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Article Excerpt
This article analyzes tacit collusion in infinitely repeated multiunit uniform price auctions in a symmetric oligopoly with capacity-constrained firms. Under two popular definitions of the uniform price, when each firm sets a price-quantity pair, perfect collusion with equal sharing of profit is easier to sustain in the uniform price auction than in the corresponding discriminatory auction. Moreover, capacity withholding may be necessary to sustain this outcome. Even when firms may set bids that are arbitrary finite step functions of price-quantity pairs, in repeated uniform price auctions maximal collusion is attained with simple price-quantity strategies exhibiting capacity withholding.
1. Introduction
* This article contributes to the study of tacit collusion by analyzing infinitely repeated multiunit uniform price auctions with capacity-constrained firms. As in our earlier work on discriminatory auctions, we modify the Bertrand-Edgeworth approach by allowing each firm to simultaneously set a price-quantity pair specifying the firm's minimum acceptable price and the maximum quantity the firm is willing to sell at this price. (1) Using this game, we analyze the feasibility of perfect collusion using two different rules for determining the uniform price. Under the first rule, which we call the Market Clearing Price rule, the uniform price is equal to the minimum price at which the quantity offered by the firms is greater than or equal to demand. Under the second rule, called the Maximum Accepted Price rule, the uniform price is equal to the highest submitted price at which the residual demand left over from supply provided at strictly lower prices is strictly greater than zero. Both definitions have been used extensively in the literature (see, for example, Green and Newbery, 1992; von der Fehr and Harbord, 1993).
When each firm sets a price-quantity pair, there exists a range of discount factors for which the monopoly outcome with equal sharing is sustainable in either of the uniform price auctions, but not in the corresponding discriminatory auction. Moreover, capacity withholding may be necessary to sustain this outcome.
We extend these results to the case where firms may set bids that are arbitrary step functions of price-quantity pairs with any finite number of price steps. Surprisingly, under the Maximum Accepted Price rule, firms need employ no more than two price steps to minimize the value of the discount factor above which the perfectly collusive outcome with equal sharing is sustainable on a stationary path. Under the Market Clearing Price rule, only one step is required. That is, within the class of step bidding functions with a finite number of steps, maximal collusion is attained with simple price-quantity strategies exhibiting capacity withholding.
These results are particularly relevant for markets such as electricity markets in which uniform price and discriminatory auctions govern exchange. Our simple model captures some of the basic features of operating electricity markets, such as the UK spot market, the Spanish wholesale market, or the Victoria Power Exchange. In these markets, capacity-constrained firms compete by offering step bidding functions that vary in their complexity depending on the market.
The theoretical literature on capacity-constrained uniform price auctions applied to electricity markets can be traced back to Green and Newbery (1992) and von der Fehr and Harbord (1993). (2) The former assumes that capacity-constrained firms offer continuous supply functions, whereas the latter assumes that firms submit discrete step functions similar to those in this article. In both papers the analysis is static, and thus ignores the strategic implications of repeated interaction. Although, as Borenstein, Bushnell, and Wolak (2002) note, most electricity markets provide favorable conditions for firms to collude, surprisingly, little attention has been paid to the theoretical modelling of collusion in electricity markets. An exception is Fabra's (2003) comparison of the uniform price and discriminatory auctions in Bertrand-Edgeworth duopoly supergames.
Fabra (2003) has shown that under Bertrand-Edgeworth (B-E) duopoly, divisions of the monopoly profit can be supported in the infinitely repeated uniform price auction for strictly lower discount factors than in the infinitely repeated discriminatory auction. However, this result is only valid for a subset of symmetric capacities for which nonstationary paths with bid rotation can be sustained as perfect equilibria of the uniform price auction. For example, in the duopoly, if each firm's capacity is large enough to supply the monopoly output, incentives to deviate from perfectly collusive paths in the uniform price auction are no less than in the discriminatory auction. Furthermore, on the nonstationary paths with bid rotation that minimize incentives to deviate in the uniform price auction, firms do not equally share monopoly profit. Expanding the strategy space to price-quantity pairs, thereby allowing for physical withholding, has important implications for the sustainability of perfect collusion in the uniform price auction. A direct implication of capacity withholding is that, in contrast to B-E competition, when capacity is such that n - 1 firms can supply the monopoly output, the monopoly outcome can be supported for a strictly wider range of discount factors in the uniform price auction than in the discriminatory auction. Moreover, this result holds even if we restrict attention to stationary paths on which each firm obtains an equal share of the monopoly profit.
In the discriminatory auction, the incentive to deviate from perfect collusion is minimized on a stationary path on which each firm sets the monopoly price and offers its whole capacity. On the other hand, in the uniform price auction, if the uniform price is given by the Market Clearing Price rule, the stationary path on which each firm withholds capacity to offer its share of monopoly output at a price below some critical level (strictly lower than the monopoly price) minimizes firms' incentives to deviate in the class of stationary paths with equal sharing of the monopoly profit. If the uniform price is given by the Maximum Accepted Price rule, then incentives to deviate from perfect collusion are minimized when n - 1 firms withhold capacity to offer their share of the monopoly output. The remaining firm acts as the price setter and offers capacity at the monopoly price. Together, these two results provide a conclusive theoretical link between equilibrium capacity withholding and the ability to support tacitly collusive outcomes.
The remainder of the article is organized as follows. In Section 2, we describe the model and the simultaneous move price-quantity uniform price auction under two alternative definitions of the uniform price and characterize the Nash equilibria of the game. In Section 3, we introduce notation and definitions used in analyzing the price-quantity supergame. In Section 4, we show that under both formulations of the uniform price, capacity withholding relaxes incentives to deviate on perfectly collusive stationary perfect equilibrium paths with equal sharing. On such paths, incentives to deviate are minimized when n firms withhold capacity under the Market Clearing Price rule and when n - 1 firms withhold capacity under the Maximum Accepted Price rule. Section 5 extends the results in Section 4 to L-step bidding functions, L [greater than or equal to] 1, and shows that bidding functions with at most two steps are sufficient in order to minimize firms' incentives to deviate from a perfectly collusive path. One step is required under the Market Clearing Price rule and two steps under the Maximum Accepted Price rule. Section 6 concludes.
2. The simultaneous move price-quantity game
* The model. Consider a market for a homogeneous good. There are n firms in the industry. Let N = {1, ..., n} denote the set of firms. Firm i's cost function is such that unit cost [c.sub.i] is constant up to capacity [k.sub.i]. Firms are symmetric: [k.sub.i] = k and [c.sub.i] = c = for all i. Let d(p) be market demand and assume that it satisfies the following assumptions.
Assumption 1. d(p) is continuous on [0, [infinity]). [there exists] [bar.p] > such that d(p) = if p [greater than or equal to] [bar.p] and d(p) > if p < [bar.p]. d(p) is twice continuously differentiable and d'(p) < on (0, [bar.p]). Finally, pd(p) is strictly concave on [0, [bar.p]] with maximizer [p.sup.m].
These assumptions guarantee that there exists a unique unconstrained monopoly price, [p.sup.m]. Inverse demand exists and is denoted by P(y), where y is output. To ensure that there exists a unique Cournot equilibrium with a strictly positive price in the quantity-setting game with n symmetric firms (without capacity constraints), demand given by d(p), and zero marginal cost, we further assume (3)
Assumption 2. d'(p) + pd"(p) < on (0, [bar.p]).
Under assumptions analogous to Assumption 1 for P(y), this is equivalent to assuming that log P(y) is strictly concave over the relevant range and implies that Cournot quantity best-response functions are downward sloping (see Deneckere and Kovenock, 1999). Denote by r(z) a firm's Cournot best response to an aggregate quantity z set by other firms. That is, r(z) maximizes P(x + z)x with respect to x. Let [y.sup.c] be the quantity set by each firm in the Cournot equilibrium with strictly positive price.
In the one-shot simultaneous move price-quantity game, firms simultaneously set price-quantity pairs, (p, q), where p [member of] [R.sub.+] and q [member of] [0, k]. Firm i's strategy space is thus [S.sub.i] = [R.sub.+] x [0, k]. A strategy profile (p, q) = (([p.sub.1], [q.sub.1]), ..., ([p.sub.n], [q.sub.n])) is an element of [x.sup.n.sub.i = 1] = [S.sub.i]. In this article, we restrict the analysis to pure strategies.
Define [[??].sub.i] = min{[q.sub.i], d(0)} to be the effective quantity offered by firm i. Given a strategy profile (p, q) and a coordinate p [member of] [R.sub.+] of the price vector p, define the set L (p | p, q) [equivalent to] {i [member of] N | [p.sub.i] = p}. L(p | P, q) is the set of firms setting price p. We have L(p | p, q) = [empty set] if for all i, [p.sub.i] [not equal to] p. Let [L.sup.-](p | p, q) [equivalent to] [U.sub.z < p] L(z | p, q) be the set of all firms charging a price strictly less than p. To simplify notation, we often drop the argument (p, q).
We assume efficient rationing. Hence, given a strategy profile (p, q), the residual demand faced by firms in L(p) is
R(p | p, q) = max {d(p) - [summation over (j[member of][L.sup.-](p)] [q.sub.j], 0}.
If [L.sup.-](p) is empty, then we define R(p | P, q) = d(p). Note that here the residual demand is the demand left over from supply provided at strictly lower prices.
If, in case of a tie in price at p, we assume that firms share residual demand in proportion to their effective quantities offered, then for i [member of] L(p | p, q), sales are
[s.sub.i] (p | p, q) = min {[[??].sub.i], [[??].sub.i]/[[summation].sub.l[member of]L(p)][[??].sub.l] R(p | p, q)}.
In this context, the literature has defined a uniform price auction in two distinct ways. We will examine each in turn. In the first definition, we follow Green and Newbery (1992), who use a specification in which the uniform price is the price at which the quantity demanded is equal to the quantity supplied (see also Boom, 2003; Ubeda, 2004; Fabra, vonder Fehr, and Harbord, 2006). This formulation leaves open the possibility that the uniform price will not be one of the submitted bids. See Figure 1 for an illustration.
Definition 1 (market clearing price). Given a strategy profile (p, q) in the uniform price auction, the uniform price [P.sup.e](p, q) is the unique price that solves
min{p | [summation over (i[member of][laplace](p | p, q)] [[??].sub.i] [greater than or equal to] d(p)}.
where [laplace](p | p, q) = [L.sup.-] (p | p, q) [union] L(p | p, q).
Definition 2 is the approach used by yon der Fehr and Harbord (1993) (see also Crampes and Creti, 2005; Fabra, 2003; Le Coq, 2002). The price each firm receives in the uniform price auction is equal to the maximum accepted price, where the maximum accepted price is the highest submitted price at which the residual demand left over from supply provided at strictly lower prices is strictly positive. Note that in this definition, the uniform price must be one of the submitted prices, and thus may not clear the market. See Figure 1 for an illustration. For Definition 2, we require slightly more notation. Let p = ([p.sub.1, ..., [p.sub.n]) and define
P(p, q) = {p [member of] {[p.sub.1], ..., [p.sub.n]}] R(p | p, q) > 0}.
P(p, q) is the set of submitted prices with R(p | p, q) > 0.
[FIGURE 1 OMITTED]
Definition 2 (maximum accepted price). Given a strategy profile (p, q) in the uniform price auction, the uniform price [P.sup.a](p, q) is equal to the maximum accepted price, that is
[P.sup.a](p, q) = max P(p, q) if P(p, q) [not equal to] [empty set],
and
[P.sup.a](p, q) = [bar.p] if P(p, q) = [empty set].
For u [member of] {e, a}, firm i's payoff, i = 1, ..., n, under the two alternative definitions is simply
[[pi].sub.i](p, q) = [P.sup.u](p, q)[s.sub.i]([p.sub.i] | p, q).
Before proceeding with the characterization of equilibria, we first justify our assumption of efficient rationing. Note that our choice of the efficient rationing rule does not play a role in Definition 1. The market clearing price does not depend on the specific rationing rule used, but only on the vector of price-quantity pairs submitted by the firms. Moreover, every consumer that obtains a unit of the good pays the same price per unit no matter which firm supplies it. It follows that without cross-subsidization between consumers, any consumer who obtains a unit must be willing to pay at least the uniform price for that unit.
When the uniform price is defined as in Definition 2, the firms' prices are ranked in increasing order, with the lowest-price firms selling first. If demand were not rationed efficiently, at some strategy profiles, there would exist consumers who would be required to pay more than their reservation value at the uniform price. Implementation of other rationing rules would therefore require cross-subsidization of consumers. (4)
[] Pure strategy equilibria. We now define critical prices that are useful in characterizing a firm's profit from deviating from a given profile. We also characterize a firm's minmax payoff.
First, for q < d(0), define the residual demand monopoly price for a firm with capacity k, [p.sup.r](k, q):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
[p.sup.r](k, q) is unique for every pair (k, q) given our assumptions on demand. From the strict concavity of pd(p), it is clear that whenever [p.sup.r](k, q) is strictly positive, it is strictly decreasing in q. A firm's profit from setting [p.sup.r](k, q) after lower-price firms have sold a quantity q is [[pi].bar](k, q) [equivalent to] [p.sup.r](q)[d([p.sup.r](q)) - q]. For q [greater than or equal to] d(0), a firm's residual demand after other firms have sold a quantity q is zero for all p. In this case, we define [p.sup.r](k, q) [equivalent to] and it follows that [[pi].bar.](k, q) = for all q [greater than or equal to] d(0).
Defining [p.sup.r] [equivalent to] [p.sup.r](k, (n - 1)k), it is straightforward to show that if (n - 1)k 0. If (n - 1)k [greater than or equal to] d(0), then by definition, [p.sup.r]((n - 1)k) = and each firm's minmax payoff is [[pi].bar] = 0.
Following Deneckere and Kovenock (1992), let [p.bar](k, q) be the unique price less than or equal to [p.sup.r](k, q) at which a firm is indifferent between being the low-price firm at [p.bar](k, q) and being a monopolist on residual demand left after q is sold and earning [[pi].bar](k, q). [p.bar](k, q) is equal to the smallest solution to
p x min{d(p), k} = [[pi].bar](k, q).
If q 0. If q [greater than or equal to] d(0), by definition [[pi].bar](k, q) = 0, and thus [p.bar](k, q) = 0. In the continuation, we will use the notation [p.bar] to denote [p.bar](k, (n - 1)k). Moreover, because [k.sub.i] = k for every i, when there is no ambiguity we use [p.sup.r](q) to denote [p.sup.r](k, q), [[pi].bar](q) to denote [[pi].bar] (k, q), and [p.bar](q) to denote [p.bar](k, q).
We can now state the following proposition describing equilibrium in the one-shot price-quantity uniform price auction with common capacities [k.sub.i] = k and common unit costs [c.sub.i] = 0, for every i, which we denote by [[GAMMA].sup.u](k, 0), where u [member of] {e, a} indicates the definition of the uniform price that is employed.
Proposition 1. The sets of pure strategy equilibria, [E.sup.u](k, 0), of the one-shot uniform price auctions [[GAMMA].sup.u](k, 0), u = e, a, are completely characterized as follows.
(i) Suppose k [less than or equal to] [y.sup.c]. Then [E.sup.a](k, 0) = {([p.sup.*], [q.sup.*]) | [p.sup.*.sub.i] [less than or equal to] P(nk) and [q.sup.*.sub.i] = k, [for all]i [member of] N, with [p.sup.*.sub.j] = P(nk) for at least one j [member of] N} and [E.sup.e](k, 0) = {([p.sup.*], [q.sub.*]) | [p.sup.*.sub.i] [less than or equal to] P(nk) and [q.sup.*.sub.i] = k, [for all]i [member of] N}.
(ii) Suppose k [greater than or equal to] d(0)/n - 1. Then [E.sup.a](k, 0) = {([p.sup.*], [q.sup.*]) | [there exists] i, j, i [not equal to] j, such that [p.sup.*.sub.i] = [p.sup.*.sub.j] = and [for all]h [member of] L(0 | [p.sup.*], [q.sup.*]), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Define C(0) [equivalent to] {([p.sup.*], [q.sup.*]) | [p.sup.*.sub.i] [less than or equal to] [p.bar]((n - 1)[y.sup.c]) and [q.sup.*.sub.i] = [y.sup.c], [for all]i [member of] N}. Then [E.sup.e](k, 0) = [E.sup.a](k, 0) [union] C(O).
(iii) Suppose k [member of] ([y.sup.c], d(0)/n - 1). Define [y.bar] to be the unique y [member of] (d([p.sup.r]) - (n - 1)k, k) such that [[pi].bar] ((n - 2)k + y) = [p.sup.r]k. Then [E.sup.a](k, 0) = {([p.sup.*], [q.sup.*]) | [there exists] j [member of] N such that [p.sup.*.sub.j] = [p.sup.r] and [q.sup.*.sub.j] [member of] [[y.bar], k] and [for all]i [not equal to] j, [p.sup.*.sub.i]...
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