Discriminatory Nonlinear Pricing, Fixed
Costs, and Welfare in Intermediate-Goods
Markets∗
Fabian Herweg† and Daniel Müller‡
January 26, 2016
We investigate the welfare effects of third-degree price discrimination in
input markets when nonlinear wholesale tariffs are feasible. After accepting their respective wholesale contracts, two downstream firms have
to pay a fixed cost in order to become active in the downstream market.
If the downstream firm with lower marginal cost has significantly higher
fixed cost, uniform pricing leads to lower marginal wholesale prices for
all downstream firms and thus higher quantities of the final product being produced. This in turn implies that banning price discrimination
improves welfare and consumer surplus. If the downstream firm with
lower marginal cost has only weakly higher (or even lower) fixed cost,
banning price discrimination deteriorates welfare and consumer surplus.
JEL classification: D43; K21; L11; L42
Keywords: Fixed Cost; Third-Degree Price Discrimination; Two-Part Tariffs; Vertical Relations
∗
We have benefited from comments made by the co-editor, Philipp Schmidt-Dengler, two anonymous referees, Magdalena Helfrich, Heiko Karle, and Johannes Muthers. Pekka Sagner has
provided excellent research assistance in preparing the manuscript. All errors are of course our
own.
†
University of Bayreuth, Faculty of Law, Business and Economics, Universitätsstr. 30, D-95440
Bayreuth, Germany, E-mail address: [email protected]
‡
University of Würzburg, Chair for Information Economics and Contract Theory, Sanderring 2,
D-97070 Würzburg, Germany, E-mail address: [email protected].
1
1. Introduction
A classic topic of antitrust economics—dating back to Robinson (1933)—is the welfare effect of third-degree price discrimination in markets for intermediate goods.
Legal conduct regarding price discrimination by big manufacturers is governed by
major legal enactments, like the Robinson-Patman Act in the US and Article 102
TFEU in the EU. A manufacturer who charges different wholesale prices from different retailers may be found liable of an infringement of these laws. In particular,
discriminatory quantity discounts are generally considered as a violation of antitrust
laws. For instance, in the European sugar industry decision from 1973, the Commission ruled that “the granting of a rebate which does not depend on the amount
bought [...] is an unjustifiable discrimination [...].” (Recital II-E-1 of Commission
decision 73/109/EC) Nondiscriminatory quantity discounts, on the other hand, are
commonly regarded as a justifiable pricing strategy of manufacturers by antitrust
authorities both in the EU as well as in the US, which is hardly surprising in light
of the well-known double markup problem (Spengler, 1950).1
By now, there exists a sizable literature investigating the welfare effects of price
discrimination when wholesale tariffs are nonlinear and thus allow for quantity discounts. These articles, which we will review in more detail below, abstract from
downstream production involving fixed costs. We add to this literature by allowing
for a fixed cost component of the production process downstream, which retailers
have to incur in order to be active in the downstream industry.2 Allowing for a fixed
cost component downstream has crucial implications from a welfare perspective,
even if all downstream firms are active irrespective of whether price discrimination
is banned or not. While in the existing literature it is always the participation constraint of a downstream firm with high marginal cost that is binding under uniform
pricing, with fixed costs the participation constraint of a downstream firm with low
marginal cost can become binding. If this is the case, previous findings regarding the welfare effects of banning third-degree price discrimination under nonlinear
wholesale contracts are turned upside down.
1
For instance, in the Vitamins, Hoffmann – La Roche ruling from 1976, the European Court
of Justice stated that a company has the right to offer volume discounts as long as they are
extended to all customers. (Commission decision 76/642/EEC) For more detailed information
and additional cases where this view becomes apparent see Russo, Schinkel, Gnster, and Carree
(2010).
2
The fixed cost component can be interpreted in various ways. It could be a fixed cost of production, e.g. the firm’s lease of the production plant. The fixed cost component could also be some
form of entry cost, e.g. the fee for acquiring or renewing a business license. Alternatively, the
fixed cost component can be regarded as a retailer’s outside option, that it forgoes the moment
it accepts the manufacturer’s contract offer. In this sense, the fixed cost in our model might
also reflect a retailer’s profits from turning to an alternative source of supply (i.e., demand side
substitution) or from producing the input in house (i.e., backward integration).
2
We model a vertically related industry where a monopolistic manufacturer supplies
an essential input to two downstream firms. These downstream firms differ not only
in their marginal cost of transforming the manufacturer’s input into a final product,
but also with regard to a fixed cost component, that they have to incur in order to
become active in the downstream industry. The manufacturer has all the bargaining
power and makes an observable two-part tariff offer to each downstream firm. Under
price discrimination these tariffs can be different, whereas under uniform pricing the
manufacturer is forced to offer the same tariff to both downstream firms. In our
baseline model, we focus on the case where downstream firms are local monopolists,
i.e, they operate in independent markets. Then, under price discrimination the unit
wholesale price equals upstream marginal costs of production and the fixed fees are
set in order to extract all the rents from the downstream firms. Under uniform
pricing, the manufacturer cannot extract all the rents because the fixed fee has to
be the same for both downstream firms. If the downstream firm with the lower
marginal cost has only a slight disadvantage (or even an advantage) regarding the
fixed cost component, then the participation constraint of the downstream firm with
the higher marginal cost determines the fixed fee. Here, the optimal uniform unit
wholesale price is set above upstream marginal costs, which allows the manufacturer
to extract larger rents from the downstream firm with the lower marginal cost. Thus,
in contrast to price discrimination, uniform pricing leads to double marginalization
and banning discriminatory wholesale contracts decreases welfare—in accordance
with the widespread opinion in the extant literature. If, on the other hand, the
downstream firm with the higher marginal cost has a substantial fixed cost advantage
but it is still optimal to serve both downstream firms, the participation constraint
of the downstream firm with lower marginal costs determines the fixed fee. Here, a
unit wholesale price below upstream marginal costs is optimal because it significantly
relaxes the participation constraint of the downstream firm with lower marginal costs
and thus allows the manufacturer to charge a higher fixed fee. In this case, banning
price discrimination leads to lower unit wholesale prices for all downstream firms.
These lower unit wholesale prices translate into higher quantities produced of the
final good, which boosts welfare and consumer surplus. We show robustness of this
insight with respect to allowing for (i) more general nonlinear pricing schedules and
(ii) downstream competition in differentiated products.
Related literature: By now, there is a large literature investigating the welfare
effects of price discrimination in intermediate-goods markets. Most of the early
contributions focused on linear wholesale contracts, e.g., Katz (1987), DeGraba
3
(1990) and Yoshida (2000). Here, the main finding is that the wrong firm—i.e.,
the less efficient firm—receives a discount under price discrimination, such that
banning price discrimination improves allocative efficiency and typically also welfare.
While the aformentioned articles presume that it is optimal to contract with all
downstream firms, this decision is explicitly analyzed by Herweg and Müller (2012)
and Dertwinkel-Kalt, Haucap, and Wey (2016). These contributions show that
price discrimination fosters entry, which often improves welfare. However, price
discrimination may also reduce welfare because it may lead to a severe inefficiency
in production. Entry may restrict the manufacturer in its price setting similar as
the threat of demand-side substitution, i.e., if the manufacturer faces competition
by a competitive fringe. This case is analyzed by Inderst and Valletti (2009), who
show that a ban on price discrimination benefits consumers in the short run but
reduces consumer surplus in the long run.3
The contributions that investigate the welfare effects of a ban on third-degree
price discrimination when quantity discounts are feasible can be decomposed into
two strands. The first strand assumes that tariff offers made by the manufacturer
cannot be observed by other retailers—cf. O’Brien and Shaffer (1994) and Rey and
Tirole (2007).4 Here, once a retailer has signed a contract, the manufacturer has
an incentive to offer better wholesale conditions—selling a higher amount—to the
competing retailer. Due to this commitment problem, the manufacturer loses part of
its monopoly power, whereas this monopoly power is restored under uniform pricing.
Hence, a ban on price discrimination reduces total quantity supplied, which in turn
reduces consumer surplus and welfare.5
Our paper is more closely related to the second strand that assumes observable
wholesale tariffs.6 Inderst and Shaffer (2009) show that the optimal discriminatory
two-part tariff amplifies downstream firms’ relative competitiveness. Thus, price
discrimination enhances productive efficiency, which—at least for the case of independent downstream markets—translates into higher welfare and consumer surplus.
A similar finding is obtained by Arya and Mittendorf (2010), who assume that one
downstream firm operates in multiple markets while the other downstream firm op3
While almost all contributions assume that the manufacturer has all the bargaining power,
O’Brien (2014) investigates the welfare effects of price discrimination when the manufacturer
negotiates the wholesale price with the retailers.
4
The assumption of nonobservable contract offers goes back to Hart and Tirole (1990), O’Brien
and Shaffer (1992), and McAfee and Schwartz (1994).
5
Caprice (2006) shows that the welfare findings may be reversed if the manufacturer competes
against a competitive fringe.
6
Similar models are analyzed in the regulatory economics literature on two-part input prices
(often called two-part access pricing). For instance, Valletti (1998) shows that the optimal
two-part access price set by a regulator discriminates between downstream firms. For an early
contribution to this literature see Panzar and Sibley (1989).
4
erates only in a single market. These findings may not be robust if downstream
firms have private information about demand or retailing costs, as is shown by Herweg and Müller (2014). In this paper, we abstract from private information and
analyze a model similar to the one used by Inderst and Shaffer (2009). The crucial
difference is that we allow for a fixed cost component of the downstream production
process. For the case where the downstream firm with the lower marginal cost has
only a slight disadvantage (or even an advantage) regarding the fixed cost component, we show robustness of the findings from Inderst and Shaffer (2009). For the
case where the downstream firm with the lower marginal cost has a substantial fixed
cost disadvantage, however, the findings regarding welfare and consumer surplus are
reversed: A ban on price discrimination leads to lower unit wholesale prices for all
downstream firms and thus to higher consumer surplus and total welfare.
The paper is organized as follows. Section 2 introduces the baseline model where
downstream firms operate in separate markets. The optimal wholesale tariffs under
both pricing regimes, price discrimination and uniform pricing, are derived in Section
3. The implications of a ban on price discrimination for consumer surplus and
welfare are investigated in Section 4. We illustrate our findings and assumptions
in Section 5 with a linear demand example. Robustness of our main findings with
regard to downstream competition and more general nonlinear wholesale tariffs is
shown in Section 6 and Section 7, respectively. The final Section 8 summarizes our
main findings and discusses extensions and shortcomings of our model. Except for
the proofs concerning Section 7, proofs are relegated to Appendix A. Appendix B
contains a detailed derivation of the results presented in Section 7.
2. The Model
Consider a vertically related industry where the upstream market is monopolized
by manufacturer M . This manufacturer produces an essential input at constant
marginal cost K > 0, which is supplied to a downstream sector. There are potentially two downstream firms, i ∈ {0, k}, that transform one unit of input into one
unit of a final good. Downstream firm i produces at constant marginal cost ki and
with sunk fixed cost Fi . Without loss of generality, we assume that downstream firm
0 produces with lower marginal cost than downstream firm k. Specifically, we set
kk = k > 0 = k0 so that k denotes the production cost disadvantage of downstream
firm k. The fixed costs are weakly positive, Fi ≥ 0, and downstream firm 0 can
have higher or lower fixed cost than firm k. Let F := F0 − Fk denote the fixed
cost advantage of downstream firm k, which may be a disadvantage; i.e. F may be
5
negative.
We posit that the downstream firms serve independent markets. Both markets
are symmetric and characterized by the same inverse demand function P (q), which
is strictly decreasing and three times continuously differentiable where P > 0.
Moreover, we impose the standard assumptions that P (0) > K + k and P 0 (q) <
min{0, −qP 00 (q)} where P > 0.7 Additionally, we impose the following assumption
that guarantees that the optimal wholesale tariffs are always well defined.
Assumption 1. Downstream marginal revenue is concave, i.e., 3P 00 (q) + qP 000 (q) ≤
0, whenever P > 0.
The sequence of events is as follows: First, M makes a take-it-or-leave-it offer to
each downstream firm.8 Under price discrimination, M offers each downstream firm
a possibly different two-part wholesale tariff Ti (q) = Li + wi q, where q ≥ 0 denotes
the quantity of the input, wi is the unit wholesale price, and Li represents a fixed fee.
Under uniform pricing, on the other hand, the same two-part wholesale tariff applies
to both downstream firms, Ti = T . Thus, upon accepting M ’s offer, downstream
firm i’s effective marginal cost is ci = wi + ki . In stage two, after observing the
contracts offered by M , each downstream firm i decides whether to become active
in the downstream market by paying the fixed cost Fi .9 In stage three, all active
firms in the downstream industry purchase a non-negative quantity of the input
from M and decide how much of this input to transform into the final good and to
sell to consumers. We assume free disposal so that in equilibrium w ≥ 0 and all
acquired units of the input are transformed into the final good.
Before proceeding with the analysis, we investigate the benchmark of an integrated
structure comprising of manufacturer M and downstream firm i. The integrated
structure generates a joint surplus (JS)—exclusionary of the fixed cost—of
π JS (ki ) ≡ max{q[P (q) − ki − K]}
q≥0
by selling the amount q JS (ki ) ≡ arg maxq≥0 {q[P (q) − ki − K]}. Note that 0 <
π JS (k) < π JS (0).
For the sake of conciseness, we restrict attention to situations where both downstream firms are served in equilibrium under either pricing regime. The following
assumptions are sufficient to ensure this.
7
See, for example, Vives (1999).
As argued by Inderst and Shaffer (2009, p. 660), the assumption of the manufacturer having
all the bargaining power “can be justified on the grounds that for antitrust purposes the considerations of price discrimination in intermediate-goods markets is primarily relevant if the
supplier enjoys a dominant position.”
9
We abstract from any commitment problems and assume that M can credibly commit to the
tariffs quoted in this first stage.
8
6
Assumption 2. The following holds:
(i) π JS (ki ) > Fi for all i ∈ {0, k};
(ii) π JS (0) − π JS (k) − [π JS (k) − Fk ] < F < π JS (0) − π JS (k) + [π JS (0) − F0 ].
Assumption 2(i) ensures that the joint surplus in any market exceeds the necessary
fixed cost in that market; i.e., it is efficient that a firm is active in any market. This
assumption ensures that under price discrimination both markets are served. Under
uniform pricing it is optimal to serve both firms if they do not differ by too much
regarding their efficiency. In particular, for a given marginal cost advantage k, the
fixed cost advantage F = F0 − Fk can neither be too small nor too large. This is
ensured by Assumption 2(ii). If Assumption 2(ii) is violated, we may observe the
usual entry promoting effect of price discrimination, which in turn improves welfare
according to the classic Chicago School argument (Bork, 1978). It is important to
note that Assumption 2 allows for negative F . In particular when the maximum
surplus from serving market k, π JS (k)−Fk , is large and the marginal cost advantage
is low, such that π JS (0) − π JS (k) is small, then the lower bound on F is negative.
The equilibrium concept employed is subgame-perfect Nash equilibrium in pure
strategies. As a tie-breaking rule, we assume that M serves both downstream firms
when indifferent between serving only one or both downstream firms. Lastly, superscripts d and u refer to the discriminatory and the uniform pricing regime, respectively.
3. Optimal Wholesale Pricing
Letting
q(c) ≡ arg max{q[P (q) − c]},
q≥0
(1)
an active firm that produces with effective marginal costs c in stage three realizes
profits π(c) − L, where
π(c) ≡ q(c)[P (q(c)) − c].
(2)
Both q(c) and π(c) are strictly decreasing in c where q(c) > 0.10 Moreover, by
Assumption 1, q 00 (c) ≤ 0.
Firm i will accept M ’s offer in stage one only if its profits are non-negative, i.e.,
if
π(ci ) − Li ≥ Fi .
10
Note that π JS (ki ) ≡ π(ki + K) for all ki ∈ {0, k}.
7
(PCi )
The above participation constraint requires that firm i is able to recoup its fixed cost
under M ’s contract offer. Notice that the manufacturer faces a contracting problem
with type-dependent outside options. This is the crucial difference to the analysis
of Inderst and Shaffer (2009) and drives the diverging results.
3.1. Price Discrimination
If not restricted in her price setting, M solves two independent maximization problems. By offering individualized two-part tariffs, M can perfectly disentangle the
two goals of avoiding double marginalization and rent extraction. Therefore, given
that M prefers firm i ∈ {0, k} to be active, M fully extracts the surplus with the
individualized fixed fee, allowing firm i just to break even. The discriminatory
marginal price then is set to maximize the profits from the integrated structure and
thus equals upstream marginal cost, wid = K. Finally, by Assumption 2(i) we have
Fi < π JS (ki ), such that M indeed prefers both firms to be active.
Lemma 1. Suppose Assumptions 1 and 2 hold. The optimal discriminatory unit
wholesale prices are wkd = w0d = wd = K.
3.2. Uniform Pricing
Under uniform pricing M has to offer the same tariff to both firms. If serving both
firms is optimal, M maximizes
max (w − K)[q(w + k) + q(w)] + 2L
w,L
subject to
(P C0 ), (P Ck ).
(3)
Depending on the differences between the fixed costs, in the optimum either (PCk ),
(PC0 ), or both are binding. Irrespective of which constraint binds, with the same
two-part tariff being offered to both firms, M cannot extract the generated surplus
via the fixed fee in both markets. The optimal marginal price therefore balances the
trade-off between efficiency and rent extraction.
Suppose that only the participation constraint of the downstream firm with higher
marginal cost, (PCk ), is binding. In this case, M ’s profit as a function of w is
Π(w) = (w − K)[q(w + k) + q(w)] + 2[π(w + k) − Fk ],
(4)
where Π0 (w) > 0 for w ≤ K and Π00 (w) < 0 for w > K. Hence, the unit price that
maximizes (4), w̄, is larger than marginal cost upstream, w̄ > K, and determined
by the first-order condition. Intuitively, with the participation constraint of the firm
8
with high marginal cost binding, M can extract a larger share of the surplus generated by the firm with low marginal cost by charging a higher unit price than under
price discrimination. With the corresponding wholesale tariff being independent of
firm 0’s fixed cost F0 , (PC0 ) then indeed is satisfied only for low differences in fixed
costs, i.e., if
F ≤ π(w̄) − π(w̄ + k) =: F̄1 .
Next, suppose that only the participation constraint of the firm with low marginal
cost (PC0 ) is binding. The profit of M as a function of w is
Π(w) = (w − K)[q(w + k) + q(w)] + 2[π(w) − F0 ],
(5)
where Π0 (w) < 0 for all w ≥ K. Thus, the unit price that maximizes (5), w, is
¯
lower than the upstream marginal cost of production, w ∈ [0, K). By hypothesis,
¯
(PC0 ) is binding. Lowering the unit wholesale price below marginal cost relaxes this
participation constraint but reduces the rents that can be extracted from k. With
firm 0 having lower marginal costs, the increase in the fixed fee which is associated
with the decrease in the unit wholesale price, more than compensates M for the
reduction in profits from contracting with k. Given unit price w and the associated
¯
fixed fee—which is independent of Fk —(PCk ) is satisfied only for high differences in
fixed costs, i.e., if
F ≥ π(w) − π(w + k) =: F̄2 .
¯
¯
0
0
Note that F̄1 < F̄2 because π (x) − π (x + k) = −q(x) + q(x + k) < 0.
For intermediate differences in fixed costs, F ∈ (F̄1 , F̄2 ), both constraints are
binding and the optimal uniform wholesale price, w(F ), is implicitly characterized
by
π(w(F )) − π(w(F ) + k) ≡ F.
(6)
Implicit differentiation of equation (6) with respect to F yields
−1
dw(F )
=
< 0.
dF
q(w) − q(w + k)
(7)
Moreover, notice that w(F ) = K for
F = π JS (0) − π JS (k) =: F̄.
Lemma 2. Suppose Assumptions 1 and 2 hold. Then, the optimal uniform unit
wholesale price exceeds marginal cost upstream if and only if the difference between
fixed costs is sufficiently low; i.e.,
wu R K iff F Q F̄.
9
wu
wd
K
wu
0
F̄1
F̄
F = F0 − Fk
F̄2
Figure 1: Optimal wholesale prices.
It is important to note that F̄ is between the upper and lower bound on F provided
by Assumption 2(ii). In other words, all three cases of Lemma 2 indeed occur under
the considered parameters. The unit wholesale price wu as a function of F is depicted
in Figure 1, which also depicts the unit wholesale price under price discrimination.
Figure 1 facilitates the welfare analysis, which is conducted next.
4. Welfare Analysis
Welfare is defined as the unweighted sum of consumer and producer surplus. Welfare
and consumer surplus in market i ∈ {0, k} under pricing regime r ∈ {d, u} is denoted
by Wir and CSir , respectively. If the firm in market i is active under both pricing
regimes, then the change in welfare and consumer surplus due to a regime shift from
uniform pricing to price discrimination amounts to
∆Wi ≡
Wid
−
Wiu
Z
q(wd +ki )
=
P (x)dx − (ki + K) q(wd + ki ) − q(wu + ki ) (8)
q(wu +ki )
and
∆CSi ≡ CSid − CSiu
Z q(wd +ki )
Z
d
=
[P (x) − P (q(w + ki ))]dx −
0
q(wu +ki )
[P (x) − P (q(wu + ki ))]dx, (9)
0
respectively. Accordingly, the overall change in welfare and consumer surplus due
P
P
to a regime shift is ∆W = i∈{0,k} ∆Wi and ∆CS = i∈{0,k} ∆CSi , respectively.
10
By Lemmas 1 and 2, the unit wholesale price is higher under price discrimination
than under uniform pricing if differences in fixed costs are relatively large (see Figure
1). With higher wholesale prices translating into higher final good prices and thus
lower quantities, it seems reasonable to expect that permitting price discrimination
harms welfare and consumer surplus if differences in fixed costs are high. For consumer surplus this is trivial because consumer surplus in market i is decreasing in
the final-good price P (q(wr + ki )) in that market. For total welfare, on the other
hand, the argument is somewhat more involved because higher unit prices lead to
lower quantities sold and thus reduce production costs. However, even if wu < K,
the resulting price for final consumers is always at least as large as joint marginal
costs—i.e., P (q(wu +ki )) ≥ ki +K. In other words, quantities under uniform pricing
are never too high from a welfare point of view and thus permitting price discrimination harms welfare if wd > wu . Notice that if P (q(wu + ki )) < ki + K, joint profits
from the contractual relationship of M and i are negative. If this would be the case,
M makes losses by serving firm i and thus would benefit from excluding firm i.11
We are now able to state our main result that—in contrast to the prevailing
opinion in the literature—banning price discrimination may actually improve welfare
even under nonlinear wholesale tariffs.
Proposition 1. Suppose Assumptions 1 and 2 hold. Then, permitting price discrimination reduces welfare and consumer surplus if and only if the difference in
fixed costs is large, i.e., ∆W < 0 and ∆CS < 0 if and only if F > F̄ .
The above proposition confirms the finding of Inderst and Shaffer (2009) for low
differences in the fixed costs. For high differences in the fixed costs, however, we
obtain the opposite welfare finding: A ban on price discrimination improves welfare
and consumer surplus. The reason is that in Inderst and Shaffer (2009) the fixed
fee is always determined by the profits made by the firm with high marginal cost,
whereas in our model—due to the type-dependent fixed cost—the fixed fee may be
determined by the profits made by the firm with low marginal cost.
At first glance, the above proposition seems to be similar to the findings of Inderst
and Valletti (2009) and Herweg and Müller (2012), who allow for type-dependent
outside options but focus on linear wholesale prices.12 Both these contributions
find that, irrespective of the pricing regime, if the outside option imposes a binding
11
Instead of offering a two-part tariff so that it makes losses in market i, M could offer w = K
and L = π JS (kj ) − Fj with j 6= i. This contract extracts the maximum profit from market j
and is rejected by firm i.
12
While Herweg and Müller (2012) posit that one downstream firm has to incur a fixed entry cost
in order to become active in the downstream market, Inderst and Valletti (2009) assume that
downstream firms have access to an alternative source of supply.
11
restriction, then the linear wholesale price is completely determined by the binding
participation constraint. Under uniform pricing, if the manufacturer wants to serve
both firms, it has to pass on the low constrained wholesale price, which has to be
offered to ensure participation of the firm with lower marginal cost and higher outside
option, also to the firm with higher marginal cost and a lower outside option. For the
case of separate markets this immediately implies that a ban on price discrimination
improves welfare. Importantly, uniform pricing leads to higher welfare due to a lower
wholesale price only in one market, while welfare in the other market is unaffected.
In our model, with two-part tariffs, in the cases where a ban on price discrimination
improves welfare, the unit wholesale price is lower in both markets under uniform
pricing and thus both markets benefit from a ban on price discrimination. This is a
crucial difference to the literature that focuses on linear pricing.
5. Application with Linear Demand
We now illustrate our findings for the case of linear demand, i.e., P (q) = 1 − q for
q ∈ [0, 1]. Assumption 2, which ensures that both downstream firms are always
served in equilibrium, translates into:
Assumption 3.
(i)
(ii)
1
(1 − K − k)2 ≥ Fk and 41 (1 − K)2 ≥ F0 ;
4
− 14 (1−K)2 +k(1−K)− 21 k 2 +Fk < F0 −Fk
< 14 (1−K)2 + 21 k(1−K)− 14 k 2 −F0 .
Furthermore, we assume that k < 2K, which ensures that free disposal never
imposes a binding restriction. Notice that the assumption that joint surplus in each
market is positive implies that 1 − K − k > 0.
The optimal quantity produced by a retailer with effective marginal cost c and
the associated profits are given by q(c) = 21 (1 − c) and π(c) = 41 (1 − c)2 , respectively.
Price discrimination: The optimal tariffs under price discrimination follow immediately from Lemma 1. Moreover, by Assumption 3(i), it is optimal for the
manufacturer to contract with both downstream firms.
Observation 1. Suppose Assumption 3 holds and price discrimination is permitted.
Then, the optimal wholesale contracts specify wkd = w0d = K, Lk = 14 (1−K −k)2 −Fk ,
and L0 = 41 (1 − K)2 − F0 . The prices for final consumers in the two markets are
pdk = 21 (1 + k + K) and pd0 = 12 (1 + K).
12
Uniform pricing: If price discrimination is banned and the manufacturer contracts
with both downstream firms, the optimal uniform wholesale contract solves
1
1
max
(w − K) (1 − w − k) + (1 − w) + 2L
w,L
2
2
(10)
subject to
1
(1 − w − k)2 − L ≥ Fk
(PCk )
4
1
(1 − w)2 − L ≥ F0
(PC0 )
4
Depending on how large the difference in fixed costs is, either only (PCk ) or only
(PC0 ) or both constraints are binding. Provided that Assumption 3(ii) holds, it is
indeed optimal for the manufacturer to contract with both downstream firms.
Observation 2. Suppose Assumption 3 holds and price discrimination is banned.
Then, the optimal unit wholesale price

1

 K + 2k
wu =
1 − 12 k − 2F
k


1
K − 2k
is given by
if F ≤ k2 (1 − K − k) =: F̄1
if F̄1 < F < F̄2
(11)
if F ≥ k2 (1 − K) =: F̄2
The optimal fixed fee is
 1
3 2

 4 [1 − K − 2 k] − Fk
1 2F
Lu =
[ − 12 k]2 − Fk
4 k

 1
[1 − K + 12 k]2 − F0
4
The prices for final consumers are
 1
3

 2 (1 + K + 2 k)
1
puk =
(2 + 12 k − 2F
)
2
k

 1
(1 + K + 12 k)
2
if F ≤ F̄1
if F̄1 < F < F̄2
(12)
if F ≥ F̄2
if F ≤ F̄1
if F̄1 < F < F̄2
(13)
if F ≥ F̄2
and
 1
1

 2 (1 + K + 2 k)
1
pu0 =
(2 − 12 k − 2F
)
2
k

 1
1
(1 + K − 2 k)
2
if F ≤ F̄1
if F̄1 < F < F̄2
(14)
if F ≥ F̄2
For the subsequent welfare comparison it is important to note that the prices
faced by final consumers are always too high from a welfare point of view. For the
sake of the argument, consider the price charged in market 0 for the case of high
differences in the fixed cost, pu0 =
1
[1
2
+ K − 21 k], which is the lowest final good
price possibly charged. The welfare optimal price equals the joint marginal cost of
production, which in market 0 is pW
0 = K. For all k so that the joint surplus in
market k is positive it holds that pu0 > pW
0 .
13
Welfare: From the previous analysis we know that ∆W < 0 and ∆CS < 0 if and
only if wd > wu . Thus, the next result—which is a corollary of Proposition 1—is
readily obtained from Observations 1 and 2.
Corollary 1. Suppose Assumption 3 holds. Permitting price discrimination reduces
welfare and consumer surplus if and only if the difference in the fixed cost is high;
i.e. ∆W < 0 and ∆CS < 0 if and only if F > k2 (1 − K − 12 k) =: F̄ .
6. Downstream Competition
So far, with downstream firms operating in separate markets, we dealt with the
important case of geographic price discrimination.13 Nevertheless, major legal enactments, like the Robinson-Patman Act in the US and Article 102(c) TFEU in the
EU, treat price discrimination as an infringement of the law only if a downstream
firm is placed at a competitive disadvantage. Therefore it seems warranted to prove
that our findings are robust towards introducing competition downstream. Suppose both downstream firms compete in the same market but produce differentiated
goods. The utility of a representative consumer is given by
U = qk + q0 −
1
(q 2 + q02 + 2γqk q0 ) + x,
2(1 − γ 2 ) k
with 0 ≤ γ < 1 measuring the degree of product differentiation and x denoting
the consumption of a numraire good. Maximizing U yields the demand for each
downstream firm’s product as a function of both retail prices. The demand for
product i ∈ {0, k} is given by
d(pi , pj ) = α − pi + γpj ,
(15)
where α = 1−γ. For γ = 0 we are back in the case of downstream firms operating in
separate markets. More precisely, if γ = 0 and thus α = 1, the model is equivalent
to the linear demand example discussed in Section 5.14
Downstream firms compete la Bertrand and set prices. The wholesale tariffs are
observable, i.e., each downstream firm knows not only its own but also its rival’s
effective marginal cost when setting the price for its own final good. We impose
13
In the EU, geographic price discrimination is per se prohibited. The reason is that price discrimination along national borders is regarded as incompatible with one of the most basic primitives
of EU competition policy, which is to promote free movement of goods and services within a
common market.
14
A more detailed description of this representative consumer approach is provided by Inderst and
Shaffer (2009). They investigate the welfare effects of a ban on price discrimination for the
special case F0 = Fk = 0.
14
additional restrictions on k and K that ensure that both the quantities and the unit
wholesale prices in equilibrium are strictly positive. These assumptions also imply
that the two downstream firms’ products are not too close substitutes, i.e., that γ
is not too large. Otherwise, it would be optimal for the manufacturer to contract
only with a single downstream firm—i.e., only with the firm that is overall more
efficient.15
Solving for the Nash equilibrium prices, quantities, and profits of downstream
firm i ∈ {0, k} as functions of the effective marginal costs yields
α(2 + γ) + 2ci + γcj
,
4 − γ2
α(2 + γ) − (2 − γ 2 )ci + γcj
qi = q(ci , cj ) =
4 − γ2
(16)
pi = p(ci , cj ) =
(17)
and
πi = π(ci , cj ) =
α(2 + γ) − (2 − γ 2 )ci + γcj
4 − γ2
2
,
(18)
respectively.
The manufacturer solves the following maximization problem:
max
{w0 ,wk ,L0 ,Lk }
(wk − K)q(wk + k, w0 ) + (w0 − K)q(w0 , wk + k) + Lk + L0
subject to
π(wk + k, w0 ) − Lk ≥ Fk
(PCk )
π(w0 , wk + k) − L0 ≥ F0
(PC0 )
Under uniform pricing the manufacturer faces the additional constraints wk = w0 =
w and Lk = L0 = L. In this section we presume that it is optimal for the manufacturer to serve both downstream firms.
Price discrimination: Under price discrimination both constraints, (PCk ) and
(PC0 ), always bind. Solving the above problem yields the optimal discriminatory
15
Formally, we assume that
k < min
2(2 − γ − γ 2 )(1 − K) 2(2 + γ)[γ + K(2 − γ)]
,
6 + γ − 3γ 2
4 + 4γ − γ 2
.
For fixed marginal cost parameters k and K, this condition imposes an upper bound on γ. The
precise role that this assumption plays becomes apparent in the proof of Proposition 2.
15
unit wholesale prices:
αγ + K(2 − 3γ + γ 2 )
2(1 − γ)
γ[α − k(1 − γ)] + K(2 − 3γ + γ 2 )
=
.
2(1 − γ)
wkd =
(19)
w0d
(20)
Note that wkd > w0d , i.e., the downstream firm with low marginal cost receives a
discount under price discrimination.
Uniform pricing: Under uniform pricing, as before, either only (PCk ) or only (PC0 )
or both constraints are binding. Which of the three cases arises depends on the
difference in fixed costs between the two downstream firms, F = F0 − Fk .
For low differences in the fixed costs only (PCk ) is binding. The optimal unit
wholesale price in this case is given by
w̄ =
αγ
(4 − 3γ 2 )k (2 − γ)K
+
+
.
2(1 − γ)
4(2 + γ)
2
(21)
It is important to note that w̄ > wkd > w0d . Thus, for low values of F , banning price
discrimination leads to higher unit wholesale prices for both downstream firms.
For high differences in the fixed costs, only (PC0 ) is binding and the optimal unit
wholesale price is
w=
¯
2(2 + γ)[αγ + K(2 − 3γ + γ 2 )] − k(4 − 5γ 2 + γ 3 )
.
4(2 − γ − γ 2 )
(22)
Tedious but straightforward calculations reveal that w < wEd < wId . Hence, if F is
¯
large, a ban on price discrimination leads to lower unit wholesale prices for both
downstream firms.
For intermediate differences in the fixed costs both participation constraints are
binding. The optimal unit wholesale price w(F ) is implicitly characterized by the
two binding constraints:
π(w(F ), w(F ) + k) − π(w(F ) + k, w(F )) ≡ F.
(23)
Implicit differentiation of (23) yields
dw(F )
4 − γ2
=−
< 0.
dF
2k(1 − γ 2 )
(24)
In the appendix we show that the optimal unit wholesale price under uniform pricing
wu is a continuous function in F . For low values of F we have wu = w̄, which is
independent of F . For intermediate levels of F the unit wholesale price is wu = w(F )
and thus decreasing in F . For high F the unit wholesale price is wu = w and thus
¯
again independent of F .
16
Welfare and consumer surplus: Let qir denote the quantity produced by downstream firm i ∈ {k, 0} under pricing regime r ∈ {d, u}. Formally qir = q(wir +ki , wjr +
kj ) with i 6= j. Consumer surplus under pricing regime r amounts to
CS r ≡ qkr + q0r −
(qkr )2 + (q0r )2 + 2γqkr q0r
− p(wkr + k, w0r ) qkr − p(w0r , wkr + k) q0r .
2(1 − γ 2 )
(25)
Let ∆CS ≡ CS d − CS u . With final good prices being increasing in both effective
marginal costs and the utility of the representative consumer being lower for higher
final good prices, a comparison of consumer surplus under the two pricing regimes
is readily obtained.
Proposition 2. Suppose that it is optimal for the manufacturer to serve both downstream firms. Then, permitting price discrimination reduces consumer surplus if and
only if the difference in fixed cost is sufficiently large; i.e., there exits F̄CS > 0, such
that ∆CS < 0 if and only if F > F̄CS .
As we focus on situations where the manufacturer serves both downstream firms
under either pricing regime, welfare under pricing regime r ∈ {d, u} is
W r ≡ qkr (1 − k − K) + q0r (1 − K) −
(qkr )2 + (q0r )2 + 2γqkr q0r
− F0 − Fk .
2(1 − γ 2 )
(26)
Let ∆W ≡ W d − W u . A welfare comparison is more intricate than the previous
analysis regarding consumer surplus because lower unit wholesale prices for both
downstream firms do not necessarily lead to higher welfare. With downstream competition the allocation of the total quantity sold matters; i.e., how much is produced
by the firm with lower marginal costs. Under price discrimination the firm with
lower marginal costs obtains a discount and thus produces relatively more than the
firm with higher costs compared to the case of uniform pricing. This positive effect
of price discrimination on allocative efficiency may make a ban on price discrimination undesirable from a welfare point of view in cases where this ban is desirable
when a consumer standard is applied.
In the following we discuss the welfare effects of a ban on price discrimination
in more detail. It is important to note that under price discrimination—with both
participation constraints binding—the manufacturer maximizes aggregate producer
surplus; i.e., industry profits are maximized. Under uniform pricing the manufacturer is restricted and industry profits are not maximized, even though downstream
firms may make a positive profit. Thus, if price discrimination is beneficial for consumer surplus, then it also improves social welfare; i.e., if ∆CS > 0, then ∆W > 0.
17
On the other hand, if price discrimination is harmful for consumer surplus, it still
improves industry profits. Now, a welfare analysis is less clear cut due to opposing
effects. In the appendix, we show that the effect on consumer surplus is larger than
the effect on industry profits if only (PC0 ) is binding under uniform pricing. In order
to derive this result we rely on numerical methods and the details are provided in
the appendix. Observing that W u is decreasing in the uniform unit wholesale price
and thus increasing in F , allows us to establish the next result.
Proposition 3. Suppose that it is optimal for the manufacturer to serve both downstream firms and that k < 2[1 − K 2−γ
]. Then, permitting price discrimination
1−γ
reduces welfare if and only if the difference in fixed cost is sufficiently large; i.e.,
there exits F̄W ≥ F̄CS , such that ∆W < 0 if and only if F > F̄W .
Propositions 2 and 3 confirm our previous findings regarding welfare and consumer surplus for the case of downstream competition. The novelty compared to
Proposition 1 is that the threshold of F may be lower for a consumer standard than
a total welfare standard.16 Propositions 2 and 3 provide a justification for nondiscrimination clauses when markets are concerned where firms with low marginal
costs tend to have high fixed costs. This holds true in particular if the welfare
measure is consumer surplus—as with the consumer standard in the EU.17
7. General Nonlinear Tariffs
So far, we presumed that the wholesale contracts offered by the manufacturer take
the form of a two-part tariff. While this is without loss in generality under price
discrimination, it imposes a binding restriction on the manufacturer under uniform
pricing. In this section, we extend the model of Section 2 by allowing for general
nonlinear tariffs, thereby investigating robustness of our previous welfare findings.
Under each pricing regime there is an optimal mechanism that belongs to the class
of menus of quantity-transfer pairs (q, t). If a downstream firm selects pair (q, t) out
of the offered menu, it receives amount q of the input and has to pay transfer t.
16
With independent downstream markets, all three unit wholesale prices (the two discriminatory
and the one uniform unit wholesale price) coincide for F = F̄ . This implies that industry profits
under the two pricing regimes are identical for F = F̄ . With downstream competition, the
three prices never coincide because wkd > w0d . Thus, industry profits are always strictly higher
under price discrimination than under uniform pricing. This explains why with downstream
competition the two thresholds, F̄W and F̄CS , typically do not coincide.
17
Interestingly, the qualitative results of Proposition 2 and 3 do not depend on the degree of
competition/ product differentiation γ. The thresholds F̄W and F̄CS , however, are functions of
γ. We are able to establish the existence of these thresholds but do not solve for their explicit
expressions.
18
Under price discrimination the manufacturer can offer different menus to the two
downstream firms. Under uniform pricing, on the other hand, the manufacturer has
to offer the same menu to both downstream firms.
Under price discrimination the menu offered to downstream firm i contains only
a single quantity-transfer pair with qid = q JS (ki ) and tdi = π JS (ki ) − Fi .
Under uniform pricing the offered menu contains at most two quantity-transfer
pairs, one designated to firm k and the other to firm 0. The manufacturer has to ensure that each downstream firm selects the quantity-transfer pair that it is supposed
to choose; i.e., when designing the menu, the manufacturer has to take into account incentive compatibility constraints just as in a screening problem. Therefore,
despite asymmetric information being absent, under uniform pricing the manufacturer faces a two-type screening problem with type-dependent outside options.18 If
only the participation constraint of firm k and the incentive constraint of firm 0 are
binding, we obtain the usual no-distortion-at-the-top and downward-distortion-atthe-bottom result. Firm 0 obtains the joint surplus maximizing quantity, whereas
the quantity procured by firm k is distorted downwards compared to the joint surplus maximizing quantity. This case occurs for low differences in the fixed costs. For
high differences in the fixed costs, on the other hand, the only binding constraints
are the participation constraint of firm 0 and the incentive constraint of firm k. In
this case there is no distortion at the bottom but an upward distortion in quantity
at the top; i.e., firm k obtains the joint surplus maximizing quantity and firm 0
receives a quantity that exceeds the joint surplus maximizing quantity.
A formal analysis can be found in Appendix B. The main conclusion of this
analysis is summarized in the following result.
Proposition 4. Suppose Assumptions 1 and 2 hold and that K ≥ k. Then:
(i) If
F
<
k
JS
q JS (k), then ∆W > 0 and ∆CS > 0.
(ii) If q (k) ≤
JS
(iii) If q (0) <
F
≤ q JS (0), then
k
F
, then ∆W < 0
k
∆W = 0 and ∆CS = 0.
and ∆CS < 0.
The above proposition reveals that the qualitative welfare findings of Proposition
1 do not hinge on the restriction to two-part tariffs. The only qualitative difference
is that with menus of quantity-transfer pairs there exists a range of F values where
both pricing regimes lead to the same allocation.
18
For a textbook treatment of a two-type screening model with type-dependent outside options
see Laffont and Martimort (2002).
19
8. Conclusion
In the extant literature on third-degree price discrimination in intermediate-goods
markets, several contributions that allow for nonlinear wholesale tariffs voice concerns about the efficacy of the Robinson-Patman Act or its analogue in EU competition law. In contrast to this opinion, we have shown that the reservation toward
discriminatory pricing embodied in these legal enactments may well be warranted
even under nonlinear wholesale tariffs if the downstream production process involves
a fixed cost component.
We find that uniform pricing leads to lower marginal wholesale prices for all downstream firms than price discrimination if the individual rationality constraint of the
downstream firm with the lower marginal cost is binding due to a substantial fixed
cost disadvantage. This fixed cost disadvantage may arise, for example, if the downstream firm with the lower marginal cost is a potential entrant who has to incur
an entry fee (e.g., the acquisition of a business license), whereas the downstream
firm with the higher marginal cost is already incumbent in the downstream industry. Alternatively, if the fixed cost is interpreted as a downstream firm’s outside
option, the fixed cost disadvantage might as well reflect that backward integration
or demand-side substitution leads to higher profits for the downstream firm with
the lower marginal costs than for the one with higher marginal costs. Thus, several scenarios may lead to the fixed fee under uniform pricing being determined by
the individual rationality constraint of the downstream firm with lower marginal
cost, which implies that marginal wholesale prices are lower for all firms under uniform pricing than under price discrimination. These lower marginal wholesale prices
translate into higher welfare and consumer surplus under uniform pricing.19
Our analysis focused on the welfare effects of a ban on price discrimination in the
short run. What are the welfare effects in the long-run, i.e., if downstream firms can
invest in a reduction of their costs before the manufacturer makes its offers?20 For
the sake of the argument, imagine that the downstream firm with lower marginal cost
can invest resources in order to reduce its fixed cost component. Moreover, suppose
that its initial fixed cost is sufficiently low, such that without investments a ban
on price discrimination is detrimental for welfare and consumer surplus. The firm
with lower marginal cost then has no incentives to invest in a reduction of its fixed
19
While Katz (1987) allows for backward integration and Inderst and Valletti (2009) allow for
demand-side substitution via a competitive fringe, both contributions focus on linear pricing,
whereas our focus is on nonlinear wholesale tariffs.
20
Such a long-run analysis was first conducted by DeGraba (1990), who showed that a ban on
price discrimination does not only improve welfare in the short run but also in the long run by
enhancing downstream firms’ investment incentives.
20
costs under price discrimination, where its individual rationality constraint always is
binding. It may invest a positive amount under uniform pricing, however, where its
individual rationality constraint is slack. Hence, uniform pricing can lead to lower
fixed costs which often improves welfare. Consumers, however, are still harmed by
a ban on price discrimination. As an alternative scenario, one can imagine that
the downstream firm with higher marginal cost can invest in a reduction of its
marginal cost. Again, under price discrimination there is no incentive to invest in
cost reduction, whereas making such an investment might be optimal under uniform
pricing. The investment reduces that downstream firm’s effective marginal cost
and increases the quantity it sells to final consumers. Hence, uniform pricing will
typically increase consumer surplus and total welfare in the long run. A detailed
analysis of the long-run welfare effects of a ban on price discrimination is left for
future research.
A. Proofs of Propositions and Lemmas
Proof of Lemma 1. The manufacturer solves two independent maximization problems. When contracting with downstream firm k and 0, the fixed fee is given by
Lk = π(wk + k) − Fk and L0 = π(w0 ) − F0 , respectively. Thus, the contract M offers
to downstream firm i solves:
max (wi − K)q(wi + ki ) + π(wi + ki ) − Fi
wi
(A.1)
By noting that π 0 (·) = −q(·), from the first-order condition of profit maximization
it follows immediately that wid = K. Moreover, by Assumption 1, M ’s profit as a
function of wi is strictly concave and thus the first-order condition is necessary and
sufficient.
Lastly, M strictly prefers both downstream firms to be active: M ’s profit from
contracting with firm k and 0 is π JS (k) − Fk and π JS (0) − F0 , respectively. Both
terms are strictly positive by Assumption 2(i). Thus, it is more profitable for M to
serve both downstream firms than contracting with only one firm.
Proof of Lemma 2. If it is optimal to serve both downstream firms under uniform
pricing, then M solves the following program:
max
w≥0,L
Π := (w − K)[q(w) + q(w + k)] + 2L
21
subject to
π(w + k) − L − Fk ≥ 0,
(PCk )
π(w) − L − F0 ≥ 0.
(PC0 )
We distinguish three cases. First, suppose that only (PCk ) is binding in optimum.
Thus, the optimal uniform wholesale price maximizes (4). The change in M ’s profit
due to a marginal increase in w is,
Π0 (w) = q(w) − q(w + k) + (w − K)[q 0 (w) + q 0 (w + k)].
(A.2)
Note that Π0 (w) > 0 for w ≤ K. Moreover, Π00 (·) < 0 for w > K. Thus, the optimal
unit wholesale price is characterized by the first-order condition and larger than
K. Let w̄ be defined by Π0 (w̄) = 0. The participation constraint (PC0 ) is indeed
satisfied if
π(w̄) − [π(w̄ + k) − Fk ] ≥ F0
(A.3)
⇐⇒ F ≤ π(w̄) − π(w̄ + k) =: F̄1 .
(A.4)
Now, suppose only (PC0 ) is binding in the optimum. In this case, M ’s profit is
given by (5). Differentiation of (5) with respect to w yields
Π0 (w) = q(w + k) − q(w) + (w − K)[q 0 (w) + q 0 (w + k)].
(A.5)
Note that Π0 (w) < 0 for all w ≥ K. Moreover, due to free disposal, w is bounded
from below by zero—if w < 0, then any downstream firm could make infinite profits
by ordering an infinite quantity of the input, which would imply infinite losses for
M . Thus, the optimal unit wholesale price in this case satisfies w ∈ [0, K). Under
¯
the corresponding wholesale tariff the participation constraint of k is satisfied if
π(w + k) − [π(w) − F0 ] ≥ Fk
¯
¯
⇐⇒ F ≥ π(w) − π(w + k) =: F̄2 .
¯
¯
(A.6)
(A.7)
With π(x) − π(x + k) being decreasing in x, it follows that F̄1 < F̄2 .
Finally, both constraints can be binding. In this case, the unit wholesale price is
uniquely determined by the two binding constraints and implicitly characterized by
(6). Implicit differentiation of equation (6) with respect to F yields
−1
dw(F )
=
< 0.
dF
q(w) − q(w + k)
(A.8)
Moreover, note that w(F ) = K iff F = π JS (0) − π JS (k) =: F̄ , with F̄1 < F̄ < F̄2 .
This completes the first part of the proof. It remains to be shown that it is indeed
optimal for M to serve both downstream firms.
22
If M decides to serve only one downstream firm, it would like to set w = K and to
extract the whole surplus via the fixed fee. If F = F0 − Fk ≤ F̄ , then the maximum
profit from serving only 0 is at least as large as from serving only k. Since k rejects
the contract w = K and L = π JS (0) − F0 , M generates profits of π JS (0) − F0 .
Likewise, if F = F0 − Fk > F̄ , the maximum profit from serving only k is larger
than from serving only 0. Since 0 rejects the contract w = K and L = π JS (k) − Fk ,
M generates profits of π JS (k)−Fk . In the following we show that M strictly benefits
from serving both firms in both cases. We do so by deriving a lower bound on the
profits from serving both downstream firms. If F ≤ F̄ , the contract with w = K
and L = π JS (k) − Fk is accepted by both firms.21 Hence, M prefers to serve both
firms if its profit from offering this (non-optimal) tariff exceeds its profits from the
optimal tariff if it serves only firm 0—i.e., if
2[π JS (k) − Fk ] ≥ π JS (0) − F0
⇐⇒ F = F0 − Fk ≥ π JS (0) − π JS (k) − [π JS (k) − Fk ].
(A.9)
(A.10)
By Assumption 2(ii), (A.10) is always satisfied. Likewise, for F > F̄ , the contract
w = K and L = π JS (0) − F0 is accepted by both firms. If M ’s profit under this
(non-optimal) tariff is higher than the profit obtained from contracting only with k,
then serving both firms is optimal. This is the case if
2[π JS (0) − F0 ] ≥ π JS (k) − Fk
⇐⇒ F = F0 − Fk ≤ π JS (0) − π JS (k) + [π JS (0) − F0 ],
(A.11)
(A.12)
which holds by Assumption 2(ii).
Proof of Proposition 1. Consumer surplus in market i is a strictly decreasing function in wi . Thus, from Lemmas 1 and 2 it follows immediately that






 > 

 < 

∆CS
= 0 if F
= F̄.








<
>
(A.13)
Now, we investigate the change in welfare due to a regime shift. If F = F̄ and
thus wu = wd = K, then ∆W = 0. For F < F̄ we have wu > wd = K and thus
q JS (ki ) = q(wd + ki ) > q(wu + ki ) for all i ∈ {0, k}. Moreover, q JS (·) is too low
from a welfare point of view, i.e., welfare in market i is strictly increasing in q for
q ≤ q JS (ki ). Thus, ∆W > 0 if F < F̄ .
21
Under this contract both downstream firms produce a positive amount. Thus, if this contract
is better than the optimal contract from serving only firm 0, the optimal contract is such that
both firms produce a strictly positive amount.
23
For F > F̄ , we have wu < wd = K and thus q(wu + ki ) > q(wid + ki ) = q JS (ki )
for all i ∈ {0, k}. Thus,
Z
q(wu +ki )
P (x)dx + (ki + K) q(wu + ki ) − q JS (ki )
∆Wi = −
q JS (ki )
< − q(wu + ki ) − q JS (ki ) {P (q(wu + ki )) − (ki + K)} < 0, (A.14)
where the last inequality holds because P (q(wu +ki ))−(ki +K) > 0 for i ∈ {0, k}. To
see this, remember that for F > F̄ we have wu < K and π JS (k) − Fk > π JS (0) − F0 .
Hence, in neither market M makes profits as large as π JS (k) − Fk . Therefore, if
profits in any market are less or equal than zero, then M makes profits strictly less
than π JS (k) − Fk and it could profitably deviate to serving only firm k by offering
w = K and L = π JS (k) − Fk , thereby making profits equal to π JS (k) − Fk . Hence,
M ’s profits in both markets have to be strictly positive. These profits, in turn, are
bounded above by the profits made by an integrated structure comprising of M and
i when selling quantity q(wu + ki ), which are given by
q(wu + ki )[P (q(wu + ki ) − (ki + K)].
(A.15)
These profits thus have to be strictly positive, which requires P (q(wu + ki )) − (ki +
K) > 0—just as claimed above.
Proof of Observation 1. The result is readily obtained from Lemma 1.
Proof of Observation 2. From the proof of Lemma 2 we know that it is optimal
for M to serve both downstream firms under the imposed assumptions. Thus, the
manufacturer solves
max
w,L
1
1
(w − K) (1 − w − k) + (1 − w) + 2L
2
2
(A.16)
subject to
1
(1 − w − k)2 − L ≥ Fk
4
1
(1 − w)2 − L ≥ F0
4
(PCk )
(PC0 )
In order to solve the manufacturer’s problem, three cases can be distinguished.
Case I: F is low and thus only (PCk ) is binding.
The manufacturer’s profit can be written as
1
1
1
ΠI (w) = (w − K) (1 − w − k) + (1 − w) + (1 − w − k)2 − 2Fk .
2
2
2
24
(A.17)
From the first-order condition the optimal wholesale price is readily obtained
1
w̄ = K + k.
2
The imposed assumptions ensures that w̄ + k < 1 and thus q(w̄ + k) > 0. The
optimal fixed fee is
1
L̄ =
4
2
3
1 − K − k − Fk .
2
The participation constraint of firm 0 is indeed satisfied if and only if F ≤ F̄1 , where
2
2
1
1
1
3
1
1
F̄1 =
1−K − k −
1 − K − k = (1 − K)k − k 2 .
4
2
4
2
2
2
Given Assumption 3(ii), we then must have
F̄1 > −
k2
1−K
2Fk
(1 − K)2
+ k(1 − K) −
+ Fk ⇐⇒ k <
−
.
4
2
2
1−K
As this implies k < 23 (1 − K), we have qku =
1−k−wu
2
> 0.
Case II: F is large and thus only (PC0 ) is binding.
The manufacturer’s profit is
1
1
1
ΠII (w) = (w − K) (1 − w − k) + (1 − w) + (1 − w)2 − 2F0 .
2
2
2
(A.18)
Thus, the optimal wholesale price is
1
w=K− k
¯
2
and the optimal fixed fee is
1
L=
¯
4
2
1
1 + k − K − F0
2
Firm k’s participation constraint holds if and only if F ≥ F̄2 , where
2
2
1
1
1
1
1
F̄2 =
1−K + k −
1 − K − k = (1 − K)k.
4
2
4
2
2
Note that F̄1 < F̄ < F̄2 . Given Assumption 3(ii), we must have
F̄2 <
p
(1 − K)2 k(1 − K) k 2
+
−
− F0 ⇐⇒ k < (1 − K)2 − 4F0 .
4
2
4
As this implies k < 2(1 − K), we have qku =
1−k−wu
2
> 0.
Case III: F is intermediate and thus both constraints, (PCk ) and (PC0 ), are binding.
25
If both constraints are satisfied with equality, we have
1
(1 − k − w)2 − Fk = L
4
1
(1 − w)2 − F0 = L.
4
(A.19)
(A.20)
The above conditions imply that
1
1
(1 − k − w)2 − Fk = (1 − w)2 − F0 .
4
4
Solving for the wholesale price as a function of F = F0 − Fk leads to
1
2F
w(F ) = 1 − k −
.
2
k
The corresponding fixed fee is
2
2
1 2F
1
1 2F
1
L=
− k − Fk =
+ k − F0 .
4 k
2
4 k
2
(A.21)
(A.22)
It can be shown that w(F̄1 ) = w̄ and w(F̄2 ) = w. Finally, given Assumption 3(i),
¯
we have
p
p
k < 1 − K − 2 Fk and 1 − K < 2 F0 ,
√
√
√
√
√
which implies k < 2( F0 − Fk ). As F0 − Fk < F0 − Fk , we thus have
qku =
1−k−wu
2
> 0.
Proof of Corollary 1. First, note that wu is continuous in F and strictly decreasing
in F for F ∈ (F̄1 , F̄2 ). By noting that wu = K for F = k2 [1 − K − k2 ] =: F̄ the result
follows directly from Proposition 1.
Proof of Proposition 2.
Wholesale contracts: The optimal wholesale contracts under price discrimination are fully characterized in the main text. If price discrimination is banned, three
cases need to be distinguished.
First, suppose that (PC0 ) is slack and only (PCk ) is binding, such that L =
π(w + k, w) − Fk . The manufacturer’s choice of w maximizes
Π(w) = (w − K)
2α(2 + γ) − 2w(2 − γ − γ 2 ) − k(2 − γ − γ 2 )
4 − γ2
2
α(2 + γ) − w(2 − γ − γ 2 ) − k(2 − γ 2 )
+2
− 2Fk (A.23)
4 − γ2
From the first-order condition, we obtain the optimal unit wholesale price
w̄ =
αγ
(4 − 3γ 2 )k (2 − γ)K
+
+
.
2(1 − γ)
4(2 + γ)
2
26
(A.24)
For w = w̄, the participation constraint of firm 0 is satisfied if and only if
F ≤ π(w̄, w̄ + k) − π(w̄ + k, w̄) =: F̄1 .
(A.25)
Secondly, suppose that (PCk ) is slack and only (PC0 ) is binding, such that L =
π(w, w + k) − F0 . The manufacturer’s choice of w maximizes
Π(w) = (w − K)
2α(2 + γ) − 2w(2 − γ − γ 2 ) + k(2 − γ − γ 2 )
4 − γ2
2
α(2 + γ) − w(2 − γ − γ 2 ) − kγ
− 2F0 (A.26)
+2
4 − γ2
From the first-order condition, we obtain the optimal unit wholesale price
w=
¯
2(2 + γ)[αγ + K(2 − 3γ + γ 2 )] − k(4 − 5γ 2 + γ 3 )
.
4(2 − γ − γ 2 )
(A.27)
For w = w, the participation constraint of firm k is satisfied if and only if
¯
F ≥ π(w, w + k) − π(w + k, w) =: F̄2 .
¯ ¯
¯
¯
(A.28)
It holds that w < w̄ and F̄1 < F̄2 . To see the latter define L(w) ≡ π(w, w + k) −
¯
π(w + k, w) and note that
L0 (w) = −k
2(2 − γ − γ 2 )(2 + γ − γ 2 )
< 0.
(4 − γ 2 )2
(A.29)
Thirdly, if both constraints (PCk ) and (PC0 ) are binding, the unit wholesale price
solves
π(w, w + k) − π(w + k, w) = F.
(A.30)
The left-hand side of (A.30) is strictly decreasing in w and thus there is a bijective
mapping from F to w. In other words, the optimal unit wholesale price is a function
of F , w = w(F ), and implicitly characterized by (A.30).
From the definitions of F̄1 and F̄2 , (A.25) and (A.28) respectively, it follows that
w(F̄1 ) = w̄ and w(F̄2 ) = w, respectively. These observations together with the fact
¯
that dw(F )/dF < 0—as shown in the text—establish that the unit wholesale price
is a continuous and weakly decreasing function in F .



w̄
if F ≤ F̄1


u
w = w(F ) if F ∈ (F̄1 , F̄2 ) .



w
if F ≥ F̄2
¯
(A.31)
Consumer surplus: We are now ready to establish the result regarding a change
in consumer surplus due to a shift from uniform pricing to price discrimination,
27
∆CS = CS d −CS u . It is important to note that that final good prices are increasing
in both effective marginal costs and that the utility of the representative consumer
is lower for higher prices. For F ≤ F̄1 we have w0d < wkd < wu = w̄ and thus
∆CS > 0. For F ≥ F̄2 we have w = wu < w0d < wkd and thus ∆CS < 0. For
¯
the remaining case, F ∈ (F̄1 , F̄2 ), the argument is somewhat more complicated.
Consumer surplus under price discrimination does not depend on the fixed costs F0
and Fk . Consumer surplus under uniform pricing depends on the fixed costs solely
via the unit wholesale price. The unit wholesale price wu = w(F ) is a function only
of the differences in fixed costs, F = F0 − Fk . Consumer surplus under uniform
pricing is strictly decreasing in the unit wholesale price and thus strictly increasing
in F . We know that wu and thus also ∆CS is continuous across all three cases.
Thus, there exists a critical F̄CS ∈ (F̄1 , F̄2 ) such that ∆CS = 0, which concludes
the proof.
Positive quantities: Finally, we establish that under our parameter restrictions
the demanded quantities by the two downstream firms are always positive. The
lowest procured quantity is the quantity of firm k under unit wholesale price w̄,
q(w̄ + k, w̄). Recall that α = 1 − γ and thus q(w̄ + k, w̄) > 0 if and only if
2(2 − γ − γ 2 )[1 − K] − k(6 + γ − 3γ 2 ) > 0.
(A.32)
For γ ∈ [0, 1] the above condition can be written as
k<
2(2 − γ − γ 2 )
(1 − K).
6 + γ − 3γ 2
(A.33)
The right-hand side is decreasing in γ and thus it has always to hold that k <
2
(1
3
− K) (necessary condition for (A.33) to hold).
Positive unit wholesale prices: Due to free disposal negative unit wholesale
prices cannot be optimal. The lowest unit wholesale price is w. It holds that w > 0
¯
¯
if and only if
2(2 + γ)(1 − γ)γ + K2(2 + γ)(2 − 3γ + γ 2 ) − k(4 − 5γ 2 + γ 3 ) > 0
⇐⇒ k <
2(2 + γ)[γ + K(2 − γ)]
.
4 + 4γ − γ 2
(A.34)
Proof of Proposition 3. As outlined in the text, ∆CS > 0 implies ∆W > 0. Hence,
for F < F̄CS ∈ (F̄1 , F̄2 ) it holds that ∆W > 0.
Next, suppose that F ≥ F̄2 . This implies that under uniform pricing constraint
(PC0 ) is binding and constraint (PCk ) is slack. Let q0r and qkr be the amount produced
28
under pricing regime r ∈ {d, u} by firm 0 and firm k, respectively. Welfare under
pricing regime r amounts to
W r = (1 − K − k)qkr + (1 − K)q0r −
(qkr )2 − (q0r )2 − 2γqkr q0r
− F0 − Fk . (A.35)
2(1 − γ 2 )
First, we address welfare under price discrimination. In equilibrium, the quantities
produced by the two downstream firms are
1
[1 − K − k − γ(1 − K)]
2
1
=
[1 − K − γ(1 − K − k)]
2
qkd =
(A.36)
q0d
(A.37)
Hence, welfare under price discrimination is given by
3 3
3
W d = (1 − K)2 (1 − γ) − k(1 − K)(1 − γ) + k 2 − F0 − Fk .
4
4 8
(A.38)
Under uniform pricing, the quantities produced by the two downstream firms are
qku =
q0u =
1
2
3
8(1
−
K)(1
−
γ)
−
4k
−
2γ
(1
+
k
−
K)
+
γ
(2
+
3k
−
2K)
4(4 − γ 2 )
(A.39)
1
8(1 − K)(1 − γ) + 4k(1 + γ)
4(4 − γ 2 )
− γ 2 (2 + 5k − 2K) + γ 3 (2 + 3k − 2K)
(A.40)
Welfare under uniform pricing is given by
1
8 − 10γ + γ 2 − γ 3
W u = (1 − K)2 (1 − γ) − k(1 − K)
4
4(4 − γ 2 )
48 + 48γ − 8γ 2 + 52γ 3 − 27γ 4 − 34γ 5 + 21γ 6
− F0 − Fk . (A.41)
− k2
16(1 − γ)(4 − γ 2 )2
The change in welfare caused by a regime shift away from uniform pricing to price
discrimination, ∆W ≡ W d − W u , amounts to
−k
∆W =
(1 − K)(64 − 96γ − 48γ 2 + 152γ 3 − 56γ 4 − 32γ 5 + 16γ 6 )
2
2
16(1 − γ)(4 − γ )
2
3
4
5
6
− k(48 − 48γ − 56γ + 100γ − 21γ − 40γ + 21γ ) (A.42)
Numerically it can be shown that the term after k is positive for all γ ∈ [0, 1). To
derive an upper bound on ∆W , we use that k < (4 − 2γ − 2γ 2 )/(6 + γ − 3γ) (see
29
Figure 2: Plot of Γ(γ) for γ ∈ [0, 1].
footnote 15 and the proof of Proposition 2). Inserting the highest feasible k in the
second line of (A.42) leads to
∆W < −
k
Γ(γ),
16(1 − γ)(4 − γ 2 )2
(A.43)
with
Γ(γ) := 192 − 224γ − 352γ 2 + 544γ 3 + 132γ 4 − 386γ 5 + 26γ 6 + 74γ 7 − 6γ 8 .
(A.44)
Numerical simulations reveal that Γ > 0 for all γ ∈ [0, 1), as depicted in Figure 2.
This implies that ∆W < 0 for all γ ∈ [0, 1) if F ≥ F̄2 .
Finally, note that the change in welfare does not depend on the fixed costs, F0
and Fk , per se but only on the difference in fixed cost via the unit wholesale price
under uniform pricing. With the wholesale prices and thus welfare being continuous
at F = F̄1 and F = F̄2 , we can conclude that ∆W is a continuous function in F . If
∆W is monotonic in F , there exists a F̄W



>0


∆W = 0



< 0
∈ (F̄1 , F̄2 ) such that
if F < F̄W
if F = F̄W .
(A.45)
if F > F̄W
Note that from the above discussion it follows immediately that F̄W ≥ F̄CS .
It remains to be shown that W u is a monotonic function in F for F ∈ (F̄1 , F̄2 ).
For F ∈ (F̄1 , F̄2 ) the unit wholesale price wu = w(F ) is strictly decreasing in F . In
the following we will show that welfare under uniform pricing is a strictly increasing
function in the uniform unit wholesale price w, which then completes the proof.
30
For an arbitrary uniform unit wholesale price w, the quantities produced by the
two downstream firms are
γ
1−γ
+
k−
2 − γ 4 − γ2
1 − γ 2 − γ2
qk =
−
k−
2 − γ 4 − γ2
q0 =
1−γ
w
2−γ
1−γ
w.
2−γ
(A.46)
(A.47)
Welfare as function of the uniform unit wholesale price—ignoring the fixed costs F0
and Fk —is
W u (w) = (1 − K)(q0 + qk ) − kqk −
q02 − qk2 − 2γq0 qk
− F0 − Fk .
2(1 − γ 2 )
(A.48)
Inserting the explicit quantities and taking the derivative with respect to w yields
dW
1−γ
1−γ
1
1
=−
2
− 2K + 2w
−k
.
(A.49)
dw
2−γ
2−γ
2−γ
2−γ
Thus dW/dw < 0 for all w ≥ 0 if
K<
1−γ
(2 − k).
4 − 2γ
(A.50)
B. Supplementary Material to “General Nonlinear
Tariffs” (Proof of Proposition 4)
Instead of two-part tariffs, the manufacturer now offers menus of quantity-transfer
pairs to the downstream firms. Here, downstream firms acquire a given quantity at
a pre-specified price, such that they indeed might have an incentive not to convert
all units of the input into the final product. In this regard, the assumption of
free disposal gives leeway in decision-making to the downstream firms and weakens
the position of the manufacturer. In what follows, we initially abstract from free
disposal of the input good; i.e., we consider quantity forcing contracts, under which a
downstream firm sells the same amount of the final consumption good as it acquired
from the input good. In the end, we will show that—under the additional assumption
that K > k—the allocation implemented by the optimal quantity forcing contract
also prevails under free disposal, such that this contract must also be optimal if
downstream firms can freely dispose of the input.
Slightly abusing the notation introduced in the paper, let
π(q, ki ) = q[P (q) − ki ]
31
(B.1)
denote the “gross” profits (i.e., profits before subtracting any transfer payment and
the fixed cost) of a firm from selling quantity q when operating at marginal cost
ki ∈ {0, k}.
Price Discrimination.—If price discrimination is allowed, the manufacturer can
offer each retailer an individualized contract and therefore, effectively, solves two
independent optimization problems. Consider the manufacture’s contract offer to
downstream firm i ∈ {0, k} that operates at marginal cost ki , where k0 = 0 < k = kk ,
and associated fixed cost Fi . The logic of the revelation principle implies that
maximum upstream profits can be achieved by an individually rational contract of
the form {(qi , ti )}. In consequence, the manufacturer solves the following problem:
max ti − Kqi
(qi ,ti )
subject to
(PCi ) π(qi , ki ) − ti ≥ Fi
Under the optimal contract, the (PCi ) constraint has to bind. In consequence, the
manufacturer effectively chooses quantity qi to maximize the profits of a vertically
integrated firm, which leads to the following observation:
Lemma 3. Let qid denote the optimal quantity offered to retailer i ∈ {0, k} under
price discrimination. Then, qid = q JS (ki ).
Uniform Pricing.—If price discrimination is banned, the manufacturer has to offer one and the same contract to both downstream firms. As there are two types
of downstream firms, the logic of the revelation principle implies that maximum
upstream profits can be achieved by an incentive compatible and individually rational menu of the form {(q0 , t0 ), (qk , tk )}, where (qi , ti ) is the quantity-transfer pair
designated to downstream firm i ∈ {0, k}. In consequence, the manufacturer solves
the following problem:
max
(q0 ,t0 ),(qk ,tk )
[t0 − Kq0 ] + [tk − Kqk ]
subject to
(PC0 ) π(q0 , 0) − t0 ≥ F0
(PCk ) π(qk , k) − tk ≥ Fk
(IC0 ) π(q0 , 0) − t0 ≥ π(qk , 0) − tk
(ICk ) π(qk , k) − tk ≥ π(q0 , k) − t0
32
We start with some basic, yet important, observations. First, both (IC0 ) and (ICk )
being simultaneously satisfied implies that the following monotonicity requirement
is satisfied:
qk ≤ q0 .
(MON)
Second, under the optimal contract, either (PCi ) or (ICi ) (or both) has to be binding. Third, if the monotonicity requirement is satisfied and one retailer’s incentive
compatibility constraint binds, then the other retailer’s incentive compatibility constraint is automatically satisfied. Hence, under the optimal contract, at most one
incentive compatibility constraint imposes a binding restriction.
Which constraints actually impose a binding restriction is determined by the ratio
of the difference in fixed costs to the difference in marginal costs,
F
k
with F := F0 −Fk .
We next present a detailed analysis of the cases that have to be distinguished.
Case I: (IC0 ) and (PCk ) bind.
If (IC0 ) and (PCk ) are binding, transfers are given by
t0 = π(q0 , 0) − kqk − Fk
and
tk = π(qk , k) − Fk .
(B.2)
Ignoring (ICk ) and (PC0 ) for the moment, the manufacturer’s problem amounts to
max [π(q0 , 0) − Kq0 ] + [π(qk , k) − (K + k)qk ] − 2Fk
q0 ,qk
(B.3)
From the manufacturer’s objective function it becomes apparent that the optimal
quantity to offer the retailer with low marginal cost is q0I = q JS (0). Differentiation
w.r.t. qk reveals that the optimal quantity to offer the retailer with high marginal
cost is qkI = 0 if P (0) ≤ 2k+K. If P (0) > 2k+K, on the other hand, the retailer with
high marginal cost is offered a strictly positive quantity qkI > 0, which is implicitly
characterized by
P (qkI ) + qkI P 0 (qkI ) = K + 2k.
(B.4)
It remains to check whether the neglected constraints are satisfied. By assumption
the left-hand side of (B.4) is decreasing in q, such that qkI < q JS (k). This, in turn,
implies that the monotonicity requirement (MON) is satisfied. As (IC0 ) binds, (ICk )
then is automatically satisfied. Finally, (PC0 ) is satisfied as long as
F0 ≤ kqkI + Fk ⇐⇒ qkI ≥
F
.
k
(B.5)
Case II: (IC0 ), (PC0 ), and (PCk ) bind.
If (PC0 ) and (PCk ) are binding, transfers are given by
t0 = π(q0 , 0) − F0
and
33
tk = π(qk , k) − Fk .
(B.6)
Inserting these transfers into the binding (IC0 ) constraint pins down the quantity
optimally offered to the retailer with high marginal cost:
F0 = π(qk , 0) − π(qk , k) + Fk ⇒ qkII =
F0 − Fk
.
k
(B.7)
Ignoring (ICk ) for the moment, the manufacturer’s problem amounts to
max [π(q0 , 0) − Kq0 ] + [π(qkII , k) − KqkII ] − F0 − Fk .
q0
(B.8)
Hence, the optimal quantity to offer the retailer with low marginal cost is q0II =
q JS (0).
The monotonicity requirement (MON) is satisfied as long as
q JS (0) ≥
F
,
k
(B.9)
in which case also (ICk ) is satisfied because (IC0 ) binds.
Case III: (PC0 ) and (PCk ) bind.
If (PC0 ) and (PCk ) are binding, transfers are given by
t0 = π(q0 , 0) − F0
and
tk = π(qk , k) − Fk .
(B.10)
Ignoring (IC0 ) and (IC0 ) for the moment, the manufacturer’s problem amounts to
max [π(q0 , 0) − Kq0 ] + [π(qk , k) − Kqk ] − F0 − Fk
q0 ,qk
(B.11)
From the manufacturer’s objective function it becomes apparent that the optimal
quantity to offer the retailer with low marginal cost is q0III = q JS (0). Likewise, the
optimal quantity to offer the retailer with high marginal cost is qkIII = q JS (k).
It remains to check whether the neglected constraints are satisfied. First, (IC0 ) is
satisfied as long as
F0 ≥ kq JS (k) + Fk ⇐⇒ q JS (k) ≤
F
.
k
(B.12)
Likewise, (ICk ) is satisfied as long as
Fk ≥ −kq JS (0) + F0 ⇐⇒ q JS (0) ≥
F
.
k
(B.13)
Case IV: (PC0 ), (PCk ), and (ICk ) bind.
If (PC0 ) and (PCk ) are binding, transfers are given by
t0 = π(q0 , 0) − F0
and
34
tk = π(qk , k) − Fk .
(B.14)
Inserting these transfers into the binding (ICk ) constraint pins down the quantity
optimally offered to the retailer with low marginal cost:
Fk = π(q0 , k) − π(q0 , 0) + F0 ⇒ q0IV =
F
.
k
(B.15)
Ignoring (IC0 ) for the moment, the manufacturer’s problem amounts to
max [π(q0IV , 0) − Kq0IV ] + [π(qk , k) − Kqk ] − F0 − Fk .
qk
(B.16)
Hence, the optimal quantity to offer the retailer with high marginal cost is q0IV =
q JS (k).
The monotonicity requirement (MON) is satisfied as long as
F
,
k
q JS (k) ≤
(B.17)
in which case also (IC0 ) is satisfied because (ICk ) binds.
Case V: (PC0 ) and (ICk ) bind.
If (PC0 ) and (ICk ) are binding, transfers are given by
t0 = π(q0 , 0) − F0
and
tk = π(qk , k) + kq0 − F0 .
(B.18)
Ignoring (IC0 ) and (PCk ) for the moment, the manufacturer’s problem amounts to
max [π(q0 , 0) − (K − k)q0 ] + [π(qk , k) − Kqk ] − 2F0
q0 ,qk
(B.19)
From the manufacturer’s objective function it becomes apparent that the optimal
quantity to offer the retailer with high marginal cost is qkV = q JS (k). Differentiation
w.r.t. q0 reveals that the optimal quantity to offer the retailer with low marginal
cost is q0V > 0, which is implicitly characterized by
P (q0V ) + q0V P 0 (q0V ) = K − k.
(B.20)
It remains to check whether the neglected constraints are satisfied. The left-hand
side of (B.20) is decreasing in q, such that q0V > q JS (0). This, in turn, implies
that the monotonicity requirement (MON) is satisfied. As (ICk ) binds, (IC0 ) then
is automatically satisfied. Finally, (PCk ) is satisfied as long as
Fk ≤ −kq0V + F0 ⇐⇒ q0V ≤
F
.
k
(B.21)
Noting that the manufacturer will always be (weakly) better off in a situation
where only two (Cases I, III and V) rather than three constraints (Cases II and IV)
impose a binding restriction, the following result summarizes the optimal quantities
under uniform pricing.
35
Lemma 4. Let qiu denote the optimal quantity offered to downstream firm i ∈ {0, k}
under uniform pricing. Then:
(i) If
F
k
≤ qkI , then qku = qkI and q0u = q JS (0).
(ii) If qkI <
F
k
< q JS (k), then qku =
F
k
and q0u = q JS (0).
(iii) If q JS (k) ≤
F
k
≤ q JS (0), then qku = q JS (k) and q0u = q JS (0).
(iv) If q JS (0) <
F
k
< q0V , then qku = q JS (k) and q0u =
(v) If q0V ≤
F
,
k
F
.
k
then qku = q JS (k) and q0u = q0V .
Free Disposal.—Lemmas 3 and 4 characterize the optimal quantities for the case of
quantity forcing. Now suppose that downtream firms can freely dispose of the input.
Given downstream firm i ∈ {0, k} obtains quantity q̃ of the input for free, it will sell
min{q̃, q(ki )} units of the final product, where q(ki ) is defined in (1) and satisfies
P (q(ki )) + q(ki )P 0 (q(ki )) = ki . With q JS (k) and q0V being defined by P (q JS (k)) +
q JS (k)P 0 (q JS (k)) = K + k > k and P (q0V ) + q0V P 0 (q0V ) = K − k > 0, respectively,
and P (q) + qP 0 (q) bing strictly decreasing, it follows that qku ≤ qkd = q JS < q(k)
and q0d ≤ q0u ≤ q0V < q(0). In consequence, free disposal leaves the quantities sold
by each downstream firm under each pricing regime unaffected. The quantities that
the manufacturer offers to the two downstream firms under the respective pricing
regime, as characterized in Lemmas 3 and 4, are depicted in Figure 3
Welfare.—Social welfare does not depend on the specifics of the contractual form,
but on the quantities of the final consumption good. As P (0) > K + k by assumption, each market should be served from a welfare perspective. The quantity
that maximizes welfare in the market served by downstream firm i ∈ {0, k}, qiW ,
Rq
maximizes Wi = 0 P (z)dz − (K + ki )q and thus is characterized by
P (qiW ) = K + ki .
(B.22)
Regarding the market served by the downstream firm with marginal cost k, as
P (qkW ) = K + k < K + k − q JS (k)P 0 (q JS (k)) = P (q JS (k)) and P 0 < 0 (whenever
P > 0), we must have q JS (k) < qkW . Regarding the market served by the downstream
firm with marginal cost 0, note that we must have tV0 − Kq0V ≥ 0, otherwise the
manufacturer would be better off in Case V by offering only the quantity-transfer
pair (qkJS , tJS
k ), which would be rejected by the downstream firm with marginal cost
0. With tV0 = q0V P (q0V ) − F0 , this implies q0V [P (q0V ) − K] ≥ F0 . As, in Case V,
JS
W
0
F0 > Fk ≥ 0 , we must have P (q0V ) > K = P (W
0 ), such that q (0) < qk as P < 0
36
q0V
q0u
q JS (0)
q0d
qkd
q JS (k)
qku
qkI
qkI
JS
q (k)
JS
q0V
q (0)
F0 −Fk
k
Figure 3: Optimal quantities under general nonlinear contracts.
37
(whenever P > 0). With Wi being strictly concave in q (whenever P > 0), welfare in
market i ∈ {0, k} is higher under the pricing regime that leads to the larger quantity
being sold in that market. The welfare comparison across pricing regimes stated in
Proposition 4 then is immediate.
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