BUSINESS (NO PLAGARISM A+ WORK, ON TIME)

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The Oligopoly Lucas Tree

Winston Wei Dou University of Pennsylvania

Yan Ji Hong Kong University of Science and Technology

Wei Wu Texas A&M University

This paper proposes a novel quantitative framework with endogenous strategic competition in heterogeneous concentrated industries. Oligopolies compete strategically for profit margins in repeated games, trading off the benefits of future cooperation against those of reaping higher short-run profits by undercutting their rivals. Cross-industry dispersions in market leadership persistence and cash flow loadings on expected growth, as primitive characteristics, simultaneously determine the relationships among profitability, book-to- market ratios, and systematic risk exposures, thereby quantitatively rationalizing the gross profitability and value premium across industries and, importantly, their interactions. Controlling for the book-to-market ratio (gross profitability) makes the gross profitability (value) premium more pronounced. (JEL G12, L13, O33, C73)

Received August 12, 2020; editorial decision August 30, 2021 by Editor Stefano Giglio. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

We are especially grateful to Hengjie Ai, Frederico Belo, Bob Dittmar, Itamar Drechsler, Itay Goldstein, Joao Gomes, Bryan Kelly, Lukas Schmid, and Jessica Wachter for detailed discussions and comments. We also thank Jules van Binsbergen, M. Cecilia Bustamante, Hui Chen, Zhi Da, Lorenzo Garlappi, Xavier Giroud, Vincent Glode, Lars Peter Hansen, David Hirshleifer, Harrison Hong, Urban Jermann, Ken Judd, Ron Kaniel, Don Keim, Richard Kihlstrom, Leonid Kogan, Jun Li, Xiaoji Lin, Andrey Malenko, Gustavo Manso, Robert Novy-Marx, Alan Moreira, Thien Nguyen, Christian Opp, Marcus Opp, Jonathan Parker, Ľuboš Pástor, Thomas Philippon, Raghu Rajan, Uday Rajan, Krishna Ramaswamy, Michael Roberts, Nick Roussanov, Larry Schmidt, Stephanie Schmitt-Grohé, Enrique Schroth, Rob Stambaugh, Luke Taylor, David Thesmar, Robert Townsend, Harald Uhlig, Gianluca Violante, Neng Wang, Pengfei Wang, Wei Xiong, Amir Yaron, John Zhang, and Lu Zhang; seminar participants at City University of Hong Kong, Federal Reserve Bank at Dallas, Peking University (Guanghua and PHBS), University of Rochester (Simon), University of Southern California, University of Texas at Dallas, and Wharton; and conference participants at Asian Finance Association, CICM, COAP Conference, European Finance Association, Midwest Finance Association, Hong Kong University of Science and Technology, HKUST Finance Symposium, HKUST-JINAN Workshop, Mack Institute Workshop, MIT Junior Faculty Conference, Mays Innovation Research Center Workshop, PNC Kentucky Finance Conference, Northeastern Finance Conference, 6th SAFE Asset Pricing Workshop, SFS Cavalcade North America, Stanford SITE, Western Finance Association, and Young Scholars Finance Consortium (YSFC). This paper is partly based on the manuscript previously circulated under the title “Competition, Profitability, and Risk Premia.” We are also grateful for the comments and insights from Britt Harris, the President, CEO, and CIO of the University of Texas/Texas A&M Investment Company (UTIMCO). Supplementary data can be found on The Review of Financial Studies web site. Send correspondence to Winston Wei Dou, [email protected].

The Review of Financial Studies 35 (2022) 3867–3921 © The Author(s) 2021. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For permissions, please e-mail: [email protected]. doi: 10.1093/rfs/hhab120 Advance Access publication November 6, 2021

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This paper proposes a quantitative equilibrium model that extends the standard Lucas-tree asset pricing framework to reserve an explicit role for endogenous strategic competition in heterogeneous concentrated industries.1 By calibrating the model to the data without using information from stock returns, we show that our model can quantitatively reconcile the gross profitability and value premium across industries, especially their intriguing interactions. In particular, after controlling for the book-to-market ratio and gross profitability, we find the gross profitability and value premium become more pronounced, respectively. Jointly addressing these two “anomalies” and their interactions is challenging because profitable industries are associated with low book-to-market ratios and are therefore likely to generate a counterfactual “growth premium” across industries. Moreover, the interaction patterns of the two anomalies cannot be easily rationalized by single-factor models, especially in the presence of a strong correlation between gross profitability and the book-to-market ratio across industries. Similar puzzling empirical patterns across firms, not industries, were first documented and highlighted by Novy-Marx (2013).

Many capital market anomalies are firm-level phenomena that are likely to vanish at the industry level. However, the gross profitability and value premium, as well as their interactions, survive across industries. In particular, we show that, in the data, the patterns survive across industries at the level of the four-digit Standard Industrial Classification (SIC4) codes, with statistical significance and economic magnitude comparable to the same patterns across firms within industries and those across all firms. Rationalizing industry- level anomalies adds another layer of complexity — in general, financial economists find it difficult to connect industry-level risk exposures to the fundamental characteristics of an industry (e.g., Fama and French 1997; Dittmar and Lundblad 2017), and the theory of a firm for explaining stock returns cannot be directly extended to that of an industry (e.g., Dou, Ji, and Wu 2021). To connect stock returns and fundamental characteristics across industries, we must explicitly consider several key industrial organizational features: (a) industries are highly concentrated; (b) the market leadership of major players is persistent; and (c) market leaders compete strategically for market share within an industry.2 Despite these features, the asset pricing literature has paid

1 The closest quantitative general-equilibrium models with heterogeneous firms in an endowment economy include those proposed by Menzly, Santos, and Veronesi (2004) and Santos and Veronesi (2006, 2010), who adopt a similar top-down modeling approach. Our paper differs from these studies by explicitly highlighting the crucial role of endogenous strategic competition in asset pricing.

2 See, for example, Grullon, Larkin, and Michaely (2019), Gutiérrez, Jones, and Philippon (2019), Autor et al. (2020), Loecker, Eeckhout, and Unger (2020), Corhay, Kung, and Schmid (2021), and Dou, Ji, and Wu (2021) for evidence related to industry concentration; see, for example, Sutton (2007) and Bronnenberg, Dhar, and Dubé (2009) for evidence related to market leadership persistence; and see Chen et al. (2020) and the references therein for evidence related to strategic competition, such as tacit collusion.

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The Oligopoly Lucas Tree

little attention to the effect of endogenous strategic competition. This paper fills this gap by taking a first step toward the development of a full-fledged quantitative asset pricing framework with endogenous strategic competition.

Our explanation of the joint patterns of industry-level stock returns emphasizes the endogeneity of strategic competition and its interplay with two dimensions of cross-industry heterogeneity. In the proposed economic mechanism of endogenous competition, market leaders in a given industry compete strategically through tacit collusion or cooperation sustained by punishment for deviation in repeated games. They optimally set profit margins by trading off the long-run benefits of tacitly colluding with their rivals against the short-run benefits of reaping higher profits by departing from the collusive agreement and undercutting their rivals. The incentive to collude (i.e., capacity to collude) is endogenously driven by fluctuations in the discount rate and expected growth. Both a rise in the discount rate and a decline in expected growth reduce the present value of future cooperation, causing firms to compete more fiercely for short-run profits by undercutting each other. Moreover, this endogenous competition mechanism interacts with primitive industry characteristics in important ways. In the model, cross-industry heterogeneity is introduced through cross-sectional differences in two primitive industry characteristics, the market leadership turnover rate and the cash flow loading on expected growth. Cross-sectional differences in the market leadership turnover rate reflect differences in the average life span of market leaders (i.e., the effective patience of market leaders) in an industry, while cross-sectional differences in the cash flow loading on expected growth reflect differences in the riskiness of firms’ growth options in an industry. Intuitively, the dispersion of the market leadership turnover rate, together with discount rate shocks, generates the gross profitability premium, as industries with lower market leadership turnover rates are effectively more patient; thus, they have stronger endogenous competition effects; their market leaders have the capacity to collude with one another on higher profit margins, while being more negatively exposed to discount rate shocks. This is the endogenous competition channel for the industry-level gross profitability premium proposed by Dou, Ji, and Wu (2021). Meanwhile, the dispersion of the cash flow loading on expected growth, together with expected growth shocks, generates the value premium, as industries with higher cash flow loadings on expected growth have riskier growth options and thus have higher book-to-market ratios, while being more positively exposed to expected growth shocks. This is the cash flow duration channel for the value premium similar in spirit to that proposed by Campbell and Vuolteenaho (2004), Lettau and Wachter (2007, 2011), Da (2009), Santos and Veronesi (2010), and Croce, Lettau, and Ludvigson (2014), among others.3

3 In the appendix, we discuss in detail the connection between our mechanism for generating the value premium and the mechanisms in the literature.

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At first glance, it may not be very surprising that, by combining two cross- sections and two systematic shocks, our model is able to quantitatively explain two anomalies simultaneously in a unified framework. However, the goal of our model extends beyond merely generating the gross profitability and value premium separately in their own cross sections. Importantly, the main contribution of our model is to advance our understanding of the important and nontrivial interactions between the two cross-sections. In doing so, this paper differs considerably from Dou, Ji, and Wu (2021) and contributes to the literature along three respects. First, our model shows how gross profitability and book-to-market ratios are jointly determined by two primitive industry characteristics, the market leadership turnover rate and the cash flow loading on expected growth. Second, our model shows how industry-level stock return exposures to these two systematic shocks are jointly determined by the two primitive industry characteristics. Third, our model establishes the cross- industry interdependence between gross profitability, book-to-market ratios, and systematic risk exposures, thereby rationalizing the interactions between the gross profitability and value premium across industries.

The seemingly complex interdependence between two industry characteris- tics, two financial ratios, risk exposures to two systematic shocks, and stock returns can be explained in a fairly transparent way because we show that a “nearly separating property” holds in our model, a fact that is also verified in the data. More precisely, we show that cross-industry dispersions in both gross profitability and exposure to discount rate shocks, as well as their association, are mainly determined by the dispersion of the market leadership turnover rate. Meanwhile, cross-industry dispersions in both the book-to-market ratio and exposure to expected growth shocks, as well as their association, are mainly determined by the dispersion of the cash flow loading on expected growth. This “nearly separating property” is not obvious ex ante. As a main contribution, our model quantitatively confirms this property and uses it to guide our empirical tests of the main theoretical results in the data. The “nearly separating property” of our model ensures that the cross-industry correlation between gross profitability and book-to-market ratios, as well as the interaction between the gross profitability and value premium, ultimately boil down to the correlation parameter of the two primitive industry characteristics. By calibrating the correlation parameter to match its empirical counterpart measured directly in the data, we show that our model can quantitatively match the data in many dimensions, including the complex interactions between the two cross-sections of industries.

We also conduct counterfactual analyses to shed light on the importance of each ingredient in our model. The model implies that the time-varying discount rate plays a major role in generating the gross profitability premium. By contrast, fluctuations in expected growth are necessary and important to generate the value premium. We quantitatively isolate the contribution of the two industry characteristics. The dispersion of gross profitability across

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industries is mainly attributed to the difference in the market leadership turnover rate. The model cannot generate a gross profitability return spread if the turnover rate of market leaders is the same across industries. Meanwhile, the dispersion of book-to-market ratios across industries is mainly attributed to the difference in the cash flow loading on expected growth. Without dispersion in the cash flow loading on expected growth across industries, the model would generate a growth premium rather than a value premium. Furthermore, we quantitatively prove that the correlation between the two primitive characteristics across industries does play a vital role in shaping the interactions between the gross profitability and value premium. Finally, we extend the benchmark model by incorporating fixed costs of production and thus operating leverage. We calibrate the extended model to the data and show that the main quantitative results remain unchanged after taking operating leverage into account.

Our paper contributes to the literature on the interaction between the gross profitability and value premium. Novy-Marx (2013) presents the gross profitability premium. Moreover, the gross profitability and value premium become more pronounced after controlling for the book-to-market ratio and gross profitability, respectively, a puzzling pattern that lies at the heart of both anomalies. Feng, Giglio, and Xiu (2020) show that the profitability factor, unlike most recently discovered factors, is useful in explaining asset returns, even after accounting for a large set of other factors. Despite mounting evidence, only a few papers offer risk-based theoretical explanations of the profitability premium. Kogan and Papanikolaou (2013) highlight the role of the investment- specific technology (IST) shock as a systematic risk factor priced in the cross-section. More recently, Li, Kogan, and Zhang (2020) and Li et al. (2020) develop models in which more profitable firms are riskier because they benefit less from the operating hedge offered by intermediate inputs. Our paper differs from these papers in two main aspects. First, their models focus on competitive equilibrium and ignore the heterogeneity of product market competition across industries, which is the focus of our paper. Second, their mechanisms can help explain the within-industry gross profitability premium, but not the cross- industry gross profitability premium. Both the heterogeneity of competition across industries and the cross-industry gross profitability premium are the main focuses of our paper.

Our paper also contributes to the growing literature on industry returns. Previous studies have examined the relationship between industry returns and industry information leads and lags (e.g., Moskowitz and Grinblatt 1999; Croce, Marchuk, and Schlag 2019), demographics (e.g., DellaVigna and Pollet 2007), industry concentration (e.g., Hou and Robinson 2006; Ali, Klasa, and Yeung 2009; Giroud and Mueller 2011; Bustamante and Donangelo 2017; Corhay, Kung, and Schmid 2020), durability of industry output (e.g., Gomes, Kogan, and Yogo 2009), network concentration and sparsity of an industry (e.g., Herskovic 2018), expected inflation (e.g., Boudoukh, Richardson, and Whitelaw 1994), consumption risk exposures (e.g., Dittmar and Lundblad 2017), and political

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connectedness of an industry (e.g., Belo, Gala, and Li 2013; Addoum and Kumar 2016). We contribute to the literature by shedding light on the connection between industry returns and two primitive industry characteristics, the market leadership turnover rate and the cash flow loading on expected growth.

Moreover, our paper contributes to the burgeoning literature at the intersection of industrial organization (IO), customer base, and finance. Early contributions to this line of research, including those of Fershtman and Judd (1987), Bolton and Scharfstein (1990), and Aggarwal and Samwick (1999), focus on the interaction between competition and contracting. Recently, many studies have focused on the interaction among competition, customer base, asset pricing, and industry dynamics (e.g., Basak and Pavlova 2004; Garlappi 2004; Hou and Robinson 2006; Novy-Marx 2007; Aguerrevere 2009; Carlin 2009; Gârleanu, Kogan, and Panageas 2012; Carlson et al. 2014; Opp, Parlour, and Walden 2014; Bustamante 2015; Koijen and Yogo 2015; Loualiche 2016; Bustamante and Donangelo 2017; Corhay 2017; Garlappi and Song 2017; Andrei and Carlin 2018; Chen et al. 2020; Corhay, Kung, and Schmid 2020; Corhay, Kung, and Schmid 2021; Crouzet and Eberly 2020; Babenko, Boguth, and Tserlukevich 2021; Dou and Ji 2021; Dou et al. 2021b). Most of these papers focus on one-shot noncollusive Nash equilibria, whereas we consider collusive Nash equilibria. Two exceptions are Chen et al. (2020) and Dou et al. (2021b), who study the feedback effect between endogenous competition intensity and distress risk, as well as the distress spillover effects through the endogenous strategic competition mechanism within and across industries. Another exception is Opp, Parlour, and Walden (2014), who investigate how competition intensifies endogenously as the discount rate increases. Their model focuses on identical firms producing homogeneous goods within an industry and on industries with different number of firms. By contrast, our model allows firms within an industry to differ and focuses on industries with different turnover rates of market leaders. Moreover, their model is qualitative, whereas ours is quantitative.

More broadly, an increasing number of works study how time-varying discount rates endogenously alter cash flows of firms and thus stock returns by affecting agents’ strategic interactions. For example, Garlappi (2004) analyzes the impact of competition on the risk premiums of R&D ventures engaged in a multiple-stage patent race. Pástor and Veronesi (2012) develop a novel model with learning to study the asset pricing implications of political uncertainty, in which the firms’ investment decisions and the government’s policy decision are made simultaneously: the government takes into account the firms’ anticipated response, and each firm considers the actions of the other firms as well as the government. The authors investigate price dynamics in the Nash equilibrium. More recently, Pástor and Veronesi (2020) provide an explanation for the “presidential puzzle” by developing a model with endogenous election outcomes driven by voters’ time-varying risk aversion. Agents play a simultaneous-move game in deciding which party to elect. Our mechanism

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relies on the collusive Nash equilibrium and folk theorem (Fudenberg and Maskin 1986), thus is fundamentally different from those above.

1. Model

In this section, we develop a dynamic asset pricing model to rationalize the gross profitability and value premium across industries, especially to reconcile their important interactions. Our model applies the theoretical machinery on endogenous competition developed by Dou, Ji, and Wu (2021) to a quantitative general-equilibrium endowment-based model (i.e., a quantitative Lucas-tree framework). It extends the model of Dou, Ji, and Wu (2021) in three important ways. First, in addition to the dispersion of market leadership persistence across industries, we introduce the dispersion of firms’ cash flow exposures to expected growth shocks across industries. By calibrating these two primitive industry characteristics and their correlation based on the data, we show that our model connects two industry-level financial metrics, namely, gross profitability and the book-to-market ratio, and industry-level stock return exposures to priced economy-wide shocks through their endogenous interactions with the two primitive industry characteristics. Accordingly, our calibrated model generates the joint pattern of the gross profitability and value premium observed in the data. Second, the two dimensions of industry heterogeneity interact in economically interesting and nontrivial ways in our model as in the data, which cannot be generated by the model of Dou, Ji, and Wu (2021). Specifically, the interaction between the gross profitability and value premium and that between competition intensity and the book-to-market ratio in a given industry are missing in the model of Dou, Ji, and Wu (2021). Third, rather than exogenously specify a stochastic discount factor (SDF), we build on the habit- formation framework of Campbell and Cochrane (1999), augmented with a predictable component of consumption growth as in Santos and Veronesi (2006, 2010) and Lettau and Wachter (2007). The tight connection between the model-implied SDF and aggregate consumption dynamics further disciplines the model quantitatively, thereby substantially strengthening the quantitative justification of the model’s core mechanism, which goes beyond the scope of Dou, Ji, and Wu (2021).

1.1 Basic environment There is a continuum of atomistic and homogeneous households with access to complete financial markets. The corporate sector comprises a continuum of industries indexed by i∈I≡ [0,1] and is owned by these households. Each industry i has n oligopolies (market leaders) and many followers with measure zero.4 We set n=2 so that each industry is essentially a duopoly. Market leaders

4 In Internet Appendix 3.3, we extend the model to allow a nonzero measure of followers and microfound their competition with leaders. Doing so does not change the main implications of the paper.

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are indexed by j ∈{1,2}. We denote a generic firm by ij , referring to firm j

in industry i, and its competitor by ij̄ . Firms produce differentiated perishable goods and set their profit margins strategically to maximize shareholder value.

Although firms optimally choose their own outputs (i.e., firm-level output is highly endogenous), aggregate output is exogenously specified, effectively making our model an endowment economy. A similar top-down modeling approach has been adopted by Menzly, Santos, and Veronesi (2004) and Santos and Veronesi (2006, 2010). In particular, we denote the aggregate endowment in terms of final goods at time t by Et and assume that the log aggregate endowment, denoted by et ≡ ln(Et ), evolves as follows:

det =gtdt +σedZe,t , and (1)

dgt =−κ(gt−g)dt +σg √ gt−ςdZg,t , (2)

where gt is the time-varying expected growth rate, Ze,t and Zg,t are two inde- pendent standard Brownian motions, and ς is the theoretical lower bound for gt .

We assume that the growth of the aggregate endowment has a predictable low- frequency component gt for three reasons. First, this assumption is consistent with the literature on long-run risk, which emphasizes that a small component of consumption growth is persistent and predictable (e.g., Kandel and Stambaugh 1991; Bansal and Yaron 2004; Hansen, Heaton, and Li 2008; Bansal, Kiku, and Yaron 2012; Müller and Watson 2018). Second, fluctuations in the expected growth in demand and output can affect the intensity of competition among industry rivals, as emphasized in the macroeconomics and IO literature (e.g., Haltiwanger and Harrington 1991; Bagwell and Staiger 1997; Galeotti and Schiantarelli 1998; Ivaldi et al. 2007; Nekarda and Ramey 2013). The primary goal of our model is to investigate the asset pricing implications of endogenous strategic competition; thus, it is necessary for the model to incorporate the low-frequency component gt in expected growth. Third, the cross-sectional heterogeneity of firms’ exposures to fluctuations in expected future cash flow growth is important to generate the value premium (e.g., Cohen, Polk, and Vuolteenaho 2002; Campbell and Vuolteenaho 2004; Santos and Veronesi 2010). Incorporating expected growth gt is a natural way to model long-run cash flow risk in an endowment economy, as in Menzly, Santos, and Veronesi (2004), Santos and Veronesi (2006, 2010), and Lettau and Wachter (2007, 2011), among others.

1.2 Preferences 1.2.1 External habits, differentiated goods, and customer bases. There is a representative agent who has a two-level constant elasticity of substitution (CES) preference. In particular, the utility of final goods consumption, denoted by Ct , is characterized by

U0 =E0

[∫ ∞

0 ut (Ct,Ht )dt

] , (3)

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where the instantaneous utility function is given by

ut (Ct,Ht )=e−ρt (Ct−Ht )1−γ

1−γ . (4)

In Equation (4), the variables Ht , γ , and ρ denote an external habit level, the agent’s risk aversion, and the subjective discount rate, respectively. The preference falls into the class of external habit formation utilities. Similar to Menzly, Santos, and Veronesi (2004) and Santos and Veronesi (2006, 2010), our specification is a continuous-time analog of the preference adopted by Campbell and Cochrane (1999). The external habit level Ht depends on past aggregate consumption. That is, the representative agent derives utility from its consumption relative to the past aggregate consumption path. The external habit levelHt captures a subsistence level of consumption or social externality.

The preference in our model differs from that in the asset pricing literature because the final good Ct is determined by a two-level CES aggregation.5

First, the demand for Ct is determined through the aggregation of industry composites:

Ct =

[∫ 1

0 M

1 ε i,tC

ε−1 ε

i,t di

] ε ε−1

, (5)

where Ci,t is the demand for industry i’s composite, and the parameter ε>1 captures the elasticity of substitution among industry composites. The weight Mi,t captures the representative agent’s taste for industry i’s composite. A higherMi,t reflects a higher utility gain from consuming industry i’s composite.

Second, industry i’s composite Ci,t is further determined by aggregating firm-level differentiated products:

Ci,t =

⎡⎣ 2∑ j=1

( Mij,t

Mi,t

) 1 η

C η−1 η

ij,t

⎤⎦ η η−1

, (6)

where Mi,t = ∑2

j=1Mij,t also appears in Equation (5) as the taste for industry i’s composite, Cij,t is the demand for firm ij ’s goods, and the parameter η>1 captures the elasticity of substitution among goods in the same industry. The weight Mij,t /Mi,t captures the representative agent’s taste for firm ij ’s goods. Consistent with the literature (e.g., Atkeson and Burstein 2008; Corhay, Kung, and Schmid 2020), we assume that η>ε>1, which means that goods within the same industry are more substitutable than those across different industries.

The taste coefficient Mij,t in the CES aggregator (6) can be interpreted as consumers’ tendency to continue to buy firm ij ’s products either because of brand loyalty or because of consumer inertia (Klemperer 1995). From a firm’s perspective, consumers’ taste Mij,t can be seen as firm ij ’s customer base, which affects the demand for the firm’s goods.

5 The two-level CES preference is a standard modeling device for agents’ preferences and demand system in the international trade literature (e.g., Armington 1969; Anderson 1979).

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1.2.2 External habit evolution. The habit level Ht depends on the past consumption process. The effect of habit persistence on risk aversion can be conveniently summarized by the surplus consumption ratio, St ≡ (Ct−Ht )/Ct , defined as percentage difference between consumption and habit. Following the ideas of Campbell and Cochrane (1999, 2000), Menzly, Santos, and Veronesi (2004), Santos and Veronesi (2006, 2010), and Lettau and Wachter (2007), among others, we directly postulate the evolution of st ≡ ln(St ) as follows:6

dst =−φs(st−s)dt +ψ(st )(dct−Et [dct ])+π (dgt−Et [dgt ]), (7)

where ct ≡ ln(Ct ) is the log aggregate consumption of final goods. The sensitivity function ψ(st ) determines how the habit level is formed from past consumption and is given by

ψ(st )=

{ S

−1√ 1−2(st−s)−1, when st ≤ ŝ,

0, when st > ŝ, (8)

where s≡ ln(S), with S =σe √ γ /φs being the deterministic steady state of St

as in Campbell and Cochrane (1999), and the threshold is defined as ŝ≡s+( 1−e2s

) /2. According to Equation (8), the sensitivity function ψ(st )=0 if and

only if st ≥ ŝ. In our specification (7), the log consumption surplus ratio st and the shock

to contemporaneous log consumption growth (dct−Et [dct ]) are not perfectly conditionally correlated because of the term π (dgt−Et [dgt ]), which differs from the model of Campbell and Cochrane (1999). This specification is similar in essence to that of Brandt and Wang (2003), who allow st to be correlated with other business cycle variables, such as inflation, and to that of Lettau and Wachter (2007) and Bekaert, Engstrom, and Xing (2009), who introduce shocks

to preferences. We set π = √

2/(γ σ 2 g ) to ensure a constant equilibrium interest

rate. A positive π is consistent with the evidence for a negative correlation between expected growth rates and discount rates (e.g., Chen 2010).

1.2.3 Equilibrium SDF. The marginal utility under preference (3) is strictly positive, and thus the aggregate endowment is equal to the aggregate consumption of final goods in equilibrium:

Et =Ct . (9)

It is straightforward to derive the equilibrium SDF as follows:

�t =e −ρt (Ct−Ht )−γ =e−ρtS−γ

t C −γ t . (10)

6 Our specification does not lead to the linear habit formation of Constantinides (1990) and Detemple and Zapatero (1991): Ht =φs

∫ t−∞e−φs (t−τ )Cτ dτ . However, Li (2015) shows that linear habit persistence has quantitative implications similar to the nonlinear habit persistence of Campbell and Cochrane (1999).

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By plugging the equilibrium condition (9) into Equation (10) and applying Ito’s lemma, we derive the dynamics of �t as follows:

d�t

�t

=−rf dt−η(st )dZe,t−ς (gt )dZg,t , (11)

where rf is the equilibrium risk-free rate and is equal to

rf =ρ+γ ς− γφs

2 , (12)

and η(st ) and ς (gt ) are the equilibrium market prices of risk for Ze,t and Zg,t , respectively, and are equal to

η(st )≡γ σe[1+ψ(st )] and ς (gt )≡γ σgπ√ gt−ς. (13)

The market price of risk forZe,t has the same functional form as in the nonlinear external habit formation model of Campbell and Cochrane (1999). Moreover, the market price of risk for Zg,t is positive and sizeable as in Bansal and Yaron (2004) and Ai and Bansal (2018), where the agent has Epstein-Zin-Weil (EZW) utility with a preference for early resolution of uncertainty.7 Under relevant calibrations, the market price of risk for Zg,t is much less volatile than that of Ze,t . Moreover, consistent with the model solution of Bansal and Yaron (2004), the market price of risk for the long-run growth shock Zg,t is approximately a constant, with ς (gt )≈γ σgπ√

g−ς . Therefore, fluctuations in the discount rate are almost entirely driven by variations in η(st ), which are caused by changes in the log surplus consumption ratio st .

Importantly, through the specifications in Equations (2) and (7), our model emphasizes the persistent components of both expected returns and expected future dividend growth, indicating the predictability of returns and dividend growth consistent with empirical findings in the literature (e.g., Binsbergen and Koijen 2010; Koijen and Nieuwerburgh 2011).8

1.2.4 Demand system for differentiated products. We derive the represen- tative agent’s demand system for differentiated goods from the CES preference in Equations (5) and (6). Let Pi,t be the price of industry i’s composite. Given Pi,t and Ct , we obtain Ci,t by solving a standard expenditure minimization problem:

Ci,t =Mi,t

( Pi,t

Pt

)−ε Ct , with Pt =

(∫ 1

0 Mi,tP

1−ε i,t di

) 1 1−ε , (14)

where Pt is the price index of final goods. Without loss of generality, we normalize Pt ≡1 so that the final goods are the numeraire. Industry-level

7 Dew-Becker (2012) embeds habit formation in the EZW utility, in order to generate movements in risk aversion.

8 Moreover, the model specification in Equation (7) implies the upward-sloping pattern of the positive covariance between expected excess returns and subsequent anticipated consumption growth across different horizons, highlighted by Gârleanu, Panageas, and Yu (2012).

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demand Ci,t arises endogenously from the CES aggregation of industry composites in Equation (5), which provides a micro foundation for the industry- level demand curve exogenously postulated in the model of Dou, Ji, and Wu (2021).

Next, given Ci,t , the demand for firm ij ’s goods is given by

Cij,t = Mij,t

Mi,t

( Pij,t

Pi,t

)−η Ci,t , with Pi,t =

⎛⎝ 2∑ j=1

Mij,t

Mi,t

P 1−η ij,t

⎞⎠ 1

1−η

, (15)

which arises endogenously from the CES aggregation of differentiated products at the firm level described in Equation (6). In Equation (15), the demand for firm ij ’s goods Cij,t increases with Mij,t /Mi,t , while the price Pij,t and industry- level demand Ci,t are kept constant. Thus, a natural consideration is Mij,t as firm ij ’s customer base and Mi,t as industry i’s total customer base. In other words, the customer base determines the demand for firm ij ’s goods Cij,t and industry i’s composite Ci,t for given prices (e.g., Gourio and Rudanko 2014; Dou et al. 2021a). Moreover, Equation (15) implies that firm ij has a greater influence on the price index Pi,t when its share Mij,t /Mi,t is higher.

By combining Equations (14) and (15), the demand curve faced by firm ij

is given by Cij,t ≡Cij,t (Pij,t ,Pij̄ ,t )=P−η

ij,t P η−ε i,t Mij,tCt . (16)

Equation (16) shows that a firm’s pricing decision creates externalities to its rival’s cash flows through the industry’s price index Pi,t . When η−ε is large (i.e., the cross-industry elasticity of substitution is significantly greater than the within-industry elasticity of substitution) andMij,t /Mij̄,t is close to 1 (i.e., market shares are balanced in industry i), the competitor’s price Pij̄,t has a significant impact on firm ij ’s demand Cij,t through the effect of Pij̄,t on the industry-level price index Pi,t .

We define effective customer capital by M̃ij,t ≡Mij,tCt . Then, the demand curve faced by firm ij in Equation (16) can be rewritten as

Cij,t ≡Cij,t (Pij,t ,Pij̄ ,t )=P−η ij,t P

η−ε i,t M̃ij,t , (17)

Clearly, the demand for firm ij ’s goods Cij,t increases with effective customer capital M̃ij,t for given prices.

1.3 Two dimensions of heterogeneity across industries In our model, the customer base Mij,t fluctuates over time, driven by Poisson displacement shocks, Brownian shocks, and slow-moving fluctuations in expected growth. We consider two sources of heterogeneity across industries, both of which are reflected in how firms’ customer bases evolve in a given industry. One primitive industry characteristic is the displacement rate of industry i’s market leaders, denoted by λi , and the other is industry i’s cash flow exposure to the aggregate expected growth rate, denoted by ϕi . Our analysis

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The Oligopoly Lucas Tree

focuses on the asset pricing patterns in an economy with heterogeneous λi and ϕi . As we show in Section 2.3, the dispersion of these two primitive industry characteristics plays a key role in quantitatively accounting for the joint patterns of the gross profitability and value premium.

We use the Poisson process Ni,t with industry-specific intensity λi to characterize the occurrence of displacement shocks in industry i. A lower λi indicates that market leadership is more resilient in industry i. If displacement occurs over [t,t +dt] (i.e., if dNi,t =1), the two current market leaders are displaced by two new market leaders who were previously followers. To generate a nondegenerate distribution of customer bases across firms in the same industry, we assume that the customer bases of the two new market leaders in the industry are “reset” to equal values when displacement occurs over [t,t +dt]. That is, Mij,t =Mij̄,t =Mi,t/2 for the two new market leaders when dNi,t =1. Because the economy comprises a continuum of industries, industry-specific changes in market leaders are idiosyncratic events for the representative agent. The industry-specific market leadership turnover rateλi captures the first source of heterogeneity across industries.

The growth rates of different industries have different loadings on the aggregate expected growth rate gt . If displacement does not occur over [t,t +dt] (i.e., if dNi,t =0), the existing market leaders hold their ground, and firm ij ’s customer base (i.e., the representative agent’s taste for firm ij ’s products) evolves over [t,t +dt] according to

dMij,t

Mij,t

=ϕi(gt−g)dt +σMdWij,t , (18)

where the term ϕi(gt−g)dt captures the sensitivity of the growth rate of firm ij ’s cash flow to the aggregate expected growth rate gt ,9 and the standard Brownian motionWij,t captures idiosyncratic shocks to firm ij ’s customer base. The industry-specific loading ϕi captures the second source of heterogeneity across industries.

Integrating both sides of Equation (18) leads to the evolution equation of the aggregate customer base Mt ≡

∫ 1 0 Mi,tdi, as follows:

dMt

Mt

=ϕM (gt−g)dt, (19)

where ϕM is the average loading across industries (i.e., ϕM is the mean of the distribution of ϕi across industries). According to Equation (19), the total customer base Mt follows a stationary process in equilibrium.

9 For example, Bansal, Dittmar, and Lundblad (2005) show that firms’ expected future cash flow growth loads differently on aggregate expected growth, and Da and Warachka (2009) provide further empirical support by deriving a measure of expectations regarding firms’ future cash flows using analysts’ consensus earnings forecasts.

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By combining Equations (1), (9), and (18), we can derive the evolution equation of firm ij ’s effective customer capital M̃ij,t as follows:

dM̃ij,t

M̃ij,t

=[g+(ϕi +1)(gt−g)]dt +σedZe,t +σMdWij,t . (20)

The heterogeneity of λi endogenously generates the gross profitability premium across industries as in Dou, Ji, and Wu (2021). The heterogeneity of ϕi generates the value premium as in Da (2009), Santos and Veronesi (2010), Croce, Lettau, and Ludvigson (2014), and Li and Zhang (2016), who show that the value premium can be quantitatively explained by the estimated loadings on low-frequency consumption risk: value firms compared with growth firms are more exposed to low-frequency consumption risk.

1.4 Firms’ optimization under strategic rivalry Each firm ij faces two choice variables: its product price Pij,t and output Yij,t . Firm ij chooses them simultaneously to maximize its value, considering the externalities of its competitor’s price Pij̄,t and output Yij̄ ,t .

Our model takes a top-down approach similar to those of Menzly, Santos, and Veronesi (2004) and Santos and Veronesi (2006, 2010). Specifically, in our model, aggregate consumption dynamics are exogenously specified; however, the shares of individual firms’ outputs and cash flows as fractions of aggregate consumption are endogenously determined by their customer bases and competition intensity. The firm incurs production costs with intensityωYij,t to produce a flow of goods with intensity Yij,t over [t,t +dt]. These production costs are not a deadweight cost incurred by the representative agent. Rather, the production costs incurred by each firm ij can be seen as nonfinancial income received by the representative agent. Linear production technology is commonly adopted in the macroeconomics and IO literature (e.g., Garcia- Macia, Hsieh, and Klenow 2018; Aghion et al. 2019; Bils, Klenow, and Ruane 2020). In Section 2.6, we extend our model by incorporating fixed costs of production (and thus the implied operating leverage) as in Carlson, Fisher, and Giammarino (2004), and we show that the main quantitative implications are not changed in the extended model.

Let the firm-level and industry-level profit margins be θij,t =(Pij,t−ω)/Pij,t and θi,t ≡ (Pi,t−ω)/Pi,t , respectively. We focus on profit margins rather than price levels to increase the transparency of the central economic mechanisms; our choice of focus does not change the main insights or results of this paper.10

The demand function in Equation (17) can be rewritten in terms of profit margins as follows:

Cij,t (θij,t ,θij̄ ,t )=ω−ε(1−θij,t )η(1−θi,t )ε−ηM̃ij,t , (21)

10 A detailed discussion of the economic reasons for focusing on profit margins rather than price levels when studying product competition can be found in Dou, Ji, and Wu (2021).

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The Oligopoly Lucas Tree

where M̃ij,t is firm ij ’s effective customer capital, defined in Equation (17). To maximize its market value, the firm would never produce more than

demand Cij,t =Cij,t (θij,t ,θij̄ ,t ), because goods are immediately perishable and production is costly. Thus, following Gourio and Rudanko (2014), Corhay, Kung, and Schmid (2020), and Dou et al. (2021a), the demand constraint Yij,t ≤ Cij,t (θij,t ,θij̄ ,t ) must hold forYij,t ,Yij̄ ,t , θij,t , and θij̄ ,t in equilibrium. Moreover, it is optimal for the firm to choose θij,t >0, so the firm will produce up to Cij,t (θij,t ,θij̄ ,t ) in equilibrium:

Yij,t =Cij,t (θij,t ,θij̄ ,t ). (22)

Because the financial market is frictionless, the firm has no incentive to hoard cash as in Bolton, Chen, and Wang (2011), and the operating profit of firm ij is entirely paid out as dividendsDij,tdt over [t,t +dt]. That is, the firm’s dividend flow intensity is given by

Dij,t =(Pij,t−ω)Yij,t =θij,t (1−θij,t )−1ωYij,t , with θij,t >0. (23)

Plugging Equation (22) into Equation (23) and rearranging terms, we have

Dij,t =�ij,t (θij,t ,θij̄ ,t )M̃ij,t , (24)

where �ij,t (θij,t ,θij̄ ,t ) reflects the profitability per unit of firm ij ’s effective customer capital M̃ij,t and has the following expression:

�ij,t (θij,t ,θij̄ ,t )=ω1−εθij,t (1−θij,t )η−1(1−θi,t )ε−η. (25)

Firm ij optimally and strategically chooses θij,t and Yij,t to maximize its shareholder value as follows:

Vij,0 = sup θij,t ,Yij,t

E0

[∫ τi

0

�t

�0 Dij,tdt

] , subject to Yij,t ≤Cij,t (θij,t ,θij̄ ,t ) (26)

=sup θij,t

E0

[∫ τi

0

�t

�0 �ij,t (θij,t ,θij̄ ,t )M̃ij,tdt

] , (27)

where τi is the random stopping time at which the market leaders are displaced. We follow Dou, Ji, and Wu (2021) and consider the collusive equilibrium.11

The two firms in the same industry play a dynamic game, in which the stage games of setting profit margins are played continuously and repeated infinitely with exogenous and endogenous state variables changing over time. There exists a Markov-perfect noncollusive Nash equilibrium, which is the repetition of the one-shot Nash equilibrium. Importantly, there also exist multiple subgame-perfect collusive Nash equilibria in which profit-margin strategies

11 Extensive evidence shows that industries are highly concentrated and (tacit) collusion among market leaders is widespread, which has a significant economic impact (e.g., Ivaldi et al. 2007). We discuss evidence of (tacit) collusion in Internet Appendix 3.4.

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are sustained by conditional punishment strategies. Intuitively, the punishment for deviation is to switch from (tacit) cooperation to a full-blown price war. Theoretically, the punishment for deviation is to switch from the collusive Nash equilibrium to the noncollusive Nash equilibrium, which leads to strictly lower profit margins. When deviation occurs at time t , the punishment is implemented with probability ξdt over [t,t +dt]. The intensity ξ can be viewed as a parameter governing the credibility of the punishment for deviating behavior. A higher ξ reduces the incentive to deviate. Firms’ profit-margin strategies depend on both the “payoff-relevant” physical states xi,t =

{ Mi1,t ,Mi2,t ,Ct ,st ,gt

} in state

space X, as in Maskin and Tirole (1988a,b), and a set of indicator functions that track whether any firm has previously deviated from a collusive profit-margin agreement, as in Fershtman and Pakes (2000, p. 212).12 The characterization of the noncollusive and collusive equilibria is similar to that of Dou, Ji, and Wu (2021) and is discussed in Internet Appendix 3.5.

1.5 Discussions of the model’s ingredients 1.5.1 Homogeneity. By exploiting the model’s homogeneity in terms of industry-level effective customer capital M̃i,t =M̃i1,t +M̃i2,t =Mi,tCt for all firms in each industry i∈I, we can reduce the state space of the model to three state variables to characterize industry i’s equilibrium. The three state variables are Mi1,t /Mi,t , st , and gt . In particular, the value function of firm ij in the collusive equilibrium, denoted by V Cij (Mi1,t ,Mi2,t ,Ct ,st ,gt ), can be represented by

V Cij (Mi1,t ,Mi2,t ,Ct ,st ,gt )≡vCij (Mi1,t /Mi,t ,st ,gt )M̃i,t . (28)

We numerically solve the normalized firm values vCij (Mi1,t /Mi,t ,st ,gt ) and profit margins θCij (Mi1,t /Mi,t ,st ,gt ) in the collusive equilibrium.13 Here, the superscript C denotes the value and policy functions in the collusive equilibrium. Because we focus only on the collusive equilibrium, we omit the superscriptC throughout the rest of the paper to ease the notational burden.

1.5.2 Aggregate and idiosyncratic shocks. The aggregate state variable gt determines expected consumption growth. Its evolution is given by Equation (2), where the aggregate shockZg,t can be interpreted as the aggregate expected growth shock. The industry characteristic ϕi reflects heterogeneous cash flow loadings on expected growth across industries, as shown in Equation (18). The expected growth shock Zg,t has a positive market price of risk ς (gt ) in Equation (11).

The aggregate state variable st determines the aggregate discount rate. Its evolution is given by Equation (7), which incorporates aggregate shocks

12 For notational simplicity, we omit the indicator states of historical deviations.

13 See Internet Appendix 4 for numerical algorithms.

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The Oligopoly Lucas Tree

Zg,t and Ze,t , where the latter shock enters Equation (1) to determine the evolution of the aggregate endowment (consumption). Becauseσe σg

√ g−ς ,

the evolution of st is determined mainly by dZe,t in Equation (7). Thus, we can interpret Ze,t as the aggregate discount rate shock; that is, a positive shock, dZe,t >0, results in a higher log surplus consumption ratio st and a lower discount rate. The shock Ze,t has a positive market price of risk η(st ) in Equation (11).

The evolution of firm ij ’s customer base share Mij,t /Mi,t is driven by two idiosyncratic shocks, Wij,t and Ni,t . The idiosyncratic shocks Wij,t (j =1,2) in Equation (18) are not crucial for the central mechanisms; however, they are necessary to ensure stationary and nondegenerate industry dynamics in the long run. Intuitively, they can be interpreted as idiosyncratic demand (or “taste”) shocks. When the leadership turnover shock occurs (dNi,t =1), industry i’s total customer base Mi,t remains unchanged, while the two new market leaders start with the same customer base, that is, Mij,t =Mij̄,t =Mi,t/2. This model specification ensures that no firm will dominate its industry, even in the long run.

1.5.3 Profitability and valuation ratios. Industry-level gross profitability is defined as industry i’s gross profits normalized by its assets (i.e., effective customer capital M̃i,t =M̃i1,t +M̃i2,t in the model):

GPi,t ≡ (Pi,t−ω)Ci,t M̃i,t

=θi,t

( ω

1−θi,t )1−ε

. (29)

In equilibrium, the industry-level profit margin θi,t is always less than or equal to the monopolistic profit margin 1/ε, and thus it is straightforward to show that GPi,t is strictly increasing in θi,t . This implies that the profit margin θi,t captures information similar to gross profitability GPi,t in our model. Consistent with empirical patterns, profitability ratios �ij,t (θij,t ,θij̄ ,t ) and GPi,t are stationary in equilibrium.

Similar to endowment-based models for value and growth firms (e.g., Lettau and Wachter 2007; Tsai and Wachter 2016), we use the decomposition of assets in place and growth options as a proxy for the book-to-market ratio.14 One way to approximate the value of assets in place is to focus on the value of existing assets without accounting for any growth effects. Specifically, we consider the value of existing effective customer capital that does not include any growth opportunities, denoted by M̃a

ij,t , and we assume that it decays stochastically over time:

dM̃a ij,t

M̃a ij,t

= dM̃ij,t

M̃ij,t

−[g+(ϕi +1)(gt−g)]dt,

14 In production-based models for value and growth firms (e.g., Papanikolaou 2011; Kogan and Papanikolaou 2013; Kogan and Papanikolaou 2014; Dou 2017), the firm’s book-to-market ratio is used to measure the relative contribution of assets in place and growth opportunities to firm value, because firms with lower book-to-market ratios are likely to derive most of their value from growth opportunities.

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where g+(ϕi +1)(gt−g) is the drift of dM̃ij,t /M̃ij,t (see Equation (20)). The value of assets in place in the collusive equilibrium, denoted by V aij,0,

is the present value of cash flows paid based on the profits generated by effective customer capital M̃a

ij,t over time. Thus, given the optimal collusive profit margins θij,t , it holds that

V aij,0 =E0

[∫ τi

0

�t

�0 �ij,t (θij,t ,θij̄ ,t )M̃

a ij,tdt

] . (30)

The value of growth opportunities in the collusive equilibrium is defined by

V oij,0 ≡Vij,0 −V aij,0, The difference between the total market value and the value of assets in place gives a generic definition of the value of growth options, which captures the present value of dividends owing to the growth of capital in the future (e.g., Gomes, Kogan, and Zhang 2003, equation (25)).

1.5.4 Market-clearing condition. Last, we discuss the market-clearing condition in Equation (9). From Equations (5) and (6), it follows that Ct = PtCt =

∫ 1 0 Pi,tCi,tdi and Pi,tCi,t =

∑2 j=1Pij,tCij,t . These relationships, together

with the product-level market-clearing conditions Cij,t =Yij,t and Equation (9), lead to

Et = ∫ 1

0

⎛⎝ 2∑ j=1

Pij,tYij,t

⎞⎠di, (31)

where the revenue of firm ij isPij,tYij,t =Dij,t +ωYij,t ; that is, the firm’s revenue is the sum of its dividends and production costs (see Equation (23)). In other words, the aggregate endowment is equal to the total revenue of the corporate sector. Thus, although we exogenously specify the evolution of the aggregate endowment Et in Equation (1), the way in which it is split into firm-level revenue Pij,tYij,t and further into firm-level dividends Dij,t and production costs ωYij,t is endogenous. This top-down Lucas tree modeling approach is similar in spirit to that adopted by Menzly, Santos, and Veronesi (2004) and Santos and Veronesi (2006, 2010); however, firm-level revenue in our model evolves endogenously according an economic mechanism different from that used in these papers.

2. Economic Mechanisms and Quantitative Analyses

In this section, we explain the central economic mechanisms and investigate the quantitative capacity of our model to match the data. Section 2.1 describes the data and empirical measures. Section 2.2 calibrates the model. Section 2.3 discusses the central economic mechanisms. Section 2.4 shows the main asset pricing results. Finally, Section 2.5 quantitatively inspects the model’s mechanisms and dissects the effects of the main model ingredients by turning them off one at a time and checking the changes incurred.

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The Oligopoly Lucas Tree

2.1 Data and empirical measures We here describe the data and empirical measures used in our quantitative analyses.

2.1.1 Data and industry portfolio returns. We obtain consumption data from the U.S. Bureau of Economic Analysis (BEA) and stock return data from the Center for Research in Security Prices (CRSP). Consumption growth is measured by the log growth rate of per-capita real personal consumption expenditures on nondurable goods and services.

We compute profitability and the book-to-market ratio based on financial data from Compustat. Industry-level gross profitability is constructed as gross profits (revenue minus cost of goods sold) scaled by assets, as defined by Novy- Marx (2013). The industry-level book-to-market ratio is the ratio of book equity to market equity in an industry. Industry-level revenue, cost of goods sold, book assets, book equity, and market equity are the sum of the corresponding firm-level measures for firms in the same industry.

Our model focuses on strategic competition among a few oligopolistic firms whose products are close substitutes. Therefore, we use the SIC4 codes to define industries, following the literature (e.g., Hou and Robinson 2006; Gomes, Kogan, and Yogo 2009; Frésard 2010; Giroud and Mueller 2010; Giroud and Mueller 2011; Bustamante and Donangelo 2017). We use Compustat segment data to improve the precision of industry classifications (see Internet Appendix 2.1).

Our analysis focuses on the gross profitability and value premium across industries. We compute industry-level stock returns as the value-weighted average stock returns of individual firms in a given industry weighted by their market capitalization lagged by 1 month. We exclude financial and utility firms from the analysis and use CRSP delisting returns to adjust for delisted stocks. To ensure that the cross-industry gross profitability and value premium do not simply reflect the firm-level premium, we exclude industries that contain fewer than three firms when computing the cross-industry premium.

2.1.2 Leadership turnover rates. We construct an industry-level measure of market leadership turnover rates following Dou, Ji, and Wu (2021). In particular, we define market leaders as the top two firms ranked by sales in a given industry, which includes both public and private firms. Similar to estimating the probability of corporate events (e.g., Shumway 2001; Campbell, Hilscher, and Szilagyi 2008), we estimate the leadership turnover rate using a logistic regression model. Specifically, we assume that the marginal probability of market leadership turnover from year t to year t +1 follows a logistic regression model given by

P(1t→t+1 turnover,i =1)=1/

[ 1+exp(−b0 −b1xi,t−δi−θt )

] , (32)

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where 1t→t+1 turnover,i is an indicator that equals 1 if the market leaders of industry i in

year t +1 are different from those in year t .15 The term xi,t is a column vector of explanatory variables whose values are known at the end of year t . Following the IO literature (e.g., Geroski and Toker 1996; Sutton 2007; Kato and Honjo 2009), the vector of explanatory variables xi,t includes industry asset growth rate, industry advertising intensity (i.e., advertising expenses scaled by revenue), industry research and development (R&D) intensity (i.e., R&D expenses scaled by revenue), and the industry-level innovation similarity measure (see appendix B in Dou, Ji, and Wu, 2021). The terms δi and θt are industry and time fixed effects, respectively.

The leadership turnover measure, denoted by λ̂i,t , is the predicted probability that one or more existing market leaders are replaced in year t +1:

λ̂i,t =1/ [ 1+exp(−b̂0 − b̂1xi,t− δ̂i− θ̂t )

] , (33)

where b̂0, b̂1, δ̂i , and θ̂t are estimated using specification (32). We use the average of λ̂i,t over time to measure the market leadership

turnover rate of industry i, denoted by λ̂i . To construct gross-profitability-sorted portfolio-level measures for market leadership turnover rates, we first sort all industries into quintiles (and deciles) according to their gross profitability, and then measure the leadership turnover rate for each portfolio, denoted by λ̂p, using the median value of the leadership turnover rates in the industry-year panel within portfolio p. We tabulate λ̂p in panel B of Table 2, which shows that the leadership turnover rate is lower in more profitable industries, consistent with the theoretical and quantitative implications of our model.

2.1.3 Loadings on expected growth. We estimate the industry-level cash flow loading on expected growth by running the following time-series regression for each industry i separately using data from 1951 to 2018:

2∑ j=0

φjROEi,t−j =α+ϕi

2∑ j=0

φjct−j +εi,t , (34)

where ∑2

j=0φ jROEi,t−j is the accumulated return on equity (ROE) of industry

i from year t−2 to year t . Following the definition of ROE in Santos and Veronesi (2010), we calculate industry-level ROE in year t as the ratio of industry-level clean-surplus earnings in year t and industry-level book equity in year t−1, where clean-surplus earnings in year t are the changes in

15 We define the turnover indicator based on the market leaders in years t and t +1. The turnover indicator from years t to t +1 is equal to one if (a) the largest firm ranked by sales in the industry in year t +1 is neither of the two largest firms in year t or (b) the second largest firm ranked by sales in the industry in year t +1 is neither of the two largest firms in year t , and it is large enough so that its sales are greater than 80% of the sales of the largest firm in year t +1. We impose this size requirement to ensure that the second largest firm is a market leader in the industry in year t +1.

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book equity from year t−1 to year t plus dividends in year t . The term∑2 j=0φ

jct−j is the accumulated consumption growth from year t−2 to year t , a proxy for latent expected future consumption growth used in studies involving low-frequency consumption risk (e.g., Bansal, Dittmar, and Lundblad 2005; Dittmar and Lundblad 2017). We set φ = 0.87 to be consistent with the yearly persistence coefficient of the surplus consumption ratio estimated by Campbell and Cochrane (1999). The coefficient ϕ̂i estimated using specification (34) is the industry-level cash flow loading on expected growth.

To measure the book-to-market-sorted portfolio-level cash flow loading on expected growth, we first sort all industries into quintiles (and deciles) according to their book-to-market ratios, and then follow Santos and Veronesi (2010) to estimate the portfolio-level loading by running the following time series regression for each industry portfolio p separately using data from 1951 to 2018:

2∑ j=0

φjROEp,t−j =α+ϕp

2∑ j=0

φjct−j +εp,t , (35)

where ∑2

j=0φ jROEp,t−j is the accumulated ROE of industry portfolio p

from year t−2 to year t . We measure portfolio-level ROE by computing the value-weighted average industry-level ROE based on industry-level market capitalization lagged by 1 year. For each industry portfolio p, we denote the estimated portfolio-level cash flow loading on expected growth by ϕ̂p. We tabulate ϕ̂p in panel B of Table 2, which shows that the cash flow loading on expected growth is higher in industries with higher book-to-market ratios, consistent with the theoretical and quantitative implications of our model.

2.1.4 Discussions of λi and ϕi . Our model treats both the leadership turnover rate λi and cash flow loading on expected growth ϕi as primitive industry characteristics and takes them as exogenously given. Yet both characteristics are endogenous in reality and likely reflect more fundamental economic primitives. Although it goes way beyond the scope of this paper to endogenize λi and ϕi , we find it useful to provide additional discussions on the potential determinants of λi and ϕi by examining their associations with other industry characteristics, such as innovation similarity, investment reversibility, and operating leverage, as motivated by the previous studies.16 We also provide a few examples to further illustrate what industries with different λi and ϕi typically look like.

Specifically, we find that λi is negatively associated with lagged innovation similarity and operating leverage (with regression coefficients −0.174 and −0.080, as well as t-statistics −6.684 and −2.380, respectively), while it is

16 The industry-level measure for investment reversibility is similar in spirit to the measures used by Bai et al. (2019). We relegate the detailed description to Internet Appendix 2.5. The measure for innovation similarity is first described below Equation (32), and that for operating leverage will be described in Equation (40).

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positively associated with lagged investment reversibility (with a regression coefficient 0.421, as well as a t-statistic 6.427). The detailed results are reported in panel A of Table OA.14 of Internet Appendix 2.5. These findings are consistent with the intuitions in the literature: market leaders are typically displaced through followers’ distinctive innovation (e.g., Christensen 1997) or rapid business expansion that usually requires rapid fixed asset expansion. Thus, a lower market leadership turnover rate λi is probably due to a lower likelihood of success in distinctive and radical innovation, which helps market leaders retain their leadership positions by preventing creative destruction (e.g., Dou, Ji, and Wu 2021), or due to a higher fixed cost of production (i.e., higher operating leverage) and lower investment reversibility, which also help market leaders retain their leadership positions by eliminating small competitors (e.g., Baumol and Willig 1981; Rasmusen 1988).

Moreover, we find that ϕi is positively associated with innovation similarity and operating leverage (with regression coefficients 16.839 and 7.458, as well as t-statistics 2.679 and 1.063, respectively), while it is negatively associated with investment reversibility (with a regression coefficient −32.426, as well as a t-statistic −2.366). The detailed results are reported in panel B of Table OA.14 of Internet Appendix 2.5. These findings are consistent with the intuitions in the literature: industries’ cash flows are more exposed to fluctuations in expected growth if their firms have less flexibility in operations. Thus, an industry’s cash flow loading on expected growth ϕi is relatively high probably because the firms in the industry are relatively lack of distinctive innovation opportunities, controls over operating costs, or disinvestment capacity, all of which amplify the sensitivity of their cash flows to changes in economic growth prospects.

To make the intuitions above more concrete, we provide a few examples illustrating what industries with a high or low λi typically look like. “Carpets and Rugs,” “Paints and Allied Products,” and “Dolls and Stuffed Toy” are three examples of industries that feature relatively high innovation similarity and low investment reversibility. Their market leaders’ positions are relatively persistent according to our empirical measure λ̂i , and consistent with the predictions of our model, their gross profitability is relatively high in the data. By contrast, “Crude Petroleum and Natural Gas,” “Television Broadcasting Stations,” and “Commercial Physical and Biological Research” are three examples of industries that feature relatively low innovation similarity and high investment reversibility. Their market leaders’ positions are relatively fragile according to our empirical measure λ̂i , and consistent with the predictions of our model, their gross profitability is relatively low in the data.

We also provide a few examples illustrating what industries with a high or low ϕi typically look like. “Silver Ores,” “Household Audio and Video Equipment,” and “Management Consulting Services” are three examples of industries that feature relatively high investment reversibility and low operating leverage. Their market leaders’ cash flow loadings on expected growth are relatively low according to our empirical measure ϕ̂i , and consistent with the predictions

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Table 1 Calibration and parameter choice

Parameter Symbol Value Parameter Symbol Value

A. Externally determined parameters Risk aversion γ 2 Volatility of consumption growth σe 0.021 Persistence of surplus ratio φs 0.13 Average consumption growth g 0.0189 Lower bound of growth ς −0.02 Persistence of expected growth κ 0.3 Volatility of expected growth σg 0.014 Cross-industry elasticity ε 1.1 Volatility of customer base σM 0.01 Within-industry elasticity η 17 Marginal cost of production ω 1 B. Internally calibrated parameters Subjective discount factor ρ 0.185 Punishment rate ξ 0.037 Copula coefficient of λi and ϕi ϑ 0.159 Range of loadings on gt [ϕ, ϕ] [−1, 7] Range of turnover rates [λ, λ] [0.019, 0.125]

of our model, their book-to-market ratios are relatively low in the data. By contrast, “Air Courier Services,” “Groceries,” and “Department Stores” are three examples of industries that feature relatively low investment reversibility and high operating leverage. Their market leaders’ cash flow loadings on expected growth are relatively high according to our empirical measure ϕ̂i , and consistent with the predictions of our model, their book-to-market ratios are relatively high in the data.

2.2 Calibration We determine some model parameters based on external information without simulating the model (see panel A of Table 1), while calibrating the remaining model parameters internally by matching important features of the data (see panel B of Table 1).

2.2.1 Externally determined parameters. We follow Campbell and Cochrane (1999) and set g=1.89%, φs =0.13, and γ =2. We set the volatility of consumption growth at σe =2.1%, which is close to the value calibrated by Bansal, Kiku, and Yaron (2012). We set the persistence of expected growth at κ =0.3, the volatility of expected growth at σg =1.4%, and the lower bound of aggregate consumption growth at ς =−2%, ensuring that 2κ(g−ς )>σ 2

g ; that is, the square root process (2) is well defined. The consumption process implied by these dynamic parameters is consistent with the data in terms of average growth rates, autocorrelations, and variance ratios (see panel A of Table 2).

Importantly, we calibrate our model to capture the characteristics of U.S. industries. We set the within-industry and cross-industry elasticity of substitution at η=17 and ε=1.1, respectively. They are broadly consistent with the calibration and estimation in the IO and international trade literature (e.g., Harrigan 1993; Head and Ries 2001; Atkeson and Burstein 2008). We set customer base volatility at a low constant value of σM =0.01 to capture the idea of a sticky customer base (e.g., Gourio and Rudanko 2014; Gilchrist et al. 2017). Without loss of generality, the marginal cost of production is normalized to one, that is, ω=1.

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Table 2 Moments in the data and model

A. Consumption growth, risk-free rate, and profit margin

Moments Data Model Moments Data Model

Average consumption growth (%) 1.89 1.92 VR(2) of consumption growth 1.47 1.39 [1.51,2.26] [1.10,1.86]

AC(1) of consumption growth 0.46 0.40 VR(4) of consumption growth 1.89 1.79 [0.18,0.70] [0.85,3.15]

AC(4) of consumption growth 0.11 0.09 VR(6) of consumption growth 2.21 2.04 [−0.20,0.27] [0.88,4.00]

AC(6) of consumption growth 0.05 0.05 VR(8) of consumption growth 2.42 2.41 [−0.35,0.14] [0.92,4.37]

Average real risk-free rate (%) 0.68 0.68 Average gross profit margin (%) 31.39 27.99 [−0.21,1.65] [29.98,33.00]

B. Industry characteristics Portfolios sorted on gross profitability D1 (low) Q1 Q2 Q3 Q4 Q5 D10 (high) λ̂p in the data 0.126 0.121 0.089 0.075 0.070 0.044 0.013 λ̂p in the model 0.125 0.119 0.096 0.072 0.049 0.025 0.019

Portfolios sorted on the book-to-market ratio D1 (low) Q1 Q2 Q3 Q4 Q5 D10 (high) ϕ̂p in the data 0.62 2.77 3.81 4.75 3.50 5.95 7.98 ϕ̂p in the model 0.53 0.96 2.79 4.75 6.25 7.55 7.89 Correlation between λ̂i and ϕ̂i in the data 0.15 Correlation between λi and ϕi in the model 0.15

This table tabulates the moments in the data and model. The quarterly consumption data are constructed using U.S. BEA data and cover the postwar period from 1948 to 2017. The moments in panel A are computed following Beeler and Campbell (2012), who focus on the sample period from 1948 to 2008 (our moments replicate theirs when we focus on the same sample period). Numbers in the brackets represent the [2.5%,97.5%] confidence interval. AC(k) of consumption growth refers to the autocorrelation of consumption growth with a k-year lag. VR(k) of consumption growth refers to the variance ratio of consumption growth with a k-year horizon. Real risk-free interest rate is the average difference between the annual returns of 1-month Treasury bills from CRSP and the rate of change in the consumer price index from 1948 to 2017. The estimation of λ, ϕ, and their correlation in the data is explained in Section 2.1. When constructing the model moments, we simulate a sample of 1,000 industries for 150 years with an 80-year burn-in period. We then compute the model-implied moments as we do for the data. For each moment, the table reports the average value of 2,000 simulations.

2.2.2 Internally calibrated parameters. We calibrate the remaining parameters by matching the real risk-free rate, the average gross profit margin, the ranges of λi and ϕi across industry portfolios, and the correlation between λi and ϕi across industries in the data, which are summarized in Table 2. We run 2,000 independent parallel simulations. For each simulation, we generate a sample of 1,000 industries for 150 years according to the model solution. The first 80 years are dropped as the burn-in sample; thus, we retain a 70- year simulated panel, putting the length of our simulated sample in a range to mimic the data. We then compute the model-implied moments and adjust the parameters until the moments are in line with the data.

We set the subjective discount factor at ρ =0.185 to match the average real risk-free rate between 1948 and 2018. We set the punishment rate at ξ =0.037 to match the average gross profit margin of all industries.

Crucially, we calibrate the bivariate joint distribution of λi and ϕi across industries entirely based on the “baseline moments,” which only involve the empirical measures of these two primitive industry characteristics without resorting to model-implied information from asset prices. Our calibration approach is largely free from the potential overfitting issues caused by inferring

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the key structural parameters directly from the potentially misspecified “cross- equation asset pricing restrictions” implied by the model. This is similar in spirit to the recursive estimation procedure advocated by Hansen (2007, 2012) and Chen, Dou, and Kogan (2021) for structural asset pricing models. As the main results of this paper, all the asset pricing moments summarized in Table 3, can be interpreted as untargeted moments. In general, the goodness of fit of additional untargeted moments, particularly the moments that a model attempts to explain, is emphasized as a useful criterion for assessing a model’s external validity in the literature on structural estimation.17

In particular, we specify the marginal distribution of an industry’s leadership turnover rate λi and cash flow loading on expected growth ϕi , in stationary equilibrium, as a uniform distribution over the intervals [λ, λ] and [ϕ,ϕ],

respectively.18 We calibrate λ=0.019 and λ=0.125 so that the median values of the market leadership turnover rates in portfolios D1 (the 1st decile) and D10 (the 10th decile) of all industries sorted on gross profitability in the model are in line with those of the data (see panel B of Table 2). Similarly, we calibrate ϕ=−1 and ϕ=7 so that the model-implied values of industry portfolios’ cash flow loadings on expected growth in portfolios D1 (the 1st decile) and D10 (the 10th decile) of all industries sorted on the book-to-market ratio are in line with those of the data (see panel B of Table 2).19 Importantly, as an additional “out-of- sample” validation, we report the industry characteristics λi and ϕi for different quintile portfolios (i.e., Q1 to Q5) in the data and model for comparison. Overall, the similarity between the data and the model shown in columns Q1 to Q5 of panel B suggests that the distributions and functional forms assumed in the model are reasonable.

We use a flexible parametric method to capture the interdependence between λi and ϕi across industries. We denote by F1(λi) and F2(ϕi) the marginal cumulative distribution functions of λi and ϕi , respectively, and describe the statistical interdependence between λ and ϕ using the Gaussian copula:

CGaussϑ (x1,x2)≡�ϑ

( �−1(x1),�−1(x2)

) , (36)

where�(·) is the cumulative distribution function of a standard normal variable, and �ϑ (·,·) is the joint cumulative distribution function of a bivariate normal distribution with zero mean, unity variance, and correlation coefficient ϑ .

The parameter ϑ governs the dependence between the two marginal distributions, F1(λ) and F2(ϕ). A higher ϑ implies that λi and ϕi are more

17 The structural estimation literature has a long tradition of using one set of moments to estimate a model and another set of untargeted moments to test the model’s out-of-sample fit. Recent examples that explicitly stress the importance of untargeted moments include Dou et al. (2021c) and Akcigit, Hanley, and Serrano-Velarde (2021).

18 When simulating the model, we discretize the values of λi and ϕi in N =10 grids with λ1 =λ, λN =λ, ϕ1 =ϕ, and ϕN =ϕ.

19 The industry’s cash flows are proportional to its effective customer capital M̃i,t , whose cash flow loading on expected growth is ϕi +1 (see Equation (20)).

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Table 3 Industry-level gross profitability and value premium in the data and model

Q1 (low) Q2 Q3 Q4 Q5 (high) Q5 − Q1

A. Gross profitability return spreads (sorted on gross profitability) Data (EW) 7.23∗∗∗ 7.35∗∗∗ 9.26∗∗∗ 9.16∗∗∗ 10.03∗∗∗ 2.81∗∗

[2.73] [3.01] [3.55] [3.75] [4.24] [2.04] Data (VW) 6.47∗∗∗ 9.02∗∗∗ 8.77∗∗∗ 9.32∗∗∗ 9.97∗∗∗ 3.50∗∗

[2.99] [3.93] [3.31] [3.99] [4.55] [2.14] Model (EW) 9.78 9.92 10.25 11.14 12.92 3.14 Model (VW) 9.43 9.56 9.90 10.78 12.60 3.17

B. Gross profitability return spreads after controlling for the book-to-market ratio Data (EW) 6.32∗∗ 6.28∗∗ 9.66∗∗∗ 8.47∗∗∗ 10.11∗∗∗ 3.79∗∗∗

[2.43] [2.21] [3.91] [3.14] [4.20] [3.22] Data (VW) 5.50∗∗ 6.58∗∗ 10.87∗∗∗ 8.35∗∗∗ 9.94∗∗∗ 4.44∗∗∗

[2.55] [2.46] [4.90] [3.19] [4.44] [2.98] Model (EW) 9.01 9.56 10.25 11.55 13.64 4.63 Model (VW) 8.99 9.55 10.24 11.55 13.66 4.66

C. Value return spreads (sorted on the book-to-market ratio) Data (EW) 5.28∗∗ 7.55∗∗∗ 9.04∗∗∗ 9.02∗∗∗ 10.07∗∗∗ 4.79∗∗∗

[2.01] [3.02] [3.70] [3.67] [3.94] [3.01] Data (VW) 7.52∗∗∗ 8.13∗∗∗ 8.85∗∗∗ 9.17∗∗∗ 11.07∗∗∗ 3.55∗

[3.13] [3.77] [4.25] [4.28] [4.97] [1.87] Model (EW) 8.08 9.63 11.49 12.24 12.56 4.48 Model (VW) 8.22 9.82 11.70 12.46 12.59 4.37

D. Value return spreads after controlling for gross profitability Data (EW) 5.76∗∗ 6.93∗∗ 7.93∗∗∗ 8.90∗∗∗ 11.43∗∗∗ 5.66∗∗∗

[2.19] [2.53] [3.23] [3.28] [4.53] [3.93] Data (VW) 6.65∗∗∗ 7.94∗∗∗ 7.78∗∗∗ 8.11∗∗∗ 11.68∗∗∗ 5.03∗∗∗

[2.81] [3.32] [3.69] [3.73] [5.22] [2.74] Model (EW) 7.93 9.21 10.67 12.33 13.87 5.94 Model (VW) 7.97 9.25 10.72 12.37 13.91 5.94

Panel A shows the equal- and value-weighted average excess returns of industry portfolios sorted on gross profitability. In the data, in June of each year t , we sort all industries into quintiles based on their gross profitability in year t−1. Once the portfolios are formed, their monthly returns are tracked from July of year t to June of year t +1. In the model, we simulate a sample of 1,000 industries for 150 years with an 80-year burn-in period. The 70-year simulated panel places our model sample within the range of the observed data. We then perform portfolio sorting as we do for the data. The excess returns of the portfolios in the model are the average values of 2,000 simulations. In panel B, we perform a double-sort analysis in which we first sort all industries into ten groups based on their book-to-market ratios, and then we sort the industries in each group into quintiles based on their gross profitability. Panel C shows the equal- and value-weighted average excess returns of the industry portfolios sorted on the book-to-market ratio. In panel D, we perform a double-sort analysis in which we first sort all industries into ten groups based on their gross profitability and then we sort the industries in each group into quintiles based on their book-to-market ratios. The sample period for the data is from July 1951 to June 2018. We exclude financial firms and utility firms from the analysis. To ensure that the cross-industry gross profitability and value premium do not simply reflect the firm-level premium and avoid potential problematic data points, we exclude all industries with fewer than three firms when computing the cross-industry premium. Newey-West standard errors are estimated with one lag. We annualize the average excess returns by multiplying them by 12. We include t-statistics in brackets. ∗p<.1;∗∗p<.05;∗∗∗p<.01.

positively associated with each other in the cross-section of industriesgn. When ϑ =0, the two variablesλi andϕi are independent across industries. We calibrate ϑ =0.159 to match the correlation between λi and ϕi across industries in the data, which is equal to 0.15. Based on the definition of the copula, the joint distribution of (λi,ϕi), denoted by F (λ,ϕ), can be expressed as follows:

F (λ,ϕ)=CGaussϑ (F1(λ),F2(ϕ)). (37)

The correlation between λi and ϕi in the data is statistically significant, with a p-value of .024. Intuitively, the positive correlation between λi and ϕi at the

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industry level is consistent with the results of the finance literature. On the one hand, market leaders in industries with higher book-to-market ratios are more subject to displacement threats from followers within the same industries through disruptive innovations (e.g., Gârleanu, Kogan, and Panageas 2012; Kogan and Papanikolaou 2013; Kogan and Papanikolaou 2014; Kogan et al. 2017; Kogan, Papanikolaou, and Stoffman 2020). That is, industries with higher book-to-market ratios tend to have a higher λi . On the other hand, firms in industries with higher book-to-market ratios are more exposed to fluctuations in expected growth in aggregate consumption (e.g., Bansal, Dittmar, and Lundblad 2005; Parker and Julliard 2005; Hansen, Heaton, and Li 2008; Santos and Veronesi 2010; Li and Zhang 2016). That is, industries with higher book-to- market ratios tend to have a higher ϕi . Taken together, these two results suggest that λi and ϕi should be positively correlated across industries, which we find based on our estimation in Section 2.1.

2.3 Central economic mechanisms 2.3.1 Overview of challenges and contributions. Our model combines two economic mechanisms. The first economic mechanism is the one proposed by Dou, Ji, and Wu (2021), through which industries with lower leadership turnover rates λi have higher profit margins, and both of their profit margins and stock returns are more negatively exposed to fluctuations in the discount rate η(st ). Thus, our model can rationalize the cross-industry gross profitability premium through this mechanism. The second economic mechanism is the one suggested by the empirical findings of Parker and Julliard (2005), Hansen, Heaton, and Li (2008), Santos and Veronesi (2010), and Li and Zhang (2016), among others. Through this mechanism, industries with higher cash flow loadings on expected growth ϕi are endogenously associated with higher book-to-market ratios and return exposures to fluctuations in expected growth. Thus, our model also rationalizes the cross-industry value premium.

It may not be surprising that by combining two cross-sections (i.e., λi and ϕi) and two systematic fluctuations (i.e., changes in η(st ) and gt ), our model is able to quantitatively explain two cross-sectional equity premia (the gross profitability and value premium) simultaneously in a unified framework. Our main objective goes substantially beyond separately rationalizing two cross- sectional equity premia within a single model. Motivated by the important insights in the literature,20 we aim to advance our understanding of the complex and intriguing interactions between the two cross-sections, as illustrated in Figure 1. Specifically, by focusing on the interactions between the two cross sections, this paper differs considerably from Dou, Ji, and Wu (2021) and contributes to the literature in the following three respects. First, our model shows how gross profitability and book-to-market ratios are endogenously

20 Influential studies in the literature have stressed the importance of understanding the interactions between the gross profitability and value premium (e.g., Novy-Marx 2013; Kogan and Papanikolaou 2013).

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λ

φ

Gross profitability

Book-to-market ratio

Return loadings on ( )

Return loadings on

Gross profitability

return spreads

Value return spreads

Figure 1 Illustration of the relationships and interactions implied by the model This figure illustrates the relationships and interactions among the key variables in the model. The solid link between λi and ϕi represents the only exogenously calibrated relationship in our quantitative analysis. The heavy-colored dashed links and the light-colored dotted links with arrows indicate the significant and insignificant structural causal relationships among the key endogenous variables in the model, respectively. The dash-dotted links represent the correlations between the variables that endogenously arise from the model-implied structural causal relationships.

and jointly determined by the two primitive industry characteristics λi and ϕi . Second, our model shows how industry-level stock return exposures to η(st ) and gt are endogenously and jointly determined by the two primitive industry characteristics λi and ϕi . Third, based on the two results above, our model demonstrates the interdependence between gross profitability, book-to-market ratios, and industry-level stock return exposures toη(st ) andgt across industries, thereby rationalizing the interactions between the gross profitability and value premium across industries.

To help visualize the above discussion, Figure 1 illustrates the key variables and their relationships implied by the model and verified by the data. Each of the dashed, dotted, or dash-dotted links represents a relation endogenously implied by the model, and the solid link between the two primitive industry characteristics λi and ϕi represents the only exogenously calibrated relation. The arrow links represent the model-implied structural causal relationships between the key variables: the heavy-colored dashed links and the light-colored dotted links with arrows represent the significant and insignificant endogenous structural causal relationships in the model, respectively. The heavy-colored dash-dotted links represent the correlations between the variables that endogenously arise from the model-implied structural causal relationships.

As illustrated in Figure 1, only the correlation and marginal distributional moments of (λi,ϕi) across industries, which are represented by the solid link connecting λi and ϕi , are calibrated to match their empirical counterparts, and all other relationships and interactions, which are represented by the dashed, dotted, and dash-dotted links, are endogenously generated by the economic mechanisms of our model. Put differently, in our quantitative analysis, the

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correlation and marginal distributional moments of (λi,ϕi) across industries are the only targeted moments; all other cross-industry relationships, such as equilibrium asset pricing relationships, are untargeted moments used to test the economic mechanisms.

Overall, the above discussion related to Figure 1 suggests that jointly explaining the gross profitability and value premium across industries, especially their interactions, is a difficult theoretical and quantitative task. We tackle this challenge by showing that a “nearly separating property” holds in the model and verify this property in the data. Specifically, we show that (a) cross-industry dispersions in gross profitability and stock return exposures to the discount rate η(st ), as well as their correlation, are mainly determined by the cross-industry dispersion of λi , but not of ϕi , and (b) cross-industry dispersions in book-to-market ratios and stock return exposures to expected growth rate gt , as well as their correlation, are mainly determined by the cross- industry dispersion of ϕi , but not of λi . This “nearly separating property” is not obvious ex ante. One of our main contributions is to develop a quantitative model that confirms this property and guides our empirical tests of the key theoretical results in the data. Owing to the “nearly separating property,” the interdependence between gross profitability and the book-to-market ratio is endogenously determined by the calibrated correlation between the two primitive industry characteristics (the market leadership turnover rate λi and the cash flow loading on expected growth ϕi), and so is the interdependence between the gross profitability and value premium.

The rest of this subsection is organized as follows. In Section 2.3.2, we first show and discuss the structural causal relationship between the primitive industry characteristic λi and an industry’s endogenous return exposure to aggregate shocks (i.e., the heavy-colored dashed and the light-colored dotted links with arrows that represent a strong connection between “λi” and “Return loadings on η(st )” and a weak connection between “λi” and “Return loadings on gt ,” respectively, in Figure 1). In Section 2.3.3, we show and discuss the structural causal relationship between the primitive industry characteristic ϕi and an industry’s endogenous return exposure to aggregate shocks (i.e., the heavy-colored dashed and the light-colored dotted links with arrows that indicate a strong connection between “ϕi” and “Return loadings on gt” and a weak connection between “ϕi” and “Return loadings on η(st ),” respectively, in Figure 1). In Section 2.3.4, we show and explain the structural causal relationship between the primitive industry characteristicλi and the endogenous profitability/valuation ratios (i.e., the heavy-colored dashed and the light- colored dotted links with arrows that indicate a strong connection between “λi” and “Gross profitability” and a weak connection between “λi” and “Book-to- market ratio,” respectively, in Figure 1). In Section 2.3.5, we show and explain the structural causal relationship between the primitive industry characteristic ϕi and the endogenous profitability/valuation ratios (i.e., the heavy-colored dashed and the light-colored dotted links with arrows that represent a strong

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connection between “ϕi” and “Book-to-market ratio” and a weak connection between “ϕi” and “Gross profitability,” respectively, in Figure 1). Taken together, these relationships show the “nearly separating property” of the model. Last, in Section 2.3.6, we show that the endogenous correlation between gross profitability and the book-to-market ratio and the endogenous interaction between the gross profitability and value premium (i.e., the heavy-colored dash- dotted links in Figure 1) are consistent with the data when the correlation between λi and ϕi is calibrated on the data (i.e., the solid link that connects “λi” and “ϕi” in Figure 1). This is because the “nearly separating property” allows us to associate the variation of each sorting variable (gross profitability and the book-to-market ratio) with a distinct cross-section of industry primitive characteristics (λi and ϕi).

To ensure that the experiments and illustrations are quantitatively meaningful, we adopt the calibrated parameter values from Table 1 in Section 2.2 when presenting the results of the numerical experiments below.

2.3.2 Risk exposures of returns in the cross-section of λi . Panels A and B of Figure 2 illustrate the role of the market leadership turnover rate λi in determining an industry’s stock return exposures to η(st ) and gt . In the numerical experiment, we consider two industries with the same ϕ but different λi , with a low value λL or a high value λH . We set ϕ=3, the median of the calibrated distribution of ϕi in the cross-section of industries, and λH =0.07, the median of the calibrated distribution of λi in the cross-section of industries, based on the model calibration in Section 2.2. We set λL=0.03 to capture an industry with a low market leadership turnover rate (i.e., an industry with highly persistent market leadership).

Panels A and B show how industries’ stock return betas to fluctuations in η(st ) and gt , respectively, depend on their market leadership turnover ratesλi . In particular, panel A shows that the industry with a lower leadership turnover rate λL (the solid line) has a more negative stock return beta to fluctuations in η(st ) than the industry with a higher leadership turnover rate λH (the dashed line). Moreover, panel B shows that the industry with a lower leadership turnover rate λL (the solid line) has a more positive stock return beta to fluctuations in gt than the industry with a higher leadership turnover rate λH (the dashed line).

In our model, market leaders’ capacity to collude decreases as the discount rate η(st ) increases or expected growth gt decreases. Intuitively, in the presence of higher discount rates or lower expected growth, market leaders care less about future cooperation and the threat of punishment for deviation; consequently, their capacity to collude is affected by the discount rate η(st ) and expected growth gt , and thus their profit margins fluctuate endogenously with η(st ) and gt , which further amplifies the direct effects of η(st ) and gt on firms’ stock returns. In particular, when the discount rate η(st ) increases or expected growth gt decreases, firm value will decline, not only because of their direct effects,

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Figure 2 Stock return betas in two cross-sections This figure is plotted using the calibrated parameter values in Table 1. Panels A and B plot two industries with different leadership turnover rates (λL and λH ), but the same cash flow loading on expected growth (ϕ). We set λH =0.07 and ϕ=3, corresponding to the median values of λ and ϕ across all industries, respectively. We set λL =0.03. Panels C and D plot two industries with different cash flow loadings on expected growth (ϕL and ϕH ) but the same market leadership turnover rate λ. We set ϕH =3 and λ=0.07, corresponding to the median values of ϕ and λ across all industries, respectively. We set ϕL =0. Panels A and C plot the industries’ stock return betas to fluctuations in η(st ), and we consider a shock that increases η(st ) from its median value (ηL≡η(sL)) to the value of the 95th percentile of the distribution (ηH ≡η(sH )). Panels B and D plot the industries’ stock return betas to fluctuations in gt , and we consider a shock that increases gt from its median value (gL) to the value of the 95th percentile of the distribution (gH ). Industry-level stock return betas are value-weighted firm-level betas. In panels A and C, industries’ stock return betas to fluctuations

in η(st ) are given by β V,η(s) i,t

= ∑2 j=1wij,t β

V,η(s) ij,t

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=Vij,t (sH ,g)/Vij,t (sL,g)−1 and wij,t =

Vij,t (sL,g)/ ∑2 j ′=1

Vij ′,t (sL,g). In panels B and D, industries’ stock return betas to fluctuations in gt are given by

β V,g i,t

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V,g ij,t

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=Vij,t (s,gH )/Vij,t (s,gL)−1 and wij,t =Vij,t (s,gL)/ ∑2 j ′=1

Vij ′,t (s,gL).

which can be observed from the Gordon valuation formula, but also because of declining profit margins amid intensified industry competition caused by market leaders’ lower capacity to collude.

In the cross-section, the extent to which firms’ capacity to collude is driven by fluctuations in η(st ) and gt may vary substantially from one industry to another. Specifically, in industries with higher leadership turnover rates λi , market

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leaders are effectively more impatient, and thus, the endogenous competition mechanism is less pronounced, and market leaders’ capacity to collude is less responsive to variations in η(st ) and gt . Therefore, in industries with higher λi , market leaders’ profit margins are less exposed to fluctuations in the discount rate η(st ) and expected growth gt , so their stock return betas to variations in η(st ) and gt are smaller in magnitude. By showing how competition intensity and profit margins are endogenously driven by fluctuations in expected growth gt , the above theoretical and quantitative results extend those of Dou, Ji, and Wu (2021), who focus on the discount rate shock that drives fluctuations in η(st ).

More importantly, the difference between the solid and dashed lines is very small in panel B but significantly larger in panel A. This stark contrast indicates that as λi changes across industries, an industry’s the stock return beta to fluctuations in the discount rate η(st ) varies considerably, whereas its stock return beta to fluctuations in expected growth gt almost stays constant. In fact, the stark contrast between panels A and B highlights an important theoretical and quantitative result of this paper: the endogenous competition channel works for both the discount rate η(st ) and expected growth gt (i.e., both η(st ) and gt endogenously drive market leaders’ capacity to collude). However, the calibrated model shows that the impact of gt on stock returns through the endogenous competition channel remains almost unchanged as the market leadership turnover rateλi changes across industries, whereas the impact of η(st ) increases significantly from one industry to another as λi decreases. These theoretical and quantitative results further strengthen and extend the main result of Dou, Ji, and Wu (2021).

2.3.3 Risk exposures of returns in the cross-section of ϕi . Panels C and D of Figure 2 illustrate the role of the cash flow loading on expected growth ϕi in determining an industry’s stock return exposure to η(st ) and gt . In the numerical experiment, we consider two industries with the same λ but different ϕi , with a low value ϕL or a high value ϕH . We set λ=0.07, the median of the calibrated distribution of λi , and ϕH =3, the median of the calibrated distribution of ϕi in the cross-section of industries, based on the model calibration in Section 2.2. We set ϕL=0 to capture an industry with a low cash flow loading on expected growth.

In the cross-section, the extent to which firms’ stock returns are affected by fluctuations in η(st ) and gt vary across industries with different cash flow loadings on expected growth ϕi . Panel C shows that the industry with a higher cash flow loading on expected growth ϕH (the dashed line) has a less negative stock return beta to fluctuations in η(st ) than the industry with a lower cash flow loading on expected growth ϕL (the solid line). Indeed, a higher ϕi results in a higher risk premium, which curbs firms’ incentive to collude on average. Therefore, the endogenous competition mechanism is indirectly weakened by a higherϕi because market leaders’ capacity to collude and profit margins are less

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responsive to fluctuations in η(st ). Moreover, panel D shows that the industry with a higher cash flow loading on expected growth ϕH (the dashed line) has a more positive stock return beta to fluctuations in gt than the industry with a lower cash flow loading on expected growth ϕL (the solid line). The reason is that a firm’s cash flow is equal to its profitability per unit of effective customer capital �ij,t multiplied by its effective customer capital M̃ij,t (see Equation (24)). In the industry with a higher cash flow loading on expected growth ϕi , firms’ customer baseMij,t responds more strongly to variations in gt , which in turn results in a more positive loading of �ij,t to fluctuations in gt through the endogenous competition channel and a more positive loading of M̃ij,t ’s growth to fluctuations in gt . Thus, a higher ϕi leads to a higher stock return beta to fluctuations in gt . By showing how industries’ stock return betas to fluctuations in η(st ) and gt are endogenously affected by the primitive industry characteristic ϕi , the above theoretical and quantitative results are beyond the scope of Dou, Ji, and Wu (2021), who focus on one primitive industry characteristic, the market leadership turnover rate λi .

More importantly, the difference between the solid and dashed lines is very small in panel C but significantly larger in panel D. This stark contrast indicates that as ϕi changes across industries, an industry’s stock return beta to fluctuations in expected growth gt varies dramatically, whereas its stock return beta to fluctuations in the discount rate η(st ) almost stays constant. Indeed, ϕi directly determines an industry’s cash flow loading on expected growth gt ; thus, it strongly affects the dependence of stock returns on expected growth gt . By contrast, ϕi only indirectly affects how profit margins depend on the discount rate η(st ) through the endogenous competition mechanism; thus, it only weakly affects the dependence of stock returns on the discount rate η(st ). These theoretical and quantitative results are further nontrivial extensions not covered by Dou, Ji, and Wu (2021).

2.3.4 Profit margins and book-to-market ratios in the cross section of λi . In panel A of Figure 3, the solid line shows that industries with a higher λi are associated with significantly lower profit margins. As discussed above for panels A and B of Figure 2, the higher turnover rate λi makes the market leaders in these industries more impatient, and thus they are less incentivized to collude to set high profit margins. By contrast, the dashed line in panel A of Figure 3 shows that an industry’s book-to-market ratio increases with λi , but only weakly. This weak relationship across industries is mainly due to the weak association between an industry’s exposure to gt and λi under our baseline calibration (see panel B of Figure 2). Intuitively, the difference in industries’ book-to-market ratios is mainly caused by industries’ heterogeneous exposures to fluctuations in expected growth gt (see the detailed discussion in Section 2.3.5), which is only weakly associated with λi across industries. Although the dispersions of industry-level profit margins and exposures to the discount rate η(st ) are large in the cross-section of λi (see panel A of Figure 2),

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Figure 3 Profit margins and book-to-market ratios across industries This figure is plotted using the calibrated parameter values in Table 1. In panel A, we set the industries’ loadings on gt at ϕi =3, the median value of the calibrated distribution of ϕ in the cross-section of industries. In panel B, we set the industries’ leadership turnover rates at λi =0.07, the median value of the calibrated distribution of λ in the cross-section of industries. An industry’s market and book values correspond to the sum of the market and book values of its two firms, respectively. The market value of firm ij is given by Equation (27). The book value of firm ij , defined as the value of assets in place, is calculated using Equation (30).

they do not lead to large variations in book-to-market ratios. The reason is that profit margins and discount rates affect both the value of assets in place and the value of growth options in roughly the same proportion, having little effect on their ratios.

2.3.5 Profit margins and book-to-market ratios in the cross-section of ϕi . In panel B of Figure 3, the solid line shows that an industry’s profit margin

decreases with its cash flow loading on expected growth ϕi , but only slightly. Intuitively, the stock returns of industries with a higher ϕi are more exposed to expected growth gt , so these industries compensate their shareholders with a higher risk premium. This higher risk premium (i.e., discount rate) effectively makes market leaders more impatient, leading to lower collusive profit margins. However, quantitatively, the variation in profit margins across industries with different ϕi is much smaller than that across industries with different λi , mainly because λi directly affects the incentive to collude on profit margins, but ϕi only indirectly affects this incentive through the change in the risk premium. The first direct effect of λi dominates under the baseline calibration.

By contrast, the dashed line in panel B of Figure 3 shows that industries with a higher ϕi are associated with significantly higher book-to-market ratios. Intuitively, a higher ϕi directly increases the exposure of an industry’s growth to expected growth gt and thus makes growth options riskier, which reduces the value of these growth options because the future profits created by these options are discounted more aggressively; however, the value of assets in place is not affected by ϕi . Consequently, the book-to-market ratio varies significantly in the cross-section of ϕi , which is in sharp contrast to the small change in profit margins in the same cross-section of industries. Similar to Lettau and Wachter

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(2007, 2011), we refer to the effect of ϕi on the book-to-market ratio as the cash flow duration channel, a discussion of which is provided in the appendix.

2.3.6 Interactions between the two cross-sections of λi and ϕi . As a key point of our model, we now show that the cross-section of industries sorted on gross profitability, as well as the endogenous competition mechanism, interacts with the cross-section of industries sorted on book-to-market ratios in economically interesting ways.

Specifically, to rationalize the intriguing correlation between the value return spread and the gross profitability return spread across industries, it is important that the two industry-level sorting variables (i.e., profitability and the book-to-market ratio) have differential sensitivity to changes in the two primitive industry characteristics λi and ϕi . Such differential sensitivity arises endogenously in our model, consistent with the discussion of the “nearly separating property.” This allows us to associate the variation of each sorting variable with a distinct cross-section of industry characteristics. Specifically, sorting industries on gross profitability mainly captures the cross-section of industries with different λi because profit margins are substantially more sensitive to λi than to ϕi across industries (see Figure 3). Industries with lower leadership turnover rates λi are associated with higher profit margins (see panel A of Figure 3) and are more negatively exposed to fluctuations in the discount rate η(st ) (see panel A of Figure 2), generating the gross profitability premium. By contrast, sorting industries on the book-to-market ratio mainly captures the cross-section of industries with different ϕi because book-to-market ratios are substantially more sensitive to ϕi than to λi across industries (see Figure 3). Industries with a higher ϕi are associated with higher book-to-market ratios (see panel B of Figure 3) and are more positively exposed to fluctuations in expected growth gt (see panel D of Figure 2), generating the value premium.

Furthermore, given the single-sorted portfolios based on the book-to- market ratio, which mainly reflect the cross-section of the primitive industry characteristic ϕi , a further sort based on gross profitability within each book- to-market portfolio (i.e., double-sort) almost purely reflects the variation in the primitive characteristic λi because the first sort essentially controls for the other industry characteristic ϕi . Given that the two primitive industry characteristics λi and ϕi are positively correlated as in the data (see panel B of Table 2), our model implies that the double-sort naturally generates a more pronounced gross profitability premium because λi and ϕi have opposite effects on expected equity returns. The same reason explains why the value premium also becomes more pronounced after controlling for gross profitability.

Crucially, the “nearly separating property” ensures that the intriguing correlation between gross profitability and the book-to-market ratio, as well as the interaction between the gross profitability and value premium, ultimately boil down to the correlation between the two cross-sections of the primitive industry characteristics λi and ϕi , as illustrated in Figure 1. The following

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hypothetical scenario provides a counterfactual example to illustrate the importance of the correlation between λi and ϕi . Suppose that λi and ϕi are perfectly negatively correlated across industries. Then sorting on either gross profitability or book-to-market ratio would capture the same cross-section of industries. Doing so would still allow the model to generate both the gross profitability premium and the value premium because industries with a higher λi (or equivalently, a lower ϕi) are associated with lower expected returns, lower gross profitability, and lower book-to-market ratios. However, the model- implied interactions between different cross-sections are totally off the mark in matching the data. Specifically, the gross profitability premium and the value premium would become less pronounced, rather than more pronounced as in the data, after controlling for the book-to-market ratio and gross profitability, respectively. The main reason is that the cross-section of industries captured by the control variable is the same as that captured by the sorting variable. In other words, double-sort analysis essentially sorts twice on the same variable, and the difference among the portfolios created by double sorting is substantially smaller than that among the portfolios created by the first sort.

The above discussion emphasizes the importance of the correlation between λi and ϕi to ensure reasonable quantitative performance of the model. As a key result, we show in Section 2.4 that by calibrating the parameter ϑ to match the correlation of 0.15 between the market leadership turnover rate λi and the cash flow loading on expected growth ϕi across industries in the data (see panel B of Table 2), our model can quantitatively reproduce the double-sort results observed in the data (see Table 3). To further highlight the key role of the correlation between λi and ϕi , we present a systematic discussion of how different correlations between λi and ϕi affect the interactions between the industry-level gross profitability and value premium in Table 8 of Section 2.5.

2.4 Asset pricing implications 2.4.1 Results based on the value-weighted average of all firms in the industry. Using the calibrated model, we now investigate whether our theory can quantitatively rationalize the industry-level gross profitability and value premium, as well as their interactions. We first compute the industry-level stock returns, gross profitability, and book-to-market ratios based on the value- weighted average of all firms in the industry. We then sort all SIC4 industries into quintiles in June of each year t based on their gross profitability or book- to-market ratios in year t−1. Once the portfolios are formed, their monthly industry returns are tracked from July of year t to June of year t +1. We compute the portfolio returns by weighting the industry returns in each portfolio with equal weight (EW) and by weighting the industry returns based on the 1 month lagged industry-level market capitalization (VW). We emphasize that the industry-level stock returns, gross profitability, and book-to-market ratios are always the value-weighted average of all firms in the industry, regardless of whether the portfolios of the industries in the sorting analyses are constructed

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based on the equal or value weighting scheme. These value-weighted industry- level stock returns and financial ratios (based on all firms in the industry) enable us to capture the dynamics of market leaders of different industries while maintaining the stability of the industry-level measures in our empirical analyses. Doing so is crucial for the purpose of connecting the theoretical predictions and empirical tests, because the dynamic oligopoly model is mainly concerned with the market leaders in the industry. We conduct a similar sorting analysis for the equal- and value-weighted portfolios of the industries using the simulated counterparts from the calibrated model.

Panel A of Table 3 presents the average excess returns of industry portfolios sorted on gross profitability. In both the data and the model, the portfolio consisting of industries with the highest gross profitability (i.e., portfolio Q5) exhibits significantly higher average excess returns than that comprising industries with the lowest gross profitability (i.e., portfolio Q1). The equal- and value-weighted gross profitability return spreads (i.e., Q5 − Q1) are 2.81% and 3.50% in the data, respectively, and the model-implied equal- and value- weighted return spreads are 3.14% and 3.17%, respectively. Furthermore, the capital asset pricing model (CAPM) alphas of these quintile portfolios in the data are economically and statistically significant and are reported in Table OA.5 of Internet Appendix 2.2.

Panel B of Table 3 presents a double-sort analysis by sorting first on the book- to-market ratio and then on gross profitability. In both the data and the model, the magnitude of the gross profitability return spread increases after controlling for the book-to-market ratio. The equal- and value-weighted gross profitability return spreads increase to 3.79% and 4.44% in the data, respectively, and to 4.63% and 4.66% in the model, respectively.

Panel C of Table 3 presents the average excess returns of industry portfolios sorted on the book-to-market ratio. In both the data and the model, the industry portfolio with the highest book-to-market ratio (i.e., portfolio Q5) has significantly higher average excess returns than that with the lowest book- to-market ratio (i.e., portfolio Q1). The equal- and value-weighted value return spreads (i.e., Q5 − Q1) are 4.79% and 3.55% in the data, respectively. Correspondingly, the model-implied equal- and value-weighted value return spreads (i.e., Q5 − Q1) are 4.48% and 4.37%, respectively. Moreover, the double-sort analysis in panel D indicates that, after controlling for gross profitability, the equal- and value-weighted value return spreads increase to 5.66% and 5.03% in the data, respectively, and to 5.94% and 5.94% in the model, respectively.

Our paper focuses on the cross-industry gross profitability and value premium, especially their intriguing interactions. Both premia are also prevalent within industries.21 Tables 4 and 5 present the cross-industry, within-industry,

21 In fact, some empirical studies have shown that the gross profitability and value premium cannot be fully explained by industry effects (e.g., Lewellen 1999; Cohen, Polk, and Vuolteenaho 2003; Novy-Marx 2013). As

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Table 4 Cross-industry, within-industry, and firm-level gross profitability premium

Q1 (low) Q2 Q3 Q4 Q5 (high) Q5 − Q1

A. Cross-industry gross profitability return spreads EW 7.23∗∗∗ 7.35∗∗∗ 9.26∗∗∗ 9.16∗∗∗ 10.03∗∗∗ 2.81∗∗

[2.73] [3.01] [3.55] [3.75] [4.24] [2.04] VW 6.47∗∗∗ 9.02∗∗∗ 8.77∗∗∗ 9.32∗∗∗ 9.97∗∗∗ 3.50∗∗

[2.99] [3.93] [3.31] [3.99] [4.55] [2.14]

B. Within-industry gross profitability return spreads VW 5.08∗∗ 7.44∗∗∗ 8.23∗∗∗ 8.67∗∗∗ 9.15∗∗∗ 4.08∗∗∗

[2.04] [3.18] [3.67] [4.06] [4.76] [2.94]

C. Firm-level gross profitability return spreads VW 5.66∗∗∗ 7.21∗∗∗ 7.61∗∗∗ 7.39∗∗∗ 9.89∗∗∗ 4.23∗∗∗

[2.59] [3.56] [3.74] [3.59] [5.14] [3.11]

Panel A is based on panel A of Table 3. In panel B, we sort all individual firms within each industry (with at least five firms) into quintiles based on their gross profitability lagged by 1 year. In panel C, we sort all firms into quintiles based on their gross profitability lagged by 1 year. The sample period is from July 1951 to June 2018. We exclude financial firms and utility firms from the analysis. Newey-West standard errors are estimated with one lag. We annualize the average excess returns and alphas by multiplying them by 12. We include t-statistics in brackets. *p<.1; **p<.05; ***p<.01.

Table 5 Cross-industry, within-industry, and firm-level value premium

Q1 (low) Q2 Q3 Q4 Q5 (high) Q5-Q1

A. Cross-industry value return spreads EW 5.28∗∗ 7.55∗∗∗ 9.04∗∗∗ 9.02∗∗∗ 10.07∗∗∗ 4.79∗∗∗

[2.01] [3.02] [3.70] [3.67] [3.94] [3.01] VW 7.52∗∗∗ 8.13∗∗∗ 8.85∗∗∗ 9.17∗∗∗ 11.07∗∗∗ 3.55∗

[3.13] [3.77] [4.25] [4.28] [4.97] [1.87]

B. Within-industry value return spreads VW 6.75∗∗∗ 8.07∗∗∗ 8.95∗∗∗ 8.48∗∗∗ 10.63∗∗∗ 3.88∗∗∗

[2.97] [3.80] [4.32] [4.11] [4.71] [2.69]

C. Firm-level value return spreads VW 6.51∗∗∗ 7.41∗∗∗ 8.46∗∗∗ 9.31∗∗∗ 11.67∗∗∗ 5.16∗∗∗

[3.04] [3.82] [4.46] [4.62] [5.20] [3.07]

Panel A is based on panel C of Table 3. In panel B, we sort all individual firms within each industry (with at least five firms) into quintiles based on their book-to-market ratios lagged by 1 year. In panel C, we sort all firms into quintiles based on their book-to-market ratios lagged by 1 year. The sample period is from July 1951 to June 2018. We exclude financial firms and utility firms from the analysis. Newey-West standard errors are estimated with one lag. We annualize the average excess returns and alphas by multiplying them by 12. We include t-statistics in brackets. *p<.1; **p<.05; ***p<.01.

and firm-level gross profitability and value spreads in average excess returns. The magnitude of the cross-industry premium is comparable to that of the within-industry premium, indicating that cross-industry and within-industry variations are equally important to account for the firm-level premium. The gross profitability spreads and value spreads in CAPM alphas are presented in Tables OA.6 and OA.7 of Internet Appendix 2.2, respectively, and the patterns

a complement, our cross-industry single- and double-sort empirical results show that the relationship between the premia at the industry level is similar to that at the firm level as documented in the literature.

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of the CAPM alphas are similar to those of the average excess returns in Tables 4 and 5.

2.4.2 Results based only on the value-weighted average of top firms in the industry. Strictly speaking, our model is mainly concerned with the dispersion of market leaders’ expected returns across industries; in other words, the asset pricing implications of the model mainly apply to large firms, rather than all firms, in different industries. In the asset pricing results presented thus far (Tables 3 to 5), we have focused on the industry returns and financial ratios that are constructed based on the value-weighted average of all firms in the industry. We use the industry-level stock returns, gross profitability, and book-to-market ratios based on the value-weighted average of all firms in the industry in our main asset pricing results, rather than a few top firms in the industry, for four main reasons. First, similar to using a few top firms, using the value-weighted average of all firms in the industry enables us to connect the theoretical predictions and empirical tests because the value- weighted industry-level measures mainly reflect the dynamics of market leaders (or large firms) of different industries. Second, using the value-weighted average of all firms in the industry enables us to better maintain the stability of the industry-level measures, ensure better representation of the industry dynamics, and provide results that are more robust against misspecification of the number of market leaders, than only using a few top firms.22 Third, by using the value-weighted average of all firms in the industry, we exploit the common pattern that the profit margins and stock returns of small followers usually comove positively with those of market leaders in the same industry, driven by the fluctuation in competition intensity of the industry. Fourth, by using the value-weighted average of all firms in the industry, we relate the industry-level gross profitability and value premium more closely to the firm- level patterns documented in the literature, because the firm-level patterns are computed based on all firms, not only large firms.

However, a potential concern for our main asset pricing analysis is that small followers may significantly affect the empirical results. To address this concern and further strengthen the empirical results, we perform additional robustness tests focusing on a few top firms in each industry, the results of which are summarized in Table 6. We define the top four firms, ranked by sales in each industry, as the market leaders, and use this definition to compute the industry- level stock returns, financial ratios, leadership turnover rates, and cash flow loadings on expected growth in a consistent manner. Although we consider a duopoly in the model for tractability, it makes sense for us to consider more

22 For example, using the top two firms likely provides very noisy representation of the dynamics of market leaders in the data, not to mention the dynamics of the industry. The number of market leaders is latent and varies across industries in the real world.

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Table 6 Industry-level gross profitability and value premium, as well as industry characteristics, computed based on the top four firms in the industry

Q1 (low) Q2 Q3 Q4 Q5 (high) Q5-Q1

A. Gross profitability return spreads (sorted on gross profitability) EW 7.28∗∗∗ 7.45∗∗∗ 9.81∗∗∗ 9.85∗∗∗ 10.05∗∗∗ 2.77∗∗

[2.75] [3.05] [3.81] [4.08] [4.24] [2.01] VW 7.30∗∗∗ 7.02∗∗∗ 9.68∗∗∗ 8.06∗∗∗ 10.66∗∗∗ 3.36∗∗

[3.55] [3.26] [4.03] [3.47] [5.00] [2.10]

B. Gross profitability return spreads after controlling for the book-to-market ratio EW 6.35∗∗ 7.51∗∗∗ 9.11∗∗∗ 9.20∗∗∗ 10.37∗∗∗ 4.02∗∗∗

[2.45] [2.71] [3.73] [3.35] [4.37] [3.37] VW 6.10∗∗∗ 6.56∗∗∗ 9.69∗∗∗ 7.69∗∗∗ 10.34∗∗∗ 4.24∗∗∗

[2.93] [2.71] [4.46] [3.04] [4.69] [2.85]

C. Value return spreads (sorted on the book-to-market ratio) EW 5.52∗∗ 7.95∗∗∗ 9.37∗∗∗ 9.55∗∗∗ 9.94∗∗∗ 4.43∗∗∗

[2.15] [3.23] [3.87] [3.86] [3.88] [2.86] VW 7.49∗∗∗ 9.28∗∗∗ 9.20∗∗∗ 9.57∗∗∗ 9.67∗∗∗ 2.19

[3.28] [4.47] [4.49] [4.68] [4.18] [1.49]

D. Value return spreads after controlling for gross profitability EW 5.98∗∗ 7.35∗∗∗ 8.53∗∗∗ 9.28∗∗∗ 11.35∗∗∗ 5.37∗∗∗

[2.34] [2.69] [3.54] [3.37] [4.52] [3.82] VW 7.45∗∗∗ 7.76∗∗∗ 8.14∗∗∗ 8.16∗∗∗ 10.80∗∗∗ 3.35∗

[3.40] [3.28] [3.92] [3.62] [4.73] [1.92]

E. Industry characteristics computed based on top four firms Portfolios sorted on D1 (low) Q1 Q2 Q3 Q4 Q5 D10 (high)

gross profitability λ̂p based on turnovers 0.116 0.120 0.077 0.063 0.059 0.051 0.029

of the top four firms Portfolios sorted on the D1 (low) Q1 Q2 Q3 Q4 Q5 D10 (high)

book-to-market ratio ϕ̂p based on ROEs 0.58 2.23 2.52 3.49 5.13 5.50 7.26

of the top four firms

We use the top four firms in each industry to compute the industry-level stock returns, gross profitability, book- to-market ratios, leadership turnover rates, and cash flow exposures to expected growth shocks. Similar to Table 3, panels A to D present the gross profitability and value premium in both the single- and double-sort analyses. *p<.1; **p<.05; ***p<.01. Different from Table 3, we compute the industry-level measures using the value-weighted average of the top four firms, not all firms. We then sort industries into quintiles based on the industry-level gross profitability and book-to-market ratios computed based on the top four firms. We present the equal-weighted (EW) and value-weighted (VW) portfolio returns in panels A to D. Panel E presents λ̂p across the groups of industries sorted on gross profitability and ϕ̂p across the groups of industries sorted on the book-to-market ratio. To estimate λ̂p , we define the turnover indicator based on the market leaders in year t and year t +1. The turnover indicator from year t to year t +1 is equal to one if (a) the largest firm ranked by sales in the industry in year t +1 is none of the four largest firms in year t , or (b) if any of the second to fourth largest firm ranked by sales in the industry in year t +1 is none of the four largest firms in year t and it is large enough so that its sales are greater than 60% of the sales of the largest firm in year t +1. We impose this size requirement to ensure that the turnover of the second to fourth largest firm represents a turnover of market leaders. Different from Table 3, we reduce the size requirement from 80% of the sales of the largest firm to 60% because we define the top four firms rather than the top two firms as market leaders here.

market leaders in the empirical analyses because in reality, the number of market leaders can greatly exceed two.23

Panels A to D of Table 6 present the industry-level gross profitability and value premium computed based on the top four firms of each industry in both

23 In Table OA.8 of Internet Appendix 2.3, we also consider the top ten firms of each industry as market leaders and find similar results.

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the single- and double-sort analyses. As in Table 3, the magnitude of the gross profitability (value) return spreads increases after controlling for the book-to- market ratio (gross profitability). The magnitude of the return spreads is also comparable to that in Table 3. The only exception is the VW value return spreads (see the last column of panels C and D in Table 6), which are smaller than those in panels C and D of Table 3. This exception is perhaps not surprising because the resultant returns of the industry portfolios are likely to be dominated by the largest few firms in the economy when we focus on the top four firms of a few largest industries.24 Therefore, the results based on the EW portfolios of the industries should be more informative and reliable when we use the top four firms as market leaders.

Panel E of Table 6 presents the estimated market leadership turnover rate, λ̂p, across the groups of industries sorted on gross profitability and the estimated cash flow exposure to expected growth shocks, ϕ̂p, across the groups of industries sorted on the book-to-market ratio. The distributions of λ̂p and ϕ̂p are very similar to those shown in panel B of Table 2. Taken together, we find that the industry-level gross profitability and value premium, as well as industry characteristics, computed based on the top four firms of each industry exhibit similar patterns to those computed based on all firms, which provides additional empirical support for our model’s predictions.

2.5 Quantitative inspection of the central mechanisms In this section, we conduct counterfactual analyses based on the calibrated model.

2.5.1 Effects of the key model ingredients. We first conduct several counterfactual analyses to shed light on the central economic mechanisms and evaluate the quantitative effects of the key ingredients of our model by turning them off one at a time. Column 3 of Table 7 presents the implications of the model for the noncollusive equilibrium. The average profit margin across industries is lower than that of the baseline collusive equilibrium (column 2) because of the lack of collusion. The growth rate of average net profits is less volatile because profit margins do not vary with η(st ) or gt in the noncollusive equilibrium, as shown by Dou, Ji, and Wu (2021). The equity premium and the volatility of market excess returns are, respectively, about 1.88% (=8.79%−6.91%) and 3.92% (=18.37%−14.45%) lower in the noncollusive equilibrium than in the baseline collusive equilibrium, because there is no amplification effect from additional endogenous competition risk.

24 Intuitively, the VW portfolios of the industries with each industry represented by its top four firms are effectively portfolios of a few largest firms in the economy because of the power law of firm size (e.g., Gabaix 2011). The value premium sharply declines to nearly zero when shifting the focus onto the subsample of the largest few firms (e.g., Fama and French 1993), whereas the gross profitability premium remains at a similar positive level when shifting the focus onto the subsample of the largest few firms (e.g., Novy-Marx 2013).

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Table 7 Inspection of the model mechanisms

(1) (2) (3) (4) (5) (6) (7) Data Model No collusion gt ≡g st ≡s ϕi ≡ϕ λi ≡λ

Average gross profit margin (%) 31.39 27.71 11.25 28.11 27.00 27.66 23.67 [29.98, 33.00]

Volatility of the growth rate 16.22 12.01 5.15 8.55 10.53 12.00 9.60 of net profits (%) [11.11, 19.88] Equity premium 6.68 8.79 6.91 6.71 3.69 9.12 8.33 (E(r−rf ),%) [2.34, 10.88] Volatility of market excess returns 16.89 18.37 14.45 11.65 10.70 19.49 17.17 (σ (r−rf ),%) [13.21, 19.39] Sharpe ratio 0.40 0.48 0.48 0.58 0.34 0.47 0.48 (E(r−rf )/σ (r−rf )) [0.13, 0.77] Gross profitability premium (sorted on 3.50 3.17 −0.01 2.99 −0.65 4.20 −0.96 gross profitability E(RQ5 −RQ1), %) [1.22, 6.06] Gross profitability premium controlling 4.44 4.66 0.00 1.43 0.67 0.45 0.08 for the book-to-market ratio (%) [1.85, 6.56] Value premium (sorted on the 3.55 4.37 5.21 2.61 6.77 −4.23 5.93 book-to-market ratio E(RQ5 −RQ1), %) [1.61, 6.33] Value premium controlling 5.03 5.94 5.09 0.00 6.77 −0.92 5.78 for gross profitability (%) [3.27, 6.70] Correlation between gross profitability -0.34 −0.32 0.00 0.76 −0.22 −0.99 −0.03 and the book-to-market ratio [-0.53, -0.13]

The sample period is from 1950 to 2018 in the data. R and r are annualized simple returns and log returns, respectively. When constructing the model moments, we simulate a sample of 1,000 industries for 150 years with an 80-year burn-in period. We then compute the model-implied moments as we do for the data. For each moment, the table reports the average value of 2,000 simulations. Numbers in the brackets represent the [2.5%,97.5%] confidence interval.

The cross-industry gross profitability premium becomes negligible when the endogenous competition channel is shut down because industry-level profit margins are no longer correlated with λi or industry-level risk exposure in the noncollusive equilibrium. The cross-industry value premium is 5.21% in the noncollusive equilibrium, slightly higher than the premium of 4.37% in the collusive equilibrium. The main reason is that the market leadership turnover rate λi does not affect industry-level risk exposure in the noncollusive equilibrium, so the positive correlation between ϕi and λi does not dampen the value premium. Moreover, the value premium remains roughly unchanged after controlling for gross profitability because gross profitability and the book- to-market ratio are not correlated in the noncollusive equilibrium (see the last row of Table 7).

In columns 4 and 5, we separately quantify the contributions of fluctuations in the discount rate η(st ) and expected growth gt to generate the gross profitability and value premium across industries. Specifically, to quantify the contribution of fluctuations in the discount rate, in column 4, we set expected growth gt at its long-run mean g, while keeping everything else as in the baseline calibration. Comparing columns 2 and 4, we find that fluctuations in the discount rate alone generate a gross profitability premium of 2.99%, whereas the model of baseline calibration with fluctuations in both the discount rate η(st ) and expected growth gt generates a gross profitability premium of 3.17%. This indicates that the gross profitability premium is mainly due to the time-varying discount rate. Although

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the value premium is 2.61% in column 4, it is caused by the strong positive correlation between gross profitability and the book-to-market ratio (0.76; see the last row of Table 7), but not by the dispersion of industry-level cash flow exposures to expected growth gt . However, the correlation is strongly negative in the data, which explains why, after controlling for gross profitability, the value premium in column 4 decreases to zero, whereas it increases in both the data and the baseline calibration (columns 1 and 2). Therefore, the results suggest that fluctuations in expected growth gt play a key role in jointly generating the value premium and the negative correlation between gross profitability and the book-to-market ratio in our model.

In column 5, we quantify the contribution of fluctuations in expected growth by setting the log surplus consumption ratio st at its long-run mean s, while keeping everything else as in the baseline calibration. The cross-industry gross profitability premium is −0.65% in the absence of fluctuations in the discount rate, which clearly contradicts the data. Intuitively, the gross profitability premium should be negative because of the positive correlation of 0.15 between λi and ϕi across industries. A lower λi leads to a higher profit margin and higher exposure to fluctuations in expected growth gt through the endogenous competition mechanism. In addition, a lower λi is associated with a lower ϕi , leading to a lower risk premium due to less exposure to fluctuations in expected growth gt . These two channels imply opposite relationships between gross profitability and risk premium across industries, with the latter channel quantitatively dominating under the calibration of column 5. Once we control for the book-to-market ratio (i.e., approximately control for ϕi), the gross profitability premium increases from −0.65% to 0.67%, which mainly reflects the effect of λi through the endogenous competition channel described above. The cross-industry value premium is 6.77% in column 5, higher than the premium of 4.37% in column 2, because in the absence of fluctuations in the discount rate, the dispersion of λi does not lead to a significant dispersion of risk premia (see panel B of Figure 2) or a significant dispersion of book-to-market ratios (see panel A of Figure 3). Therefore, the positive correlation between λi and ϕi , which is 0.15, does not dampen the value premium reflected mainly in the cross-section of ϕi alone (see panel D of Figure 2 and panel B of Figure 3).

In columns 6 and 7, we investigate the role played by the two primitive industry characteristics in the cross-section. In column 6, we assume that industries have the same loading, ϕi≡ϕ, on expected growth gt , but differ in their leadership turnover rates λi . Column 6 shows that the gross profitability premium remains significant at 4.20%, but the value premium is negative, with a value of −4.23%, which strongly contradicts the data. Indeed, when ϕi is the same across industries, sorting on the book-to-market ratio reflects the cross- section of λi but not that of ϕi . However, panel A of Figure 3 indicates that a higher λi is associated with a higher book-to-market ratio but lower gross profitability. Thus, the positive gross profitability premium is naturally coupled with a negative value premium. In column 7, we assume that industries have

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the same λi≡λ but different ϕi . Based on the above discussion, unsurprisingly, we find that the value premium remains significant at 5.93%, but the gross profitability premium disappears.

Overall, our counterfactual analyses in columns 4 to 7 clearly suggest that both the time-varying discount rate η(st ) and expected growth gt , as well as the dispersion of the two primitive industry characteristics λi and ϕi across industries, are necessary to simultaneously account for the value premium, the gross profitability premium, the cross-industry correlations between these two premia, and the cross-industry correlation between gross profitability and the book-to-market ratio. The absence of any one of these key model ingredients would prevent the model from matching the data.

2.5.2 Effects of the correlation between λi and ϕi . We now conduct additional counterfactual analyses to investigate the role of the correlation between the two primitive industry characteristics in explaining the key interaction patterns observed in the data. As discussed above, the theoretical and quantitative model property—“nearly separating property”—arises endogenously. This property ensures that the intriguing correlation between gross profitability and the book-to-market ratio, as well as the interactions between the gross profitability and value premium ultimately, boil down to the correlation between the two cross-sections of the primitive industry characteristics λi and ϕi . That is, accounting for the complex interactions between the gross profitability and value premium depends on the appropriate calibration of the correlation ϑ between λi and ϕi across industries. We now elucidate the role of the key structural parameter ϑ , which governs the correlation between these two primitive industry characteristics.

Column 2 of Table 8 tabulates the baseline calibration where we setϑ =0.159 to match the positive correlation of 0.15 between the industry-level leadership turnover rate λi and cash flow loading on expected growth ϕi . Using this value of ϑ leads to a negative correlation between gross profitability and the book- to-market ratio (−0.32) close to its empirical counterpart in the data.

In column 3, we set ϑ =−0.7, leading to a significantly negative correlation between λi and ϕi of −0.67. As a result, the endogenous correlation between gross profitability and the book-to-market ratio becomes 0.47. The cross- industry gross profitability premium is significantly higher in column 3 than in column 2. This is because a negative correlation betweenλi andϕi indicates that industries with higher gross profitability are associated with lower λi and higher ϕi , both of which contribute to a higher risk premium. For a related reason, the value premium is also significantly higher in column 3 than in column 2 because industries with higher book-to-market ratios are associated with a higher ϕi and a lower λi . More importantly, the value premium decreases sharply after controlling for gross profitability, and the gross profitability premium decreases significantly after controlling for the book-to-market ratio. Both phenomena

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Table 8 Inspection of the correlation between λi and ϕi

(1) (2) (3) (4) (5) Data Model (ϑ =0.159) ϑ =−0.7 ϑ =0 ϑ =0.7

Correlation between λi and ϕi 0.15 0.15 −0.67 0.00 0.67 [0.01,0.28]

Correlation between gross profitability −0.34 −0.32 0.47 −0.19 −0.75 and the book-to-market ratio [−0.53, −0.13] Gross profitability premium (sorted on 3.50 3.17 8.49 4.13 0.35 gross profitability, E(RQ5 −RQ1), %) [1.22, 6.06] Gross profitability premium controlling 4.44 4.66 4.81 4.92 3.18 for the book-to-market ratio (%) [1.85,6.56] Value premium (sorted on the 3.55 4.37 6.31 4.90 3.37 book-to-market ratio, E(RQ5 −RQ1), %) [1.61, 6.33] Value premium controlling 5.03 5.94 4.15 6.16 5.39 for gross profitability (%) [3.27, 6.70]

The sample period is from 1950 to 2018 in the data. All returns are expressed as annualized simple returns. When constructing the model moments, we simulate a sample of 1,000 industries for 150 years with an 80-year burn-in period. We then compute the model-implied moments as we do for the data. For each moment, the table reports the average value of 2,000 simulations. Numbers in the brackets represent the [2.5%,97.5%] confidence interval.

clearly contradict our observations in the data and the baseline calibration, as summarized in columns 1 and 2, respectively.

In column 4, we set ϑ =0, indicating that λi and ϕi are independent. In this case, the endogenous correlation of −0.19 between gross profitability and the book-to-market ratio remains negative, but the magnitude is much smaller than the correlation of −0.34 in the data (−0.19 in column 4 but −0.34 in column 1). It is important to understand why this endogenous correlation remains negative with a value of −0.19, even when ϑ =0. The main reason is that profit margins decrease monotonically in both λi and ϕi , and book- to-market ratios increase monotonically in both λi and ϕi (see Figure 3). Thus, within both the cross-sections of λi and ϕi , profit margins and book- to-market ratios are negatively correlated. This implies that even when λi and ϕi are not correlated, the correlation between profitability and the book- to-market ratio remains endogenously negative across industries. With this negative correlation, the model can replicate the pattern whereby the gross profitability and value premium become more pronounced after controlling for the book-to-market ratio and gross profitability, respectively. However, the difference in the risk premia with and without controls is clearly smaller than that in our baseline calibration in column 2.

In column 5, we set ϑ =0.7, which generates a strong negative correlation of −0.75 between gross profitability and the book-to-market ratio; this result is very far from its empirical counterpart and from the implication of the model with the baseline calibration (see columns 1 and 2). The difference in the risk premia with and without controls is much larger than in our baseline calibration (see column 2). Specifically, the gross profitability premium increases from 0.35% to 3.18% after controlling for the book-to-market ratio; and the value premium increases from 3.37% to 5.39% after controlling for gross profitability.

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2.6 Extended model with operating leverage Operating leverage can play a role in generating the gross profitability and value premium. The benchmark model in Section 1 abstracts from operating leverage to preserve tractability and highlight the central mechanism. Below we extend the benchmark model with fixed costs of production, and show that the main quantitative results remain robust when we calibrate the extended model to the data. See Internet Appendix 1 for more details and discussions.

We assume that a firm incurs a flow cost with intensity ωYij,t +fiM̃ij,t to produce an output flow with intensity Yij,t over [t,t +dt]. The term ωYij,t captures firm ij ’s variable cost of production as in our benchmark model, and the term fiM̃ij,t captures the additional fixed cost of production, where M̃ij,t is the effective customer capital, defined in (17). A similar approach to modeling fixed costs has been used in the literature (e.g., Carlson, Fisher, and Giammarino 2004). With fixed costs, the dividend flow intensity in (23) is modified as follows:

Dij,t =θij,t (1−θij,t )−1ωYij,t−fiM̃ij,t , with θij,t =(Pij,t−ω)/Pij,t . (38)

The industry-level gross profitability GPi,t in (29) is modified as follows:

GPi,t =θi,t

( ω

1−θi,t )1−ε

−fi. (39)

The fixed cost of production fi is not directly observable in the data. Thus, we calibrate the model to match the distribution of operating leverage �i,t ≡∑2

j=1fiM̃ij,t / ∑2

j=1�ij (θCij,t ,θ C

ij̄ ,t )M̃ij,t across industries and its pairwise

correlations with the industry characteristics λi and ϕi (see Tables OA.1 and OA.2 in Internet Appendix 1 for calibration details). We construct the measure of operating leverage based on the idea that operating leverage is mainly due to the fixed component in the labor costs of hiring ordinary employees for production (e.g., Favilukis, Lin, and Zhao 2020):

�i,t =αXLRi,t / ( EBITDAi,t +XLRi,t

) , (40)

where the variableEBITDAi,t is earnings before interest, taxes, depreciation, and amortization aggregated over all firms in industry i in year t , and XLRi,t is total staff expenses aggregated over all firms in industry i in year t . Of the total staff expenses, a fraction α accounts for the fixed cost of maintaining employees, and the rest accounts for the variable cost. We set α=0.35 to be consistent with the wage-resetting frequency adopted in the New Keynesian literature (e.g., Uhlig 2007; Galí 2008).

Importantly, the extended model implies a positive association between operating leverage and gross profitability, which is consistent with the ideas of Li, Kogan, and Zhang (2020), because in both the data and the model, the correlation between λi and �i is negative (equal to −0.14), and industries with a higher λi are associated with lower gross profitability. Intuitively, high

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fixed costs of production can create an entry barrier for new market leaders. Facing high fixed costs, small competitors are discouraged from expanding their businesses to enter the “club” of market leaders. The high fixed costs thus help market leaders retain their leadership positions by eliminating small competitors (e.g., Baumol and Willig 1981; Rasmusen 1988). As a result, the correlation between �i and λi should be negative in the cross-section of industries.

The extended model also implies a positive association between operating leverage and the book-to-market ratio, consistent with Carlson, Fisher, and Giammarino (2004), because in the data and the model, the correlation between ϕi and �i is mildly positive (equal to 0.04), and industries with higher cash flow loadings on expected growth tend to have higher book-to-market ratios.

Not only do the main asset pricing results—the gross profitability and value premium across industries and their interactions—remain unchanged after accounting for the fixed costs of production (and thus operating leverage), in fact, the results for these two cross-sectional premia are strengthened by the amplification effects of operating leverage (see Tables OA.3 and OA.4 in Internet Appendix 1 for details). Intuitively, the negative correlation between λi and �i makes the profit margin of industries with lower λi more sensitive to discount rate shocks, thereby further amplifying the gross profitability premium (see columns 2 and 4 of Table OA.4). Furthermore, the positive correlation between ϕi and �i increases the exposure of the cash flow of industries with higher ϕi to expected growth shocks, thereby further amplifying the value premium (see columns 2 and 5 of Table OA.4). More important, the interactions between the two cross-sections are not significantly affected by the inclusion of the fixed costs of production (see the quantitative results in Table OA.4 in Internet Appendix 1 for details). This is because the correlation between gross profitability and the book-to-market ratio across industries, as well as the interactions between the industry-level gross profitability and value premium, ultimately boil down to the correlation between λi and ϕi , as discussed in Section 2.3.6. To further support the model’s implications, in Internet Appendix 2.4.2, we provide evidence suggesting that operating leverage is unlikely to be the channel through which the interaction patterns of the gross profitability and value premium arise at the industry level.

3. Conclusion

This paper provides a quantitative explanation of the joint patterns of the gross profitability and value premium across industries. As widely acknowledged in the literature, jointly rationalizing the gross profitability and value premium, especially their interactions, is a difficult task because profitable industries share common characteristics with growth industries, despite their high expected returns. To this end, we develop a novel general equilibrium framework with heterogeneous concentrated industries, consumer inertia, and endogenous

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strategic competition. Heterogeneity across industries is introduced through cross-sectional differences in two primitive industry characteristics, the market leadership turnover rates and the cash flow loadings on expected growth. Industries with lower market leadership turnover rates are more profitable and responsive to fluctuations in the discount rate through the endogenous competition channel; an increase in the discount rate reduces the present value of future cooperation, causing firms to compete more fiercely for short-run profits by undercutting each other. Meanwhile, industries with higher cash flow loadings on expected growth endogenously have higher book-to-market ratios through the cash flow duration channel and are more responsive to fluctuations in expected growth. The “nearly separating property” of our model ensures that the cross-industry correlation between gross profitability and the book- to-market ratio, as well as the interaction between the gross profitability and value premium ultimately boil down to the correlation parameter of the two primitive industry characteristics. By calibrating the correlation parameter to match its empirical counterpart measured directly in the data, we show that the model has many implications, including the complex interactions between the two cross-sections of industries, which is quantitatively consistent with our observations in the data.

Appendix. Discussion of the Cash Flow Duration Channel

Columns 4 and 6 of Table 7 show that dispersion of the cash flow loading on expected growth is crucial for our model to generate the value premium, as well as the interactions between the value and gross profitability premium across industries. An extensive literature has attempted to rationalize the observed value premium. In our model, industries with higher book-to-market ratios tend to have shorter cash flow durations (i.e., their cash flows are weighted more toward the present), and they have higher expected returns as the equity term structure slopes down. This is referred to as the cash flow duration channel for the value premium (e.g., Campbell and Vuolteenaho 2004; Dechow, Sloan, and Soliman 2004; Lettau and Wachter 2007; Lettau and Wachter 2011; Santos and Veronesi 2010). Intuitively, the longer the duration, the longer it takes for shareholders to recover the cash from their investment. Specifically, cash flow duration depends on not only shareholders’ expected cash flows over a long time frame but also the risk-adjusted rate of return at which these cash flows are discounted. In fact, to stress the importance of the effect of the discount rate in determining cash flow durations, Da (2009), Santos and Veronesi (2010), Croce, Lettau, and Ludvigson (2014), and Li and Zhang (2016) point out the importance of not only a firm’s temporal cash flow patterns but also its cash flow’s covariance with consumption in explaining the observed cross-sectional patterns of stock returns, especially the value premium.25 Our model also relies on the effect of the discount rate on cash flow durations to generate the value premium. Industries with higher cash flow loadings on expected growth have riskier growth options and thus have higher book-to-market ratios. In addition, they have shorter cash flow duration despite being more sensitive to expected growth shocks, if the market price of risk of expected growth shocks (i.e., ς (gt )≈γ σgπ√

g−ς in Equation (11)) is sufficiently large and the cash flow effect sufficiently reinforces the discount effect (i.e., the coefficient π in Equation (7) is positive and sufficiently large). In fact, the driving force of the value premium in the models of Zhang (2005) and

25 Additional evidence includes Bansal, Dittmar, and Lundblad (2005), Parker and Julliard (2005), and Hansen, Heaton, and Li (2008), among others.

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Li and Zhang (2016) is also the reinforcement effect of cash flows on the discount effect. Therefore, as in these papers, the downward-sloping equity term structure implies the value premium in our model.

Moreover, three main approaches are used in the literature to microfound cash flow durations and the equity term structure. The first approach emphasizes the importance of growth options in generating the value premium (e.g., Berk, Green, and Naik 1999; Gomes, Kogan, and Zhang 2003). Similar to these models, growth options are riskier than assets in place in our model. However, unlike these models, the driving force of the value premium in our model is that value firms have shorter cash flow durations than growth firms. The second approach builds on the idea of the creative destruction of innovation and shows that positive IST shocks hurt assets in place, while being hedged by growth options (e.g., Papanikolaou 2011; Gârleanu, Kogan, and Panageas 2012; Kogan and Papanikolaou 2013; Kogan and Papanikolaou 2014; Kogan et al. 2017; Kogan, Papanikolaou, and Stoffman 2020). Ai and Kiku (2013) also argue that growth options provide a hedge against risks in assets in place because the cost of option exercise is procyclical. A similarity between these models and ours is that value firms have shorter cash flow durations than growth firms. However, unlike these models, growth options are riskier than assets in place in our model. As suggested by Pástor and Veronesi (2006, 2009) and Campbell et al. (2018), uncertainty should play an important role in generating the value premium. In an incomplete-market growth- option framework, Dou (2017) shows that fluctuations in cash flow and investment uncertainty generate a discount effect and an effective IST shock, respectively; thus, the value premium arises in equilibrium. The third approach emphasizes inflexibility caused by the fixed cost of production and the asymmetric adjustment cost of investment (e.g., Carlson, Fisher, and Giammarino 2004; Zhang 2005), in which the book-to-market ratio is mainly determined by the riskiness of assets in place. Unlike these models, the book-to-market ratio is determined by the riskiness of growth options in our model. In Section 2.6, we analyze an extended model and show that incorporating a fixed cost of production and its implied operating leverage into our model does not affect the main quantitative results. Indeed, operating leverage exerts an amplification effect to strengthen the mechanisms that generate the gross profitability and value premium. Furthermore, in Internet Appendix 2.4.2, we provide evidence showing that operating leverage is unlikely to be the channel through which the interaction patterns of the gross profitability and value premium at the industry level arise in the data.

References

Addoum, J. M., and A. Kumar. 2016. Political sentiment and predictable returns. Review of Financial Studies 29:3471–518.

Aggarwal, R. K., and A. A. Samwick. 1999. Executive compensation, strategic competition, and relative performance evaluation: theory and evidence. Journal of Finance 54:1999–2043.

Aghion, P., A. Bergeaud, T. Boppart, P. J. Klenow, and H. Li. 2019. A theory of falling growth and rising rents. Working Paper, London School of Economics.

Aguerrevere, F. L. 2009. Real options, product market competition, and asset returns. Journal of Finance 64:957–83.

Ai, H., and R. Bansal. 2018. Risk preferences and the macroeconomic announcement premium. Econometrica 86:1383–430.

Ai, H., and D. Kiku. 2013. Growth to value: Option exercise and the cross section of equity returns. Journal of Financial Economics 107:325–49.

Akcigit, U., D. Hanley, and N. Serrano-Velarde. 2021. Back to basics: basic research spillovers, innovation policy, and growth. Review of Economic Studies 88:1–43.

Ali, A., S. Klasa, and E. Yeung. 2009. The limitations of industry concentration measures constructed with compustat data: Implications for finance research. Review of Financial Studies 22:3839–71.

3915

D ow

nloaded from https://academ

ic.oup.com /rfs/article/35/8/3867/6422548 by Jnls C

ust Serv on 19 July 2022

[15:01 5/7/2022 RFS-OP-REVF210139.tex] Page: 3916 3867–3921

The Review of Financial Studies / v 35 n 8 2022

Anderson, J. E. 1979. A theoretical foundation for the gravity equation. American Economic Review 69:106–16.

Andrei, D., and B. I. Carlin. 2018. The risk of schumpeterian competition. Working Paper, McGill University.

Armington, P. S. 1969. A theory of demand for products distinguished by place of production. Staff Papers (International Monetary Fund) 16:159–78.

Atkeson, A., and A. Burstein. 2008. Pricing-to-market, trade costs, and international relative prices. American Economic Review 98:1998–2031.

Autor, D., D. Dorn, L. F. Katz, C. Patterson, and J. V. Reenen. 2020. The fall of the labor share and the rise of superstar firms. Quarterly Journal of Economics 135:645–709.

Babenko, I., O. Boguth, and Y. Tserlukevich. 2021. Dynamic competition and expected returns. Working Paper, Arizonal State University.

Bagwell, K., and R. Staiger. 1997. Collusion over the business cycle. RAND Journal of Economics 28:82–106.

Bai, H., E. X. Li, C. Xue, and L. Zhang. 2019. Does costly reversibility matter for us public firms? Working Paper, University of Connecticut.

Bansal, R., R. F. Dittmar, and C. T. Lundblad. 2005. Consumption, dividends, and the cross section of equity returns. Journal of Finance 60:1639–72.

Bansal, R., D. Kiku, and A. Yaron. 2012. An empirical evaluation of the long-run risks model for asset prices. Critical Finance Review 1:183–221.

Bansal, R., and A. Yaron. 2004. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59:1481–509.

Basak, S., and A. Pavlova. 2004. Monopoly power and the firm’s valuation: a dynamic analysis of short versus long-term policies. Economic Theory 24:503–30.

Baumol, W. J., and R. D. Willig. 1981. Fixed costs, sunk costs, entry barriers, and sustainability of monopoly. Quarterly Journal of Economics 96:405–31.

Beeler, J., and J. Y. Campbell. 2012. The long-run risks model and aggregate asset prices: An empirical assessment. Critical Finance Review 1:141–82.

Bekaert, G., E. Engstrom, and Y. Xing. 2009. Risk, uncertainty, and asset prices. Journal of Financial Economics 91:59 – 82.

Belo, F., V. D. Gala, and J. Li. 2013. Government spending, political cycles, and the cross section of stock returns. Journal of Financial Economics 107:305–24.

Berk, J. B., R. C. Green, and V. Naik. 1999. Optimal investment, growth options, and security returns. Journal of Finance 54:1553–607.

Bils, M., P. J. Klenow, and C. Ruane. 2020. Misallocation or mismeasurement? Working Paper, University of Rochester.

Binsbergen, J. H. V., and R. S. J. Koijen. 2010. Predictive regressions: A present-value approach. Journal of Finance 65:1439–71.

Bolton, P., H. Chen, and N. Wang. 2011. A unified theory of Tobin’s q, corporate investment, financing, and risk management. Journal of Finance 66:1545–78.

Bolton, P., and D. S. Scharfstein. 1990. A theory of predation based on agency problems in financial contracting. American Economic Review 80:93–106.

Boudoukh, J., M. Richardson, and R. F. Whitelaw. 1994. Industry returns and the fisher effect. Journal of Finance 49:1595–615.

Brandt, M. W., and K. Q. Wang. 2003. Time-varying risk aversion and unexpected inflation. Journal of Monetary Economics 50:1457 – 98.

3916

D ow

nloaded from https://academ

ic.oup.com /rfs/article/35/8/3867/6422548 by Jnls C

ust Serv on 19 July 2022

[15:01 5/7/2022 RFS-OP-REVF210139.tex] Page: 3917 3867–3921

The Oligopoly Lucas Tree

Bronnenberg, B. J., S. K. Dhar, and J.-P. H. Dubé. 2009. Brand history, geography, and the persistence of brand shares. Journal of Political Economy 117:87–115.

Bustamante, M. C. 2015. Strategic investment and industry risk dynamics. Review of Financial Studies 28:297–341.

Bustamante, M. C., and A. Donangelo. 2017. Product market competition and industry returns. Review of Financial Studies 30:4216–66.

Campbell, J., J. Hilscher, and J. Szilagyi. 2008. In search of distress risk. Journal of Finance 63:2899–939.

Campbell, J. Y., and J. H. Cochrane. 1999. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107:205–51.

———. 2000. Explaining the poor performance of consumption-based asset pricing models. Journal of Finance 55:2863–78.

Campbell, J. Y., S. Giglio, C. Polk, and R. Turley. 2018. An intertemporal capm with stochastic volatility. Journal of Financial Economics 128:207–33.

Campbell, J. Y., and T. Vuolteenaho. 2004. Bad beta, good beta. American Economic Review 94:1249–75.

Carlin, B. I. 2009. Strategic price complexity in retail financial markets. Journal of Financial Economics 91:278–87.

Carlson, M., E. J. Dockner, A. Fisher, and R. Giammarino. 2014. Leaders, followers, and risk dynamics in industry equilibrium. Journal of Financial and Quantitative Analysis 49:321–49.

Carlson, M., A. Fisher, and R. Giammarino. 2004. Corporate investment and asset price dynamics: implications for the cross-section of returns. Journal of Finance 59:2577–603.

Chen, H. 2010. Macroeconomic conditions and the puzzles of credit spreads and capital structure. Journal of Finance 65:2171–212.

Chen, H., W. W. Dou, H. Guo, and Y. Ji. 2020. Feedback and contagion through distressed competition. Working Paper, MIT.

Chen, H., W. W. Dou, and L. Kogan. 2021. Measuring the ’dark matter’ in asset pricing models. Working Paper, MIT.

Christensen, C. 1997. The innovator’s dilemma: When new technologies cause great firms to fail. Cambridge, MA: Harvard Business Review Press.

Cohen, R. B., C. Polk, and T. Vuolteenaho. 2002. Does risk or mispricing explain the cross-section of stock prices. Working Paper, Harvard University.

———. 2003. The value spread. Journal of Finance 58:609–41.

Constantinides, G. M. 1990. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98:519–43.

Corhay, A. 2017. Industry competition, credit spreads, and levered equity returns. Working Paper, University of Toronto.

Corhay, A., H. Kung, and L. Schmid. 2020. Competition, markups, and predictable returns. Review of Financial Studies 33:5906–39.

———. 2021. Q: Risk, rents, or growth? Working Paper, University of Toronto.

Croce, M. M., M. Lettau, and S. C. Ludvigson. 2014. Investor Information, Long-Run Risk, and the Term Structure of Equity. Review of Financial Studies 28:706–42.

Croce, M. M., T. Marchuk, and C. Schlag. 2019. The leading premium. Working Paper, Bocconi University.

Crouzet, N., and J. Eberly. 2020. Rents and intangible capital: A Q+ framework. Working Paper, Northwestern University.

3917

D ow

nloaded from https://academ

ic.oup.com /rfs/article/35/8/3867/6422548 by Jnls C

ust Serv on 19 July 2022

[15:01 5/7/2022 RFS-OP-REVF210139.tex] Page: 3918 3867–3921

The Review of Financial Studies / v 35 n 8 2022

Da, Z. 2009. Cash flow, consumption risk, and the cross-section of stock returns. Journal of Finance 64:923–56.

Da, Z., and M. C. Warachka. 2009. Cashflow risk, systematic earnings revisions, and the cross-section of stock returns. Journal of Financial Economics 94:448–68.

Dechow, P. M., R. Sloan, and M. T. Soliman. 2004. Implied equity duration: A new measure of equity risk. Review of Accounting Studies 9:197–228.

DellaVigna, S., and J. M. Pollet. 2007. Demographics and industry returns. American Economic Review 97:1667–702.

Detemple, J. B., and F. Zapatero. 1991. Asset prices in an exchange economy with habit formation. Econometrica 59:1633–57.

Dew-Becker, I. 2012. A model of time-varying risk premia with habits and production. Working Paper, Northwestern University.

Dittmar, R. F., and C. T. Lundblad. 2017. Firm characteristics, consumption risk, and firm-level risk exposures. Journal of Financial Economics 125:326–43.

Dou, W. W. 2017. Embrace or fear uncertainty: Growth options, limited risk sharing, and asset prices. Working Paper, University of Pennsylvania.

Dou, W. W., and Y. Ji. 2021. External financing and customer capital: A financial theory of markups. Management Science 67:5569–85.

Dou, W. W., Y. Ji, D. Reibstein, and W. Wu. 2021a. Inalienable customer capital, corporate liquidity, and stock returns. Journal of Finance 76:211–65.

Dou, W. W., Y. Ji, and W. Wu. 2021. Competition, profitability, and discount rates. Journal of Financial Economics 140:582–620.

Dou, W. W., S. Johnson, M. A. Shao, and W. Wu. 2021b. Competition networks, distress propagation, and stock returns. Working Paper, University of Pennsylvania.

Dou, W. W., L. A. Taylor, W. Wang, and W. Wang. 2021c. Dissecting bankruptcy frictions. Journal of Financial Economics. Advance Access published June 24, 2021, 10.1016/j.jfineco.2021.06.014.

Fama, E. F., and K. R. French. 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33:3–56.

———. 1997. Industry costs of equity. Journal of Financial Economics 43:153 – 93.

Favilukis, J., X. Lin, and X. Zhao. 2020. The elephant in the room: The impact of labor obligations on credit markets. American Economic Review 110:1673–712.

Feng, G., S. Giglio, and D. Xiu. 2020. Taming the factor zoo: A test of new factors. Journal of Finance 75:1327–70.

Fershtman, C., and K. L. Judd. 1987. Equilibrium incentives in oligopoly. American Economic Review 77:927–40.

Fershtman, C., and A. Pakes. 2000. A dynamic oligopoly with collusion and price wars. RAND Journal of Economics 31:207–36.

Frésard, L. 2010. Financial strength and product market behavior: The real effects of corporate cash holdings. Journal of Finance 65:1097–122.

Fudenberg, D., and E. Maskin. 1986. The folk theorem in repeated games with discounting or with incomplete information. Econometrica 54:533–54.

Gabaix, X. 2011. The granular origins of aggregate fluctuations. Econometrica 79:733–72.

Galeotti, M., and F. Schiantarelli. 1998. The cyclicality of markups in a model with adjustment costs: econometric evidence for US industry. Oxford Bulletin of Economics and Statistics 60:121–42.

Galí, J. 2008. Introduction to monetary policy, inflation, and the business cycle: An introduction to the new Keynesian framework. In Monetary policy, inflation, and the business cycle: An introduction to the New Keynesian

3918

D ow

nloaded from https://academ

ic.oup.com /rfs/article/35/8/3867/6422548 by Jnls C

ust Serv on 19 July 2022

[15:01 5/7/2022 RFS-OP-REVF210139.tex] Page: 3919 3867–3921

The Oligopoly Lucas Tree

framework, introductory chapters. Princeton, NJ: Princeton University Press.

Garcia-Macia, D., C.-T. Hsieh, and P. J. Klenow. 2018. How destructive is innovation? Working Paper, International Monetary Fund.

Garlappi, L. 2004. Risk premia and preemption in R&D ventures. Journal of Financial and Quantitative Analysis 39:843–72.

Garlappi, L., and Z. Song. 2017. Capital utilization, market power, and the pricing of investment shocks. Journal of Financial Economics 126:447–70.

Gârleanu, N., L. Kogan, and S. Panageas. 2012. Displacement risk and asset returns. Journal of Financial Economics 105:491–510.

Gârleanu, N., S. Panageas, and J. Yu. 2012. Technological growth and asset pricing. Journal of Finance 67:1265– 92.

Geroski, P., and S. Toker. 1996. The turnover of market leaders in uk manufacturing industry, 1979-86. International Journal of Industrial Organization 14:141 – 58.

Gilchrist, S., J. Sim, R. Schoenle, and E. Zakrajsek. 2017. Inflation dynamics during the financial crisis. American Economic Review 107:785–823.

Giroud, X., and H. M. Mueller. 2010. Does corporate governance matter in competitive industries? Journal of Financial Economics 95:312 – 31.

———. 2011. Corporate governance, product market competition, and equity prices. Journal of Finance 66:563–600.

Gomes, J., L. Kogan, and L. Zhang. 2003. Equilibrium cross section of returns. Journal of Political Economy 111:693–732.

Gomes, J. F., L. Kogan, and M. Yogo. 2009. Durability of output and expected stock returns. Journal of Political Economy 117:941–86.

Gourio, F., and L. Rudanko. 2014. Customer capital. Review of Economic Studies 81:1102–36.

Grullon, G., Y. Larkin, and R. Michaely. 2019. Are US industries becoming more concentrated? Review of Finance 23:697–743.

Gutiérrez, G., C. Jones, and T. Philippon. 2019. Entry costs and the macroeconomy. Working Paper, New York University.

Haltiwanger, J., and J. E. Harrington. 1991. The impact of cyclical demand movements on collusive behavior. RAND Journal of Economics 22:89–106.

Hansen, L. P. 2007. Generalized method of moments estimation. In The new Palgrave dictionary of economics, eds. S. N. Durlauf and L. E. Blume. London, UK: Palgrave Macmillan.

———. 2012. Proofs for large sample properties of generalized method of moments estimators. Journal of Econometrics 170:325 – 30.

Hansen, L. P., J. C. Heaton, and N. Li. 2008. Consumption strikes back? measuring long-run risk. Journal of Political Economy 116:260–302.

Harrigan, J. 1993. OECD imports and trade barriers in 1983. Journal of International Economics 35:91–111.

Head, K., and J. Ries. 2001. Increasing returns versus national product differentiation as an explanation for the pattern of U.S.-Canada trade. American Economic Review 91:858–76.

Herskovic, B. 2018. Networks in production: Asset pricing implications. Journal of Finance 73:1785–818.

Hou, K., and D. T. Robinson. 2006. Industry concentration and average stock returns. Journal of Finance 61:1927–56.

3919

D ow

nloaded from https://academ

ic.oup.com /rfs/article/35/8/3867/6422548 by Jnls C

ust Serv on 19 July 2022

[15:01 5/7/2022 RFS-OP-REVF210139.tex] Page: 3920 3867–3921

The Review of Financial Studies / v 35 n 8 2022

Ivaldi, M., P. Rey, P. Seabright, and J. Tirole. 2007. The economics of tacit collusion: Implications for merger control. In The political economy of antitrust, contributions to economic analysis, eds. V. Ghosal and J. Stennek, 217–40. Bradford, UK: Emerald Group Publishing.

Kandel, S., and R. F. Stambaugh. 1991. Asset returns and intertemporal preferences. Journal of Monetary Economics 27:39–71.

Kato, M., and Y. Honjo. 2009. The persistence of market leadership: evidence from japan. Industrial and Corporate Change 18:1107–33.

Klemperer, P. 1995. Competition when consumers have switching costs: An overview with applications to industrial organization, macroeconomics, and international trade. Review of Economic Studies 62:515–39.

Kogan, L., and D. Papanikolaou. 2013. Firm characteristics and stock returns: The role of investment-specific shocks. Review of Financial Studies 26:2718–59.

———. 2014. Growth opportunities, technology shocks, and asset prices. Journal of Finance 69:675–718.

Kogan, L., D. Papanikolaou, A. Seru, and N. Stoffman. 2017. Technological innovation, resource allocation, and growth. Quarterly Journal of Economics 132:665–712.

Kogan, L., D. Papanikolaou, and N. Stoffman. 2020. Left behind: Creative destruction, inequality, and the stock market. Journal of Political Economy 128:855–906.

Koijen, R. S., and S. V. Nieuwerburgh. 2011. Predictability of returns and cash flows. Annual Review of Financial Economics 3:467–91.

Koijen, R. S. J., and M. Yogo. 2015. The cost of financial frictions for life insurers. American Economic Review 105:445–75.

Lettau, M., and J. Wachter. 2011. The term structures of equity and interest rates. Journal of Financial Economics 101:90–113.

Lettau, M., and J. A. Wachter. 2007. Why is long-horizon equity less risky? a duration-based explanation of the value premium. Journal of Finance 62:55–92.

Lewellen, J. 1999. The time-series relations among expected return, risk, and book-to-market. Journal of Financial Economics 54:5–43.

Li, J., L. Kogan, and H. H. Zhang. 2020. Operating hedge and gross profitability premium. Working Paper, University of Texas at Dallas.

Li, J., L. Kogan, H. H. Zhang, and Y. Zhu. 2020. Operating leverage and hedging: A tale of two production costs for asset pricing. Working Paper, University of Texas at Dallas.

Li, J., and H. H. Zhang. 2016. Short-run and long-run consumption risks, dividend processes, and asset returns. Review of Financial Studies 30:588–630.

Li, Y. 2015. Expected returns and habit persistence. Review of Financial Studies 14:861–99.

Loecker, J. D., J. Eeckhout, and G. Unger. 2020. The rise of market power and the macroeconomic implications. Quarterly Journal of Economics 135:561–644.

Loualiche, E. 2016. Asset pricing with entry and imperfect competition. Working Paper, University of Minnesota.

Maskin, E., and J. Tirole. 1988a. A theory of dynamic oligopoly, i: Overview and quantity competition with large fixed costs. Econometrica 56:549–69.

———. 1988b. A theory of dynamic oligopoly, ii: Price competition, kinked demand curves, and edgeworth cycles. Econometrica 56:571–99.

Menzly, L., T. Santos, and P. Veronesi. 2004. Understanding predictability. Journal of Political Economy 112:1–47.

Moskowitz, T. J., and M. Grinblatt. 1999. Do industries explain momentum? Journal of Finance 54:1249–90.

3920

D ow

nloaded from https://academ

ic.oup.com /rfs/article/35/8/3867/6422548 by Jnls C

ust Serv on 19 July 2022

[15:01 5/7/2022 RFS-OP-REVF210139.tex] Page: 3921 3867–3921

The Oligopoly Lucas Tree

Müller, U. K., and M. W. Watson. 2018. Long-run covariability. Econometrica 86:775–804.

Nekarda, C. J., and V. A. Ramey. 2013. The cyclical behavior of the price-cost markup. Working Paper, Federal Reserve Board of Governors.

Novy-Marx, R. 2007. An equilibrium model of investment under uncertainty. Review of Financial Studies 20:1461–502.

———. 2013. The other side of value: the gross profitability premium. Journal of Financial Economics 108:1–28.

Opp, M. M., C. A. Parlour, and J. Walden. 2014. Markup cycles, dynamic misallocation, and amplification. Journal of Economic Theory 154:126–61.

Papanikolaou, D. 2011. Investment shocks and asset prices. Journal of Political Economy 119:639–85.

Parker, J. A., and C. Julliard. 2005. Consumption risk and the cross section of expected returns. Journal of Political Economy 113:185–222.

Pástor, v., and P. Veronesi. 2006. Was there a nasdaq bubble in the late 1990s? Journal of Financial Economics 81:61–100.

———. 2009. Technological revolutions and stock prices. American Economic Review 99:1451–83.

———. 2012. Uncertainty about government policy and stock prices. Journal of Finance 67:1219–64.

———. 2020. Political cycles and stock returns. Journal of Political Economy 128:4011–45.

Rasmusen, E. 1988. Entry for buyout. Journal of Industrial Economics 36:281–99.

Santos, T., and P. Veronesi. 2006. Labor income and predictable stock returns. Review of Financial Studies 19:1–44.

———. 2010. Habit formation, the cross section of stock returns and the cash-flow risk puzzle. Journal of Financial Economics 98:385–413.

Shumway, T. 2001. Forecasting bankruptcy more accurately: a simple hazard model. Journal of Business 74:101–24.

Sutton, J. 2007. Market share dynamics and the "persistence of leadership" debate. American Economic Review 97:222–41.

Tsai, J., and J. A. Wachter. 2016. Rare booms and disasters in a multi-sector endowment economy. Review of Financial Studies 29:1377–408.

Uhlig, H. 2007. Explaining asset prices with externa habits and wage rigidities in a DSGE model. American Economic Review 97:239–43.

Zhang, L. 2005. The value premium. Journal of Finance 60:67–103.

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