Roger Guesnerie

Strategic Substitutabilities versus Strategic Complementarities: Towards a General Theory of Expectational Coordination?

This paper contrasts the views of expectational coordination in a stylised economic model under two polar assumptions: Strategic Complementarities (StCo) dominate or on the contrary are dominated by Strategic Substitutabilities (StSu). Although in the StCo case, 'uniqueness' often 'buys' 'eductive stability', the two issues are strikingly different in the second case. Furthermore if, in the first case, incomplete information often improves 'expectational coordination', it may have the converse effect in the StSu case. It is finally argued that, in macroeconomic contexts, StSu often unambiguously dominate StCo, even in a large class of models with Keynesian features, and even in an aggregate framework that magnifies the StCo effects. The 'remains' of StCo in general cases are discussed.

Résumé : Cet article compare l'analyse de la stabilité des anticipations dans un modèle économique stylisé dans deux cas polaires : Les Complémentarités Stratégiques (CoSt) dominent ou sont dominées par les Substituabilités Stratégiques (SuSt). Alors que, dans le premier cas, l'unicité va souvent de pair avec la "stabilité divinatoire", les deux questions sont largement indépendantes sous SuSt. Qui plus est, l'information incomplète, qui souvent contribue positivement à la coordination sous hypothèse de SuSt, a souvent l'effet inverse sous CoSt. Enfin, l'examen d'une classe de modèles macroéconomiques, même dans un cadre agrégé qui amplifie les CoSt, et même sous des hypothèses néokeynésiennes, suggère que les SuSt y dominent les CoSt. Les "vestiges" des CoSt subsistant cependant dans les modèles désagrégés sont finalement discutés.

1 Introduction 1 For a time, economic theory was tempted to view expectations as fixed data (this is the view taken in the so-called 'temporary equilibrium' literature, where expectations, like preferences, are unexplained) or moving data (adaptive expectations), but in both cases as partly exogenous data. The Rational Expectations Hypothesis, which has become dominant in formal economic theory, views expectations as endogenous: expectations, conforming the predictions of the so-called 'relevant economic theory', tend to be 'self-fulfilling'. Although the Rational Expectations Hypothesis has brought a major and fruitful conceptual breakthrough in the way we think about expectations, only zealots of the Hypothesis would believe that the problem of expectational coordination has disappeared from the agenda of economic theory. Indeed, the title of this conference, 'Coordination failures', where coordination, if I infer from the program, refers mainly to expectational coordination, echoes this remark.

When and why expectational coordination succeeds or fails (success or failure being associated with conformity to, or divergence from, the rational expectations predictions)? In order to think about this problem, a line of reflection, broadly represented in the program of the conference, has attracted a lot of attention and produced a number of related and consistent results that have generated a clear advance in our understanding. This line of research has exploited successfully the analytical regularities generated by a certain type of assumptions concerning economic interactions, the so-called Strategic Complementarities (from now on, StCo) assumptions (a subject pioneered by [START_REF] Topkis | Equilibrium Points in Nonzero-Sum n-Person Submodular Games[END_REF]. Early influential contributions are associated with the names of [START_REF] Cooper | Coordinating Coordination Failures in Keynesian models[END_REF], [START_REF] Milgrom | Rationalizability, Learning and Equilibrium in Games with Strategic Complementarities[END_REF] and [START_REF] Vives | Nash Equilibria with Strategic Complementarities[END_REF]. Carlsson and Van Damme (1993) have introduced new ideas on the role of incomplete information in StCo contexts, that will be scrutinized later. 2 The reflection on expectational coordination is however far from been confined to the Strategic Complementarities literature. As an example, my own main subjects of interest over the last twenty years, endogenous fluctuations and 'eductive stability' of expectations, 3 designate two different lines of attack 1 This article is based on my talk at the Conference Cofail (Coordination Failures) held at Nanterre in june 2004. I thank the discussant Patrick Pintus, and Antoine d'Autume, Jean-Charles Rochet and Olivier Tercieux for their useful remarks on a preliminary version. My understanding of the issues has greatly benefited from conversation with Christophe Chamley within the framework of a 2004 EHESS seminar that we jointly organised. I am also grateful to Steve Morris who found the analysis of noisy strategic susbstitutabilities intriguing. The computations he then undertook with Hyun Song Shin (personal correspondence : "notes on strategic substitutes and complements in global games" (January 2005)) indeed support and shed additional light on the argument of Section 4-2. Finally, I am particularly indebted to Christian Bidard who made many suggestions : most of them have been implemented and have improved the text. I remain responsible of any shortcomings.

2 These ideas have been exploited in economic contexts by (without claim to exhaustivity) [START_REF] Morris | Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks[END_REF], [START_REF] Morris | Contagion[END_REF], [START_REF] Chamley | Coordinating regime switches[END_REF], [START_REF] Frankel | Resolving Indeterminacy in Dynamic Settings: The Role of Shocks[END_REF] and [START_REF] Rochet | Coordination Failures and the Lender of Last Resort: Was Bagehot Right After All?[END_REF].

3 A number of papers I authored or co-authored are reproduced in Assessing Rational Expectations (MIT Press), which has two volumes with subtitles: Sunspot Multiplicity and 2 for understanding expectational coordination. The present paper is particularly concerned with the last line of research, that relative to 'eductive stability' or, to follow my previous terminology on the topic, 'strong rationality' [START_REF] Guesnerie | On the 'Eductive' Justifications of the Rational Expectations Hypothesis[END_REF][START_REF] Guesnerie | An Exploration of the Eductive Justifications of the Rational Expectations Hypothesis[END_REF] this terminology is adopted in my work with co-authors: for example, [START_REF] Desgranges | Do Prices Transmit Rationally Expected Information?[END_REF][START_REF] Evans | Rationalizability, Strong Rationality and Expectational Stability[END_REF][START_REF] Gauthier | Dynamic Equivalence Principle in Linear Rational Expectations Models[END_REF], Guesnerie-Rochet, 1992). 4 These two lines of research, let us call them strategic complementarities and eductive stability, have common references, such as iterated dominance. 5 They superficially exploit similar types of argument, based on 'eductive reasoning', and refer to Common Knowledge (from now on CK) considerations. However, and quite surprisingly, they seem to have few connections. When comparing the two approaches, one might argue that the first evokes a highway whereas the second is rather a mountain trail. The comfort of the highway benefits from the power and generality of well developed analytical tools in situations where they are appropriate. The diversity of situations met in the second line of research requires, like strolling on a mountain trail, hesitating and winding approaches, and stresses solutions of doubtful universality. However, the experience from the mountain trail that will be described later suggests that, if the view of the landscape from the highway is neater and more organized, it concerns a small part of the whole landscape.

The purpose of the present paper is to give some content to the above image by revisiting the connections of the two lines of approach. The ordered landscape that the dominance of strategic complementarities allows us to discover will be first appraised (Section 2): the emphasis will successively be put on the connections that StCo establish between uniqueness and expectational stability and, then, on the now well established role of incomplete information in triggering both properties. This may shed some light on why the first one has developed in a rather autonomous way.

When strategic substitutabilities dominate (Section 3), the previous two messages are basically affected: uniqueness and expectational stability as assessed through the 'eductive stability' criterion evoked above are distinct properties; and incomplete information is no longer a powerful deus ex machina for the analysis of expectational stability (Section 4).

Finally (Section 5), it will be argued by means of examples that in many macroeconomic contexts, strategic substitutabilities tend to dominate strategic complementarites. The 'remains' of strategic complementarities in the explanation of expectational coordination will be discussed.

Economic Fluctuations (2001) and Eductive Stability in Economics (2005).

4 Contributions along similar lines includes [START_REF] Borgers | Iterated Elimination of Dominated Strategies in a Bertrand-Edgeworth Model[END_REF], [START_REF] Dekel | Rationalizable outcomes of large private-value first-price discrete auctions[END_REF], Desgranges (1999), Heinemann F. (1995), Heinemann M. (2002) and [START_REF] Gauthier | Dynamic Equivalence Principle in Linear Rational Expectations Models[END_REF]. 5 The game-theoretical ideas of 'iterative dominance' are already present in [START_REF] Luce | Games and Decisions[END_REF]. Early economic applications are found in [START_REF] Gabay | On the Uniqueness and Stability of Nash Equilibria in Noncooperative Games, Applied Stochastic Control in Econometrics and Management Science[END_REF] and [START_REF] Bernheim | Rationalizable Strategic Behavior[END_REF]. The framework considered here is rather specific, when compared to an abstract game-theoretical framework (see [START_REF] Topkis | Equilibrium Points in Nonzero-Sum n-Person Submodular Games[END_REF][START_REF] Milgrom | Rationalizability, Learning and Equilibrium in Games with Strategic Complementarities[END_REF][START_REF] Vives | Nash Equilibria with Strategic Complementarities[END_REF] in which the most general version of the properties I am describing can be expressed. Let there be a continuum of agents i ∈ [0, 1], each one being concerned only with his own action a(i) and with aggregate data, denoted A, a vector of R p , on which everybody has only an infinitesimal influence. Each individual's best response depends on the subjective probability distribution of the aggregate data, denoted p(i) (with p(i) belonging to the set ∆(R p ) of probability distributions over the space of aggregate data). In the reduced form version of the model (see [START_REF] Evans | Rationalizability, Strong Rationality and Expectational Stability[END_REF], for a general discussion of reduced forms of this type), agent i 's best response only depends on the probability distribution of the aggregate. It is then denoted b(i, p(i)). The aggregate situation B depends itself on the collection of actions taken by individual agents. The outcome can be formally written as B(.)=Φ [Π i [b(i, .)]], 6 where Φ is some aggregation map, so that B is a mapping which associates an aggregate situation B(P ) ∈ R p with a profile of individual distributions (each of them is over R p ) denoted P = Π i p(i). When agents have similar point-expectations on the aggregate state A, the outcome is denoted B(A) where, by abuse of notation, A denotes the collection of point-expectations (Dirac measures) concentrated on A. With these notations, in equilibrium, beliefs on the aggregate state are self-fulifilling, i.e. A * = B(A * ). 7 Here, the strategic complementarity (StCo) assumption means that each individual best response function increases monotonically in the sense that, given b(i, A), we have b(i, A 0 ) > b(i, A), 8 whenever A 0 is either a deterministic point-expectation with A 0 > A or a stochastic aggregate with a support S(A 0 ) such that S(A 0 ) > A. In this basic framework, let us consider the equilibria and the 'rationalizable' outcomes. We will provide below some simple examples illustrating this abstract framework.

The basic StCo framework: equilibria and rationalizable outcomes

A one-dimension basic story Consider the simplest version of the problem, with a one-dimension aggregate here denoted a (instead of A in the general version of the model just sketched). The graph of B(a), which describes the aggregate state of the system as a function of point expectations a, is increasing (Fig. 1). If B(0) is positive and 6 A local approximation of the type B(.)= R B(i, .)di is often used in the analysis [START_REF] Evans | Rationalizability, Strong Rationality and Expectational Stability[END_REF][START_REF] Gauthier | Dynamic Equivalence Principle in Linear Rational Expectations Models[END_REF]. 7 Equilibria in the present framework are 'deterministic', at least at the aggregate level. 8 Symbol > designates the vector inequality:

x > y ⇐⇒ x h ≥ y h ∀h F ig u re 1 B (a ) a * 1 a * 2 a * 3 a Figure 1:
B(+∞) finite, then the number of fixed points {a; B(a) = a}, i.e. of equilibria, is 'normally' odd (three in Figure 1). What was called earlier, in the introduction, the 'eductive' viewpoint, puts emphasis on game-theoretical 'rationalizability' arguments, the logic of which in the present context is explained in [START_REF] Guesnerie | Anchoring Economic Predictions in Common Knowledge[END_REF]. The 'eductive' viewpoint leads here to select the set of rationalizable outcomes. In Figure 1, this set simply consists of the segment [a 1 , a 3 ]. The eductive argument sustaining the conlusion is extremely simple:

-Step 0: From the StCo assumption and the fact that a cannot be negative, the final state of the system, whatever the individual expectations may be, will be such that a > B(0). We assume B(0) > 0.

-Step 1: Everybody knowing that a > B(0) (we start using the Common Knowledge assumption), and given StCo, the final state of the system a will be such that :

a > B • B(0) (B • B or B 2 denotes the second iterate of B).
-Step 2: Everybody taking into account the outcome of step 1 (they know and know that the others know), the situation will be such that a > B•B •B(0).

-The argument goes on, with step n taking into account the conclusion of step n -1, and so on. Then, as illustated by Figure 1, the iterative argument eliminates the whole segment [0, a 1 [. A symmetric argument, starting from the CK fact that B(+∞) is finite, eliminates ]a 3 , +∞[ . Hence, the conclusion.

It is worthwhile noting that, as it is implicit in the 'rationalizability' construct,9 the argument relies on the fact that the model is CK and that Bayesian rationality is also CK. A simple one-dimension example A simple example of a StCo situation, used now and later, has been analyzed systematically by [START_REF] Chamley | Coordinating regime switches[END_REF][START_REF] Chamley | Rational Herds[END_REF]. In this model, agents either join or do not join. The state of the system is characterized by the mass m of joining agents (0 ≤ m ≤ 1); the profitability of joining is supposed to be equal to m. At equilibrium, an agent whose cost c is smaller than m joins. Let F be the cumulative distribution of costs, associated with density f . If the collective belief on the mass of joining agents is m, all agents of cost smaller than m, i.e. F (m) agents, would like to join.

The model displays StCo: the more, the better for everybody. Figure 2, with the cost parameter on the horizontal axis, shows the graph of the cumulative distribution. At equilibrium, the cost c of the highest cost joining agent is such that c = F (c). We now argue that F plays the role of the aggregate best response function of previous theory. Indeed, there are one (Figure 2a) or three equilibria (Figure 2b), such that c = F (c).10 Moreover, the rationalizable set follows from an iterative argument exploiting the same simple intuition as above: The zerocost people join anyway;11 the highest-cost people (c higher than one) never join. This triggers the eductive process: at intermediate levels, agents whose cost is strictly below c 1 join if they believe that all agents below them have joined; symmetrically, agents whose cost strictly exceeds c 3 do not join if they believe that all agents above them do not join.

A flavor of the general argument Under StCo, the above line of argument extends to the case where A is n-dimensional. This is visualized in Figure 3, where A = [a 1 , a 2 ] is a 2dimension aggregate and where the aggregate best response (vector) function

B(a 1 , a 2 ) is increasing in (a 1 , a 2 ).
The iterative dominance argument starts by considering the South-West boundary: (a 1 , a 2 ) > (0, 0) =⇒ B(a 1 , a 2 ) > B(0, 0) = A 1 . Whatever the initial beliefs of the agents may be, the aggregate state will have coordinates above those of A 1 . Next, as in step 1 above, we use the fact that the previous fact is known. We derive the implication:

A > A 1 =⇒ B(A) > B(A 1 ) = B • B(0) = A 2 . As A 1 > 0, StCo implies that B(A 1 ) > B(0), i.e. A 2 > A 1 .
But this new inequality is known by everybody, because it is known that the previous inequality is known. Pursuing the argument, and using further implications of the CK assumption, generates an increasing sequence

A 1 , A 2 , ...A n which, if B(R 2 +
) is bounded from above, converges. Symmetrically, an iterative argument trigrered by the fact that B(R 2 + ) has an upper bound generates a decreasing sequence A 0 1 , A 0 2 , ...A 0 n which converges. Hopefully, the argument and its diagrammatic support have made clear the following key properties of the eductive coordination process leading to the set of rationalizable outcomes (those that survive the consequences of CK, as progressively derived in the above iterative process):

-The limit points of the process are equilibria, i.e. they are states such that A * = B(A * ) (this was obvious in the one-dimension case);

-The set of rationalizable outcomes is a product set (here, a square) which contains all equilibria.

StCo with incomplete information

It has been repeatedly argued in the literature on StCo that incomplete information, i.e. the fact that people do not know exactly the game they are playing, makes the analysis both more convincing and in a sense neater. This idea originates in the work of Carlsson-Van Damme (1993) and has been exploited by a number of authors (see footnote 2). I will introduce the flavor of the argument by relying on Chamley's model. The idea has two distinct components.

Incomplete information helps 'buying' uniqueness of equilibrium

Consider a model in which: (i) each agent observes his true cost; (ii) the costs are taken from a distribution with a known variance but an unknown mean, which is taken form a random distribution. 12Assume that joining is determined by a trigger strategy depending on the signal received: join if c < c, do not join if not. In these conditions, the mass of agents whose cost is smaller than yours is a random variable, whose mean is a function of the observed cost. Such a function is the cumulative of a new distribution which, under a normality hypothesis on the two underlying distributions, has a higher variance than the initial distribution of costs (Chamley, 2004, chapter 11-2). This property is not generally sufficient to conclude to uniqueness. However, if the distribution of costs, conditional on the mean, is symmetric and close to a Dirac measure, everybody thinks that a proportion close to 1/2 has received a signal lower than his own signal, therefore the relevant distribution is almost flat, and a flat enough distribution involves a unique equilibrium.

Incomplete information does not preclude iterative elimination

Take the last version of Chamley's model. It is a dominant strategy to join if your cost is very low. Assume that all agents who have received a (true) cost signal less than c have joined, and that their expected mass is F (c) > c. If, being risk neutral, you receive c or c + e, e being very small, you still join. But if e is not very small, and even if c + e > F (c), you will rationally infer from your signal c + e that the mass of people who have received a signal smaller than c is smaller than F (c) (because your subjective evaluation of the actual mean is below the evaluation of somebody who has received the signal c). Therefore, the profitability of joining, as considered in the mental process of elimination (under the assumption that everybody below c has joined) will not stay as flat as in Figure 1, but will drop more quickly as suggested in Figure 3. The process will still be convergent, although in longer virtual time (see [START_REF] Chamley | Rational Herds[END_REF], Figure 11-7).

Strategic Substitutabilities

A one-dimension view of StSu

Let us come back to the one-dimension version of the model of section 2.2, but assume that B satisfies the property dB/da < 0, as in Figure 5: We have strategic substitutabilities. The Muthian model, considered in [START_REF] Guesnerie | On the 'Eductive' Justifications of the Rational Expectations Hypothesis[END_REF][START_REF] Guesnerie | An Exploration of the Eductive Justifications of the Rational Expectations Hypothesis[END_REF] for assessing expectational coordination in a partial equilibrium context, provides such an example of one-dimension strategic substituabilities. Let the variable a denote the size of the crop. As the price P (a) decreases with a, the size B(a) of the supplied crop, associated with expectations a, decreases with a. There is a unique equilibrium that can be 'educed' through arguments of the following type: If some farmers with high costs do not join, then the crop has a maximal size for which the price is rather low, but not so low as to prevent a few efficient enough farmers to be willing to plant their land (the argument is the same as in step one of the previous argument). But, when only these low-cost farmers participate, the size of the crop is such that the price will be lower than some (high) treshold. As a consequence, some farmers, who a priori would have liked to join initially, drop. Therefore, the price will be higher than assumed in the previous step, so that more low-cost farmers will join, etc. The process is described more formally in [START_REF] Guesnerie | An Exploration of the Eductive Justifications of the Rational Expectations Hypothesis[END_REF] and its steps are illustrated in Figure 4.

Let us reformulate the model in order to make it directly comparable to the above StCo cost model: Each farmer produces only one unit at cost c. I denote B(u) the mass of farmers that produce if they expect that the mass of active farmers is u. Clearly, B(u) = F (P (u)), where F is the cumulative distribution of costs; B(u) is also the mass of farmers who want to produce if all the farmers of cost smaller than u, and only them, do produce. It is left to the reader to rewrite the just sketched 'eductive' argument in the modified setting. The reader will also verify that the analysis (either the equilibrium analysis or the 'eductive' one) is not basically modified in the presence of risk, with risk neutral farmers (u being a signal, define B(u) as B(u) = F (E(P (u))), where E designates the expectation operator.

Note that, in this model: -Equilibrium is unique.

-When the eductive process can be started (this condition may require some hypothesis on CK at the boundary), the set of rationalizable equilibria generally consists of a segment [u 1 , u 2 ] such that u 1 = B(u 2 ), u 2 = B(u 1 ), i.e. its limits are a 2-cycle of B.

-The unique equilibrium is globally eductively stable if B has no cycle. It is locally eductively stable iff |dB/du(u)| = |F 0 (P 0 (u)| < 1, i.e. if the price We also have B(a 1 , a 2 ) < B(A 0 ) if A 0 is any distribution the support of which is 'above' [a 1 , a 2 ], and B(a 1 , a 2 ) > B(A 0 ) if A 0 is any distribution the support of which is 'below' [a 1 , a 2 ]. 13 -Assume that B(0) = A 0 1 and (this is not entirely innocuous) A 1 = B(A 0 1 ) > 0. The iterative dominance argument can be started, at the North-East boundary, whenever it is CK that (a 1 , a 2 ) < A 0 1 . This implies B(a 1 , a 2 ) > B(A 0 1 ) = A 1 , where A 1 is as in Figure 6.

The n-dimension version of the story

-As in step 1 above, the CK belief

A > A 1 implies that B(A) < ΛB(A 1 ) = B • B(A 0 1 ) = A 0 2 . -Suppose (that is indeed an assumption) that A 0 2 < A 0 1 . Pursuing the argu- ment generates both an decreasing sequence A 0 1 , A 0 2 , ..., A 0 n such that A 0 n = B • B(A 0 n-1 ) and an increasing sequence A 1 , A 2 , ..., A n such that A n = B•B(A n-1 ).
The procedure suggests the following key properties of the eductive coordination process leading to the set of rationalizable equilibria: 13 'above', in the sense that the support belongs to [a 1 , a 2 ] + R 2 + , 'below' when the support belongs to

[a 1 , a 2 ] -R 2 + a 1 0 F igure 6 a 2 A '1 A '2 =B (A 1 )= B oB (A '1) A 1 =B (A '1) A 2 =B oB (A 1 )
Figure 5:

-The set of rationalizable outcomes is a product set (at least, under the assumptions introduced in the above argument).

-This set contains all equilibria.14 As in the one-dimension case, the limit points of the processes are cycles of order 2 of B: they are not equilibria, unless they coincide.

Comparing worlds: StCo versus StSu

We are now in a position to assess the somewhat striking differences in the conditions of expectational coordination in the polar contexts of StCo and StSu. We first consider the complete information case, in which our theory is rather general.

Key differences and implications

Main insights from the complete information model

The results obtained in the complete information framework sustain different views on expectational coordination.

Under StCo, uniqueness is not guaranteed but uniqueness implies the success of the elimination process. In my terminology, a unique equilibrium is 'strongly rational' [START_REF] Guesnerie | An Exploration of the Eductive Justifications of the Rational Expectations Hypothesis[END_REF] or 'eductively stable' [START_REF] Evans | Rationalizability, Strong Rationality and Expectational Stability[END_REF]. And stability is global (in case of mutiplicity, local eductive stability may however obtain: it is the case for the extreme two equilibria of Figure 1). In short, uniqueness is the Graal: the unique equilibrium is a focal point which is furthermore singled out by the eductive mental process. Rational expectations are not, as some initial enthusiasts argued, the consequence of the rationality hypothesis. But they follow from the Common Knowledge of both the model and rationality.

Under StSu, uniqueness is still obtained in the one-dimension case, but it does not necessarily hold in the n-dimension version of the model. But uniqueness does not solve the expectational coordination problem neatly: the equilibrium may not be globally eductively stable or 'strongly rational'; the convergence property may not even hold locally (in the sense that no hypothetical CK conjecture that restricts the outcome of the situation to a close neighbourhood of the equilibrium may exist, so that there is no way to trigger, even locally, the mental process towards the equilibrium, see [START_REF] Guesnerie | Anchoring Economic Predictions in Common Knowledge[END_REF]). For example, in the Muth model, local eductive stability requires that supply is less elastic than demand.

If one considers that the case for expectational coordination is serious whenever the equilibrium under consideration is locally eductively stable (which I tend to believe), uniqueness is neither a sufficient condition nor a necessary one for creating favourable conditions to 'expectational coordination'.

In a sense, the views suggested by the analysis of StSu models are embarrassing to the dominant intellectual sensitivity of the profession that adheres to the precepts of philosophical determinism, according to which the ideal of scientificity is a univoque prediction. The strong rationality criterion advocated here is at variance with such a view. The criterion may reject a unique equilibrium and, in the presence of several equilibria, it may pick none, one or several of them!15

Mixed worlds

Let me now stress, although this point is rarely noted, that the opposition just raised between the orderly world of StCo and the anarchist outside world is not as clear cut as suggested above. The reason is that StCo in the aggregate may coexist with some elements of StSu at the micro-level. Let us adopt the notation of footnote 5, i.e. B (.) = R B (i, .), but assume dB/da > 0 (this amounts to saying that R [dB(i, .)/da]di > 0 does not rule out dB(i, a)/da < 0 for a subset I SS ). In this mixed world, uniqueness obtains when the aggregate reaction is flat enough. But, now, even uniqueness does not buy 'eductive stability': Indeed, the argument developed in [START_REF] Evans | Rationalizability, Strong Rationality and Expectational Stability[END_REF] and the posterior discussion in [START_REF] Guesnerie | Anchoring Economic Predictions in Common Knowledge[END_REF], show that local 'eductive stability' requires

R ISS [-dB(i, a)/da]di+ R I\ISS [dB(i, a)}di < 1.
Pessimism in I SS combined with optimism in I NSS may perturb considerably the coordination of beliefs. 16This example leads us to wonder if there is any basis for the Graal!

The case of incomplete information

Up to now, complete information has been assumed. To examine the case of incomplete information, let us return to the above version of the Muth problem, but suppose now that the agents have a known cost drawn from an unknown distribution. Under the same assumptions as above, the relevant distribution of costs for the analysis has a higher variance than the actual distribution of costs. Naturally, as uniqueness still holds, a flatter distribution of costs does not make the equilibrium more unique. Does it make it more eductively stable? Consider the limit case where the mean of the distribution is unknown but the distribution of costs very concentrated (remind this is the favourable case for eductive stability in the presence of StCo). Is it still true that, if they know that some high-cost agents drop, some low-cost agents are induced to join, so that other high-cost agents may drop? It is certainly true that a Bayesian rational agent receiving his cost signal will believe that about half of the agents have received a higher signal. But the hypothesis, similar to the one used in the triggering argument of the StCo iterative process, that all agents with higher cost signals have dropped, has no much grasp on the other agents' actions, in the present case. Assume indeed that high-cost agents (i.e., above some cutoff cost c) drop. Because you believe that the distribution of costs is highly concentrated, you can only infer, if you have, say, average cost (and, a fortiori, if you are a low-cost agent), that only a small number of high-cost people will drop: then only a small number of lower-cost agents are induced to join. This is the first iterate of process depicted by the dotted curve of Figure 6. The process goes on: knowing the first part of the argument, high-costs agents will know that only very-low-cost agents join under the initial conjecture; but they rationally believe that a very small number of low cost agents are induced to join, and this weakens the rationale for the initial drop decision (as suggested by the upper dotted curve). This makes reasonably clear why the equilibrium will not be eductively stable.17 Note that the graphical argument that extends the one used in the StCo case suggests a rather general "destabilizing" role for incomplete information!

The conclusion is: The circumstances in which incomplete information generates the Graal (uniqueness plus eductive stabilty) do not contribute to destroying uniqueness but tend to destroy eductive stability. 185 Is macroeconomics dominated by StCo or StSu?

The question that arises from the previous analysis is the plausibility of StCo in economic analysis. I do not deny that the topics studied in the StCo literature (crisis, agglomeration) are relevant. However, in the field of macroeconomics, where the StCo tools have been seen as potentially useful by some authors, the case is disputable. Let me indeed consider a set of macroeconomic models for which the StCo apparatus seems relevant (they are aggregate models in which the role of StCo is a priori magnified, thus having a Keynesian flavour). I want to argue that, for these models, StSu are likely to dominate StCo, even if the analysis will also point out to 'remains' of StCo.

One-dimension macroeconomics with Keynesian flavour

It has been argued that macroeconomic situations give rise to strategic complementarities. The argument goes back at least to [START_REF] Diamond | Aggregate Demand Management in Search Equilibrium[END_REF]: in his model, trade results from random meetings and trade frequency increases with the number of active participants (this is the source of standard StCo). The incentive to participate increases with the number of active participants, therefore the corresponding diagram is similar to the one of Figure 1 or 2, with one or several equilibria. However, this story leaves in the shadow some key features of exchange such as price competition. 19 In order to stress the limits of the StCo story in a macroeconomic context, let us take a variant of the model I have investigated in Guesnerie (2001a).

Let there be a large number of firms, each one producing one unit of good, so that, in the limit, total production is Q = R I di, the integral being taken on the set of active firms I. But firms are more or less productive, so that firm i needs to hire θ(i) workers (in other words, its cost is c(i) = θ(i)w, where w is the wage). All firms have to decide on production, in the morning of the first day. Income in the afternoon is pQ + M, 20 where M is the money held. If the afternoon price p and the morning wage w are fixed, all firms i ∈ I(p) such that θ(i)w < p would like to produce. However, effective demand is D(p, pQ + M ), notional supply is Q = R I(p) di, and supply is supposed to exceed demand when p is large (a 'Keynesian' situation). In that case there is rationing, assumed to be such that the most efficient firms remain in operation. Hence, there is a relationship between the expected production Q e and the actual production D(p, pQ e + M ). At equilibrium Q * = D(p, pQ * + M ). We are in a somewhat standard StCo case: actual production increases with anticipated production and the slope of the function is p(∂D/∂R), i.e. the marginal propensity to consume π governs the speed of the eductive process. An exogenous shock of 1 on demand would push production by 1/(1π), the standard Keynesian multiplier. There is no doubt that this type of Keynesian macroeconomics enters the class of standard one-dimension StCo models.

However, assume now, as in my paper (Guesnerie, 2001a), that the afternoon price is the market clearing price. The model remains Keynesian, with the morning wage fixed at a level that generates an excess supply of labour. However, the equilibrium price

ψ(Q) satisfies Q = D(ψ(Q), ψ(Q)Q + M ), so that 21 ψ 0 (Q) = (∂D C )/(1 -π(Q))
where ∂D C is a 'compensated elasticity' of price demand and π(Q) is the marginal propensity to consume, calculated at the point (ψ(Q), ψ(Q)Q + M ). The (Keynesian ?) fixed wage equilibrium

Q * satis- fies S(ψ(Q * )) = D(ψ(Q * ), ψ(Q * )Q * + M )
where S is the 'notional' competitive supply.

The point to be stressed is that the limited-price-flexibility hypothesis has led us away from the StCo world. Indeed, the model under consideration displays strategic substitutabilities. Hence, it has a unique equilibrium which turns out to be locally eductively stable whenever (S 0 /D 0 ) * < m * where S 0 /D 0 is, as in the Muth model, the ratio of supply over demand elasticities, and m * is the Keynesian mutiplier (all evaluated at equilibrium). The set of rationalizable equilibria is included in the segment (ψ(Q 0 ), Q 0 ), where Q 0 is the full employment activity level, and only in good cases may coincide with the equilibrium... And there is no clear sense in which incomplete information would make things better. 22Introducing now wage flexibility in the morning brings us back to a Walrasian setting closer to the presently standard macroeconomic models, and further away from the StCo world of fully fixed price models. In the simplest setting considered in section 5 of Guesnerie (2001a), equilibrium is unique and, in a sense, 'often' eductively stable: the reason is that wage flexibility guarantees full employment and this is a powerful anchor to expectation coordination. Still, the set of rationalizable equilibria is not always reduced to the Walrasian equilibrium. It may happen, for example, that low-cost firms (those whose cost is smaller than the Walrasian cut-off cost) do not produce and are replaced by less efficient firms (those whose cost exceeds the Walrasian cut-off cost). These latter firms are optimistic in the sense they expect a price higher than the Wal-rasian price; however, since they are less efficient than the Walrasian firms, the market-clearing wage will be mechanically higher than the Walrasian wage. However, the expectations associated with this situation are rationalizable in the following sense: price expectations of the most efficient firms (the producing firms at the Walrasian equilibrium) coincide with the Walrasian price and, both in the intuitive sense and in the formal sense, this is 'rationalizable', even if it induces some of them to drop at the actual market-clearing wage. The price expectations of the optimistic firms are also rationalizable, for the simple reason that the price they expect does finally prevail! The fact that this price is above the Walrasian price is due to the lower productivity of these firms, which also leads to a lower actual production.

Can we conclude that the StCo story is entireley irrelevant when one leaves the fully rigid price world? In fact, strategic complementarities associated with income effects generated by increased economic activity are still present in the two models I have briefly presented. Although they are dominated by strategic substitutabilities, strategic complementarities alleviate strategic substitutabilities and play a positive role in improving expectational coordination. This is illustrated by the role of the Keynesian multiplier in the local eductive stability of the equilibrium: roughly speaking, as suggested by the above formula, a higher Keynesian multiplier makes eductive stability more plausible. In a more surprising way, the Keynesian multiplier turns out to be an ingredient of some conditions that are sufficient for eductive stability of the Walrasian equilibrium (formula (4-13) in Guesnerie (2001a)): The intuitive explanation is that a high Keynesian multiplier favours the expectational stability of the Walrasian equilibrium!

To sum up, the basic insights of the standard StCo models disappear, but with significant remains of the StCo story. The next subsection attempts to evaluate the importance of these remains.

n-dimension world: the remains of StCo in macroeconomics?

In a loose sense, the previous analysis suggests that strategic complementarities, although they are dominated by StSu (outside the extreme polar world we first considered) still coexist with them and alleviate StSu in a sense favourable to expectational coordination, even in a fully Walrasian world. 23 However, the level of aggregation of the model leaves a doubt on the robustness of the analysis: are the 'remains' of StCo an artefact due to the aggregation of the model? In order to examine this question, I have considered in [START_REF] Guesnerie | On the Robustness of the Analysis of Expectational Coordination : From 3 to n+2 goods[END_REF], an n-dimension ('Keynesian') version of the three-goods model of the previous subsection. There is a market clearing price for each of these goods, but the morning wage is fixed at a 'Keynesian' level. Now, local eductive stability of the equilibrium is obtained whenever the eigenvalue of highest modulus of the matrix {(∂S)(∂Z) -1 (I -A)} is smaller than 1, where ∂S is the Jacobian matrix of supply (as a function of prices), (∂Z) -1 is the inverse of the Jacobian matrix of demand, and A is the matrix of marginal propensities to consume. The reader will note that this formula generalizes the previous one-dimension condition. Is it the case that a 'bigger' matrix A might improve expectational coordination? In that case, there are unambiguous StCo remains in the problem: higher propensities to consume, i.e. a higher (Keynesian-like) income multiplier, would be, in a reasonable sense, good for expectational coordination. Such a positive answer can only be given when Z satisfies the gross substitutability hypothesis. This is the case where the 'remains' of the StCo intuitions, with the positive role of income effects they stress, are significant and identifiable. In the general case, the intuition drawn from the StCo approach is difficult to disentangle from the other aspects of the problem.

Conclusion

This paper has made, hopefully convincingly, three main points:

-It has compared and opposed two analyses of expectational coordination. In the StCo paradigm, a neat route consists in searching for uniqueness. In the opposite polar framework (StSu), global or even local stability involves a variety of tools and approaches. However, StSu is a more relevant framework for economic analysis.

-It has suggested that the interplay of StCo and incomplete information opens only a rather tiny territory of reflection. In particular, incomplete information does not seem to facilitate the analysis of expectational coordination in StSu situations.

-It has finally argued, from the examination of a class of models, that the StCo paradigm is likely to account for a very limited set of phenomena in the field of macroeconomics. However, even when StCo are dominated by StSu, detecting the 'remains' of StCo may help understanding expectational coordination.

  Figure 2 F (c )

  Figure 3

  Figure 4

Figure 6

 6 Figure 6 visualizes the two-dimension counterpart of Figure 2 when StSu replaces StCo. Now B(a 1 , a 2 ) is (coordinate by coordinate) decreasing in [a 1 , a 2 ].We also have B(a 1 , a 2 ) < B(A 0 ) if A 0 is any distribution the support of which is 'above' [a 1 , a 2 ], and B(a1 , a 2 ) > B(A 0 ) if A 0 is any distribution the support of which is 'below' [a 1 , a 2 ].13 -Assume that B(0) = A 0 1 and (this is not entirely innocuous) A 1 = B(A 0 1 ) > 0. The iterative dominance argument can be started, at the North-East boundary, whenever it is CK that (a 1 , a 2 ) < A 0 1 . This implies B(a 1 , a 2 ) > B(A 0 1 ) = A 1 , where A 1 is as in Figure6.-As in step 1 above, the CK beliefA > A 1 implies that B(A) < ΛB(A 1 ) = B • B(A 0 1 ) = A 0 2 . -Suppose (that is indeed an assumption) that A 0 2 < A 0 1 . Pursuing the argument generates both an decreasing sequence A 0 1 , A 0 2 , ..., A 0 n such that A 0 n = B • B(A 0 n-1 ) and an increasing sequence A 1 , A 2 , ..., A n such that A n = B•B(A n-1 ).The procedure suggests the following key properties of the eductive coordination process leading to the set of rationalizable equilibria:

F

  Figure 6:

See Tan and Werlang (1988), who discuss rationalizability à la[START_REF] Bernheim | Rationalizable Strategic Behavior[END_REF] and[START_REF] Pearce | Rationalizable Strategic Behaviour and the Problem of Perfection[END_REF]. Similar analyses apply to dominance solvability à la[START_REF] Luce | Games and Decisions[END_REF], see[START_REF] Moulin | Dominance Solvability and Cournot stability[END_REF].

It is assumed that F (0) > 0, i.e. there is an 'atom' of zero-cost agents.

We assume that some zero-cost people, with a positive global weight though as small as desired, do join.

One could also consider a variant of this model in which the agents do not know exactly their cost c (although the distribution of costs is known) but receive a noisy signal s. At a symmetric equilibrium (which is a priori plausible since every agent uses the same criterion), the cumulative distribution to be taken into account for the equilibrium analysis is the distribution of signals, that depends on the noise and the objective distribution of costs. The signal distribution is obtained as a mean preserving spread of the cost distribution. Is it 'flatter' than the non-noisy distribution of costs? Assume that the noisy signal is symmetric and of zero mean. The left-side cumulative distribution of the signals will be above that of the costs. This phenomenon often involves that, in the middle area, the derivative of the cumulative distribution of signals is smaller than that of the costs. As the equilibria correspond to the intersections of the cumulative distribution curve with the diagonal, the uniqueness of equilibrium is somewhat more likely. However the argument remains loose and, in general, is not sufficient to 'buy' uniqueness.

In the one-dimension model, uniqueness obtains, but not necessarily in this 2-dimension version of the model.

In terms of the metaphor sketched in the introduction, the highway displays an orderly landscape, the mountain trail is synomym of disorder.

This is likely to have consequences for the model of crisis à la Morris-Shin, where the homogeneity of agents, in the expectatioanl terms introduced here, is doubtful.

S. Morris found this argument striking, but has developed a formal calculation with H. Shin, in one of the models they previously studied (personal correspondence). These calculations clearly support the above analysis. Our analysis has not the status of a formal proof but is likely to provide the basic ingredients for it.

 18 Walking on the mountain trail requires a behaviour at odds with driving on the highway

...19 Although the specific story told by Diamond make the price issue actually

irrelevant. 20 I assume rather unrealistically that agents receive not only wage income but also the ex-post realized profit. This simplifies the algebra without changing the conclusions.

This is a simple computation left to the reader who will check that ∂D c = ∂D +Q∂D/∂R.

StSu may even worsen the result.

Note, however, that when going towards a world closer to the RBC-like model, the multiplier we have pointed at would become smaller, as a result of the decrease of the marginal propensity to consume present income, if a life-cycle model of consumption is adopted.


Kristian Orsini
email: kristian.orsini@econ.kuleuven.ac.be

Amedeo Spadaro
email: amedeo.spadaro@uib.es

Vincenzo Atella

André Decoster

Sharing resources within the household: a multi-country microsimulation analysis of the determinants of intrahousehold "strategic weight" differentials and their distributional outcomes

Keywords: C70, D1, J16, H31 intra-household sharing, tax-benefit systems, microsimulation intra-household sharing, tax-bene…t systems, microsimulation JEL Classi…cation : C70, D1, J16, H31

Equal intra-household sharing is still assumed by the vaste majority of applied analyses in welfare economics. Few pieces of work have tried to depart from the equal sharing hypothesis, but their impact has been limited by lack of data or restricted application to special cases. This paper proposes a new framework to derive sharing rules based on individual bargaining power. The latter is de…ned for each household member as the share of resources gained by the household due to his/her presence. The causes of power di¤erentials and their impact on income distribution are analysed in four EU countries presenting signi…cantly di¤erent tax-bene…t systems: Finland, Italy, Germany and the United Kingdom.

Introduction

The traditional approach to household decision making and intra household allocation processes has been challenged in the past 15 years by several contributions attempting to study such processes from both a theoretical and empirical perspective. Yet no agreement seems to emerge over a framework for modelling multi member household decision making and resource allocation. The analyses continue to be polarized between partisans of the unitary approach, who conceive the household as a single utility maximizing agent [START_REF] Samuelson | Social Indi¤erence Curves[END_REF] or a group of individuals headed by an altruistic individual maximizing the collective welfare [START_REF] Becker | A Theory of Marriage: Part II[END_REF], and partisans of a collective approach who prefer treating the household as a set of individuals with diverging as well as converging interests (Manser and [START_REF] Brown | Marriage and Household Decision-Making: A Bargaining Analysis[END_REF][START_REF] Horney | Nash-Bargained Household Decisions: Towards a Generalisation of the Theory of Demand[END_REF][START_REF] Bourguignon | Rationalité individuelle ou rationalité stratégique: le cas de l'o¤re familiale de travail[END_REF][START_REF] Chiappori | Rational Household Labor Supply[END_REF][START_REF] Chiappori | Collective Labor Supply and Welfare[END_REF][START_REF] Bourguignon | Intrahousehold Allocation of Consumption: Some Evidence on French Data[END_REF], Lundberg and Pollack, 1993).

The principal appealing feature of the collective model is that it provides a single framework for the analysis of decision making process and intra-household allocations. The spouses engage in a bargaining process which not only a¤ects their behaviour, but also each spouse's well being. On the contrary, in the unitary approach, the intra-household allocations rely on exogenous (to the household decision process) assumptions concerning equivalence scales and equal sharing. These assumptions rely often on the equal sharing hypothesis that, as shown by several authors (i) has no theoretical foundation, (ii) does not a priori descend from the unitary model itself and (iii) has been rejected by statistical evidence (for a review, see [START_REF] Behrman | Intrahousehold Distribution and the Family[END_REF].

Another appeal of collective approach is particularly evident inasmuch as the coexistence of converging/diverging interests and preferences within the household allows, di¤erently from the unitary model, for an explanation of dynamic aspects of household formation and household dissolution.

Both unitary and collective models provide testable restrictions that guarantee the consistency of data with the underlying theoretical framework for decision-making. In studies adopting a unitary approach symmetry and negative semide…niteness of the Slutsky matrix have been systematically rejected using both labor supply [START_REF] Blundell | Consumer Behaviour: Theory and Empirical Evidence -A survey[END_REF][START_REF] Blundell | Selection Criteria for a Microeconometric Model of Labour Supply[END_REF][START_REF] Blundell | Consumer Behaviour: Theory and Empirical Evidence -A survey[END_REF][START_REF] Blundell | A Life-cycle Consistent Empirical Model of Family Labour Supply Using Cross-Section Data[END_REF] and household consumption data [START_REF] Blundell | Consumer Behaviour: Theory and Empirical Evidence -A survey[END_REF]. The 'income pooling hypothesis'(implied by the unitary approach) has also been empirically rejected [START_REF] Thomas | Intra-household Resource Allocation: an Inferential Approach[END_REF][START_REF] Schultz | Testing the Neoclassical Model of Labor Supply and Fertility[END_REF][START_REF] Fortin | A Test of the Unitary and Collective Models of Household Labour Supply[END_REF].

Collective models, on the other hand, tend to have less stringent Slutsky matrix implications [START_REF] Vermeulen | Collective Household Models: Principles and Main Results[END_REF]. Empirical tests on restrictions have so far not been rejected [START_REF] Browning | E¢ cient Intra-Household Allocations: a General Characterization and Empirical Tests[END_REF], and although this result is far from conclusive, some authors advocate that it is time to shift the burden of the proof back to those favoring unitary approaches [START_REF] Alderman | Unitary Versus Collective Models of the Household: Is It Time to Shift The Burden of Proof ?[END_REF].

Recently several empirical researches have explicitly adopted a collective household approach to analyze labour supply and welfare distribution e¤ects of reforms in the tax-bene…t system. [START_REF] Laisney | Collective Models of Household Labour Supply with Non-Convex Budget Sets and Non-Participation: a Calibration Approach[END_REF] have developed an estimation technique that allows the identi…cation of a collective model with caring preferences and non participation. [START_REF] Laisney | Collective Models of Household Labour Supply with Non-Convex Budget Sets and Non-Participation: a Calibration Approach[END_REF] use this methodology to simulate a tax reform on French data, [START_REF] Carrasco | Does the Representation of Family Decision Process Matter? A Collective Model of Household Labour Supply for the Evaluation of a Personal Tax Reform in Spain[END_REF] analyze the impact of the 1999 Spanish tax reform and [START_REF] Laisney | Collective Models of Household Labour Supply with Non-Convex Budget Sets and Non-Participation: a Calibration Approach[END_REF] simulate the labor supply and welfare impact of introducing the French tax-bene…t system in Germany.

These recent papers represent an important contribution to the di¤usion of alternative frameworks. Particularly in the …eld of welfare evaluation, where the unitary approach and the equal sharing hypotheses remain fundamentally unchallenged. Yet the proposed approach is not totally convincing: one of the crucial hypothesis in the model is that individuals in couples and singles have the same preference parameters, so that identi…cation of household members' individual utility parameters relies on estimations on sub sets of single male and female households and on a calibration procedure [START_REF] Laisney | Collective Models of Household Labour Supply with Non-Convex Budget Sets and Non-Participation: a Calibration Approach[END_REF].

This paper provides an attempt to depart from the intra-household equal sharing hypothesis, using a very intuitive idea of intra-household power di¤erentials, which is based on microsimulation techniques. Indeed the latter are powerful instruments whose analytical potentials in the di¤erent spheres of economic research have not yet been fully explored [START_REF] Bourguignon | Microsimulation as a Tool for Evaluating Redistribution Policies: Theoretical Background and Empirical Applications[END_REF].

The crucial issue is to derive the strategic weight of each household member, and hence its power in the resource sharing game. More importantly, we look at how power di¤erentials depend on the tax-bene…t systems across countries. To this extent we consider four European countries with profoundly di¤erent tax bene…t systems.

Our approach is similar to what has been done in the game theory literature by [START_REF] Shapley | A Value for n-Person Games[END_REF]. His index (the Shapley value) captures the importance of adding (or subtracting) a player in a winning coalition of a game (and hence its strategic weight). In the same way, we are concerned with a de…nition of the strategic importance of each of the individuals in a given household 1 .

The bene…ts of making use of such a measure are multiple: on the one hand it allows for a possible construction of an intra-household resources sharing rule (or at last of its boundary threshold); on the other hand it allows for a comparative analysis of the performance of redistribution systems in equalizing/disequalizing the "bargaining power" of the household members both within and across countries.

Both aspects have indeed crucial implications in terms of equality, gender issues, social justice, inequality measurement and poverty analysis. More interestingly, it may also reveal social planners' preferences about intra-household resources allocation.

The structure of the paper is the following. Section 2 introduces our de…nition of household members'strategic weight. Section 3 describes data selection and EUROMOD, the microsimulation model used to derive strategic weights. Section 4 presents the results: it analyses power di¤erentials, focusing in par-ticular on the role of the tax bene…t systems. Section 5 analyzes the potential e¤ects of a resource allocation based on strategic weight di¤erentials and section 6 concludes.

Determining individual strategic weight

In what follows, let us assume that households simply exist because it is convenient for individuals to aggregate into households, whatever the source of that convenience. Let us for the moment assume that there is no public good and that agents behave in purely egoistic terms. Agents will continue to be part of the household only to the point that this represents a "convenient strategy". In other terms, household members would not accept to "command" a share of resources which is inferior to their marginal contribution to global household welfare. The "power" of each individual within the household is hence determined by a hypothetical counterfactual: it corresponds to the share of resources that would be lost if he or she where to "withdraw" from the household.

In formal terms the power of an individual i may be de…ned as:

i = Y D(n) Y D(n i) Y D(n)
where Y D(n) and Y D(n i) are household disposable income with and without household member i:

Clearly, the individual power depends on two major factors: his/her own original income and the weight attached to him/her by the tax-bene…t system. Since disposable income may be divided into gross income and net transfers, we have that:

i = GY (n) + N T (n) (GY (n i) + N T (n i)) Y D(n)
or simply:

i = i + i where: i = GY (n) GY (n i) Y D(n) i = N T (n) N T (n i) Y D(n)
Here, we are not concerned with the power per se, but rather the power of each household member relative to the other household members. To this extent power indexes must be normalized in order to be interpreted as a sharing rule:

i = i P n k=1 k
The following relation also holds: i = i + i where and have also been normalized with respect to P n k=1 k . It is not the purpose of this paper to provide a plausible sharing rule. What we propose is rather a lower boundary to every sharing rule, in the sense that whatever sharing rule is actually applied between household members, no sharing rule is likely to produce a more unequal outcome than the one resulting from the pure application of sharing based on power2 -no matter the degree of egoism of households members.

The previous decomposition allows us to capture the weight that a tax bene…t system attaches to each individual in the household, given the prevailing roles in a society in terms of age and gender.

Obviously, the proposed approach su¤ers from several shortcomings. In the …rst place we assume that there are no public goods. The share of income devoted to the purchase of public goods is likely to vary across households and to decrease as income increases. Further, research on sharing rules should explicitly recognize the di¤erence between public and private goods and adapt equivalence scales and sharing rules accordingly. Signi…cant insights, in these respects, could come from household expenditure surveys.

Secondly, the treatment of children is not fully satisfactory. The possibility of terminating the household contract is in fact an option available to adult household members, but not to children, especially the younger.

Further research should probably address the issue of how parents bargain over the children's power. It seems a priori likely that the parent who is most likely to obtain the parental responsibility would in some way incorporate children's power. Yet, this opens the issue of how much is actually given to the children. Alternatively, children may be conceived as a sort of public good into which both parents pour resources, before bargaining over how to share residual income.

Data selection and microsimulation software

As explained in the previous section, the sharing rule is based on a counterfactual situation. Therefore, in order to determine the sharing rule we need a set of disposable incomes that correspond to household disposable income once each member has been dropped. For this purpose we use EUROMOD, an integrated microsimulation model for the EU-15 countries, which allows the simulation of tax systems and most of those bene…ts which are not related to past employment records (mostly family bene…ts, housing allowances and income maintenance schemes) 3 . Table 1: Total population and selected sub-sample (weighted)

The present paper focuses on four EU countries, namely Finland, Germany, Italy and the UK. The selection of the countries was mainly inspired by the desire to have a su¢ ciently large variation of tax bene…t systems and social models, intended as gender distributions of market and home production roles. For each country, we have selected a sample of married and cohabiting adult couples (i.e. aged at least 18) with and without children, irrespective of their activity status. The latter are de…ned as single persons living with their parents and aged less than 30. The very broad de…nition is meant not to exclude a signi…cant number of households with grown-up children in Italy. For simplicity we excluded single parents and three-generation households. tab.1 shows the sample size for the three countries before and after selection. The share of individuals included into the selection varies from 71:6% in Italy to 59:9% in Finland. The latter is in fact the country with the highest share of single households.

Table 2: Descriptive statistics (selected weighted sample) Tab. 2 shows some descriptive statistics for the selected samples in the four considered countries. Having selected only heterosexual couples the number of females correspond to the number of males. The average age appears to be very similar across the panel, with females aged around two years less than their male partners. When it comes to the share of males and females in employment, we notice signi…cant variation across the di¤erent "social models". Finland's male employment rate is almost 10% higher than that in Italy and in the UK. However, it is in female employment rate that di¤erences are most striking: in Finland the rate of female employment is almost twice than that of Italy, while Germany and the UK are in an intermediate position. Let us recall that the above data refer to a period from the mid to the late 90ties, and that female employment rates have signi…cantly increased over the past years in all countries but Finland. When it comes to household typologies, we notice that childless households are the dominant household typology in all countries but Italy4 . Indeed, Italy is characterized by a particularly high incidence of households with grown-up children. Finland, Germany and the UK have similar shares of household with one and two children. Finland and Italy, moreover, have a signi…cant share of households with three or more children (above 11%).

Power index: some results

Tab 3 shows the average (normalized) power index for females, males and children (average power per child). The male-female power di¤erential appears to be lowest in Finland and highest in Italy (the normalized power index for females and males is respectively 0:426 and 0:573 in Finland and 0:345 and 0:655 in Italy), which is broadly in line with our expectations, given the di¤erential in employment rates. The results are more surprising for Germany and the UK. Employment rates in the two countries were quite similar for males, whereas the British female employment rate is somewhat higher than the German one and yet the relative power index for German females is always higher than that of the British women. This is especially true in households with one and two children, where the average power of female spouses is :302 and :265 against :252 and :211 respectively in Germany and in Britain. Evidently other features in the system play at least as an important role as employment rate in explaining gender power di¤erentials. This is especially evident when it comes to children. With the signi…cant exception of Italy, in fact, the latter have almost no original income, so their power is essentially derived from the weight assigned to them by the tax-bene…t system. German children enjoy the highest degree of power, whereas Italian children have a power index which is less than half that of German children, in the case of a two-children household (:128 against :059). For a similar household, the power of British and Finnish children lies between such extreme values, with Finland just slightly above Italy. Indeed, British children seem to enjoy a signi…cant degree of power in single-child households, whereas in two and three children household they have signi…cantly less power. This is also the case in Italy, where children in single-child household "command" a share of resources which is almost twice that of children in two or more child households. Germany and Finland show more of a constant pattern.

Average power index di¤erentials, nevertheless, tend not to be very informative, given fundamental heterogeneity of employment statuses and earning capacities in the sampled households. An interesting question concerns the pattern of power di¤erentials with respect to total income. …g.1 and …g.2 show respectively the pattern of power indexes by household disposable income in households without and with children. The pro…le appears ‡atter than expected: male spouses in Germany and UK present slightly N-shaped pattern, matched by a slight U-shaped pattern of female spouses. The latter is probably due to means tested bene…ts in the bottom of the distribution (which takes into account the number of dependants), and to progressively higher female employment rates, as household income increases. Finland, is characterized by a somewhat ‡atter pro…le, possibly linked to the homogeneous distribution of female employment rates across all income deciles, except for the last one. Italy, on the other hand, starts with a particularly low female power index, probably due to the lack of income support scheme and low employment rates. The power increases in the second and third deciles, then it decreases, and then it increases again, converging in the last two income deciles towards the level in the other When children are present in the household, di¤erences across countries become more evident, and at the same time di¢ cult to interpret. Children's power tend to be quite high in the very bottom income deciles, probably due to a combination of means tested child bene…ts and tax allowances which represent a signi…cant share of income when the latter is low. As income increases, however, the role of net transfers (reduced tax liabilities and child allowances) come to play a smaller role, and the power of children decreases.

The analysis of the pattern of power indexes across deciles reveals some interesting features:

1. Italy shows the greatest variation of gender power di¤erential: starting from very low levels of power, Italian females recover some power starting from the 5th decile, determining an inverse trend in male power. However, male power in the bottom of the distribution is extremely high, probably due to the lack of public transfers targeting poor households;

2. Finland has an almost constant pattern of power indexes for both males, females and children. The power of the children appears to be somewhat lower than that of children in the other countries, probably owing to greater equality in household income distribution. The gender power di¤erential, on the other hand, is the lowest across the examined countries;

3. Germany is also characterized by a rather ‡at pro…le in power indexes. The greatest variation is in the power of children which starts very high and decreases constantly, mainly to the advantage of males. The power of females spouses, on the other hand, increases only slightly across deciles;

4. UK shares a similar pattern with Germany: here however, the progressive loss of power by children goes hand in hand with a widening of the power gap between male and female partners, probably owing to a fundamentally individualized tax system. As we can see, such features are mainly driven by di¤erences in the tax ben-e…t system. However, di¤erences in employment rates are also very signi…cant, for both males and females -although di¤erences in the employment rates of the latter are also remarkable. In order to better separate the role of market and state institutions in determining power di¤erentials, it is of interest for households of working age only, to look at power di¤erentials related to female employment status.

Gender power di¤erentials and female employment

As shown in tab.4, although the relative weight of di¤erent household types varies signi…cantly across the countries, spouses and children in similar situations enjoy signi…cantly di¤erent degrees of power. For example, females out of employment in Finland enjoy a signi…cant share of power (:356), although the latter could of course be related to previous activity on the labour market. In Germany, also the power index of inactive women is relatively high (:282), whereas in Italy and in the UK, who share an individualized tax system, the power of inactive women is modest (around :20). As expected, when both spouses are in employment, the pattern of power di¤erentials in childless couples is very similar across countries. Observed di¤erences are probably due to gender di¤erences in working hours as well as in the hourly wage, which may penalize women.

In households with children the relative power of spouses is reduced. Interestingly, however, the presence of children seem to have more of a negative impact on the female than on male spouse. In households where the female spouse is inactive the negative impact is probably due to the relative generosity of the tax bene…t system with respect to dependent spouses and dependent children, whereas it is at least likely that in households where females are active, the female partner reduces to some extent her labour supply when children are born, thus reducing her relative power in the household. The …gures are relatively similar for Germany and the UK (just above :30), whereas in Finland and especially in Italy females enjoy a somewhat higher degree of power (:363 and :407 respectively), which is consistent with data showing lower part-time female employment rates in the above countries.

The previous table is further disaggregated in the appendix: average power indexes are decomposed according to equivalent household income decile. The following paragraph will look into some detail at the role of net transfers in altering the power di¤erentials that arise from the market.

Net public transfers and power

An interesting question at this stage is how much power di¤erentials are affected by the original distribution of incomes, and how much by the tax and bene…t system. Following the framework set out above, normalized power indexes have been decomposed for each group into a market component (original income) and into public transfers component (net transfers) 5 . tab.5 shows such decomposition for the four countries.

Italy stands out for the signi…cant role of net transfers in de…ning the power index in households with no children. This does not come as a surprise: as children tend to stay longer with their families than in the rest of Europe, and family formation tends to be considerably delayed, households with no children are on average older than in the other European countries considered. The net transfer here is positive (on average) for both female and male spouses, although the size of the transfer tend to reinforce power di¤erential of original income. Again, this is not surprising: as old age bene…ts are employment related they tend to reproduce similar power di¤erential patterns based on original income. This seems to be the case also in Finland. This intuition, nevertheless, should be con…rmed by further analysis. In Germany and in the UK, on the other hand, net transfers tend to have a very small average e¤ect.

When it comes to households with children, net transfers tend to be negative. Also in this case, the age structure of the two populations is likely to have an e¤ect, since adults in households with children tend to be active on the labour market.

Germany stands out for the signi…cant role of net transfers in de…ning the power index. In particular it appears that taxes strongly reduce the relative power of males, whereas the reduction of the relative power of females is much lower, owing to lower labour market participation. At the other opposite, we …nd Italy, where public transfers seem to play a marginal role in households with children. The size of relative power index basically corresponds to that based on market incomes. This is also consistent with Italian welfare state system, which is highly biased towards pensions. The United Kingdom and Finland, on the other hand, share similar patterns when it comes to households with children: in both cases transfers reduce the relative power of females and males to increase that of children, but females are much less a¤ected than men. Again this is probably due to the interaction of employment rate and earning di¤erentials with progressive taxation.

As in the previous paragraph, it is possible to analyze the role of transfers and original income across income deciles. Fig. 3 and …g.4 and 5 show the pro…le of power indexes by income decile before and after public transfers for households without and with children respectively.

Each decile power index has been decomposed into a market and net transfer component. The …gures show how the power index is modi…ed by net transfers: the dotted line represents power index as computed on gross income, whereas the solid line represents the power index as computed on disposable income, i.e. gross income plus net public transfers. When looking at …g.3, the pattern is very similar across all countries: public transfers "stabilize" individual power by increasing the power of individuals in the bottom deciles and slightly decreasing the power in the top deciles. The decile point where the switch in the e¤ect takes place is di¤erent across countries: in Italy, for example, net transfers are positive for both men and women up to the ninth decile, this household typology being on average older than the correspondent typologies in the other countries analyzed. In the UK, on the other hand the switch in the e¤ect is in the 5th and 6th decile -probably as a consequence of smaller role of old age bene…ts. In Finland and Germany the switch come between the 6th and the 8th decile. In a gender perspective, net public transfers have an ambiguous e¤ect. As shown in tab.5, the increase in power is positive for both males and females, but except for the UK, the increase for male spouses is greater than that of female spouses, both in absolute and relative terms. When we analyze the decile patterns, we see that net public transfers constantly increase power di¤erentials based on original income in Finland and in Germany. Original labour market di¤erences are hence replicated through employment related bene…ts. In the case of Italy, however, net transfers tend to contrast power di¤erentials based on original income in the very bottom and top deciles. In the UK, on the other hand, net transfers reduce gender power di¤erentials in the …rst two income deciles and increases it in the rest of the distribution.

When it comes to …g.4 and 5, we observe a similar pattern for male and female spouses. Except that now the e¤ect of transfers switches from positive to negative earlier in the distribution. The latter is again due to the di¤erent age structure of households with and without children. Net transfers tend to increase the power of female spouses in all income deciles. Indeed the pattern of female power index is only slightly modi…ed by net transfers, the only exception being Finland, probably due to the high female employment rate.

In the case of children, …nally, the previous intuitions are con…rmed: in Finland, Germany and the UK public transfers signi…cantly a¤ect the power of children in the …rst income deciles. However the importance of family bene…ts shrink in relation to household income as we move across higher income deciles, thus reducing the power of children. Italy has a di¤erent pattern: transfers play indeed a limited role in de…ning their power, which is mostly driven by their own gross income.

Yet, it is not clear how and to what extent each speci…c element of the tax bene…t system is responsible for the observed pattern. This aspect will be clari…ed in the following section.

The role of taxes and bene…ts

In this sub-section we will explore in details how the di¤erent instruments in the considered tax bene…t systems a¤ect within household power di¤erentials. To this extent we have recurred extensively to the microsimulation model. Instruments have been classi…ed into broad groups: (i) taxes and social security contributions, (2) social assistance and housing bene…ts, (3) family bene…ts, (4) old age and sickness bene…ts and (5) unemployment bene…ts. For each group of measures we have simulated what the power di¤erentials within the household would be if the measures did not exist. This allows us to estimate the speci…c contribution of each element of the tax bene…t system. Again the analysis was performed on households with and without children.

Tab.6 and tab.7 present the results of such decomposition for households with and without children respectively. The tables have di¤erently shaded ares: pale gray corresponds to positive and negative variations in the interval [0, 0.05], gray corresponds to the interval (0.05,1] and dark grey intervals correspond to variations in the interval [START_REF] Alderman | Unitary Versus Collective Models of the Household: Is It Time to Shift The Burden of Proof ?[END_REF][START_REF] Alderman | Unitary Versus Collective Models of the Household: Is It Time to Shift The Burden of Proof ?[END_REF]. This allow us to immediately see which instruments play a signi…cant role in reshaping intra-household power di¤erentials.

The tax system does not play such a central role: in Finland it appears to be totally neutral, whereas even in Germany, where a joint tax system is in place, the e¤ects seem to be quite modest. This is probably due to an age e¤ect: as this groups include quite a large number of households with pension income, income taxation and especially social security contributions do not play a major role in determining power di¤erentials.

Housing and social assistance bene…ts also play a smaller role in households without children. As expected, the e¤ect is concentrated in bottom deciles and tend to favor the female spouse (probably due to lower earnings). When it comes to sick, invalidity and old age bene…ts, strong di¤erences across countries Table 5: Impact of original income and transfers on the average power of household members per household typology 7: E¤ect of di¤erent instruments of the tax bene…t system on intrahousehold power di¤erentials (households with children) emerge: Italy appears to be a true "pension state": pension bene…ts signi…cantly increase the power of the male spouse, especially in the very bottom deciles (where other sources of income are less likely to be found). Germany has a similar structure, but Finland and the UK di¤er substantially: here pension transfers are less important. Pension bene…ts continue to favour male spouses, but the their impact is rather marginal and concentrated in the bottom deciles. Finally, unemployment bene…ts do not have a clear cut e¤ect and their overall impact is indeed quite small.

When it comes to households with children the picture gets more complicated. Taxes and social security contributions now do play a larger role: the power of female spouses is slightly increased in Italy in the UK and, especially, in Germany. The power increase is somewhat stronger in bottom deciles, which is probably due to increasing female employment rates when moving towards higher deciles of household disposable income. Children also bene…t from the tax system. This is particularly true in Germany for households in the bottom deciles. In Finland the picture is partially di¤erent: the tax system has a negative e¤ects on female power in upper deciles, while the e¤ect of the tax system is positive in bottom deciles.

The e¤ect of family bene…ts is not surprising: Italy's means tested bene…ts clearly come out, as the impact is concentrated in the bottom deciles. Furthermore, in Italy and Germany the increase in children's power, related to the presence of family bene…ts, mainly reduces the power of males (probably due to the very small share of female spouses' earnings in some households with children).

For old age and sickness bene…ts, as well as for unemployment bene…ts, the conclusion reached above still hold, except that for this age group the relative importance of the two bene…ts is reversed. Apart from Italy, very few households with children receive pension incomes. In Finland and in Germany unemployment bene…ts play a more signi…cant role: they increase the power of both spouses, mainly to the disadvantage of children. [START_REF] Becker | A Theory of Marriage: Part II[END_REF] The "strategic weight" sharing rule: outcomes

In this section we turn to the consequences of intra-household power di¤erentials. In particular, we have computed inequality and poverty indexes. As stated in the introduction we are not claiming that intra-household sharing is realistically based on power di¤erentials only. Solidarity between household members generates some reallocation from the most powerful to the less powerful individuals. It is nevertheless realistic to interpret the sharing rule based on power di¤erentials as an upper boundary: no matter the degree of egoism of households members, it is unlikely that the actual sharing rule agreed between household members will be more unequal than the sharing of resources With this caveat in mind, it is interesting to look at di¤erentials across countries. Tab.8 and tab.9, present FGT indexes for households without and with children. The poverty line has been computed once as 60% of median individual income when income is shared equally across members. In …g.8, male and female head count ratios, income gap and income gap squared all have the same value under the equal sharing hypothesis. Poverty rates di¤er signi…cantly from o¢ cial statistics as incomes have not been equivalized. When shifting to unequal income sharing, poverty rates go up in all countries. In absolute terms the increase is modest in Finland, but very high in Germany and in the UK and especially in Italy. Interestingly, in Finland, the shift to unequal distribution would make both male and female partners worse o¤ -on average. Poverty risk increases for both members. In the UK average poverty risk is almost unchanged (although the severity of poverty actually increases for men as well), whereas in Germany the increase is hardly signi…cant. In Italy, on the other hand, average poverty risk for male adults decreases, whereas that of females increases substantially.

As it is shown in tab.9, poverty risk increases substantially in households with children: poverty rates vary from 3:8% in Germany to 18:4% in the UK. In the UK and in Italy children present a signi…cant poverty risk even under the equal sharing hypothesis, whereas Finland and Germany have particularly low child poverty rates.

When it comes to unequal sharing, poverty risk and intensity of poverty are highest in Italy, which is followed closely by Finland and the UK. Higher poverty rates are linked to the extremely high poverty rates faced by children. Only the German system seems to attach higher weights to children: poverty risk here is 65%, against almost 90% in Finland and 80% in Italy and in the UK. What is also surprising, is the poverty risks faced by female adults in Germany: despite the lower participation rates, females face a lower poverty risk in Germany than in the UK. Yet females adults face from three (Finland) to thirteen times (Italy) the poverty risk of male partners.

While in theory the e¤ect of unequal sharing of resources within the household may have ambiguous e¤ects on poverty rates, inequality indexes always increase when the intra-household distribution of resources is unequal. Tab.10 presents Gini indexes under the two sharing hypothesis for both the whole population and population subgroups. Germany presents the lowest level of income inequality, whereas the UK, Finland and Italy have signi…cantly higher Gini coe¢ cients. It is interesting to analyze inequality amongst children: when incomes are shared unequally, inequality amongst German and British children is very high: this is probably due to the relatively important share of resources that children "command" in worse o¤ households (due to the generous income assistance supplements for children). In Finland inequality amongst children is lower, but in Italy it is extremely high: this is probably linked to the heterogeneity of situations amongst children: some of them indeed continue to live with their parents although economically active.

Conclusions

This paper has introduced a new framework to analyze intra-household power di¤erentials and sharing rules. The concept of power as strategic weight that is developed is intrinsically connected to the interest of each individual in forming a household: if the level of egoism of some household member were to push towards a more unequal sharing rule than that produced by the power di¤erential, the household member could threaten to leave the household. In this sense the power concept may be useful in determining a lower bound to the inequality that may arise from unequal distribution within the household. Yet the framework is not totally satisfying: not all children have the option of leaving the households, so that one of the parents could end up bene…ting from the power of his/her child. Also the proposed framework is completely static: when calculating the power of one of the partners, for example, we did not consider that the other partner could adjust his/her behavior on the labour market. Also, economies of scale in consumption have not been accounted for: although income is shared unequally, it is likely that a part of it will be spent on public goods. It would be reasonable to assume that one part of total household income is consumed for the purchase of non-private goods and services, and that only the residual share is allocated in accordance to power di¤erentials.

With all this limitations in mind, the proposed approach has allowed us to cast some additional light on how resources could potentially be shared within a household, and how sharing arrangements might be in ‡uenced by external/internal conditions. Internal conditions mostly concern individuals'labour supply strategies. These de…nitively play a signi…cant role in determining earning capacity and hence power di¤erentials. However, di¤erences in employment rates, especially female employment rates, are only one of the factors a¤ecting intra-household sharing. Net transfers, positive or negative, also play a significant role in reshaping power di¤erentials. Tax bene…t systems play a positive role in re-balancing power di¤erentials. While some measures are substantially neutral, others tend to reduce existing inequalities and others yet tend to exacerbate the power di¤erentials within the household.

The framework we have developed may therefore represent in itself a straightforward tool to analyze the impact of tax bene…t systems on relative strategic weight, compare their e¤ect across countries and -for example -assess the direction of reforms in tax bene…t systems that may di¤erently a¤ect the power of individuals within the household.

This approach could also be implemented as a possible way to test and validate sharing rules obtained through estimates from consumption data. If econometric estimates produces more unequal output than the power index (in principle, this should never happens), then it would be interesting to understand the reasons of that.

In a more ambitious perspective, the framework we have developed could be used as a starting point for a more realistic sharing rule that account for dynamic strategies (i.e. responses of individuals to the threat of household splitting), adults' control over younger children and economies of scale in the purchase of public goods and services. These aspects are left for further research.

  Finnish data are provided by the Income Distribution Survey, which contains a combination of register data and information gathered through interviews by Statistics Finland. The dataset refers to 1998 and contains detailed socioeconomic information for 25; 010 individuals living in 9; 345 households. German data come from the German Socioeconomic Panel (GSOEP) initiated by the German Institute for Economic Research (DIW) in 1984. Unlike Finland, the data are collected yearly through interviews only. The 1998 dataset contains information on 18; 772 individuals living in 7; 677 households. Italian data are collected each two year in the Survey of Household Income and Wealth (SHIW) by the Bank of Italy. In this paper we use the 1995 dataset which contains information for 23; 924 individuals living in 8; 135 households. Finally, data for the UK comes from the Family Expenditure Survey, and is provided by the Of-…ce for National Statistics. It collects information over 15; 586 individuals and 6; 797 households over the period 1995-1996.

Figure 1 :Figure 2 :

 12 Figure 1: Pattern of power di¤erentials by household disposable income (households without children)

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Power indexes computed on gross market income and disposable income, by income decile (households without children)

Table 3 :

 3 Average power index according to household typology European countries considered.

Table 4 :

 4 Power indexes (and frequencies) by female employment status

Table 10 :

 10 Gini indexes under equal and unequal sharing according to power di¤erentials.

The Shapley value has been also applied to the decomposition of inequality by[START_REF] Shorrocks | Decomposition procedures for distributional analysis: a uni…ed framework based on the Shapley value[END_REF] and[START_REF] Sastre | Shapley inequality decomposition by factor components: some methodological issues[END_REF].

Of course, the previous statement must be interpreted cautiously: young children are not likely to easily withdraw from the household, meaning that one of the parents might incorporate their power, in accordance to which of them is most likely to hold the parental responsibility.

For an detailed description of EUROMOD see[START_REF] Sutherland | Final report EUROMOD : an integrated European tax-bene…t model[END_REF]

It should be noted that childless household may be composed younger couples as well as older couples where children have already left the household.

Replacement incomes in this case have been treated as net transfers, although arguably, they could be considered as deferred wages.

Table 9: FGT indexes for equal and unequal intra-household sharing (households with children)

project, …nanced by the Improving Human Potential programme of the European Commission (SERD-2001-00099). For the countries considered in this paper, EUROMOD relies on the following micro-data: the Income Distribution Survey made available by Statistics Finland; the German Socio-Economic Panel Study made available by DIW; the Survey of Household Income and Wealth, provided by the Bank of Italy and the Family Expenditure Survey provided by the British O¢ ce for National Statistics. We are indebted to our present and former colleagues of the EUROMOD team. Amedeo Spadaro acknowledges …nancial support of Spanish Government-MCYT, (Programa de Acciones Integradas HP2002-0031, Programa Nacional de Promoción General del Conocimiento SEC2002-02606).