Firms Costs, Profits, Entries, and Innovation under Optimal Privatization Policy

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1 MPRA Munch Personal RePEc Archve Frms Costs, Profts, Entres, and Innovaton under Optmal Prvatzaton Polcy Junch Haraguch and Toshhro Matsumura 22 August 2017 Onlne at MPRA Paper No , posted 26 August :23 UTC

2 Frms Costs, Profts, Entres, and Innovaton under Optmal Prvatzaton Polcy Junch Haraguch Faculty of Economcs, Gakushun Unversty and Toshhro Matsumura Insttute of Socal Scence, The Unversty of Tokyo August 22, 2017 Abstract We nvestgate how cost condtons of prvate frms affect optmal prvatzaton polcy and prvate frms profts. We fnd that the optmal degree of prvatzaton s decreasng wth the costs of prvate frms unless the publc frm s fully prvatzed n equlbrum. A cost reducton n a prvate frm ncreases the degree of prvatzaton and benefts for all prvate frms. Therefore, each prvate frm s proft s ncreasng wth ts rval prvate frms costs, whch s n contrast to the result when the degree of prvatzaton s gven exogenously. Ths nterestng property yelds two mportant results. The proft of each prvate frm can ncrease wth the number of prvate frms, and the postve externalty of nnovaton accelerates prvate frms R&D. JEL classfcaton numbers: D43, H44, L33 Key words: partal prvatzaton, cost-reducng R&D, asymmetrc prvate frms, constant margnal costs We are grateful to Akfum Ishhara, Hrokazu Ishse, Norak Matsushma, Masak Nakabayash, Cong Pan, Naok Wakamor, and the partcpants of semnars at Shnshu Unversty and the Unversty of Tokyo for ther helpful comments and suggestons. We acknowledge fnancal support from JSPS KAKENHI (Grant Numbers 15K03347, 17J07949) and the Zengn Foundaton for Studes on Economcs and Fnance. Any errors are our own. Correspondng author. Faculty of Economcs, Gakushun Unversty, 1-5-1, Mejro, Toshma-ku, Tokyo , Japan. Phone: (81) Fax: (81) E-mal: jyunch.haraguch@gmal.com Insttute of Socal Scence, the Unversty of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo , Japan. Phone: (81) Fax: (81) E-mal: matusmur@ss.u-tokyo.ac.jp 1

3 1 Introducton For more than 50 years, we have observed a worldwde wave of the prvatzaton of state-owned publc enterprses. Nevertheless, many publc and partally prvatzed enterprses are stll actve n planned and market economes n developed, developng, and transtonal countres. Whle some publc enterprses are tradtonal monopolsts n natural monopoly markets, a consderable number of publc and partally prvatzed enterprses competes wth prvate enterprses n a wde range of ndustres. 1 Optmal prvatzaton polces n such mxed olgopoles have attracted extensve attenton from economcs researchers n such felds as ndustral organzaton, nternatonal economcs, publc economcs, fnancal economcs, and development economcs. 2 Specfcally, the lterature on mxed olgopoles has nvestgated optmal prvatzaton polcy n dfferent stuatons. Matsumura (1998) nvestgated Cournot mxed duopoles and showed that the optmal degree of prvatzaton s never zero unless full natonalzaton yelds a publc monopoly. Ln and Matsumura (2012) and Matsumura and Okamura (2015) found that the optmal degree of prvatzaton ncreases wth the number of prvate frms and decreases wth the foregn ownershp share n prvate frms. In free-entry markets, Matsumura and Kanda (2005) showed that the optmal degree of prvatzaton s zero when prvate compettors are domestc, whle Cato and Matsumura (2012) found that the optmal degree of prvatzaton s strctly postve when prvate compettors are foregn and ncreases wth the foregn ownershp share n prvate frms. In addton, Chen (2017) showed that the optmal degree of prvatzaton s postve even n free-entry markets f prvatzaton mproves producton effcency. Fujwara (2007) showed a nonmonotonc (monotonc) relatonshp between the degree of product dfferentaton and optmal degree of prvatzaton n a non-free-entry (free-entry) market. Cato and Matsumura (2015) dscussed the relatonshp between optmal trade and prvatzaton polces, showng that a hgher tarff rate reduces the optmal degree of prvatzaton n free-entry markets. Lee et al. (2017) showed that the optmal degree 1 Examples nclude Unted States Postal Servce, Deutsche Post AG, Areva, Nppon Telecom and Telecommuncaton (NTT), Japan Tobacco, Volkswagen, Renault, Électrcté de France, Japan Postal Bank, Kampo, Korea Development Bank, and Korea Investment Corporaton. 2 For examples of mxed olgopoles and recent developments n ths feld, see Heywood and Ye (2009a), Ishda and Matsushma (2009), Chen (2017), and the works cted theren. 2

4 of prvatzaton depends on the tmng of prvatzaton. Heywood et al. (2017) nvestgated how asymmetrc nformaton on demand condtons affects the optmal degree of prvatzaton. One common assumpton n the abovementoned studes s that all prvate frms share the same cost functon. However, n mxed olgopoles, t s often the case that prvate frms are not symmetrc. In the Japanese fnancal ndustry, publc fnancal nsttutons compete wth mega banks, such as the Bank of Tokyo-Mtsubsh UFJ, and smaller regonal banks, such as Aozora Bank. In the overnght delvery ndustry, Japan Post competes wth Yamato Transport, Nppon Express, Sagawa Express, and Seno Transportaton, whch are far from symmetrc. In the telecommuncaton ndustry, NTT group competes wth large carrers, such as Softbank and KDDI Corp., as well as many of smaller companes, such as Japan Communcaton. In the automoble ndustry, VW and Rouault compete wth huge prvate frms, such as General Motors and Toyota Motor Corp., as well as smaller prvate frms, such as Hyunda Motor Company, Honda Motor Co. Ltd., and Mazda Motor Corp. Thus, t s realstc to assume that prvate frms are not always symmetrc. In ths study, we allow cost asymmetry among prvate frms. Ths enrches the analyss of mxed olgopoles. For example, suppose that a decrease n prvate frms costs ncreases these frms profts. If we allow asymmetrc costs among prvate frms, we can decompose ths costreducton effect nto the followng two effects: the effect of the reducton of a frm s own cost and that of ts rval s cost. Then, we can nvestgate how the rval s cost affects profts and thus, the behavor of other prvate frms. In ths study, we use the model of Pal (1998) wth lnear demand and constant margnal costs, and we allow cost dfferences among prvate frms. We adopt the partal prvatzaton approach of Matsumura (1998) and endogenze the degree of prvatzaton. We fnd that under optmal prvatzaton polcy, the reducton of a prvate frm s margnal cost ncreases the profts of all prvate frms. Under the optmal prvatzaton polcy, a reducton of a prvate frm s margnal cost ncreases the degree of prvatzaton, whch makes the publc frm less aggressve. Ths s benefcal for all prvate frms and thus, the reducton of a prvate frm s margnal cost s benefcal for all prvate frms. By contrast, f the degree of prvatzaton s gven 3

5 exogenously, the reducton of a prvate frm s margnal cost ncreases ts own proft but reduces the other prvate frms proft. Ths basc prncple can apply to mxed olgopoles wth symmetrc prvate frms, and we derve mportant mplcatons even n models wth these frms. Frst, we nvestgate the relatonshp between the new entry of a prvate frm and the optmal degree of prvatzaton. We fnd that the new entry of a prvate frm ncreases the degree of prvatzaton, whch ncreases the profts of all prvate frms. By contrast, f the degree of prvatzaton s gven exogenously, the new entry of a prvate frm decreases the profts of all ncumbent prvate frms. Next, we nvestgate an nnovaton ncentve for prvate frms. We formulate a model n whch prvate frms engage n cost-reducng R&D nvestments wth externalty among prvate frms. We fnd that prvate frms more ntensvely engage n nnovaton when the degree of prvatzaton s endogenous. In addton, we fnd that R&D expendture s ncreasng wth the degree of spllover effect among prvate frms and the number of prvate frms when the degree of prvatzaton s endogenous. These fndngs are because a decrease of one prvate frm s cost ncreases the profts of all prvate frms. These results suggest that the tmng of prvatzaton affects the entry decson and nnovaton actvtes of prvate frms. The rest of ths paper s organzed as follows. In Secton 2, we present the basc model. Secton 3 presents an equlbrum analyss and derves the optmal prvatzaton polcy. Secton 4 dscusses the relatonshp between prvate frms profts and ther costs. Secton 5 nvestgates the relatonshp between prvate frms profts and new entres. Secton 6 endogenzes the costs of prvate frms by consderng cost-reducng R&D. Secton 7 concludes. 2 The Model We consder a mxed olgopoly model n whch one publc frm (frm 0) competes wth n prvate frms (frms 1, 2,...,n). These frms produce homogeneous products for whch the nverse demand functon s p(q) = a Q, 4

6 where p denotes prce, a s a postve constant, and Q := n =0 q s the total output. The margnal costs are constant. Let c 0 be the frm s margnal cost. Each frm s proft s gven by π = (p(q) c )q. We assume that c < c 0 for = 1, 2,..., n. In other words, we assume that the publc frm s less effcent than the prvate frm. 3 The socal surplus W s gven by W = Q 0 p(q)dq pq + n π = =0 Q 0 p(q)dq n c q. Followng Matsumura (1998), the publc frm s objectve Ω s convex-combnaton of socal surplus and ther own proft, Ω = απ 0 + (1 α)w, where α [0, 1] represents the degree of prvatzaton. In the case of full natonalzaton (.e., α = 0), frm 0 maxmzes socal welfare. In the case of full prvatzaton (.e., α = 1), frm 0 maxmzes ts proft. Each prvate frm s objectve s ts proft. The complete nformaton game runs as follows. In the frst stage, the government chooses =0 the degree of prvatzaton α to maxmze the socal surplus. smultaneously chooses ts output to maxmze ts objectve. In the second stage, each frm We solve ths game by backward nducton and the equlbrum concept s subgame perfect Nash equlbrum. Throughout ths study, we assume that a s suffcently large. It guarantees that the solutons n the second-stage subgames are nteror. In other words, the publc frm produces a postve output n equlbrum regardless of α. 3 The assumptons of lnear demand and constant margnal costs wth cost dsadvantage of a publc frm over prvate frms s popular n the lterature on mxed olgopoles. See Pal (1998), Capuano and De Feo (2010), and Matsumura and Ogawa (2010). For a dscusson on the endogenous cost dsadvantage of publc frms, see Matsumura and Matsushma (2004). 5

7 3 Equlbrum Frst, we solve the second stage game gven α. The frst-order condtons of publc and prvate frms are Ω q 0 = a (1 + α)q 0 n q c 0 = 0, π = a 2q q j c = 0 ( = 1,...n), q j respectvely. The second-order condtons are satsfed. These frst-order condtons yeld the followng reacton functons of publc and prvate frms R 0 (q ) = a n q c 0, 1 + α R (q j ) = a j q j c 2 ( = 1, 2,..., n j ), respectvely. These reacton functons yeld the followng equlbrum quanttes of publc and prvate frms q0(α) = a (n + 1)c 0 + n 1 + (n + 1)α c q (α) = α(a + n c ) + c 0 (1 + (n + 1)α)c 1 + (n + 1)α, (1) ( = 1, 2,..., n), (2) respectvely. We obtan the followng equlbrum total output, prce, prvate frm s proft, and welfare Q (α) = (na n c )α + a c 0, (3) 1 + (n + 1)α p (α) = (a + n c )α + c 0, (4) 1 + (n + 1)α π (α) = W (α) = ( α(a + n c ) ) + c 0 (1 + (n + 1)α)c 2 ( = 1, 2,..., n), (5) 1 + (n + 1)α X 1 2(1 + (n + 1)α) 2, (6) respectvely, where X 1 := (a(1+nα) c 0 α n c ) 2 +2α(a (n+1)c 0 + n c ) 2 +2 n (α(a+ n c ) + c 0 (1 + (n + 1)α)c ) 2. 6

8 Next, we dscuss the government s welfare maxmzaton problem n the frst stage. Let α be the equlbrum degree of prvatzaton. Lemma 1 () α > 0. () α = mn{α, 1}, where α := nc 0 n c a (n + 1) 2 c 0 + (n + 2) n c. () α s decreasng n c for = 1, 2,..., n and ncreasng n c 0. Proof See the Appendx. Lemma 1() was shown by Matsumura (1998) n duopoles and by Matsumura and Kanda (2005) n olgopoles wth symmetry among prvate frms. Lemma 1() states that as long as the soluton s nteror (.e., full prvatzaton s not optmal), the optmal degree of prvatzaton s decreasng wth the cost of each prvate frm. An ncrease of the degree of prvatzaton makes frm 0 less aggressve, because t s less concerned wth consumer surplus. Through the strategc nteracton, the less aggressve behavor of frm 0 makes prvate frms more aggressve. In other words, producton substtuton from the publc frm to the prvate frms takes place. Because the margnal cost of the publc frm s hgher than that of each prvate frm, ths producton substtuton mproves welfare (welfare-mprovng effect). 4 However, because the total output s decreasng n α, an ncrease of the degree of prvatzaton reduces welfare (welfarereducng effect). Ths trade-off determnes the optmal degree of prvatzaton. The hgher (lower) c 0 (c for = 1, 2,..., n) s, the stronger s the abovementoned welfare mprovng effect of the producton substtuton. Therefore, the optmal degree of prvatzaton s ncreasng n c 0 and decreasng n c for = 1, 2,..., n. From Lemma 1(), we fnd that the optmal degree of prvatzaton remans unchanged as long as n c remans unchanged. Ths ncludes an mportant polcy mplcaton. Gven the average productvty among prvate frms, the dstrbuton of the costs among prvate frms (or the degree of heterogenety among prvate frms) does not affect the optmal prvatzaton polcy. 4 For an excellent dscusson on the welfare-mprovng producton substtuton, see Lahr and Ono (1988). 7

9 4 Prvate Frms Profts Suppose that the soluton of the frst stage s nteror (.e., α < 1). By substtutng α nto π (α), we obtan the followng equlbrum proft of prvate frms: ( π = (n + 1)c 0 We now present our man result. ) 2 n c c ( = 1, 2,..., n). (7) Proposton 1 If the optmal prvatzaton polcy s not full prvatzaton (.e., α < 1), prvate frm s proft s decreasng n c j for, j = 1, 2,..., n and ncreasng n c 0. Proof See Appendx. In order to hghlght the mplcaton and ntuton of ths result, we present a supplementary result as a benchmark. Proposton 2 Suppose that the degree of prvatzaton α s gven exogenously. () Prvate frm s proft s decreasng n c and ncreasng n c 0. () Prvate frm s proft s nondecreasng n c j for = 1, 2,..., n, j =, 1,..., n and j. () If α > 0, prvate frm s proft s ncreasng n c j for = 1, 2,..., n, j =, 1,..., n and j. Proof See the Appendx. In both exogenous and endogenous α cases, prvate frm s proft s decreasng wth ts own cost and ths result s ntutve. A reducton of c drectly ncreases frm s proft even when all frms outputs are gven exogenously. In addton, a decrease n c mproves the compettve advantage of frm and ncreases the equlbrum output of frm. Through the strategc nteracton, a reducton of c reduces the total output of other frms, whch further ncreases frm s proft. Suppose that α s gven exogenously. Prvate frm s proft s ncreasng wth ts rvals costs, whch s an ntutve result. A reducton of c j (j ) ncreases frm j s output (drect effect) and though strategc nteracton, decreases the other frms outputs, ncludng frm s output (strategc effect). Ths reducton of frm s output reduces ts proft. In addton, a reducton of c j ncreases the total output, and further reduces frm s proft. 8

10 However, when α s endogenous, an addtonal effect exsts. A reducton of c j (j ) ncreases α, and makes the publc frm (frm 0) less aggressve. Ths effect s so strong that a reducton of c j (j ) ncreases the profts of all prvate frms. Ths result suggests that as long as the soluton s nteror (.e., α < 1), prvate frms that compete wth a publc frm have ncentves to reduce the prvate rvals costs as well as ther own costs. Ths fact mght suggest that a recent open strategy of Tokyo-Mtsubsh UFJ (Nkke Newspaper, May 3, 2017) to reduce ts own cost as well as those of other prvate frms s reasonable. From Proposton 1, we guess that the new entry of a prvate frm mght ncrease the profts of ncumbents because t also ncreases the degree of prvatzaton. In addton, we guess that prvate frms engage n cost-reducng R&D more ntensvely when the prvatzaton polcy s determned after R&D actvty and when there s a stronger spllover effect that reduces the prvate rvals costs. We dscuss these problems n the followng two sectons. 5 New Entry 5.1 New entry of a prvate frm We consder how the entry of a prvate frm affects the profts of ncumbent prvate frms. Suppose that a new entrant, frm n+1, enters the market. We assume that c n+1 < c 0. Frst, we consder the stuaton n whch α s determned before the entry. Lemma 2 Suppose that α s gven exogenously. The new entry of a prvate frm reduces the profts of all ncumbent prvate frms. Proof See the Appendx. Gven α, the new entry of a prvate frm accelerates competton and reduces the market share of each ncumbent, and thereby reduces the profts of all frms. We now consder the stuaton n whch α s determned after the entry. Proposton 3 Suppose that α < 1. The new entry of a prvate frm ncreases α. Proof See the Appendx. 9

11 The new entry of a prvate frm strengthens the welfare-mprovng effect of prvatzaton (the effect of producton substtuton from the publc frm to the prvate frm), whereas t reduces the welfare-reducng effect of prvatzaton (the output-reducton effect). Therefore, ths new entry ncreases the optmal degree of prvatzaton. frms. We now dscuss how the new entry of a prvate frm affects the profts of ncumbent prvate Proposton 4 Suppose that α s endogenous. Suppose that α < 1 even after the entry of frm n + 1. Then, the entry of frm n + 1 ncreases the profts of all ncumbent prvate frms. Proof See the Appendx. A new entry ncreases the degree of prvatzaton, whch makes the publc frm less aggressve. Ths s benefcal for all prvate frms and thus, a new entry ncreases the profts of all prvate frms Free entry We now dscuss a free-entry market. We assume that all potental entrants (prvate frms) share the same margnal cost c and entry cost F. In the frst stage, each frm ( = 1, 2,.., n) chooses whether to enter the market. In the second stage, after observng n, the government chooses α. 6 In the thrd stage, all frms choose ther outputs ndependently. Let q be the output of each prvate frm. In the thrd stage the frst-order condtons of frm 0 and frm ( = 1, 2,...n) are p + αp q 0 c 0 = 0, (8) 5 Matsumura and Sunada (2013) nvestgated a mxed olgopoly wth msleadng advertsng competton. They showed that the new entry of a prvate frm mght ncrease the profts of the ncumbent prvate frms, because t ncreases (decreases) advertsng of the publc frm (prvate frms). Some studes on prvate olgopoles have showed that a new entry could ncrease the profts of ncumbents. Mukherjee and Zhao (2009) consdered an asymmetrc Stackelberg settng n whch there are two ncumbent frms (leader) wth dfferent margnal costs and a potental entrant (follower) wth a hgher margnal cost. The authors then show that the exstence of an neffcent follower can ncrease the proft of the more effcent leader. Ishda et al. (2011) consdered a model n whch a domnant frm competes wth mnor frms and showed that an ncrease of the number of mnor frms accelerates the R&D and proft of the domnant frm. Chen and Rordan (2007) showed that n a dfferentated market, an ncrease of varety by a new entry mght soften competton and ncrease the profts of ncumbent frms. The drvng force of our study s dfferent from that n these studes. 6 For examples supportng ths tmelne (entry then prvatzaton), see Lee et al. (2017). 10

12 p + p q c = 0, (9) respectvely. In the second stage, the government chooses α such that { α n(c 0 c) } = mn a (n + 1) 2 c 0 + n(n + 2)c, 1. (10) In the frst stage, each frm ( = 1, 2,...n) enters the market as long as the proft s nonnegatve. Therefore, as long as n > 0, By defnton (p c)q F = 0. (11) q 0 + nq = Q. (12) These fve equatons determne the equlbrum values of q 0, q, Q, n, and α. The free-entry equlbrum s locally stable f the proft of each prvate frm s decreasng n n. As we showed n the prevous subsecton, the proft of each frm s ncreasng n n as long as α < 1. Therefore, as long as α < 1, the equlbrum must be locally unstable. Under these condtons, α must be one (full prvatzaton) at the locally stable equlbrum. These dscussons lead to the followng proposton. Proposton 5 If n > 0, at the locally stable equlbrum α = 1. If F s large enough, no prvate frm enters the market. In ths case, α = 0. Therefore, n the constant margnal cost model, the free-entry equlbrum yelds ether a publc monopoly or a prvate olgopoly n whch the government fully prvatzes frm R&D In ths secton, we endogenze the margnal costs of prvate frms. We consder cost-reducng R&D actvtes by prvate frms. 8 We assume that all prvate frms are symmetrc at the begnnng of 7 Most studes on mxed olgopoles n free-entry homogeneous product markets such as Matsumura and Kanda (2005), Cato and Matsumura (2012, 2015) and Lee et al. (2017), have assumed ncreasng margnal costs because the constant margnal cost model yelds ths techncal problem. 8 For dscussons on R&D n mxed olgopoles, see Nshmor and Ogawa (2002), Matsumura and Matsushma (2004), Ishbash and Matsumura (2006), Zkos (2010), Gl-Moltoet al. (2011), and Lee and Tomaru (2017). How- 11

13 the game. The margnal cost of frm s gven by c (x ; x j ) = C (x + β j x j ) (, j = 1, 2,..., n, j) where C denotes the prvate frm s margnal cost before R&D, x s frm s R&D level, and β [0, 1] s the degree of spllover from other prvate frms R&D. 9 We assume that C < c 0. Ths assumpton guarantees that the publc frm s less effcent, whch s also assumed n the prevous sectons. Each prvate frm s proft and socal welfare are W = Q 0 π = ( p(q) c (x ; x j ) ) q γx 2, p(q)dq pq + n π = =0 Q 0 p(q)dq n c q γ =0 n x, respectvely, where γ s a postve constant. We assume that γ s suffcently large. Ths assumpton guarantees the second-order condton and nteror solutons n the quantty competton stage. The game runs as follows. In the frst stage, each prvate frm chooses x ndependently. In the second stage, the government chooses α. In the thrd stage, all frms choose ther outputs ndependently. We restrct our attenton to the symmetrc equlbrum n whch all prvate frms choose the same x. We have already solved the second and thrd subgames. We now solve the frst stage game. The frst-order condton of each prvate frm s 10, π x = 2((n + 1)c 0 n c (x ; x j ) c (x ; x j ))((n 1)β + 2) 2γx = 0. ( = 1, 2,..., n). Ths leads to the followng equlbrum R&D level. x = (n + 1)(βn β + 2)(c 0 C) γ (n + 1)(βn β + 2)(βn β + 1). (13) ever, none of these studes dscuss optmal prvatzaton polcy. Heywood and Ye (2009c) s the excepton. They nvestgated the optmal degree of prvatzaton but assumed that prvatzaton occurs before R&D. Therefore, the authors dscussed only our benchmark tmelne. Moreover, they dscussed a duopoly (only one prvate frm) and thus, dd not dscuss how prvate rvals costs affected the proft of the prvate frm. 9 Ths type of cost-reducng R&D s ntensvely dscussed n the lterature on prvate olgopoles. See Spence (1984), d Aspremont and Jacquemn (1988), and Suzumura (1992). 10 The second-order condton s satsfed f γ > 1 2((n 1)β+2) 2 12

14 Then, we obtan the followng equlbrum margnal cost c c = γc (n + 1)(βn β + 2)(βn β + 1)c 0. (14) γ (n + 1)(βn β + 2)(βn β + 1) By substtutng (14) nto α n Lemma 1(), we obtan α = nγ(c 0 C) (a (n + 1) 2 c 0 + n(n + 2)C)γ (n + 1)(βn β + 2)(βn β + 1)(a c 0 ) (15) as long as α < 1. We now dscuss the case wth exogenous α as a benchmark. Suppose that α s gven exogenously. The frst-order condtons of each prvate frm ( = 1, 2,..., n) s 11, π x = 2 ( α(a + j c j(x j ; x ) c 0 ) + c 0 (1 + nα)c (x ; x j ) 1 + (n + 1)α Ths leads to the followng equlbrum R&D level. )( (n βn + β)α + 1 ) 2γx = (n + 1)α x (α) = (1 + α (n 1)αβ)(α(a C) + c 0 C) (1 + (n + 1)α) 2 γ (1 + α)(1 + (n 1)β)(1 + nα (n 1)αβ). (16) We now present our result on R&D. Proposton 6 Suppose that α < 1. () x (α ) < x. () x s ncreasng n β, whereas x (α) s decreasng n β. () x s ncreasng n n, whereas x (α) s decreasng n n. Proof See the Appendx. Proposton 6 states that the tmng of prvatzaton s mportant for nnovaton. Expectng future prvatzaton, prvate frms competng wth a publc frm engage n R&D more ntensvely to ncrease the degree of prvatzaton. 12 Because the cost reducton of prvate rvals s benefcal for all prvate frms when α s endogenous, the exstence of spllover effect among prvate frms stmulates R&D. By contrast, when the degree of prvatzaton s gven exogenously, each prvate frm mght less ntensvely engage n 11 The second-order condton s satsfed f γ > ( 1+nα (n 1)αβ 1+(n+1)α ) 2 12 However, t mght not be welfare-mprovng. Suppose that the government, not each prvate frm, chooses x. Let x S be ths second-best R&D level. We obtan that x S < x as long as α < 1 (.e., the equlbrum R&D nvestment level s excessve for socal welfare). 13

15 R&D, because the reducton of prvate rvals cost reduces the profts of other prvate frms n ths case. Smlarly, expectng future prvatzaton, the new entry of a prvate frm accelerates prvate frms R&D, because t affects the optmal prvatzaton polcy. An ncrease n β (or n) ncreases x as long as α < 1. A further ncrease n β (or n) ncreases x and α wll eventually reach one. Thereafter, the equlbrum x s x (1). Therefore, the equlbrum nvestment level s nonmonotone wth respect to β and n (nverted U-shape). Fnally, we dscuss the lmtaton of our results. We assume that the publc frm s cost s gven exogenously. If the spllover effect reduces the publc frm s cost, Proposton 6 does not hold. A decrease n c 0 reduces the profts of all prvate frms, and thus, each prvate frm has a weaker ncentve for nnovaton when the spllover effect s stronger. 7 Concludng remarks In ths study, we ntroduce cost dfferences among prvate frms and nvestgate how a prvate frm s cost affects the optmal degree of prvatzaton and profts of prvate frms. We fnd that a cost reducton of a prvate frm reduces the optmal degree of prvatzaton, whch s benefcal for all prvate frms. Therefore, a cost reducton of a prvate frm ncreases the profts of all prvate frms, whch never holds when the degree of prvatzaton s gven exogenously. In addton, we fnd that the new entry of a prvate frm s benefcal for all ncumbent prvate frms because t ncreases the degree of prvatzaton. Fnally, we dscuss nnovaton of prvate frms and fnd that expectng future prvatzaton accelerates nnovaton actvtes of prvate frms. In ths study, we assume that prvate frms are domestc. In the lterature on mxed olgopoles, t s known that the natonalty of prvate frms often affects the behavor of a publc frm and the optmal prvatzaton polcy. To extend our analyss n ths drecton s dffcult work and remans for future research Whether the prvate frm s domestc or foregn often yelds contrastng results n the lterature on mxed olgopoles. See Corneo and Jeanne (1994), Fjell and Pal (1996), Pal and Whte (1998), Bárcena-Ruz and Garzón (2005a, 2005b), and Heywood and Ye (2009b). The optmal degree of prvatzaton s decreasng wth the foregn ownershp rate n prvate frms when the number of prvate frms s gven exogenously (Ln and Matsumura, 2012), whle t s ncreasng wth the foregn ownershp rate n prvate frms n free-entry markets (Cato and Matsumura, 2012). 14

16 Appendx Proof of Lemma 1 From (6), we obtan W α = ( a (n + 1)c0 + n c )( α(a (n + 1) 2 c 0 + (n + 2) n c ) nc 0 + n c ) ( ) (n + 1)α By substtutng α = 0 nto ths, we obtan W α Ths mples Lemma 1(). = ( a (n + 1)c 0 + α=0 n )( c nc0 By solvng W / α = 0 wth respect to α, we obtan The second-order condton α = n ) c > 0. nc 0 n c a (n + 1) 2 c 0 + (n + 2) n c. (17) (a (n + 1)2 c 0 + (n + 2) n c ) 4 (a (n + 1)c 0 + n c ) 2 < 0 s satsfed. Therefore, the optmal degree of prvatzaton s α as long as α < 1. Ths mples Lemma 1(). From (17), we obtan = (a (n + 1)2 c 0 + (n + 2) n c ) + (n + 2)(nc 0 n c ) c (a (n + 1) 2 c 0 + (n + 2) n c ) 2 < 0 ( = 1, 2,..., n), = n(a (n + 1)2 c 0 + (n + 2) n c ) + (n + 1) 2 (nc 0 n c ) c 0 (a (n + 1) 2 c 0 + (n + 2) n c ) 2 > 0. α α These mply Lemma 1(). Proof of Proposton 1 From (7), we obtan π ( = 4 (n + 1)c 0 c π ( = 2 (n + 1)c 0 c j n ) c c < 0 ( = 1, 2,..., n), n ) c c < 0 (, j = 1, 2,..., n j ), π ( = 2(n + 1) (n + 1)c 0 c 0 n ) c c > 0 ( = 1, 2,..., n). 15

17 These results mply Proposton 1. Proof of Proposton 2 Because we assume nteror solutons n the quantty competton stage, from (2) we obtan α(a + n c ) + c 0 (1 + (n + 1)α)c ) > 0. From (5), we obtan π (α) = 2(α(a + n c ) + c 0 (1 + (n + 1)α)c ) c 0 (1 + (n + 1)α) 2 > 0 ( = 1, 2,..., n), (18) π (α) 2(1 + nα)(α(a + n = c ) + c 0 (1 + (n + 1)α)c ) c (1 + (n + 1)α) 2 < 0 ( = 1, 2,..., n), (19) π (α) c j = 2α(α ( a + n c ) + c 0 (1 + (n + 1)α)c ) (1 + (n + 1)α) 2 0 (, j = 1, 2,..., n j ). (20) The strct nequalty n (20) holds f and only f α > 0. These mply Proposton 2. Proof of Lemma 2 Let π (n+1) (α) and π n (α) be the equlbrum proft of frm ( = 1, 2,..., n) wth and wthout the entry of frm n + 1, respectvely, gven α. We obtan π (n+1) (α) = ( α(a + ) n+1 2 c ) + c 0 (1 + (n + 2)α)c ( = 1, 2,..., n + 1). (21) 1 + (n + 2)α Let q (n+1) (α) be the equlbrum output of frm ( = 1, 2,..., n + 1) wth the entry of frm n + 1, gven α. We obtan q (n+1) (α) = α(a + n+1 c ) + c 0 (1 + (n + 2)α)c 1 + (n + 2)α ( = 1, 2,..., n + 1). (22) Because we assume nteror solutons n the quantty competton stage, from (2) and (22), we obtan α(a + n c ) + c 0 (1 + (n + 1))α)c > 0 and α(a + n+1 c ) + c 0 (1 + (n + 2)α)c ) > 0. From (21) and (5), we obtan π (n+1) (α) π n αx 2 (α) = (1 + (n + 2)α)(1 + (n + 1)α) 0 ( = 1, 2,..., n) (23) where X 2 := (1+(n+1)α)(α(a+ n+1 c )+c 0 (1+(n+2)α)c )+(1+(n+2)α)(α(a+ n c )+ c 0 (1 + (n + 1)α)c )))(α(a + n c ) + c 0 (1 + (n + 1)α)c n+1 ) > 0. The strct nequalty n (23) holds f and only f α > 0. These mply Lemma 2. 16

18 Proof of Proposton 3 Let α (n+1) and α n be the equlbrum degree of prvatzaton wth and wthout the entry of frm n + 1, respectvely. We obtan α (n+1) = (n + 1)c 0 n+1 c a (n + 2) 2 c 0 + (n + 2) n+1 c (24) as long as α (n+1) < 1. From (17) and (24), we obtan α (n+1) α n = X 3 (a (n + 2) 2 + (n + 1) n+1 c )(a (n + 1) 2 c 0 + (n + 2) n c ) > 0, where X 3 := (a (n + 1) 2 c 0 + (n + 2) n c )(c 0 c n+1 ) + ((n + 2)(c 0 c n+1 ) + (n + 1)c 0 )(nc 0 n c ) > 0. Ths mples Proposton 3. Proof of Proposton 4 Let π (n+1) and π n of frm n + 1, respectvely. be the equlbrum proft of frm ( = 1, 2,..., n) wth and wthout the entry By substtutng α (n+1) nto π (n+1) (α), we obtan ( π (n+1) = From (7) and (25), we obtan π (n+1) π n Ths mples Proposton 4. Proof of Proposton 6 ) n+1 2 (n + 2)c 0 c c ( = 1, 2,..., n + 1). (25) = 2(c 0 c n+1 )((n + 1)c 0 By substtutng (15) nto (16), we obtan n c c ) + (c 0 c n+1 ) 2 > 0. x (α ) = (c 0 C)(n + 1)((a c 0 n(c 0 C))γ (a c 0 )(βn β + 1)(βn β + 2))X 4 (γ (n + 1)(βn β + 1)(βn β + 2))X 5, (26) where X 4 := (a c 0 n(c 0 C)(βn β + 2))γ (n + 1)(a c 0 )(βn β + 1)(βn β + 2), and X 5 := (a c 0 n(c 0 C)) 2 γ 2 (a c 0 )(βn β + 1)((a c 0 )(βn 2 β + 2n + 3) n(c 0 C)(n + 17

19 βn + 3 β))γ + (a c 0 ) 2 (n + 1)(βn β + 1)(βn β + 2), and both X 4 and X 5 are postve because we assume that a and γ are suffcently large. From (13) and (26), we obtan x x (α ) = (a c 0)(c 0 C)(n + 1)(βn β + 1)γX 6 (γ (n + 1)(βn β + 1)(βn β + 2))X 5 > 0, where X 6 := (a c 0 n(c 0 C))γ (a c 0 )(n + 1)(βn β + 1)(βn β + 2), and ths s postve because we assume that a and γ are suffcently large. These mply Proposton 6(). From (13), we obtan x β = (n + 1)(n 1)(c 0 C)(γ + (n + 1)(βn β + 2) 2 ) (γ (n + 1)(βn β + 2)(βn β + 1)) 2 > 0. Ths mples the former part of Proposton 6(). From (16), we obtan x (α) β ((a C)α + c 0 C)X 7 = ((1 + (n + 1)α) 2 γ (1 + α){1 + (n 1)β)(1 + nα (n 1)αβ)) 2 < 0 where X 7 := α(1 + (n + 1)) 2 γ (1 + α)(1 + nα (n 1)αβ) 2, and ths s postve because we assume that γ s suffcently large. Ths mples the latter part of Proposton 6(). From (13), we obtan x n = (c 0 C)(2(1 + nβ)γ + β(n + 1) 2 (βn β + 2) 2 ) (γ (n + 1)(βn β + 2)(βn β + 1)) 2 > 0. Ths mples the former part of Proposton 6(). From (16), we obtan x (α) n ((a C)α + c 0 C)X 8 = ((1 + (n + 1)α) 2 γ (1 + α){1 + (n 1)β)(1 + nα (n 1)αβ)) 2 < 0 where X 8 := α(αn + α + 1)(2(1 + α) + β + 3αβ nαβ)γ + α 2 (1 + α)(1 β)β 2 n 2 2αβ(1 + α)(1 β)(1+α+αβ)n αβ 2 (1+α)(2+α+αβ) (1+α) 2 (α +β αβ), and ths s postve for suffcently large γ. Ths mples the latter part of Proposton 6(). 18

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