Via Order Markets Towards Price-Taking Equilibrium

Abstract

Can order markets lead participants towards price-taking equilibrium? Viewing market sessions as steps of iterative algorithms, this paper indicates positive prospects for convergence. Mathematical arguments turn on convolution, efficiency and generalized gradients. Economic arguments revolve around reservation costs, derived from indifference or threshold payments for quantities supplied or demanded.

1 Introduction

This paper relates order markets to price-taking alias competitive equilibrium. Linking the dynamics of the former to the steady state nature of the latter, the paper has four purposes:

first, to compare and contrast the trade process with its stationary points,

second, to emphasize the stability and welfare properties of order markets,

third, to stress the endogenous selection of equilibrium, and

fourth, to permit effects of agents’ endowments and inventories.

Motivation for focusing on order markets stems from their

$\star$:

growing importance [21] and noticed efficiency [2, 11, 14, 24],

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links to agent-based algorithms and computations [8],

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connections to classic concepts of competitive market equilibrium [9],

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use of money to dispense with transferable utility [17], and

$\star$:

accommodation of parties having less than perfect foresight or knowledge.

The paper presumes no knowledge of economic theory. For backdrop on order markets, note that some participants therein are price-proposers who seek to learn from repeated trades. By contrast, in competitive equilibrium, all parties are price-takers who need neither propose nor learn anything.¹

These glaring gaps reflect that order markets feature institutions and mechanisms whereas competitive equilibrium fully dispenses with all such auxiliaries—and even with money. Also, while real trade is a process—taking time and proceeding step by step—market equilibrium seems a steady state, fixed after trade. This paper makes a modest attempt to bridge the said gaps, linking dynamics to fixed points.

No issue is taken with various concepts of market equilibrium; they remain references. However, since transactions alter endowments, Debreu’s valuation equilibrium [5] fits better, as focal point, than would the Walrasian version—which keeps them fixed [17].²

Money enters as commodity here—and plays a chief role [15]. It doesn’t only describe rates of exchange and substitution in equilibrium—where they are common.³ Rather, while disequilibrium reigns, some rates still differ, whence drive transactions.

For simplicity, suppose just one commodity or asset be traded for money. On that premise, it’s shown below that the dynamics of limit order markets have valuation equilibria as steady states. If Pareto efficiency doesn’t prevail already, another market session brings Pareto improvement. Moreover, by iterated sessions, the agents may eventually solve a fixed point problem—one which no party ever addressed or stated.

Main novelties are two. First, Theorem 5.1 captures and quantifies convergence as iterative depletion of value added—seen here as potential or Lyapunov function. Second, Theorem 6.1 identifies each limit with Pareto efficiency and valuation equilibrium.

By emphasizing market microstructure [12], the paper connects to trade of financial securities, since long the object of many studies [2]. Here, however, the emphasis differs in several respects. The item traded could be an ordinary commodity. And, instead of equilibrium analysis [13], disequilibrium dynamics take center stage. Differences in valuations drive short-term trade and ensure long term stability. For that, construe the setting as sure and time-invariant; no exogenous information flow (private or public) impacts decisions.

No empirical inquiries are made or mentioned here [1]. I believe though, that doors open to numerical experiments and computer simulations.

The paper is planned as follows. Section 2 describes the market, featuring its orders, “book”, supply–demand curves and welfare measure. Section 3 invokes the endowment and preference relation of any generic agent—alongside money—to define his reservation curve in terms of indifference payments. Using one such curve for each agent, Sect. 4 formalizes the clearing of a market session. Section 5 analyzes convergence of iterated sessions. Section 6 identifies any limit with Pareto efficiency and valuation equilibrium. Section 7 briefly concludes.

The paper resides at the interfaces of convex analysis, fixed point theory and microeconomics. It addresses mathematicians, optimizers, operation researchers or computer scientists—concerned with links between convex criteria, decentralized procedures and duality [7, 22, 23]. Also addressed are economic theorists of market mechanisms [3], auctions [16] and agent-based behavior [20].

Notations and preliminaries: Any proper (cost) criterion $\chi \mapsto c(\chi )$, which maps ${\mathcal {X}}:={\mathbb {R}}$ into $\mathbb {R\cup } \left\{ +\infty \right\}$ has non-empty effective domain $domc:=\left\{ \chi \in {\mathcal {X}}:c(\chi )\in {\mathbb {R}}\right\} .$ Cost $c(\cdot )$ has a subgradient$\chi ^{*}$ at $\chi \in domc$—as conveyed by writing

$$\begin{aligned} \chi ^{*}\in \partial c(\chi )\text { iff }\chi \in \arg \max \left\{ \chi ^{*}\hat{\chi }-c(\hat{\chi }):\hat{\chi }\in {\mathcal {X}} \right\} . \end{aligned}$$

(1)

To stress symmetry and unburden notations, a starred item signals price; a non-starred variable stands for “physical” quantity. Accordingly, in (1), regard $\chi ^{*}$ as the unit price for the best quantity $\chi$. (1) holds iff $\chi$ realizes price-taking profit or value added:

$$\begin{aligned} c^{*}(\chi ^{*}):=\sup \left\{ \chi ^{*}\hat{\chi }-c(\hat{\chi }): \hat{\chi }\in {\mathcal {X}}\right\} . \end{aligned}$$

(2)

Equivalently, (1) is satisfied iff revenue $\chi ^{*}\chi$ splits into profit $c^{*}(\chi ^{*})$ plus cost $c(\chi )$:

$$\begin{aligned} \chi ^{*}\chi =c^{*}(\chi ^{*})+c(\chi ). \end{aligned}$$

Mirror-imaging cost is payoff or utility $u:\mathbb {R\rightarrow R\cup }\left\{ -\infty \right\}$ with supgradient

$$\begin{aligned} \chi ^{*}\in \hat{\partial }u(\chi )\text { iff }\chi ^{*}\in -\partial [-u](\chi ). \end{aligned}$$

(3)

Equivalently, (3) happens iff $\chi \in \arg \max \left\{ u-\chi ^{*}\right\} .$ While Fenchel conjugation (2) prices output, its counterpart (3) prices input, but both report price-taking profit.

2 Single-Good Order Markets

As prerequisites, this section describes submissions to an idealized, anonymous order market.⁴ It defines limit orders, stacked and stored in “the book”—to which a standard welfare measure applies.

Considered is exchange of just one good or asset for money. By assumption, either item is perfectly divisible, storable and transferable. No fees are charged, no frictions affect trade, and nobody bargains. Transactions proceed—often at high speed—on some platform(s), possibly electronic and internet-based. Execution or matching of orders might be automated and computerized. No participants have special privileges or roles. Readers familiar with order markets can skip this section or return to it later.⁵

A limit order$(\bar{\chi }^{*},\bar{\chi })\in {\mathcal {X}}^{*}\times {\mathcal {X}}:={\mathbb {R}}\times {\mathbb {R}}$ is a commitment, on the part of the anonymous submitter, to transact, at unit price $\bar{\chi }^{*}$, or better, any quantity $\chi \in [0,\bar{\chi } ],$ $\bar{\chi }>0.$⁶ Upon posting a limit ask or sell order, he precludes whatever sale at unit price $\chi ^{*}<\bar{\chi }^{*}$. Similarly, upon placing a limit bid or buy order, he will make no purchase at unit price $\chi ^{*}>\bar{\chi }^{*}$.⁷ By idealized hypothesis, each participant can change, post, split or withdraw limit orders, as many and often as he wants—to his heart, and for free.⁸

A limit order may have to wait—“in the book”—for execution. If executed, fully or partly, it’s matched then with a limit order from the other side of the market, or —more likely and rather swiftly—with a market order, mentioning no limit price $\bar{\chi }^{*}$, but a limit quantity $\bar{\chi }$.⁹

While some agents post limit orders, others sit on the fence, waiting for attractive matches, quickly executed as market orders. Who does what may vary. But invariably, agents remain anonymous. They sign or name no orders and behave in non-cooperative, non-coordinated manner.

The temporal resolution of “stage play” deserves emphasis: If entering first, submitters of limit orders act like “Stackelberg leaders”; market orders are “followers” [18]. Complying thus with Bertrand’s view on markets, this arrangement favors and foreshadows competitive outcomes [18].¹⁰

Some limit orders, being “outliers”—but nonetheless, in the book—can fail to fill before market clearing.¹¹ These may survive until trade resumes. Additional orders could arrive out of market hours and queue for the subsequent opening.¹²

To simplify, neither trade nor valuation is affected by uncertainty—apart maybe from random matching.¹³

The “book” displays anonymous, still standing limit orders—often in monotone manner, easily readable or visible. All participants see the same book and market conditions. For arguments and description, use lexicographic ordering on ${\mathbb {R}}^{2}$:

$$ \begin{aligned} (\chi ^{*},\chi )\preccurlyeq (\hat{\chi }^{*},\hat{\chi }) \Longleftrightarrow \chi ^{*}<\hat{\chi }^{*}\text { or otherwise }(\chi ^{*}=\hat{\chi }^{*}{ \& }\chi \le \hat{\chi }). \end{aligned}$$

(4)

Arrange posted limit ask or sell orders in a finite sequence

$$\begin{aligned} (\bar{\chi }_{1\uparrow }^{*},\bar{\chi }_{1\uparrow })\preccurlyeq (\bar{ \chi }_{2\uparrow }^{*},\bar{\chi }_{2\uparrow })\preccurlyeq \cdots . \end{aligned}$$

(5)

That sequence generates an inverse supply “order curve”

$$\begin{aligned} \chi \mapsto s(\chi ):=\min _{n}\left\{ \bar{\chi }_{n\uparrow }^{*}:\bar{ \chi }_{1\uparrow }+\cdots +\bar{\chi }_{n\uparrow }\ge \chi \right\} . \end{aligned}$$

(6)

$s(\chi )$ is well defined from $\chi =0$ up to aggregate supply $\bar{\chi }^{s}:=$ the sum of still standing, limit sell quantities. Thus emerges a stair-case, each step being the next jump upwards in price asked,¹⁴ Extend or fill in the steps to have inverse supply as increasing, maximal monotone correspondence $\chi \in [0,\bar{\chi }^{s}]\rightrightarrows s(\chi )\subset {\mathbb {R}}$ [25]. The associated “anti-derivative”

$$\begin{aligned} \chi \in {\mathbb {R}}\mapsto S(\chi ):=\left\{ \begin{array}{cl} \int _{0}^{\chi }s &{} \quad \text {when }\chi \in [0,\bar{\chi }^{s}], \\ +\infty &{} \quad \text {otherwise,} \end{array} \right. \end{aligned}$$

(7)

generated by Riemann integration, is convex with subdifferential $s=\partial S$ (1).¹⁵

Likewise, arrange posted limit bid or buy orders as a lexicographically ordered sequence $(\bar{\chi }_{1\downarrow }^{*},\bar{ \chi }_{1\downarrow })\curlyeqsucc (\bar{\chi }_{2\downarrow }^{*},\bar{ \chi }_{2\downarrow })\curlyeqsucc \cdots$ (4). What comes up thereby is a piece-wise constant, inverse demand “order curve”

$$\begin{aligned} \chi \mapsto d(\chi ):=\min _{n}\left\{ \bar{\chi }_{n\downarrow }^{*}: \bar{\chi }_{1\downarrow }+\cdots +\bar{\chi }_{n\downarrow }\ge \chi \right\} , \end{aligned}$$

(8)

well defined from $\chi =0$ up to aggregate demand $\bar{\chi } ^{d}:=$ the sum of all still standing, limit buy quantities.¹⁶ Again, extend or fill in the steps to have inverse demand as decreasing, maximal monotone correspondence $\chi \in \mathbb {[}0, \bar{\chi }^{d}]\rightrightarrows d(\chi )\subset {\mathbb {R}}$, with $d(\chi )=\emptyset$ iff $\chi \notin [0,\bar{\chi }^{d}].$ Its antiderivative

$$\begin{aligned} \chi \in {\mathbb {R}}\mapsto D(\chi ):=\left\{ \begin{array}{cl} \int _{0}^{\chi }d &{} \quad \text {when }\chi \in [0,\bar{\chi }^{d}], \\ -\infty &{} \quad \text {otherwise}, \end{array} \right. \end{aligned}$$

(9)

is concave with supdifferential $d=\hat{\partial }D$ (3).

While $d(0)\le s(0),$ and no market orders intervene, the trading venue is closed, dead or inactive. Otherwise, if $d(0)>s(0)$, and $d(\cdot ),$ $s(\cdot )$ extend beyond 0,  both “curves” change rapidly, parts being “picked off”—maybe in monotone manner—say, by matching highest bid with lowest ask, up to smallest quantum limit.

No attempt is made here to trace the evolution of the book. It suffices to note that, when $0<\min \left\{ \bar{\chi }^{s},\bar{\chi }^{d}\right\} <+\infty$ and $d(0)>s(0)$, swift trade will realize a clearing pair $(\chi ^{*},\chi )$ with $\chi >0:$

Definition 2.1

(market clearing). Any pair $(\chi ^{*},\chi )\in d\cap s$—that is, any price $\chi ^{*}$ and related quantity $\chi >0$ such that $\chi ^{*}\in d(\chi )\cap s(\chi )$—is said to clear the market.

Neither $\chi ^{*}$ nor $\chi$ needs be unique.¹⁷ Anyway, the function

$$\begin{aligned} \chi \mapsto W(\chi ):=\int _{0}^{\chi }(d-s)=D(\chi )-S(\chi ) \end{aligned}$$

(10)

is well defined and commonly used as welfare measure:

Proposition 2.1

(on welfare). Welfare $W(\cdot )$ (10) is maximal iff a total quantity $\chi$ clears the market. Then, any clearing price $\chi ^{*}\in d(\chi )\cap s(\chi )$ splits $\max W$ into consumers’ surplus $\int _{0}^{\chi }(d-\chi ^{*})=D(\chi )-\chi ^{*}\chi \ge 0$alongside producers’ surplus $\int _{0}^{\chi }(\chi ^{*}-s)=\chi ^{*}\chi -S(\chi )\ge 0$.$\square$

Clearing means end of market session—as is explained later.

To facilitate analysis, and maintain fewer curves, it’s expedient to balance supply against demand, letting respective amounts be endogenous. So, as convention, henceforth record proper supply as positive quantity and demand as negative. Thus, limit sell orders generate inverse market supply $0\le \chi ^{+}\mapsto S(\chi ^{+})=:C^{+}(\chi ^{+})$ (7). Likewise, limit buy orders generate inverse market demand $0\le -\chi ^{-}\mapsto D(-\chi ^{-})=:-C^{-}(\chi ^{-})$ ( 9). Let $\chi :=\chi ^{+}+\chi ^{-}$ denote excess supply. In these terms,

$$\begin{aligned} C(\chi ):=C^{+}(\chi ^{+})+C^{-}(\chi ^{-})=-W(\chi ){ (10) \ and \ }C(0)=-\max W. \end{aligned}$$

(11)

Diverse remarks conclude this section. None are essential; all can be skipped.

(On algorithmic trading). Order markets for financial assets have seen automated trading for at least two decades. Algorithms, aimed at high-frequency trading, easily out-compete manual traders on reaction time, speed, and information processing [4].(On intermediation). An arbitrager (broker or middleman) might offer the “highest” bidder markdown or discount $\Delta _{1\downarrow }\ge 0$ and the “lowest” seller markup or premium $\Delta _{1\uparrow }\ge 0$ per unit. If $\bar{\chi } _{1\downarrow }^{*}-\bar{\chi }_{1\uparrow }^{*}\ge \Delta _{1\downarrow }+\Delta _{1\uparrow },$ and the arbitrager matches those two orders, he might appropriate, for himself, value $\bar{\chi }_{1\downarrow }^{*}-\bar{\chi }_{1\uparrow }^{*}-\Delta _{1\downarrow }-\Delta _{1\uparrow }$ per unit transacted. Provided “the book” be open and readable—and the market likewise—this observation indicates that positive surplus W (10), if any, tends to dwindle rapidly. Put differently: no two-sided session—during which both curves $d(\cdot ),$ $s(\cdot )$ are visible with $d(0)>s(0)$—will last long. Thus, the most frequent scenario displays just one curve $s(\cdot )$ or $d(\cdot ),$ parts of which being rationally and sequentially picked off by market orders from the opposite side.(On matching and sharing). Matching and posting of limit orders allow manifold protocols and strategic opportunities. None are explored, singled out, or specified here. Consequently, it’s hard to identify how surplus or welfare be shared. In hindsight, various agents may appreciate some deals but regret others. However, as explained below, no transaction entails any set-back for the affected parties; it complies with agents’ incentives.(On “slopes” of order curves). The staircase form of (6) and (8) causes problems in defining slopes or elasticities. For numerical surrogates and their significance, see [6] and [21].(On fragmented markets). There might be several platforms, competing on reaction times, technology or visibility. For an idealized setting, suppose agents can switch cost-free from one platform to another.

3 The Agent’s Reservation and Order Curve

To prepare the ground, this section considers just one generic agent. He contemplates various exchanges of the commodity (or asset) for money. To each quantity corresponds an indifference amount of money. Such pairs, coupling quantity to threshold value, form an idiosyncratic reservation curve c—a function with extended real values. As argued below, from that function the agent can derive limit prices.

Formal arguments to this end invoke the agent’s actual endowment, his permanent preference relation, and the special role of money. Together these “items”—introduced next in that order—generate the said reservation curve. Finally, suitable subgradients (1) to the latter defines his limit order curve C.

The leading idea is that any quantity $\chi \in [0,\bar{\chi } ]$, traded at limit price $\bar{\chi }^{*}$, or better, should offer the submitter some "improvement"—and preferably, a strict one. That is, each transaction should be voluntary and comply with incentives.

The agent’s actual endowment is a pair $x=(r,\chi )\in {\mathbb {R}} \times {\mathcal {X}}=:\mathbb {X=R}^{2}$ of money reserves (or liquid bank roll) $r\in {\mathbb {R}}$ alongside his actual holding $\chi \in {\mathcal {X}}={\mathbb {R}}$ of the commodity in question. The “bundle” x affects his market behavior—as clarified next. Occasionally, x is updated—as discussed later.

The agent’s preference is fixed and represented by a binary relation $\succsim$ on ${\mathbb {X}}$. His last updated endowment $x\in {\mathbb {X}}$ belongs to the effective domain of $\succsim$—denoted $dom\succsim$ and supposed non-empty—iff the preferred set

$$\begin{aligned} \left\{ \hat{x}\in {\mathbb {X}}:\hat{x}\succsim x\right\} \end{aligned}$$

(12)

contains x but differs from ${\mathbb {X}}$. Consequently, relation $\succsim$ is reflexive on its domain, but not necessarily complete. By standing assumption, $\succsim$ is transitive. Define strict preference $\succ$ by

$$ \begin{aligned} \hat{x}\succ x\Longleftrightarrow \hat{x}\succsim x~ \& ~x~{\textit{not }}\succsim \hat{x}. \end{aligned}$$

Money serves as payment for the commodity in question. Being numeraire, money always commands unit price 1; see Proposition 5.2. So,

$$\begin{aligned} \text {any pair }x^{*}=(r^{*},\chi ^{*})\text { intended or used for valuation, has }r^{*}=1. \end{aligned}$$

(13)

The component $\chi ^{*}$ prices the commodity. Conversely, any such price $\chi ^{*}$ extends to a unique valuation regime $x^{*}=(1,\chi ^{*})$ on endowments $x=(r,\chi )\in {\mathbb {X}}$:

$$\begin{aligned} x\mapsto x^{*}x:=r+\chi ^{*}\chi . \end{aligned}$$

(14)

As in [15], following Keynes, money is a strictly desirable commodity in itself, meaning that for any $x\in dom\succsim$,

$$\begin{aligned} \left. \begin{array}{ccl} r>0 &{}\quad \Longrightarrow &{} (r,0)+x\succ x\text {, and} \\ &{} \quad &{} \\ \hat{x}\succ x &{}\quad \Longrightarrow &{} (-r,0)+\hat{x}\succsim x\text { for small enough }r>0. \end{array} \right\} \end{aligned}$$

(15)

The agent’s reservation curves are defined as follows. While holding endowment $x\in dom\succsim$ and contemplating to supply commodity quantity $\chi \in {\mathcal {X}}$, he would ask no less money than

$$\begin{aligned} a(\chi \left| x\right. ):=\inf \left\{ r\in {\mathbb {R}}:(r,-\chi )+x\succsim x\right\} . \end{aligned}$$

(16)

Similarly, upon demanding quantity $\chi \in {\mathcal {X}}$, he would bid no more money than

$$\begin{aligned} b(\chi \left| x\right. ):=\sup \left\{ r\in {\mathbb {R}}:(-r,\chi )+x\succsim x\right\} . \end{aligned}$$

(17)

Neither curve—ask $a(\cdot \left| x\right. )$ or bid $b(\cdot \left| x\right. )$—needs be known or reported to others. Note that

$$\begin{aligned} a(\chi \left| x\right. )=-b(-\chi \left| x\right. )=:c(\chi \left| x\right. ). \end{aligned}$$

(18)

So, from here onwards, to simplify and synthesize: demand is negative supply, and expense is negative revenue. Accordingly, the function $\chi \mapsto c(\chi \left| x\right. )$ (18) is henceforth chosen as unifying object, called the agent’s "theoretical" reservation curve. As concept, it reflects his (typically hidden) indifference or threshold payments—all denominated in money and affected by his actual endowment x. As convention: proper supply is recorded as positive quantity, demand as negative.

Henceforth assume that each preferred set (12) comes closed convex. Also, by hypothesis

$$ \begin{aligned} c(\cdot \left| x\right. )>-\infty ~ \& ~\chi \in domc(\cdot \left| x\right. )\Longrightarrow \partial c(\chi \left| x\right. )\ne \varnothing ~{ (1).} \end{aligned}$$

(19)

Criterion $c(\cdot \left| x\right. )$ was defined on ${\mathcal {X}}= {\mathbb {R}}$. As many risk measures [10]—and like (14)—it extends quasi-linearly from ${\mathcal {X}}$ to ${\mathbb {X}}= {\mathbb {R}}\times {\mathcal {X}}$ as follows:

$$\begin{aligned} \begin{array}{lll} \hat{x} &{}:= &{} (\hat{r},\hat{\chi })\in {\mathbb {X}}\mapsto c(\hat{x}\left| x\right. ) \\ &{}:= &{} \inf \left\{ r: (r,0)-\hat{x}+x\succsim x\right\} =\hat{r}+c( \hat{\chi }\left| x\right. ). \end{array} \end{aligned}$$

(20)

The following easy result is stated without proof:

Proposition 3.1

(on indifference criteria). Since each preferred set (12) is closed convex, so is also the function $\chi \mapsto c(\chi \left| x\right. )$ (18).¹⁸$\square$

It simplifies arguments and notations to begin with the frequent and realistic situation in which the agent has posted merely one limit order:

Proposition 3.2

(on the agent’s first and leading limit order). Suppose he has posted just one limit order $(\bar{\chi } ^{*},\bar{\chi })$ subject to

$$\begin{aligned} \bar{\chi }^{*}\in \partial c(\bar{\chi }\left| x\right. )+{\mathbb {R}} _{+}{ \textit{ and } }\bar{\chi }>0. \end{aligned}$$

(21)

Then, any price $\chi ^{*}\ge \bar{\chi }^{*}$ and quantity $\chi \in (0,\bar{\chi }]$ “improves” his position. That is, $(\chi ^{*}\chi ,-\chi )+x\succsim x$—or equivalently: $c(\chi \left| x\right. )\le \chi ^{*}\chi .$

The proof is preceded by some comments. First, when $c(\cdot \left| x\right. )$ is differentiable at $\bar{\chi },$ inclusion (21) simply means that $\bar{\chi }^{*}\ge c^{\prime }(\bar{ \chi }\left| x\right. )$.

Recall that supply $\chi >0$ is furnished for threshold ask $a(\chi \left| x\right. )=c(\chi \left| x\right. )$ (16). By convexity, if the agent offers limit supply $\bar{\chi }>0,$ any price $\bar{\chi }^{*}\in \partial c(\bar{\chi }\left| x\right. )$ exceeds every counterpart $\chi ^{*}\in \partial c(0\left| x\right. )$. That is, $\bar{\chi }^{*}\ge \chi ^{*}.$

Turning the table, recall the convention that demand $\chi$ is negative supply. It would be satisfied by threshold bid $b(-\chi \left| x\right. )=-c(\chi \left| x\right. )$ (18). So, if a consumer who worships concave “utility” $u(\cdot \left| x\right. )=-c(\cdot \left| x\right. )$, quotes limit demand $-\bar{\chi }>0,$ then any margin $\bar{\chi }^{*}\in \hat{\partial }u(-\bar{\chi }\left| x\right. ):=-\partial [-u](-\bar{\chi }\left| x\right. )$ underestimates every $\chi ^{*}\in \hat{\partial }u(0\left| x\right. )$ (3). That is, $\bar{\chi } ^{*}\le \chi ^{*}.$

Proof of Proposition 3.2

When $\bar{\chi }>0,$ by convexity of the function $c(\cdot \left| x\right. )$ (Proposition 3.1), its secant over the interval $[0,\bar{\chi }]$ has slope

$$\begin{aligned} \frac{c(\chi \left| x\right. )-c(0\left| x\right. )}{\chi }\le \bar{ \chi }^{*}\text { when }\chi \in (0,\bar{\chi }]. \end{aligned}$$

In turn, this implies $c(\chi \left| x\right. )\le c(0\left| x\right. )+\bar{\chi }^{*}\chi .$ So, from $c(0\left| x\right. )\le 0$ it follows the desired inequality $c(\chi \left| x\right. )\le \bar{ \chi }^{*}\chi$. $\square$

Nothing prevents though, the agent from submitting several limit orders. Each being construed as ask or sell, these form a finite increasing sequence $(\bar{\chi }_{1\uparrow }^{*},\bar{\chi }_{1\uparrow })\preccurlyeq (\bar{\chi }_{2\uparrow }^{*},\bar{\chi }_{2\uparrow })\preccurlyeq \cdots$ (4), forming his practical limit supply order curve (5)—visible whenever he has posted such orders. If some of his orders are executed , then naturally, in sequential, monotone manner. First hit is the leading one $(\bar{\chi }_{1\uparrow }^{*},\bar{\chi }_{1\uparrow })$. If fully filled, $(\bar{\chi }_{2\uparrow }^{*},\bar{\chi }_{2\uparrow })$ is picked up next—and so on.

After any order execution, full or partial, the agent holds a new endowment. Immediately thereafter, he updates his holding—say, to x—and his limit orders. Consequently, extending (21), he invariably cares that

$$\begin{aligned} \bar{\chi }_{n\uparrow }^{*}\in \partial c(\bar{\chi }_{1\uparrow }+\cdots +\bar{\chi }_{n\uparrow }\left| x\right. )+{\mathbb {R}}_{+}{ \textit{ for } }n=1,2,.. \end{aligned}$$

(22)

Thus, conditioned by his last updated endowment x, he posts elementary limit orders

$$\begin{aligned} C_{n}(\chi \left| x\right. ):=\left\{ \begin{array}{ll} \bar{\chi }_{n}^{*}\chi &{} \quad \text {if }\chi \in conv\left\{ 0,\bar{\chi } _{n}\right\} \\ +\infty &{}\quad \text {otherwise.} \end{array} \right. \end{aligned}$$

(23)

These convolute to form his personal order curve:

$$\begin{aligned} \left. \begin{array}{rl} C(\chi \left| x\right. ):= &{} \inf \left\{ \sum _{n}C_{n}(\chi _{n}\left| x\right. ):\sum _{n}\chi _{n}=\chi \right\} \\ &{} \\ = &{} \inf \left\{ C_{1\uparrow }(\chi _{1}\left| x\right. )+\cdots +C_{n\uparrow }(\chi _{n}\left| x\right. ):\chi _{1}+\cdots +\chi _{n}=\chi \right\} . \end{array} \right\} \end{aligned}$$

(24)

In view of (11), this curve equals the integral over conv$\left\{ 0,\chi \right\}$ of his limit orders, lexicographically arranged by (4).

Granted (21), or more generally (22), it follows forthwith that the agent’s order curve $C(\cdot \left| x\right. )$ (24) stays above his reservation curve $c(\cdot \left| x\right. )$ ( 18):

Proposition 3.3

(on one-sided order curves). The agent’s order curve (24) is closed, convex and piece-wise linear. By (22) he runs no the risk of being picked off at “wrong” prices. In fact, at any time,

$$\begin{aligned} C(\chi \left| x\right. )\ge c(\chi \left| x\right. ){ \textit{ for all }\ }\chi . \end{aligned}$$

(25)

Inequality (25) is strict, whence strict improvement obtains, over stretches where the reservation curve $c(\cdot \left| x\right. )$ (18) is strictly convex. $\square$

For any single order $(\bar{\chi }^{*},\bar{\chi })$ inclusion ( 21) links a prudent limit price $\bar{\chi }^{*}$ to a limit quantity $\bar{\chi }$. (22) allows multiple orders. Yet, in the first place, how might the agent choose his limit quantity $\bar{\chi }$?

This question bears on strategic behavior, not explored—and on eventual order costs, ignored throughout. It brings up issues as to continuous trading, "ticks" in price-quantities, and batch versus single order arrivals. None of these issues are taken up here. It suffices to notice that a finer grid on quantities tightens the gap $C-c\ge 0$ (25), if any. Indeed, it follows from the convexity of c that using more secants gives closer approximation:

Proposition 3.4

(on ticks in quantities). Suppose the agent contemplates to replace his list L (4) of limit orders with a refined list $\hat{L}\supset L.$ That refinement generates an order curve $\hat{C}$ (24) which satisfies$C\ge \hat{C}\ge c.$ $\square$

The subsequent section considers one trading session. Crucial there is the last limit order $(\bar{\chi }_{n\uparrow }^{*},\bar{\chi } _{n\uparrow })$ the agent submits during the session—say, just before clearing or closure. Then, for execution and profitable fill, it appears reasonable that (22) be “tightened” to

$$\begin{aligned} \bar{\chi }_{n\uparrow }^{*}\in \partial c(\bar{\chi }_{1\uparrow }+\cdots +\bar{\chi }_{n\uparrow }\left| x\right. ). \end{aligned}$$

Thus, after his last update, having “supplied” altogether $\chi :=\bar{\chi }_{1\uparrow }+\cdots +\bar{\chi }_{n\uparrow }$, the agent ultimately trades at a margin of his. That is,

(26)

4 A Market Session

It’s time now to accommodate an ensemble I of agents. I is fixed and finite, not necessarily large, but certainly, $\#I\ge 2$. Member $i\in I$ has a time-invariant preference relation $\succsim _{i}$.

Trade is a punctuated process. It proceeds in discrete, endogenous time steps—and unfolds during repeated but non-overlapping intervals. This section considers just one such interval, called a session. The main issue is here: in what manner might a session end?

Agent $i\in I$ enters the session with some endowment $x_{i}\in dom\succsim _{i}$ and exits with some updated version $x_{i}^{+1}\succsim _{i}x_{i}$. Most likely, neither his endowment nor his reservation curve needs be known or tracked by others. During the session, agent i may have traded several times. His step-wise transactions yield a chain

$$\begin{aligned} x_{i}\precsim _{i}\cdots \precsim _{i}x_{i}^{(k)}\precsim _{i}\cdots \precsim _{i}x_{i}^{+1} \end{aligned}$$

(27)

of steadily “improved” holdings. After any or some interim update—say, the k-th of these—agent i withdraws all orders, revises his reservation curve to $c_{i}(\cdot \left| x_{i}\right. )$, with $x_{i}=x_{i}^{(k)}$, and submits new orders, if any.¹⁹

As said, at any time, he may post several limit orders $(\bar{\chi } _{i1\uparrow }^{*},\bar{\chi }_{i1\uparrow })\preccurlyeq (\bar{\chi } _{i2\uparrow }^{*},\bar{\chi }_{i2\uparrow })\preccurlyeq \cdots$ each anonymously. Construed as sell, these define his order curve $C_{i}(\cdot \left| x_{i}\right. )$ as in (23), (24). It invariably satisfies $C_{i}(\cdot \left| x_{i}\right. )\ge c_{i}(\cdot \left| x_{i}\right. )$ (25).

To avoid cumbersome notations, let $x_{i}$ denote the endowment on which agent i last based his order and reservation curve $C_{i}(\cdot \left| x_{i}\right. )$ and $c_{i}(\cdot \left| x_{i}\right. )$ respectively.

For given endowment profile ${\textbf{x}}:=(x_{i})$, a reallocation $(\chi _{i})\in {\mathcal {X}}^{I}$ is declared feasible if $\sum _{i\in I}\chi _{i}=0$ and each $\partial c_{i}(\chi _{i}\left| x_{i}\right. )$ is non-empty.

Assumption 4.1

(on clearing and closure of a session). The order market clears, and the session closes, with a feasible reallocation $(\chi _{i})$ supported by some common price

$$\begin{aligned} \chi ^{*}\in \partial C_{i}(\chi _{i}\left| x_{i}\right. )\subseteq \partial c_{i}(\chi _{i}\left| x_{i}\right. ) \forall i. \end{aligned}$$

(28)

(28) reflects a simple economic feature: At clearing and closing time, presumably known or predicted, the commodity trades at one price which equals the marginal valuation of every party. Otherwise, by first economic principles, astute traders would execute profitable, last-second transactions. (28) also reflects another, more subtle feature, namely: the last limit order is executed to the full. This feature bears on agents’ short-term foresight. Through the session it might be imperfect, albeit not at the ultimate moment.

Assumption 4.1 links common pricing to efficiency—as is explained next by convoluting individual order curves:

Proposition 4.1

(on convolution of agents’ order curves). When the endowment profile ${\textbf{x}}=(x_{i})$ prevails, and agent i has submitted his order curve $C_{i}(\cdot \left| x_{i}\right. )$ (24), the resulting aggregate curve becomes an inf-convolution

$$\begin{aligned} \chi \in {\mathcal {X}}\mapsto C_{I}(\chi \left| {\textbf{x}}\right. ):=\inf \left\{ \sum _{i\in I}C_{i}(\chi _{i}\left| x_{i}\right. ):\sum _{i\in I}\chi _{i}=\chi \right\} . \end{aligned}$$

(29)

The latter is closed, convex, piece-wise linear and has $0\in domC_{I}(\cdot \left| {\textbf{x}}\right. )$.

Conversely, if some aggregate order “curve” $C:\mathbb { R\rightarrow R\cup }\left\{ +\infty \right\}$ is closed, convex and piece-wise linear—with pieces of minimal length $>0$ - on a bounded domain which contains 0,  then for some finite index set I and elementary orders (23), $C=C_{I}(\cdot \left| {\textbf{x}}\right. )$ (29).²⁰$\square$

Convolution (29) models efficient allocation across the members of I. In fact, if $i\in I^{+}\subseteq I$ supplies $\chi _{i}^{+}\ge 0$, then $C_{I^{+}}(\chi ^{+}\left| {\textbf{x}}\right. )$ $=S(\chi ^{+})$ equals the aggregate threshold compensation for total supply $\chi ^{+}:=\sum _{i\in I^{+}}\chi _{i}^{+}$ (7). Likewise, if $i\in I^{-}\subseteq I$ consumes $-\chi _{i}^{-}\ge 0$, $-C_{I^{-}}(-\chi ^{-}\left| {\textbf{x}} \right. )=D(-\chi ^{-})$ equals the aggregate threshold expense for total consumption $-\chi ^{-}:=-\sum _{i\in I^{-}}\chi _{i}^{-}$ (9).

Let $I:=I^{+}\cup I^{-}$ be a disjoint union. Posit $\chi :=\chi ^{+}+\chi ^{-}$ for excess supply and

$$\begin{aligned} C_{I}(\chi \left| {\textbf{x}}\right. ):=C_{I^{+}}(\chi ^{+}\left| {\textbf{x}}\right. )+C_{I^{-}}(\chi ^{-}\left| {\textbf{x}}\right. ). \end{aligned}$$

In view of Proposition 2.1 and (11), it holds

$$\begin{aligned} C_{I}(\chi \left| {\textbf{x}}\right. )=-W(\chi )\text { with } C_{I}(0\left| {\textbf{x}}\right. )=-\max W. \end{aligned}$$

The following result says that efficient allocation happens iff all terms have “equal margins”:

Proposition 4.2

(on market clearing and equal subgradients [9]). If commodity reallocation $(\chi _{i})$ clears the order market, meaning it solves (29) for $\chi =0$, then

$$\begin{aligned} \partial C_{I}(0\left| {\textbf{x}}\right. )\subseteq \cap _{i\in I}\partial C_{i}(\chi _{i}\left| x_{i}\right. ). \end{aligned}$$

(30)

Conversely, whenever $\sum _{i\in I}\chi _{i}=0$, it holds the turned-around inclusion

$$\begin{aligned} \partial C_{I}(0\left| {\textbf{x}}\right. )\supseteq \cap _{i\in I}\partial C_{i}(\chi _{i}\left| x_{i}\right. ). \end{aligned}$$

If moreover, $\cap _{i\in I}\partial C_{i}(\chi _{i}\left| x_{i}\right. )$ is non-empty, then $(\chi _{i})$ efficiently clears the market.

In particular, suppose every agent $i\in I$ submits one elementary limit order$(\bar{\chi }_{i}^{*},\bar{\chi }_{i})$ (23) for which the prices $\bar{\chi }_{i}^{*}$ differ. Then, if an efficient allocation $(\chi _{i})$ has some interior $\chi _{i}\notin \left\{ 0,\bar{\chi }_{i}\right\}$, the subdifferential $\partial C_{I}(0\left| {\textbf{x}}\right. )$ reduces to $\bar{\chi }_{i}^{*}$, and each agent $j\ne i$ makes a boundary choice $\chi _{j}\in \left\{ 0,\bar{\chi } _{j}\right\} .$ $\square$

Remarkably, Proposition 4.2 presumed no convexity, just presence of a shadow price $\chi ^{*}\in \partial C_{I}(0\left| {\textbf{x}} \right. )$ alongside some efficient reallocation $(\chi _{i})$ of assets. As upshot, the session clears and closes with a price which equals a common marginal value of holding the commodity:

Proposition 4.3

(on clearing and closing price). Suppose $C_{I}(0\left| {\textbf{x}}\right. )$ be attained by some feasible reallocation $(\chi _{i})$, and that $\partial C_{I}(0\left| {\textbf{x}}\right. )$ be non-empty. Then, market clearing and closure happens at some common price

$$\begin{aligned} \chi ^{*}\in \cap _{i\in I}\partial c_{i}(\chi _{i}\left| x_{i}\right. ). \end{aligned}$$

(31)

Proof

By Proposition 4.2, for any $\chi ^{*}\in \partial C_{I}(0\left| {\textbf{x}}\right. )$ it holds $\chi ^{*}\in \cap _{i\in I}\partial C_{i}(\chi _{i}\left| x_{i}\right. )$ (30). Now (31) follows from Assumption 4.1. $\square$

5 Convergence by Depletion of Value Added

By (31), any clearing price $\chi ^{*}$ belongs to $\partial c_{I}(0\left| {\textbf{x}}\right. ).$ Consequently, the price-taking aggregate profit or total value added (2) equals

$$\begin{aligned} V_{I}:=V_{I}({\textbf{x}}):=\sup _{\chi }\left\{ \chi ^{*}\chi -c_{I}(\chi \left| {\textbf{x}}\right. )\right\} . \end{aligned}$$

(32)

Together with individual values $i\mapsto V_{i}:=V_{i}(x_{i}):=c_{i}^{*}(\chi ^{*}\left| x_{i}\right. ),$ the said total satisfies

$$\begin{aligned} V_{I}=c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. )=\sum _{i\in I}c_{i}^{*}(\chi ^{*}\left| x_{i}\right. )=\sum _{i\in I}V_{i}. \end{aligned}$$

(33)

Since $c_{i}(0\left| x_{i}\right. )\le 0$, it holds $V_{i}\ge 0$ $\forall i$ (2). Thus, $V_{I}\ge 0$ comes up as a decomposed, monetary measure of total improvement. If strictly positive, it foreshadows that another market session might be worth the agents’ while.

On this ground, it’s tempting to ask: does aggregate value added $V_{I}=c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. )$ qualify as potential (alias Lyapunov function) for iterated order market sessions? That question is explored next. Its basis is already reinforced by the minimality of $V_{I}$:

Proposition 5.1

(on smallest value added). The above order market session, which clears at some price $\chi ^{*}$ (31), adds minimal value

$$\begin{aligned} V_{I}=c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. )=\inf _{\hat{ \chi }^{*}}c_{I}^{*}(\hat{\chi }^{*}\left| {\textbf{x}}\right. )=\inf _{\hat{\chi }^{*}}\sum _{i\in I}c_{i}^{*}(\hat{\chi }^{*}\left| x_{i}\right. ). \end{aligned}$$

Proof

$\chi ^{*}\in \partial c_{I}(0\left| {\textbf{x}} \right. )$ implies $0\in \partial c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. ).$ Hence $c_{I}^{*}(\cdot \left| {\textbf{x}} \right. )$ is minimal at $\chi ^{*}.$ The last equality derives from $c_{I}^{*}(\cdot \left| {\textbf{x}}\right. )=\sum _{i\in I}c_{i}^{*}(\cdot \left| x_{i}\right. )$. $\ \square$

It facilitates the subsequent convergence analysis that, for any economic agent, price-taking value added (2) relates to his minimal expenditure:

Proposition 5.2

(on linear pricing and minimal expenditure). Under any linear price regime $x^{*}=(r^{*},\chi ^{*})$ on ${\mathbb {X}}$, the price-taking value added (2 ):

$$\begin{aligned} c^{*}(x^{*}\left| x\right. ):=\sup \left\{ x^{*}\hat{x}-c( \hat{x}\left| x\right. ):\hat{x}\in {\mathbb {X}}\right\} \end{aligned}$$

cannot be finite unless $r^{*}=1$.²¹Then,

$$\begin{aligned} c^{*}(x^{*}\left| x\right. )=\sup \left\{ x^{*}(x-\hat{x}): \hat{x}\succsim x\right\} . \end{aligned}$$

(34)

$\hat{x}$ is a best choice in (34) iff it solves the problem of minimal expenditure:

$$\begin{aligned} {\mathcal {E}}(x^{*}\left| x\right. ):=\inf \left\{ x^{*}\hat{x}: \hat{x}\succsim x\right\} . \end{aligned}$$

(35)

Further,

$$\begin{aligned} c^{*}(x^{*}\left| x\right. )=x^{*}x-{\mathcal {E}}(x^{*}\left| x\right. ). \end{aligned}$$

(36)

Proof

Recall extension (20) and observe that

$$\begin{aligned} c^{*}(x^{*}\left| x\right. )&=\sup \left\{ x^{*}\tilde{x}-r \left| \hat{x}:=(r,0)-\tilde{x}+x\succsim x\text {, }r\in {\mathbb {R}}\text {, }\chi \in {\mathcal {X}}\right. \right\} \\&=\sup \left\{ x^{*}(x-\hat{x})+r(r^{*}-1):\hat{x}\succsim x\text {, }r\in {\mathbb {R}}\right\} \\&=\left\{ \begin{array}{ll} \sup \left\{ x^{*}(x-\hat{x}):\hat{x}\succsim x\right\} &{} \quad \text {hence} (34) \text {holds iff }r^{*}=1, \\ +\infty &{} \quad \text {otherwise} \end{array} \right. \\&=\left\{ \begin{array}{ll} x^{*}x-{\mathcal {E}}(x^{*}\left| x\right. ) &{} \quad \text {if }r^{*}=1, \\ +\infty &{}\quad \text {otherwise. \ } \end{array} \right. \end{aligned}$$

Agent i entered the session with some endowment $x_{i}\in dom\succsim _{i}$. If party to trades, he may—at least at clearing (27)—update $x_{i}$ by the rule

$$\begin{aligned} x_{i}^{+1}\leftarrow \cdots \leftarrow (r_{i},-\chi _{i})+x_{i}\succsim _{i}x_{i}. \end{aligned}$$

(37)

$r_{i}$ equals his reward for supplying commodity amount $\chi _{i}$ during the said trade. Chain (37) reflects that each transaction “improves” his position. Hence, by voluntary trade and the transitivity of $\succsim _{i},$ (37) holds, step by step, throughout the session; see (27). In particular, (37) remains valid for the agent’s holding $x_{i}$, with which he entered, and the endowment $x_{i}^{+1}$ with which he exited. Further, $\sum _{i\in I}(r_{i},\chi _{i})=(0,0)$ at each time.

Proposition 5.3

(on reduced value added). Another session of the order market adds less value. That is, with consecutive clearing prices $\chi ^{*+1}$ and $\chi ^{*},$ at endowment profiles ${\textbf{x}}^{+1}$ and ${\textbf{x}}$ respectively, it holds

$$\begin{aligned} V_{I}^{+1}:=c_{I}^{*}(\chi ^{*+1}\left| {\textbf{x}}^{+1}\right. )\le c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. )=:V_{I}. \end{aligned}$$

(38)

Proof

By (27) and (37) $x_{i}^{+1}\succsim _{i}x_{i}$ for each $i\in I.$ So, granted transitive $\succsim _{i},$ expenditures increases: ${\mathcal {E}}_{i}(\cdot \left| x_{i}^{+1}\right. )\ge {\mathcal {E}}_{i}(\cdot \left| x_{i}\right. )$ for all $i\in I;$ see (35). Consequently, writing $x_{I}:=\sum _{i\in I}x_{i},$ and $x^{*}=(r^{*},\chi ^{*})=(1,\chi ^{*})$ (13), inequality (38) follows from Propositions 4.3, 5.1 &2 and (36) by

$$\begin{aligned} c_{I}^{*}(\chi ^{*+1}\left| {\textbf{x}}^{+1}\right. )&=\inf _{ \hat{x}^{*}}\left\{ \hat{x}^{*}x_{I}-\sum _{i\in I}{\mathcal {E}}_{i}( \hat{x}^{*}\left| x_{i}^{+1}\right. )\right\} \\&\le \inf _{\hat{x}^{*}}\left\{ \hat{x}^{*}x_{I}-\sum _{i\in I} {\mathcal {E}}_{i}(\hat{x}^{*}\left| x_{i}\right. )\right\} =c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. ). \end{aligned}$$

The updated profile ${\textbf{x}}^{+1}=(x_{i}^{+1})$ (37) belongs to the “algorithmic” set

$$ \begin{aligned} {\textbf{A}}({\textbf{x}}):=\left\{ {\textbf{x}}^{+1}=(x_{i}^{+1})\in {\mathbb {X}} ^{I}:\text {conditions } (31) \& (37) \text { hold}\right\} . \end{aligned}$$

(39)

For any initial endowment profile $i\mapsto x_{i}^{0}\in dom\succsim _{i}$ let correspondence ${\textbf{A}}$ (39) operate on the non-empty, closed convex set of feasible allocations:

$$\begin{aligned} {\textbf{X}}:=\left\{ {\textbf{x}}=(x_{i})\in {\mathbb {X}}^{I}:x_{i}\succsim x_{i}^{0} \forall i\text { and }\sum _{i\in I}x_{i}=\sum _{i\in I}x_{i}^{0}\right\} . \end{aligned}$$

(40)

Theorem 5.1

(on steady depletion of value added). Let each preference relation $\succsim _{i}$ form a closed subset of $\mathbb {X\times X}$. Suppose the correspondence $\mathbf {x\in X\rightrightarrows \partial }c_{I}(0\left| {\textbf{x}}\right. )$ has non-empty values and compact graph. Then, each cluster point ${\textbf{x}}$ of a sequence ${\textbf{x}}^{k+1}\in {\textbf{A}}({\textbf{x}}^{k}),$ $k=0,1,...,$ (39), emanating from ${\textbf{x}}^{0}\in {\textbf{X}}$, satisfies

$$\begin{aligned} V_{I}=\inf \left\{ c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. ):\chi ^{*}\in \partial c_{I}(0\left| {\textbf{x}}\right. )\right\} =0. \end{aligned}$$

Hence each $V_{i}=0$ (33).

The proof is coached by Zangwill’s convergence theorem [26]. Consider a sequence $\left( \chi ^{*k},{\textbf{x}}^{k}\right)$ with $\chi ^{*k}\in \partial c_{I}(0\left| {\textbf{x}}^{k}\right. ).$ Posit $V^{k}:=c_{I}^{*}(\chi ^{*k}\left| {\textbf{x}} ^{k}\right. )$. From (38) follows that $V^{k}\searrow V$ for some limit $V\ge 0.$ By compactness, some subsequence $\left( \chi ^{*k},{\textbf{x}}^{k}\right) ,$ $k\in K,$ converges to $\left( \chi ^{*}, {\textbf{x}}\right)$ with $\chi ^{*}\in \partial c_{I}(0\left| {\textbf{x}}\right. ).$ Suppose $V>0$ and consider a subsequence ${\mathcal {K}} \subseteq K$ such that $\left( \chi ^{*k+1},{\textbf{x}}^{k+1}\right) \rightarrow _{{\mathcal {K}}}\mathbf {(}\chi ^{*+1},{\textbf{x}}^{+1})$ with $\chi ^{*+1}\in \partial c_{I}(0\left| {\textbf{x}}^{+1}\right. )$ and $c_{I}(\chi ^{*+1}\left| {\textbf{x}}^{+1}\right. )<V.$ This contradicts the fact that $V^{k+1}\rightarrow _{{\mathcal {K}}}V$. $\square$

6 Pareto Efficiency and Valuation Equilibrium

If $(\chi _{i})$ solves (29) with $\chi =0$, and $\chi ^{*}\in \partial C_{I}(0\left| {\textbf{x}}\right. )\subseteq \cap _{i\in I}\partial c_{i}(\chi _{i}\left| x_{i}\right. )$, then any monetary reward profile

$$\begin{aligned} i\in I\mapsto r_{i}\in [0,c_{i}^{*}(\chi ^{*}\left| x_{i}\right. )], \end{aligned}$$

on top of costs $c_{i}(\chi _{i}\left| x_{i}\right. )$, generates improvements: strict for those who receive $r_{i}>0$—and it ensures indifference for others. Thus, as long as $V_{I}>0$ (32), an order session enhances efficiency. Moreover, when $V_{i}=c_{i}^{*}(\chi ^{*}\left| x_{i}\right. )>0$ (33) for some $i\in I,$ if a part—say, $V_{i}/2$—of his added value be taken from him and distributed equally among all others, strict Pareto improvement remains over ${\textbf{x}}$. So, the following result verges on the obvious:

Proposition 6.1

(on Pareto efficiency). Suppose $\chi ^{*}\in \mathbf {\partial }c_{I}(0\left| {\textbf{x}}\right. )$. Then, $V_{I}=c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. )=0$ iff ${\textbf{x}}$ is Pareto efficient. The common desire for more money (15) obviates the distinction between weak and strong Pareto optima. $\square$

Beyond Pareto optimality, in view of the first welfare theorem [17], this section takes a larger, more significant step: It identifies limits of iterated order markets with market equilibria of price-taking sort. For this concept, recall from Proposition 5.2 that any linear valuation regime $\chi ^{*}\in {\mathcal {X}}^{*}$ extends to linear price $x^{*}=(1,\chi ^{*})$ on ${\mathbb {X}}.$

Definition 6.1

(valuation equilibrium [5]). A linear price $x^{*}$ cum feasible allocation $(x_{i})\in {\textbf{X}}$ (40) constitutes a valuation equilibrium in a pure exchange economy iff each agent $i\in I$ has already made a best choice within his actual budget $\beta _{i}:=x^{*}x_{i}.$ Formally,

$$\begin{aligned} \left\{ x_{i}^{+1}\succ _{i}x_{i}:x^{*}x_{i}^{+1}\le \beta _{i}\right\} { \textit{is empty for each }}i\in I{\textit{.}} \end{aligned}$$

(41)

This concept differs from that of Walras where initially specified endowments $x_{i}^{0}\in {\mathbb {X}}$ determine terminal budgets $\beta _{i}^{0}:=x^{*}x_{i}^{0}.$ Put differently: except by chance, the agent’s wealth in valuation equilibrium doesn’t equal its Walrasian version. Since order market sessions change holdings, it’s hard to find Walrasian outcomes realistic—unless, of course, out-of-equilibrium trade be precluded.

Theorem 6.1

(on efficiency and equilibrium). Suppose inf-convolution $c_{I}(0\left| {\textbf{x}}\right. )$ be attained for each ${\textbf{x}}\in {\textbf{X}}$. Then, a limit point $(\chi ^{*},\mathbf {x)}$ of iterated order sessions is a Pareto optimum iff it’s a valuation equilibrium as well.

Proof

As seen in Prop. 6.1, $(\chi ^{*},\mathbf {x)}$ qualifies as Pareto optimum iff

$$\begin{aligned} \chi ^{*}\in \partial c_{I}(0\left| {\textbf{x}}\right. )\text { and }V_{I}=c_{I}^{*}(\chi ^{*}\left| {\textbf{x}}\right. )]=0. \end{aligned}$$

(42)

In turn, (42) holds iff any efficient reallocation $(\chi _{i})$ entails

$$\begin{aligned} \chi ^{*}\in \partial c_{i}(\chi _{i}\left| x_{i}\right. )\text { and }c_{i}^{*}(\chi ^{*}\left| x_{i}\right. )=0 \forall i\in I. \end{aligned}$$

Now, let $x^{*}=(1,\chi ^{*})$. Further, consider one agent at a time, and invoke the next proposition to conclude. $\square$

Proposition 6.2

(on budget-constrained best choice). If $x\in dom\succsim$ and $c^{*}(x^{*}\left| x\right. )>0,$ then the affordable, strict upper level set $\left\{ \hat{x}\succ x:x^{*}\hat{x}\le x^{*}x\right\}$ can not be empty. Conversely, if that set isn’t empty, then, $c^{*}(x^{*}\left| x\right. )>0.$ In short, given budget $\beta :=x^{*}x,$ the set

$$\begin{aligned} \left\{ \hat{x}\succ x:x^{*}\hat{x}\le \beta \right\} { \textit{ is empty iff } }c^{*}(x^{*}\left| x\right. )=0. \end{aligned}$$

(43)

Proof

From (17) follows that

$$\begin{aligned} \sup \left\{ b(\check{x}\left| x\right. )-x^{*}\check{x}:\check{x} \in {\mathbb {X}}\right\} =c^{*}(x^{*}\left| x\right. ). \end{aligned}$$

(44)

So, if $c^{*}(x^{*}\left| x\right. )>0$, some $\check{x}\in {\mathbb {X}}$ satisfies $b(\check{x}\left| x\right. )-x^{*}\check{x}>0$, and thereby,

$$\begin{aligned} x^{*}[\check{x}-b(\check{x}\left| x\right. )(1,0)+x]<\beta =x^{*}x. \end{aligned}$$

Hence, for sufficiently small $r>0,$ the bundle $\hat{x}:=\check{x}-(b( \check{x}\left| x\right. )+r,0)+x$ commands cost $x^{*}\hat{x}\le \beta$. At the same time, from (15),

$$\begin{aligned} \hat{x}\succ \check{x}-b(\check{x}\left| x\right. )(1,0)+x\succsim x. \end{aligned}$$

Consequently, $\hat{x}$ is affordable (within budget $x^{*}x$) and strictly preferred to x.

For the converse, suppose some $\hat{x}$ is such a bundle. Then, by (15), for sufficiently small $r>0$, it holds $-(r,0)+(\hat{x}-x)+x\succ x$ and thereby $b(\hat{x}-x\left| x\right. )>0=b(0\left| x\right. )$. Then,

$$\begin{aligned} b(\hat{x}-x\left| x\right. )-x^{*}(\hat{x}-x)>b(0\left| x\right. )-x^{*}0\ge 0. \end{aligned}$$

In turn, by (44), this implies $c^{*}(x^{*}\left| x\right. )>0.$$\square$

In essence an order market models a pure exchange economy. Yet one may ask: Could producers also be accommodated—as in [5]? For indication, consider a generic one and let $c(\cdot \left| x\right. )$ be his (contingent) cost function. In valuation equilibrium there is a common price $x^{*}\in \partial c(x\left| x\right. )$. In view of (43), if $x^{*}\hat{x}$ strictly exceeds maximal profit $\beta =x^{*}x$, then $\hat{x}$ is infeasible.

7 Concluding Remarks

Pareto optima exist under weak assumptions as to compactness and continuity. Adding suitable convexity conditions (in the aggregate), valuation equilibria also obtain [17]. But existence proofs hardly explain whether or how agents could get thereto. On this, Theorems 5.1 and 6.1 indicate that order markets are constructive and expedient. Such institutions serve as mechanisms and "algorithms."

Market theory has long faced difficulties, and it still struggles, with disequilibrium dynamics. At the root of its problems lies the query: Why should every party always be a perfectly foresighted, price-taking optimizer [8]? Order markets sidestep this lack of realism. At such venues some agents propose prices and quantities.²² Off equilibrium, every asset or good, except money, can, in principle, come with as many prices as there are agents. Deals are decentralized, driven by the agents themselves, and executed out of steady states. No auctioneers, price-setters, resource dispatchers or system operators are required.

Moreover, utility functions need not be pegs on which formal analysis hangs. “Utility” hardly qualifies as primitive concept—and less so if construed as transferable. Instead, modulo a money commodity, preferences can substitute for utility functions, and reservation costs can stand in for transferability.

Some queries remain though. Can agents run out of money—or into debt? Why is money necessary? Answers may relate to stochastics. To wit, how would money-less agents fare on an unforeseen, rainy day? This issue points to the amenities provided by perfectly liquid cash [15].

It also appears that an “improved” second welfare theorem lurks in the background [17]—one which takes advantage of money. Arguments to that fall, however, outside the scope of this paper.