Emergence of Price-Taking Behavior Sjur Didrik Flm Univ. Bergen. - - PowerPoint PPT Presentation

Emergence of Price-Taking Behavior

Sjur Didrik Flåm

• Univ. Bergen. Norway

CMS conf. Chemnitz, March 27-29, 2019

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 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?

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 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?

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 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.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 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) λ and λ(β − px) = 0. (opt. cond.) Non-standard form: with a money good g (gold), the actual x yields money margin λ = λ(x) := ∂u(x) ∂xg > 0, assuming u is C 1 in money good g. Posit p ∈ P(x) := ∂u(x) λ and β := px to get the same (opt. cond.) once again.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

The producer’s problem; same music

Standard form: given price p and technology X ⊂ X, maximize pˆ x subject to ˆ x ∈ X. Production plan x is optimal ⇐ ⇒ pˆ 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

• bjective u(·), and x an already committed plan, let

λ = λ(x) := ∂u(x) ∂xg > 0, (again with u C1 in money). Posit p := P(x) = ∂u(x) λ to get the same (variational ineq.) once again.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

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;

• ff equilibrium they don’t.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

Out-of-equilibrium trade, bilateral version

Agent i actually * holds some specific xi ∈ Xi ⊂ X, * operates (locally) with a personal price vector pi ∈ Pi(xi) - a scaled generalized gradient ∂ui(xi)/λi. * He trades with agent j, who holds xj ∈ Xj ⊂ X and has personal price vector pj ∈ Pj(xj). * Trade largely affected/determined by the price difference pi − pj.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

A two-agent direct deal

Agent i updates his holding: x+

i

:= xi + ∆ Likewise, for his interlocutor: x+

j

:= xj − ∆ where the transfer ∆ = sd for some step-size s > 0 along some direction d ∼ pi − pj with d ≤ 1 Repeated trade, evolving (in discrete time) over stages k = 1, 2..........

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

Feasibility?

For agent is update x+1

i

:= xi + sd d must be a feasible direction in the convex cone Fi(xi) := {d : xi + sd ∈ domui for small s ≥ 0} assumed closed. Likewise, x+1

j

:= xj − sd = ⇒ d ∈ −Fj(xj). Hence d ∈ Fij(xi, xj) := Fi(xj) ∩ [−Fj(xj)] Best slope σij(xi, xj) := sup

d

inf {(pi − pj)d : pi − pj ∈ Pi(xi) − Pj(xj) and d ∈ Fij(xi, xj) ∩ B} .

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

Incentives to trade?

Agents i, j shoud trade if their best slope σij(xi, xj) > 0. Otherwise, they shouldn’t! Then, they either see some common price: p ∈ Pi(xi) ∩ Pj(xj),

• r no fesible direction:

Fij(xi, xj) = ∅.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

Convergence?

Step-sizes sk > 0 should dwindle, but not too fast - say, sk ∼ 1/k. Directions ∼ price differences dk ∼ pk

i − pk j

with d ≤ 1 Which protocols (matching mechanisms)? Who meets/trades next with whom? Alternatives: (quasi-) Cyclic, periodic, randomly, or predicated by bid-ask spreads.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

Convexity? constraints? compactness? differentiability?

Convex preferences: each ui is concave. Constraints: xi ∈ closed convex Xi - but more general than orthants. Compact convex feasible domain. Differentiability: No objective needs be smooth in real goods. But, each ui must be continuously differentiable wrt money.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

Why non-smooth objectives?

Leontief type objectives. Diverse tariffs or production lines. Boundary choice. Objectives that arise from underlying programs - say LP.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 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.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

Existence: A constructive approach

• Theorem. (Existence of comp. eq.) Let

x = [xi] → λ = [λi] := [∂ui(xi)/∂xig ] ∈ RI

++.

For λi > 0, posit Ui(·) := ui(·)/λi X(x) := arg max

• ∑

i∈I

Ui(ˆ xi) : ∑

i∈I

ˆ xi = aggregate endowment

• Then there is a fixed point - and a competitive equilibrium allocation -

x ∈ X(x), supported by any price. p ∈ ∩i∈I ∂Ui(xi). (1) A constructive approach: Two agents (two vector components) at a time.

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19

On disequilibrium and personal prices

Out of equilibrium agent i has (under concavity) a personal price pi ∈ ∂Ui(xi). But, none is common: ∩i∈I ∂Ui(xi) is empty. In equilibrium: there is some p ∈ ∩i∈I ∂Ui(xi).

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 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?

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 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)

Sjur Didrik Flåm (Univ. Bergen. Norway) Emergence of Price-Taking Behavior CMS conf. Chemnitz, March 27-29, 2019 / 19