Does SFE correspond to expected behavior at the uniform price auction?

1. Social Choice and Welfare Higher School of Economics Moscow, 23 July 2010 Does SFE correspond to expected behavior at the uniform price auction? Alexander Vasin Lomonosov Moscow State University

2. Introduction • Electricity and gas markets develop in several countries for about 20 years. Typically they perform as a uniform price auction where a producer’s bid is a monotonous function that determines the supplied quantity depending on the price. An important problem for such markets is limitation of large producers’ market power. One way to solve the problem is to choose such kind of the auction that minimizes this power. Another important feature of the markets is uncertainty of the demand. It relates to random changes of the environment and also to variations of the demand within a day for which the bids are submitted. • In this context, Klemperer and Meyer (1989) propose a promising auction model and theoretical results. They assume a bid to be a monotone smooth function and the demand function to depend on a random parameter. Thus, the cut-off price that equalizes the total supply and the demand is random. A bid profile is called a supply function equilibrium (SFE) if, for any parameter value, the bid of each firm maximizes its profit under fixed bids of other producers. For a symmetric oligopoly, the authors derive a differential equation for an equilibrium bid and describe the set of SFE. The SFE price is always less than the Cournot oligopoly price. For some cases, the price reduction is valuable (see Newbery, 1998, Green, 1997), so the supply function auction limits the market power in this sense.

3. However, computation of the SFE bids is rather sophisticated mathematical problem. In general, its solution requires full information on the demand function and the cost functions of all competitors. Why should one expect that the actual behavior at the auction corresponds to this concept? • A similar question for Nash equilibria of normal form games is considered in the framework of adaptive and learning mechanisms’ investigation (see Milgrom, Roberts, 1990, Vasin, 2005). The study shows that for some classes of games rather simple mechanisms provide convergence of strategy profiles to stable NE for players with bounded rationality and incomplete information. • The present paper aims to consider best reply dynamics for two variants of a symmetric oligopoly with a linear demand function: A) with a linear marginal cost, B) with a fixed marginal cost and a limited production capacity. Our purpose is to find out for each case if the dynamics converges to any SFE.

4. Formal model of the auction and SFE concept • Klemperer and Meyer (1989) consider the following model of the market: is the set of players (producers) for C(q) is the total cost function depending on the production volume q (symmetric oligopoly). Properties: • Consumers are characterized by demand function depending on the price p and random factor t (another interpretation: t is the day time). This function meets conditions: • A strategy of a player is a supply function that determines the supplied amount of the good depending on the market price p. The players set simultaneously without information about t. • For a given strategy profile , the price for a given t proceeds from the balance of the market supply and demand: . Each producer i aims to maximize his profit:

5. Strategy profile is called an SFE if • Proposition 1,2 Klemperer and Meyer (1989): If then is SFE if and only if and meets equation (1):

6. A) The market with a linear marginal cost function. • Consider a symmetric duopoly with cost function , c1 > 0, where , , and demand function , where and is a maximal demand depending on random parameter t with a given distribution function. According to Klemperer and Meyer (1989), an equilibrium supply function for this case should meet differential equation • If then there exists a unique SFE and the bid function is linear: (2). • Consider best reply dynamics for the repeated auction in this case. At every stage =1, 2, ... each firm sets bid that is the best reply to its competitor’s bid at the previous stage. (We assume S(p,0)=0).

7. Formally, is the best reply to if that is a solution to also provides the maximal profit: Though the demand depends on random parameter t, at every stage there exists a bid that maximizes the profit under any value of this parameter. • Proposition 1.Bid is the best reply to bid under any . • Thus, the best reply bid at stage τ is where The unique fixed point for this equation corresponds to the SFE (2) of the auction. • Proposition 2.Best reply dynamics for model A converges to the SFE (2). Moreover, .

8. B) The market with a fixed marginal cost and a limited production capacity. • In this case , , and . SFE bid for this case is a continuous monotone function that meets equation until for some p. • A general solution for this equation depends on integration constant A. • We define function and inverse function proceeding from the system: . In particular, for • Proposition 3.If then a unique SFE of the model corresponds to . The equilibrium bid is If then the set of equilibrium bids depends on the ratio . • Proposition 4.If then there exists a unique SFE, the same as for .

9. Picture 1.

10. If then for any A such that bid determines SFE and for any , where meets , bid also determines SFE. If , then corresponds to the Walrasian equilibrium. If then the capacity constraint is not essential. In this case, the bid determines SFE. In particular, for the SFE coincides with Walrasian equilibrium. Now consider best reply dynamics. Proposition 5.Best reply dynamics for converges to the SFE corresponding to Walrasian equilibrium;

11. Now consider the case . Let . Then , • Thus, at the 3rd stage a monotonous BR function does not exist, and the BR dynamics does not converge to the SFE.

12. What happens if we limit the set of strategies with simple bids of the form ? Then the 3rd stage BR does not exist: for the optimal k=3 while for the optimal k=1. • If we consider the BR dynamics for the class of bids under a fixed , then for we obtain cycles with different periods depending on . In particular, for . so that period of the cycle is T=4. so the BR dynamics converges to Cournot supply function. For . Thus, in this class we also do not find convergence to any equilibrium.

13. Picture 3.

14. References • Paul D. Klemperer, Margaret A. Meyer, Supply function equilibria in oligopoly under uncertainty, Econometrica 57 (6),1243-1277, 1989. • R. Green, D. Newbery, Competition in the British Electricity Spot Market, Journal of Political Economy 100 (5), 929-953,1992. • А.А. Васин, В.В Морозов, Теория игр и модели математической экономики. - М.:МАКС Пресс, 2005. • A.A. Vasin, N. Durakovich, P.A. Vasina, Cournot Equilibrium and Competition via Supply Functions, 2003. • А.A. Васин, П.A. Васина, Модели конкуренции функций предложения и их приложение к сетевым аукционам. -Москва: EERC, 2005. • D. Newbery, M. David, Competition, Contracts and Entry in the Electricity Spot Market, Rand Journal of Economics, 29,p. 726-749, 1998. • R. Green, J. Richard, The Electricity Contract Market in England and Wales. Journal of Industrial Economics 47(1), p.107-123, 1997. 4 • В.А. Ильин, В.А. Садовничий,Бл.Х. Сендов, Математический анализ.-Ч.1.-М.:ТК Велби, Изд-во Проспект,2006. • А.A. Васин, П.С. Краснощеков, В.В. Морозов, Исследование операций. - М.: Издательский центр "Академи 2008. • D. Newbery, Analitic Solutions for Supply Function Equilibria: Uniqueness and Stability.-EPRG Wprking Paper 0824,2008.