Forward Vertical Integration: The Fixed-Proportion Case Revisited. Abstract

Transcription

1 Forward Vertical Integration: The Fixed-roortion Case Revisited Olivier Bonroy GAEL, INRA-ierre Mendès France University Bruno Larue CRÉA, Laval University Abstract Assuming a fixed-roortion downstream roduction technology, artial forward integration by an ustream monoolist may be observed whether the monoolist is advantaged or disadvantaged cost-wise relative to fringe firms in the downstream market. Integration need not induce cost-redation and the rofits of the fringe may increase. The outut rice falls and welfare unambiguously rises. Senior authorshi is not assigned. Funding from FDTA is graciously acknowledged. Citation: Bonroy, Olivier and Bruno Larue, (2007) "Forward Vertical Integration: The Fixed-roortion Case Revisited." Economics Bulletin, Vol. 12, No Submitted: Setember 6, Acceted: October 5, URL: htt://economicsbulletin.vanderbilt.edu/2007/volume12/eb-07l20009a.df

2 1. Introduction Many textbooks (e.g., Church and Ware, 2000) begins their treatment of forward integration by showing that it is ointless when downstream fringe firms and an ustream monoolist have access to the same downstream fixed-roortions constant-return technology. It is then shown that if the ustream monoolist was able to secure a better downstream technology, it would eject the fringe. Given this razor s edge effect, also found in Quirmbach s (1992) model, one must aeal to regulations to rationalize the existence of a artial integration outcome. We analyze a fixed-roortion case of vertical integration that suorts artial integration over a range of downstream cost advantages and disadvantages on the art of the ustream monoolist. A concern with forward integration is what Salo and Scheffman (1987) call costredation. This arises when a dominant firm sulying inuts to its cometitors urosely raises the inut rice to reinforce its dominant osition on the downstream market. Our results suggest that forward integration need not induce cost-redation and may even increase both the margin of fringe firms and the fringe s outut. The model is resented in the next section. The third section begins by showing that the ustream monoolist may be artially integrated over a range of cost disadvantages and advantages in downstream roduction relative to fringe firms. Then, the imlications of artial forward integration on outut and inut rices, the margin of fringe firms, rofits and welfare are analyzed. 2. The model A dominant/redator firm, referred to as, is a monoolist in an ustream market and a dominant firm in a downstream market. It is assumed that the integration is a forward one, with firm being the sole inut sulier for the downstream firms with which it cometes in the final good s market. Thus, firm and a cometitive fringe, denoted by f, face a demand curve D( ) for a homogenous roduct sold at rice and D < 0. 1 Each firm relies on a fixed-roortion technology that requires one unit of inut to roduce one unit of outut. The cost functions for firm and the firms in the fringe are resectively C () q = F() q λq+ R( q+ x) and f C ( x) = F( x) + rx, where oututs are given by q and x, such that D q+ x, r is the rice of the inut sold by firm to the fringe firms and R (.) is the cost to roduce that inut for firm. It is assumed that 0 i F( q) j F >, R > 0, C > 0 with i= x, q and j = f,. We refer to the ortion D i + as its inut manufacturing cost. The redator firm can be more cost-efficient (inefficient) than the fringe firms by setting λ > 0 ( λ < 0 ). 2 λq as the outut manufacturing cost of the redator firm and to R( q x) For simlicity, there is no fixed cost on the downstream market. The fringe s suly curve is denoted by S(, r ). It is derived from the fringe s cost function: 1 The demand must not be too convex for the second order conditions to hold. To simlify the exosition, it is assumed in most instances that D = 0. 2 Riordan (1998) relies on the same assumtion in a backward vertical integration model. 1

3 S(, r) = Argmax f (1) f where [ x F( x) rx]. 3 The fringe s suly deends on -r. As such, ()./ = 1/, () S S F γ + 1 xx x Sr S./ r = 1/ Fxx and S = Sr. We assume that γ γ F( x) = x, which imlies that S(). = ( r) γ and S = Srr = Sr < 0 when 0< γ < 1. γ + 1 The second order condition requires that 2 f / x 2 F xx < 0 and it is satisfied when γ > 0. The redator firm maximizes its rofit : max = [ q F( q) + λq+ rx R( q+ x)] (2) r, subject to q D x and x S(, r). 4 The residual demand facing firm is such that q/ = D S, q/ r = Sr, where () D D./. The first order conditions of firm with resect to and r can be exressed in D S terms of the elasticity of its residual demand on the downstream market, ε ( D S) q, S and the elasticity of demand on the ustream market, ε Sr r S: S D Fq + λ + r RD D S ( D S) 1 = (3) D S ε ( r Fq + λ) 1 = (4) S r ε From (3) and (4), we can deduce that: RD < Fq + λ < r. rovided that Fqq = Fxx > 0 q= x and λ > 0, the last inequality and the fact that F x = r jointly imly that in equilibrium Fq > Fx, which in turn imlies q> x. This clearly shows how the dominant osition of the redator firm on the downstream market is related to its cost advantage. The second order condition is resected if: > 0 (5) rr r r with, rr < 0. If D = 0, (5) holds when economies of scale in inut manufacturing are not too strong: R > 2 / D (see Aendix 1). 5 3 As in Salo and Scheffman (1987) and Riordan (1998), we assume that F xx > 0 which imlies S > 0. 4 We refer to q D S(, r) as the dominant firm s residual demand in the downstream market. 5 It is imortant to note that R > 2/ D is sufficient, but not necessary, for the second order condition to hold. 2

4 3. Forward Integration The redator firm finds it rofitable to artially integrate the downstream market as long as (3) holds. When it is active in the inut and outut markets, it chooses equilibrium inut and outut rices r * ( λ ) and * ( λ ). However, equilibria for which either the fringe firms or the redator c firm are not active in the downstream market are ossible. Defining as the cometitive outut rice when the redator is not integrated and min ( λ ) as the outut rice when the fringe is just * min c about to be ejected, then ( λ) ( ( λ), ). Because there are no fixed costs, the fringe does not roduce any outut when its average cost is minimized and min ( λ) = r * ( λ). If λ was sufficiently low to ermit * c ( λ), the fringe would suly the whole market demand. * * Conversely, if the redator s cost structure was such that ( λ) r ( λ), then it would eject the fringe. We imlicitly define a lower bound λ such that * c ( λ ) =. In this case the equilibrium m rices * c r r ( λ) and * ( λ) are consistent with (3) and (4), but also with the first order condition of the rofit maximizing redator firm when it is only an ustream monoolist: max[ rs( r, ) RS ( ( r, ))] (6) r subject to D( *) = S( *, r*). Thus, for λ = λ, then * m r ( λ ) = r, * c * * c ( λ ) =, q = 0 and x = D( ). As long as λ < λ, increases in λ have no effect because it is more rofitable for the redator firm to restrict its activities to the ustream market. * * By the same token, we may imlicitly define an uer bound λ such that ( λ) = r ( λ). When λ λ, it is rofitable for the redator firm to monoolize the outut market. Consequently, at λ = λ, the equilibrium rices are consistent with (3) and (4) as well as the first order condition of the rofit maximization roblem of the redator firm when it is a downstream monoolist: max[ D F( D) + λ D R( D)] (7) When the inut rice is equal to its chosen outut rice, the fringe is ejected and the outut rice m is the monooly rice, ( λ ). For λ = λ, then r * ( λ) = * ( λ), * m * m ( λ) = ( λ), q = D( ) and * x = 0. The condition under which artial integration takes lace is: 6,7 6 Imlicitly defining λ max such that 0 max C q =, the domain for artial integration is λ ( λ, λ ) max integration equilibrium cannot be observed if λ < λ. 7 A different motivation for entry can be found in Blair, Cooer and Kaserman (1985). as full 3

5 Lemma 1: λ ( λλ, ) then downstream market. λ < λ < and the redator firm is artially integrated on the * c r Unlike in Quirmbach s (1992) model in which the dominant firm would choose full integration unless it is arbitrarily restricted from doing so, the dominant firm does not necessarily want to monoolize the downstream market when it is free to choose its degree of integration. Figure 1 ortrays the redator s rofit as a function of λ when D (.) is linear, both R (.) and F (.) are cubic functions and R > 0 in the neighborhood of the equilibria. On the left of λ = 0.226, the ustream monoolist chooses not to be integrated, but it is artially integrated between λ and λ = For λ > λ, the fringe is ejected. 8 Thus, an ustream monoolist with a cost disadvantage in the downstream market may rofitably integrate. Likewise, a redator that enjoys a technological advantage over the fringe firms may refer a artial integration scenario to a monooly scenario. The intuition behind these results is simle. Going back to the textbook case involving downstream cometitive firms with constant unit costs (i.e. F xx = 0 and F qq = 0 ), the redator is indifferent between integrating or not when λ = 0, but would (not) integrate when λ > 0 ( λ < 0 ). In our model, the downstream market is an increasing cost one from the ersective of the firms in the cometitive fringe as well as the redator firm (i.e., F xx > 0 and F qq > 0 ). As such, it is rofitable for the redator to enter the market even when it has a cost disadvantage and not to force the exit of the fringe when it has a cost advantage. As mentioned before, an increase in λ can cause a non integrated ustream monoolist to artial integrated the downstream market Hence, we may use static comarative to comare an equilibrium without integration ( λ = λ ) to one characterized by artial integration ( λ > λ ). Before dwelling on the welfare imlications, we analyze the imact of artial integration uon outut and inut rices. dr > roosition 1: A) d < 0. B) 0. A necessary, but not sufficient, condition for a costredation effect, 0 < dr >, is 0 d dr R >. C) > 0 if and only if R < 2/ D < 0. roof: See Aendix 2. Integration unambiguously induces a lower outut rice to the benefit of consumers, but it has ambiguous effects on the inut rice and the fringe s margin. If the ustream technology is characterized by increasing returns such that R < 2 / D, then the integration makes the inut rice fall enough to increase the fringe s margin! To make sense of this result, note that the condition on the ustream technology is derived under the assumtion of decreasing returns in the downstream market. As a result, the redator firm may lower the inut rice to avoid moving u too high on its outut manufacturing marginal cost curve. Finally, the fringe s outut, like its margin, may decrease or increase with λ. 9 8 max For λ > λ = 0.157, the redator s cost is non-ositive. Naturally, such cases are dismissed as non-ertinent. 9 We obtain ( ) sign ds./ > 0if and only if R < 2/ D. 4

6 f roosition 2: A) λ > 0, B) λ > 0 if and only if R < 2/ D < 0, C) CS λ > 0, and D) W λ > roof: See Aendix 3. roosition 2 states that integration increases the redator s rofit, consumer surlus and welfare. The fringe s surlus can increase rovided that there are sufficient economies of scale in ustream roduction. 4. Conclusions In this aer, we show that relaxing constant return to scale in roducing the ustream inut and the downstream outut modifies the classic result about forward integration with fixed roortion technology discussed in most textbooks. artial forward integration by an ustream monoolist may be observed in equilibrium as long as the integrated firm is not too advantaged or disadvantaged relative to a downstream cometitive fringe. Interestingly, economies of scale in the roduction of the ustream inut imly that the forward integration is never imlemented by a cost-redation strategy and the rofits of the fringe can even increase. 5. References Blair, R. D., Cooer, T. E. and D. L. Kaserman (1985) A Note on Vertical Integration As Entry International Journal of Industrial Organization, 3, Church, J. and Ware, R. (2000) Industrial Organization: A Strategic Aroach, Toronto: Irwin- McGraw-Hill. Quirmbach, H. C. (1992) Sequential Vertical Integration Quarterly Journal of Economics, 107, Riordan, M. H. (1998) Anti-Cometitive Vertical Integration by a Dominant Firm American Economic Review, 88, Salo, S.C. and Scheffman, D.T. (1987) Cost-Raising Strategies The Journal of Industrial Economics, 36, CS and W reresent resectively the consumer surlus and total welfare. 5

7 Figure 1 Figure 1. The imact of λ on the redator s rofit when D = 1, R( D) 3 = D and 1 γ =. 2 6

8 Aendix Aendix 1: roof of the second order condition. The derivatives of the dominant firm s rofit are: = ( F + λ)( D S ) + ( D S) + rs R D = 0 q D r = ( Fq+ λ r)( Sr) + S = 0 2 = ( F + λ)( D S ) + 2 F ( D S ) ( D S ) + rs R D R D < 0 q qq D = ( F + λ r) S + 2 F S S < 0 rr q rr qq r r = ( F + λ r) S + S S 1 F ( D S ) > 0 r q r r qq = > 0 r r λ D S rλ Sr 0 = ( ) < 0 = > Assuming D = 0, the second order condition for the dominant firm s otimization roblem ( > 0 ) is reduces to: rr r r ( 2 2 ( 2 + ( + )) + ( 2 + ) + ( + λ )( 2 + ( + )) ) D D F R S F D R S r F D F R S qq qq q qq rr Given that Fqq > 0, Srr < 0, S > 0, D < 0, it can readily be seen that this exression is unambiguously ositive when R 0. However, economies of scale in inut manufacturing are allowed, as 0> R > ( 2/ D) Fqq. Hence, 0 > R > 2 / D is sufficient, but not necessary for the second order condition to hold. Aendix 2: roof of roosition 1. The imact of λ on inut and outut rices can be determined by totally differentiating the first order conditions, = = 0, and by alying Cramer s rule to the resulting equations. Then, it can be shown that: r dr = r λ rλ rr r r 2 where ij = / ijwith ij, = r,. From (A8), the denominator is ositive and hence : dr sign sign( r λ rλ ) = (A9) Alying the same stes for the outut rice, we obtain: d sign sign( rλ r λ rr) = (A10) (A8) 7

9 art A): The roof directly follows from (A10), the definitions rovided in the aendix 1 and the restrictions on the derivatives of the fringe s suly curve arising from the fixed roortion technology, secifically ( S = S > 0, S = S = S < 0 ). From this, it can be shown that: r rr r sign = sign D 2S ( r λ F ) S λ + + q rr. The bracketed exression is clearly ositive and given that the demand for the final good is negatively sloed, it turns out that the outut rice is unambiguously decreasing in λ. art B): The roof follows from (A9). It can be shown that the sign of = sign of ( λ ) ( λ ) r λ rλ D DRS + r+ Fq Srr DS Fq + RD. Obviously, setting D = 0 does not resolve the ambiguity about dr and the cost redation result hinges on the bracketed exression being negative. This requires that R > 0. (( ) ( )) r r rr r r d dr art C): It can be shown that: sign = sign λ λ λ λ = d dr sign( S ( D ( 2 + D R ) D ( F + λ R ))) as D 0 r q D = we have > 0 if and only if R < 2/ D < 0. We have demonstrated in aendix 1 that R > 2/ D is only a sufficient condition for the second order condition to hold. The second order condition holds if R = 2/ D ε as long as ε is not too high. QED Aendix 3: roof of roosition 2. art A): By the enveloe theorem (, r, ) λ λ λ λ ( ( λ) r( λ) λ) = q,, > 0. art B): The decrease or increase in the fringe s surlus comes from the facts that the fringe s λ r λ and that the outut may shrink or increase as surlus and outut are conditioned by ds (). ( 2 ( 2 ) ) < λ increases, sign = sign S 2D D R 0. Thence, for R < 2/ D then > d dr > 0 and sign( ds (./ ) ) > 0, the fringe s surlus is increasing in λ. Inversely, for d dr R > 2/ D then < 0 and sign( ds (./ ) ) < 0, the fringe surlus is decreasing in λ. d art C): As < 0, the consumer surlus is unambiguously decreasing in λ. 8

10 art D): Welfare is defined as the sum of consumer surlus, the fringe s surlus and the redator 1 firm s rofit D (, ) and x S(, r) q D S r 0 λ W = D q+ x d q+ x F x F q + q R q+ x where. The enveloe theorem can again be invoked to show that welfare increases in resonse to artial integration, but instead we rely on a more intuitive roof. If the quantity marketed of the final good remained constant, consumer surlus would be unaffected by the artial integration and welfare changes would be driven solely by sourcing cost considerations: R ( D ) + F( D S(, r) ) + F( S(, r) ) λ ( D S(, r) ). The fringe could roduce the same outut at the same cost, but the redator firm would benefit from its cost advantage to roduce the same outut. Thus, welfare would increase if total outut and each firm s outut were to remain constant. But the oututs do change. At the new equilibrium, total outut is higher because the redator firm fully exloits its cost advantage. As a result the more efficient firm roduces more and the less efficient fringe roduces less. 11 Therefore, there is a cost rationalization and an increase in consumer surlus as -r gets smaller. Consequently, welfare unambiguously increases uon artial integration. QED 11 The increase in total outut and in the outut of the dominant firm cannot increase cost too much because from the first order conditions, it must be a ositive margin on the roduct sold in the downstream market: > RD + Fq λ. 9