Emergence of price-taking Behavior

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Emergence of price-taking Behavior Sjur Didrik Flåm1 Received: 17 December 2018 / Accepted: 18 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract Price-taking behavior is the bedrock of much market theory. How might such behavior emerge? Addressing that old but still intriguing question, this paper uses a money commodity to denominate all rates of exchange and substitution. Out of equilibrium, some rates differ between agents, thereby driving trade. The simplest form of trade is bilateral; it needs no broker, center or supervisor. Yet, under broad conditions, that elementary institution can take the economy to competitive equilibrium. Proving convergence, this paper invokes minimal hypotheses as to agents’ behaviors, competences and deals. Moreover, agents may make boundary choices, appreciate relatively few commodities, choose within general domains and deploy non-smooth preferences. Convex analysis provides the chief tool kit. Keywords Money commodity · Bilateral exchange · Market equilibrium. JEL Classification C63 · D03 · D51

1 Introduction Price-taking behavior is a key assumption in much market theory—and fundamental for many of its conclusions. In competitive equilibrium, such behavior is rational and standard because prices persist, and they are unique. But off such equilibrium, some still change, or they differ—whence common versions are hard to take or predict. Then, how might prices evolve toward market-clearing levels?

No outside support was given to this project. Preceding support from Røwdes Fond is gratefully acknowledged. Sincere thanks are due a referee for constructive criticism and thorough comments.

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Sjur Didrik Flåm [email protected] Informatics Department, University of Bergen, Bergen, Norway

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S. D. Flåm

Since long, many studies have considered this important question.1 Addressing it anew, this paper invokes one special good, referred to as money (Alchian 1977; Starr 2003). Following Keynes, suppose each agent always retains some money— of which he invariably wants more.2 Accordingly, let all marginal rates of exchange and substitution be denominated in the money good. Thus, any agent’s rates reflect indifference payments or reservations values, depending on his actual holding. Stacked together, these rates constitute his personal price or valuation vector, local in nature and not necessarily unique.3 As shown later, when valuation vectors coincide, pricetaking equilibrium prevails. Heuristics and preview. Any personal price system pi , used by agent i, attaches a unit value tag pic to marketable commodity c. If pi is not unique, let pic be minimal among such tags. Similarly, for another agent j, let p¯ jc be maximal among his corresponding tags. Provided agent i bids more than agent j asks, meaning Bi j = pic − p¯ jc > 0,

(1)

then the first could buy some c amount from the second. If such transfer is the only one feasible, (1) amounts to a most elementary form of bilateral bid-ask spread. More generally, to allow simultaneous transfer of severa

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