Assuming a fixed-proportion downstream production technology, partial forward integration by an upstream monopolist may be observed whether the monopolist is advantaged or disadvantaged cost-wise relative to fringe firms in the downstream market. Integration need not induce cost predation and the fringe firms’ margin may even increase. The output price falls and welfare unambiguously rises.Vertical integration; cost predation; cost asymmetries

Bonroy, Olivier

Larue, Bruno

English

Research Papers in Economics

  1
Forward Vertical Integration:  
The Fixed-Proportion Case Revisited 
 
 
 
 
Olivier Bonroy 
 
Department of Rural Economy and Management 
Agrocampus Rennes 
 
Center for Research on the Economics of Agri-Food (CREA) 
Université Laval 
 
olivier.bonroy@agrocampus-rennes.fr 
 
 
and 
 
 
Bruno Larue 
 
Canada Research Chair in International Agri-Food Trade 
CREA, Université Laval 
 
bruno.larue@eac.ulaval.ca 
 
 
September 2006  
 
 
 
 
 
Abstract: Assuming a fixed-proportion downstream production technology, partial forward 
integration by an upstream monopolist may be observed whether the monopolist is advantaged or 
disadvantaged cost-wise relative to fringe firms in the downstream market.  Integration need not 
induce cost predation and the fringe firms’ margin may even increase. The output price falls and 
welfare unambiguously rises.   
 
Keywords: Vertical integration, cost predation, cost asymmetries.  
 
J.E.L. Classification: L22 
 
 
*Senior authorship is not assigned.  Funding from FDTA is graciously acknowledged.    2
 
1. Introduction 
 
Many textbooks (e.g., Church and Ware, 2000) begins their treatment of forward integration by 
showing that it is pointless when downstream fringe firms and an upstream monopolist have 
access to the same downstream fixed-proportions constant-return technology.  It is then shown 
that if the upstream monopolist 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 
appeal to regulations to rationalize the existence of a partial integration outcome.  We analyze a 
fixed-proportion case of vertical integration that supports partial integration over a range of 
downstream cost advantages and disadvantages on the part of the upstream monopolist.         
A concern with forward integration is what Salop and Scheffman (1987) call cost 
predation.  This arises when a dominant firm supplying inputs to its competitors purposely raises 
the input price to reinforce its dominant position on the downstream market.  In their model and 
in Riordan (1998), integration induces an increase in the output price and fringe firms end up 
getting “squeezed”.  In our model, integration reduces the output price and it may even increase 
both the margin of fringe firms and the fringe’s output.    
The model is presented in the next section.  The third section begins by showing that the 
upstream monopolist may be partially integrated over a range of cost disadvantages and 
advantages in downstream production relative to fringe firms. Then, the implications of partial 
forward integration on output and input prices, the margin of fringe firms, profits and welfare are 
analyzed.   
 
   3
2. The model 
 
A dominant/predator firm, referred to as P, is a monopolist in an upstream market and a dominant 
firm in a downstream market. It is assumed that the integration is a forward one, with firm P 
being the sole input supplier for the downstream firms with which it competes in the final good’s 
market. Thus, firm P and a competitive fringe, denoted by f, face a demand curve  () D p  for a 
homogenous product sold at price  p and  0 p D < .
1 Each firm relies on a fixed proportion 
technology that requires one unit of input to produce one unit of output.  The cost functions for 
firm  P and the firms in the fringe are respectively  () () ( )
P Cq F q qR qx λ = −+ +  and 
() ()
f Cx F x r x =+ , where outputs are given by q  and  x, such that Dqx ≡ + , r  is the price of the 
input sold by firm P to the fringe firms and  (.) R  is the cost to produce that input for firm P. It is 
assumed that  0 i F > ,  0 D R > ,  0
j
i C >  with  , ix q =  and  , jf P = .  We refer to the portion 
() Fq q λ −  as the output manufacturing cost of the predator firm and to  () R qx +  as its input 
manufacturing cost.  The predator firm can be more cost-efficient (inefficient) than the fringe 
firms by setting  0 λ >  ( 0 λ < ) as in Riordan (1998).  Our model shares important elements with 
Salop and Scheffman (1987) and Riordan (1998), but the differences are not innocuous.   
 
For simplicity, there is no fixed cost on the downstream market. The fringe’s supply curve 
is denoted by  (,) Sp r. It is derived from the fringe’s cost function: 
  (,) m a x
f
x
Sp r A r g π =  (1) 
where [ ( ) ]
f pxF x r x π ≡− − .
2  The fringe’s supply depends on p-r.  As such, 
() ./ 1 / px x SS pF ≡∂ ∂ = ,  () ./ 1 / rx x SS r F ≡∂ ∂ =−  and  pr SS = − .  We assume that 
()
1
1
Fx x
γ
γ γ
γ
+
=
+
, which implies that  () ( ) . Sp r
γ =− and  0 pp rr pr SS S = =− <  when 01 γ < < .  
                                                 
1 The demand must not be “too convex” for the second order conditions to hold.  To simplify the exposition, it is 
assumed in most instances that  0 pp D = . 
2 As in Salop and Scheffman (1987) and Riordan (1998), we assume that  0 xx F >  which implies  0 p S > .   4
The second order condition requires that 
22 /0
f
xx xF π ∂ ∂≡ − <  and it is satisfied when  0 γ > .  
The predator firm maximizes its profit 
P π : 
 
, max [ ( ) ( )]
P
pr pqF q qr xR qx πλ =− + + −+  (2) 
subject to  () qD p x ≡ −  and  (,) x Spr ≡ .
3 
 
The residual demand facing firm P is such that  / pp qpD S ∂ ∂= − , / r qr S ∂∂ = −, where 
() ./ p DDp ≡∂ ∂ .  The first order conditions of firm P with respect to p and r can be expressed in 
terms of the elasticity of its residual demand on the downstream market,  ()
DS
pp DSp q ε
− ≡− − , 
and the elasticity of demand on the upstream market, 
S
r SrS ε ≡− : 
 
() 1
pp
qD
pp pP
DS
SD
pF r R
DS DS
p
λ
ε
−
−+ + −
−−
=  (3) 
 
() 1 q
S
prF
r
λ
ε
− −+ −
=  (4) 
 
From (3) and (4), we can deduce that:  Dq R pF r λ < −+ < .  Provided that  0 qq xx FF q x => ∀ =  
and  0 λ  , the last inequality and the fact that  x Fp r = −  jointly imply that in equilibrium 
qx FF > , which in turn implies qx > . This clearly shows how the dominant position of the 
predator firm on the downstream market is related to its cost advantage.  The second order 
condition is respected if: 
  0
PP PP
pp rr pr rp ππ ππ − > , (5) 
with  ,0
PP
pp rr ππ < .  If  0 pp D = , (5) holds when economies of scale in input manufacturing are not 
too strong:  2/ DDp R D > .
4 
 
 
 
                                                 
3 We refer to  () (,) qD p S p r ≡−  as the dominant firm’s residual demand in the downstream market. 
4 It is important to note that   2/ DDp RD >  is sufficient, but not necessary, for the second order condition to hold.   5
   
3. Forward Integration 
 
The predator firm finds it profitable to partially integrate the downstream market as long as (3) 
holds.  When it is active in the input and output markets, it chooses equilibrium input and output 
prices 
*() r λ  and 
*() p λ .  However, equilibria for which either the fringe firms or the predator 
firm are not active in the downstream market are possible.  Defining 
c p  as the competitive output 
price when the predator is not integrated and 
min() p λ  as the output price when the fringe is just 
about to be ejected, then  ( )
*m i n () () ,
c p pp λλ ∈ .  Because there are no fixed costs, the fringe does 
not produce any output when its average cost is minimized and 
min * () () pr λ λ = .  If  λ  was 
sufficiently low to permit 
*()
c p p λ ≥ , the fringe would supply the whole market demand. 
Conversely, if the predator’s cost structure was such that 
** () () pr λ λ ≤ , then it would eject the 
fringe.  We implicitly define a lower bound λ such that 
*()
c p p λ = . In this case the equilibrium 
prices 
*()
m rr λ ≡  and 
*()
c pp λ ≡  are consistent with (3) and (4), but also with the first order 
condition of the profit maximizing predator firm when it is only an upstream monopolist: 
  max[ ( , ) ( ( , ))]
r r S pr RS pr −  (6) 
subject to  ( *) ( *, *) D pS p r = . 
Thus, for λ λ = , then 
*()
m rr λ = , 
*()
c p p λ = , 
* 0 q =  and 
* ()
c x Dp = .  As long as λ λ < , increases 
in λ  have no effect because it is more profitable for the predator firm to restrict its activities to 
the upstream market.   
 
By the same token, we may implicitly define an upper bound λ such that 
** () () pr λ λ = . 
When λ λ ≥ , it is profitable for the predator firm to monopolize the output market. Consequently, 
at λ λ = , the equilibrium prices are consistent with (3) and (4) as well as the first order condition 
of the profit maximization problem of the predator firm when it is a downstream monopolist: 
  max[ ( ) ( ( )) ( ) ( ( ))]
p pDp FDp Dp RDp λ − +−  (7)   6
When the input price is equal to its chosen output price, the fringe is ejected and the output price 
is the monopoly price,  ()
m p λ . For λ λ = , then 
** () () rp λ λ = , 
*() ()
m pp λ λ = , 
* ()
m qD p =  and 
* 0 x = . 
The condition under which partial integration takes place is:
5,6 
 
Lemma 1:  ( ) , λ λλ ∀∈  then 
* () ()
c rp p λλ << and the predator firm is partially integrated on the 
downstream market. 
 
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 
monopolize the downstream market when it is free to choose its degree of integration.    Figure 1 
portrays the predator’s profit as a function of λ when  (.) D  is linear, both  (.) R  and  (.) F  are cubic 
functions and  0 DD R >  in the neighborhood of the equilibria.  On the left of  0.226 λ =− , the 
upstream monopolist chooses not to be integrated, but it is partially integrated between λ and 
0.111 λ = .  For λ λ > , the fringe is ejected.
7 Thus, an upstream monopolist with a cost 
disadvantage in the downstream market may profitably integrate.  Likewise, a predator that 
enjoys a technological advantage over the fringe firms may prefer a partial integration scenario to 
a monopoly scenario. The intuition behind these results is simple. Going back to the textbook 
case involving downstream competitive firms with constant unit costs (i.e.  0 xx F =  and  0 qq F = ), 
the predator is indifferent between integrating or not when  0 λ = , but would (not) integrate when 
0 λ >  ( 0 λ < ).  In our model, the downstream market is an increasing cost one from the 
perspective of the firms in the competitive fringe as well as the predator firm (i.e.,  0 xx F >  and 
0 qq F > ).  As such, it is profitable for the predator 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.     
 
                                                 
5 Implicitly defining 
max λ  such that  () 0
P Cq = , the domain for partial integration is  ( )
max , λλ λ ∈  as full 
integration equilibrium cannot be observed if 
max λ λ < .   
6 A different motivation for entry can be found in Blair, Cooper and Kaserman (1985).   
7 For
max 0.157 λλ >= , the predator’s cost is non positive.  Naturally, such cases are dismissed as non-pertinent.   7
As mentioned before, an increase in λ .can cause a non integrated upstream monopolist to 
partial integrated the downstream market    Hence, we may use static comparative to compare an 
equilibrium without integration (λ λ = ) to one characterized by partial integration (λ λ > ). 
Before dwelling on the welfare implications, we analyze the impact of partial integration upon 
output and input prices. 
 
Proposition 1: A)  0 dp dλ < .  B)   0
dr
dλ
>
<
.  A necessary, but not sufficient, condition for a cost 
predation effect,  0
dr
dλ
> , is  0 DD R > . C)  . 0
dp dr
dd λλ
− >  if and only if  2/ 0 DD p RD < < . 
Proof: Available upon request from the authors 
 
Integration unambiguously induces a lower output price to the benefit of consumers, but it 
has ambiguous effects on the input price and the fringe’s margin.  If the upstream technology is 
characterized by increasing returns such that  2/ DDp R D < , then the integration makes the input 
price fall enough to increase the fringe’s margin!  To make sense of this result, note that the 
condition on the upstream technology is derived under the assumption of increasing returns in the 
downstream market.  As a result, the predator firm may lower the input price to avoid moving up 
too high on its output manufacturing marginal cost curve.  Finally, the fringe’s output, like its 
margin, may decrease or increase with λ .
8  
 
Proposition 2:A)  0
P πλ ∂∂ > , B)  0
f πλ ∂∂ > if and only if  2/ 0 DD p RD < < , C)  0 CS λ ∂∂ > , and 
D)  0 W λ ∂∂ > .
9 
Proof:  Available upon request from the authors. 
 
Proposition 2 states that integration increases the predator’s profit, consumer surplus and 
welfare.  The fringe’s surplus can increase provided that there are sufficient economies of scale in 
upstream production. 
 
                                                 
8 We obtain  ( ) ( ) ./ 0 sign dS dλ >  if and only if  2/ DD p R D < .     
9 CS and W represent respectively the consumer surplus and total welfare.   8
6. References 
Blair, R. D., Cooper, T. E. and Kaserman, D. L.  1985.  “A Note on Vertical Integration As 
Entry.” International Journal of Industrial Organization, 3:219-229. 
Church, J. and Ware, R. 2000.  Industrial Organization: A Strategic Approach. Toronto: Irwin-
McGraw-Hill. 
Quirmbach, H. C. 1992. “Sequential Vertical Integration.”  Quarterly Journal of Economics, 107: 
1101-11. 
Riordan, M. H. 1998. “Anti-Competitive Vertical Integration by a Dominant Firm.” American 
Economic Review, 88:1232-1248. 
Salop, S.C. and Scheffman, D.T. 1987. “Cost-Raising Strategies.” The Journal of Industrial 
Economics, 36:19-34. 
   9
Figure 1 
 
Figure 1. The impact of λ on the predator’s profit when  () 1 D pp = − , 
3 () R DD =  and 
1
2
γ = . 
 
 

Forward Vertical Integration: The Fixed-Proportion Case Revisited

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