4. Price Formation: Overview of the Theory

A buyer’s shopping basket contains a quart of milk, a pound of butter, a pound of bacon, and so on.

The interval being closed implies that a seller (buyer) is willing to vend at her cost (pay his value) rather than forego a trade. Consequently, the \([c_{j} ,v_{i} ]\) include a profit sufficient to induce a trade at either limit.

Throughout this book, we will insist on the generic, qualitative, and “nonparametric” characterization of classical competitive price dynamics. A more detailed, “quantitative,” approach would require more careful case-by-case specification, which is not undertaken here. Thus, the overall theory does not depend on the specifics of the distribution of reservation prices, nor the speed of price adjustment, which involve parameters that likely depend on the specifics of the market institution. An extensive experimental literature has developed on the rules of trading for a host of institutions, modern and ancient (Holt, in Holt, 1995, pp. 360–377) Among the market institutions, the double auction, described in Chapter 3, Sect. 3.5, is particularly interesting, in that it involves all forms of competition discussed under 4 in the text below: buyer-buyer, seller-seller, and buyer-seller. Various models of this specific market institution have been offered in the experimental literature (Anufriev et al., 2013; Asparouhova et al., 2020; Cason & Friedman, 1996; Friedman, 1991; Friedman & Rust, 2018; Gjerstad & Dickhaut, 1998).

Experiments using the continuous double auction appear to approximate this ordering probabilistically. The more profitable the unit the more eager is the subject to consummate a contract. But we know of no systematic empirical examination of trade order.

At this intuitive and heuristic stage of the discussion, we need not know explicitly the maximum achievable number of transactions, although one should expect it to be given by the intersection of the demand and supply curves. The theoretical rationale for this association is not trivial, however. It is easy to establish under short-side rationing (Fig. 4.6).

Throughout this book, the superscript “T” denotes vector or matrix transposition, following standard notation.

The price p in (4.1)–(4.7) is merely a variable at this stage: it is a dummy variable and is perhaps more rigorously (but less revealingly) denoted by another variable, say x. However, speaking precisely in terms of process, there are no transaction prices except as they are discovered in markets, so we can only speak of demand or supply contingent on price as a variable found in the market. This is of importance in understanding two central ideas: (1) that it is only in market realizations that buyers and sellers collectively aggregate demand, supply and find determinant contract prices; (2) the information function and interpretation of markets. “Aggregate” here has nothing to do with adding up total amounts consumed and produced if price is fixed at a given level—the exercise that defined neoclassical econimc theory. Rather, aggregation is a “vertical” process of matching buyers and sellers at the prices they negotiate in a multilateral exchange process and that occurs naturally in profit space as buyers strive to buy cheap and sellers to sell dear. The characteristic functions we use to describe the market process have neoclassical interpretation inimical to that process.

Affordability usually applies to buyers, and profitability, to sellers. But the distinction is of no substance: a distinctive aspect of the present (classical) formulation of price theory is the symmetry of demand and supply treatments.

As originally tested in the early experiments (V. L. Smith, 1962), the minimization of the function V (then denoted as R, for buyers’ and sellers’ rent) is referred to as the Excess Rent Hypothesis, “the hypothesis found to be most successful in these experiments” (p. 112). “The principle is this: in static equilibrium a competitive market minimizes the total virtual rent received by buyers and sellers. By “virtual rent” I mean the rent that would be enjoyed if all buyers and sellers could be satisfied at any given disequilibrium price” (p. 132).

The Austrian tradition, however, preserved the classical view of competition, notably after the socialist calculation debate. See, for example, Hayek (1948).

We list buyer-seller confrontation or “higgling” first because it is the only meaningful form of competition in bilateral bargaining. But the second is essential in understanding the role of short-side rationing in classical price formation theory. Both, however, contribute to gains from trade and wealth creation.

We have used the identity \((a_{h} )^{k} = a_{k} ,k \ge 1,\) shared by all indicator functions.

Compare our description of buyer/seller competition in the market with that of Adam Smith ([1776] 1904, pp. 58–59).

These contradictory conditions led J. Bertrand to point out the path-dependency problems in Walras, in turn leading Walras to exclude disequilibrium trades in his theory, and to eventually reformulate his tatonnement theory towards the modern interpretation, as a virtual price adjustment process. See Bertrand (1883, p. 505) On the evolution of Walras’ tatonnement, see Walker and van Daal’s introduction to Léon Walras (1874b [1896, 2014]).

It is perhaps regrettable that the modern discussions on price dynamics tend to focus on the Walrasian formulation, and attempts to solve its intrinsic difficulties, rather than going back to the original classical understanding of competitive price adjustment. Like other early marginalists, Walras initially took for granted the classical competition process, as evidenced by various passages in his early memories that form the substance of his famous Elements (Walras, 1874a).

Thus did Adam Smith warn us earlier, concerning his following three chapters (v, vi, and vii) that “I must very earnestly entreat both the patience and attention of the reader: his patience in order to examine a detail which may perhaps in some places appear unnecessarily tedious; and his attention in order to understand what may, perhaps, after the fullest explication which I am capable of giving it, appear still in some degree obscure. I am always willing to run some hazard of being tedious in order to be sure that I am perspicuous; and after taking the utmost pains that I can to be perspicuous, some obscurity may still appear to remain upon a subject [which is] in its own nature extremely abstracted” (A. Smith ([1776] 1904, pp. 30–31). Remarkably, his warning was both justified and ignored in the great diversion from these principles that evolved in the form of neoclassical general-equilibrium theory.

In the market experiments, at the end of each period, T, the supply and demand are refreshed before going into period T + 1, so that S and D are flows of goods or services into and out of the market. In each market we have bilateral transactions t’ = 1,…,T’, where T’ is the maximum number of units that can be profitably traded in a period. Hence, any inefficient transactions in an early period will tend not to be repeated, because either the buyer paid too much, or the seller received too little, and being motivated to buy cheap and sell dear they seek better terms in later transactions, and this assures empirically that the surplus, V(p(t'  + T)) will get better across transactions.

The discontinuity of demand and supply (owing to the discreteness of the commodity space and the fickle and fitful nature of economic behaviors) used to be taken for granted in economics, and the assumption of smooth demand and supply was relegated to large markets. This intuition was perhaps first articulated by Cournot (1838 [1897], pp. 49–50). It will be acknowledged throughout the marginal revolution, where many authors invoke it to justify the treatment of demand and supply as smooth functions, but usually fall back on the average-agent simplification, and not by considering explicitly the distribution of agents’ characteristics (Jevons, 1871 [1888], pp. 89–90; Marshall, 1890 [1920], p. 83; Pareto, 1897, p. 9; Walras, 1926, pp. 57–58).

The concept of large market goes back to Cournot: “We will assume that the function F(p), which expresses the law of demand or of the market, is a continuous function […]. It might be otherwise if the number of consumers were very limited: thus in a certain household the same quantity of firewood will possibly be used whether wood costs 10 francs or 15 francs the stere, and the consumption may suddenly be diminished if the price of the stere rises above the latter figure. But the wider the market extends, and the more the combinations of needs, of fortunes, or even of caprices, are varied among consumers, the closer the function F(p) will come to varying with p in a continuous manner. However little may be the variation of p, there will be some consumers so placed that the slight rise or fall of the article will affect their consumptions, and will lead them to deprive themselves in some way or to reduce their manufacturing output, or to substitute something else for the article that has grown dearer, as, for instance, coal for wood or anthracite for soft coal” (Cournot, 1838 [1897], pp. 49–50).

Thus, the value center is obvious to identify graphically. One should not, however, understate the theoretical and conceptual importance of this generalization of competitive equilibrium: it calls into question the centrality in economic theory of abstract assumptions suggested by a priori belief in the necessity of “perfect competition” and “market clearing,” which greatly narrow the domain of validity of competitive price theory: “large number of traders, infinitely divisible commodity, continuous preference relations,” and so on. The present classical formulation of competition should invite a redefinition of the concept of “market failure.” It will be shown that no special advantage is gained in the limit of a “large number of traders, etc.,” besides mathematical simplicity, since in this case, demand and supply can be assumed smooth, and, as we shall see in the next chapter, the value center shrinks to a single (market-clearing point). Moreover, if Coasian transactions cost increase with these idealized conditions, the effect is to escalate dead weight economic loss.

Intuitively, the continuity-by-integration principle is an instance of a general regularity by aggregation principle that has well-known specific instances; all share in common their exhibiting or articulating a subtle, and at times conflictual, relationship between the whole and its parts, whereby the properties of the whole sometimes are opposite to those of an individual part. Thus, certainty emerges (trivially) by integration over uncertainty (namely a random variable, whose integral, the mean or expectation, is of course a number); more strikingly, certainty also emerges by summation of individual randomness under well-known conditions (the laws of large numbers).

An important step in the unified view of statistics was taken by the mathematician M. Fréchet (1948), who generalizes the concept of summary statistic to abstract mathematical spaces (an optimal summary of a distribution by one point). Summary statistics thus abstractly generalized (although usually assuming constant weights) is known as “Fréchet mean” (lambda = 2) or “Fréchet median” (lambda = 1). It is of note that the ordinary mathematical notion of “distance” is too specific for the purpose of statistics (e.g., a square distance, common in statistics, does not necessarily obey the tringular inequality); thus the term “distance” is used here in the more general sense of “statistical distance.”

Median-based statistics (based on minimum absolute-deviations criterion or the “L₁ norm”) preceded historically the now more popular mean-based regression (based on minimum least-square deviations or the “L₂ norm”): it first appeared in the eighteenth century in the median regression work by Boscovich (1757), but was eclipsed by the simpler to implement and still more popular mean-based statictics (based on square deviations), through the infuencial contribution of Legendre (1805) and Gauss (1809) in the early nineteenth century. Despite a brief resurgence in late nineteenth century through (Edgeworth, 1888a, 1888b) “L₁ statistics” will truly enjoy an important revival only in the 1950s–1960s in (Hampel, 1968; Huber, 1964; Tukey, 1960), aided with the development of computational techniques, notably the linear programming formulation of (Wagner, 1959), and it expanded enormously in the late 1970s due notably to the quantile regression extension (Koenker & Bassett Jr, 1978; Koenker, 2005), and is now of course a mature industry, included in standard statistical packages. On the fascinating early development of statistics, see Stigler (1986).

As we have emphasized in experimental economics, the concept of WTP (or WTA) is fundamentally an association of individual values with commodity units, i.e., value (or reservation price) is functionally dependent on commodity units. Since additional units may have the same value, “demand” is always written mathematically as quantity a function of price to avoid nonexistence of the inverse. But it is units that have dependent reservation values and that figure into classical “higgling and bargaining” out of which prices emerge, and that determine aggregate market demand. Thus: for individuals coming to market, values (expressing maximum payment intentions) are dependent on units; after interacting for a sufficient-enough (equilibrating) period of time, individuals leaving the market have found the price or the center of value to which we relate their demand and aggregate quantity. When they return to the market, they trade at that center of value unless some of their reservation price values have changed in which case the center has shifted, and it must be discovered again through their interactions.

The efficient-market literature is well-known: see, the famous review by Fama (1970). Owing to the famous No-Trade Problem of neoclassical finance, however, the efficient-market hypothesis is not easy to derive as an equilibrium condition of supply and demand, hence the standard recourse to No-Arbitrage Condition erected as the fundamental principle of this subfield of neoclassical economics. On the No-Trade Theorem, see notably Milgrom and Stokey (1982)and Tirole (1982). On a simple overview of the difficulty of articulating the Efficient Market Hypothesis as a competitive equilibrium condition in view of the No-Trade Theorem, and the appeal to No-Arbitrage Condition as a way out, see Ross (2005, ch. 3). Financial markets are the subject of Chapter 7 of this book.

It is in fact in terms of the ratio of supply and demand that the law of supply and demand was originally formulated in the classical school: “The opinion that the price of commodities depends solely on the proportion of supply to demand, or demand to supply, has become almost an axiom in political economy […]” (Ricardo, 1817 [2004], Ch. XXX, p. 382). Adam Smith visualized the market price as “regulated by the proportion between the quantity which is actually brought to market, and the demand of those who are willing to pay the natural price of the commodity…” (A. Smith [1776] 1904, p. 58).

Mathematically, we have a singularity if S = 0 or D = 0, respectively, that is, when price < min(cost) or price > max(value). This is not a problem, however, as will be clear soon.

Standard information theory focuses instead on the average uncertainty or entropy, obtained by integration over a whole probability distribution.

The Austrian tradition has a complex relationship with the classical school. Although it is most vocal in defending the subjectivity of value against the alleged objectivist classical view, yet on competitive market price formation, the Austrian theory is perhaps the closest to the original classical view on competition within the marginalist school; that is, in comparison with the Jevons-Walras tradition based on “price-taking behavior” (Böhm-Bawerk, [1888] 1891, bk. IV; Menger, [1871] 1976, ch. V; von Wieser, [1889] 1893, bk. II, ch. I). To understand this subtlety of the Austrian school vis-a-vis the classical school requires some understanding of an old articulation of value theory that runs from Adam Smith to Alfred Marshall in terms of: (1) the problem of the nature and origin of value (pertaining notably to the ultimate cause of value: utility or cost); (2) the problem of the measure of value (or the possibility of an absolutely invariable value standard); and (3) price theory proper (about market price formation). A great deal of the controversies on value, first within the classical school (thus Ricardo and Say debated on the origin of value; and Ricardo and Malthus, on the measure of value); or opposing the early marginalists to the classical school (on the origin of value) pertain mostly to the first two of the three value problems. There is widespread consensus across both schools on the fact that market price is determined in a market competition of supply and demand (hence by both utility and cost): “when prices are said to be determined by demand and supply, it is not meant that they are determined either by the demand alone, or by the supply alone, but by their relation to each other” (Malthus, 1820 [1836], p. 62). “Almost all writers have agreed substantially, and have rightly agreed, in founding exchangeable value upon two elements: power in the article valued to meet some natural desire or some casual purpose of man [utility], in the first place, and, in the second place, upon difficulty of attainment [cost]. These two elements must meet, must come into combination, before any value in exchange can be established” (De Quincey, 1844, p. 13). The notion that the classical school reduced value to the supply side (labor, cost) is probably due to a reduction of the classical school to Ricardo’s formulation, who indeed can be said to have downplayed the role of demand to short-run market price fluctuations; Ricardo was, in fact, thereby opposing a widespread supply-and-demand view: “The opinion that the price of commodities depends solely on the proportion of supply to demand, or demand to supply, has become almost an axiom in political economy, and has been the source of much error in that science” (Ricardo, 1817 [1821], Ch. XXX, pp. 460–461); in fact, Malthus (1820 [1836], p. 68) and Say (1828 [1852] pp. 361–362, 397), rectifying Ricardo’s reduction, restated the importance of demand, even regarding long-run value determination, and thus reformulating price theory closer to its original formulation by A. Smith (1776 [1904], ch. VII, pp. 58–59). However, the controversy on the origin of value is a different one than is often presumed: it was a debate over the original or ultimate cause of value, consisting of knowing which of the two causes (utility or cost) is the most primitive cause of value: labor (Adam Smith, Ricardo), utility (J.-B. Say), or marginal utility (the marginalists). Thus, similarly to most authors from Adam Smith to Alfred Marshall, it was Marshall who in effect reduced value theory to price formation theory. The Austrian economists articulated their value theory in two parts, broadly speaking. The first (“value theory” narrowly defined) deals with the rather philosophical problems about the nature and origin of value, (and this is where lies in this literature the well-known insistence on the subjectivity of value and the centrality of marginal utility): it is on this aspect where the Austrian opposition with the old school is perhaps the most pronounced, on the other hand, stands the Austrian theory of competitive market price formation (“price theory”), which takes as given the traders’ valuations of a commodity (as emphasized in Böhm-Bawerk’s example in the text), and explains how a competitive price emerges from these subjective valuations. (Compare Jevons-Walras formulation based on the utility function as the primitive concept.) On this important aspect, the Austrian tradition builds on the old, classical, view on competition. In this respect, the most significant shift that occurred during the marginal revolution was when Jevons and Walras, following Cournot’s ([1838] (1897) innovation, redefine competition as passive price-taking behavior, a reformulation that fits well the conception of supply and demand as optimal quantity choices at given prices, but which is hardly compatible with rivalrous “higgling and bargaining” market behavior.

To simplify the notation, we will not use a vector transpose except to avoid the risk of confusion.

The correlation condition can be written in terms of other correlation measures; for example, rank correlations or cosine similarity.

From the narrow perspective of the sellers, those most capable of selling low would want to limit the entry of other sellers and to command a higher demand price. Symmetrically, from the narrow perspective of the buyers, those whose circumstances enable them to buy high would want to limit the entry of other buyers and to command a lower supply price. Hence, if the market is competitive as required in classical economics, neither of these interests will get their way, causing an increase in the wealth of nations.

It was in this more familiar form, which goes back to Marshall (who named it consumers’ and producers’ surplus), that the minimization principle was invoked in the early market experiments, where it was called “virtual” rent or surplus (V. L. Smith, 1962, p. 132). The qualification “virtual” or “potential” (traditionally missing) is again crucial.

Throughout the neoclassical school, from the earliest treatises (Walras, 1874b [1896, 2014], lesson 37, pp. 451–454) to the modern textbooks (where the reservation-price, or “Marshallian” view, tends to be reduced to a quasilinear utility function), the reservation price is often regarded as a particular instance of utility theory, applicable when wealth effects are negligible, thus viewed as too specific for a general price theory and relegated to the status of an approximate welfare measure. See, e.g., Walras’ critique of Dupuit’s memoirs, which Walras considered to ignore wealth and related goods: “In general, the maximum pecuniary sacrifice that a consumer is willing to make to obtain a unit of a product depends not only on the utility of the product, but also on the utility of all the other products that are in the market, and also, finally, on the consumer’s means [wealth]. […] Let us just say that it is appropriate to put entirely aside all the considerations in Dupuit’s two memoirs relative to the variation of utility with the prices and with the quantities demanded at these prices” (Walras, 1874b [1896, 2014], p. 454). Actually Dupuit acknowledged the dependence of reservation price on taste, wealth (whose distribution is an important determinant of classical market demand, as we emphasize in Chapter 6), and related goods: “Ce prix [a person’s reservation price for a kg of meat] sera-t-il le même pour toutes les personnes? évidemment non. Car non-seulement ce prix dépend de la fortune [wealth] de cette personne, comme le fait remarquer M. Bordas, mais de son goût [taste] pour la viande, de sa faim, du prix des autres denrées alimentaires et de mille autres circonstances imposs ibles à énumérer d’une manière complète; mais toutes ces circonstances n’empêchent pas que ce prix n’existe pour chaque objet, pour chaque personne et à chaque instant” (Dupuit, 1849, p. 184). Dupuit was taking wealth and related goods as given merely for simplicity. Of course Marshall also, who formalized the classical methodology, showing that economics needs not go beyond monetary expression of economic motive (Marshall, 1890 [1920], bk. I, ch. II), nonetheless contributed inadvertently to downplaying the demand-price/supply-price (Marshall’s names for willingness to pay/willingness to accept) approach to price theory, through his famous clause of marginal utility of wealth, for which he made an adjustment within a utility-function view (Math Appendix, Note I and VI). Yet Marshall actually argues that (1) variations of marginal-utility of wealth at the individual consumer’s level are relatively minor effects that “belong to the second order of small quantities” (bk. III, ch. VI, §4, footnote 1, p. 132); (2) moreover, these effects are inconsequential on the aggregate of many consumers, poor and rich combined, their overall effect “seldom needed in considering large groups of people” (Marshall, 1890 [1920], marginal summary, p. 132): “if we take averages sufficiently broad to cause the personal peculiarities of individuals to counterbalance one another, the money which people of equal incomes will give to obtain a benefit or avoid an injury is a good measure of the benefit or injury” (bk. I, ch. II, §2, p. 18); “such differences between individuals might be neglected, since we were considering in either case the average of large numbers of people” (bk. III, ch. VI, §3, footnote 2, p. 130).

Qualitative versus quantitative: this distinction is understood as used in the theory of dynamical systems: the qualitative approach to dynamics, originating in the major contribution of H. Poincaré, but also Lyapunov (1892) studies the long-run overall behavior of a dynamical system (all solutions of the dynamics treated as a whole), without having to solve explicitly for a trajectory of the system, whereas the quantitative analysis investigates details of the behavior of a specific trajectory, investigating specific aspects such as the speed of convergence.

This should not be confused with the classical definition of “free competition,” a market situation when no barriers (natural or artificial) constrained entry in the market, so that supply is potentially so large (compared to demand) that only the lowest-cost firms will eventually survive the intense rivalry on the supply side, hence price converges to the minimum possible price: the minimum production cost across all firms.

Bertrand’s review, to be more precise, pertains more specifically to Cournot’s monopoly and duopoly theories in his Researches (Cournot, 1838 [1897], ch. V–VII) and Walras’ exposition of competitive market price formation in one of his early memoires, (Walras, 1883: memoire I, 1873, secs. III–IV, pp. 12–18) Crucially, Walras’ exposition of price formation under review pertains to partial equilibrium, based on supply and demand curves as given, as primitive concepts, operationalized in terms of market or limit orders to buy and sell, as in finance, and therefore using more or less explicitly, limit, or reservation, prices as primitive concepts. (It is in the following section that Walras derives the demand and supply curves from a utility function.) Thus, there is no mention of “wealth effects” in Bertrand’s important objection. Bertrand is even more explicitly presuming none other than a situation in which orders to buy and sell within limit prices, are taken as given. (One can check that throughout the neoclassical school, realistic explanations of the price mechanism invoke none other than reservation prices, not utility functions, as we emphasized earlier for the Austrian marginalists, and for the case of Marshall also.) However, in trying to solve the specific path-dependency problem emphasized by Bertrand, Walras, most likely, realized the full scope of the path-dependency problem (e.g., due to wealth effects), and, even more importantly, the fundamental problem of reconciling disequilibrium price adjustment with price-taking utility maximization behavior, leading him to assume away disequilibrium dynamics.

To see this, argues Bertrand, imagine that all the buyers (sellers) in a market are replaced by a single one whose is willing to buy (sell) the same as the actual buyers (sellers) combined; then the market situation is changed (although the market demand and supply schedules are the same), because the buyer-seller matching will most likely be affected (Bertrand, 1883, p. 505).

Notice that the allocation vector q involves only the integer numbers −1, 0, and 1. One can show from linear programming theory that that the competitive equilibrium allocation thus defined is indeed integral: technically, the trade, namely a pairing of buyers and sellers (or the allocation problem), can be viewed as a matching in a bipartite network (buyers versus sellers) and the corresponding linear program yields indeed an integral solution (Birkhoff, 1946; Dantzig, 1951). In fact, even more simply, the complementary slackness condition associated.

In the modern commentaries, it is of course the third, wealth effects, that are more particularly emphasized in economic theory, although the partial-equilibrium version also is well-recognized very early in the experimental literature.

This view found support, previously unrecognized, in the first market experiments based on fixed individual wtp (wta) reservation price distributions (V. L. Smith, 1962). But it is implied more generally in experiments reported by Crockett et al. (2021), who carefully vary prices and budget endowments and directly observe people’s quantity responses (revealed demand) for two goods (Arrow securities) for which people are presumed to have coherent preferences, based on their attitudes toward risk. The revealed individual preferences are diverse, noisy, unsystematic, and even incoherent. Put these same people, with their measured badly-behaved preferences, into a market and they are quite effective in finding equilibrium prices, sales, and purchases in the same environment in which the preferences were measured. Hence, prices consistent with quantity “preferences” are discovered in the socio-economic context of market interaction. This strength, general truth, and property of markets are studied narrowly within the context of the abstract Walrasian market mechanism, a theorist’s mechanism, unnatural and foreign to economic practice. We claim that the principle demonstrated by these authors has far wider implications than an exercise by theorists for theorists in the artificial Walrasian context. What is unsatisfactory, unproven, and in doubt, we believe, is the neoclassical-maintained hypothesis that market stability and efficacy depend on and require pre-existing well-behaved individual utility preferences. The latter is an imaginary neoclassical construct, that the Crockett, et al. (2021) paper shows is inconsistent with what real people can systematically reveal, but who nevertheless are not thereby handicapped in markets. This suggests a knowledge-how in markets, that is analogous to the inarticulable knowledge of the craftsman, transferable only through the apprenticeship of experience-based learning.