THE JOURNAL OF INDUSTRIAL ECONOMICS 0022-1821 Volume 0 October 2023 No. 0 WHOLESALE PRICING WITH ASYMMETRIC INFORMATION ABOUT A PRIVATE LABEL* JOHANNES PAHA†,‡ A monopolistic manufacturer produces a branded good that is sold to final consumers by a monopolistic retailer who also sells a private label. The costs of the private label are unobserved by the manufacturer, which affects the terms of the contract offered by the manufacturer to the retailer. Given the revelation principle, the manufacturer distorts the quantity of the branded product downwards to learn those costs. The manufacturer can further reduce the retailer’s information rent by distorting the quantity of the private label upwards—but this quantity is typically beyond its control. The optimum can nonetheless be achieved when combining a quantity discount with an end-of-year repayment. I. INTRODUCTION THIS APPLIED THEORY ARTICLE PRESENTS A mechanism design analysis of the wholesale contract proposed by the monopolistic manufacturer of a branded product to a monopolistic retailer if the retailer also sells a private label, whose costs are, however, unobserved by the manufacturer. Given the revelation principle, the manufacturer can learn the costs of the pri- vate label by distorting the quantity of the branded product downwards compared to the complete information benchmark, leaving the retailer an information rent. Based on this result, Yehezkel [2008] showed in a related model with asymmetric information about demand how the manufacturer can further reduce the retailer’s information rent by conditioning the con- tract also on the quantity of the private label. These so-called market share contracts were studied, for example, by Majumdar and Shaffer [2009], *I would like to thank Alessandro Bonatti and an anonymous associate editor of the Journal of Industrial Economics, as well as Georg Götz, Thomas Wagner, the audiences and discussants at the EARIE 2017 (Maastricht), Verein für Socialpolitik 2017 (Vienna), and MaCCI 2018 (Mannheim) conferences, as well as the participants of my habilitation talk at Justus-Liebig-University for their helpful comments on this article. Open Access funding enabled and organized by Projekt DEAL. †Authors’ affiliations: Department of Business & Economics, Justus-Liebig-University, Giessen, Germany. e-mail: johannes.paha@wirtschaft.uni-giessen.de ‡Centre for Competition Law and Economics, Stellenbosch University, Stellenbosch, South Africa. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 1 http://crossmark.crossref.org/dialog/?doi=10.1111%2Fjoie.12350&domain=pdf&date_stamp=2023-10-06 2 JOHANNES PAHA Inderst and Shaffer [2010], Mills [2010], and Chen and Shaffer [2019]. While in the model of Yehezkel [2008] the quantity of the private label should be distorted downwards compared to the complete information benchmark, the present model with asymmetric information about costs suggests the opposite: The quantity of the private label should be distorted upwards. This study was motivated by two observations: Brand manufacturers of fast-moving consumer goods have increasingly become subject to com- petition from private label products over the last decades. Private label “products encompass all merchandize sold under a retailer’s brand. That brand can be the retailer’s own name or a name created exclusively by that retailer.”1 The availability of private labels has allowed retailers to appropriate a greater share of industry profits, that is, the sum of profits made by retailers and manufacturers. This is the first observation motivating this study. As the second observation, retail supermarkets have been reported to receive substantial payments from the manufacturers of branded products (Villas-Boas [2007]). Sometimes, those payments are easy to rationalize, such as slotting allowances that provide preferred retail shelf space, or merchan- dising support (Kim and Staelin [1999]; Klein and Wright [2007]). Those well-explicable payments are often made at the beginning of the period. Other payments are made at the end of the period, and they cannot be rationalized that easily. For example, marketing fees have been found to exceed the retailers’ expenses for advertising so that part of those fees lacks an adequate service in return.2 Courts, policymakers, and authorities have sometimes expressed skepti- cism toward payments made by manufacturers to retailers as they might constitute unfair trading practices, which “are a collective name for very heterogeneous practices [… such as] receiving benefits without provid- ing adequate services in return’; (European Commission [2017]). They are hypothesized to be particularly harmful because of the increas- ing bargaining power of retailers. Indeed, end-of-year payments are often paid especially to big retailers, whose bargaining power may have been enhanced by their use of private labels.3 While these concerns 1 Private Label Manufacturers’ Association. “The Store Brands Story.” https://goo.gl/ NrMmEJ (accessed on 15 September 2022). 2 Lebensmittelzeitung. “Wege aus der Sackgasse.” 20 September 2002, https://goo.gl/LrMFkP (accessed on 15 September 2022). 3 Lebensmittelzeitung. “So schnell geben wir nicht auf.” 09 October 2015, https://goo.gl/ pmL186 (accessed on 15 September 2022). Alexander Italianer. “The Devil is in the Retail.” Speech held at the conference on the study of the economic impact of modern retail on choice and innovation in the EU food sector, Brussels, 2 October 2014, https://goo.gl/ax7M5J (accessed on 15 September 2022). © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense https://goo.gl/NrMmEJ https://goo.gl/NrMmEJ https://goo.gl/LrMFkP https://goo.gl/pmL186 https://goo.gl/pmL186 https://goo.gl/ax7M5J ASYMMETRIC INFO IN WHOLESALE PRICING 3 about anticompetitive conduct may be valid in certain cases, the present article suggests that end-of-year payments can have procompetitive effects, too. To show this, the article extends the seminal model of Mills [1995], who established that competition from a private label reduces double marginal- ization by strengthening the position of the retailer vis à vis the brand manufacturer, which lowers the wholesale and retail price of the branded good, raises the retailer’s profit, lowers that of the manufacturer, and raises consumer surplus. These predictions are in line with the empirical findings of, for example, Draganska et al. [2010], Meza and Sudhir [2010], and Narasimhan and Wilcox [1998]. Mills [1999] later extended his model to non-linear pricing, as is in line with evidence provided by Villas-Boas [2007] for the yogurt market in the U.S., or by Bonnet and Dubois [2010] in the French market for bottled water. Yehezkel [2008], to which the present article relates most closely, extended the Mills [1995] model by analyzing certain effects of asymmetric information. In his model, demand is observable by the retailer but not by the manufacturer (with costs being observable by both firms). The retailer thus has an incentive to understate demand to mislead the manufacturer into believing that the benefit to the retailer of accepting the contract and selling the branded product is low. At the same time, the retailer also has an incentive to overstate demand to mislead the manufacturer into believing that the retailer’s profit from selling just the private label is high. Yehezkel [2008] shows that the first effect always dominates the second. Given the revelation principle, the manufacturer can learn demand by distorting the quantity of the branded product downwards. Moreover, if the manufacturer can also control the quantity of the private label, it can diminish the retailer’s information rent by distorting the quantity of the private label downwards, too. This causes higher prices and harms final consumers. The market share contract thus has exclusionary effects.4 The present model, however, yields a very different result. It assumes asym- metric information about the costs of the private label that can be observed by the retailer but not by the manufacturer. Now, the retailer has an incen- tive to understate the costs of the private label to mislead the manufacturer into believing that the retailer’s profit from selling just the private label is high. The manufacturer would also in this case optimally distort the quan- tity of the branded product downwards to learn the costs of the private label. But contrary to Yehezkel [2008], the manufacturer would want to diminish the retailer’s information rent by distorting the quantity of the private label 4 The exclusionary effects of market-share contracts were also studied by Chen and Shaf- fer [2019] in a “naked exclusion’ model. By analyzing entry deterrence in a model with stochastic entry costs and a homogeneous good (that is, without a vertically differentiated private label) their study is, however, focused on a setting different from the one analyzed here. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 4 JOHANNES PAHA upwards. The higher aggregate quantity benefits final consumers by causing prices weakly below those in the complete information benchmark, as will be demonstrated in this article.5 A practitioner might, however, argue that the menu derived from the mechanism design analysis is quite different from contracts observed in the industry. The menu derived from theory entails, for example, a lump-sum pay- ment made by the retailer at the beginning of the period. But such payments have been observed in practice only infrequently (Villas-Boas [2007]). The menu also requires third-party enforcement, which is atypical for contractual relationships in markets for fast-moving consumer goods. The manufacturer must also control the quantity of the private label that, however, is typically beyond the manufacturer’s command in practice. Yet, a deeper investigation reveals that the contracts observed in practice are just variants of the optimal menu that, interestingly, possess some further desirable properties: The application of a quantity discount scheme may help avoiding the lump-sum payment from the retailer to the manufacturer, which is empirically uncommon. Moreover, the pricing function is optimally spec- ified by the manufacturer to serve two further purposes. As its first purpose, the pricing function allows the manufacturer to collect an excess payment that is only refunded to the retailer at the end of the period upon observing the upwards-distorted sales of the private label. The manufacturer can, thus, incentivize the retailer to choose the “correct” quantity of the private label without being able to control it. Because the quantities of the two products are strategic substitutes, the retailer responds by distorting the quantity of the branded product down- wards. Therefore, as its second purpose, the manufacturer can choose the curvature of the pricing function such that it is a best response for the retailer to set the level of the branded product desired by the manufacturer, even if the retailer is formally free to choose also any other quantity. As the opti- mal menu benefits consumers, this article presents one reason why end-of-year payments may—contrary to the concerns of some courts, policymakers, and authorities—be considered procompetitive under the circumstances analyzed here. The article is structured as follows. The model is presented in Section II. Section III demonstrates the complete information benchmark before turn- ing to incomplete information about costs. Section IV demonstrates how the optimal mechanism can be implemented realistically. Section V concludes the article. Proofs are provided in the Appendix. 5 Acconcia et al. [2008] study a related model where the upstream manufacturer can neither observe downstream demand nor the retailer’s sales efforts. They compare contracts where only sales or sales and the retail price are contractible. Although conceptually related, the specific problem studied by Acconcia et al. [2008] differs from the one investigated here. The retailer in their model does not sell a private label so that the firms cannot condition the tariff on its sales, which is central to solving the asymmetric information problem analyzed in the present article. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 5 II. THE MODEL Consider a static, bilateral monopoly model with one upstream manu- facturer and one downstream retailer, as was proposed by Mills [1995]. The downstream retailer sells two vertically differentiated products to final customers. One product is produced by the upstream manufacturer and the other by the downstream retailer; they are thus indexed by u and d. The sales in the downstream, retail market are made at prices pu, pd . The products have exogenously determined qualities su and sd with 0 < sd < su. Hence, product u is thought of as a high-quality, branded product whereas product d is a lower quality private label. The quality differential is defined as Δs = su − sd > 0. The qualities su and sd are observed by the firms and the final customers. To specify downstream demand, final customers’ preference for quality is measured by the variable 𝜙 that is uniformly distributed in the interval 𝜙 ∈ [0, 1] with mass 1. Consumers’ indirect utility function for the high-quality product is given by equation (1), and by equation (2) for the low-quality prod- uct. vu = 𝜙su − pu,(1) vd = 𝜙sd − pd .(2) The demand model was introduced by Mussa and Rosen [1978] and is in line with the discrete choice specifications used in the empirical studies of, for example, Draganska et al. [2010] or Meza and Sudhir [2010]. Other than in Yehezkel [2008], who assumed that the demand parameter 𝜙 is observed by the retailer but not by the manufacturer, 𝜙 is common knowledge in the present model. The high-quality product is produced by the upstream manufacturer at con- stant marginal costs cu. The downstream retailer obtains the private label at marginal costs cd with cd ≤ cu (Inderst and Shaffer [2010]). As an exten- sion to Mills [1995], the marginal costs cd(𝜃) = 𝜃 are a parsimonious function of the retailer’s continuous type 𝜃 ∈ [0, 1]. The cost differential is defined as Δc(𝜃) = cu − cd(𝜃) ≥ 0 for all 𝜃. Production does not require fixed costs. The type 𝜃 is distributed according to the density function g(𝜃) with g(𝜃) > 0. The cumulative distribution function is denoted by G(𝜃) with G(0) = 0 and G(1) = 1. The variable H(𝜃) denotes the inverse hazard rate. H(𝜃) ≡ 1 − G(𝜃) g(𝜃) .(3) The inverse hazard rate H(𝜃) is assumed to be non-increasing in 𝜃, as is stan- dard. It is an important element of the optimal tariff if the manufacturer is incompletely informed about 𝜃. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 6 JOHANNES PAHA Therefore, let all parameters of the model be common knowledge except for the type 𝜃, which is private information to the retailer. Consider a direct revelation mechanism and the timing of the game as follows: 1. The type 𝜃 is realized and observed by the retailer only. 2. The manufacturer (the principal) offers a menu ⟨qu(𝜃), qd(𝜃),T(𝜃)⟩ to the retailer (the agent) that specifies combinations of the quantity qu(𝜃) of the branded product, the quantity qd(𝜃) of the private label, and a lump-sum payment T(𝜃) made at the beginning of the period. The menu includes ⟨0, ⋅, 0⟩, that is, the retailer may choose not to deal with the manufacturer, in which case it decides freely about qd . As a benchmark, the article also analyzes the menu ⟨qu(𝜃),T(𝜃)⟩ that is conditional on the quantity of the branded product only. 3. The retailer reports ̂𝜃 ∈ [0, 1] and receives ⟨qu( ̂𝜃), qd( ̂𝜃),T( ̂𝜃)⟩. Whenever necessary, I will denote the retailer’s report ̂𝜃 to distinguish it from the true 𝜃. 4. The retailer chooses the downstream prices pu, pd such that the market clears at these quantities. The payments are made, and the profits 𝜋d ( ̂ 𝜃|𝜃 ) of the downstream retailer and 𝜋u ( ̂ 𝜃|𝜃 ) = T ( ̂ 𝜃 ) − cuqu ( ̂ 𝜃 ) of the upstream manufacturer are realized. The manufacturer chooses the menu ⟨qu(𝜃), qd(𝜃),T(𝜃)⟩ pursuing the objective of maximizing (4) subject to (IC) and (IR). max T(⋅),qu(⋅),qd (⋅) ∫ 1 0 [ T(𝜃) − cuqu(𝜃) ] g(𝜃)d𝜃,(4) 𝜋d ( 𝜃|𝜃 ) ≥ 𝜋d ( ̂ 𝜃|𝜃 ) ∀𝜃, ̂𝜃 ∈ [0, 1],(IC) 𝜋d ( 𝜃|𝜃 ) ≥ 𝜋d,n𝓁(𝜃) ∀𝜃 ∈ [0, 1].(IR) Condition (IC) represents the retailer’s incentive constraint. If it is satisfied, the retailer prefers ⟨qu(𝜃), qd(𝜃),T(𝜃)⟩ after reporting the true 𝜃 to any other ⟨qu( ̂𝜃), qd( ̂𝜃),T( ̂𝜃)⟩ after reporting an incorrect ̂𝜃. The profit 𝜋d( ̂𝜃|𝜃) is shown by (5). 𝜋d( ̂𝜃|𝜃) = R( ̂𝜃|𝜃) − qd( ̂𝜃|𝜃)cd(𝜃) − T( ̂𝜃).(5) The retailer’s revenue R(qu, qd) is presented in (6), where the second line makes use of the indirect utility functions (1) and (2). Further information on determining the inverse demand functions pu and pd is provided in the Appendix. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 7 R(qu, qd) = qu pu + qd pd = qu ( su − suqu − sdqd ) + qd ( sd − sdqu − sdqd ) . (6) This function assumes qu, qd > 0. This assumption will be justified below in this section, where conditions will be presented that ensure positive demand. In equation (5), the revenue R( ̂𝜃|𝜃) is denoted as a function of ̂𝜃 and 𝜃 only instead of being denoted as a function of qu and qd , which are fully specified by ̂ 𝜃 and 𝜃: The manufacturer chooses qu based on the retailer’s report ̂𝜃. If the retailer is free to choose the quantity of the private label, it sets qd based on the manufacturer’s choice of qu ( ̂ 𝜃 ) and its type 𝜃. The retailer’s reaction function (7) is found by maximizing 𝜋d ( ̂ 𝜃|𝜃 ) w.r.t. qd . qd(qu, ̂ 𝜃|𝜃) = { sd−cd (𝜃) 2sd − qu ( ̂ 𝜃 ) if qu( ̂𝜃) < qu ≡ sd−cd (𝜃) 2sd . 0 otherwise (7) The retailer optimally sets qd(qu, ̂ 𝜃|𝜃) = 0 if the manufacturer offers a quantity of the branded product weakly above qu. Condition (IR) represents the retailer’s individual rationality constraint. The high-quality product is listed if the profit 𝜋d(𝜃|𝜃) of the downstream retailer when selling both products is weakly greater than its reservation profit 𝜋d,n𝓁(𝜃) when not listing (n𝓁) the branded product, that is, selling the private label only. The profit 𝜋d,n𝓁(𝜃) is shown in (8). 𝜋d,n𝓁(𝜃) = max qd [R(0, qd(𝜃)) − qd(𝜃)cd(𝜃)](8) = [ sd − cd(𝜃) ]2 4sd . The profit 𝜋i(𝜃) of a vertically integrated firm (see equation (9)) provides a benchmark when solving the model. 𝜋i(𝜃) = R ( qu, qd ) − qdcd(𝜃) − qucu.(9) Maximizing the industry profit 𝜋i(𝜃) w.r.t. qu and qd gives the optimal quan- tities q∗u(𝜃) and q∗ d (𝜃) as are shown in (10) and (11), as well as the optimal industry profit 𝜋∗i (𝜃) shown in (12). q∗u(𝜃) = 1 − Δs + Δc(𝜃) 2Δs ,(10) q∗d(𝜃) = Δs + Δc(𝜃) 2Δs − sd + cd(𝜃) 2sd ,(11) © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 8 JOHANNES PAHA 𝜋 ∗ i (𝜃) = 𝜋d,n𝓁(𝜃) + [ Δs − Δc(𝜃) ]2 4Δs .(12) Assumption (13) ensures that all relevant equilibrium and off-equilibrium quantities qu and qd , which will be explored in this article, are greater than zero, as is proven in the Appendix. Δs sd cd(𝜃) < Δc(𝜃) < Δs −H(𝜃) ∀𝜃 ∈ [0, 1].(13) This precludes cases where the retailer maximizes profits by selling only one of the two products. Foreclosure concerns played a great role in the model of Yehezkel [2008] with asymmetric information about demand. In his model, the manufacturer requires the retailer to sell too little of the private label. Sometimes, the manufacturer would foreclose the private label altogether even if the product would be sold in the complete information benchmark. It will, however, be shown that in the present model with asymmetric information about the private label’s marginal costs, the incompletely informed manufac- turer requires the retailer to sell too much of the private label. Hence, foreclo- sure is not a concern in this case, and I will concentrate on a situation where both goods are sold. III. THE MENUS Section III(i) establishes the complete information benchmark. Section III(ii) presents the adverse selection problem occurring if the cost type 𝜃 is private information to the retailer. The manufacturer must leave the retailer an information rent when offering a menu ⟨qu(𝜃),T(𝜃)⟩ that is conditional on qu only. Section III(iii) demonstrates how the manufacturer can diminish the retailer’s information rent by offering a menu ⟨qu(𝜃), qd(𝜃),T(𝜃)⟩ that places an additional restriction on the quantity qd of the private label. III(i). Complete Information Assume that the manufacturer observes the retailer’s type 𝜃 (complete information; indexed by c) and offers ⟨qu,c(𝜃),Tc(𝜃)⟩ so that the retailer earns 𝜋d,c(𝜃) as defined in (14). 𝜋d,c(𝜃) = R ( qu,c(𝜃), qd,c(𝜃) ) − qd,c(𝜃)cd(𝜃) − Tc(𝜃).(14) The manufacturer optimally sets the fixed fee Tc(𝜃) shown in (15) such that the retailer’s individual rationality constraint binds in equality (𝜋d,c(𝜃) = 𝜋d,n𝓁(𝜃)). Tc(𝜃) = qu,c(𝜃)cu + [ 𝜋i ( qu,c(𝜃), qd,c(𝜃) ) − 𝜋d,n𝓁(𝜃) ] .(15) © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 9 This results in the manufacturer’s profit shown in (16). 𝜋u,c(𝜃) = 𝜋i ( qu,c(𝜃), qd,c(𝜃) ) − 𝜋d,n𝓁(𝜃).(16) Because 𝜋d,n𝓁(𝜃) is independent of qu,c the manufacturer chooses the quantity qu,c(𝜃) = q∗u(𝜃) that maximizes industry profits 𝜋∗i (𝜃). The retailer uses reaction function (7) to determine the quantity qd,c(𝜃) = q∗ d (𝜃) as in the vertically inte- grated situation (Yehezkel [2008]). Assumption (13) ensures 0 < qu,c(𝜃) < 1 and 0 < qd,c(𝜃) < 1. III(ii). Incomplete Information: Conditioning on qu Now, assume that the manufacturer does not observe the retailer’s type 𝜃. The manufacturer offers a menu ⟨qu,i1(𝜃),Ti1(𝜃)⟩ that conditions the tariff on the quantity of one product, that is, qu only. This situation is indexed i1 (incom- plete information, conditional on one product). It serves as a benchmark for the situation where the manufacturer controls the quantities of both goods as is analyzed in Section III(iii). In a direct revelation mechanism, the retailer reports ̂𝜃 and receives the pair qu,i1( ̂𝜃) and Ti1( ̂𝜃). It earns the profit 𝜋d,i1( ̂𝜃|𝜃) shown in (17), where qd,i1( ̂𝜃|𝜃) denotes its best response to qu,i1( ̂𝜃) given its type 𝜃. 𝜋d,i1( ̂𝜃|𝜃) = R(qu,i1( ̂𝜃), qd,i1( ̂𝜃|𝜃)) − qd,i1( ̂𝜃|𝜃)cd(𝜃) − Ti1( ̂𝜃).(17) An adverse selection problem arises for all cost types but the lowest. The retailer has an incentive to understate the costs of the private label ( ̂𝜃 < 𝜃) and, thus, exaggerate the reservation profit 𝜋d,n𝓁( ̂𝜃), which would result in a lower payment Ti1( ̂𝜃). Let Ui1( ̂𝜃|𝜃) denote the retailer’s additional profits earned over its complete information profits 𝜋d,n𝓁(𝜃) in this case. Ui1( ̂𝜃|𝜃) = 𝜋d,i1( ̂𝜃|𝜃) − 𝜋d,n𝓁(𝜃).(18) Following the revelation principle, the manufacturer may choose ⟨qu,i1(𝜃),Ti1(𝜃)⟩ to induce truth-telling ( ̂𝜃 = 𝜃) and minimize the information rent Ui1(𝜃|𝜃), which will be abbreviated as Ui1(𝜃). Using the functional forms of 𝜋d,i1( ̂𝜃|𝜃) and 𝜋d,n𝓁(𝜃), and applying the envelope theorem, one obtains the marginal information rents as are shown in (19). 𝜕Ui1(𝜃) 𝜕𝜃 = qu.(19) The sign of 𝜕Ui1(𝜃)∕𝜕𝜃 > 0 shows that the retailer’s information rent from understating its cost type 𝜃 rises in 𝜃. Lemma 1 characterizes the fully reveal- ing menu ⟨qu,i1(𝜃),Ti1(𝜃)⟩ chosen by the manufacturer. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 10 JOHANNES PAHA Lemma 1. The manufacturer chooses qu,i1(𝜃) and Ti1(𝜃) as are shown by (20) and (21). qu,i1(𝜃) = q∗u(𝜃) − H(𝜃) 2Δs ,(20) Ti1(𝜃) = qu,i1(𝜃)cu + [ 𝜋i(qu,i1(𝜃), 𝜃) − 𝜋d,n𝓁(𝜃) − ∫ 𝜃 0 𝜕Ui1( ̂𝜃|𝜃) 𝜕 ̂ 𝜃 d ̂𝜃 ] .(21) For all 𝜃 < 1, the manufacturer distorts qu,i1(𝜃) downwards in comparison to the complete information quantity q∗u(𝜃). The retailer reveals ̂𝜃 = 𝜃 and sets qd,i1 according to best response function (7), which gives qd,i1(𝜃) = q∗d(𝜃) + H(𝜃) 2Δs .(22) Proof. See the Appendix. ◾ In comparison to the complete information solution, the manufacturer distorts qu,i1(𝜃) downwards for all but the retailer’s highest cost type 𝜃 = 1 (no distortion at the top), who demands most of the manufacturer’s branded product. The downward distortion of qu for 𝜃 < 1 reduces the retailer’s infor- mation rent that, however, still takes a positive value for all but the lowest cost type 𝜃 = 0 (no information rent at the bottom). A downward distortion of qu was also found by Yehezkel [2008] if the manufacturer cannot observe demand. The predictions of his model and the present one, where the man- ufacturer cannot observe cd , differ however perceptibly if the manufacturer can also control qd , which is shown next. III(iii). Incomplete Information: Conditioning on qu and qd The menu ⟨qu,i1(𝜃),Ti1(𝜃)⟩ induces truth-telling ( ̂𝜃 = 𝜃). The retailer is, how- ever, free to choose qd . The firm selects qd,i1(𝜃) to maximize its profit 𝜋d,i1(𝜃|𝜃) and thereby also its information rent. If, however, the manufacturer is able to control both qu and qd through a menu ⟨qu,i2(𝜃), qd,i2(𝜃),Ti2(𝜃)⟩ it can dimin- ish the retailer’s information rent Ui2(𝜃) ≤ Ui1(𝜃) and charge a weakly higher payment Ti2(𝜃) ≥ Ti1(𝜃), which lowers the retailer’s profits 𝜋d,i2( ̂𝜃|𝜃) as defined by (23). 𝜋d,i2 ( ̂ 𝜃|𝜃 ) = R ( qu,i2 ( ̂ 𝜃 ) , qd,i2 ( ̂ 𝜃 )) − qd,i2 ( ̂ 𝜃 ) cd ( 𝜃 ) − Ti2 ( ̂ 𝜃 ) .(23) This result, which was shown by Yehezkel [2008], follows naturally from the fact that the contract space of the menu ⟨qu,i2(𝜃), qd,i2(𝜃),Ti2(𝜃)⟩ is a super-set of the menu ⟨qu,i1(𝜃),Ti1(𝜃)⟩. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 11 Yet, it is not obvious how the optimal contract should look like and whether consumers gain or loose from it. For example, Yehezkel [2008] assumed the retailer to possess private information about demand for the two products. In his model, the manufacturer imposes a maximum restriction on the quantity of the private label, or the manufacturer even forecloses the product alto- gether. This harms consumers. Should we expect a similar result also in the present model with asymmetric information about the production costs of the private label? Proposition 1 demonstrates the properties of the tariff ⟨qu,i2(𝜃), qd,i2(𝜃), Ti2(𝜃)⟩. The analysis is done under the assumption that the manufacturer has the power to control qd . Section IV shows under what conditions this may be the case. Proposition 1. If the manufacturer offers the menu ⟨qu,i2(𝜃), qd,i2(𝜃),Ti2(𝜃)⟩, it optimally sets the same quantity of the branded product as in the case of ⟨qu,i1(𝜃),Ti1(𝜃)⟩, which is shown by (24). qu,i2(𝜃) = qu,i1(𝜃).(24) The manufacturer, however, chooses a higher quantity for the private label. qd,i2(𝜃) = qd,i1(𝜃) + H(𝜃) 2sd .(25) It charges Ti2(𝜃) as defined in (26), so that Ti2(𝜃) ≥ Ti1(𝜃). Ti2(𝜃) = qu,i2(𝜃)cu + [ 𝜋i(qu,i2(𝜃), qd,i2(𝜃)) − 𝜋d,n𝓁(𝜃) − ∫ 𝜃 0 𝜕Ui2( ̂𝜃|𝜃) 𝜕 ̂ 𝜃 d ̂𝜃 ] . (26) The fully revealing mechanism ⟨qu,i2(𝜃), qd,i2(𝜃),Ti2(𝜃)⟩ diminishes the retailer’s information rents (Ui2(𝜃) ≤ Ui1(𝜃)) with Ui2( ̂𝜃|𝜃) = 𝜋d,i2( ̂𝜃|𝜃) − 𝜋d,n𝓁(𝜃).(27) Proof. See the Appendix. ◾ Part of Proposition 1 is in line with Yehezkel [2008]: The manufacturer chooses the same qu,i2(𝜃) = qu,i1(𝜃) for both menus. Intuitively, when using ⟨qu,i1(𝜃),Ti1(𝜃)⟩ the manufacturer has an incentive to lower qu in order to reduce the retailer’s information rent. This objective is no different when using ⟨qu,i2(𝜃), qd,i2(𝜃),Ti2(𝜃)⟩ so that the manufacturer would not want to raise qu above qu,i1(𝜃). The firm also has no incentive to lower qu below qu,i1(𝜃) because this reduces Ti2(𝜃) by lowering the industry profit 𝜋i, which can be distributed among the firms. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 12 JOHANNES PAHA The manufacturer, however, modifies the quantity qd of the private label. And this result differs from those of Yehezkel [2008]. In his model, the manufacturer distorts qd downwards, whereas in my model the manufac- turer distorts qd upwards. This upward distortion depresses the retailer’s profit 𝜋d,i2( ̂𝜃|𝜃) and information rent Ui2( ̂𝜃|𝜃) because of two effects. Firstly, the high quantity qd,i2(𝜃) can only be sold at a low price. Secondly, pro- ducing qd,i2(𝜃) instead of the retailer’s best response qd,i1(𝜃) raises its costs. Both effects diminish the retailer’s profit and, thus, information rents (Ui2(𝜃) ≤ Ui1(𝜃)). This allows the manufacturer to extract a higher payment Ti2(𝜃) ≥ Ti1(𝜃) despite a lower joint profit. As is standard, one does not find a quantity distortion at the top (for 𝜃 = 1) and no information rent at the bottom (for 𝜃 = 0). The differences between the predictions of Yehezkel [2008] and the present model can be explained as follows: Given that in his model the manufacturer cannot observe demand, the retailer has an incentive to understate demand. That way, the manufacturer would set the payment T suboptimally low, leav- ing the retailer with a profit above its reservation profit 𝜋d,n𝓁(𝜃). The man- ufacturer, however, reduces the retailer’s information rents by setting a low quantity qd that prevents the retailer from exploiting the, in fact, high demand. For certain parameter constellations, the manufacturer would even use an exclusive dealing contract to foreclose the private label. This is the case even if both products would be sold under complete information. In my model, however, the manufacturer cannot observe the production costs of the private label, and the retailer has an incentive to understate these costs. That way, the manufacturer would overestimate the reservation profit that the retailer could earn when selling the private label only. The manufac- turer would therefore set the payment T suboptimally low, leaving the retailer with a profit above its reservation profit 𝜋d,n𝓁(𝜃). The manufacturer reduces the retailer’s profit and, thus, information rents by setting a high quantity qd that can only be sold at low prices while being produced at high costs. Hence, other than in the model with asymmetric information about demand, asym- metric information about the private label’s costs lowers foreclosure concerns instead of raising them. In other words, in Yehezkel [2008], a retailer has a high willingness to pay for the branded product if the retailer is of the “high demand type”, so that it wants a high qd . In the present model, a retailer has a high willingness to pay for the branded product if it is of the “high cost type”, so that it wants a low qd . This affects the retailer’s best response qd(qu, ̂ 𝜃|𝜃)when it is not part of the contract; see equation (7). The sign of 𝜕qd(qu, ̂ 𝜃|𝜃)∕𝜕𝜃, which is negative in the present model, flips depending on the nature of the private information; see equation (6) in Yehezkel [2008]. Therefore, the finding of Yehezkel [2008] (downward distortion of qd) can easily be reversed depending on the nature of the private information. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 13 TABLE I EQUILIBRIUM PRICES pu = su − suqu − sd qd pd = sd − sd qu − sd qd q∗u(𝜃), q ∗ d (𝜃) su+cu 2 sd+cd (𝜃) 2 qu,i1(𝜃), qd,i1(𝜃) su+cu 2 + H(𝜃) 2 sd+cd (𝜃) 2 qu,i2(𝜃), qd,i2(𝜃) su+cu 2 sd+cd (𝜃) 2 − H(𝜃) 2 III(iv). Welfare Analysis The contracts ⟨qu,c(𝜃),Tc(𝜃)⟩ under complete information, ⟨qu,i1(𝜃),Ti1(𝜃)⟩ under incomplete information when conditioning on qu only, and ⟨qu,i2(𝜃), qd,i2 (𝜃),Ti2(𝜃)⟩ when conditioning on both qu and qd affect consumer surplus. This can be seen from Table I showing the equilibrium prices pu, pd for all three cases.6 When the menu is conditional on qu only, the retailer chooses qd,i1(𝜃) ≥ q∗ d (𝜃) such that this increase just balances the manufacturer’s downward distortion of qu,i1(𝜃) ≤ q∗u(𝜃), so that qu,i1(𝜃) + qd,i1(𝜃) = q∗u(𝜃) + q∗ d (𝜃) applies. The price pu of the branded product weakly increases while pd remains at the same level as in the complete information benchmark. Those weakly higher prices lower consumer surplus. A similar effect can be seen in the model of Yehezkel [2008] where, despite the different nature of the information asymmetry, the manufacturer also distorts the quantity qu of the branded product downwards. When the menu is conditional on both qu and qd , the manufacturer chooses qu,i2(𝜃) and qd,i2(𝜃) such that the total output rises above the com- plete information benchmark (qu,i2(𝜃) + qd,i2(𝜃) ≥ q∗u(𝜃) + q∗ d (𝜃)). The price pu is the same as in the complete information benchmark, but pd is weakly lower. Those weakly lower prices raise consumer surplus. This result is quite different from those obtained by Yehezkel [2008]. In his model, where the manufacturer is incompletely informed about demand, the manufacturer distorts both qu and qd downwards, which causes higher prices and lowers consumer surplus. IV. APPLICATION Industry observers might argue that the contracts observed in reality look quite different than the optimal menu. This section demonstrates how the- ory and practice can be reconciled: The observed contracts are actually vari- ants of the menu ⟨qu,i2(𝜃), qd,i2(𝜃),Ti2(𝜃)⟩ that are used to cope with certain 6 Changes in the exogenous parameters cu, su, and sd have intuitive effects on the equilib- rium outputs. One finds 𝜕qu∕𝜕cu < 0, 𝜕qd∕𝜕cu > 0, 𝜕qu∕𝜕su > 0, 𝜕qd∕𝜕su < 0, 𝜕qu∕𝜕sd < 0, and 𝜕qd∕𝜕sd > 0 in all three cases. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 14 JOHANNES PAHA difficulties in practice. For example, the manufacturer may not be able to control qd , which can be solved by end-of-year repayments, as is shown in Section IV(i). Evidence also suggests that a lump-sum payment T is uncom- mon in practice (potentially due to the retailer’s financing constraints), and that the retailer may, additionally, be free to choose quantities other than qu,i2(𝜃), qd,i2(𝜃). Section IV(ii), therefore, illustrates how T can be replaced by empirically observed quantity discount schemes that – at the same time – incentivize the retailer to choose the “correct” quantities. IV(i). End-Of-Year Repayment While the manufacturer can restrict deliveries of the branded product to distort the quantity downwards to qu,i2(𝜃), the manufacturer typically lacks the power to control the retailer’s sales of the private label to distort its quantity upwards to qd,i2(𝜃). Lemma 2, therefore, establishes how the man- ufacturer can overcome its inability to control qd if the contract is enforced by a third party. Instead of requiring the retailer to set qd = qd,i2(𝜃), the manufacturer uses a tariff Tf (qd , 𝜃) conditional on 𝜃 where a third party imposes a fine F(𝜃) on the retailer in case it sets qd ≠ qd,i2(𝜃). The index f stands for fine. Choosing F(𝜃) sufficiently high incentivizes the retailer to set qd = qd,i2(𝜃). This obvious result serves as a stepping stone for the further analysis. Lemma 2. The retailer adheres to qd,i2(𝜃) if the manufacturer proposes a menu ⟨qu,i2(𝜃),Tf (qd , 𝜃),F(𝜃)⟩ with Tf (qd , 𝜃) = Ti2(𝜃) + F(𝜃) ( 1 − I(qd , 𝜃) ) ,(28) F(𝜃) > F(𝜃) ≡ H(𝜃)2 4sd , and(29) I(qd , 𝜃) = { 1 if qd = qd,i2(𝜃), 0 if qd ≠ qd,i2(𝜃), (30) although the retailer is free to choose quantities other than qd,i2(𝜃). Proof. Lemma 1 showed that the retailer chooses its best response to qu,i2(𝜃), which is qd,i1(𝜃), if the manufacturer does not impose a restriction on qd . Now, setting qd ≠ qd,i2(𝜃), so that I(qd , 𝜃) = 0, is punished by a fine F(𝜃). The retailer then earns the profit shown in the second row of (31). R(qu,i2(𝜃), qd,i2(𝜃)) − qd,i2(𝜃)cd(𝜃) − Ti2(𝜃) >(31) R(qu,i2(𝜃), qd,i1(𝜃)) − qd,i1(𝜃)cd(𝜃) − Ti2(𝜃) − F(𝜃). © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 15 The first row of (31) shows the retailer’s profit from setting the quantity qd,i2(𝜃) as desired by the manufacturer. Using the functional forms of qu,i2(𝜃), qd,i2(𝜃), and qd,i1(𝜃), it is straightforward to show that inequality (31) applies if F(𝜃) satisfies (29). ◾ There is not much evidence that manufacturers actually relied on such fines. One observes, however, that manufacturers of fast-moving consumer goods frequently make end-of-year (re)payments to retailers (Kim and Staelin [1999]; Bloom et al. [2000]; Klein and Wright [2007]; Villas-Boas [2007]). Based on this evidence, Lemma 3 suggests that instead of using a fine, the manufacturer may also use a scheme where it collects F(𝜃) as a deposit at the beginning of the period, which is part of the total payment Te(qd , 𝜃) and refunded at the end of the period upon observing qd = qd,i2(𝜃). The index e stands for end-of-year repayment. Lemma 3. The retailer adheres to qd,i2(𝜃) if the manufacturer proposes a menu ⟨qu,i2(𝜃),Te(qd , 𝜃),F(𝜃)⟩ with Te(qd , 𝜃) = [Ti2(𝜃) + F(𝜃)] − F(𝜃)I(qd , 𝜃) and F(𝜃) > F(𝜃),(32) although the retailer is free to choose quantities other than qd,i2(𝜃). The proof of Lemma 3 is straightforward and can be omitted. One sees that Te(qd , 𝜃) = Tf (qd , 𝜃) and, therefore, 𝜋d,e(𝜃) = 𝜋d,f (𝜃) apply for all values of qd so that the retailer adheres to qd,i2(𝜃) as follows from Lemma 2.7 Note that both contracts must be enforced by a third party. In case of ⟨qu,i2(𝜃),Tf (qd , 𝜃),F(𝜃)⟩, this party ensures that the retailer pays the fine upon setting qd ≠ qd,i2(𝜃). In case of ⟨qu,i2(𝜃),Te(qd , 𝜃),F(𝜃)⟩, it ensures that the manufacturer makes the end-of-year repayment upon observing qd = qd,i2(𝜃). It is well-known that agreements can be self-enforcing if the parties interact repeatedly (see Telser [1980], for example). Repeated interaction is indeed a common feature of markets for fast-moving consumer goods. To provide one example where the menu ⟨qu,i2(𝜃),Te(qd , 𝜃),F(𝜃)⟩ with the end-of-year repayment is self-enforcing, assume that after the current period the manufacturer-retailer pair interacts in one additional period. The man- ufacturer has learned the retailer’s type by then so that the firms earn their complete information profits 𝜋u,c(𝜃) and 𝜋d,c(𝜃) in this future period. Recall that, in this period, the retailer earns 𝜋d,c(𝜃) = 𝜋d,n𝓁(𝜃) whether it sells the branded product or not. Therefore, assume that the retailer lists the branded product if the manufacturer has made the repayment F(𝜃) in the first period, 7 An application of deposit-refund schemes to public good games was provided by Gerber and Wichardt [2009]. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 16 JOHANNES PAHA and it de-lists the branded product otherwise. After normalizing the man- ufacturer’s discount factor to 1, one finds that the manufacturer makes the repayment if inequality (33) applies. F(𝜃) < 𝜋u,c(𝜃).(33) Inequality (33) shows that the manufacturer refunds F(𝜃) if it earns a higher profit 𝜋u,c(𝜃) in the second period from continuing the business relationship than from withholding F(𝜃) in the first period. The menu ⟨qu,i2(𝜃),Te(qd , 𝜃),F(𝜃)⟩ is self-enforcing in this case.8 Otherwise, the firms must rely on third-party enforcement. Or they have to resort to menu ⟨qu,i1(𝜃),Ti1(𝜃)⟩, which is conditional on qu only, if third-party enforcement is not an option. IV(ii). Quantity Discount Lemmas 2 and 3 demonstrated how the manufacturer can incentivize the retailer to sell qd,i2(𝜃), instead of playing its best response qd,i1(𝜃) to qu,i2(𝜃), even if the manufacturer cannot formally control the retailer’s choice of qd . The menus ⟨qu,i2(𝜃),Tf (qd , 𝜃),F(𝜃)⟩ with the fine, and ⟨qu,i2(𝜃),Te(qd , 𝜃),F(𝜃)⟩ with the end-of-year repayment, which are variants of the optimal menu ⟨qu,i2(𝜃), qd,i2(𝜃),Ti2(𝜃)⟩, required the retailer to sell the quantity qu,i2(𝜃) of the branded product. In reality, however, retailers are often free to select also quantities other than qu,i2(𝜃). Therefore, Lemma 4 shows how the manufacturer can make it a best response (hence the index b) for the retailer to choose qu,i2(𝜃), which requires two small modifications of the menu ⟨qu,i2(𝜃),Te(qd , 𝜃),F(𝜃)⟩: The threshold for the deposit F(𝜃) is raised to ̃F(𝜃) > F(𝜃). The combination of the quantity restriction qu = qu,i2(𝜃) and the lump-sum payment Te(qd , 𝜃) is replaced by a two-part tariff Tb(qu, qd , 𝜃) where the retailer may purchase any quantity qu of the branded product at a price equaling the marginal production costs cu. Lemma 4. The retailer adheres to qd,i2(𝜃) and qu,i2(𝜃) if the manufacturer proposes a menu ⟨Tb(qu, qd , 𝜃), ̃F(𝜃)⟩ with Tb(qu, qd , 𝜃) = [ Ti2(𝜃) + F(𝜃) − qu,i2(𝜃)cu ] + qucu − F(𝜃)I(qd , 𝜃) = Te(qd , 𝜃) + (qu − qu,i2(𝜃))cu, and (34) F(𝜃) > ̃F(𝜃) ≡ H(𝜃)2 4sd ⋅ su Δs ,(35) 8 Conditions other than (33) will, of course, emerge under different assumptions. If, for example, the cost-type 𝜃 follows a random walk so that the manufacturer faces the same asymmet- ric information problem in every consecutive period, an infinitely repeated game may be needed to prevent the manufacturer from withholding the repayment. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 17 although the retailer is free to choose quantities other than qd,i2(𝜃) and qu,i2(𝜃). Proof. The manufacturer uses a two-part tariff Tb(qu, qd , 𝜃), according to which it charges a fixed fee Ti2(𝜃) + F(𝜃) − qu,i2(𝜃)cu and a variable payment qucu. The fixed fee collects the payment Ti2(𝜃) and the deposit F(𝜃) but is net of the production costs of the branded product. The retailer purchases the branded product at a price equaling the manufacturer’s marginal costs cu. The deposit F(𝜃) is returned upon observing qd = qd,i2(𝜃), in which case I(qd , 𝜃) = 1 applies. The tariff is specified such that Tb(qu, qd , 𝜃) = Ti2(𝜃) and 𝜋d,b(𝜃) = 𝜋d,i2(𝜃) apply if the retailer chooses qu,i2(𝜃), qd,i2(𝜃). Recall that, in Proposition 1, qu,i2(𝜃) was determined as the best response to qd,i2(𝜃) if the branded product is obtained at marginal costs cu. Therefore, in a first step, I prove that this is also the case now because Tb(qu, qd , 𝜃) is speci- fied such that 𝜕Tb(qu, qd , 𝜃)∕𝜕qu = cu applies. Setting qu,i2(𝜃) is a best response if the mechanism incentivizes the retailer to set qd,i2(𝜃), and if the mechanism is fully revealing. Maximizing 𝜋d,b(𝜃) = R(qu, qd) − cd(𝜃)qd − Tb(qu, qd , 𝜃) with respect to qu gives the retailer’s best response function (36). qu,b(qd(𝜃))= { 1 2su [ su− 𝜕Tb(qu,qd ,𝜃) 𝜕qu −2sdqd(𝜃) ] if qd(𝜃)< 1 2sd [ su− 𝜕Tb(qu,qd ,𝜃) 𝜕qu ] 0 otherwise . (36) Plugging qd,i2(𝜃) and 𝜕Tb(qu, qd , 𝜃)∕𝜕qu = cu in (36) proves qu,b(qd,i2(𝜃)) = qu,i2(𝜃). The second-order condition of the maximization problem is given by (37). 𝜕 2 𝜋d,b(𝜃) 𝜕q2 u = −2su − 𝜕 2Tb(qu, qd , 𝜃) 𝜕q2 u .(37) Because of 𝜕2Tb(qu, qd , 𝜃)∕𝜕q2 u = 0, (37) is negative for all qu. In a second step, I prove that the retailer finds it optimal to set qd,i2(𝜃) if the mechanism is fully revealing. This is the case if the deposit F(𝜃) is set high enough to prevent double deviations from qu,i2(𝜃), qd,i2(𝜃): In Lemmas 2 and 3 the manufacturer controlled qu, so that there could only be single deviations, meaning that the retailer chose qd according to best response function (7). Now, there may be double deviations, where the retailer chooses qd and qu according to best response functions (7) and (36). Given 𝜕Tb(qu, qd , 𝜃)∕𝜕qu = cu, those functions intersect at the quantities q∗u(𝜃) and q∗ d (𝜃) that maximize the firms’ joint profits. If the retailer chooses q∗ d (𝜃) ≠ qd,i2(𝜃) the manufacturer will, however, withhold the deposit F(𝜃) > ̃F(𝜃). The threshold ̃F(𝜃) in (35) was chosen such that 𝜋d,b(q∗u(𝜃), q ∗ d (𝜃), 𝜃) < 𝜋d,b(qu,i2(𝜃), qd,i2(𝜃), 𝜃) applies, which makes double deviations unprofitable for the retailer. Note that ̃F(𝜃) > F(𝜃) applies, with © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 18 JOHANNES PAHA F(𝜃) being defined in (29). The deposit for preventing double deviations is, therefore, higher than the deposit required for preventing single deviations. As a third step, recall that ⟨Tb(qu, qd , 𝜃), ̃F(𝜃)⟩ was specified such that Tb(qu, qd , 𝜃) = Ti2(𝜃) and 𝜋d,b(𝜃) = 𝜋d,i2(𝜃) apply if the retailer chooses qu,i2(𝜃) and qd,i2(𝜃). The retailer, thus, reveals its type 𝜃 truthfully as follows from Proposition 1. This proves Lemma 4. ◾ The menus proposed in Lemmas 2 to 4 assume the payment of a lump-sum fee at the beginning of the period. This is, however, at odds with the observa- tion that in “mainstream retail sectors such as grocery retailing or departmen- tal stores, retailers do not seem to pay lump-sum fees to manufacturers” (Iyer and Villas-Boas [2003]). Similarly, Villas-Boas [2007] establishes “that retail supermarkets do not often pay fixed fees to their manufacturers”, whereas the “existence of quantity discounts is common practice in [the food retail] industry.” Therefore, Draganska et al. [2010] proposed that a “fruitful avenue for future research would be to explore how to incorporate quantity discounts into the negotiation process” between a manufacturer and a retailer. This is what Lemma 5 does. Based on the contributions of Jeuland and Shugan [1983] and Kolay et al. [2004], who showed that a lump-sum payment may be replaced by a quantity discount, Lemma 5 demonstrates how the man- ufacturer can collect Ti2(𝜃) + F(𝜃) by charging a high price on the initial units of the branded product while granting a quantity discount on the additional units. The index r thus stands for rebate. Lemma 5. The retailer adheres to qd,i2(𝜃) and qu,i2(𝜃) if the manufacturer proposes a menu ⟨Tr(qu, qd , 𝜃), ̃F(𝜃)⟩ with Tr(qu, qd , 𝜃) = ⎧ ⎪ ⎨ ⎪ ⎩ x(𝜃) ( qu ⋅ qu,i2(𝜃) − q2 u 2 ) + qucu − F(𝜃)Ir(qd , 𝜃) if qu≤qu,i2(𝜃), Tb(qu, qd , 𝜃) if qu>qu,i2(𝜃), (38) Ir(qd , 𝜃) = { 1 if qd = qd,i2(𝜃) and qu > 0, 0 otherwise, (39) x(𝜃) = 2 [ Ti2(𝜃) + F(𝜃) − cuqu,i2(𝜃) ] qu,i2(𝜃)2 ,(40) and F(𝜃) > ̃F(𝜃), although the retailer is free to choose quantities other than qd,i2(𝜃) and qu,i2(𝜃), and although the menu does not entail a lump-sum payment. © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 19 Proof. The payment function satisfies Tr(0, qd , 𝜃) = 0 for qu = 0, so that it does not entail a lump-sum payment. For qu > 0, the shape of Tr(qu, qd , 𝜃) is determined by (41) and (42). 𝜕Tr(qu, qd , 𝜃) 𝜕qu = { x(𝜃)(qu,i2(𝜃) − qu) + cu if qu ≤ qu,i2(𝜃), cu if qu > qu,i2(𝜃). (41) 𝜕 2Tr(qu, qd , 𝜃) 𝜕q2 u = { − x(𝜃) if qu ≤ qu,i2(𝜃), 0 if qu > qu,i2(𝜃). (42) For qu ≤ qu,i2(𝜃), the manufacturer charges a positive price 𝜕Tr(qu, qd , 𝜃)∕𝜕qu that falls linearly with a slope of −x(𝜃) toward cu as qu approaches qu,i2(𝜃). This decrease constitutes a quantity discount. Given 𝜕2Tr(qu, qd , 𝜃)∕𝜕q2 u < 0, the payment function rises concavely in qu, and it reaches Tr(qu,i2(𝜃), qd , 𝜃) = Tb(qu,i2(𝜃), qd , 𝜃) at qu,i2(𝜃), which is ensured by choosing x(𝜃) according to (40). For qu > qu,i2(𝜃), the function satisfies Tr(qu, qd , 𝜃) = Tb(qu, qd , 𝜃), so that it rises linearly because qu is sold at a price cu. As before, the payment is combined with a refund F(𝜃) > ̃F(𝜃) for setting qd = qd,i2(𝜃), which is only paid in cases with qu > 0. I will show that the retailer’s profit is maximized when choosing qu,i2(𝜃), qd,i2(𝜃). To see this, consider the retailer’s profit function (43) that results if the retailer is assumed to choose qd according to best response function (7). 𝜋d,r(qu, 𝜃) = qu ( Δs − Δsqu + cd ) + 𝜋d,n𝓁(𝜃) − Tr(qu, qd , 𝜃).(43) For the moment, assume qd ≠ qd,i2(𝜃) so that Ir(qd , 𝜃) = 0. The first and sec- ond derivatives of the profit function are given by (44) and (45). 𝜕𝜋d,r(qu, 𝜃) 𝜕qu = Δs − 2Δsqu + cd − 𝜕Tr(qu, qd , 𝜃) 𝜕qu ,(44) 𝜕 2 𝜋d,r(qu, 𝜃) 𝜕q2 u = −2Δs − 𝜕 2Tr(qu, qd , 𝜃) 𝜕q2 u ,(45) Because of the discontinuity in 𝜕Tr(qu, qd , 𝜃)∕𝜕qu at qu = qu,i2(𝜃), one must distinguish the cases qu > qu,i2(𝜃) and qu ≤ qu,i2(𝜃). For qu > qu,i2(𝜃), the payment function was specified such that Tr(qu, qd , 𝜃) = Tb(qu, qd , 𝜃) applies. Given 𝜕Tr(qu, qd , 𝜃)∕𝜕qu = cu, the relevant deviation point is the same as in Lemma 4, which is q∗u(𝜃), q ∗ d (𝜃). Lemma 4 already proved that the retailer receives a higher profit when setting qu,i2(𝜃), qd,i2(𝜃) as compared to setting q∗u(𝜃), q ∗ d (𝜃), which is because of F(𝜃) > ̃F(𝜃). © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 20 JOHANNES PAHA For qu ≤ qu,i2(𝜃), solving 𝜕𝜋d,r(qu, 𝜃)∕𝜕qu = 0 for qu gives qdev u,r (𝜃) = qu,i2(𝜃) + H(𝜃) 2Δs − x(𝜃) .(46) Re-arranging qdev u,r (𝜃) ≤ qu,i2(𝜃) yields 2Δs − x(𝜃) ≤ 0. This weak inequality, however, implies 𝜕2 𝜋d,r(qu, 𝜃)∕𝜕q2 u = − (2Δs − x(𝜃)) ≥ 0, so that the deviation point with qdev u,r (𝜃) is a minimum of the retailer’s profit function. Because both the retailer’s revenue and the payment function Tr(qu, qd , 𝜃) are concave in qu, a deviation point with qdev u,r (𝜃) exists if Tr(qu, qd , 𝜃) is “more concave” than the revenue function. Starting from qu = 0, the profit initially falls when increasing qu, before it rises again once qu is raised beyond qdev u,r (𝜃). The retailer would optimally choose one of the two corner solutions. One is found at qu,i2(𝜃), qd,i2(𝜃), which is the combination of quantities desired by the manufacturer. The other corner solution is found at qu = 0, where qd would be set according to best response function (7), which gives (47). qd,n𝓁(𝜃) = sd − cd(𝜃) 2sd .(47) The retailer would make the profit 𝜋d,r(0, qd,n𝓁 , 𝜃) = 𝜋d,n𝓁(𝜃). However, once the retailer chooses qd,i2(𝜃) and qu,i2(𝜃), it receives the refund F(𝜃) and earns the profit 𝜋d,r(qu,i2(𝜃), qd,i2(𝜃), 𝜃) = 𝜋d,i2(𝜃) ≥ 𝜋d,n𝓁(𝜃). This proves that the retailer selects qd,i2(𝜃) and qu,i2(𝜃) although the firm is free to choose also quantities other than that, and although the menu does not entail a lump-sum payment. Because of 𝜋d,r(qu,i2(𝜃), qd,i2(𝜃), 𝜃) = 𝜋d,i2(𝜃) the retailer reveals its type truthfully, which follows from Proposition 1. This proves Lemma 5. ◾ After characterizing the optimal menu in Proposition 1, Lemmas 2 and 3 showed how a fine or refund F(𝜃) can be used so that the retailer sets the opti- mal qd,i2(𝜃). Lemma 4 demonstrated how the condition on qu can be dropped when using a two-part tariff with a fixed and a variable payment, where the branded product is sold at a price of cu. Following Jeuland and Shugan [1983] and Kolay et al. [2004], Lemma 5 demonstrated how the fixed fee can be replaced by a quantity discount with decreasing prices. The equivalence of both types of tariffs may help to explain the observations made, for example, by Iyer and Villas-Boas [2003], Villas-Boas [2007], or Draganska et al. [2010] according to which quantity discount schemes are more common in retail markets than tariffs with a fixed fee. V. CONCLUSION This article presents a mechanism design analysis of the optimal wholesale contract proposed by the monopolistic manufacturer of a branded product © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 21 to a monopolistic retailer if the retailer also sells a private label whose costs are unobserved by the manufacturer. The retailer has an incentive to under- state the costs of the private label and benefit from a lower payment to the manufacturer. Given the revelation principle, the manufacturer can induce truth-telling by requiring the retailer to sell a suboptimally low quantity of the branded product. Yehezkel [2008] showed that the manufacturer can fur- ther reduce the retailer’s information rent by imposing a restriction on the quantity of the private label, too. Such market share contracts, which are conditional on the quantities of both goods, can have exclusionary effects. If the manufacturer cannot observe aggregate demand (as in Yehezkel [2008]) the manufacturer would want to distort the quantity of the branded product downwards along with that of the private label. While this raises concerns that the private label might be foreclosed, a quite different effect is suggested by the model analyzed in the present article where the manufacturer cannot observe the costs of the private label. In this case, the manufacturer would want to distort the quantity of the private label upwards. This is an effect contrary to exclusion, and it even creates a benefit for consumers by contributing to lower prices. One might, however, be concerned that this suggests anticompetitive effects of a different type. Because the manufacturer is typically unable to control the quantity of the private label, it incentivizes the retailer to distort the quantity of the private label upwards. This is done by collecting an excess payment through high prices of the branded product, which is only repaid to the retailer at the end of the period after observing that the retailer had indeed sold the high quantity of the private label. Courts, policymakers, and authorities have sometimes expressed skepticism toward payments that manufacturers had to make to retailers. This is especially the case if the manufacturer did not receive a specific service in return, and if these payments were made to a powerful retailer with a strong private label, as evidenced by high sales of this product and low sales of the branded product. Such payments might be considered unfair trading practices. Yet, it “is often difficult to distinguish [unfair trading practices] from what might be considered normal competitive behavior” (European Com- mission [2017]). And indeed, the model proposed in this article suggests an efficiency rationale for such repayments. Here, the repayment is part of a fully-revealing, self-enforcing mechanism that allows the manufacturer to discriminate between different retailer types and to raise its profit by dimin- ishing the retailer’s information rents. The model shows that the aggregate quantity is even higher and prices lower than in the complete information benchmark, so that the market share contract with end-of-year repayments benefits final consumers. Future research might analyze how these results must be modified when assuming an upstream and/or downstream oligopoly—a conceptually straightforward but analytically demanding extension. Such a model would © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 22 JOHANNES PAHA also allow analyzing the effects of an exchange of information about the costs of the private label among the producers of branded products. This is relevant for competition policy because, for example, the conduct of several producers of drugstore products, who had exchanged information about retailers in the downstream market, was considered a violation of competition laws by the German Federal Cartel Office. APPENDIX Determining revenue function (6). Equating vu = vd as defined in (1) and (2) yields the location ̂ 𝜙 of the indifferent consumer. All consumers with preferences 𝜙 ∈ [ ̂𝜙, 1] demand the branded product as is shown by demand function (A1). qu = 1 − ̂ 𝜙 with ̂ 𝜙 = pu − pd Δs .(A1) Let 𝜙0,d define the critical value of 𝜙 where vd(𝜙0,d) = 0 applies. Hence, all consumers with preferences 𝜙 ∈ [𝜙0,d , ̂ 𝜙) demand the private label as is shown by demand func- tion (A2) qd = ̂ 𝜙 − 𝜙0,d with 𝜙0,d = pd sd .(A2) Inverting the system of demands (A1) and (A2) yields the inverse demand functions shown in (6). ◾ Derivation of Assumption (13). A vertically integrated firm sets q∗u, q ∗ d as defined in (10) and (11), which are also chosen in competition if the manufacturer observes the retailer’s type 𝜃 (see Section III(i)). q∗u(𝜃) = 1 − Δs + Δc(𝜃) 2Δs .(10) q∗d(𝜃) = Δs + Δc(𝜃) 2Δs − sd + cd(𝜃) 2sd .(11) If the manufacturer does not observe the retailer’s type and conditions the menu on qu only, it optimally sets qu,i1(𝜃), and the retailer responds by setting qd,i1(𝜃) as are shown in (20) and (22) (see Lemma 1). qu,i1(𝜃) = q∗u(𝜃) − H(𝜃) 2Δs ,(20) qd,i1(𝜃) = q∗d(𝜃) + H(𝜃) 2Δs ,(22) If the manufacturer can also control qd , it sets qu,i2(𝜃) and qd,i2(𝜃) as defined by (24) and (25) (see Proposition 1). qu,i2(𝜃) = qu,i1(𝜃),(24) © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ASYMMETRIC INFO IN WHOLESALE PRICING 23 qd,i2(𝜃) = qd,i1(𝜃) + H(𝜃) 2sd ,(25) These are also the equilibrium quantities relevant in Lemmas 2 to 5. The off- equilibrium quantities are q∗u(𝜃), q ∗ d(𝜃) in Lemmas 4 and 5. In Lemma 5, the retailer also considers setting qdev u,r (𝜃) ∈ [0, qu,i2(𝜃)], which results in qdev d,r (𝜃) ≤ qd,n𝓁(𝜃) as is defined in (47). qd,n𝓁(𝜃) = sd − cd(𝜃) 2sd .(47) One sees that qu,i2(𝜃) = qu,i1(𝜃) ≤ q∗u(𝜃) and q∗d(𝜃) ≤ qd,i1(𝜃) ≤ qd,i2(𝜃) < qd,n𝓁(𝜃) apply, so that qu, qd > 0 require (A3) and (A4). qu,i2(𝜃) > 0 ⇔ Δc(𝜃) < Δs −H(𝜃),(A3) q∗d(𝜃) > 0 ⇔ Δs sd cd(𝜃) < Δc(𝜃).(A4) Combining (A3) and (A4) gives (13). ◾ Proof of Lemma 1. At the optimum, the IR-constraint must be binding for a retailer of the lowest type so that Ui1(0) = 0. When using Ui1(0) = 0 and plugging 𝜋d,i1( ̂𝜃|𝜃) from (17) in Ui1( ̂𝜃|𝜃) from (18), one can solve for Ti1(𝜃) as is shown in (21). Plugging (21) in the manufacturer’s maximization problem (4) and integrating by parts yields the manufacturer’s expected profit (A5). max qu ∫ 1 0 [ 𝜋i(qu, qd , 𝜃) − 𝜋d,n𝓁(𝜃) −H(𝜃) 𝜕Ui1(𝜃) 𝜕𝜃 ] g(𝜃)d𝜃.(A5) Optimizing (A5) w.r.t. qu gives the optimal output as is shown in (20). Assumption (13) ensures 0 < qu,i1(𝜃) < 1 and 0 < qd,i1(𝜃) < 1. ◾ Proof of Proposition 1. Given the revelation principle, the menu ⟨qu,i2(𝜃), qd,i2(𝜃), Ti2(𝜃)⟩ is fully revealing. Therefore, write the information rents Ui2( ̂𝜃|𝜃) as Ui2(𝜃), and determine the marginal information rents (A6) using 𝜋d,i2( ̂𝜃|𝜃) from (23), 𝜋d,n𝓁(𝜃) from (8), and the envelope theorem. 𝜕Ui2(𝜃) 𝜕𝜃 = [ sd − cd(𝜃) 2sd − qd ] .(A6) Using the definition of Ui2(𝜃) from (27), the functional forms of 𝜋d,i2( ̂𝜃|𝜃) and 𝜋d,n𝓁(𝜃), along with condition Ui2(0) = 0, one can solve for Ti2(𝜃) as was stated in (26). The manufacturer’s profits and maximization problem are then given by max qu ,qd ∫ 1 0 [ 𝜋i(qu, qd , 𝜃) − 𝜋d,n𝓁(𝜃) −H(𝜃) 𝜕Ui2(𝜃) 𝜕𝜃 ] g(𝜃)d𝜃.(A7) © 2023 The Authors. The Journal of Industrial Economics published by The Editorial Board of The Journal of Industrial Economics and John Wiley & Sons Ltd. 14676451, 0, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/joie.12350 by Justus L iebig U niversitaet G iessen, W iley O nline L ibrary on [22/11/2023]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 24 JOHANNES PAHA Solving the first-order conditions of (A7) yields qu,i2(𝜃) and qd,i2(𝜃) as were shown in (24) and (25). Because the menu ⟨qu,i2(𝜃), qd,i2(𝜃),Ti2(𝜃)⟩ is a super-set of the menu ⟨qu,i1(𝜃),Ti1(𝜃)⟩, the manufacturer’s choice qd,i2(𝜃) ≥ qd,i1(𝜃) implies 𝜋u,i2(𝜃) ≥ 𝜋u,i1(𝜃). 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See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense WHOLESALE PRICING WITH ASYMMETRIC INFORMATION ABOUT A PRIVATE LABEL&AST; I INTRODUCTION II THE MODEL III THE MENUS III(i) Complete Information III(ii) Incomplete Information: Conditioning on [[math]] III(iii) Incomplete Information: Conditioning on [[math]] and [[math]] III(iv) Welfare Analysis IV APPLICATION IV(i) End-Of-Year Repayment IV(ii) Quantity Discount V CONCLUSION APPENDIXvspace *{-12pt} REFERENCES