Free and Second-best Entry in Oligopolies with Network

Transcription

1 Free and Second-best Entry in Oligopolies with Network Effects Adriana Gama Mario Samano September 7, 218 Abstract We establish an important difference between Cournot oligopolies with and without positive network effects. When there is a single-network with perfectly compatible products, and there are business-stealing effects (individual production decreases with entry), there may be under or excessive entry with respect to the social planner s (second-best) solution. This contrasts with the standard Cournot oligopoly model (without network effects) result, where excessive entry is the norm (under-entry might happen but only by one firm). Our results have important implications for competition policy: allowing for free entry when there are network effects may make consumers worse-off than with regulated entry. JEL codes: D43, L13, L14. Keywords: Cournot competition, positive network effects, complete compatibility, underentry, excessive entry. Corresponding author. Centro de Estudios Económicos, El Colegio de México, Carretera Picacho-Ajusco 2, Col. Ampliación Fuentes del Pedregal, Tlalpan 1411, Mexico City, Mexico. agama@colmex.mx HEC Montreal, Department of Applied Economics. mario.samano@hec.ca 1

2 1 Introduction It is well known that in a standard Cournot oligopoly with business-stealing effects (that is, individual output decreases with more firms), entry in the absence of regulation is excessive compared to the second-best number of firms (Mankiw and Whinston [1986], henceforth MW). 1 If entry of competitors increases the production by firm (business-enhancing effect), then, free entry is insufficient to achieve the second-best (Amir et al. [214], henceforth ACK). However, business-enhancing competition is rare in a standard Cournot oligopoly, and can be globally achieved only in the absence of variable costs (Amir [1996]). Our paper shows that the introduction of positive network effects to a Cournot oligopoly results in under-entry for a broader set of industries, even in the presence of variable costs or more surprisingly, under business-stealing competition. By industries with positive network effects we mean industries where the consumers willingness to pay increases with the number of people the network that will, or are expected to purchase a compatible good. That is, consumers benefit from the size of the network, an effect also known as demand-side economies of scale. We consider oligopolies with positive network effects and complete compatibility. As defined by Katz and Shapiro [1985] (henceforth KS), this corresponds to goods that are compatible no matter which firm produced them, and therefore, there is a single-network. Some examples are the telephone (landline and cell phones), instant messaging, DVDs, fashion, 1 Such result holds when one ignores the integer constraint; accounting for it implies that free entry might be lower than the second-best solution, but by no more than one firm. For simplicity, we refer to the latter result as excessive entry. Second-best refers to the regulation of entry, not production. 2

3 and goods created under the umbrella of an SSO (standard-setting organization) 2. Amir and Lazzati [211] (henceforth AZ) provide a thorough study of industries with network effects and complete compatibility. They analyze the important issue of viability (when an equilibrium with strictly positive production exists) and the effects of entry on the industry, while providing a comparison with the standard Cournot oligopoly (without network effects) for an exogenous number of firms. In this paper, we endogenize the number of firms in the industry with network externalities in the same fashion as Mankiw and Whinston [1986]. That is, we compare the second-best number of firms with the free entry outcome for a symmetric oligopoly with a single-network. Besides other relatively standard assumptions detailed in the next section, we assume that the cost of production is linear, for simplicity and because it is enough to provide insightful results about the oligopoly with positive network externalities. AZ show that in network industries, per-firm profits may increase or decrease with entry of new competitors. In the absence of network effects, such profits always decrease. 3 Then, we divide our main results into two parts: i) when equilibrium individual profits decrease with entry, and ii) when they increase. When profits decrease with the number of firms, we prove that under-entry is plausible under the business-stealing competition, while in standard Cournot, it only happens when competition is business-enhancing. Intuitively, if the network effect is strong enough, it might offset the business-stealing effect, mimicking the benefit that firms have on each other when individual production increases with entry. 2 See Shapiro [2]. 3 See Amir and Lambson [2] (henceforth AL) for a thorough study on existence and comparative statics of the oligopoly without network effects, also referred to as standard Cournot oligopoly in this work. 3

4 On the other hand, with business-enhancing competition, the standard result of underentry prevails whenever individual profits decrease with entry. If instead of decreasing, individual profits increase with competition (together with the business-enhancing effect), both the number of firms under free and second-best entry coincide. An important consequence of our results for competition policy is that by allowing for free entry when there are network effects, it is possible (depending on parameter values and the nature of the demand function) that consumer surplus is lower than under regulation. The next section provides the setting of the industry and its assumptions, Section 3 presents the results divided by the effects of entry on individual profits, and Section 4 concludes. All the comparisons ( increasing, decreasing, under-entry, etc.) refer to weak notions, unless it is specified that they are strict. 2 Preliminaries We consider a Cournot oligopoly with n firms producing homogenous goods that are perfectly compatible in an industry with positive network effects. In other words, a consumer values a good more when more people use this or a compatible product, which is any product of the industry. We assume that every consumer buys exactly one unit of the good, therefore, the expected size of the network is equivalent to the expected number of buyers. Given the positive externalities in demand, consumers form a belief on the size of the network that cannot be directly influenced by the firms, hence the use of the term expected size of the network. As we detail in the next section, these beliefs are confirmed in equilibrium. The firms in the industry are symmetric, that is, all of them face the same inverse de- 4

5 mand function P (z, s) and the same technology C(x), where z denotes total output, s is the expected size of the network, and x is the production of the firm under consideration. Then, π(x, y, s) = xp (x + y, s) cx is the profit of a given firm, before it pays the cost of entry K >. Hence, for a fixed number of firms n N in the industry, each firm solves max{π(x, y, s) K}, (1) x where y denotes the production by the other n 1 firms, that is, z = x + y. Let x n denote the solution to problem (1) when there are n firms, z n the equilibrium total output (z n = nx n ), π n the optimal profits (before paying the entry cost), and W n the social welfare without the entry costs, nk. Then, if individual profits decrease with n, the number of firms under free entry is n e N such that π n e K and π n e +1 < K. The second-best number of firms is the solution n N to the regulator s problem: max {W n nk}. n Throughout the paper we make the following assumptions. We use subindices for the function P to denote partial derivatives with respect to the first and second arguments. (A1) P : R + R + R + is twice continuously differentiable, P 1 (z, s) < and P 2 (z, s) >. (A2) C(x) = cx, c. (A3) Each firm s individual output is bounded by X >. (A4) P (z, s)p 12 (z, s) P 1 (z, s)p 2 (z, s) >, for all (z, s). (A5) π 1 > K, K >. 5

6 Assumptions (A1)-(A4) imply that a symmetric equilibrium exists in both oligopolies (with and without network effects) and that no asymmetric equilibria exist in either of the two (AL and AZ). Since there may be multiple equilibria, all of our results refer to the extremal equilibria with respect to total output. The reason is that the comparative static results used in this work, due to AL and AZ, apply to extremal equilibria only. In particular, P 1 < reflects the law of demand, and P 2 > the demand-side economies of scale (willingness-to-pay increases with the expected size of the network). We restrict our study to a technology with constant returns to scale (A2) because this is enough to show crucial differences between the standard Cournot oligopoly model and the one with network effects that are due only to the network and not to the production technology. The firms capacity constraint given by (A3) is a technical requirement that deals with the existence of equilibria and prevents technical problems with unbounded production, but X can be as large as needed. Assumption (A4) is used to guarantee the existence of the equilibrium in the oligopoly with compatible networks. It has the precise interpretation that the elasticity of demand increases with the expected size of the network, which is a characteristic of the industries with network effects (see AZ for further discussion). In addition, it allows the inclusion of industries with pure network goods, that is, where P (z, ) =. Finally, (A5) states that at least one firm can survive in the market and guarantees that the industry is viable, that is, an equilibrium with strictly positive production exists. 6

7 3 Results With complete compatibility, there is only one single-network composed by all the consumers of the compatible products. Although they compete in quantity, the firms cannot influence the consumers expectation of the network size, so that they are network-size taking. The firms work together to build a common network, yet, they compete to each other to serve it. This concept of equilibrium was introduced by KS and it is known as fulfilled or rational expectations Cournot equilibrium (henceforth RECE or simply equilibrium ); in equilibrium, the firms choose the production that maximize their profits, and the industry s production coincides with the expected size of the network (s = z n ). In other words, both the consumers and the firms correctly predict the market outcome, and the expectations of the consumers are fulfilled. Definition 1 A symmetric RECE of the one-network industry consists of a vector of individual outputs (x 1, x 2,..., x n ) and an expected network size s such that: (1) x i arg max{xp (x + j i x j, s) cx K : x } for each firm i, and (2) n i=1 x i = s. In the equilibrium of the industry without network externalities (and taking s as a parameter), π n (s) always decreases with the number of firms (AL), which is not the case in the industry with a single-network; as noted by AZ, per-firm profit might increase or decrease with the entry of firms. But as in the standard Cournot model, total output always increases with the number of firms n. It is important to keep in mind those results to compare the industries with and without network effects. We summarize them in Table 1 together with our main findings: the comparison of the number of firms from the free-entry (n e ) and second-best (n ) equilibria. 7

8 w/o net. effects w. net. effects π n π n x n x n not n e n n e n possible x n x n x n x n n e n n e n n e = n n e n Table 1: Summary of results For a better exposition of our results, we separate this study in two cases: (i) when per-firm profits in the industry with network effects decrease with entry, and (ii) when they increase. 3.1 Equilibrium per-firm profits decrease with entry We start with the situation where equilibrium individual profits decrease with entry, π n π n+1 for all n, because it provides a similar scenario to the one in the standard Cournot model in which individual profits always decrease with n. In this context we establish an important difference between the industries with and without network effects: with business-stealing competition, under-entry may occur by more than one firm in the presence of demand-side economies of scale Business-stealing competition As brought up by MW and confirmed by ACK, in the absence of network effects, excessive entry (under-entry is possible, but at most by one firm) is a direct consequence of the businessstealing effect. Under business-stealing competition, the regulator allows an additional firm in the market whenever its profits compensate for the social loss generated by the decline in the production of the rest of the firms. With a compatible network, even if entry reduces 8

9 individual output, it also creates a social gain given by the expansion of the network, allowing for the possibility of under-entry by more than one firm. For a better understanding of this phenomenon, consider equation (2) derived by MW that expresses the derivative of social welfare W n in the absence of network effects. Ignoring the integer constraint and taking n as continuous they get W n = π n + n[p (z n ) c]x n. (2) The second term of the right hand side of equation (2) is negative due to the business-stealing effect, and because in equilibrium, price exceeds the marginal cost of production, P (z n ) c. With a compatible network, the derivative of social welfare with respect to n incorporates a third term that is always positive and accounts for the effect of the expansion of the network (recall that P 2 > and that in equilibrium, total output increases in n), that is, W n = π n + n[p (z n, z n ) c]x n + z n zn P 2 (t, z n )dt. (3) Therefore, the sign of the sum of the last two terms is ambiguous when x n decreases in n. In particular ignoring the integer constraint and assuming that the solution is interior, if such sum is positive with n firms, there is under-entry; if it is negative, there is excessive entry (such as in the industry with no network effects). Consequently, the comparison between free and second-best entry cannot be predicted in general, and it depends on the primitives of the industry. Later on, in Remark 1, we summarize the previous finding. First, we show with a simple setting that under-entry with business-stealing competition is easily achieved in the presence of network effects. A particular industry with linear inverse demand function (in z and 9

10 s) and costless production provides insightful results about this particular characteristic of the oligopolies with network effects. Then, under-entry might be more common that it is considered to be. To facilitate the presentation of our first proposition, we invoke Result 1 that is due to AZ; to this end, we define the following functions: 1 (z) P 1 (z, z) + P 2 (z, z), and 2 (z) [P (z, z) c][p 11 (z, z) + P 12 (z, z)] P 1 (z, z)[p 1 (z, z) + P 2 (z, z)]. Notice that 1 (z) accounts for the total change in price when aggregate output changes along the fulfilled expectation path. 2 (z) refers to the fact that, as we discuss in the following section, business-enhancing competition is broader in the oligopoly with network effects (for a detailed discussion, see AZ). Throughout this paper, all the results refer to a Cournot oligopoly with positive network effects and complete compatibility. Result 1 (AZ) If 1 (z) and 2 (z) on [z n, z n+1 ] for all n, then, π n π n+1 and x n x n+1 for all n. Result 1 provides sufficient conditions to attain the situation under study in this section: decreasing equilibrium individual output and profits with respect to n. Now consider a particular industry with a compatible network, inverse demand function given by P (z, s) = a + bs z, with a > and < b < 1, and costless production, c =. 4 Observe that 1 (z) = 2 (z) = 1 + b <, then, by Result 1, x n and π n globally decrease in n. The reader can easily verify that the equilibrium output is given by 5 z n = na 1 + n(1 b), (4) 4 This setting is equivalent to have costly production, c >, and P (z, s) = (a c) + bs z, with a > c. 5 See AZ-Example 2 for details of the calculation. 1

11 and then, we have the following result (this and the rest of the proofs are provided in the Appendix). The constraint on K guarantees that at least one firm can always remain in the market, (assumption (A5)). Proposition 1 Let P (z, s) = a+bs z and c = ; a >, < b < 1, and < K < a 2 /(2 b) 2. a Then x n = and π a 1+n(1 b) n = 2 decrease in n; moreover [1+n(1 b)] 2 i) If b <.5, then, n e n ; ii) If b.5, then, n e n. ACK state that in the absence of network effects, and whenever per-firm output and profits decrease in n (such as in our Proposition 1), we have n e n 1. Hence, we establish a clear difference between the two oligopolies: provided that per-firm profits globally decrease with entry, under-entry by more than one firm is not possible without network effects (b = ) if individual output decreases, yet, under-entry by more than one firm is possible with the existence of a compatible network. This departure from the Cournot model without network effects is due only to the network effects, not to the primitives of the model. Proposition 1 clearly reflects the relevance of the network effect in establishing the relationship between n e and n. In fact, b measures the sensitivity of the consumers willingness to pay to the expected size of the market, an b is the only determinant to whether there is under or excessive entry. When the sensitivity is low (part i), the business-stealing effect is stronger than the network effect (see equation (3)), and more firms than the socially optimal level decide to enter. On the other hand, if b is high (part ii), the positive network effect prevails and there is under-entry. Example 1 illustrates Proposition 1. Example 1. Let P (z, s) = a + bs z and c = ; a >, < b < 1, and < K < a 2 /(2 b) 2. According to Proposition 1, x n and π n globally decrease in n. Table 2 shows 11

12 that the comparison between free and second-best entry changes for three different values of b; similarly, we show the social welfare achieved under each setting. The parameters a = 8 and K = 4 are fixed at those levels only for illustrative purposes. Table 2: Free and second-best entry under complete compatibility with P (z, s) = a + bs z, a = 8, K = 4, c =, and three different values of b. b W n e n e K W n n K n e n A B C In line with Proposition 1, cases A and B correspond to the standard Cournot result of over-entry given the business-stealing effect. Case C presents the departure from the standard model, since there is under-entry by more than one firm. An important consequence of Proposition 1 is that the consumers might be better off with a regulator than with free entry, contrary to what one would think. In the absence of network effects, consumer surplus increases with the number of firms, consequently, the consumers prefer the free entry scheme that leads to more firms. In the industry considered in Proposition 1, consumer surplus also increases with the number of firms (AZ, Theorem 11-i), hence, when b.5, the consumers prefer to be in a regulated industry. Our findings in this section are generalized in Remark 1, when we ignore the integer constraint. The hypothesis is written in terms of the equilibrium variables to consider all the 12

13 industries whose firms in equilibrium, have decreasing profits and production with respect to entry. Yet, the condition π n π n+1 and x n for all n can be replaced by the hypothesis of Result 1. 6 Remark 1 Let n be interior; π n π n+1 and x n for all n; ignore the integer constraint. i) If n [P (z n, z n ) c]x n + zn z n P 2 (t, z n )dt, then, n e n ; ii) If n [P (z n, z n ) c]x n + zn z n P 2 (t, z n )dt, then, n e n. If n is a corner solution, the comparison is straightforward. Otherwise, if n is interior, Remark 1 provides sufficient conditions that lead to excessive or under-entry, that follow immediately from equation (3). Observe that the conditions in Remark 1 are local (for the second-best number of firms); they can be replaced by their global version, but that would considerably reduce the number of industries that satisfy them. Next we study the setting where profit decreases with entry, π n π n+1, but per-firm output expands, x n x n Business-enhancing competition When individual profits decrease with competition but individual production increases, the result in the industry with network effects is equivalent to that of the industry without network effects: there is under-entry. With the network effects and business-enhancing competition, both the increase of individual production and the expansion of the network have the same direction and therefore, there is always under-entry. Specifically, the sum of the last two terms of equation (3) is positive whenever x n increases in n. 6 For ease in the presentation, the condition on equilibrium individual output is globally established, x n. As in ACK, it can be replaced by its local version, x n, to include all the industries that yield to non-monotonic individual output but satisfy x n. 13

14 Recall that in the oligopoly without network effects, individual profits always decrease with n, which is not the case in the presence of network effects. Thus, we make explicit in Proposition 2 that profits decrease with entry. 7 Proposition 2 Let π n π n+1 for all n. If x n x n+1, then, n e n. The industries with and without network effects share the characteristic of under-entry with the business-enhancing competition, but this kind of competition is more common in industries with network effects (AZ). For instance, when P (z, s) is log-concave in z or there are variable costs, business-enhancing is impossible in the standard Cournot model (see AL and ACK), but it is feasible with network effects. However, together with decreasing individual profits, business-enhancing competition under network effects (hypothesis of Proposition 2) might be hard to attain. For instance, it is unfeasible in the industry described in Proposition 1. There, 1 = 2 and according to AZ, whenever these variables are both positive (negative), per-firm output and profits increase (decrease) with n (see Results 1 and 2). Then, if 1 = 2, per-firm output and profits cannot have an opposite direction of change with entry, as required by Proposition 2. The following example presents an industry that satisfies the hypothesis of Proposition 2, it is due to AZ with a slight change on the inverse demand function. 8 Example 2. Consider an industry with a single-network, costless production, and inverse ( ) z demand function given by P (z, s) = exp. The symmetric RECE is implicitly exp(1 1/s) given by x n = exp(1 1/z n ); taking the derivative of the previous expression with respect to 7 As in Remark 1, x n x n+1 can be replaced by x n ( x n ) +1 (see Footnote 6). 8 2z See Example 1 of AZ. There, P (z, s) = exp and the industry is not viable for n = 1 (then, exp(1 1/s) π 1 = ). Changing the coefficient 2 by 1 in P (z, s), makes the monopoly viable (such as in our Example 2). 14

15 n and using the fact that z n is increasing, we have that x n also increases in n. Observe that z 1 = x 1 = 1, and then, the industry is viable for all n 1 (see AZ-Theorem 7). Similarly, it can be verified that π n decreases with n. In particular, if K =.5, we have that n e = 3 and n = 7, in line with Proposition 2. Although AZ provide an example for which per-firm profits decrease in n and x n increases in n under network effects (as required by Proposition 2), they do not provide general conditions for this to happen. To finalize this section, the next proposition establishes such conditions. Proposition 3 Let 1 (z) and 2 (z) on [z n, z n+1 ] for all n. If in addition, 1 (z) [P (z, z) c]/z on [z n, z n+1 ], then, π n π n+1 and x n x n+1. In particular, observe that 1 (z n ) implies that in equilibrium, the demand effect is stronger that the network effect (the magnitude of P 1 dominates that of P 2 ). Intuitively, even though individual output increases with entry, the market effect is so strong that individual profits decrease. 3.2 Equilibrium per-firm profits increase with entry Finally, we study the case where the firms increase their profits with entry, π n π n+1. Although this case might seem strange, AZ discuss its plausibility. In a nutshell, the firms cannot mimic the production of more firms to increase their profits because they cannot influence the consumers expectation of the network size. One justification is that they cannot credibly commit to a given level of production. As discussed earlier, this inability of the firms to influence the consumers beliefs is the spirit of the RECE. 15

16 Intuitively, regardless of the number of firms set by the regulator, firms will improve their profits with a new competitor, and as a consequence, there will always be excessive entry. Yet, it turns out that whenever individual profits and output increase with entry, free and second-best entry will always coincide. As we will see later, free and second-best entry might be finite or infinite according to the primitives of the industry. Proposition 4 Let π n π n+1 for all n, then, n e n. If in addition, x n x n+1 for all n, we have n e = n. The particular case studied in Proposition 1 illustrates Proposition 4 with a finite number of firms entering the market. But first, we state the following result due to AZ (Lemma 9-i and Theorem 1-i), that provides conditions such that both the individual production and profits expand with entry. Result 2 (AZ) If 1 (z) and 2 (z) on [z n, z n+1 ] for all n, then, π n π n+1 and x n x n+1 for all n. Recall that in the industry analyzed in Proposition 1, the inverse demand function is linear in both z and s, P (z, s) = a+bs z, a >, and c =. From 1 (z) = 2 (z) = 1+b, it follows that when b > 1, individual output and profits expand with entry (Result 2). But now, the denominator in equation (4) may become negative if n is big enough, hence, total output becomes na z n = if n < 1 ; 1+n(1 b) b 1, otherwise. Notice that whenever b 2, equilibrium total output is zero, then, we focus on the values 1 < b < 2. Let n denote the maximum number of firms for which the industry is viable (that is, for which z n > ); then, n = 1/(b 1) 1 = min{n N : 1/(b 1) n} 1. Since π n 16

17 increase in n n and π 1 > K, free entry leads to n e = n firms. Similarly, it can be shown that the regulator will pick n = n e = n. Proposition 5 Let P (z, s) = a + bs z and c = ; a >, 1 < b < 2, < K < a 2 /(2 b) 2, and n = 1/(b 1) 1. Then, n e = n = n. Proposition 5 provides an example of an industry where n e = n is finite. Example 3 shows that n e = n might be infinite. Example 3. Consider an industry with a compatible network, inverse demand function P (z, s) = a + bs α z, with a, b and α strictly positive, and costless production (c = ). Observe that if 1 (z) = 2 (z) = 1 + αbz α 1 on [z n, z n+1 ], then, x n and π n globally increase with n (Result 2). For instance, such is the case when a = 1.3, b =.99, α = 1.2 and K = 2, then, the number of firms that enter the market is unbounded, which coincides with the regulator s decision. Note that 1 (z) = 2 (z) increases in z provided α > 1. By AZ, z n increases in n and thus, 1 (z) = 2 (z) take their lowest value when n = 1. Hence, z 1 = 1.24 establishes the lower bound of 1 (z) = 2 (z), which is equal to Final Remarks We have shown that a simple modification to the standard Cournot oligopoly model the existence of network effects is enough to change the conventional wisdom that excessive entry (under-entry) is a direct consequence of the business-stealing (business-enhancing) competition. With a single-network, under-entry occurs under business-enhancing competition, but may also happen with the business-stealing competition. Moreover, under-entry as 17

18 a consequence of business-enhancing competition is more common when network effects exist (provided its plausibility discussed in Proposition 3), then, under-entry in oligopolies may be more common than one would think. Recall that without network effects, the firms decrease their profits with competition, and in general, they also decrease their production with entry (business-stealing competition) hence, there is excessive entry. Consequently, the firms obtain less profits than in the socially optimal scenario and the consumers benefit from it by getting lower prices. With the existence of network externalities, such as in telecommunications industries, there might be under-entry even if the firms hurt each other with competition, then, the firms are better off with free entry and the consumers decrease their surplus. Hence, one has to be very careful when allowing free entry of firms if the objective is to maximize consumer surplus; although it is natural to assume that free entry benefits the consumers, it might be the opposite in the presence of network effects. 5 Appendix Proof of Proposition 1. The assumption < K < a 2 /(2 b) 2 implies that at least one firm is active, π 1 > K. < b < 1 implies that 1 (z) = 2 (z) = 1 + b <, so that x n and π n decrease in n (Result 1). In general, the problem of the regulator is zn max [W n nk = P (t, z n )dt ncx n nk], n 18

19 with FOC: π n K + n[p (z n, z n ) c]x n + z n zn P 2 (t, z n )dt =. (5) When P (z, s) = a + bs z and c =, the left-hand side of the FOC (5) becomes and the second order condition (SOC) is a 2 (1 + bn) (1 + n(1 b)) K = π n(1 + bn) K, (6) n(1 b) a 2 (4b 3 2bn(1 b)) (1 + n(1 b)) 4 <, which holds for b.75 or for n sufficiently large, n > (4b 3)/[2b(1 b)]. i) Assume that < b <.5. At n e + 1, equation (6) becomes π n e +1(1 + b(n e + 1)) 1 + (n e + 1)(1 b) K < K(1 + b(ne + 1)) 1 + (n e + 1)(1 b) K = K(ne + 1)(2b 1) 1 + (n e + 1)(1 b) <. The first inequality follows by the free entry condition, π n e +1 < K, and the second one, by < b <.5. Then, provided that W n is strictly concave in n (given b.75), the regulator must decrease the number of firms in order to optimize social welfare, n < n e + 1, which implies n n e. ii) Now suppose that.5 b < 1. Analogous to part i), but using the fact that π n e K, we have π n e(1 + bn e ) 1 + n e (1 b) K K(1 + bne ) 1 + n e (1 b) K = Kne (2b 1) 1 + n e (1 b). If.5 b.75, the result follows immediately by the strict concavity of W n. Otherwise, if.75 < b < 1, the second order condition holds when n > (4b 3)/[2b(1 b)]. Similarly, note that when n = 1, the regulator can (weakly) improve social welfare with an additional firm, 19

20 which can be seen from equation (6): π 1 (1 + b) 2 b K > K(1 + b) 2 b K = K(2b 1) 2 b (the first inequality follows by (A5), π 1 > K, and the second one, by.5 b < 1). Hence, social welfare is first increasing and convex, then changes curvature and finally decreases. Altogether, we have n e n. Proof of Remark 1. Let n be an interior solution of the regulator, then, ignoring the integer constraint, it satisfies the FOC (5). i) If n [P (z n, z n ) c]x n +z n zn P 2 (t, z n )dt, then π n K, and since π n π n+1 for all n, we have n e n. The proof of ii) is analogous to that of i). Proof of Proposition 2. First notice that since P 1 < and x n +1 > (if x n +1 = the result is immediate, n = n e = ), x n +1P (z n +1, z n +1) Also, by optimality of n, (n +1)x n +1 n x n +1 P (t, z n +1)dt < P (t, z n +1)dt. (7) W n n K W n +1 (n + 1)K, 2

21 that is zn P (t, z n )dt n cx n n K zn +1 P (t, z n +1)dt (n + 1)cx n +1 (n + 1)K. Then, we have the following expressions π n +1 K x n +1P (z n +1, z n +1) + zn zn x n +1P (z n +1, z n +1) + n x n +1 < P (t, z n +1)dt + P (t, z n )dt P (t, z n +1)dt zn zn +1 zn +1 P (t, z n +1)dt n c(x n x n +1) n (x n x n +1)P (z n +1, z n +1) n c(x n x n +1) = n [P (z n +1, z n +1) c](x n x n +1). P (t, z n +1)dt n c(x n x n +1) P (t, z n +1)dt n c(x n x n +1) The first inequality follows by π n +1 = x n +1P (z n +1, z n +1) cx n +1; the second one by z n +1 z n (AZ) and P 2 >, and the third one by (7). The fourth inequality is given by P 1 < and z n +1 n x n +1 n x n = z n. The last one follows by P (z n +1, z n +1) c and x n +1 x n. Hence, π n +1 < K, and by π n π n+1, we have the result. Proof of Proposition 3. By AZ Lemma 9i, 2 (z) on [z n, z n+1 ], implies that x n x n+1. Now observe that π n = x n [P (z n, z n ) c], taking derivative with respect to n, we have π n = x n[p (z n, z n ) c] + z n[p 1 (z n, z n ) + P 2 (z n, z n )]x n. 21

22 Since z n = nx n, we have z n = nx n + x n, and then π n = x n[p (z n, z n ) c + z n [P 1 (z n, z n ) + P 2 (z n, z n )]] + x 2 n[p 1 (z n, z n ) + P 2 (z n, z n )]. If [P 1 (z, z) + P 2 (z, z)] [P (z, z) c]/z and P 1 (z, z) + P 2 (z, z) on [z n, z n+1 ] for all n, we have, π n π n+1 for all n. Proof of Proposition 4. First, since π n increases in n and π 1 > K, it is clear that the firms always have incentives to enter, and hence, n e n. Now suppose that x n x n+1 for all n. Then, W n K = π n K + n[p (z n, z n ) c]x n + z n > n[p (z n, z n ) c]x n + z n zn P 2 (t, z n )dt. zn P 2 (t, z n )dt The first inequality follows by π 1 > K and π n increasing in n; the second one by x n and z n increasing in n, P (z n, z n ) c, and P 2 >. As a consequence, the regulator s objective function is also increasing in n and therefore, she will allow as many firms as possible, which coincides with the firms behavior without regulation. Proof of Proposition 5. Recall that n denotes the maximum number of active firms in the industry; with more than n firms, they will stop to produce. From 1 (z) = 2 (z) = 1 + b >, it follows that π n increases in n for all n < n (Result 2), and so does π n K. Then, (A5) π 1 > K implies that π n > K; as a consequence n e = n. 22

23 On the regulator s side, from the proof of Proposition 1, we have that W n nk is strictly convex in n whenever b > 1, specifically, the SOC of the regulator is then, it chooses between 1 and n firms. a 2 (4b 3 2bn(1 b)) (1 + n(1 b)) 4 > ; Since π 1 > K and π n increase in n, we have that π n > K for all n n; together with b > 1 and equation (6), we have (W n nk) n = π n(1 + bn) K(1 + bn) nk(2b 1) K > K = 1 + n(1 b) 1 + n(1 b) 1 + n(1 b) > for all n < n, then, social welfare is maximized at n = n. References Amir, R. (1996). Cournot Oligopoly and the Theory of Supermodular Games. Games and Economic Behavior, 15: Amir, R., De Castro, L., and Koutsougeras, L. (214). Free Entry versus Socially Optimal Entry. Journal of Economic Theory, 154: Amir, R. and Lambson, V. E. (2). On the Effects of Entry in Cournot Markets. The Review of Economic Studies, 67(2): Amir, R. and Lazzati, N. (211). Network Effects, Market Structure and Industry Performance. Journal of Economic Theory, 146(6): Katz, M. L. and Shapiro, C. (1985). Network Externalities, Competition, and Compatibility. The American Economic Review, 75(3):

24 Mankiw, N. G. and Whinston, M. D. (1986). Free Entry and Social Inefficiency. The RAND Journal of Economics, pages Shapiro, C. (2). Navigating the Patent Thicket: Cross Licenses, Patent Pools, and Standard Setting. Innovation Policy and the Economy, 1: