PARTIAL PRIVATIZATION OF STATE HOLDING CORPORATIONS

Transcription

1 PARTIAL PRIVATIZATION OF STATE HOLDING CORPORATIONS by Quan Dong, Juan Carlos Bárcena-Ruz and María Begoña Garzón 2016 Workng Paper Seres: IL. 97/16 Departamento de Fundamentos del Análss Económco I Ekonom Analsaren Onarrak I Sala Unversty of the Basque Country

2 Partal Prvatzaton of State Holdng Corporatons* Quan Dong South Chna Normal Unversty, School of Economcs and Management, Hgher Educaton Mega Center, Guangzhou , P. R. Chna. Juan Carlos Bárcena-Ruz Unversdad del País Vasco UPV/EHU. Avenda Lehendakar Agurre 83; Blbao, Span. Mara Begoña Garzón Unversdad del País Vasco UPV/EHU. Avenda Lehendakar Agurre 83; Blbao, Span. Abstract We consder a state holdng corporaton wth two plants that may produce complement or substtute goods and that compete wth one or two prvate frms. We fnd that the government partally prvatzes the two plants of the state holdng corporaton and s ndfferent between sellng the m partally to a sngle nvestor or to dfferent nvestors. However, n the former case the government retans a greater (lower) stake n the state corporaton f goods are substtutes (complements). Keywords: partal prvatzaton; state holdng corporaton; multproduct frms; olgopoly. JEL classfcaton: L22, Q56, Q58. * Acknowledgments: Fnancal support from Mnstero de Cenca y Tecnología (ECO , ECO P), Basque Government (IT ) and the Unversty of the Basque Country (EHU14/05) s gratefully acknowledged. Correspondng author: Juan Carlos Bárcena-Ruz, Departamento de Fundamentos del Análss Económco. Facultad de Economía y Empresa. Unversdad del País Vasco UPV/EHU. Avenda Lehendakar Agurre 83; Blbao, Span. Phone: Fax: E-mal: juancarlos.barcena@ehu.es. 1

3 1. INTROCTION One of the ssues analyzed by the lterature on mxed olgopoly s the decson by governments whether to prvatze a sngle publc frm (see, for example, De Fraja and Delbono 1989, 1990; Corneo and Jeanne 1994). 1 These papers have been extended to consder, among other factors, partal prvatzaton of publc frms (Matsumura, 1998; Ln and Matsumura, 2012), strategc prvatzaton under nternatonal trade (Bárcena- Ruz and Garzón 2005a, 2005b), sequental prvatzaton of publc frms (Matsumura and Shmzu 2010), prvatzaton when the publc frm s as effcent as prvate frms (Bárcena-Ruz 2012), and prvatzaton under an nterdependence payoff structure (Matsumura and Okamura 2015). The papers cted above usually assume that the publc frm produces a sngle good at a sngle producton plant. However, n practce governments own frms that produce varous types of goods at varous producton plants, and they are manly organzed as state holdng corporatons (see Kumar 1992). 2 As far as we know, the theoretcal lterature on mxed olgopoly has hardly analyzed prvatzaton of state holdng corporatons (henceforth referred to as state corporatons). One excepton s the paper by Bárcena-Ruz and Garzón (2016), who consder a state corporaton wth two plants that produce dfferentated goods. They study whether the government wants to prvatze the state corporaton, and n that case whether the two plants are sold to dfferent prvate nvestors or to a sngle nvestor. They assume that f a plant of the state corporaton s prvatzed t s fully sold to prvate nvestors. Ths has happened n many cases of prvatzaton of publc frms ntegrated nto state corporatons. 3 However, on other occasons state corporatons partally prvatze ther frms. Ths ssue has not been 1. The OECD (2005) ponts out that n the EU governments are the largest shareholders n many partally prvatzed frms. State control s also sgnfcant n Japan, Chna, Indonesa, Korea, Malaysa, Sngapore, and Thaland. In many ndustres n these countres there s nteracton between prvate and publc frms, as n the markets for cars, shps, and steel manufactures (see De Fraja, 2009). 2. Holdngs comprsng domestc publc frms have been set up by European governments such as, for e xa mple, the Socedad Estatal de Partcpacones Industrales -SEPI- n Span (see the Isttuto per la Rcostruzone Industrale -IRI- n Italy (see Cafferata, 2010), the Agence des Partcpatons de l'état -APE- n France ( and the State-owned Assets Supervson and Admnstraton Commsson of the State Councl (SASAC) n Chna ( 3. Examples of fully prvatzed Spansh publc frms nclude: the telecommuncatons frm Telefónca, the shpbuldng frm Iza and the nsurance company Musn (see 2

4 studed by the relevant lterature, so n order to fll ths gap ths paper analyzes the decson whether to partally prvatze state corporatons. Our paper relates to the lterature on partal prvatzaton that began wth the semnal paper by Matsumura (1998). He consders a mxed duopoly where a publc frm and a prvate frm compete and fnds partal prvatzaton n equlbrum under moderate condtons. Ths paper has been extended to analyze factors that affect partal prvatzaton of publc frms. Among other factors, the lterature has consdered partally foregn-owned prvate frms (Han and Ogawa, 2008), foregn nvestment n partally prvatzed frms (Ln and Matsumura, 2012), cross-ownershp of frms (Jan and Pal, 2012; Cha and Karasawa-Ohtashro, 2015), trade polces (Chao and Yu, 2006; Long and Stähler, 2009), product dfferentaton (Fujwara, 2007; Lu and Poddar, 2007), free entry (Matsumura and Kanda, 2005; Wang and Chen, 2010), endogenous tmng of decsons (Bárcena-Ruz and Garzón, 2010), envronmental problems (Kato, 2006; Ohor, 2006), and merger problems (Bárcena-Ruz and Garzón, 2003; Mendez-Naya, 2008). However, ths lterature has not consdered partal prvatzaton of state corporatons that produce more than one type of goods at more than one producton plant. State corporatons are usually multproduct, multplant frms that produce dfferent products, whch may be substtutes or complements. Therefore, we consder an ndustry made up by a state corporaton and a prvate sector. The state corporaton owns two producton plants each of whch produces a dfferentated good, and these goods may be substtutes or complements. The prvate sector comprses two prvate plants that produce dfferentated goods. These plants may be owned by dfferent nvestors (henceforth unplant frms) or by a sngle nvestor (henceforth the multplant frm). 4 The government may partally prvatze the two plants of the state corporaton, so t has two optons: It may sell a percentage of the ownershp of both plants to a sngle prvate nvestor, or t may sell a percentage of the ownershp of each plant to a dfferent prvate nvestor. 4. Multproduct frms are omnpresent n modern economes (Eckel and Neary, 2010). Lterature on ths ssue has analyzed, for example, market structure (Shaked and Sutton, 1990), product choce and the determnants of product varety (Anderson and De Palma, 2006), the effect of frm heterogenety on ndustry proftablty and welfare (Sy meond s, 2009), nternatonal trade and productvty (Bernard et al., 2010), and envronmental polces mplemented by governments (Bárcena-Ru z and Garzón, 2014). 3

5 We fnd evdence supportng the dea that the analyss conducted n the paper s mportant. Renault and Volkswagen are good examples of frms producng substtute goods. Renault produces two brands of cars, Daca and Renault, and the French government owns 15% of the frm. Smlarly, the Volkswagen Group produces several brands of car (such as Volkswagen, Aud, Seat, and Skoda) and the State of Lower Saxony owns at least 15% of the frm. 5 An example of publc frms that produce complementary goods can be found n telecommuncaton frms, whch usually have several busness lnes: Telephone, televson and the Internet. In the European Unon several governments have partally prvatzed ther telecommuncaton frms, such as Deutsche Telekom n Germany and France Télécom n France (see The German government currently owns 31.7% of the shares n Deutsche Telekom and the French government owns 13.4% of the shares n Orange, 6 the brand under whch France Télécom operates. 7 We fnd n the paper that the government partally prvatzes the two plants of the state corporaton but s ndfferent between sellng them to a sngle prvate nvestor or to dfferent prvate nvestors. When goods are complements, the government keeps a larger stake n the state corporaton f ts plants are sold to dfferent prvate nvestors than f they are sold to a sngle prvate nvestor. Moreover, the government retans a larger stake f prvate frms are multplant than f they are unplant. If goods are substtutes the contrary result s obtaned. Fnally, f goods are ndependent n demand the government keeps the same stake n the state corporaton n all cases. From ths t can be concluded that the stake that the government retans n the state corporaton depends on the type of goods produced by the state corporaton and the prvate frms, and on whether prvate frms are unplant or multplant. 5. See (for Renault), and /bnarystoragetem/fle/gb+2014_e.pdf (for Volkswagen). 6. See (for Deutsche Telekom), and traders.com/orange-sa-4649/company/ (for Orange). 7. Another example s Chna State Shpbuldng Corporaton (CSSC), a state corporaton that has many subsdares. In 2015 one of ts core subsdares, Guangzhou Shpyard Internatonal Company (GSCI), whose actvty manly covers shp buldng, offshore marne constructon and heavy machne manufacturng, sold 4.83 % of ts shares to prvate frm Yangzhou Kejn Shpbuldng Co. Ltd. Now GSCI s known as Offshore & Marne Engneerng Company (COMEC) and the Chnese government owns more than 60 percent of the shares n ths company ( 4

6 The rest of the paper s organzed as follows. Secton 2 presents the model. Secton 3 analyzes the prvatzaton of the state corporaton assumng unplant prvate frms. Secton 4 extends the analyss by consderng a multplant prvate frm. Secton 5 compares the two cases, and Secton 6 concludes. 2. THE MODEL We consder an economy made up of a publc sector and a prvate sector that produce dfferentated goods, denoted by 1 and 2, whch may be substtutes or complements. The publc sector comprses a state holdng corporaton, denoted by frm A, whose objectve functon s socal welfare f t s fully publc. It owns two plants producng goods 1 and 2, denoted by 1A and 2A respectvely. The prvate sector may comprse two unplant prvate frms or a sngle multplant frm wth two plants. We denote the prvate unplant frms and the plants of the prvate multplant frm whch produce good 1 and good 2 by 1B and 2B respectvely. The objectve functon of a unplant prvate frm s ts own proft and the objectve functon of the multplant prvate frm s the jont profts of ts two plants. On the consumpton sde, there s a contnuum of consumers of the same type. The representatve consumer maxmzes U(q 1, q 2 ) p 1 q 1 p 2 q 2, where p s the prce of good, q q A +q B s the quantty of good and, q k s the output produced by frm or plant k, 1, 2; ka, B. The functon U(q 1, q 2 ) s assumed to be quadratc, strctly concave and symmetrc n q 1 and q 2 : U(q 1, q 2 ) (q 1 +q 2 ) ((q 1 ) 2 +2bq 1 q 2 +(q 2 ) 2 )/2, 1<b<1. The nverse demand functons are gven by: p 1 (q A +q B ) b(q ja +q jb ), j;,j1,2; 1<b<1, where goods are substtutes f b>0, complements f b<0 and ndependent n demand f b0. Followng De Fraja and Delbono (1989) and Bárcena-Ruz and Garzón (2005a), 5

7 we assume that frms have dentcal technologes represented by the followng quadratc cost functon: C(q k ) cq k 2 /2, 1, 2; k A, B. Therefore, the proft functon of plant or frm k s: π k p q k cq k 2 /2, 1, 2; k A, B. (1) The proft of multplant frm k s the jont proft of the two plants: π k π 1k +π 2k, k A, B. (2) The producer surplus s the sum of the profts of frms and s gven by PS π 1A + π 2A + π 1B + π 2B. The consumer surplus s gven by: CS ((q 1A +q 1B ) 2 + 2b(q 1A +q 1B )(q 2A +q 2B ) + (q 2A +q 2B ) 2 )/2. The government ams to maxmze the socal welfare,.e. the sum of the producer surplus and the consumer surplus: W CS + PS. (3) To ncrease socal welfare the government can sell off part of the plants of the state corporaton to prvate nvestors. In ths case the government has two optons: to sell to a sngle prvate nvestor or to dfferent prvate nvestors. The state corporaton s then jontly owned by the publc and prvate sectors. We assume that the government owns β percent of the shares and the prvate nvestor owns the remanng (1 β) percent, so the partally prvatzed frm maxmzes the weghted average of socal welfare and frm profts (see Matsumura, 1998). Therefore, f the government sells part of the two plants of the state corporaton to the same nvestor t owns the same stake n the two plants and thus the objectve functon of the frm s gven by: 6

8 V βw + (1 β)(π 1A +π 2A ), 0 β 1. (4) When the government sells part of each plant to a dfferent prvate nvestor, the objectve functon of plant s gven by: V β W + (1 β ) π A, 0 β 1, 1, 2. (5) Clearly, f β 1 plant A of the state corporaton remans publc, and f β 0 plant A s fully prvatzed. To analyze the government s decson on the optmal degree of partal prvatzato n of the state corporaton, we propose a two-stage game. In the frst stage the government decdes what percentage of the shares n the state corporaton t wll sell to prvate nvestors. In ths case, the government has two optons: sell off part of both plants to a sngle prvate nvestor, or sell part of each plant to a dfferent nvestor. In the second stage the frms make producton decsons smultaneously. We solve the game by backward nducton to obtan a subgame perfect equlbrum. We consder two cases: n the frst case prvate frms are unplant, and n the second there s a sngle multplant prvate frm. Henceforth we assume that c1 to smplfy the presentaton of results. It can be shown that results are robust to changes n ths parameter. 3. UNIPLANT PRIVATE FIRMS Denote the case where prvate frms are unplant by superscrpt U. The government may partally prvatze the two plants of the state corporaton, sellng shares n each plant to a dfferent prvate nvestor (denoted by the superscrpt D) or sellng shares n the state corporaton to a sngle prvate nvestor (denoted by superscrpt S) SINGLE PRIVATE INVESTOR In ths case the two plants of the state corporaton are sold off n part to a sngle prvate nvestor. In the second stage of the game sempublc frm A chooses the output levels 7

9 q 1A and q 2A that maxmze ts objectve functon, gven by expresson (4). Prvate frm B sets the output level q B that maxmzes ts proft, gven by expresson (1), 1, 2. Solvng these problems, we obtan the followng frst order condtons: (1 3q A q B b(2q Aj + q Bj ))(1 β) + (1 2q A q B b(q Aj + q Bj ))β 0, 1 q A 3q B b q Aj + q Bj 0, j;, j1, 2. (6) From (6) we obtan the followng output of frms and socal welfare as a functon of β: 2 2+b β(1+b) q A, q, 8+b 2 (1 β) 3β +b(7 4β) B 8+b 2 (1 β) 3β +b(7 4β ) (7) W (40+44b +12b2 +b 3 2(1+b)(14+b(8+b))β+(1+b) 2 (4+b)β 2 ), 1, 2. (8+b 2 (1 β) 3β +b(7 4β)) 2 In the frst stage of the game the government chooses the optmal value of β that maxmzes socal welfare, gven by expresson (7). Solvng ths problem we obtan the followng result. Lemma 1. Under unplant prvate frms, when the government sells part of the state corporaton to a sngle prvate nvestor, n equlbrum: SU β 4+b 5+b q B 3+b (5+b)(7+3b ), SU q, SU, SU 5+b A π 2(7+6b +b 2 ) 2(7+6b +b 2 ) A 8(7+6b +b 2 ) 2, SU 8(7+6b +b 2 ) 2, SU CS (1+b)(4+b) 2, SU PS (7+6b +b 2 ) 2 (31+20b +3b 2 ) 2 (7+6b +b 2 ) 2, SU W 9+2b 2(7+6b +b 2 ) π B, 1, 2. 3(3+b) 2 It s easy to see that 0<β SU <1, so the government partally prvatzes the state corporaton. Consumer surplus decreases whle producer surplus ncreases wth the degree of prvatzaton of the state corporaton. These two effects balance for a value of β between 0 and 1, β β SU. We also fnd that the percentage of shares that remans publc ncreases wth b ( β SU / b > 0). To explan why β SU ncreases wth b two effects must be taken nto account. The frst s the nternalzaton effect: a multplant frm nternalzes how the output of one of ts plants affects that of ts other plant. When such a frm produces substtute goods (b>0) t 8

10 takes on board that ts two plants compete wth each other, whch encourages t to reduce the output of ts plants. Thus, wth substtute goods multplant frms produce less than unplant frms. Wth complement goods (b<0) the opposte result s obtaned snce a multplant frm takes on board that ts two plants cooperate, so the output of one plant ncreases wth that of the other plant. Thus, wth complement goods multplant frms produce more than unplant frms. As the state corporaton produces two goods, one n each plant, t nternalzes how the output of one of ts plants affects that of the other. The second effect s the objectve functon effect: the state corporaton produces more than prvate frms snce t takes consumer surplus nto account. If the state corporaton s sempublc, ts producton decreases wth the stake owned by the prvate sector, 1-β. When b<0, goods become less complementary as b ncreases so the nternalzaton effect becomes weaker. In ths case the output of the two plants of the state corporaton decreases wth b. When b>0, goods become closer substtutes as b ncreases and thus the nternalzaton effect becomes stronger. In that case the output of the two plants of the state corporaton also decreases wth b. Therefore, as b ncreases the output of the frms decreases due to the nternalzaton effect, regardless of whether goods are substtutes or complements. Snce the government wants to attan a certan output from frms t can acheve the desred result by choosng the rght stake n the state corporaton due to the objectve functon effect. The greater β s, the greater the weght of socal welfare s n the objectve functon of the sempublc state corporaton and, thus, the greater ts output s. Therefore, the government retans a greater stake the greater the value of b ( β SU / b > 0) s, to ncrease the output of the plants of the state corporaton. SU Fnally, q A > SU q and > SU B π A SU π B for all values of parameter b. Ths s because frm A s sempublc and therefore takes nto account the consumer surplus, whch ncreases wth the output of the frms. As a result frm A produces more than prvate frms and obtans greater market share and profts DIFFERENT PRIVATE INVESTORS In the second stage of the game, prvate frm B sets the output level q B that maxmzes ts proft, gven by expresson (1), 1, 2. In ths case, part of each plant of the state 9

11 corporaton s sold to a dfferent prvate nvestor. Each sempublc frm chooses the output level q A that maxmzes ts objectve functon gven by expresson (5). Solvng these problems, we obtan the followng frst order condtons: (1 3q A q B b(2q Aj + q Bj ))(1 β ) + (1 2q A q B b(q Aj + q Bj ))β 0, (8) 1 q A 3q B b q Aj + q Bj 0, j;, j1, 2. From (8) we obtan the followng output of the frms and socal welfare as a functon of β 1 and β 2 : q A 2(8 b(4 β j ) 3β j ) q B, b 2 (4 β )(4 β j ) (8 3β )(8 3β j ) (2 β )(8 b(4 β j ) 3β j ) b 2 (4 β )(4 β j ) (8 3β )(8 3β j ), j;, j1, 2, W ((b 3 (4 β 1 ) 2 (4 β 2 ) 2 2b 2 (β 1 48β 2 7β β β 2 + β (96 48β 2 + 7β 2 2 )) + b(β β 2 15β β β β 2 2 (9) 8( β β 2 2 )) + 2(β 1 (672 β 2 111β ) + β 2 1 ( β β 2 2 ) + 2( β β 2 2 ))))/ (b 2 (4 β 1 )(4 β 2 ) (8 3β 1 )(8 3β 2 )) 2. In the frst stage of the game the government chooses the optmal value of β 1 and β 2 that maxmzes socal welfare, gven by expresson (9). Solvng ths problem we obtan the followng result. Lemma 2. Under unplant prvate frms, when the government sells part of each plant of the state corporaton to a dfferent prvate nvestor, n equlbrum: β 4 q A 5+b,, q 5+b 2(7+6b +b 2 ) B CS (1+b)(4+b) 2 (7+6b+b 2 ) 2, PS 3+b 2(7+6b +b 2 ), π (5+b )(7+3b) A π B 8(7+6b+b 2 ) 2, 31+20b +3b 2 2(7+6b +b 2 ) 2, W 9+2b 2(7+b(6+b)), 1, 2. 3(3+b) 2 8(7+6b +b 2 ) 2, As n the case of a sngle nvestor t s obtaned that the government partally prvatzes the state corporaton: 0< β <1. Gven the symmetry of the model, the government retans the same stake n each plant of the state corporaton. The percentage of shares that remans publc, β, decreases wth b ( β b < 0). As the state corporaton s partally prvatzed by sellng the plants to dfferent prvate nvestors, 10

12 there are two sempublc unplant frms and there s no nternalzaton effect. When goods are substtutes, as b ncreases goods become closer substtutes and market competton ncreases. When goods are complements the output of the frms s lower than when goods are substtutes, but goods become less complementary as b ncreases, so the output of the frms ncreases. As a result, as b ncreases, and the output of the frms ncreases whether goods are substtutes or complements. Thus, due to the objectve functon effect, the stake n the sempublc frms retaned by the government becomes smaller the greater the value of b s. Fnally, q A > q and > B π A π B for all values of parameter b snce frm A s sempublc and takes consumer surplus nto account COMPARISON OF RESULTS We frst compare the degree of prvatzaton of the two plants of the state corporaton when they are sold to a sngle nvestor wth that whch results when they are sold to dfferent nvestors. From Lemmas 1 and 2 the followng s obtaned. Proposton 1. Under unplant prvate frms, n equlbrum: β SU > β f goods are substtutes (b>0), β SU < β f goods are complements (b<0), and β SU β f goods are ndependent n demand (b0). Regardless of whether goods are substtutes or complements, the nternalzaton effect s stronger f the plants of the state corporaton are sold off n part to a sngle prvate nvestor than f they are sold to dfferent prvate nvestors. Ths s because n the second case sempublc frms are unplant so there s no nternalzaton effect. Gven a stake-holdng n the state corporaton by the government, when goods are substtutes producton and market competton s lower f the two plants are sold n part to a sngle prvate nvestor. In ths case, due to the objectve functon effect, the government retans a greater percentage of the shares n the state corporaton (β SU > β ) to ncrease producton. However, when goods are complements producton and market competton are greater f the two plants are sold off n part to a sngle prvate nvestor due to the 11

13 nternalzaton effect. In that case, the objectve functon effect mples that the government retans a lower stake n the state corporaton (β SU < β ). Fnally, f goods are ndependent n demand (b0) there s no nternalzaton effect so a multplant frm produces the same output as unproduct frms, whch means that β SU β. By comparng the welfare levels shown n Lemmas 1 and 2 the followng result s obtaned. Proposton 2. Under unplant prvate frms, n equlbrum, W SU W. Ths means that the government s ndfferent between sellng part of the two plants of the state corporaton to a sngle nvestor and sellng part of each plant to a dfferent prvate nvestor. When the government sells part of the state corporaton the producer surplus ncreases at the expense of consumer surplus. Thus, the government partally prvatzes the state corporaton to balance these two effects. Proposton 1 shows that f goods are substtutes (complements), the government retans a greater (lower) stake n the state corporaton when the plants are partally sold to a sngle prvate nvestor than when they are sold to dfferent prvate nvestors. In ths way, the government can acheve the same welfare n both cases. As a result, the frms produce the same output SU n both cases ( q A SU q, A q B q B ), so the consumer and producer surpluses and SU welfare are also the same ( CS SU CS, PS W SU PS, W ) MULTIPLANT PRIVATE FIRMS Up to now we have consdered that the state corporaton competes n the product market wth unplant prvate frms. However, n modern economes multplant frms are omnpresent and state corporatons may compete wth multplant prvate frms rather than wth unplant prvate frms. Thus, we now consder that the state corporaton 8. Ths s not possble when the government has to fully prvatze the state corporaton (as n Bárcena- Ruz and Garzón, 2016), so welfare s not equal n both cases under full prvatzaton. 12

14 competes n the product market wth a multplant prvate frm that owns two plants producng dfferentated goods. We denote ths case by superscrpt M. We now compare the degree of prvatzaton of the two plants of the state corporaton when they are sold to a sngle nvestor and when they are sold to dfferent nvestors. From Lemmas A1 and A2 (see Appendx) the followng s obtaned. Proposton 3. Under a multplant prvate frm, n equlbrum: β SM > β f goods are substtutes (b>0), β SM < β f goods are complements (b<0), and β SM β DM f goods DM are ndependent n demand (b0). DM The explanaton of ths result s smlar to that gven n Proposton 1 so we omt t. The man dfference s that there s one multplant prvate frm rather than two unproduct prvate frms. Due to the nternalzaton effect, a multplant prvate frm produces less (more) wth substtute (complement) goods than unplant prvate frms. By comparng the welfare levels shown n Lemmas A1 and A2 the followng result s obtaned. Proposton 4. Under a multplant prvate frm, n equlbrum: W SM W DM. The explanaton of ths result s smlar to that gven n Proposton 2 so we omt t. 5. COMPARISON OF RESULTS By comparng Propostons 1 and 3 the followng result s obtaned. Proposton 5. In equlbrum: f b<0 we obtan that β > β >β SM >β SU, f b>0 we DM obtan that β SU >β SM > β > β DM, and f b0 we obtan that β SU β SM β β DM It can be shown that the man result obtaned n ths proposton holds that there f t s assumed that there are n unplant or multplant prvate frms competng n the product market. The state corporaton s never fully prvatzed even when n s hgh, and the degree of prvatzaton ncreases wth n. Ths s a well known result n the relevant lterature (see, for example, Fujwara, 2007). 13

15 Ths proposton shows that when goods are complements (b<0), the government keeps a larger stake n the state corporaton f ts plants are sold to dfferent prvate nvestors than f they are sold to a sngle prvate nvestor. Moreover, the government retans a larger stake f prvate frms are multplant than f they are unplant. If goods are substtutes (b>0) the contrary result s obtaned. Fnally, f goods are ndependent n demand (b0) the government keeps the same stake n the state corporaton n all cases. From ths t can be concluded that the stake that the government retans n the state corporaton depends on the type of goods produced by the frms and on whether prvate frm are unplant or multplant. To explan the result obtaned n Proposton 5, note that the nternalzaton effect and the objectve functon effect depend on two factors: The frst s whether the two plants of the state corporaton are sold n part to a sngle prvate nvestor or to dfferent nvestors. The second one s whether prvate frms are unplant or multplant. Both factors affect the nternalzaton effect snce only multplant frms (ether sempublc or prvate) take nto account how the output of one plant affects that of the other. However, only the frst factor affects the objectve functon effect snce only a sempublc frm takes socal welfare nto account. Thus, the frst factor s more mportant snce t s relevant to both effects. When goods are complements, the nternalzaton effect means that, the two plants produce more f they are sold to a sngle prvate nvestor than f they are sold to dfferent nvestors. Thus, as shown n Propostons 1 and 3, the objectve functon effect means that the government retans a lower stake n the state corporaton n the former case. Moreover, compared wth the case when there are two prvate unplant frms, the prvate sector produces more when there s a multplant frm because of the nternalzaton effect. Ths may cause sempublc plants or frms to reduce output and market competton. In ths case, to counterbalance the output reducton the government retans a larger stake n the state corporaton due to the objectve functon effect. Thus, ndependently of whether the state corporaton s sold to a sngle nvestor or to dfferent nvestors, the government retans a greater percentage of the shares n the state corporaton f prvate frms are multplant than f they are unplant. Therefore, f goods are complements t s obtaned that β > β >β SM >β SU. DM 14

16 When goods are substtutes the nternalzaton effect means that the two plants produce less f they are sold to a sngle prvate nvestor than f they are sold to dfferent nvestors. Thus, as shown n Propostons 1 and 3, due to the objectve functon effect the government retans a lower stake n the state corporaton n the latter case. Moreover, compared wth the case when there are two prvate unplant frms, the nternalzaton effect means that the prvate sector produces less when there s a multplant frm, whch may encourage partally prvatzed plants to ncrease ther output. In ths case, the government has less ncentve to ncrease output by retanng a hgher stake n the state corporaton due to the objectve functon effect. Thus, ndependently of who buys part of the state corporaton, the government retans a lower percentage of the shares f prvate frms are multplant than f they are unplant. Therefore, f goods are substtutes t s obtaned that β SU >β SM > β > β DM. Fnally, f goods are ndependent n demand there s no nternalzaton effect, so β SU β SM β DM β. 6. CONCLUSIONS The theoretcal lterature on mxed olgopoly has hardly analyzed the prvatzaton of state corporatons. One excepton s the paper by Bárcena-Ruz and Garzón (2016) who assume that f a plant of the state corporaton s prvatzed t s fully sold to prvate nvestors. However, there s evdence showng that state corporatons often partally prvatze ther frms. In ths paper we analyze a government s decson on whether to partally prvatze a state holdng corporaton wth two plants that may produce complement or substtute goods. The state holdng corporaton competes wth two prvate plants, whch may belong to dfferent frms or to a multplant frm. To prvatze the state holdng corporaton the government may sell off part of ts two plants to a sngle prvate nvestor or to dfferent prvate nvestors. The result obtaned n ths paper helps to understand the dfferent degrees to whch state holdng corporatons have been prvatzed by governments n practce dependng on whether the goods produced by frms are substtutes or complements. We fnd n the paper that the government partally prvatzes the two plants of the state corporaton but 15

17 s ndfferent between sellng them to a sngle prvate nvestor or to dfferent prvate nvestors. When goods are complements, the government keeps a larger stake n the state corporaton f ts plants are sold to dfferent prvate nvestors than f they are sold to a sngle prvate nvestor. However, socal welfare n equlbrum s the same n the two cases and thus the government s ndfferent between them. Moreover, the government retans a larger stake f prvate frms are multplant than f they are unplant. If goods are substtutes the contrary result s obtaned. The government keeps a lower stake n the state corporaton f ts plants are sold to dfferent prvate nvestors than f they are sold to a sngle prvate nvestor. Socal welfare n equlbrum s the same n the two cases so the government s ndfferent between them. Moreover, the government retans a lower stake f prvate frms are multplant than f they are unplant. Fnally, f goods are ndependent n demand the government keeps the same stake n the state corporaton n all cases consdered n ths paper. From ths t can be concluded that the stake that the government retans n the state corporaton depends on the type of goods produced by the state corporaton and the prvate frms, and on whether prvate frm are unplant or multplant. 16

18 APPENDIX: MULTIPLANT PRIVATE FIRM ) Sngle prvate nvestor. In ths case, the two plants of the state corporaton are sold n part to a sngle prvate nvestor. In the second stage of the game sempublc frm A chooses the output levels q 1A and q 2A that maxmze ts objectve functon, gven by expresson (4). Prvate frm B sets the output levels q 1B and q 2B that maxmze ts proft, gven by expresson (2). Solvng these problems, we obtan the followng frst order condtons: (1 3q A q B b(2q Aj + q Bj ))(1 β) + (1 2q A q B b(q Aj + q Bj ))β 0, 1 q A 3q B b q Aj + 2q Bj 0, j;, j1, 2. From the above frst order condtons we obtan the followng output of the frms and socal welfare as a functon of β: q A 2+b q B, 8+5b (2 β) 3β+b 2 (3 2β ) 2+b β(1+b), 8+5b (2 β) 3β +b 2 (3 2β ) W (2(2+b)2 (5+4b ) 2(1+b)(2+b)(7+5b )β+(1+b) 2 (4+3b)β 2 ) (8+5b(2 β) 3β +b 2 (3 2β )) 2, 1, 2. In the frst stage of the game the government chooses the optmal value of β that maxmzes socal welfare. Solvng ths problem we obtan the followng result. Lemma A1. Under multplant prvate frms, when the government partally sells the state corporaton to a sngle prvate nvestor, n equlbrum: SM π A SM β (2+b) b+b 2 5+5b+b 2, SM q A 14+22b +10b 2 +b 3, SM q B (5+5b+b 2 )(7+9b+3b 2 ) (3+2b ) 3 3+2b 14+22b +10b 2 +b 3, (1+b)(8+7b +b 2 ) 2, SM π 2(14+22b +10b 2 +b 3 ) 2 B 2(14+22b +10b 2 +b 3 ) 2, SM CS (14+22b +10b 2 +b 3 ) 2, SM PS b +103b 2 +32b 3 +3b 4 SM (9+8b +b (14+22b +10b 2 +b 3 ) 2, W 2 ) 14+22b +10 b 2 +b 3, 1, 2. ) Dfferent prvate nvestors. In ths case, each plant of the state corporaton s sold n part to a dfferent prvate nvestor. In the second stage of the game, prvate frm B sets the output level q B that maxmzes ts proft, gven by expresson (1), 1, 2. Each sempublc frm chooses the output level q A that maxmzes ts objectve functon gven by expresson (5). Solvng these problems, we obtan the followng frst order condtons: (1 3q A q B b(q Aj + q Bj ))(1 β ) + (1 2q A q B b(q Aj q Bj ))β 0, 17

19 1 q A 3q B b q Aj + 2q Bj 0, j;, j1, 2. From the above frst order condtons we obtan the followng output of the frms and socal welfare as a functon of β 1 and β 2 : q A (2+b)(8+b 2 3β j b(7 2β j )), b 4 +b 2 (β (11 4β j ) 11(3 β j ))+(8 3β )(8 3β j ) q B (b 2 (2 β j )+(8 3β j )(2 β )+b(β j (5 2β )+6β 14)) b 4 +b 2 (β (11 4β j ) 11(3 β j ))+(8 3β )(8 3β j ), j;, j1, 2, W (b 7 2b 6 + b 5 (β 1 (22 9β 2 ) + 22β 2 59) + b 4 (92 + 2β 1 2 (1 β 2 ) 41β 2 + 2β β 1 (26β 2 2β )) + b 3 ( β β β 1 (454β 2 71β ) + β 2 1 (101 71β β 2 2 )) + 2 β 1 672β 2 111β β β β β β b 2 (1078 β 2 173β β 2 1 (125β 2 20β ) + β 1 ( β β 2 2 ))+ b(β 2 1 (127β 2 21β ) 4( β β 2 2 ) + β 1 ( β β 2 2 ))))/ (b 4 + b 2 (β 1 (11 4β 2 ) 11(3 β 2 )) + (8 3β 1 )(8 3β 2 )) 2. In the frst stage of the game the government chooses the optmal value of β 1 and β 2 that maxmzes socal welfare. Solvng ths problem we obtan the followng result. Lemma A2. Under a multplant prvate frm, when the government sells each plant of the state corporaton to a dfferent prvate nvestor, n equlbrum: DM π A DM β 4+3b 5+5b +b 2, DM q A 14+22b +10b 2 +b 3, DM q B (5+5b +b 2 )(7+9b +3b 2 ), DM π 2(14+22b +10b 2 +b 3 ) 2 B DM PS 5+5b +b 2 3+2b 14+22b +10b 2 +b 3, (3+2b) 3 2(14+22b +10 b 2 +b 3 ) 2, DM CS (1+b)(8+7b +b 2 ) b +103 b 2 +32b 3 +3b 4 DM 9+b (8+b) (14+22b +10b 2 +b 3 ) 2, W 14+b(22+b(10+b)) (14+22b +10b 2 +b 3 ) 2,, 1, 2. 18

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