Equilibria with Coordination Failures

No. 2004–107

EQUILIBRIA WITH COORDINATION FAILURES By P.J.J. Herings, G. van der Laan, A.J.J. Talman

October 2004

Equilibria with Coordination Failures

P. Jean-Jacques Herings

Gerard van der Laan

Dolf Talman

October 29, 2004

1

P.J.J. Herings, Department of Economics, Maastricht University, P.O. Box 616, 6200 MD

Maastricht, The Netherlands, [email protected]

2

G. van der Laan, Department of Econometrics and Tinbergen Institute, Vrije Universiteit,

De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands, [email protected]

3

A.J.J. Talman, Department of Econometrics & Operations Research and CentER, Tilburg

Abstract

This paper extends the recent literature on equilibria with coordination failures to arbi-trary convex sets of admissible prices. We introduce a new equilibrium concept, called quantity constrained equilibrium (QCE), giving a unified treatment to all cases considered in the literature so far. At a QCE the expected trade opportunities on supply and demand are completely determined by a rationing vector satisfying that the prevailing price system maximizes the value of the rationing vector within the set of admissible prices. When the set of admissible prices is compact, we show the existence of a connected set of non-trivial QCEs. This set connects two trivial no-trade equilibria, one with completely pessimistic expectations concerning supply opportunities and one with completely pessimistic expec-tations concerning demand opportunities. Moreover, the set contains for every commodity a generalized Drèze equilibrium, being a QCE at which for that commodity no bind-ing trade opportunities on both supply and demand are expected, and also a generalized supply-constrained equilibrium at which no binding constraints on demand opportunities are expected and for at least one commodity also not on supply. We apply this main result to several special cases, and also discuss the case of an unbounded set of admissible prices. Key words: exchange economy, rationing, coordination failures, price rigidities.

1

Introduction

This paper is motivated by the recent interest in equilibria with coordination failures. This interest can be traced back to the work of Roberts (1987ab, 1989ab). These papers establish the existence of a continuum of equilibria with quantity rationing of supply at competitive prices in a class of economies characterized by homothetic preferences and constant returns to scale. The important point made in these papers is that rationing is not due to price distortions, but rather to pessimestic expectations concerning trade opportunities. Even at competitive prices, there is a continuum of equilibria with rationing. Rationing is therefore not due to wrong relative prices of commodities but is caused by coordination failures.

Several authors contributed further to the coordination failure literature by gener-alizing the work of Roberts in a number of directions. Herings (1996ab, 1998) extends Roberts’ results to an exchange economy with standard assumptions on the primitives and allows for non-competitive prices. The set of admissible prices is obtained by specifying, for each commodity, a lower bound and an upper bound on its price. Drèze (1997) and Citanna, Crès, Drèze, Herings and Villanacci (2001) treat the combination of fixed and flexible prices and consider the framework of an economy with production. Drèze (2001) focuses on explaining downward real price rigidities and understanding the nature and the sources of the coordination failures. His arguments are based on the combination of un-certainty and incomplete markets. When markets are incomplete, price regulations may be used to generate Pareto improvements as stressed also by Drèze and Gollier (1993) and Herings and Polemarchakis (2004).

In this paper we extend the results on equilibria with coordination failures to a setting where the set of admissible prices is not given by a simple combination of fixed and flexible prices or by a simple specification of a lower and an upper bound on the prices of all commodities. We are interested in sets of admissible prices that may reflect any kind of price indexation or price linkages. Special cases of such sets of admissible prices have been studied before in the fix-price literature by Chetty and Nayak (1978), Kurz (1982), Dehez and Drèze (1984), van der Laan (1984) and Weddepohl (1987). To deal with general cases of price restrictions, we allow for an arbitrary convex set of prices and will define a unifying equilibrium concept, called Quantity Constrained Equilibrium (QCE). Whereas the literature of the previous paragraph takes an ad hoc approach in defining an equilibrium appropriate for the set of admissible prices studied, we pin down a common principle behind all these definitions. This common principle is that pessimistic expectations concerning supply or demand opportunities may only emerge at a certain price system, if that price system maximizes the value of the rationing vector within the set of admissible prices. An immediate implication of this principle is that pessimistic expectations concerning supply or demand opportunities of a commodity are excluded when the price of that commodity

is flexible and that pessimistic expectations concerning supply (demand) possibilities of a commodity may only occur when there is a downwards (upwards) price rigidity.

We show that in case the set of admissible prices is compact and only contains positive price vectors, there exists a continuum of non-trivial Quantity Constrained Equi-libria connecting two trivial equiEqui-libria. One trivial QCE is a fully supply-constrained no-trade equilibrium with completely pessimistic expectations concerning supply opportu-nities for all commodities and prices such that the value of the total initial endowments is minimized within the set of admissible prices. The other trivial QCE is a trivial fully demand-constrained no-trade equilibrium with completely pessimistic expectations con-cerning demand opportunities for all commodities and prices such that the value of the total initial endowments is maximized.

The connected set of non-trivial QCEs contains for every commodity a generalized Drèze equilibrium, being a QCE at which for that commodity no binding constraints on supply and on demand are expected, a generalized supply-constrained equilibrium, at which no binding constraints on demand are expected and for at least one commodity also not on supply, and a generalized demand-constrained equilibrium, at which no binding constraints on supply are expected and for at least one commodity also not on demand.

In Section 5 we consider some applications and extensions. First we consider the case of price indexation and show that our main theorem extends to all kinds of ad hoc results that were obtained in the literature for special cases of price indexation. Second, we consider cases in which the set of admissible prices is unbounded, allowing for models in which some prices are fully flexible, some prices are bounded, and other prices are linked. In case the set of prices is unbounded, the connected set of QCEs is also unbounded with some of the prices going to infinity, while simultaneously for any commodity with price bounded from above, its relative price tends to zero and eventually all consumers will have complete pessimistic expectations concerning demand opportunities of such a commodity, implying no trade for this commodity. In the limit the economy reduces to an economy where trade only takes place in the commodities for which the prices are unbounded from above. When the prices of these commodities are not tied to prices of other commodities, these prices tend to Walrasian equilibrium values for this reduced economy. In case this holds for all commodities, there is a connected set of QCEs leading from a fully supply-constrained no-trade equilibrium to a Walrasian equilibrium. Since all consumers keep their initial endowments at a fully supply-constrained equilibrium and the Walrasian equilibrium is Pareto efficient, this case is most striking as far as the potential detrimental effects of coordination failures are concerned.

The paper has been organized as follows. Section 2 describes the model and dis-cusses equilibria with coordination failures in case the prices are fixed or restricted by a

lower and upper bound. Section 3 introduces the general concept of Quantity Constrained Equilibrium. For the compact, convex case with positive prices the existence result is given in Section 4. In Section 5 we consider the application of general price indexation and the extension to unbounded admissible sets of prices.

2

Equilibria with Coordination Failures

We consider an exchange economy E = ({Xi_,i_{, w}i}mi_{=1, P})_. In this economy there are

m consumers, indexed _i = 1_{, . . . , m}, and _n commodities, indexed _j = 1_{, . . . , n}. For _k a positive integer, we denote _Ik = {1_{, . . . , k}}. Each consumer _i ∈ _Im is characterized by a consumption setXi, a preference preorderingi onXi_,and a vector of initial endowments

wi. The total endowment _w is defined by _w = _i∈Imwi. We assume that the admissible price systems in the economyE are described by the set _P ⊂IRn_+. Several specifications of

P that have been analyzed in the literature will be considered in this section.

The following standard assumptions X, U and W with respect to the economyE are made.

Assumption X

For every consumer _i ∈ _Im_, the consumption set Xi is a closed and convex subset of IRn₊ andXi + IRn₊⊂ Xi.

Assumption U

For every consumer _i ∈ _Im_, the preference preordering i on Xi is complete, continuous, strongly monotonic, and strictly convex.1

Assumption W

For every consumer_i∈_Im, the vector of initial endowments_wibelongs to the interior ofXi. The assumption of strict convexity allows us to work with demand functions instead of demand correspondences and thereby simplifies our notation. All our results carry over to the case of convex preferences. Also the assumption of strongly monotonic preferences is made for the sake of simplicity and can be relaxed considerably.

Suppose that trade takes place at prices_p∈_P.It is not necessarily the case that _p corresponds to a competitive equilibrium price system. As a consequence, there might be

1

A preference preordering

is said to be strongly monotonic if

and

implies

A preference preordering

is said to be strictly convex when for any pair

markets with excess demand and markets with excess supply. In both cases, one needs a distributional rule to determine the final allocation that will result. Such a distributional rule is called a rationing system. For example, in case of excess demand, some households could expect to be rationed on their net demand opportunities, and in case of excess supply some households could expect to be rationed on their net supply opportunities. In this paper we consider the case where the rationing system is uniform. In a uniform rationing system, all households that experience supply (demand) rationing in a certain market, will end up with the same net supply (demand) of the commodity concerned. Our results carry over to a variety of other rationing systems, see Herings (1996a) for a general treatment of rationing systems.

The expectations of available supply opportunities for a household on the various markets are described by a vector ∈ −IRn_+, and the expectations of available demand opportunities by a vector _u ∈ IRn_+. In equilibrium, the expected trade opportunities are required to be rational. The expectations of trade opportunities should therefore match the amounts allocated by the rationing system.

Given a price system_p∈_P and expected trade opportunities for supply and _ufor demand, the constrained budget set of consumer _i∈_Im is given by

Bi(p, , u) ={xi ∈ Xi |p·xi ≤p·wi and k≤_xik−_wik ≤_uk_, ∀_k ∈_In}_.

The corresponding constrained demand _di(_{p, , u}) of consumer _i is defined as the best element for i in _Bi(_{p, , u}). Because of the strict convexity and strong monotonicity assumptions, this element is unique and lies on the budget hyperplane, i.e. _p·_di(_{p, , u}) =

p·wi.

A well-known case discussed in the literature, see for instance Herings (1998) or Citanna, Crès, Drèze, Herings and Villanacci (2001), is when the set_P is equal to{_p}_,with

p∈ IRn++ any price system, possibly a Walrasian equilibrium price system. A constrained equilibrium may in this case be defined as follows.

Definition 2.1 A Constrained Equilibrium for the economy E = ({Xi_,i_{, w}i}mi_=1,{_p}) is a price system _p∗ = _p, expected trade opportunities (∗_{, u}∗) ∈ −IRn₊×IRn₊, and, for every consumer _i∈_Im, a consumption bundle _x∗i ∈ Xi such that

(i) for all _i∈_Im,_x∗i =_di(_p∗_, ∗_{, u}∗); (ii) mi_=1x∗i =_w;

(iii) for all _k∈_In: _xk∗h−_whk =∗_k for some_h ∈_Im implies _x∗ik−_wik _{< u}∗_k,∀_i∈_Im, and, analogously, _x∗_kh−_whk =_u∗_k for some _h∈_Im implies _x∗ik−_wik _> ∗_k, ∀_i∈_Im.

Condition (i) requires that the consumption of each consumer equals his constrained de-mand, while Condition (ii) is the market clearing condition. Condition (iii) says that there can not be (binding) pessimistic expectations simultaneously on both sides of a market, i.e. markets are frictionless.

At a constrained equilibrium, pessimistic expectations are self-confirming. Observe that there exist two trivial equilibria. One is the fully supply-constrained equilibrium, with completely pessimistic expectations concerning the supply of all commodities, so ∗ = 0n and allocation_x∗i =_wi, _i∈_Im_.With completely pessimistic expectations concerning sup-ply opportunities, no consumer will express a demand for commodities because of lack of income. The other trivial equilibrium is the fully demand-constrained equilibrium, with completely pessimistic expectations concerning the demand of all commodities, so_u∗ = 0n and allocation _x∗i = _wi_{, i} ∈ _Im_. With completely pessimistic expectations concerning demand opportunities, no consumer will express any supply for commodities. As a conse-quence, the completely pessimistic expectations concerning lack of trade opportunities are in both cases confirmed and are an extreme case of coordination failure.

It has been shown in Herings (1998) that there exists a connected set of non-trivial constrained equilibria that connects both trivial equilibria. For the case where _p cor-responds to a Walrasian price vector, there is a clear incidence of coordination failures. Although a first-best Pareto efficient allocation is achievable, i.c. the Walrasian equilib-rium allocation, self-confirming negative expectations can lead to an arbitrarily depressed state of the economy. In case the expectations are completely pessimistic, trade will not even take place.

The continuum result also admits very simple proofs of a number of special cases that were treated in the fix-price literature before, like the existence of a Drèze equilibrium with respect to any a priorily chosen commodity (Drèze (1975)) and the existence of a supply-constrained equilibrium (van der Laan (1980, 1982)). A Drèze equilibrium with respect to commodity _k is a constrained equilibrium without rationing in the market for commodity _k, so there are no pessimistic expectations concerning trade opprtunities for commodity _k. For example, commodity _k could be the numeraire commodity. A supply-constrained equilibrium is a supply-constrained equilibrium without rationing on any demand, i.e. expectations concerning demand opportunities are never binding, and without rationing on the market of at least one commodity, so that in at least one market neither expectations concerning supply opportunities nor expectations concerning demand opportunities are binding. An algorithm to compute the connected set of constrained equilibria has been proposed in Herings, Talman and Yang (1996).

In Drèze (1975), the set_P is given by the cube

where_pkand_pk are a priorily given lower and upper bounds for the price_pk of good_k ∈_In satisfying0_{< pk} ≤_pk.For this case, the definition of a constrained equilibrium is as follows (Drèze (1975)).

Definition 2.2 A Constrained Equilibrium for the economy E = ({Xi_,i_{, w}i}mi_{=1, C}n) is a price system _p∗ ∈_Cn, expected trade opportunities (∗_{, u}∗)∈ −IRn₊×IRn₊, and, for every consumer _i∈_Im, a consumption bundle _x∗i ∈ Xi such that

(i) for all _i∈_Im,_x∗i =_di(_p∗_, ∗_{, u}∗); (ii) mi_=1x∗i =_w;

(iii) for all _k∈_In: _xk∗h−_whk =∗_k for some_h ∈_Im implies _x∗ik−_wik _{< u}∗_k,∀_i∈_Im, and, analogously, _x∗_kh−_whk =_u∗_k for some _h∈_Im implies _x∗ik−_wik _> ∗_k, ∀_i∈_Im;

(iv) for all _k ∈_In, if there is _i∈ _Im such that ∗_k =_x∗ik −_wik, then _p∗_k = _pk and if there is_i∈_Im such that _u∗_k =_x∗ik−_wik, then _p∗_k=_pk.

The motivation for Conditions (i)-(iii) is as before. Condition (iv) precludes pessimistic expectations concerning supply opportunities in the market of some commodity as long as its price is not on its lower bound, whereas pessimistic expectations concerning demand opportunities in the market of a commodity do not arise if its price is not on its upper bound. Pessimistic expectations are only allowed when the price mechanism of falling prices in case of excess supply and rising prices in case of excess demands is not allowed to operate.

Observe that there again exist two trivial equilibria. One is the fully supply-constrained equilibrium, with completely pessimistic expectations concerning the supply of all commodities, so∗ = 0n_,price system_p∗ =_pand allocation _x∗i =_wi,_i∈_Im_.The other is the fully demand-constrained equilibrium, with completely pessimistic expectations con-cerning the demand of all commodities, so _u∗ = 0n, price system _p∗ = _p and allocation

x∗i =wi, i∈Im.It follows from the results of Herings (1998) that there exists a connected set of non-trivial constrained equilibria that connects both trivial equilibria. Again, there is a clear incidence of coordination failures, in particular for the case where_Cn includes a Walrasian equilibrium price system.

Constrained equilibria with different expected supply and demand opportunities may well yield the same constrained equilibrium allocation. When expectations concern-ing supply opportunities are not bindconcern-ing, the precise specification of the expected supply opportunity ∗_k is immaterial, as long as ∗_{k < x}∗ik−_wik for all _i ∈ _Im. Analogously, when expectations concerning demand opportunities are not binding, any expected demand op-portunity _u∗_k satisfying _u∗_{k > x}∗ik −_wik for all _i ∈ _Im is compatible with a constrained

equilibrium. The freedom in the specification of non-binding opportunities can be used to simplify notation and proofs by making a particular choice. We focus on expected trade opportunities(_{l, u})that are represented by a single vector_r∈[−_{w, w}]_,called the rationing vector, where [−_{w, w}] ={_x ∈IRn| −_w≤ _x≤_w}. A rationing vector_r ∈[−_{w, w}] induces expected trade opportunities (_{l, u}) given by

l=−w−r≤0n and _u=_w−_r≥0n_.

Taking use of the single vector_rallows us to redefine the concept of a constrained equilib-rium on_Cn as follows.

Definition 2.3 A Constrained Equilibrium for the economy E = ({Xi_,i_{, w}i}mi_{=1, C}n) is a price system _p∗ ∈ _Cn, a rationing vector _r∗ ∈ [−_{w, w}]_, and, for every consumer _i∈ _Im, a consumption bundle _x∗i ∈ Xi such that

(i) for all _i∈_Im,_x∗i =_di(_p∗_,−_w−_r∗_{, w}−_r∗); (ii) mi_=1x∗i =_w;

(iii) for all _k∈_In, if there is _i∈_Im such that −_wk−_rk∗ =_x∗ik−_wik, then _p∗_k =_pk and if there is _i∈_Im such that _wk−_r∗_k =_x∗ik−_wik, then _p∗_k =_pk.

Since 0n ≤ _x∗i ≤ _w and 0n _wi _w and therefore −_{w x}∗i −_wi _w for any _i, a negative value of _r∗_k implies that only expected trade opportunities concerning the supply of commodity _k can be binding, in which case _pk is on its lower bound, while the opposite holds in case of a positive value of _r∗_k. When_r∗_k = 0_, then expected trade opportunities on neither the demand side nor the supply side of the market for commodity_kcan be binding. Therefore Condition (iii) replaces both Condition (iii) and Condition (iv) of Definition 2.2. The fix-price literature has analyzed many sets of admissible prices_P that cannot be described by a lower bound and an upper bound on the price of each commodity. Important examples concern cases where some of the prices are linked to prices of other commodities, an obvious example being price indexation. For example, consider the case

P ={p∈IRn+ | pj ≤pj ≤pj, j ∈I, πj(pI)≤pj ≤πj(pI), j ∈J} (1) where_I ⊂_In is a set of index commodities with prices bounded from below and above, _pI are the prices of the index commodities and _J =_In\_I is the set of indexed commodities with prices bounded from below and above by the index functions _πj(·) and_πj(·), _j ∈ _J. Because _P is not a cube or a singleton, and thus there are no independently given lower and upper bounds for the prices, we have to modify the complementarity Condition (iii) of Definition 2.3. Under certain additional assumptions on_P, this issue was raised already by

Chetty and Nayak (1978), Kurz (1982), Dehez and Drèze (1984), van der Laan (1984) and Weddepohl (1987). For example, Dehez and Dréze assume that for fixed values _pj it holds that _pj = _pj = _p, _j ∈ _I, implying that _πj(_pI) = _πj(_pI) and _πj(_pI) =_πj(_pI) are constant for all _p∈ _P, so that_P reduces to _P ={_p ∈IRn₊ |_pj = _pj_{, j} ∈ _{I, pj} ≤_pj ≤ _pj)_{, j} ∈ _J}, with _pj = _πj(_pI) and _pj = _πj(_pI), _j ∈ _J. In van der Laan the set _P in (1) is considered under the assumption that_pj = 0 and_pj =∞ for _j ∈ _I and that_πj(_pI) =_pjj∈Ipj and

πj(pI) = p_{j j}∈Ipj, j ∈ J. Then after normalising j∈Ipj = 1, the set P defined in (1) reduces to

P ={p∈IRn+|

j∈Ipj

= 1, p_j ≤pj ≤p_j, j ∈J}, (2) For this _P van der Laan (1984) shows that there exists a supply-constrained equilibrium such that at least one of the two following conditions hold: (i) there are no binding expected trade opportunities for all commodities in _I or (ii) there is no binding expected trade opportunities for at least one of the commodities in_J. When the latter condition holds in equilibrium, there may be binding expected trade oppotunities for the index commodities in _I, illustrating the problem that Condition (iii) of Definition 2.3 does not need to be satisfied. More precisely, if at least one of the commodities in _I is constrained, then typically there are binding expected supply opportunities concerning all commodities in_I, although due to the fact that _j∈Ipj = 1, at least one of the prices _pj, _j ∈ _I, is positive and is not downwards rigid. This calls for a more general equilbrium concept for general sets of admissible prices.

3

The General Case

An equilibrium concept for general price restrictions that allows for coordination failures should contain the concept of Definition 2.2 as a special case and should also deal with a set of prices as in (2) in a satisfactory way. To do so, the expected trade opportunities

(, u)will be again induced by a rationing vector_r ∈[−_{w, w}]and given by=−_w−_r ≤0n for supply and_u=_w−_r≥0n for demand.

It is clear that Conditions (i) and (ii) of Definition 2.2 should be imposed. Every consumer maximizes utility given his budget and expected trade opportunities, and total constrained demand equals total initial endowment on each market. The description of expected trade opportunities by a rationing vector _r ∈ [−_{w, w}] will take automatically care of these two conditions as in Definition 2.3 and also of Condition (iii) of Definition 2.2 that in equilibrium expected trade opportunities concerning the supply and the demand of a particular commodity can never be binding simultaneously. When in equilibrium

commodity _k, and when in equilibrium _rk _< 0 there can be only binding expected trade opportunities concerning the supply of commodity_k. Moreover, when in equilibrium_rk= 0 there can be no binding expected trade opportunities at all concerning commodity_k.

It is more difficult to generalize Condition (iv) of Definition 2.2. Consistent with Walrasian tatonnement, when prices can not be adjusted in such a way that the uncon-strained excess demands and supplies are expected to decrease (in absolute value) according to the Law of Supply and Demand, pessimistic expected trade opportunities take over and will constrain individual excess demand or supply accordingly. Formally, we require that at a constrained equilibrium the rationing vector _r∈[−_{w, w}] is such that it points outwards to _P at the prevailing price vector _p, i.e. _p·_r = max{_pˆ·_r|_pˆ∈ _P}. In the sequel a pair

(p, r) satisfying this property will be called stable.

In our equilibrium concept for the general case of a set _P of admissible prices to be defined below we will require that the equilibrium pair (_p∗_{, r}∗) is stable. This implies that rationing is not binding when the equilibrium price vector_p∗ lies in the interior of the set_P, because then the pair (_p∗_{, r}∗) can only be stable when _r∗ = 0n. Since _p∗ lies in the interior of _P, prices are fully flexible and can be adjusted according to the Law of Supply and Demand and therefore excess demand or excess supply need not be constrained by rationing. If _p∗ does not lie in the interior of the set _P, which for example is always the case when_P is not full-dimensional, then the stableness property implies that the rationing vector _r∗ points outwards to_P at_p∗, being a direction into which the price vector cannot be further adjusted according to the Law of Supply and Demand. The collection of all these directions is called the normal cone at _p∗ to_P. For _p∈ _P, the normal cone _G(_p) to

P at_p is given by

G(p) = {y∈IRn|pˆ·y≤p·y for any _pˆ∈_P}_.

So, the normal cone _G(_p) of _P at _p determines precisely the rationing vectors that may occur at _p and thus the stability concept is therefore equivalent to the requirement that

r ∈ G(p). This gives us the following definition of the general concept of a Quantity Constrained Equilibrium.

Definition 3.1 (Quantity Constrained Equilibrium)

A Quantity Constrained Equilibrium (QCE) for the economyE = ({Xi_,i, _wi}mi_{=1, P})is a price system _p∗ ∈ _P, a rationing vector _r∗ ∈ [−_{w, w}], and, for every consumer _i ∈ _Im, a consumption bundle _x∗i ∈ Xi satisfying

(i) for all _i∈_Im,_x∗i =_di(_p∗_,−_w−_r∗_{, w}−_r∗); (ii) mi_=1x∗i =_w;

Condition (iii) links the rationing vector to the price restrictions. Expected trade oppor-tunities are completely determined by a vector _r∗ in the normal cone _G(_p∗) to the set of admissible prices_P at the equilibrium price system_p∗. In case that_P equals_Cn_,consider a pair (_p∗_{, r}∗)with_rk∗ positive. Then such a pair can only be stable when_p∗_k =_pk. Similarly, when _r∗_k is negative, the pair can only be stable when_p∗_k =_pk. Therefore, in this case the requirement that (_p∗_{, r}∗) is stable yields Condition (iii) of Definition 2.3. For the set _P given in (2), it holds for any positive price vector _p and any _r ∈ _G(_p) that _rk = _rh for every two indices _k and _h in the index set _I. Hence, if expected trade opportunities con-cerning these commodities are binding, then according to Condition (iii) of Definition 3.1, the expected trade opportunities are the same for all commodities in_I.

In general, since 0n ≤ _x∗i ≤ _w and 0n _wi _w, in a QCE it holds that −_wk _<

x∗ik −wik < wk for all _k ∈ _In and _i ∈ _Im. Hence, in a QCE, a consumer _i ∈ _Im can only face binding expected demand opportunities for good _k if _rk∗ =_wk−(_x∗ik−_wik) _>0_. Analogously, some consumer _i ∈ _Im can only face binding expected supply opportunities for good_k if_rk∗ =−_wk−(_x∗ik−_wik)_<0_. Notice that the more the expectations concerning supply (demand) opportunities of commodity _k are pessimistic, the closer its rationing level _rk∗ is to −_wk (to _wk). In particular, expectations concerning supply opportunities of commodity_k are completely pessimistic if _r∗_k =−_wk and expectations concerning demand opportunities of commodity _k are completely pessimistic if _rk∗ = _wk. When _p∗ is in the interior of_P, then_r∗ = 0n, in which case the expected demand opportunities equal_w and the expected supply opportunities equal −_w, implying that in equilibrium the expected trade opportunities are not binding and thus _p∗ is a Walrasian equilibrium price vector. Summarizing, in a QCE consumers maximize their utility in their constrained budget sets, total demand equals total supply, there are no binding expected trade opportunities on both supply and demand simultaneously, and the rationing vector points into a direction such that the value of the rationing can not be increased by moving the price vector within

P.

From an economic point of view, it is of crucial importance that the equilibrium concept is independent of the units of measurement that are used in the definition of a commodity. Suppose that the unit of measurement used in the definition of commodity_kis multiplied by a positive constant_αk_{, k}= 1_{, . . . , n}. Let_α= (_{α1, . . . , α}n). An economyE(_α) with initial endowments _wi(_α) given by _wik(_α) = _wk_/αk_{, k} = 1_{, . . . , n,} set of admissible prices

P(α) ={p∈IRn |pk =αkpk, k= 1, . . . , n, for some_p∈_P}

and appropriately redefined consumption setsXi(_α)and preference relationsi (_α)should have an equivalent set of QCEs as the economyE_.More precisely, for each QCE(_p∗_{, r}∗_{, x}∗) of E there should be a QCE (_p∗(_α)_{, r}∗(_α)_{, x}∗(_α)) of E(_α) and vice versa, where _p∗(_α) is

obtained from_p∗ by componentwise multiplication by_α,and_r∗(_α)and_x∗(_α)are obtained from_r∗and_x∗by componentwise division by_α.The following result claims that equivalence indeed holds.

Theorem 3.2

For any choice of _α 0, it holds that (_p∗_{, r}∗_{, x}∗) is a QCE of E = ({Xi_,i_{, w}i}mi_{=1, P}) if and only if (_p∗(_α)_{, r}∗(_α)_{, x}∗(_α)) is a QCE of E(_α) = ({Xi(_α)_,i (_α)_{, w}i(_α)}mi_{=1, P}(_α)).

Proof.

It is obvious that (_p∗_{, r}∗_{, x}∗) satisfies Conditions (i) and (ii) of Definition 3.1 of a QCE of E if and only if (_p∗(_α)_{, r}∗(_α)_{, x}∗(_α)) satisfies Conditions (i) and (ii) of Definition 3.1 of a QCE of E(_α). It is easily verified that_r∗ ∈_G(_p∗) if and only if _r∗(_α)∈ _Gα(_p∗(_α)), where

Gα(p∗(α)) denotes the normal cone to _P(_α)at _p∗(_α). This shows that(_p∗_{, r}∗_{, x}∗) satisfies Condition (iii) of the definition of a QCE of E if and only if (_p∗(_α)_{, r}∗(_α)_{, x}∗(_α)) satisfies Condition (iii) of the definition of a QCE of E(_α)_. Q.E.D.

4

Existence results

In this section we consider the case that_P is a compact, convex set containing only positive prices. In the next section we also allow for an unbounded set of admissible prices.

Assumption P

The set _P of admissible prices is a non-empty, convex and compact subset in the interior of IRn₊.

We first show that there are two different trivial no-trade QCEs, one with completely pessimistic expectations concerning supply opportunities, implied by _r∗ = −_w, and one with completely pessimistic expectations concerning demand opportunities, implied by

r∗ =−w. Let_P0 and_P1 be given by

P0 ={p∈P|w·p≤w·pfor all _p∈_P} and

P1 ={p∈P|w·p≥w·pfor all _p∈_P}_.

Under Assumption P, both _P0 and _P1 are non-empty, convex and compact and their intersection is either empty or equal to _P. Prices in _P0 are such that the value of total initial endowment_p·_wis minimized, and prices in_P1are such that this value is maximized. The next result claims that there is a trivial no-trade QCE with completely pessimistic

expections concerning supply opportunities at any price in_P0 and a trivial no-trade QCE with completely pessimistic expectations concerning demand opportunities at any price in

P1.

Theorem 4.1

For any_p∈_P0 there is a fully supply-constrained equilibrium with rationing vector_r∗ =−_w and allocation _x∗i = _wi_{, i} ∈ _Im_. For any _p ∈ _P1 there is a fully demand-constrained equilibrium with rationing vector _r∗ =_w and allocation _x∗i =_wi_{, i}∈_Im_.

Proof.

Take any _p ∈ _P0_. By definition of _P0 we have that −_w ∈ _G(_p)_. Taking_r =−_w gives for any_i ∈_Im that _Bi(_p,0n_,2_w) ={_wi} and hence _di(_p,0n_,2_w) =_wi_, implying that markets clear and thus a no-trade equilibrium with completely pessimistic expectations concerning supply opportunities is obtained.

Analogously, take any _p∈ _P1_. It holds that _w ∈ _G(_p)_. Taking _r = _w, we have for any _i ∈ _Im that _Bi(_p,−2_w,0n) = {_xi ∈ Xi|_xi ≤ _wi} and hence by Assumption U that

di(p,−2w,0n) = wi, implying that markets clear and thus a no-trade equilibrium with completely pessimistic expectations concerning demand opportunities is obtained. Q.E.D. If at an QCE it holds that_r∗ =−_w or_r∗ =_w, we say that the QCE is trivial. Otherwise, we call the QCE non-trivial. The existence of a continuum of non-trivial QCEs follows from the next result, which says that there exists a connected set of QCEs containing both a trivial no-trade equilibrium with completely pessimistic expectations concerning supply opportunities and a trivial no-trade equilibrium with completely pessimistic expectations concerning demand opportunities.

Theorem 4.2

Let E = ({Xi_,i, _wi}mi_{=1, P}) be an economy satisfying Assumptions X, U, W and P. Then there exists a connected set _D of Quantity Constrained Equilibria of the economy E_, containing a fully supply-constrained equilibrium (_p0_,−_{w, w}1_{, . . . , w}m) for some _p0 ∈ _P0_, and a fully demand-constrained equilibrium (_p1_{, w, w}1_{, . . . , w}m) for some _p1 ∈_P1_.

Since the rationing vector is equal to −_w at a fully supply-constrained equilibrium and equal to _w at a fully demand-constrained equilibrium, the connected set _D contains a continuum of non-trivial QCEs.

We continue this section with the proof of Theorem 4.2. We first focus on the equilibrating mechanism to find a QCE. To do so, we introduce a set _Q ⊂IRn containing

P and define for every _q ∈ _Q an admissible price vector _p(_q) ∈ _P and a rationing vector

r(q)∈[−w, w]. The set_Q is taken to be

so_Qis the set of elements inIRn lying at most at a distance 1 from_P, using the Euclidean norm, and thus includes the set _P. In the sequel bnd(_Q) denotes the boundary of _Q and int(_Q)its interior. For any _q ∈_Q, we define the corresponding price vector _p(_q) to be the projection of_q on_P, i.e.

p(q) =arg min

p∈P p−q2.

Since by Assumption P, the set_P is convex and compact, for every_q ∈_Qit holds that_p(_q) is uniquely defined and is continuous in_q,and_q−_p(_q)∈_G(_p(_q))_.Moreover,_q−_p(_q)2 ≤1_, with equality if and only if _q ∈ bnd(_Q)_. It also holds that _p(_q) = _q if and only if _q ∈ _P. The set_Q has the following properties.

Lemma 4.3

(i) The set _Q is a convex, compact, full-dimensional subset of IRn and contains _P in its interior;

(ii) The boundary of _Q is smooth.

Proof. The compactness follows from the compactness of_P.That_Qis full-dimensional and contains _P in its interior follows immediately from its definition. To prove convexity, take any_q1_{, q}2 ∈_Qand0≤_λ ≤1_,and let_q(_λ) =_λq1+(1−_λ)_q2and_p(_λ) =_λp(_q1)+(1−_λ)_p(_q2)_. Since _P is convex, we know that _p(_λ)∈ _P. Moreover, _q(_λ)−_p(_λ)2 ≤_λq1−_p(_q1)2+

(1−λ)q2−p(q2)2 ≤1.Therefore, _q(_λ)∈_Q.

Property (ii) follows from the use of the 2-norm in the definition of _Q. Take any

q∗ ∈bnd(_Q)_.Then_q∗−_p(_q∗)2 = 1and any_qwith_q−_p(_q∗)2 ≤1belongs to_Q.Hence, the normal cone at _q∗ contains at most one vector with length one. Since _Q is convex, we know that the normal cone at _q∗ is non-empty and upper-semicontinuous. Hence, the normal cone at any boundary point _q of _Q contains a unique vector with length one and this vector is continuous in _q, i.e. the boundary of_Q is smooth. Q.E.D.

We define the rationing vector _r(_q) corresponding to _q by the function_r :_Q→IRn, where

r(q) = 0n, q∈P

and

r(q) = min_{j∈In |} wj

qj−pj(q)|q−p(q)2(q−p(q)), q ∈Q\P.

Since_q−_p(_q)₂ goes to zero when_q−_p(_q)converges to0nand−_wk≤minj_∈In _|qj−wj_pj(q)|(_qk−

pk(q))≤wk, for all _k∈_In, it follows that _r(_q) goes to0n when_qconverges to a point in_P. This implies that _r(_q) is continuous in _q. The other properties below immediately follow from the definition of the function _r.

Corollary 4.4

The function _r:_Q→IRn satisfies the following properties: (i) _r is continuous;

(ii) _r(_q)∈[−_{w, w}] for all _q∈_Q;

(iii) _rk(_q) =_wk if and only if _qk−wk_pk(q) = minj_∈In _|qj−wj_pj(q)| and _q∈bnd(_Q); (iv) _rk(_q) =−_wk if and only if −_qk−wk_pk(q) = minj_∈In _|qj−wj_pj(q)| and _q∈bnd(_Q); (v) _r(_q)∈_G(_p(_q)) for all _q∈_Q.

We now define for any consumer_i ∈_Im the reduced budget correspondence _Bi : _Q→IRn by

Bi(q) =Bi(p(q),−w−r(q), w−r(q)), q∈Q.

Finally, for any consumer _i ∈ _Im we define the reduced demand function _di : _Q → IRn by _di(_q) = _di(_p(_q)_,−_w−_r(_q)_{, w} −_r(_q)). Because the set _P is in the interior of IRn₊, we have that for any _q ∈ _Q the price vector _p(_q) ∈ _P is strictly positive. Notice that since −w−r(q) can be equal to zero, the cheaper point assumption that is usually required to show continuity of the budget correspondence, is violated. Nevertheless, it follows from Theorem 2.2 in Herings (1996b) that_Bi is continuous at any_q ∈_Q.With Assumption U it then follows that_di is a continuous function and so is the reduced excess demand function

z :Q→IRn defined by

z(q) =

i∈Imd

i_(q)−w.

By Assumption U the budget constraint _p(_q)·_di(_q) ≤ _p(_q)·_wi is always satisfied with equality and hence Walras’ law holds, i.e. _p(_q)·_z(_q) = 0 for all _q ∈ _Q. The function _z satisfies the following properties.

Lemma 4.5 Under Assumptions X, U, W and P, the reduced excess demand function

z :Q→IRn satisfies the following: (i) _z is continuous;

(ii) Walras’ law holds: _p(_q)·_z(_q) = 0 for all _q∈_Q;

(iii) _rk(_q) =−_wk implies _zk(_q)≥0 and _rk(_q) =_wk implies _zk(_q)≤0_. The next lemma shows that a zero point of _z on_Q induces a QCE.

Lemma 4.6

Let _q∗ be a zero point of _z on _Q, i.e. _z(_q∗) = 0n. Then (_p(_q∗)_{, r}(_q∗)_{, d}1(_q∗)_{, . . . , d}m(_q∗)) is a Quantity Constrained Equilibrium.

Proof.

We have to show that Conditions (i), (ii) and (iii) of Definition 3.1 hold. Clearly,_p(_q∗)∈_P,

r(q∗)∈ [−w, w], and Conditions (i) and (ii) hold by construction of the reduced excess de-mand function. By property (v) of Corollary 4.4 we have that _r(_q∗) is an element of

G(p(q∗)), which shows that Condition (iii) of Definition 3.1 holds. Q.E.D. From Lemma 4.6 it follows that the question of existence of a QCE reduces to the existence of a zero point of_z. Before stating our general result on the connected set of equilibria in terms of the parameter _q, we first show that there are two different kinds of trivial zero points of_z, one corresponding to a trivial no-trade QCE with completely pessimistic expec-tations concerning supply opportunities and one with completely pessimistic expecexpec-tations concerning demand opportunities. Let_Q0 and_Q1 be given by

Q0 ={q∈Q|w·q≤w·qfor all_q∈_Q} and

Q1 ={q∈Q|w·q≥w·qfor all_q∈_Q}_.

Clearly, since _Q is compact, the sets _Q0 and _Q1 are both non-empty. Moreover, the intersection of_Q0 and_Q1 is empty, since _Qis full-dimensional.

Lemma 4.7

Each element in _Q0 or _Q1 is a zero point of _z and yields a trivial no-trade quantity con-strained equilibrium:

(i) Any _q ∈ _Q0 induces the fully supply-constrained QCE (_p(_q)_,−_{w, w}1_{, . . . , w}m) with

p(q)∈P0.

(ii) Any _q ∈ _Q1 induces the fully demand-constrained QCE (_p(_q)_{, w, w}1_{, . . . , w}m) with

p(q)∈P1.

Proof.

Take any _q ∈ _Q0_. Clearly, _q ∈ bnd(_Q)_. By definition of _Q0 we have that _r(_q) = −_w and

p(q)∈P0.It follows that, for any_i∈_Im_{, B}i(_q) ={_wi}and hence_di(_q) = _wi, implying that

z(q) = 0n and thus_q induces a no-trade equilibrium with completely pessimistic expected supply opportunities.

Analogously, take any_q ∈_Q1_.Then_r(_q) =_wand_p(_q)∈_P1_.It follows that, for any

i∈ Im, Bi(q) ={xi ∈ Xi|xi ≤wi} and hence by Assumption U that_di(_q) =_wi, implying that _z(_q) = 0n and thus _q induces a no-trade equilibrium with completely pessimistic

From this lemma it follows immediately that all trivial QCEs are induced by _Q0 and _Q1 and that every zero point of _z not in _Q0 or _Q1 induces a non-trivial QCE. Theorem 4.2 now follows from the next result saying that _Q contains a connected set _C of zero points of _z having a non-empty intersection with both_Q0 and _Q1.

Theorem 4.8

Let E = ({Xi_,i,_wi}mi_{=1, P}) be an economy satisfying Assumptions X, U, W and P. Then there exists a connected set _C⊂_Qof zero points of_z such that_C∩_Q0 =∅and_C∩_Q1 =∅.

Proof.

Let_γ0 and_γ1 be such thatnk_=1wk_qk=_γ0 when_q ∈_Q0 andnk_=1wk_qk =_γ1 when_q ∈_Q1. Notice that_γ0 and_γ1 are well-defined, _{γ1 > γ0} and_γ0 ≤nk_=1wk_qk ≤_γ1 for all_q ∈_Q. For some _{M >}0, define _X0 ⊂IRn by X0 ={x∈IRn| n k=1wkxk =γ0, max_k∈Inxk≤M} and, for 0_{< α}≤1, _Xα ⊂IRn by Xα ={x∈IRn|x=x0+_nkα =1wk2(γ1−γ0)w, x 0 _∈X0_}.

Clearly, for _x ∈ _Xα, 0 ≤ _α ≤ 1, we have that nk_=1wk_xk = (1−_α)_γ0 +_αγ1, and thus _nk

=1wkxk=γ0 when x∈X0 andnk=1wkxk =γ1 when x∈X1. Further, define

X =∪_α∈[0,1]Xα

and take_M sufficiently large that_Q⊂_X and that any_q∈_Q\(_Q0∪_Q1)lies in the interior of _X. Notice that _Q0 =_Q∩_X0 and_Q1 =_Q∩_X1.

We define the set _X0 = {_x ∈ IRn|nk_=1wk_xk = _γ0}_. Let _τ(_x) be the orthogonal projection of _x∈ IRn on_X0, i.e. _τ(_x) =_x−_λw for some _λ ∈ IR. Next, let the set_X0 be defined by

X0 ={x∈X0|x=τ(q+z(q)), q ∈Q} ∪ {x∈X0|x=τ(p(q)), q ∈bnd(_Q)}_.

Since _Qis compact and _z and_τ are continuous functions, it follows that _X0 is a bounded subset of_X0 and thus_M can be taken so large that_X0 contains_X0 in its relative interior.

For_x∈_X0 and_α∈[0_,1], denote _xα =_x+nα

k=1w2k(γ1−γ0)w∈X

α _{and define the}

point-to-set mapping _ϕ:_X0×[0_,1]→_X0 by ϕ(x, α) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ {τ(xα+z(xα))} if _xα ∈int(_Q)_, Conv({_τ(_xα+_z(_xα))} ∪ {_τ(_p(_xα))}) if _xα ∈bnd(_Q)_, {τ(p(q(xα)))} if _xα ∈_X\_Q,

for all(_{x, α})∈ _X0×[0_,1]_, where Conv(.) denotes the convex hull of a set and _q(_x) is the orthogonal projection of _x∈IRn on_Q. Clearly, _X0 is compact and convex, the mapping_ϕ is upper semi-continuous, and for every(_{x, α})∈_X0×[0_,1]it holds that_ϕ(_{x, α})is compact, convex and non-empty. According to Browder’s theorem, see Browder (1960), there exists a connected set _C of fixed points of _ϕ in _X0 ×[0_,1] such that _C ∩(_X0 × {0}) = ∅ and

C∩(X0 × {1})=∅, i.e. there exists a connected set_C in_X0×[0_,1]satisfying

x∈ϕ(x, α), for all(_{x, α})∈_C,

containing both a point in _X0 × {0} and a point in _X0 × {1}. Hence, the set _C ⊂ _X defined by

C ={x∈X|x=xα, (x, α)∈C}

is a connected set in _X such that _C∩_X0 = ∅ and _C ∩_X1 = ∅. It remains to be shown that every element of_C lies in _Q and is a zero point of_z.

Take any _x∈ _C, then _x=_xα for some (_{x, α})∈_C. Suppose first that _xα ∈ _X\_Q. From the definition of_ϕ it then follows that

x=τ(p(q(xα))).

Since _τ(_x) is the projection of _x on _X0_, there exists _λ ∈ IR such that _x = _τ(_p(_q(_xα))) =

p(q(xα))−λw. Since_xα−_q(_xα) =_β(_q(_xα)−_p(_q(_xα))) for some _{β >}0_, it follows that

(β+ 1)(q(xα)−p(q(xα))) = xα−p(q(xα))

= xα−(λw+x)

= (_nkα

=1wk2(γ1−γ0)−λ)w. This implies that _q(_xα)−_p(_q(_xα)) = _δw for some _δ∈ {−_w1

2,

1

w2}. Hence, either α = 0

and _xα ∈ _Q0_, or _α = 1 and _xα ∈ _Q1_. This contradicts that _xα ∈ _X \_Q. Consequently,

xα ∈Qfor every _xα∈_C.

Next, suppose that_xα∈int(_Q). Then_xα−nα

k=1wk2(γ1−γ0)w=x=τ(x

α_+z(xα_)).

Since _τ(_xα+_z(_xα)) = _xα+_z(_xα)−_λw for some_λ ∈IR_, it follows that

z(xα) = (λ− _nkα

=1wk2(γ1 −γ0))w. Because of Walras’ law, we obtain

p(xα)·z(xα) = (λ− _nkα

=1wk2(γ1−γ0))p(x

α_{)·w= 0.}

Hence, since_p(_xα)·_{w >}0_,we have that_λ = nα

k=1w2k(γ1−γ0)and thusz(x

α_{) = 0}n_{, showing}

Finally, suppose_xα ∈bnd(_Q). Then there exists_β1 ≥0and_β2 ≥0with_β1+_β2 = 1 such that

β1τ(xα+z(xα)) +β2τ(p(xα)) =x=xα−nkα

=1wk2(γ1−γ0)w.

Using that_τ(_x)is the projection of _x on_X0_, it follows that this can be rewritten as

β1xα+β1z(xα) +β2p(xα)−µw=xα− nkα

=1w2k(γ1−γ0)w for some _µ∈IR. With_β1+_β2 = 1 and letting _λ=_µ−nα

k=1w2k(γ1−γ0)

we obtain

β1z(xα) =β2(xα−p(xα)) +λw.

First, suppose _β1 = 0. Then, as before, it follows that _xα −_p(_xα) = _δw for some _δ ∈ {− 1

w2,

1

w2}, implying thatxα ∈Q

0_∪Q1_{. Hence, according to Theorem 4.7, z(x}α_{) = 0}n and thus _xα is a zero point of _z. Second, suppose _{β1 >}0. Then

z(xα) = β2 β1(x α_−p(xα_{)) +} λ β1w= δ β1r(x α_{) +} λ β1w,

for some _δ≥0. Since_xα ∈bnd(_Q), we have that there is_k such that|_rk(_xα)|=_wk_.When

rk(xα) =−wk it follows that

zk(xα) = − δ β1wk+

λ

β1wk ≥0, so _λ≥_δ. Hence, −_δ+_{λ >}0and thus

zh(xα) = δ β1rh(x α_{) +} λ β1wh ≥ − δ β1wh+ λ β1wh ≥0

for all _h∈_In and therefore_z(_xα) = 0n due to Walras’ law. Similarly, when_rk(_xα) =_wk it follows that

zk(xα) = δ β1wk+

λ

β1wk ≤0,

for some _δ≥0 and thus _δ+_λ≤0_. Then

zh(xα) = δ β1rh(x α_{) +} λ β1wh ≤ δ β1wh+ λ β1wh ≤0

for all_h∈_In and therefore _z(_xα) = 0n due to Walras’ law. Q.E.D. The theorem says that there is a connected set _C in _Q of zero points of the reduced ex-cess demand function_z, containing at least a trivial zero point _q0 in _Q0 and a trivial zero point _q1 in _Q1. Since _Q0 and _Q1 have a non-empty intersection, there must be a con-nected set of non-trivial zero points between _Q0 and _Q1. According to Lemma 4.6 every

non-trivial zero point _q∗ ∈ _C of _z induces a non-trivial quantity constrained equilibrium

(p∗, r∗, x∗) = (p(q∗), r(q∗), d(q∗)), where _d(_q∗) = (_d1(_q∗)_{, . . . , d}m(_q∗)) is the demand allo-cation induced by _q∗_. In order to prove Theorem 4.2 we still have to show that the set {(p(q), r(q), d(q))|q∈C} is a connected set.

Proof of Theorem 4.2

Take the function_f:_Q→_P ×IRn× _i∈ImXi defined by

f(q) = (p(q), r(q), d1(q), . . . , dm(q)), q ∈Q.

Consider the connected set _C of zero points of_z as defined in Theorem 4.8. It follows by Lemma 4.6 that every point in _C induces a QCE of E_. Since _f is a continuous function and by Theorem 4.8 _C is connected, it follows that the set _D = {_x|_x = _f(_q)_{, q} ∈ _C} is connected. By Lemma 4.7, it follows that _f(_q) is a fully supply-constrained equilibrium whenever _q ∈ _Q0 and _f(_q) is a fully demand-constrained equilibrium whenever _q ∈ _Q1_. Since _C intersects both _Q0 and _Q1_, the set _D contains both a fully supply-constrained equilibrium and a fully demand-constrained equilibrium. Q.E.D. We have proved now that there exists a connected set of non-trivial quantity constrained equilibria connecting a no-trade equilibrium with completely pessimistic expectations con-cerning supply opportunities and a no-trade equilibrium with completely pessimistic ex-pectations concerning demand opportunities. Each quantity constrained equilibrium is in-duced by some_q∈_Q. Any_qin_Q0or in_Q1yields a trivial quantity constrained equilibrium, whereas any_q∗ in_C not in_Q0 or_Q1 yields a non-trivial quantity constrained equilibrium. If _q∗ ∈ _P, then _p(_q∗) = _q∗ is a Walrasian equilibrium price vector, the rationing vector

r(q∗) = 0n,and the induced expected trade opportunities given by (∗_{, u}∗) = (−_{w, w}) are not binding, i.e. no consumer faces binding expected trade opportunities on either the demand side or the supply side and there is no coordination failure. If _q∗ is not in _P and so _p(_q∗) = _q∗_, then the price vector _p∗ = _p(_q∗) lies on the boundary of the set _P of admissible prices and the rationing vector _r∗ = _r(_q∗) lies in the normal cone _G(_p∗) at _p∗ to_P, and so the prices cannot be adjusted further at_p∗ in the direction_r∗. The vector _r∗ then determines the pessimistic expected trade opportunities (∗_{, u}∗) = (−_w−_r∗_{, w}−_r∗). Expected trade opportunities on supply can only be binding for commodity_k when_r∗_k<0 and expected opportunities on demand can only be binding for commodity_hwhen _rh∗ _>0. In these cases coordination failure takes place if some of the expected trade opportunities are binding.

Having proved that there exists a connected set of QCEs connecting a trivial fully supply-constrained no-trade equilibrium and a trivial fully demand-supply-constrained no-trade equilib-rium, it is easy to show that this set contains several non-trivial QCEs with some specific

properties. These properties are most easily stated in terms of the parametrization by the set _Q, with each _q ∈ _Q inducing a price _p(_q)_, a rationing vector _r(_q)_, and an allocation

(d1(q), . . . , dm(q)).

First of all, recall that for any _q0 ∈ _C∩_Q0 it holds that _r(_q0) = −_w and that for any_q1 ∈_C∩_Q1 it holds that_r(_q1) =_w. Therefore, since_C connects_Q0 and_Q1 and_r is a continuous function on_Q, for any a priori chosen_k ∈_In there exists a_q(_k)∈_C satisfying

rk(q(k)) = 0. Such a _q(_k) induces expected trade opportunities (_{, u}) with k = −_wk and

uk =wk.The point _q(_k)in_Qtherefore induces a QCE at which no consumer faces binding expected opportunities concerning commodity_k. We call such an equilibrium a generalized Drèze equilibrium with respect to commodity_k.

Corollary 4.9

For any _k ∈_In, there exists a generalized Drèze equilibrium with respect to commodity _k. The corollary implies that if commodity _k is the numeraire, there exists a QCE at which there are no binding expected trade opportunities concerning the numeraire.

The connected set _C also contains a point _q− satisfyingmaxj_rj(_q−) = 0_. This ele-ment of_C induces a QCE where for at least one commodity there are no binding expected trade opportunities and for all other commodities there may only be binding expected supply opportunities and therefore is a generalized supply-constrained equilibrium. Sim-ilarly, there exists a QCE at which consumers may only face binding expected demand opportunities and for at least one commodity there are no binding expected trade oppor-tunities, and is therefore what we call a generalized demand-constrained equilibrium. Such an equilibrium is induced by a point _q+ in_C for which minj_rj(_q+) = 0_. Notice that both types of equilibria are QCEs for which there are no binding expected trade opportunities concerning at least one commodity, but it cannot be said a priorily which commodity.

Corollary 4.10

There exists a generalized supply-constrained equilibrium and there exists a generalized demand-constrained equilibrium.

5

Applications and extensions

5.1

Price indexation

In this subsection we apply the concept of QCE as defined in Definition 3.1 to the set of admissible prices given in formula (1) of Section 2, i.e.

This set satisfies the so-called noncircularity condition of Weddepohl (1987), saying that the prices may not be indexed directly or indirectly by themselves. As discussed in Section 2, Dehez and Drèze (1984) and van der Laan (1984) have considered special cases of this set by making additional assumptions on the upper and lower bound of the index commodities or on the index functions. Here we only make some weak assumptions by assuming that for any_j ∈_I it holds that 0_{< pj} ≤_{pj <}∞ and that for any_j ∈_J, the lower bound function

π_j is convex, the upper bound function_πj is concave and that for all feasible _pI it holds that0_{< πj}(_pI)≤_πj(_pI)_<∞, so that_P satisfies Assumption P. To give a characterization of a QCE, for simplicity we assume that all index functions are continuously differentiable. Suppose _q ∈ _Q induces a QCE and thus _z(_q) = 0n. Since _r(_q)∈ _G(_p(_q)), we have that _p(_q)·_r(_q) = maxp_∈P _p· _r(_q). From the first-order Kuhn-Tucker conditions it then follows that there exists nonnegative numbers _λj and_λj, _j ∈_In, such that

rj(q) =λj −λ_j + h∈Jλh ∂π_h(pI(q)) ∂pj −_h∈Jλh ∂πh(pI(q)) ∂pj , j ∈I, (4) rj(q) =λj −λ_j, j ∈J, (5) and λ_j(pj(q)−p_j) = 0, λj(p_j−pj(q)) = 0, λ_jλj = 0, j ∈I, (6) λ_j(pj(q)−π_j(pI(q))) = 0, λj(πj(pI(q))−pj(q)) = 0, λ_jλj = 0, j ∈J. (7) Recall that in a QCE Condition (iii) of Definition 3.1 is satisfied, but not necessarily Condition (iv) of Definition 2.2. So, when there are binding expected supply (demand) opportunities concerning a commodity, its price is not necessarily on its lower (upper) bound. To make this more precise, take an indexed commodity _j ∈ _J. From (5) and the last complementarity condition in (7) it follows that

λ_j =−rj(q)>0 if _rj(_q)_<0_, and_λj =_rj(_q)_>0if_rj(_q)_>0_.

From the first two complementarity conditions in (7) it then follows that_pj(_q) = _πj(_pI(_q)) if there are binding expected supply opportunities for commodity _j ∈_J and that_pj(_q) =

πj(pI(q)) if there are binding demand opportunities for commodity _j ∈ _J, i.e. in case of binding expected supply (demand) opportunities the price of an indexed commodity is on its lower (upper) bound given the prices _pI(_q) of the index commodities. For an index commodity _j ∈ _I, in general such a clear link does not exist. For instance, suppose that all index functions are monotonically increasing in _pI and all the derivatives are positive.