Oligopoly Games under Asymmetric Costs and an Application to Energy Production

Transcription

1 Oligopoly Games under Asymmetric Costs and an Application to Energy Production Andrew Ledvina Ronnie Sircar First version: July 20; revised January 202 and March 202 Astract Oligopolies in which firms have different costs of production have een relatively understudied. In contrast to models with symmetric costs, some firms may e inactive in equilirium. With symmetric costs, the results trivialize to all firms active or all firms inactive.) We concentrate on the linear demand structure with constant marginal ut asymmetric costs. In static one-period models, we compare the numer of active firms, i.e. the numer of firms producing a positive quantity in equilirium, across four different models of oligopoly: Cournot and Bertrand with homogeneous or differentiated goods. When firms have different costs, we show that, for fixed good type, Cournot always results in more active firms than Bertrand. Moreover, with a fixed market type, differentiated goods result in more active firms than homogeneous goods. In dynamic models, asymmetric costs induce different entry times into the market. We illustrate with a model of energy production in which multiple producers from costly ut inexhaustile alternative sources such as solar or wind compete in a Cournot market against an oil producer with exhaustile supply. JEL classification: C72; D43; L; L22 Keywords: Oligopolies; Bertrand model; Cournot model; Differentiated goods; Dynamic games; Exhaustile resources. Introduction The main focus of this paper is to study the numer of firms in an oligopoly who actively produce in equilirium when they have different costs of production. Asymmetric costs are commonplace in markets of large and small producers for example a major chain such as Barnes & Nole in the U.S. vs. an independent local ookstore), where the larger firm can achieve economies of scale. In energy markets, producers from different fuels and technologies have widely different costs of production: for example, oil and coal sources are much cheaper to produce from than renewales such as solar ORFE Department, Princeton University, Sherrerd Hall, Princeton NJ 08544; aledvina@princeton.edu. Work partially supported y NSF grant DMS ORFE Department, Princeton University, Sherrerd Hall, Princeton NJ 08544; sircar@princeton.edu. Work partially supported y NSF grant DMS

2 or hydropower. However, most analyses assume symmetric costs, or restricted cost structures that guarantee all firms are active in equilirium. Whether a firm produces a positive quantity in an equilirium is an important question in the study of oligopolistic competition ecause it can e used to explain market variety as a cost to the consumer. We first focus on oth the Cournot and Bertrand static games within which we have two cases: when the goods are homogeneous, and when they are differentiated. We will show that, within a fixed type of market, Cournot or Bertrand, differentiated goods result in more active firms in Nash equilirium than homogeneous goods. These results depend crucially on cost asymmetries etween the firms, as with symmetric costs the results trivialize to all firms active or all firms inactive. Our results, e.g. Theorem 4., partially explain the relative differences to consumers etween these two types of markets, as consumers have more choice in Cournot markets ut with higher prices, whereas they have less choice in Bertrand markets as the numer of firms in equilirium is less) ut they are compensated for this fact y reduced prices. Then we show the impact of cost differences in a dynamic Cournot game etween energy producers, one of which has a cheap exhaustile source such as oil, while the others have inexhaustile costly technologies, such as solar or wind. Here, the Markov perfect equilirium is characterized y nonlinear differential equations which we can solve to determine the entry points of the alternative producers into the market. We show how entry leads to jumps in the rate of change over time of the equilirium energy price and derive a modified Hotelling s rule for exhaustile resources under oligopolistic competition Propositions 5.6 and 5.7).. Background The study of non-cooperative oligopolistic competition originated with the seminal work of Cournot [6]. His original model assumes firms choose quantities of a homogeneous good to supply and then receive profit ased on the single market price as determined through a linear inverse demand function of the aggregate market supply. Moreover, marginal costs of production were assumed constant and equal across firms. Throughout this paper, we also assume constant marginal costs, however we drop the cost symmetry assumption. The result of Cournot s analysis is what one would expect: with equal marginal costs across firms, every firm chooses the same quantity to supply and the market price is aove cost y an amount that is inversely proportional to one plus the numer of firms in the market. Hence, as the numer of firms tends to infinity, the price approaches marginal cost, ut with a finite numer of firms, prices are aove cost and firms earn positive profits. Following this, Bertrand [3] argued to change the strategic variale from quantity to price. Firms producing a homogeneous good were assumed to set prices and produce to meet any demand of the market. As the goods are homogeneous, there can only e one market price, as oserved in the Cournot model aove, and therefore only the firm quoting the minimum price receives any demand. This feature of winner takes all leads to very harsh and contradictory results relative to common market oservations. If all firms have equal cost, then as long as there are two or more firms in the market, all firms price at cost and have zero profit. This perfectly competitive outcome differs sustantially from the Cournot outcome and is commonly referred to as the Bertrand paradox. To paraphrase from [2], the Bertrand model results in perfect competition in all cases esides monopoly, which is unrealistic in most settings, leading one to conclude that the correct set-up leads to the wrong result. On the other hand, as most firms seem to set their prices, not their 2

3 quantities, many economists have argued that the Cournot model gives the right answer for the wrong reason. Since these two original papers, there has een much interest in modifying these models in various ways to otain more realistic results. Furthermore, there is a significant literature comparing the two modes of competition when some of the assumptions in the original papers are altered. These modifications are usually rought aout to reconcile the Bertrand paradox. If one considers constraints on the capacity of the firms, see Edgeworth [8], then the result of the Bertrand model can e rought closer to that of the Cournot model. Kreps and Scheinkman [8] use capacity constraints to otain the Cournot outcome from Bertrand competition. As such, the Bertrand model cannot really e thought of as complete without more realistic assumptions. One of the original assumptions that is commonly not satisfied is that goods are homogeneous. This leads one to consider differentiated goods, which can mean either sustitute or complementary goods, although we concentrate on the case of sustitute goods. The original work in this area is Hotelling [6], where consumers were assumed to associate a cost of travel depending on the location of the firms, therey differentiating their otherwise identical goods ased on the location of the firms relative to the location of the consumer. For classical results on differentiated goods and other models of oligopoly, we refer to the ooks y Friedman [0] and Vives [24]..2 Asymmetric Costs Very often, modifications of the original models are made under the assumption of symmetric marginal costs across firms. When one considers an asymmetric cost structure, the issue of the numer of firms who are active in an equilirium ecomes crucially important. This fact is often overlooked or assumed away in studies of asymmetric cost, see for example [7] and []. As we will see though, it is not always the case that inactive firms can simply e ignored their presence may affect equilirium quantities and prices. We refer also to [5] and [22] for analysis of consumer surplus under Bertrand and Cournot modes of competition, ut again with symmetric and zero) costs. The issue of asymmetric cost in oligopolies has een addressed efore. In the work of Singh and Vives [23], they consider a game in which firms in a differentiated duopoly can choose to offer either price or quantity contracts to consumers. Essentially this allows firms to select either Bertrand or Cournot competition. They show for sustitute goods that it is always a dominant strategy for oth firms to choose the quantity competition. However, they derive their results under the assumption that oth firms are active in equilirium. In a related work, Zanchettin [25] compares prices and outputs in a duopoly with linear demand where the firms can have asymmetric costs and the goods are differentiated. The main idea is to remove the assumption that oth firms are active in equilirium and to consider the effect of this on prices and outputs. The main results of this paper are that prices are lower in Bertrand compared to Cournot, however outputs of oth firms may not always e greater in the Bertrand market relative to the Cournot market. Furthermore, industry profits can e higher in Bertrand than in Cournot for certain parameter values. This work is closest to ours although, as it is a duopoly model, the possile numer of active firms is restricted to one or two which simplifies the analysis consideraly. Furthermore, additional asymmetries are tractale in a duopoly that are essentially unworkale in an N-firm oligopoly. 3

4 Static Cournot/Bertrand games also arise as an intermediary in their dynamic counterparts, where asymmetric shadow costs encode firms scarcity values as their resources or capacities deplete. See [3] for a Cournot dynamic game with exhaustile resources and [9] for a dynamic Bertrand game in which firms of different sizes egin competing with asymmetric lifetime capacities. Study of these leads to the issue of lockading where some firms may e inactive due to high shadow costs until the resources of their competitors fall and their shadow costs rise) accordingly. Hence, a thorough analysis of the activity levels in the static Cournot and Bertrand games is needed, which is the purpose of Sections 3-4. The numer of active firms is generally studied in terms of entry and exit from a market in a dynamic setting, see for example [9] and [2]. We compute explicitly the entry points as a function of the asymmetric costs in a dynamic Cournot energy production model in Section 5. 2 Models of Oligopolistic Competition We start with a market that has N firms. Let q i e the quantity produced y Firm i, and similarly let p i e the price charged for their respective good. Dropping the suscript, we denote y p and q the vector of all prices or quantities, respectively. We define Q = q i and P = p i as the aggregate quantity and price, respectively. 2. Sustitutaility of Goods We focus on two types of markets: those where the goods eing sold are not identical, and those where the goods are identical. Differentiated goods means that the goods are not perfect sustitutes for one another. Each firm can in principle then receive demand even if they are not the lowest price firm or the firm producing the highest quantity. Homogeneous goods means that the goods sold y the different firms are perfect sustitutes for one another. Therefore, only one price can prevail in the market. In the case of firms choosing quantities, this does not add undue urden at all, as it simply implies that the price a firm receives will depend on the aggregate market supply and their profit will then depend on that price and their chosen quantity. The case of firms setting prices is more complicated. Each firm can offer a price, ut only one price can ultimately prevail in the market as goods are deemed identical. Thus, only the firms setting the lowest price will receive demand from the market. The rest can post a price, ut as they will not receive demand at this higher price, it is irrelevant. The only issue that arises is in a tie for the lowest price. To accommodate this important case, one must define precisely how the lowest price firms share demand in equilirium. These rules are commonly referred to as sharing rules and can have important equilirium consequences, see for example [4]. We shall make explicit in what follows which sharing rule we use and our reason for doing so. 4

5 2.2 Linear Oligopolies We concentrate on a linear demand specification in order to present explicit calculations, particularly for the numer of active firms in equilirium. However, the existence of an equilirium in each of the four models under asymmetric costs can e estalished in more general settings: for the homogeneous Cournot case, see [3, Section 2], and [9, Section 2] for the differentiated Bertrand case. We derive demand from the ehavior of a representative consumer with the following quadratic utility function: N Uq) = α q i N 2 β N N qi 2 + γ q i q j, α, β > 0. ) i= i= i= j= j i Our representative consumer solves the utility maximization prolem max Uq) q R N + N p i q i, from which we can derive the inverse demand functions for the firms as i= p i q) = U = α βq i γ q j, i =,, N. Differentiated Cournot 2) q i j i This gives actual quantities provided p i > 0 for all i. In the case that the quantities result in any of the prices not eing positive, we must remove the individual firm with the highest quantity from the system and then consider a market with one less firm. If any of the prices are still not positive, then we repeat this procedure, one firm at a time, until we have a market with positive prices for all firms. The parameter γ can e positive, negative or zero depending on whether the goods are sustitutes, complements or independent. Definition 2.. Within the linear demand set-up, the following concepts can e characterized: Independent goods γ = 0; Sustitute goods γ > 0; Differentiated goods γ < β; Homogeneous goods γ = β. We will not deal with the case of complementary goods, which corresponds to γ < 0. The quantity γ/β expresses the degree of product differentiation, which ranges from zero independent goods) to one perfect sustitutes, or homogeneous goods). When γ < β, System 2) can e inverted to otain q i p) = a N N p i + c N p j, i =,, N, Differentiated Bertrand 3) j i 5

6 where for n N, we define a n = α β + n )γ, n = β + n 2)γ β + n )γ)β γ), c n = γ β + n )γ)β γ). 4) As in the case of inverse demands, System 3) gives the actual quantities if they are all strictly positive. If some of them are not positive, then we consider the case where the firms are ordered y price such that p p 2 p N. Then, we remove the price and quantity of firm N from the utility function and repeat the aove procedure. If some remain that are not positive, we continue removing firms in this manner one at a time until we have only firms with positive quantities remaining. The equivalent manner to remove them at the level of the demand functions is to again consider the firms ordered y price, and to remove the price and quantity of Firm N from the system and use Expression 4) with n equal to N. If firms still remain with negative quantity in this reduced demand system, then we again continue this procedure one firm at a time until only firms with positive quantity remain. If firms are not ordered y price, we can re-lael them for this procedure and then restore their correct laels once the demand system is determined. For differentiated goods, System 2) gives the inverse demand functions relevant to the Cournot market, and System 3) gives the demand functions relevant to the Bertrand market. For homogeneous goods, we need to consider the limit of the utility function in Equation ) when γ = β. This is no prolem for the inverse demand functions and we arrive at p H i q) = α β N q j, i =,, N, Homogeneous Cournot 5) j= where the H stands for Homogeneous, which is to distinguish this inverse demand function from that expressed in System 2). This is the same price for all of the firms and it depends only on the aggregate supply in the market. The difficulty with the Bertrand market is that we cannot invert this relationship. The issue that arises in the Bertrand case can est e presented with a two-firm duopoly example. If we suppose p is fixed and p 2 = p +ɛ then q 2 p) = a 2 2 c 2 )p 2 ɛ, and q = a 2 2 c 2 )p +c 2 ɛ. As γ approaches β, we have 2 c 2 ). Thus, for γ sufficiently close to β we will have q 2 < 0. However, when this occurs, the demand function for Firm changes in a consistent manner to reflect the fact that Firm 2 is out of the market. This is accomplished y using Expression 4) with n =. If p = p 2 = p, then we have from System 5) that q + q 2 ) = α β β p. In other words, the quantities of the two firms must add up to the quantity that a single firm would receive if they set the lowest price alone. Furthermore, as firms are distinguishale only y the prices they set, equal prices must imply they get equal demand. Moreover, as we have estalished what the sum of their demands is, we know that each gets an equal share of this demand. In this way we otain α β β i) p q H i p) = # {k : p k p j j} {pi =minp)}. Homogeneous Bertrand 6) For an N-firm oligopoly, the procedure for determining demand is identical as that aove, where firms are removed from the demand system one-y-one as their quantities go negative. We thus arrive at the same conclusion regardless of the numer of firms. This is the classical winner-takeall Bertrand model with monopoly demand α β β p. 6

7 3 Nash Equiliria of the Four Static Games We assume that each firm has constant marginal cost, which we denote y s i for Firm i. We denote y s the vector of such costs. Our main goal is to demonstrate how cost asymmetries, coupled with the degree of product differentiation and the mode of competition, affect the structure of the market. We egin with static Cournot and Bertrand games, and in Section 5, we will study a dynamic Cournot game. Each firm chooses its price or quantity to maximize profit in a non-cooperative manner, although they do so taking into account the actions of all other firms. Therefore, we assume firms make decisions to maximize profit in the sense of Nash equilirium. The argument of the profit function that each firm maximizes over depends on the mode of competition, ut in all cases the profit function is given y Π i = q i p i s i ). 7) Remark 3.. If we started with firms having individual α i s in Equation ), then this could e reduced to the identical α case y asoring each α i into the cost s i in Equation 7). However, we do not treat the case of individual β s or γ s here. We suppose that the N firms are ordered y cost such that, possily after a suitale relaeling, 0 < s s 2 s N. Further, we assume that s N < α, that is every firm s cost is lower than the maximum possile market price. In the next two susections, we present the Nash equiliria results for the Cournot and Bertrand games for markets with differentiated goods. The results for the homogeneous goods markets under Cournot and Bertrand competition are given in Appendix A. 3. Differentiated Cournot Competition Throughout this section, we roughly follow [3] wherein the results of the homogeneous Cournot game with general price functions are given. Although here we are dealing with the differentiated Cournot model, the method for constructing the Nash equilirium is very similar for the case of linear inverse demands. We shall present here the full details of the construction in the differentiated goods setting. The profit functions of the firms are given in Equation 7), and here each firm i maximizes over q i with price given as the function of q in System 2). If all the equilirium quantities are strictly positive, then the first-order conditions give the est-response function for a given firm: qi BR = α γ q j s i, i =,..., N. 8) 2β j i We can sum these equations over i to otain 0 = 2βQ + Nα γn )Q S N), 9) where we let S n) = n j= s j, the sum of the costs of the first n firms. We define the candidate equilirium total production Q,N, as the solution to the scalar equation f N Q) = S N), where f N Q) = Nα 2β + N )γ)q, Q > 0. 7

8 Let us define the following effective market price function: P Q) = α γq. This is not an actual price in the market, ut it is a useful mathematical tool. We use the description effective as we shall see that this price serves the purpose of the single price in the market that all costs must e compared to. With this market price function, we can rearrange Equation 8) to otain a candidate Nash equilirium q,n i = P Q,N ) s i, i =,..., N. 0) 2β γ Utilizing Equations 9) and 0), we can express the candidate equilirium as ) q,n P Q,N ) s i i = N ) Q,N. ) j= P Q,N ) s j Equation ) is convenient for expressing the candidate equilirium quantities as it has the interpretation that once the total equilirium quantity Q,N is determined, each player produces the fraction which is the deviation of his cost s i from the effective market price P Q,N ) relative to the total deviation of all players costs from that price. However, it may occur that q,n i < 0 for some i, which is not an admissile solution and we must thus consider equiliria with less than N firms. For n N, we define f n Q) = nα 2β + n )γ)q, Q > 0. 2) Oserve that f n Q) is strictly decreasing in Q for all n. For a fixed n {,..., N}, there is a unique Q,n such that f n Q,n ) = S n), which is given y Q,n = For each n, we have the following n-player candidate equilirium: nα Sn) 2β + n )γ. 3) q,n i = { P Q,n ) s i 2β γ for i n, 0 for n + i N, 4) where Q,n is given in Equation 3). We denote the corresponding prices y p,n i using System 2). In order to determine which of the candidate equiliria is the true equilirium of the game, we first compute, for a given firm, the net revenue per unit as p,n i s i = β P Q,n ) ) s i. 5) 2β γ The following lemma then follows directly from Equations 4) and 5). Lemma 3.. q,n i < 0 if and only if p,n i < s i. A given candidate equilirium can fail to e a Nash equilirium of the game if i) q,n i < 0 for some i n, or ii) s i < p i q,n,..., q,n n, 0,..., 0 ) for some n + i N. 8

9 Case i) holds if and only if s i > P Q,n ). So the effective market price is too low for this player and they would e etter off not producing at all. Thus, we should look for a Nash equilirium with a smaller numer of active players. We provide the following lemma efore discussing the second case. ) Lemma 3.2. s n < p n q,n,..., q,n n, 0,..., 0 if and only if s n < P Q,n ). In other words, Firm n wishes to participate in the n )-firm equilirium if and only if s n < P Q,n ). Proof. The result follows from the equivalence ) n p n q,n,..., q,n n, 0,..., 0 = α β0) γ q,n j = α γq,n = P Q,n ). j= Case ii) means that some firm i {n +,..., N} could produce a strictly positive quantity and would receive a price strictly aove cost from the market and therefore make a positive profit. Lemma 3.2 shows that this case occurs if and only if s i < P Q,n ). In this case we should look for a Nash equilirium with a larger numer of active players. The procedure for determining the equilirium is to start with the one firm equilirium that consists of the lowest cost firm producing as a monopoly. We then consider whether Firm 2 wishes to participate. That is, we check if Case ii) holds with i = 2. If so, we ask if oth wish to participate in the two-player candidate equilirium. We continue in this manner until we run out of firms to add or additional firms do not wish to enter. This construction has similarities to the equilirium constructions found in [4] and [], in that there is a cutoff point elow which firms produce and aove which firms are costed out. We must, however, take care that this procedure terminates and that this is sufficient to determine a unique Nash equilirium. We estalish in the following lemma the necessary inductive step. Lemma 3.3. Fix some n < N. We have n and n + )-player candidate equiliria with total production quantities Q,n and Q,n+, respectively, and the individual firm production quantities given y Equation 4) with the corresponding Q. Then, Firm n + will want to e active in the n-firm equilirium if and only if they want to e active in the n + )-firm equilirium. Proof. From Lemma 3.2 Firm n + wants to e active in the n-firm equilirium if and only if s n+ < P Q,n ). Recall the definition of Q,n as the unique Q which satisfies f n Q,n ) = S n), where the f n are defined in Equation 2) and are strictly decreasing for Q > 0. One can show for n < N that f n+ Q,n ) = S n) + P Q,n ), and therefore f n+ Q,n+ ) f n+ Q,n ) = s n+ P Q,n ). 6) In a similar manner, one can show f n Q,n+ ) = S n+) P Q,n+ ), and hence f n Q,n+ ) f n Q,n ) = s n+ P Q,n+ ). 7) 9

10 We then find s n+ < P Q,n ) f n+ Q,n+ ) < f n+ Q,n ) from Equation 6), Q,n+ > Q,n as f n+ is decreasing, f n Q,n+ ) < f n Q,n ) as f n is decreasing, s n+ < P Q,n+ ) from Equation 7), from which the conclusion follows. Corollary 3.. The total candidate equilirium quantity is strictly increasing from n firms to n+) firms if and only if Firm n + ) is active in the n + )-firm candidate equilirium. Proof. The corollary states: Q,n+ > Q,n if and only if q,n+ n+ > 0. The conclusion follows from the proof of Lemma 3.3 as we have s n+ < P Q,n ) Q,n+ > Q,n, and the lemma itself gives that this is equivalent to Firm n + eing active in the n + )-firm candidate equilirium. Proposition 3.. There exists a unique Nash equilirium to the differentiated Cournot game. The unique equilirium quantities are given y ) q = q,n,..., q,n n, 0,..., 0, where q,n i is given in Equation 4) and n is the numer of active firms in equilirium which is given y n = min { n {,..., N} : Q,n = Q }, with Q = max {Q,n : n N}. Proof. We know y assumption that Firm will participate in the one-firm candidate equilirium as s < α. Hence Q, > 0. Suppose that for some n < N we have a candidate Nash equilirium in which the first n firms are active. If s n+ < P Q,n ), or equivalently if Q,n+ > Q,n, then from Lemma 3.3, Firm n + wishes to enter, and y the same lemma, they will participate in the n + )-firm equilirium. Furthermore, as all the other n firms have costs lower than or equal to Firm n +, they will also e active in the n + )-firm equilirium. Therefore, every candidate equilirium with n or fewer players cannot e a true equilirium due to entry. We can proceed adding players until either no further players wish to enter or there are no further players. We have uniqueness y construction. The explicit characterization of n, the numer of active firms in equilirium, comes from the fact that Q,n is strictly increasing in n as firms enter. At some point, we may have Q,n = Q,n ) if a firm is exactly indifferent etween entering and not entering, i.e. qn,n = 0. Such a firm is not considered active in the equilirium. It may stay flat for a range of n if there is a range of firms with identical costs all of which are indifferent etween entering and not entering. Then, the sequence will e decreasing. Thus, n will e given y the first n where Q,n attains its maximum Q. In other words, n = min { n {,..., N} : Q,n = Q }. We illustrate the ehavior of Q,n in Figure. In this example, the numer of active firms in equilirium, n, is 5, while N = 9. This figure was generated with α =, β = 0.5, γ = 0.2. We also set the vector of costs to s = 0.25, 0.27, 0.35, 0.6, 0.6, , , 0.72, 0.85). Firms 4 and 5 have equal cost and as Q,5 > Q,4, we know that oth are active in equilirium. We also note that 0

11 Firms 6 and 7 have equal cost, greater than the cost of Firm 5, and are exactly indifferent etween entering and not entering. We see this as Q,5 = Q,6 = Q,7. The shape of the curve appears to always e concave, although we have not explored whether this holds in general. The main important feature to note is that the curve is strictly increasing up to n and flat or decreasing after n max Q *,n Q *,n.2. n * Firm numer Figure : Illustrative ehavior of Q,n and the numer of active firms in equilirium. 3.2 Differentiated Bertrand Competition The profit function that the firms seek to maximize is again given in Equation 7), where the firms now choose p i and quantities are given y q i p) in System 3). In addition, here we assume that if a firm receives zero demand in equilirium, they set price equal to cost. The method of solution for this model is quite similar on the surface to the differentiated Cournot model. We can solve for the firms est-response functions and otain candidate Nash equiliria from their intersection. The reason for considering different candidate equiliria is that, again, in the game with heterogeneous costs, the candidate equilirium with all N firms may result in negative demands for some firms. We must thus consider sugames which, for n =,..., N, involve only the first n players. Let p,n = p,n,..., p,n n, s n+,..., s N ), where the first n components solve the Nash equilirium prolem with profit functions as descried aove, and with N replaced y n in the coefficients of the demand functions q i p) given in System 3). These are given explicitly in Equation 8). Let p denote the vector of prices in equilirium. It is shown in [9] that the Nash Equilirium of the Bertrand game will e one of three types: I All N firms price aove cost. In this case, p i > s i for all i =,..., N, and the Nash ) equilirium is simply the N-player interior Nash equilirium given y p = p,n,..., p,n N, where the p,n i are as defined aove.

12 II For some 0 n < N, firms,..., n price strictly aove cost and the remaining firms set price equal to cost. In other words, p i > s i for i =,..., n, and p j = s j for j +,..., N. The first n firms play the interior n-player su-game equilirium as if firms n +,..., N did not exist. These firms are completely ignorale ecause their costs are too high. The high cost firms receive zero demand and have prices fixed at cost y our definition of Nash equilirium. The Nash equilirium is p = p,n,..., ) p,n n, s n+,..., s N. III For some k, n) such that 0 k < n N, firms,..., k price strictly aove cost if k = 0 then no firms price strictly aove cost), and the remaining firms set price equal to cost. In other words, p i > s i for i =,..., k and p j = s j for j = k +,..., N. Type III differs from Type II in that firms k +,..., n are not ignorale: their presence is felt in the pricing decisions of firms,..., k, and we say that firms k +,..., n are on the oundary. Similar to Type II, firms n +,..., N are still completely ignorale. If firms,..., k ignore firms k+,..., n and set k-player Nash equilirium prices, then the latter firms will set a price aove cost which contradicts their eing ignored. On the other hand, if we consider firms k+,..., n to e fully active in equilirium, then their optimal prices in the n-player Nash candidate equilirium would not e aove their costs. Therefore, there is the Type III equilirium where firms k +,..., n are not active in the equilirium, ut firms,..., k set prices that take the presence of the former firms into consideration. Only an equilirium of Type I or Type II is possile in the Cournot game. There we had Lemma 3.2 which resolved the fact that a firm whose quantity is negative in a candidate equilirium does not wish to enter the game with a positive quantity if the remaining firms ignore this firm and play the candidate Nash equilirium with one less player. However, in the Bertrand game, such a result does not hold. The unique Nash equilirium is given explicitly in the following. Proposition 3.2 Prop 2.6 in [9]). There exists a unique equilirium to the Bertrand game with linear demand. The type I and II candidate solutions are given y [ p,n na n + n n m= i = a n + c s ] m n 2 n + c n ) 2 n n )c n ) + ns i. 8) The type III candidate solutions are given y p,n+,n+ k i = [ ) n+ a n+ + c n+ s m 2 n+ + c n+ ) m=k+ +c n+ n ) a n+ + c n+ n+ m=k+ s m + k n+ m= s m + n+ s i, 2 n+ k )c n+ where the superscript, n +, n + k) stands for oundary, n + ) firms entering into the demand function, and n + k firms on the oundary, i.e. not ignorale. The Nash equilirium is constructed as follows: If s > a, then p = s,..., s N ). Type II with n = N). Else, find n < N such that p,n i > s i, i =,..., n, and p,n+ n+ s n+. 2

13 If s n+ n+ an+ + c n ) n+ i= p,n i, then p = p,n,..., ) p,n n, s n+,..., s N, Type II with n < N), Else, if p,n+, n > s n, then p = else, find k < n such that p,n+,,..., p,n+, n, s n+,..., s N ), Type III ), p,n+,n+ k i > s i for all i =,..., k, and p,n+,n+ k+) k+ < s k+. Then p = p,n+,n+ k,..., p,n+,n+ k k, s k+,..., s N ). Type III ). The Nash equiliria for the homogeneous Cournot and Bertrand games are given in the Appendix. 4 How many firms are active? In this section we consider the question of whether a firm is active or inactive in equilirium. The main result of this section is an ordering of the numer of active firms in equilirium across the four different types of games. Following the analytic result, we provide numerical examples to demonstrate the various prospective outcomes. 4. The General Case: N-Firm Games Fix the total numer of firms N, the parameters α, β, and γ, and the vector of costs s = s,..., s N ) such that 0 < s s 2 s N < α. We shall denote y n Bh, n Bd, n Ch, n Cd, the numer of active firms in equilirium in the homogeneous Bertrand game, differentiated Bertrand game, homogeneous Cournot game and differentiated Cournot game, respectively. The purpose of this section is to show specifically how these numers relate to one another. This will e shown in Theorem 4., ut first we ) estalish three helpful lemmas. In the process we shall need the quantity θ n = β γ). 2β+n )γ β+n 2)γ Recall the fact that in oth Cournot type games, a firm is active in equilirium, if and only if the total equilirium supply strictly increases when this firm enters the market. In the homogeneous Cournot game, this amounts to the fact that Firm n is active if and only if Q,n H > Q,n H. In the differentiated Cournot game, the condition is Q,n > Q,n if and only if Firm n is active in equilirium. We characterize these conditions in terms of the cost of the firms in the following lemma. Lemma 4.. In the homogeneous Cournot game, we have Q,n H H s n < α + Sn ). n > Q,n In the differentiated Cournot game, we have Q,n > Q,n s n < α2β γ) + γsn ) 2β γ) + n )γ. 3

14 Proof. For the homogeneous case, the definition of Q,n H is given in Equation 58) in Appendix A. Then the result follows y writing down Q,n H Q,n H and rearranging. The differentiated case can e found in Corollary 3., where we note that s n < P Q,n ) is equivalent to the given condition. Lemma 4.2. The following expression is increasing in h provided the denominator is not zero. αh + γs n ) h + n )γ, Proof. Take a derivative with resect to h to otain [ ] αh + γs n ) = h h + n )γ h + n )γ) 2 αγn ) γs n )). As γ is strictly positive, we find that this expression is strictly positive provided n )α > S n ). This is clearly true ased on the assumption that s i < α for all i. Lemma β γ) > θ n for all n >. Proof. ) 2β + n )γ 2β γ θ n = 2β γ β γ) β + n 2)γ ) n 2)β + γ = γ. 9) β + n 2)γ Oviously the Expression 9) is positive for n >. Theorem 4.. n Bh n Ch n Cd. Further, n Bd n Cd. Finally, n Ch n Bd if θ n Ch < γ. Proof. n Bh n Ch : First, we note that n Bh = # {k : s k = min i s i }. Thus, this is equal to one unless we have for some j, 0 < s = = s j < s j+ s j+2 s N. In such a case, we would have n Bh = j. Our assumption s < α implies that Q, H > 0. Furthermore, with the aove specification for the costs, simple algera shows ) Q,j j H = 2 Q, H j +. ) Moreover, 2 j j+ > if and only if j >. Hence, Q,j H > Q, H for all j >. Therefore at least all firms with cost equal to the lowest cost firm are active in the homogeneous Cournot equilirium. In this case, this means n Ch j. Therefore, n Bh n Ch. n Ch n Cd : Lemma 4. specifies that firm n participates in the homogeneous Cournot equilirium if s n < α + Sn ) n = αγ + γsn ) γ + n )γ, 20) 4

15 and this firm participates in the differentiated Cournot equilirium if s n < α2β γ) + Sn ). 2β + n 2)γ Using Lemma 4.2, we have α2β γ) + S n ) 2β γ) + n )γ > αγ + γsn ) γ + n )γ, 2) ecause 2β γ) > γ. Hence, if Firm n is active in the homogeneous Cournot equilirium, i.e. if their cost is low enough, then the firm is also active in the differentiated Cournot equilirium y Expression 2). Therefore, n Ch n Cd. n Bd n Cd : The differentiated Bertrand equilirium can e of three types. In Type I, we have that all firms price aove cost according to their interior candidate equilirium prices p,n i. The condition p,n N > s N is necessary and sufficient for this equilirium to hold. This condition on s N is equivalent to s N < αθ N + γs N ) θ N + N )γ. Using Lemmas 4.2 and 4.3, we find that αθ N + γs N ) θ N + N )γ < α2β γ) + SN ) 2β γ) + N )γ. 22) This implies that if there is a Type I equilirium in the differentiated Bertrand game, i.e. n Bd = N, then we also have n Cd = N. If we a priori know that we have an equilirium of Type II, then we know that Firm n is active in the differentiated Bertrand equilirium if and the firm is active in the differentiated Cournot equilirium if s n < s n < αθ n + γs n ) θ n + n )γ, 23) α2β γ) + Sn ) 2β γ) + n )γ. 24) Again using Lemmas 4.2 and 4.3, we find the same inequality as in Expression 22) with N replaced y n. Hence, a firm eing active in the Type II differentiated Bertand equilirium implies the firm is active in the differentiated Cournot equilirium. So for a Type II equilirium, we have n Bd n Cd. Finally, the situation of a Type III equilirium could arise. However, this serves only to decrease the numer of active firms in the market from the maximum numer of firms that could e sustained in a Type II equilirium. That is, a Type III equilirium starts with a Type II equilirium and then possily removes some firms from eing active. Therefore, in all cases we have n Bd n Cd. n Ch vs n Bd : 5

16 Let us ignore for now the case of a Type III equilirium in the differentiated Bertrand game. We have estalished the conditions on s n for a firm to e active in the two types of equiliria. First, in the homogeneous Cournot case we have the condition given in Expression 20). Second, in a Type I or Type II equilirium in the differentiated Bertrand game, we have the condition given in Expression 23). We find y Lemma 4.2 that Firm n eing active in the differentiated Bertrand game implies the firm is active in the homogeneous Cournot game if θ n Ch < γ. Therefore, this implies n Bd n Ch. This result continues to hold for a Type III equilirium as again such an equilirium only serves to reduce n Bd. Remark 4.. If θ n Bd > γ then the relative size of the conditions on cost reverses and therefore one expects n Ch n Bd to hold. However, this does not account for the possiility of a Type III equilirium. The numer of active firms in a Type III equilirium is highly sensitive to the costs and it is quite difficult to ascertain what occurs in this case. However, Type III equiliria are rare in the sense that they only occur for very special specifications of the costs. Therefore, in most situations, one should have n Ch n Bd if θ n > γ. Bd Remark 4.2. The condition θ n > γ essentially oils down to γ eing small enough relative to β. In fact, one can show that θ n γ = 2β + nγ)β 2γ) + 3γ 2. Thus, a sufficient condition for this to e positive is β > 2γ. Moreover, what happens as γ approaches β is that the homogeneous Cournot and the differentiated Cournot approach one another, and similarly with the two Bertrand models. However, as the homogeneous Bertrand model always has only one active firm in the market when there is a unique low cost firm) and the differentiated Cournot model has the most active, we find that the differentiated Bertrand and homogeneous Cournot have to cross over at some point to account for this lack of differentiation among the goods. Thus, informally, one can claim that for sufficiently differentiated products we have: n Bh n Ch n Bd n Cd. Remark 4.3. If we let γ = 0, then the differentiated Bertrand and Cournot models coincide in terms of the numer of active firms. This can e seen from Expressions 23) and 24), as setting γ = 0 causes θ n = 2β and the conditions on cost for a firm to e active in equilirium in the two cases then coincide. Also, ased on the assumption that s i < α for all i, we will have all firms active in the equilirium that results with γ = 0. This is all fairly ovious ut at least worthwhile to note as the ehavior of the numer of active firms in equilirium depends highly on γ and there is no difference etween differentiated Bertrand and Cournot when goods are independent. 4.2 Numerical Examples Fix s = 0.25, 0.27, 0.35, 0.4, 0.5), α = and β = 0.5. We then consider the addition of another firm to the market with cost s 6 varying from 0.00 to We consider three different possile values of γ, a low, ase and high case. The low and high cases for the four different oligopoly models are displayed in Figure 2 a) and ). The results for the ase case of γ = 0.3 are displayed in Figure 3. In the high case, we see that increasing γ this much towards β causes the numer of active firms in the differentiated Bertrand market to e less than that of the homogeneous Cournot market. We also see that the homogeneous and differentiated Cournot markets have similar numers of active firms, coinciding over a large region of the cost of the additional firm, contrary to the result in Figure 3 where they are completely separated over the whole space of costs. 6

17 Numer of Active Firms Numer of Active Firms Cournot Diff.) Cournot Hom.) Bertrand Diff.) BertrandHom.) Numer of Active Firms Cournot Diff.) Cournot Hom.) Bertrand Diff.) BertrandHom.) Numer of Active Firms Cost of additional firm a) Low, γ = Cost of additional firm ) High, γ = 0.45 Figure 2: Numer of Active Firms - 6 firm example Numer of Active Firms 6 5 Numer of Active Firms Cournot Diff.) Cournot Hom.) Bertrand Diff.) BertrandHom.) Cost of additional firm Figure 3: Numer of Active Firms - 6 firm example, γ = 0.3 In the low case representing the other extreme point of view where we move γ towards zero, we see that the numer of firms in the differentiated Bertrand and Cournot models the top two lines in the figure which are on top of one another except etween 0.7 and 0.8) are almost identical, with Cournot having more firms over a small region of the space of costs. We also see that the homogeneous Cournot is completely separated from the three other models of oligopoly. Another point of view from which we can study the ehavior of the effect of the degree of product differentiation on the market equilirium is to plot the numer of active firms in the four different market types versus the ratio γ/β. We fix the total possile numer of firms at 40, we set α =, and finally we fix β = 0.5. We then vary γ so that γ/β ranges from 0 to. First, we set s = 0. and s 40 = 0.99, and specify the costs of the intermediate firms to e linearly increasing from s to s 40 so that the difference etween the costs of adjacent firms is constant. We display this in Figure 4 a). Next, we put firms into 5 groups so that there are 8 firms with equally spaced costs within each 7

18 group, the costs varying from lowest to highest y 0.005, then with a gap of 0.8 etween groups. For example, the first group has s = 0., s 2 = 0.05, s 3 = 0.0,..., s 8 = 0.35; then s 9 = We continue in this manner until we get to s 40 = We display the results with this cost structure in Figure 4 ) Numer of Active Firms Cournot Diff.) Cournot Hom.) Bertrand Diff.) BertrandHom.) Numer of Active Firms Cournot Diff.) Cournot Hom.) Bertrand Diff.) BertrandHom.) Numer of Active Firms Numer of Active Firms Degree of Product Differentiation Degree of Product Differentiation a) Evenly spaced costs ) Five different cost clusters Figure 4: Numer of Active Firms - 40 firm example One can clearly see from oth figures that the numer of active firms in equilirium is decreasing in the degree of product differentiation. That is, with γ/β = 0, meaning independent goods, we have the maximum numer of active firms, and with γ/β =, meaning homogeneous goods, we have the minimum numer of active firms. We can see the point where the differentiated Bertrand crosses over the homogeneous Cournot is at a high degree of product differentiation, therey supporting our previous remark that for low values of γ/β we expect n Bh n Ch n Bd n Cd. Comparing the two figures, we see that with a nice linearly interpolated cost structure we get relatively smooth ehavior of the numer of active firm curves. However, with clustered costs, we can see that essentially each cluster is either wholly in or wholly out. This leads to the large flat regions of space with no change in structure, along with sudden large drops in the numer of firms. In a realistic dynamic situation as market conditions change, the numer of active firms in equilirium will vary over time. We illustrate this issue in a dynamic homogeneous Cournot model of energy production in the following section. 5 A Dynamic Game With Asymmetric Costs Consider an energy market with one oil producer with exhaustile reserves and cost of production equal to zero. There are also N producers of energy using alternative, costly ut inexhaustile technologies such as solar or wind. These firms are identified and ordered y their unit costs of production, s i, such that 0 s s 2 s N <, and we recall the notation S k) = k j= s j. 8

19 The oil producer Player 0) has reserves xt) at time t, with the dynamics dx dt = q 0xt)) {xt)>0}, 25) where q 0 xt)) is his production rate. When his reserves run out, he no longer participates in the market. Let q i xt)), i =,..., N e the production rates of the alternative technology producers. Then, the price of energy in a homogeneous Cournot market is given y N P t) = q 0 xt)) where in the notation of Section A. we have taken α = β =. i= q i xt)), 26) In the static Cournot oligopoly game, the Nash equilirium q i s 0, s) as a function of the costs s 0 for Player 0 and the vector s = s,..., s N ) for the others was given in Proposition A.. We denote y G i s 0, s), i = 0,..., N, the static game equilirium profit, and in fact G i s 0, s) = q i s 0, s)) 2. The firms are maximizing lifetime profit, and once the oil producer runs out of oil, and goes out of usiness, the remaining firms with their inexhaustile resources repeatedly play a static game with profit flow G i, s) over an infinite time horizon. The first argument in G i is set to the choke price,, which means that Player 0 no longer participates, and it is an N player game. Given initial oil reserves x0) = x, the value functions, v 0 x) for Player 0 and w i for Players i =,..., N are v 0 x) = sup q 0 w i x) = sup q i 0 0 e rt q 0 xt)) q 0 xt)) e rt q i xt)) q 0 xt)) N j= N j= q j xt)) {xt)>0} dt, q j xt)) s i {xt)>0} dt + r G i, s), with v 0 0) = 0 and w i 0) = G i, s)/r. The case N = 2 was studied in [3] except there, the oil producer ecame a renewale energy producer with the same cost s upon exhaustion). Here we analyze the case of a genuine N-player oligopoly. See also [20] for the two-player game with exploration; and [2] for a related prolem in the context of mean-field games. We look for a Markov Perfect Nash equilirium q i xt)), for i = 0,..., N. Hamilton-Jacoi equation for v 0 is given y The associated rv 0 = G 0 v 0, s), v 0 0) = 0. 27) Here s 0 = v 0 x) is the shadow cost encoding the scarcity value for the oil producer when his reserves are at level x. The equilirium production rates in the dynamic game are given y q i xt)) = q i v 0xt)), s), i = 0,..., N. As these are determined y the value function v 0, we will not study the other players value functions w i. 9

20 5. Blockading Points When oil is plentiful, that is, x is large, we expect the cost of energy production from alternative technologies to e too cost prohiitive to compete with the oil producer, and so he is the only active player. This is ecause the oil producer s scarcity cost v 0 x) 0 as x, and so, when reserves are large, it is small compared to the cost of the alternative technologies. At the other extreme, when x is small, we expect all N producers to e active ecause the scarcity cost rises as reserves fall. Therefore, as reserves decline, we will go from monopoly, through duopoly to the full oligopoly with N players. The reserve level at which Player n enters is denoted x n. Similarly, the time at which Player n enters is denoted y t n. We illustrate this intuition in Figure 5. Reserves 0 All Active x N x N 2 x 2 Duopoly # Active N N N x Oil Monopoly Time t N t N 2 t 2 t Figure 5: Blockading intuition 0 In other words, we expect for x 0, x N ) that there are N active firms in the sense that they all produce a positive quantity. For x x n, xn ), there are n active firms and N n inactive firms in the sense that in equilirium they produce a zero quantity over this range of reserves. As reserves decline, x n denotes the point at which Firm n egins production and enters the market. Or, coming from the left, x n is the level of reserves aove which Firm n is lockaded from the market. We shall see that the lockading points x n depend on the costs of the alternative technologies, and in fact, if these costs are small enough, it is possile to have x n = for some values of n. We first introduce the notation ρ n = + Sn ). n Assumption 5.. The vector of costs s is such that s N < ρ N. This is the condition from Lemma 4. that implies that Firm N is active in the equilirium at x = 0. Furthermore, it is clear that, as Firm N has the highest cost, all N alternative technology firms will e active in the equilirium at x = 0. In fact as we assume for the N -player game that is played repeatedly after oil runs out is such that n = N, then Proposition A. tells us that the sequence {Q,n H } is strictly increasing and so, from Lemma 4., we have s n < ρ n, for all n =,..., N. 28) 20