Appendix B Available online – Not for publication Supplemental material for “Contracting, Exclusivity and the Formation of Supply Networks with Downstream Competition” Paolo Ramezzana This online appendix fully solves the model of duopolistic (Cournot and Bertrand) downstream competition with linear demand introduced in Section 4, verifies that it satisfies Assumption 1, and derives exact expressions for the vertical profits that form the basis for the payoffs of the contracting game in Sections 5 through 7. Consider a representative consumer that maximizes the following quasi-linear utility function 1 U = m + ∑ ∑ v − (qsr + aqs0 r + bqsr0 + abqs0 r0 ) qsr (B-1) 2 s=1,2 r =1,2 subject to the budget constraint m + ∑s=1,2 ∑r=1,2 psr qsr = I, with v > c and a, b ∈ [0, 1].
B.1
Cournot competition
The inverse demand for product s at retailer r resulting from maximization of (B-1) is psr = v − (qsr + aqs0 r ) − b (qsr0 + aqs0 r0 ) .
(B-2)
The profits of retailer r in supply network g are h i g g g g g g g g g πr = ∑ `sr psr qsr , qs0 r , qsr0 , qs0 r0 − wsr qsr
(B-3)
s=1,2
g
g
In stage 2, each retailer r chooses the quantities qsr of every product s = 1, 2 for which `sr = 1 to maximize (B-3), given the supply network g that resulted from contracting in stage 1, the g
(privately observed) wholesale prices wsr that she faces for every product s = 1, 2 for which g
g
g
`sr = 1, and the quantities q1r0 and q2r0 chosen by the other retailer, r 0 . The first order condition g
of this profit maximization problem with respect to every product s for which `sr = 1 is1 # " g g g ∂πr ∂psr g g ∂ps0 r g g g g = psr − wsr + g qsr + `s0 r g q 0 = 0. ∂qsr ∂qsr ∂qsr s r 1 It
(B-4)
can be verified that, with the linear demand in (B-2), the second order conditions are always satisfied. 2 g g g g g g g g Specifically, ∂2 πr /∂ qsr = −2`sr and ∂2 πr /∂qsr ∂qs0 r = − a `sr + `s0 r , which implies that the Hessian of the profit maximization problem is negative semidefinite both when retailer r sells only one product (and g g g g thus `sr `s0 r = 0) and when it sells both products (and thus `sr `s0 r = 1).
B-1
One can use (B-4) to verify that the linear inverse demand system in (B-2) satisfies Assumption 1 in the paper. Specifically, by totally differentiating the first order conditions in (B-4) with g
g
g
g
g
g
respect to wsr , qsr and, only if `s0 r = 1, qs0 r , and using the fact that with the linear demand system g
g
in (B-2) ∂psr /∂qsr = −1 and ∂ps0 r /∂qsr = − a, one obtains g
1 dqsr < 0. g =− g 2 dwsr 2 1 − `s0 r a
(B-5) g
g
g
Together with the fact that retailer r 0 does not observe wsr , and thus dqsr0 /dwsr = 0, this g
g
g
g
g
g
implies that dqs /dwsr = dqsr /dwsr + dqsr0 /dwsr < 0, as in Assumption 1. g
Proposition 1 in the paper implies that in a CPNE it must be wsr = c for all s and r and all g. Setting v − c ≡ υ > 0, the total vertical profits Π g generated by each individual retailer with supply network g when all wholesale prices are equal to c are Πdca
=
2υ2 (1 + a)(2 + b)2
Π pe
=
υ2 (2 + ab)2
Πum
=
υ2 (2 + b )2
Πdm
=
υ2 2(1 + a )
=
υ2 , 4
Πmix1 =
υ2 (2 + b )2
Πbm
Πmix2 =
(B-6)
[8 + (1 − a ) (4 + b ) b ] υ2 . 4 (1 + a ) (2 + b )2
Inspection of (B-6) reveals that Πdca ≥ Πmix1 = Πum , a relation of which I make use in a number of proofs in the paper. Finally, one can use the analysis above to confirm the welfare result in Proposition 4 of the main paper. In particular, using the utility function in (B-1), consumer utility in symmetric equilibria with g = dca, g = pe and g = dm can be written, respectively, as U dca U pe U dm
1 dca = I + 4 v − (1 + a ) (1 + b ) q qdca , 2 1 pe pe pe = I − 2p q + 2 v − (1 + ab) q q pe , 2 1 dm dm dm = I − 2p q + 2 v − (1 + a) q qdm . 2
− 4pdca qdca
(B-7)
Since qdca = υ/ [(1 + a) (2 + b)] and pdca = c + υ/ (2 + b) in an equilibrium with g = dca, q pe = υ/ (2 + ab) and p pe = c + υ/ (2 + ab) in an equilibrium with g = pe, and qdm = υ/2 (1 + a) and
B-2
pdm = c + υ/2 in an equilibrium with g = dm, one has U dca = I +
2 (1 + b ) υ2
(1 + a ) (2 + b )2
U pe
= I+
(1 + ab) υ2 . (2 + ab)2
U dm
= I+
υ2 4 (1 + a )
, (B-8)
It can be verified that U dca > U pe > U dm for all a, b ∈ (0, 1), and therefore a fortiori when g = pe and g = dm constitute equilibria in Proposition 3 of the paper.
B.2
Bertrand competition
To obtain an expression for the direct demand for product s at retailer r when all four products are consumed in positive quantities one can invert the inverse demand system in (B-2) and obtain qsr = ρ [(1 − a)(1 − b)v − psr + aps0 r + bpsr0 − abps0 r0 ] ,
(B-9)
where ρ ≡ 1/ (1 − a2 )(1 − b2 ) ≥ 1. When some of the products are not available for consumption, the demand for the remaining products must be adjusted by setting the prices of the missing products at that level for which their demand is equal to zero, and the resulting coefficients are different from those in (B-9) (the details are straightforward and are omitted here). In a supply network with g = dca in which all four products are available, as in the demand system in (B-9), qsr depends negatively on ps0 r0 (note that this does not happen with other supply networks). This is due to significant diversion from product s0 r 0 towards products s0 r and sr 0 in response to an increase in ps0 r0 , which crowds out sales of product sr. This notwithstanding, as shown below, this direct demand specification satisfies Assumption 1 on derived demand, and the game analyzed in the main paper is thus well defined.2 2 Other
authors, such as Inderst and Shaffer (2010) and Rey and Verg´e (2010), specify direct demand in a two-supplier two-retailer environment like mine as qsr = v − psr + aps0 r + bpsr0 + abps0 r0 . Such an ad-hoc demand specification poses, however, a number of issues. First, it is not clear whether this direct demand specification has been derived from a utility function representing reasonable consumer preferences. In fact, by inverting it, one can see that the resulting inverse demand, and thus consumer marginal utility, for product sr is increasing qs0 r0 , which suggests complementarity, not substitutability, between the two products in the underlying consumer preferences. Second, and related to the previous issue, in order for the demand specification in Rey and Verg´e (2010) to be well behaved one needs to impose ad-hoc restrictions on the parameters a and b (e.g., a + b + ab < 1) to ensure that demand for a given product decreases when all prices increase. These restrictions reduce the range of a and b over which one can conduct comparative statics and make it, for example, impossible to analyze the limit case of perfectly substitutable suppliers (a = 1) and/or retailers (b = 1). This is instead possible with a specification that is explicitly derived from a utility function, like the one that I adopt in this paper.
B-3
The profits of retailer r in supply network g are h i g g g g g g g g g πr = ∑ `sr psr − wsr qsr psr , ps0 r , psr0 , ps0 r0 .
(B-10)
s=1,2
g
g
In stage 2, each retailer r chooses the prices psr of every product s = 1, 2 for which `sr = 1 to maximize (B-10), given the supply network g that resulted from contracting in stage 1, the g
(privately observed) wholesale prices wsr that she faces for every product s = 1, 2 for which g
g
g
`sr = 1, and the prices p1r0 and p2r0 charged by the other retailer, r 0 . The first order condition of this profit maximization problem with respect to the retail price of every product s for which g
`sr = 1 is # " g ∂q g0 ∂q g ∂πr g g g g g g g sr sr = ` + ` q p + p − w − w sr sr sr sr g g g = 0. s0 r s0 r s0 r ∂psr ∂psr ∂psr
(B-11)
One can use (B-11) to verify that the linear direct demand system in (B-9) satisfies Assumption 1 in the paper. Specifically, by totally differentiating the first order conditions in (B-4) with respect g
g
g
g
to wsr , qsr and, only if `s0 r = 1, qs0 r one can show that for any symmetric linear demand system g
g
g
g
g
g
g
g
in which ∂qsr /∂psr = ∂qs0 r /∂ps0 r and ∂qs0 r /∂psr = ∂qsr /∂ps0 r (and thus a fortiori for the linear demand system in (B-9)) and for all g: g
g dps0 r dpsr 1 and g = g = 0, 2 dwsr dwsr
(B-12)
with the prices psr0 and ps0 r0 charged by the other retailer remaining unchanged because changes in wsr are not observed by that retailer. The total effect on the total quantity of product s sold is " g " g g # g # g g 1 ∂qsr dqs dpsr ∂qsr g ∂qsr 0 g ∂qsr 0 = < 0, (B-13) g = g g + `sr 0 g g + `sr 0 g 2 ∂psr dwsr dwsr ∂psr ∂psr ∂psr g
g
where the inequality follows from the fact that, with the preferences in (B-1), ∂qsr /∂psr < 0 g g g g and ∂qsr /∂psr > ∂qsr0 /∂psr in any network g. This demonstrates that Bertrand downstream competition with the direct demand system obtained from (B-1) yields derived demand functions that are consistent with Assumption 1. The total vertical profits Π g generated by individual retailers for different supply networks g
when wsr = c for all s and r and all g and downstream competition (if any) is in prices (differen-
B-4
tiated Bertrand) are Πdca
=
2(1 − b ) υ2 , (1 + a)(2 − b)2 (1 + b)
Π pe
=
(1 − ab)υ2 , (2 − ab)2 (1 + ab)
Πum
=
(1 − b ) υ2 , (2 − b )2 (1 + b )
Πdm
=
υ2 , 2(1 + a )
Πbm
=
Πmix2 =
(1 − b ) υ2
υ2
, Πmix1 = 4 (1 − a ) b3 − 3(1 − a ) b2 − 4(1 + a ) b + 8 υ2 . 4( a + 1)(2 − b)2 (b + 1)
(2 − b )2 (1 + b )
(B-14) ,
As in the case of Cournot competition discussed above, also in the case of differentiated Bertrand competition one has Πdca ≥ Πmix1 = Πum . Finally, the welfare implications of exclusive contracts can be assessed by relying on the same expression for equilibrium utility under g = dca, g = pe and g = dm in (B-7) above. In the case of differentiated Bertrand competition one has qdca = υ/ [(1 + a) (1 + b) (2 − b)] and pdca = c + υ (1 − b) / (2 + b) in an equilibrium with g = dca, q pe = υ/ (1 + ab) (2 − ab) and p pe = c + υ (1 − ab) / (2 − ab) in an equilibrium with g = pe, and qdm = υ/2 (1 + a) and pdm = c + υ/2 in an equilibrium with g = dm. Therefore consumer utility in these equilibria is U dca = I + U pe
= I+
U dm
= I+
2υ2
(2 − b )2 (1 + a ) (1 + b ) υ2
(2 − ab)2 (1 + ab)
,
,
(B-15)
υ2 . 4 (1 + a )
As in the case of Cournot competition analyzed above, also in this case U dca > U pe > U dm for all a, b ∈ (0, 1) and thus, a fortiori, when g = pe and g = dm constitute equilibria in Proposition 3 of the paper.
References Inderst, Roman and Greg Shaffer (2010), “Market-share Contracts as Facilitating Practices,” RAND Journal of Economics, Vol. 41 No. 4, Winter, pp. 709-729. Rey, Patrick and Thibaud Verg´e (2010), “Resale Price Maintenance and Interlocking Relationships,” Journal of Industrial Economics, 58(4) December, pp. 928-961.
B-5
