Increased Uncertainty Drives Firms to Merge

Transcription

1 Working Paper Series No.112, Faculty of Economics, Niigata University Increased Uncertainty Drives Firms to Merge Kojun Hamada Series No.112 Address: Faculty of Economics, Niigata University 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata City Japan address: Tel. and fax:

2 Increased Uncertainty Drives Firms to Merge Kojun Hamada Faculty of Economics, Niigata University January 23, 2010 Abstract We examine the effect of uncertainty on horizontal mergers, in the view that firms must decide whether to merge before observing the realized cost of the merged firm. The existing literature clarifies that merging firms facing cost uncertainty decide to merge if expected efficiency gains or informational advantages exist. We show that even if there is no uncertain efficiency gain, as the variance of uncertainty grows, horizontal mergers generate large expected profits than before the merger and improve consumer surplus and welfare in expected terms. The findings suggest that increased uncertainty itself provides the incentive for firms to merge. JEL classification: D43; L13; L22 Keywords: Horizontal merger; Uncertainty; Bayesian Cournot/Nash equilibrium Faculty of Economics, Niigata University Address: 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata City , Japan Tel. and fax:

3 1 Introduction In recent decades, the trends of globalization and the information and communication technology (ICT) development have led to drastic and unforeseeable changes in business. Moreover, the global financial crisis since 2007, which was in part amplified by globalization and the ICT development, has been intensifying uncertainty in world economy. In view of business journalism, it is argued that the growing uncertainty promotes firms to engage in mergers and acquisitions (M&A) activities. For instance, in the semiconductor industry, the global DRAM market tends to raise the concentration ratios and has experienced the restructuring of the industry including not only technical cooperation but also M&A activities, mainly due to the increase in market volatility and technological uncertainty. 1 Likewise, the recent consolidations in biotechnology and oil industries reflect the growing uncertainty caused by the rising R&D expenditures and raw material costs, respectively. 2 This paper aims to examine whether the increased uncertainty promotes firm mergers. One of the conventional ideas explaining the effects of uncertainty on horizontal mergers is that the enlargement of the market share enhances the firm s capacity to cope with the increased uncertainty. If horizontal mergers are convenient tools to immediately increase the market share, they should have been more prevailing along with the increased uncertainty. However, existing literature under complete 1 The DRAM industry has intensified the degree of oligopolization for a decade. In isuppli News Release, Samsung Electronics with the largest share in the DRAM market had 22% share in 2001, and it increased to 34.1% in the second quarter of C3 (three-firm concentration ratio) increased from 62% in 2001 to 72% in HHI3 (Herfindahl-Hershman Index by the top 3 firms) increased from 1,292 in 2001 to 1,906 in In sd quarter of 2009, Hynix (No.2 share), Elpida (No.3), and Micron (No.4) had 21.7%, 16.5%, and 13.8% shares, respectively. Due to the significant demand shrinking of the semiconductor caused by the global financial crisis, Qimonda, which was a top German semiconductor company with the fourth largest world share (11.5%) in 2007, bankrupted in January 2009, and consolidation in the DRAM industry is becoming more active. At present (in October 2009), Hynix is considering the sale of its own company. Elpida and Micron have independently been negotiating to integrate with Taiwan semiconductor makers. 2 Banal-Estañol (2007) points in his introduction that the recent wave of consolidation in oil and biotechnology industries has been related to the large scale uncertainty in technological or market volatility. 2

4 information without uncertainty shows that unless the merger acquires sufficiently large market share, it is unprofitable. In a seminal paper, using a model in a symmetric Cournot oligopoly with linear demand and costs, Salant et al. (1983) showed that the merger is profitable only if the merged firm occupied a premerger market share of at least 80%, which is referred to as merger paradox. In the deterministic market without any demand and cost uncertainty, horizontal mergers are profitable only if the industry is already very concentrated (Perry and Porter, 1985). In order to resolve the merger paradox and give satisfactory explanation to the profitable merger, the existing literature allows merging firms to exploit production synergies such that the merger reduces their post-merger costs (Farrell and Shapiro, 1985). As other explanations, for example, the concavity of demand function and market structures other than Cournot quantity competition are derived (Fauli Oller, 1997 and Deneckere and Davidson, 1985). Once the uncertainty or the informational asymmetry is derived in the model, most of the theoretical analysis succeeds in explaining the profitability of mergers caused mainly by reducing the post-merger cost. Amir et al.(2009) explore the model to explain profitable merger under uncertainty, and they clarify that the scope of the profitable merger enlarges with uncertainty. They presume uncertain efficiency gains, in the sense that the merged firm s expected cost exceeds the nonmerged outsider firms costs (which is same as the premerger firms costs). The uncertain efficiency gain affects the ex ante beliefs on the merged firm s cost by outsiders and elicits the competitive advantage to the merged firm from strategic aspects. Choné and Linnemer (2008) explain the role of uncertainty over efficiency gains in a very general model with multiproduct firms. Banal-Estañol (2007) shows that there are added incentives for mergers because the merged firms can aggregate private information among the premerged firms and it can properly adjust production levels. In a cost structure different from that of Banal-Estañol (2007), Stennek (2003) also shows that the mergers with cost uncertainty and private information increase consumer welfare if consumers are sufficiently risk-averse. The existing literature explains profitable merger by uncertain efficiency gains or informational advantages by aggregating private information. In contrast, this paper takes a different approach to explain 3

5 whether or not increased uncertainty promotes firm mergers. We show that even if there is neither expected efficiency gains nor informational advantage under private information, the increased uncertainty itself is possible to urge firms to merge. A merged firm can properly adjust the production level in response to the realized costs after uncertainty is resolved. The ex post ability for the merged firm to adjust production level raises the expected profit of the merged firm. Although the (expected) level of cost of the merging firms remains unchanged between premerger and post-merger, if uncertainty is sufficiently large, the merit of the merger by properly adjusting production levels exceeds the loss by losing the market share by shrinking production under strategic substitutes, as is shown in the existing literature in the deterministic model. Our result suggests that as the uncertainty grows, the firms have more incentive to merge. This result provides a consistent explanation to real business phenomenon that cases of M&A activities increase with increased uncertainty. As regarding the welfare analysis, when we compare the consumer surplus and the social welfare before and after merger, as the increased uncertainty grows, the post-merger expected consumer surplus and social welfare exceeds those of the premerger. Although it may seem paradoxical at first glance that the increase in cost uncertainty after merger raises the post-merger social welfare and consumer surplus, this result can be explained by similar reason that the increased uncertainty raises profitability of the merged firm. After deciding to merge, the merged firm observing the realized cost can properly adjust the quantity level. When low (resp. high) cost is realized, the merged firm with a cost advantage (resp. disadvantage) enlarges (resp. shrinks) production levels. This ability of the merged firm to adjust quantity levels to reflect the realized true cost generates the additional expected surplus by causing the proper production in the industry that reflect the cost difference between the merged and nonmerged firms. In other words, when uncertainty is sufficiently large, appropriate response to realized cost level by the merged firm provides for appropriate allocation of resource about production activities in the industry, and it results in the increase of not only the additional expected profit but also the social welfare. If we turn our attention to the relationship between merger incentives and private information, moral 4

6 hazard by the firm s manager as an agent might present one of the reasons why uncertainty promotes mergers. As is corroborated by a lot of articles in the agency theory, when the manager whose objective diverges from that of the principal, choosing between a high-risk and high-return project and low-risk and low-return one, the manager is apt to choose the high-risk and high-return project. 3 If under uncertainty, horizontal mergers are riskier projects for the firm s manager, the agency theory might suggest that the increased uncertainty urges the firm s manager to choose the merger. In contrast, this paper puts aside from consideration any agency problem under private information. However, by using a very simple setting, our paper clarifies that even if there is no incentive problem caused by a conflict of interest within the merged firm, cost uncertainty itself creates an incentive for firms to merge under incomplete information; further, it concludes that the uncertainty increases the expected social welfare. The remainder of the paper is organized as follows. Section 2 introduces the basic model. Section 3 analyzes the effects of uncertainty on mergers. Section 4 presents the welfare analysis. Section 5 extends the results in the previous sections. Section 6 concludes the paper. 2 The model Most of the basic setting in our model utilizes the model of Amir et al.(2009); the difference is in the cost structure. In the premerger situation, we consider a homogeneous good market in which n + 1 (n 2) identical firms engage in quantity competition in a Cournot fashion. Each firm has a constant marginal cost c and chooses quantity level q in the market with inverse demand P = a bq,wherea > c > 0and b > 0. Each firm s premerger output and profit in the Cournot equilibrium are as follows: q = a c b(n + 2) and π = (a c)2 b(n + 2) 2. (1) In the post-merger situation, we focus on only a bilateral merger. (n 1) nonmerged firms and a merged firm engage in a Bayesian Cournot game. At the ex ante stage before two firms decide 3 As a typical paper, see Stiglitz and Weiss (1981). 5

7 whether to merge all firms in the market are uncertain over the marginal cost of the merged firm. Thus, any two merging firms must decide whether or not to merge without observing the true cost of the merged firm. We assume a binomial distribution under which the firms believe that with probability p (resp. (1 p)), the merged firm will have a lower (resp. higher) marginal cost c l < c (resp. c h > c). Thus, with probability p (resp. (1 p)), the merged firm has experienced efficiency gains (resp. loss). In order to focus on the effect of uncertainty on mergers without any uncertain efficiency gains, we assume that the expected marginal cost of the merged firm is equal to the nonmerged firm s cost (which is the same as the premerger firm s one), i.e., c pc l +(1 p)c h = c. It should be noted that unlike Amir et al. (2009), there is no ex ante expectation about efficiency gains by the merger. The variance of the binominal distribution is denoted by σ 2 p(1 p)(δc) 2,whereΔ c h c l. The probability p (0,1) is common knowledge amongst all firms. Irrespective of whether or not to merge, all firms recognize that in expected terms, the firm s cost remains unchanged before and after merger. 4 At the ex post stage, after two firms decide to merge, only the merged firm observes its cost (or type) and conditions its output level on its realized type. Each outsider firm chooses its output level to maximize its ex ante expected profit. Let q l m and qh m be the merged firm s quantities conditional on being the low-cost type and high-cost type, respectively, and Eq m pq l m +(1 p)qh m be its expected quantity. q o denotes each outsider s quantity. By solving the intersection of reaction functions, as is delineated in Figure 1, the Bayesian Nash equilibrium quantities are obtained as follows: q l m = 2(a c) (n + 1)(c l c) 2b(n + 1) q o =, q h m = 2(a c) (n + 1)(c h c), (2) 2b(n + 1) a c b(n + 1), and Eq m = a c b(n + 1). (3) 4 This paper has changed one of the specific assumptions that Amir et al. (2009) impose. They assume that it does not allow for the possibility that a merger might increase the final cost of the merged firm, because the merging firms have the option of keeping the premerger production set up unchanged in both their plants. However, from both theoretical and empirical viewpoints, it is not confirmed that the merged firm can always produce with a lower cost. For example, as Williamson (1985) has suggested through the thought of selective intervention, the merged firm cannot necessarily replicate the performance of merging firms. This paper allows for the merged firm to increase the cost by considering the possibility of the failure of merger. 6

8 q h m < q o = Eq m < q l m is satisfied. We make the following assumption for all these quantities to be strictly positive. Assumption 1: Δc < 2(a c) p(n+1). The corresponding market prices are as follows: P l = 2(a + nc)+(n + 1)(c l c) 2(n + 1), P h = 2(a + nc)+(n + 1)(c h c), 2(n + 1) and EP = a + nc n + 1. (4) Figure 1 around here The expected equilibrium profits of the merged firm, conditional on its cost type, c l or c h,andthe profits of each outsider firm are, respectively, as follows: π l m = (2(a c) (n + 1)(c l c)) 2 4b(n + 1) 2, π h m = (2(a c) (n + 1)(c h c)) 2 4b(n + 1) 2, (5) and Eπ o = (a c)2 b(n + 1) 2. (6) It should be noted that π h m < Eπ o < π l m is satisfied. There are no uncertain efficiency gains in the cost structure of this paper, which differs from that in Amir et al. (2009). As outsider firms recognize no efficiency gains created by the merger in expected terms, the informational asymmetry about the merged firm s type between the merged firm and the outsiders does not work in favor of the merged firm. Therefore, when the merger succeeds with low cost, the merged firm obtains larger profit than the expected profits of outsiders only because of its cost advantage, while the adverse result is obtained when the merger fails with high cost. In the cost structure in which the expected cost of the merged firm is same as the outsider firm s cost, there is no strategic advantage to the merged firm. 7

9 However, although there are no expected efficiency gains, it is possible that the merger is profitable under incomplete information, which differs from the existing literature under complete information. In the following section, we investigate the possibility that two premerger firms have the incentive to merge. 3 Merger analysis This section examines the incentives to merge and compares outputs, price, and the profit of the outsider firms before and after merger. As the merging firms must decide whether or not to merge when cost uncertainty is not resolved, we compare the expected profit of the merged firm in the Bayesian Cournot equilibrium with the premerger firm s profit in order to derive the condition for the mergers to be profitable. 5 The ex ante expected profit of the merged firm is as follows: Eπ m pπ l m +(1 p)π h m = 4(a c)2 +(n + 1) 2 σ 2 4b(n + 1) 2. (7) The merger is profitable if Eπ m > 2π. By simple calculation, the following proposition is obtained. Proposition 1. Suppose Assumption 1 holds. If the variance of the uncertainty is sufficiently large, i.e., with σ 2 4(a c)2 (n 2 2) (n+1) 2 (n+2) 2, the merger is profitable. More exactly, if p < (n+2)2 2(n+1) 2, the pair of cost difference and number of firms (Δc, n) exists in which the merger is profitable. In particular, for all p 1 (n+2)2 2, there exists (Δc,n) in which the merger is profitable. On the other hand, if p, any 2(n+1) 2 merger is unprofitable. Proof. By simple calculation, it is shown that if and only if Eπ m > 2π, σ 2 p(1 p)(δc) 2 4(a c)2 (n 2 2) (n+1) 2 (n+2) 2. In order for σ 2 4(a c)2 (n 2 2) (n+1) 2 (n+2) 2 2(a c) ( n 2 2 (n+1)(n+2) p(1 p), 2(a c) p(n+1) ). to be satisfied under Assumption 1, it is necessary that Δc lies in Δc n 2 2 p(1 p) < 2(a c) (n+2)2 p(n+1) if and only if p <. In the case of 2(n+1) 2 2(a c) (n+1)(n+2) 5 Amir et al. (2009) also investigate the worst and the best case scenarios wherein the merged firm fails to achieve high cost and low cost, respectively. However, any firm including the merging firms cannot recognize the future scenarios ex ante. We limit to the argument of only the expected profits. 8

10 p < (n+2)2,ifδc (0, 2(n+1) 2 ( 2(a c) n 2 2 (n+1)(n+2) p(1 p), 2(a c) p(n+1) 2(a c) (n+1)(n+2) n 2 2 p(1 p) ), the merger is unprofitable. If Δc (n+2)2 ), the merger is profitable. As is a decreasing function with respect to n 2(n+1) 2 n 2 2 (n+1)(n+2) p(1 p), 2(a c) p(n+1) ). (n+2) and lim 2 n = 1 2(n+1) 2 2,ifp 1 2(a c) 2, there always exists (Δc,n) that satisfies Δc ( On the other hand, in the case of p (n+2)2, for all regions of Δc (0, 2(a c) 2(n+1) 2 p(n+1) ), the merger is unprofitable. Proposition 1 implies that even when the expected cost is same before and after merger, as the variance of cost uncertainty grows larger, firms have the incentive to merge. In (7), the expected profit of the merged firm increases with the variance σ 2. Thus, as the variance becomes larger, the expected profit of the merger grows. When the size of the variance exceeds a threshold, as shown in Proposition 1, the expected profit of the merged firm becomes larger than the sum of the premerger firm s profit, and firms facing cost uncertainty choose to merge facing cost uncertainty. Proposition 1 suggests that even if there are no uncertain efficiency gains, the ex ante cost uncertainty itself will urge the firms to merge. The intuition of Proposition 1 is explained by the convexity of profit function with respect to the cost of the merged firm, that is, 2 π i m c 2 i = qi m c i > 0,i {l,h}. As πm i (qi m ) is a convex function with respect to q i m, π m (Eq) < Eπ i m is satisfied, as is shown in Figure 2. Therefore, as the variance increases, Eπ m becomes larger than π m (Eq m ). If the variance is sufficiently large, the case in which Eπ m > π m (Eq m ) can take place. Figure 2 around here Under strategic interaction in the oligopolistic competition, the decrease in unit cost increases the quantity level, due to both the direct effect on cost advantage and the indirect effect by making the outputs of outsider firms shrink, and results in the enlargement of the profit of the merged firm. These effects enlarge as the cost difference increases, that is, as the uncertainty grows, as shown in (7). This result is derived from the adjusting effect on production level. In response to the realization of cost, the merged firm can properly adjust production level under low cost or high cost. If this adjustment effect 9

11 exceeds the profit loss by the shrinking effect of production under strategic substitutes, the increased uncertainty drives the firms to merge. In this setting of the model, the merging firms are risk-neutral in that they decide whether to merge only on comparing the expected profits. However, as Eπ m increases with respect to σ 2, as shown in (7), it seems that the firms behave as if they are risk-lovers, as shown in Figure 2. Amir et al. (2009) showed that when the uncertain efficiency gains are expected under the merger, the ex ante belief about the merged firm s cost makes the outsider firms shrink their output levels under strategic substitutes, and the merged firm acquires the strategic advantage through the belief formation. In contrast, in this paper, there is no such strategic advantage that is generated by the ex ante belief that outsider firms form. Instead, it is shown that unless there is no expected cost advantage, the adjustment effect on production in response to the different cost raises the expected profit of the merged firm. In the deterministic model, as Salant et al. (1983) first showed, unless the market share is sufficiently large, most of the horizontal mergers are unprofitable. In Proposition 1, as the variance is sufficiently close to zero (σ 2 0), the merger is unprofitable, as is shown by Salant et al. (1983) that π(n 1) < 2π(n), unless at least 80% market share is attained by the merger. However, in the stochastic model of this paper, even when the merger is unprofitable in the deterministic case, as the variance grows larger, the expected profit also increases because the gain of the optimal quantity adjustment enlarges, and it is possible that the expected profit Eπ m exceeds the sum of profits of the premerger firms 2π. Therefore, Proposition 1 presents one of the explanations of the merger paradox. Let us present a numerical example with the cost difference Δc unchanged. As we can normalize a = 1andc = 1 2 by converting units without loss of generality, and the variance is maximized when p = 1 2, we specify a = 1, c = 1 2,andp = 1 (n+2)2 2.Asp < 2(n+1) 2 is always satisfied under this example, there exists (n,δc) for mergers to be profitable. Assumption 1 is Δc < 2 n+1, and the condition for the profitable merger is Δc 2 n 2 2 (n+1)(n+2). Thus, if Δc lies in Δc ( 2 n 2 2 (n+1)(n+2), 2 n+1 ), the merger is profitable. We present the ranges of Δc wherein the merger is profitable in Table 1 and the region of Δc in Figure 3. 10

12 Table 1 around here Figure 3 around here By Table 1, for example, when (n,δc) =(5, 1 4 ), the merger is profitable. It should be noted that unlike the deterministic model under complete information, under incomplete information, even if the number of firms in the market is sufficiently large, two firms have the incentive to merge. This result is in sharp contrast with that in Salant et al. (1983), under which the market share of at least 80% is needed for profitable mergers, and when n = 5, at least four firms decide to merge. We investigate the effects of the merger on the (expected) outputs and outsiders profits. Proposition 2. (i ) The merged firm contracts the (expected) output, and the outsider firms expand output, i.e., Eq m < q < q o. (ii ) The merger decreases (expected) total output, increases industry price, and increases an outsider firm s (expected) profit, i.e., (n 1)q o + Eq m < (n + 1)q, EP > P, and Eπ o > π. Proof. All these are obtained immediately by simple calculation. The result of Proposition 2 is the one that is forecasted by the existing literature. The decrease in the number of firms decreases the total output and increases the price. The merged firms reduce their output in order to internalize the external effect between the merging firms, and the outsider firms obtain the fish in troubled waters under strategic substitutes. 4 Welfare analysis Consumer surplus (CS), producer surplus (PS), and social welfare (SW) before the merger are as follows: CS = (n + 1)2 (a c) 2 (n + 1)(a c)2 (n + 1)(n + 3)(a c)2 2b(n + 2) 2, PS = b(n + 2) 2, and SW = 2b(n + 2) 2. (8) We evaluate the welfare effect of a merger in expected terms. In merging, the expected CS is ECS pcs l +(1 p)cs h,wherecs i = b 2 (Qi ) 2,i {l,h}. As the ex post PS conditional on the realized cost 11

13 isps i =(n 1)π i o +π i m,i {l,h}, the expected PS is EPS pps l +(1 p)ps h =(n 1)(pπ l o +(1 p)π h o )+Eπ m. The expected SW is ESW ECS+ EPS. By simple calculation, we obtain the following expected values of CS, PS, and SW when the merger is chosen. ECS = 4n2 (a c) 2 +(n + 1) 2 σ 2 8b(n + 1) 2, EPS = 4n(a c)2 +(n + 1) 2 σ 2 4b(n + 1) 2, and ESW = 4n(n + 2)(a c)2 + 3(n + 1) 2 σ 2 8b(n + 1) 2. (9) It is worth noting that the surpluses are increasing functions with respect to the variance σ 2. Thus, as the uncertainty increases, the surpluses grow in expected terms. By comparing CS, PS, and SW before and after merger in expected terms, we obtain the following proposition. Proposition 3. The merger increases the expected PS, irrespective of the size of the variance. When the variance becomes sufficiently large, the merger increases CS and SW in expected terms. That is, if σ 2 > 4(2n2 +4n+1)(a c) 2,ECS> CS. If σ 2 > 4(2n+3)(a c)2,esw> SW. (n+1) 2 (n+2) 2 3(n+1) 2 (n+2) 2 Moreover, if p > , the CS cannot rise after merger. If p 1 3, there exists the pair of cost difference and number of firms (Δc,n) in which the merger raises the consumer surplus. If p < , there exists the pair of cost difference and number of firms (Δc,n) in which the merger raises the social welfare. Proof. EPS > PS if and only if σ 2 > 4(n(n+1) 1)(a c)2 (n+1) 2 (n+2) 2. As the variance is positive and the righthand side of the above inequality is negative, EPS > PS is always satisfied, irrespective of the sizes of variance and the number of firms. By simple calculation, it is obtained that ECS > CS if and only if σ 2 > 4(2n2 +4n+1)(a c) 2 (n+1) 2 (n+2) 2 2(a c) (n+1)(n+2) 2(a c) 2n 2 +4n+1 (n+1)(n+2) p(1 p), 2(a c) p(n+1) 2(a c) 2n 2 +4n+1 (n+1)(n+2) p(1 p),andesw > SW if and only if σ 2 > 4(2n+3)(a c)2. σ 2 > 4(2n2 +4n+1)(a c) 2 is ar- 3(n+1) 2 (n+2) 2 (n+1) 2 (n+2) 2 2n 2 +4n+1 p(1 p). Under Assumption 1, if Δc lies in ranged as Δc > ( CS. As ), there exists the cost difference in which ECS exceeds the premerger < 2(a c) p(n+1) there exists (Δc,n) that satisfies ECS > CS. if and only if p < (n+2)2 (n+2) 2 (n+1)(3n+5) (n+1)(3n+5),ifp < (n+2)2 (n+1)(3n+5) is satisfied, is a decreasing function. When n = 2, this 12

14 value is and lim n (n+2)2 = (n+1) 2 3. Thus, if p , for all n, the merger decreases the CS. On the other hand, if p 1 3, there always exists (Δc,n) that satisfies ECS > CS. Likewise, σ 2 > 4(2n+3)(a c)2 is arranged as Δc > 2(a c) 2n+3 3(n+1) 2 (n+2) 2 (n+1)(n+2) 3p(1 p). Under Assumption 1, if Δc lies in ( 2(a c) 2n+3 (n+1)(n+2) 3p(1 p), 2(a c) p(n+1) As 2(a c) (n+1)(n+2) 2n+3 3p(1 p) < 2(a c) p(n+1) (n+3)(3n+5),ifp < 3(n+2)2 (n+3)(3n+5) is satisfied, there exists (Δc,n) that satisfies ESW > SW. ), there exists the cost difference in which ESW exceeds the premerger SW. if and only if p < 3(n+2)2 3(n+2) 2 (n+3)(3n+5) is an increasing function. When n = 2, this value is and lim n 3(n+2)2 48 (n+3)(3n+5) = 1. Thus, if p < , for all n, there always exists (Δc,n) that satisfies ESW > SW. It seems at first glance that the result that the expected consumer surplus and social welfare increase as the degree of uncertainty enlarges is counter intuitive. However, the similar logic is applied as that when the expected profit of the merged firm increases along with the size of the variance. In other words, when the uncertainty grows, the ability of the merged firm to properly adjust the quantity level, conditional on the realized cost, generates the additional surplus. Let us present a numerical example. Like in Section 3, under a = 1, c = 1 2,andp = 1 2,theCS decreases by the merger, as shown in Proposition 3. As Assumption 1 and the condition for the social welfare to enhance are calculated as Δc < 2 n+1 and Δc > 2 6n+9 3(n+1)(n+2), respectively, if Δc lies in ( 2 6n+9 3(n+1)(n+2), 2 n+1 ), the mergers raise the social welfare. We present the ranges of Δc wherein the ESW increases in Table 2 and the region of Δc in Figure 4. Table 2 around here Figure 4 around here As Proposition 3 shows, when p > , the CS cannot raise by merger. Under the above example, as p = 1 2, the CS necessarily exaggerates by merger. As another numerical example, let us consider the case in which a = 1, c = 1 2,andp = 1 3. In this case, as Proposition 3 suggests, there exists (Δc,n) in which the merger raises the consumer surplus. As Assumption 1, the condition for the consumer 13

15 surplus to enhance, and the condition for the social welfare to enhance are calculated as Δc < 3 n+1, Δc > 3 4n 2 +8n+2 2(n+1)(n+2),andΔc > 12n+18 2(n+1)(n+2), respectively. 3 4n 2 +8n+2 ( 12n+18 2(n+1)(n+2), 3 4n 2 +8n+2 2(n+1)(n+2) 2(n+1)(n+2) > 12n+18 2(n+1)(n+2) is satisfied. If Δc lies in ), the mergers raise the social welfare, while they decrease the consumer surplus. If Δc lies in ( 3 4n 2 +8n+2 2(n+1)(n+2), 3 n+1 ), the mergers raise the consumer surplus as well as the social welfare. We present the ranges of Δc wherein the ECS and/or the ESW raise in Table 3 and the region of Δc in Figure 5. Table 3 around here Figure 5 around here 5 Extensions 5.1 the uncertain efficient loss In the previous section, we showed that even if there are no uncertainty efficiency gains in that the expectation of the post-merger marginal cost is the same as the premerger cost, it is possible for the merger to be profitable. As we analogize from this conclusion, although the uncertain efficiency losses are forecasted, as the variance of the uncertainty becomes larger when these losses are relatively small, it is expected that the merger is possible to be profitable. In this section, by modifying the model in Section 2, we show that even under the expectation of the uncertain efficiency losses, the result that horizontal mergers are profitable takes place. In the post-merger situation, the firms believe that with probability p the merged firm will have a lower marginal cost c l < c, while with probability (1 p) the merged firm will have a higher marginal cost c h > c. Unlike the setting in Section 2, we assume that the expected value of the post-merger firm s cost is higher than the premerger cost c pc l +(1 p)c h > c. The cost variance remains unchanged, i.e., σ 2 = p(1 p)(δc) 2. 14

16 The Bayesian Nash equilibrium quantities are as follows: q l m = 2a (n + 1)c l +(n 1)(2c c) 2b(n + 1) q o = a 2c + c b(n + 1), and Eq m =, q h m = 2a (n + 1)c h +(n 1)(2c c), 2b(n + 1) (10) a nc +(n 1)c. b(n + 1) (11) q h m < q l m, q h m < q o,andeq m < q o are satisfied. 6 All these quantities are strictly positive if one assumes 2(a 2c + c) > (n + 1)(c h 2c + c). The corresponding market prices are as follows: P l = 2a +(n + 1)c l +(n 1)(2c c), P l = 2a +(n + 1)c h +(n 1)(2c c), 2(n + 1) 2(n + 1) a +(n 1)c + c and EP =. (12) n + 1 The expected equilibrium profits of the merged firm, conditional on its cost type, c l or c h,andthe profits of each outsider firm are, respectively, as follows: π l m = (2a (n+1)c l+(n 1)(2c c)) 2 4b(n+1) 2, π h m = (2a (n+1)c h+(n 1)(2c c)) 2 4b(n+1) 2, (13) and Eπ o = (a 2c + c)2 b(n + 1) 2. (14) Note that π i m (q)=b(qi m )2 ; i {h,l} and Eπ o = bq 2 o. By combining them with qh m < ql m and qh m < q o,it is satisfied that π h m < πl m, πh m < Eπ o. Moreover, π l m > Eπ o, if and only if q h m > q o. The merger is profitable if Eπ m > 2π. By tedious calculation, as Eπ m = 4((a+(n 1)c)2 n(2a+2(n 1)c nc)c)+(n+1) 2 σ 2 4b(n+1) 2 is satisfied, Eπ m > 2π if and only if σ 2 4((n2 2)(a c) 2 + 2n(n + 2) 2 (a c)(c c) n 2 (n + 2) 2 (c c) 2 ) (n + 1) 2 (n + 2) 2. (15) By the condition for profitable mergers (15), the following proposition is obtained. Proposition 4. Even if c > c, when the variance of the uncertainty is sufficiently large so that (15) is satisfied, it is possible for profitable merger to take place. 6 q l m > q o if and only if c c l > c c. 15

17 Proof. Define ε c c > 0. The right-hand side of (15) is rewritten as 4((n 2 2)(a c) 2 +2n(n+2) 2 (a c)ε n 2 (n+2) 2 ε 2 ). When ε = 0, Proposition 1 shows that there always exists the (n+1) 2 (n+2) 2 pair of cost difference and number of firms, (Δc, n) in which the merger is profitable, when the variance is sufficiently large. The function of the left-hand side of (15) is continuous with respect to ε. By the continuity of the function, when ε is sufficiently smaller, there exists σ 2 that satisfies (15). Although the expected cost of the post-merger firm is less than the cost of the premerger firms, if the size of the variance is sufficiently large, the firms decide to merge. Proposition 4 suggests that even if the firm manager who engages in the merger activities forecasts the uncertain efficiency losses in expected terms, it is possible for the manager to have the incentive to merge. As a numerical example, consider a = 1, c = 0.5, c l = 0.34, c h = 0.68, p = 0.5, and c = 0.51 (Δc = 0.34 and σ 2 = ). Under the assumption that the quantities are positive under low cost, n < c c h+1 c+c h must be satisfied. By tedious calculation, it is shown that inequality (15) is satisfied for all n {2,3,4}. In order for (15) to be satisfied, the cost difference ε must be sufficiently small. Because the right hand side of (15) depends on ε with the squared order, as ε grows larger, the right hand side increases rapidly, and this inequality is not satisfied. Moreover, the cost difference between the low and high costs must be sufficiently large in order for (15) to be satisfied. Therefore, only if ε is sufficiently small and Δc is sufficiently large, the merger is profitable although the post-merger expected cost is higher than the premerger deterministic cost. This result is in sharp contrast with the existing literature, which emphasizes the synergy effects on horizontal mergers, including the uncertain efficiency gains. While we omit the detailed argument, by the similar logic, as in the previous section, it is shown that the expected consumer s surplus and social welfare are possible to rise in expected terms. 5.2 when the merged firm with high cost withdraws from the market Returning to the setting in Section 2, in which the expected cost remains unchanged before and after merger, we consider the case in which the merged firm withdraws from the market when it has high 16

18 cost. When the cost difference Δc is sufficiently large (Δc 2(a c) p(n+1) ), the merging firm with the high cost withdraws from the market. If the corner solution is considered, the reaction functions of the merging firms and the outsider firms are, respectively, as follows: q l m = Rl m (q o) max{ a c l b(n 1)q o,0}, 2b q h m = R h m(q o ) max{ a c h b(n 1)q o,0}, and 2b q o = R o (Eq m ) max{ a c beq m,0}. (16) bn As the nonmerged firms are not aware of the true cost of the merging firm, they decide their quantity levels q o with the expectation of the reaction function of the merging firm. The expectation of the reaction function is arranged as follows: If q o a c h b(n 1), Eq m = pr l m(q o )+(1 p)r h m(q o )= a c b(n 1)q o ; 2b if q o > a c h b(n 1), Eq m = pr l m(q o )= p(a c l b(n 1)q o ). (17) 2b As the quantity of high-cost type is zero when Δc 2(a c) p(n+1), the Bayesian Nash equilibrium is obtained by solving q o = R o (Eq m ) and Eq m = pr l m (q o) with respect to (q o,eq m ), as shown in Figure 6. Figure 6 around here The Bayesian Nash equilibrium quantities are as follows: q l m = a nc l +(n 1)c b(2n p(n 1)), qh m = 0, q o = 2(a c) p(a c l) b(2n p(n 1)), and Eq m = pq l m. (18) q l m > q o is satisfied. The corresponding market prices are as follows: P l = a +(n 1)c +(n p(n 1))c l 2n p(n 1) and,p h = 2a + 2(n 1)c p(n 1)c l, 2n p(n 1) EP = (2 p)a +(2 p)(n 1)c + pc l. (19) 2n p(n 1) The equilibrium profits of the merged firm and the expected profits of each outside firm are, respectively, as follows: π l m = (a nc l +(n 1)c) 2 b(2n p(n 1)) 2, π h m = 0, and Eπ o = (2(a c) p(a c l)) 2 b(2n p(n 1)) 2. (20) 17

19 As πm l = b(q l m) 2 and Eπ o = bq 2 o, πm l > Eπ o is satisfied. The condition for the merger to be profitable is Eπ m = pπm l > 2π. This condition is rewritten as follows: p(n + 2) 2 (a c + n(c c l )) 2 2(2n p(n 1)) 2 (a c) 2. (21) By (21), the following proposition is obtained. Proposition 5. We consider the case in which the high-cost merging firm withdraws from the market. If p < (n+2)2 2(n+1) 2, the firms have the incentive to merge, despite the possibility that the inefficient merging firm withdraws from the market. Proof. The assumption that the high-cost merging firm withdraws, Δc > 2(a c) p(n+1), is arranged as c c l > 2(1 p)(a c) p(n+1). Under this assumption, the left hand side of (21) satisfies that p(n+2) 2 (a c+n(c c l )) 2 > (n+2) 2 (2n p(n 1)) 2 (a c) 2. p < (n+1)2 if and only if (n+2)2 (2n p(n 1)) 2 (a c) 2 > 2(2n p(n 1)) 2 (a c) 2. p(n+1) 2 2(n+1) 2 p(n+1) 2 Thus, if p < (n+1)2 2(n+1) 2, the left hand side of (21) exceeds the right hand side of (21). Proposition 5 implies that although the post-merger firm exits from the market because the realized cost is sufficiently high, the merger is attractive in expected terms before the uncertainty is resolved. It seems at first glance that as the probability of low-cost type, p, becomes smaller, the merger is likely to be unprofitable. However, Proposition 5 shows that when p is relatively small, (21) is likely to be satisfied, and the possibility that the merger is profitable becomes larger. The reason is as follows: As c = pc l +(1 p)c h if and only if p = c c l Δc,largevalueofp corresponds with lower cost c l. When the high-cost firm withdraws from production, the decrease in p causes the trade off on the firm s profit. On the one hand, the decrease in p directly decreases the expected value of the merger profit. On the other hand, as lower p implies that c l has lower value, the expected profit rises when the cost is low. The former effect is the effect on the expected profit and the latter effect is that on the proper production adjustment. Proposition 5 suggests that when p is relatively small, the latter effect exceeds the former, and when the inefficiently merged firm withdraws from the market, more expected profit of the merged 18

20 firm is acquired than otherwise. By combining Proposition 1 with Proposition 5, we can conclude that when the uncertainty increases, irrespectively of whether or not the high-cost firm operates, there is the possibility that the merger is profitable. 6 Conclusion In this paper, we clarified that if uncertainty increases, even if it is not accompanied with any expected efficiency gains, the firms are likely to choose the mergers. Moreover, this paper presents the result that even if there is the ex ante uncertain efficiency losses, as the uncertainty increases, the merger can be possibly profitable. Amir et al. (2009) has already shown that when there is the uncertain efficiency gain after merger, firms tend to choose the horizontal merger. Their result presupposes that the post-merger firm never produces in an inefficient manner in that even if there is no efficiency after merger, the post-merger firm has the option to produce by at least the same technology as that of the premerger firms. In this presumption, the expectation with regard to the efficiency yields the competitive advantage, and the merged firm realizes higher profit. In contrast, our paper does not presume any uncertain efficiency gains. We assume that the ex ante expected cost to the post-merger firm is the same as that to the premerger firms. In this case, the merged firm has no competitive advantage to the nonmerger outsider firms. Nevertheless, through appropriate adjustment of production, as the uncertainty increases, the merger firms are expected to acquire higher profit than when the firms do not merge. This characteristic of firms risk taking is closely related to the conclusion arrived at by Choné and Linnemer (2008), which shows that the attitude towards uncertainty depends on the curvature of the objective function. They show that under linear demand, the consumer surplus is globally convex in quantity. As the CS is convex in quantity even in this model, the expected CS is larger than the CS under expected quantity. Thus, as uncertainty grows, the expected CS becomes larger. Likewise, as the firm s profit is convex in quantity, 19

21 it is possible that the expected merger profit is higher than the sum of the premerger firms profit through appropriate quantity adjustment. Finally, we note the remaining extensive issues. The paper deals with the specified linear demand and cost function with the binominal distribution of the cost uncertainty in order to explain the effect of uncertainty on horizontal mergers. It seems that the basic logic of the results in the paper remains unchanged even though the probability distribution is generalized in more general ways, including the continuous probability distribution. However, whether or not the similar conclusion is derived under generalized demand functions has not been clarified. It is probable that the convexity or concavity of the demand function affects the risk attitude toward the merging firm, and the risk premium must be calculated in a more complicated manner. Moreover, the paper focuses on the Cournot quantity competition. It is not obvious that the similar result can be obtained under different competitive forms including the Bertrand price competition. Acknowledgements This study was partially supported by a grant-in-aid from the Zengin Foundation for Studies on Economics and Finance. All errors are mine. References Amir R, Diamantoudi E, Xue L. Merger performance under uncertain efficiency gains. International Journal of Industrial Organization 2009;27; Choné P, Linnemer L. Assessing horizontal mergers under uncertain efficiency gains. International Journal of Industrial Organization 2008;26; Deneckere R, Davidson C. Incentives to form coalitions with Bertrand competition. Rand Journal of Economics 1985;16; Farrell J, Shapiro C. Horizontal mergers: An equilibrium analysis. American Economic Review 1990;80; Faulí-Oller R. On merger profitability in a Cournot setting. Economics Letters 1997;54;

22 Perry MK, Porter RH. Oligopoly and the incentive for horizontal merger. American Economic Review 1985;75; Salant SW, Switzer S, Reynolds RJ. Losses from horizontal merger: The effects of an exogenous change in industry structure on Cournot-Nash equilibrium. Quarterly Journal of Economics 1983;98; Stennek J. Horizontal mergers without synergies may increase consumer welfare. Topics in Economic Analysis & Policy 2003;3. Stiglitz JE, Weiss A. Credit rationing in markets with imperfect information. American Economic Review 1981;71; Williamson OE. The economic institutions of capitalism. New York: Free Press;

23 n Δc ( 2 n 2 2 (n+1)(n+2), 2 n+1 ) 2 (0.2357, ) 3 (0.2646, 0.5) 4 (0.2494, 0.4) 5 (0.2284, ) 6 (0.2083, ) 7 (0.1904, 0.25) n Δc ( 2 n 2 2 (n+1)(n+2), 2 n+1 ) 8 (0.1750, ) 9 (0.1616, 0.2) 10 (0.1500, ) 20 (0.0864, ) 50 (0.0377, ) 100 (0.0194, ) Table 1: Ranges of Δc wherein the merger is profitable (a = 1, c = 1 2,andp = 1 2 ) 2 6n+9 n Δc ( 3(n+1)(n+2), 2 n+1 ) 2 (0.2546, ) 3 (0.1732, 0.5) 4 (0.1277, 0.4) 5 (0.0991, ) 6 (0.0799, ) 7 (0.0661, 0.25) 2 6n+9 n Δc ( 3(n+1)(n+2), 2 n+1 ) 8 (0.0559, ) 9 (0.0481, 0.2) 10 (0.0420, ) 20 (0.0164, ) 50 (0.0044, ) 100 (0.0016, ) Table 2: Ranges of Δc wherein the merger raises the ESW (a = 1, c = 1 2,andp = 1 2 ) 22

24 n Δc ( 12n+18 2(n+1)(n+2), 3 4n 2 +8n+2 2(n+1)(n+2) ) (ESW increases but ECS decreases) Δc ( 3 4n 2 +8n+2 2(n+1)(n+2), 3 n+1 ) (ESW and ECS increase) 2 (0.2700, ) (0.7289, 1) 3 (0.1837, ) (0.5906, 0.75) 4 (0.1354, ) (0.4950, 0.6) 5 (0.1051, ) (0.4256, 0.5) 6 (0.0847, ) (0.3731, ) 7 (0.0701, ) (0.3320, 0.375) 8 (0.0593, ) (0.2991, ) 9 (0.0510, ) (0.2720, 0.3) 10 (0.0445, ) (0.2495, ) 20 (0.0174, ) (0.1363, ) 50 (0.0047, ) (0.0577, ) 100 (0.0017, ) (0.0294, ) Table 3: Ranges of Δc wherein the merger raises ECS and/or ESW (a = 1, c = 1 2,andp = 1 3 ) 23

25 Eq m Q o (Eq m ) q l m Eq m q h m 0 (n-1)q Eq m (Q o ) low cost high cost Q o Figure 1: Reaction curves of ex ante merged firm and aggregate outsiders π m π m (q i ) π l m Eπ m π m (Eq m ) π h m 0 q h m Eq m q l m q m Figure 2: Expected profit and profit of the expected output 24

26 Δc n Figure 3: Region of Δc wherein the merger is profitable (a = 1, c = 1 2,andp = 1 2 ) Δc n Figure 4: Region of Δc wherein the ESW increases (a = 1, c = 1 2,andp = 1 2 ) The dotted line denotes the lower bound of Δc in which the merger is profitable. 25

27 1 Δc 0.75 Both ESW and ECS increase ESW increases but ECS decreases n Figure 5: Region of Δc wherein ESW and/or ECS increase (a = 1, c = 1 2,andp = 1 3 ) Eq m R o (Eq m ) q l m Eq m (q o ) Eq m q h m = 0 0 q o high cost low cost q o Figure 6: Reaction curves of ex ante merged firm and an outsider firm 26