MIXED OLIGOPOLIES AND THE PROVISION OF DURABLE GOODS. Baranovskyi Volodymyr. MA in Economic Analysis. Kyiv School of Economics

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1 MIXED OLIGOPOLIES AND THE PROVISION OF DURABLE GOODS by Brnovskyi Volodymyr A thesis submitted in prtil fulfillment of the requirements for the degree of MA in Economic Anlysis Kyiv School of Economics 0 Thesis Supervisor: Professor Prokopovych Pvlo Approved by Hed of the KSE Defense Committee, Professor Wolfrm Schrettl Dte

2 Kyiv School of Economics Abstrct MIXED OLIGOPOLIES AND THE PROVISION OF DURABLE GOODS by Brnovskyi Volodymyr Thesis Supervisor: Professor Prokopovych Pvlo This pper studies models in which socilly concerned firm is present on the durble goods mrket nd competes with privte or nother socilly concerned firm. We compre results of these models with investigted erlier pure duopoly cse, when only privte firms re present on the mrket. The previous results for the pure duopoly cse show the presence of Prisoners Dilemm. In such sitution every firm prefers selling to renting, but (renting, renting outcome gives more profit for ech firm thn (selling, selling one. We consider models: in the first one socilly concerned firm competes with privte firm nd in the second one socilly concerned firms re competing. We received tht in both cses Prisoners Dilemm still determines behvior of mrket prticipnts. Therefore the ttitude of the firm towrds socil welfre is not crucil in mking renting/selling decisions.

3 TABLE OF CONTENTS Chpter. INTRODUCTION... Chpter. LITERATURE REVIEW...3 Chpter 3. METHODOLOGY...7 Chpter 4. CONPETITION BETWEEN A PRIVATE FIRM AND A SOCIALLY CONCERNED ONE...9 Cse. Renting/Renting...9 Cse. Renting/Selling... Cse 3. Selling/Renting...3 Cse 4. Selling/Selling...7 The extended gme...0 Chpter 5. CONPETITION BETWEEN TWO SOCIALLY CONCERNED FIRMS...5 Cse. Renting/Renting...5 Cse. Renting/Selling...7 Cse 3. Selling/Selling...30 The extended gme...33 Chpter 6. CONCLUSIONS...36

4 LIST OF FIGURES Number Grph. Difference between selling nd renting decisions for privte firm if SC firm chooses renting...0 Grph. Difference between selling nd renting decisions for privte firm if SC firm chooses selling... Grph 3. Difference between selling nd renting decisions for SC firm if privte firm chooses renting... Grph 4. Difference between selling nd renting decisions for SC firm if privte firm chooses selling... Grph 5. Dependence of CS on prmeter θ in different cses...3 Grph 6. Difference between selling nd renting outcomes for firm if firm chooses renting...33 Grph 7. Difference between selling nd renting outcomes for firm if firm chooses selling...34 Grph 8. Dependence of CS on prmeter θ in different cses...35 Pge ii

5 ACKNOWLEDGMENTS The uthor wishes to express his grtitude to his dvisor, Prof. Pvlo Prokopovych for his supervision, useful comments, interesting ides, vluble dvices, which helped lot. The uthor is grteful to Pr. Denys Nizlov for his help while reviewing this pper. Also the uthor is grteful to Pr. Hnn Vkhitov nd Pr. Olen Nizlov for their useful comments during RW presenttions. iii

6 Chpter INTRODUCTION Cose (97 ws the first one to notice tht monopoly selling durble good hs different behviour from nother one selling non-durble good. In his seminl pper he showed tht due to its profit-mximizing behviour monopoly firm prefers renting the durble good to selling it. The simple explntion of it corresponds to the ssumption of consumers rtionl expecttions. Consumers predict tht monopoly selling durble goods would fce smller potentil mrket size in the next periods nd therefore will be forced to decrese prices. These expecttions reduce the price tht consumers re redy to py in the first period when the monopoly offers the product for sle. Due to the ltter demnd for durble good flls. As consequence, the monopoly would be forced to set up price for the durble good close to mrginl cost (the Cose conjecture. Lter, Bulow (98 showed the firms might prefer to decrese the durbility of their product. Poddr (004 rised the following question: How does the sitution chnge when durble goods mrket moves from monopoly to oligopoly? He considered pure duopoly mrket structure (two privte firms re competing in durble goods mrket. In his pper he shows tht mentioned bove competition leds to some kind of Prisoners Dilemm: selling is preferred ction for both firms, but in this equilibrium firms mke lower profits compring to the sitution when both firms re renting. This finding cn prtilly explin co-existence of selling nd renting strtegies on the mrket.

7 But not only pure oligopolies cse is present on the mrket. In mny countries privte firms hve to compete with public non-profit firms. Oil industry nd mnufcturing re the typicl exmples of so-clled mixed oligopolies. In recent yers non-profit orgniztions (NPO hve become more ctive nd moved into trditionlly commercil mrkets. Cse (005 stressed tht these NPOs hve got billions in revenues nd ctively continue to grow. Currently, NPOs re not very ctively present on durble goods mrket. But Benz (005 climed tht NPOs should be very successful in the hi-tech industries, in which durble output is produced. In my thesis I m trying to nswer the following question: Wht will be the consequences of more ctive prticiption of socilly concerned firms on durble good mrkets? The impct of their presence nd behviour on the renting/selling decisions of privte firms is lso investigted. The reminder of the pper is orgnized in the following wy. The literture review is presented in Chpter. Chpter 3 provides methodology nd model setup. The mixed duopoly cse, when privte firm competes with the socilly concerned one is exmined in chpter 4. Competition of socilly concerned firms is investigted in chpter 5. Finlly, the pper is summrized, nd concluding remrks re mde in Chpter 6.

8 Chpter LITERATURE REVIEW In his seminl pper Cose (97 showed tht for durble goods monopolist renting is more profitble thn selling. The min reson behind it is tht in the cse of selling durble good is used by customer in the following periods, therefore decresing demnd tht monopolist fces. As consequence, monopolist is forced to decrese price nd profit from the next periods. Moreover, from the hypothesis of rtionl expecttions (RE hypothesis consumers re willing to py less even in the first period. As result monopolist loses substntil prt of its profits in the selling cse. Bulow (98, 986 investigted the cse when monopoly firm cnnot convince customers tht it will not chnge prices when demnd flls. In this cse the selling monopolist chooses smller durbility for goods thn those produced by renting monopolists. This sitution, in which goods re produced with indequtely short lives to force buyers to repet purchses, is clled Plnned Obsolescence. Bucovetsky nd Hilton (986 investigted the cse when durble goods monopolist is fced with some thret of entry. They showed tht in contrst to pure cse when monopolist prefers renting under new circumstnces it will choose some kind of mixed strtegy: both selling nd renting. Kuhn nd Pdill (996 considered the sitution when durble good monopolist is lso present on non-durble goods mrket, demnd for which is strongly correlted with demnd for durble goods. They mde conclusion tht presence of 3

9 competitor on non-durble goods mrket increses willingness to sell for monopolist on the durble goods mrket. The other reson why monopolists prefer selling is the discrimintion in qulity. Kumr (00 investigted the cse when monopoly cn discriminte consumers nd provide high-demnd nd low-demnd buyers with different pckges of qulity of product nd prices. In cse of resle, trding monopolist will try to increse qulity of production over time. This behviour stimultes highly sophisticted consumers to buy products with the highest qulity vilble nd resle it to less sophisticted consumers s soon s new qulity of production is vilble. Moreover, it does not even imply tht the price of the product goes up over time. Kumr (00 shows tht prices cn be even decresed in the future. In rel world, pure monopoly cse is not the most frequent one. As consequence other mrket structures were nlysed. Sggi nd Vetts (000 investigted pure duopoly cse, when firms re doing both ctivities (selling nd renting in ech period. Their min result ws tht renting to selling rtio is determined by vlues of production costs. For prticulr firm n increse in production costs implies increse in renting/selling rtio. Poddr (004 considered pure duopoly structure for -period model. He showed tht for every firm selling is preferred to renting lthough (renting, renting gives higher profits thn (selling, selling for them. This finding cn help prtly explin the fct tht selling is dominting on the durble goods mrket in the rel world. Another possible sitution on the mrket is the cse when privte firm is competing with non-profit socilly concerned firm. The cpcity choice in 4

10 such mixed oligopolies ws investigted by Lu nd Poddr (005. They showed tht on the non-durble goods mrket the privte firm chooses overproduction while public firm chooses underproduction. Lu nd Poddr (006 investigted the cse of uncertinty in demnd. They showed tht public firm might lso prefer to choose excess cpcity in such conditions. Ogw (006 considered influence of different product differentition. He showed tht in cse when products produced by firms re complements both firms choose to oversupply. Goering (006 nlysed simple liner demnd -period durble goods for the cse where only non-profit orgniztion is present on the mrket. He showed tht socilly optiml (here it is equivlent to cost minimizing durbility would be provided by NPO in cse of renting. Also he proved tht NPO seller would not provide this durbility if it hd no commitment power with buyers. Moreover, in such cse NPO seller is producing less durble good thn profit-mximizing firm. Also he climed tht mixed oligopoly model fits better for exploring the impct of NPO. He ssumed tht full comprison of socil impcts of NPOs would likely to led to interesting lthough complicted results. To sum up, only cse of pure duopoly, when privte firms re competing, ws investigted for the durble goods mrket. In this thesis we re investigting other cses. The first one rises when privte firm competes with socilly concerned firm within the -period durble goods frmework. The second cse we consider is competition of eqully socilly concerned firms. We re trying to investigte whether Prisoners dilemm, which is described by Poddr (004 for the pure duopoly cse, still is present for highly mentioned mrket situtions. We prove tht it still exists, nd we cn conclude tht sitution on durble goods duopoly is not strongly ffected by 5

11 firms ttitude to socil welfre. Therefore, in such cse the following conclusion cn be mde. If monopoly is present on the mrket monopolist chooses renting strtegy, which is not socilly optiml. But if new firm enters the mrket its ttitude to socil welfre does not ply importnt role: (selling, selling is the only Nsh equilibrium, lthough (renting, renting gives higher profits for both firms. Therefore the fctor of competition is more importnt here thn ttitude of firms to socil welfre. 6

12 Chpter 3 METHODOLOGY Consider mrket in which consumers hve different reservtion prices for some durble good. Suppose tht consumers live for two periods. The good does not deprecite from the first to the second period. Also we ssume tht secondhnd mrket does not exist, so consumers tht buy product in the first period cnnot resell it in the second period. Producers fce liner demnd curve. At period the ggregte demnd of consumers is given by p( Q Q, where p is the price of the good t period, nd Q is the ggregte supply of this good. We consider two types of ctions: renting nd selling. In our model, we should distinguish between renting nd selling in the first period only. If consumer buys good in the first period he will lso use it during the second period. Since the lst one is the end of consumers lifetime both ctions (renting nd selling re equivlent. Now consider firms behviour. Suppose two firms re producing the sme durble good. We consider cse when privte firm competes with socilly concerned firm in Chpter 4, while cse when both firms re socilly concerned is nlysed in Chpter 5. Assume both firms hve zero production costs. Although the cse of nonzero costs cn lso be considered, s our purpose is to determine the influence of mrket structure on production nd renting vs. selling decisions this kind of specifiction is pproprite. 7

13 Consider model of Cournot competition where firms re competing in quntities nd due to the ltter ones the mrket price is determined. Further, we consider tht gme between firms is going in the following wy (in the wy of describing gme I follow Poddr (004, who considered the similr cse, but for privte firms only. Before mking decisions of how much good to produce in the first period firms should decide simultneously which ction renting or selling to choose. This might be considered s pre-ply stge. After decision ws mde firms choose quntities in periods nd with the purpose of mximizing its own profit functions, which will be introduced further. As we noted erlier, the choice of ctions mtters only for the first period since the second period is the end of lifetime. Now describe the specific profit functions, which both firms re trying to mximize. The gol of privte firm is to mximize its own totl profits gined in periods nd. On the contrry, the gol of SC firm is to mximize the sum of its own totl profits nd consumers surplus, discounted by fctor θ (0 θ < (but it does not cre bout other privte firm. Fctor θ shows interest of SC firm in the welfre of society: θ 0 corresponds to cse of pure privte firm, if θ is close to we cn interpret it s SC firm cres bout society lmost in the sme wy s bout its own profits. 8

14 Chpter 4 COMPETITION BETWEEN A PRIVATE FIRM AND A SOCIALLY CONCERNED ONE In this chpter we investigte competition between privte firm nd socilly concerned firm, the gol of the ltter one is to mximize the sum of its own totl profits nd totl consumers surplus, discounted by fctorθ. We strt solving the gme by the method of bckwrd induction. In this gme, in the pre-ply period 4 scenrios my occur:. Both firms decide to rent. Privte firm rents nd NPO sells 3. Privte firm sells nd NPO rents 4. Both firms decide to sell. We investigte ech of these 4 cses (subgmes seprtely while solving this gme. Cse. Renting/Renting In cse of renting/renting consumers re obligted to return goods fter the first period. Therefore, in the second period firms fce the sme demnd curve s in the first period. As result we cn investigte -period cse nd it will be sufficient to receive the nswer for it. Let x - mount of good tht st (privte firm produces, y - mount of good tht nd (SC firm. 9

15 Then price for the good is determined from the liner demnd curve p( Q Q, where Q is the ggregte supply. Therefore p x y. Profit functions for the firms re determined s the following: Privte x( x y NPO y ( x y + θ* CS, where CS is consumer surplus. For the liner demnd curve: ( (. x y CS v p dv x + y Therefore NPO y( x y + θ*0.5( x + y. Write down first-order conditions to determine Nsh equilibrium of this subgme: y x 0 x x y+ θ ( x+ y 0 y Solving this system we receive: θ x 3 θ 3 + θ y 3 θ 3 x+ y 3 θ 3 θ p x y 3 θ 3 Therefore we come to the following conclusion. 0

16 Proposition. Compring with the pure duopoly cse in -period gme, in mixed duopoly cse privte firm underproduces ( x, wheres NPO 3 overproduces ( y. Totl production increses ( x+ y, while the 3 3 price nturlly becomes lower ( p. 3 Substituting obtined x nd y into consumers surplus profit functions we obtin: R CS ( x + y (3 θ R θ px 3 θ R R + θ* + θ θ py CS (3 θ Tking into ccount tht renting/renting cse is equivlent to sequentil - period gmes we receive: θ 3 θ 9 RR R to the cse when it competes with nother privte firm RR RR ( + θ θ CS RR (3 θ R 4 CS (3 θ (privte firm receives less profit compring

17 Cse. Renting/Selling Let x - mount of good tht st (privte firm rents in the first period, y - mount of good tht nd (SC firm sells in the first period. x - mount of good tht st (privte firm sells in the second period, y - mount of good tht nd (SC firm produces in the second period. Let s use bckwrd induction to nlyze this cse. In the second period both firms fce the following demnd curve p( Q y Q (s y units were lredy sold in the first period. Therefore, s ws shown in renting/renting cse firms choose the following quntities in the second period: θ + θ x ( y y ( y 3 θ 3 θ Price, which is set up by firms in the second period: θ p ( y 3 θ Then profits received by firms in the second period re given by the following formuls: RS CS( ( y (3 θ ( p x θ ( y 3 θ ( ( + θ θ ( y (3 θ Now move bck to the first period. For this very period both renting nd S selling re present, there re prices on the mrket: price for selling p nd R price for renting p. The necessry condition for co-existence of both ctivities: S R p p + p

18 (otherwise ll consumers prefer one ctivity to nother. The second one is the condition for mrginl consumer, which is indifferent between buying in the first period t higher price, then using the good during second period nd buying it in the second period t lower price: [ x y ] p x y p S From these conditions we obtin tht: p x y R 4 θ p ( y x 3 θ S Substituting received prices into profit functions we get: R ( θ p x+ x( x y + ( y p y + θ* CS + S ( 3 θ Consumer surplus when both renting nd selling re present y RS S R R ( y y x y x y CS ( v p dv+ ( v p dv ( v p dv+ ( v p dv ( x + y + ( y y + y 3 θ Then S ( 4 θ p y+ θ* CS+ y[ ( y x] + θ*( ( x+ y + 3 θ (+ θ θ + ( y y+ y + ( y 3 θ (3 θ Write down first-order conditions to determine Nsh equilibrium of this subgme: 3

19 y x 0 x 4 θ ( y x+ θ ( x y ( y y y 3 θ θ ( + θ θ ( y 0 (3 θ Solving this system we receive: 3 θ + θ θ x θ + θ 3θ 3 θ 3θ + θ y θ + θ 3θ Plugging these vlues we receive the following results: x 4( θ( θ 35 45θ + θ 3θ 3 p p 4( θ( θ 35 45θ + θ 3θ 3 ( θ (5 3 θ( θ 35 45θ + θ 3θ S 3 y 4( θ ( θ( + θ 35 45θ + θ 3θ 3 p θ + θ θ 35 45θ + θ 3θ 3 R 3 Substituting received prices into profit functions we get: RS RS RS CS CS( + CS( ( x + y + y ( y + y + ( y 3 θ (3 θ θ + 99θ 38θ + 5θ 4θ + θ 3 (35 45θ + θ 3 θ 4(3 4θ + 5 θ ( θ ( θ (35 45θ + θ 3 θ RS θ 770θ + 067θ 640θ + 93θ 6θ + θ (35 45θ + θ 3 θ RS 3 4

20 Cse 3. Selling/Renting Let x - mount of good tht st (privte firm sells in the first period, y - mount of good tht nd (SC firm rents in the first period. x - mount of good tht st (privte firm sells in the second period, y - mount of good tht nd (SC firm sells in the second period. Let s use bckwrd induction to nlyze this cse. In the second period both firms fce the following demnd curve p( Q x Q (s x units were lredy sold in the first period. Therefore, s ws shown in renting/renting cse firms choose the following quntities in the second period: x θ ( 3 θ x y + θ ( 3 θ x Price, which is set up in the second period: p θ ( 3 θ x Then profits received by firms in the second period re given by the following formuls: RS CS( ( x (3 θ ( p x θ ( x ( ( + θ θ ( x 3 θ (3 θ Now move bck to the first period. As during this period both renting nd S selling re present, there re prices on the mrket: price for selling p nd R price for renting p. 5

21 By full nlogy with renting/selling cse we obtin tht: p x y R 4 θ p ( x y 3 θ S Substituting received prices into profit functions we get: S ( 4 θ θ p x+ x[ ( x y] + ( x 3 θ 3 θ R RS ( p y + θ* CS( + y[ y x] + θ( ( x+ y + ( x x+ 3 θ (+ θ θ + x + ( x (3 θ Write down first-order conditions to determine Nsh equilibrium of this subgme: 4 θ ( θ ( x y ( x 0 x 3 θ (3 θ x y+ θ ( x+ y 0 y Solving this system we receive: ( θ( 5 θ x θ + 3θ θ 4θ y θ + 3θ θ Plugging these vlues we receive the following results: x ( θ(8 5 θ + θ 3 p p 35 39θ + 3θ θ ( θ(8 5 θ + θ θ + 3θ θ S ( θ(0 8 θ + θ θ + 3θ θ y ( + θ(8 5 θ + θ θ + 3θ θ p R ( θ( θ + θ θ + 3θ θ 6

22 Substituting received prices into profit functions we get: SR SR SR CS CS( + CS( ( x + y + x ( x + x + ( x 3 θ (3 θ θ + 678θ 98θ + 5θ 3 (35 39θ + 3 θ θ (4 07θ + 8 θ θ ( θ( θ (35 39θ + 3 θ θ 3 SR θ 997θ + 88θ 86θ + 39θ θ (35 39θ + 3 θ θ SR 3 Cse 4. Selling/Selling Let x - mount of good tht st (privte firm sells in the first period, y - mount of good tht nd (SC firm sells in the first period. x - mount of good tht st (privte firm sells in the second period, y - mount of good tht nd (SC firm sells in the second period. Let s use bckwrd induction to nlyze this cse. In the second period both firms fce the following demnd curve p( Q x y Q (s x y + units were lredy sold in the first period. Therefore, s ws shown in renting/renting cse firms choose the following quntities in the second period: x θ ( 3 θ x y y + θ ( 3 θ x y Price, which is set up by firms in the second period: 7

23 p θ ( 3 θ x y Then profits received by firms in the second period re given by the following formuls: SS CS( ( x y (3 θ ( p x θ ( x y 3 θ ( + θ θ ( (3 θ ( x y Now move bck to the first period. Mrginl consumer should be indifferent between buying good t higher price in the first period, then using it for both periods nd buying good t lower price in the second period: ( x y p ( x y p Therefore p 4 θ ( 3 θ x y Substituting prices into profit functions, we receive: 4 θ θ px+ px ( x y x+ ( x y 3 θ 3 θ 4 θ + θ θ 3 θ (3 θ SS py+ py + θ* CS( ( x y y+ ( x y + θ + θ *( x+ y[ + ( x+ y] 3 θ 3 θ Write down first-order conditions to determine Nsh equilibrium of this subgme: 8

24 4 θ ( θ ( x y ( x y 0 x 3 θ (3 θ 4 θ θ ( y x + θ + ( x + y ( + θ θ ( x y 0 (3 θ y 3 θ 3 θ 3 θ Solving this system we receive: ( θ(5 3 θ x 3 6 8θ + 7θ θ 5+ θ θ y 3 6 8θ + 7θ θ Plugging these vlues we receive the following results: x ( θ ( θ ( + θ y 8 5θ + θ 8 5θ + θ p ( θ ( θ ( θ p 8 5θ + θ 8 5θ + θ Substituting received prices into profit functions we get: θ CS CS CS x y x y x y 3 θ 3 θ (3 θ SS SS SS ( + ( ( + [ + ( + ] + ( 37 6θ + 5θ (8 5 θ + θ SS ( θ ( 8 θ + θ (8 5 θ + θ 3 4 SS + 5θ 8θ + 9θ θ (8 5 θ + θ 9

25 The extended gme. Now we clculte the best response (BR functions for both plyers. For privte firm SR RR If SC firm rents we should check the difference between ( θ nd ( θ. Using Wolfrm we come to conclusion tht for every θ [0, : ( θ ( θ > 0 (see Grph, therefore for every θ [0, : BR( R S SR RR Grph. Difference between selling nd renting decisions for privte firm if SC firm chooses renting. SS RS If SC firm sells we should check the difference between ( θ nd ( θ. Using Wolfrm we come to conclusion tht for every θ [0, : ( θ ( θ > 0(see Grph, therefore for every θ [0, : BR ( S S SS RS 0

26 Grph. Difference between selling nd renting decisions for privte firm if SC firm chooses selling. Proposition. In the mixed duopoly durble goods mrket the best rection of privte firm is to sell. This decision does not depend on the ction of SC firm nd prmeter θ. For SC firm RR If privte firm rents we should check the difference between ( θ nd RS ( θ. Using Wolfrm we come to conclusion tht (see Grph 3 * RS RR if 0 θ θ 0.345: ( θ ( θ > 0 nd BR ( R S * RS RR if > θ θ : ( θ ( θ < 0 nd BR ( R R

27 Grph 3. Difference between selling nd renting for SC firm if privte firm chooses renting. SS If privte firm sells we should check the difference between ( θ SR nd ( θ. Using Wolfrm we come to conclusion tht for every θ [0, : ( θ ( θ > 0(see Grph 4, therefore for every θ [0, : BR ( S S SS SR Grph 4. Difference between selling nd renting decisions for SC firm if privte firm chooses selling.

28 Proposition 3. In mixed duopoly durble goods mrket the best rection of the SC firm is to sell if privte firm sells. But if privte firm chooses renting, the decision of SC firm does depend on the prmeter θ : if θ it chooses selling, but if SC firm hs strong benefits from socil welfre ( θ > it chooses renting. Joining together BR functions of privte firm nd SC firm we come to the following conclusion. Proposition 4. In mixed duopoly durble goods mrket (S,S is the only Nsh equilibrium for ny prmeter θ. At the sme time for every θ (R,R gives better outcomes for both plyers thn (S,S. Moreover, if θ > (R,R is the best outcome for SC firm nd therefore it hs no incentive to devite from coopertive outcome (R,R. Now we cn compre which decision is optiml for the society. Using Wolfrm (see Grph 5 we cn conclude tht for every θ [0, : CS RR ( θ < CS SR ( θ < CS RS ( θ < CS SS ( θ Grph 5. Dependence of CS on prmeter θ in different cses 3

29 Proposition 5. In mixed duopoly durble goods mrket the only Nsh equilibrium (S, S is socilly optiml. If firms mke other decisions, consumer surplus decreses. 4

30 Chpter 5 COMPETITION BETWEEN TWO SOCIALLY CONCERNED FIRMS Now we consider tht insted of two firms tht hve different concerns bout socil welfre (0 ndθ we hve two firms tht cre bout society in the sme θ θ wy ( nd. We ll try to investigte this sitution, when firms re the sme in their ttitude to socil welfre. We strt solving the gme by the method of bckwrd induction. In this gme, in the pre-ply period 3 scenrios re considered:. Both firms decide to rent. Firm rents nd firm sells (As firms hve the sme preferences, cse in which firm sells nd firm rents gives the sme results 3. Both firms decide to sell. We investigte ech of these 3 cses (subgmes seprtely while solving this gme. Cse. Renting/Renting In the cse of renting/renting consumers will be obligted to return goods fter the first period. Therefore, in the second period firms will fce with the sme demnd curve s in the first period. As result we cn investigte -period cse nd it will be sufficient to receive the nswer. Let x - mount of good tht st firm produces, y - mount of good tht nd firm produces. 5

31 Then price for the good will be determined from the liner demnd curve p( Q Q, where Q is the ggregte supply. Therefore p x y. Profit functions for the firms will be determined s the following: θ x( x y + * CS θ y ( x y + * CS, where CS is consumer surplus. As we lredy know for the liner demnd curve ( CS x + y. Therefore θ x( x y + * ( x+ y θ y ( x y + * ( x+ y. Write down first-order conditions to determine Nsh equilibrium of this subgme: θ y x+ ( x+ y 0 x θ x y+ ( x+ y 0 y Solving this system we receive: x 3 θ 3 y 3 θ 3 x+ y 3 θ 3 p x y θ 3 θ 3 6

32 Substituting obtined x nd y into profit functions we obtin: θ px+ * ( x+ y (3 θ R θ py+ * ( x+ y (3 θ R Tking into ccount tht renting/renting cse is equivlent to sequentil -period gmes we receive: (3 θ RR R (3 θ RR R CS CS ( x y RR R + 4 (3 θ Cse.Renting/Selling Let x - mount of good tht st (privte firm rents in the first period, y - mount of good tht nd firm (NPO sells in the first period. x - mount of good tht st (privte firm sells in the second period, y - mount of good tht nd firm (NPO produces in the second period. Let s use bckwrd induction to nlyze this cse. In the second period both firms fce the following demnd curve p( Q y Q (s y units were lredy sold in the first period. Therefore, s ws shown in renting/renting cse firms choose the following quntities in the second period: 7

33 x ( y 3 θ y ( y 3 θ Price, which is set up by firms in the second period: θ p ( y 3 θ Then profits received by firms in the second period re given by the following formuls: RS CS( ( y (3 θ ( y (3 θ ( ( ( y (3 θ Now move bck to the first period. For this very period both renting nd S selling re present, there re prices on the mrket: price for selling p nd R price for renting p. S R The necessry condition for co-existence of both ctivities: p p + p (otherwise ll consumers prefer one ctivity to nother. The second one is the condition for mrginl consumer, which is indifferent between buying in the first period t higher price, then using the good during second period nd buying it in the second period t lower price: [ x y ] p x y p S From these conditions we obtin tht: p x y R 4 θ p ( y x 3 θ S 8

34 Substituting received prices into profit functions we get: R θ ( θ θ p x+ * CS+ x( x y + * CS+ ( y 3 θ S θ ( 4 θ θ θ p y+ * CS+ ( y x+ * CS+ ( y 3 θ 3 θ Consumer surplus when both renting nd selling re present y RS S R R ( y y x y x y CS ( v p dv+ ( v p dv ( v p dv+ ( v p dv ( x + y + ( y y + y 3 θ Write down first-order conditions to determine Nsh equilibrium of this subgme: θ y x+ ( x+ y 0 x 4 θ θ ( y x + ( x + y + ( y + y ( y 0 (3 θ y 3 θ 3 θ Solving this system we receive: 4 0θ + θ x θ + θ θ 6θ y θ + θ θ Plugging these vlues we receive the following results: 9

35 x p p 4( θ( θ 35 45θ + θ 3θ θ + 0θ θ 70 49θ + θ θ 3 8 0θ + θ 4 7θ + θ S y p 4( θ ( θ( + θ 35 45θ + θ 3θ θ + 0θ θ 70 49θ + θ θ 3 R 3 Substituting received prices into profit functions we get: 83 38θ + 3θ 3θ + 9θ θ (70 49θ + θ θ RS θ + 73θ + 3θ 9θ + θ (70 49θ + θ θ RS 3 RS RS RS CS CS( + CS( ( x + y + ( y y + y + ( x + y 3 θ 3 4 4(69 508θ + 54θ 0 θ + θ 3 (70 49θ + θ θ Cse 3. Selling/Selling Let x - mount of good tht st firm sells in the first period, y - mount of good tht nd firm sells in the first period. x - mount of good tht st firm sells in the second period, y - mount of good tht nd firm sells in the second period. Let s use bckwrd induction to nlyze this cse. In the second period both firms fce the following demnd curve p( Q x y Q (s x y + units were lredy sold in the first period. Therefore, s ws shown in renting/renting cse firms choose the following quntities in the second period: 30

36 x ( x y y ( x y 3 θ 3 θ Price, which is set up by firms in the second period: θ p ( x y 3 θ Then profits received by firms in the second period re given by the following formuls: SS CS( ( x y (3 θ ( x y (3 θ ( ( x y (3 θ ( Now move bck to the first period. Mrginl consumer should be indifferent between buying good t higher price in the first period, then using it for both periods nd buying good t lower price in the second period: ( x y p ( x y p Therefore p 4 θ ( 3 θ x y Substituting prices into profit functions, we receive: θ SS ( 4 θ θ px + * CS( + ( x y x+ *( x+ y[ + 3 θ 3 θ θ + ( x+ y] + ( x y 3 θ (3 θ θ SS ( 4 θ θ py + * CS( + ( x y y+ *( x+ y[ + 3 θ 3 θ θ + ( x+ y] + ( x y 3 θ (3 θ 3

37 Write down first-order conditions to determine Nsh equilibrium of this subgme: 4 θ θ ( y x + θ + ( x + y ( x y 0 (3 θ 4 θ θ θ x 3 θ 3 θ 3 θ ( y x + + ( x+ y y 3 θ 3 θ 3 θ ( x y 0 (3 θ Solving this system we receive: 5 θ x 6 0θ + θ 5 θ y 6 0θ + θ Plugging these vlues we receive the following results: x θ θ y 8 5θ + θ 8 5θ + θ p ( θ ( θ ( θ p 8 5θ + θ 8 5θ + θ Substituting received prices into profit functions we get: θ CS CS CS x y x y x y 3 θ 3 θ (3 θ SS SS SS ( + ( ( + [ + ( + ] + ( 37 6θ + 5θ (8 5 θ + θ 3 3 θ 4 θ + SS SS θ (8 5 θ + θ 3

38 The extended gme. Now it is sufficient to clculte the best response (BR function for firm (s both firms hve the sme preferences. SR RR If firm rents we should check the difference between ( θ nd ( θ. Using Wolfrm we come to conclusion tht for every θ [0, : ( θ ( θ > 0 (see Grph 6, therefore for every θ [0, : BR( R S SR RR Grph 6. Difference between selling nd renting outcomes for firm, if firm chooses renting. SS RS If SC firm sells we should check the difference between ( θ nd ( θ. Using Wolfrm we come to conclusion tht for every θ [0, : ( θ ( θ > 0(see Grph 7, therefore for every θ [0, : BR( S S SS RS 33

39 Grph 7. Difference between selling nd renting outcomes for firm, if firm chooses selling. Proposition 6. In the cse of durble goods mrket, where both firms re socilly concerned, the best rection of ny firm is to sell. This decision does not depend on the ction of nother firm nd prmeterθ. Joining together BR functions for both firms we come to the following conclusion. Proposition 7. In durble goods duopoly, where both firms re socilly concerned, (S, S is the only Nsh equilibrium for ny prmeter θ. Now we cn compre which decision is optiml for the society. Using Wolfrm (see Grph 8 we cn conclude tht for every θ [0, : RR RS SS CS ( θ < CS ( θ < CS ( θ 34

40 Grph 8. Dependence of CS on prmeter θ in different cses. Proposition 8. If firms prticipting on the mrket re eqully socilly concerned, the only Nsh equilibrium (S, S is socilly optiml. If firms mke other decisions, consumer surplus decreses. 35

41 Chpter 6 CONCLUSIONS In this pper we investigte the influence of socilly concerned firms on durble goods mrket. We cme to conclusion tht the influence of presence of SC firm on selling/renting decisions on the mrket does not differ from the clssicl cse, when only privte firms were considered. We introduced crucil prmeterθ, which cn be interpreted s shre of socil welfre tht goes (directly or indirectly to firm. We dmit tht cse θ 0 corresponds to the cse of purely privte firm. In Chpter 5 we considered the cse when privte firm (with nonzeroθ competes with socilly concerned firm. Then for every vlue ofθ, kind of clssicl Prisoners Dilemm exists: (selling, selling is the only Nsh equilibrium in this gme, wheres (renting, renting strictly domintes this equilibrium. The only difference with pure duopoly cse is tht for θ > 0.3 one of the prisoners, socilly concerned firm hs no incentive to devite from the coopertive outcome. In Chpter 6 we considered sitution when the shre of socil welfre θ is divided eqully between firms, which re prticipting on the mrket. The results show presence of the sme Prisoners Dilemm for every vlue ofθ. The only Nsh equilibrium (selling, selling is lso socilly optiml outcome, lthough in the cse of (renting, renting both firms receive higher profits. Now we cn compre monopoly results with different kinds of duopolies for durble goods mrket (pure nd mixed. We cn conclude tht regrdless the chrcter of duopoly there is chnge in selling/renting decisions. Wheres monopolist is better off under renting strtegy, if competitor ppers, it does not mtter whether it is privte or socilly concerned, mrked 36

42 chnge in behviour occurs: former monopolist switches to selling. Even for socilly concerned competitor the best rection is to follow ex-monopolist nd choose selling. As result both firms sell nd therefore they produce lrger mount of production together s it is better for them to produce more t price, which is higher thn in the second period. As result firms produce nonoptimlly high output nd receive lower price for it under Cournot competition compring with coopertive outcome. This leds to lower firms profits, but higher consumers welfre. To sum up, we cn dmit the fct tht selling/renting decisions lmost do not depend on the kind of duopoly (only renting/renting my become more preferble. The reson behind it tht even socilly concerned firm cres more bout own profit thn bout socil welfre (it is reflected by conditionθ <. In prticulr, we cn conclude more ctive presence of NPO on durble goods mrket will not cuse drmtic chnges on current mrket ctivities. Therefore the following policy recommendtions my be proposed. If government wnts to chnge renting behvior of the monopolist, it should encourge competitors to move onto the mrket (of course if it is possible nd should cre less bout the ttitude to the socil welfre, s the chrcter of ccepted renting/selling decisions will be similr. 37

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