## 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.

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## Notes

- 1.
- 2.
“..

*money enters into the economic scheme in an essential and peculiar manner.*” J. M. Keynes,*General Theory*, p. vii. - 3.
If some agent has a boundary position or kinky objective, he holds an ensemble of price vectors—apt to generate bid-ask spreads on his part alone.

- 4.
Mertens (2003) refers to this feature as part of Shapley’s “textitwindows model.”

- 5.
- 6.
Claims concerning agents’

*bounded rationality*or*zero intelligence*appear condescending. - 7.
Coordinate methods have a long history in numerical analysis. Typically, they presume continuously differentiable (\(C^{1})\) objectives—as required in Eckalbar (1986) and Feldman (1973). Here, convergence obtains under close to minimal conditions. Unlike Necoara et al. (2017) and Nesterov (2012), Lipschitz continuous gradients are never needed.

- 8.
Broadly, provided preferences and technologies be closed convex, such an equilibrium exists (Arrow and Hahn 1971).

- 9.
This objection not withstanding, a central auctioneer could change prices (by tâtonnement) to reduce imbalances in aggregate demand. However, the prospects for convergence or stability are not generally good (Mandel and Gintis 2016).

- 10.
While the customer just implements

*market orders,*the shopkeeper has already unveiled*limit sell orders*on various commodities (Flåm 2016b). Most likely, \(\min \left\{ \bar{q} _{ic},\underline{q}_{jc}\right\} =0\) for many \(c\in C\). - 11.
Just window-shopping, meaning no trade \((q=0),\) is always possible. Note that a middleman might offer discount \(\delta _{ic}\ge 0\) to the buyer and markup \(\mu _{jc}\ge 0\) to the seller, thereby cutting off a slice \((\bar{p}_{ic}-\delta _{ic})-(\underline{p}_{jc}+\mu _{jc})>0\) per unit for himself.

- 12.
Indeed, either party could display some degree of autism. It is left open whether and how their behaviors are maximizing or based on common price perceptions.

- 13.
The transaction, just outlined, could be

*one**equilibrium move*in some non-cooperative, extensive-form game which features dynamic matching and bargaining. Gale (2000) goes considerable distance in that direction, thereby providing strategic foundations for competitive equilibrium. This paper is complementary; it only considers*disequilibrium*behavior. - 14.
This issue relates to discussion of “

*money illusion/neutrality*” and “*real balances*” (Patinkin 1965). - 15.
This paper may be read as depicting a spot market. It is open-ended though, if—as in finance or insurance—some of the objects traded are contracts written on future contingencies \(c\in C\).

- 16.
Extension to Hilbert space is immediate.

- 17.
A household

*H*may constitute a subset of*I*. Various members \(h\in H,\#H\ge 2,\) can act then in different capacities to form a joint business. Alternatively,*H*could refer to a single agent, coming forward in complementary roles—as consumer, producer, shopkeeper or trader —each specializing in some reduced set of commodities (Howitt 2005). - 18.
Agent

*i*may care about a relatively small subset \(C_{i}\subsetneq C\) of commodities. Then, quite likely, \(domu_{i}\) is part of \(\mathbb {R}^{C_{i}}\)—whence \(int(domu_{i})\) is empty;*each*\(x_{i}\in domu_{i}\) resides at the boundary, and the feasible cone (10) \(F_{i}(x_{i})=F(u_{i},x_{i})\) is “slim.” - 19.
- 20.
Money facilitates transactions when endowments or tastes are not complements. It suffices that agents differ in their willingness to substitute some good for money.

- 21.
A component \(q_{c}\ge 0\) stands for a

*c*-quantity which*i*consumes or demands. When \(q_{c}<0,\) he offers, produces or supplies the amount \(\left| q_{c}\right| \). - 22.
In particular, if \(x_{i}\in int(domu_{i})\) and \(p_{ic}\ne p_{jc},\) agent

*i*might propose that \((p_{ic}-p_{jc})d_{c}>0\). - 23.
Money is perfectly liquid; it never displays any bid-ask spread.

- 24.
Most likely, the protocol is affected by randomness (Flåm 2018), spatial separations or network topology. Such features are not explored here.

- 25.
Feldman (1973) let each matching produce a

*core solution*to a two-person, non-transferable utility, cooperative game. Thereby, he presumed—on the part of players—non-negligible competence or effort. The present approach avoids—and dispenses with—these restrictive presumptions. Unlike Howitt (2005) exchange is not organized by intermediaries or “shops,” and no cash is demanded in advance. - 26.
- 27.
Conditions (41) mention no initial holdings \(x_{i}^{0}\). Competitive equilibrium is declared

*Walrasian*if final “wealth” \(px_{i}\) equals \(px_{i}^{0}\). It seems improbable that an economic agent, after repeated out-of-equilibrium trades, in face of changing prices, preserves his fortune as valued ex post. - 28.
By the first welfare theorem, each equilibrium is Pareto optimal. However, only the uniform allocation maximizes the egalitarian welfare function \(\sum _{i\in I}u_{i}(x_{i})\).

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Flåm, S.D. Emergence of price-taking Behavior.
*Econ Theory* **70, **847–870 (2020). https://doi.org/10.1007/s00199-019-01232-5

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### Keywords

- Money commodity
- Bilateral exchange
- Market equilibrium

### JEL Classification

- C63
- D03
- D51