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Price-Setting and Attainment of Equilibrium: Posted Offers Versus an Administered Price

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Price-Setting and Attainment of Equilibrium: Posted Offers Versus an Administered Price Munich Personal RePEc Archive Price-Setting and Attainment of Equilibrium: Posted Offers Versus An Administered Price Collins, Sean M. and James, Duncan and Servátka, Maroš and Woods, Daniel Fordham University, Fordham University, Macquarie Graduate School of Management, Purdue University 19 September 2017 Online at https://mpra.ub.uni-muenchen.de/81489/ MPRA Paper No. 81489, posted 21 Sep 2017 23:27 UTC Price-Setting and Attainment of Equilibrium: Posted Offers Versus An Administered Price Sean M. Collins, Duncan James, MaroˇsServ´atka and Daniel Woods September 19, 2017 1 PRICE-SETTING AND ATTAINMENT OF EQUILIBRIUM: POSTED 1 2 OFFERS VERSUS AN ADMINISTERED PRICE 2 3 3 4 Sean M. Collins, Duncan James, Maroˇs Servatka´ and Daniel 4 5 Woods 5 6 The operation of the posted offer market with advance production environ- 6 7 ment (Mestelman and Welland, 1988), appropriately parameterized, differs from 7 that of the market entry game (Selten and G¨uth, 1982), appropriately presented, 8 8 only in terms of price-setting. We establish the effect of this difference in price- 9 9 setting on attainment of the competitive equilibrium allocation while controlling 10 for effects relating to the presentation of the market entry game and to the 10 11 stationarity or non-stationarity of environment. Free posting of prices promotes 11 convergence to the competitive equilibrium allocation, while the typical market 12 12 entry game data can be characterized as displaying cycling prices. 13 13 14 14 15 How do markets equilibrate? What is responsible when they do not? We 15 16 generate insight on these questions by setting up a comparison of the mar- 16 17 ket entry game (Selten and G¨uth, 1982) and a posted offer with advance 17 18 production environment (Mestelman and Welland, 1988), hereafter denoted 18 19 as the POAP. We demonstrate that the POAP can be thought of as a non- 19 20 isomorphic relaxation of the market entry game, where the market entry 20 21 Collins: Fordham University, 113 W 60th St, New York, NY 10023, 21 22 [email protected]. James: Fordham University, 441 E Fordham Road, Bronx, 22 23 NY 10458, [email protected]. Serv´atka: Macquarie Graduate School of Manage- 23 ment, 99 Talavera Road, Macquarie Park, New South Wales 2113, Sydney, Australia, 24 24 [email protected]. Woods: Purdue University, 403 West State Street, West 25 25 Lafayette, IN, 47907, [email protected]. Funding was provided by University of 26 Canterbury. The Erskine Programme supported this research with a Visiting Erskine 26 27 Fellowship awarded to Duncan James to visit the University of Canterbury. This study 27 28 involved human subjects and received IRB approval from Fordham University. The 28 authors are grateful for comments from M. Battaglini, M. Isaac, B. K˝oszegi, C. Plott, 29 29 T. Tassier, an anonymous associate editor, and two anonymous referees. 1 workingpaper.cls ver. 2006/04/11 file: paper.tex date: September 19, 2017 2 1 game appears conversely as a market with advance production environ- 1 2 ment restricted to have an administered pricing rule—specifically a uniform 2 3 price that allows ex post market clearing—instead of freely and individu- 3 4 ally posted offers. This insight then allows the construction of experiments 4 5 which isolate the marginal effects of different design features, by means of a 5 6 sequence of incrementally varying designs. Empirically, we find different out- 6 7 of-equilibrium dynamics associated with the administered ex post market 7 8 clearing price rule versus posted offers, and more evidence of convergence to 8 9 the competitive equilibrium outcome given use of posted offers. Stationarity 9 10 of environment also aids equilibration. 10 11 The above results are demonstrated by data from our study. We generate 11 12 these data by implementing a sequence of treatments, beginning with the 12 13 market entry game in its original format. In our experiments, as in the prior 13 14 empirical literature, the market entry game generates volatile outcomes that 14 15 are generally inconsistent with complete adoption of pure strategy play, al- 15 16 though perhaps tempting to describe as “equilibrium plus noise”. From 16 17 there we alter the exogenous control variable from “capacity” (i.e. a pa- 17 18 rameter of the demand schedule) to marginal cost. We then build on that 18 19 by altering the presentation of the game (in previous literature, presented 19 20 as an algebraic payoff function) to make explicit the (previously implicit) 20 21 numerical demand schedule and the accompanying administered price rule, 21 22 i.e. ex post market clearing, both inherent in the market entry game. Each 22 23 of the experimental treatments listed so far introduces a single change in 23 24 design only, isolating the marginal effect of each change. Each change in 24 25 format and/or control variable as just described also preserves isomorphism 25 26 with the original implementation of the market game. 26 27 27 However, we then break with isomorphism by introducing a further treat- 28 28 ment, which introduces a second stage in which each subject nominates 29 29 his or her own price subsequent to entry. Individual posting of prices thus workingpaper.cls ver. 2006/04/11 file: paper.tex date: September 19, 2017 3 1 replaces the uniform ex post market clearing price rule embedded in the 1 2 immediately prior transformation of the market entry game; our sequence 2 3 of treatments thus terminates at a particular version of the POAP. 3 4 While the market entry game and the POAP are not isomorphic, it is 4 5 however the case that given pricing “via the demand curve”1 in the second 5 6 stage of the POAP (when prices are posted) the payoff function in the first 6 7 (advance production) stage of the POAP is exactly equivalent to the pay- 7 8 off function in the market entry game. In consequence there are subgame 8 9 perfect pure strategy equilibria in the POAP that have the same observ- 9 10 able outcomes, in quantities and prices, as the pure strategy equilibria in 10 11 the market entry game, in number of entrants and prices implicit to its ad- 11 12 ministered price rule. (In Appendix A, we demonstrate the preceding and 12 13 also delineate additional equilibria in the POAP which are not possible in 13 14 the market entry game; those additional equilibria are not exhibited by our 14 15 data.) 15 16 16 Does restricting the pricing possibilities, thereby reducing the number of 17 17 pure strategy equilibria relative to the POAP, allow the market entry game 18 18 to more quickly attain the competitive equilibrium allocation common to 19 19 both? Quite the opposite: we find that the POAP converges more rapidly 20 20 to the competitive equilibrium allocation than does the market entry game. 21 21 Additionally, outcomes in the market entry game appear not to be evidence 22 22 of mixed strategy use by the subjects, but rather an out-of-equilibrium 23 23 phenomenon, en route to an equilibrium in pure strategies (consistent on 24 24 this point with results from Duffy and Hopkins, 2005). We are also able to 25 25 advance understanding of the market entry game by identifying something 26 26 1 27 Pricing “via the demand curve” means that each seller nominates a price that is equal 27 28 to the price coordinate of the point on the demand curve where the quantity coordinate 28 is given by the units produced (i.e., number of sellers who have decided to produce one 29 29 unit) in that round. workingpaper.cls ver. 2006/04/11 file: paper.tex date: September 19, 2017 4 1 that it would seem is going on instead of mixing: cycling. 1 2 2 1. THE GAME, THE MARKET, AND THEIR PREDICTIONS 3 3 4 Introduced by Selten and G¨uth (1982), the market entry game is an n- 4 5 player simultaneous game where players decide between two strategies: enter 5 6 the market (IN) or stay out (OUT). Empirically, the game has been studied 6 7 with linear payoffs. We consider a specification that nests earlier work, where 7 8 player i’s payoff is 8 9 9 v, if player i chooses OUT, (1) πi = 10 10 v + r(c − m) − h, if player i chooses IN. 11 11 In this specification, m is the number of entrants, the parameters v, r, 12 12 and c, are positive integers, and h is a non-negative integer that satisfies 13 13 0 <h ≤ r(c−1). Following the literature, v may be interpreted as an outside 14 14 option or entry subsidy, c as the capacity of the market to support entrants, 15 15 and r as a parameter determining the scale of the surplus captured from 16 16 entry, i.e. r(c − m). The parameter h may be interpreted as a cost incurred 17 17 to enter the market. 18 18 Alternatively, one might present the payoffs in Equation 1 as the conse- 19 19 quence of entry or not when demand is P (m) = r(c − m) with an ex post 20 20 market clearing price, P , enforced based on a realized m; entry or not each 21 21 attract the same subsidy, v; and marginal cost of production is h. 22 22 For our discussion of Nash equilibria, we definec ˆ ≡ c − h/r. One might 23 23 think ofc ˆ as market capacity adjusted for the presence of an entry cost.
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