Bloc formation in international monetary policy coordination

(1)

EUI WORKING PAPERS

(2)

European University Institute I im iim --- - rJIFe ~m 3 0001 0035 5456 7 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(3)

EU R O PEA N U N IV ERSITY IN STITU TE, F L O R E N C E ECONOMICS DEPARTMENT

EUI Working Paper ECO No. 97/31

Bloc Formation in

International Monetary Policy Coordination

Marion Kohler

WP 3 3 0 EUR

BADIA FIESO LA N A , SAN DO M EN ICO (FI)

(4)

No part of this paper may be reproduced in any form without permission of the author.

© Marion Kohler Printed in Italy in December 1997

European University Institute Badia Fiesolana I - 50016 San Domenico (FI)

Italy © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(5)

Bloc formation in international monetary

policy coordination *

Marion Kohler

European University Institute, Florence

Abstract

In a standard framework of international monetary policy games we show that countries will prefer to split up into several coalition blocs of a smaller size rather than forming one big coalition. Depending on the strategic position between the coalitions in an equilibrium there will be either only coalitions of the same size or the ‘leading’ coalition will be smaller than the ‘follower’ coalition.

A possible application of these results is the formation of currency blocs. Large currency blocs, though Pareto Optimal compared with non-cooperative situa tions, are not sustainable because there is an incentive for individual countries to stay outside a bloc. However, the formation of several smaller blocs may be the outcome of individually optimal decisions. The result is not dependent on model asymmetries in the underlying trade structures or asymmetric shocks.

*1 want to thank Michael Artis and Mark Salmon for helpful discussions and com ments. Of course, all remaining errors axe mine. Comments are most welcome. E-mail: kohler@datacomm.iue.it . © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(6)

(7)

1 In trod u ctio n

The theoretical literature on monetary policy coordination has mostly only considered full coordination through the cooperation of all countries invol ved as an alternative to non-cooperative policy making. Cooperation of all countries is pareto-superior to non-coordinated policy-making in many situations. This argument has been considered to be sufficient to enforce1 cooperation which forces countries to deviate from their individually opti mal reactions. However, another possible outcome has been neglected, the existence of partial coordination through coalition formation. This latter possibility seems to match reality more where we do in fact observe bloc formation rather than the two extreme cases described above.

Motivated in particular by the discussion on a hard core EMU, some aut hors have considered the possibility of partial coordination that is, coalition formation in monetary policy. I shall discuss four of these contributions in more detail.

Canzoneri and Henderson [7] use a three country setup where two countries form a coalition by assumption and play Nash against a third country. They show that when assuming an optimal response of the non-member the result is not necessarily pareto-preferred to the non-cooperative Nash- outcome. This can be seen as “partial coordination might not pay off” in the debate whether coordination is beneficial. Like in most of the models which consider coalition formation, the result of Canzoneri and Henderson is essentially due to asymmetries in the model.

’ In repeated games, coordination can be enforced through trigger-strategies.

(8)

Martin [14] shows in a dynamic model with three countries - two of which are low-inflation countries, the third is a high-inflation country - that waiting for convergence in order to form a three-country union carries a danger. The third country has a free-rider incentive not to undergo the monetary discipline within the union but to stay outside once it has converged in reputation. This paper sheds light on the main force which drives the result in Kohler [13], the positive spillovers from the coalition formation itself. However, Martin’s result holds only in the asymmetric case and, hence, his policy recommendation is to offer more voting power to the smaller - third - country, based on the solution offered by Casella [8]. Martin’s model differs from the model used in Kohler [13] which is as well the basis of the model used here in that there is ‘world-wide’ only one good which is produced in all countries and international spillovers occur on the supply-side, the production sector. Methodologically, he does not use a game-theoretic framework based on the notation in the cartel literature.

Finally, Buiter et al. [3] extend the model of Canzoneri and Henderson to the n country case. They distinguish between a ‘centre’ country which does not participate in coordination and n symmetric periphery countries which have entered a cooperative agreement in a fixed exchange rate regime. They analyse the break-down of the cooperative agreement in the case of an asymmetric aggregate demand shock which hits the centre country. The peripheral countries trade goods and services only with the centre country. This simplifying assumption, however, neglects one aspect. When there are underlying economic spillovers between the peripheral countries it may

(9)

not be optimal for all of them to stay in a cooperative agreement even in the case of a symmetric shock. The result in Buiter et al. is also based on asymmetry features like an asymmetric shock and asymmetries in the objectives of the policy makers.

Alesina and Grilli [1] draw attention to questions of reputation and credi bility. They investigate coalition formation in the sense of a ‘multispeed Europe’ in a five country model. Although the title of the paper implies international monetary policy issues, it highlights rather the question of commitment technology. Countries differ in their degree of “conservativen ess” , i.e. in their emphasis on the objective of price stability relative to employment. They show that a coalition might include only two or three countries. A country outside the Union will join the Union if it can gain credibility and improve upon the Nash equilibrium. In that sense the paper is perfectly in line with the results established by Rogoff [17], who shows that a government might be better off appointing an (independent) con servative central banker in order to avoid the time-inconsistency “trap” . Alesina and Grilli provide a theoretical argument for the current political discussions about the European central bank, its degree of independence and its influence on price stability in member countries. Since we use a static game we do not consider issues like credibility and reputation. Unfortunately, the model of Alesina and Grilli is not very useful for our analysis since it lacks links between the economies which transmit spill overs. This lack of underlying economic links creates a problem of another nature: the union has no reason to allow in additional members since they would deteriorate the union’s losses. Hence, one has to add a free-entry

(10)

assumption in order to allow for coalition formation. Underlying econo mic links may make an extension of a coalition profitable for all countries involved as we will show below.

When determining a coalition we have to ensure that it is credible in the sense that the decision whether to join a coalition or not is individually optimal for each country. Asymmetries are one way to ensure that some countries have an incentive to stay in a coalition or that some countries want to leave it. In Kohler [13] we have chosen a different approach: we have shown in a symmetric model that there are forces in the underly ing economic structure and during the coalition formation at work which restrict stable coalition sizes. These forces may be best worked out in a symmetric model where asymmetries do not force the system into the desi red stable equilibrium. In Kohler [13] we have shown with the model used by Canzoneri and Henderson and by Buiter but with a symmetric setup that coalition formation might stop at three countries since the formation process leads to pareto-superior results only for up to three countries. A fourth country would be worse off joining the coalition than remaining in the fringe. In this paper we will ask a different question. We will in vestigate whether the countries outside the coalition would rather form a second coalition than play noncooperatively against all other countries and the coalition. In this sense, we aim at providing a model which can explain the co-existence of currency blocs.

(11)

2 T h e underlying econom y

The individual country’s economy is described by a standard model of monetary policy which is based on Canzoneri and Henderson [7]. We will extend their model to the n country case.

All variables represent deviations of actual values from zero-disturbance equilibrium values and are expressed in terms of logarithms except for the interest rate. The domestic country’s variables are indexed by i; j = 1 .. . n, j ^ i denote the foreign countries. We will focus our attention on a symmetric model which allows us an analytical solution of the equilibrium for n countries2. Hence, the values of the model parameters do not vary over countries3.

Each country is specialized in the production of one good. Output «/, increases in employment 1; and decreases with some (world) productivity disturbance x (independently distributed with mean 0). The following

2It is not only considerations of analytical tractability which make it appealing to analyze the symmetric. We will show that the equilibrium can have asymmetric features although the model structure is symmetric. It is often argued that real world coalitions Eire based on hegemonic or at least asymmetric structures. But the existence of an asymmetric equilibrium in a symmetric model implies that eliminating asymmetries in the structures amongst the individual countries (through e.g. policies aiming at conver gence of important economic indicators) does not necessarily eliminate the asymmetric structure of the outcome.

3In Kohler [13] we solve the reduced form for a model where only some parameters are not indexed and thus not allowed to vary over countries. In particular, the openness of a country /3 and the weights in the policymakers’ loss functions might be different for different countries.

(12)

equation describes the supply of the domestic good, based on the production function.

y{ = (1 - a)li - x (1)

with a between 0 and 1.

Profit-maximizing firms hire labour up to the point at which real wages

(w — p) axe equal to the marginal product of labour. Labour demand is

therefore determined by:

Monetary policy is effective because of contractually fixed nominal wages. Home wage setters set w at the beginning of the period so as to fix em ployment at a full-employment level (1; = 0) if disturbances are zero and expectations axe fulfilled. They minimize the expected deviation of actual employment fxom full-employment by setting the nominal wage:

with m\ the expected money supply deviation and wt the deviation fxom the full-employment wage-level4. Actual labour demand might differ due to unexpected disturbances. It is assumed that the wage setters guarantee that labour demanded is always supplied.

The consumer price index qi is a weighted average of the home country’s and the foreign countries’ price levels where all foreign countries are weigh ted equally (according to the structure of the demand equation below). /3

4Equations 1, 2 and 8 give m = w + l. Home wage setters solve the optimization problem minwE[n2\ = minwE[(m — w)2]. This is obviously minimized by setting w equal to me. For the time being we will set m ' = 0.

U)i — Pi — —ali — X (2) (3) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(13)

is the fraction of imported goods in total consumption.

Qi = (1 — P)Pi + 0 ~ z T E ( ev + Pi) (4)

n 1 j=i 3*i

Pi is the price of domestic output and e,;- is the nominal exchange rate, i.e.

the price of the currency of country j in terms of the domestic currency. Price increases abroad raise the domestic price level through the share of imported goods.

The real exchange rate z defined as the relative price of the foreign good in terms of the domestic good is:

zij = (eij + Pi ~ Pi) (5)

The demand for the good produced in the home country is:

Vi = 6 E za+ ( ! -P )m + —3 T E P£ib - C1 - P)vri — t t E Pvri (6)

Consumers spend the fraction e of their income yj on consumption. They spend the share /? of their expenses on foreign goods and the share (1 —

P) on the domestic good. Demand for the domestic good rises with yj, j = l , ... ,n. A rise in the relative price of a foreign good shifts world

demand from the foreign good to the home good by 6. The demand for all goods decreases with expected real interest rates, r*. The residents in each country spend the amount u less for each percentage point increase in the expected real interest rate.

The expected real interest rate is:

r, = ii ~ q- + Qi (7) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(14)

where i,- is the nominal interest rate and q\ is the expected value of the consumer price index tomorrow based on the information available today. Goods market equilibrium is obtained by equating demand and supply. The market equilibrium for money is realized when the money supply sa tisfies a simple Cambridge equation:

mi - Pi + yi (8)

International capital mobility and perfect substitutability of bonds give an

additional condition:

ii = ij + eij — eij (9)

for all i , j — 1 Only with this condition will private agents hold

positive amounts of both bonds.

In our model, equation (9) assumes a floating exchange rate regime whilst governments use the money supplies in order to optimize explicitly their individual loss function or the joint loss function in a coalition. However, this does not contradict a general notion of a currency union (or, in the terminology used here, a coalition) by which we mean often a common currency or a fixed exchange rate regime5. Exchange rate pegging can be viewed as a viable alternative to full-fledged coordination, see Giavazzi and Giovannini [11]. Then, governments use the money supply to stabilize the exchange rate and, hence, monetary policy is constrained. In the union we have to stabilize (n — 1) exchange rates, but we have n money supplies. This leaves one money supply free for optimizing the common loss function. The other countries will have to adapt their money supplies so as to peg

5Equation (9) is then reduced to equality of the interest rates.

(15)

their exchange rates. In our symmetric model there should be no problem in realizing the optimum even in a system of pegged exchange rates since in equilibrium all exchange rates and all money supplies within the coalition axe the same. For the time being, we will use the assumptions underlying equation (9): a floating exchange rate regime and perfect international capital markets.

2.1 Policymakers’ objectives

Welfare is captured in the policy objective function. This is typically a Tinbergen type quadratic loss function over the deviation of macroecono mic variables (employment and inflation6) from some target values (natural rate of employment and often zero inflation).

The objective function is:

Li = \ { ° £ + q2i) (10)

The parameter c.- denotes the relative weight the policymaker gives to the objective ‘full-employment’. A low <r,- denotes a ‘conservative’ monetary authority for whom price stability is the ultimate goal. Policymakers mi nimize this function subject to the restrictions arising from the economy.

2.2 Reduced form of the economy’s behaviour

We can reduce equations (1) to (9) in the symmetric case to two equations for each country. They determine the constraints for the policymaker’s

6A rise in the level q of the CPI is the same thing as inflation if we assume that g_i equals zero. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(16)

optimization problem. The money supply m, is free as an instrument for optimizing the loss function.

The reduced forms for g,- are7:

= ra, _{( i i )} q{ = Am,- — k rrij + x (12) j=i (13) with: A K / ? ( ! - < * ) ( ! - e ( l - ^ ) ) “ 6n + u( l - ^ ) 2 A — a

I will briefly explain the reduced form. Each country’s employment Z; rises one for one with the domestic money supply. Output rises with employment. Since real wages have to ‘balance’ the increase in employment (i.e. fall), the price of the domestic good rises. Hence, the price level rises with the money supply. The exchange rate depreciates. Consequently, the consumer price level, which is a weighted sum of the domestic good price and the prices of the imported goods, rises. This is reflected in equation (13) with A being positive.

Now, let us consider a symmetric world productivity disturbance, which gives rise to a stabilization game. Without policy intervention a negative disturbance (x > 0) would have no effect on employment because nomi nal output is unaffected. According to equation (1) and (2), a negative productivity disturbance lowers real output and raises the output price

7The reduced form is explicitly derived in Kohler [13], Appendix A.

(17)

by the same amount, since employment only remains constant if the real wage falls, i.e. the price of domestic goods rises. There is no change in the real exchange rate since real output falls in all countries by the same amount. Consequently, the consumer price index rises. Real interest rates have to rise in order to equilibrate the goods markets8. Since the real and the nominal exchange rate do not change, perfect sustitutability on the international capital markets requires that the nominal interest rates in all countries change by the same amount.

In short, a negative productivity shock will leave employment unchanged and increase CPI inflation. Each policymaker - facing a loss function which increases in the square of employment and CPI deviations - now has an incentive to contract the money supply a little bit in order to lower infla tion. He accepts the small loss from reducing employment below the full employment level in favour of the significant gain from lowering inflation. Contractionary monetary policy in the home country improves the terms of trade, lowers the price of imports and thus lowers inflation. Abroad, the price of imports is increased, thus causing inflation. Thus, monetary policy creates a negative externality which is reflected in the negative co

efficient (k is positive) of foreign monetary policy in equation (13). If

all policymakers perform anti-inflationary policy, they enter in a competi

8Whether nominal interest rates fall or rise depends on the size in the model para meters. When the real interest rate elasticity of goods demand is lower than the income elasticity of savings, nominal interest rates will rise; if it’s the other way around, nomi nal interest rates will fall. This can be checked with eqn. 7, 6, 4, bearing in mind that the change in real exchange rate is zero and the fall in real output is matched by the rise in prices. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(18)

tive appreciation which leads to a contractionary bias in the losses. The exchange rate in the end remains unchanged but all policymakers have con tracted too much with respect to their optimal money supply. This could be avoided if all countries coordinated on a less contractionary monetary policy9.

3 B loc form ation in a non-cooperative gam e

In the previous section we have outlined how policy makers will react to a negative productivity shock if they do not cooperate at all. Since they impose negative externalities on each other there is scope for improvement through cooperation. For this reason, the literature on international mone tary policy coordination has - starting with the seminal work of Hamada [12] - argued that coordination is beneficial for all parties involved. In Kohler [13], we have argued that countries may prefer forming a coalition to full coordination. While other models, for instance the model of Alesina and Grilli [1], focus on the question of whether entry will be limited by the insiders, we focussed on the question whether outsiders will refuse entry. We will see that in our model coalition members will always want other countries to join them. Hence, we do not need an explicit assumption which ensures free entry in the coalitions. The main result in Kohler [13] was that coalition formation will stop when it reaches a size of three

9Hamada [12] pioneered the studies that uncoordinated policy making across coun tries may be inefficient. The result of shock stabilization after a negative productivity shock was first formalized by Canzoneri and Gray [5] and was then used as a work-horse by Canzoneri and Henderson [6] and [7], Persson and Tabellini [16], among others.

(19)

countries. The reason is that the coalition formation process itself causes positive spillovers for the outsiders: the increased discipline within the coalition reduces the negative externalities the coalition countries create for all countries, independent of whether they are ‘in’s or ‘out’s. Countries will decide whether to join the union or not on the basis of whether it pays more to reduce imported inflation or to be able to export inflation. We will now ask whether countries which decided not to join the ‘first’ coalition would prefer to form a second ‘competing’ coalition. Each country can join an existing coalition or form a new one. We will see that the hierarchical structure between the coalitions becomes important for the size of the stable coalitions. For analytical convenience we will restrict to two the maximal number of coalitions. However, the result can be extended to more coalitions analogously.

3.1 The game structure

A coalition is defined as a subset of countries which optimize a common loss function. The common loss function is a weighted average of the indi vidual countries’ loss functions. The relative weights are denoted a,- with

£jfc'=i a i = 1 are typically determined in a (cooperative) bargaining

process. Since we have a symmetric model structure we will assume that the individual countries’ weights are equal, hence, we will set this weight (a,) equal to ^ for all i = 1 , . . . , k{.

• Coédition 1 consists of the countries i — and optimizes:

ki k\ 1 £ = $ > £ , • = £ -L ,-.=l i=i © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(20)

• Coalition 2 includes the countries i = 1 , . . . , &2 and optimizes:

5 k2 i=l

• The remaining n — kx — k2 countries play a non-cooperative Nash strategy against all other countries by minimizing their individual loss functions.

In order to solve the optimization problem of the coalition members we have to clarify a further element of the structure of the game: the coalitions can be involved in a Nash game or in a Stackelberg game with the non members and with each other.

The coalition outcome represented by a Nash equilibrium cannot be achie ved without a commitment technology since the countries which play co operatively within a coalition are off their individual Nash reaction func tions. We have to assume that the coalition members can enter into a binding agreement that is known about by all players. But then, it is not clear why the coalition would not use this commitment technology to behave as a Stackelberg leader in a Stackelberg game and realize a Stackel berg leader profit.

(21)

The Stackelberg concept10, on the other hand, creates a problem when loo king at the coalition formation process. With a symmetric model structure there is no obvious reason for the Stackelberg leadership of a single coun try. Hence, when there is only one country in the coalition we should get the non-cooperative Nash equilibrium as the outcome. However, assuming a Stackelberg leadership for the coalition implies assuming a Stackelberg leadership for the single country at the ‘early’ stage of the coalition forma tion. A Stackelberg leadership of a single country has to be explained by a structural difference which we have not modelled. Since it is not clear which of the two concepts should be chosen we will perform the analysis for both structures.

We can distinguish several game structures by combining the three groups- coalition one, coalition two and the fringe - and the two behavioural as sumptions - Nash or Stackelberg. In this paper three model structures will be analyzed11:

10The Stackelberg concept gives in general a time-inconsistent result, that is the Stackelberg leader would ex post like to change his strategy and, hence, does not play an optimal response. A structural difference in the timing of the decision making could explain such behaviour. One could argue that the coalition has to announce its policy at an early stage because all its members have to coordinate on the optimal policy. It sets its money supply before the non-members react or it can credibly commit itself to its monetary policy.

11 The hierarchical relation between groups are illustrated in the graphs. Two groups which lie on the same horizontal line do not have hierarchical differences and, hence, play a Nash game against each other. Groups which are connected by a vertical line play a Stackelberg game against each other where the top group is the Stackelberg leader while the lower group represents the Stackelberg follower.

(22)

• In the N ash-N ash gam e we assume that both coalitions play a Nash game against each other and against the fringe. One might identify such a structure with a situation where no country has an a-priori advantage and both coalitions are formed simultaneously.

• The N ash-Stackelberg gam e takes account of the commitment structure within the coalition. This opens the opportunity for a coalition to play a Stackelberg game against the outsiders. Yet, we have to clarify the strategic behaviour between the two coalitions. In this game we assume that the two coalitions are strategically in the same position i.e. they play a Nash game against each other. The idea behind such a structure is that both coalitions are ‘opened’ at the same time, that is a country has from the beginning the choice to enter any of them. This way, no structural difference in the stra tegic position of one coalition can develop since the countries would immediately switch to the more profitable coalition.

( Coalition 1 ) Nash ( Coalition 2 ) Nash (.Non-member)

( Coalition 1 ) ^ as~ ( Coalition 2 )

(23)

• In the Stackelberg-Stackelberg gam e we will consider the situa tion where the “first” coalition plays Stackelberg against the “secon- d” . The idea of a time structure for the coalition formation may help to explain the different strategic positions of the two coalitions. The situation we have in mind is some time structure which forms a certain commitment structure: in the beginning, coalition one is opened and increased until it reaches a stable size. The remaining countries axe given the choice to form a second coalition or to remain outside. They will form a second coalition until it reaches its stable size. For stability of the entire equilibrium, however, we will ensure that in the stable situation no country would like to leave its group.

C Coalition 1 ) - Stackelberg \ ( Coalition 2 ) Stackelberg \ (/Von-member^ Stackelberg

3.2 Equilibrium strategies and losses

T he countries outside the coalition In order to solve the policyma ker’s optimization problem when he is outside the coalition, we calculate the Nash strategy. We replace /,• and g,- in the loss function by the reduced form equations. This function is minimized with respect to m,- subject to given strategies of the other countries nij = rn~~c for all j ^ i if j is an out sider, rrij = rrij Cl for all j if j is a member of coalition one and rrij — rfTJ^

(24)

for all j if j is a member of coalition two. The symmetric setup implies that all countries have the same degree of conservativeness a. Since we have a symmetric structure in every respect, we can assume that all countries outside the coalition have the same optimal money supply m*nc. We can derive the money supply of a non-member as a function of the coalition’s money supply12: k\ &2 mnc = d + 9 Y . ~ dx (14) j= l j = 1 with: 9 d Xk a + A2 — A/c(n — ki — — 1) > 0 > 0

The optimal policy outside the coalition depends positively on the coalition policy i.e. the money supplies of a non-member and a coalition member are strategic complements13. This means that a less contractionary monetary policy of the coalition members triggers a less contractionary response from the non-members. The reasoning is as follows: the coalition creates less competitive appreciation for the non-members by contracting less. Hence, the countries outside the coalition also need to contract less, because they face less ‘imported’ inflation.

12The results axe derived in Appendix A.l.

13Strategic complements imply upward sloping reaction functions, see Bulow et al. [4J. The reaction function of a non-member is upward sloping since 9 is positive. The slopes of the reaction functions of the coalition members are positive, as derived in Appendix A.l. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(25)

The equilibrium in the N ash -N ash gam e The reaction functions of the coalitions are derived in Appendix A.l. They are upward sloping with respect to the other countries’ money supplies. This means that the money supplies are strategic complements for all players: a less contractive money supply from coalition two or from the fringe triggers a less contractive reaction from coédition one.

Equating of all three reaction functions gives the Nash equilibrium:

mcl - -PlX (15) mc2 = - p 2x (16) Wine — — QX (17) with: Pi P2 P - (1 + K.k2ip2)(l + /c(n - ki - k2)it))ipi P —(1 + /cfciV'iXl + k (n — kx — k2)d)ip2 P —(1 + nkiipi)(l + K,k2ip2)ti > 0 > 0 > 0

The equilibrium strategies are linear functions of the shock x. If there is no shock, the optimal policies are zero since there is no need for a stabilization game and therefore no need for a deviation from the zero- disturbance equilibrium values. If the shock is negative, i.e. x > 0, the optimal policies for all countries are .a contractionary monetary policy. The losses in equilibrium are determined through the equilibrium policies:

Lci - <rm*i + (Am’j - n(ki — 1 )m*x — /tfc2m*2 — k(ti— k\ — k2)m*nc + x) 2 Lc2 = am*c2 + (A m*2 — Kk\m*cl — n(k2 — l)m*2 — n{n — k\ — fc2)m*c + x)2 Lnc = <rm*nc + (Am*nc - Kk\m*cl - K.k2m*c2 - n(n - ki - k2 - l)m *c + x)2 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(26)

Figure 1: Losses of a member of Figure 2: Losses of an outsider coalition one

The losses varying with k\ and k2 are shown in figures 1 and 214. Los ses decrease with the number of coalition members. The explanation is as follows: the more members a coalition has, the more externalities are internalized. Without cooperation the countries impose negative exter nalities on each other. The coalition provides partial coordination of the monetary policies and eliminates the negative externalities the coalition members impose on each other. Hence, the coalition members conduct a less contractionary, and thus less deflationary, policy. But if the coalition members contract less, the inflation in the non-member countries is lower

14In order to evaluate the behaviour of losses with varying coalition sizes a simulation analysis was performed. The figures show the loss functions for n = 22 countries and with the parameter values: a = 0.5, /3 = 0.5, t = 0.8, v = 0.05, <J — 1 and 6 = 0.3. Figure 1 shows the loss function for the members of coalition one. The figure for the members of coalition two is an exact mirror due to the symmetry of the game.

(27)

as well, since the currency of a coalition member appreciates less against

all currencies. The process of the coalition formation thus produces posi

tive spillovers for the non-members. The country outside the union will be able to increase employment without increasing inflation. This will lower the losses for both parties. Members of one coalition profit more from an increase in the size of the other coalition than from an expansion of their own coalition. In figure 1, which shows the losses of a coalition one member, this can be seen in the steeper decline of the losses towards the right side (increase of coalition two) than towards the left side. The lowest losses are reached when all other countries are in coalition two. The reason is clear: the more coalition members, the stronger is “coalition discipline” from which all countries profit; the smaller the coalition size, the closer is the coalition policy to the individually optimal Nash response.

N u m b e r o f c o a l i t i o n m e m b e r s k j ( k 2 = 3 )

Figure 3: Lossfunctions of insiders and outsiders

(28)

Figure 3 shows the relative positions of the three groups with increasing coalition size fcj for n = 22 and k2 — 3. All groups profit from the increasing coordination within coalition one which can be seen in the downward slo ping functions. The lowest losses are always realized by the non-members. This is a general result in games with strategic complements since non- members gain from the increased ‘discipline’ within the coalition but are able to play an individually optimal response themselves. For the same reason coédition two profits from the increasing size of coalition one more

than the members of coalition one themselves. This is reflected by a steeper decline of the loss function of coalition two than that of coalition one.

The equilibrium in the N ash-Stackelberg gam e The optimization problems of the three groups and the resulting reaction functions are de rived in Appendix A.2. Equating the reaction functions yields the equili brium solution: mc i = -<£ \x (18) mc2 = -<t>2x (19) TYln c — —ipx (20) with: h = <t> 2 — <P = ?7i(l + *72^2*0 1 - T]iT]2kik2K2 7?2(1 +??1^1/C) 1 - T h ^ k ^ K 2 d ( l -(- T)ikiK)(l + 7 /2 ^ 2 /c) > 0 > 0 > 0 1

-Again, the equilibrium policies are linear (negative) functions of the shock

x. The same explanations as above apply. The qualitative analysis of the

(29)

loss functions with respect to increasing coalition size is identical to the Nash-Nash game. Though losses differ only slightly from the Nash-Nash case they are slightly lower for all groups15. The reason is intuitively clear: the Stackelberg leader will always be better off than in the Nash case since he could always realize the Nash outcome by playing his Nash strategy. In a game with strategic complements the outsiders profit as well from the improvement of the Stackelberg leader, see Dowrick [10].

The equilibrium in the Stackelberg-Stackelberg gam e The opti mization problems and the reaction functions are derived in Appendix A.3. The reaction function of the first coalition (the Stackelberg leader) is inde pendent of the other countries’ reaction functions and, hence, is identical with the equilibrium policy. It determines successively the equilibrium policies of coalition 2 and of the fringe.

mc 1 = —UJiX (21) mc2 = (22) ^nc = —vx (23) with: w _ (1 + f6 )( 1 + r)2k2K)(\ + k - kiK( 1 + f0 ){ 1 + T]2k2K)) a + (A + k - kiK,(l + f6)( 1 + r)2k2n))2 w2 = 772(1 + huiK,) > 0 v — 9(k\uj\ -f- £2^2) + 7?) > 0

15Obviously, there are equilibria where both games give the same result. Amongst these are all cases where either no country is in the fringe or where all countries are in the fringe. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(30)

As above, equilibrium policies are linear, negative functions of the shock x. Losses are reduced with increasing coalition size the same way as described above. All three groups are now better off than in the two previous cases16, profiting from the improvement of the Stackelberg leader upon his Nash- Nash and Nash-Stackelberg strategies.

3.3 The stability of coalitions in equilibrium

In the previous section we have seen that the equilibrium losses vary with the coalition sizes k\ and k2. The coalition formation produces positive spillovers for all countries and lowers their losses. In this section we will analyze whether these spillovers prevent countries from joining the coali tion. We will determine when a country would like to remain in the coali tion, join another coalition or join the fringe.

The decision whether a country would like to join or leave a coalition and, if so, which coalition it would like to join depends on the sizes of the re spective groups, ki, k2 and n — k\ — k2. We will analyze the dynamics of coalition formation and we will determine whether there exists a “sta ble” equilibrium in the sense that no country would like to change group (coalition one, two or fringe) unilaterally. The idea behind this is that a equilibrium with a coalition size where the coalition members find it profitable to change from the coalition(s) to the fringe and vice versa is not sustainable and in this sense not credible. We will adopt a stability

16Again, we can find cases where the two respective games give the same outcome. In particular, all countries being in the fringe or all countries being in one coalition reduce all three games to the same game.

(31)

concept used to examine the stability of cartels in models of industrial organization17.

The loss function of e.g. a non-member is denoted by Lnc(n, k\, k2). If it joins the coalition one (and no other country changes from one group to another), it will have the loss Lc(n, k\ + 1, k2). If L nc(n, k\,k2) is smaller than Lc(n, kx + 1 , k2), the country has no incentive to join the coalition. A similar condition can be applied for members of coalition two. If no country from the fringe or from coalition two has an incentive to join coalition one, then coalition one is called “externally stable” . If, on the other hand no member from coalition one has an incentive to leave the coalition, coalition one is “internally stable” . If both conditions are fulfilled, the coalition one is stable, with size Aq18. The equilibrium is stable when both coalitions are stable.

Hence, the coalitions are internally stable with size k^, k2 if:

Lel (kj , k2, n) ^ Lnc(k1 1, k2 > ti) and Lc\ (Aq, k2, n) < Tc2( Aq 1, k2

-t-l,n)

-I,c2(^ i) k2, n) ^ Lnc(k1, k2 1, n) and Tc2(A:^, k2, n) ^ L c\(Z:^ 1, k2

l , n )

17The stability condition used here is based on the one proposed by D’Aspremont et al. [9].

18This algorithm assumes that only the country under consideration takes a decision; all other countries remain in their ‘group’. If the result is stability, there is no problem since no one actually will change. But if the result is instability, this algorithm might give an incorrect signal since all members of a group will taike the decision to chamge and not only the country under consideration.

(32)

They axe externally stable if:

-^c2 ( ^ l ) ^2 5 ^ 0 ^ M * 1 "f" ^5^2 1 ? ^ ) -^ n c (^ l ? ^2 5 ^ - ^ c l ( ^ l

“t-l , ^ , n )

L d ( ^ 1 i ^2 5 ^ -^c2 (^ 1 f j ^2 "f“ ^ ^ n c ( ^ 1 j ^2 > ^ ) ^ ^ c 2 (^ 1 j ^2 "f"

l,n)

They are stable when they are internally and externally stable19. Our stability conditions do not allow the coalition to block a further extension of the coalition. However, the coalition in our game would never want to limit entry since the coalition members’ losses decrease when new countries enter the coalition, see figure 1. Hence, we do not need a condition which ensures “free entry” 20.

The factors p, g, <f>, ip and ui in the optimal policies are non-linear in n, ki, k2 and it is difficult to analyze analytically how the model parameters affect

19The stability condition has to be modified for the case when there Eire no countries in one of the coalition. The argument for external stability i.e. whether a country would like to join the coalition has to be made with two countries vs. zero countries. If we allow for only one country being in a coalition we get either a behaviour like a country in the fringe or a behaviour of a Stackelberg leader in the Nash and the Stackelberg cases, respectively. In the first case nothing changes since the losses are equal to the non-member countries. In the second cases losses change through the switch in the strategic position. However, we lack an explanation for this switch of the strategic position of a single country.

20In contrast to our model, coalition formation which is based on reputational consi derations like in Alesina and Grilli [1] faces this problem. It is not in the interest of the coalition to admit a ‘weaker’ member which would deteriorate the ‘stronger’ members’ positions. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(33)

the outcome. One possible approach is to perform numerical simulations with specific values for the model parameters whilst varying n, kj and The results reported here are based on a simulation where n varies from 3 to 22 and k\ and k2 vary from 1 to 2221.

3.3.1 The N ash -N ash gam e

The detailed results are described in appendix B .l. Here, we summarize the main results. The Nash-Nash game has a unique stable equilibrium where each coalition has three members if there are more than five coun tries. In a Nash-Nash game with three countries full coordination i.e. all countries in one coalition is the stable equilibrium. When there are four countries there will be two countries in each coalition in a stable equi librium22. For n = 5 countries the ‘last’ country to enter is indifferent between both coalitions, and, hence, may switch in equilibrium between the two coalitions. Stability is here only fulfilled with equality. However, the last country will clearly prefer joining one of the coalitions to remai ning in the fringe. These results are independent of the total number of countries, in particular of the number of countries in the fringe, once the total number of countries exceeds five. Subsequently, we will explain the features of these results in detail.

21The parameter values for the simulation are: a = 0.5, fi = 0.5, e = 0.8, v = 0.05, <r = 1 and 8 = 0.3. A sensitivity analysis was performed; the results can be checked in appendix B.

22For n = 4 countries we have few cases where k is very high such that it pays off to extend the first coalition to the stable size of three rather than having two countries in each coalition. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

(34)

One may be surprised by the feature that the stable coalition size remains small even as the number of countries increases. However, this result has an equivalent in the cartel literature in a space of strategic complements, see D’Aspremont et al. [9]. It pays for countries to join a coalition up to three members and undergo the costs of coalition discipline while they gain from the additional discipline of the other members against itself. Above three members the costs imposed through the discipline within the coalition are too high. The reasoning is twofold. First the discipline costs within the coalition increase with the number of members because a coalition member has to adapt its monetary policy not only to two others but to three, four, ... etc. which means that the country is off its Nash reaction function, i.e. its optimal response, for all these countries (and - having only one strategic variable - it drives the country off its individual reaction function towards the outsiders, as well). Secondly, the gains of joining the coalition decrease with the number of coalition members since we are acting in a space with strategic complements that, with each ‘step’ towards more discipline within the coalition, lowers losses outside, as well. Hence, there is less and less to gain by joining as the coalition size increases.

Increasing costs of joining and decreasing gains from joining for an indivi dual country explain as well why the stable coalition size is unique. Gains from joining the coalition decrease monotonically and costs of joining it increase monotonically with the coalition size (a graphical illustration is given below). Once the stable size is reached, that is the costs equal the gains, costs will always exceed the gains if the coalition size increases. By the same token, we can explain why it is individually optimal to form

(35)

two coalitions of three countries but not one coalition of six which may be pareto-superior to the two 3-country-blocs. For the answer we have to bear in mind that we have only a stable coalition when it is individually optimal for each single country not to join the coalition (for a country in the fringe or in the other coalition) or not to leave the coalition (for a country in the coalition under consideration). Countries would undergo the increase of discipline from a two to a three country coalition, however, increasing coalition discipline and increasing free-riding possibilities at the same time prevents them from joining a coalition which has more than three countries. Therefore, the step from a three to a six country coalition is not incentive compatible and countries would leave a coalition of six countries.

Figures 4, 5 and 6 illustrate23 the stability conditions (that is, the costs and gains from changing the group) as well as the dynamics of coalition forma tion. The curves represent the different stability conditions for coalition one. The two lower graphs represent the external stability conditions, that is Lci{ k \ ,k l,n ) - L cl(k\ + l ,& J - l ,n ) andLnc(k \,k z,n )- L ci(k\ + \,k l,n ).

The other two graphs describe internal stability, that is L c\(k\,k,2,n) <

L„c(kl —lyk^,n) and Lci(kl,k2,n) < Lci{k\ — l,k^ + l,n ) 24. Coalition one

is stable if all differences are negative. Hence, for negative values of the graphs changes do not pay off and the corresponding equilibrium is stable

23All graphs are based on specific parameter values. Unless otherwise noted, the parameter values are a = 0.5,/3 = 0.5, e = 0.8, v = 0.05, a = 1,6 = 0.3 and n = 22, ki + &2 from 0 to 22.

24Unfortunately, the losses are nonlinear in the coalition size k\. Hence, it is difficult to determine analytically the monotonicity of the two conditions in ki.

(36)

in our sense. Positive values indicate that a country will gain from chan ging the group (coalition one, coalition two or the fringe) and consequently the equilibrium is not stable.

St:»fc>ili-fcy o f co a litio n one: gains from cha ng ing ■fctie gro u p

Figure 4: Stability of coalition one with varying Aq (for k2 — 2)°

“Negative “gains from changing the group” imply that the group is stable. The convex graphs show the internal stability conditions. The concave graphs show the external stability conditions. Here, no coalition size of coalition one fulfils all stability criteria.

Figure 4 shows the “gains from changing the group” for different sizes of coalition one when there are only two countries in coalition two. When coalition one has more than two members there are no incentives to join coalition one any more. Members of coalition two prefer to remain in their own coalition which has only two members since they can profit from the discipline in coalition one even if they axe not members and they have to undergo less discipline themselves in their own smaller coalition. Members of the fringe would prefer, if at all, to join the smaller coalition two. The increasing decline of the graphs Coal2 => Coall and Fringe =>

(37)

Coal 1 indicates that with increasing coalition size ki these disincentives to

join coalition one become even larger since the coalition-induced discipline which a joining member would have to undergo increases and outsiders profit from the increasing coalition size. For three members in coalition one these members are indifferent between staying where they are or joining coalition two (and being there the third member). For fc* above three coalition one members find it profitable to switch to the smaller coalition two or to join the fringe. Here, the internal stability condition Coall =>

Coal2 fails to hold with inequality for k > 3 and this is the reason why

there is no stable coalition size for coalition one when there are only two members in coalition two.

Figure 6 shows, on the other hand, the situation where there are = 5

countries in coalition two i.e. two more than the stable coalition size. Whereas the graphs of the switches between the fringe and coalition one are qualitatively the same than in figure 4, the graphs for the switches with coalition two have shifted. Coalition two members will find it now profitable up to ki = 4 members profitable (or are indifferent) to switch to coalition one and, hence, will prevent a stable coalition size below this number. On the other hand, coalition one members will prefer to switch to the fringe above three members in coalition one. Again, we have no stable equilibrium.

Figure 5 finally shows a stable equilibrium for coalition one. It is exactly for three countries in both coalitions and all other countries in the fringe where no country wishes to switch between fringe and coalition one, as well as between the two coalitions. The three figures show only the stability

(38)

I 1

0'1

S tab ility of coalition one: gains from changing the group

Figure 5: Stability of coalition one with varying kx (for k2 = 3)“

S t a b i l i t y o f c o a l i t i o n o n e : g a i n s f r o m c h a n g i n g t h e g r o u p

Figure 6: Stability of coalition one with varying ki (for k2 = 5)6

“Only coalition size three fulfils all stability criteria. *No coalition size fulfils all stability criteria.

(39)

conditions for coalition one. Since we have a symmetric model the respec tive graphs for coalition two have exactly the same shape. As we have a symmetric stable equilibrium for coalition one, we will have the same stable equilibrium for coalition two and, therefore, for the entire model. Comparison of all three pictures shows that two graphs shift with incre

asing k‘2 and two graphs do not change their values very much. The two

graphs which remain unchanged are the potential gains of changes between the fringe and coalition one. The reason is evident: both groups profit ex actly the same way from an enlargement of coédition two against whom both groups play a Nash game. Hence, the relative positions which deter mine the gains from switches between the groups remain unchanged. The two graphs which determine the profitability of switching between coali tion one and two, however, shift with the size of coalition two. The more countries there are in coalition two, the lower the incentive for a member of coalition one to join coalition two. When coalition two has a size of two members a coalition one member for ki = 3 will be indifferent between joining either of the coalitions. Only when coalition one has more than three members will it be profitable to switch to coalition two. A similar argument applies for the gains from switches between coalition two and coalition one. Only when there are fewer members in coalition one them in coalition two will a member of coalition two not lose by switching to coali tion one. Hence, both graphs Coall => Coal2 and Coal2 => Coal 1 shift to the right with increasing k\. Whereas the first permits stable coalition sizes only for a ki which have less or equal than k2 members, the latter shows stable coalition sizes which have at least k2 members. In figure 4

(40)

it is Coal 1 => Coal2 which prohibits a stable coalition size of three which is the stable size implied by the conditions not to switch to the fringe. In figure 6 it is Coal2 => Coa/1 which is negative only above k\ = 4 and therefore fails to give stability for a lower kx.

The analysis above shows another feature: the symmetry of the game implies that the potential switches between the countries force a stable equilibrium to have the same coalition sizes for both coalitions (if we insist on strict inequality for stability). If not, it would pay off for a country to switch between the coalitions. Hence, every combination which gives the same coalition sizes for both countries is stable against switches between the coalitions. It is the trade-off with the fringe which forces the system into a unique stable equilibrium of kx — k2 — 3. This explains as well why we have a stable coalition size of 3 here and in Kohler [13] where we have a fringe and only one coalition.

Although we have only elaborated the case of two coalitions we may specu late what happens if we allow for three coalitions, four coalitions etc. The answer seems to be straightforward in the case where the coalitions play a Nash game against each other: the stable coalition size is determined by the trade-off between the fringe and cooperation within one of the coalitions. In our model, the stable equilibrium of three countries in each coalition is ‘dynamically’ stable in the sense that it always pays for two countries to ‘open’ a new coalition and for a third one to join them. It pays off as well for a fourth country which might have accidentally joined the coalition to leave it again. Hence, we can expect an extension to more coalitions to be straight forward, that is countries will prefer to split up in blocs of three

(41)

countries to both options, staying in the fringe or forming a big coalition. In the former case, they gain from free-riding on the coalition discipline but they loose from incurring negative spillovers from non-cooperative po licies with the other fringe countries. In the latter case, they gain from internalizing the externalities with the other coalition members, hwoever, they suffer too much coalition discipline. To join a smaller bloc seems to offer a ‘balanced’ solution to this cost-benefit analysis. Hence, there are mechanisms which are in effect in a symmetric world and are intrinsic to the process of coalition formation which can explain the existence of blocs which coordinate monetary policies. The literature on optimum currency areas, e.g. explains this phenomenon with asymmetries in the economic structures of the countries belonging to different blocs25.

3.3.2 T he N ash-Stackelberg gam e

Since the analysis of the coalition formation process does not change quali tatively very much in the Stackelberg games we will restrict our discussion in the following to the differences from the Nash case. The detailed results for all three games can be found in appendix B.

Here, like in the Nash-Nash case, the stable coalition size is the same for both coalitions. We will see in the Stackelberg-Stackelberg game that the strategic position between the two coalitions is crucial for this result. In the two games where the coalitions play a Nash game against each other the stable coalition sizes axe the same and determined by the trade-off with

25For a more detailed account on this literature, see e.g. Masson and Taylor [15]. A recent (empirical) study is e.g. Bayoumi and Eichengreen [2].

(42)

the fringe. Only when we assume a difference in strategic positions of the coalitions this feature changes.

Since it is the trade-off between the coalitions and the fringe which deter mines the unique stable equilibrium we can see this model as the equivalent to the Stackelberg case in the model with only one coalition in Kohler [13]. For three countries one coalition of three, for four countries two coalitions of two, for five countries one coalition of two and one of three and for six countries and more two coalitions of three constitute the unique stable equilibrium. Although the coalition has now another strategic position as a Stackelberg leader, this does not pay off in terms of the stable coalition size which takes account of the relative losses. The reason is that - in a framework of strategic complements - the Stackelberg follower is always relatively better off than the Stackelberg leader since he profits from the Stackelberg leader’s discipline as well as from the own possibility to play an optimal response, see Dowrick [10]. Though the coalition has lower losses than in the Nash case, the fringe has lower losses, too. Hence, the relative position which determines the stability of the equilibrium has not changed enough in order to change the stable coalition size.

3.3.3 The Stackelberg-Stackelberg gam e

In the Stackelberg-Stackelberg game we have now a slightly different stable constellation: coalition one consists of only two countries, coalition two of three countries in the stable equilibria. This is the result for more than four countries. For n — 3 countries we get a stable coalition with two countries in coalition two and none in coalition one, for n — 4 countries we