Abstract
This chapter restates more formally and works through all proofs for the theory of classical competitive market price formation as it applies to a single market. Goods and services are demanded (supplied) in discrete units, where to each unit demanded corresponds a consumer’s valuation (producer’s cost), which are reservation values expressed in units of a monetary medium of exchange (for example, dollars). We take seriously the discreteness of the commodity, treating the case of smooth supply and demand merely as a simplified, specific case of the theory, as a model of a large market. Throughout this chapter the distribution of reservation values (potential prices) is treated as data, that are fixed and given for a period and during static sequential replications of a period, as in typical laboratory market experiments. The theory is generalized to multiple markets in Chapter 6, where, as in classical economics, wealth is explicitly treated as one of the determinants of reservation prices in market realizations.
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Notes
- 1.
Multiple units may belong to the same trader.
- 2.
We clarify underlying issues concerning (5.4) by inserting two remarks: (1) although profitable contracts are contained in this interval, trial bids and asks in the higgling message space often fall outside it because traders have only private value information; hence, their higgling message space must be larger than their contract space to allow them to learn the feasibility limits on their desire to buy cheap and sell dear; each person’s choice is constrained by what others will voluntarily permit in their own interest. (2) Allowing the interval to be empty permits recognition that there are markets representing boundary solutions with no gains from trade (zero contracts), but with end-states of higgling revealed in a Bid = max (v) < Ask = min(c) interval that is informative in identifying which items are not goods, and how near or far they are from being goods. In this way, our generalization of the classical model formally recognizes, and incorporates, Hayek’s critique of neoclassical price theory that “it starts from the assumption of a ‘given’ supply of scarce goods. But which goods are scarce goods, or which things are goods, and how scarce or valuable they are—these are precisely the things that competition has to discover” (Hayek 1984, p. 256). Thus, a market, in which (5.4) is empty, identifies a thing that is not a good; the width of the bid-ask spread measures “how scarce” it is, in the sense that the greater the spread the more remote from becoming a good. Moreover, this interval signals the potential effect of an increase in WTP demand in expanding the set of goods and/or the potential reward to cost reducing innovation.
- 3.
Following standard custom in information theory, we take the base-2 logarithm and hence measure information in bits. Of course, this is merely a convention.
- 4.
For future reference, notice that \(V(0) = \sum\nolimits_{i} {v_{i} } ,\) the sum of buyers’ values.
- 5.
In fact, the invariance principle allows for a finer specification of the competitive attractor: notably, only the invariant subsets of \(V^{ - 1} (\mu )\) are relevant. For our purpose, however, Proposition 5.1 suffices.
- 6.
Theorem 5.1 might be extended, we believe, to a more general case of constrained competition, with appropriate proviso such as a so-called constraint qualification, as developed in the theory of convex optimization.
- 7.
To save space, we adopt a simplified integral notation in the proof, omitting dx, for example.
- 8.
In mathematical jargon, this property is sometimes referred to as “radial unboundedness.”
- 9.
On can go further in the analysis of \(C\) as generalizing the clearing set \(Z^{ - 1} (0)\) through the concept of generalized inverse. Embrechts, P., & Hofert, M. (2013). A note on generalized inverses. Mathematical Methods of Operations Research, 77(3), 423–432.
- 10.
- 11.
The first function is sometimes referred to in probability theory as the “mean excess function” or “mean residual life” function. Assume \(p\) in \(\Theta = [c_{\min } ,v_{\max } ]\) to avoid zero as denominator.
- 12.
Throughout, “smooth” means “differentiable with continuous derivative.”
- 13.
We are implicitly assuming bounded supports; otherwise, if \(c_{\max } = \infty\) or \(v_{\max } = \infty ,\) then the left and right limits are excluded, and we assume then \({\mathbb{E}}(c) < \infty\) and \({\mathbb{E}}(v) < \infty .\)
- 14.
- 15.
In the experiments, a buyer (seller) earns zero at price equal to their value (cost). The observations in the swastika experiments yield a value minus price distance measure of the minimum profit a buyer requires, and similarly for a seller. Uncertainty as to this profit implies that the attractor price can only be specified subject to this same uncertainty.
- 16.
It is not considered rational for a buyer to raise their own bid, or to “jump-bid” (raise the standing bid by more than the minimum bid), but it occurs in experiments. Such action hastens the end of the auction; if time is valuable, then such bids are not irrational. Transactions cost matter.
- 17.
The “English” auction has a colorful history. Its origin is not English; rather, it was introduced into England by the Roman occupation. Roman auctioneers followed and served the soldiers who, upon seizing any spoils-of-war property, received the proceeds from the auction conversion of the property into gold or currency. It was an ascending bid procedure, as suggested by the Latin auctiō, from the Latin verb augere, “to increase.” Further emphasizing their connection with warfare, auctions were conducted sub hasta, “under the spear” in old English. Thus, for example, even today in Spanish, “subhasta” refers to an auction or auction house (Cassady, 1967, pp 26–30).
- 18.
The literature on the game-theoretic approach to auctions is vast: for a review, see, e.g., the background to The Nobel Prize in Economics (2020),Committee, N. (2020). Improvements to auction theory and inventions of new auction formats: Scientific Background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2020 (The Committee for the Prize in Economic Sciences in Memory of Alfred Nobel, Issue., awarded to P. R. Milgrom and R.B. Wilson.
https://www.nobelprize.org/uploads/2020/09/advanced-economicsciencesprize2020.pdf
- 19.
Reminder: \(I(A)\) denotes the indicator of \(A,\) namely \(I(A) = 1\) if \(A\) is true, \(I(A) = 0\) if \(A\) is false.
- 20.
For a comparative study of multiple-unit Dutch versus English auctions see McCabe, K. A., Rassenti, S. J., & Smith, V. L. (1990). Auction institutional design: Theory and behavior of simultaneous multiple-unit generalizations of the Dutch and English auctions. The American economic review, 80(5), 1276–1283..
- 21.
This aggregation procedure also is invoked anytime there is a major news event affecting a stock that triggers a halt in trading. A call reopens trading.
- 22.
Notice that here we have not followed the notational convention of treating individually each value and cost unit; but rather we wrote the multiplicities or frequencies explicitly: this is merely to conform to ordinary practice here, for we will not carry for long the heavier notation including the frequencies.
- 23.
Hint: for demand, e.g., apply iterated expectations to \(I(b_{i} \ge p) =\)\(I(v_{i} \ge p + \varepsilon ).\)
- 24.
Buyer–seller direct haggling over price still survives for “big ticket” items such as houses, automobiles, expensive appliances, and antique furniture.
- 25.
One can have a quick overview of the much misunderstood but deep and rigorous classical vocabulary in Inoua, & V. L. Smith (2020). Adam Smith’s Theory of Value: A Reappraisal of Classical Price Discovery (To Appear in The Adam Smith Review, 2023). ESI Working Papers, 2020 (20–10). https://digitalcommons.chapman.edu/esi_working_papers/304/.
- 26.
These early experimental posted offer markets implement price with less flexibility than commonly associated with inventory management in large retail establishments. Thus, when slow moving items allow the inventory of perishables like milk to rise, the items are offered at a temporary discount from the accustomed posted price. Similarly, the rapid inventory stock-out of an item may lead to its being replenished at higher posted label prices. Classical higgling is thereby expressed in the form of rules for real-time price adjustment.
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Inoua, S.M., Smith, V.L. (2022). Price Formation: Partial Equilibrium. In: Economics of Markets. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-031-08428-7_5
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