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In this article we study how to model situations of non-clearing markets in a full general equilibrium framework. As we shall see from the historical discussion at the end, the theories we obtain synthesize three major schools of thought: (a) the Walrasian school, as Walras was the first to study a fully fledged general equilibrium system; (b) the Keynesian school, as Keynes emphasized the importance of quantity adjustments in reaching a macroeconomic equilibrium with at least one non-clearing market (that is, the labour market); and (c) the imperfect competition school, which endogenized prices through explicit price making by agents internal to the system.

This synthesis is notably achieved by introducing quantity signals into the traditional general equilibrium model. These quantity signals are quantity constraints which tell each agent the maximum quantity he can trade in each market. As we shall see, the introduction of these quantity signals in addition to price signals considerably enlarges the scope of traditional general equilibrium since they allow us not only to treat equilibria with various price rigidities, but also to endogenize prices in a decentralized imperfect competition framework.

The plan of the entry is the following. In the next three sections we describe the general concepts. The fourth section gives a brief historical outline of this line of thought.

Non-clearing Markets and Quantity Signals

In this section and the next two we describe various concepts in the framework of a monetary exchange economy where one good, money, serves as numéraire, medium of exchange and reserve of value (similar concepts have been developed for barter economies – see Bénassy 1975b, 1982 – but the formalization gets quite clumsy). There are ℓ markets in the period considered, where non-monetary goods indexed by h = 1, … ,ℓ; ‘are exchanged against money at the price ph. We call p the vector of these prices.

Agents are indexed by i = 1, … ,n; n. In market h agent i may make a purchase dih ≥ 0 or a sale sih ≥ 0. Define his net transaction of good h, zih = dih − sih, and zi the ℓ-dimensional vector of these net transactions.

At the beginning of the period agent i holds quantities \( {\overline{m}}_i \) of money, and ωih of good h. Call ωi the vector of the ωih. As a result of his trades zi, agent i ends up with final holdings of non-monetary goods and money, xi and mi, given respectively by:

$$ {x}_i={\omega}_i+{z}_i\ {m}_i={\overline{m}}_i-{pz}_i $$

We assume that agent i has a utility function on these final holdings Ui(xi, mi) = Ui(wi + zi, mi), which we assume throughout strictly concave in its arguments.

Walrasian Equilibrium

In order to contrast it with the non-Walrasian equilibrium concepts that will follow, let us describe briefly the Walrasian equilibrium of this economy (Arrow and Debreu 1954; Debreu 1959). Each agent i receives (from the implicit auctioneer) a price signal p. As a response he expresses a Walrasian net demand given by the function zi(p), solution in zi of the following program:

$$ {\displaystyle \begin{array}{l}\mathrm{Maximize}\kern1em {U}_i\left({\omega}_i+{z}_i,{m}_i\right)\mathrm{s}.\mathrm{t}.\\ {}\operatorname{}{pz}_i+{m}_i={\overline{m}}_i\end{array}} $$

A Walrasian equilibrium price vector p* is defined by the condition that all markets clear, that is:

$$ \sum \limits_{i=1}^n{z}_i\left({p}^{\ast}\right)=0 $$

The vector of transactions realized by each agent i is zi(p*).

Demands and Transactions

As we will be studying non-clearing markets, we must now make an important distinction, that between demands and supplies on the one hand, and the resulting transactions on the other.

Transactions, that is, purchases or sales of goods, denoted \( {d}_{ih}^{\ast } \) and \( {s}_{ih}^{\ast } \), are exchanges actually made, and must thus identically balance on each market, that is:

$$ {D}_h^{\ast }=\sum \limits_{i=1}^n{d}_{ih}^{\ast }=\sum \limits_{i=1}^n{s}_{ih}^{\ast }={S}_h^{\ast}\kern1em \mathrm{for}\;\mathrm{all}\ h $$
(1)

On the other hand, demands and supplies, denoted \( {\tilde{d}}_{ih} \) and \( {\tilde{s}}_{ih} \), are signals transmitted to the market (that is, to the other agents) before exchange takes place. They represent as a first approximation the exchanges the agents wish to make on each market. So they do not necessarily match in a specific market, and no identity like (1) applies to them:

$$ {\tilde{D}}_h=\sum \limits_{i=1}^n{\tilde{d}}_{ih}\ne \sum \limits_{i=1}^n{\tilde{s}}_{ih}={\tilde{S}}_h $$

In order to shorten notation, we often work in what follows with net demands and net transactions defined respectively by:

$$ {\tilde{z}}_{ih}={\tilde{d}}_{ih}-{\tilde{s}}_{ih}\kern0.5em {z}_{ih}^{\ast }={d}_{ih}^{\ast }-{s}_{ih}^{\ast } $$

The equality of aggregate purchases and sales Eq. (1) is then rewritten:

$$ \sum \limits_{i=1}^n{z}_{ih}^{\ast }=0\kern0.5em \mathrm{for}\;\mathrm{all}\ h $$
(2)

Rationing Schemes

In each market h the exchange process must generate consistent transactions (that is, transactions satisfying Eq. (1) or (2)) from any set of possibly inconsistent demands and supplies. Some rationing will necessarily occur, which may take various forms, such as uniform rationing, queuing, priority systems, proportional rationing, and so forth . . . depending on the particular organization of each market. We call rationing scheme the mathematicalrepresentation of each specific organization. To be more precise, the rationing scheme in market h is defined by a set of n functions:

$$ {z}_{ih}^{\ast }={F}_{ih}\left({\tilde{z}}_{1h},\dots, {\tilde{z}}_{nh}\right)i=1,\dots, n $$
(3)

such that:

$$ \sum \limits_{i=1}^n{F}_{ih}\left({\tilde{z}}_{1h},\dots, {\tilde{z}}_{nh}\right)=0\kern0.50em \mathrm{for}\;\mathrm{all}\kern0.50em {\tilde{z}}_{1h},\dots, {\tilde{z}}_{nh} $$

We assume that Fih is continuous, non-decreasing in \( {\tilde{z}}_{ih} \) and non-increasing in the other arguments. Before examining the possible properties of these rationing schemes, let us take a most simple example with two agents. Agent 1 emits a demand \( {\tilde{d}}_{1h} \), agent 2 a supply \( {\tilde{s}}_{2h} \). Then a natural rationing scheme, implicit in most macroeconomic models, is to take the level of transactions as equal to the minimum of demand and supply, that is:

$$ {d}_{1h}^{\ast }={s}_{2h}^{\ast }=\min\ \left({\tilde{d}}_{1h},{\tilde{s}}_{2h}\right) $$
(4)

Properties of Rationing Schemes

We first study two possible properties that a rationing scheme may satisfy: voluntary exchange and market efficiency.

The first property is actually an extremely natural one in a free market economy: We shall say that there is voluntary exchange in market h if no agent can be forced to purchase more than he demands, or to sell more than he supplies, which is expressed by:

$$ {d}_{ih}^{\ast}\le {\tilde{d}}_{ih}{s}_{ih}^{\ast}\le {\tilde{s}}_{ih}\kern0.5em \mathrm{for}\;\mathrm{all}\kern0.62em i $$

or equivalently in algebraic terms:

$$ \mid {z}_{ih}^{\ast}\mid <\mid {\tilde{z}}_{ih}\mid {\tilde{z}}_{ih}\cdot {\tilde{z}}_{ih}\ge 0\kern0.5em \mathrm{for}\;\mathrm{all}\ i. $$

Most markets in reality meet this condition, and we henceforth assume that it always holds. This allows us to classify agents in a market h in two categories: unrationed agents for which \( {z}_{ih}^{\ast }={\tilde{z}}_{ih} \), and rationed agents who trade less than they wanted.

The second property we study here is that of market efficiency, or absence of frictions, which corresponds to the idea of exhaustion of all mutually advantageous exchanges: a rationing scheme on a market h is efficient, or frictionless, if one cannot find simultaneously a rationed demander and a rationed supplier in market h. The intuitive idea behind this is that in an efficiently organized market a rationed buyer and a rationed seller would meet and exchange until one of the two is not rationed. Together with voluntary exchange, it implies the ‘short-side rule’, according to which agents on the ‘short side’ of the market can realize their desired transactions:

$$ {\tilde{D}}_{\hbar}\ge {\tilde{S}}_h\Rightarrow {s}_{ih}^{\ast}\kern0.5em \mathrm{for}\kern0.15em \mathrm{all}\kern0.5em i{\tilde{S}}_h\ge {\tilde{D}}_h\Rightarrow {d}_{ih}^{\ast }-{\tilde{d}}_{ih}\kern0.5em \mathrm{for}\;\mathrm{all}\kern0.62em i $$

This rule also implies that the global level of transactions on a market h will be equal to the minimum of aggregate demand and supply:

$$ {D}_h^{\ast }={S}_h^{\ast }=\min\ \left({\tilde{D}}_h,{\tilde{S}}_h\right) $$

We should note that the market efficiency assumption may not always hold, notably if one considers a fairly wide and decentralized market, because some demanders and suppliers might not meet pairwise. In particular, the market efficiency property is usually lost by aggregation of sub-markets, whereas the voluntary exchange property remains intact in the aggregation process. So we must keep in mind that it does not always hold. Fortunately, this hypothesis is not necessary for most of the microeconomic concepts presented in the next sections.

Quantity Signals

Now it is clear that at least rationed agents must perceive a quantity constraint in addition to the price signal. As it turns out, these quantity signals appear quite naturally in the formulation of a number of rationing schemes called non-manipulable, which can be written under the form:

$$ {d}_{ih}^{\ast }=\min \left({\tilde{d}}_{ih},{\tilde{d}}_{ih}\right){s}_{ih}^{\ast }=\min \left({\tilde{s}}_{ih},{\tilde{s}}_{ih}\right) $$
(5)

where the quantity signals \( {\overline{d}}_{ih} \) and \( {\overline{s}}_{ih} \) are functions only of the demands and supplies of the other agents. As an example, we can note that the rationing scheme corresponding to Eq. (4) above is of this type with:

$$ {\overline{d}}_{1h}={\tilde{s}}_{2h}{\overline{s}}_{2h}={\tilde{d}}_{1h} $$

For non-manipulable schemes the relation between \( {z}_{ih}^{\ast } \) and \( {\tilde{z}}_{ih} \) looks as in Fig. 1, in which we see where the term ‘non-manipulable’ comes from: once rationed, the agent cannot increase, or ‘manipulate’, the level of his transactions by increasing his demand and supply.

figure 1417figure 1417

Non-clearing Markets in General Equilibrium, Fig. 1

To make things a little more precise, let us rewrite the rationing scheme in market h Eq. (3) under the form:

$$ {z}_{ih}^{\ast }={F}_{ih}\left({\tilde{z}}_{ih},{\tilde{z}}_{- ih}\right) $$
(6)

where \( {\tilde{z}}_{- ih} \) is the set of all net demands on market h, except that of agent i, that is, \( {\tilde{z}}_{- ih}=\left\{{\tilde{z}}_{jh}|j\ne i\right\} \). The rationing scheme is non-manipulable if it can be rewritten as in Eq. (5), or algebraically:

$$ {F}_{ih}\left({\tilde{z}}_{ih},{\tilde{z}}_{- ih}\right)=\left\{\begin{array}{c}\hfill \min \left({\tilde{z}}_{ih},{\tilde{d}}_{ih}\right){\tilde{z}}_{ih}\ge 0\kern1.9em \hfill \\ {}\hfill \max \left({\tilde{z}}_{ih},-{\overline{s}}_{ih}\right)\kern1em {\tilde{z}}_{ih}\ge 0\hfill \end{array}\right. $$

where \( {\overline{d}}_{ih} \) and \( {\tilde{s}}_{ih} \) are functions of all demands and supplies in market h, except that of agent i, which we shall write as:

$$ {\tilde{d}}_{ih}={G}_{ih}^d\left({\tilde{z}}_{- ih}\right)\ge 0{\tilde{s}}_{ih}={G}_{ih}^s\left({\tilde{z}}_{- ih}\right)\ge 0 $$
(7)

Note that the functions \( {G}_{ih}^d\left({\tilde{z}}_{- ih}\right) \) and \( {G}_{ih}^s\left({\tilde{z}}_{- ih}\right) \) are not arbitrary, but are related to the rationing scheme Fih through:

$$ {G}_{ih}^d\left({\tilde{z}}_{- ih}\right)=\max \left\{{\tilde{z}}_{ih}|{F}_{ih}\left({\tilde{z}}_{ih},{\tilde{z}}_{- ih}\right)={\tilde{z}}_{ih}\right\} $$
(8)
$$ {G}_{ih}^s\left({\tilde{z}}_{- ih}\right)=-\min \left\{{\tilde{z}}_{ih}|{F}_{ih}\left({\tilde{z}}_{ih},{\tilde{z}}_{- ih}\right)={\tilde{z}}_{ih}\right\} $$
(9)

where it appears clearly that these quantity constraints are indeed the maximum purchase and sale that agent i can make in market h.

We may note that some rationing schemes, called manipulable, such as the proportional rationing scheme, cannot be written under this form. The phenomenon of manipulation through demand and supply leads then to a perverse phenomenon of overbidding, and to the non-existence of an equilibrium unless additional constraints are put on demands and supplies (Bénassy 1977b, 1982).

Most rationing schemes in the real world are actually non-manipulable through demand and supply, and we thus from now on study only such rationing schemes as can be characterized by Eq. (5) or (7). The variables \( {\overline{d}}_{ih} \) and \( {\overline{s}}_{ih} \) in (5) and (7) are quantity constraints. These are the quantity signals that each agent receives, and they play a fundamental role in both quantity and price determination, as we see in the next two sections. Before moving to the study of these problems and to the definition of non Walrasian equilibria, it is useful to rewrite Eqs. (6) and (7) pertaining to an agent i under vector form:

$$ {z}_i^{\ast }={F}_i\left({\tilde{z}}_i,{\tilde{z}}_{-i}\right){\overline{d}}_i={G}_i^d\left({\tilde{z}}_{-i}\right){\overline{s}}_i={G}_i^s\left({\tilde{z}}_{-i}\right) $$
(10)

where \( {\tilde{z}}_i \) is the vector of \( {\tilde{z}}_{ih},h=1,\dots, \mathrm{\ell} \), and \( {\tilde{z}}_{-i} \) is the set of all such vectors, except that of agent i himself, i.e. \( {\tilde{z}}_{-i}=\left\{{\tilde{z}}_j|j\ne i\right\} \).

Fixprice Equilibria

We now study a first concept of non-Walrasian equilibrium, that of fixprice equilibrium. This concept is of interest for several reasons. First, it gives us a very large class of consistent market allocations, since we shall find that under very standard conditions a fixprice equilibrium exists for every positive price system and every set of rationing schemes (we may note that Walrasian allocations are particular fixprice allocations, specifically those corresponding to a Walrasian price vector). Second, as we see in the next section, fixprice equilibria are a very important building block in constructing other non-Walrasian equilibrium concepts with flexible prices.

We thus assume that the price system p is given. As indicated, we assume that the rationing schemes in all markets are non-manipulable. Accordingly, transactions and quantity signals are generated in all markets according to the formulas seen above Eq. (10). We immediately see that all that remains to be done in order to obtain a fixprice equilibrium concept is to determine how demands themselves are formed, a task to which we now turn.

Effective Demands and Supplies

Demands and supplies are signals that agents send to the ‘market’ (that is, to the other agents) in order to obtain the best transactions. Consider thus an agent i faced with a price vector p and vectors of quantity constraints, \( {\overline{d}}_i \) and \( {\overline{s}}_i \). He knows that his transactions will be related to his demands and supplies by formulas (5) seen above, namely,

$$ {d}_{ih}^{\ast }=\min \left({\tilde{d}}_{ih},{\overline{d}}_{ih}\right){s}_{ih}^{\ast }=\min \left({\tilde{s}}_{ih},{\overline{s}}_{ih}\right) $$

Now the problem is to choose a vector of net effective demands \( {\tilde{z}}_i \) which will lead him to the best possible transactions. As it turns out, there exists a simple and workable definition which generalizes Clower’s (1965) original ‘dual decision’ method: the effective demand of agent i on market h is the trade which maximizes his utility subject to the budget constraints and to the quantity constraints on the other markets. Formally the effective demand \( {\tilde{z}}_{ih} \) is solution in zih of the following programme:

$$ {\displaystyle \begin{array}{l}\mathrm{Maximize}\kern0.50em {U}_i\left({\omega}_i+{z}_i,{m}_i\right)\mathrm{s}.\mathrm{t}.\\ {}\operatorname{}\left\{\begin{array}{l}{pz}_i+{m}_i={\overline{m}}_i\hfill \\ {}-{\overline{s}}_{ik}\le {z}_{ik}\le {\overline{d}}_{ik}k\ne h\hfill \end{array}\right.\end{array}} $$

Because of the strict concavity of Ui, we obtain a function, denoted \( {\tilde{\xi}}_{ih}\left(p,{\tilde{d}}_i,{\tilde{s}}_i\right) \). Repeating the operation for all markets h = 1,…ℓ, we obtain a vector function of effective demands \( {\tilde{\xi}}_i\left(p,{\overline{d}}_i,{\overline{s}}_i\right) \). This vector of effective demands has two good properties. First, it leads to the best transactions that it is possible to attain given the price vector p and the quantity constraints \( {\overline{d}}_i \) and \( {\overline{s}}_i \). Second, whenever a constraint is binding on a market h, the corresponding demand or supply is greater than the quantity constraint, which thus ‘signals’ to the market that the agent trades less than he would want. Such signals are useful to avoid trivial equilibria where no one would trade because nobody else signals that he wants to trade.

Fixprice Equilibrium

With the above definition of effective demand, we are now ready to give a first definition of a fixprice equilibrium, found in Bénassy (1975a, 1982).

Definition 1

A fixprice equilibrium associated with a price system p and rationing schemes represented by functions Fi , i = 1, … , n, is a set of effective demands\( {\tilde{z}}_i \), transactions\( {z}_i^{\ast } \)and quantity constraints\( {\overline{d}}_i \)and\( {\overline{s}}_i \)such that:

(a)\( {\tilde{z}}_i={\tilde{\xi}}_i\left(p,{\overline{d}}_i,{\overline{s}}_i\right)\kern0.5em i=1,\dots, n \)

(b)\( {z}_i^{\ast }={F}_i\left({\tilde{z}}_i,{\tilde{z}}_{-i}\right)\;i=1,\dots, n \)

(c)\( {\overline{d}}_i={G}_i^d\left({\tilde{z}}_{-i}\right)\;{\overline{s}}_i={G}_i^s\left({\tilde{z}}_{-i}\right)\;i=1,\dots, n \)

Equilibria defined in this way exist for all positive prices and all rationing schemes satisfying voluntary exchange and non-manipulability (Bénassy 1975a, 1982). The ‘exogenous’ data are the price system p and the rationing schemes Fi, i = 1,…n. One may wonder whether for given such exogenous data the equilibrium is likely to be unique. A positive answer has been given by Schulz (1983), who showed that the equilibrium is globally unique, provided the ‘spillover’ effects (there is a spillover effect when a binding constraint in one market modifies the effective demand in another market) are less than 100 per cent in value terms. For example in the simplest Keynesian model this would amount to a propensity to consume strictly smaller than 1.

In what follows we assume that the Schulz conditions hold, and denote by \( {\tilde{Z}}_i(p),{Z}_i^{\ast }(p),{\overline{D}}_i(p) \) and \( {\overline{S}}_i(p) \) the functions giving the values of \( {\tilde{z}}_i,{z}_i^{\ast },{\overline{d}}_i \), and \( {\overline{s}}_i \) at a fixprice equilibrium corresponding to p (the market organization, and thus the rationing schemes, being assumed invariant).

An Alternative Concept

We shall now present an alternative concept of fixprice equilibrium, due to Drèze (1975) (who actually dealt with the more general case of prices variable between fixed limits), and which we shall recast using our notations. That concept does not separate demands from transactions, and thus considers directly the vectors of transactions \( {z}_i^{\ast } \) and quantity constraints \( {\overline{d}}_i \) and \( {\overline{s}}_i \). The original concept actually assumed uniform rationing, so that the vectors di and si were the same for all agents.

Definition 2

A fixprice equilibrium for a given set of prices p is defined as a set of transactions\( {z}_i^{\ast } \) and quantity constraints \( {\overline{d}}_i \) and \( {\overline{s}}_i \) such that:

(a)\( \kern1em {\sum}_{i=1}^n{z}_{ih}^{\ast }=0\forall h \)

(b) The vector\( {z}_i^{\ast } \)is solution in ziof:\( {\displaystyle \begin{array}{l}\kern3.55em Maximize\kern0.5em {U}_i\left({\omega}_i+{z}_i,{m}_i\right)\mathrm{s}.\mathrm{t}.\\ {}\operatorname{}\left\{\begin{array}{l}{pz}_i+{m}_i={\overline{m}}_i\hfill \\ {}-{\overline{s}}_{ih}\le {z}_{ih}\le {\overline{d}}_{ih}\forall h\hfill \end{array}\right.\end{array}} \)

(c) \( \forall h\;{z}_h^{\ast }={\overline{d}}_{ih} \)for some i implies\( {z}_{jh}^{\ast }>-{\overline{s}}_{jh}\kern1.9em \forall {jz}_{ih}^{\ast }=-{\overline{s}}_{ih} \)for some i implies\( {z}_{jh}^{\ast }<-{d}_{jh}\forall j \)

Let us now interpret these conditions. Condition (a) is the natural requirement that transactions should balance in each market. Condition (b) says that transactions must be individually rational, that is, they must maximize utility subject to the budget constraint and the quantity constraints on all markets. We may note at this stage that using quantity constraints under the form of upper and lower bounds on trades implicitly assumes rationing schemes which exhibit voluntary exchange and non-manipulability, as we saw when studying rationing schemes. Condition (c) says that rationing may affect either supply or demand, but not both simultaneously. We recognize here with a different formalization the condition of market efficiency which is thus built into this definition of equilibrium, whereas it is not in the previous definition.

Drèze (1975) proved that an equilibrium according to def 2 exists for all positive price systems and for uniform rationing schemes under the standard concavity assumptions for the utility functions. The concept is easily extended to non-uniform bounds (Grandmont and Laroque 1976; Greenberg and Muller 1979), but in this last case it is not specified in the concept how shortages are allocated. Because of this there will be usually an infinity of equilibria corresponding to a given price vector, as soon as there are two rationed agents on one side of a market.

As we noted above, the two concepts of fixprice equilibrium we described in this section are based, implicitly or explicitly, on a representation of markets under the form of rationing schemes satisfying voluntary exchange and non-manipulability. This suggests that, if in the first definition we further assume that all rationing schemes are efficient or frictionless, the two definitions should yield similar sets of equilibrium allocations for a given price system. This was indeed proved by Silvestre (1982, 1983) for both exchange and production economies. The relation between the two concepts has been further explored by D’Autume (1985).

Price Making and Equilibrium

As this stage we still need a description of price making by agents internal to the system. We describe in this section a concept dealing with that problem and we shall see that, just as in demand and supply theory, quantity signals play a prominent role. It is indeed quite intuitive that quantity constraints must be a fundamental part of the competitive process in a decentralized economy: it is the inability to sell as much that they want which leads suppliers to propose, or accept from other agents, a lower price, and conversely it is the inability to purchase as much as they want that leads demanders to propose, or accept, a higher price.

Various modes of price making integrating these aspects can be envisioned. We deal here with a realistic organization of the pricing process where agents on one side of the market (most often the suppliers) quote prices and agents on the other side act as price takers. The general idea relating the concepts in this section to those of the previous one is that price makers change their prices so as to ‘manipulate’ the quantity constraints they face (that is, so as to increase or decrease their possible sales or purchases). As we shall see, this model of price making is quite reminiscent of the imperfect competition line (Chamberlin 1933; Robinson 1933; Triffin 1940; Bushaw and Clower 1957; Arrow 1959), and more particularly of the theories of general equilibrium with monopolistic competition, as developed notably by Negishi (1961).

The Framework

We thus now assume that agent i controls the prices of a (possibly empty) subset Hi of goods. Goods are distinguished both by their physical characteristics and by the agent who sets their price. We thus consider two goods sold by different sellers as different goods, a fairly natural assumption since these goods differ at least by location, quality, and so on, so that:

$$ {H}_i\cap {H}_j=\left\{\varnothing \right\}i\ne j $$

We denote by pi the set of prices controlled by agent i and p−i the rest of prices, that is:

$$ {p}_i=\left\{{p}_h|h\in {H}_i\right\}\;{p}_{-i}=\left\{{p}_h|h\kern0.50em \notin \kern0.50em {H}_i\right\} $$

Each agent chooses his price vector pi taking the other prices p−i as given. The equilibrium structure is thus that of a Nash equilibrium in prices, corresponding to an idea close to that of monopolistic competition. The basic idea behind the modelling of price making itself in such models is, as we indicated above, that each price maker uses the prices he controls to ‘manipulate’ the quantity constraints he faces. Consider the markets whose price are determined by agent i, and subdivide further Hi into \( {H}_i^d \) (goods demanded by i) and \( {H}_i^s \) (goods supplied by i). We may note in passing that, although agent i appears formally as a monopolist in markets \( h\in {H}_i^s \) or a monopsonist in markets \( h\in {H}_i^d \), his actual ‘monopoly power’ may be very low due to the fact that other agents sell or buy products which are extremely close substitutes to those he controls. Because the price makers are alone on their side of the markets where they set prices, their quantity constraints on these markets have the simple form:

$$ {\overline{s}}_{ih}=\sum \limits_{j\ne i}{\tilde{d}}_{jh}\kern0.50em h\in {H}_i^s\kern1em {\overline{d}}_{ih}=\sum \limits_{j\ne i}{\tilde{s}}_{jh}\kern0.50em h\in {H}_i^d $$

that is, the maximum quantity that price setter i can sell is the total demand of the others, and conversely if he is a buyer. All we need to know, in order to pose the problem of price setting as a standard decision problem, is the relation, as perceived by the price maker, between the quantity constraints he faces and the prices he sets. Several approaches allow us to treat this problem and to link it with the concepts seen previously. The first, based on Negishi’s (1961) subjective demand curve approach, was developed in Bénassy (1976, 1982). The second is an objective demand curve approach, developed in Bénassy (1987, 1988), and which we shall now briefly describe.

Objective Demand Curves

The implicit idea behind the objective demand curve approach (Gabszewicz and Vial 1972; Marschak and Selten 1974; Nikaido 1975) is that each price maker knows the economy well enough to be able to compute under all circumstances the actual quantity constraints he will face. Since we are considering a Nash equilibrium, he must be able to perform this computation for any set pi of prices he chooses as well as for any set p− i of the other prices; that is, he must be able to compute his constraints for any vector of prices, once all feedback effects have been accounted for.

But we know from the previous section that, for a given organization of the economy (that is, notably for given rationing schemes), and for a given set of prices p, the quantity constraints agent i faces are given by the functions \( {\overline{D}}_i(p) \) and \( {\overline{S}}_i(p) \). If the agent has full knowledge of the parameters of the economy (a strong assumption, of course, but which is embedded in the notion of an objective demand curve), then he knows this and the objective demand and supply curves will be respectively given by the functions \( {\overline{S}}_i(p) \) and \( {\overline{D}}_i(p)\kern-0.25em \). We may note that the objective demand curve \( {\overline{S}}_i(p) \) is denoted as a constraint on agent i’s supply, which is natural since the sum of all other agents’ demands acts as a constraint on the sales of agent i, and symmetrically with the objective supply curve \( {\overline{D}}_i(p) \).

Price Making and Equilibrium

If agent i knows the two vector functions \( {\overline{D}}_i(p) \) and \( {\overline{S}}_i(p) \), the programme giving his optimal price pi is the following:

$$ {\displaystyle \begin{array}{l}\mathrm{Maximize}\;{U}_i\left({\omega}_i+{z}_i,{m}_i\right)\mathrm{s}.\mathrm{t}.\\ {}\operatorname{}\left\{\begin{array}{l}{pz}_i+{m}_i={\overline{m}}_i\hfill \\ {}-{\overline{S}}_i(p)\le {z}_i\le {\overline{D}}_i(p)\hfill \end{array}\right.\end{array}} $$

which yields the optimum price pi chosen by agent i as a function of the other prices p−i.

$$ {p}_i={\psi}_i\left({p}_{-i}\right) $$

This naturally leads us to the definition of an equilibrium with price makers:

Definition 3

An equilibrium with price makers is characterized by a set of prices\( {p}_i^{\ast } \), net demands\( {\tilde{z}}_i \), transactions\( {z}_i^{\ast } \)and quantity constraints\( {\overline{d}}_i \)and\( {\overline{s}}_i \)such that:

(a) \( \kern1.0em {p}_i^{\ast }={\psi}_i\left({p}_{-i}^{\ast}\right) \)

(b) \( {\tilde{z}}_i,{z}_i^{\ast },{\tilde{d}}_i,{\overline{s}}_i \) are equal respectively to \( \kern3.25em {\tilde{Z}}_i\left({p}^{\ast}\right) \), \( {Z}_i^{\ast}\left({p}^{\ast}\right) \), \( {\overline{D}}_i\left({p}^{\ast}\right) \), \( {\overline{S}}_i\left({p}^{\ast}\right) \).

Condition (a) indicates that we have a Nash equilibrium in prices, given each agent’s optimal price responses. Condition (b) says that the various quantities form a fixprice equilibrium (according to def 1) for the price vector p*. Further discussion and conditions for existence can be found in Bénassy (1988, 1990).

Bibliographical References

So far we have concentrated in this entry on the microeconomic concepts allowing us to deal with non-clearing markets at a general equilibrium level. We now indicate further bibliographical references both on the early history of the domain and on macroeconomic applications.

History

The field we described in this entry has a triple ancestry. On one hand Walras (1874) developed a model of general equilibrium with interdependent markets where adjustment was made through prices. This model, in its modern reformulation (Arrow and Debreu 1954; Arrow 1963; Debreu 1959) has become the basic benchmark concept in microeconomics. On the other hand Keynes (1936) and Hicks (1937) built, at the macroeconomic level, a concept of equilibrium where adjustment was made by quantities (the level of national income) as well as by prices. Finally, following the contributions by Chamberlin (1933) and Robinson (1933), progress was made on the treatment of imperfect competition. Notably, Negishi (1961) formalized imperfect competition with subjective demand curves in a general equilibrium framework.

A few isolated contributions in the post-war period made some steps towards modern theories of non-clearing markets. Bent Hansen (1951) introduced the ideas of active demand, close in spirit to that of effective demand, and of quasi-equilibrium where persistent disequilibrium created steady inflation. Patinkin (1956, ch. 13) considered the situation where firms might not be able to sell all their Walrasian output. Hahn and Negishi (1962) studied non-tâtonnement processes where trade could take place before a general equilibrium price system was reached.

A stimulating impetus came from the contributions of Clower (1965) and Leijonhufvud (1968), who reinterpreted Keynesian analysis in terms of market rationing and quantity adjustments. These insights were included in the first fixpricefixwage macroeconomic model by Barro and Grossman (1971, 1976).

The main subsequent development was the construction of rigorous microeconomic concepts allowing us to deal with non-clearing markets and imperfect competition in a full multi-market general equilibrium setting, as described above. Notably, Drèze (1975) and Bénassy (1975a, 1977b, 1982) bridged the gap between the Walrasian and Keynesian lines of thought by generalizing the Walrasian equilibrium concept to integrate non-clearing markets and quantity signals. The link between this new line of work and the imperfect competition equilibrium concepts in the Negishi (1961) line was made in Bénassy (1976, 1977a, 1988). These contributions led to the unified framework we set out in the previous sections. Of course, since one of the main goals of this line of research was to bridge the gap between microeconomics and macroeconomics, there were a number of macroeconomic applications of the above concepts, which we now briefly describe.

Macroeconomic Applications

As indicated above, the first fully worked out fixprice-fixwage macroeconomic model embedding the notions set out above is that of Barro and Grossman (1971, 1976). Early attempts are found in Glustoff (1968) and Solow and Stiglitz (1968). Further developments of the model were made in Bénassy (1977a, 1982, 1986), Malinvaud (1977), Hildenbrand and Hildenbrand (1978), Muellbauer and Portes (1978), Honkapohja (1979), Neary and Stiglitz (1983), and Persson and Svensson (1983). Most of these models concentrated on the problem of employment and policy. Other problems have been treated with this methodology, including notably foreign trade (Dixit 1978; Neary 1980; Cuddington et al. 1984), growth (Ito 1980; Picard 1983; D’Autume 1985), business cycles (Bénassy 1984), as well as the specific problems of planned socialist economies (Portes 1981).

An important part of this line of macroeconomic modelling is that concerned with the explicit introduction of price making and imperfect competition in the macro-setting. Models of that type can be found notably in Bénassy (1977a, 1982, 1987, 1990, 1991), Negishi (1977, 1979), Hart (1982), Snower (1983) Weitzman (1985), Svensson (1986), Blanchard and Kiyotaki (1987), Dixon (1987), Sneessens (1987), Silvestre (1988) and Jacobsen and Schultz (1990).

Now the concepts described in this entry are full general equilibrium models in the tradition of, say, Arrow and Debreu (1954) and Debreu (1959). Contemporaneously to these developments, other authors developed, under the initial name of real business cycles, dynamic stochastic models based on the hypothesis of rational expectations. At some point these two lines of work were synthesized, and the result of this synthesis is described in the dictionary article ‘dynamic models with non-clearing markets’.

See Also