Keywords: JEL Classification: E17, E63, H63 monetary and fiscal policies, EMU, Stability Pact, fiscal theory of inflation, public debt

We use a two-country dynamic model in a monetary union in which we introduce a wealth private behaviour. We study the implications of public debt on monetary and fiscal policies in the short and the long run. We analyse the effects of asymmetric fiscal policies in Euroland: the first country has fiscal room for manoeuvre whereas the second country is fettered by the Stability Pact. We show that such a situation creates two feedback effects which reduce the efficiency of economic policies. First, because of the inability of one government to implement an expansionist fiscal policy, the other government has to substitute for it in order to stabilise inflation and production. Second, the ECB's involvement in macroeconomic stabilisation will also be exacerbated. The more substantial these effects, the more co-operation is needed between European governments and the ECB.

Résumé : Nous nous inscrivons dans le cadre d'un modèle dynamique à deux pays formant une union monétaire, avec introduction de comportements patrimoniaux de la part des agents privés. Nous étudions ainsi spécifiquement les implications de l'accumulation de la dette publique sur les politiques monétaire et budgétaires à court et à long terme. Nous analysons les effets du caractère dissymétrique de la politique budgétaire dans la zone euro, c'est-à-dire le fait qu'un pays puisse bénéficier de marges de manoeuvre tandis que l'autre est contraint par les dispositions du Pacte de stabilité. On montre alors que cette situation provoque deux types d'externalité qui réduisent le degré d'efficacité des politiques économiques. D'une part, l'incapacité d'un pays à mener une politique budgétaire oblige les autres ; d'autre part, la BCE devra s'impliquer davantage dans une politique de stabilisation de l'inflation et de la production à son potentiel. Une coordination entre les gouvernements et la BCE est d'autant plus souhaitable que ces externalités sont importantes.

THE STABILITY PACT AND FEEDBACK POLICY EFFECTS

Jérôme Creel

The treaties of Maastricht and Amsterdam did not either resolve the old problem regarding economic policies' co-operation in the EU, nor organise fiscal federalism. If co-operation has been tackled, it has been in a quite questionable way. First, the socalled co-operation is asymmetric: in order to prevent any drift in public finances, any temptation or risk to monetise public debts, any loss of credibility by the independent Central bank1 , fiscal policy has been compelled during the convergence stage to stick to the absolute norm of a public deficit inferior to 3% of GDP. The Stability (and Growth) Pact has moreover reinforced the convergence process for public finances in the EU: it will limit public deficits after the arrival of the euro. Second, the Council (Euro 11) is not a European government. The future should eventually demonstrate its efficiency.

Two macroeconomic implications of the Pact are already clear. On the one hand, it will be more difficult or maybe impossible for European governments to provide an optimal regulation for economic fluctuations. In the present situation, though the Maastricht's norm on public debt has been wiped out as regards the entry of some countries in the Euroland, we nonetheless think that the public finances in these countries (Belgium, Italy) will be carefully supervised by the Commission or the European Central Bank (ECB). Any deviation in the way to fiscal contraction will be penalised by financial markets operators (according to the discrimination principle developed by [START_REF] Buiter | Reflections on the Fiscal Implications of a Common Currency[END_REF], households (Ricardian equivalence along [START_REF] Barro | Are Government Bonds Net Wealth ?[END_REF], or Giavazzi & Pagano's, 1990, analyses), the Commission or the ECB (in order to preserve its credibility). In this sense, fiscal policies in these countries will be pro-cyclical.

On the other hand, the ECB will strategically dominate governments in the short run: it will set its policy in line with its goal of price stability and if fiscal policies are judged too expansionist, it will further increase the nominal European interest rate. Governments will have to choose between reducing their spending, i.e. a domination by the ECB; increasing them to compensate the restrictive impact of monetary policy (see Capoen et al., 1994); last, trying to co-operate with the ECB. Based on the circumstances during the convergence stage, governments might well choose the first solution.

Two other consequences of the Pact have been neglected so far, as far as we know. First, the inability of a government to implement a fiscal policy after a shock has occurred may impinge largely on its EU partners: these governments should be compelled to use their own fiscal expenditures to offset the shock. This situation resembles that of the free rider. If expansionist fiscal policies can produce favourable effects which spill over among EU's countries, they also have costs to the country which has increased the deficit: implementation costs, sunk costs, financial costs through the interest charges which impede the future capacities of governments to implement fiscal policies.

Second, the ECB will also suffer from the constraints of the Pact. It will have to substitute for the absence of fiscal policy in a country under the rule of the Pact. In the long run, if we oppose to the Barro-Ricardian equivalence between tax and public borrowing a theory of wealth effects linked to public debt holding by households, along the lines of [START_REF] Patinkin | Money, Interest and Prices[END_REF], the ECB will have to adjust households' private wealth projects to actual public debt and net foreign assets levels: if, for example, public debt increases in relation to GDP after a negative private demand shock, the ECB will have to choose between reducing the interest rate to curb debt's accumulation or increasing it to make private wealth grow faster. Depending on its reaction in the long run, the ECB will be more or less fettered in its policy choices and its ability to smooth economic fluctuations2 .

As an illustration of these two feedback effects, we use a two-country model in the EMU with dynamic behaviours. We assume that one of the two countries has no fiscal room for manoeuvre because it has already reached the deficit ceiling of the Stability Pact. Its government must strictly meet its budget constraint. We further assume that the other government and the ECB implement economic policies which are consistent with the minimisation of their own loss functions. Policies are not exogenous; rather, they respond to a private demand shock. This policy framework differs from that found usually in two-country models dealing with the EMU: it is necessary, as far as the stability of wealth accumulation is concerned, that public debt over GDP be stable in the long run, but not in the short run. Hence, fiscal policies in our framework may well reveal macroeconomic stabilisation properties which are absent from other models (see Jensen and Jensen, 1995, for example).

The shock in the economy (or economies) is supposed to be permanent. We study Nash equilibria between the three autonomous policymakers and then compare them to co-operative equilibria which we compute according to the Nash-bargaining procedure3 .

The theoretical model

Despite the Stability Pact, it is not at all sure that countries in the Euroland will be fettered by its provisions after a supply or demand shock. This will be all the more true if they have already recovered fiscal room for manoeuvre. However, some countries will surely have to spare any additional increase in their public debt over GDP ratio in the following years. These countries will have to pursue balanced fiscal policies in the best case, or to keep on reducing their deficits, in the worst.

Aggregate demand and wealth effect

Our model is dynamic and can be characterised by the wealth behaviour of private agents in the economy4 . It has Keynesian features in the short run (output is driven by the level of demand, prices adjust slowly to their steady state levels) but Wicksellian features in the long run: output is determined according to the real equilibrium interest rate, which depends on the effects of monetary and fiscal policies on the inflation rate5 . Aggregate demand is similar to that used in Mundell-Fleming models, except that we introduce a wealth effect. Aggregate supply follows a Phillips curve.

We study a polar case in the EMU: two countries, identical as far as private behaviours are concerned, form a monetary union6 . Households in both countries hold wealth which is the sum of public debt and net foreign assets. We assume that private agents form wealth plans which positively depend on the real interest rate (ρ) and disposable income ( ( ) 1-τ y )7 . This wealth effect is introduced according to the following channels. In the short run, an increase in the real interest rate has two opposite effects on consumption in theory: substitution and wealth effects. We suppose that the former dominates the latter: private savings increases, i.e. the part of disposable income which is devoted to the accumulation of wealth w y / [( ) ] 1 -τ increases, and consumption decreases. Aggregate demand thus has to satisfy:

∂ ∂ρ µβ τ y w y / ( ) = - - < 1 0 .
In the long run, wealth increases with the real interest rate and consumption is higher (see equation 1). If actual wealth (w) differs from desired (or planned) wealth, private agents help it to reach its desired level8 at speed µ. This wealth effect thus resembles the Pigouvian or real balance effect: if actual wealth is beneath its desired level, because of an increase in the real interest rate, private consumption will be reduced until savings has reached the desired equilibrium level. In the long run, households will use this savings in order to boost their own consumption.

In short, aggregate demand (equation 1) depends on the gap between actual and planned wealth, but also on disposable income, interests on wealth (ρw)9 , trade balance (bc), public expenditures (g) and a private demand shock (x). Aggregate supply (equation 3) is derived according to a Phillips curve with a cost effect due to the real interest rate. π represents the inflation rate, z a supply shock. We assume that there is a delay in the adjustment of prices because of staggered contracts in the economy. Wealth (equation 4) is the sum of public debt and net foreign assets which grow after a trade deficit. Equations 5 and 6 describe the growth of these two assets. The model is supposed to be quarterly. Shocks occur in the first quarter of 1999 and are permanent.

(1) y y w bc g w y x = -

+ + + + -+ - + ( ) [ ( )( ) ] 1 1 τ ρ µ α βρ τ (2) bc y y = -+ - η ηδ π π ( * ) ( * ) (3) π π λ θ ρ η π π = + + + + -+ -1 1 log ( ) ( * ) y z (4) w b f = + (5) b b g y = + + - - ( ) 1 1 ρ τ (6) f f bc = + + - ( ) 1 1 ρ

Governments and the ECB

In the EMU, i.e. in the absence of any exchange rate risk, a target for public debt is needed in order to determine the division of wealth between net foreign assets and public debt. Without this target, private wealth could be balanced with unstable and symmetric levels of public debt and external assets 10 .

In country 1, this target is such that the government is stabilising public debt over GDP in the long run; this long term steady state level for debt is unknown in the short run. Government 1 thus controls public spending and debt with no limitations in the short run. This is not the case in country 2. In this country, the government must stabilise its debt over GDP at a pre-determined level and must do it in the short run since it has already reached the ceiling of the Stability Pact. This government must thus follow a fiscal balance rule in each period.

We analyse the reactions of the two governments and the ECB in these conditions after a negative private demand shock of 1% of GDP. The shock is either symmetric or asymmetric. Since countries are heterogeneous in terms of government policies, an asymmetric shock in country 1 does not give the same results as an asymmetric shock in country 2.

We consider in this model that the macroeconomic behaviour of private agents is represented by equations 1 (wealth and disposable income effects) and 3 (wage-price loop); they do not have any strategic behaviour. This is not the case for policymakers: they are represented by their reaction functions 11 . For example, government 1 uses its expenditures in order to minimise its loss function each quarter: (7) LG a y a a g y a b y

= + + + 0 1 2 3 ² ² ( / )² ( / )² π .
This government is assumed to have targets for output and inflation. Variables in equations 7 to 10 represent deviations from the initial steady state. Government 1 incurs losses when it uses its spending.

Government 2 is fettered by the Stability Pact: it is using its tax rate τ to minimise its loss function but has to pursue meanwhile a strict balanced-budget rule 12 :

(8) LG a y a a * * ² * ² ' * ² = + + 0 1 2 π τ
10 In a flexible exchange rate regime or in the EMS, the uncertainty regarding the future value of external assets denominated in foreign currencies or risk aversion by households are sufficient conditions for determining the division between domestic and foreign assets.

11 See appendix 4 for more details.

12 See Barrel & Sefton (1997), Capoen & Villa (1997), [START_REF] Jensen | Debt, Deficits and Transition to EMU: a Small Country Analysis[END_REF], van der Ploeg (1995) for a similar framework. Public expenditures are used to stabilise debt, rather than tax rates, since it is now well known that fiscal adjustments are more efficient when expenditures are reduced.

= - + - + - - 1 1 χ χ τ ρ µ Φ
The Φ letter represents the public debt target of government 2; it is exogenous and has the same form as the public debt norm introduced in the Maastricht treaty. The χ µ g parameter represents the speed of adjustment of the public deficit to the level required to reach Φ. In order for stability conditions of the model to hold, this speed has to be high (see appendix 1).

The EMU can be characterised by the uniqueness of the nominal short run interest rate (i) and the independence of the ECB. We will assume that the ECB sets this rate to implement its monetary policy13 . We avoid the difficulties regarding the definition and level of money supply in the EU and the complications due to the instability of money demand in financial economies. The ECB minimises its loss function:

(10) LM k y y k k M = + + + + 0 1 2 2 2 ( * )² ( * )² ² π π ρ , where ρ π π M i = - + * 2 .
The ECB disregards the ratio of public debt over GDP and has henceforth no specific target regarding it.

Three remarks on monetary and fiscal policies are worth mentioning. First, if the ECB had controlled a monetary aggregate, the long term inflation rate would have been unchanged after a shock, following the so-called 'quantitative theory of money'. Output would have also been unchanged. If the interest rate is determined through a reaction function, this time, the long run inflation and real interest rates are not constant, and the output level can reach a new steady state.

Second, macroeconomic models usually give the priority to inflation in the central bank's loss function, according to the credibility argument: the government inflation bias needs a tough reaction by Central bankers. With no costs for the use of fiscal or monetary instruments, k 1 should therefore be superior to k 0 and (k 1 /k 0 ) should be superior to (a 1 /a 0 ). In our formulation with costs, this latter condition can be rewritten:

(11) ) a a a /( a k / ) k k ( 3 2 0 1 0 2 1 - - > -
, since the cost of using the interest rate reduces the capacity of the ECB to curb inflation, and the cost of using public expenditures (or the tax rate) and the cost of increasing public debt reduce the capacity of governments to stabilise output. We will consider that the weights for output and public debt are similar in government 1 loss function: condition (11) will then obviously hold.

Last, the behaviours of monetary and fiscal authorities are such that policies cannot be considered as exogenous shocks to the economy. In our model, the rationale of policymakers through the minimisation of loss functions make these policies be endogenous reactions to private demand shocks.

Calibrations

Parameter values are presented in table 1. Two considerations intervened in their choice14 : first, their realism which means either that they fit available data (the degree of openness, for example) or econometric results; second, the necessity that the model be stable.

Stability conditions are discussed in appendix 1, and the robustness of results to changing parameters is evaluated in appendix 3. We only concentrate on some important parameters: those affecting the wealth effect; the speeds of adjustment of wealth to its desired level and of public spending to the level required to reach a balanced budget; and the influence of the interest rate on inflation. We check that simulation results are weakly sensitive to the three latter parameters. As could be expected, the model is more sensitive to the β parameter: output results can be reversed if the wealth effect is reduced. We justify the β value we used for our simulations in appendix 2.

Demand shocks

The effects of demand shocks and reactions by the different policymakers are considered next. Above all, we are interested in the variations of the public debt over GDP ratio in country 1 since we think these have been widely ignored when establishing a fiscal asymmetry in the provisions of the Stability Pact. The demand shock can be characterised as an increase in the planned wealth to GDP ratio. It is a negative shock since consumption is reduced in the short run.

Symmetric shock

After a negative symmetric private demand shock, at Nash equilibrium between the three policymakers, the nominal interest rate decreases in the Union and government 1 implements an expansionist fiscal policy. Output is kept slightly over its initial steady state level and deflation is curbed (table 2). In country 2, the tax rate, which is reduced to comply with the smoothing of fluctuations, sees its effects on output partly cancelled because public expenditures have also been decreased. The output and inflation gaps between country 1 and country 2 create an external deficit in country 1 (hence, an external surplus in country 2). The external indebtedness of country 1 will gradually dampen its output increase.

In the medium run, output in country 2 is lower than in the initial steady state and the fall in the inflation rate is more pronounced than in country 1. Since government 2 is fettered by the Stability Pact, government 1 has to implement a more expansionist policy 15 ; in the long run, public debt increases by 2 points of GDP in country 1. Still, the improvement in the output in country 1 also crowds in to country 2. The game between monetary and fiscal authorities prevent key variables (output, inflation rates) from converging to their initial steady state levels: the real long run interest rate has been decreased. However, the rise of public debt impinges on monetary policy: it has to be less expansionist in order to commit governments to more restrictive fiscal policies.

Co-operative policies, in the Nash-bargaining framework 16 , give the following results: monetary policy in the medium and long run is more expansionist whereas fiscal policy in country 1 is less in the medium run in comparison with Nash equilibrium. Loss functions for the three policymakers are reduced and this equilibrium is Pareto-optimal. In the long run, the fall in the interest rate reduces public debt over GDP growth; to cope with the negative shock, government 1 does increase its spending more than in a Nash equilibrium. Note that policy co-operation, may it be Pareto-optimal, does not mean that public spending should be lessened. 15 Government 1 implements a less expansionist policy when it has to bear alone the burden of the shock (table 3). 16 The co-operative equilibrium is reached after the product of the game earnings for the three players has been maximised.

Asymmetric shocks

After an asymmetric negative private demand shock, the economic situation in each country not only depends on the fact that it has been hit by the shock, but also on its ability to react to it.

If the shock occurs in country 1 (table 3), the relatively large use of public expenditures by government 1 at Nash equilibrium is such that the ECB is encouraged to implement a very weak expansionist policy; output falls in country 1 and its public debt soars. Country 2 is isolated from the effects of the shock in the short run since monetary policy is not tough. In the medium run, the accumulation of an external surplus in country 1 (the income and price effects on the trade balance go in the same direction) increases its external wealth and is favourable to output growth in this country. In the long run, effects of fiscal and monetary policies cancel each other: with the small reduction of the interest rate, government 1 just slightly increases its spending; there is therefore a large fall in prices which spreads over country 2.

At co-operative equilibrium, monetary and fiscal policies are more expansionist in the short run to keep outputs nearest to their steady state levels; this provokes a larger increase of the public debt over GDP ratio than at Nash equilibrium. In the medium run, this substantial debt leads the ECB to keep its interest rate at a higher level than at Nash equilibrium; as a consequence, public spending in country 1 is dampened. Output levels in both countries are kept near to their initial levels and the fall in prices is slowed down. If the shock occurs in country 2 (table 4), the large price decrease in this country is such that the ECB has to reduce its interest rate more toughly than in the preceding case since government 2 is unable to react to the shock. Government 1 increases its spending to compensate the deflationary shock; this can be implemented more easily than in the preceding case because public debt has dropped with interest charges. In the medium and long run, the decrease in country 2 output will be dampened by the growth of external wealth.

In the medium run, the economic situation in country 1, as described by government losses, is more favourable in this case than in the preceding one. Government 1 is however more involved in this shock than government 2 was when the shock occurred in country 1: government 1 losses in the present case are superior to government 2's in the preceding. In the long run, government 1 losses are superior to its own losses after country 1 had faced the shock. This can be explained by the fact that the sharp fall in the real interest rate after the shock has hit country 2 has given government 1 the opportunity to implement a more and more expansionist policy in order to curb deflation. In fine, public debt has soared because the ECB has decided to reduce the drop in the 17 Both governments have a loss function of the form given in equation ( 7). Only government 1 and the ECB situations in our two variants can be compared since their loss functions are similar from one variant to the other. interest rate after public expenditures have been kept on increasing. Co-operative equilibrium does not reverse the feedback effect: the country which manages its fiscal policy without restraint suffers from the fiscal policy burden on its partner. As a conclusion, co-operation is always favourable for the three policymakers and it does not automatically lead to a reduction in public deficits and debts (see the case of a symmetric or asymmetric shock in country 1). We can also check that the Stability Pact places a heavy weight on government 1 and the ECB. In tables 3 and 4, results for the simulations of asymmetric shocks are presented in the same model as that used before except that the two countries control their fiscal instruments (they can implement whatever fiscal policy they want); hence, both countries are in the situation of country 1 in the previous model.

Country 1, whose government can implement fiscal policies in the two models, is largely burdened when government 2 is fettered by the Pact. If the shock occurs in country 2, the loss of government 1 is divided by ten if government 2 is free to choose its policy, in comparison with the situation with the Pact. Government 2 would also prefer to use its spending without restraint: macroeconomic stabilisation would not depend to a large extent on a "medium" monetary policy and the slow spillover effects of foreign policies. As for the ECB, its policy is always less expansionist when it does not have to substitute for a government as far as stabilisation policies in the EU are concerned. Its losses are reduced. This result is not the consequence of the reduced costs for the use of the monetary instrument (see equation 10). If we do not take these costs into account, ECB's losses when both governments implement policies (countries are symmetric) will always be equal or inferior to losses when one of them falls under the rules of the Pact (countries are asymmetric) (table 5).

Conclusion

Since the Amsterdam treaty in 1997, the compelling respect of the public deficit criterion has been dictated to countries willing to enter the Euroland. The so-called norm of 3% of GDP has been confirmed by the Stability Pact and will force countries either to implement pro-cyclical fiscal policies if this limit has already been reached or exceeded, or to have a very weak structural deficit in order to gain rooms for manoeuvre.

In this paper, we have dealt with the feedback effects of the Stability Pact on the ECB and the country which is not directly subject to the provisions of the Pact. The three policymakers in the model see their stabilisation capacities reduced by the existence of the Pact. With our dynamic and patrimonial model, we have been able to distinguish between the short-run and long-run effects of economic policies. Hence, we have shed light on the constraints falling on indebted countries. Debt implications are substantial: they change the temporal feature of monetary policy which, by way of consequence, modify fiscal policies. Note also that in this model, countries do not have to follow fiscal balance rules. The change in fiscal policy in the long run, in order to limit debt growth in percentage of GDP, is a sufficient condition to reach a new steady state.

We show that a stringent application of the provisions of the Stability Pact will substantially fetter the framework for economic policies in the whole EU. More noteworthy, the Stability Pact impinges negatively on fiscally 'virtuous' policymakers, as well as on the ECB. A softer respect of the fiscal norm would be more appropriate: the 3% of GDP would be considered as a nominal anchor; countries would be free to stabilise their economies after a downturn and they would have to take benefit of economic expansions to recover rooms for manoeuvre for future crises 18 . One condition should also be met: governments in the Euroland would rather co-operate with each others and co-operate with the ECB.

the risks with the norm promoted by Buiter et al. (no credibility, possible drift…). He prefers that the structural public deficit be zero.

Under a matrix form, we get:

(A5) π ρ χ ρ µ χ π π • • •             = - + -           - - -           + w g w w g g Y g 0 0 0 0 1 0 ( ) * * * , or V AU Y = + .
The determinant of matrix A is always positive since χµ g > 0 . The trace of matrix A is negative if and only if: ρ χ < . The stability of the model, without any macroeconomic stabilisation policies, thus necessitates that public expenditures be highly sensitive to public deficit variations. The introduction of a policy, even a fiscal-balance one, is a sufficient stability condition for this model. It is nonetheless crucial that this fiscal rule be stringent. After any increase in public debt or decrease in taxes, the government has to reduce its spending to make debt converge to its stable arbitrarily chosen level, Φ .

Appendix 2: Wealth effect

Our specification for private consumption can be compared with the one used in the Multimod model of the IMF (see [START_REF] Masson P | MULTIMOD Mark II: a Revised and Extended Model[END_REF]. In this model, real private wealth W is the sum of human wealth, capital, money, public debt and net financial assets. Recall that in our model, wealth is the sum of public debt and net foreign assets only.

The consumption function in the short run in Multimod is written:

(A6) 80 DUM C DEM C ] y ) 1 log[( C C ) C W log( C C C log 5 4 3 2 1 1 1 0 + + τ - ∆ + ρ + + = ∆ - -
where C is private consumption, DEM the dependency ratio which takes into account the link between young and old generations, DUM80 a dummy variable for 1980, and ∆ the first difference operator. Equation (A6) has been tested and results are :

. 02 . 0 C ; 19 . 0 C ; 35 . 0 C ; 59 . 0 C ; 09 . 0 C ; 4 . 0 C 5 4 3 2 1 0 = = = - = = - =
Consumption growth increases with wealth and disposable income's growth but decreases with the real interest rate.

In the long run, consumption takes the following form:

(A7) DEM 2 2 . 6 W log 09 . 0 C C log 0 + ρ - + = .
These results do not explain much on the reasons why households want to accumulate wealth for their consumption. Moreover, is long term consumption increasing or decreasing with the interest rate? This question is unanswerable so long as wealth accumulation is not taken into account in equation (A7). This is the reason why we introduced the real wealth effect and supposed planned or desired wealth W would be growing with the real interest rate and disposable income. Further, we considered that the propensity to save (or increase wealth) from disposable income would be growing with the real interest rate, so that we gave the priority to the following formulation:

(A8) y ) 1 )( ( W ~τ - βρ + α = .
For a zero real interest rate, the wealth over disposable income ratio is supposed to be positive ( % 30 = α

). At the initial steady state, we assumed debt over GDP was equal to 25% and net foreign assets were zero. With an initial real interest rate equal to 1%, β equals 2.5. With these parameter values, our consumption formulation does not differ much from that in Masson et al., except in the long run.

In the short run, consumption in our model follows:

(A9) ) W W ( W y ) 1 ( C - µ + ρ + τ - = .
Wealth has two effects on consumption: households consume interests and there exists a real wealth effect.

Using equations (A8) and (A9) in first difference, we obtain:

(A10) ρ ∆ τ - µβ - + ∆ µ + ρ + ∆ τ - βρ + α µ - = ∆ ] y ) 1 ( W [ W ) ( y ) 1 )]( ( 1 [ C 0 0 0 0 0 0 ;
where variables with subscript 0 are initial values (see table 1).

With the chosen values, private consumption in the short run is such that:

(A11) ρ ∆ - ρ + ∆ τ - = ∆ - 35 . 0 W 31 . 0 y ) 1 ( 9 . 0 C 1 0 .
This formulation differs slightly from Masson et al.'s since in our model, variations and not only the level of the real interest rate impinge on consumption growth. In the long run, differences are more substantial since wealth can be replaced by its value in equation (A8). We thus get:

(A12) ρ ∆ + ∆ τ - = ∆ 27 . 0 y ) 1 ( C 0 .
Consumption growth hence depends positively on the real interest rate growth. Inflation hence reduces private consumption in the long run whereas a restrictive monetary policy increases it. Therefore, the government has to implement a fiscal contraction to reduce aggregate demand.

Appendix 3: Model robustness to some parameters

In simulated models, and more importantly in dynamic models, results can depend on parameter values. We present in this section the time path of the output level at Nash equilibrium in the country which faces the demand shock, using three different values for each parameter we study. These values meet the stability conditions of the model.

In our model, four parameters deserve peculiar attention; the two firsts are related to the private wealth effect: the sensitivity to the interest rate, β, and the speed of adjustment to the desired wealth level, µ. Next, because of the introduction of a fiscal balance rule, the sensitivity of public spending to public deficit variations, χ, becomes a crucial parameter. Last, inflation dynamics depends on the effect of the interest rate whose sensitivity on prices is measured by (1+θ).

The β β parameter

A low level for the sensitivity of private wealth to the interest rate means that this wealth effect is weakened: if the negative demand shock has led to a public debt increase, the interest rate has to rise substantially (or be lessened to a smaller extent) to maintain the long run patrimonial balance. The output impact is therefore more restrictive than with larger values of β (see figures A1 to A3). More precisely, with a weak wealth effect, fiscal policy is very active in the short run (in comparison with a situation with a higher β) because public debt has no substantial effect on aggregate demand. In the medium and long run, however, this higher debt modifies monetary and fiscal policies: monetary policy, which was expansionist, and fiscal policy become very tight. The ECB moves first; afterwards, the long-lasting effect on debt of the rise in the interest rate compels the government to reduce spending; the falls in output and prices are very tough in the long run. The µ µ parameter

The higher the speed of adjustment of actual to planned wealth, the speeder economies reach their new steady states. In the short run, a higher speed of adjustment has more substantial stabilisation effects: the gap between actual and planned wealth has more impact on aggregate demand if µ is large. Output deviations from the initial steady state are smaller. The forms of the time path between initial and final steady states are not radically changed if values of this parameter are modified (see figures A7 to A9). The θ θ parameter

The different values for parameter θ do not have any influence on the output steady state path. The model is therefore not sensitive to this parameter (see figures A10 to A12). To determine a co-operative equilibrium, one usually distinguishes between two methods: first, the Nash-bargaining equilibrium which consists in the maximisation of the product of earnings due to co-operation, i.e. the product of the differences between losses at Nash equilibrium and at the co-operative equilibrium for each player; the resulting bargaining equilibrium is henceforth Pareto-optimal. We have, with L CG et L CM the government and central bank loss functions at the co-operative equilibrium respectively: The negotiation framework will be Pareto-optimal if the following condition is met:

Max L L L L NG CG NM CM τ ρ , ( )( ) - - , hence:
( ) : E L L L L NM CM NG CG 3 1 - - = - λ λ .
Thus, the Nash-bargaining equilibrium can be described as the search for the value of λ which maximises the product of earnings due to the co-operation. It is therefore impossible that the negotiation comes up to a counter-productive (or sub-optimal) equilibrium. Moreover, the condition (E3) ensures that if one player is in a more favourable situation than at the Nash equilibrium, it is also the case for the second player since λ λ / ( ) 1 0 -> : equilibrium is Pareto-optimal.

Values for the economic instruments at the co-operative equilibrium are resulting from equations (E1') and (E2'):

τ

Figure A1 :Figure A2 :-Figure A3 :-

 A1A2A3 Figure A1: Output sensitivity in country 1 to beta, symmetric shock

Figure A4 :Figure A5 :-Figure A6 :-

 A4A5A6 Figure A4: Output sensitivity in country 1 to χ, symmetric shock

Figure A7 :Figure A8 :-Figure A9 :-

 A7A8A9 Figure A7: Output sensitivity in country 1 to µ, symmetric shock

Figure A10 :Figure A11 :-Figure A12 :-

 A10A11A12 Figure A10: Output sensitivity in country 1 to θ, symmetric shock

  have already reached Nash equilibrium which means, by definition, that each authority has assumed that the policy of the other authority is constant, we also have: follows a negotiation framework. Co-operative equilibrium is reached after a weighted average of both policymakers' loss functions has been minimised. With the negotiation parameter λ, where 0

Table 1 : Parameters and initial steady state values

 1

	Parameters	α	β	χ	δ	λ	η	µ	µ g	θ
		0,3	2,5 0,85	0,2	8	0,2	0,3	0,3	0,05
	Loss functions	a 0	a 1	a 2	a 3	a' 2	k 0	k 1	k 2
		0,5	1,5 0,16	0,5	0,16	1,5	3	0,5
	Initial values	b/y	f/y	w/y	ρ	τ	Φ/y
		0,25	0	0,25 0,01	0,2	0,25

Table 2 : Effects of a symmetric private demand shock

 2

			Nash equilibrium			Co-operation
		Ctry 1		Ctry 2		Ctry 1		Ctry 2
		2000	2010	2000	2010	2000	2010	2000	2010
	Output (%)	0.118	0.223	-0.128	0.261	0.230	0.332	0.048	0.355
	Inflation (points)	-0.502	-1.283	-0.631	-1.261	-0.372	-0.817	-0.474	-0.803
	Public expenditures	0.269	0.100	-0.047	-0.128	0.255	0.154	0.033	-0.027
	(points of GDP)
	Tax rate (points)	0.000	0.000	-0.225	-0.244	0.000	0.000	-0.156	-0.191
	Public debt	0.535	1.768	0.006	0.000	0.435	0.995	0.001	0.000
	(points of GDP)
	Losses	0.540	4.059	0.614	2.427	0.339	1.555	0.342	1.035
		2000	2010			2000	2010
	ECB's losses	1.214	5.049			0.845	2.358
	Real interest rate (points)	-0.707	-0.465			-0.747	-0.655

N.B.1: results are deviations from initial values. N.B.2: country 2 is fettered by the Stability Pact.

Table 3 : Effects of an asymmetric private demand shock in country 1

 3

			Nash equilibrium			Co-operation
		Ctry 1		Ctry 2		Ctry 1		Ctry 2
		2000	2010	2000	2010	2000	2010	2000	2010
	Output (%)	-0.149	0.139	0.088	0.115	-0.086	0.113	0.130	0.098
	Inflation (points)	-0.214	-0.651	-0.093	-0.667	-0.169	-0.339	-0.049	-0.349
	Public expenditures	0.095	0.055	0.064	-0.076	0.104	0.040	-0.065	-0.235
	(points of GDP)
	Tax rate (points)	0.000	0.000	0.009	-0.138	0.000	0.000	-0.121	-0.285
	Public debt	0.773	0.825	-0.003	0.000	0.757	1.060	0.006	0.000
	(points of GDP)
	Losses	0.380	0.987	0.017	0.677	0.334	0.741	0.014	0.200
	(Losses*)	(0.42)	(0.50)	(0.02)	(0.63)	(0.35)	(0.25)	(0.03)	(0.31)
		2000	2010			2000	2010
	ECB's losses	0.094	1.358			0.060	0.392
	(ECB's losses*)	(0.10)	(0.61)			(0.03)	(0.10)
	Real interest rate (points)	-0.212	-0.248			-0.217	-0.201
	(Real interest rate*)	(-0.17)	(-0.09)			(-0.13)	(-0.11)

N.B.1: results are deviations from initial values. N.B.2: country 2 is fettered by the Stability Pact. *: simulations with the same model, except that both countries can implement fiscal policies with no constraint in the short-medium run 17 .

Table 4 : Effects of an asymmetric private demand shock in country 2

 4

			Nash equilibrium			Co-operation
		Ctry 1		Ctry 2		Ctry 1		Ctry 2
		2000	2010	2000	2010	2000	2010	2000	2010
	Output (%)	0.267	0.084	-0.215	0.146	0.336	0.143	-0.187	0.195
	Inflation (points)	-0.289	-0.632	-0.538	-0.594	-0.152	-0.408	-0.439	-0.376
	Public expenditures	0.175	0.045	-0.111	-0.051	0.169	0.075	-0.061	-0.005
	(points of GDP)
	Tax rate (points)	0.000	0.000	-0.235	-0.106	0.000	0.000	-0.168	-0.086
	Public debt	-0.238	0.942	0.009	0.000	-0.050	0.518	0.006	0.000
	(points of GDP)
	Losses	0.194	1.047	0.467	0.541	0.097	0.396	0.311	0.232
	(Losses*)	(0.02)	(0.63)	(0.42)	(0.50)	(0.03)	(0.31)	(0.35)	(0.25)
		2000	2010			2000	2010
	ECB's losses	0.637	1.170			0.363	0.556
	(ECB's losses*)	(0.10)	(0.61)			(0.03)	(0.10)
	Real interest rate (points)	-0.495	-0.217			-0.431	-0.322
	(Real interest rate*)	(-0.17)	(-0.09)			(-0.13)	(-0.11)
	N.B.1: results are deviations from initial values.
	N.B.2: country 2 is fettered by the Stability Pact.

*: simulations with the same model, except that both countries can implement fiscal policies with no restraint in the short-medium run

Table 5 : ECB's losses at Nash equilibrium* Shock on country 1 Shock on country 2

 5

		Asymmetric	Symmetric	Asymmetric	Symmetric
		countries	countries	countries	countries
		2000	2010	2000	2010	2000	2010	2000	2010
	ECB's losses	0.07	1.33	0.08	0.61	0.51	1.15	0.08	0.61
	*: losses are computed with	0 k 2 = in equation 10.

Cf. Eichengreen & Wyplosz (1998).

Note that in the multi-country Multimod model (see[START_REF] Masson P | MULTIMOD Mark II: a Revised and Extended Model[END_REF], for more details), there is no distinction between effective and desired wealth in the short run. Nothing therefore explains why households want to hold net foreign and public assets. Furthermore, it is not easy to understand how assets accumulation gets to an end. We compare our consumption function to that used byMasson et al. in appendix 2.

Computations are described in appendix

For a first approach in the EMU, see[START_REF] Creel J | Du Bon Usage des Politiques Monétaire et Budgétaire en Europe[END_REF]. See also notes 2 and 8.

For more details on the fiscal theory of inflation, see[START_REF] Woodford | Public Debt and the Price Level[END_REF].

Equations in country 2 are obtained by circular permutation; this country's variables are starred.

The consumption function is detailed in appendix 2.

The introduction of a wealth effect with a slow adjustment to the long term desired level comes from aCreel & Sterdyniak (1995) model. Fair (1997), for example, also takes wealth into account in a multi-country model. Unlike him however, we do not introduce stocks in households and firms behaviours.

In case of perfect substitutability between domestic and foreign assets, we assume that interests on wealth are not taxed.

There is no monetary targeting in the sense that the Central bank sets monetary aggregates to a certain level or controls that they reach the appropriate level. On this topic, see[START_REF] Svensson | Monetary Policy Issues for the forthcoming in Carnegie-Rochester Conference Series on Public Policy[END_REF].

See the advice of[START_REF] Cooley | Calibrated Models[END_REF] on calibrated models.

[START_REF] Buiter | Excessive Deficits : Sense and Nonsense in the Treaty of Maastricht[END_REF] have proposed that the 3% norm be applied to the structural public deficit, in order to satisfy to the so-called golden rule for public finances.[START_REF] Eichengreen | Saving Europe's Automatic[END_REF] insists however on

See Svensson (1999).

Appendix 1: Stability conditions

To make understanding easier, we present the model dynamics in the closed economy and assume that there is just one asset: public assets. We consider two cases in the model.

First, the government is able to implement any policy, whether expansionist or restrictive, in the short or in the long run: the government controls its instruments. The model dynamics if we do not make fiscal policies explicit is of the following form:

Under a matrix form, we get:

The model is intrinsically unstable if a fiscal or monetary policy is not implemented. The economy has to be stabilised by a government, which sets public expenditures, and/or a Central banker, which sets the nominal interest rate. These two policies impinge on long term inflation and the wealth dynamics. Hence, fiscal and monetary policies have symmetric influences on inflation. Our model differs obviously and voluntarily from a monetarist model and give henceforth the priority to a realist 19 and patrimonial view of monetary policy.

In the second case, the government follows a fiscal-balance rule. The model dynamics if we do not make monetary policy explicit becomes:

(A4) g g w y

g g

.

After a small variation in one of the two instruments, the government loss function varies in the sense:

After any deviation of one instrument from its value at the co-operative equilibrium, loss functions of the two policymakers vary in opposite directions: the co-operative equilibrium is then Pareto-optimal.


Hervé Crès
email: cres@hec.fr

Dominique Lepelley

Matthieu Martin

Vincent Merlin

Hervé Moulin

Maurice Salles

Aggregation of Coarse Preferences *

Keywords: Super majority voting, condorcet cycles, weak preference orderings. JEL Classification: C7, D7

research documents, whether they are published or not. The documents may come

Introduction

The aggregation of individual preferences through majority voting may yield Condorcet cycles. The simplest example of such a cycle involves three collective choices 1, 2, and 3 and three agents whose individual preferences are respectively 1 2 3, 2 3 1 and 3 1 2. Pairwise comparisons of these alternatives under majority rule show that there is always a two-third majority to prefer 1 to 2, 2 to 3, and 3 to 1.

Condorcet cycles are the main obstacle to the ability of majority voting to aggregate individual preferences. Moreover, it is a simple exercise to construct preference profiles in which some cycle appears. To what extent Condorcet cycles really jeopardize the efficiency of majority voting as a collective decision-making device is a question that has been addressed from the early fifties with the pioneering work of [START_REF] Guilbaud | Les théories de l'intérêt général et le problème logique de l'agrégation[END_REF], who computes the probability of the occurrence of such cycles when there are three alternatives and a large number of voters. His work was followed by a line of others, among them:

Niemi and Weisberg (1968); DeMeyer and [START_REF] Demeyer | The probability of a cyclical majority[END_REF]; [START_REF] Kelly | Voting anomalies, the number of voters, and the number of alternatives[END_REF]; [START_REF] Gehrlein | The probability of the paradox of voting: A computable solution[END_REF]. They all tend show that, under the simple majority rule, the probability of occurrence of Condorcet cycles converges to one when both the numbers of alternatives and voters become large. All these results are obtained through the assumption of impartial culture, which states that voters draw at random their preferences in the set of all possible orderings of the n alternatives, i.e., the group of all permutations of n elements,

S n .
The oldest problem of the theory of Social Choice is to determine conditions under which such cycles are ruled out. This question has been solved along two main lines.

The first one, initiated by [START_REF] Black | The theory of committees and elections[END_REF] through the concept of single-peakedness gives sufficient conditions on individual preferences that rule out cycles. The family of spatial voting models globally pertains to this line: Alternatives are taken on some underlying continuous space (usually taken to be Euclidean); individual preferences over the set of alternatives are 'structured' by classical assumptions (like continuity and convexity or symmetry); and/or the profiles of individual preferences show some features which can be given natural interpretations in terms of 'consensus' of the society; see, e.g., [START_REF] Plott | A notion of equilibrium and its possibility under majority rule[END_REF], [START_REF] Kramer | On a class of equilibrium conditions for majority rule[END_REF], [START_REF] Grandmont | Intermediate preferences and the majority rule[END_REF]. This line usually suffers from restricted assumptions on the 'domain' of such preferences: the set of allowed preferences is sometimes drastically shrunk. The second strand of research focuses on super majority rules. It was shown that for super majority rules of rates higher than 1 -1/q, there cannot exist cycles of length smaller than q [START_REF] Ferejohn | On a class of rational decision procedures[END_REF]. But, if alternatives are numerous, such an argument guarantees the aggregation process to be devoid of cycles only for the unanimity rule, which is highly intractable: the acyclicity of the aggregated preference is obtained at the expense of a lot of incompleteness in this collective preference. [START_REF] Caplin | On 64% majority rule[END_REF] yields a very beautiful result along both lines. Balasko andCrès (1997, 1998) introduce, along these two lines, in a mostly discrete But one could argue that it is difficult, even for a very clever and rational voter, to completely order a large set of social alternatives. Reasons could be the difficulty of gathering full information about the alternatives or the lack of time as well as the lack of interest to compute this complete ranking. It makes sense to assume that people usually have their subset of most preferred alternatives, their subset of least preferred alternatives and some intermediate subsets, the number of which depends on the concern of the voter about the poll, his ability to make up his mind in a limited amount of time about what is at stake and which alternative is truly better. Hence it is not unreasonable to claim that voters' preferences, although transitive, may exhibit a lot of ties between the alternatives:

individual preferences are coarse. A tie between two alternatives is not meant to be a real indifference. It only means that the voter finds it too costly either to figure out what is his real preference between the two alternatives, or to express this preference through voting: he is undecided. He then prefers to abstain from voting. A preference is said to be of refinement ≤ n when voters cannot distinguish, and then order, more than groups of alternatives. Hence, in the case of three alternatives, the only preference of refinement 1 is the fully indecisive ranking: 1 ∼ 2 ∼ 3. It is of refinement 2, or dichotomous, if the individual can distinguish only its most preferred alternatives and its least preferred ones; in the case of three alternatives: 1

(2 ∼ 3); (2 ∼ 3) 1; 2 (1 ∼ 3); (1 ∼ 3) 2;
3 (1 ∼ 2) and (1 ∼ 2) 3.

There has been in recent years a strand of research supporting the idea that ties within individual preferences makes their aggregation easier and surer: beyond the early work of [START_REF] Inada | A note on simple majority decision rule[END_REF] and [START_REF] Fishburn | The paradox of voting: Effects of individual indifference and intransitivity[END_REF], one can refer to [START_REF] Gehrlein | Condorcet paradox and the Condorcet efficiency of voting rules[END_REF], [START_REF] Gehrlein | Condorcet Efficiency: A preference for indifference[END_REF], Jones et alii (1995), [START_REF] Lepelley | Condorcet's paradox for weak preference orderings[END_REF]Martin (1998), Saari (1995), [START_REF] Tataru | On the relationship of the Condorcet winner and positional voting rules[END_REF] and Van Deemen (1997).

It is shown in the present paper that, if the refinement of preferences does not exceed , a super majority rule with rate 1 -1/ is necessary and sufficient to rule out Condorcet cycles of any length, a generalization of [START_REF] Ferejohn | On a class of rational decision procedures[END_REF]. Hence the key factor to rule out cycles is not the number of alternatives, but the maximal degree of refinement of the preferences expressed in the society. This is the object of Section 3. It is moreover argued (in Section 4) how the coarser the individual preferences, the lower the rate of super majority necessary to reduce to almost zero the relative size of the set of profiles for which Condorcet cycles appear, hence their probability under various cultures. The drawback of super majority rules is that it makes it rather improbable that pairwise comparisons of alternatives be possible. It is shown that, for a fixed super majority rule, the probability of a pairwise comparison between two alternatives increases toward 1 when the number of preferences exhibiting a strong ranking between these two alternatives decreases. Hence the coarser the individual preferences, the more likely pairwise comparisons, for a fixed super majority rule.

2 The Setup

Individual Preferences

We consider a set of n political alternatives A n = {1, 2, . . . , n}. Let us first define individual preferences over A n . A fine individual preference is a complete strict and transitive ranking over A n , hence can be identified with a permutation of {1, 2, . . . , n}: to each alternative is associated its rank in the individual preference; e.g., the permutation 231 of A 3 stands for the individual preference 3 1 2.

A voter v associates a rank to each alternative in A n . Weak preference orderings are allowed: two different alternatives can be given the same rank, which means that in the voter's opinion, there is a tie between them. Hence an individual preference over the set of alternatives can be conceptualized through the following two-stages process. A voter will be said to have a preference of refinement if it can first partition the set of alternatives

A n into subsets, and then rank the subsets within a linear ordering. His preference is coarse to the extent that he might not exhibit a strong preference ordering between a pair of alternatives taken in the same subset. But it is transitive:

i v or ∼ v j and j v or ∼ v k      =⇒ i v or ∼ v k ,
and at least one strict preference in the left-hand side of the assertion entails the strict preference in the right-hand side.

To illustrate this point, take n = 7 and consider an individual having the following preference: his first-best alternatives are {3, 5}, but he cannot make up his mind whether alternative 3 is better or worse than alternative 5, and decides to stick to a tie; his second-best choices consist in the subset of alternatives {1, 4, 7}, and finally his third-best alternatives are {2, 6}; he ends up exhibiting the preference of refinement 3:

(3 ∼ 5) (1 ∼ 4 ∼ 7) (2 ∼ 6).
In this paper individual preferences will be represented as sequences (see below): the

preference (3 ∼ 5) (1 ∼ 4 ∼ 7) (2 ∼ 6) is identified to the sequence 2 3 1 2 1 3 2 ,
listing the ranks given to each alternative: alternative 3 and 5 are ranked first, therefore the third and fifth element of the sequence is 1, etc. A preference is represented as a sequence σ : σ(1) σ(2) . . . σ(n) . The integer σ(i) ∈ {1, 2, . . . , } is the rank the voter gives to alternative i.

Sequences

Take an integer and let (e 1 , e 2 , • • • , e ) be strictly positive integers such that

e 1 + e 2 + • • • + e = n .
Consider the sequences

σ : σ(1) σ(2) . . . σ(n) ,
where for all i, 1 ≤ i ≤ n, σ(i) ∈ {1, 2, . . . , } and σ reaches the value m exactly e m times, 1 ≤ m ≤ . The sequence σ is said to be of refinement and specification (e 1 , e 2 , . . . , e ) (for technical points see [START_REF] Carlitz | Enumeration of sequences by rises and falls: a refinement of the Simon Newcomb problem[END_REF].

Let S n denote the set of sequences of refinement and

S ≤ n = m= m=1
S m n the set of sequences of refinement at most ; S n = S ≤n n is the set of all sequences on A n . A sequence σ of refinement is merely a surjective mapping from A n into {1, . . . , }.

Then the number of sequences of refinement exactly on a set of n elements is the number of such surjective mappings: it is well known that S n = l!S n, , where S n, the Stirling number of the second kind at entry (n, ). Notice that sequences are also permutations of the multiset

{1 e 1 , 2 e 2 , . . . , e } = { 1, . . . , 1 e 1 times , 2, . . . , 2 e 2 times , . . . , , . . . , e times }.

The voting procedure

Let σ be a sequence of refinement ; denote S the domain of individual preferences (it is the set S n of all sequences if there is no domain restriction). Let m σ denote the number of voters whose preference is σ. Then m = σ∈S m σ is the total number of voters and λ σ = m σ /m is the proportion of voters whose preference is σ. The parameter space considered in this paper in the (b S -1)-simplex (where b S = S, the cardinal of S) of profiles:

∆ S = (λ σ ) σ∈S | σ∈S λ σ = 1 .
Any profile of individual preferences can be represented by a point in ∆ S . Conversely, any point in ∆ S can represent a profile of individual preferences provided the number of agents is allowed to tend to infinity or even be infinite. This makes ∆ S a suitable parameter space.

Define the two following characteristic functions on the pairs (i, j) of alternatives, i = j:

χ σ (i, j) =    1 if σ(i) < σ(j), 0 otherwise, and δ σ (i, j) =    1 if σ(i) = σ(j), 0 otherwise.
It is obviously the case that for all σ and pair (i, j), one has:

χ σ (i, j) + χ σ (j, i) + δ σ (i, j) = 1.
When comparing the pair of alternatives (i, j) through the voting procedure, the proportion of the voting population which prefers alternative i to alternative j, and then votes for i against j, is σ∈S λ σ χ σ (i, j).

Definition 1 Consider the super majority rule with rate τ , 0.5 ≤ τ < 1. Alternative i is collectively preferred to alternative j (denoted i j) for the super majority rule τ if and

only if σ∈S λ σ χ σ (i, j) > µ σ∈S λ σ χ σ (j, i), where µ = τ 1 -τ .
Voters who tie i and j abstain from voting. Alternative i defeats alternative j if it rallies a proportion higher than τ of the expressed votes. The simple majority rule, τ = 0.5 corresponds to µ = 1. The majority rule will be called indifferently τ or µ, with τ = µ 1 + µ .

Lemma 1 Alternative i defeats alternative j for the majority rule µ if and only if

σ∈S λ σ (1 -χ σ (i, j) + µχ σ (j, i)) < 1 . (1)
Proof: Straightforwardly obtained from the inequality of definition 1 by adding the quan-

tity σ∈S λ σ (1 -χ σ (i, j)) to both sides. 2

Cycles of alternatives

The voting procedure defines a binary relation ( ) on the set of alternatives. A q-cycle of alternatives, or cycle of length q, for the binary relation ( ) is an ordered q-tuple a = (a 1 , a 2 , . . . , a q ) of the set A n , that satisfy a 1 a 2 . . . a q a 1 .

A cycle of alternatives is clearly defined up to a circular permutation of the q-tuple.

Therefore, a cycle for ( ) can be identified with an equivalence class of ordered sets, two ordered sets being equivalent if one is obtained from the other by a circular permutation.

Inequality (1) is applied to get a necessary condition for the existence of cycles. Define

a q+1 = a 1 . Lemma 2 The inequality σ∈S λ σ ζ σ (a) < q , (2)
where

ζ σ (a) = q - q i=1 χ σ (a i , a i+1 ) s σ (a) +µ q i=1 χ σ (a i+1 , a i ) t σ (a)
, is a necessary condition for the occurrence of the cycle a.

Proof: Obtained readily by adding the q inequalities (1) (one for each pair (a i , a i+1 ))

defining the cycle a. 2

Remark: The quantity s σ (a) measures how many of the q pairwise comparisons defining the cycle a the sequence σ strongly agrees with. The quantity t σ (a) measures how many times the sequence σ strongly disagrees with the cycle. Hence ζ σ (a) can be interpreted as a sort of distance of the preference σ to the cycle.

3 The 1 -1 -super majority rule

It is shown in this section that, if the refinement of preferences does not exceed , a super majority rule with rate 1 -1/ is necessary and sufficient to rule out Condorcet cycles of any length (see Corollary 1 below). Hence the key factor to rule out cycles is not the number of alternatives, but the maximal degree of refinement of the preferences expressed in the society. This will be done by proving the following result.

Proposition 1 When the refinement of individual preferences does not exceed , the super majority rule with rate

1 - 1 min (q, )
is necessary and sufficient to rule out Condorcet cycles of length q.

This result is well known in the case where there are only fine individual preferences, in which case = n and min (q, ) = q (cf. [START_REF] Ferejohn | On a class of rational decision procedures[END_REF]. Two immediate corollaries obtain.

Corollary 1 When the refinement of individual preferences does not exceed , the super majority rule with rate 1 -1/ is necessary and sufficient to rule out Condorcet cycles of any length.

Corollary 2 When the refinement of individual preferences does not exceed 2 (individual preferences are dichotomous), there is no Condorcet cycles through the simple majority rule.

This last corollary is identical to the result of aggregation obtained with approval voting [START_REF] Brams | Approval voting[END_REF]. The difference is that approval voting is a mechanism that forces voters to express only dichotomous preferences.

Before proving Proposition 1, it is useful to introduce preliminary comments and notation. The strong properties of symmetry of the setup allows us, without loss of generality, to the study to the cycle ι = (1, 2, . . . , q): only the length matters (see Balasko and Crès, lemma 7). Let us introduce a useful concept. In the sequence σ(1) σ(2) . . . σ(n) , a pair of consecutive elements (σ(i),

σ(i + 1)) is called 1. a rise if σ(i) < σ(i + 1) (we denote it σ(i) ↑ σ(i + 1)), 2. a fall if σ(i) > σ(i + 1) (we denote it σ(i) ↓ σ(i + 1)), 3. a level if σ(i) = σ(i + 1) (we denote it σ(i) → σ(i + 1)).
If s, t, and u denote the number of rises, falls and levels in σ respectively, then s + t + u = n -1, and the following obvious lemma holds.

Lemma 3 Let σ be a sequence of length n. Let sq , tq and ūq denote respectively the numbers of rises, falls and levels of the sequence of length q + 1: σ : σ(1) . . . σ(q)σ(1) .

Then sq + tq + ūq = q and t σ (ι) = tq and s σ (ι) = sq .

Proof of Proposition 1: Consider the case < q; for the other case, cf. [START_REF] Balasko | Condorcet cycles and super-majority rules[END_REF], lemma 13. To prove that the 1 -1 -super majority rule is sufficient to rule out Condorcet cycles, it is sufficient to show that any preference σ of refinement smaller or equal to satisfies: ζ σ (ι) ≥ q when µ ≥ -1. Indeed, then

σ∈S λ σ ζ σ (ι) ≥ q σ∈S λ σ = q ,
an inequality that rules out the cycle ι as lemma 2 states.

For µ ≥ -1, one has ζ σ (ι) ≥ (qsq ) + ( -1) tq . Let us first cope with the case where ūq = q, sq = tq = 0, which means that the preference σ is totally indecisive between the alternatives {1, . . . , q}. Then ζ σ (ι) = q and we are home. Otherwise, then sq and tq are both bigger or equal to 1 (since, in the sequence σ, we depart from and arrive to σ(1), if there is a rise, there must be a fall, and reciprocally). And a sufficient condition for the property to hold is (qsq ) + ( -1) tq ≥ q which is equivalent to proving that sq tq ≤ -1 .

Obviously, if this last inequality does not hold, the relative proportions of rises versus falls is higher than -1, then one would have, along the loop

σ(1) σ(q) σ(2) • σ(3) • • • •
• at least one series of consecutive rises (ignoring levels); by consecutive is meant: uninterrupted by any fall. This would entail that the refinement of the preference σ is bigger or equal to + 1, which cannot be. Hence the sufficiency.

As far as necessity is concerned, the standard example of a profile composed with circular individual preferences la Condorcet immediately allows to conclude. 2

The probability of Condorcet cycles

This section provides rates of super majority ruling out Condorcet cycles 'in probability', i.e., rates of super majority for which the relative volume, in the simplex ∆ S , of profiles giving rise to Condorcet cycles is very small. A probabilistic interpretation of the Lebesgue measure on the simplex of profiles can be constructed: it is the classical Impartial Anonymous Culture (IAC) for an infinite number of agents; see [START_REF] Gehrlein | Condorcet paradox and the Condorcet efficiency of voting rules[END_REF]. In this section, wherever an interpretation in probabilistic terms is given, this culture (or any other resulting in a probability distribution that is absolutely continuous with respect to the Lebesgue measure) is implicitly assumed. It is argued that the coarser the individual preferences, the lower the required rates.

An upper bound

Only an upper bound of these volumes will be provided here, and a very brutal one: it is the volume of the set described by inequality1 (2). But yet an additional argument is then given that reinforces the idea that the coarser the preferences, the easier and surer it is to aggregate them through majority voting.

The key feature to measure the relative volume of the set of profiles (λ σ ) such that σ∈S λ σ ζ σ (ι) < q is to rewrite this inequality by aggregating the weights λ σ having the same coefficient ζ σ (ι); the dimension of the simplex is thus reduced. To do that, let us partition the set of individual preferences S by means of the characteristic function ζ σ (ι), into the subsets:

B S (r, t; q) = {σ ∈ S | ζ σ (ι) = r + µt},
where (r, t) ∈ {q, 0} × {(r, t) | 1 ≤ t ≤ r ≤ q -1}, therefore defining 1 + q(q -1) 2 such subsets. Hence the new simplex of aggregated weights is of dimension q(q -1) 2 , and is endowed with this measure that is the projection of the Lebesgue measure on the original simplex.

The leading example: n = 3. The only cycles are of length 3, so that q = 3 is omitted.

The domain S contains 13 individual preferences: the six fine preferences 7), (8), ( 9), ( 10), ( 11), ( 12)} B S (3, 0) = {(13)}

.

Back to the general case, let b S (r, t; q) be the cardinal of B S (r, t; q); b S = r,t b S (r, t; q) be the total number of individual preferences. Define finally bS (r, t; q) = b S (r, t; q) b S .

Proposition 2 Fix the q-cycle ι = (1, . . . q). Define the continuous function F S (q; µ)

from N × R + into R + : F S (q; µ) = r,t
r + µt q bS (r,t;q)

.

The quantity

1 F S (q, µ) b S
is an upper bound of the relative Lebesgue measure, in the (b S -1)-simplex, of the set of profiles giving rise to the cycle ι.

Proof: Standard argument reproduced from Balasko and Crès (1997), proposition (15).

2

F S (q; µ) is an increasing function of the rate µ and lim µ→∞ F S (q; µ) = +∞ readily obtains. Furthermore, F S (q; 0) ≤ 1 comes from r ≤ q. As soon as F S (q; µ) > 1, proposition 2 guarantees, for a big enough number of admissible preferences b S , the volume of 'bad' profiles to be very small (even when multiplied by the number of possible cycles).

Definition 2 The function F S (q; •) is the characteristic function of the set of preferences S with respect to q-cycles, 3 ≤ q ≤ n. The (unique) positive real µ S (q) such that F S (q; µ S (q)) = 1 is the critical rate of the set of preferences S with respect to q-cycles.

It is said critical because it is a threshold: for µ > µ S (q), the probability of the cycle ι is very small as soon as there are enough preferences in S. In the case of the leading example (n = 3), the characteristic function is

F S (µ) =   1 + µ 3 3 2(1 + µ) 3 3 2 + µ 3 6   1 13
, and the critical value is µ = 1.0718, corresponding to a super majority rule with rate τ = 51.73%.

The sequel aims at showing that the coarser the individual preferences, the lower the critical rate. Therefore, if individual preferences are coarse, one needs a lower rate of super majority than for fine preferences to rule out cycles in probability.

Computations of the critical rate of S

The present subsection is rather technical and can be skipped at first reading. It aims at showing that the whole problem reduces to compute the critical rates, with respect to q-cycles, of the set of preferences S q (of refinement on q alternatives). The argument goes through two steps:

1. by showing that one can study the behavior of the critical rate of S, µ S , through the behavior of the critical rates of any partition of S: this is done in lemma 4; and the chosen partition is (S n ) n whose critical rates with respect to q-cycles are denoted (µ n (q)) n ;

2. then by showing that one can study the behavior the (µ n (q)) n,q through the behavior of the 'diagonal' critical rates µ q (q) (denoted µ q ): this is done in corollary 3.

The following lemma allows us to focus only on a well-chosen partition of S, namely the subsets (S n ) n of preferences of refinement exactly .

Lemma 4 Consider two subsets of preferences S and S , whose critical rates with respect to q-cycles are µ S (q) and µ S (q) respectively, with µ S (q) ≤ µ S (q). Let µ S∪S (q) denote the critical rate, with respect to q-cycles, of the set of preferences S ∪ S . Then µ S (q) ≤ µ S∪S (q) ≤ µ S (q).

Proof: Let F S (q; •), F S (q; •) and F S∪S (q; •) denote the characteristic functions, with respect to q-cycles, of the set of preferences S, S and S ∪ S respectively. Then one has

F S∪S (q; µ) = [F S (q; µ)] b S b S +b S [F S (q; µ)] b S b S +b S .
We have

F S∪S (q; µ S (q)) = [F S (q; µ S (q))] b S b S +b S
which is ≤ 1 because µ S (q) ≤ µ S (q) and F S (q; ) is strictly increasing. Symmetrically, we have

F S∪S (q; µ S (q)) = [F S∪S (q; µ S (q))] b S b S +b S which is ≥ 1. Hence the result. 2
Let us now focus on the set of preferences S n of refinement exactly , whose critical rate with respect to is denoted µ l n (q). The corresponding partition through the function ζ σ (ι) will be denoted (B n (r, t; q)) r,t and the related coefficient (b n (r, t; q)) r,t ; the total number of elements of S n is b n . The following lemma and its corollary allow us to restrict the study to the cycles of full length: n = q.

Lemma 5 We have:

b n+1 (r, t; q) = b n (r, t; q) + b -1 n (r, t; q) .
Proof: The proof goes as in [START_REF] Balasko | Condorcet cycles and super-majority rules[END_REF], lemma 8. Adding the (n + 1)-th alternative has no effect on the property of a given individual preference σ with respect the studied cycle ι, i.e., with respect to the value of s σ (ι) and t σ (ι). On the other hand, an individual preference of B n (r, t; q) defines preferences of B n+1 (r, t; q), as many as there are different subsets of tied alternatives, and an individual preference of B -1 n (r, t; q) defines preferences of B n+1 (r, t; q), as many as there are places between (and at both ends of) different subsets of tied alternatives. 2 Corollary 3 The critical rate µ n+1 (q) is contained between µ -1 n (q) and µ n (q).

Proof: The proof goes exactly as in lemma (4) and exploits the preceding proposition. 2

Thanks to the results of this subsection, we know that we have all the required information about the rates (µ n (q)) ,n,q (e.g., whether the sequence (µ n (q)) decreases with ) as soon as we can compute the rates (µ q (q)) ,q ; denoted from now on (µ q ) ,q .

Results on critical rates

The only critical rate that are computed are µ q q and µ 2 q , the rates of the sets S q q of fine and dichotomous individual preferences.

Proposition 3 The critical rate of the set of complete preferences S q q with respect to q-cycles is

µ q q = V (q -1) -1 , where 2 V (q) = q + 1 1 q 1 q! 2 q 2 q! . . . q q q q! . 2
The quantity q p is the Eulerian number at entry (q, p).

Probabilities of pairwise comparisons

The major weakness of the results provided in this section was raised by [START_REF] Tovey | Probabilities of preferences and cycles with super majority rules[END_REF]: a rate of super majority, along with ruling out cycles, makes it also highly improbable that two alternatives can be compared. In fact, one should compare the probability of a cycle with the probability of a pairwise comparison.

Here is provided the relative measure of the subset of parameters for which alternatives i and j are comparable through simple majority voting, i.e., the probability that the pairwise comparisons i j or j i occur. Define b i j

S = σ∈S χ σ (i, j) (resp. b i∼j S = σ∈S δ σ (i, j
)), i.e. the number of preferences in S for which alternative i is better than (resp. as good as) alternative j. Of course, for all (i, j),

b i j S + b i∼j S + b j i S = b S .
Proposition 6 The relative Lebesgue measure, in ∆ S , of the subset of parameters for which there is a pairwise comparison between i and j through simple majority voting is

2 1 µ + 1 b i j S +b j i S -1 b j i S k=1   b i j S + b j i S -1 b j i S -k   µ b j i S -k .
Proof: Following the principles exposed in Balasko andCrès (1997, 1998), the relative volume of profiles for which i j is:

(b S -1)! 1 µ+1 t=0 t b i∼j S -1 (b i∼j S -1)! (1 -t) b i j S -1 (b i j S -1)! dt .
It is the same for the profiles for which j i, hence the coefficient 2. 2

The important fact is that this probability does not depend on b i∼j S , the number of undecided preferences on the pair (i, j). For values of µ close to 1 and small values of b i j S +b j i S , it remains non-trivial, whereas the probability of a cycle involving i and j is shown to be very small when b i∼j S is big.

Example: Consider n = 3 and assume that dichotomous preferences are 9 times more frequent than fine preferences (it amounts to assume the impartial anonymous culture with an initial urn where dichotomous preferences are 9 times more likely to be chosen than fine preferences); then the characteristic rate is τ = 50.4%, and for a super majority rule with rate 53.5%, the upper bound gives 0.066 for the probability of a cycle (the actual probability is certainly much smaller), whereas the probability of a pairwise comparison is 0.65.

For a fixed super majority rule, the probability of a pairwise comparison between two alternatives increases toward 1 when the number of preferences exhibiting a strong ranking between these two alternatives decreases. Hence the coarser the individual preferences, the more likely pairwise comparisons, for a fixed super majority rule.

Concluding comments

The paper gives several arguments tending to show how the coarser the individual preferences, the surer it is that simple or super majority voting rules can aggregate them.

The approach chosen in Section 4 is probabilistic and based on the assumption of impartial anonymous culture (i.e. the Lebesgue measure on the simplex of profiles). A simple artifact allows to include much more distributions on the simplex of profiles: simply by allowing the domain S to include the same individual preference more than once 4 .

As far as the logic of Section 3 is concerned (condition securing the total elimination of cycles), another line could be investigated: this is the impact of the assumption that profiles of preferences contain an irreducible weight of voters with coarse preference. This can be simply illustrated by the leading example where n = 3. Denote Λ 1 = λ 1 + λ 4 + λ 5 , Λ 2 = λ 2 + λ 3 + λ 6 and Λ 3 = 13 i=7 λ i ; one easily gets, as a necessary condition for the cycle (1, 2, 3) to occur:

1 + Λ 1 > µ(1 + Λ 2 ) . (3)
Under the assumption that at least a ratio ν of the voting population exhibits coarse preferences (which translates in

Λ 3 ≥ ν ⇐⇒ Λ 1 + Λ 2 < 1 -ν), then inequality (3) is impossible if µ ≥ 2 -ν.
One of course gets back the result that for dichotomous preferences (i.e., ν = 1) cycles do not occur under the simple majority rule. But moreover 4 In the example at the end of Section 4, where n = 3, S contains 60 preferences: the 6 fine ones and 9

times the 6 dichotomous ones; the Lebesgue measure on the simplex of dimension 60 then projects into a measure on the usual simplex of dimension 12 with a heavier density on the dichotomous preferences:

35! 12 i=7 λ 8 i 8! .
See [START_REF] Balasko | Condorcet cycles and super-majority rules[END_REF] for details.

if at least half of the voting population exhibits coarse preferences (i.e., ν = 0.5), then τ = 60% is enough to exclude cycle, instead of 66% in the general case.

It is a step toward proving the conjecture since if, for example, F -1 q (1) ≥ F q (1) , and the two curves do not cross too early, then F -1 q crosses the value 1 before F q .

Proof: Fix k, 1 ≤ k ≤ , < q, and consider the two sets S and S of sequences of respective specification (e 1 , . . . , e ) and (e 1 , . . . , e k-1 , e k -1, 1, e k+1 , . . . , e ), e 1 + . . . + e = q.

Take a sequence σ of S; it contains e k times the integer k: σ -1 (k) = {i 1 , . . . , i e k } is a subset of {1, . . . , }. If we replace, in σ, the integer σ(i 1 ) = k by the integer k + 1, and then all the integers k , k + 1 ≤ k ≤ , by their immediate successor k + 1, then we obtain a sequence of S : denote it σi1 . Such a process associates with each sequence of S e k sequences of S . We get then a correspondence Γ k from S into S : Γ k : S -→ S σ -→ {σ i1 , . . . , σi l }

It is easy to check that all sequences of S can be constructed from the right sequence of S through this operation. Recall that S = q!/e 1 ! . . . (e k -1)! . . . e ! and S = q!/e 1 ! . . . e ! which entails S = e k × S.

We want to look at the effect of such transformations through Γ k on the characteristic functions of S and S respectively. We need compare the respective values of ζ σ (ι) and ζ σi 1 (ι). To do so, thanks to lemma (3) we only have to check how Γ changes the properties of σ in terms of rises, falls and levels.

Consider an integer i ∈ σ -1 (k), σ being an element of the set B S (r, p; q). There can be nine configurations in terms of rises, falls and levels around σ(i) = k. We only care about those configurations that are changed by the operation Γ k described above. They are:

1. σ(i -1) → σ(i)(= k) → σ(i + 1) which becomes σi (i -1) ↑ σi (i)(= k + 1) ↓ σi (i + 1); then σi ∈ B S (r -1, t + 1; q), 2. σ(i -1) → σ(i) ↑ σ(i + 1) which becomes σi (i -1) ↑ σi (i) ↑ σi (i + 1); then σi ∈ B S (r -1, t; q), 3. σ(i -1) → σ(i) ↓ σ(i + 1) which becomes σi (i -1) ↑ σi (i) ↓ σi (i + 1); then σi ∈ B S (r -1, t; q), 4. σ(i -1) ↑ σ(i) → σ(i + 1) which becomes σi (i -1) ↑ σi (i) ↓ σi (i + 1); then σi ∈ B S (r, t + 1; q), 5. σ(i -1) ↓ σ(i) → σ(i + 1) which becomes σi (i -1) ↓ σi (i) ↓ σi (i + 1); then σi ∈ B S (r, t + 1; q), Denote Ŝi the set S after transformation of the sequence σ as described above. Then we have:

• F Ŝi = F S × (r-1)+(t+1)µ r+tµ 1/ S in case 1,

• F Ŝi = F S × (r-1)+tµ r+tµ 1/ S in cases 2 and 3,

• F Ŝi = F S × r+(t+1)µ r+tµ 1/ S in cases 4 and 5.

It is clear that for µ = 1, case 1 is neutral with respect to the characteristic function. In order to get the proposition, it is sufficient to prove that to each transformation of type 4 or 5 corresponds a transformation of type 2 or 3. Indeed, the association of a case of type 2 or 3 with a case of type 4 or 5 amounts to multiply the characteristic function F S by (r + tµ) 2 + (r + tµ)(µ -1)µ (r + tµ) 2 1/ S

  indecisive one, (13) : 111. The partition of S through the characteristic function ζ gives four subsets: B S (1, 1) = {(1), (4), (5)} B S (2, 2) = {(2), (3), (6)} B S (2, 1) = {(

  setup, milder assumptions than usual (i.e., respectively, super majority rules with a low rate and a domain restriction -based on some consensus of individual preferences-for which a large ratio of all rankings are admissible) which rule out Condorcet cycles, if not completely, at least with a high probability. The same simple framework is used in the present paper. Let there be n alternatives. There are n! ways of ordering them: one

can identify the preference orderings with the permutations of the n alternatives. To each preference σ is associated the percentage λ σ of the voting population having this particular individual preference. The vector (λ σ ) σ∈S (where S is the domain of individual preferences; e.g. S = S n , the whole symmetric group of cardinality n!, if there are no restrictions) is then a point in the ( S -1)-simplex; it represents a profile of individual preferences, i.e., a particular distribution of voters among preferences. It appears that for certain profiles, at least one Condorcet cycle obtains. Hence some regions give rise to Condorcet cycles, and some others do not. The authors compute a brutal upper bound of the relative Lebesgue measure of the regions in which cycles appear: these measures are very small.

This inequality is obtained by addition of the q inequalities defining the cycle, therefore a lot of information is lost in the process.

This is a stronger property than the one obtained by[START_REF] Balasko | Condorcet cycles and super-majority rules[END_REF], theorem 16. This is nevertheless checked on computers up to q = 1000, which is way above the highest values that are interesting in the framework of this paper. These computations suggest that in fact V (q) can be approximated by 2 + 1/(3q) -5/(36q 2 ).

Proof: The preferences of S q q are fine; therefore, for r = t, bq q (r, t; q) = 0. Then the characteristic function of S q q becomes F q q (q; µ) = q-1 t=1

t(1 + µ) q bq q (t,t;q)

= (1 + µ) bq q (q) q q-1 t=1 t bq q (t,t;q) , where bq q (q) = 1, and bq q (t, t; q) = q -1 t /(q -1)! thanks to proposition (11) in [START_REF] Balasko | Condorcet cycles and super-majority rules[END_REF]. Then the characteristic function F q q (q; •) can be rewritten

Proposition 4 For all n ≥ q, the critical rate with respect to q-cycles of the set S 2 n of preferences of refinement 2 is 1:

Proof. This is an immediate corollary of proposition 2. 2

The following table gives the values of µ q , and corresponding rate τ q , for 3 ≤ q ≤ 6 and These first values suggest the following conjecture.

Conjecture:

Hence the coarser the individual preferences, the lower the rate of super-majority necessary to rule them out in probability. We can prove the conjecture for = q.

Proposition 5 For all q s. t

Proof: See Appendix. 2

Appendix

Proposition 7 For all q and l such that ≤ q, then we have

which is strictly smaller than 1 for µ = 1. The final argument to get the proposition is that the full transformation of S through Γ k gives e k times S , which has the same characteristic function as S.

Assume that case 4 arises, i.e., we start from a sequence σ ∈ B S (r, p; q) such that k = σ(i) whose environment is σ(i -1) ↑ σ(i) → σ(i + 1). It entails in particular that σ(i + 1) = k. Consider the ordered set {i, i + 1, . . . , q, 1, 2, . . . , i -1}. Reading this set from left to right, pick the first integer (call it j) such that σ(j) = k. That such a j exists is obvious, otherwise σ is not a surjection on {1, 2, . . . , }. We then know that the environment of k = σ(j) is σ(j -1) → σ(j) ↓ σ(j + 1) or σ(j -1) → σ(j) ↑ σ(j + 1) depending on whether σ(j +1) is smaller or bigger than k, meaning that case 2 or 3 automatically follows case 4. The same line of reasoning holds if case 5 arises. 2

Proof of proposition ( 5): We reproduce the preceding proof in the case where e j = 1 for all 1 ≤ j = k ≤ q -1 and e k = 2. For this set, case 1 cannot occur, because an integer appears at most twice in a sequence. On top of that if σ is a sequence of S included in B S (r, t; q), then necessarily, since there is at most one level, i.e., we have: t ≤ r ≤ t + 1. Only when r = t + 1 the transformation through Γ k as described in the preceding proof is not neutral with respect to F S : we know that it then amounts to multiply F S by the quantity

Associate to σ ∈ B S (t + 1, t; q) the reverse sequence Opp(σ) : σ(n) . . . σ(1). It has also one level and as many falls as σ has rises, i.e., qt -1, so that we have Opp(σ) ∈ B S (qt -1, qt; q). And the transformation Γ k applied to Opp(σ) amounts to multiply F S by the quantity

If we conjugate the transformations of σ and Opp(σ), we obtain a multiplication by the quantity [f q (t; µ)] 1/ S , where the function f q is given by:

It can be easily proved that f q (•; µ) reaches its maximum for x = (q -1)/2, and that f q ((q -1)/2; µ) is smaller than 1 when µ ≤ 1 + 2/(q -1). Indeed f q ((q -1)/2; µ) = (q + 1)(q -1)(1 + µ) (q + 1) + (q -1)µ 4 , so that it is smaller than 1 if and only if [ (q + 1)(q -1) -(q -1)]µ ≤ (q + 1) -(q + 1)(q -1) which is equivalent to √ q -1( √ q + 1-√ q -1)µ ≤ √ q + 1( √ q + 1-√ q -1) hence µ ≤ q+1 q-1 = 1 + 2/(q -1). The proposition holds true, first because the argument can be reproduced for all k, 1 ≤ k ≤ q -1we obtain the whole set S q q for each k-, and second because for µ = V (q -1) ≤ 1 + 2/(q -1), F S q q (q; V (q -1)) = 1 and then F S q-1 q (q; V (q -1)) > 1 which implies µ q-1 q < µ q q = V (q -1). 2