Social Status, Reputation, Financing, and Commitment∗

Preliminary and incomplete - Please do not make it available without the authors’ permission

Rog´erioMazali† Jos´eA. Rodrigues-Neto‡ April 25, 2016

Abstract We develop a dynamic model of a horizontally differentiated product duopoly. The purpose is to investigate the relationship between the market niche each firm targets, the social status of its brand, its project financing and ownership structure. Firms’ financial decisions can be used as signals of the social status of each firm’s brand name, and thus, the market niche that is targeted. We provide conditions under which a firm is not able to produce high status goods credibly, but by restricting its ability to adopt an industrial technology it may. A firm that produces high status goods may purposely restrict its own access to capital markets, remaining private, using this decision as a commitment device. In this case the firm has the credibility it needs to convince other parties that it will keep producing a high status good and not “invade” its rival’s market niche. Following a stream of literature that started with Titman (1984) and Brander and Lewis (1986), the present paper provides a novel link between financial and capital markets: concerns with “brand reputation” regarding the social status of a firm’s products. Keywords: brand, competition, reputation, commitment, status, capital structure. JEL classification: G32, C78, L13, L15.

∗We thank Steﬀen Lippert, Simona Fabrizi, and seminar participants at the Australasian Economic Theory Workshop, Australian National University, University of Bras´ıliaand Catholic University of Bras´ıliafor helpful comments. All remaining errors are our own responsibility. †Catholic University of Brasilia, Graduate Program in Business Economics, SGAN 916 M´oduloB, Bloco B sala A-120, Bras´ılia,DF, CEP 70790-160, Brazil. Phone: +55 (61) 3448-7192. Email: [email protected] ‡Australian National University, College of Business and Economics, Research School of Economics, H.W. Arndt Building 25A, ANU, Canberra ACT - 0200, Australia. Phone: +61 (2) 612-55633. Email: [email protected] Contents

1 Introduction 1

2 Setup 7 2.1 Firms and Investors ...... 7 2.2 Consumers ...... 8 2.3 Game Dynamics ...... 9 2.4 Dynamic Game Setup and Equilibrium Concept ...... 9

3 Stage-Game Analysis 10 3.1 Consumer Demands ...... 10 3.2 Hotelling Equilibrium Under the Same Technology and No Status ...... 11 3.3 Hotelling Equilibrium Under Diﬀerent Technologies and No Status ...... 11 3.4 Hotelling Equilibrium Under the Same Technology and Status ...... 12 3.5 Hotelling Equilibrium Under Diﬀerent Technologies and Status ...... 13 3.6 Stage Game Along the Equilibrium Path ...... 13 3.7 One-Shot Deviation: Firm 0 ...... 15 3.8 One-Shot Deviation: Firm 1 ...... 15 3.9 No Punishment After Deviation: Hotelling Competition with Reputation Loss . . 16 3.10 Punishment After Deviation: Generic Pricing ...... 17 3.11 Punishment After Deviation: Stackelberg Equilibrium ...... 18 3.12 Punishment After Deviation: Minmax Punishment ...... 19 3.13 Firm 0 deviates from punishing Firm 1 ...... 19 3.14 Firm 1 punishes Firm 0 ...... 20

4 Dynamic Game Equilibria 21 4.1 Equilibrium with Product Diﬀerentiation on Status ...... 21 4.1.1 Financial Markets ...... 25 4.2 Dynamic Strategy ...... 26 4.3 Incentive Compatibility Constraints ...... 28 4.4 Deviations from the Equilibrium Path: Customer Punishment Only ...... 29 4.5 Deviations from the Equilibrium Path: Customer and Rival Punishment . . . . . 30 4.6 Incentives for Firm 0 to Punish a Deviation from Firm 1 ...... 31 4.7 Incentives For Firm 1 to Return to the Equilibrium Path ...... 34 4.8 Incentives For Firm 1 to Punish a Deviation from Firm 0 ...... 35

5 Optimal Punishment 36 5.1 Case 1 (δ ≥ bδ1): “Customer punishment only” can enforce equilibrium with status goods...... 36 5.1.1 Comparing the two types of punishment ...... 37 5.2 Case 2 (δ < bδ1): “Customer punishment only” cannot enforce equilibrium with status goods ...... 38

2 6 Social Status, Private Firms and Commitment 39 6.1 Subgame in which Firm 1 chooses no commitment ...... 42 6.2 Equilibrium of the Dynamic Game with Commitment ...... 44 L NP 6.2.1 Eﬀect of commitment when p0 ≥ p0 ...... 44 L NP 6.2.2 Eﬀect of commitment when p0 < p0 ...... 48

7 Discussion 50

A Appendix: Proofs 51 A.1 Stage Game ...... 51 A.1.1 Consumer Demands ...... 51 A.1.2 Hotelling Equilibrium Under the Same Technology and No Status . . . . . 51 A.1.3 Hotelling Equilibrium Under Different Technologies and No Status . . . . . 52 A.1.4 Hotelling Equilibrium Under the Same Technology and Status ...... 53 A.1.5 Hotelling Equilibrium Under Different Technologies and Status ...... 54 A.1.6 Stage Game Along the Equilibrium Path ...... 54 A.1.7 One-Shot Deviation: Firm 0 ...... 56 A.1.8 One-Shot Deviation: Firm 1 ...... 56 A.1.9 No Punishment After Deviation: Hotelling Competition with Reputation Loss ...... 57 A.1.10 Punishment After Deviation: Generic Pricing ...... 57 A.1.11 Punishment After Deviation: Stackelberg Equilibrium ...... 58 A.1.12 Punishment After Deviation: Minmax Punishment ...... 58 A.1.13 Firm 0 deviates from punishing Firm 1 ...... 58 A.1.14 Firm 1 punishes Firm 0 ...... 59 A.2 Dynamic Game Equilibria ...... 59 A.2.1 Equilibrium with Product Differentiation on Status ...... 59 A.2.2 Dynamic Strategy ...... 62 A.2.3 Incentive Compatibility Constraints ...... 62 A.2.4 Deviations from the Equilibrium Path: Customer Punishment Only . . . . 62 A.2.5 Deviations from the Equilibrium Path: Customer and Rival Punishment . 63 A.2.6 Incentives for Firm 0 To Punish a Deviation from Firm 1 ...... 63 A.2.7 Incentives For Firm 1 to Return to the Equilibrium Path ...... 64 A.2.8 Incentives For Firm 1 to Punish a Deviation from Firm 0 ...... 65 A.3 Optimal Punishment ...... 65 A.3.1 Case 1 (δ ≥ bδ1): “Customer punishment only” can enforce equilibrium with status goods ...... 65 A.3.2 Case 2 (δ < bδ1): “Customer punishment only” cannot enforce equilibrium with status goods ...... 66 A.4 Social Status, Private Firms and Commitment ...... 67 A.4.1 Subgame in which Firm 1 chooses no commitment ...... 67 A.4.2 Equilibrium of the Dynamic Game with Commitment ...... 69

B Appendix: Diluting Share Ownership of Regular Goods Firm 70 B.1 Proofs ...... 71 1 Introduction

In the last decades, many economists have dedicated their eﬀorts in trying to understand the market for status or luxury goods. Unlike regular goods, these goods might have little or no intrinsic value, in the sense that their consumption provides little or no direct utility. In the terminology of Pollack (1967, 1976), Koopmans (1960), among others, these goods provide relatively low “cardinal utility”. The main component of the value these goods provide come from their “ordinal utility”, that is, the utility given by the position one occupies in a rank after its consumption. Because of this feature, Frank (1985, 2005) refers to these goods as positional goods.

It is not just the positional goods themselves that have special features that characterize them. The companies that produce them also have special characteristics that go beyond the production of status goods. Firms that produce high-end status goods tend to be private. For example, most watchmakers of high status, such as Cartier, and Rolex, are also private companies, as well as fashion giants such as Armani, Versace, and jewelers H. Stern and Swarovski. Of the world’s top ten jeweler companies according to Luxe High Life’s catalogue, only one (Tiffany & Co.) is publicly traded. Another is a subsidiary in a public conglomerate (Harry Winston, part of the Swatch Group), and the other eight are all privately held companies (Cartier, Van Cleef & Arpels, Piaget, Bvlgari, Mikimoto, Graff, Buccelati and Chopard).1 Most of the su- persport car manufacturers are completely private, 100% independent endeavors, or a part of a conglomerate with significant restrictions on access to funds from other companies in the conglomerate. McLaren, Aston Martin, W Motors and Koenigsegg are in the first group while Bugatti and Lamborghini are in the second. Out of the top 10 world’s most expensive cars of 2015, according to Digital Trends2, six are manufactured by private independent companies (Zenvo ST1, Pagani Huayra, Aston Martin One-77, Koenigsegg One:1, W Motors Lycan Hyper- sport, and Koenigsegg CCXR Trevita), while three are manufactured by private firms within public conglomerates (Mansory Vivere, Bugatti Veyron, and Lamborghini Veneno). Only two, both manufactured by Ferrari N.V., are manufacuted by a public company (Ferrari LaFerrari and Ferrari F60 America). Nevertheless, Ferrari has traded as a private company for most of its life, only announcing its intention to go public in 2015.3 The decision of Ferrari going public is part of a spin-off operation that dismembered Ferrari from the Fiat-Chrysler Automobiles conglomerate. An article published in Fortune magazine in January 2016 seems to imply that being part of the conglomerate was not beneficial for Ferrari in any way and that Ferrari was not able to obtain

1http://luxehighlife.com/jewellery/the-world-s-top-10-jewellery-brands 2http://www.digitaltrends.com/cars/the-top-ten-most-expensive-cars-in-the-world/ 3See http://fortune.com/2015/07/23/ferrari-ﬁles-for-ipo-nyse/.

1 financing from the conglomerate, given the high leverage of the conglomerate. 4 Other high-end status goods that are part of conglomerates have restrictions in obtaining financing from parent companies. Lamborghini S.p.A, for example, was acquired by the Volkswagen group in 2011. Instead of maintaining it under direct control of Volkswagen, the group’s management decided to keep it under control of one of its higher luxury subsidiaries, Audi A.G.. This ownership structure made it very difficult for Lamborghini to obtain financing from Volkswagen, since it would need aproval from Audi management as well. These restrictions in internal financing make many of these high-status car makers that are part of large conglomerates behave as if they are completely independent private companies with respect with financing. Why is it the case that so many high-end status good producers have preferred to remain private? By retaining private status, these firms choose to give up valuable financing sources for their projects. Hence, these firms could be forced to pass on positive net present value (NPV) projects due to financial constraints. We argue that high status goods might have something to gain out of retaining private status, namely, reputation for their products. High status is closely related to the exclusivity of its consumption. A high status good has to commit to producing few units of its good in order to charge a high price for its status value. Once it has acquired such status with its customers, the high-status good producer has incentives to deceive consumers and sell additional units to consumers of lower social strata. The producer might have to switch technologies and acquire industrial machinery in order to mass-produce its good. If high social strata consumers anticipate that firms will do that, they will never pay such a high price for the firm’s good in the first place. As a result, there will be no market for status goods. Indeed, many of the high-end status goods are almost completely manufacted by hand. Watches, haute couture, jewelry, oriental tapestry are examples of industries that are notori- ously known for the handicraft of the goods they produce. Even the high-tech supercar factories have many “hands on the job” procedures in assembling one of their expensive car models. Those processes are extensively documented in two series of documentaries produced by the National Geographic (“Ultimate Factories: Car Collection” and “Mega Factories”) and TLC (“Rides”). In order to acquire the trust of high-end consumers, a firm producing high status goods would have to “burn the bridges” and restrict its ability to acquire the funds necessary to obtain industrial machinery. By doing so, the firm makes it virtually impossible for it to produce “cheap goods” and sell them as if they were of high status. Customers can then trust that the firm will not sell additional units and thus they attribute high status to the good the firm produces.

We develop a dynamic stylized model of a horizontally diﬀerentiated product duopoly. The model investigates the relationship between the market niche each ﬁrm targets, the social status

4See http://fortune.com/2016/01/04/ferrari-spinoﬀ-ﬁat-chrysler/.

2 of its brand and its project financing. Two duopolistic firms offer a horizontally differentiated product that can yield their consumers extra utility from social status. Consumers use firms’ financial decisions as a signal of the social status that the good each firm produces will have. In our model, firms produce a pure status good that gives no direct utility to its consumers. In a monopolistic competition setup, firms choose the price of their goods. Prices can include a status premium for the status utility they give to their consumers if they good they produce is the more exclusive. Firms also choose the technology with which they produce their goods (handicraft or industrial manufacturing) and how they finance the production of their goods. We provide conditions under which there is an equilibrium of the dynamic game in which one firm produces high status goods using a handicraft technology while the other adopts the industrial technology, obtaining a larger market share than its rival. Such strategy profile is implementable as a Subgame Perfect Equilibrium if fixed costs for the acquisition of industrial machinery are sufficiently large, if marginal costs for producing using the handicraft technology are sufficiently small, and if the common discount factor is sufficiently large. Whenever the high status producer deviates from its prescribed actions, its product loses its high status reputation. This loss of reputation works as a punishment carried out by consumers. If this punishment is able to deter deviation from the status good producer, then the producer of the “no status” good can increase its profit by “softening punishment”. In this case, the pivotal condition to implement the equilibrium turns out to be incentive conditions for a firm deviating from handicraft production of a status good to return to the equilibrium path. In the case consumers alone are not able to deter deviation, and the regular good producer has to incur in short-term losses to punish a deviant status good producer, the pivotal condition to implement the equilibrium depends on the punishment prices charged by the regular good producer to punish a deviant status good producer, and the punishment prices charged by the status good producer to punish the regular good producer when it fails to punish devious behavior. In the case the above strategy profile is not an equilibrium, firms can commit to producing a high status product by restricting its ability to adopt an industrial, mass-production technology. We show that in many cases firms that produce high status goods will purposely adopt more restrictive financing policies, such as retaining private status. Retaining private status works as a commitment device that provides the high status product firm the credibility it needs to convince customers and rivals that it will keep producing high status products and not deviate to obtain short-term gains by “invading” its rival’s market niche. Although for a firm that seeks to produce high status goods, its technology decision affects its financing, for a firm producing using the industrial technology, financing decisions are irrelevant, and the classical Modigliani and Miller (1958) result holds. Following a stream of literature that started with Titman (1984) and Brander and Lewis (1986), our paper provides a reason why product and capital markets

3 are linked and aﬀect one another through status social and reputation concerns.

We assume the social status component of the utility of consumers is given by the exclusivity of its consumption. In particular, we assume the status value of the good is inversely proportional to the number of individuals consuming it, and that only the most exclusive good has this extra status utility component. Even if consumers do not care directly about the number of individuals consuming a particular good, matching concerns might give rise to behavior that looks “as if” consumers cared about how exclusive their goods are. As examples of such models, we can cite Mazali and Rodrigues-Neto (2013) and Koford and Tschoegl (1998), models that generate equilibria in which consumers care ex post about the rarity or exclusivity of its consumption. One way or another, this status component is only learned through experience. We make assumptions regarding consumers’ beliefs regarding the social status of a particular brand that are, in essence, equivalent to assuming the status component of the consumption good is a pure experience good, as defined in Nelson (1970), and analyzed by Shapiro (1983) and Villas-Boas (2004, 2006), among others. We assume, however, that consumers update their beliefs about the true status value of the good through social learning instead of individual learning, as in Bergemann and Välimäki (2006).

The market for positional or status goods has been extensively studied by the economic literature. Hopkins and Kornienko (2004, 2009, 2010) showed how the market for status goods could be modeled in a game theoretic fashion, on which status goods are used as signaling devices of hidden abilities, and the resulting market will have similar characteristics as those of ordinal utility goods. Pesendorfer (1995) studied the market for fashion, showing that it has characteristics of status goods, and that its dynamics implies new designs will appear whenever the current design is copied by sufficiently many people in the lower strata of society so that the current design loses its status value. Rayo (2013) provided conditions under which a monopolist facing no fixed costs would choose to offer the same variety (or brand, or model) of status good to a positive measure of consumers instead of fully customizing status goods in that vicinity. Mazali and Rodrigues-Neto (2013) showed that the presence of fixed costs in creating new varieties implies full customization will never occur in equilibrium, and the market equilibrium will be stratified in a similar fashion as in Burdett and Coles (1996). Many other papers have been written pointing one characteristic or another of the market for status goods, and it is impossible to make justice to all of them here. A summary of theoretical studies on social status in economics can be found in Truyts (2010). For empirical studies related to social status in economics, refer to Heffetz and Frank (2010).

The present study is related to a stream of literature relating a ﬁrm’s ﬁnancial decisions to

4 its product market decisions that started with Titman (1984) and Brander and Lewis (1986). Brander and Lewis show that firms can use debt to commit themselves to an aggressive strategy in a duopolistic product markets. The resulting equilibrium had with lower prices and higher quantities than obtained in the traditional Cournot model of quantity competition with no debt. Titman (1984) showed that customer warranty concerns can influence a firm’s capital structure decision because when a firm that produces a durable good goes into financial distress customers stop purchasing from the firm increasing the probability of asset liquidation. Firms that want to avoid this warranty concerns trap have to limit the amount of debt they issue. Many other papers followed showing other relations between a firm’s capital structure and product and in- put markets, in what one might call the “stakeholder theory of capital structure” (Titman, 1984; Bronars and Deere, 1994; Klein and Leffler, 1981; Maksimovic and Titman, 1991). These studies consider the fact that the firm’s non-financial stakeholders, such as customers, workers, suppliers, etc., affect and are affected by debt of a firm they have a stake in.5 More recently, researchers published studies linking a firm’s capital structure to product development and product differentiation. Holmström(1989) argues that, as firms grow and evolve into complex conglomerates whose shares are publicly traded, they become more intolerant to failure and therefore tend to shy away from risky innovative R&D projects. Manso (2011) shows that managerial compensa- tion schemes can be set to give manager the appropriate incentives to innovate. Those incentives include great tolerance for failure and long-term stability, and can be obtained with the appropriate combination of stock options with long vesting periods, option repricing, golden parachutes, and managerial entrenchment. Ferreira et al. (2012) show that uncertainties regarding how innovative a project is compounded with the presence of liquidity traders in the market cause firms with operational profitable innovative projects to remain private, while firms with conservative projects go public. Our study shows that, even if a project is not that innovative, if its intended market niche is most upscale, firms will also have incentives to remain private.

Our study is also closely related to a stream of literature that relates reputation and incentives to produce high quality products. Kreps and Wilson (1982) and Milgrom and Roberts (1982) were among the first to point out that reputation concerns could motivate many economic phenomena. Shapiro (1982, 1983, 1983a) was one of the first to apply the concept of reputations to analyze a firm’s incentive to provide high quality goods. Holmström(1999) applies it to managerial incentives to show that the implicit contract defined by career concerns might not be sufficient to overturn short-turn incentives to shirk. Mailath and Samuelson (2000) build a model in which firms have a short-term incentive to exert low effort, but could obtain higher operational profits if they could commit to exerting high effort in every period. Competent firms exert high effort

5For a broad review of this literature, see Abdelaziz and Rodr´ıguez-Fern´andez(2006).

5 while incompetent firms shirk. As there is occasional exit in the market, incompetent firms may acquire good reputation firms and then deplete it, while competent firms tend to acquire firms with average reputation and build it up. Tadelis (1999) shows that, in a dynamic adverse selection model in which firms can trade their brand names, there will be trade involving firms with bad reputations. Tadelis (2002) finds conditions under which finitely lived firms will have incentives to exert high effort until the end. Maksimovic and Titman (1991) show that high debt can compromise a firm’s ability to commit to producing high quality products and thus reduce firm value, unless the firm has assets with high salvage value in liquidation. In this case, debt may increase the firm’s ability to commit to producing high quality products and increase firm value. They also show that appropriate dividend restrictions can eliminate this incentive to produce low quality goods. Klein and Leffler (1981) also evaluate a firm’s incentives to honor contracts (including implicit contracts firmed with consumers) as a function of its salvage value. They show that the condition for a firm to continuously honor contracts with consumers is that the price they obtain is significantly above salvageable production costs, and thus firms would make positive operational profits at every period.

The results shown here are closely related to Maksimovic and Titman (1991). However, there are important differences. In our case, the access to public financial markets destroys the firm’s ability to commit to producing high status goods, regardless of the nature of the firm’s assets (industrial machinery). If firms cannot commit to selling high status goods, the only alternative left is to limit their own access to funds so that it becomes impossible for the firm to obtain access to industrial machinery. Unlike in Maksimovic and Titman (1991), debt does not compromise the firm’s ability to commit to producing high-end goods any more than equity does. What matters to compromise the firm’s credibility is the firm’s ability to raise enough funds to finance the acquisition of industrial machinery, and any publicly traded asset has the same effect on the firm’s credibility. The remainder of the paper is organized as follows. Section 2 describes the basic setup of the model and the definition of equilibrium used throughout the paper. Section 3 describes many stage-game outcomes that are used in the analysis of the dynamic game. Section 4 characterizes the equilibrium of the dynamic game and proves that the profile described is indeed a Subgame Perfect Equilibrium. Section 5 describes optimal punishment strategies used in equilibrium. Section 6 discusses commitment devices and their relation to financial constraints purposely set up by firms. Section 7 discusses some of the economic implications of the results developed in the current study. Appendix A has all the proofs. Appendix B has a discussion on stock ownership dilution.

6 2 Setup

2.1 Firms and Investors

There are two firms, j ∈ {0, 1}, each living for infinitely many periods, t = {0, 1, 2,...} . They play an infinitely repeated game. There is a common discount factor, 0 < δ < 1. The firms produce a single physical indivisible good that carries the firm’s brand. This homogeneous physical good is differentiated due to consumer preferences over brands. Let good j be the good produced by Firm j. This terminology is independent of the technology adopted by Firm j. By adopting a handicraft technology, each firm may produce its good at zero fixed costs and marginal cost c > 0. Alternatively, firms may produce using an industrial (mass production, manufacturing) technology, at zero marginal cost and positive fixed cost K > 0. Industrial machinery is completely depreciated after one period. After then, if the firm wants to keep mass-producing its product, it needs to purchase the industrial machinery again by paying the same fixed cost K. The two firms can exit and re-enter the market freely. An industrial technology firm needs to finance the acquisition of industrial machinery. We assume that firms are financially constrained, so that they do not have internal funds to acquire the industrial technology. Each firm decides what type of company it will be in each period: public or private. If a firm decides to go public, it obtains access to financial capital markets, and is able to issue publicly traded stocks and bonds. Public firms may finance the cost K of industrial machinery through the issuance of stocks or debt. Private firms cannot issue any stocks or debt. If firms issue debt to finance part or all of the fixed cost K, they repay in the same period, after operational profits are realized. Assume that there is no intra-period cost of capital; the firm repays the same amount when paying off its debt. Assume that investors are numerous, homogeneous and rational, so that they pay the fair price for the firms’ shares. Also, assume investors have no market power and invest in the firm only if the firm repays them accordingly, that is, if the firm is able to generate non-negative value. Firms’ managers act in the interest of a firm’s current shareholders. Therefore, managers maximize the present value Vj of the discounted cash flows of the share of current shareholders on future profits. Let α be the fraction of the firm current shareholders give up to new shareholders that occurs if the firm issues stock to finance a fraction of K. If a firm issues bonds, assume that in each period it borrows Kγ, for some 0 ≤ γ ≤ 1, to finance a fraction γ of the cost of machinery.

7 2.2 Consumers

Each consumer lives for one period and may purchase good j ∈ {0, 1} by paying price pj. Only the first unit of the good provides utility to its consumers. Thus, consumers have no incentives to purchase additional units of good j. If a consumer purchases a unit of good j, he/she obtains no additional utility by purchasing a unit of good −j. Consumers are indexed in the unit interval by i ∈ [0, 1]. Index i can represent geographical location, as in the standard Hotelling (1929) model, endowment, as in Cole, Mailath, and Postlewaite (1992), individual ability, as Mazali and Rodrigues-Neto (2013), individual “pizazz”, as in Burdett and Coles (1997), consumer switching cost, as in Klemperer (1987, 1995), income, as in Hopkins and Kornienko (2004), among others. If i represents location, for example, then a consumer’s position defines his transportation costs and preferences according to the distance to the different shops. At the end of each period, another generation of consumers is born and replaces the previous. The previous generation dies and thus leaves the game. The direct utility of consumer i from consuming the good j in a single period is given by H(i, j), where:

u − i, if j = 0; H(i, j) = u − 1 + i, if j = 1. where u > 0 is a constant. The function H(i, j) is the component of the utility indicating intrinsic preferences. In the absence of status concerns, this static model is identical to Hotelling’s (1929) model of spatial competition.

Besides direct utility, a good might also have a status-enhancing component. Let St(j) ∈ {0, 1} be an indicator variable that represents the beliefs of consumers at time t regarding whether good j has high status. This variable assumes the value St(j) = 1 if consumers think that good j provides social status to its consumers at period t. If consumers think that good j does not provide any social status to its consumers at period t, then St(j) = 0. An exogenous social norm determines which good, if any, provides social status. The status component is an experience good. The social norm assigns St(j) = 1 if and only if Firm j was the only firm to produce a handcrafted good in period t − 1. On the other hand, if at time t − 1 Firm j uses the industrial technology, then the social norm assigns St(j) = 0. If both firms use the handicraft technology at t − 1, then St(j) = 0. This hypothesis is equivalent to assuming that good j is an experience good, and that only after its consumption individuals will be able to infer it’s true social status value. Each good lasts exactly one period, and cannot be passed on from one generation to another. In the Hotelling (1929) model, consumers on the left (sufficiently small index i) buy good 0 and ∗ consumers on the right consume good 1. Let it be the consumer that, at time t, is indifferent between goods 0 and 1; that is, the consumer on the border of the two market niches. The

8 magnitude of the status component of the utility function depends on the current exclusivity of the status good. We adopt a “word of mouth goes quickly” hypothesis, as in Ahn and Suominen (2001). Once a particular consumer experiences a particular status level for this good, it becomes ∗ public information. Let Dt(j, it ) be the demand for the good produced by Firm j at time t when a ∗ ∗ ∗ ∗ ∗ mass of it ∈ [0, 1] consumers buys from Firm 0. Demands are Dt(0, it ) = it and Dt(1, it ) = 1−it . ∗ The status component of the time t utility is St(j)[1 − Dt(j, i )]. The more exclusive a good ∗ perceived as of high status, the more utility its consumers obtain from it. Let Ut(i, j; it ) denote the time t utility of a consumer with index i ∈ [0, 1] from buying the good from Firm j ∈ {0, 1} ∗ when a mass of it ∈ [0, 1] consumers buys from Firm 0. This utility is:

∗ ∗ Ut(i, j; it ) = H(i, j) + St(j)[1 − Dt(j, it )] − pj.

2.3 Game Dynamics

The game is repeated inﬁnitely. At every stage-game, events occur in the following order:

• 1. Firms simultaneously choose oj, τ j, and pj; 2. Consumers observe prices and simultaneously choose a good to buy; 3. Period payoﬀs are realized, period t+1 consumers and ﬁrms observe status and update their beliefs.

2.4 Dynamic Game Setup and Equilibrium Concept

Let ht0 be the one-period history of the game outlying the possible outcome of a stage game occur- 0 ring in time t . Let Ht = h0h1 . . . ht be the ﬁnite history of the game up to time t. Let Ht be the set of possible histories up to time t. Thus Ht ∈ Ht. A strategy σj for Firm j consists of a sequence in which each element is a vector aj (Ht−1) = (pj (Ht−1) , τ j (Ht−1) , αj (Ht−1) , γj (Ht−1) , oj (Ht−1)) whose coordinates are: price pj (Ht−1), technology τ j (Ht−1) ∈ {m, h} (where m represents the industrial technology, and h represents the handicraft technology), quantity of stocks issued

αj (Ht−1), proportion of debt γj (Ht−1) and the ownership structure oj (Ht−1) ∈ {public, private}, 0 for each possible history Ht−1. The period t history is the set of actions played in that period, ht0 = (a0 (Ht0−1) , a1 (Ht0−1)). Let H = h0h1h2 ... be an infinite-history, an element of H, the set of all possible infinite histories of the game. A strategy profile σ = (σ0, σ1) is a pair of strategies, one for each firm.

After each ﬁnite history Ht , there is a subgame starting at t+1. Let G(Ht) be the continuation subgame starting at period t + 1 after the occurrence of history Ht. A continuation strategy

σj (Ht) is the restriction of a strategy from t + 1 onwards such that the complete history of the

9 game has finite history Ht from periods 0 to t. A continuation profile is a pair of continuation strategies (σ0 (Ht) , σ1 (Ht)) where each player plays a continuation strategy with the same finite history Ht. A Nash Equilibrium of the continuation subgame is a profile of continuation strategies such that no player has a unilateral profitable deviation. A Subgame Perfect Equilibrium is a strategy profile such that, in every subgame that is reached after finite history Ht, the continuation profile induced by σ is the Nash Equilibrium of the subgame. The focus of the analysis is on profiles that characterize Subgame Perfect Equilibria.

Example 1 As an example of a strategy profile that may be implementable in equilibrium is one in which both firms choose the handicraft technology, go public, finance 100% of their capital needs K by issuing stocks, issue no debt, and play Hotelling prices in every period. Any deviation from this profile is ignored by the players. Subsection 3.2 describes the stage-game payoffs that each player obtains if both firms play such a strategy.

Another example of a strategy profile that may be implementable is one in which each firms choose different technologies. This strategy profile and its implementability as a Subgame Perfect Equilibrium will be discussed in the remainder of this study.

3 Stage-Game Analysis

3.1 Consumer Demands

Consider the index i∗ such that consumers with this index are indiﬀerent between buying from Firm 0 or from Firm 1. In this case, the demands of the ﬁrms are D(0, i∗) = i∗ and D(1, i∗) = ∗ 1 − i . The following lemma states consumer demands as functions of prices p0 and p1.

Lemma 1 If consumers do not attribute status value to good 1 at period t, that is, if St(1) = 0, then the aggregate demands for goods 0 and 1 are given, respectively, by 1 + p − p i∗ = 1 0 (1) 2 1 − p + p 1 − i∗ = 1 0 . (2) 2

If otherwise consumers do attribute status value to good 1 at period t, that is, if St(1) = 1, then the aggregate demands for goods 0 and 1 are given, respectively, by 1 + p − p i∗ = 1 0 (3) 3 2 − p + p 1 − i∗ = 1 0 . (4) 3

10 The good produced by Firm 1 is the most exclusive when i∗ > 1/2. This is true if and only if its price is signiﬁcantly greater than the price of the other good, in the sense that p1 − p0 > 1/2. In the following subsections, we will consider a series of stage-games that are relevant for the equilibrium of the Dynamic Game.

3.2 Hotelling Equilibrium Under the Same Technology and No Status

Suppose that both ﬁrms adopt the same technology at time t, that is, τ 0 (Ht−1) = τ 1 (Ht−1) and that they chose the same technology in period t − 1, that is, τ j (Ht−1) = τ j (Ht−2), j = 0, 1. In this case, the model degenerates into a model that is analogous to the regular Hotelling model of spatial competition. The next result describes the outcome of the stage-game in this case.

Lemma 2 In the Nash Equilibrium of the stage game in which both firms use the same technology and consumers attribute no status to either good, both firms charge the same price. Prices are H H H H given by p0 = 1 + c = p1 , if both firms adopt the handicraft technology, and p0 = 1 = p1 if both firms adopt the industrial technology. Each firm sells its good to half of the market, that is, H H H H i = 1 − i = 1/2. Their operational profits are π0 = π1 = 1/2.

3.3 Hotelling Equilibrium Under Diﬀerent Technologies and No Sta- tus

Suppose that ﬁrms adopt diﬀerent technologies at time t. Assume, for example, that Firm 0 adopts the industrial technology, that is, τ 0 (Ht−1) = m, while Firm 1 adopts the handicraft technology, that is, τ 1 (Ht−1) = h, and that they both chose the same technology in period t − 1, that is, τ 0 (Ht−2) = τ 1 (Ht−2). The next result describes the outcome of the stage-game in this case.

Lemma 3 Suppose that, at period t, Firm 0 adopts the industrial technology, that is, τ 0 (Ht−1) = m, while Firm 1 adopts the handicraft technology, that is, τ 1 (Ht−1) = h, and suppose they both have chosen the same technology in period t − 1, that is, τ 0 (Ht−2) = τ 1 (Ht−2). In the Nash Equilibrium of the stage game, prices, demands, and operational proﬁts are, respectively: 1 pHDT = 1 + c, (5) 0 3 2 pHDT = 1 + c, (6) 1 3

3 + c iHDT = , (7) 6 3 − c 1 − iHDT = , (8) 6 11 (3 + c)2 πHDT = , (9) 0 18 (3 − c)2 πHDT = , (10) 1 18 Because the game is symmetric in the absence of status effects, in order to obtain the stage- game payoffs firms obtain when Firm 1 adopts the industrial technology while Firm 0 adopts the handicraft technology, just switch the subindices 0 and 1 in equations (5) through (10) to obtain the answer.

3.4 Hotelling Equilibrium Under the Same Technology and Status

Suppose that both ﬁrms adopt the same technology at time t, that is, τ 0 (Ht−1) = τ 1 (Ht−1) and that they chose diﬀerent technologies in period t − 1, with τ 0 (Ht−2) = m and τ 1 (Ht−2) = h. In this case, consumers attribute status value to good 1 and no status value to good 0. The next result describes the outcome of the stage-game in this case.

Lemma 4 In the Nash Equilibrium of the stage game in which both ﬁrms use the handicraft technology and consumers attribute status value to good 1 only, prices, demands, and operational proﬁts are, respectively: 4 pHS = + c, (11) 0 3 5 pHS = + c, (12) 1 3

4 iHS = , (13) 9 5 1 − iHS = , (14) 9

16 πHS = , (15) 0 27 25 πHS = , (16) 1 27 and, in the Nash Equilibrium of the stage game in which both ﬁrms use the industrial technology and consumers attribute status value to good 1 only, prices are, respectively: 4 pHS = , (17) 0 3 5 pHS = , (18) 1 3 while demands and operational proﬁts are given by (13)and (14), and (15) and (16), respectively.

12 3.5 Hotelling Equilibrium Under Diﬀerent Technologies and Status

Suppose that firms adopt different technologies at time t. Assume, for example, that Firm 1 adopts the industrial technology, that is, τ 1 (Ht−1) = m, while Firm 0 adopts the handicraft technology, that is, τ 0 (Ht−1) = h, and that they chose different technologies in period t−1, with

τ 0 (Ht−2) = m and τ 1 (Ht−2) = h. In this case, consumers attribute status value to good 1 and no status value to good 0. The case in which Firm 0 adopts the industrial technology, that is,

τ 0 (Ht−1) = m, while Firm 1 adopts the handicraft technology that is, τ 1 (Ht−1) = h, is identical to the industry conﬁguration of the equilibrium path of the dynamic game, and therefore we analyze separately. The next result describes the outcome of the stage-game in the case where

τ 1 (Ht−1) = m and τ 0 (Ht−1) = h.

Lemma 5 Suppose that, at period t, Firm 0 adopts the handicraft technology, that is, τ 0 (Ht−1) = h, while Firm 1 adopts the handicraft technology, that is, τ 1 (Ht−1) = m, and suppose that they chose diﬀerent technologies in period t − 1, with τ 0 (Ht−2) = m and τ 1 (Ht−2) = h. In the Nash Equilibrium of the stage game, prices, demands, and operational proﬁts are, respectively: 4 2 pHSDT = + c, (19) 0 3 3 5 1 pHSDT = + c, (20) 1 3 3

4 − c iHSDT = , (21) 9 5 + c 1 − iHSDT = , (22) 9

(4 − c)2 πHSDT = , (23) 0 27 (5 + c)2 πHSDT = , (24) 1 27 Because the game is symmetric in the absence of status effects, in order to obtain the stage- game payoffs firms obtain when Firm 1 adopts the industrial technology while Firm 0 adopts the handicraft technology, just switch the subindices 0 and 1 in equations (19) through (24) to obtain the answer.

3.6 Stage Game Along the Equilibrium Path

Suppose that Firm 0 adopts the industrial technology, while Firm 1 adopts the handicraft technology. Suppose also that consumers attribute social status utility to good 1 and no social status to good 0. The next result describes the stage-game along the equilibrium path.

13 Lemma 6 Equilibrium path prices, demands, and operational proﬁts are, respectively: 4 1 p∗ = + c, (25) 0 3 3 5 2 p∗ = + c, (26) 1 3 3

4 + c i∗ = , (27) 9 5 − c 1 − i∗ = , (28) 9

(4 + c)2 π∗ = , (29) 0 27 (5 − c)2 π∗ = , (30) 1 27 Furthermore, the value functions, evaluated at the equilibrium values, are:

(4 + c)2 K V ∗ = − , (31) 0 27(1 − δ) (1 − δ)

(5 − c)2 V ∗ = . (32) 1 27(1 − δ)

If the equilibrium prices are given by equations (25) and (26), consumer i∗’s utilities become:

∗ ∗ 5 − c 4 + c U ∗ (p ) = U ∗ (p ) = u − − . i 0 i 1 9 3

∗ ∗ ∗ For consumers with index i , it is individually rational to buy the good if Ui∗ (p0) = Ui∗ (p1) ≥ 0, or: 5 − c 4 + c u ≥ + . 9 3 At the solution, the operational profit of Firm 0 is larger than the operational profit of Firm ∗ ∗ 1, or, mathematically, π0 > π1, if and only if c > 1/2. Firm 1 produces the high status good. In order for this good to have smaller demand than the regular good (produced by Firm 0), it is necessary and sufficient that i∗ > 1/2. This condition is equivalent to c > 1/2. This implies ∗ ∗ that Firm 0’s operational profits are larger than Firm 1’s, that is, π0 > π1. Thus, if c < 1/2, this social norm ensures that, in any equilibrium path, the status good is the most exclusive. Exclusiveness has long been considered a key factor in defining whether a particular product is a status good or not.6 Status goods being the most exclusive imply that

6See, for instance, Silverstein et al (2008), Chapter 3.

14 operational profits of the firm producing status goods must be smaller than those of the firm producing a good of no status. The good produced by Firm 1 has positive demand when i∗ < 1. This condition is equivalent to c < 5. Firm 1’s operational profits are higher in this equilibrium with product differentiation ∗ H 2 than in the Hotelling equilibrium or, mathematically, π1 > π1 , if and only if (5 − c) /27 > 1/2. ∗ H The profit π1 of Firm 1 is larger than its Hotelling profit, π1 , if only if the marginal cost is sufficiently small in the sense that c < 5 − p27/2. Henceforth, we assume that 1/2 < c < 5 − p27/2.

3.7 One-Shot Deviation: Firm 0

In order to verify that the allocations described in the previous sections are indeed equilibrium outcomes, we need to check whether there are operational profitable one-shot deviations from the D0 D0 prescribed actions. We need to calculate the off-equilibrium path payoffs. Let π0 and π1 be the one-period, stage-game payoffs of Firms 0 and 1 it obtain, respectively, when Firm 0 deviates D0 D0 from the equilibrium path. Analogously, define p0 and p1 as the prices Firms 0 and 1 charge in this situation.

Lemma 7 The stage-game outcome when Firm 1 deviates is such that the prices, demands, and D D operational proﬁts π0 and π1 are given by: 4 1 iD0 = − c, (33) 9 18 5 1 1 − iD0 = + c, (34) 9 18

4 5 pD0 = p∗ = + c, (35) 0 0 3 6 5 2 pD0 = p∗ = + c, (36) 1 1 3 3

4 1 4 1 (8 − c)2 πD0 = − c − c = , (37) 0 3 6 9 18 108 5 1 5 1 πD0 = − c + c . (38) 1 3 3 9 18

3.8 One-Shot Deviation: Firm 1

In order to verify that the allocations described in the previous sections are indeed equilibrium outcomes, we need to check whether there are proﬁtable deviations from the prescribed actions.

15 D D We need to calculate the off-equilibrium path payoffs. Let π0 and π1 be the one-period, stage- game payoffs that Firms 0 and 1 obtain, respectively, when Firm 1 deviates from the equilibrium D D path. Analogously, define p0 and p1 as the prices Firms 0 and 1 charge in this situation. If the status good firm deviates, it will raise K in order to make the low quality good at D zero marginal cost, but still sell it for a price p1 to maximize its current operational profit given consumers’ beliefs. Consumers base their beliefs on (social) experience, and thus take Firms 1’s good as being of high status with exclusivity utility given by 1 − i∗. Firm 1 sells these goods to the 1 − iD consumers on the right end of the unit interval.

Lemma 8 The stage-game outcome when Firm 1 deviates is such that the prices, demands, and D D operational proﬁts π0 and π1 are given by: 4 1 iD = + c, (39) 9 18 5 1 1 − iD = − c, (40) 9 18

4 1 pD = p∗ = + c, (41) 0 0 3 3 5 1 pD = − c, (42) 1 3 6

4 1 4 1 πD = + c + c , (43) 0 3 3 9 18 5 1 5 1 (10 − c)2 πD = − c − c = . (44) 1 3 6 9 18 108

In order for the demand for the good produced by Firm 0 to be positive when Firm 1 deviates, we need iD > 0. This condition is equivalent to c < 8. This latter condition is automatically satisﬁed because of the previous assumption that c < 5 − p27/2.

3.9 No Punishment After Deviation: Hotelling Competition with Reputation Loss

Suppose both firms have used the industrial technology in the previous period, and now Firm 1 wants to build a reputation for high status products. This situation can occur when Firm 1 deviates from the differentiated product equilibrium path and Firm 0 chooses not to punish Firm 1. Although not punished by Firm 0, Firm 1 suffers a loss due to the loss of reputation of its product. Although consumers do not act strategically in our model, we can interpret this as a form of “consumer punishment”.

16 If Firm 1 reverts to the handicraft technology, then the two ﬁrms play a simultaneous Hotelling game. Consumers attribute low status to both goods 0 and 1. Firm 1, however, uses the handicraft technology, while Firm 0 uses the industrial technology. The next result establishes the equilibrium of the resulting stage-game.

Lemma 9 The stage-game equilibrium occurring in the period immediately after Firm 1 had NP NP deviated has prices, demands, and operational proﬁts π0 and π1 given by, respectively:

1 pNP = 1 + c (45) 0 3 2 pNP = 1 + c, (46) 1 3

1 1 iNP = + c (47) 2 6 1 1 1 − iNP = − c, (48) 2 6

1 1 2 πNP = 1 + c (49) 0 2 3 1 1 2 πNP = 1 − c . (50) 1 2 3

3.10 Punishment After Deviation: Generic Pricing

Suppose Firm 0 has deviated from the equilibrium path and Firm 0 wants to punish Firm 1 by L setting up a punishment price p0 in the first period after deviation. For now, let us consider that this price can be arbitrarily fixed. This situation is equivalent to a Stackelberg model, in which Firm 0 is the leader, and Firm 1 is the follower. Consumers once again attribute low status to both goods 0 and 1. Firm 1, however, uses the handicraft technology, while Firm 0 uses the L industrial technology. Firm 1 reacts to the arbitrary price p0 fixed by Firm 0 by playing a best L reply p1 (p0):

L L p1 = arg max(p1 − c)(1 − i ) (51) p1 1 + p0 − p1 = arg max(p1 − c) , p1 2 where equation (2) is used to calculate Firm 1’s demand. The following lemma gives us the outcomes of this stage-game.

17 Lemma 10 The stage-game outcome occurring when Firm 0 chooses a (generic) punishment L price p0 in the period immediately after Firm 1 had deviated has prices, demands, and operational L L proﬁts π0 and π1 given by, respectively:

1 + pL + c pL = 0 (52) 1 2 3 − pL + c iL = 0 (53) 4 1 + pL − c 1 − iL = 0 (54) 4 pL(3 − pL + c) πL = 0 0 (55) 0 4 (1 + pL − c)2 πL = 0 . (56) 1 8 Remark 1 If the two ﬁrms are using the industrial technology, then the outcome of the stage- game can be obtained by making c = 0 in the equations of Lemma 10. In this case, we have Firm LK L LK L LK L 1’s price p1 = 1 + p0 /2, the demands are i = 3 − p0 /4 and 1 − i = 1 + p0 /4, L LK L LK L 2 respectively, and proﬁts are given by π0 = p0 (3 − p0 )/4 and π1 = (1 + p0 ) /8, respectively.

3.11 Punishment After Deviation: Stackelberg Equilibrium

Let us consider now the situation in which the two ﬁrms play a Stackelberg equilibrium, in which Firm 0 is the leader, and Firm 1 is the follower. Firm 0 maximizes its operational proﬁt taking into consideration Firm 1’s best reply to be played afterwards. This objective function is given by equation (55). The equilibrium values of this stage-game are shown in the following lemma.

Lemma 11 The stage-game equilibrium occurring when ﬁrms play sequentially, Stackelberg style, ST with Firm 0 as leader and Firm 1 as follower, has prices, demands, and operational proﬁts π0 ST and π1 given by, respectively:

c + 3 pST = (57) 0 2 3c + 5 pST = (58) 1 4 c + 3 iST = (59) 8 5 − c 1 − iST = (60) 8 (3 + c)2 πST = (61) 0 16 (5 − c)2 πST = . (62) 1 32

18 Remark 2 If the two ﬁrms are using the industrial technology, then the outcome of the stage- game can be obtained by making c = 0 in the equations of Lemma 11. In this case, we have Firm LK LK LK 1’s price p1 = 5/4, the demands are i = 3/8 and 1 − i = 5/8, respectively, and proﬁts are LK LK given by π0 = 9/16 and π1 = 25/32, respectively.

3.12 Punishment After Deviation: Minmax Punishment

Firm 0 may choose as punishment, in case of deviation, a price that will inflict the highest possible loss to Firm 1. This most severe punishment is carried out through a minmax punishment. Firm 0 chooses the price that minimizes the one-period payoff Firm 1 obtains, given that Firm 1 plays a best reply to Firm 0’s price. Because Firm 1 deviated in the previous period, consumers will take both firms as producing goods of no status value. After observing a deviation, Firm 0 solves:

MM L p0 = arg minπ1 p0, p1 (p0) ,S(0) = S(1) = 0 (63) p0 L L = arg min(p1 (p0) − c)(1 − i ). p0

The following lemma summarizes the outcome of this stage-game.

Lemma 12 The stage-game outcome occurring when Firm 0 chooses a minmax punishment price in the period immediately after Firm 1 had deviated has prices, demands, and operational MM MM proﬁts π0 and π1 given by, respectively:

MM p0 = c − 1 (64) MM p1 = c (65) iMM = 1 (66) 1 − iMM = 0 (67) MM π0 = c − 1 (68) MM π1 = 0. (69)

3.13 Firm 0 deviates from punishing Firm 1

If Firm 0 refuses to punish a deviating Firm 1, Firm 1 punishes Firm 0. Suppose that Firms 0 L and 1 are such a punishment period. We expect Firm 0 to play the established value p0 . Both L L L goods have no status and Firm 1 plays a best reply to p0 ; that is, p1 = (1 + c + p0 )/2. However, L Firm 0 deviates and instead of carrying out the punishment by playing p0 as expected, it plays

19 L a best reply to p1 . The proﬁt of Firm 0 is: p (1 + pL − p ) π (p ) = 0 1 0 (70) 0 0 2 p 3 + c + pL = 0 0 − p 2 2 0

Outcomes of this stage game are given by the lemma below:

Lemma 13 The stage-game outcome when Firm 0 deviates, in the punishment path, from punishing Firm 1, is such that the prices, demands, and operational proﬁts πˇ0 and πˇ1 are given by:

3 + c + pL pˇ = 0 , (71) 0 4 1 + pL + c pˇ = pL = 0 , (72) 1 1 2

3 + pL + c ˇi = 0 , (73) 4 1 − c − pL 1 − ˇi = 0 , (74) 4

(3 + c + pL)2 πˇ = 0 . (75) 0 32 (1 − c + pL)(9 − c − pL) πˇ = 0 0 . (76) 1 24 3.14 Firm 1 punishes Firm 0

Suppose Firm 1 has status and plays as a leader in a leader-follower stage-game by setting a S (punishment) price p1 to be determined later. We will show that this conﬁguration occurs when Firm 1 wants to punish Firm 0 from deviating from the punishment path. Suppose that Firm 0 uses industrial technology and Firm 1 uses handcraft technology. The Firm 0’s proﬁt is: p (1 + pS − p ) π (p ) = 0 1 0 . (77) 0 0 3 The following lemma establishes the outcomes of this stage-game:

Lemma 14 The stage-game outcome occurring when Firm 1 chooses a (generic) punishment S price p1 in the period immediately after Firm 0 had deviated has prices, demands, and operational proﬁts π¨0 and π¨1 given by, respectively:

20 1 + pS pS = 1 (78) 0 2 3 − pL + c iL = 0 (79) 4 1 + pL − c 1 − iL = 0 (80) 4 (1 + pS)2 π¨ = 1 , (81) 0 12 (pS − c)(5 − pS) π¨ = 1 1 . (82) 1 6 4 Dynamic Game Equilibria

4.1 Equilibrium with Product Diﬀerentiation on Status

This section shows that if economic agents are sufficiently patient, then there is a strategy profile σ∗ such that a status good is produced at every period along the equilibrium path. Firm 1 chooses to use the handicraft technology and charge prices according to equation (26). Along the equilibrium path, Firm 0 uses the industrial technology and charges prices according to equation (25). Firm 0 actively punishes Firm 1 in case of deviation. Firm 1 punishes Firm 0 if Firm 0 refuses to punish it after deviation. Along the equilibrium path, consumers attribute status value for good 1, no status value to good 0, and demand these goods according to equations (27) and (28). As long as no firm deviates from the equilibrium path of the strategy profile σ∗, good 1 has status value to its consumers, while good 0 does not. Either firm may have incentives to deviate and deceive consumers to obtain larger short-term profits. Consider now a situation in which Firm 0 does not apply any punishment. Instead, it simply plays a best reply to Firm 1’s price. In the period following Firm 1’s deviation, good 1 will lose its high status value. If Firm 1 decides to return to handmaking its product, it will produce at a higher marginal cost and sell it at a lower price, obtaining reduced profits. This “consumer punishment” may provide sufficient incentives to deter Firm 1 from ever deviating from the prescribed actions. If this consumer punishment cannot deter deviation, Firm 0 can then actively engage in punishment actions that may deter Firm 1 from ever deviating. In fact, Firm 0 might engage in active punishment even when consumer punishment can deter Firm 1’s deviation. From subsection 4.2 onwards, we show under which conditions this profile can be implemented as a Subgame Perfect Equilibrium and describe precisely the dynamic strategies that can implement it. The punishment that deters Firm 1’s deviation might envolve actions that hurt Firm 0 in the short term. More precisely, Firm 0 might charge a punishment price that gives Firm 0 profits that are lower than those it would obtain by simply playing a best response to Firm 1’s action in

21 that period. Firm 0 is compensated for going through with the punishment by obtaining higher profits in the future, and it does indeed go through with the punishment if it is sufficiently patient. Off this punishment path, we have to describe what Firm 1’s behavior would be. In the eventuality that Firm 0 does not go through with the punishment, Firm 1 can punish Firm 0 for it. This punishment would imply choosing a punishment action that reduces Firm 0’s profits in the period following deviation from punishment with a subsequent return to the equilibrium path profile σ∗. ∗ ∗ ∗ ∗ ∗ ∗ ∗ Call aj = pj , τ j , αj , γj , oj the action taken by Firm j along strategy profile σ . Let Firm 0’s ∗ ∗ ∗ ∗ ∗ ∗ ∗ action along σ be a0 = (p0, m, α0, γ0, public), where p0 is set according to equation (25), and α0 ∗ and γ0 are any combination of stock and debt issuance rules satisfying the financing constraint that the total amount raised with assets issuance is at least K. Analogously, let Firm 1’s action ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ along σ be a1 = (p1, h, α1, γ1, o1), where p1 is set according to equation (26), α1 and γ1 are any combination of stock and debt issuance rules satisfying the financing constraint that the total ∗ amount raised with assets issuance is at least K in case it decides to go public, and o1 can be any decision from Firm 1 regarding ownership structure (public, private or mixed action).7 ∗ ∗ At strategy profile σ , Firm 0 plays according to the following dynamic strategy σ0:

∗ 1. At t = 0, Firm 0 plays a0;

∗ ∗ 2. If Firm 0 observes action a1 at period t − 1, then Firm 0 responds with action a0 in period t;

∗ 3. If Firm 0 observes action a1,t−1 6= a1 at period t − 1, then Firm 0 responds, in period t, by starting the punishing phase:

L L L L L (a) Play action a0 ≡ p0 , m, α0 , γ0 , public at period t, where p0 ∈ [c − 1, (c + 3) /2] is L L a punishment price picked to satisfy incentive constraints, and α0 and γ0 are any combination of stock and debt issuance rules that clears ﬁnancial markets and raises L L L L L proceeds of at least K, and require Firm 1 to play action a1 ≡ p1 , h, α1 , γ1 , o1 , L L L where p1 is set according to (127) and α1 and γ1 are any combination of stock and debt issuance rules that clears ﬁnancial markets and raises proceeds of at least K, L while consumers take both brands as being of no status, and o1 can be any decision from Firm 1 regarding ownership structure (public, private or mixed action);8

L ∗ (b) If Firm 0 observes action a1 at period t, then Firm 0 responds with action a0 in period t + 1;

7In Subsection 4.1.1, we show that ﬁnancing K through the issuance of stocks and debt are equally optimal. 8 L We show in Subection 4.2 the precise restrictions p0 must satisfy.

22 L (c) If Firm 0 observes action a1,t 6= a1 at period t, then Firm 0 responds, in period t + 1, by returning to item (a) and restarting the punishing phase;

L L (d) If Firm 0 observes action profile (a0, a1) 6= a0 , a1 at period t, then: S S S S S i. Play a0 ≡ p0 , m, α0 , γ0 , public at period t + 1, where p0 is a one-period best- S S S response to a punishment price p1 charged by Firm 1, and α0 and γ0 are any combination of stock and debt issuance rules that clears financial markets and raises proceeds of at least K; S S ii. If Firm 0 observes action profile a0 , a1 in period t + 1, then Firm 0 responds ∗ with action a0 in period t + 2; S iii. Otherwise, play a0 again in period t + 2.

∗ ∗ At strategy proﬁle σ , Firm 1 plays according to the following dynamic strategy σ1:

∗ 1. At t = 0, Firm 1 plays a1;

∗ ∗ ∗ ∗ 2. If Firm 1 observes action proﬁle (a0, a1) was played at period t−2 and action proﬁle (a0, a1) L L ∗ or a0 , a1 was played at period t − 1, then Firm 1 plays action a1 in period t;

∗ ∗ 3. If Firm 1 observes action proﬁle (a0, a1) was played at period t − 2 and action proﬁle ∗ ∗ L (a0, a1) 6= (a0, a1) was played at period t − 1, then Firm 1 plays action a1 .

∗ ∗ 4. If Firm 1 observes action proﬁle (a0, a1) 6= (a0, a1) was played at period t − 2 and action L L proﬁle (a0, a1) 6= a0 , a1 was played at period t − 1, then:

S S S S S S (a) Play a1 ≡ p1 , h, α1 , γ1 , o1 at period t, where p1 ∈ [0, (5 + 2c) /3] is a punishment S S price picked to satisfy incentive constraints, α1 and γ1 are any combination of stock and debt issuance rules that clears ﬁnancial markets and raises proceeds of at least S K, and o1 can be any decision from Firm 1 regarding ownership structure (public, private or mixed action);

S S (b) If Firm 0 observes action proﬁle a0 , a1 in period t, then Firm 0 responds with action ∗ a1 in period t + 1; S (c) Otherwise, play a1 again in period t + 1.

∗ ∗ ∗ Strategy profile σ = (σ0, σ1) is the candidate equilibrium of the dynamic game for sufficiently patient agents. In the next few sections we show that σ∗ is indeed a Subgame Perfect Equilibrium. Figure 1 portraits the strategies firms 0 and 1 use in equilibrium, along with off-equilibrium transitions between states. In the following subsections we analyse profile σ∗ in further detail, showing that it indeed constitutes a subgame perfect equilibrium of the dynamic game.

23 ∗ ∗ (a0, a1) with a0 6= a0

S (a0, a1 ) ∗ ∗ ∗ ∗ S S (a0, a1) a0, a1 a , a with (aS, aS) 0 1 S 0 1 a0 6= a0

L L L (a0 , a1 ) (a0, a1 ) with a0 6= L a0 ∗ ∗ S S (a0, a1) with a1 6= a1 (a0 , a1) with a1 6= a1

L L a0 , a1

L L (a0 , a1) with a1 6= a1

Figure 1: Strategy Proﬁle σ∗: The superior left node represents the equilibrium path; the inferior node represents the state in which Firm 0 punishes a deviant Firm 1; in the superior right node, Firm 1 punishes a deviant Firm 0.

24 4.1.1 Financial Markets

To raise the capital amount K to purchase industrial machinery, at every period, Firm 0’s shareholders must issue shares and bonds. The total proceeds of this issuance must yield at least K. Old shareholders must give away a fraction α to new shareholders to finance (1 − γ)K, while the remaining γK are financed through the issuance of debt. Hence, Firm 0’s value function is a fraction 1 − α of the present value of current and future cash flows, that is:

∗ ∗ ∗ V0 = (1 − α)[π0 − Kγ + δV0 ] (83) (1 − α)[π∗ − Kγ] V ∗ = 0 . (84) 0 1 − δ + αδ Firm 0’s value function represents the value that shares of current shareholders attain in equilibrium, excluding shareholders who purchase shares issued at the current period. Similarly, Firm 1’s value function is given by:

∗ ∗ ∗ V1 = π1 + δV1 (85) π∗ V ∗ = 1 . (86) 1 1 − δ This is the value of Firm 1’s equity to current shareholders. Firm 1 shareholders do not give away any share of their firm. ALL Let V0 denote the total value generated by Firm 0. Firm 0’s old shareholders take share ∗ ALL ALL 1 − α of this value; that is, V0 = (1 − α)V0 , while surrendering share αV0 of the firm’s value to new shareholders. If Firm 0’s managers act in the interest of old shareholders, then ∗ they choose α and γ that maximize V0 , subject to the constraint that they must raise at least K with their stock and bond issuance. If the company decides to finance γK by issuing bonds, there remains (1 − γ)K to be funded by the issuance of new shares. By non-arbitrage, the value paid by new shareholders for a share α of Firm 0 must coincide with their discounted flow of gains. Hence:

ALL αV0 = (1 − γ)K. (87)

ALL Using this as well as equations (84) and V0 = (1 − α)V0 leads to: α[π∗ − Kγ] 0 = (1 − γ)K. 1 − δ + αδ

We now proceed to show that the classic Modigliani and Miller (1958) result holds in our model: Firm 0’s value is independent of its ﬁnancing.

25 Proposition 1 (Modigliani and Miller, 1958) The value of the industrial technology firm is independent on how it finances its capital investment K. This value is: π∗ − K V ∗ = 0 . (88) 0 1 − δ This result is intuitive because none of the basic assumptions of the original Modigliani-Miller model is broken along the equilibrium path. The fact that Firm 0’s value to old shareholders is equal to the present value of the company’s cash flows comes as a no arbitrage condition. Despite the dilution process that takes place every time the firm has to raise capital for new machinery, old shareholders are not expropriated in any way, because they receive their “fair value” for the shares they give away. Because Firm 0’s value is independent of financing, we can ignore financing strategies in calculating the firm’s values. The validity of Modigliani and Miller’s theorem is also another point in which the present study differs from Maksimovic and Titman (1991). In their study, firms are never indifferent between the two sources of financing. Increasing debt always change incentives for firms to offer high quality products, because it increases the probability of financial distress. In our study, public companies are indifferent between debt and equity financing. We have chosen Firm 0 to show Modigliani-Miller’s proposition because, as we show in subsequent sections, Firm 0 always chooses to be public in equilibrium. However, all results hold for Firm 1 if it chooses to go public and, as we show ahead, for sufficiently patient firms, there are Subgame Perfect Equilibria in which Firm 1 goes public as well. Modigliani-Miller holds in our model because the form of financing for a public company does not interfere with the Firm 1’s incentives to offer high status goods, only access to capital markets does, and access to capital markets is obtained in the ownership structure of the firm, defined by its decision to go public or private.

Remark 3 The equilibrium value of Firm 0 is at least as large as the value of Firm 1 when ∗ ∗ ∗ ∗ V0 ≥ V1 . This inequality holds if and only if π0 − K ≥ π1. This is true when the ﬁxed cost is ∗ ∗ ∗ ∗ suﬃciently small in the sense that K ≤ π0 − π1. Because π0 − π1 = (2c − 1) /3, then

∗ ∗ V0 ≥ V1 if and only if 3K ≤ 2c − 1.

4.2 Dynamic Strategy

This section analyzes the dynamic features of the game. We start by discussing ﬁrms’ incentives to obtain one-period gains if they deviate from the equilibrium path actions described in subsection 3.6. We start by considering Firm 0’s incentives to deviate from the equilibrium path by switching to the handicraft technology. If Firm 0 remains in the equilibrium path, it obtains

26 ∗ ∗ D0 2 V0 = (π0 − K) / (1 − δ). If it deviates, it obtains π0 = (8 − c) /108 in the ﬁrst period, and H Hotelling payoﬀs of π0 = 1/2 afterwards. Firm 0 obtains one-period gains from deviating if D0 ∗ π0 > π0 − K, that is, if: (8 − c)2 (4 + c)2 > − K (89) 108 27 which implies 108K > 48c + 3c2

D ∗ Firm 1 may obtain higher short-term gains if it deviates whenever π1 − K > π1, depending on values of δ. Assume that this inequality holds, and suppose that Firm 1 deviates from its D prescribed actions by switching to the industrial technology and charging price p1 as deﬁned in equation (42). Substituting (30) and (44) into this condition yields

108K < 20c − 3c2 (90)

If this condition does not hold, Firm 1 never has incentives to deviate from the equilibrium path, and the analysis that follows is irrelevant. Thus, in what follows, we assume that this condition holds. Comparing conditions (89) and (90) allows us to say that if Firm 0 is able to obtain short-term gains by deviating, then Firm 1 is not able to obtain short-term gains by deviating, and vice-versa. The following lemma formalizes this claim.

Lemma 15 If Firm 0 can obtain one-period gains by deviating from the equilibrium path, that is, if inequality (89) holds, then Firm 1 cannot obtain one-period gains by deviating from the equilibrium path, that is, inequality (90) does not hold. Likewise, if Firm 1 can obtain one-period gains by deviating from the equilibrium path, that is, if inequality (90) holds, then Firm 0 cannot obtain one-period gains by deviating from the equilibrium path, that is, inequality (89) does not hold.

Henceforth, let us assume that Firm 1 can obtain short term profits by deviating, that is, inequality (90) holds. Because, in this case, Firm 0 never has incentives to deviate, we do not have to worry about Firm 0’s short-term incentives to deviate. If Firm 1 deviates, it makes a higher operational profit in the current period, because consumers still believe they are purchasing a ∗ status good. Firm 0 still charges p0. With these prices, Firm 1 is able to catch a much larger market share, as part of Firm 0’s former market niche will now be served by Firm 1. Firm 0’s ∗ D operational profit will be reduced from π0 of equation (29) to π0 of equation (43). Firm 1 not only reduces Firm 0’s current operational profits, by occupying part of its market niche, it can also reduce Firm 0’s future prospects because, if this industrial configuration remains the same for the following periods, then firms will compete in the same category (goods with no status value), reducing total surplus to be shared by the firms.

27 By the end of the deviation period, Firm 0 realizes that Firm 1 had deviated. In the subsequent period, Firm 0 may choose to punish the deviating firm by choosing a punishment price L p0 . In this stage-game, as already mentioned in subsection 3.10, Firm 0 will act as a leader in Stackelberg competition, obtaining an extra market power that comes from its first-mover advantage. Consider the strategy profile σ∗. We need to verify whether: (i) It is credible for Firm 1 not to deviate from the equilibrium path; and (ii) provided that Firm 1 has deviated from the equilibrium path, it is credible for Firm 0 to go through with the punishment, that is, whether it would not prefer to simply play a one-stage game Nash Equilibrium every period after the deviation; (iii) once Firm 1 has been punished, if it has incentives to comply with the punishment and return to the equilibrium path; and (iv) if Firm 0 does not go through with the punishment, whether Firm 1 has incentives to punish it. L Firm 0’s punishment price p0 has two boundaries. First, it is bounded from below by the MM minmax value p0 defined in equation (64) in which, during the punishment phase, Firm 0 charges a sufficiently low price such that Firm 1’s demand collapses, driving its profits to zero. This is the strategy yielding Firm 1 the least one-period payoff. If Firm 0 chooses prices this way, it will be using the harshest punishment to deviation. This strategy is the one that induces cooperation from Firm 1 for the widest set of discount rate parameters. Second, the punishment L ST price p0 is bounded from above by the Stackelberg price p0 defined by equation (57). This punishment maximizes Firm 0’s one-period profits given that Firm 1 has deviated in the previous period, and is the least costly punishment to Firm 0.

4.3 Incentive Compatibility Constraints

L The punishment price p0 must be set so that Firm 1 will not have incentives to deviate from the equilibrium path, that is: ∗ D L 2 ∗ V1 ≥ π1 − K + δπ1 + δ V1 . (91) If Firm 1 deviates, this punishment price has to be such that Firm 0 is willing to punish Firm 1. There are two possible paths after deviation in case Firm 1 deviates. Firm 1 can continue using the industrial technology (and compete Hotelling style with Firm 0) or switch back to the handicraft technology, in which case both ﬁrms return to the equilibrium path. Therefore, in order for σ∗ to be a subgame perfect equilibrium, we must have:

H  π0 1−δ , if Firm 1 chooses the industrial technology; L ∗  π0 + δV0 ≥ (92)  NP ∗ π0 + δV0 , if Firm 1 chooses the handicraft technology.

28 We can now verify whether the ﬁrms would have incentives to comply with the equilibrium and punishment prescriptions.

4.4 Deviations from the Equilibrium Path: Customer Punishment Only

If Firm 1 deviates, consumers attribute no status value to Firm 1’s product in the following period, which reduces Firm 1’s profits, working as a “consumer punishment” for deviation. It might be the case that this “non-strategic punishment” that consumers give Firm 1 could be sufficient to deter deviation, and no active punishment from Firm 0 is necessary to implement the cooperative equilibrium. Let us assume that Firm 0 does not punish a deviant Firm 1 and just plays a stage-game best response to Firm 0’s actions. In this case, deviation is not profitable if: π∗ π∗ 1 ≥ πD − K + δπNP + δ2 1 , 1 − δ 1 1 1 − δ D NP where π1 is given by equation (44), and π1 is given by (50). This inequality is equivalent to ∗ D NP (1 + δ)π1 ≥ π1 − K + δπ1 , or δ ≥ bδ1, where bδ1 is defined as:

D ∗ π1 − K − π1 bδ1 = ∗ NP . (93) π1 − π1 Therefore, the following lemma holds.

Lemma 16 Suppose the only punishment Firm 1 suffers if it deviates is consumer punishment. Then, deviation is not profitable for Firm 1 for sufficiently large values of δ and K, and sufficiently small values of c. More precisely, deviation is not profitable if and only if δ ≥ bδ1, where bδ1 is given by equation (93) or, alternatively, by: −6c2 + 40c − 216K bδ1 = . −4c2 − 8c + 92

For relatively small K or small δ and for relatively large c, the inequality δ ≥ bδ1 may not hold. To see this, consider the following example.

Example 2 Assume c = 1 and δ = 0. Then condition δ ≥ bδ1 holds if and only if K ≥ 17/108. Hence, if K < 17/108, customer punishment is not suﬃcient to deter deviation from Firm 1.

Thus, for sufficiently low values of K (or sufficiently low values of δ, or sufficiently large values of c), Firm 0 needs to actively punish deviations to prevent Firm 1 from doing it.

29 4.5 Deviations from the Equilibrium Path: Customer and Rival Pun- ishment

Let us consider now the situation in which a deviant Firm 1 is punished by Firm 0. We need to check whether Firm 1 has incentives to deviate from the equilibrium path. Without such incentives, inequality (91) holds. Inequality (91) then becomes: π∗ π∗ 1 ≥ πD − K + δπL + δ2 1 . (94) 1 − δ 1 1 1 − δ Solving for the discount factor yields:

∗ D L (1 + δ)π1 ≥ π1 − K + δπ1 L δ ≥ δ1(p0 ),

L where δ1(p0 ) is deﬁned as:

D ∗ L π1 − K − π1 δ1(p0 ) = ∗ L (95) π1 − π1 −6c2 + 40c − 216K = . 2 L L L −19c + 2 (27p0 − 13) c + 173 − 27p0 (2 + p0 ) Proposition 2 Firm 1 has incentives to play according to the actions prescribed by the equilibrium path; that is, inequality (94) holds, if and only if it is suﬃciently patient in the precise L sense that δ ≥ δ1(p0 ).

L If δ ≥ δ1(p0 ), then Firm 1 will not have incentives to deviate from the equilibrium path. The

L L threshold δ1(p0 ) is a function of the punishment price picked by Firm 0. The higher is p0 , the L higher δ1(p0 ) will be, and thus the smaller is the set of parameters δ such that Firm 1 is willing to comply with the equilibrium with diﬀerentiated products. There are parameter values such that inequality (94) holds, but also parameter values such that inequality (94) does not hold. To see that, consider the example below.

Example 3 Assume c = 1. Then equation (95) becomes:

L 34 − 216K δ1(p0 ) = 2 128 − 27p0

L For the harshest punishment, the minmax punishment, we have p0 = 0. In this case, we have 17 ≤ 64δ + 108K and inequality (94) might hold even for δ = 0, as long as K ≥ 17/108. For L the softest punishment, the Stackelberg punishment, we have p0 = 2. In this case, we have 17 ≤ 10δ + 108K and inequality (94) might hold even for δ = 0, as long as K ≥ 17/108.

30 In general, higher values of δ are needed to implement cooperation when punishment becomes softer. If, for example, K = 1/54, we need δ ≥ 15/64 to implement cooperation with the minmax punishment, but cooperation is not implementable using the Stackelberg punishment, since we would need δ ≥ 15/10 for equation (95) to be satisﬁed.

Example 3 shows that the incentive compatibility constraint may not be satisfied for many punishment prices and parameter values. Therefore, in many economically relevant situations, it is impossible to induce firms to coordinate in an equilibrium with product differentiation in which one good has status value.

4.6 Incentives for Firm 0 to Punish a Deviation from Firm 1

In order for sigma star to be an SPE, it must also be the case that the punishing firm prefers to go through with the punishment rather than playing a best-response action at this period. In this case, we first need to find what the continuation payoffs would be in case Firm 0 decides not to go through with the punishment. These payoffs depend on Firm 1’s actions subsequent from deviation. Two possible candidates arise:

Case 1 Firm 1 keeps the industrial technology, and the two firms compete then ”Hotelling style” at every period from then on. In the first case, Firm 1’s payoffs in each stage game would be given by 1/2 − K. The continuation payoffs are then (1/2 − K)/(1 − δ).

Case 2 Firm 1 returns to using a handicraft technology and the two firms revert to the situation in which each company uses a different technology. If Firm 1 reverts to the handicraft technology, then, in the first period, Firm 0 just plays a best reply to Firm 1’s price. Firm 1 plays a best reply to Firm 0’s price using the handicraft technology, taking into consideration the fact that consumers will take his product as being of low status, which is given by equation (127). The operational profits are given by equations (49) and (50). In subsequent periods, Firm 1 recovers its reputation as a status good producer, and both firms return to their pre-deviation industry configuration.

Firm 1 prefers to return to using the handicraft technology if: δ πH − K πNP + π∗ ≥ 1 (96) 1 1 − δ 1 1 − δ which occurs if δ > δNP , where H NP NP π1 − K − π1 δ ≡ ∗ NP π1 − π1 18c + 3c2 − 54K = 23 − 2c + c2

31 Claim 1 Firm 1 may prefer keeping the industrial technology, and the two ﬁrms compete then ”Hotelling style” at every period from then on (Case 1) over situation in which each company uses a diﬀerent technology (Case 2) or vice-versa, depending on parameter values.

To see that, consider the following example:

Example 4 Suppose c = 1 and K = 17/108. Then if δ > 2251/2376, Firm 1 prefers strictly to return to the equilibrium path, while if δ < 2251/2376, then Firm 1 prefers the Hotelling continuation payoﬀ, and if δ = 2251/2376, then Firm 1 is indiﬀerent.

Assuming Firm 1 returns to the equilibrium path, let us ﬁrst consider a situation in which Firm 0 is always willing to punish a deviant Firm 1, regardless of the discount factor. We would have that Firm 0 would obtain, by not engaging in punishment:

δ 1 1 2 δ (4 + c)2 K πNP + π∗ = 1 + c + − . 0 1 − δ 0 2 3 1 − δ 27 1 − δ

Firm 0 would rather not engage in punishment if the payoﬀ above is higher than the punishment path payoﬀ. It follows that a necessary condition for σ∗ to be a subgame perfect equilibrium is:

L L NP π0 p0 ≥ π0 (97)

L L L L ST Because π0 p0 is an increasing function of p0 for p0 < p0 = (c + 3) /2, the inequality above establishes an upper bound on the punishment prices Firm 0 charges in case of deviation.

Lemma 17 Suppose Firm 1 prefers to return to the equilibrium path rather than to play Hotelling L using the industrial technology henceforth. Then, Firm 0 has incentives to enforce punishment p0 L for any discount factor, that is, the punishment price p0 satisfies constraint (92) for all δ ∈ [0, 1], if and only if it satisfies: 3 + c 2 (3 + c) ≤ pL ≤ (98) 3 0 3 Lemma 17 states the punishment prices under which Firm 0 would always want to engage in active punishment, regardless of the discount factor. For Firm 0 to be willing to punish Firm 1 when it deviates, Firm 0 must have a profit that is at least as high as what it would obtain from NP just playing a best response to Firm 1’s current prices, that is, by charging p0 = (c + 3) /3. This NP NP is achieved by charging prices that lie above p0 , up until 2p0 . These prices actually “soften” NP punishment when compared to the situation in which Firm 0 just plays a best response p0 L to Firm 1 and let consumers do all the punishment, because, as p0 increases, the loss inflicted L onto Firm 1 becomes smaller. Larger values of p0 imply that Firm 1 is only willing to stick

32 to the equilibrium path for higher values of δ. It follows that punishment prices satisfying (98) cannot help implement σ∗ as a subgame perfect equilibrium if σ∗ is not already a subgame perfect NP equilibrium with punishment price p0 . In other words, if δ < bδ1, there are no subgame perfect equilibria with punishment prices satisfying (98) and strategy profile σ∗. L Firm 0 can, however, charge a punishment price p0 < (c + 3) /3. In this case, Firm 0 only has incentives to go through with the punishment if it is sufficiently patient. Charging L p0 < (c + 3) /3 hurts the firm in the short-term, as it can obtain a higher one-period profits by L charging p0 = (c + 3) /3. However, it will be compensated with higher future profits, as firms return to action profile a∗. Therefore, if Firm 0 is sufficiently patient, it will go through with the punishment. Firm 0 has the correct incentives to carry out the prescribed punishment if:

δ2 δ2 πL(pL) + δπ∗ + π∗ ≥ πˇ (pL) + δπ¨ (pS) + π∗. (99) 0 0 0 1 − δ 0 0 0 0 1 1 − δ 0 This is equivalent to: ´ L S δ ≥ δ0(p0 , p1 ),

L S whereπ ˇ0(p0 ) is given by (75),π ¨0(p1 ) is given by (81), and:

L L L ´ L S πˇ0(p0 ) − π0 (p0 ) δ0(p0 , p1 ) ≡ ∗ S (100) π0 − π¨0(p1 ) 1 L2 32 (3 + c) − 3p0 = 2 . 2 S (4+c) (1+p1 ) 27 − 12

´ L S Lemma 18 The threshold value δ0(p0 , p1 ) of the discount factor that determines in which cases L Firm 0 has incentives to punish a deviating Firm 1 is a decreasing function of p0 and an increas- S L ´ L S ´ NP S ing function of p1 . Moreover, at p0 = (c + 3) /3, we have that δ0(p0 , p1 ) = δ0(p0 , p1 ) = 0.

S We have that p1 should be larger than Firm 1’s minmax value. Firm 1’s minmax value is S,MM given by p1 = 0. This price drives Firm 0 proﬁts to zero. Because of our free exit hypothesis, this is the maximum damage one ﬁrm can impose on another. The lower bound is given by Firm 1’s Stackelberg value: 5 + c pS,ST = 1 2 Also, since the denominator in (100) must be positive, we must have that:

2 1 + pS (4 + c)2 1 ≤ 12 27 which implies 5 + 2c pS ≤ . 1 3 33 Because c ≤ 5 − p27/2, we have that

5 + 2c 5 + c < 3 2

S S,ST Therefore, the constraint that p1 ≤ p1 is not binding. S The value of punishment price p1 that enforces Firm 0 punishing a deviant Firm 1 for the S S,MM largest possible set of parameters δ is the minmax price p1 = p1 = 0. This occurs because ´ L S S ´ L S L δ0(p0 , p1 ) is increasing in p1 and thus, if we plot δ0(p0 , p1 ) in the space of δ × p0 as in Figure S ´ L S 2, a larger p1 implies in a shift to the right in the curve representing δ0(p0 , p1 ). Because we ´ L S L L L need δ to be above the curves representing δ0(p0 , p1 ), δ1(p0 ) and eδ1(p0 ), and both curves δ1(p0 ) L L ´ L S ´ L S and eδ1(p0 ) are increasing in p0 , shifting δ0(p0 , p1 ) to the right implies that δ0(p0 , p1 ) and the L L maximum between δ1(p0 ) and eδ1(p0 ) will intercept further to the right. Hence, this interception will occur at a higher value of δ. The values of δ that satisfy all constraints must lie above this S interception, and therefore increasing p1 reduces the values of δ satisfying all constraints in the model.

4.7 Incentives For Firm 1 to Return to the Equilibrium Path

The analysis made in the previous sections assumes Firm 1 uses the handicraft technology in the punishment period, and best-replies to the Stackelberg leader punishment prices given their technologies. Firm 1, however, may choose to continue adopting the industrial technology. Firm 1 has incentives to return to using the handicraft technology if and only if:

π∗ π∗ πL pL + δ 1 ≥ πLK pL − K + δ πL pL + δ 1 1 0 1 − δ 1 0 1 0 1 − δ

L Solving this inequality for δ yields δ ≥ eδ1 p0 , where

LK L L L L π1 p0 − K − π1 p0 eδ1 p0 ≡ ∗ L L (101) π1 − π1 (p0 )

L It turns out that, in the case where p0 ≥ (c + 3) /3, condition (101) is stricter than condition L L (94), which is redundant. Thus condition δ ≥ eδ1 p0 is the relevant one when p0 ≥ (c + 3) /3. The next proposition states this result formally.

Proposition 3 If Firm 1 is suﬃciently patient so that it has incentives to return to the equilibrium path once it ﬁnds itself in the punishment stage, then it also has incentives not to deviate from the equilibrium path. The reciprocal result does not hold, in general. Mathematically, for L L L every p0 in the relevant interval, we have that eδ1 p0 > δ1 p0 .

34 4.8 Incentives For Firm 1 to Punish a Deviation from Firm 0

Consider now the situation in which Firm 0 fails to punish Firm 1 after Firm 1 deviates. Firm 1 S S S S S S is supposed to punish this deviation by playing strategy a1 = p1 , h, α1 , γ1 , o1 , where α1 and S γ1 clear financial markets and generate proceeds of at least K, the firm is indifferent between S public and private ownership structures, and p1 is a punishment price chosen by Firm 1 to deter L S Firm 0 from deviating from a0 in the punishment path. Firm 0’s best response to p1 is given by S p0 = (1 + p1 )/2. Firm 1’s profits are thus given by:

S 1+p1 ! 2 − p1 + 2 πb1 = (p1 − c) b 3

Maximizing this function yields: 5 + pS + 2c p = 1 1 4 Therefore S2 5 − 2c + p1 πb1 = . b 48 L Firm 1 has incentives to punish Firm 0’s deviation from a0 if:

2 2 S ∗ δ ∗ S L L δ ∗ π¨1(p ) + δπ + π ≥ πb1(p ) + δπ (p ) + π . (102) 1 1 1 − δ 1 b 1 1 0 1 − δ 1

¨ L S that is, if δ ≥ δ1(p0 , p1 ), where:

S S ¨ L S πb1(p1 ) − π¨1(p1 ) δ1(p0 , p1 ) ≡ ∗ L L . π1 − π1 (p0 ) Substituting the proﬁt functions yields:

2 (5−2c+pS ) (pS −c)(5−pS ) 1 − 1 1 ¨ L S 48 6 δ1(p0 , p1 ) = 2 (103) 2 L (5−c) (1+p0 −c) 27 − 8

¨ L S Lemma 19 The threshold value δ0(p0 , p1 ) of the discount factor that determines in which cases S Firm 1 has incentives to punish a deviating Firm 0 is a decreasing function of p1 and an in- L creasing function of p0 .

¨ L S These features of δ1(p0 , p1 ) can be seen in Figure 2, where the dashed curves plot values ¨ L S S L S of δ1(p0 , p1 ) for diﬀerent values of p1 . All dashed curves are increasing with p0 . Also, as p1 ¨ L S ¨ L S increases, the δ1(p0 , p1 ) curves shift downwards, ilustrating the fact that δ1(p0 , p1 ) is decreasing S with p1 .

35 L L ´ L S ¨ L S Figure 2: Values of δ1 p0 , eδ1 p0 , δ0 p0 , p1 and δ1 p0 , p1 for c = 1 and K = 0.05, for values S of p1 changing 0.25 units at every curve, varying from 0 (dark red) to 2.25 (blue). The thick line n L Lo S shows max δ1 p0 ,eδ1 p0 . The purple vertical line represents the upper bound for p1 given by S 5+2c p1 = 3 .

36 L L ´ L S ¨ L S S Figure 3: Values of δ1 p0 , eδ1 p0 , δ0 p0 , p1 and δ1 p0 , p1 for c = 1, K = 0.05 and p1 = 0.75. n L Lo The thick line shows max δ1 p0 ,eδ1 p0 .

37 5 Optimal Punishment

In order for proﬁle σ∗ to be a subgame perfect equilibrium of the repeated game, all incentive contraints (91), (98)/(99), (101), and (102) need to hold. Incentive constraint (91) implies that L ´ L S δ ≥ δ(p0 ). Incentive constraints (98) and (99) impliy that δ ≥ δ(p0 , p1 ). Incentive constraint L ¨ L S (101) implies that δ ≥ eδ(p0 ). Incentive constraint (102) implies that δ ≥ δ1(p0 , p1 ). Taken together, these constraints imply that:

L S n L ´ L S L ¨ L S o δ ≥ δ(p0 , p1 ) ≡ max δ(p0 ), δ(p0 , p1 ),eδ(p0 ), δ1(p0 , p1 ) (104)

In the case in which δ ≥ bδ1, customer punishment can enforce the actions necessary to implement σ∗ as a subgame perfect equilibrium. In this case, Firm 0 can choose its punishment price to maximize proﬁts, even if it softens punishment. In this case, some of the constraints listed above are redundant. In particular, because in this case Firm 0 always has incentives to carry ´ L S ¨ L S out punishment, constraints δ ≥ δ(p0 , p1 ) = 0 and δ ≥ δ1(p0 , p1 ) always hold. The following subsection analyzes this case in detail.

5.1 Case 1 (δ ≥ bδ1): “Customer punishment only” can enforce equilibrium with status goods

Firm 0 can implement its punishment in case of deviation in an optimal way. In order to do it, it maximizes its current period proﬁt after deviation subject to (94) and (98) and to Firm 1’s ST best reply (124). It also needs to take into consideration the upper and lower bounds p0 and MM L p0 to punishment prices p0 . First, Firm 0 will never choose minmax punishment. This occurs because inequality (98) L L MM implies that p0 > (3 + c) /3, and this lower bound on p0 is larger than p0 = c − 1. This is a consequence of the assumption that c > 5 − p27/2. Thus, the following lemma holds:

Lemma 20 Minmax punishment prices are lower than the best-reply price under “customer only MM NP punishment”, that is, p0 < p0 = (c + 3) /3. Therefore, if δ ≥ bδ1, then minmax punishments are not used in any Subgame Perfect Equilibrium of the dynamic game.

Proposition 3 shows that the incentive condition to make Firm 1 return to the equilibrium path after it has deviated is the strictest of all the incentive conditions when δ ≥ bδ1. Hence, a necessary and suﬃcient condition for strategy proﬁle σ∗ to be a subgame perfect equilibrium is that incentive condition (101) holds. The following result thus holds.

38 Proposition 4 Let δ ≥ bδ1. Then a necessary and sufficient condition for the strategy profile described in subsection 4.2 to be a Subgame Perfect Equilibrium is that Firm 0 has incentives to return to the equilibrium path once it has deviated from it and suffered one period of punishment. In other words, the strategy profile described in subsection 4.2is a subgame Perfect Equilibrium L of the dynamic game if and only if δ ≥ eδ1 p0 .

L For a given δ, Firm 0 will pick a price p0 satisfying both (101) and (136). In the relevant L interval in which both of these constraints are satisﬁed, Firm 0’s proﬁts are increasing in p0 . Therefore, the optimal punishment involves choosing a price such that (101) holds with equality, L whenever the value of p0 that makes (101) hold with equality is below the Stackelberg price ST L L L ST L ST p0 = (3 + c)/2. Let pe0 denote such value of p0 . If pe0 > p0 , Firm 0 just chooses p0 = p0 . Therefore, the following proposition holds.

∗L Proposition 5 Let δ ≥ bδ1. Then the optimal punishment price p0 Firm 0 chooses is the smallest between the Stackelberg price and the price that makes Firm 1 indiﬀerent between complying with the punishment and returning to the equilibrium path and playing a Hotelling equilibrium for another period. Formally: L ∗ L ST p0 = min pe0 , p0 .

∗L ∗L Firm 0 will only carry out this punishment if p0 ≥ (c + 3) /3. If p0 < (c + 3) /3, then it might not be worth for Firm 0 to go through with the punishment, because equation (98), which states the condition under which Firm 0 would always prefer to carry out the punishment, regardless of the discount rate δ, is violated. Subsection 5.2 below analyzes this case.

5.1.1 Comparing the two types of punishment

We can now compare which form of punishment is more eﬃcient in providing incentives to Firm 1: punishment by consumers alone or punishment by Firm 0. The following proposition formalizes the discussion.

L Proposition 6 Fix p0 in the interval [(c + 3) /3, (c + 3) /2] so that Firm 0 is willing to carry out such punishment for all δ. Suppose that individuals are suﬃciently patient so that Firm 1 L has no incentives to deviate from the equilibrium path when Firm 0 sets punishment price at p0 , L that is, suppose that δ ≥ δ1 p0 . Thus, Firm 1 has no incentives to deviate from the equilibrium path when punishment is carried out only by consumers either, that is, we must have δ ≥ bδ1. L L Summarizing, we have that δ1 p0 ≥ bδ1, with equality holding if and only if p0 = (c + 3) /3.

In the case in which δ ≥ bδ1, we can summarize conditions for the existence of an equilibrium in which Firm 1 will not deviate with the equilibrium path in the following proposition.

39 D ∗ Lemma 21 Assume that short-term deviation is profitable for Firm 1 that is, if π1 −K > π1. In L ST the relevant region of parameter values, in which π0 ≤ π0 and equations (91) and (92) hold, the L following statements are equivalent: (i) Firm 0 is willing to carry out the punishment at price p0 L for all 0 < δ < 1, that is, p0 ≥ (c + 3) /3; (ii) Consumer punishment is at least as severe as any punishment Firm 0 agrees to implement, that is, bδ1 ≤ δ1 (p0); (iii) the stage-game payoffs Firm 1 obtains in the punishment period is at least as large as the payoff it obtains in the punishment L L NP period of the stage-game when only consumers punish Firm 1, that is, π1 p0 ≥ π1 ; (iv) the stage-game payoffs Firm 0 obtains in the punishment period is at least as large is the payoff it obtains in the punishment period of the stage-game when only consumers punish Firm 1, that is, L L NP π0 p0 ≥ π0 .

5.2 Case 2 (δ < bδ1): “Customer punishment only” cannot enforce equilibrium with status goods

In this case, Firm 0 cannot count on consumers to do the punishment and maximize current ∗ period profits as in the case in which δ ≥ bδ1. In this case, in order to enforce σ , Firm 0 has L to choose a punishment price p0 that reduces its current profits in exchange for higher future profits. In this case, constraints (99) and (102) are not redundant. Furthermore, unlike the case where δ ≥ bδ1, any of the constraints can be the strictest, and therefore, the relevant one. Figure S 3 depicts an example with c = 1, K = 0.05 and Firm 1’s punishment price given by p1 = 0.75. L For low Firm 0’s punishment prices p0 , we have that the relevant constraint is Firm 0 must be ´ L S L willing to punish Firm 1 for its deviant behavior, that is, that δ ≥ δ p0 , p1 . As p0 increases, ´ L S ¨ L S δ p0 , p1 decreases and δ p0 , p1 increases. Eventually, the two curves cross each other and the relevant constraint is now that Firm 1 must be willing to punish Firm 0 from not punishing Firm ¨ L S L ¨ L S L 1, that is, that δ ≥ δ p0 , p1 . As p0 continues to incresase, δ p0 , p1 crosses eδ p0 , and the relevant constraint is now that Firm 1 must be willing to comply with Firm 0’s punishment and L return to the equilibrium path, that is, that δ ≥ eδ p0 . S ´ L S ¨ L S Figure 2 shows how changes in Firm 1’s punishment price p1 affects δ p0 , p1 and δ p0 , p1 . S ´ L S ¨ L S S As p1 increases, δ p0 , p1 shifts northeast, while δ p0 , p1 shifts southeast. If p1 = 0, the ¨ L S L ∗ constraint δ ≥ δ p0 , p1 cannot be satisfied for any p0 , and σ cannot be sustained as a subgame ¨ L S L perfect equilibrium. After a certain level, δ p0 , p1 crosses δ1 p0 and the constraint δ ≥ ¨ L S δ p0 , p1 becomes redundant. L Because Firm 0’s profits are increasing with punishment price p0 , Firm 0 chooses the high- L est punishment price that satisfies all incentive constraints. Let p0 be the largest of Firm 0’s L S punishment prices such that inequality (104) holds with equality, that is, p0 (p1 ) is chosen such

40 that:

L S S δ = δ(p0 (p1 ), p1 ) n o L S ´ L S S L S ¨ L S S = max δ(p0 (p1 )), δ(p0 (p1 ), p1 ),eδ(p0 (p1 )), δ1(p0 (p1 ), p1 ) (105)

L S L The punishment price p0 (p1 ) is the largest value of p0 that makes all incentive constraints hold. This is thus the optimal punishment for Firm 0 in the case where δ < bδ1. The following proposition summarizes this result.

∗L Proposition 7 Let δ < bδ1. Then the optimal punishment price p0 Firm 0 chooses is the largest L value of p0 that satisﬁes all incentive constraints, given by (91), (99), (101), and (102). Formally:

L ∗ L S p0 = p0 (p1 ).

L S L where p0 (p1 ) is chosen as the largest value of p0 to satisfy equation (105).

6 Social Status, Private Firms and Commitment

L S There are, however, combinations of parameter values such that no punishment price p0 and p1 satisfy all incentive constraints necessary for σ∗ to be a subgame perfect equilibrium. This occurs L S L S when δ < δ p0 , p1 for all values of p0 and p1 . In this case, even the most severe punishments are not suﬃcient to deter deviations from σ∗ and, therefore, the proﬁle σ∗ is not a subgame perfect equilibrium of the repeated game.

However, there are alternative setups of the game in which an equilibrium with product differentiation on status can be implemented. One such alternative setup is a modified version of the game presented in the previous sections in which the firm can impose private financing only on its constitution. By doing so, it can guarantee that it will not have access to the industrial technology and therefore insure production of status goods to their customers. In this case, the “burning the bridges” firm constitution provision works as a commitment device. By committing to a private financing policy, the firm can insure a higher operational profit than if they always adopt the industrial technology and compete Hotelling style with its rival. Suppose now that Firm 1 can commit to permanently retaining private status (through a written statement on its constitution or some other mechanism), forbidding its management from ever going public. This would make capital markets inaccessible to Firm 1, making deviation from the equilibrium path not feasible. Suppose further that this commitment is publicly observable.9

9If commitment is not publicly observable, we are back in the same case as Section 5. For a suﬃciently impatient private ﬁrm, that is, δ < δ, there is an incentive to deviate and produce the regular good claiming it to be the status good.

41 Firms then have to choose at time zero which path they will follow: if they do not commit, the subgame after this action reduces to the game analyzed in the previous sections, in which case for low values of delta choosing this path means choosing Hotelling competition with no status goods being produced in every period; or, for Firm 1, producing the status good forever while Firm 0 produces the regular good forever. The sequence of events at every stage-game are the same as before. The only difference is now that Firm 1 has to choose between committing and not committing at the beginning of the game, before any interaction between firms and consumers or between firms and potential new investors occur. Assume, as before, that, in the first period of play, consumers attribute positive status value to good 1 and no status value to good 0.

If δ < bδ1 = eδ1 ((c + 3) /3) and Firm 1’s shareholders decide to commit by forbidding the firm’s managers from ever making the firm public, the firm plays along the same equilibrium path described in previous sections. If Firm 1’s shareholders decide, on the other hand, not to make this commitment, then firms will not play along the equilibrium path of Subsection 4.2, because δ < bδ1 implies that the actions along that path cannot be enforced. A strategy for Firm 1 in the new game is now characterized by vectors of actions that specify the decision in the beginning of period 0 regarding commitment and the decisions at every period regarding prices, technology, financing and company ownership structure (private or public) for every possible history of the game. Formally, a vector σ ≡ ρ , (p (H ) , τ (H ) , α (H ) , γ (H ) , o (H )) M,1 1 1 t−1 1 t−1 1 t−1 1 t−1 1 t−1 Ht−1∈H = ρ , (σ (H )) , 1 M,1 t Ht−1∈H where ρ1 ∈ {commit, do not commit} represents the firm’s decision at period 0 regarding commitment to a private ownership structure, defines a Firm j strategy in the modified game with commitment. A strategy for Firm 0 in the new game is now characterized by vectors of actions that specify actions in case Firm 1 commits and in case Firm 1 does not commit, that is, it is a 0 pair σM,0 = (σ1 if Firm 1 commits, σ1 if Firm 1 does not commit), where each of these subgame strategies follow the same definitions as the continuation game strategies for the original game.

After each finite history Ht , there is a subgame starting at t + 1. Let GM (Ht) be the continuation dynamic subgame of the modified game starting at period t+1 after the occurrence of history Ht. A continuation strategy σM,j (Ht−1) is the restriction of a strategy from t + 1 onwards such that the complete history of the game has finite history Ht from periods 0 to t.

A continuation proﬁle is a pair of continuation strategies (σM,0 (Ht−1) , σM,1 (Ht−1)) where each player plays a continuation strategy with the same ﬁnite history Ht.

42 A Nash Equilibrium of the continuation game is a continuation profile such that no player has a unilateral profitable deviation. A Subgame Perfect Equilibrium is a strategy profile such that, in every subgame that is reached after finite history Ht, the continuation profile induced by σ is the Nash Equilibrium of the subgame. The focus of the analysis in this study is on profiles that characterize Subgame Perfect Equilibria. The equilibrium of the modified game is now defined ∗ ∗ ∗ ∗ ∗ ∗ ∗ by a profile σM = ((ρ0, σ0 (Ht−1)) , (ρ1, σM1 (Ht−1))), in which (σM0 (Ht−1) , σM1 (Ht−1)) is a Nash

Equilibrium of every continuation game, and firms choose ρj optimally. ∗ We claim that the following strategy profile σM constitutes a Subgame Perfect Nash Equilib- rium of the modified game with commitment for sufficiently patient agents. At strategy profile ∗ ∗ ∗ H H σM , Firm 0 plays σ0,M = σ0 if Firm 1 commits, σ0 if Firm 1 does not commit , where σ0 con- H H H H sists in playing action a0 = p1 (m), m, α1 , γ1 , public in every period regardless of the history H of the game. Action a0 is the action Firm 0 plays if playing a Hotelling stage-game Nash Equi- H H H librium at every stage-game. Thus p1 (m) = 1, and α1 and γ1 are determined such that the amount raised yields at least K. At strategy profile σ∗, Firm 1 plays according to the following ∗ dynamic strategy σ1,M :

∗ ∗ 1. At t = 0, Firm 1 plays aM = commit, a1,M , where

∗ ∗ ∗ H a1,M = a1 with o1 = private if it commits; and a1 if it does not commit ,

H H where a1 is deﬁned analogously to a0 ;

∗ ∗ 2. Firm 1 plays action a1 in period t if Firm 1 played action a1 at period t − 1 or if action L L proﬁle a0 , a1 was played at period t − 1 if Firm 1 has committed at t = 0 and keeps H playing action a1 if it has not committed at t = 0;

L 3. Firm 1 plays action a1 for all other action proﬁles (a0,t−1, a1,t−1) played at period t − 1 if H it has committed at t = 0 and plays action a1 if it has not committed at t = 0.

∗ ∗ ∗ Strategy proﬁle σM = σ0,M , σ1,M is the candidate equilibrium of the dynamic game for ∗ suﬃciently patient agents. In the next few subsections we show that σM is indeed a Subgame L S Perfect Equilibrium. We also show that for many punishment price choices p0 and p1 in the “no commitment” subgame, the commitment path allows for the status good to be provided in equilibrium for a much wider set of parameter values δ than would be possible in the original game. In fact, for many values of K and c, commitment would enlarge the set of values of δ for which status goods are provided in subgame perfect equilibria even when Firms 0 and 1 choose L S the harshest possible punishment prices p0 and p1 .

43 6.1 Subgame in which Firm 1 chooses no commitment

How do firms play in the subgame in which they do not commit if they cannot implement the equilibrium with product differentiation on status? One possible equilibrium of this subgame is that firms just play a Nash Equilibrium of the stage-game at every period. Subsections 3.2, 3.3, 3.4, 3.5 and 3.6 show the payoffs obtained in Nash equilibria of the subgames that result at every stage-game after firms have chosen their technologies. We can use straightforward backward-induction to determine which technology firms 0 and 1 choose in each stage-game. In period 0, consumers give status value to good 1 and no status value to good 0. Subsection 3.4 analyzes the resulting subgame when firms adopt the same technology. Subsection 3.5 analyzes the resulting subgame when Firm 0 adopts the handicraft technology while Firm 1 adopts the industrial technology. Finally, Subsection 3.6 analyzes the resulting subgame when Firm 1 adopts the handicraft technology while Firm 0 adopts the industrial technology. The one-period payoffs that occur in the equilibrium of each subgame resulting from each pair of technology choices can be summarized in the payoff matrix below:

Firm 1 h m 16 25 (4−c)2 (5+c)2 (106) Firm 0 h 27 , 27 27 , 27 − K (4+c)2 (5−c)2 16 25 m 27 − K, 27 27 − K, 27 − K Computing the Nash Equilibrium of this stage-game generates the following result.

Lemma 22 If Firm 1 has a short-term incentive to deviate from the equilibrium path with product differentiation on status, that is, inequality (90) holds, then choosing the handicraft technology is a strictly dominated strategy of the stage-game with status with payoffs summarized by the payoff matrix (106) for both firms. It follows that the stage-game in which consumers attribute status value to good 1 has a unique Nash Equilibrium in which both firms choose the industrial technology.

The lemma above shows that the stage-game that starts at the first period has a unique Nash Equilibrium in which both firms choose the industrial technology, obtaining payoffs 16/27 − K and 25/27 − K, respectively. In the following period, because both firms had previously used the industrial technology, consumers will attribute no status value to either good. The resulting stage-game was analyzed in subsections 3.2 and 3. The one-period payoffs that occur in the equilibrium of each subgame resulting from each pair of technology choices can be summarized

44 in the payoff matrix below: Firm 1 h m 1 1 (3−c)2 (3+c)2 (107) Firm 0 h 2 , 2 18 , 18 − K (3+c)2 (3−c)2 1 1 m 18 − K, 18 2 − K, 2 − K Computing the Nash Equilibrium of this stage-game generates the following result. Lemma 23 If Firm 1 has a short-term incentive to deviate from the equilibrium path with product differentiation on status, that is, inequality (90) holds, then choosing the handicraft technology is a strictly dominated strategy of the stage-game with no status with payoffs summarized by the payoff matrix (107) for both firms. It follows that the stage-game in which consumers attribute no status value to either good has a unique Nash Equilibrium in which both firms choose the industrial technology. The lemma above shows that the stage-game played subsequently to a game in which both firms use the industrial technology also has a unique Nash Equilibrium in which both firms use the industrial technology. It follows that, in the subgame following a firm’s decision not to commit to private financing, if firms play a stage-game Nash equilibrium at every period, at the equilibrium path of this subgame, both firms will always use the industrial technology. Regardless of whether consumers attribute status to any of the goods produced, the Nash Equilibrium of the stage-game always produces an outcome that is, from the point of view of the firms alone, Pareto-dominated by other outcomes of the game. In particular, if firms could coordinate in using the handicraft technology, they would both be strictly better off than at the Nash Equilibrium. This “Prisoners’ Dilemma” feature of the stage-game occurs because the marginal cost of handicraft production is entirely passed on to consumers, since we are implicitly assuming all consumers purchase one good or the other. Can firms, in the subgame in which there is no commitment, find an alternative arrangement in which they can coordinate in using the technology that generates the higher aggregate payoff, the handicraft technology? One possible arrangement includes the use of Nash Reversion strategies that penalize a deviation from the use of handicraft technology with switching to the industrial technology once-and-for-all. Let −j denote the rival of Firm j. Such a strategy would involve:

H 1. If Firm j observes price p−j = 1 + c at t − 1, then Firm j responds by using the handicraft H technology and charging price pj = 1 + c in period t;

H 2. If Firm j observes a price of good −j diﬀerent from p−j = 1+c at period t−1, then Firm 0 LH responds, in period t, by switching to the industrial technology and charging price pj = 1 afterwards.

45 The following result shows that the Nash Reversion strategy just described cannot implement coordination in the subgame following a decision by Firm 1’s stockholders not to commit to private ﬁnancing.

Lemma 24 If Firm 1 has a short-term incentive to deviate from the equilibrium path with product differentiation on status, then, in the subgame following a decision by Firm 1’s stockholders not to commit to private financing, firms always have incentives to deviate from coordination by switching to the industrial technology and charging a one-period best-response to pH = 1 + c.

6.2 Equilibrium of the Dynamic Game with Commitment

Lemma 24 states that, if Firm 1 does not commit to private financing in period 0, then the best payoff it can obtain in subsequent periods is the Nash Equilibrium of the stage-game at every period. In the first period, Firm 1 enjoys a higher payoff due to the fact that consumers attribute status value to good 1. Afterwards, it just obtains the payoff 1/2 − K of the Hotelling game without status in which both firms adopt the industrial technology. Firm 1 shareholders prefer commitment over no commitment if the firm value under commitment is higher than under no commitment, that is, if: (5 − c)2 25 δ 1 ≥ − K + − K (108) 27(1 − δ) 27 1 − δ 2

C Let us deﬁne δ1 as the value of δ such that (108) holds with equality. Solving the inequality above for δ yields: 20c − 2c2 − 54K δC ≡ (109) 1 23

L NP 6.2.1 Eﬀect of commitment when p0 ≥ p0 Let us consider the case in which Firm 0 only plays punishment prices in which it is always willing to participate in punishment, regardless of δ and regardless of Firm 1’s punishment. This situation occurs when Firm 0’spunishment price is larger than its simultaneous best response to L NP Firm 1’s post-deviation price, that is, when p0 ≥ pe0 = p0 = (c + 3) /3. If we are considering S only strategies of this kind, Firm 1’s punishment price p1 is irrelevant, and the most severe L punishment that yields an equilibrium with a status good being provided is p0 = pe0 = (c + 3) /3.

L ∗ Because eδ1 (p0) is increasing in p0 , the strategy profile σ is not a Subgame Perfect Equilibrium if, even the most severe punishment pe0 is not sufficient to prevent Firm 1 from deviating. When ∗ δ < eδ1 ((c + 3) /3), the strategy profile σ is not implementable as an equilibrium. We can now state the following result.

46 Proposition 8 Suppose that Firm 0 only plays punishment prices in which it is always willing L NP to participate in punishment, that is, p0 ≥ p0 . Suppose further that Firm 1 can commit to private financing, and suppose that δ < eδ1 ((c + 3) /3), so that there is no Subgame Perfect C Equilibrium with product differentiation on status of the original game. If δ1 < eδ1 ((c + 3) /3), C and δ1 < δ < eδ1 ((c + 3) /3), then there exists a Subgame Perfect Equilibrium of the dynamic game with commitment in which the private firm produces the status good while the public firm ∗ ∗ produces the regular good. In the equilibrium, the overall payoffs of the firms are V0 and V1 , as ∗ ∗ stated by equations (31) and (32). Firms sell their goods at prices p0 and p1, given by (25) and (26), respectively, and sell their goods to market niches i∗ and 1 − i∗, given by (27) and (28), respectively. The regular good firm can sustain its production through public financing by issuing shares and/or debt.

C Is there a case such that parameter values yield δ1 < eδ1 ((c + 3) /3)? If not, then the set ∗ of parameters for which σM is a subgame perfect equilibrium of the modiﬁed game is empty.. Fortunately, there are such cases. To see that, consider the following example.

Example 5 Consider the game with commitment with parameters c = 1 and K = 17/108. Con- L straint (90) holds with equality in this case. Then p0 = (c + 3) /3 = 4/3 and the threshold level L eδ1 p0 for the equilibrium with product differentiation on status of the game without commit- C ment to hold is eδ1 (4/3) = 13/16. The value of the threshold level δ1 for the equilibrium with C product differentiation on status of the game with commitment to hold is δ1 = 19/46 < 13/16. Therefore, for values of δ in the interval (19/46, 13/16), there can only be equilibria with product differentiation on status in the game with commitment.

C Is there a case such that parameter values yield δ1 > eδ1 ((c + 3) /3)? Figure 4 shows that, in C C the area where δ1 > eδ1 ((c + 3) /3), we also have that both δ1 and eδ1 ((c + 3) /3) are negative, C which violates (90). Therefore, we must have δ1 < eδ1 ((c + 3) /3) in the economically relevant region. Hence, the following result follows:

Proposition 9 Suppose that Firm 0 only plays punishment prices in which it is always willing L NP to participate in punishment, that is, p0 ≥ p0 . If Firm 1 has a short-term incentive to deviate from the equilibrium path with product differentiation on status, then the set of discount factor values such that there is an equilibrium with product differentiation on status in the game without commitment is strictly contained in the set of discount factor values such that there is an equilibrium with product differentiation on status in the game with commitment. Formally, if (90) holds, then we must have: C δ1 < eδ1 ((c + 3) /3) .

47 Figure 4: Threshold values of δ for games with and without commitment. The red-to-blue surface gives the value of eδ1 ((c + 3) /3) as a function of K and c, while the orange-to-brown surface gives C the value of δ1 as a function of K and c. The colorless grid represents the plan δ = 0.

The proposition above states that there are always values of δ such that an equilibrium with L NP p0 ≥ p0 and product differentiation on status does not exist because either Firm 1 would have incentives to deviate or, if it did deviate, it would not have incentives to comply with the punishment and return to the equilibrium path, but an equilibrium with product differentiation of status in the game with commitment does exist. Thus, in economies with high values of δ, we should observe public companies providing both types of goods, with and without status value. In economies with low values of δ, status goods cannot be provided and all companies, regardless of their financing, provide only goods of no status value. For economies with intermediate values of C δ, namely values between δ1 and eδ1 ((c + 3) /3), we should observe private firms using handicraft technology offering status goods and public firms using industrial technologies.

Alternative commitment devices It is not common, however, for stockholders of a private firm to be able to completely forbid management from ever taking the firm public. However, there are mechanisms that stockholders of a private firm can use to make issuance of new shares and new bonds very costly, virtually taking the option of accessing public capital markets “off the table”. Among these mechanisms, are debt covenants between firms and banks that might provide

48 a company with short-term loans for working capital, and supermajority vote requirements for the approval of issuance of new shares. These requirements might imply that a firm cannot issue new assets without incurring in extra costs that go beyond the cost of capital of the resources being brought into the firm. Breaking debt covenants might mean that the firm would have to pay contractual fees and fines. In order to be able to obtain an IPO/SEO approval from the Board of Governors and/or the General Shareholders’ Meetings, managers might have to engage in costly proxy fights to obtain the necessary votes. One way or another, obtaining resources from public capital markets becomes a more expensive endeavor. Let us then reinterpret the variable K as being the total cost of issuing new securities for the issuing firm. There are two components in this cost: a cost of capital rK that is completely exogenous to the firm, whose value is determined by the equilibrium of supply and demand in capital markets; and an extra cost F determined in period 0, representing procedural costs of issuing new corporate securities that is determined internally, say, in the firm’s constitution. These costs represent the costs associated with all internal procedures necessary for issuing new securities in public markets, such as IPOs, SEOs, and publicly traded bonds. Now we have that

K = rK + F . In terms of formalization of strategies, the only thing that changes in comparison with the modified game analyzed in the beginning of this section is that ρ1 is now a continuous decision variable defined by the choice of F . We will refer to this game with a continuous commitment variable as the “dynamic game with costly issuance procedures”. Since Proposition 1 showed that Modigliani-Miller holds, a public company’s cost of capital is also independent of the firm’s capital structure, so we can assume that rK is independent of firms’ decisions regarding financing. So firms need to choose F in order to maximize firm value. √ 3 If c ≤ 5 − 2 6, Firm 1 shareholders would prefer to differentiate good 1 from good 0 in terms of status. Therefore, they need to set F such that all the incentive constraints hold, that is, Firm 1 does not have incentives to deviate from the equilibrium path (equation (95)), Firm 0 has incentives to carry out the punishment in case Firm 1 does deviate (equation (98)), and Firm 1 has incentives to comply with the punishment and return to the equilibrium path once it has deviated and accordingly punished (equation (101)). We have shown in Proposition 4 that constraint (101) is the strictest of the three constraints. Therefore, F needs to be set such that equation (101) holds, that is, such that δ ≥ eδ1 (p0). Thus,

∗ L L LK L L L δ π1 − π1 p0 ≥ π1 p0 − rK − F − π1 p0

Substituting the equilibrium proﬁt values and solving for F yields:

(1 + pL)2 (1 + pL − c)2 (5 − c)2 F ≥ 0 − (1 − δ) 0 − δ − r (110) 8 8 27 K 49 Therefore, we can enunciate the following proposition: √ 3 Proposition 10 Suppose that c ≤ 5− 2 6, and that one firm can choose procedures that lead to an increase F in the costs of issuance of stocks and bonds. Then, there exists a Subgame Perfect Equilibrium of the dynamic game with costly issuance procedures in which the private firm (Firm 1) produces a status good while the public firm (Firm 0) will produce a good with no status. In the ∗ ∗ equilibrium, the overall payoffs of the firms are V0 and V1 , as stated by equations (31) and (32). ∗ ∗ Firms sell their goods at prices p0 and p1, given by (25) and (26), respectively, and sell their goods to market niches i∗ and 1 − i∗, given by (27) and (28), respectively. Firm 0 can sustain its production through public financing by issuing shares and/or debt. In this equilibrium, Firm 1 shareholders fix procedures to issue new stocks and bonds leading to procedural costs F ∗ satisfying (110).

Proposition 10 shows that Firms can purposely make it more costly and difficult to access capital markets, because these restrictions create credibility with rivals and customers that the firm will keep producing the high status good, and not deviate to obtain short-term gains. The credibility created through F allows for a subgame perfect equilibrium with the provision of a status good to exist in the dynamic game with costly issuance procedures. The costly procedures established by the private firm in order to restrain the firm’s ability to access capital markets work as a commitment device that allows the firm to credibly offer a good with status value that would otherwise not be offered. L Because eδ1 (p0) is increasing in p0 , the smallest level of F that is able to implement the L equilibrium is obtained when Firm 0 uses the most severe punishment, that is, p0 = (c + 3) /3. Substituting this value in the equation above yields:

108c − 9c2 + δ [4c2 + 8c − 92] F ≥ − r 216 K

This lower bound for F is undoubtedly increasing in c, given that c < 5 − p27/2. Therefore, as the marginal cost of producing with the handicraft technology increases, so does the level of procedural costs that Firm 1 shareholders need to set for the issuance of new corporate securities.

L NP 6.2.2 Effect of commitment when p0 < p0 The effect of the existence of the commitment option on the set of values of δ for which there is a L NP subgame perfect equilibrium in which Firm 1 offers a status good is unanbiguous when p0 ≥ p0 . C C There is an equilibrium in which Firm 1 will commit if δ > δ1 , and because δ1 > eδ ((c + 3) /3), the set of values for which there is an equilibrium with status goods has a larger measure when the commitment option is available.

50 L L ´ L S ¨ L S S Figure 5: Values of δ1 p0 , eδ1 p0 , δ0 p0 , p1 and δ1 p0 , p1 for c = 1, K = 0.05 and p1 = 0.75. n L Lo C The thick curve shows max δ1 p0 ,eδ1 p0 .The thick line shows δ .

When the punishment price chosen by Firm 0 is such that it will only agree to punish Firm L NP 1 if it is sufficiently patient, that is, if p0 < p0 , then this effect is no longer guaranteed. To see that, reconsider the example portrayed in Figure 3, when c = 1, K = 0.05, and Firm 1 S C chooses a punishment price p1 = 0.75. In Figure 5, we added a thick line representing δ1 for L S these parameters and punishment price. The value of δ p0 , p1 is the maximum of the curves L NP in the graph. One can see that, for values of p0 close to 0 and close to p0 = 4/3, we have C L S that δ1 < δ p0 , p1 and the introduction of the commitment option helps to enforce cooperation and the provision of a status good for a wider set of values of the parameter δ. However, for L C L S intermediate values of p0 , we have that δ1 > δ p0 , p1 and commitment does not help implement the provision of status goods when punishment alone does not. Also, if Firm 0 chooses its punishment to provide the best incentives for both firms given the S L L L choice of p1 = 3/4 in Figure 5, that is, if it chooses p0 to minimize δ p0 , 3/4 , it will choose p0 in the intermediate area where commitment does not help improving the firms’ incentives. One can wonder if this is always the case. In other words, if Firms 0 and 1 can somewhow L S coordinate their punishments to provide the best incentives to both firms by choosing p0 and p1 L S to minimize δ p0 , p1 , does it completely offset any incentives benefit generated by the intro-

51 duction of the commitment option? Tables 1 through 5 show several variables for combinations L S L S of K and c in which p0 and p1 are chosen to minimize δ p0 , p1 . The values in bold letters represent parameter combinations for which Firm 1 has a short term to deviate from σ∗ in the L “no commitment” subgame. Table 1 portrays the values of p0 obtained, Table 2 portays the S L S C values of p1 obtained, Table 3 shows δ p0 , p1 and Table 4 shows δ1 . Table 5 shows a dummy C L S variable that takes a value of 1 if δ1 < δ p0 , p1 and zero otherwise, indicating whether there is any incentives improvement in having the option of commitment if Firms 0 and 1 can coordinate their choices of punishment prices. Actual incentives improvement is obtained in the cells in bold (where Firm 1 has short-term incentives to deviate from prescribed actions) marked with a value of 1. This occurs because in the cells not in bold, where Firm 1 does not have short-term C L S incentives to deviate, having δ1 < δ p0 , p1 does not change incentives, since Firm 1 always complies with prescribed acions in this case. As one can see from Table 5, for combinations of parameters along the short-term incentives threshold, where bold character numbers meet normal character numbers, there is incentive improvement. This threshold forms an imperfect diagonal in the matrix of the table. Along this diagonal, with both K and c increasing, we ﬁnd combination of parameters for which Firm 1 has C L S a short-term to deviate and δ1 < δ p0 , p1 , meaning that introducing the commitment option improves incentives. As we move away from this diagonal, it becomes harder to ﬁnd parameter combinations for which commitment improves incentives.

7 Discussion

The results in Sections 4 and 5, taken together, have the following implication. If the economic and institutional environment is such that the discount factor is relatively high, that is, if firms are sufficiently patient and care a great deal about future operational profits, then there is an equilibrium in which firms specialize and each produce a good designed for a particular market niche. The firm producing the status good may be private, but there is nothing constraining it to remain private in the future. The threat that consumers will punish the firm by forcing it to produce a high quality good at a low price, if the firm changes its status and produces a low quality good for one period may be sufficient to enforce this equilibrium. If firms are not that patient, then the firm’s rival might be able to retaliate from deviation by switching to a Stackelberg price or a constrained Stackelberg price in which the deviant’s rival chooses the highest price such that Firm 1 is able to comply with the punishment, Firm 0 is willing to punish Firm 1 in case it deviates, Firm 1 is willing to comply with the punishment, and willing to punish Firm 0 in case it fails to punish one of its deviations.

52 If the deviant’s rival cannot find such a price, or if the institutional environment is such that firms do not care as much about future operational profits as they do about present operational profits, that profile does not hold as a subgame perfect equilibrium. This occurs because consumers cannot give the status good firm a punishment sufficiently severe to give it the proper incentives for it to continue producing the status good, or either firm is unable to produce sufficiently severe punishments to provide the other firm the correct incentives for them to play the appropriate actions in each contingency of the game. If neither consumers nor rivals are able to enforce the cooperation of the high status good firm, then the game described in section 4 has no equilibrium with status goods production. However, if firms can commit to a strategy in which they produce the status good from the beginning, namely, permanently retaining private status, these goods will be produced in an equilibrium of the game with commitment, and the outcome will be similar to that obtained in the game without the commitment option. This commitment can be implemented through a written statement in the firm constitution precluding it to ever going public. Without public funding, the firm would not have access to the amount of capital it needs to purchase the industrial technology to mass-produce the regular good. The status good producing firm will have incentives to commit because it can obtain higher operational profits if it focuses on producing the high status good when compared to the alternative, in which firms engage in Hotelling price competition with both producing goods of no status value.

A Appendix: Proofs

A.1 Stage Game A.1.1 Consumer Demands

Proof. (of Lemma 1) The indiﬀerence of consumers having index i∗ means that their utility is equal if they buy the good from either ﬁrm; that is:

∗ ∗ ∗ ∗ (u − 1 + i ) + St(1) [1 − Dt(1, i )] − p1 = (u − i ) + St(0) [1 − Dt(0, i )] − p0.

Cancelling the common term u on both sides:

∗ ∗ ∗ ∗ −1 + i + St(1) [1 − Dt(1, i )] − p1 = −i + St(0) [1 − Dt(0, i )] − p0.

When no ﬁrm has reputation for producing a high status good, then St(0) = St(0) = 0. ∗ ∗ ∗ ∗ Hence, −1 + i − p1 = −i − p0. Solving this last equation for i and 1 − i yields equations (1) and (2). When only Firm 1 has reputation for producing a high status good, then St(0) = 0 and

53 ∗ ∗ ∗ St(1) = 1. In this case, −1 + i + [1 − Dt(1, i )] − p1 = −i − p0. Because Firm 1’s demand is ∗ ∗ ∗ ∗ ∗ ∗ ∗ Dt(1, i ) = 1 − i , and thus 1 − Dt(1, i ) = i . Hence, −1 + i + i − p1 = −i − p0. Solving this equation for i∗ and 1 − i∗ yields equations (3) and (4).

A.1.2 Hotelling Equilibrium Under the Same Technology and No Status

Proof. (of Lemma 2) Firms can, in principle, use either technology to produce their goods. To start with, let us analyze the case in which both firms use the handicraft technology. Firm 0 H H maximizes its operational profits π0 = (p0 − c)i . Firm 0’s operational profit function is: 1 + p − p πH = p iH = (p − c) 1 0 . 0 0 0 2

Maximizing this function yields ﬁrst order conditions:

(1 + p1 − p0) − (p0 − c) = 0.

Firm 0’s best reply is: 1 pH,BR = [1 + p + c] . (111) 0 2 1 We can now turn to Firm 1’s operational proﬁt maximization problem. By producing the status good, its operational proﬁt is:

∗ π1 = (1 − i )(p1 − c) 1 − p + p = 1 0 (p − c). 2 1

The first order condition for Firm 1’s operational profit maximization problem is −2p1 + p0 + c + 1 = 0. Firm 1’s best reply is: p + c + 1 pH,BR = 0 . (112) 1 2 H,BR H,BR Solving the system of best reply equations in p0 and p1 to find the equilibrium price leads to: H H p0 = 1 + c = p1 . Substituting these solutions back into (1) and (2), we obtain the threshold index and the demands: 1 iH = = 1 − iH . 2 Each firm obtains operational profit equal to: 1 πH = πH = . (113) 0 1 2

54 The equilibrium of the Hotelling model with both ﬁrms using the industrial technology can be obtained easily by making c = 0 in the equilibrium described above. In this case, the prices, market niche frontiers and operational proﬁts are:

IH IH p0 = 1 = p1 , 1 iIH = = 1 − iIH , 2 1 πH = πH = . (114) 1 1 2 We have thus completely characterized the equilibrium of the Hotelling setup.

A.1.3 Hotelling Equilibrium Under Diﬀerent Technologies and No Status

Proof. (of Lemma 3) Firm 0’s operational proﬁt function is:

1 + p − p πHDT = p iHDT = p 1 0 . 0 0 0 2

Maximizing this function yields ﬁrst order conditions:

1 + p1 − 2p0 = 0.

Firm 0’s best reply is: 1 pHDT,BR = [1 + p ] . (115) 0 2 1 We can now turn to Firm 1’s operational proﬁt maximization problem. By producing the status good, its operational proﬁt is:

∗ π1 = (1 − i )(p1 − c) 1 − p + p = 1 0 (p − c). 2 1

The ﬁrst order condition for Firm 1’s operational proﬁt maximization problem is −2p1 + p0 + c + 1 = 0. Firm 1’s best reply is: p + c + 1 pHDT,BR = 0 . (116) 1 2

HDT,BR HDT,BR Solving the system of best reply equations in p0 and p1 to find the equilibrium price leads to (5) and (6). Substituting these solutions back into (1) and (2), we obtain the threshold index and the demands (7) and (8). Substituting (5), (6), (7) and (8) into the profit functions of Firms 0 and 1 gives us the operational profits (9) and (10).

55 A.1.4 Hotelling Equilibrium Under the Same Technology and Status

Proof. (of Lemma 4) Firms can, in principle, use either technology to produce their goods. To start with, let us analyze the case in which both firms use the handicraft technology. Firm 0 HS HS maximizes its operational profits π0 = (p0 − c)i . Firm 0’s operational profit function is: 1 + p − p πHS = p iHS = (p − c) 1 0 . 0 0 0 3 Maximizing this function yields first order conditions:

(1 + p1 − p0) − (p0 − c) = 0.

Firm 0’s best reply is: 1 pHS,BR = [1 + p + c] . (117) 0 2 1 We can now turn to Firm 1’s operational proﬁt maximization problem. By producing the status good, its operational proﬁt is:

∗ π1 = (1 − i )(p1 − c) 2 − p + p = 1 0 (p − c). 3 1

The first order condition for Firm 1’s operational profit maximization problem is −2p1 + p0 + c + 2 = 0. Firm 1’s best reply is: p + c + 2 pHS,BR = 0 . (118) 1 2 HS,BR HS,BR Solving the system of best reply equations in p0 and p1 to find the equilibrium price leads to (11) and (12). Substituting these solutions back into (3) and (4), we obtain the threshold index and the demands (13) and (14). Substituting prices (11) and (12) and demands (13) and (14) into the profit functions of firms 0 and 1 gives us the operational profits (15) and (16). The equilibrium of the Hotelling model with both firms using the industrial technology can be obtained easily by making c = 0 in the equilibrium described above.

A.1.5 Hotelling Equilibrium Under Diﬀerent Technologies and Status

Proof. (of Lemma 5) Firm 0’s operational proﬁt function is:

1 + p − p πHSDT = p iHSDT = (p − c) 1 0 . 0 0 0 3 Maximizing this function yields ﬁrst order conditions:

1 + c + p1 − 2p0 = 0.

56 Firm 0’s best reply is: 1 pHSDT,BR = [1 + c + p ] . (119) 0 2 1 We can now turn to Firm 1’s operational proﬁt maximization problem. By producing the status good, its operational proﬁt is:

∗ π1 = (1 − i )p1 2 − p + p = 1 0 p . 3 1

The first order condition for Firm 1’s operational profit maximization problem is −2p1+p0+2 = 0. Firm 1’s best reply is: p + 2 pHSDT,BR = 0 . (120) 1 2 HSDT,BR HSDT,BR Solving the system of best reply equations in p0 and p1 to find the equilibrium price leads to (19) and (20). Substituting these solutions back into (3) and (4), we obtain the threshold index and the demands (21) and (22). Substituting (19), (20), (21) and (22) into the profit functions of Firms 0 and 1 gives us the operational profits (23) and (24).

A.1.6 Stage Game Along the Equilibrium Path

Proof. (of Lemma 6) Let us analyze an equilibrium with product differentiation on status in which one firm, say, Firm 1, adopts the handicraft technology to produce its good while Firm 0 adopts the industrial technology.10 Then, good 1 will be the more expensive good to produce and will be the more exclusive. Consumers attribute high status to this good if both goods are ∗ ∗ sold at the equilibrium prices (calculated below) p0 and p1 and low status to good 0. If things remain unchanged in the next period, the learned status level of goods for the next period are the same as their current status levels. There is a consumer i∗ indifferent between 0 and 1. The demand for good 0 is D(0, i∗) = i∗ and the demand for good 1 is D(1, i∗) = 1 − i∗. Hence, for consumer i∗: ∗ ∗ ∗ u − 1 + i + i − p1 = u − i − p0 Solving for i∗, we obtain equation (3) and (4). The operational profit function of the firm ∗ producing regular good is π0 = p0i . Substituting equation (3) into Firm 0’s operational profit function yields: 1 + p − p π = p 1 0 . (121) 0 0 3

10Because the game is symmetric, all results with ﬁrm 1 adopting the industrial technology are a mirror image of the results displayed here.

57 ∗ By equation (31), arg maxV0 = arg maxπ0. Substituting the operational proﬁt function (121) p0 p0 into the old shareholder’s value function yields:

(1 − α)p [1 + p − p ] V ∗ = 0 1 0 . 0 3(1 − δ + αδ) which is the objective function to be maximized by Firm 0’s manager. The first order condition for this firm’s value maximization problem is 1 + p1 − 2p0 = 0. Firm 0’s best reply is: 1 pBR = [1 + p ] . (122) 0 2 1 We can now turn to Firm 1. The operational profit of Firm 1 when producing the status good is:

∗ π1 = (1 − i )(p1 − c) 2 − p + p = 1 0 (p − c). (123) 3 1

∗ By equation (32), arg maxp1 V1 = arg maxp1 π1. The first order condition for Firm 1’s operational profit maximization problem is −2p1 + p0 + c + 2 = 0. Firm 1’s best reply is: 1 pBR = [p + c + 2] . (124) 1 2 0 We can now calculate the market equilibrium prices and market niches of each firm which will constitute the equilibrium in the product market. The system of best replies of each firm is: 2p0 = 1 + p1 2p1 = p0 + c + 2

Solving this system leads to equations (25) and (26). The diﬀerence between these equilibrium prices are: 1 + c p∗ − p∗ = . 1 0 3 Substituting these prices into the demand equations yield equations (27) and (28). Substituting these into the operational proﬁt functions yield the results of equations (29) and (30). Substi- tuting these solutions into the value functions yield equations (31) and (32).

A.1.7 One-Shot Deviation: Firm 0

Proof. (of Lemma 7) Firm 0’s operational proﬁts are given by:

D0 D0 D0 π0 = p0 i (125)

58 Substituting the demand iD0 in equation (3) results in:

1 + p − pD0 πD0 = 1 0 pD0. 0 3 0 Firm 0 maximizes this expression to obtain maximum current operational proﬁt. The ﬁrst-order condition to this problem is 1 + p1 + c − 2p0 = 0. Firm 0’s best reply is: 1 + p + c pD0,BR = 1 0 2 Firm 1 does not observe Firm 0’s action until it is played. Therefore, in this period, it plays D0 ∗ p1 = p1. The system of equations that give us the deviation period prices is:

D0 ∗ p1 = p1 D0 D0 2p0 = 1 + p1 + c

D0 D0 The prices solving this system are given by equations (35) and (36).Substituting p1 and p0 into (3) and (4) yields demands in equations (33) and (34). Substituting (35), (36), (33), and (34) back into the operational proﬁt functions lead to equations (37) and (38).

A.1.8 One-Shot Deviation: Firm 1

Proof. (of Lemma 8) Firm 1’s operational proﬁts are given by:

D D D π1 = (1 − i )p1 (126)

Substituting the demand 1 − iD in equation (4) results in:

2 − pD + p πD = 1 0 pD. 1 3 1 Firm 1 maximizes this expression to obtain maximum current operational proﬁt. The ﬁrst-order condition to this problem is 2 − p0 − 2p1 = 0. Firm 1’s best reply is: 1 pD,BR = 1 + p 1 2 0 Firm 0 does not observe Firm 1’s action until it is played. Therefore, in this period, it plays D ∗ p0 = p0. The system of equations that give us the deviation period prices is:

D ∗ p0 = p0 D D 2p1 = 2 + p0

D D The prices solving this system are given by equations (41) and (42).Substituting p1 and p0 into (3) and (4) yields demands in equations (39) and (40). Substituting (41), (42), (39), and (40) back into the operational proﬁt functions lead to equations (43) and (44).

59 A.1.9 No Punishment After Deviation: Hotelling Competition with Reputation Loss

In the equilibrium of this stage-game, Firm 0 just plays a best reply to Firm 1’s price. Firm 1 plays a best reply to Firm 0’s price using the handicraft technology, taking into consideration the fact that consumers consider its product as having no status value, which is given by (127). Firm 0 chooses prices according to equation (122). Proof. (of Lemma 9) Solving the resulting system of equations yields the equilibrium prices (45) and (46). Substituting then back into (3) gives us stage-game equilibrium values of the demands (47) and (48). Substituting these values back in the proﬁts functions give us the proﬁt values (49) and (50).

A.1.10 Punishment After Deviation: Generic Pricing

Proof. (of Lemma 10) The ﬁrst order condition to problem (51) is:

1 − p1 + p0 − p1 + c = 0 which yields: 1 + p + c pL = 0 (127) 1 2 L Substituting Firm 0’s chosen punishment price p0 into (127) gives us (52). Substituting (52) L into (1) gives us (53) and (54). Substituting p0 , (52), (53) and (54) into the proﬁt functions yields (55) and (56).

A.1.11 Punishment After Deviation: Stackelberg Equilibrium

Proof. (of Lemma 11) Solve Firm 0’s maximization problem to obtain (57). Substituting this expression into (52) gives us (58). Substituting (57) and (58) into (53) gives us (59) and (60). Substituting (57), (52), (59) and (60) into (55) and (56) yields (61) and (62).

A.1.12 Punishment After Deviation: Minmax Punishment

Proof. (of Lemma 12) Substituting (127) and (2) into Firm 0’s problem (63) gives us: MM 1 + p0 + c 1 + p0 − p1 p0 = arg min − c p0 2 2 ! 1 + p + c 1 + p − 1+p0+c = arg min 0 − c 0 2 p0 2 2 1 + p − c 1 + p − c = arg min 0 0 p0 4 2 (1 + p − c)2 = arg min 0 p0 8

60 The ﬁrst order condition to this problem is:

2(1 + p0 − c) = 0.

MM MM MM Solving for p0 yields the solution p0 = p0 , where p0 = c − 1. Substituting p0 = p0 into the best reply of Firm 1 yields: MM p1 = c. The demand for good 1 is then given by: 1 + p − p 1 + (c − 1) − c 1 − iMM = 0 1 = = 0. 2 2 Hence, iMM = 1. Operational proﬁts become:

MM MM MM π0 = p0 i = (c − 1); MM MM MM π1 = (p1 − c)(1 − i ) = 0.

A.1.13 Firm 0 deviates from punishing Firm 1

Proof. (of Lemma 13) Firm 1 plays the prescribed action, but Firm 0 deviates by maximizing its current period proﬁts, given by equation (70). The ﬁrst order condition to this problem is: 3 + c + pL 0 − 2p = 0 2 0

Thus, Firm 0 maximizes its operational proﬁt by setting price according to (71), that is, p0 = L (3 + c + p0 )/4. Substituting (71) into (1) and (2) yields (73) and (74). Substituting (71), (73) and (74) into the proﬁt functions yields (75) and (76).

A.1.14 Firm 1 punishes Firm 0

Proof. (of 14) Firm 0 maximizes its profit, given by equation (77). The first order conditions are: S 1 + p1 − 2p0 = 0 S Thus, Firm 0 maximizes its profits by choosing p0 = (1 + p1 )/2. Substituting equation (78) into equations (1) and (2) yields demand equations (79) and (80). Substituting (78), (79) and (80) yields the operational profitsπ ¨0 andπ ¨1 given by equations (77) and (82).

A.2 Dynamic Game Equilibria A.2.1 Equilibrium with Product Diﬀerentiation on Status

No proofs.

61 Financial Markets Proof. (of Proposition 1) The proof is carried out in three steps. First, we show that Modigliani and Miller’s result holds at the equilibrium path. We then show that, if in the current period we observe actions that are different from the equilibrium prescribed ones, but after then we observe equilibrium actions, Modigliani and Miller’s result still holds. Then, we extend this result by finite induction to any finite number of periods of actions played different from the equilibrium prescribed ones. Step (i): We first show that, at the equilibrium path, the objective function depends neither on α nor on γ. The new shareholders pay (1 − γ)K for a fraction α of the firm, expecting to obtain π0 − Kγ in the present and a continuation payoff of V0. Making the new shareholders payment equal to their payoff means that:

(1 − γ)K = α [π0 − Kγ + δV0].

Using this and equation (83) results in the following no arbitrage condition: αV (1 − γ)K = 0 . (128) 1 − α Substituting equation (84) into this expression leads to:

α(π − Kγ) (1 − γ)K = 0 . 1 − δ + αδ Cross multiplying and solving for Kγ:

(1 − δ + αδ)K − (1 − δ + αδ)γK = απ0 − αKγ

(1 − δ + αδ)K − απ0 = (1 − δ + αδ)γK − αKγ

(1 − δ + αδ)K − απ0 = [1 − δ + αδ − α]Kγ (1 − δ + αδ)K − απ Kγ = 0 . (1 − δ)(1 − α) Substituting this formula for Kγ into the objective function of Firm 0’s managers yields:

(1 − α)[π − Kγ] (1 − α) (1 − δ + αδ)K − απ 0 = π − 0 1 − δ + αδ 1 − δ + αδ 0 (1 − δ)(1 − α) (1 − α) [(1 − δ + αδ − α)π − (1 − δ + αδ)K + απ ] = 0 0 (1 − δ + αδ)(1 − δ)(1 − α) [(1 − δ + αδ)π − (1 − δ + αδ)K] = 0 (1 − δ + αδ)(1 − δ) (1 − δ + αδ)[π − K] = 0 (1 − δ + αδ)(1 − δ) π − K = 0 . 1 − δ

62 This proves that equation (88) holds regardless of the choice of γ. Step (ii): This part shows that, if during the first period, profits differ from their equilibrium values, the Modigliani-Miller result still holds. Let π0(t) be Firm 0’s operational profit in period ∗ t, and that π0(t) may be different from π0. Suppose that in period t + 1 and beyond, Firm 0’s ∗ operational profits are equal to π0. Let αt be the proportion of the firm that old shareholders sell at time t. Let γt be the proportion of capital finance through the issuance of debt at time ∗ t. Let V0(t) and V0(t + 1) = V0 be Firm 0’s value at times t and t + 1. This notation omits the ∗ price p0 and focusses on the time. Then, by equation (88): π − K V (t + 1) = V ∗ = 0 . 0 0 1 − δ The value of the Firm 0 at time t is:

V0(t) = (1 − αt)[π0(t) − γtK] + δ(1 − αt)V0(t + 1) ∗ = (1 − αt)[π0(t) − γtK] + δ(1 − αt)V0 . (129)

Assuming the constraint (87) holds with equality, we ﬁnd the following no arbitrage condition that makes the new shareholders (those buying stocks at time t) payment equal to their payoﬀ:

αtV0(t) = (1 − γt)K. 1 − αt

Solving this equation for γtK leads to:

αtV0(t) γtK = K − . 1 − αt

Substituting this expression for γtK into equation (129) yields: αtV0(t) ∗ V0(t) = (1 − αt) π0(t) − K + + δ(1 − αt)V0 1 − αt αt(1 − αt) ∗ 1 − V0(t) = (1 − αt)[π0(t) − K] + δ(1 − αt)V0 . 1 − αt

∗ Solving for V0(t) and replacing the equilibrium value V0 by the expression in equation (88) yields:

∗ V0(t) = (π0(t) − K) + δV0 π − K = π (t) − K + δ 0 . 0 1 − δ

∗ This proves that V0(t) is not a function of either γ, αt or α . Therefore, Firm 0’s value is independent of ﬁnancing.

63 Step (iii): Suppose now that profits differ from the equilibrium path for the first T periods, 0 ∗ 0 0 starting at t. Let V0(t + t ) and V0(t + T ) = V0 be Firm 0’s value at times t + t , for t ∈ {0, 1, ··· ,T − 1}, and t + T . By equation (88): π − K V (t + T ) = V ∗ = 0 . 0 0 1 − δ 0 0 Firm 0’s values V0(t + t ), for t ∈ {0, 1, ··· ,T − 1}, are: V0(t + T − 1) = (1 − αt+T −1) π0(t + T − 1) − γt+T −1K + δ(1 − αt+T −1)V0(t + T ) (130) ∗ = (1 − αt+T −1) π0(t + T − 1) − γt+T −1K + δ(1 − αt+T −1)V0 (131) ······················································ 0 0 0 V0(t + t ) = (1 − αt+t0 ) π0(t + t ) − γt+t0 K + δ(1 − αt+t0 )V0(t + t + 1) (132) ······················································

V0(t) = (1 − αt)[π0(t) − γtK] + δ(1 − αt)V0(t + 1). (133)

Assuming the constraint (87) holds with equality, the no arbitrage condition at each time t + t0, for t0 ∈ {0, 1, ··· ,T }, is: 0 αt+t0 V0(t + t ) γt+t0 K = K − . 1 − αt+t0 Substituting this into equations (131) to (133) yields:

0 0 0 αt+t0 V0(t + t ) V0(t + t ) = (1 − αt+t0 ) π0(t + t ) − K + (134) 1 − αt+t0 0 0 αt+t0 V0(t + t ) + δ(1 − αt+t0 )(1 − αt+t0+1) π0(t + t + 1) − K + + 1 − αt+t0 T −1 t+t0−1 Y αt+t0 V0(t + T − 1) + ... + δ (1 − αt+t0+J ) π0(t + T − 1) − K + 1 − αt+T −1 j=t+t0 T −1 T −t0 Y ∗ + δ (1 − αt+t0+J )V0 . j=t+t0 This is the same as: 0 0 0 0 αt+t0 V0(t + t ) V0(t + t ) = [π0(t + t ) − K] + δ(1 − αt+t0+1) π0(t + t + 1) − K + + (135) 1 − αt+t0 T −1 T −1 Y αtV0(t + T − 1) ∗ + ··· + δ (1 − αt+J ) π0(t + T − 1) − K + + δV0 . 1 − αt+T −1 j=t+t0

0 The term in between keys is V0(t + T − 1) by making t = T − 1 in (134). By step (ii), this is given by ∗ V0(t + T − 1) = [π0(t + T − 1) − K] + δV0 .

64 Similarly, for each j ∈ {1, ··· ,T }, we have:

V0(t + j) = [π0(t + j) − K] + δV0(t + j + 1).

Substituting these expressions into (135), we obtain:

T −1 ∗ V0(t) = [π0(t) − K] + δ [π0(t + 1) − K] + ... + δ {[π0(t + T − 1) − K] + δV0 } .

This expression is independent of all αt. So, Firm 0 value is independent of ﬁnancing.

A.2.2 Dynamic Strategy

Proof. (of Lemma 15) In the main text.

A.2.3 Incentive Compatibility Constraints

No proofs.

A.2.4 Deviations from the Equilibrium Path: Customer Punishment Only

Proof. (of Lemma 16) Substitute the formulas (44) and (50) into the equation (93) to ﬁnd the result.

A.2.5 Deviations from the Equilibrium Path: Customer and Rival Punishment

Proof. (of Proposition 2) The inequality (94) implies:

∗ D L (1 + δ)π1 ≥ π1 − K + δπ1 D ∗ π1 − K − π1 δ ≥ ∗ L . π1 − π1 Substituting the values of (30), (44) and (50) into the equation above yields:

2 2 (c+10) − K − (5−c) 8(c + 10)2 − 864K − 32(5 − c)2 δ ≥ 108 27 = . (136) (5−c)2 (1+pL−c)2 2 L 2 0 32(5 − c) − 108(1 + p0 − c) 27 − 8 This is the value of δ such that the above inequality holds as an equality.

A.2.6 Incentives for Firm 0 To Punish a Deviation from Firm 1

NP ∗ H Proof. (of Claim 1). Using the formulas (50) and (30) for π1 and π1, as well as π1 = 1/2, then inequality (96) is equivalent to:

(3 − c)2 δ (5 − c)2 1 1 + ≥ − K . 18 1 − δ 27 1 − δ 2

65 Multiplying both sides by 1 − δ leads to: (3 − c)2 (5 − c)2 1 (1 − δ) + δ ≥ − K. 18 27 2 2 Because c < 5 − p27/2, if δ = 1, then (5 − c)2/27 > p27/2 /27 = 1/2 > 1/2 − K. Because c > 1/2, if δ = 0, then (3 − c)2/18 < (5/2)2/18 = 25/72, which is smaller than 1/2 − K for all values of K such that K < 11/72. Therefore, both continuation payoffs can occur, depending on the parameters. Proof. (of Lemma 17): Substituting (49) and (55) into equation (97) yields: pL(3 + c − pL) (3 + c)2 0 0 ≥ . (137) 4 18 Define the auxiliary variable x as follows: pL x = 0 . 3 + c Under this notation, inequality (137) becomes simply −9x2 + 9x − 2 ≥ 0. The roots of the equation −9x2 + 9x − 2 = 0 are 1/3 and 2/3. Therefore, the punishment price lies between (3 + c)/3 and 2(3 + c)/3. ´ L S L Proof. (of Lemma 18) Hence, δ0(p0 , p1 ) is increasing in p0 iff:

L −6 (3 + c) − 3p0 ≥ 0 that is, if L (3 + c) − 3p0 ≤ 0 c + 3 pL ≥ 0 3 L ´ L S L Because in this case we have p0 < (c + 3) /3, it follows that δ0(p0 , p1 ) is decreasing in p0 . Because S p1 only appears in the denominator of equation (100), with a negative sign, it is straghtforward ´ L S S L to verify that δ0(p0 , p1 ) is increasing in p1 . At p0 = (c + 3) /3, we have: 1 2 c + 3 S 32 [(3 + c) − 3 (3 + c) /3] δ0 , p1 = 2 = 0 2 S 3 (4+c) (1+p1 ) 27 − 12

A.2.7 Incentives For Firm 1 to Return to the Equilibrium Path

L L Proof. (Of Proposition 3) From (95) and (101), we have that eδ1 p0 > δ1 p0 occurs if and only if LK L L L D ∗ π1 p0 − K − π1 p0 π1 − K − π1 ∗ L L > ∗ L L . π1 − π1 (p0 ) π1 − π1 (p0 )

66 Because the denominators on the left and right hand-sides of the above inequality are strictly L positive and equal, it is necessary and suﬃcient to compare the numerators. Hence, eδ1 p0 > L δ1 p0 holds if and only if: LK L L L D ∗ π1 p0 − π1 p0 > π1 − π1 Substituting (30), (44), and (56) into the above inequality yields:

(1 + pL)2 (1 + pL − c)2 (10 − c)2 (5 − c)2 0 − 0 > − 8 8 108 27 After straightforward calculations, we obtain that this inequality is equivalent to:

L 54p0 > 21c − 14. (138)

L If p0 = (c + 3) /3, this inequality becomes

68 > 3c

L which always holds. Because p0 ≥ (c + 3) /3, and the left-hand side of the above inequality is L increasing in p0 , it follows that condition (138) is always satisﬁed.

A.2.8 Incentives For Firm 1 to Punish a Deviation from Firm 0

¨ L S S Proof. (of 19) The denominator of δ1(p0 , p1 ) is not a function of p1 . Therefore, if the numerator S ¨ L S on the right hand side of (103) is a non-decreasing function of p1 , then δ1(p0 , p1 ) is also a non- S decreasing function of p1 . The numerator on the right hand side of (103) is: 2 5 − 2c + pS 8(pS − c)(5 − pS) ¨δ (pL, pS) π∗ − πL(pL) = 1 − 1 1 1 0 1 1 1 0 48 48 1 h 2 i = 5 − 2c + pS − 8(pS − c)(5 − pS) 48 1 1 1 1 h 2 2 i = 25 − 10 −2c + pS + −2c + pS − 8 5pS − pS − 5c + cpS 48 1 1 1 1 1 1 h 2 2 i = 25 + 20c − 10pS + pS − 4cpS + 4c2 − 40pS + 8 pS + 40c − 8cpS 48 1 1 1 1 1 1 1 h 2 i = 25 + 60c − 50pS + 9 pS − 12cpS + 4c2 48 1 1 1 1 h 2i = 25 + 60c + 4c2 − (50 + 12c) pS + 9 pS 48 1 1 S The numerator on the right hand side of (103) is a non-decreasing function of p1 if:

S − (50 + 12c) + 18p1 ≥ 0 S 18p1 ≥ 50 + 12c 25 + 6c pS ≥ 1 9 67 S ∗ Because we are considering, in this situation, only values of p1 < p1, and 25 + 6c 5 + 2c > = p∗, 9 3 1 ¨ L S S it follows that we have that δ1(p0 , p1 ) is decreasing in p1 . L Also, because p0 only appears in the second term in the denominator of equation (103), which is a quadratic form, it follows that an increase in this term decreases this second term, which decreases the denominator of equation (103), given that there is a negative sign preceding this L ¨ L S ¨ L S term. As a consequence, an increase in p0 increases δ1(p0 , p1 ). Hence, δ1(p0 , p1 ) is an increasing L function of p0 .

A.3 Optimal Punishment

A.3.1 Case 1 (δ ≥ bδ1): “Customer punishment only” can enforce equilibrium with status goods

MM Proof. (of Lemma 20) From equation (64), we have that p0 = c − 1, which is larger than NP p p p0 = (c + 3) /3 if and only if c > 3. Since we are assuming that c < 5− 27/2 and 5− 27/2 < MM NP L 3, it follows that p0 < p0 . Lemma 10 shows that Firm 0’s proﬁts are increasing in p0 . It MM follows that p0 is never chosen in equilibrium if δ ≥ bδ1. Proof. (of Proposition 4) In the main text. Proof. (of Proposition 5) In the main text.

Comparing the Two Types of Punishment Proof. (of Proposition 6) We need to show that the set of parameters δ such that both Firm 1 is willing to comply with the equilibrium and Firm 0 is wiling to punish a deviant Firm 1 is contained in the set of parameters that implements L the equilibrium with customer punishment only. This occurs if δ1(p0 ) ≥ bδ1. Hence, we need to L show that δ1(p0 ) ≥ bδ1. Substitute (93) and (95) into this inequality to obtain: D ∗ D ∗ π1 − K − π1 π1 − K − π1 ∗ L ≥ ∗ NP (139) π1 − π1 π1 − π1 All terms are the same in the left and right-hand sides of the above inequality, except for the last term in the denominator. The numerators in both left and right hand sides are both strictly positive. Both the left and right hand sides of the inequality above are increasing functions of this denominator last term. Therefore, the inequality above holds if and only if

L NP π1 ≥ π1 (140) . Substituting (50) and (56) into this inequality yields: (1 + pL − c)2 (3 − c)2 0 ≥ 4 9 68 which is equivalent to: L 3 + 3p0 − 3c ≥ 6 − 2c which is equivalent to L p0 ≥ (c + 3) /3, (141) which must hold by equation (98). Proof. (of Lemma 21) From Lemma 17, we have that statements (i) and (ii) are equivalent. In the proof of Proposition 6, we established that statements (iii) and (iv) are equivalent (from the beginning until equation (140)), and that statements (ii) and (iii) are equivalent (from equation (140) to (141)).

A.3.2 Case 2 (δ < bδ1): “Customer punishment only” cannot enforce equilibrium with status goods

Proof. (of Proposition 7) In the main text.

A.4 Social Status, Private Firms and Commitment A.4.1 Subgame in which Firm 1 chooses no commitment

Proof. (of Lemma 22) If Firm 1 chooses m, Firm 0 chooses h if:

(4 − c)2 16 ≥ − K, 27 27 which implies 108K ≥ 32c − 4c2. (142)

If Firm 1 chooses h, Firm 0 chooses h if:

16 (4 + c)2 ≥ − K 27 27 which implies 108K ≥ 32c + 4c2

From (90), we have that 108K < 20c − 3c2 h i For values of c in the interval 1/2, 5 − p27/2 , we have that 32c + 4c2 > 32c − 4c2 > 20c − 3c2. Therefore, Firm 0 never chooses h and thus h is strictly dominated for Firm 0. Now let us evaluate whether Firm 1 ever chooses h. If Firm 0 chooses m, Firm 1 plays h if:

(5 − c)2 25 ≥ − K, 27 27 69 which implies 108K ≥ 40c − 4c2. (143)

If Firm 0 chooses h, Firm 1 plays h if:

25 (5 + c)2 ≥ − K, 27 27 which implies 108K ≥ 40c + 4c2. h i Once again, for values of c in the interval 1/2, 5 − p27/2 , we have that 40c+4c2 > 40c−4c2 > 20c − 3c2. Therefore, Firm 1 never chooses h and thus h is strictly dominated for Firm 1. Proof. (of Lemma 23) If Firm 1 chooses m, Firm 0 chooses h if:

(3 − c)2 1 ≥ − K, 18 2 which implies 108K ≥ 36c − 6c2. (144)

If Firm 1 chooses h, Firm 0 chooses h if:

1 (3 + c)2 ≥ − K 2 18 which implies 108K ≥ 36c + 6c2.

From (90), we have that 108K < 20c − 3c2 h i For values of c in the interval 1/2, 5 − p27/2 , we have that 36c + 6c2 > 36c − 6c2 > 20c − 3c2. Therefore, Firm 0 never chooses h and thus h is strictly dominated for Firm 0. Because the game is symmetric, it follows that choosing h is also a strictly dominated strategy for Firm 1. Proof. (of Lemma 24) First, let us calculate the payoffs of the stage-game that occur if one firm, let us say, Firm 0, deviates and uses the industrial technology. It will choose a price DCH H p0 that is a best response to the price p1 = 1 + c charged by Firm 1. In this case, Firm 1’s operational profits are given by:

1 + pH − p π = p 1 0 0 0 2 2 + c − p = p 0 0 2

70 The ﬁrst-order condition to this problem is 2 + c − 2p0 = 0, which yields c pDCH = 1 + 0 2 H Substituting this result and p1 = 1 + c into (1) and (2) gives us the demands: 2 + c iDCH = 4 2 − c 1 − iDCH = 4 and substituting these demands and prices into the proﬁt functions yields:

(2 + c)2 πDCH = 0 8 2 − c πDCH = 1 4 Firm 0 cooperates with Firm 1 if:

1 (2 + c)2 16 δ2 1 ≥ + δ − K + − K 2 (1 − δ) 8 27 1 − δ 2 which implies 216K ≥ 20δ (1 − δ) + 27 (1 − δ) c (4 + c) (145) The equation above implies a region in the interval [0, 1] such that for values of δ in this region, cooperation in technology adoption can be achieved through the use of Nash Reversion strategies. Equation (145) is quadratic in δ and therefore has at most two real roots. In the area satisfying both (90) and c < 5 − p27/2, one of the roots is negative and we thus have the usual result in DCH DCH which δ ≥ δ , in which δ is the highest of the roots of the equation that appears when (145) holds as an equality, given by

DCH 1 27 27 1 √ δ ≡ − c − c2 + 4320c − 17 280K + 12 744c2 + 5832c3 + 729c4 + 400. 2 10 40 40

DCH We can now compare the values of δ with the values of eδ1 ((c + 3) /3) to see whether there are situations in which the following conditions hold: (i) the equilibrium with product differentiation on status shown in Subsection 4.2 cannot be implemented in the original game; and (ii) in the modified game with commitment, firms would have incentives to coordinate their technologies on handicraft by using Nash Reversion strategies in the subgame following a decision not to DCH commit to private financing. Cumbersome but straightforward comparison shows that δ > eδ1 ((c + 3) /3) in the relevant region. Therefore, whenever firms are willing to cooperate in the subgame of the game with commitment following the decision not to commit, they are also willing to cooperate in the game without commitment to produce product differentiation on status.

71 A.4.2 Equilibrium of the Dynamic Game with Commitment

L NP Effect of commitment when p0 ≥ p0 Proof. (of Proposition 8) If the private firm follows the prescribed equilibrium actions, committing to retaining its private status permanently, and selling only the status good in every period, it will obtain (5 − c)2 V ∗ = . 1 27(1 − δ) If it deviates, it will obtain the Bertrand payoff from competing with the regular good firm for H the unit interval demand, obtaining operational profits π1 = 1/2 in every period, which has present value of 1/ [2(1 − δ)]. Therefore, we need to have: (5 − c)2 1 V ∗ = ≥ . (146) 1 27(1 − δ) 2(1 − δ) Therefore, we must have: (5 − c)2 1 ≥ . 27 2 This inequality holds iff: 23 c2 − 10c + ≥ 0 2 √ 3 that is, if c ≤ 5 − 2 6, which holds given the technical hypotheses made earlier. Therefore, the incentive compatibility constraint (146) always holds and thus the high status good firm has always something to gain from committing to the private financing strategy. C L Proof. (of Proposition 9) We need to show that δ1 < eδ1 ((c + 3) /3). Substituting p0 = (c + 3) /3 into (101) yields 108c − 9c2 − 216K eδ1 ((c + 3) /3) = 92 − 8c − 4c2 (80c − 8c2) + (18c − c2) − 216K = (147) 92 − (8c + 4c2) From equation (109), we have that 20c − 2c2 − 54K δC = 1 23 80c − 8c2 − 216K = (148) 92 When c = 0, the expressions (147) and (148) coincide, for all values of K. As c increases, the numerator on (147) grows by a larger sum than the numerator in (148), because 18c − c2 > 0 for c < 18. The denominator on (147) is reduced as c grows, while the denominator on (148) remains h p i C constant. Since c must fall in the interval 1/2, 5 − 27/2 , it follows that eδ1 ((c + 3) /3) > δ1 .

72 Alternative Commitment Devices Proof. (of Proposition 10) In the main text.

L NP Eﬀect of commitment when p0 < p0 No proofs.

B Appendix: Diluting Share Ownership of Regular Goods Firm

We have shown that Firm 0 is indiﬀerent between ﬁnancing K with the issuance of debt and equity. Therefore, any choice of α is equally good. Therefore, we will assume onwards that shareholders retain as much as possible of their shares, and that all of the investment in industrial machinery will be funded by the issuance of stock. In other words, Firm 0 will sell just enough shares to cover the cost of K and issue no debt. If Firm 0 chooses its capital structure this way, then the following result holds.

Lemma 25 The old shareholder sells a share 0 < α∗ < 1 of Firm 0, and this raises exactly K to the Firm 0, where: ∗ (1 − δ)K K − δK α = ∗ = ∗ . (149) π0 − δK π0 − δK As Firm 0 issues new shares to finance the purchase of K, the old shareholders’ stake at the firm is diluted. Because capital only lasts one period, it needs to be replaced at every period, and thus we have permanent ownership dilution process. The dilution rate is given by α∗, the fraction of the firms’ cash flows old shareholders need to give to new shareholders. This dilution process has to occur at a higher speed as the fixed cost K increases and the operational profit ∗ π0 decreases.

Lemma 26 The ownership dilution process (defined by the parameter α∗) speeds up as the fixed ∗ cost K increases and the profit π0 decreases. Formally: ∂α∗ > 0, ∂K ∂α∗ ∗ < 0. ∂π0

B.1 Proofs

Proof. (of Lemma 25) Substituting equation (31) into (87) and making γ = 0 leads to:

απ∗ 0 = K. [1 − (1 − α)δ]

73 Expressing this as a polynomial of degree two in α yields:

∗ (π0 − δK)α − (1 − δ)K = 0. (150)

Solving (150) for α yields equation (149). Proof. (of Lemma 26) By diﬀerentiating (149), we obtain:

∗ ∗ ∂α (1 − δ)(π0 − δK) + δ(1 − δ)K = ∗ 2 > 0, ∂K (π0 − δK) ∂α∗ −(1 − δ)K ∗ = ∗ 2 < 0. ∂π0 (π0 − δK)

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