Emergence of Price-Taking Behavior Sjur Didrik Flåm Univ. Bergen. Norway CMS conf. Chemnitz, March 27-29, 2019 CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
From observations to the question Observations: * Off market equilibrium , prices differ. * Bid-ask spreads drive trade. * Prices are common (and unique) only in equilibrium. Question: How might common prices emerge? CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
A critique of microeconomic market theory Standard presentations presume throughout that * prices are common and posted, and * behavior is price-taking, perfectly optimizing. Thereby, theory gets off on the wrong foot. Where do prices come from? No room for adaptive, myopic and step-wise behavior? CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
The consumer’s problem - turned around Standard frame : Given already exogenous common price vector , find rational choice! Alternative frame : Given already endogenous choice, find rational price vector! That is, which prices could rationalize actual holding/ position? For the analysis, following Keynes : assume that money be a good in itself. CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Formal version - enters a money good Standard form of a consumer’s problem: given exogenous price p and budget β , maximize concave usc proper u ( ˆ x ) subject to pˆ x ≤ β . An optimal x yields budget/income margin λ > 0 such that p ∈ P ( x ) : = ∂ u ( x ) and λ ( β − px ) = 0 . (opt. cond.) λ Non-standard form : with a money good g (gold), the actual x yields money margin λ = λ ( x ) : = ∂ u ( x ) > 0 , ∂ x g assuming u is C 1 in money good g . Posit p ∈ P ( x ) : = ∂ u ( x ) and β : = px λ to get the same (opt. cond.) once again. CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
The producer’s problem; same music Standard form : given price p and technology X ⊂ X , maximize p ˆ x subject to ˆ x ∈ X . Production plan x is optimal ⇐ ⇒ p ˆ x ≤ px for all ˆ x ∈ X . That is, iff p ( ˆ x − x ) ≤ 0 for all ˆ x ∈ X . (variational ineq.) Non-standard form : given a money good g (gold), a more general objective u ( · ) , and x an already committed plan, let λ = λ ( x ) : = ∂ u ( x ) > 0 , ∂ x g (again with u C 1 in money). Posit p : = P ( x ) = ∂ u ( x ) λ CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19 to get the same (variational ineq.) once again.
Upshot so far Locally, each agent rationalizes his actual choice by a personal price vector . That vector is a scaled utility gradient (possibly generalized). The scale factor = inverse of money margin - a partial derivative (predicated on partial C 1 ) . Any personal commodity price = the substitution rate: commodity for money = an idiosyncratic money price. Intuition : In equilibrium price vectors coincide ; off equilibrium they don’t . CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Out-of-equilibrium trade, bilateral version Agent i actually * holds some specific x i ∈ X i ⊂ X , * operates (locally) with a personal price vector p i ∈ P i ( x i ) - a scaled generalized gradient ∂ u i ( x i ) / λ i . * He trades with agent j , who holds x j ∈ X j ⊂ X and has personal price vector p j ∈ P j ( x j ) . * Trade largely affected/determined by the price difference p i − p j . CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
A two-agent direct deal Agent i updates his holding: x + : = x i + ∆ i Likewise, for his interlocutor: x + : = x j − ∆ j where the transfer ∆ = sd for some step-size s > 0 along some direction d ∼ p i − p j with � d � ≤ 1 Repeated trade, evolving (in discrete time) over stages k = 1 , 2 .......... CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Feasibility? For agent i � s update x + 1 : = x i + sd i d must be a feasible direction in the convex cone F i ( x i ) : = { d : x i + sd ∈ domu i for small s ≥ 0 } assumed closed. Likewise, x + 1 : = x j − sd = ⇒ j d ∈ − F j ( x j ) . Hence d ∈ F ij ( x i , x j ) : = F i ( x j ) ∩ [ − F j ( x j )] Best slope σ ij ( x i , x j ) : = sup inf { ( p i − p j ) d : p i − p j ∈ P i ( x i ) − P j ( x j ) and d ∈ F ij ( x i , x j ) ∩ B } . d CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Incentives to trade? Agents i , j shoud trade if their best slope σ ij ( x i , x j ) > 0 . Otherwise, they shouldn’t! Then, they either see some common price : p ∈ P i ( x i ) ∩ P j ( x j ) , or no fesible direction : F ij ( x i , x j ) = ∅ . CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Convergence? Step-sizes s k > 0 should dwindle, but not too fast - say, s k ∼ 1 / k . Directions ∼ price differences d k ∼ p k i − p k � d � ≤ 1 with j Which protocols (matching mechanisms)? Who meets/trades next with whom? Alternatives: (quasi-) Cyclic, periodic, randomly, or predicated by bid-ask spreads. CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Convexity? constraints? compactness? differentiability? Convex preferences: each u i is concave. Constraints : x i ∈ closed convex X i - but more general than orthants. Compact convex feasible domain. Differentiability: No objective needs be smooth in real goods. But , each u i must be continuously differentiable wrt money. CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Why non-smooth objectives? Leontief type objectives. Diverse tariffs or production lines. Boundary choice. Objectives that arise from underlying programs - say LP. CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
What limits will obtain? Proposition : In each limit point no two agents have any incentive to further trade; they see a common price vector. When might all agents see a common price vector? Proposition : They do if * at least one agent makes interior choice and has diff. objective there, or if * for every good at least one agent has a unique partial derivative in that good. Proposition: If all see a common price, competitive equilibrium obtains. Equilibrium isn’t necessarily unique - and there can be path-dependence. CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Existence: A constructive approach Theorem. (Existence of comp. eq.) Let x = [ x i ] �→ λ = [ λ i ] : = [ ∂ u i ( x i ) / ∂ x ig ] ∈ R I ++ . For λ i > 0 , posit U i ( · ) : = u i ( · ) / λ i � � x i ) : ∑ ∑ X ( x ) : = arg max U i ( ˆ x i = aggregate endowment ˆ i ∈ I i ∈ I Then there is a fixed point - and a competitive equilibrium allocation - x ∈ X ( x ) , supported by any price. p ∈ ∩ i ∈ I ∂ U i ( x i ) . (1) A constructive approach: Two agents (two vector components) at a time. CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
On disequilibrium and personal prices Out of equilibrium agent i has (under concavity) a personal price p i ∈ ∂ U i ( x i ) . But, none is common: ∩ i ∈ I ∂ U i ( x i ) is empty. In equilibrium: there is some p ∈ ∩ i ∈ I ∂ U i ( x i ) . CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
Hindsight How can players arrive at equilibrium - if any? While underway, how much competence, coordination, and foresight is required? What are the roles of cognition, information, and perception? CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
References Feldman, Bilateral trading processes, pairwise optimality, and Pareto optimality, Rev Econ Studies (1973) Flåm & Gramstad, Direct exchanges in linear economies, Int. J. Game Th (2013) Flåm, Blocks of coordinates, stoch. programming, and markets, Comp. Manag. Science 16, 3-16 (2019) Flåm, Generalized gradients, bid-ask spreads, and market equilibrium, to appear Optimization (2019) Gode & Sunder, Allocative efficiency of markets with zero intelligence traders: markets as a partial substitute for individual rationality, J. Pol. Econ (1993) Shapley & Shubik, Trade using one commodity as a means of payment, J Pol Econ (1977) CMS conf. Chemnitz, March 27-29, 2019 Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior / 19
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