We introduce R&D activity and R&D subsidies in the context of a mixed oligopoly and evaluate the effects of privatization on welfare. We show that when R&D subsidies are employed, privatization is welfare and R&D promoting provided that the number of competitors is sufficiently large

Joanna Poyago-Theotoky (7195115)

Maria Jose Gil-Molto (7195238)

Vasileios Zikos (7195418)

English

Loughborough University Institutional Repository

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For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/           ISSN 1750-4171            DEPARTMENT OF ECONOMICS  DISCUSSION PAPER SERIES    R&D Policy and Privatization in a Mixed Oligopoly  Maria Jose Gil Molto, Joanna Poyago-Theotoky and Vasileios Zikos  WP 2006 - 22          Dept Economics Loughborough University Loughborough  LE11 3TU  United Kingdom Tel:  + 44 (0) 1509 222701 Fax: + 44 (0) 1509 223910 http://www.lboro.ac.uk/departments/ec            R&D policy and privatization in a mixed oligopolyMaria José Gil Moltó, Joanna PoyagoTheotoky andVasileios ZikosyLoughborough University, Department of EconomicsAbstractWe introduce R&D activity and R&D subsidies in the context ofa mixed oligopoly and evaluate the e¤ects of privatization on welfare.We show that when R&D subsidies are employed, privatization is wel-fare and R&D promoting provided that the number of competitors issu¢ ciently large.Keywords: mixed oligopoly, process innovation, R&D subsidy, pri-vatization.JEL Classication: L31, L32, O38, L13, L50.1 IntroductionThere is a large body of literature on mixed oligopolies analyzing the ef-fects on welfare of privatization. Interestingly, White (1996) and PoyagoTheotoky (2001) showed that when policy makers use output subsidies asa policy instrument, the issue of privatization is not welfare related. Fur-ther, Fjell and Heywood (2004) proved that privatization will bear negativeconsequences on welfare if the public rm remains as a leader in the post-privatization regime.The analysis in these papers has been conned to output production andconsequently, subsidies to output. However, the study of the R&D activityWe would like to thank the audience at the 33rd EARIE Conference (Amsterdam,August 25-27, 2006) as well as seminar participants at Alicante and Leicester for helpfulcomments. The last author gratefully acknowledges nancial support by LoughboroughUniversity under its grant scheme.yCorresponding author: Department of Economics, Loughborough University, Lough-borough LE11 3TU, United Kingdom. Email: V.Zikos@lboro.ac.uk, tel.: + 44 (0) 1509222708, fax: + 44 (0) 1509 26317.1and R&D subsidies in the context of mixed oligopolies has not yet beenconsidered, despite the signicant empirical evidence citing the importanceof public funding towards R&D (Katz, 2001) and the substantial presenceof public rms in innovative industries (examples are the health-care sector,Aanestad, 2003, and bioagriculture, Oehmke, 2001).In this paper, we study the use of R&D subsidies in the context of amixed oligopoly and evaluate the e¤ects of a privatization. We show that,apart from addressing the market failures arising from the R&D activity, theuse of R&D subsidies corrects (to some extent) the ine¢ cient distributionof production costs which arises in mixed industries. In that sense, an R&Dsubsidy may (at least partially) serve the same purpose as an output subsidy.Our results indicate that the optimal subsidy to R&D output is non-monotone in the number of private rms both in the private and the mixedmarkets and it is always lower for the former than for the latter. Further,when R&D subsidies are employed, privatization may increase total R&Dand welfare provided that the number of private rms is su¢ ciently large.The latter contrasts with the results obtained in previous contributions,where output subsidies are employed.2 The modelConsider an industry consisting of n identical private rms and a public rmproducing a homogeneous good. The inverse demand function is linear andgiven by p(Q) = a Q; Q is aggregate output, Q = q0+Pni=1 qi, q0 denotesthe output of the public rm and qi, i = 1; ::; n, is the output of the i-thprivate rm. We postulate that all rms engage in cost-reducing (process)R&D and there are no spillovers.1 Thus, the production cost of each rmis represented by the quadratic function Cj(xj ; qj) = (c   xj)qj + q2j , j 2f0; 1; :::; ng, where xj is the cost reduction of the j-th rm and a > c > 0.2We also make the standard assumption that R&D spending is subject todiminishing returns to R&D expenditure,  j(xj) = x2j , j 2 f0; 1; :::; ng.1 In other words, the patent system is fully e¤ective.2The presence of the quadratic term is standard in the mixed oligopoly literatureand rules out the possibility of a public monopoly by introducing diminishing returns inproduction.2A rms prot function is given byj = qj(a nXj=0qj)  (c  xj)qj   q2j   x2j + sxj , j 2 f0; 1; :::; ng, (1)where s denotes the (per unit) subsidy to R&D output. Social welfare,dened as the sum of consumer surplus, CS = (1=2)Q2 and producer surplusnet of R&D subsidies is given bySW = CS +nXj=0j   snXj=0xj . (2)The timing of the game is as follows: In stage one, the government com-mits to a subsidy on R&D output so as to maximize welfare. In stage two,rms make their R&D decisions and in the last stage, a standard Cournotgame is played. We solve the entire game by backward induction to ob-tain the Subgame Perfect Nash Equilibria (SPNE henceforth) for both amixed and a private oligopoly and compare the results across the two mar-ket arrangements.3 Mixed oligopolySolving the last stage of the game, the respective equilibrium quantities ofthe public and the private rms areqm0 (x0; xi) =3(a  c) + (3 + n)x0  Pni=1 xi2n+ 9(3)qmi (x0; xi) =2(a  c)  x0 + (3=n)Pni=1 xi2n+ 9(4)In the second stage, the associated equilibrium R&D output levels are3xm0 (s) =(a  c)[ 3 + n(47 + 6n)]  2n2(6 + n)s 15 + n[235 + 102n+ 12n2] , (5)xmi (s) =3(a  c)(3 + n)(1 + 2n) + n[135 + n(56 + 6n)]s 15 + n[235 + 102n+ 12n2] ; i 2 f1; :::; ng.(6)As it can be seen from xm0 and xmi , the subsidy exerts a positive e¤ect on3The second order condition for the public rm requires 135 + 56n + 6n2 > 0 and foreach private rm, 9  21n+ 75n2 + 36n3 + 4n4 > 0. Indeed, both conditions are fullled.3the R&D output of a private rm, whereas the reverse holds for the publicrm. This implies that, similarly to an output subsidy (see White 1996;PoyagoTheotoky 2001), a subsidy to R&D has a cost redistribution e¤ect.Hence, we can state that R&D subsidies may serve (at least partially) thesame purpose as output subsidies.Substituting (3)   (6) into (2) and performing the maximization withrespect to s, we obtain the optimal subsidy4sm =(a  c)(5n  3)n(5 + n)(7 + 2n). (7)The SPNE solutions of the entire game are the followingqmi =(a  c)(9 + 2n)(5 + n)(7 + 2n); qm0 =(a  c)(14 + 3n)(5 + n)(7 + 2n)xmi =(a  c)(6 + n)(5 + n)(7 + 2n);xm0 =(a  c)(7 + n)(5 + n)(7 + 2n)mi =(a  c)2[ 18 + n(153 + 65n+ 7n2)]n(5 + n)2(7 + 2n)2m0 =(a  c)2[ 21 + n(179 + 75n+ 8n2)]n(5 + n)2(7 + 2n)2CSm =2(a  c)2[7 + n(6 + n)]2(5 + n)2(7 + 2n)2; (8)SWm =(a  c)2[7 + n(7 + n)](5 + n)(7 + 2n): (9)It is important to note that although the optimal R&D subsidy im-proves the distribution of total costs, it does not restore cost e¢ ciency.Thus, it attains a second best in the sense that complete equalization ofproduction costs would require an additional instrument a subsidy to out-put quantity at the governments disposal.4 Private oligopolyThe industry now consists of (n+1) prot-maximizing (private) rms. TheSPNE outcomes of the game are54With requirement for second order condition 6n3(5+n)(7+2n)(135+56n+6n2) > 0.5The associated second order conditions are all satised and available upon request.4sp =4(a  c)(3n  2)3n(3 + n)(9 + 2n)qpi = qp0 =2(a  c)(4 + n)(3 + n)(9 + 2n)xpi = xp0 =(a  c)(5 + n)(3 + n)(9 + 2n)pi = p0 =(a  c)2[ 40 + n(361 + 174n+ 21n2)]3n(3 + n)2(9 + 2n)2CSm =2(a  c)2(1 + n)2(4 + n)2(3 + n)2(9 + 2n)2SW p =(a  c)2(1 + n)(5 + n)(3 + n)(9 + 2n): (10)5 Results and DiscussionComparing the results obtained in the mixed oligopoly and the privateoligopoly cases, we can state the following results:6Proposition 1 The optimal subsidy to R&D output in the mixed oligopolyis always greater than the subsidy in the private oligopoly, sm(n) > sp(n).The intuition behind proposition 1 follows: In the case of a privateoligopoly, two sources of market failure exist: (i) the imperfect competition,which will lead to underproduction (and hence, allocative ine¢ ciency), and(ii) the R&D undervaluation e¤ect (as dened by Ulph, 19997), which willlead to under-investment in R&D by private rms. In the case of a mixedoligopoly a further source of market failure exists, the di¤erent nature (pub-lic or private) of the rms in the market. As a result, the production costsare ine¢ ciently distributed. Hence, it is optimal for the decision-maker tosubsidize more heavily a mixed market.Proposition 2 The optimal subsidy to R&D output, s i(n), i = m, p, isalways positive and increasing in the number of private rms n, as n goesfrom 1 to 2 , but decreasing in n if n > 2 .6All proofs are relegated to the appendix.7Private rms do not take into account the increases in Consumers Welfare as a con-sequence of the investment on R&D (as Consumers Welfare does not belong to theirobjective function). This will result in underinvestment in R&D.5Proposition 2 shows that, in contrast to the monotonic nature of an out-put subsidy identied in previous contributions, the optimal R&D subsidy isinitially increasing and after a threshold value of the number of rms (n = 2)decreasing. This is a result of the interaction of the market failures identi-ed above. The e¤ect of imperfect competition becomes less important asthe number of rms increases whereas the undervaluation e¤ect will havean inverted U-shape with respect to the number of rms, as identied bySuzumura (1992). The combination of these two e¤ects (plus the ine¢ ciencyin the distribution of the production costs in the mixed case) results in aninverted U-shape of the optimal subsidy.The next proposition compares total R&D output, output quantity andprots between the mixed and the private oligopolies:Proposition 3 (i) Total R&D output in the private oligopoly is higher thanin the mixed oligopoly if n > 4 ; (xm0 +nxmi ) < (n + 1 )xpi . (ii) Total outputquantity in the mixed oligopoly always exceeds total output quantity in theprivate oligopoly; (qm0 +nqmi ) > (n + 1 )qpi . (iii) Total prot in the privateoligopoly always exceeds total prot in the mixed one; (m0 +nmi ) < (n + 1 )pi .With regard to proposition 3, it is relevant to note that the public rmwill tend to reduce its R&D investment more than a private rm as n in-creases, i.e., j @xm0@n j> j@xmi@n j, leading therefore to higher levels of totalR&D output in the private oligopoly than in the mixed one for su¢ cientlylarge n (n > 4). It turns out, however, that the public rms behaviorwill not impact total output quantity in the same way and output will bealways higher in the mixed oligopoly than in the private one. Regardingequilibrium prots, the underproduction problem will be more serious inthe private oligopoly than in the mixed one as a result of the lower inten-sity of competition, leading thus to higher oligopoly rents and allocativeine¢ ciency.The next proposition contains a welfare assessment of privatization poli-cies and is largely a consequence of Proposition 3:Proposition 4 When government policy takes the form of an optimal sub-sidy to R&D output, then privatization enhances total welfare if n > 4 ,SWm(n) < SW p(n).6The intuition for the above proposition follows: First, we can state thattypically privatization improves productive e¢ ciency. The reason is that inthe move from the mixed to the private oligopoly optimum, the ine¢ ciencyin the distribution of production costs vanishes. However, privatizationworsens allocative e¢ ciency as it promotes higher oligopoly rents. It turnsout that the gains in terms of productive e¢ ciency will outweigh the lossesin terms of allocative e¢ ciency only if n is su¢ ciently large.6 ConclusionThis paper aims at lling a gap in the literature on mixed oligopolies andprivatization by introducing R&D activity and R&D subsidies. Our resultsindicate that the optimal subsidy to R&D output is non-monotone in thenumber of private rms both in the private and the mixed markets andthat the mixed industry should be more heavily subsidized than the privateone. Similarly to an output subsidy, a subsidy to R&D can address theine¢ cient distribution of costs. However, in contrast to the welfare resultsof privatization when output subsidies are provided, privatization is welfareenhancing if the number of rms in the industry is su¢ ciently high. Further,under the same condition, privatization yields increases in the total R&Dlevels. In industries with a small number of rms, privatization would resultin a loss of surplus and decreases in the R&D activity.References[1] Aanestad, M., Mork, E. B., Grisot, M., Hanseth, O., Syvertsen, M. C.,2003. Knowledge as a Barrier to Learning: A Case Study from MedicalR&D, paper presented at the 4th European Conference on Organisa-tional Knowledge, Learning and Capabilities, April.[2] Fjell, K. and Heywood, J., 2004. Mixed oligopoly, subsidization and theorder of rmsmoves: the relevance of privatization, Economics Letters83, pp. 411-416.[3] Katz, J., 2001. Structural reforms and technological behavior. Thesources and nature of technological change in Latin America in the1990s, Research Policy 30, pp. 1-19.7[4] Oehmke, F. J., 2001. Biotechnology R&D races, industry structure, andpublic and private sector research orientation, AgBioForum 4, pp. 105-114.[5] Poyago-Theotoky, J., 2001. Mixed oligopoly, subsidization and the orderof rmsmoves: an irrelevance result, Economics Bulletin, 12, pp. 1-5.[6] Suzumura, K., 1992. Cooperative and noncooperative R&D in anoligopoly with strategic commitments, American Economic Review 82,pp. 1307-1320.[7] White, M.D., 1996. Mixed oligopoly, privatization and subsidization,Economics Letters 53, pp. 189-195.7 AppendixProof of Proposition 1: It is immediate to show that sm(n) > 0 as a  c > 0and  3 + 5n > 0 8n 2 Z+. Next, sm(1) < sm(2) because a c27 < a c22 . Toshow that sm is decreasing in n i¤ n > 2, ignore that n is an integer; it mustbe shown that dsmdn < 0 i¤ n > 2. This derivative has the same sign as theexpression 105 + 102n  67n2   20n3. The latter is negative i¤ n > 2. (Theproof for sp(n) is similar and hence is omitted). QEDProof of Proposition 2 : It will be shown that sm(n) sp(n) > 0 8n 2 Z+.To this end, ignore that n is an integer. Then it su¢ ces to show that37  14n+ 19n2 + 6n3 > 0 8n, which in turn is always true. QEDProof of Proposition 3 : Let   = (3+n)(9+2n) > 0 and  = (5+n)(7+2n) > 0. We have the following (ignoring that n 2 Z+):(i) (xm0 +nxmi )  (n+1)xpi = (a c)(14 n n2)  < 0 i¤ 14 n n2 < 0. Theinequality holds i¤ n > 4.(ii) (qm0 + nqmi )  (n+ 1)qpi = 2(a c)(49+24n+3n2)  > 0 8n.(iii) (m0 + nmi )  (n+ 1)pi =   (a c)2	3n 22< 0 i¤ 	 > 0, where	 =  3073 + 44548n + 151318n2 + 139764n3 + 59759n4 + 13400n5 +1540n6 + 72n7. It is not di¢ cult, although tedious, to show that 	 > 0 8n,which completes the proof. QED8Proof of Proposition 4 : In proving that SWm(n) < SW p(n) i¤ n > 4,it su¢ ces to show that  14 + n + n2 > 0 i¤ n > 4. The result followsimmediately. QED9

R&D policy and privatization in a mixed oligopoly

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R&D policy and privatization in a mixed oligopoly

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