The commitment eect in belief evolution

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1 The commitment eect in belief evolution Thomas Gehrig Werner Guth Rene Levnsky September 003 Abstract In this note we establish that rational demand expectations will typically not evolve in an evolutionary model In an evolutionary model beliefs act like a commitment device to more aggressive behavior This commitment eect has the same direction for strategic substitutes and complements and fades away in large markets JEL Classication: C70, D43, D81 A preliminary draft of this paper was presented at Max Planck Institute in Jena and at Freiburg University We appreciate valuable comments by the seminars participants Institut zur Erforschung der wirtschaftlichen Entwicklung, Albert-Ludwigs-Universitat, Kollegiengebaude II, Platz der Alten Synagoge, Freiburg im Breisgau, Germany Max Planck Institute for Research into Economic Systems, Kahlaische Strasse 10, Jena, Germany Corresponding author renelevinsky@vwluni-freiburgde

2 1 Introduction It has been established that rational cost expectations will typically not emerge in an evolutionary model (Guth, 1998) In the context of a simple market model individuals tend to entertain over-optimistic beliefs about their rival's costs inducing more aggressive behavior relative to a rational expectations model Thus the evolving beliefs act as if the rivals could strategically commit to more aggressive beliefs, what is individually but not collectively rational for the competing rms A strategic choice of beliefs would, however, be rather questionable: one would rst choose one's beliefs and then denitely accept those just chosen beliefs when deciding about market behavior (see also Frank, 1997, and Bar-Hillel and Budescu, 1995, for a discussion of such wishful thinking) In this note we perform a similar analysis for demand expectations We want to analyze, whether the strategic properties (strategic complementarity vs strategic substitutability) aect the nature of the commitment eect Moreover we want to analyze the relation between evolutionarily stable and rational expectations in large markets Not surprisingly, we conrm non-convergence to rational expectations also in the case of demand uncertainty More interestingly, we nd that the commitment eect always has the same direction for strategic complements as well as strategic substitutes In both cases rms tend to \commit" to more aggressive behavior in the evolutionary model The commitment eect, however, declines as the number of (symmetric) competitors grows large Hence on large markets rational expectations seem (asymptotically) justied by an evolutionary process 1

3 The Basic Model Consider a (horizontally) dierentiated market environment with n rms Market demand is implicitly dened by a system of linear and symmetric inverse demand functions p i = y x i n 1 x j, (1) The parameter denes the strategic properties of the price instrument Products are economic and strategic complements (see Bulow, Geanakoplos, Klemperer, 1985) when π i x j > 0, ie < 0 and substitutes and strategic substitutes when π i x j < 0, ie > 0 Otherwise the products are independent and competition becomes monopolistic (Chamberlin, 1933, Robinson, 1933) The true realization of y is unknown and determined by nature according to the density function ϕ having mean µ Individual rms entertain homegrown beliefs f i about the probability distribution ϕ So, the expected prot function of the vector x = (x 1,, x n ) of individual sales amounts is π i (x) = + 0 [y x i n 1 x j ]x i f i (y)dy = [ µ i x i n n x j ]x i () Due to the linear structure of the market model, only the rst moments µ, µ i of distribution functions ϕ, f i matter We therefore will analyse the (evolution of) rst moments only In our evolutionary analysis this can be justied by an innite population of sellers who are randomly matched to n- seller markets Given their beliefs rms select prices rationally by maximizing expected prots Note that these beliefs may be quite dierent across rms In a rational expectations model in contrast all competitors are forced to entertain the same rationally expected demand parameter µ In our model rms can entertain whatever beliefs they please However, evolutionary pressure will ultimately eliminate beliefs that can be improved upon by changing beliefs Whether an alternative belief type leads

4 to an improvement is, however, judged by the true prot expectation, ie changing beliefs directly imply dierent expected utilities, but can determine reproductive success (expected true prots) only indirectly via market behavior In evolutionary language, belief mutants will enter a given population of beliefs as long as these mutants earn more than average success To keep the model simple we abstract from production costs Remark We restrict ourselves to economically suitable 's, ie we suppose that in the case of strategic substitutes the cross-impact of rm j's output x j on the price of p i is not bigger than the impact of own output x i, ie n 1 In other words, more (less) similar goods rely on larger (smaller) parameters with the border cases = 0 of monopolistic competition and = n 1 of identical goods, ie the homogeneous market 3 Results Given their beliefs µ i rms maximize their individually expected prot functions () as determined by their own idiosyncratic belief type µ i The necessary rst order conditions read µ i x i n 1 x j = 0 i N (3) From (3) we get the equilibrium (for derivation see Appendix, Subsection 41) what, of course, presupposes that individual beliefs are commonly known: x i = µ i[(n 1) + (n )] µ j ( + )[(n 1) ] (4) 3

5 Remark An extreme point is a maximum if the Hessian matrix is negative denite Therefore, the matrix of the system given in Appendix, equation (6) has to be positive denite, because it equals to Hessian matrix multiplied by minus one Since its principal minors equals to H n = ( + )( ) and we presuppose that < n 1 the second order condition holds for >, otherwise the extreme point is a minimum and the prot maximizing amounts x diverge to innity Let us note that the same condition holds in the standard rational expectation model, where the structure of the Hessian matrix is identical Naturally, we restrict ourselves to condition > Finally, we abstract from the border case = 0 of no strategic interaction or monopolistic competition Thus the set of all possible heterogeneity parameters is = (, 0) (0, n 1] With the help of these solutions we can now dene the evolutionary game for studying the evolution of beliefs Here the strategies are the possible belief types and the reproductive success is the true expected prot Although µ i determines seller i's market behavior, as derived above, the evolution of beliefs is governed by the true demand (parameter) µ, ie by material success only For the one-parametric mutant space µ i [0, + ) we get a unique symmetric equilibrium Proposition For all n there exists a critical n (, 1) such that for all ( n, n 1] the unique evolutionarily stable belief type or equilibrium belief is characterized by µ i = (1 + ρ)µ, where ρ = (n 3)( + ) , (5) For n the revenue is unbounded and the beliefs diverge Proof See Appendix, Subsection 4 Corollary For all suitable all rms produce more than in the rational expectation case (with µ i = µ for all i = 1,, n) 4

6 Proof For > n the statement follows from the fact that ρ > 0, whenever θ > 0 (see for the denition of θ the paragraph \Second Order Condition" in Subsection 4 below) For < n the amounts x i diverge to innity, while they are nite in the rational expectation model Note that beliefs µ i exceed µ for all, ie the direction of the commitment aect is not aected by the strategic properties of prices, ie strategic complementarity or strategic substitutability Finally notice 1 that for all lim n ρ = 0 and lim n n = So, with increasing n, the system converges to the standard rational expectation case Hence, over-optimism is especially pronounced in small markets Put dierently, rational expectations become evolutionarily stable whenever the commitment advantage fades away due to increasing competition as measured by n References [1] Bar-Hillel, M and D Budescu (1995), The Elusive Wishful Thinking Eect, Thinking and Reasoning, 1, 71{104 [] Bulow, JI, Geanakoplos, JD and PD Klemperer (1985), Multimarket Oligopoly: Strategic Substitutes and Complements, Journal of Political Economy, 93, 488{511 [3] Chamberlin, E (1933), The Theory of Monopolistic Competition Cambridge: Cambridge University Press [4] Frank, RH (1997), If Homo Economicus Could Choose His Own Utility Function, Would He Want One with a Conscience?, in: Casson, M, Culture, social norms and economics, Elgar Reference Collection, 486{97 1 For homogeneous markets where is a function of n as = one gets ρ = n +1 > 0 for nite n and lim n ρ = 0 5

7 [5] Guth, W (1998), Are rational cost expectations evolutionarily stable?, Ifo-Studien - Zeitschrift fur empirische Wirtschaftsforschung, 1, 1{13 [6] Robinson, J (1933), The Economics of Imperfect Competition, London: Macmillan 4 Appendix 41 Derivation of equation (4) The rst order conditions (3) generates the following system of linear equations x = The system can be solved, eg, by Cramer's rule, or directly via solving two equations in two unknowns since as the matrix on the left hand side also its inverse is symmetric The determinant of the system is = = = ()(+) = = ()(+) = ( + )( ) µ 1 µ µ n (6) 6

8 The determinant of the matrix obtained by replacing the i-th column by the righthand side is µ 1 µ 1 µ µ µ i = µ i µ n + + µj + µ 1 µ = ()(+) µ i µ j ()(+) = ( + )( )n [ µ i µ j ()(+) ], and therefore x i = µ i µj ()(+) = ( + )(n 1)µ i µ j ( + )[(n 1) ] (7) 4 Proof of Proposition First Order Condition As usual in evolutionary analysis, the true expected prot is the measure of reproductive success The true expected prot of belief µ i against an otherwise µ j -monomorphic population is ( R i = µ x i ) x j x i, (8) n 1 where the choices x j and x i are dened by (7) Substituting x j = ( + )(n 1)µ j µ k ( + )[(n 1) ] = (n 1)( µ j µ i ) ( + )[(n 1) ] for the sum and (6) for x i in (7), we get 1 { R i = ( + )[(n 1) ]µ [(n 1)+ ( + ) [(n 1) ] + (n ) ]µ i (9) }{ µ j [(n ) + (n 1)]µ i } µ j 7

9 Belief µ i is an optimal response to µ j if the rst order necessary condition ( + )[(n 1) ][(n ) + (n 1)]µ [(n ) + (n 1) ][(n ) + (n 1)]µ i {[ (n ) (n 1)] + [(n ) + (n 1)]} µ j = 0, for an interior value µ i holds In an evolutionarily stable monomorphism the best reply µ i to the µ j -monomorphism satises µ i = µ j Considering the symmetry of the problem, we can set µ i = µ j = µ and obtain ( + )[(n 1) ][(n ) + (n 1)]µ (10) [(n ) + (n 1) ][(n ) + (n 1)]µ 3 (n 1)µ = 0 Finally dividing by [(n 1) ] yields ( + )[(n ) + (n 1)]µ = [(n 3) + (n 3) + 4(n 1)]µ, ie µ = (n ) + (n 3) + 4(n 1) (n 3) + (n 3) + 4(n 1) µ (11) Second Order Condition The principal minors of the system equal α β β β α β = [α + (n 1)β](α β), (1) β β α }{{} n where α = [(n ) + (n 1) ][(n ) + (n 1)] and β = 3 The matrix of the system corresponds to the Hessian matrix multiplied by minus one, so the matrix has to be positive denite For α β we get (α β) = ( + )[(n 1)(n 1)( + ) + ], that is positive for all 8

10 Since for [α + (n 1)β] we get { see the coecient of µ in equation (10): [α + (n 1)β] = [(n 1) ][(n 3) + (n 3) + 4(n 1)] and [(n 1) ] > 0, the critical condition to be discussed is θ = [(n 3) + (n 3) + 4(n 1)] > 0 For n = we get θ = [(1 + 5) ][ (1 5)] and that θ is positive, ie second order condition holds if ( > = 1 5) For n = 3 one has θ = (6 + 8) and that the principal minors are positive, ie second order condition holds for ( 3, ], where 3 = 4 3 For n 4, the variable θ can be factorized as [ θ = (n 3) ][ 4n ] 4n 3 n 3 n 3 The rst to elements are positive, so the last one has to be negative, ie for the second order condition is satised if 4n 3 3 > n = + n 3 One can easily see that for all n we get n (, 1), and lim n + n = 9

In classical competitive equilibrium, the rms in a market maximize prots taking price as given, each rm produces the quantity at which price equals ma

In classical competitive equilibrium, the rms in a market maximize prots taking price as given, each rm produces the quantity at which price equals ma Relative Prots and Competitive Equilibrium Cheng-Zhong Qin and Charles Stuart Department of Economics University of California at Santa Barbara Santa Barbara, California 93106 August 30, 2001 Abstract