WORKING 

PAPERS 

All-units Discounts and Double Moral Hazard

Daniel P. O’Brien 

WORKING PAPER NO. 316 

Original Release: March 2013 

Revised: July 2014 

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BUREAU OF ECONOMICS 

FEDERAL TRADE COMMISSION 

WASHINGTON, DC 20580
All-units Discounts and Double Moral Hazard

Daniel P. O’Brien1

June, 2014

Abstract

An all-units discount is a price reduction applied to all units purchased if the customer’s

total purchases equal or exceed a given quantity threshold. Since the discount is paid

on all units rather than marginal units, the tariff is discontinuous and exhibits a negative marginal price (“cliff”) at the threshold that triggers the discount. This paper

shows that all-units discounts arise in optimal agency contracts between upstream and

downstream firms that face double moral hazard. I present conditions under which allunits discounts dominate two-part tariffs and other continuous tariffs. I also examine

these tariffs when the upstream market faces a threat of entry. In the case considered, all-units discounts deter entry by less efficient rivals without distorting price and

investment, whereas continuous tariffs either accommodate such entry or deter it by

distorting price and investment. These findings begin filling the gap in economists’

understanding of the equilibrium effects of all-units discounts in intermediate markets

in which contract design affects incentives for pricing, investment, and competitive entry.

Keywords: All-units discounts, retroactive rebates, double marginalization, double moral hazard,

principal-agent

JEL Classifications: D42, D86, L12, L42

1Visiting Professor of Economics, Department of Business, Economics and Public Policy, Kelley School of Business,

Indiana University and Senior Economic Policy Advisor, U.S. Federal Trade Commission. The views expressed herein

are my own and do not purport to represent the views of the Federal Trade Commission or any Commissioner. Contact

the author at obriend@indiana.edu.
I. Introduction

An all-units discount is a price reduction offered on all units purchased if the customer’s total

purchases equal or exceed a given quantity threshold. These tariffs are quite common in vertical

contracts and have received significant attention from antitrust authorities in recent years. In some

cases the discounts are offered at the time of purchase in the form of off-invoice allowances; in

other cases they are offered as periodic rebates as quantity thresholds are met. In the latter form,

all-units discounts are sometimes called retroactive rebates.

2 They have also been defined as one

type of loyalty discount in the antitrust policy literature.3

Since the discount applies to all units rather than only additional units, the tariff exhibits a

discontinuous, downward jump—a negative marginal price, or “cliff”—at the threshold that triggers

the discount. The negative marginal price property has raised concerns among antitrust authorities

that these tariffs might harm competition. Consider a downstream firm that currently purchases all

of its supplies from an incumbent supplier. If the firm’s decision to purchase some amount from a

rival supplier would cause its purchases from the incumbent to fall below a discount threshold, the

rival may need to compensate the buyer for discounts lost on inframarginal units purchased from

the incumbent. Based on this logic, the European Union treats all-units discounts by dominant

firms as a potential abuse of dominance aimed at excluding competitors.4

While all-units discounts are viewed suspiciously by antitrust authorities, the economic rationale

for their use is not well-understood. The practice is not mentioned in the two chapters on price

discrimination in the Handbook of Industrial Organization.5 The early agency literature identifies

abstract examples in which the optimal payment from a principal to an agent may be a discontinuous

function of the principal’s revenue,6 but there has been little systematic work connecting these

results to actual tariffs that arise between upstream and downstream firms.7

2For a discussion of different rebate forms, see European Commission (2009).

3See, e.g., Greenlee and Reitman (2005).

4All-units discounts are considered a potential abuse of dominance under Article 102 (formerly Article 82) of the

Treaty on the Functioning of the European Union. In a review of Article 102, the European Commission states:

“In general terms, retroactive rebates may foreclose the market significantly, as they may make it less attractive for

customers to switch small amounts of demand to an alternative supplier, if this would lead to loss of the retroactive

rebates. ... The higher the rebate as a percentage of the total price and the higher the threshold, the greater the

inducement below the threshold, and, therefore, the stronger the likely foreclosure of actual or potential competitors.”

(European Commission, 2009, paragraph 40).

5See Varian (1989) and Stole (2005).

6Examples include Lewis (1980) and Singh (1983).

7An exception is Kolay et al (2004), discussed below. In canonical models of moral hazard (e.g., Holmstrom, 1979,

1982) and screening (e.g., Maskin and Riley, 1984), payment schedules are typically continuous functions of revenue

or quantity.

1
In this paper I offer an explanation for all-units discounts that does not involve exclusionary

motives. I consider a vertical relationship in which an upstream and a downstream firm make

non-contractible decisions that affect both firms’ profits (“double moral hazard”). Prior to these

decisions, firms agree to supply terms that may depend on output, but not on investment or pricing

decisions. If firms can divide profits with a fixed transfer payment, their objective is to write a

contract that induces investment and pricing decisions that maximize joint profits. If the tariff

is linear in quantity (a two-part tariff), a conflict arises in attempting to promote both upstream

and downstream incentives. A low wholesale price (equal to upstream marginal cost) is required to

eliminate double-marginalization and induce efficient downstream investment, but a higher marginal

price is required to give the upstream firm an incentive to invest. This conflict prevents two-part

tariffs from solving both incentive problems simultaneously.8

An intuitive, but incomplete, argument suggests that an all-units discount tariff might be a

useful incentive device in this environment. All-units discounts have the merit of providing strong

incentives for downstream output expansion while preserving a positive upstream margin that

encourages upstream investment. A potential problem with this logic, recognized by Romano

(1994)9, is that the downward jump in an all-units discount tariff introduces an additional moral

hazard problem—the upstream firm may be able to shirk on its investment just enough to prevent

the downstream firm from reaching the threshold that triggers the discount. When this additional

moral hazard is present, the downstream firm will recognize the upstream firm’s incentive to shirk

and will optimize against the undiscounted price. If upstream investment is a continuous decision

variable and the effects of investment on demand are known with certainty (as in Romano’s model),

all-units discounts are ineffective in the provision of downstream incentives except at the lowest

possible level of upstream investment.

In this paper, I consider three variants of the double moral hazard problem in which the positive

incentive effects of all-units discounts outweigh the cost of the additional moral hazard they introduce. In the first environment, investment returns are deterministic, as in Romano, but upstream

investment is lumpy. Under this assumption, the slight shirking by the upstream firm that undermines all-units discounts when the investment choice is continuous is not feasible. In the second

environment, firms are uncertain about the prospects for upstream investment that might arise after

contracts are signed (“uncertain prospects”). In the third environment, firms are uncertain about

8This conflict was first identified formally by Holmstrom (1982), who showed that sharing rules that do not break

the budget (e.g., by paying a penalty to a third party) typically cannot maximize joint profits in team production.

9See also Nandeibam (2002).

2
upstream investment returns (“uncertain returns”). Under both uncertain prospects and uncertain

returns, the upstream firm’s incentive to exploit discontinuities in the tariff is limited because the

equilibrium quantity generally differs from the quantity at which the tariff is discontinuous. In all

three environments, all-units discounts can arise in equilibrium.

Under lumpy upstream investment with deterministic returns, all-units discounts are optimal

tariffs. They support the vertically integrated outcome when upstream costs are sufficiently low,

and they distort downstream pricing and investment decisions less than two-part tariffs when

investment costs are high. In the optimal all-units discount, the wholesale price exceeds marginal

cost at all output levels. The tariff works by giving the retailer the incentive to expand output

by enough to receive the discount, while giving the manufacturer sufficient margin to support its

investment.

When investment returns are deterministic, declining block tariffs (or the un-dominated portion

of a menu of piece-wise linear tariffs) are equivalent to all-units discounts, and thus are also optimal.

While both price schedules dominate two-part tariffs, the model is not rich enough to distinguish

between continuous and discontinuous tariffs when investment returns are deterministic.

However, when investment prospects or returns are uncertain, the tariffs are no longer equivalent. Under these conditions, I identify two cases in which all-units discounts achieve the first

best outcome and dominate continuous tariffs: when upstream investment causes an iso-elastic

shift in demand, or when the downstream firm’s only decision is price. The sufficiency of all-units

discounts in these cases does not rely on lumpy upstream investment. The basic logic for the benefits of all-units discounts in these cases is similar to that in the deterministic case: the discounts

simultaneously encourage output expansion by both firms. However, all-units discounts support

the efficient outcome in a wider range of cases when investment prospects or returns are uncertain

by exploiting the risk experienced by the retailer that it may fail to receive a discount if it prices

too high or invests too little to reach the quantity threshold.

The antitrust policy question surrounding all-units discounts is whether they can be used to

exclude competitors in a way that harms competition. To begin addressing this question in a model

in which all-units discounts arise in equilibrium, I introduce the potential for entry in the upstream

market, focusing on the deterministic, lumpy upstream investment case. In this environment, I

find that the incumbants accommodate entry by a more efficient competitor with either all-units

discounts or continuous tariffs. Thus, if continuous tariffs are feasible, restricting the use of allunits discounts does not increase the incidence of entry by more efficient competitors in the model

3
considered here. On the other hand, all-units discounts deter entry by less efficient competitors

without distorting price and investment, whereas continuous volume discounts either accommodate

entry by a less efficient competitor or deter it by distorting price and investment.

A related paper is Romano (1994), which examines the role of resale price maintenance under

double moral hazard. My focus is on all-units discounts rather than RPM, and I consider the cases

of lumpy investment and uncertain investment prospects or returns, whereas Romano examined

continuous investment returns that are known with certainty. Under Romano’s assumptions, twopart tariffs are optimal contracts. Under the assumptions here, more complex tariffs generally

dominate two-part tariffs. Romano also does not address entry, which this paper does.

The literature on moral hazard in teams and partnerships10 identifies conditions under which

sharing rules exist that achieve or approximate the first best outcome in problems with N-sided

(N ≥ 2) moral hazard. Legros & Matthews (1993) study the case of deterministic investment

returns and show that a partnership can attain full efficiency with pure strategies if the partners’

decision sets are finite or the investment technology is Leontief. In this paper, the upstream firm’s

decision set is finite (invest or not), but downstream firm’s decision set is continuous and smooth,

and is not Leontief. Full efficiency is generally not possible in pure strategies, but all-units discounts

are optimal, second best contracts in pure strategies.11 Williams & Radner (1988) and Legros &

Matsushima (1991) provide conditions for efficiency under stochastic returns when action spaces

are finite. In this paper, I present conditions in which all-units discounts achieve efficiency under

stochastic returns when the action space is continuous. Rasmusen (1987) shows that risk aversion

also increases the scope for efficiency in the stochastic returns case. In this paper, agents are risk

neutral.

Several papers have shown that penalty schemes can be used to approximate or achieve the first

best outcome in one-sided moral hazard problems.12 The use of all-units discounts in this paper is

related to the role of penalties in those papers, although here the incentive contract must also deal

with the manufacturer’s moral hazard. In the cases of uncertain investment prospects or returns

that I examine, the penalty imposed by the difference between the wholesale prices in the upper and

lower tiers in a two-price all-units discount aligns the retailer’s pricing and investment incentives

10Examples include Holmstrom (1982), Rasmusen (1987), Williams & Radner (1988), Legros & Matsushima (1991),

and Legros & Matthews (1993).

11Legros & Matthews also consider mixed strategies and show that firms can achieve approximate efficiency in

mixed strategies in a broad class of cases. This paper does not consider mixed strategies.

12See, e.g., Mirrlees (1974, 1975, 1999), Gjesdal (1976), Holmstrom (1979, footnote 7), Lewis (1980), and Singh

(1984). For a recent example, see Wang (2009).

4
with joint incentives. This allows firms to set the wholesale price level to make the manufacturer

the residual claimant to the joint effects of its investment.

In the first formal analysis of all-units discounts, Kolay et al. (2004) examined their role as a

screening device. They found that a price-discriminating monopolist selling to buyers with discrete

types does better with all-units discounts than with a menu of self-selecting two-part tariffs. The

motivation for all-units discounts here is quite different than it is in their paper. In this paper,

there is only one buyer and no hidden information, so monopoly screening is not an issue.

The remainder of this paper is organized as follows. Section II presents the model. Section

III examines the deterministic lumpy investment case. Section IV examines the case of uncertain

upstream investment prospects and returns. Section V compares all-units discounts and two-block

tariffs when the upstream firm faces potential entry, focusing on the case of deterministic returns.

Section VI concludes the paper with a discussion of implications and some thoughts on future

research.

II. Basic Model

An upstream firm (the “manufacturer”) distributes a single product through a downstream firm

(the “retailer”). The final demand for the retailer’s product is Q(P, x, I), where P is the retail

price, I is the manufacturer’s investment, and x is the retailer’s investment. Assume that Q is

decreasing in P and increasing in x and I. For all (x, I), there exists a finite choke price P(x, I)

above which demand is zero. In the first part of the paper, I assume that demand is known with

certainty. Later I consider two types of uncertainty and introduce the additional notation when it

is needed.

The investment costs are m(I) for the manufacturer and r(x) for the retailer, both of which

are increasing in the levels of investment, with m(0) = r(0) = 0. For any levels of investment, the

manufacturer and retailer produce at variable costs C(Q) and V (Q), respectively. Variable costs

are increasing in Q, with C(0) = V (0) = 0. All functions are twice continuously differentiable.

To simplify notation in what follows, let c(Q) = CQ(Q) and v(Q) = VQ(Q) be the upstream and

downstream marginal costs.

Production and contracting are described by a two-stage game. In stage 1, the firms negotiate

a supply contract. The contract specifies a fixed fee S that the retailer pays the manufacturer

to stock the product (S may be negative, a slotting allowance), and an additional tariff T(Q)

5
that the retailer pays the manufacturer to purchase and resell Q units. This tariff depends on

the quantity purchased, but it cannot be conditioned on price or investment levels unless otherwise

noted.13 In stage 2, given the contract terms (S, T(Q)), the manufacturer chooses I to maximize its

profit, and the retailer simultaneously chooses P and x to maximize its profit. The manufacturer’s

variable profit is U = T(Q(P, x, I)) − C(Q(P, x, I)) − m(I), and the retailer’s variable profit is

π = P Q(P, x, I) − V (Q(P, x, I)) − T(Q(P, x, I)) − r(x). I look for sub-game perfect equilibria.

The joint profits of the manufacturer and retailer are Π = U +π = P Q(P, x, I)−C(Q(P, x, I))−

V (Q(P, x, I)) − m(I) − r(x). Let (P

∗

, x∗, I∗) maximize Π. I will refer to (P

∗

, I∗, x∗) as the “integrated” outcome.

III. Lumpy Investment and Deterministic Returns

If profits are continuous in own investment and demand is known by both firms at the time of

contracting, the equilibrium contract must be continuous at the optimal quantity. If it were discontinuous, either the manufacturer or the retailer could adjust its investment slightly up or down

and cause a discrete jump in its profit.14 In this case, a binding all-units discount tariff—one that

induces the retailer to purchase the minimum quantity required to receive a discount—cannot arise

in equilibrium.

However, many investment technologies are not continuous. Some investments are lumpy by

their nature. Examples include process innovations that enhance quality and discrete marketing

projects like participation in trade shows.15 Some investments are effectively lumpy due to friction

in the business decision-making process. For example, the life cycle of a typical investment project

involves a development stage at lower levels within the firm, evaluation by senior management,

and an up or down decision on whether to proceed. The investment decision in this context is

more about whether to pursue a discrete project proposed to management than it is about minor

adjustments of investment levels at the margin.

13There are many reasons why price and investment may be non-contractible. For example, RPM may be illegal,

or it may be costly for the manufacturer to monitor the retail price. Similarly, it may be difficult for the retailer

and/or a court to verify manufacturer and retailer investments.

14See Romano (2004).

15Even advertising can have an element of discreteness if it rotates demand about a price, as in Johnson and Myatt

(2006). They show that if demand is ordered by a sequence of rotations in the amount of an attribute (it often is)

and the marginal cost of production is constant, then variable profit is quasiconvex in the attribute when factor prices

are linear. In this case, if the fixed costs of the attribute are not too concave, then the profit-maximizing choice of

the attribute is an extreme: either the maximum possible amount (e.g., an entire advertising budget) or zero. Of

course, discontinuous tariffs eliminate the quasiconvexity and may generate incentives to shirk. However, firms may

be inclined to treat certain choices as discrete if this assumption works in the majority of the firm’s experiences in

making such choices.

6
In this section, I focus on such lumpy investment:

Assumption 1 (Lumpy Upstream Investment) The manufacturer chooses investment I ∈ {0, I∗},

i.e., it makes the investment I

∗

, or it invests zero.

To simplify notation under Assumption 1, let D(P, x) ≡ Q(P, x, I∗) be demand when the upstream

firm invests I

∗

, and let D0(P, x) = Q(P, x, 0) be demand when it invests zero.

Since the the manufacturer and retailer can exchange a lump sum transfer at the time of

contracting, the general contracting problem can be described as choosing a tariff T(·), retail price

P, and investment levels x and I to maximize joint profits subject to the constraints that I and

(P, x) are mutual best responses for the manufacturer and retailer given T(·). If the firms choose

not to induce upstream investment, only the retailer’s incentives matter, and joint profits can be

maximized (conditional on no upstream investment) with a two-part tariff that makes the retailer

the residual claimant to joint profits. There is no role for more complex contracts. The more

interesting case examined in this paper is when firms decide to induce upstream investment. In

this case, the optimal contract solves the “general contracting problem:”

(GCP) max

P,x,T(·)∈T

Π = P D(P, x) − C(D(P, x)) − V (D(P, x)) − r(x) − m(I

∗

) s.t.

(1) (P, x) = arg max

(P0,x0)

P

0D(P0

, x0) − V (D(P

0

, x0)) − T(D(P

0

, x0)) − r(x

0

),

(2) T(D(P, x)) − C(D(P, x)) − m(I

∗

) ≥ T(D0(P, x)) − C(D0(P, x)).

where T is the set of all feasible contracts.

I will compare the solution to GCP with the solution to the contracting problem when tariffs are

restricted to take three simple, commonly-observed forms: two-part tariffs (TP), declining block

tariffs with two blocks (TB), and all-units discounts with two prices (TA). These tariffs are written

as follows:

T

T P (Q) = 

0 if Q = 0,

F + wQ if Q > 0,

T

T B(Q) =







0 if Q = 0,

F + w1Q if 0 < Q < q,

F + w2Q + (w1 − w2)q if Q ≥ q,

T

T A(Q) = (

w1Q if 0 ≤ Q < q,

w2Q if Q ≥ q

7
where w, w1, and w2 are wholesale prices, F is a fixed fee, and q is a quantity threshold that

determines the applicable per-unit price.16

The two-part tariff is the standard “continuous” tariff17 that appears in much of the literature

on vertical control. The two-block tariff is a slightly more flexible continuous tariff, charging two

different marginal prices depending on whether quantity falls in the first block (Q < q) or second

block (Q ≥ q). In most of the literature on vertical control, customer-specific two-block tariffs are

equivalent to customer-specific two-part tariffs, because a customer purchasing in the second block

will view the extra payment (w1 −w2)q for quantities in the first block as part of the fixed fee. The

all-units discount tariff is similar to the two-block tariff in that it specifies two prices that depend

on whether the quantity purchased is above and below a quantity threshold q. However it differs in

two key respects: (1) customers that purchase in the second block (Q ≥ q) do not pay an implicit

fixed fee; and (2) if w1 > w2, the all-units discount tariff is discontinuous at q. As I have noted,

all-units discounts have received little formal attention in the literature on vertical control.18

The following preliminary result motivates the potential role for all-units discounts and twoblock tariffs in this model.

Proposition 1 Two-part tariffs support the integrated outcome if and only if the manufacturer’s

incremental quasi-rents from investment at wholesale price w

∗ = c(D(P∗

, x∗)) are sufficiently large.

Proof: Under a two-part tariff, the retailer will choose the fully integrated price and investment

only if it faces the same marginal incentives as an integrated firm. This requires the wholesale price

w

∗ = c(D(P∗

, x∗)). The upstream firm’s incremental profit from investing is then

(3) ∆ = Z D(P

∗,x∗)

D0(P∗,x∗)

[w

∗ − c(q)]dq − m(I∗

)

The integral represents the manufacturer’s incremental quasi-rents from investment at the wholesale

price w

∗

. The integrated outcome is supported if and only if ∆ ≥ 0, which requires sufficiently

large quasi-rents. Q.E.D.

Proposition 1 is the lumpy investment analog of Proposition 1 in Romano (2004), which established that two-part tariffs cannot support the integrated outcome when the manufacturer chooses

investment from a continuous set.19 In the remainder of this paper, I assume that the manufac16Of course, T

T B and TT A only exhibit marginal price “discounts” if w2 < w1.

17It is continuous except at zero.

18Kolay et al. (2004) is the primary exception.

19Although he assumed constant marginal cost, a two-part tariff would not support the integrated outcome in his

model even with high quasi-rents because the manufacturer would distort its continuous investment choice at the

margin.

8
turer produces at constant marginal cost c (no quasi-rents). This rules out the possible efficiency

of two-part tariffs due to high manufacturer quasi-rents, focusing attention on cases in which more

complex contracts might do better.

A. Optimal All-Units Discounts

Next I characterize the optimal all-units discount tariff. Define an effective all-units discount as

one in which w1 > w2, and the retailer elects to sell enough to reach the discount threshold q and

pay the lower price w2. (An ineffective all-units discount would have the same incentive effects

as a two-part tariff with wholesale price w1.) Under an effective all-units discount that induces

upstream investment, there are three constraints on the firms’ investment and pricing decisions.

First, the retailer will choose P and x to maximize its profit given the all-units discount quantity

threshold:

(4) (P, x) = arg max

(P0,x0)

(P

0 − w2)D(P0

, x0) − V (D(P

0

, x0)) − r(x

0

) s.t. D(P

0

, x0) ≥ q.

Second, the retailer must earn more by selling at least q units at price w2 than by “defecting” from

the all-units discount and optimizing against the higher wholesale price w1:

(5) (P −w2)D(P, x)−V (D(P, x))−r(x) ≥ πˆ(w1) ≡ max

(P0,x0)

(P

0−w1)D(P0

, x0)−V (D(P

0

, x0))−r(x

0

).

Third, the manufacturer must find it profitable to invest:

(6) (w2 − c)D(P, x) − m(I

∗

) ≥ Uˆ

where Uˆ is the profit the manufacturer earns if it “defects” by choosing not to invest. This profit

is

Uˆ =



(w1 − c)D0(P, x) if D0(P, x) < q,

(w2 − c)D0(P, x) if D0(P, x) ≥ q.

The optimal all-units discount maximizes joint profits Π subject to (4), (5), and (6).

The following assumption simplifies the analysis.

Assumption 2 (First Order Sufficiency) The first order conditions to the maximization problem

in (4) are sufficient for (P, x) to solve (4).

Assumption 2 holds if π is concave and D(P, x) is quasiconcave, though it may also hold with

weaker conditions on demand.

9
Given Assumption 2, an effective all-units discount that induces upstream investment will solve

(AUDT) max

(P,x,w1,w2,q,ξ)

(P − c)D(P, x) − V (D(P, x)) − r(x) − m(I

∗

) s.t.

(7) (P − w2)D(P, x) − V (D(P, x)) − r(x) ≥ πˆ(w1),

(8) (w2 − c)D(P, x) − m(I

∗

) ≥ U, ˆ

(9) D(P, x) + (P − v(D(P, x)) − w2)DP (P, x) + ξDP (P, x) = 0,

(10) (P − v(D(P, x)) − w2)Dx(P, x) − rx(x) + ξDx(P, x) = 0,

(11) D(P, x) ≥ q,

(12) ξ(D(P, x) − q) = 0

where conditions (9) through (12) are the first order conditions for (P, x) to maximize the retailer’s profit, and ξ is the Lagrangian multiplier in the retailer’s maximization problem (4). The

Lagrangian for (AUDT) is20

L = (P − c)D − V − r − m + λ[(P − w2)D − V − r − πˆ] + η[(w2 − c)D − m − Uˆ]

+γ1[D + (P − v − w2 + ξ)DP ] + γ2[(P − v − w2 + ξ)Dx − rx] + γ3[D − q] + γ4ξ[D − q].

The following lemmas characterize the role of the quantity constraint in the all-units discount.

Lemma 1 In any effective all-units discount that improves upon a two-part tariff, q ≥ D0(P, x),

and thus Uˆ = (w1 − c)D0(P, x).

Proof: Suppose q < D0(P, x). Then ξ = 0, and the quantity constraint does not affect the

manufacturer’s investment decision. It is then optimal to set w1 arbitrarily high to relax (7) as

much as possible, which sets ˆπ(w1) = 0. The contracting problem is then equivalent to choosing a

two-part tariff with fixed fee S and wholesale price w2. Q.E.D.

Lemma 2 In characterizing the optimal retail price and investment levels under an all-units discount that improves upon two-part tariffs, it is sufficient to consider only cases in which (11) is

binding (i.e., D(P, x) = q).

20Arguments of functions are omitted for brevity except when this could cause confusion.

10
Proof: Suppose constraint (11) does not bind. Then γ3 = 0, and since the constraint in the

retailer’s optimization problem is slack, ξ = 0. The derivative of the Lagrangian with respect to

q is then Lq = −γ3 − γ4ξ = 0, and the other derivatives of the Lagrangian do not depend on q.

Therefore increasing q until q = D(P, x) does not affect the maximized joint profits or the optimal

investment levels. Q.E.D.

Using D(P, x) = q and Uˆ = (w1 − c)D0(P, x) from Lemmas 1 and 2, the first order conditions

for w1, w2, and ξ are

Lw1 = −λπˆw1 − ηD0(13) (P, x) = 0,

(14) Lw2 = −(λ − η)D − γ1DP − γ2Dx = 0,

(15) Lξ = γ1DP + γ2Dx = 0.

Substituting Lξ into Lw2

implies λ = η. So the constraints (7) and (8) are either both binding or

both slack. If the constraints are binding, substituting λ = η into Lw1

and applying the envelope

theorem yields

(16) −πˆw1

(Pˆ(w1), xˆ(w1)) = D(Pˆ(w1), xˆ(w1)) = D0(P, x)

where (Pˆ(w1), xˆ(w1)) = arg maxP0

,x0(P

0 − w1)D(P0

, x0) − V (D(P

0

, x0)) − r(x

0

).

Condition (16) has an intuitive interpretation that illuminates the key factors in play at the

optimal all-units discount. To see this, add constraints (7) and (8) to get

(17) m(I

∗

) ≤ (P − c)D(P, x) − V (D(P, x)) − r(x) −



πˆ(w1) + (w1 − c)D0(P, x)



.

The bracketed term in (17) is sum of (i) the retailer’s profit if it defects by choosing output too low

to receive the discount and optimizes against w1, and (ii) the manufacturer’s profit if it defects by

choosing not to invest. Condition (17) indicates that the highest upstream investment cost such

that all-units discounts can support (P, x) in equilibrium is found by choosing w1 to minimize the

sum of the defection profits. The minimum occurs where D(Pˆ(w1), xˆ(w1)) = D0(P, x), as in (16).

To determine whether all units discounts can support the integrated outcome, fix the retail

price and investment at their integrated levels, (P

∗

, x∗). For any given wholesale price w2 in the

low price tier, the retailer incentive constraints (9) and (10) can be satisfied by choosing ξ so that

w2 − ξ = c. This gives the retailer the same effective marginal cost (including the shadow cost ξ)

as an integrated firm. All-units discounts will support the integrated outcome if there exist values

of w1 and w2 that also satisfy constraints (7) and (8) when evaluated at (P

∗

, x∗).

11
Let w

R

2

(w1) be the value of w2 that solves the retailer’s participation constraint (7) with equality.

That is, w

R

2

(w1) is the retailer’s iso-profit contour representing the set of wholesale prices over which

it is just indifferent between pricing and investing to reach the discount threshold q and defecting

by optimizing against w1. Similarly let wM

2

(w1) solve the manufacturer’s participation constraint

(8) with equality; wM

2

(w1) is the manufacturer’s iso-profit contour along which it is just indifferent

between investing I

∗ and investing zero. Using these definitions, the participation constraints (7)

and (8) evaluated at (P

∗

, x∗) can be rewritten as

(18) w2 ≤ w

R

2

(w1) = P

∗D(P∗

, x∗) − V (D(P

∗

, x∗)) − r(x

∗

) − πˆ(w1)

D(P∗, x∗)

,

(19) w2 ≥ w

M

2

(w1) = w1D0

(P

∗

, x∗) + m(I

∗

) + c[D(P

∗

, x∗) − D0(P

∗

, x∗)]

D(P∗, x∗)

.

Let w1 be the wholesale price that induces the retail choke price in the event the retailer defects.

That is, D(Pˆ(w1), xˆ(w1)) = 0. The functions w

R

2

(w1) are wM

2

(w1) are plotted in Figure 1 using

the following facts, which are easy to confirm:

w

R

2

(c) = c,

∂wR

2

(c)

∂w1

=

−πˆw1

(c)

D(P∗, x∗)

=

D(Pˆ(c), xˆ(c))

D(P∗, x∗)

= 1,

∂wR

2

(w1)

∂w1

= 0,

w

M

2

(c) = c +

m(I

∗

)

D(P∗, x∗)

,

∂wM

1

(w1)

∂w1

=

D0(P

∗

, x∗)

D(P∗, x∗)

< 1 ∀ w1.

The retailer’s iso-profit contour intersects the point (w1, w2) = (c, c) with a slope of one. The

manufacturer’s iso-profit contour crosses the line w1 = c above the retailer’s by an amount equal

to m(I

∗

)/D(P

∗

, x∗) and with a slope less than one. It follows that if the upstream investment

cost m(I

∗

) is sufficiently small, there exist wholesale prices (w1, w2) such w2 ≤ w

R

2

(w1) and w2 ≥

wM

2

(w1). These are the prices that lie in the shaded area in Figure 1 (the value mA is defined in

Proposition 2 below). This proves the following proposition.

Proposition 2 Under lumpy upstream investment with deterministic returns, there exists an upstream investment cost threshold mA > 0 such that, for all investment costs below mA, a two-price

all-units discount supports the integrated outcome.

I now show that all-units discounts cannot support all investments an integrated firm would

make. The highest investment cost such that all-units discounts support the integrated outcome is

represented geometrically in Figure 1 as the investment cost associated with the point of tangency

12
_____

w2

(w1,w2)

w2 (w1)

M*

* *

c w

_

w2(w1)

mA

D(P*,x*)

c + _____

w1

R

w2(w1)

M

Figure 1: Equilibrium wholesale prices under all-units discounts.

between the iso profit contours. Analytically, mA is the right hand side of (17) evaluated at (P

∗

, x∗).

Using condition (16), mA can be written

mA = {(P

∗ − c)D(P∗

, x∗) − V (D(P

∗

, x∗)) − r(x

∗

)} − n(Pˆ − c)D0(P

∗

, x∗) − V (D0(P

∗

, x∗)) − r(ˆx)

o

.

Let m∗ be the maximum upstream investment a fully integrated firm would make. This is

m∗ = {(P

∗ − c)D(P∗

, x∗) − V (D(P

∗

, x∗)) − r(x

∗

)}−max

P,x

(P − c)D0(P, x) − V (D0(P, x)) − r(x)



.

Subtracting mA from m∗ gives

m∗−mA =

n

(Pˆ − c)D0(P

∗

, x∗) − V (D0(P

∗

, x∗)) − r(ˆx)

o

−



max

P,x

(P − c)D0(P, x) − V (D0(P, x)) − r(x)



.

If m∗ − mA > 0, an integrated firm would make investments that cannot be supported by all-units

discounts.

A simple example shows that m∗ − mA may be positive, which means that all-units discounts

may not support the integrated outcome. Suppose demand is unaffected by downstream investment

(fix x at zero), assume VQ = v is constant, and let D0(P, 0) = αD(P, 0) for some α < 1. Then

the integrated price P

∗ also maximizes joint profit when there is no upstream investment. The

13
difference between m∗ and mA is then

(Pˆ − c − v)D0(P

∗

, 0) − (P

∗ − c − v)D0

(P

∗

, 0) > 0,

since P > P ˆ ∗.

A remaining question is whether all-units discounts improve upon two-part tariffs whenever

they support upstream investment. The following assumption is required to establish this.

Assumption 3 D(Pˆ(w1), xˆ(w1)) is decreasing in w1.

Assumption 3 is the standard case, but it is not guaranteed by the other assumptions. It holds

if the cross partials of D are not too large. It implies that the retailer’s defection profit ˆπ(w1) is

convex and the retailer’s iso-profit contour w

R

2

(w1) is concave in w1. The curve w

R

2

(w1) is drawn

this way in Figure 1, although Assumption 3 is not required for Proposition 2.

Proposition 3 Suppose Assumption 3 holds. Under lumpy upstream investment with deterministic

returns, if all units discounts support upstream investment, they yield higher joint profits than twopart tariffs.

Proof: Note that the general character of Figure 1 does not change when the iso-profit contours

are evaluated at any (P, x) 6= (P

∗

, x∗). In particular, the manufacturer’s iso-profit contour has a

slope between zero and one, while the slope of the the retailer’s iso-profit contour runs from one

to zero as w1 runs between c and w1. The wholesale prices in the optimal all-units discount occur

at a point of tangency between wM

2

and w

R

2

. Since w

R

2

(w1) is concave by Assumption 3, the point

of tangency yields w2 < w1. This tariff dominates any two-part tariff, since the firms could have

chosen w1 = w2. Q.E.D

B. All-units Discounts are Optimal Contracts

I return now to the general contracting problem (GCP) introduced at the beginning of this section

and show that the optimal all-units discount solves GCP. Let (P

e

, xe, Te(·)) solve (GCP), and let

(P

A, xA, wA

1

, wA

2

, qA) solve (AUDT).

Proposition 4 Under lumpy upstream investment with deterministic returns, a two-price all-units

discount is an optimal contract, i.e., (P

A, xA) = (Pe

, xe).

14
Proof: The equilibrium contract T

e

(·) is chosen from the set of all feasible contracts, T . Let

F ⊂ T be the set of all two-point forcing contacts of the form

T

F

(Q) =







T1 if Q = D0(P

0

, x0)

T2 if Q = D(P

0

, x0)

∞ otherwise.

The method of proof is to show that (GCP) can be solved by restricting attention to contracts from

the set F, and that the solution to (AUDT) yields the same price and investment levels as when

contracts from the set F are employed.

Consider the specific two-point forcing contract

T

F e(Q) =







T

e

(D0(P

e

, xe)) if Q = D0(P

e

, xe)

T

e

(D(P

e

, xe)) if Q = D(P

e

, xe)

∞ otherwise.

Under this contract, the retailer will choose either (P

e

, xe), or some (P

0

, x0) such that D(P

0

, x0) =

D0(P

e

, xe). Any other choice would be unprofitable. Since (P

e

, xe, Te) solves (GCP), it follows

that for all (P

0

, x0),

P

eD(Pe

, xe) − V (D(P

e

, xe)) − T

F e(D(Pe

, xe)) − r(x

e

)

= P

eD(Pe

, xe) − V (D(P

e

, xe)) − T

e

(D(P

e

, xe)) − r(x

e

)

≥ P

0D(P0

, x0) − V (D(P

0

, x0)) − T

e

(D(P

0

, x0)) − r(x

0

(20) ).

Since (20) is true for all (P

0

, x0), it is also true for any (P

0

, x0) such that D(P

0

, x0) = D0(P

e

, xe).

Therefore, for all (P

0

, x0) such that D(P

0

, x0) = D0(P

e

, xe),

P

eD(Pe

, xe) − V (D(P

e

, xe)) − T

F e(D(Pe

, xe)) − r(x

e

)

≥ P

0D0

(P

e

, xe) − V (D0(P

e

, xe)) − T

e

(D0(P

e

, xe)) − r(x

0

)

= P

0D0

(P

e

, xe) − V (D0(P

e

, xe)) − T

F e(D0

(P

e

, xe)) − r(x

0

(21) ).

This implies that the retailer optimality constraint (1) is satisfied at (P

e

, xe) under T

F e(·). Similarly, by the definition of (P

e

, xe, Te),

T

F e(D(Pe

, xe)) − cD(P

e

, xe) − m(I

∗

) = T

e

(D(P

e

, xe)) − cD(P

e

, xe) − m(I

∗

)

≥ T

e

(D0(P

e

, xe)) − cD0(P

e

, xe)

= T

F e(D0

(P

e

, xe)) − cD0(P

e

, xe),

which means the manufacturer’s optimality constraint (2) is also satisfied at (P

e

, xe) under T

F e(·).

Therefore, the two-point forcing contract T

F e(·) solves (GCP).

15
Next I argue that the solution to (AUDT) yields the same retail price, investment levels, and

transfers as an optimal two-point forcing contract and therefore solves (GCP). For any candidate

solution (P

0

, x0, w0

1

, w0

2

, q0) to (AUDT), D(Pˆ(w

0

1

), xˆ(w

0

1

)) = D0(P

0

, x0) by (16). Therefore, the retailer’s decision whether to price and invest as expected under the all-units discount or optimize

against w

0

1

is effectively a decision whether to produce D(P

0

, x0) or D0(P

0

, x0). The manufacturer is

effectively choosing between the same two points. Thus, there is no loss of generality in restricting

attention to two-point forcing contracts of the form

T

F A(Q) =







w1D0(P

0

, x0) if Q = D0(P

0

, x0)

w2D(P

0

, x0) if Q = D(P

0

, x0)

∞ otherwise.

Note that T

F A takes the same form as TF e, with w1D0

(P

0

, x0) = T1 and w2D(P

0

, x0) = T2. Since

the optimal contract and the optimal all-units discount can both be characterized using the same

two-point forcing contract, they yield the same outcome. Q.E.D.

The use of all-units discounts in this model bears a relationship to the use of quantity forcing

to eliminate double marginalization. In addressing double marginalization, there is no loss of

generality in restricting attention to a single point forcing contract that specifies one price for the

efficient quantity and an infinite price (or any price above the choke price) for any other quantity.

It happens that a traditional forcing contract, which specifies one price for quantities that equal or

exceed the efficient quantity and an infinite (or very high) price for quantities below the efficient

level can achieve the same outcome as a single point forcing contract, because quantities other than

the efficient quantity are dominated under such a contract. Here, a single point forcing contract

would also be sufficient if the manufacturer did not make an investment decision. However, because

the manufacturer can choose not to invest, the contract must specify prices for two points, D and

D0, and the prices must be set so that choosing D dominates choosing D0for both the manufacturer

and the retailer. It happens that an all-units discount can support the same outcome as a two-point

forcing contract, because under the optimal all-units discount, choices other than D and D0 are

dominated.

C. Two-Block Tariffs

All-units discounts work by offering an incentive that lowers the retailer’s shadow cost of producing output to w

A

2 − ξ + v, while keeping the wholesale price w

A

2 high enough to compensate the

manufacturer for investment. One might conjecture that a continuous tariff, say a declining block

16
tariff, might accomplish the same objective with a wholesale price in the low-price block equal to

w

A

2 − ξ and an inframarginal price in the high-price block that compensates the manufacturer for

investing. I now show that this conjecture is correct when investment returns are deterministic.

Define an effective two-block tariff as one in which the retailer purchases in the low-price block

and pays the marginal price w2 < w1. A two-block tariff that supports upstream investment I

∗

will solve

(TBT) max

(P,x,w1,w2,q)

(P − c)D(P, x) − V (D(P, x)) − r(x) − m(I

∗

) s.t.

(22) (P − w2)D(P, x) − V (D(P, x)) − (w1 − w2)q − r(x) ≥ πˆ(w1),

(23) (w2 − c)D(P, x) + (w1 − w2)q − m(I

∗

) ≥ Uˆ T,

(24) D(P, x) + (P − v(D(P, x)) − w2)DP (P, x) = 0,

(25) (P − v(D(P, x)) − w2)Dx(P, x) − rx(x) = 0,

(26) D(P, q) ≥ q

where

Uˆ T =



(w1 − c)D0(P, x) if D0(P, x) ≤ q,

(w2 − c)D0(P, x) + (w1 − w2)q if D0(P, x) > q.

The Lagrangian is

L

T = (P − c)D − V − r − m + λ[(P − w2)D − V − (w1 − w2)q − r − πˆ]

+η[(w2 − c)D + (w1 − w2)q − m − Uˆ T]

+γ1[D + (P − v − w2)DP ] + γ2[(P − v − w2)Dx − rx] + γ3[D − q].

Lemma 3 In characterizing the optimal retail price and investment levels under an effective twoblock tariff, it is sufficient to consider only cases in which q ≥ D0

(P, x), and thus Uˆ T = (w1 −

c)D0(P, x).

Proof: Suppose q < D0(P, x), so that Uˆ T = (w2 − c)D0(P, x) + (w1 − w2)q. Then adjusting q does

not affect constraint (23). Constraint (26) is slack, since q < D0(P, x) < D(P, x), and q does not

enter constraints (24) and (25). The first order condition with respect to q is L

T

q = −λ(w1 − w2) =

0. Since w1 > w2 for any effective two-block tariff, this implies that λ = 0, which means that

constraint (22) is slack. Thus, raising q does not affect the constraint set and therefore leaves

optimal investment levels unchanged. Q.E.D

17
Proposition 5 Under lumpy upstream investment and deterministic returns, two-block tariffs and

all-units discounts are equivalent tariffs. Both are optimal contracts.

Proof: Let (P

A, xA, wA

1

, wA

2

, qA, ξA) solve (AUDT). I will show that there exists a vector (w1, w2, q)

such that constraints (22) through (26) are satisfied when evaluated at (P

A, xA). This means that

(P

A, xA) is feasible under two-block tariffs, and since two-block tariffs cannot do better than allunits discount tariffs (by Proposition 3), two block tariffs will yield the same outcome as all-units

discounts.

Let w2 = w

A

2 − ξ

A and w1 = wA

1

. Constraints (24) and (25) are then identical to (9) and (10)

and are therefore satisfied. Constraints (22) and (23) (using Lemma 3) can be written

(27) (P

A−wA

2

)D(P

A, xA)−V (D(PA, xA))−r(xA)+

ξ

A[D(PA, xA) − q] − (wA

1 − w

A

2

)q

	

≥ πˆ(w

A

1

),

(28) (w

A

2 − c)D(P

A, xA) − m(I∗

) −



ξ

A[D(PA, xA) − q] − (wA

1 − w

A

2

)q

	

≥ (w

A

1 − c)D0

(P

A, xA).

Constraints (27) and (28) are identical to (7) and (8) except for the term in curly braces. These

conditions will be satisfied if this term equals zero, i.e., if

(29) q = q

T ≡

ξ

A

wA

1 − wA2 + ξA

D(P

A, xA).

We need to show that q

T ∈ [D0

(P

A, xA), D(PA, xA)], consistent with Lemma 3.

Because ξ

A > 0 and wA

1 −w

A

2 > 0, q

T < D(PA, xA). To see that qT ≥ D0

(P

A, xA), define πT

(q)

as the retail profit from choosing (P

A, xA) under two-block tariffs when the price in the upper block

is w

A

1

and the price in the lower block is w

A

2 − ξ

A. Define ∆T

(q) = π

T

(q) − πˆ(w

A

1

) as the increase

in the retailer’s profit from choosing (P

A, xA) rather than optimizing against wA

1

in the high price

block. The retail profits under two-block tariffs can then be written as

π

T

(q) = ˆπ(w

A

1

) + ∆T(q).

Now suppose q = D0. This would mean that retailer pays a constant marginal price w

A

2 − ξ

A to

expand sales from D0to D. Since expanding to D maximizes retail profits at the marginal price

w

A

2 −ξ

A, it follows that ∆T

(D0) ≥ 0. Note that ∆T(q) is the curly-braced term in (27), and that it

is decreasing in q. It follows that the value of q such that ∆T(q) = 0 occurs when q ≥ D0. Q.E.D.

The equivalence between all-units discounts and two-block tariffs in the perfect certainty case is

not completely surprising given the intuition following Proposition 3. Just as a tariff with a single

18
marginal price can replicate a one-point forcing contract for the case in which only retail incentives

matter, a two-block tariff with two marginal prices can replicate a two-point forcing contract for

the case in which both retail and manufacturer incentives matter.

IV. Uncertain Investment Prospects and Returns

The previous section established the equivalence of two-price all-units discounts and two-block

tariffs when upstream investment is lumpy and investment returns are certain. In this section, I

introduce two notions of uncertainty and show that all-units discounts and declining block tariffs

are no longer equivalent. The two cases are described as follows:

Definition 1 (Uncertain Prospects). At the time of contracting, firms are uncertain whether a

productive upstream investment project exists. The prospects for investment are revealed to the

manufacturer before its investment decision, but after contracts have been signed.

Definition 2 (Uncertain Returns). The random returns from investment are unknown to both the

manufacturer and the retailer at the time of contracting.

The case of uncertain prospects might arise if the manufacturer is engaged in ongoing research that

randomly yields prospects for productive investment projects, or if specific marketing opportunities

might arise during the period covered by the contract. Uncertain returns is the standard case that

appears in most of the agency literature where investment yields a noisy return.21

Assume that firms have prior beliefs that investment I

∗ will yield demand D(P, x) with probability θ and demand D0

(P, x) with probability 1 − θ. For the case of uncertain prospects, θ is the

probability that an upstream investment project becomes available to the manufacturer that would

increase demand from D0to D. For the case of uncertain returns, θ is the probability that upstream

investment will increase demand from D0to D. This formulation retains the lumpy investment

assumption for ease of exposition and to permit comparison with the deterministic returns case.

However, lumpiness is not required for the results, as explained in the discussion in §IV.B below.

The contracting game is similar to that in the previous section, except that investment returns

are uncertain when contracts are signed. In stage one, firms negotiate (S, T(·)). In stage two, the

manufacturer decides whether to invest zero or I

∗

, and the retailer simultaneously chooses P and

21A third case of interest is when the manufacturer has private information about investment returns when contracts

are signed. In this case, which I have not analyzed, the contract would have elements of signalling and screening.

19
x. Uncertain prospects and returns differ according to whether the manufacturer knows whether a

productive investment opportunity exists before making its investment decision.

In either case, the best firms can hope to achieve is to maximize joint profits conditional on

P and x being chosen before the resolution of uncertainty. Let (P

∗

, x∗, I∗) be this “first best”

outcome.22 In both cases I assume that investing I

∗

is jointly optimal. The optimal retail price

and investment solve

(30) max

P,x

(P − c)D(P, x) − θV (D(P, x)) − (1 − θ)V (D0

(P, x)) − m(I

∗

) − r(x)

where D(P, x) = θD(P, x) + (1 − θ)D0

(P, x) is the expected quantity if the manufacturer plans to

invest under uncertain prospects or chooses to invest under uncertain returns.

A. Sufficient Conditions for All-Units Discounts to Support the First Best Outcome

Two speical cases are of interest.

Case 1 (No Downstream Investment Returns). D0(P, x) = D0(P, 0) and D(P, x) = D(P, 0) ∀x.

Case 2 (Iso-Elastic Upstream Investment). D0(P, x) = αD(P, x) for some α ∈ (0, 1).

In Case 1, the retailer’s only decision is its choice of price. This is equivalent to assuming that the

manufacturer and retailer can contract directly over downstream investment so that the retailer

does not make an independent investment choice. Note that this assumption does not trivialize the

problem, as the downstream firm’s pricing decision is still susceptible to double-marginalization.

Technically, the problem still involves double-moral hazard, but the retailer’s moral hazard has

a single dimension (only price) rather than two (price and investment). In Case 2, upstream

investment produces an iso-elastic shift in demand that does not affect the elasticities of demand

with respect of P and x.

Let a(P, x) be the retailer’s average incremental cost of expanding output from D0(P, x) to

D(P, x):

(31) a(P, x) = V (D(P, x)) − V (D0

(P, x))

D(P, x) − D0(P, x)

.

Let a

∗ = a(P∗

, x∗) be the average incremental cost at the first best retail price and investment

levels. The following proposition provides sufficient conditions for all-units discounts to achieve the

22I am following much of the literature in referring to the optimal outcome conditional given the information

constraints as the “first best.”

20
first best outcome and indicates that two-block tariffs do not support the first best outcome under

these conditions for all parameters.

Proposition 6 Under uncertain prospects or returns:

1. If the retailer’s only decision is price (Case 1), then a two-price all-units discount supports

the first best outcome.

2. Suppose upstream investment causes an iso-elastic shift in demand (Case 2).

(a) If R D0

(P

∗,x∗)

0

[a

∗ − v(q)]dq ≥ r(x∗

), then a two-price all-units discount supports the first

best outcome.

(b) If R D0

(P

∗,x∗)

0

[a

∗ − v(q)]dq < r(x∗

), then a two-price all-units discount with a minimum

commitment and penalty for breach supports the first best outcome.

3. Two-block tariffs do not support the first best outcome for all parameter values in Cases 1

and 2.

Proof:

Parts 1 and 2. I first show by construction that all-units discounts support the first best

outcome in Cases 1 and 2. Consider the following two-price all-units discount with a minimum

commitment of Q and penalty for breach of K:

T

∗

(Q) =







K + w1Q if Q < Q,

w1Q if Q ≤ Q < D0(P

∗

, x∗),

w2Q if Q ≥ D0(P

∗

, x∗).

If K and w1 are sufficiently large, a retailer governed by T

∗ will choose (P, x) to ensure that its

sales are at least D0(P

∗

, x∗) in all states of the world. That is, it will maximize its expected profits

subject to D0(P, x) ≥ D0(P

∗

, x∗). Note that this is true under both uncertain prospects and

returns. Suppose the contract specifies K and w1 high enough to induce this behavior. (Below I

determine whether K > 0 is actually required and discuss how high w1 must be.) Then the optimal

contract of the form T

∗

solves

(AUDT-U) max

(P,x,w2,q,ξ)

(P − c)D(P, x) − θV (D(P, x)) − (1 − θ)V (D0

(P, x)) − r(x) − m(I

∗

) s.t.

(32) (w2 − c)D − m ≥ (w2 − c)D0

(Uncertain Prospects)

(w2 − c)D − m ≥ (w2 − c)D0

(Uncertain Returns),

21
(33) D + (P − w2)DP − θv(D)DP − (1 − θ)v(D0

)D0

P + ξD0P = 0,

(34) (P − w2)Dx − θv(D)Dx − (1 − θ)v(D0

)D0

x − rx + ξD0x = 0,

(35) D0(P, x) ≥ D0(P

∗

, x∗),

(36) ξ[D0(P, x) − D0(P

∗

, x∗)] = 0

where ξ is again the Lagrangian multiplier in the retailer’s optimization problem. Note that only

one of the constraints in (32) must be satisfied, depending on whether the uncertainty is over

prospects or returns. Set P = P

∗

, x = x

∗

, w2 = P

∗ − a∗

, and

ξ =

(P

∗ − c − a∗

)DP (P

∗

, x∗)

D0

P

(P∗, x∗)

.

Substituting into conditions (32) through (36) shows that (35) and (36) are satisfied trivially. After

some algebra, and using D0 = αD (Case 2), (32) through (34) become

(37)

(P

∗ − c)[D − D0

] − [V (D) − V (D0)] ≥ m (Uncertain Prospects)

(P

∗ − c)[D − D0

] − [θV (D) + (1 − θ)V (D0) − V (D0)] ≥ m (Uncertain Returns),

(38) D + (P − c)DP − θv(D)DP − (1 − θ)v(D0)D0

P = 0,

(39) (P − c)Dx − θv(D)Dx − (1 − θ)v(D0)D0

x − rx = 0.

The conditions in (37) are the same as the conditions under which an integrated manufacturer

would invest I

∗ under uncertain prospects and returns. Conditions (38) and (39) are the same as

the first order conditions for the first best problem (30). Therefore, the first best retail price and

investment satisfy (32) through (34). This establishes that all-units discounts with a minimum

commitment and sufficiently high penalty for breach achieve the first best outcome in Case 2. In

Case 1, condition (34) is irrelevant, and it is straightforward to show that the same substitutions

establish that the other constraints hold without imposing the iso-elastic investment condition

D0 = αD.

Whether a minimum commitment with a penalty for breach is required depends on the size of

the retailer’s quasi-rents under the contract. The retailer’s expected variable profit if it chooses

(P

∗

, x∗) is

E[π] = (P

∗ − w2)D − θV (D) − (1 − θ)V (D0

) − r

= (P

∗ − (P∗ − a∗

))D − θV (D) − (1 − θ)V (D0) − r

= a

∗

θD + a

∗

(1 − θ)D0 − θV (D) − (1 − θ)V (D0) − r

= a

∗D0 − V (D0

) + θ



a

∗

[D − D0] − [V (D) − V (D0)]− r

=

Z D0

0

[a

∗ − v(q)]dq − r. (Using the definition of a

∗

(40) )

22
The retailer can “defect” from choosing (P

∗

, x∗) by choosing a quantity of zero (e.g., by setting

P very high and x = 0) and earning a profit of −K, or by choosing some price and investment

levels that yield positive quantities in some states under the recognition that it will pay the higher

price w1 and potentially a penalty K when its sales are below D0(P

∗

, x∗). The expression for

the retailer’s defection profit is somewhat tedious to write. What is important is that w1 and

K can be set high enough that the retailer’s best defection is to sell zero and earn −K. Thus,

the contract T

∗ will support the first best outcome for sufficiently high w1 if E[π] ≥ −K. If the

retailer’s expected quasi-rents, R D0

0

[a

∗−v(q)]dq, equal or exceed the retailer’s investment cost r(x∗

),

then the inequality is satisfied when K = 0, and no penalty (and no minimum commitment) is

required. This is always true in Case 1, and it will be true in Case 2 if the retailer’s expected

quasi-rents exceed r(x

∗

). If the retailer’s expected quasi-rents are less than r(x

∗

), then a minimum

commitment and associated penalty is required to ensure that the retailer chooses (P

∗

, x∗). This

establishes Parts 1 and 2 of the Proposition.

Part 3. I now establish the general insufficiency of two-block tariffs. To simplify notation,

assume V = 0, r = 0, and D0 = αD (iso-elastic upstream investment). This case suffices to

establish the insufficiency of two-block tariffs. Let w

T

1

and w

T

2 be the prices in the high-price and

low-price blocks of a two-block tariff, and let q be the quantity that divides the blocks. In any first

best two-block tariff, D0 ≤ q ≤ D; otherwise the tariff would be equivalent to a two-part tariff,

which cannot yield the first best outcome. Given (w

T

1

, wT

2

, q), the retailer solves

(41) max

(P,x)

P D(P, x) − θ[w

T

1

q + w

T

2

(D(P, x) − q)] − (1 − θ)w

T

1 D0

(P, x).

The first order condition with respect to P is

(42) D + P DP − θwT

2 DP − (1 − θ)w

T

1 D0P = 0.

The first-order condition for price at the first best outcome (differentiating (30)) is

(43) D + P DP − cDP = 0.

Two-block tariffs support the first best outcome only if (42) and (43) are both satisfied at (P

∗

, x∗).

Subtracting (43) from (42) and evaluating at (P

∗

, x∗) gives

(44) θwT

2 DP + (1 − θ)w

T

1 D0P = cDP .

Under iso-elastic upstream investment, (44) can be written

(45) θwT

2 + (1 − θ)αwT1

θ + (1 − θ)α

= c.

23
Using (45), we have

(46) w

T

1 − c =

θ(w

T

1 − w

T

2

)

θ + (1 − θ)α

,

(47) w

T

2 − c = −

(1 − θ)α(w

T

1 − w

T

2

)

θ + (1 − θ)α

.

The highest investment an integrated firm would make under uncertain prospects is (P

∗−c)[D−D0

]

The manufacturer will make the same investment only if

(48) (w

T

2 − c)D + (w

T

1 − w

T

2

)q − (w

T

1 − c)D0 ≥ (P

∗ − c)[D − D0

].

Straightforward algebra shows that condition (48) is also required for the manufacturer to make

the highest investment an integrated firm would make under uncertain returns. Using (46), (47),

and D0 = αD, condition (48) becomes

(P

∗ − c)[D − D0

] ≤ (w

T

1 − w

T

2

)

[θ + (1 − θ)α]q − (1 − θ)αD − θD0

θ + (1 − θ)α

(49)

= (w

T

1 − w

T

2

)



q −

D0

θ + (1 − θ)α



(50)

≤ (w

T

1 − w

T

2

)



D −

D0

θ + (1 − θ)α



(51)

=

(w

T

1 − w

T

2

)θ[D − D0]

θ + (1 − θ)α

(52) .

Condition (51) follows from (50) because q ≤ D. Since (w

T

1 − w

T

2

) and D − D0 are bounded,23 the

inequality (49) through (52) will be violated if θ is small, and the manufacturer will not make the

first best investment. Q.E.D

B. Intuition and Discussion

The basic intuition for the benefits of all-units discounts under uncertain prospects and returns

is similar to the intuition under deterministic returns. The all-units discount (plus the minimum

commitment, if necessary) encourages the retailer to expand output while keeping the wholesale

price high enough to ensure manufacturer investment. However, the uncertainty case shows that

presence of risk provides a tool firms can exploit to achieve efficiency in a set of environments (Cases

1 and 2) in which efficiency may not be possible when investment returns are deterministic. Under

23I am assuming that wT

2 ≥ 0. If this were not true, the retailer could increase its profit by ordering an unlimited

amount of the input.

24
both uncertain prospects and returns, the retailer weighs the risk of failing to reach the quantity

threshold against the potential gains from raising price and reducing its investment. If the penalty

for failing to reach the threshold is high enough, then the retailer will price and invest to ensure that

it reaches the discount threshold even if successful investment by the manufacturer does not occur.

If price is the retailer’s only decision, the penalty can be set high enough with an all-units discount

in all cases. If the retailer also makes a demand-enhancing investment, a minimum commitment

and penalty for breach may also be required if the investment cost is large relative to the retailer’s

quasi-rents.

Given the alignment of the retailer’s incentives with joint incentives via the discontinuous tariff,

the manufacturer becomes the residual claimant to the joint profits from its investment. Therefore,

the manufacturer chooses the joint profit-maximizing level of upstream investment.

Two-block tariffs are generally not sufficient to support the first best outcome. An optimal

two-block tariff must set a measure of the average wholesale price equal to the manufacturer’s

marginal cost c to make the retailer the residual claimant to joint profits. If upstream investment

costs are sufficiently high relative to the expected returns from upstream investment, then no such

tariff exists that can also support upstream investment.

The role of the iso-elastic upstream investment assumption is not transparent from the proof of

Proposition 6. Under the all-units discount contract T

∗

, the retailer can choose any ratio of P and

x to achieve D0(P, x) = D0(P

∗

, x∗). The first best outcome requires a particular ratio that weighs

the marginal effects of P and x on both D0 and D. The assumption that D0 = αD is sufficient to

ensure that the retailer chooses the optimal ratio.

The lumpy upstream investment assumption is not required for all-units discounts to achieve

the first best outcome. The key is that an all-units discount exists that imposes a sufficiently high

penalty on the retailer for failing to reach the threshold that the retailer will have an incentive

to price and invest to ensure that it reaches the threshold even if upstream investment is not

successful or does not materialize. Once the retailer’s incentives are aligned with joint profits, the

manufacturer is the residual claimant to the joint benefits of its investment.

To see how this can work with continuous upstream investment, consider the following generalization. Demand is Q(P, x, I, θ) = D0

(P, x)[1+h(I)θ)], where h(0) = 0, h is increasing and concave

in I, and θ has a continuous distribution F(θ) on [0, ∞]. That is, upstream investment is a continuous choice variable, and the returns from investment are positive in expected value, but uncertain.

Now set q = D0(P

∗

, x∗) = Q(P

∗

, x∗, 0, 0) and w2 = P

∗ − a∗

, as in the proof of Proposition 6. If

25
the manufacturer believes the retailer will choose (P

∗

, x∗), then the manufacturer is the residual

claimant and will invest to maximize the fully integrated expected profit. If F

0

(0) > 0, then a

small price increase by the retailer induces a discrete increase in the probability that it will sell less

than q and incur an all-units discount penalty. As in the case of lumpy upstream investment, a

sufficiently large all-units discount, possibly combined with a minimum quantity commitment and

penalty for breach, will induce the retailer to price and invest to reach the threshold. If F

0

(0) = 0,

then the first best can be approached arbitrarily closely by imposing a sufficiently high penalty.

Several papers in the early agency literature identified conditions under which penalty schemes

can be used to approximate or achieve the first best outcome in various one-sided moral hazard

problems.24 The finding here is that it is possible to find an all-units discount (with a breach

penalty, if needed) that provides the retailer with the right incentives and makes the manufacturer

the residual claimant to the joint benefits of its investment.

V. Upstream Entry

The policy debate surrounding all-units discounts centers on their potential role as a device to

exclude competitors. A complete analysis of this question is beyond the scope of this paper.

However, I make a few observations about how the potential for upstream entry affects my results

when investment returns are deterministic.

Consider the following modification of the game under deterministic returns. In stage one,

the incumbent manufacturer and retailer agree to a contract, as before. In stage two, in addition

to making investment and pricing decisions, the retailer considers whether to purchase at most

qE ≤ D0(Pˆ(w

A

1

), xˆ(w

A

1

)) units from an alternative source of supply at a unit price of wE. Entry

at quantity qE ≤ D0is “small scale” in that the entrant’s quantity is no greater than what the

incumbent’s quantity would be if it defected from the equilibrium in which it does not face a threat

of entry by choosing not to invest.25 If the contract induces the retailer to purchase qE from the

alternative source, the firms “accommodate upstream entry.” Otherwise they “deter” upstream

entry. For ease of exposition, assume that the retailer incurs no production cost beyond what it

pays the manufacturer, i.e., V (Q) = 0 ∀ Q.

24See the references in footnote 12.

25One can interpret qE as arising from an entrant willing to supply qE at unit price wE. Alternatively, a competitive

fringe may be able to supply qE at price wE, or the retailer may have the ability to integrate backward and produce

qE at marginal cost wE.

26
A. Accommodating Entry

Suppose first that firms employ an effective all-units discount intended to accommodate upstream

entry. The following intuitive result establishes that firms always accommodate small scale entry

by a more efficient competitor.

Proposition 7 Suppose upstream investment is lumpy and investment returns are deterministic.

Under either all-units discounts or two-block tariffs, the incumbent manufacturer and retailer will

accommodate small scale entry by a more efficient competitor.

Proof: Since the analysis is similar to that for the case without entry, I will simply sketch the

argument for all-units discounts. The argument for two-block tariffs parallels the argument in

Proposition 5.

The retailer must earn at least as much by accommodating entry and pricing and investing as

expected under the all-units discount as it would earn by choosing not to accommodate entry and

pricing and investing the same way. That is,

(53) (P − w2)D(P, x) + (w2 − wE)qE − r(x) ≥ (P − w2)D(P, x) − r(x).

This requires wE ≤ w2. In addition, the retailer must earn at least as much by accommodating entry

and pricing and investing as expected as it would earn by accommodating entry but optimizing

against w1. That is,

(P − w2)D(P, x) + (w2 − wE)qE − r(x) ≥ max

(P0,x0)

[(P

0 − w1)D(P0

, x0) − r(x

0

)] + (w1 − wE)qE

(54) =⇒ (P − w2)D(P, x) − r(x) − (w1 − w2)qE ≥ πˆ(w1).

The retailer must also prefer accommodation over non-accommodation and optimizing against w1.

It is easy to show that this will be true when (54) is satisfied and wE < w2. The manufacturer

must earn more by investing I

∗

than by choosing not to invest:26

(55) (w2 − c)D(P, x) − m(I

∗

) + (w1 − w2)qE ≥ (w1 − c)D0(P, x).

Finally, the analog of the incentive constraints (9) through (12) must also hold to ensure profit

maximization by the retailer.

26By an argument similar to that in Lemma 1, we can restrict attention to the case when D(P, x) − qE = q. This

implies that D

0

(P, x) − qE < q which implies that if the manufacturer chooses not to invest, it will receive the price

w1 rather than w2.

27
I now explain that (P

A, xA) will be chosen if entry is accommodated. Fix (P, x) = (PA, xA).

Conditions (54) and (55) are the same as the participation constraints (7) and (8) in (AUDT)

except for the terms involving (w1 − w2)qE. Note that reducing w2 raises the left hand side of (54)

and lowers the left hand side of (55) by the same amount. Thus, for any value of w1, there exists

a value of w2 such that (54) and (55) are satisfied at (P

A, xA). In particular, they can be satisfied

by setting w1 = w

A

1

and setting w2 > c such that (54) and (55) hold. The incentive constraints

on retail pricing and investment can be satisfied by choosing the appropriate shadow price ξ of

output expansion, as in (AUDT). Thus, (P

A, xA) is feasible under all-units discounts with entry.

To do better, firms would have to adjust w1. However, it is straightforward to verify from the

first order conditions that w

A

1

is still optimal.27 Thus, (P

A, xA) are the optimal choices if entry is

accommodated.

The joint profit of the incumbent manufacturer and retailer when entry is accommodated is

Π = (P

A − c)D(PA, xA) − (wE − c)qE − r(xA) − m(I∗

). If wE < c, this will exceed the joint profits

for the case without entry, so accommodating entry is optimal. The argument for why two-block

tariffs are equivalent follows from arguments similar to those in Proposition 5. Q.E.D.

The intuitive rationale for accommodating entry by a more efficient competitor in this model

is quite simple. Accommodation increases the joint profits of the incumbent and retailer, and it

does not alter the constraint set in a way that changes the profit-maximizing price and investment

levels.

B. Deterring Entry

Next I examine the optimal entry deterring contracts. Let w

∗

1 = w

A

1 = w

T

2 be the equilibrium price

in the high price block under two-price all-units discounts and two-block tariffs.28 Observe first

that if wE ≥ w

∗

1

, then entry is automatically deterred by the optimal prices chosen in the absence

of the entry threat under both all-units discounts and two-block tariffs. Thus, the interesting cases

are when wE ∈ (c, wA

1

).

If wE < w∗

1

, deterring entry with either a two-price all-units discount or a two-block tariff will

distort prices and investment levels. Under a two-price all-units discount, the defection constraint

tightens, as a defecting retailer will purchase qE units from the entrant at wE < w∗

1

. Under a

27The sum of the defection profits falls by the fixed amount (wE − c)qE. Therefore, entry does not alter the

wholesale price that minimizes the sum of the defection profits.

28It follows from the proof of Proposition 5 that wA

1 = w

T

1 .

28
two-block tariff, the wholesale price in the first block must be lowered to wE to prevent entry. It

is not immediately clear which tariff is more profitable.

The reason entry changes the defection constraint under a two-price all-units discounts is that

the retailer can profitably purchase a quantity other than D0(P

A, xA) from the incumbent. Thinking back to the general contracting problem (GCP) and modifying it to allow for entry, it is still

true that a two-point contract is optimal. That is, the firms could easily deter entry by charging

very high prices for any quantities other than D(P

A, xA) and D0

(P

A, xA). The problem is that a

two-price all units discount does not replicate the two-point contract because it allows the retailer

to lower its purchases by qE without a penalty when it defects and optimizes against the high price

block. However, a three-price all-units discount will replicate a two-point contract in this case. In

particular, consider a contract that charges a very high wholesale price for any quantity less than

D0(P

A, xA), a price of wA

1

for any quantity from D0(P

A, xA) up to D(PA, xA), and a price of wA

2

for any quantity greater than or equal to D(P

A, xA). This contract will deter the retailer from

purchasing from the entrant if it defects, and it supports the optimal price and investment levels.

It is therefore an optimal contract.

A two-block tariff, in contrast, deters entry only if w1 ≤ wE, which distorts pricing and investment if wE < w∗

1

. If wE is close to w

∗

1

, then the distortion will be small, and firms that use

two-block tariffs will choose to deter entry and tolerate the additional distortion. However, as

wE falls, the distortion will grow, and if wE is close enough to c, firms will be better off allowing

inefficient entry. This is true because the distortion from deterring entry gets larger as wE falls,

and the cost of allowing entry falls to zero as wE approaches c.

If wE is close enough to c, no continuous tariff that promotes investment will deter sufficiently

small scale entry. To see this, observe that for wE close to c, a marginal price above wE over some

range is required to induce upstream investment, but any marginal price above wE over a discrete

range will induce the retailer to accommodate sufficiently small scale entry.

These arguments establish the following proposition.

Proposition 8 Suppose upstream investment is lumpy and investment returns are deterministic.

If entry at scale qE is feasible at price wE ∈ (c, wA

1

), then a three-price all-units discount is an

optimal contract and deters entry by a less efficient rival. If wE is sufficiently close to c, then no

continuous tariff is an optimal contract.29

29“Optimal” here means optimal for the incumbent firms. It is possible that the optimal contract may induce overinvestment, in which case a prohibition of all-units discounts would lead to accommodation that increases welfare by

reducing investment.

29
Summarizing the results in this section, all-units discounts are a stronger entry deterrent than

continuous tariffs, but they are used only to deter less efficient entrants, and they do so without

distorting price and investment relative to the case when the entry threat is absent. If firms are

restricted to continuous tariffs, they may accommodate entry by a less efficient competitor, or they

may deter entry by distorting price and investment.

VI. Implications and Conclusion

The antitrust policy debate over all-units discounts has largely lacked an economic foundation

explaining why firms use these tariffs. This paper, along with that of Kolay et al. (2004), takes

steps toward providing this foundation.

While Kolay et al. examined the role of all-units discounts by a firm offering a menu of discounts

to multiple buyers, this paper takes a step back to examine the simpler environment of bilateral

monopoly, but with the additional complication of double moral hazard. I explored three cases in

which all-units discounts arise in equilibrium: (1) lumpy upstream investment with deterministic

returns; (2) uncertain upstream investment prospects that may become available to the upstream

firm after contracts are signed; and (3) uncertain investment returns. All-units discounts and

continuous two-block tariffs are optimal contracts in the first case. I provided sufficient conditions

for all-units discounts to support a first best outcome and dominate two-block tariffs in the second

and third cases. In all cases, all-units discounts work by giving the retailer an incentive to expand

output to reach the discount threshold while keeping upstream margins high enough to encourage

upstream investment.

Since all-units discounts arise in efficient vertical contracts between bilateral monopolists that

face no threat of entry, it would be inappropriate to presume without evidence that the practice

is anticompetitive simply because the firms employing such tariffs have market power. In fact, the

benefits of all-units discounts may actually increase with the degree of market power, as this is

precisely when sophisticated contracts have the largest effect on incentives.

The antitrust concern raised by all-units discounts is that they may raise barriers to entry and

harm competition. To begin addressing this issue, I extended the model to allow for the possibility

of small scale entry into the upstream market, focusing on the case of lumpy investment and

deterministic returns. In this environment, I showed that the incumbent supplier and retailer will

always accommodate entry by an equally- or more-efficient upstream competitor. Contrary to the

30
conventional view, all-units discounts are not used in this model to deter such entrants. I also find

that all-units discounts deter entry by less efficient competitors, whereas continuous tariffs either

accommodate such entry or deter it by distorting price and investment.

The analysis of entry in this paper is limited to a special case—entry into a single market

served by a downstream monopolist, with no potential for dynamic entry effects. Nonetheless,

the analysis casts doubt on the presumption of some European Courts that all-units discounts

are anticompetitive simply because they have low (or negative) marginal prices around quantity

thresholds. The model suggests that in the presence of double moral hazard, entry-deterring allunits discounts may promote investment that may enhance efficiency.

This paper has not explicitly addressed welfare questions, in part because of the inherent ambiguities in determining whether contracts that align private incentives increase or decrease social

welfare. It is easy to construct examples in which all-units discounts that dominate two-block tariffs

expand investment, reduce double-marginalization, and increase social welfare. It is also possible

to construct examples in which all-units discounts that expand investment decrease social welfare.

The model has some special features that deserve further investigation to understand their

importance. The case of deterministic returns assumed two upstream investment choices. More

generally, with N > 2 upstream investment choices, it seems clear that an N-point contract would

be optimal, but it is not clear whether an N-price all-units discount or an N-block tariff can replicate

the optimal contract. The analysis of entry is limited to the case of deterministic investment returns,

and it would be interesting to compare all-units discounts and continuous tariffs when investment

returns are uncertain. The entry model also assumes that the entrant is not a strategic player and

that the downstream market is served by a monopolist. These assumptions likely limit the scope

for all-units discounts to have anticompetitive effects.

These limitations aside, the model here is both canonical and rich enough to establish that

all-units discounts can have benefits that two-part tariffs and more complex continuous tariffs do

not offer. The model does not support an antitrust approach that treats all-units discounts by

firms with market power as inherently likely to have harmful effects.

31
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32