Ye 2009 Bulletin

C© 2009 The Authors. Journal compilation C© 2009 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research. Published byBlackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St., Malden, MA 02148, USA. Bulletin of Economic Research 61:2, 2009, 0307-3378 DOI: 10.1111/j.1467-8586.2008.00301.x PARTIAL PRIVATIZATION IN A MIXED DUOPOLY WITH AN R&D RIVALRY John S. Heywood* and Guangliang Ye† *Department of Economics, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, USA, and †Research Institute of Economics and Management, Southwestern University of Economics and Finance, Chengdu, China ABSTRACT This paper is the first to examine the incentive for partial privatization in a mixed duopoly with R&D rivalry. We show that because mixed duopolies engage in more R&D, the optimal extent of privatization is unambiguously reduced. Yet, this reduction is often very modest. Adopting the extent of privatization that would be optimal if one ignored the R&D rivalry routinely results in greater welfare than retaining a fully public firm and ignoring partial privatization. Only when R&D has an extremely low cost would it be preferable to ignore partial privatization. Keywords: mixed duopoly, public enterprise, research and develop- ment and partial privatization JEL classification numbers: L13, L32, L33, L52 I. INTRODUCTION We demonstrate that the presence of a research and development (R&D) rivalry reduces the incentive for partial privatization in a mixed duopoly. In so doing, we combine for the first time two strands of the literature. The first strand emphasizes that partial privatization of the public firm Correspondence: J.S. Heywood, Department of Economics, University of Wisconsin- Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA. Email: heywood@uwm.edu. The authors thank the Scientific Research Foundation for Returned Overseas Chinese Scholars for Support. 165 166 BULLETIN OF ECONOMIC RESEARCH increaseswelfare. Partial privatization causes the public firm to reduce its output reducing the wasteful cost asymmetry associated with the public firm’s tendency to produce more than its private rivals (Matsumura, 1998; Matsumura and Kanda, 2005; Ohori, 2006).1 The second strand shows that when imitation is easy, the public firm invests more in R&D than private firms because the resulting production cost savings increase welfare even as they fail to increase profit (Delbono and Denicolo, 1993; Poyago-Theotoky, 1998; Ishibashi and Matsumura, 2006).2 Our duopoly model confirms that the public firm invests more in R&D than its private rival. In addition, we show that the presence of the R&D rivalry unambiguously reduces the optimal extent of partial privatization.3 The beneficial influence of the public firm’s additional R&D helps offset the excess costs associated with asymmetric outputs. Yet, the resulting decline in the optimal extent of privatization is typically small. One way to see this is to imagine a government that is unaware of the role the R&D rivalry plays on the optimal extent of privatization. Adopting the degree of privatization that ignores the role of R&D typically remains welfare superior to retaining a fully public firm. This result emerges because the additional R&D associated with the larger extent of public ownership generates a production cost saving that is small relative to the wasteful cost asymmetry associated with the public firm’s higher output. Only when R&D is very cheap – the cost of production can be dramatically lowered for small expenditures on R&D – would retaining a fully public firm be superior to adopting the degree of privation that ignores the role of the R&D rivalry. The policy implications are twofold. First, among public firms, those involved in R&D are less suitable targets for extensive privatization policies. Second, privatization becomes a less sensible policy in those cases in which R&D is more effective in lowering production costs. Such implications are important for R&D driven industries such as informa- tion technology. Thus, in China this sector includes the major firms of Lenovo, Huawei and TCL. Lenovo is a public firm, Huawei is a private firm and TCL is partially privatized with its public share reduced from 58 percent to 39.62 percent in 2002. R&D investment for Lenovo was 1 Indeed, Matsumura (1998) had 82 cites in Google Scholar as of June 2007 and his result on partial privatization remains among the best known in the literature on mixed oligopoly. 2 Ishabashi and Matsumura (2006) present an infinite horizon game in which larger innovations reduce the probability of success for a given R&D expenditure. They show both that the public firm invests too much in R&D and that its innovation size is too small. 3 These results reverse those of Matsumura and Matsushima (2004) who introduce R&D into a Hotelling-type model of mixed duopoly. Their findings that the private firm engages in less R&D and that privatization improves welfare flow from two assumptions that we relax. First, they do not allow spillovers to lower the rival’s costs. Second, they assume constant willingness to pay and so price reductions do not increase consumer surplus. These assumptions make R&D largely a strategy designed to lower price so as to take away market share. Thus, the private firm over-invests in R&D that is only privately beneficial. C© 2009 The Authors. Journal compilation C© 2009 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research. PARTIALPRIVATIZATIONINAMIXEDDUOPOLYWITHANR&DRIVALRY 167 2.8 billion RMB and for TCL was 1.9 billion RMB in 2006 (Ministry of Information Industry of China, 2007). Such circumstances are not limited to transition economies as public firms in Italy have often been at the very top of lists of R&D investors (Malerba, 1993). Apart from specific examples of public firms in R&D intensive industries, the mixed oligopoly approach is thought to have special appeal throughout technology-centred industries. Following Soubeyran and Thisse (1994), the defining notion of a ‘public firm’ is the maximization of social wel- fare. Given this, Payogo-Theotoky (1998, p. 416) argues that the public firm modelled in mixed oligopolies should be thought of as including ‘publicly funded R&D laboratories, scientific non-profit institutions and university labs’ as well as state-owned enterprises engaged in R&D. We next describe our model of an R&D rivalry in a mixed duopoly with the third section presenting the equilibrium. The major results on the influences of privatization are in the fourth section. The fifth section considers extensions and the final section concludes. II. THE MODEL We consider a duopoly in which firm 1 is a public firm maximizing social welfare and firm 2 is a private, profit-maximizing firm. The firms face a common demand function p = a − Q where a > 0 and Q = q 1 + q 2. Each firm has a production cost Ci(qi, xi, xj) that is a function of its own production, the research it undertakes, xi, and research that its rival undertakes, xj. Following the tradition from the mixed oligopoly literature, we assume quadratic production cost, Ci(qi, xi, xj) = q2i − (xi + β i xj)qi/2.4 As in D’Aspremont and Jacquemin (1988), increasing R&D serves to lower production costs. Thus, 0 ≤ β i ≤ 1 for i = 1 or 2 measures the degree of R&D spillover from the rival. Thus, when β 1 = 0, there is no spillover from the R&D of firm 2 and it remains private and therefore of no value to firm 1. On the other hand, when β 1 = 1, there is complete spillover and the R&D of firm 2 is publicly known and of equal value to both firm 1 and firm 2. We consider the case when β 1 = β 2 = 1 sometimes identified as the case of ‘easy imitation’. Following D’Aspremont and Jacquemin (1988), the direct cost of R&D is also quadratic, reflecting diminishing returns to R&D expendi- tures (increasing marginal costs). Firms choose a level of R&D and a subsequent output level. The profit function of firm i is πi = pqi − Ci (qi , xi , x j ) − Ax2i i = 1, 2 i �= j (1) 4 We note that both the linear demand curve slope and that the slope of the marginal cost curve could have been made more general. Such alterations do not change the analysis or results but do make the expressions more complicated. C© 2009 The Authors. Journal compilation C© 2009 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research. 168 BULLETIN OF ECONOMIC RESEARCH The resulting social welfare function becomes W = Q∫ 0 p(t)dt − 2∑ i=1 [ Ci (qi , xi , x j ) + Ax2i ] (2) The objective of the private firm is to maximize (1), π 2. The objective function of a public or partially privatized firm is to maximize the following: G = (1 − λ)W+λπ1 (3) where 0 ≤ λ ≤ 1. Thus, when λ = 0, the fully public firm maximizes social welfare from (2). As λ increases, the objective function gives a larger weight to profit. Matsumura (1998) thinks of this parameter as the share of a previously public firm sold to private parties. The objective function thus reflects a weighted average of the underlying ownership structure.5 We consider a three-stage game. In the first stage, the government adopts the extent of privatization, λ, that maximizes social welfare (2). In second stage, the firms make simultaneous choices of the level of R&D. In the third stage, the firms take those R&D choices as given and make simultaneous choices on output levels. The firms play non-cooperative Cournot–Nash games in both of their stages and the sub-game perfect Nash equilibrium is derived by backward induction. III. THE EQUILIBRIUM In solving for stage three, the R&D levels are assumed given and mutual best response functions in the quantities are derived from Equations (1) and (3) and solved simultaneously: q1 = 3(2a+x1 + x2) 2(11 + 4λ) q2 = (λ+2)(2a + x1 + x2) 2(11 + 4λ) (4) The quantities are now functions of R&D levels and the extent of privatization. To solve for stage two, these quantities (Equations (4)) are returned to the objective functions from which mutual best 5 The literature on delegation echoes this in part as owners of the public firm provide man- agers an objective function weighting private profit (Barros, 1995; White, 2002; Heywood and Ye, 2008). C© 2009 The Authors. Journal compilation C© 2009 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research. PARTIALPRIVATIZATIONINAMIXEDDUOPOLYWITHANR&DRIVALRY 169 response functions in R&D levels are derived and again simultaneously solved: x1 = 2a( − 5λ 3 − 21λ2+3λ+59) � x2 = 8a(λ+2) 2 � (5) where � = 5λ3+17λ2 − 19λ − 75 + (128λ2 + 704λ + 968)A. The R&D levels from Equation (5) can be returned to the expressions in Equation (4) to yield the equilibrium levels of output. q1 = 24Aa(4λ+11) � q2 = 8Aa(4λ+11)(λ+2) � (6) Thus, all equilibrium choices depend only on the extent of priva- tization. Substituting Equations (5) and (6) into Equation (2) yields equilibrium welfare: W = − 4Aa2[25λ6+194λ5+299λ4−972λ3−2597λ2+610λ+3737−8A(5λ3+21λ2−3λ−59)(4λ+11)2] �2 (7) In the next section, we characterize the equilibrium, derive the optimal extent of privatization and make critical comparisons. IV. RESULTS Our results start by confirming the importance of the R&D rivalry in differentiating the equilibriumchoices of the public and private firms.We then compare the incentive for privatization isolating the importance of the R&D cost parameterA.We show that the presence of the R&D rivalry reduces the optimal extent of privatization. We follow this by showing that the welfare associated with partial privatization that ignores the R&D rivalry typically remains greater than that associated with a fully public firm. Proposition 1. (i) In the presence of an R&D rivalry, the public firm undertakes more R&D and produces more output than the private firm. (ii) Moreover, the presence of the R&D rivalry increases the difference in output between the public and private firm. C© 2009 The Authors. Journal compilation C© 2009 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research. 170 BULLETIN OF ECONOMIC RESEARCH Proof. (i) Subtracting the equilibrium R&D levels in Equation (5) and the outputs in Equation (6) yields �x = x1 − x2 = 2a(1 − λ)(5λ 2 + 30λ + 43) � > 0 �q = q1 − q2 = 8Aa(1 − λ)(11 + 4λ) � > 0 (ii) The difference in output monotonically decreases in A and as A goes to infinity it converges to that without the R&D rivalry. d�q dA = 8a(1 − λ)(11 + 4λ)(3 + λ)(5λ 2 + 2λ − 25) �2 < 0 Also limA→∞�q = a(1 − λ)/(4λ + 11). This is exactly the�q without the R&D rivalry (as shown in Appendix 1). Thus, in the presence of the rivalry, the public firm undertakes more R&D and producesmore than the private firm. The gap in output exceeds what would exist absent the rivalry and the gap grows as the cost of R&D, A, shrinks. Proposition 2. In the presence of an R&D rivalry (i) the optimal extent of partial privatization for the public firm increases in A, dλ∗(A)/dA > 0 but (ii) remains below the extent that is optimal absent the rivalry. Proof. (i) Solving dW/dλ = 0 yields the optimal extent of privatization but the resulting expression cannot be easily solved for λ as Appendix 2 shows. Nonetheless it can be solved for A and the relationship between A and the optimal λ characterized. Thus, the first order condition can be rearranged: A = 1614 − 100λ6 − 1345λ5 − 7645λ4 − 21791λ3 − 29399λ2 − 13262λ − (12776388 − 85142088λ + 6563392λ2 + 696935408λ3 + 3338025λ10 + 269000λ11 + 10000λ12 + 1464116793λ4 + 24976050λ9 + 123765615λ8 + 420458100λ7 + 1525014210λ5 + 980556291λ6) 12 48(11 + 4λ)2(11λ − 2) (8) The numerator is negative for all λ ∈ [0, 1] and, to guarantee a positive A, λ must fall in the permissible interval of [0, 211 ] from the denominator. Then, by the implicit function theorem dλ∗(A)/dA = (dA∗(λ)/dλ)−1. When the derivative of Equation (8) with respect to λ is evaluated at λ ∈ [0, 211 ], we have dA∗(λ)/dλ > 0 and thus dλ∗(A)/dA > 0. C© 2009 The Authors. Journal compilation C© 2009 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research. PARTIALPRIVATIZATIONINAMIXEDDUOPOLYWITHANR&DRIVALRY 171 Fig. 1. The Relationship between the optimal extent of privatization and the R&D cost parameter. (ii) Note that for λ ∈ [0, 211 ], the numerator on the right hand side of Equation (8) is bounded. Thus, the only way A can go to infinity is if the denominator goes to zero, which implies λ∗ = 211 . For all A less than infinity, λ∗ < 211 . As shown in Appendix 1, 2 11 is the optimal extent of privatization in the same model but with no R&D rivalry. While this demonstrates that for any finite A the optimal extent of privatization with the rivalry is less than that without the rivalry, it does not give a flavour of the magnitude of the difference. To explore this, we can examine the relationship between λ∗ and A from the first order condition in Equation (8). Figure 1 graphs that relationship over the permissible range of λ ∈ [0, 211 ]. As is clear, for many sensible R&D cost parameters the difference is relatively small. If one assumes the marginal cost of production has the same slope as the marginal cost of R&D, A would equal 1.0 and, as shown, the optimal privatization share is 15 percentage points, not far from the 18 percent that emerges absent the R&D rivalry. Thus, the extent of optimal privatization in the face of an R&D rivalry is smaller but it asymptotically approaches that without a rivalry at a fast pace as A grows. Privatization of any degree serves to lower both production and R&D. C© 2009 The Authors. Journal compilation C© 2009 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research. 172 BULLETIN OF ECONOMIC RESEARCH Proposition 3. The total R&D level and total output at the optimal degree of privatization are lower than with the fully public firm. Proof. dX dλ = −16aA(11 + 4λ)(20λ 3 + 165λ2 + 450λ + 391) �2 < 0 dQ dλ = −16aA(640 + 1235λ + 714λ 2 + 155λ3 + 10λ4 + 4356A + 3168λA + 576λ2A) �2 < 0 Hence, the total R&D level and total output decreases as the degree of partial privatization increases. Since λ∗ > 0, X∗|λ=λ∗ < X∗|λ=0 and Q∗|λ=λ∗ < Q∗|λ=0. The traditional models of partial privatization with- out an R&D rivalry emphasize the associated reduction in output. Such a reduction persists given the rivalry and, as the public firm undertakes more R&D, privatization reduces its extent as well. While recognizing this result, we have been emphasizing that the rivalry reduces the importance of privatization. Another approach to understand the magnitude of this reduction compares the welfare asso- ciated with two cases. The first case ignores the issue of privatization and retains a fully public firm. The second case ignores the R&D rivalry and adopts the extent of privatization that would be optimal absent the rivalry. Proposition 4. For all A > 0.352 (A < 0.352), the welfare associated with a fully public firm, W |λ=0, is lower (higher) than that asso- ciated with ignoring the R&D rivalry and privatizing accordingly, W |λ=2/11. Proof. Substituting λ = 211 into Equation (7) and solving for the value of A that equates the resulting welfare with that resulting from substituting λ = 0 into Equation (7) yields 0.352. It can be confirmed that there is a single permissible solution and that W |λ=0 < W |λ=2/11 for A greater than the solution and W |λ=0 > W |λ=2/11 for A less than the solution. Thus, only for extremely low values of A, those that result in very large amounts of R&D, would it be sensible to focus on the rivalry and ignore privatization. Otherwise, it remains superior to privatize and ignore the rivalry. The intuition is that while the public firm undertakesmoreR&Dwhich lowers costs, these reductions are not sufficient to offset the higher costs associated with asymmetric outputs between the public and private firm. Put differently, privatization reduces both the cost asymmetry and the C© 2009 The Authors. Journal compilation C© 2009 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research. PARTIALPRIVATIZATIONINAMIXEDDUOPOLYWITHANR&DRIVALRY 173 TABLE 1 Four cases of imitation under the assumption that A = 1. (β1, β2) q1 q2 x1 x2 W λ (0, 0) 0.2825 0.1857 0.0664 0.0506 0.25141 0 (0, 1) 0.2882 0.2013 0.1315 0.0549 0.26351 0 (1, 0) 0.2896 0.1817 0.0683 0.0330 0.25528 0 (1, 1) 0.2956 0.1917 0.1321 0.0358 0.26773 0 (0, 0) 0.2646 0.1903 0.0638 0.0516 0.25182 0.1805 (0, 1) 0.2754 0.2032 0.1208 0.0552 0.26380 0.1130 (1, 0) 0.2664 0.1878 0.0649 0.0352 0.25599 0.2371 (1, 1) 0.2795 0.1996 0.1188 0.0370 0.26820 0.1422 (0,0) 0.2645 0.1903 0.0638 0.0516 0.25181 0.1818 (0,1) 0.2681 0.2042 0.1143 0.0554 0.26370 0.1818 (1,0) 0.2714 0.1865 0.0657 0.0347 0.25596 0.1818 (1,1) 0.2753 0.2002 0.1151 0.0372 0.26817 0.1818 All output and R&D parameters are in units of a, the demand choke price. Welfare is in units of a2. The coefficient λ is the share of privatization, set to zero in the first panel, determined optimally in the second and set to the share that ignores theR&D rivalry in the third. total amount of R&D. These have offsetting effects on overall costs and so welfare. Only when R&D is particularly cheap and effective, and so the amount undertaken is relatively large, will the optimal extent of privatization be small enough that it is better to ignore privatization than to ignore the rivalry. V. EXTENSIONS In this section, we relax the assumption of reciprocal easy imitation (β 1 = β 2 = 1) by comparing it to three alternatives, no imitation (β 1 = β 2 = 0), easy imitation by only the private firm and easy imitation by only the public firm. These three alternatives do not allow as straightforward a characterization of the relationship between the R&D cost parameter A and the optim