Pierre Salmon

The Logic of Pressure Groups and "Federa Ijrsm-11"

Constructing a complete model of the relations between politicians, bureaucrats, interest groups, and voters is not an easy task. Consequently most economic models of government have to do away with some o f these relations. One way to simplify matters is the so-called group theory o f politics (or "pluralism") which assumes that the influence o f interest groups dominates policy-making. In political scien cel , where this line of thinking has been important for a long time, the theory is not very striking, since it is not always clear that an interest group is anything more than a number of individuals, or firms, who share a common concern, Lg. an interest category. Thus, the theory can be made as empty or innocuous as one wishes In economics, the interest group theory of government is normally much stronger or bolder. The reason for this difference is that economists have been provided by Olson [1963] with a sharp distinction between interest categories (which he calls latent groups), and mobilized or organized groups. As a consequence o f the rationality o f free riding, many interest categories are not organized. At any point in time, (organized) interest groups represent only a subset of interest categories. Consequently, theories, such as the one formulated by Becker (19831. which explain policy-m aking exclusively by the interplay o f groups within this subset (hence which concede no independent influence to voters, who are assumed to be "bought", or to politicians or bureaucrats, who are assumed to be "hired"), are strong theories indeed 3.

It is true that many economic theories o f government do not ascribe such prominence to interest groups. Some models ignore them altogether, while others, such as the ones by Downs [1997], or [START_REF] Breton | The economic theory of representative government[END_REF]. give them a modest role. However, all economic models, and most political science analyses, which mention interest groups share the common assumption that they are independent actors. They may or may not be successful in influencing policy-makers; in any case they do not themselves come under any influence.

The purpose of this paper is to explore the possibility and the consequences of an alternative interpretation. It is well-known that many organized groups or associations do not really undertake to influence policy-making, but this observation is irrelevant in the present context. The groups which are discussed here are those v h ich are normally assumed to exert such influence, i.e. pressure groups or lobbies. Thus, the point which is at stake is whether pressure groups are as likely to fee. "hired" by politicians or bureaucrats as they are to "hire" them (to use Becker's words). As we shall see the view that they are "hired" is shared to some extent by a number o f political scientists 4 However, in their case, the argument seems to have been inspired by observation o f the relationship between interest groups and politicians or bureaucrats in a number o f European countries, while for our part we shall try to found the argument on theoretical grounds. More precisely, while they systematize the situation they think they observe in continental Europe, by contrast with the one they see as prevailing in the United States, we shall start our reasoning from a theoretical puzzle, originating in the combination of Olson's mobilization problem and the objection raised by Stigler [1974] to the solution Olson and others have offered for it.

The "federal dimension", defined broadly as in [START_REF] Breton | The economic constitution of federal states[END_REF], will prove helpful in two ways. First, it increases the workability o f the hypothesis proposed here.

Although no "unitary state" (in a sense that is the counterpart o f their broad definition of federalism) exists in the real world, except perhaps for tiny states like Monaco or Granada, many economic models o f government omit to take the "federal dimension'' into account. If they also gloss over other complexities o f government, such as the rivalry between bureaus, as analyzed for instance in Breton and Vintrobe [1982]. the independence o f pressure groups tends to become a natural (although, as we shall see. unexplained) assumption. Second. I think that it is a defect o f some economic theories o f government, such as the one proposed by Olson [1962]. that they imply gloomy long-term tendencies (conversely, it is an attractive feature of Becker's model that it does not). It seems wiser to include in our models a balancing mechanism o f some sort.

Although different from Olson's, the interpretation o f pressure groups suggested in this paper could also imply ultimately unchecked tendencies. "Federalism" is useful inasmuch as it might provide the counterforce that is needed in some circumstances to obtain a mechanism o f that kind.

The Olson-Stiller nuzzle

The satisfaction, or even the mere expression o f a common interest is a pure Samuelsonian public good. No member o f the interest category can be excluded from its consumption. Consequently in large homogeneous categories (see below) rational self-interested individuals or firms will free ride, that is shirk the costs o f furthering the common interest, except if they are induced to do otherwise by what Olson [ 19631 calls selective incentives. Such incentives can be negative (e.g. coercion) or positive. I shall concentrate on the latter, vh ich Stigler [1974] nicely calls "individually appropriable inducements" . Mobilization o f "large" homogeneous categories will depend on finding selective incentives. Mobilization o f "small" interest categories (including large categories in which some members are "large") and o f "intermediate" categories might take place without selective incentives but will tend, according to Olson, to be sub-optimal. Olsons's analysis has given birth to an abundant literature. Three issues deserve particular attention.

First, how do we know whether a category is "large"? To answer this question. I suggest that we should start from the relation, formulated by Olson himself, between competition and free riding. If we disregard anti-trust legislation, the difficulty o f collusion is an instance o f the difficulty o f mobilization; the rationality o f free riding entails the dominance o f competition. Thus, the way we usually deal with competition should guide us in handling the identification of large categories. On one hand, we have a theoretical construct, pure competition, the purpose o f which is to treat the seller as a price-taker, and which is founded on the assumption that the number o f firms is infinite. On the other hand, we have real world industries, say the Italian footwear industry. Then we use the theory o f competition to study a real world question, we affirm the antecedent (in the context of a hypothetical syllogism). Ve say: the Italian footwear industry is competitive. Italian shoemakers are price-takers. each o f them faces an infinitely elastic demand curve. Ve do not wait until we find in the real world an industry with an infinity o f small firms to assume that firms are price-takers. Nor do we argue that each shoemaker has some small influence on prices. In some contexts, we are even willing to assume that the steel industry is competitive, although we know that this assumption is not realistic.

Why should we not exactly proceed in the same way when we dicuss free-riding in large groups? The theoretical definition o f a large group should be that no member has any influence on outcomes, he is a an "outcome-taker". the size o f the group is infinite.

To use another word coined by Stigler, there is no "cheap riding" in large groups, just plain free riding. Then we turn to the real world and apply the theory founded on this unrealistic assumption to existing "large" groups, provided no member is "much larger" than any other. A "large" group in this sense can involve a few dozens members, or sometimes less. This means that we decide to see many large groups in existence, in the same way as we decide to see many competitive situations in the product, factor, or political markets. However. Stigler is probably right when he argues that many trade associations are not homogeneous as far as the dimension o f member-firms is concerned; they should not be treated as "large" groups in Olson's sense6. But trade associations are only a subset o f pressure groups. Essential in some contexts (e.g. for the explanation o f tariffs), they are not as important in others, such as the one studied by Becker (19831 Leaving aside for the moment the particular case o f firms, we shall assume consequently that the large group situation is very common. In particular, "group federalism" will not be considered as a general solution to the mobilization problem; even the lover-level member-groups in a federation of groups are typically "large" according to the conception adopted here.

The second issue i s : how are both the total cost of individually appropriable inducements and the cost o f producing the collective good to be financed? Olson's solution is that in successful groups, the collective good is a by-product o f inducements.

The objection raised by Stigler [1974] to this solution is the following: "Vhy should the association be able to charge more than the cost o f the services which are appropriable as private goods? I f an association seeks to add a charge for collective goods, a rival association which undertook no collective action should undersell it" (p.360).0ne possible answer to this objection is the existence o f barriers to entry. Stigler excludes this, but he considers only trade associations, and only in the context o f the United States.

If one thinks o f other kinds o f pressure groups, such as professional associations, and of other countries, barriers to entry can be found. However, they are generally the result o f government policies or regulations, or (in the case o f unions) o f decisions taken by firms. As we shall see. it is precisely one o f our main points that mobilization in large groups requires some form of support from outside. Another possible answer to Stigler is that pressure groups benefit from tax exemptions. Again, i f tax exemptions are conceded to a particular category on the condition that it gets mobilized within an approved association, it is a form o f external support for this association. If all non-profit-associations, or their members, benefit from exemptions. Stigler's objection remains valid: among associations, all o f them tax-exempted, those which show no concern, or only nominal concern, for collective action would outcompete others.

Finally, the most popular solution to the problem o f finance involves nonmaterial inducements. Among these. Olson himself mentions social, psychological, moral, and even erotic incentives. One may wish to interpret them, or some o f them, within the context o f a relaxation o f the rationality and/or self-interest assumptions. If v e do not wish to do that, it must be noted that these nonmaterial inducements can provide a solution only if they are conditional bv nature on participation in collective action, and not merely because it has been decided to offer them exclusively to those who participate in collective action. In other words, in order to solve the Olson-Stigler puzzle, they must have characteristics such that groups that do not engage in collective action must not be able to provide them at ail, or to provide inducements of the same quality. In fact, this technical link between collective action and nonmaterial inducements is plausible in many circumstances (not restricted to revolutionary periods, in which it is o f course even more likely). If I am not satisfied with this solution, adopted by many political scientists and a number o f economists, it is not because it lacks plausibility. As an individual, I think that nonmaterial motivations, and perhaps even unselfish motivations (depending on the definition given to selfishness), are important. As an economist, however, I think that the tradition, or research strategy, which consists in introducing them as parsimoniously as we can as assumptions in our models, has proved generally fruitful in economics. The problem with these assumptions is not that they are unrealistic, but rather that they make the researcher's life too easy. In other words, their heuristic value is generally negative. They prevent us from thinking deeply enough about reality.

The third issue raised in the literature that followed Olson s analysis is leadership.

Could the recognition that groups generally have leaders help in solving Olson s problem o f mobilization in large groups? Many authors have thought s o 1. However, although useful in denicting the mobilization process, and providing us with a way o f presenting things that will prove helpful in the analysis below, existing models o f leadership or o f entrepreneurship do not solve the mobilization problem *.

The theoretical difficulty we must overcome is that not enough resources are available for the mobilization o f large groups that engage in collective action in a competitive environment. The resources provided by members are just sufficient to finance the services that induce them to join the group. Many associations do not really bother about collective action. Their viability raises no problem as shown by the theory o f clubs. For those which do bother, however, if we decide to define rationality and self-interest narrowly (as we should try to do), the conclusion seems inescapable.

Insofar as they are successful, these groups necessarily receive some support or some resources from outside. Once we have decided that we have to look for mechanisms producing such support, many can be discovered. To undertake this exploration it will be convenient to start with a simple "model" o f a large group with a leader.

A model o f leadership.

Let us assume that all members o f the group are small (in the sense that price-takers are small) and that their mobilization is necessary for furthering a common interest. Le. producing a public good (the nature o f which will be considered later).

Mobilization is achieved and maintained by a "leader" (or a leadership firm), who works for money and is not a consumer of the public good. This leader is an entrepreneur as usually defined in economic theory. For the purpose o f mobilizing the group, or keeping it mobilized, he provides members with private goods.

individually appropriable inducements. Thus, his net outlay is constituted by the cost o f providing the private goods, plus the cost o f producing the public good, minus the contributions from members (it does not really matter who is actually or physically producing the public good). To this net outlay must be added the opportunity cost o f the leader himself.

At this stage, the mobilization o f the group and the production of the public good are possible, provided some "finance" from outside can be found. For this to occur, the leader must sell something to the outside world. In order to avoid an infinite regress, it must be a private good (or an excludable public good). There are several possibilities, as we shall see later. For the time-being, let us assume this private good to be "influence '. Although the leader cannot draw any net resources from the members, in spite of the utility he gives them through the production o f the public good (because they free ride), this production provides him with an "influence" o f some sort (to be considered later) over them, either in matters directly related to the public good, or in unrelated or more distant matters. By assumption, this influence has some value to the outside world. It can be sold. Figure 1 shows how a public good can be produced along these lines.

[Figure 1 about here]

The production o f the public good is measured on OY'. is a function o f the quantity o f influence that is sold on the market for influence. This function is depicted by line OV (drawn here as a straight line because our leader is assumed to be a price-taker on the "influence" market). Net profit is the vertical difference between curves OV and CT. OF is the level of "influence" that maximizes net profit. In the long run. entry will make sure that net profit is zero. Hence curves OV and Ct will be tangent at a point that will determine optimal production.

The shapes and positions o f the curves reflect usual or plausible assumptions.

Anyhow, many large categories never get organized, or do not produce a collective good.

It is sufficient that this model works in a number of cases. Of course, an implicit assumption in this model is that the provision of private goods to members gives no

S bis FIGURE I

"influence" to leaders. However, this assumption see01s to a large extent plausible: the owners or managers o f "clubs", like the producers o f all sorts o f private goods, however efficient or successful they may be, do not normally enjoy "influence" in the sense intended here.

In the context o f this paper, our "model" is simply illustrative. What we are interested in is a broader class o f exchange situations between the leader and the outside world on the one hand, and the leader and members o f the interest group on the other hand. Thus, instead o f restricting our attention to the particulars o f the exchange situation depicted above, we shall consider three broader issues. That services can be sold in general to the outside world, or more particularly to the "public sector"? How can this "public sector" pay for these services in a democracy? That is exactly the "public good" that is produced within the group?

4. The suonlv o f services to the public sector.

Then refering to influence, we do not suggest that the leader could change the preferences o f members. As usual in economics, utility functions are assumed to be given. Hence, influence is somehow related to information in a broad sense. Conveying information to members o f an interest category is not always easy for politicians and for bureaus. If they do or say something that is of value for individuals sharing a common interest, it may not be possible for them to ensure that this is fully perceived by a significant proportion o f the category. Group organizations (represented in our simplified reasoning by leaders) are in general indispensable agents for the transmission o f specialized information. However, the service provided for the public sector in this way is not a function of collective action but only of membership to the group. It could not explain curve AL in figure 1.

Now. leaders can do more than dispatch news. They can comment this news, approve or condemn policies, programmes, or declarations. They can go further and actually advise members to vote for or to join a political party. For these less passive activities to have an effect, leaders must be trusted (at least to some extent). And trust will be positive ley related to their perceived performance in promoting the common interest of the group. Thus influence is a positive function of the production o f the collective good, a relationship reflected by curve AL.

Although it could perhaps be treated separately, we can include in "influence" the power that a relatively trusted leader enjoys to define or to modify the characteristics o f common interest, or o f collective action, as perceived by members. He can emphasize one subgoal rather than another. He can induce the group to be concerned with longer-term or shorter-term perspectives than the ones it would be inclined to adopt spontaneously.

Another service that can be offered to the public sector is. once again, information, but this time offered by. the group to politicians and bureaus. This widely aknowledged service produced by interest groups includes technical information as well as information about the preferences and perceptions o f members o f the group or o f the interest category in general.

We do not have to elaborate on the services mentioned up to this point since they are so widely perceived. On the contrary, it should probably be stressed that collective action, or pressure, itself is often o f considerable value to some actor in the public sector.

This can occur either because the objectives o f the group coincide with the goal pursued by a political party, a bureau, or a government, or because mobilization o f the group, independently o f its purpose, has some value for them. Many illustrations could be found. A bureau may need pressure from a group to persuade another bureau (e.g. the ministry o f finance), or politicians, that a policy it favours should be adopted, or that a policy it does not like should be discarded. More broadly, it may welcome pressure addressed to itself as a proof that it is alive, important, fulfilling a worthy function within government, and thus that it deserves supplementary means. When in opposition, a political party will welcome expressions of discontent, or demonstrations, addressed to the government by particular groups, independently o f the content o f their demands. In some cases (a bit suspect in democracies), even ruling parties may secretely support disruptive mobilization against themselves inasmuch as this may lead to more discipline or increased powers.

The plurality o f governments implied by the "federal dimension" enlarges the market for these kinds o f services offered by pressure groups. A junior government may see the mobilization o f an interest category within its jurisdiction as a means to reinforce its bargaining position at a higher level. Te shall consider examples of this later on. Sometimes, it is the other way round: pressure at the national level is invoked to pass legislation that constrains junior governments y. These cases concern particular policy goals. However, like bureaus, governments fear to be generally ignored or bypassed. The existence o f a rivalry between them thus implies that they will often consider the pressure that is addressed to them as a means to achieve better integration and standing in the overall organization o f society.

This behaviour is particularly open when a new level of jurisdiction is created, or is provided with increased powers. In Europe, two levels have been instituted or made more significant in recent decades. One is the "regional level". As noted by Ursula Hicks10, in several countries o f Europe, large regions have received powers that were previously assigned to central governments, or to lower levels o f jurisdiction such as counties, provinces or départements. Unfortunately, only casual observation can be invoked to confirm the interest displayed by the new regional authorities in inducing pressure groups to turn to them rather than to their traditional interlocutors at the higher or lower levels.

More reliable information is available with regard to the second level which has been created in recent years, the so-called "supranational level". Before we turn to the most obvious case. Le. the European Communities, let us note that historical evidence should be sought and could probably be found on this matter. For instance, according to Sidjanski. in the nineteenth century, the Swiss federal government (Switzerland only became a federation in 1848) "founded its own interest groups on the federal level in order to build its own links with the citizenry and bypass the old channels, which were jealously defended by the cantons"

In the case o f European integration, from the start the partisans o f European unity and political scientists such as Haas [ 1948] relied upon pressure groups to provide the process with a major driving force. It was widely held that the Commission (or its predecessors) would find allies in the pressure groups. It would benefit by the influence exerted by groups within each country on their own politicians. This influence was expected to play in favour o f further integration as a result o f various "spillover" mechanisms, that is as an element in a dynamic process, seen as a productive succession o f imbalances. It is not clear whether serious consideration was given to open alliances between the Commission and pressure groups within the member states. In any case, the national governments made sure that such alliances could not develop, and in fact few occurred1*. It did not matter so much because transnational pressure groups were expected to become active directly in Brussels. This hope seems to have been shared by the national governments when the European institutions were created, since these institutions make substantial room for what [START_REF]Agrooolitics in the European Community: interest groups and the cnmmon agricultural policy[END_REF] calls the Eurogroups to form and to have a say in decision-making (the Commission is even required to deal with Eurogroups). Since then, it has been widely observed that the Commission has actually done its best for the development o f "peak groups" of this kind, and for the direct chanelling o f demands to itself.

Of course, it is generally agreed that this policy has failed. Nor has the Swiss government, in spite o f more genuine federal powers, enjoyed for a much longer time, achieved unmitigated success in inducing groups to bypass cantonal governments, so making itself the main focus o f pressure. But these observations are secondary here.

That is important is the interest revealed by overarching institutions in becoming the targets o f pressure groups.

Other services offered by pressure groups can be mentioned. Te shall not dwell on the delivery o f services that are normally produced by governments, since they have been analyzed at length in the "corporatism'* literature, and are not specific to pressure groups, and shall turn instead to a last category, constituted by what we shall call "evasion services". These generally hidden services consist in helping individuals in the public sector, as well as in helping bureaus, political parties, or junior governments, to evade regulations, rules of conductor monitoring. To take one single example, financing political campaigns is generally assumed to be done by interest groups in exchange for accepting collective demands, while it may be the case that in reality the money often ultimately comes from government (on various levels), and only transits through pressure groups, mainly as a means to evade the law. Now, the question raised by the last point as well as by others is: admitting that pressure groups can provide politicians and bureaucrats with a whole array of valuable services, how can these services be bought?

V Payments to pressure groups.

The conventional view is that pressure groups are paid in the form of compliance.

In exchange for the various services they furnish, governments decide or implement the policies that correspond to the common interests they fight for. It is not our intention to deny the importance of this kind of exchange. But, as a consequence of free riding, it cannot provide us with the financing mechanism that we decided we needed. It cannot explain that organized large groups come into existence and manage to survive.

Other goods must be delivered to the groups. We shall be brief on the most common forms of "finance" transfered to pressure groups by the public sector. Again, they are well documented in the recent literature on "corporatism" (see for instance Berger [1981]). It will be more interesting to concentrate on the role of "federalism" in these financing mechanisms.

Of course, financing must be interpreted here as direct or indirect transfers of resources, or concession of the means to acquire resources, of *nv kind Thus it includes subsidies, payments in kind, and the provision of selective incentives that are necessary for members to join the group. Compulsion to join is an extreme case. More frequent, however, is the granting of a monopoly, or at least of barriers to entry, to approved groups (for the production of the private goods which serve as selective inducements). A form of support that is less documented, but may be significant, is the deferred promotion of group leaders or their collaborators to rewarding jobs or functions in the public sector (including parliament). Some authors, such as [START_REF] Schmitter | Interest intermediation and regime governability in contemporary Westen Europe and North America[END_REF].

systematize various forms of support within a general theory of "societal corporatism", or a theory of the "institutionalization of groups". However, their analysis is partly misleading inasmuch as it tends to suggest that the basic phenomenon they describe is both novel and limited to Europe and some other parts of the world, excluding the countries, such as the United States, in which "pluralism" is still dominant. In our opinion, public support of pressure groups is much more widespread than is assumed by this neo-corporatism literature, even if it is probably true that it is more open in some countries (cf. Salisbury [1973] or [START_REF] Sabel | The internal politics of trade unions[END_REF]).

Taking into account the "federal dimension" is particularly helpful in perceiving the importance of this support. It has two kinds of implications. First, the mechanisms described above acquire incomparable legitimacy, and are consequently much easier to implement, when the public authority which provides resources can be seen itself as competing in a wider environment, especially if it can be seen as fighting for the welfare of its constituents. For instance, the collusion between farm groups, or other special interest categories, and national ministries that would be suspect to many people in a purely domestic context becomes perfectly acceptable in the perspective of negociations in Brussels. Similarly, close and open alliances between local authorities and local pressure groups do not seem improper when observed in the context of nation-wide rivalry between jurisdictions for grants or other favourable policies. Of course, broadening the scope for collusion in this way can have effects in both directions. In the context of the traditional interpretation, it helps to explain why pressure groups, although they are assumed to transfer resources to politicians or bureaucrats, are accepted as part of the democratic system. In our interpretation, it increases the plausibility of reverse flows of private goods, from public authorities to pressure groups. Taken together these two categories of flow mean that a significant proportion of the resources involved can very well only transit through pressure groups.

The second kind of implications of the "federal dimension" concerns political parties. In the context of models of "unitary" government, political parties are either in power or in the opposition. But they cannot be assumed to stay for extended periods of time in the latter, at least if these models make the usual self-interest assumptions13.

When local or junior governments are allowed for, the general viability of national political parties can be accounted for: all important parties are generally in power, either on the central level, or at least on some local or regional ones. In Europe in particular, they usually enjoy very stable local strongholds. This provides them with substantial resources. The result of this independence is that the nature of their relationship with the pressure groups that are close to them, the oarentela phenomenon made famous by LaPalombara [1964], is not to be unproblematically interpreted as a one-way transfer of resources from the pressure groups to the parties.

The last point brings us to the question of the ultimate source of funds or resources that are chanelled to the collective action of large groups. Net contribution by the "small members" of large groups is excluded under our assumptions. Public finance is most important, as suggested above. However, political action on the part of big firms is not hindered by free riding as is action by small members of interest categories. Either as individual agents, or through their membership in "small" or "intermediate" groups, they can transfer resources to governments, to political parties, or to pressure groups.

"Bigness" is of course a relative concept: modest firms can be "large" on the local level.

Thus, altogether, probably all kinds of firms contribute resources to the non-market sector, defined as including pressure groups. This means that, in aggregate terms, the ultimate source of the funds that are necessary for collective action in large interest categories is a combination of taxpayers' and firms' money. This should not necessarily be interpreted as supporting a Marxist view of the state. In some countries, political parties are probably able to ensure that the major part of these funds is extracted from the taxpayers.

6. Further remarks on the public good produced in pressure groups.

The avowed purpose of pressure groups is to farther the common interest of their members. This raises several questions. First, this common interest cannot be ascertained easily. Many policies that seem to enhance it may be illusory or detrimental. For instance, social advantages conceded to wage-earners. such as longer paid holidays, or greater security at work, are often paid for by the workers themselves in the form of lower increases in real wages, or an increased probability of lay-offs. We have seen that this lack of precision of the common interest offers particular opportunities to group leaders in the context of their relationship with politicians and bureaucrats.

Second, assuming that the common interest is not ambiguous, the point must be made that it does not exactly coincide with the public good produced in the group. It differs from it as a result of three effects. The first effect stems from the fact that a pressure group wants to influence matters. Hence, what counts is the impact its action makes, the difference it achieves. We are dealing with a theoretical variable that implies a comparison with what would have happened if the collective action had not taken place, that is with a variable which involves counterfactual propositions. No direct observation can be conclusive (as shown by the long-standing discussion on the contribution of unions to the increased welfare of workers). The second effect reflects the fact that groups do not always purport to influence matters. They may pursue more modest objectives such as pressure for its own sake or expression. At least, even if they were pursuing the objective of influencing policy-making, it is something that they actually managed to fight for it. even if they did not win. In other words, a group that has had no impact has not necessarily failed in producing a public good that is valuable to its members. The two variables which we have just identified, i.e impact and expression or pressure, concern the group as a whole. Although difficult to measure, they are in a sense objective. But there is a third variable. In the model outlined above, the influence the leader exercises is related to his personal performance as it is perceived by the members of the group.

This leads us to consideration of the time dimension. Influence is a form of capital stock. It accumulates as a result of investment, and it depreciates with the passing of time. Its value is to be explained in terms of future flows. In given circumstances, there is probably a time path of influence that is optimal for the leader. For instance, it is unlikely that performance concentrated in one period would be optimal (for the leader), except if the leader contemplates leaving the group, or transforming it into a "club" (in the theoretical sense of this word), that is providing only private services to members.

Obviously these characteristics of their position have implications when assessing the most likely behaviour of leaders. More important still, they offer politicians and bureaucrats interesting strategic opportunities. To a large extent, these actors can control the variables that are essential for the welfare of leaders. Thus they can reward or punish them by defining policies and relations in such a way that leaders will or will not be perceived by members of their groups as performing well. They can concede favours that are purely apparent because they are compensated by the use of more discrete instruments (e g monetary policy), or they can bypass the leaders and concede real favours to the interest category in such a way that collective action will appear pointless for a long time. These opportunities allow for some forms of support for leaders which cannot provide resources that are usable for mobilization under our assumptions, but which contribute in making the groups and the leaders dependent. Could be added, along this line, the opportunities offered to politicians and bureaucrats by the existence of rivalry among leaders within a group, or of rivalry between organized groups within an interest category. Thus a new object of exchange has to be considered between group leaders and various decision-makers in the public sector. In this particular market, sham battles, play-acting, or window-dressing will constitute a substantial part of the exchanged goods.

Implications.

The foregoing analysis does not lead to a clear picture of the general relationship between pressure groups, politicians, and bureaucrats. It does not imply that pressure groups have no influence on policy-making, but that their influence is not to be considered as an exogeneous variable, or even a semi-exegoneous variable, determined in a separable part of the overall politico-economic model. It is generally determined, along with other variables, in a rather complicated system, and is in any case very difficult to ascertain. In this perspective, policy-making itself is the result of a complex and opaque set of relations within a sort of "oligarchy", constituted by leaders of various organizations, such as pressure groups, political parties, bureaus, private firms, constrained and influenced from the outside. That are the conflicts and the alliances within the "oligarchy", who are the real winners and losers, whether there is stalemate or change are questions that cannot confidently be answered from outside. Given this assumed state of affairs, what problems arise for the ordinary citizen and for the model-builder?

Citizens will probably acquire a practical understanding of the system of relations that prevails in their country. For instance, they will know whether the best thing to do.

for a particular purpose, is to approach or join a union, a political party, or both.

However, the fact that they have learnt to find their way around the system does not mean that they can measure its aggregate impact on their welfare, that they are satisfied with it. or that they can contribute to its improvement. From their point of view it is particularly annoying that they cannot fully trust their representatives within unions or other pressure groups.

\

For the model-builder, to these difficulties must be added the diversity of patterns from one country to another. Depending upon traditions, bureaucracies will be strong in some countries and weak in others. The independence and authority of the executive branch of government will be very variable. Electoral systems will change matters completely. For instance, the combination of fully proportional representation, parliamentary democracy, and a veak bureaucracy will probably imply oarentela relations between political parties and pressure groups that will make them hardly distinguishable. Thus one can have some doubts as to the possibility of obtaining in this area the simple and general models that economists seek, and one can fear that many existing models that give prominence to interest groups are not really widely applicable outside North America (if they are applicable at all).

Now. in economics as in most scientific disciplines, there is a respectable tradition that when a question cannot be answered, one should try to bypass it, that is, to ask another question. I suggest that we try this strategy here. To see more precisely the nature of the problem we face, let us assume that the relations between leaders of pressure groups, politicians, and bureaucrats, are perfectly obscure for the ordinary citizen, and that nothing that members of the elites can say is to be believed with perfect confidence. The "plurality of governments" dimension will be very important under these assumptions. On the whole, studies in the field of public choice assume implicitly the existence of one unitary jurisdiction, closed to any kind of relations with the outside world. In this context, under our assumptions, there is no independent means for citizens to assess policy outcomes. Thus, whether the political system is democratic or not does not matter a great deal, although in democracies it might be the case that a number of independent institutions mitigate the ignorance of citizens14.

When we allow for a number of jurisdictions, either within a "federal state", or in the form of independent nations, even under our assumption of complete opacity, things ciian^^ substantially. Let us formulae another strong assumption: perfect information oc. dufc ¿mitt ¡A othe^ ^< .v ,r>« t « m** sa&e iev*j ) citizens are provided with a If for a moment ve relax somewhat our strong assumptions of complete opacity and of perfect information, and allow, on the "constitutional" level, a say to citizens on arrangements within the "oligarchy", we can predict that citizens will be more prone to accept the intimacy of relations within the "oligarchy" and their opacity, when they know that they can rely upon perfect information on outcomes abroad. Conversely, if they know they will not be well informed, opacity will produce discontent on their part.

Thus, we can expect small jurisdictions, exposed to maximum information from outside, to engage more openly in "corporatism" than large jurisdictions, into which information from outside flows more slowly. The role played by political responsiveness in this constitutional", or meta-political, process should of course be spelled out in a more rigorous analysis.

If we allow for easy emigration, the possibility of a perverse Hirschman exit mechanism arises *5. If citizens are dissatisfied with the performance of the oligarchy, while political change appears difficult, a number of them may emigrate. In itself, contrarily to widespread opinion, this emigration will not necessarily induce political change or improvement in policy-making. There is plenty of historical and even contemporary evidence in favour of the absence of the usual presumption that governments would respond to a loss of population by changing their policies or would suffer political sanction. Consequently, in the case of easy emigration, disparities in performance can increase without check for long periods of time. This is a real possibility in the case of independent countries as well as in the context of "federalism" within one country. But it is in the latter that it is the most likely to become significant.

S. IVrentr^li7-linn

As we noted before, the absolute size of a jurisdiction may be important. Olson [ 1982] argues that the mobilization of groups will be easier in small jurisdictions for two reasons. The first is that interest categories have fewer members. The second is that there will be more social homogeneity or identity of preferences in each category.

We can doubt the general validity of both arguments, except in the case of very tiny jurisdictions16. But it does not matter too much, for the information factor ve have stressed will tend to produce the same result, that is a higher density of pressure groups in smaller jurisdictions, provided of course that the local or junior governments have real powers or influence. Information considerations intervene in two ways. First, the opacity of a given state of oligarchic arrangements may be lower than it is in larger jurisdictions. Second, information on the outcomes of policies implemented outside the jurisdiction will generally be easier to come by. As a result of the two considerations, the tolerance of corporatism or more uninhibited corporatism will be higher, especially if.

as already argued, the junior government is seen as competing for resources in a wider environment. This increased tolerance will facilitate public financing of pressure groupsl/ .

However, emigration will tend to be less costly. Since there is no reason for political change to be easier in small jurisdictions than in large ones, while information on what happens outside is normally better, exit will tend to become more significant.

The result will be that small jurisdictions, well integrated in a high information--and high individual mobility-area, have more scope for prolonged disparities in performance, or for unchecked increasing disparity, than large ones that are not so well integrated. In itself, this will mean that regional decline, or urban decay within a metroplitan area, will be more frequently observed over long periods of time than the decline of nations as a whole.

This does not imply that "federalism" or decentralization, defined as the assignment of more powers to junior or local governments, would reduce the relative performance of the country as a whole, or would reduce on average the welfare of citizens. Again, let us take extreme cases. In each case, we assume that the country is rather isolated in terms of information, and that political change is difficult (without being impossible). In the first case, let us assume also that almost all powers are assigned to the central government. If emigration is costly, comparative performance may stabilize at a low level, as we saw. If emigration is easy, the disparity in performance can increase unchecked. A case in point may be Spain until fairly recently, a country well-known for its intellectual, Le. informational isolation from the rest of Europe, and for the high propensity of her population to emigrate. In the second case, let us assume a high level of decentralization. For any level of emigration costs (out of the country as a whole), we can expect performance to be somewhat higher than in the case of centralization. Even in an imperfectly democratic environment, a local "oligarchy" will at least sometimes fear political backlashes if its policy output is non-performing compared to neighbouring local governments. This will lead to some emulation. Although we have assumed that the country is informational isolated, this feature is more relevant on the level of citizens than it is on the level of leaders. Thus local governments, in this emulation context, may seek information from outside the country and initiate new policies. In the case of high mobility of individuals within the country, some local governments will not play the game of emulation, or will be losers in this game (we argued above that the presumption that governments generally compete in terms of population is unwarranted). However, the local jurisdictions which tend to have under-average performance will lose population, and will ultimately have decreasing importance in the country as a whole, while the above-average performance jurisdictions will become more important. Thus, altogether, performance will be higher than it would be ( ceteris paribus) in the case of centralization.

The mechanism outlined above could be applied at the sectorial level. The analytical advantage of studying decentralization in a precise sector of the public service is that it does not involve the problem raised by individual mobility to the same degree as it does in the case of overall decentralization. For instance, it is probable that the parts of the French school system which have deteriorated so visibly in recent decades would not have done so to the same extent if their management had been assigned to regional authorities instead of a central "oligarchy" of bureaucrats and union leaders within a ministry iû Paris. Comparisons between regions on this issue would have been fairly easy, while parents would not have generally responded to observed discrepancies by moving from one region to another, but would rather have exerted pressure on local politicians. Information from abroad would probably have been sought by regional governments and some experiments would have been undertaken 18.

Of course, our assumptions are too restrictive to warrant confidence in conclusions of any kind, except perhaps in the extreme cases that we have focussed on. If we take into account the various spill-overs and costs identified by [START_REF] Breton | The economic constitution of federal states[END_REF].

decentralization will have to be limited. Powers will be assigned to a variety of levels.

Concurrent authority will be pervasive. This will make the task of assessing performances more difficult for citizens. In the context of our discussion in terms of opacity, interjurisdictional information, responsiveness of the political system, and migrations, it seems tentatively attractive to consider as the most efficient the solution in which as much power as possible is assigned to junior or local governments while they are asked to coordinate their policies within a central institution that is provided with little independent power. This kind of arrangement will imply strong oligarchies on the local level and weak ones on the central level. More precisely, "peak" associations, national political parties, central bureaucracy will not gather strength, while their local or regional counterparts will try to achieve their goals, inasmuch as they imply policy-making on the national level, through inter-governmental négociation. This pattern seems to be illustrated both by Switzerland19, a very successful country indeed, and by the European Community. In both cases, as we have noted, "junior governments" have remained the focus of the main efforts of pressure groups and political parties. Of course, there are some costs, not considered in this paper, with these kinds of arrangements. Consequently, it might be more appropriate to study them in the limits of sectors of public policy rather than in the context of overall assignments as we did.

However, it should be noted that Swiss cantons are generally competent except when powers are specifically assigned to the federation. Thus real or constitutional federalism ("community states", in Albert Breton s terminology), as opposed to constitutional centralization ("unitary" and "federal" states), may be important in practice in deciding what will be the real extent of decentralization and on which level "oligarchies" will tend to become active.

The possibility of collusion between "local" oligarchies should be considered. In the context of Europe, experience shows that collusion is difficult and time-consuming.

Admittedly, there is some evidence that collusion has been significant in the case of the Monetary System, the fear that the system would allow collusion between governments or central banks and would thus result in a higher average level of inflation has not been confirmed. Admittedly, some projects, especially those supported by unions (paradoxically in the name of the fight against unemployment) seem particularly illustrative of an attempt to collude on a higher level in order to maintain arrangements endangered on the national level. But they have not be accepted for the time being. 9. Testing.

There are few convincing tests in the field of interest groups or pressure groups.

The problem is not principally that data on interest groups are scarce. It is rather that these data are by nature very ambiguous. That we are interested in is the influence of associations and not their numbers or their size, especially since most of them are not pressure groups but mere "clubs", providing their members with private services. In a recent paper, [START_REF] Murrell | An examination of the factors affecting the formation of interest groups in OECD countries[END_REF] thinks that his empirical relations determine "interest group activity" . This is rather dubious, since the explained variable is only the number of associations. But "interest group activity" is not the variable we are interested in anyway. [START_REF]Agrooolitics in the European Community: interest groups and the cnmmon agricultural policy[END_REF] observes that in Brussels, there is "intense activity" of Eurogroups. but that they have "little influence". Most observers seem to agree on this.

Thus tests should probably concern relations deduced from models in which groups play a role, but which do not explicitly include groups among the measured variables, either explained or explanatory. This is for instance the way the theory presented by [START_REF] Olson | The rise and decline of nations-economic growth, stagflation, and social rigidities[END_REF] is tested (see Mueller (1983). Our "model" might suggest a test of this kind.

The relevant variables would be income, income growth, migrations, a proxy for easiness of information on outcomes abroad, a proxy for decentralization (such as the one used by Murrell for instance), a proxy for political responsiveness (perhaps the electoral system).

Support for several arguments formulated above can be found in the existing literature. For instance, reading the empirical relations presented by [START_REF] Schmitter | Interest intermediation and regime governability in contemporary Westen Europe and North America[END_REF],

we can see that what he calls "high corporatism" happens in the small countries of Europe and not in the large ones This supports the idea that avowed corporatism is accepted provided information, allowing comparisons of outcomes, can be obtained at low cost, as is the case in small open economies. Another example is the fact that most of the cases of decline analysed by [START_REF] Olson | The rise and decline of nations-economic growth, stagflation, and social rigidities[END_REF] involve easy emigration. His theory is more convincing when applied to the decline of regions or of city-states than when applied to whole nations. To a large extent, the evidence he uses seems thus to confirm the existence of the perverse exit mechanism we have borrowed from Hirschman in the foregoing analysis.

Of course studies in depth of particular areas of policy-making, or of single associations can be useful, although most economists do not consider them as real tests.

In the case of our "model", attention should be concentrated on the flows of goods and services exchanged between a pressure group and the outside world as well as within the group. Since, in general, existing studies have taken for granted the conventional view of pressure groups, they have not sought information about the flows which appear crucial in our reasoning. Thus it is possible that the relevant information could be found if it was sought. However, it must be noted that our leader is an entrepreneur.

Consequently, real flows do not have to be matched in a single period of time. The entrepreneur and his interlocutors can invest, borrow, repay. Since property rights are not well enforced in the particular exchange relations these actors are engaged in, trust is a necessary, and complicating, factor. Finally, uncertainty is probably higher in this business than in most others. Thus equilibrium involves expected returns and risk remunerations. In other words, many leaders do not succeed but those who do may enjoy high returns. The complexity of the model to be used makes observability, and even more refutability, difficult indeed. As is often the case with others models or theories in this field, the temptation would be therefore to use monographic investigations as a means to illustrate and "verify" the main propositions rather than to submit them to a genuine confrontation with observation.

10. Normative implications.

The arguments that have been formulated here apparently favour decentralization. Inasmuch as coordination is nevertheless indispensable, these arguments seem to be more favourable to "community" states than to "federal" ones.

However, we should remain aware that many other considerations are relevant besides those that were mentioned.

In the case of Europe, the foregoing analysis suggests that current arrangements may be more appropriate than we usually think. There is no "corporatism" on the European level to speak of. Information within Europe flows swiftly. Migrations remain modest. Decision-making in Brussels is mostly an inter-governmental process. Although again, other considerations may be essential in the desire to "build" Europe, the case for more European integration should probably be more careful than it usually is about the kind of problems we have discussed here.

In the perspective adopted in this paper, and perhaps in general, a world government would be disastrous. In the long run "oligarchies" on lower levels would probably manage to collude on the world level. Powers at the centre would gradually increase. Opacity would become dangerous, not only for efficiency, but for individual freedom. The "oligarchies" we have been concerned with are rather benign, because information and emigration make them so. In a closed world society, they would become much more like the ones usually feared. The main normative implication of the paper is probably this strong anti-federalist (in the sense given to federalism by the world federalist movement) conclusion 21. Conversely, as long as societies remain open to external influences, the extent of the decentralization they engage in is not extremely important. In other words, to take the example of France, openness of the French economy and society is more important than its level of decentralization. The latter would acquire real importance only if openness were to be reduced substantially. In the case of smaller countries, in which openness is even more pronounced, and perhaps irreversible, the degree of decentralization may become almost indifferent (who cares whether Monaco is a unitary state or a federation?). This relationship between internal arrangements and the external situation seems to me to have been to a large extent overlooked by the existing economic literature on federalism.

28 11. Cf. [START_REF]Agrooolitics in the European Community: interest groups and the cnmmon agricultural policy[END_REF]. p. 54 and p. 69.

12. Cf. Averyt [1975[START_REF]Agrooolitics in the European Community: interest groups and the cnmmon agricultural policy[END_REF], and Wallace. Wallace, and Webb [1977].

13. [START_REF] Pizzorno | Interests and parties in pluralism[END_REF] sees the problem (pp. 252-254), but does not envisage its local government solution, again an instance of the heuristic harm done by relaxed assumptions (see footnote 4).

14. Cf. [START_REF] Shonfield | defense of the mi«ed economy[END_REF], especially the foreword and chapter 10.

15. Cf. [START_REF] Hirschman | Eiit. voice, and lovaltv: responses to decline in firms, organizations, and states[END_REF]Hirschman [ ] [1981]].

16. Cf. [START_REF] Olson | The rise and decline of nations-economic growth, stagflation, and social rigidities[END_REF]. pp.32-34; on the first point. I argued in the first part of the paper that even medium-sized groups should be considered as "large" in Olson s sense; on the second point let us just note that there is first a difficult problem of definition and measurement of "more homogeneous preferences", and second a real possibility, if we reason in terms of inequality in income or wealth, of cases such that some regions are more unequal than the whole country. 17. On local corporatism, see Hemes and Selvik [1981].

18. Many qualifications should of course be taken into account, for instance regarding the control of quality in the context of national degree awarding, or the supervision of teachers in the context of tenure. For some comments on decentralization in French higher education (where migrations cannot be disregarded), see Salmon [1982].

19. Cf. [START_REF] Frey | The interregional income gap as a problem of Swiss federalism[END_REF] and [START_REF] Lehner | Pressure groups and economic growth: Olson's theory and the Swiss experience[END_REF], 20. See especially table 10.1 in [START_REF] Schmitter | Interest intermediation and regime governability in contemporary Westen Europe and North America[END_REF], It is strange that openness is not seriously discussed in this paper or in the rest of the Berger [ 1981 ] volume.

21. Of course similar long run normative implications are produced by the theory of [START_REF] Olson | The rise and decline of nations-economic growth, stagflation, and social rigidities[END_REF].

  It is a function (AL) o f net outlay, measured on OX'.The total (net) cost of production o f the public good, measured on 0Y. is a function (BN) o f net outlay and the opportunity cost o f the leader (or rather the part o f this cost which is not included in the cost of private goods). The influence o f the leader, measured on OX. is a function (OS) of the quantity o f the public good that is produced. Travelling around the quadrants, we can derive curve CT. which indicates the total cost o f influence, measured on OY. Total revenue from outside, also measured on OY.

  Common Agricultural Policy (seeAveryt 11977],Senior Nello [1983] ) However, collusion should not be exaggerated. The temptation for economists is to reason backwards or indirectly: since the policy seems unreasonable, it must be the result of political or bureaucratic distorsions. However, it is not obvious that the European agricultural policy has been on the whole unreasonable. Consequently, it may be the outcome of normal intergovernmental négociations. Collusion involving pressure groups must then be limited to what is established by direct observation. In the case of the European

means of judging whether the performance of their opaque "oligarchy" is relatively good or bad. or, more importantly, whether it is relatively improving or worsening. This applies to overall performance, but also to the assessment of performance in the particular areas of policy-making given citizens are more particularly concerned with (e.g. pensions or environment).

At this stage of the argument we must consider two factors. The first one is the responsiveness of the political system to possible discontent on the part of citizens. The second factor is the possibility and the cost of emigration. In totalitarian states, the political system will not respond and citizens will not be allowed to emigrate. Thus, assuming perfect information on outcomes abroad, individuals can become very dissatisfied, but they can but resign themselves or hope that a revolution will occur. In the case of democracies, let us assume first that emigration is very costly. The important point becomes then whether political leaders can be easily rewarded or sanctioned according to their performance as seen by citizens. If they can. given perfect information on foreign outcomes data, the nature of the relations within the "oligarchy", and their opacity become largely irrelevant. If the performance of the "oligarchy" is comparatively good, its internal arrangements will be tolerated, or even approved by citizens. If it proves non-performing in relative terms, political change will ensure that the policy output is made more satisfactory or that the internal arrangements are reformed. Admittedly, in many contemporary democracies, political competition is bounded, there are barriers to entry in the political market, real political change seldom occurs (proportional representation is often responsible for this unresponsiveness to widespread discontent, as in Italy or Israel). However, it is sufficient to some extent that the probability of occurrence of political sanction is a function of the level of the disparity in performance. In the long run. in Vestern Europe, the absence of persistent disparities in growth rates, over long periods of time, is more striking than the existence of disparities in levels (that is why neither the German or the Italian "miracle", nor the British "disease" should be taken too seriously). 1. See for instance Greenstone [1976], [START_REF] Moe | The orgami^iinn nf interests[END_REF], and the literature they cite.

2. This is noted by [START_REF] Moe | The orgami^iinn nf interests[END_REF], p. 132.

3 For references to the economic version of "group theory", see chapter 1 of McCormick and Tollison [1981 ]; these authors do not seem to consider the implications of free riding as seriously as I do, and as. to a lesser extent, authors such as Landes and Posner [1975] or [START_REF] Becker | A theory of competition among pressure groups for political influence[END_REF] do; they interpret Stigler [1974] in such a vay that the problem can be bypassed. In the text. I refer to Becker s model and main line of argument, and not to the (modest) qualifications he introduces in his comments.

4. To my knovledge. no political scientist is perfectly unambiguous on this matter, the most straightforward being probably [START_REF] Offe | The attribution of public status to interest groups: observations on the Test German case[END_REF]; see in particular LaPalombara [ 1964]. Salisbury [ 1973], [START_REF] Moe | The orgami^iinn nf interests[END_REF], and the vhole Berger [1981] volume. The last three reveal a serious avareness of Olson's logic of free riding. Hovever, their excessive readiness to relax assumptions prevents them from systematizing their observation along an unorthodox main line.

3 "Homogeneous " in the vhole paper refers to the relative size of members and not to their preferences (except for a short discussion in footnote 16).

6. See Stigler [1974] and [START_REF] Moe | The orgami^iinn nf interests[END_REF]. As noted in footnote 3. Stigler is considered by some economists as having solved the free rider problem, a viev that I do not share, except perhaps as far as firms are concerned.

7. See in particular Wagner [1966]. [START_REF] Breton | An economic theory of social movements[END_REF]. [START_REF] Salisbury | An exchange theory of interest groups[END_REF], [START_REF] Fröhlich | Political leadership and collective goods[END_REF], [START_REF] Breton | The economic theory of representative government[END_REF]. and [START_REF] Moe | The orgami^iinn nf interests[END_REF] ; in the context of international relations, see for instance [START_REF] Kindleberger | Dominance and leadership in the international economy: exploitations, public goods, and free rides[END_REF].

8. Particularly helpful are [START_REF] Salisbury | An exchange theory of interest groups[END_REF] and [START_REF] Moe | The orgami^iinn nf interests[END_REF] ; for a realistic assessment of this leadership literature prior to 1971. see Olson [1971]. 9. A recent discussion in the French parliament of a bill on the financing of private schools may be a case in point: in the justification of this (aborted) project, mass demonstrations in favour of private schools vere invoked by the (leftist) government as a reason for imposing on leftist local authorities part of the burden of financing schools, vhich most of them had obstinately refiised before.

10. In the chapter she contributed to the Mueller [1983] volume.


Jan Čermák
email: cermak@ufe.cz

Alexander Kuna

Ludvík Šojdr

Patrice Salzenstein
email: patrice.salzenstein@lpmo.edu

Short-Term Frequency Stability Measurement of BVA Oscillators

This paper discusses the time-domain measurement of short-term frequency stability of ultra-stable BVA oscillators with flicker frequency modulation noise on the order of 10 -14 in terms of Allan deviation at averaging intervals from hundreds of milliseconds to tens of seconds. The stability has been measured with a highly sensitive phase-time comparator based on the dual-mixer time-difference multiplication with a background instability of ~7x10 -15 /τ. A discrepancy has been observed in the comparator background noise found with two signals from a single oscillator (comparator test) and with two signals from two oscillators (stability measurement).

I.

INTRODUCTION

The state-of-the-art BVA quartz oscillators [START_REF] Aubry | A new generation of very high stability of BVA oscillators[END_REF] exhibit, besides a very good long-term frequency stability thanks to the BVA technique, an excellent short-term stability with a flicker frequency modulation (FFM) on the order of 10 -14 in terms of Allan deviation in averaging intervals from hundreds of milliseconds to a few tens of seconds. To measure this ultra-low noise floor we need a highly sensitive measurement system in the time domain and also very good measurement conditions. In this paper we will discuss the problems associated with this challenging measurement which differs from common ones by three respects:

• The variations to be measured are comparable to variations originating from the measurement system. • The compared signals have about the same stability so that none of them can be considered a reference. • There are no signals to calibrate the measurement system because the best test signals available are the ones to be measured.

The work has been sponsored by the Czech Office for Standards, Metrology and Testing (No. III/13/07 Project).

Our experience presented in this paper is based on the time-domain measurements of 5 MHz Oscilloquartz (OSA) 8600/8607 oscillators [2], [3] carried out at the Institute of Photonics and Electronics (IPE), former Institute of Radio Engineering and Electronics.

We refer to the measurements of four oscillators (two of IPE, one of FEMTO-ST and one of OSA) during a week-long measurement campaign in February 2006 and to a great number of repeated measurements of the two IPE oscillators performed at a later time.

II. MEASUREMENT BASICS

The measured quantity is the variations in the phase-time difference, x ν (t), between two quasi-synchronous sine-wave signals at nearly equal frequency ν. To enhance the measurement sensitivity, the variation x ν (t) is first magnified to x(t) = Mx ν (t), where M is the multiplication factor, using the dual-mixer time-difference multiplication (DMTDM). The DMTDM technique has been known for years [START_REF] Allan | Picosecond time difference measurement system[END_REF] and it is still considered to be the best-suited technique for highly sensitive phase-time measurements [START_REF] Stein | Automated high-accuracy phase measurement system[END_REF], [START_REF] Stein | Frequency and time -their measurement and characterization[END_REF], [START_REF] Sze-Ming | Influence of noise of common oscillator in dual-mixer time-difference measurement system[END_REF], [START_REF] Barillet | Ultra-low noise phase comparator for future frequency standards (Comparateur de phase ultra faible bruit pour les futurs étalons de frequence)[END_REF], [START_REF] Greenhall | Common-source phase error of a dual-mixer stability analyzer[END_REF], [START_REF] Brida | High resolution frequency stability measurement[END_REF], [START_REF] Šojdr | Comparison of high-precision frequency-stability measurement systems[END_REF], [START_REF] Šojdr | Optimization of dual-mixer time-difference multiplier[END_REF]. The method is based on dual mixing the two compared signals at ν with a signal at ν ± ν B from a common oscillator (CO) to provide two beat-note signals at ν B as shown schematically in Fig. 1. The multiplication factor is thus M = ν/ν B .

A phase-time comparator (time-interval counter) then samples the process x(t) by periodically measuring the time interval between two adjacent zero-crossings of the compared signals, which occur at instances t k and t k + x k , respectively. The measurement result is the frequency stability/instability (we will use these terms interchangeably) estimated from the sequence {x k } in terms of Allan deviation σ y (τ) as a function of the averaging time interval τ [START_REF] Allan | The statistics of atomic frequency standards[END_REF], [START_REF] Barnes | Characterization of frequency stability[END_REF]. In most of the stability analysis we have used the Stable32 software package [START_REF]Stable32 version 1.35: frequency stability analysis[END_REF] that provides the uncertainty of σ y (τ) estimates based on the Chi-squared distribution with the equivalent degrees of freedom for a given τ, N and the prevailing noise type [START_REF] Riley | Confidence intervals and bias corrections for the Stable32 variance functions[END_REF]. If not mentioned otherwise, we use the 68.3% probability uncertainty throughout this text. The deviation σ y (τ) is the common overlapping Allan deviation estimated out of N samples as

σ y (τ)= ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - - - = ∑ 2 1 2 i m i m 2 i 2 2 2 2 1 m N i + + x x x τ m N (1)
where τ = mτ 0 and τ 0 is the basic sampling interval.

In modeling the variations involved in the measurement, we think it useful to introduce a concept of inherent variations as measured in near ideal conditions. Thus we presume that the oscillator pair has its inherent stability P σ y (τ), in terms of Allan deviation, and the comparator has its inherent stability C σ y (τ). In real measurement conditions we have to take into account additional variations that cause the instability M σ y (τ). These three contributions, assumed uncorrelated, give the total Allan deviation obtained from the measurement

σ y (τ) = [ P σ y (τ) 2 + C σ y (τ) 2 + M σ y (τ) 2 ] 1/2 (2)
Obviously, the minimum of (2) is

min σ y (τ) = [ P σ y (τ) 2 + C σ y (τ) 2 ] 1/2 (3)
The inherent pair stability P σ y (τ) could be approximated by measuring the oscillators with a comparator that has C σ y (τ) << P σ y (τ) in measurement conditions with M σ y (τ) << P σ y (τ). Similarly, the inherent comparator stability C σ y (τ) would be found by testing the comparator with the signals having P σ y (τ) << C σ y (τ) in conditions with M σ y (τ) << C σ y (τ). Obviously, in a good measurement we should have σ y (τ) ~ P σ y (τ).

Presuming that the inherent stability P σ y (τ) is given by pure power-law noises then P σ y (τ) is well reproducible and the FFM floor of P σ y (τ) can be clearly identifiable. Considering the same for the comparator we can expect that also C σ y (τ) is well reproducible. The difficulty is with the additional variations that originate from sources such as unstable environment and electromagnetic interference that make up the contribution M σ y (τ). These additional variations may occur irregularly thus making the short-term background process non-stationary i.e. depending on the time of its measurement. If periodic, they can distort the stability plot σ y (τ) in any region of τ.

If we measure the same pair repeatedly, an i-th measurement series provides the short-term stability σ y (τ) i which can be thought of as a sample short-term stability of the long-term continuous process {x k } whose stability is σ y (τ) for N→∞.

We can rewrite (2) for the i-th measurement series as

σ y (τ) i = [ P σ y (τ) 2 + C σ y (τ) 2 + M σ y (τ) i 2 ] 1/2 (4)
based on the previous presumption that what may vary from one measurement to another is merely the contribution M σ y (τ) while the inherent stabilities P σ y (τ) and C σ y (τ) remain unchanged.

We consider the process {x k } stationary in the sense of "short-term stability" if σ y (τ) i , i = 1, 2, … is equal within the error bars given by measurement statistics. Thus if we may presume that the process {x k } is stationary, then repeated measurements may not seem necessary. In these highly sensitive measurements, however, this cannot be presumed. Moreover, the classical DMTDM requires that the measured signals be kept quasi synchronous in order to ensure the maximum rejection of CO noise, which with free running oscillators limits the length of measurement period.

III. INSTRUMENTATION

A. Measured Oscillators

As hinted previously, we had at our disposal four 5 MHz OSA oscillators: two 8600-BC5GE with serial numbers 291 and 315 (further denoted as A and B) possessed by IPE, and two 8607-BM with serial numbers 102 and 199 (further denoted as C and D) possessed by OSA and FEMTO-ST, respectively. The A, B oscillators had been in continuous operation for more than a year before the measurement but the C, D oscillators for only 3.5 days. The C, D oscillators have the original casing while the A, B oscillators are housed in extra cases to improve the shielding and are also supplied with an arrangement for fine tuning with a resolution <1x10 -12 . The same tuning arrangement was temporarily supplied to C, D oscillators during the campaign. Each oscillator provides two sine-wave signals on SMA connectors with a power level of +7 dBm.

B. Time Domain 1) Main Comparator:

The main comparator has been developed at IPE in cooperation with LNE-SYRTE [START_REF] Riley | Confidence intervals and bias corrections for the Stable32 variance functions[END_REF]. The system is based on the classical DMTDM operating at 5 MHz. The dual mixing to ν B = 5 Hz provides the basic sampling interval τ 0 = 200 ms and M = 10 6 . The system bandwidth is given by a single-pole low-pass filter with a corner frequency of 15 Hz. The common oscillator (CO) used during the campaign was a 10 MHz HP10811 with a frequency divider providing 5 MHz at +11 dBm power level with L(f) = -152 dBc/Hz of white phase modulation (WPM) noise. The time difference x(t) from DMTDM was measured with a Stanford Research 620 time-interval counter. Further in this text it will be referred to as IPE1 comparator. Later on the IPE1 was reconstructed. The HP10811 CO was replaced with a 5 MHz Milliren MTI260-504A followed by a low-noise amplifier providing a power level of +11 dBm with L(f) = -161 dBc/Hz of WPM. In addition, all BNC connectors within the DMTDM block have been replaced with SMA connectors and the flexible cables with semi-rigid cables. We will refer to this reconstructed version as IPE2 comparator.

The noise performance of IPE1 and IPE2 comparators has been tested with two signals at +3 dBm power-split from the +7 dBm output of a single BVA oscillator (in the IPE1 test from oscillator D and in the IPE2 test from oscillator B). This is a common test based on the presumption that for the test signals originating from a single source we have P σ y (τ) << C σ y (τ) and what we get from the test is σ y (τ) = [ C σ y (τ) 2 + M σ y (τ) 2 ] 1/2 . To ensure sufficient rejection of the CO noise, the test signals were set quasi-synchronous within 2x10 -3 of the beat-note period (the same synchronization interval was maintained during stability measurements). The results of the test are shown in Fig. 2. The increase in σ y (τ) at larger τ in this specific IPE1 test is owing to a periodic variation with a period of about 600 s whose origin is not known. Still there is no significant difference between IPE1 and IPE2 in the FFM region, i.e. in both cases the slope corresponds to flicker phase modulation (FPM) of ~7x10 -15 /τ which one would suppose can be neglected against the instability of the oscillator pair.

2) Supplementary Comparator: For supplementary measurements, we used a TSC 5110A Time Interval Analyzer serial number 129064. The 5110A analyzer provides the basic sampling interval τ 0 = 10 ms though our option generates the phase data only at 1 s intervals so the stabilities at τ < 1 s are the screen shots. The multiplication factor is 5x10 6 at 5 MHz. The performance test has been made under similar conditions as that of the IPE1 and IPE2 comparators and its result is also depicted in Fig. 2 along with IPE1 and IPE2. Since +3 dBm is at the TSC specification limit (+3 dBm to +17 dBm), we also tested it at a power level of +9 dBm by making use of a lownoise amplifier before the power splitter. The equivalent noise

C. Frequency Domain

Complementary measurements in the frequency domain have been performed with a modified Femtosecond Systems FSS1000E Phase Noise Detector [START_REF]FSS1000E Phase noise detector[END_REF] connected to a SR760 FFT Spectrum Analyzer. We have modified the original FSS1000E because of its poorer noise performance (larger FPM probably due to the digital potentiometer at the FSS PLL input). We have by-passed the FSS PLL circuitry with a less noisy PLL of our own design.

D. Laboratory Conditions

The laboratory for stability measurement is housed in the underground vault which ensures a stable environment, especially concerning vibrations and mechanical shocks. The room is shielded, though far from perfect. The temperature is controlled to 23 ± 1 o C. During the reported measurements the 24 V DC voltage for the BVA oscillators was supplied from two Statron 2229 double AC-DC sources with 2 mV rms ripple (one Statron for A&B oscillators, the other for C&D oscillators). It should be noted that there was no other activity in the laboratory in addition to the measurement.

IV. PERFORMED MEASUREMENTS

During the campaign with four oscillators we carried out 29 measurements with the IPE1 comparator and 18 supplementary measurements with 5110A. The four oscillators formed six pairs henceforth designated as A-B, A-C, etc. For the stability analysis we have used the following number of measurements (in parentheses) of respective pairs: A-B (4), A-C (4), A-D (4), B-C (5), B-D (6), C-D (6) made with IPE1, and A-B (2), A-C (2), A-D (2), B-C (2), B-D (2) and C-D (8) made with 5110A. The number of samples in one measurement series was typically 5000. It should be noted that our measurement system did not allow us to measure more than one oscillator pair at the same time.

With the reconstructed IPE2 comparator, we have made around 50 additional measurements of the A-B pair in different periods of time.

V. MEASUREMENT RESULTS

A. Measurement of A, B, C, D oscillators with IPE1

Based on the IPE1 tests, we first assumed that the impact of the measurement system was negligible, i.e. σ y (τ) ≈ P σ y (τ), and that the individual variations were uncorrelated. Under this assumption we decomposed the pair stabilities into individual stabilities with the aid of the four-cornered hat technique using two different approaches.

In the analysis reported henceforth the phase data have first been checked for outliers (actually only a few outliers have been removed). The phase data used for stability calculation are residuals of the quadratic fit. The fit removal, however, has a negligible influence on the short-term results.

1) Least-square method:

To decompose the six pair stabilities into four individual stabilities we have used the common least squares method applied to pair stabilities σ y (τ) i at all τ for all pairs. Denoting the pair stabilities as σ AB (τ) i , σ AC (τ) i , σ AD (τ) i , σ BC (τ) i , σ BD (τ) i and σ CD (τ) i , and the individual stabilities σ A (τ), σ B (τ), σ C (τ) and σ D (τ), we can form a fourcornered hat as depicted in Fig. 3. Based on this hat, we have for each τ an over-determined system of equations which can be solved for the individual stabilities σ A (τ), σ B (τ), σ C (τ) and σ D (τ). We can write the equations in the form The result of the solution is shown in the above Fig. 4. It should be noted that the use of weighted least squares has no impact on the FFM region.

σ 2 AB (τ) 1 = σ 2 A (τ) + σ 2 B (τ) + ε 2 AB1 (5) σ 2 AB (τ) 2 = σ 2 A (τ) + σ 2 B (τ) + ε 2 AB2 … σ 2 AC (τ) 1 = σ 2 A (τ) + σ 2 C (τ) + ε 2 AC1 σ 2 AC (τ) 2 = σ 2 A (τ) + σ 2 C (τ) + ε 2 AC2 … etc.
2) Minimum FFM floor: This engineering approach is based on (2). In this case we are solely interested in the FFM noise which we consider the main goal of the time-domain short-term stability measurement of this type of oscillators. We assume that for a given pair any uncorrelated stochastic variations as well as periodic fluctuations make the amount of σ y (τ) i at FFM region of τ larger than the inherent FFM floor of P σ y (τ). Thus the minimum of all σ y (τ) i , i = 1, 2, … for the specific pair represents the best estimate of the inherent FFM floor. Clearly, we search for what we call a "justifiable" minimum which we define as σ y (τ) i at τ for which we have min [σ y (τ) i + u(τ) i , i = 1, 2, …]. We have searched the minima at 1 s < τ < 10 s. The results based on this approach are shown in Table I.

A B C D

Allan deviation 8.7x10 -14 9.4x10-14 1.1x10-13 6.0x10-14

B. Measurement of A,B oscillators with IPE2

Fig. 5 shows the stability plots of the A-B pair obtained from the measurements with the IPE1 comparator during the campaign (thin plots), from selected measurements made with the reconstructed IPE2 comparator at a later time (bold plots), and from selected measurements performed recently with 5110A (dashed plots). The error bars are omitted for sake of clarity. The FFM floor is about 1.3x10 -13 with IPE1, 0.9x10 -13 with IPE2 and 1.5x10 -13 with 5110A. Thus a significant improvement has been observed after the reconstruction of the IPE comparator. A question arises about which of the contributions in (2) is responsible for this improvement. We can hardly expect that it has been due to improved inherent stability of the oscillators since in both measurements with IPE1 and IPE2, the oscillators had been continuously operating for a long time. Therefore we must attribute it to better performance of the measurement system, i.e. to smaller variance C σ y (τ) 2 + M σ y (τ) 2 . Since we are not aware of any changes that may have lead to significantly smaller M σ y (τ), the responsible contribution is most likely the inherent instability of the comparator, C σ y (τ). If this is true then the contribution to the total FFM of 1.3x10 -13 from the IPE1 comparator was as large as 9x10 -14 . This is evidently in contradiction to what we would expect from the IPE1/IPE2 tests which differ little giving the FPM noise of ~7x10 -15 /τ in the short run as shown in Fig. 2. This FPM noise is negligible against the expected FFM floor of the oscillators. This implies that the comparator performs differently with the signals from a single source (as in the test) and from two different sources (as in the stability measurement). We suspect that the CO noise is rejected differently in the two modes of measurement, based on the fact that the WPM noise in the CO used in the IPE1 comparator was about three times larger than that in the reconstructed IPE2. Actually we also suspect that this effect has manifested itself in a different manner depending on the oscillator pair. At the time of the writing this text we are not able to explain this effect.

A σ A (τ) B σ B (τ) σ AB (τ) i C σ C (τ) σ BC (τ) i σ AD (τ) i D σ D (τ) σ CD (τ) i σ BD (τ) i σ AC (τ) i
As a result of this finding, the individual stabilities based on the measurement with IPE1 bear an unknown error because the pair stabilities that entered into the four-cornered hat were correlated. Provided that the A and B oscillators have approximately equal inherent FFM floor (which can be deduced from the A-D and B-D pairs), the A-B pair floor of 9x10 -14 found with IPE2 gives the individual floor of A and B at a level of 6x10 -14 (neglecting the contribution from IPE2 itself) which is considerably less than what we have obtained with IPE1.

C. Other Observations

1) Non-stationary background: In several measurements performed during the campaign, we have observed nonstationary disturbing processes. Namely, at some τ the values of σ y (τ) r ± u r (τ) and σ y (τ) s ± u s (τ), resulting from the measurement series r and s, respectively, have given the difference |σ y (τ) r -σ y (τ) s | > u r (τ) + u s (τ) even with the uncertainties extended to ±3σ (99% probability).

2) Disturbing periodicity: Fig. 6 shows an uncommon distortion of the stability plot observed in an A-D measurement series. Here the FFM floor is distorted by two periodic processes which manifest themselves in different intensity in the course of the measurement series. This effect has been detected by cutting the data into two sections. A Stable32 routine called Dynamic Stability [START_REF] Galleani | The characterization of clock behavior with the dynamic Allan variance[END_REF], [START_REF] Galleani | Tracking nonstationarities in clock noises using the dynamic Allan variance[END_REF] is a good tool to analyze this kind of phenomena. The minimum of ~1x10 -13 from the first half of data (of which, however, a significant contribution originates from the IPE1 comparator) approximates the FFM floor of the A-D pair better than the minimum from the complete data.

3) Dispersion among measurement series: In the measurements with IPE1, the standard deviation of σ y (τ) i at τ = 1 s is within the range 1.7x10 -15 (pair A-D) to 5.7x10 -15 (pair C-D). The number of measurement series is too small to draw any conclusion from this observation.

4) False phase locks:

When measuring free running highly-stable oscillators which are detuned in the order of 10 -13 and which are interconnected through the measurement system, there is always a risk for the oscillators of being pulled-in. This may end up in a false lock or just in disturbing the free-running phase. The false lock is clearly identifiable from the phase plot but it is difficult to detect a disturbing pull-in process. There are basically two ways to induce the pull-in process in these oscillators: by a reverse signal injection or through the varactor diode. Fortunately these effects are much reduced thanks to OSA's careful design. Close inspection of the phase data of all measurement series has not revealed any false phase locks. The pull-in processes, if there are any, could not be discerned from normal noise variations.

VI. CONCLUSIONS

The measurements of the four oscillators have shown that the background instability of the comparators used, i.e. IPE1 and 5110A, was too large to be neglected. This has been found in later measurements of the two of the four oscillators performed with the reconstructed IPE2 comparator which provided better results than IPE1. Thus the results from fourcornered hat shown in Fig. 4 bear unknown errors. Therefore we plan to repeat the measurement of three oscillators (A, B and D) with the IPE2 comparator in order to determine the individual stabilities more accurately. We can conclude though that the two IPE oscillators (A and B) have the inherent FFM floor at a level of 6x10 -14 and that the FEMTO-ST oscillator (D) even has the inherent FFM floor < 6x10 -14 .

Our current knowledge does not allow us to explain the finding that the DMTDM comparators perform differently in the test (with signals from a single source) and in the frequency stability measurement (with signals from different sources). This effect will be further investigated. For now it is imperative for us to consider this test as "the necessary but not sufficient" condition for a good comparator performance with uncorrelated signals. The key problem of course is that we have no less noisy uncorrelated signals to test the comparator.

It should also be noted that only recently we have found that much of the IPE1 comparator inherent instability was due to mismatches of the frequency divider used after HP10811 CO (see B 1).

Figure 1 .

 1 Figure 1. DMTDM setup.

Figure 2 .

 2 Figure 2. Performance test of IPE1/IPE2 comparators and 5110A. bandwidth (BW) of TSC is about ten times larger than that of IPE. We have verified it by measuring a 5 MHz signal with intentionally large WPM noise alternatively with IPE2 comparator and 5110A. The result was BW TSC /BW IPE = σ y (τ) 2 TSC /σ y (τ) 2 IPE = 10.3 in the WPM region.

Figure 3 .

 3 Figure 3. Four-cornered hat.

Figure 4 .

 4 Figure 4. Individual stability from least squares decomposition.

Figure 5 .

 5 Figure 5. Frequency stability of A-B pair.

Figure 6 .

 6 Figure 6. Illustration of disturbing periodicity.

TABLE I .

 I FFM FLOOR ESTIMATED FROM JUSTIFIABLE MINIMUM.

ACKNOWLEDGMENT

The Czech authors thank Dr. Roland Barillet of LNE-SYRTE for his help in the development of the IPE DMTDM comparator.