as equal to one. We then havep_,=pT, =e_ ,=q,?=qT=O. Consequently, our system becomes (9)

q=r(p*+

e-p)-pi a=->0

V

(9*)

q*=

-Ct(p*+

e-p)-pi*

/L-&&,0

l-6W (10) (11)

m-p

= /.lq- i.i P=6q

<10*> <11*>

m*-p*=

pq*-;.i*

p*=dq*

The Welfare Case

488

e=i*-i+

(12)

l=$p+

(13)

L =

European Xlonetary System

2(p-p*) (1-IL)(p*+

0.5<$ (14)

for the

(q-

< j,?+

e)

z*

(13*)

=I1/p*+(l--$)(/I-e)

1

&(I--

I)’

J_* = (q*-

( 14*>

i*)L+

E(I*-

I*)’

This simplified system will serve us throughout. Thesevenequations(9),(10),(11),(12),(9*),(10*),and(ll*)forma system in seven unknowns, q, q*, i, i*, p, p*, and e, in which M and m* are exogenous. The additional equations (13), (14), (13*), and (14*) then come into play in a related optimization exercise which sets the quantity of money. The reduced-form equations for e and q in the seven-equation system (9)(12), (9*)-(ll*), are: l+zJLS(X+& (15)

(??/- Pz*)

r=j.[l+2rrd(l-2ii)]+(~~+6)(~+2Y6)

(pf6)

(16) q= [j.+b(p+6)]{i

> i(2h-

1)

rj.(l+ap) (p+6)@+ 2x6))

226(1- 2)]+

[I+

(M-II/“)

YVe impose the condition @+6)> i@1) in order to assure For8 coefficient of m- m* in equation ( 15) will always be positive. condition is trivial (it must be satisfied if only I+ p> j.).3 These equations can be rewritten more compactly as q = (a+

(17)

e = c(m-

(18)

where

b)m-

a, 6,

and c have

the obvious q*=

(17*)

lIrn* M”)

meaning. (a+

that the 6 1, this

1 We can similarly

b)m*-

also write:

bm

To elaborate somewhat, the coefficient a refers to the positive influence of monetary expansion on demand for output resulting from the fall in interest rate. The coefficient b refers to the similar positive influence of monetary expansion on demand for output stemming from the deterioration in the terms of trade. This induced deterioration depends on a greater rise in e than p-p*, which the model assures will happen. The reduced-form equations for q, e, and together with the term for I q* ((17)> (18), and (17*)), z=b$q+

imply (19)

the reduced-form I=

equation [6(11/a+ (ac/-+ [s(l-$)a-

(1-rl/>@q*+

e)

: 1)/J)+ (l-$)C]rn (al/-

1)/J)-

(I-$>+V*

J.-\CQCES hIFLIT

489

Similarly, <19*>

I’=

[6(11/a+ (2$-+ [W-t@-

1)b)f (al/-

(1-$)c]m*

(l-4)+

I)&)-

It is important to see that in this system either country could hit both of its targets if it could control both monetary instruments. Equations (17) and ( 19) can be solved for m and m* in terms of gand f Equatjons( 17*) and ( 19*) similarly can be solved for m and m* in terms of i* and I*. In the former case, involving the home country, the solution would be:

and

(21)

m*_

-

[W-

al/w-bw-

P--4w~+ (a+ 4T

u(l-$)[6(a+ 2b)- c]

This solution would obviously yield the home country total satisfaction. Two basic points should be made. First, the total satisfaction ofeither country is only possible because n/ and m* are separate instruments, that is, because the two bear independent effects on the target variables. In our particular specification, this is true because there is an interest-rate channel of influence of home money on both targets, as we can see by closing this channel of influence or setting the coefficient a equal to zero. Then we would have: q=b(m-m*)

(22) and (23)

I=

[6(&L

1)6+ (I-$)c](m-

m- M* would therefore be a single instrument. that either country has a desired ratio of uncomfortable to sustain, since even in the country could do much about this ratio. Either Z=q=O, or they set m anywhere else relative (24)

z/q=1*/q*=

m*)

Consequently the whole notion inflation to output would be best of circumstances, neither they set m = M*, and therefore to m*, and invariably

[S(2$- l)+(l-i+Q-]/b

The interest-rate influence of home money is therefore essential. Second, the m and m* combination that satisfies either country perfectly could satisfy the other one perfectly too; but this would be sheer coincidence. Of course, it would not be coincidence if we had assumed q=i=g*=?*=O. Then m=m*=O would necessarily yield total satisfaction on both sides. But we can rule out this case as unreasonable. As Robert Barro and David Gordon have recently pointed out: ‘In the presence of unemployment compensation, income taxation, and the like, the natural unemployment rate will tend to esceed the efficient level-that is, privately chosen quantities of marketable output and employment will tend to be too low’ (1983, p. 593). Given this argument, 4 and i* are necessarily positive since the logarithm of output that corresponds to the natural rate of unemployment and constant prices, qli, is zero in the model. This

490

The Welfare

Case for the European

Monetary

System

means that there is only one special linear combination of 4, j*, < and I* that would yield identical desired combinations of m and PZ* in both countries. But the occurrence of this particular combination would then be a fluke.s The best way to see this is to assume that the two countries are identical in all respects, equals m*. But this and therefore i=j* and I=? *. In this case, m necessarily can only yield total satisfaction to both countries if the right hand sides of (20) and (21) are also equal, therefore ifs L&j

(25)

1*=&j*

and

However, this last condition will not generally obtain since the output-supply coefficient 6 obviously has nothing to do with the desired trade-off between output and inflation. Thus, as a rule, identical countries will not want m=m*, though this is what they will get. Another way to see the problem is to observe that the two countries are virtually bound to disagree about the optimal value of e since, unless (25) holds, they do not want e equal to zero (as they do not want LV=PZ*), while e enters their respective output and inflation equations with opposite signs.’ This sets the stage for the analysis. We shall first esamine the welfare losses resulting from the inability of either country to control both monetary instruments that are vital to their satisfaction in the case where the two do not cooperate. Then we shall examine these same welfare losses in the EMS, or in case of a particular sort of cooperation. Finally, we shall compare them.

II.

The

Nash

Solution

When the two countries do not cooperate, suppose that they that is, each country basis of the usual Nash assumptions; opposite one’s money as given and chooses the level of money its utility losses accordingly. The optimal quantity of money country on this assumption (as we find after differentiating function (14) with regard to m after having substituted for q (17) and (19)) is:

(26)

m= {(a+b)b-E[6(+a+ [ti-

l]b)+(l-~)r][6([1-~]a-

(n+b)*+E[6($a+ + (a+b)q+&[q~a+

[I$-

behave on the considers the that minimizes of the home the utility-loss and I based on

[al/- I]@-(I-rC/)c]jm*

l]b)+(l-$)r]2

(24b- I]b+(l-_IL)r]r

(a+b)*+E[6($II+

[2$-

l]b)+(l-$)r]’

After rearranging terms in the numerator of the coefficient rewrite this equation as follows: (A- A’)m*+ (a+ b)g+ EX? ZV= (27) A

of m*, we can

with x=6(&z+ [ac/-- 1]6)+ A= (a+ 6)2+&x2 and A’= a(a+ b+ a3 x) It is then

easily

verified

(I-$)L

that

A-A’

is positive

but

less than

A. Abroad

we

J 1CQcE.S

491

bft~rrz

m R

FIGC.RE 1.

similarly

have :

(A- A’)m+ (a+ b)$*+ &xl* A

m*= (27*)

Figure 1 provides a graphical illustration of the preceding solutions for mand m*. Equation (27) is the official reaction function RR’ (of which the constant term may be easily assumed to be positive). The slope of this reaction function, of (A- A’>/A, is necessarily less than one. R*R** is then the reaction function the foreign monetary authorities (equation (27*)). The Nash solution is at the intersection of the two curves, or point nT. Since we assume that the two countries’ targets are the same in the figure, the solution is necessarily along the 45 degree line from the origin where m=m *. In this special case, the algebraic solution is : (28) If we use (28)

m= m*=

to solve for the utility L

(29)

(a+ b)jf &XI (a+ b)q+ &xl A’ = a(a+ 6+&6X)

=

loss of either

@4*+ &(a+6)*(11-J

country,

we will then have :

i)*

(a+ b+ &ax)2

Equation (29) conforms to our earlier finding that if the two co_untries are perfectly alike, their utility loss can only be nil in the special case I=sg. If we which we substitute6 4 for Iin equation (28), we will also obtain m=m*=j/a, already know to be the right answer (see note 6).

492

The

III. The

Wel]are

EMS

Case for

the European

Solution

blonctary

System

with Two Identical

Nations

Consider the case of an exchange-rate union. This case assumes that the two countries agree to maintain a certain eschange rate, ;. In our seven-equation system, (9), (lo), (II), (12), (9*), (IO*), (ll*), e is then exogenous and m- m* endogenous. m or m* must still be given, but both of them cannot be, as we have presaged above. We will interpret an exchange-rate union to mean that the burden of fixing the exchange rate is divided between both members, so that neither of them alone determines the free quantity of money. This is very important, of course. Let us first treat the simplifying case where both countries in the eschangerate union are perfectly identical. Since m=m * in this case, the only eschangerate agreement that makes any sense is ;=O (corresponding to an exchange rate of one between the two currencies, since e is a logarithm). Furthermore, if&O, no exchange-rate pressure could ever arise since the two countries, being the same after all shocks as well as before them, are necessarily identically affected by the shocks. Thus they would always act in conformity with the eschangerate agreement. There is then not even anv need to bother about what the two countries would do in the event of eschange-rate pressure. The issue would never come up. In this special case, both countries consider q=q*=

-fii=

-pi*

and I=I*=p=p* Hence, money

based on their would be

identical

utility-loss

?Z=

(30) which

involves

a utility

function,

their

optimal

quantity

of

q+&a(1 f&(P)

loss of

(31) A comparison of (29) and (31) shows that apart from the special case6 4~ ?, the utility loss will be smaller in the ELMS than under Nash assumptions. That is, (&x)2+ &(a+ 6)s (a+ b+ ESX)’ on the RHS

of (29)

is greater

on the RHS

of (31),

b ecause

(as factoring

x2+ (a+ 6)W>

(32) given

than

that

x # (a+ b)S (since

$ # 1).

out 2(af

shows), 6)6x

J.KQCES

MELITZ

493

Figure 1 provides a graphical interpretation ofthe improvement in welfare in the EMS. Following the exchange-rate agreement, the two countries will seek the highest indifference curve possible along the 45 degree line from the origin. Hence, the two will not move to point N, or the Nash solution, but to point F, where they are necessarily on a higher indifferent curve. The only arbitrary element in this illustration is that the two points of zero utility-loss, E and E*, are shown below N along the two Nash reaction functions rather than above it. This corresponds to the situation where 64 is less than f (as can be seen by comparing equations (20) and (28)). The illustration therefore concerns the particular case where the Nash solution is overexpansionary and overinflationary, or where both countries try unsuccessfully to shift the terms of trade against themselves so as to raise Z/hq above one whereas this ratio must equal one with both countries identical in all respects. Had we shown the opposite case, however, where the two points of zero utility-loss are above N, the welfare conclusions would have been the same. The EMS solution would simply have been above N (as the indifference maps would then have centered around points above N along the two Nash reaction functions), and the basic change would have been that, withsjgreater than I, the Nash solution is overly depressionary and deflationary. Our choice of illustration of N as northeast, rather than southwest of F, thus makes no difference in the analysis. Let us now carry the argument further by comparing the results in the EMS with those in the case of alternative cooperative arrangements. The EMS solution at F is necessarily on the contract curve going from E to E *. Thus it is Pareto optimal. The solution also represents precisely a half-way compromise between the two countries. Since the two are esactly alike, this is eminently reasonable: it would be difficult to find an acceptable welfare criterion that would not lead to the same compromise. In particular, Nash (1950, 1953) has proposed a very useful set of welfare postulates which, if accepted by both parties to a bargain, would necessitate their common acceptance of a solution consisting of maximizing theproduct of their utility gains from cooperation. It is easily shown that maximizing this product would lead to the same solution as the EMS one or the solution giving exactly equal weight to the utility function of each nation. Suppose, however, any alternative cooperative arrangement leading to point F through joint agreement about m and m *. Next assume a shift in any of the parameters underlying the optimal stock of money, or equation (30) (though the two countries still stay identical). A new joint decision would then be necessary. But we can only assume that any such decision would involve some bargaining costs since the Nash bargaining equilibrium is not an ideal solution for either side. Under an exchange-rate union, however, the two countries would automatically move to the new Nash bargaining solution along the 45 degree line without even consulting one another about it. An exchange-rate union then emerges as an ideal cooperative arrangement. It enables both countries continually to remain at the optimal Nash bargaining point along their contract curve without ever debating or even talking to one another about it. There is only one fundamental qualification : a conventional fixed exchangerate arrangement where one country would set its stock of money and the other one would bear exclusive responsibility for keeping e=O, would yield precisely

494

The Welfare Case for the European ‘Clonetary System

the same results. This shows that the peculiar union have not yet emerged in the analysis. IV. The

EMS

Solution

with Two

features

Different

of an exchange-rate

Members:

1

Only if the two members of the exchange-rate union differ are we able to get into the thick of the subject. First of all, the agreement about ; no longer assumes that the countries will choose values of m and m* coinciding with the requisite value of m- m*. Hence the manner in which the two countries propose to meet any exchange-rate pressure becomes essential. There must be some provision for meeting such pressure, or an intervention rule which divides up the responsibility for intervention between the two of them, or else the system would not be properly designed. Before proposing a definition of this rule of intervention, let us separate the money stock of both countries into two parts. . one that each country controls, and another that results from exchange-rate pressure. We may refer to the first part as domestic credit, d, the second as foreign reserves, f. Hence m=d+

(33)

f

and

m*=d*+

<33*>

f *

where n and f are necessarily logarithms as well as m. This logarithmic interpretation can be grasped easily in terms of a base-multiplier framework where the monetary base consists exclusively of foreign reserves and the monetary authorities control the multiplier rather than the base. Strict11 speaking, n is then the logarithm of the multiplier, but if all money besides the base consists of domestic credit, there is no harm in identifying LJwith domestic credit. Our references to domestic credit and foreign reserves also must not be taken too literally because we assume perfect capital mobility. This means that r’, i, and i* are independent of the distribution of m and m* between a’and f. The only important distinction between d and f, we repeat, is that d is a value that the authorities decide whilefis an addition to d or a subtraction from it that exchange-rate pressure imposes upon them. We can now define the intervention rule in the exchange-rate union as follows : f=

(34)

-y(d-

d*-

r4.i)

o
1

where k is simply the reciprocal of c in the reduced-form equation for r, equation ( 18). The value of y is necessarily between zero and one since rhe burden of intervention is divided between the two countries. This is a sine pa non of an eschange-rate union, as we have stressed before. From (33) and (34), we have m=

(35) In light variable

of equation (18) (which m- m*), we also have f*

(36) and (37)

(I-y)d+;f(d*+

&)

we may now view as the reduced

=d+f-d*-ke

therefore m* = (l-_)(d-ke)+yd*

form for the

JACQCES MELITZ

495

The problem of optimization for both countries is now a matter of finding the optimal values of domestic credit dand d*. If d=d* and e=O, equations (35) and (37) come down to m=dand m*=d* irrespective of the value of?. Hence, the solutions of the previous section, where we ignored?, will follow for ;=O. Otherwise, a basic problem arises if the two sets of monetary authorities take? into account in solving for the optimal value of domestic credit: there will then be no solution at all. In order to see this, suppose we substitute (35) and (37) for m and m* in the reduced-form equations for q and q*. This yields q=a(l-y)d+ayd*+(oi’+b)ki

(38) and (38*)

q*=ayd*+a(l-y)d-

The corresponding

expressions

(39)

Z=&(l--;))d+Gayd*+

[a(+y)+b]ki

for I and I* are [6(y+II/-l)ak+b(2~-l)bk+(I--)]e

and (39*)

Z*=6ayd*+&z(l-y)d+

[6(y-Il/)uk+6(1-2$)bk-(I-$)]e-

Let the respective coefficients of ;in these last two equations we solve for the optimal values of d and d*, we get d=ay(l

+ &62)d*+ j+ &bf-

(40)

be r and r*. Then

if

[(uy + 17)k+ &6r]i

u(l-)l)(l+&62)

and

Equations (40) and (40*) clearly represent two Nash-style reaction functions, that is, two equations where the optimal value of the monetary instrument of one country depends on that of the other. As compared with the preceding Nash reaction functions (( 27) and (27*)), however, there are two important differences. First, whereas the ratios m/m* in the two preceding functions (or the slopes of these two functions) were the reciprocals of one another (with one of them less than one, the other greater than one), the slopes of d/d* in (40) and (40*) are identical. In other words, these two equations define two parallel lines which either coincide or never meet. Conformably, the simultaneous solution of these two equations yields: d=d+j+E61-9*-&E61C-(u+2b+&P(2y-l))k; (41)

u(l-y)(l+&bs)

and (41”)

d* =d*+

fl*+&SI*-+6I+(u+

2b+&62(2yuy(l+EP)

Equations (40) and (40*) either have values of d and d* in the special case (42)

j_tESi-q*-

l))&

no solution,

Eaf* = [a+ 26f

or else they

&a(27 - I)]&

hold

for all

496

The

Welfare

Case for

the European

.tlonetary

System

The other or second basic difference is that rhe reaction functions, (40) and are negatively sloping. They have a slope (40*),asopposedto(27)and(27*), of -?/(I-,/). These results have a clear interpretation. In our previous derivation of the Nash reaction functions, we treated the exchange rate as endogenous. There was then a unique solution. However, now we consider the eschange rate as exogenous. If then we persist with the assumption of two Nash reaction functions as before, the system will be overdetermined, with one endogenous variable too few. This brings us back to the essential fact that under a fixed eschange rate, the two countries can only control one stock of money between them, not two. One of the two money stocks must be endogenous. If the two countries behave as we have posited in this section, each of them optimizing while simply using ; and ‘/ as new constraints, in effect both of them refuse to relinquish any monetary control. The system therefore is unmanageable. Figure 2 depicts the situation. Let ,X,X’ be the price line which corresponds to the agreed exchange rate e. (This line, which is positive and 45 degrees, is deliberately shown as not going through the origin so as to get away from the special case of e= 0.) Suppose A to be the optimal position of country 1 along this price line and B to be the optimal position of country 2 along it. (The opposite curvatures of the two indifference curves at these two points is not essential.) The reaction function of country 1, RR’, then necessarily passes through point A : that is, if country 2 fixes d* at the level corresponding to this point, country 1 will naturally place nat the corresponding level too. Similarly,

d-2

a0

mo

FIGURE 2.

J.\CQCES

497

MELITZ

the reaction function of country 2, R*R**, passes through point 8. The two reaction functions are arbitrarily shown with a slope of - 1, corresponding to a ‘/ value of 0.5. Next, suppose that country 2 sets its domestic credit at d?. According to the figure, country 1 will reply by setting its credit at A,,. This action is precisely calculated to give rise to a solution for m and m* at point A ; that is, based on the intervention rule, for a value of d- d* - ke of d,, - d,: - 6, country 2 will self country 1 will buy m,,- d,, reserves, and country l’s desired 4: - m,*reserves, solution of rn,? and m:, will come about. But of course, country 2 will then ripost at point B, to which in turn by setting d* at dz* in order to force a solution country 1 will respond by setting deven lower, at d2. The basis for the negative slopes of the two reaction functions is now evident: the higher the domestic credit set by one country, the lower the domestic credit adopted by the other in order to induce an offsetting sale of foreign reserves by the first. It is equally evident that the slopes of the two reaction functions must depend strictly on 7. The case where all credit decisions by either country will lead to equilibrium corresponds to the situation where the optimal points along XX’ on both sides coincide, and where the two reaction functions therefore also coincide. Then no matter what either country does, the response of the other country will suit the first one just as well. After a duet of foreign eschange interventions, the two countries will attain the m- m* combination both of them desire. It follows that the system cannot work if the two countries engage in full optimization. Having shown so, I will show next that the system will work if they modifv their behavior in a certain way, which I will then assume for the remainder of the paper until the closing section, where I will return to the basic question of possible indeterminancy.

V.

The

EMS

Solution

with

Two

Different

Members:

2

Upon reflection, the previous section implies perfect sterilization since it assumes that both countries attempt to obtain the same quantity of home money independently of their foreign exchange interventions. I will show next that if the countries fail to do any sterilization whatever, the EMS will yield a determinate so1ution.s A failure to sterilize can be interpreted as a decision to set d independently of Y and f. In terms of our analysis, this means, quite simply, that the home country sets d on the assumption m=d and therefore m* =d-ki, while the foreign country ,similarly determines d* on the assumption m* =d and therefore m =d*+ Ai. If so, the home authorities consider q = (a+ b)d- b(d- ki) = ad+ bki

(43) and

I =bad+

(44)

This (45)

leads

them

[(l-$)(l--&z)+6(2&-l)bk]e

to the solution

d=4+EGI--(bk+Ed[(1-~)(1-6kxz)+6(21,-l)bk~)~ a(1 +dq

498

The Welfatr

Working

parallelly,

(as*)

d’ =

Case

the foreign

for the

European

country

Monetary

will reach

System

the result:

q*+EbI*+(b~+ES[(1-~)(1-&2)+6(2rll-l)bk.])~ a(l+&@)

Since the hypothesis of no foreign exchange generally will not, be verified, the corresponding based on equations (35) and (37), and are:

interventions need not and values of m and m* must be

(l-‘i)(q+dI)+~(q*+ci5I*) (46)

PI=

a(1 + t.432) +((2;-

l)[d(l-$)(I--6ka)+bk.(l+ti*(2~-

I))]+y~~k(lfEd?)]F

a(1 + Eij?)

(46*)

m* =

(l-y)(q+ESI)+j’(q*+dI*) a(l+tdp)

+((l-2{)[E6(1-$)(l-c56a)+6k(l+ti2(21C/-

l))]+(;‘-

l)ak(l+ti’)~i

a(ltd*)

This solution is easily interpreted in terms of Figure 2. We have simply introduced the right behavioral assumptions so that the home country will set d at the level corresponding to point A regardless of d*, the foreign country will set d* at the level corresponding to point B regardless of d, and, based on the intervention rule, the solution for m and m* will necessarily be somewhere between /I and B along the price line, depending on the esact value of y. In other words, both countries will set their single policy variable, dor d’, in order to maximize their position along the relevant 45 degree line in case of no foreign exchange interventions. By no means does this imply their belief that no foreign exchange interventions will follow. Rather they realize that if they behaved differently, the system would collapse, to their mutual disadvantage. The required cooperation in the EMS is not unknown in economics. It in\-olves the same sort of self-discipline that the individual oligopolistic firm must possess in order to adhere to the best industry solution in many an economic model. Next I propose to show that it is always possible to achieve a superior solution to the Nash noncooperative one if the nations behave in the previous way. But in order to do so, I will use strictly geometrical methods since the algebra is extremely cumbersome. Let RR’ and R*R** in Figure 3 be the two Nash reaction functions that would exist in the absence of the EMS. Along RR’ the point of zero utility-loss can be anywhere depending on the exact underlying values of i and I. More precisely, j and I do not affect the slope of RR’, but they do affect its position, since the constant term of the function is (a+ 6)?+ &xl A jand fcan differ, and (based on equation (27)): F or any value of this constant, the higher jar the lower I, the higher the point of zero utility-loss (see Note 7).g Let E and E’ then be two arbitrary, alternative points of maximum utility along the reaction function RR’. Let E* and E ** be two such similar points along R*R**. These four points of maximum-utility illustrate the four possible

JACQCES

&ft.LIn

499

combinations of pairs of maximum-utility points for the two nations: both points may be too high relative to Nash; both points may be too low; one of them may be high and the other one low; or the opposite one may be high and the opposite one low. For each of these four combinations, there is a set of points which is Pareto-superior to Nash (point N). If the two points of maximum utility are E and E*, this Pareto-superior set of points may be shown by the shaded area I; if the two are E’ and E** it may be shown by the shaded area II; if they are E and E ** bv the shaded area III; and if they are E’ and E* by the shaded area IV. The relative sizes of the four shaded areas are arbitrary, but their relative positions and general configurations are not.

FIGL Rti.3.

Wherever the shaded area may be, a whole range of exchange-rate lines runs through it. Let XX’ in Figure 4 be any such line. Suppose further that _AB is the stretch of this line that lies in the shaded area. Since I have already shown that any point between the two preferred positions along XX’ is attainable through an appropriate choice of 7, all I need to show next in order to prove that the EMS can always fine an e-7 combination that is superior to Nash is that both of the points of maximum utility along XX’ cannot lie either below A or above B. This can be done as follows. Either A and B are equally preferred by one of the two nations and this nation is then indifferent between both points and Nash (N, not shown); or else

500

The Welfare

Cast-for the Ewopcan

.\lonetary

System

A is a point ofindifference to Nash for one nation and B is such a point for the other. If A and B are equally preferred by one of the two nations, then this nation’s optimal position along XX’ is necessarily between .4 and B (in light of the concavity of indifference curves) and my basic proposition follows. If A is a point of indifference to Nash for one nation and B is such a point for the other, then let C be a preferred position of one of the two nations along the price line. I must then show that the other country’s preferred position must be above -4.

m

X

As we move away from C along XX’ in either direction, the nation with a .preferred position C is worse off. Hence this nation must prefer A to B. Moreover, since this nation also considers all points along A B to be superior to or as good as Nash, B is where the nation is indifferent to Nash along XX’. But the other nation must then be indifferent to Nash at point A. It follows that this other nation must consider all points along AB above A superior to Nash. Consequently, its preferred position along XX’ cannot possibly lie below A, but must be above it. This completes the proof that it is always possible for the EMS to do better than Nash. The same argument, it should be emphasized, is impossible to make on behalf of a conventional system of a fixed eschange rate since in that case 7 is necessarily zero or one, implying that there may be no way to make both nations better off, or even to make either one of them better off. The reasoning however leaves ground for many possible ;- ‘/ combinations

J.\CQCES hit LITZ

501

that would be superior to Nash in the EAIS, and therefore leaves open the issue of the optimal e-y combination. Once again the Nash bargaining solution is a useful point of reference. If both countries accept the Nash bargaining postulates, they will find a solution for e and y by maximizing the joint product of their utility gains from forming the EhIS with respect to both e and?. This solution will necessarily place them on their contract curve since masimizing such a product necessarily places them in a position where any movement in e or ‘/ must hurt one or the other.lO Next, assume once again a change in any of the parameters of the solution for dand d*. We can no longer suppose, as in section III, that the two countries will stay on their contract curve. However, they may nonetheless stay in the shaded area (which itself, of course, will also move with the shock). The shaded area thus assumes great importance now. It means that not every permanent shock will require a realignment in order to maintain the superiority of the EhIS to may be supposed to result in a Nash.” Any shock that does so, of course, realignment. hloreover, as we have seen, there will always be some possible realignment that would enable the two countries to return within the shaded area. Admittedly, the required realignment might necessitate a change in Y as well as e. Indeed, as long as the Nash bargaining solution serves to negotiate the new agreement, we should expect every realignment to require a change in;l as well as e. A much more balanced view of the advantages of the EhIS emerges in this section than in the preceding section III. As compared with other forms of cooperative arrangements, the EMS still retains the advantage of reducing bargaining costs. However, now the EMS may produce solutions away from the contract curve, whereas some other cooperative arrangements would not. As compared to the Nash solution, the EMS also continues to yield advantages. But these advantages may now be at the expense of an occasional realignment. The whole perspective on realignments in this section is noteworthy in admitting that an occasional realignment is not a sign of failure of the EMS.

VI. Three

or More Country

Membership

in the EMS

Consider the presence of a third country. The analytical differences are minor. There are now two exchange rates, e, and e2 not one; two terms of trade,p*+ et not one; three interest rates i, i*, and i**, not two; and -p and p **+e2-p, three separate weights in each price indes, not two, so that I=rc/tp+$a(p*+ 0.5+,<

et)+ (I-tit-+,)(P**+ 1

and

o-0,<

ea) ‘-til

the preceding analysis follows. Under Nash nonSubject to these changes, cooperative assumptions, each country has a reaction function based on the separate values of the other two countries’ moneys, and the set of these three reaction functions yields a solution for m, mx, and m**, which in turns yields an equilibrium set et, ea, i, i*, and i**, as well asy,y*,y**,p,p*, andp**. If the three an exchange-rate union countries were perfectly alike, then for i1=i2=0, would be beneficial to everyone: the presence of the third member would only

502

The We//are

Case for tbr European

iLlonetar_y Systrm

enhance everyone’s advantage. If there are differences between countries, exchange-rate union will require at least two coefficients af intervention. their simplest form, the intervention rules would say:

the In

for the first country, f = -y,(d-

n*-

k&)-&(d-

n**-

1

o
for the second

k*i’?)

country,

f * =d+f-d*-k,e, =(1-‘/1-“/2)d-(1--:,l)(d*+k:lt;)+;12(d**+kpi~)

and for the third

f **=

d-I-

country,

f - d**-

= (I-i’,-‘lJd-

k:,i, (I-y,)(d**+

+&+;l,(d*+

kj,)

As before, a limited optimization which involves no sterilization would then be beneficial to everyone with an appropriate set of eschange rates and distribution rules: it, ia, yi and ya. The only important differences would pertain to the frequency and costs of realignments. But these differences are worthy of attention. Realignments stem from a combination of two factors: changes in parameters and differences between members. The more members of the union there are, the more differences between them. Hence, presumably the more members, the more frequent the realignments will be in a dynamic environment. X convenient measure of national differences between members is the size of foreign reserve movements in the union: 1f I+ if*1 + (f **/ + etc. or / F(. Let E(C) be the expected bargaining costs in the EhIS per individual member, and let N be the number of members. Then we may stipulate:

E,(C)

(47)

= 8 FI i-1 > N,+ ,)

<‘(Id)

> 0

<‘(IV)

> 0

This hypothesis supposes that the number of .members is relevant independently of its influence on reserve flows, and therefore independently of its for the obvious reason that this influence on the frequency of realignments, number increases the bargaining costs on the occasion of any realignment. Let us stipulate further that the second derivative of N in (47) is positive while the utility gain of new entrants to each esisting member tends to decline. Then we reach the conclusion of an optimal number of members. This extension of the makes sense. But I will not pursue the argument further argument, I believe, here.

VII. We have

seen

that

an exchange-rate

Conclusion union

may yield

mutual

gains

and

why

J.+CQCES MF.LITZ

503

in official these gains may encounter limits. The gains result from changes behavior. Since the goals of the authorities do not differ in and out of the union in the treatment, the changes in official behavior strictly concern the steps that the authorities take in order to achieve their aims. What about the importance of harmonization? This question is useful, I believe, in getting a general perspective on the issues in this paper. To some extent, the analysis would suggest that harmonization matters less than has often been assumed. For regardless how different the aims of the members are, thus how little harmonization there is, there are some eschange-rate rules that would enable the members to extract some of the potential advantages of goes mostly the other way. cooperation. Yet, in my view, the argument In the case of perfect harmonization, identical nations will never need to realign, With little harmonization, any news in the environment may require a realignment even for nations that are otherwise identical. The previous issue of With any lack of harmonization possible indeterminancy is also relevant. between otherwise identical nations, there is a problem of possible indeterminancy. The experience of the EMS shows us that this problem is latent at best. This would indicate that our basic assumption of no sterilization inside the EMS is a reasonable one. But nevertheless, it is difficult to believe that in the event of major reserve changes, the previous indeterminancy problem would not rear its head and become in itself a reason for realignment, or a change in the composition of the membership, or even a break-up of the system. Suppose two members of the system were to prefer a widely different m- m* solution to the existing one, so that the only thing that keeps them together in the system are the mutual adjustments in their money stocks resulting from the distribution rule (ory value), which is around 0.5. More precisely, both of them would prefer positions along their exchange-rate line that are way outside of the shaded area and onlyy reconciles them. Each country then would have a lot to gain by exploiting the difference between domestic credit and money in deciding about domestic credit as long as the other did not do the same. The temptation on both sides to benefit from this potential gain might then be too great. There is, of course, a further reason for the possible importance of harmpnization, which falls completely outside the purview of this paper: namely, official concern about foreign reserves. The empirical application to the EMS raises a number of important questions. There has been not a word about the 2.25 per cent margins around the central rates. What is the function of these margins? Unless we can answer, we cannot say whether the margins should be wider or narrower.12 Also, the exact nature of the intervention rules inside the EMS should be spelled out. hly assumption of an intervention coefficienty is clearly an oversimplification. But intervention rules do exist. They are reflected, for example, in the automatic short-term credit facilities, the indicator of divergence, and the operating rules of thumb of the member central banks. Finally, the paper has totally disregarded the outside world in relation to the EMS. This omission has drawbacks, bufit has one saving grace: namely, it helps to see that the benefit of the system to the members does not depend on their joint responses to the outside world, in particular the United States.

504

The

Wel/nre

Case

for the European ,tlonetary System Notes

1. The fact that Germany does not retain complete monetarT control in the EAIS should be clear from the rules of the organization, which have been tully operative thus far. For more discussion, see my working paper (1984). 2. As regards the literature, the present approach to international problems of macroeconomic by Hamada (1974, 1976, policy was largely initiated by Kiehans (1968) and later developed 1979). After a lapse of a few years, a resurgence of interest in the approach took place recently: See Johansen (1982). J ones (1983). Sachs (1983), Cooper (1983). Corden (1983), de Llacedo (1983). Miller and Salmon (1983). Canzoneri and Gray (1983). Rogoff (1983). and Oudiz and Sachs (1984). For another approach to exchange-rate unions, which is inspired by Poole (1970), see Alarston (1980, 1983, 1984) and Canzoneri (1982). 3. The condition is also more stringent than necessary, but it has the advantage of simplicity. 4. These last two equations depend on identical structural parameters in the two countries. If the parameters were different, the equations would be instead q = (cI+ b)m-

b-m*

and e = rm5. To find thisparticular combination simply and r* for 1 in (20). 6. In this case, note that another implication

c*m=

equate

(20)

and (21)

after

substituting

4* for 4

is

7,

since they do The signs of 9 and fin equations (20) and (21) re q uire a word of explanation not necessarily meet the eye. In order to find them, we must go back to equations (15) and (16). It turns out thatb(a+ 2b)- c in the denominator of(20) is unambiguously negative. Since the numerator of the coefficient of j in (20) is smaller than 6 (a+ ZL)- c, both numerator and denominator of this coefficient are negative, and the coefficient therefore is positive. By the same token, the coefficient of ? is negative. This last result may seem surprising. But it can be explained by the fact that the coefficient of 1 in (21) has a larger absolute value than the coefticient of I in (20). Thus, though both m and n/* fall as igoes up, a reduction in mand m* makes m- ,* rises, which raises e and thus I. Raising m- m’ through sense since it avoids any necessary rise in qalong withp (because i goes up together with e). In a similar way, the positive coefficient of 4 in (21) is greater than that of 4 in (20). implying a with a rise in q at a constant level of I (because of fall in m - m*, which is then consistent the attendant fall in P). 8. I cannot be sure, though, that there are no other reasonable and determinate solutions to the EXIS. But I have not thought of any. This conforms with the fact that as @q diminishes, Sash equilibrium eventually becomes too low (or southwest) relative to the optimum. 10. Since the contract curve goes from one point of maximum utility to the other, it necessarily passes through the shaded area. Also, between points .-l and B along the price line in Figure 2 the slopes of the two indifference curves differ from the slope of the price line. However, this does not mean that the two indifference curves cannot be tangent fo one another. These considerations confirm the possibility of i-y combinations that enable the t.xo countries to be both on their contract curve and within the shaded area. 11. Note that based on our hypothesis about expectations, dl , may differ from ieven in the E&IS, and therefore staying in the shaded area need not mean i=i*. The easiest way to think of e:, different from ; in the E&IS, I believe, is to admit that the market conceives a certain probability of realignment. A realignment may also be required in order to stay in the shaded area from one period to the next without any news in the environment, since both countries’ output and inflation must keep changing any time e is constant but the two national inflation rates differ. Kane of this affects the argument in any way, at least not if we ignore bargaining costs. 9

12. Stochastic considerations, such as those in Marston’s treatment of exchange-rate unions, are possibly important in connection with the margins of fluctuation. Suppose the issue were one of stabilization of the target variables qand [(in a stochastic environment). A fixed exchange rate would minimize the variance of these variables only in limiting cases. We know, in general, that minimum variance requires the adoption of a feedback rule. The margins of fluctuation in the EMS then may be viewed as a way to give scope for optimal stabilization: without these margins, there would be no such scope whatever. For a recent effort to found the welfare argument for the EMS on stochastic considerations in a similar kind of strategic framework as mine, see Laskar (1984).

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506

The

POOLE, vi;., ‘Optimal

Welfare

Case

for the

European

Monetary

System

Choice of Slonetary Policy Instruments in a Simple Stochastic illacro Guarterb Journai of Economics, Alay 1970, 81: 197-210. and Counterproductive Cooperative Monetary Policies.’ Federal ROGOFF, K., ‘Productive Reserve International Finanace Section Discussion Paper no. 233, December 1983. S.+cr+s, J., ‘International Policy Coordination in a Dynamic Macroeconomic hlodel,’ NBER Working Paper no. 1166, July 1983. Alodrl.’