Profitability of Horizontal Mergers in the Presence of Price Stickiness

Esfahani, Hamideh Working Paper Profitability of Horizontal Mergers in the Presence of Price Stickiness Quaderni - Working Paper DSE, No. 747 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Esfahani, Hamideh (2011) : Profitability of Horizontal Mergers in the Presence of Price Stickiness, Quaderni - Working Paper DSE, No. 747, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4493 This Version is available at: https://hdl.handle.net/10419/159588 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Profitability of Horizontal Mergers in the Presence of Price Stickiness Hamideh Esfahani Quaderni - Working Paper DSE N° 747 Pro…tability of Horizontal Mergers in the Presence of Price Stickiness Hamideh Esfahani Department of Economics, University of Bologna Strada Maggiore 45, 40125 Bologna, Italy [email protected] First draft: December 2009 This draft: April 2011 Abstract In this paper, we investigate the pro…tability of horizontal mergers of …rms with price adjustments. We take a di¤erential game approach and both the open-loop as well as the closed-loop equlibria are considered. We show that the merger incentive is determined by how fast the price adapts to the equilibrium level. Keywords: Horizontal mergers, Di¤erential game, Sticky price JEL Classi…cation: C73, D43, L13 I would like to thank Luca Lambertini for his helpful comments and advice throughout this study. I also wish to thank Davide Dragone, Arsen Palestini and seminar participants at EBIM workshop, PhD forum at university of Bologna, ACDD, EUNIP and SAEe conferences and ASSET annual meeting for useful comments. Financial support from the University of Bologna is gratefully acknowledged. 1 Introduction When quantity-setting …rms compete in a homogenous product industry with symmetric cost and the same demand functions, horizontal merger is modelized as an exogenous change in market structure. As a result, the level of competition decreases which increases the market price and market power of …rms as well.In the case of linear demand and cost functions, the resulting anticompetitive forces are mostly to the bene…t of outsiders and mergers are advantageous to the merging …rm just in the circumstance that market share of merging …rm is extremely high, at least 80% which is almost merging to a monopoly (Salant, Switzer and Reynolds, 1983 (henceforth SSR); Gaudet and Salant, 1991, 1992). Keeping everything the same, this threshold will be reduced to 50% (which is again a considerable market share) provided that the merged entity is not restricted to remain a Cournot player after the merger (Levin, 1990) or any demand function which satis…es the second-order conditions is allowed (Cheung, 1992). There are other studies showing that mergers are privately pro…table if they are leader-generating (in industries where about less than one-third of the …rms are leaders) (Daughety, 1990), or if merger generates synergies (Perry and Porter, 1985; Farrell and Shapiro, 1990). However, the incentive to merge always exists once price is employed as the strategic variable rather than quantity. In a di¤erentiated product industry, Deneckere and Davidson (1985) demonstrate that mergers of any size are bene…cial if …rms are engaged in a price-setting game. We want to conduct an investigation into the consequences of horizontal mergers in oligopoly Cournot competition in the presence of price stickiness. When prices are sticky, for a given level of output the actual market price of a product does not adjust instantaneously to the price indicated by its demand function and price adjustment takes time. Since prices evolve over time we need a dynamic framework to investigate the e¤ect of price stickiness on the pro…tability of horizontal mergers. Using an oligopolistic di¤erential game model with sticky prices in the speci…c case of instantaneous price adjustment, Dockner and Gaunersdorfer (2001) through a numerical analysis show that, contrary to the static game, in a dynamic Cournot game where …rms use feedback strategies mergers are always pro…table independently of the number of merging …rms. Their result suggests that to analyzing merger, it is important to consider the nature of competition in the industry. Besides focusing on the same issue analytically, 1 Benchekroun (2003) shows that when …rms use open-loop strategies merger is pro…table only if the market share of the merged …rm is signi…cant enough, very similar to the SSR results, which put more emphasis on the role of feedback strategies to create incentive to merge. In this paper, we take a general approach without introducing speci…c assumptions on the degree of price stickiness to investigate the bearings of price dynamics. Scale economies as a motive for merger is ruled out by assumption because we would like to concentrate on the incentives to merge that are generated by price dynamics. To this end, we take a di¤erential game approach to price dynamics introduced by Simaan and Takayama (1978) and Fershtman and Kamien (1987). We take into consideration both the open-loop and closed-loop (memoryless)1equilibria to investigate how the speed of adjustment can a¤ect the pro…tability of horizontally merged …rms. There emerges, when price adjust with a very sticky mechanism, mergers with a small number of insiders but large number of outsiders are also privately pro…table even if …rms play open-loop. Furthermore, by …guring out the least market share required for merger to be pro…table when price adjusts instantaneously, we revisit the closed-loop e¤ect to generate incentive to merge. The remainder of the paper is organized as follows. Section 2 contains the layout of the model. Sections 3 illustrate the open-loop and closed-loop equilibria. The assessment of incentives towards mergers is given in section 4. Section 5 concludes the paper. 2 The setup Consider a dynamic oligopoly market where nsymmetric …rms, at any t2[0;1), produce quantities qi(t)0; i 2 f1;2; :::; ng;of the same homogeneous good with concave 1Broadly speaking, the main di¤erence between the open-loop equilibrium on one hand and the feedback and closed-loop equilibria on the other is that the former does not take into account strategic interaction between players through the evolution of state variables over time and the associated adjustment in controls. Under the open-loop rule, players choose their respective plans at the initial date and commit to them forever. Therefore, in general, open-loop equilibria are not subgame perfect, in that they are only weakly time consistent since players make their action ‘by the clock’only. A further distinction can be made between the closed-loop equilibrium and the feedback equilibrium, which are both strongly time consistent and, therefore, subgame perfect since, at any date , players decide ‘by the stock’of all state variables. However, while the closed-loop memoryless equilibrium takes into account the initial and current levels of all state variables, the feedback equilibrium accounts for the accumulated stock of each state variable at the current date. Hence, the feedback equilibrium is a closed-loop equilibrium, while the opposite is not true in general [2]. 2 technologies described by the quadratic cost functions Ci(t) = cqi(t) + 1 2q2 i(t); c > 0:(1) In each period, the product price, ^p(t), is determined by means of the inverse demand function ^p(t) = A n X i=1 qi(t):(2) However, since price is sticky, the actual market price does not adjust instantaneously to the price given by the demand function. That is, ^p(t)will di¤er from the current price level, p(t), and price moves according to the following equation dp(t) dt _p(t) = sf^p(t)p(t)g;(3) where s2[0;1)is a constant that determines the speed of price adjustment. The lower is s, the higher is the degree of price stickiness. When sgoes to in…nity, price is not sticky and the actual market price is equal to the price given by the demand function. The instantaneous pro…t function of …rm iis i(t) = qi(t)p(t)c1 2qi(t): Therefore, the maximization problem of …rm iis max qi(t)Ji= 1 Z0 etqi(t)p(t)c1 2qi(t)dt; (4) subject to (3), p(0) = p0and p(t)0for all t2[0;1). The factor et discounts future gains, and the discount rate is assumed to be constant and equal across …rms. We solve the di¤erential game using both the open-loop information structure where …rms choose their production plans at the initial date and stick to them for the whole time horizon and the closed-loop memoryless information structure where …rms’quantity choices at any time depend on the initial and current levels of all state variables (here, price). According to Cellini and Lambertini (2004), the steady state levels of the price and the individual output of a dynamic oligopoly game with price adjustments which are the premerger solution of our problem at the open-loop Nash equilibrium are 3 pOL =AnqOL ;qOL =(Ac)(+s) (1 + n)+ (2 + n)s;(5) and at the closed-loop Nash equilibrium are pCL =AnqCL ;qCL =(Ac)(+ns) s+ (1 + n)(+ns):(6) The corresponding single period pro…ts are OL =(Ac)2(+s)(+ 3s) 2 [(1 + n)+ (2 + n)s]2;CL =(ac)2(+ns)(+ (2 + n)s) 2 [s+ (1 + n)(+ns)]2: The superscripts OL and CL indicate the open-loop and closed-loop equilibrium level of a variable, respectively. For later reference, let us also note that in the static game where the demand and cost functions are speci…ed by (1) and (2) in turn, the equilibrium prices when …rms play à la Cournot and à la Bertrand respectively are pCN =2A+nc n+ 2 ;(7) pBN =A+nc n+ 1 :(8) 3 The merger equilibrium In this section, we consider a horizontal merger of m…rms (1 < m n)where they act collusively to maximize their discounted joint pro…ts.2nm…rms stay outside the merger. Hence, the di¤erential game becomes max qi Jm= 1 Z0 et "(p(t)c) m X i=1 qi(t)1 2 m X i=1 q2 i(t)#dt; i = 1; :::; m (9) max qj(t)Jj= 1 Z0 etqj(t)p(t)c1 2qj(t)dt; j =m+ 1; :::; n (10) subject to dp(t) dt _p(t) = s(A m X i=1 qi(t) n X j=m+1 qj(t)p(t));(11) 2Given the convex cost function, it is optimal to produce with all m…rms, and not to concentrate production on one …rm only. 4 and to the initial conditions p(0) = p0and p(t)0. qi(t)0; i 2 f1;2; :::; mgand qj(t)0; j 2 fm+ 1; :::; ngdenote, in turn, the output level of an insider and an outsider. JMand Jjrepresent the problem of the merging …rm and outsiders, respectively. According to (9), (10) and (11), the Hamiltonian functions of merging …rms and outsiders are HM(t) = et ((p(t)c) m X i=1 qi(t)1 2 m X i=1 q2 i(t)(12) + i(t)s"A m X i=1 qi(t) n X j=m+1 qj(t)p(t)#); Hj(t) = et qj(t)p(t)c1 2qj(t)(13) +j(t)s"A m X i=1 qi(t) n X j=m+1 qj(t)p(t)#); where j(t) = j(t)et and  i(t) = i(t)et and j(t)and i(t)are the co-state variables associated with p(t). 3.1 Open-loop equilibrium After the merger, at the open-loop Nash equilibrium, the steady state levels of the price and the output of merging …rm and outsiders are pOL post =AqOL M(nm)qOL O; qOL M=m (+ 2s); qOL O=(+s+ms); where =(Ac) (+s) (n+ 1) 2+ [2n+m(nm+ 2) + 3] s + [n+m(nm+ 3) + 2] s2: The subscripts Mand Oindicate the equilibrium level of a variable for the merging …rm and an outsider and subscripts post refers to the equilibrium level the price after the merger. Hence, the steady state equilibrium pro…ts are as follows OL M=2m(+ 2s)2(+s+ 2ms) 2 (+s);OL O=2(+ 3s)(+s+ms)2 2(+s): For the proof you can see Benchekroun (2003). 5 3.2 Closed-loop equilibrium Now, we look for the post-merger Nash equilibrium under the closed-loop strategies. The outcome is summarized by the following proposition: Proposition 1 At the closed-loop Nash equilibrium, the steady state levels of the price and the output of merging …rm and outsiders are pCL post =AqCL M(nm)qCL O;(14) qCL M=m (+ (nm+ 1) s)+m2m+n+ 1s;(15) qCL O=(+s(n+ 1)) + (m2m+n)s;(16) where = (Ac)=(n+ 1) 2+nm2m+ 2n+ 3+ 2s +(n+ 1) m2nmn +n2+n+ 1m4+m3s2 which yields the steady state equilibrium pro…ts CL M=1 22m(+ (nm+ 1) s)(+ (n+m+ 1) s)(+ (m2m+n+ 1)s)2 CL O=1 22(+s(n+ 1))2+ (m2m+n)s+ (m2m+n+ 2)s Proof. Taking the …rst-order conditions w.r.t. qi(t)and qj(t)and using (12) and (13), in turn, we have @HM(t) @qi(t)=p(t)cqi(t) i(t)s= 0;(17) @Hj(t) @qj(t)=p(t)cqj(t)j(t)s= 0;(18) which yields the optimal closed-loop output for, respectively, the insiders and outsiders as follows qCL i(t) = p(t)c i(t)sif p(t)> c + i(t)s; 0otherwise, (19) qCL j(t) = p(t)cj(t)sif p(t)> c +j(t)s; 0otherwise. (20) The adjoint equations for the optimum are @HM(t) @p(t) n X j=m+1 @HM(t) @qj(t) @qCL j(t) @p(t)=@ i(t) @t  i(t);(21) 6 References [1] Benchekroun, H., 2003. The closed-loop e¤ect and the pro…tability of horizontal mergers. Canadian Journal of Economics 36, 546–565. [2] Cellini, R., Lambertini, L., 2004. Dynamic oligopoly with sticky prices: Closed-loop, feedback and open-loop solutions. Journal of Dynamical and Control Systems 10, 303–314. [3] Cheung, F.K., 1992. Two remarks on the equilibrium analysis of horizontal mergers. Economics Letters 40, 119-23. [4] Daughety, A.F., 1990. Bene…cial concentration. American Economic Review, 80, 12311237. [5] Deneckere, R. and Davidson, C., 1985. Incentives to Form Coalitions with Bertrand Competition. RAND Journal of Economics 16, 473-86. [6] Dockner, E.J., Gaunersdorfer, A., 2001. On the pro…tability of horizontal mergers in industries with dynamic competition. Japan and the World Economy 13, 195–216. [7] Farrell, J. and Shapiro, C., 1990. Horizontal Mergers: An Equilibrium Analysis. American Economic Review 80, 107-26. [8] Fershtman, C. and Kamien, M. I., 1987. Dynamic duopolistic competition with sticky prices. Econometrica 55, 1151–1164. [9] Gaudet, G., Salant, S.W, 1991. Increasing the pro…ts of a subset of …rms in oligopoly models with strategic substitutes. American Economic Review 81, 658-665. [10] Gaudet, G., Salant, S.W, 1992. Thowards a theory of hrizontal mergers. In: Norman, G., La Manna, M. (Eds.), The New Industrial Economics: Recent Developments in Industrial Organization, Oligopoly and Game Theory. Elgar. [11] Levin, D., 1990. Horizontal mergers: the 50-percent benchmark. American Economic Review 80, 1238-45 [12] Perry, M.K. and Porter, R.H., 1985. Oligopoly and the Incentive for Horizontal Merger. American Economic Review. 75, 219-27. [13] Salant, S.W., Switzer, S. and Reynolds, R.J., 1983. Losses from Horizontal Merger: The E¤ects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium. Quarterly Journal of Economics 98, 185-213. [14] Simaan, M. and Takayama, T., 1978. Game theory applied to dynamic duopoly problems with production constraints. Automatica 14, 161–166. 13 