Wholesale Pricing with Asymmetric Information About the Quality of a Private Label

Abstract

This article derives the optimal wholesale tariff that is proposed by the monopolistic manufacturer of a branded product to a monopolistic retailer if the retailer also sells a private label whose quality is unobserved by the brand manufacturer. The focus is on market-share contracts where the manufacturer controls the quantities of both products. To learn the quality of the private label and diminish the retailer’s information rent, it distorts the quantity of the branded product downwards and that of the private label upwards. The manufacturer can control the quantity of the private label if it combines an excess payment with an end-of-year repayment.

1 Introduction

Over the last decades, retailers in markets for fast-moving consumer goods (FMCG) have increasingly relied on private labels. The availability of a private label enhances, for example, a retailer’s bargaining power vis à vis the manufacturer of a branded product. At the same time, bargaining is affected by various types of asymmetric information, such as information about the goods’ costs, qualities, or aggregate demand.

This applied theory article presents a mechanism design analysis of the optimal wholesale contract proposed by the monopolistic manufacturer of a branded product to a monopolistic retailer that also sells a private label. The complete information version of the model was introduced by Mills (1995). Yehezkel (2008) added asymmetric information: He analyzed a situation where the total mass of consumers is private information to the retailer. Paha (2023) examined private information about marginal production costs. The present article adds private information about the quality of the private label.

Those earlier studies showed how the manufacturer must specify a market-share contract, where the upstream firm controls the quantity of both the branded product and of the private label, such that: First, the manufacturer learns the retailer’s private information; and, second, the manufacturer diminishes the retailer’s information rent. Yehezkel (2008) demonstrates that, for these reasons, the manufacturer wants the retailer to sell a lower quantity of the private label than in the complete information benchmark, or the manufacturer forecloses the sales of the private label altogether.

Contracts that reference rivals such as market-share contracts or loyalty discounts—where a retailer receives a rebate or repayment from the upstream manufacturer if the branded product exceeds a certain market share—have been criticized by researchers and antitrust practitioners because of their potential for exclusionary effects (see Scott Morton, 2013, for a review of this literature). The present article, therefore, asks whether market-share contracts necessarily create concerns of exclusion.

The article rejects such a presumption: Because the quality of the private label is private information to the retailer, the downstream firm has an incentive to overstate this quality, and the manufacturer wants to learn the true quality. This can be achieved if the manufacturer requires the retailer to sell a lower quantity of the branded product than in the complete information benchmark. The manufacturer also wants to diminish the retailer’s information rent, which is achieved if the manufacturer requires the retailer to sell a higher quantity of the private label than in the complete information benchmark. This is the opposite of exclusionary conduct.

This result resembles a finding by Paha (2023): He also demonstrates that the consumers benefit from the market-share contract because the higher quantity can be sold only if the goods’ prices are set below those in the complete information benchmark. The present article, thus, asks whether market-share contracts necessarily benefit consumers, as in Paha (2023). Yehezkel (2008) had already shed doubt on such a view, as he showed that their effects on consumer surplus may be mixed. My article supports his skepticism.

Readers might object that it is not obvious how, in practice, the manufacturer of a branded product is able to control the quantity of a competing private label. Building on Paha (2023), the article demonstrates that the manufacturer may do so by collecting an excess payment at the beginning of the year, which is repaid to the retailer at the end only upon observing the desired quantity of the private label. The article determines the functional form for these end-of-year repayments and interprets their characteristics.

The article is structured as follows: After reviewing the literature in Sect. 2, the model is presented in Sect. 3. Section 4 derives the optimal wholesale tariff. Consumer surplus is analyzed in Sect. 5. Section 6 discusses how the optimal menu can be implemented with end-of-year repayments. Section 7 concludes.

2 Literature

This article builds on Mills (1995), who introduced a model with vertically differentiated goods where an upstream manufacturer sells a branded good at a linear wholesale price to a downstream retailer that may also offer a private label. He shows that the competition that is created by the private label reduces double marginalization by strengthening the position of the retailer vis à vis the brand manufacturer.

This reduces the wholesale and retail price of the branded good, increases the retailer’s profit, decreases the manufacturer’s profit, and increases consumer surplus. These predictions are in line with the empirical findings of, e.g., Draganska et al. (2010), Meza and Sudhir (2010), and Narasimhan and Wilcox (1998). Mills (1999, p. 125) added non-linear pricing strategies, which may be used by the manufacturer “to blunt the force of retailers’ private label programmes”. Non-linear pricing also features prominently in the present article, and in Paha (2023) and Yehezkel (2008).

Yehezkel (2008) extended the seminal model of Mills (1995) by exploring a situation with private information about aggregate demand. In his model, the retailer wants to understate demand and mislead the manufacturer into believing that the retailer’s benefit from selling the branded product is low. Given the revelation principle, the manufacturer can, however, learn demand by distorting the quantity of its branded product downwards: The manufacturer requires the retailer to sell a smaller quantity of the branded product than in the complete information benchmark, because this reduces the retailer’s profit and eliminates its incentive to understate demand.

The manufacturer must, however, leave an information rent to the retailer for revealing its private information truthfully. Yehezkel (2008) shows how the manufacturer may diminish the information rent by implementing a market-share contract. A market-share contract allows the manufacturer “to condition [the retailer’s] payment on the specific quantities of [the manufacturer’s] good and” the private label (Majumdar & Shaffer, 2009, p. 397). Yehezkel (2008) demonstrates that the information rent is diminished by also distorting the quantity of the private label downwards.

The manufacturer, thus, requires the retailer to sell a smaller quantity of the private label than in the complete information benchmark, or exclude the private label altogether. The antitrust implications of market-share contracts were highlighted by Scott Morton (2013). In line with the results of Majumdar and Shaffer (2009), Yehezkel (2008), and further literature,¹ Scott Morton (2013) points out the risk of exclusionary conduct that is created by market-share contracts:² An upstream manufacturer may create a loyalty effect and foreclose other manufacturers if it grants buyers a discount once their share of purchases from this manufacturer exceeds a previously specified threshold.

This is quite different in the present model with private information about the quality of the private label. It will be shown that, in this model, the manufacturer wants the retailer to sell more of the private label than in the complete information benchmark. One, thus, finds the opposite of exclusionary conduct. This suggests that the antitrust analysis of market-share contracts is optimally done on a case-by-case basis. A per se rule, which either allows or bans such contracts, would be likely to generate false positives and false negatives.

To justify the assumption of asymmetric information about the quality of a private label, consider the evidence that is provided, for example, by DelVecchio (2001). He elicits the determinants of consumers’ quality perceptions, which are affected by objective product characteristics as much as by consumers’ preferences and beliefs—especially if they cannot assess the product’s quality perfectly. Retailers often have superior knowledge about these quality perceptions given their direct contact to final consumers. Yet, upstream brand manufacturers are at an informational disadvantage.

Asymmetric information about product quality is also one reason why retailers temporarily delist (or cease to carry and sell) branded products, or why manufacturers refuse to supply certain retailers for some time as part of their bargaining strategies. Such conflict delistings—which allow the firms to assess customers’ substitution patterns—were studied, for example, by Van der Maelen et al. (2016).

A further assumption requires justification: My model assumes that the manufacturer of the branded product can control the quantity of the private label. As a potentially interesting finding, I argue that the manufacturer may steer the quantity of the private label if it offers a repayment to the retailer upon selling the desired private label quantity. In practice, retail supermarkets have indeed been reported to receive substantial payments from the manufacturers of branded products (Villas-Boas, 2007).

Yet, to date, those payments could mostly be rationalized if they were made at the beginning of a year, as is the case for slotting allowances that provide preferred retail shelf space, or merchandising support (Kim & Staelin, 1999; Klein & Wright, 2007). End-of-year payments are more difficult to rationalize. They might even be inefficient in terms of reducing welfare because, in practice, they are frequently paid to big retailers, whose bargaining power may have been enhanced by their use of private labels.³

Courts, policymakers, and authorities have, therefore, expressed skepticism towards those end-of-year payments as they might constitute unfair trading practices in the light of buyer power (European Commission, 2017). My article also asks whether end-of-year payments are a sign of inefficiency that is caused by buyer power.

3 Model

Consider a static, bilateral monopoly model with vertically differentiated products: For example, in an FMCG market. The upstream manufacturer u produces a branded product at constant marginal costs $c_u>0$. The product’s quality is normalized to 1. The downstream retailer d—for example, a supermarket chain—produces a private label product of a weakly lower quality $1-\theta$ with $\theta \in [0,1]$ at marginal costs $c_d \in (0,c_u]$. Assume that production does not require fixed costs. This is not realistic but keeps the model parsimonious and is innocuous in terms of the model’s results.

Sales in the downstream market are made at prices $p_u,p_d$; and consumers’ indirect utility functions are given by Eq. (1) for the branded product u and (2) for the private label d (Mussa & Rosen, 1978):

$$\begin{aligned} v_u{} & {} = \phi - p_u; \end{aligned}$$

(1)

$$\begin{aligned} v_d{} & {} = \phi (1-\theta ) - p_d. \end{aligned}$$

(2)

The variable $\phi$ measures consumers’ preference for quality and is uniformly distributed in the interval $\phi \in [0,1]$ with mass 1. Depending on the prices $p_u$ and $p_d$, a consumer with preference $\phi$ purchases product u if $v_u\ge v_d$: This is the case if her preference for quality is high with $\phi \in [(p_u-p_d)/\theta , 1]$. If $v_u<v_d$ and $v_d\ge 0$, the consumer purchases product d: That is, if her preference for quality is $\phi \in [p_d/(1-\theta ),(p_u-p_d)/\theta )$. Consumers with a much lower preference for quality—those with $v_d<0 \Leftrightarrow \phi \in [0,p_d/(1-\theta ))$—exercise their outside option and do not purchase any of the two goods. This gives the inverse demands (3) and (4), where $q_u,q_d$ denote the quantities of the two products:

$$\begin{aligned} p_u = 1-q_u-(1-\theta )q_d; \end{aligned}$$

(3)

$$\begin{aligned} p_d = (1-\theta ) (1-q_u-q_d). \end{aligned}$$

(4)

The marginal costs and product qualities were determined exogenously prior to the game. While the marginal costs and the quality of the branded product are assumed to be common knowledge, the quality $1-\theta$ of the private label is private information to the retailer: The quality disadvantage $\theta \in [0,1]$ of the private label determines the retailer’s type and is distributed according to density function $g(\theta )>0$. The cumulative distribution function is denoted by $G(\theta )$, with $G(0)=0$ and $G(1)=1$. The inverse hazard rate $H(\theta ) \equiv [1-G(\theta )]/g(\theta )$ is assumed to be non-increasing in $\theta$, as is standard. The inverse hazard rate is an important element of the optimal tariff if the manufacturer is incompletely informed about $\theta$.

This information structure complements Yehezkel (2008), who assumes asymmetric information about the consumers’ willingness to pay $\phi$, and Paha (2023) who studies asymmetric information about the costs $c_d$ of the private label. To rationalize the information structure, assume that the firms interact repeatedly even if this dynamic game is not modeled explicitly. The model presented in this section would then constitute the stage game of this dynamic game.

The assumption that the quality of the branded product is common knowledge is satisfied if the upstream manufacturer keeps quality constant over time, and if the retailer has learned this quality in an earlier period. For example, think of a soft drink, sandwich spread, or candy bar whose recipe has not been changed in years because consumers value this consistency.

The assumption that the quality of the private label product is private information to the retailer is satisfied if the quality of the private label is subject to stochastic shocks. This may be the case if the retailer chooses different suppliers in different years as part of a multi-sourcing strategy. Inadequate quality control can be a byproduct of such a procurement strategy (ter Braak et al., 2013). Hence, the manufacturer of the branded product cannot observe the exact quality of the private label while negotiating with the retailer.

Alternatively, even if the brand manufacturer can assess the objective quality of the private label, the brand manufacturer may not know exactly how this quality is perceived by consumers. It may not know either how these perceptions change over time.

In the stage game studied here, the upstream manufacturer’s and the downstream retailer’s profit are denoted $\pi _u$ and $\pi _d$. They are defined next. The retailer’s reservation profit $\pi _{d,n\ell }(\theta )$ when not listing $(n\ell )$ the branded product $(q_u=0)$ is:

$$\begin{aligned} \pi _{d,n\ell }(\theta )= & {} \underset{q_d}{\max }\left[ \left( (1-\theta ) (1-q_d)-c_d\right) q_d\right] \nonumber \\ {}= & {} \frac{(1-\theta -c_d)^2}{4(1-\theta )}. \end{aligned}$$

(5)

If, however, the retailer considers listing the branded product, assume a direct revelation mechanism and the timing of the game as follows:

• 1. The type $\theta$—the quality disadvantage of the private label product—is realized and observed by the retailer only.

• 2. The manufacturer offers a menu $\langle q_u(\theta ),q_d(\theta ),T(\theta ) \rangle$ to the retailer that entails a lump-sum payment T that is made by the retailer at the beginning of the period. The menu includes $\langle 0,\cdot ,0 \rangle$: The retailer may choose not to deal with the manufacturer, in which case it decides freely about $q_d$ and earns $\pi _{d,n\ell }(\theta )$.

• 3. The retailer reports ${\hat{\theta }} \in [0,1]$ and receives $\langle q_u({\hat{\theta }}),q_d({\hat{\theta }}),T({\hat{\theta }}) \rangle$.

• 4. The retailer chooses $p_u,p_d$ such that the market clears at $q_u({\hat{\theta }}),q_d({\hat{\theta }})$. The payments are made, and the profits $\pi _d({\hat{\theta }}|\theta )$ and $\pi _u({\hat{\theta }}|\theta )=T({\hat{\theta }}) - c_u q_u({\hat{\theta }})$ are realized.

The manufacturer chooses $\langle q_u(\theta ),q_d(\theta ),T(\theta ) \rangle$ pursuing the objective of maximizing its expected profit (6) subject to the retailer’s incentive constraint (IC; it must be profitable to report $\theta$ truthfully) and its individual rationality constraint (IR; it must be profitable to list the branded product):

$$\begin{aligned}{} & {} \underset{T(\cdot ),q_u(\cdot ),q_d(\cdot )}{\max }\int _{0}^{1} \left[ T(\theta )-c_u q_u(\theta ) \right] g(\theta ) d\theta ; \end{aligned}$$

(6)

$$\begin{aligned}{} & {} \pi _{d}(\theta |\theta ) \ge \pi _{d}({\hat{\theta }}|\theta ) \quad \forall \theta ,{\hat{\theta }} \in [0,1] ; \end{aligned}$$

(IC)

$$\begin{aligned}{} & {} \quad \pi _{d}(\theta |\theta )\ge \pi _{d,n \ell }(\theta ) \quad \forall \theta \in [0,1] . \end{aligned}$$

(IR)

The retailer’s revenue $R(q_u,q_d,\theta )$ is shown by (7), with the use of the inverse demands from (3) and (4). Its profit $\pi _d({\hat{\theta }}|\theta )$ is shown by (8):

$$\begin{aligned}{} & {} \begin{aligned} R(q_u,q_d,\theta )&= q_u p_u + q_d p_d \\&= q_u \left( 1- q_u - (1-\theta ) q_d \right) + q_d (1-\theta ) \left( 1 - q_u - q_d \right) ; \end{aligned} \end{aligned}$$

(7)

$$\begin{aligned}{} & {} \pi _d({\hat{\theta }}|\theta ) = R\left( q_u,q_d,\theta \right) - q_d({\hat{\theta }}|\theta )c_d - T({\hat{\theta }}). \end{aligned}$$

(8)

As was established by Mills (1995), it is unprofitable for the retailer to sell the private label if its quality is too low, and it is unprofitable to sell the branded product if the private label offers a comparable quality at lower costs. The current study is only meaningful if the retailer sells both products, so we focus on cases with $q_u,q_d>0$. This requires inequality (9) to hold, which is derived and interpreted in Sect. 4.4 after determining the functional forms of $q_u$ and $q_d$ in equilibrium. The difference of marginal costs $c_u-c_d$ is abbreviated as $\Delta c$:

$$\begin{aligned} \frac{\theta }{1-\theta }c_d< \Delta c < \theta - H(\theta ) \quad \forall \theta \in [0,1]. \end{aligned}$$

(9)

4 Analysis

This section studies the contracts that are signed by the upstream manufacturer and the downstream retailer both in the complete information benchmark and if the quality of the private label is private information to the retailer.

4.1 Complete Information

In the complete information benchmark, the manufacturer optimally specifies the menu $\langle q_u^*(\theta ),q_d^*(\theta ),T^*(\theta ) \rangle$ such as to maximize the industry profit $\pi _{i}(\theta ) = R(q_u,q_d,\theta ) - q_d c_d - q_u c_u$. This gives the optimal quantities $q_u^*(\theta )$ and $q_d^*(\theta )$, as are defined in (10) and (11) (Paha, 2023; Yehezkel, 2008):

$$\begin{aligned} q_{u}^*(\theta ){} & {} = 1-\frac{\theta + \Delta c}{2 \theta }; \end{aligned}$$

(10)

$$\begin{aligned} q_{d}^*(\theta ){} & {} = \frac{\theta + \Delta c}{2 \theta } - \frac{1-\theta +c_d}{2 (1-\theta ) }. \end{aligned}$$

(11)

This maximized industry profit is then shared by the firms through their choice of the fixed fee $T^*(\theta )$. The model assumes that the upstream manufacturer can appropriate all of the producer surplus that comes on top of the retailer’s outside profit $\pi _{d,n\ell }(\theta )$. The manufacturer, therefore, chooses the fixed fee according to (12) such that (IR) binds in equality:

$$\begin{aligned} T^*(\theta ) = q_u^*(\theta )c_u+\left[ \pi _i\left( q_u^*(\theta ),q_d^*(\theta )\right) -\pi _{d,n\ell }(\theta )\right] . \end{aligned}$$

(12)

The retailer earns just its outside profit ${\pi _{d}^*(\theta )=\pi _{d,n\ell }(\theta )}$. As can be seen from (5), this profit rises as $\theta$ falls towards 0, as the quality of the private label approaches that of the branded product. The higher is the quality of the private label, the higher is the retailer’s share of the industry profit.

4.2 Incomplete Information: Controlling q _u and q _d

I now turn to incomplete information: The manufacturer proposes a menu $\langle q_{u,M}(\theta ),q_{d,M}(\theta ),T_{M}(\theta ) \rangle$. The retailer reports ${\hat{\theta }}$ and either accepts or rejects the manufacturer’s offer. Because the menu is conditional on both $q_u$ and $q_d$, it constitutes a market-share contract; hence the index M. If the retailer rejects the manufacturer’s offer and exercises its outside option, it earns $\pi _{d,n\ell }(\theta )$. Upon acceptance of the offer, the retailer receives $\langle q_{u,M}({\hat{\theta }}),q_{d,M}({\hat{\theta }}),T_{M}({\hat{\theta }}) \rangle$, and it earns the profit shown by (13):

$$\begin{aligned} \pi _{d,M}({\hat{\theta }}|\theta ) = R(q_{u,M}({\hat{\theta }}),q_{d,M}({\hat{\theta }}),\theta ) - q_{d,M}({\hat{\theta }}) c_d - T_{M}({\hat{\theta }}). \end{aligned}$$

(13)

The manufacturer specifies the menu such as to induce truth-telling by the retailer and minimize the information rent $U_M({\hat{\theta }}|\theta )$ as is defined in (14):

$$\begin{aligned} U_{M}({\hat{\theta }}|\theta ) = \pi _{d,M}({\hat{\theta }}|\theta ) - \pi _{d,n\ell }(\theta ). \end{aligned}$$

(14)

To interpret the information rent $U_M({\hat{\theta }}|\theta )$: The retailer is compensated for revealing its private information truthfully, so that it earns a profit $\pi _{d,M}({\hat{\theta }}|\theta )$ weakly above its complete information profit $\pi _{d,n\ell }(\theta )$. In a fully revealing mechanism with ${\hat{\theta }}=\theta$, the retailer’s information rent will be denoted $U_{M}(\theta )$.

The information rent varies for different values of $\theta$, as can be inferred from the first derivative $\partial U_{M}(\theta )/\partial \theta$. I determine the marginal information rent (15), using $\pi _{d,n\ell }(\theta )$ from (5), $\pi _{d,M}({\hat{\theta }}|\theta )$ from (13), and the envelope theorem:

$$\begin{aligned} \frac{\partial U_{M}(\theta )}{\partial \theta } = - \left[ q_{d,M}(\theta )\left[ 1- 2 q_{u,M}(\theta )-q_{d,M}(\theta )\right] -\frac{(1-\theta )^2-c_d^2}{4 (1-\theta )^2} \right] . \end{aligned}$$

(15)

One observes that $\partial U_M(\theta )/\partial \theta >0$. The information rent rises in $\theta$ because a greater quality disadvantage $\theta$ of the private label product diminishes the profit $\pi _{d,n\ell }(\theta )$, where the retailer sells just the private label, more strongly than the profit $\pi _{d,M}({\hat{\theta }}|\theta )$, where the retailer also sells the branded product.

The marginal information rent $\partial U_{M}(\theta )/\partial \theta$ is an important element in determining the menu $\langle q_{u,M}(\theta ),q_{d,M}(\theta ),T_{M}(\theta ) \rangle$, as will be done next. Plugging the retailer’s profit (13) in the definition of the information rent (14), and solving for the fixed fee $T_M(\theta )$ yields (16):

$$\begin{aligned} \small T_{M}(\theta ) = q_{u,M}(\theta ) c_u + \left[ \pi _i(q_{u,M}(\theta ),q_{d,M}(\theta ))-\pi _{d,n\ell }(\theta )- \int _{0}^{\theta } \frac{\partial U_{M}({\hat{\theta }}|\theta )}{\partial {\hat{\theta }}} d{\hat{\theta }}\right] . \end{aligned}$$

(16)

Plugging (16) into the manufacturer’s profit function (6) and integrating by parts yields the profit that is shown in (17) (Bolton & Dewatripont, 2005, p. 87):

$$\begin{aligned} \underset{q_{u,M}(\theta ),q_{d,M}(\theta )}{\max }\int _{0}^{1} \left[ \pi _i(q_{u,M}(\theta ),q_{d,M}(\theta ))-\pi _{d,n\ell }(\theta ) - H(\theta )\frac{\partial U_{M}(\theta )}{\partial \theta } \right] g(\theta ) d\theta . \end{aligned}$$

(17)

Maximizing (17) over $q_u$,$q_d$ and jointly solving the first-order conditions yields the optimal quantities presented in (18) and (19):

$$\begin{aligned}{} & {} q_{u,M}(\theta ) = 1-\frac{[\theta -H(\theta )]+\Delta c}{2 [\theta -H(\theta )]}; \end{aligned}$$

(18)

$$\begin{aligned}{} & {} \quad q_{d,M}(\theta ) = \frac{[\theta -H(\theta )] + \Delta c}{2 [\theta -H(\theta )]} - \frac{[1-\theta +H(\theta )] +c_d}{2 [1-\theta +H(\theta )] }. \end{aligned}$$

(19)

One finds $q_{u,M}(\theta )<q_{u}^*(\theta )$ and $q_{d,M}(\theta )>q_{d}^*(\theta )$ for $\theta <1$, with ${q_{u,M}(1)=q_{u}^*(1)}$ and $q_{d,M}(1)=q_{d}^*(1)$.

Before interpreting these inequalities, note that the manufacturer leaves no information rent at the bottom of the distribution to a retailer who demands the smallest amount of the branded product. This is the case for a retailer with a private label of a quality that is equally high as that of the branded product $(\theta =0)$. Such a retailer cannot overstate the quality of the private label.

Technically, in its choice of $T_M(\theta )$, the manufacturer must select a value for the integration constant of $\int _0^{{\hat{\theta }}} (\partial U_M(\hat{\theta }|\theta )|\partial \hat{\theta })d\hat{\theta }$. Given ${\partial U_M(\theta )/\partial \theta >0}$, the manufacturer optimally makes the individual rationality constraint (IR) binding for a retailer with $\theta =0$. It chooses the fixed fee $T_{M}(0)$ such that the information rent equals zero $(U_M(0)=0)$ for this retailer, whose profit, thus, equals $\pi _{d,n \ell }(0)$.

I next turn to the properties of $q_{u,M}(\theta )$ and $q_{d,M}(\theta )$: There are no distortions at the top (${q_{u,M}(1)=q_{u}^*(1)}$, $q_{d,M}(1)=q_{d}^*(1)$) for a retailer with a low quality private label ($\theta =1$ with $H(1)=0$) that demands the largest amount of the branded product. Because the manufacturer specifies the menu with the objective of maximizing its profit, it sets the complete information quantities when dealing with its best customer.

For $\theta <1$, the manufacturer optimally distorts the quantity of the branded product downwards: The manufacturer requires the retailer to sell a quantity below the quantity that is sold in the complete information benchmark: $q_{u,M}(\theta )\le q_{u}^*(\theta )$. This downward distortion of $q_u$ is in line with Yehezkel (2008) and Paha (2023). It reduces the retailer’s profit $\pi _{d,M}({\hat{\theta }}|\theta )$, such that it becomes unprofitable for the retailer to lie about its type $\theta$.

Quite differently, the manufacturer distorts the quantity of the private label upwards: The manufacturer sets $q_{d,M}(\theta )$ weakly above the complete information quantity $q_{d}^*(\theta )$. Because the manufacturer cannot observe the quality of the private label, the retailer has an incentive to overstate this quality. By distorting $q_d$ upward, the manufacturer reduces the retailer’s information rent $U_M({\hat{\theta }}|\theta )$, because the retailer can sell this quantity only at a suboptimally low price.

Finding an upward distortion of $q_d$ is quite different from Yehezkel (2008). In his model, the manufacturer cannot observe aggregate demand, so that the retailer has an incentive to understate demand and benefit from a low payment T. The manufacturer distorts $q_d$ downwards as this prevents the retailer from exploiting the—in fact—higher demand. Therefore, while market-share contracts have exclusionary effects in Yehezkel (2008), the results of the present model refute a hypothesis according to which market-share contracts necessarily give rise to concerns of exclusion.

To summarize, the manufacturer sets a quantity $q_{u,M}(\theta )$ for the branded product below the complete information benchmark $q_u^*(\theta )$. This disincentivizes the retailer to lie about its type $\theta$. Yet, the manufacturer must leave the retailer an information rent as compensation for its private information. To reduce the information rent, the manufacturer distorts $q_d$ upwards unless the retailer is of type $\theta =1$, because a retailer with the lowest quality private label demands the largest amount of the branded product.

4.3 Incomplete Information: Controlling q _u

It proves convenient for the further analysis to study also a situation where the manufacturer proposes a menu $\langle q_{u,1}(\theta ), T_1(\theta ) \rangle$. This menu is conditional on the quantity of only one product: The branded product; hence the index 1.

In this case, the retailer chooses $q_d$ based on reaction function (20), which is found by maximizing the retailer’s profit function (8) with respect to $q_d$:

$$\begin{aligned} q_d(q_u,{\hat{\theta }}|\theta ) = \left\{ \begin{array}{ll} \frac{1-\theta -c_d}{2 (1-\theta ) }-q_u({\hat{\theta }}) &{}\quad {\text {if}}\; q_{u}({\hat{\theta }})<\frac{1-\theta -c_d}{2 (1-\theta ) } \\ 0 &{}\quad {\text {otherwise}}. \end{array}\right. \end{aligned}$$

(20)

After inserting this reaction function into the retailer’s profit function (13) and differentiating the information rent from (14), the marginal information rent is obtained as follows:

$$\begin{aligned} \frac{\partial U_1(\theta )}{\partial \theta } = q_{u,1}(\theta ) \cdot [1-q_{u,1}(\theta )]. \end{aligned}$$

(21)

If we follow the same procedure as was described in Sect. 4.2, the optimal outputs are found as in (22) and (23):

$$\begin{aligned} q_{u,1}(\theta ){} & {} = q_{u,M}(\theta ); \end{aligned}$$

(22)

$$\begin{aligned} q_{d,1}(\theta ){} & {} = \frac{[\theta -H(\theta )] + \Delta c}{2 [\theta -H(\theta )]} - \frac{1-\theta +c_d}{2 (1-\theta ) }. \end{aligned}$$

(23)

To interpret these results: Despite not controlling $q_d$, the upstream manufacturer sets the same quantity $q_{u,M}(\theta )$ as in the market-share contract, which is in line with Yehezkel (2008) and Paha (2023). The retailer then sets its best response $q_{d,1}(\theta )$ to $q_{u,M}(\theta )$. The ensuing quantity $q_{d,1}(\theta )$ is above the complete information benchmark $q_{d}^*(\theta )$, because the private label and the branded product are strategic substitutes, and the manufacturer sets $q_{u,M}(\theta )<q_u^*(\theta )$.

Yet, there is no additional upward distortion of $q_d$. The quantity $q_{d,1}(\theta )$ is weakly below the optimal $q_{d,M}(\theta )$ in the market-share contract, ${q_{d,1}(\theta )\le q_{d,M}(\theta )}$. The manufacturer leaves the retailer an information rent that is suboptimally high.

4.4 Existence

The equilibrium with $q_{u,M}(\theta )$ and $q_{d,M}(\theta )$ exists if $q_u,q_d>0$ applies both in the complete information benchmark and in the situation with incomplete information. The inequalities $q_u^*(\theta )>0$ and $q_d^*(\theta )>0$ can be written as in (24) and (25):

$$\begin{aligned}{} & {} \Delta c < \theta ; \end{aligned}$$

(24)

$$\begin{aligned}{} & {} \frac{\theta }{1-\theta } c_d< \Delta c. \end{aligned}$$

(25)

The inequalities $q_{u,M}(\theta )>0$ and $q_{d,M}(\theta )>0$ can be written as in (26) and (27):

$$\begin{aligned}{} & {} \Delta c < \theta -H(\theta ); \end{aligned}$$

(26)

$$\begin{aligned}{} & {} \frac{\theta -H(\theta )}{1-\theta +H(\theta )} c_d< \Delta c. \end{aligned}$$

(27)

Inequality (24) shows that the branded product is offered alongside the private label if the quality disadvantage of the latter is sufficiently high. Inequality (25) can be restated as $\theta <\Delta c/c_u$. The private label is offered if its quality disadvantage is sufficiently low. The retailer offers both products for intermediate values $\theta \in (\Delta c, \Delta c/c_u)$.

Given the downward distortion of $q_{u,M}(\theta )$ relative to $q_u^*(\theta )$, inequality (26) is more difficult to satisfy than (24). Because $H(\theta )$ is non-increasing in $\theta$, condition (26) gets stricter for $\theta \rightarrow 0$, that is, when the quantity distortion is stronger. Inequalities (24) to (27) are jointly satisfied if (9) applies, as was introduced in Sect. 3:

$$\begin{aligned} \frac{\theta }{1-\theta }c_d< \Delta c < \theta - H(\theta ) \quad \forall \theta \in [0,1]. \end{aligned}$$

(9)

5 Consumer Surplus

In his model with asymmetric information about the costs of the private label, Paha (2023) demonstrates that all consumers benefit from the market-share contract. The results of Yehezkel (2008), however, shed doubt on an overall positive view on market-share contracts. He showed that their effects on consumer surplus may be mixed if the retailer has private information about aggregate demand. This section asks whether market-share contracts benefit consumers in the present model with asymmetric information about the quality of the private label.

The effects of the tariffs on consumer surplus can be inferred from their effect on the prices that are charged by the retailer, as are shown in Table 1. The first row presents the prices in the complete information equilibrium with $q_u^*(\theta )$ and $q_d^*(\theta )$. The second row presents the prices if the retailer is privately informed about $\theta$, and the manufacturer controls the quantity of the branded product only. It sets $q_{u,M}(\theta )$, and the retailer responds by choosing $q_{d,1}(\theta )$. One finds that the downstream retailer charges the same price for the private label as in the complete information equilibrium but a higher price for the branded product. Consumer surplus falls because of those weakly higher prices.

Table 1 Equilibrium prices

If, however, the manufacturer implements the market-share contract and requires the retailer to sell $q_{d,M}(\theta )$, which is above the retailer’s best response $q_{d,1}(\theta )$, the downstream firm reduces the prices of both products relative to $q_{u,M}(\theta ), q_{d,1}(\theta )$ (see the third row in Table 1). Hence, consumer surplus increases if the manufacturer uses a market-share contract compared to a situation where it conditions on $q_u$ only.

Yet, there is no Pareto improvement when comparing the market-share contract with $q_{u,M}(\theta ), q_{d,M}(\theta )$ to the complete information equilibrium with $q_u^*(\theta ),q_d^*(\theta )$. This can be seen from Fig. 1, which shows consumers’ indirect utilities $v_u^*,v_d^*$ for different values of $\phi$ at the complete information prices. It also depicts $v_{u,M},v_{d,M}$ when implementing the market-share contract.

Fig. 1

Welfare effects of the market-share contract

As was shown in Sect. 3, consumers with a high valuation for quality buy the branded product: For values of $\phi$ so high that $v_u^*\ge v_d^*$. And they buy the private label otherwise. In the complete information benchmark, this gives $q_u^*(\theta )$ and $q_d^*(\theta )$ as are shown on the abscissa. If the manufacturer implements the market-share contract, the retailer raises the price of the branded product, so that $v_u^*$ shifts down to $v_{u,M}$. The retailer reduces the price of the private label, so that $v_d^*$ shifts up to $v_{d,M}$.

The effects on consumer surplus are ambiguous: (i) Because of the lower price of the private label, consumer surplus rises for consumers with a low preference for quality $\phi$ who purchase product d; (ii) because of the higher price of the branded product, consumer surplus declines for consumers with a high preference for quality $\phi$ who purchase product u; (iii) therefore, some consumers switch to the private label; (iii.a) those consumers are affected positively by the lower price of the private label if their preference for quality is low; (iii.b) yet, the surplus of switching consumers falls if their preference for quality is high.

Those ambiguous effects on consumer surplus differ from Paha (2023), who finds that the market-share contract raises the surplus for all types of consumers. In his model, the retailer has an incentive to understate the costs of the private label. To act consistently with the understated costs, the retailer must sell a sufficiently large quantity of the private label. This increased quantity must be produced at the true, higher marginal costs. This exerts a disciplining effect on the retailer and reduces its incentive to mis-state the costs of the private label.

In the optimal market-share contract that is presented in Paha (2023), the manufacturer specifies quantities $q_u$, $q_d$ that can be attained by the retailer if it merely reduces the price of the private label below the complete information price. The price of the branded product is held at the complete information level, which generates a Pareto improvement for all consumer types.

This is different in the present model, where the retailer has an incentive to overstate the quality of the private label. Doing this credibly, the retailer must sell a quantity of the private label that is sufficiently high. Yet, there is no disciplining cost effect that would reduce the retailer’s incentive to overstate quality. In the absence of such an effect, the manufacturer requires quantities $q_{u,M}(\theta )$ so low and $q_{d,M}(\theta )$ so high that the retailer reduces $p_d$ below the complete information benchmark and increases $p_u$ above this benchmark. This benefits some consumers and harms others.

Accordingly, there is no presumption according to which market-share contracts are ‘always good’ or ‘always bad’ for consumers. The welfare effects depend on the type of private information.

6 End-of-Year Repayments

Readers might object that the contracts that are observed in reality differ from the optimal menu $\langle q_{u,M}(\theta ),q_{d,M}(\theta ),T_{M}(\theta ) \rangle$. In particular, the manufacturer is typically unable to control $q_d$.

Paha (2023) showed how $\langle q_{u,M}(\theta ),q_{d,M}(\theta ),T_{M}(\theta ) \rangle$ can be modified such that the manufacturer can control $q_d$ nonetheless: The manufacturer may demand an excess payment ${T\ge T_{M}(\theta )}$ at the time of the delivery. And it makes an end-of-year repayment $T-T_{M}(\theta )$: The manufacturer refunds part of the excess payment, after observing that the retailer had sold the contractually specified quantity $q_{d,M}(\theta )$. Such end-of-year repayments are common practice in FMCG markets (Bloom et al., 2000; Kim & Staelin, 1999; Klein & Wright, 2007; Villas-Boas, 2007).

With this deposit-refund scheme, it is incentive-compatible for the retailer to set $q_{d,M}(\theta )$ instead of its best response $q_{d,1}(\theta )$ if inequality (28) is satisfied, which makes use of the retailer’s revenue $R(q_u,q_d)$ as was defined in (7):

$$\begin{aligned} \begin{aligned}&R\left( q_{u,M}(\theta ),q_{d,M}(\theta ),\theta \right) - q_{d,M}(\theta ) c_d - T + [T-T_M(\theta )] \\&\quad > R\left( q_{u,M}(\theta ),q_{d,1}(\theta ),\theta \right) - q_{d,1}(\theta ) c_d - T. \end{aligned} \end{aligned}$$

(28)

The first row presents the profit of the retailer if it sets $q_{d,M}(\theta )$. In this case, the retailer pays T at the beginning of the period, and it receives ${T-T_M(\theta )}$ at the end of the year after the manufacturer has observed $q_{d,M}(\theta )$. The second row shows the profit of the retailer if it sets its best response $q_{d,1}(\theta )$ to $q_{u,M}(\theta )$ and, thereby, forgoes the repayment.

Solving (28) for $T-T_M(\theta )$ yields (29), which demonstrates how the repayment must be specified such that the retailer finds it incentive-compatible to set $q_{d,M}(\theta )$:

$$\begin{aligned} T-T_M(\theta ) > (1-\theta ) [q_{d,M}(\theta )-q_{d,1}(\theta )]^2. \end{aligned}$$

(29)

According to (29), the manufacturer offers a larger repayment if it wants to induce a stronger quantity distortion $q_{d,M}(\theta )-q_{d,1}(\theta )$. Section 4.2 showed that this is the case for a retailer with a higher-quality private label (lower values of $\theta$) who demands less of the branded product.

Hence, retailers with a private label of a similar quality as that of the branded product receive larger repayments. One might be concerned that those retailers are demanding larger repayments as an expression of their buyer power. Such an interpretation would, however, be misleading. It is the manufacturer who specifies the repayments with this specific structure, because this allows the manufacturer to learn the retailer’s quality type.⁴

This raises the question: Are repayment schemes specific to a particular type of information asymmetry? A final answer to this question would require an analysis of all conceivable types of information asymmetry. However, a similar result as the one that is presented here was found by Paha (2023) in his model with private information about the costs $c_d$ of the private label. In both models, the combination of an excess payment with an end-of-year repayment allows the manufacturer to control $q_d$ and implement the optimal market-share contract.

This suggests that contracts that combine an excess payment with an end-of-year repayment are not specific to a particular type of information asymmetry. Only the functional form of the end-of-year repayment depends on the nature of private information.

7 Conclusion

Based on the results of Yehezkel (2008), this article asks whether market-share contracts necessarily create concerns of exclusion. It indicates that a change from asymmetric information about market demand to asymmetric information about the quality of the private label produces a markedly different result: The manufacturer optimally distorts $q_d$ upwards, which is the opposite of exclusionary conduct.

This upward distortion of $q_d$ contributes to lower prices and benefits consumers in comparison to a contract where the manufacturer controls $q_u$ only. However, consumer surplus may still be below the complete information benchmark because of the lower $q_u$.

In addition to characterizing the optimal wholesale tariff, the study also helps explain end-of-year repayments from manufacturers to retailers. Such payments are particularly hard to rationalize if the retailer does not render a clearly recognizable service in return. In the situation that is analyzed in this article, the manufacturer optimally collects an excess payment at the time of the delivery. And the end-of-year repayment, whose value depends on the revealed quality of the private label, ensures that the retailer reports this quality truthfully.

Because those repayments are made by the manufacturer so as to learn the private label’s quality, they are efficient even if higher repayments have to be made to more powerful retailers with a higher quality private label. This is a relevant result in light of a policy discussion according to which such payments have sometimes been deemed unfair.

Future research should analyze potential limits to the applicability of tariffs with excess payments and end-of-year repayments. Can they be applied to a small subset of information asymmetries only, or can they be applied more universally? Neither Paha (2023) nor the present article suggest any obvious limitations to a broad applicability of such tariffs.

The model should also be extended to an upstream and/or downstream oligopoly. Such a model allows analyzing the effects of an exchange of information about the quality of the private label among the producers of branded products. Information exchange cases have gained prominence in antitrust such as in the German drugstore products industry.