General Equilibrium Oligopoly and Ownership Structure

We develop a tractable general equilibrium framework in which firms are large and have market power, with respect to both products and labor, and in which a firm's decisions are affected by its ownership structure. We characterize the Cournot--Walras equilibrium of an economy where each firm maximizes a share-weighted average of shareholder utilities---rendering the equilibrium independent of price normalization. In a one-sector economy, if returns to scale are non-increasing then an increase in ``effective'' market concentration (which accounts for common ownership) leads to declines in employment, real wages, and the labor share. Yet when there are multiple sectors, due to an intersectoral pecuniary externality, an increase in common ownership could stimulate the economy when the elasticity of labor supply is high relative to the elasticity of substitution in product markets. We characterize for which ownership structures the monopolistically competitive limit or an oligopolistic one are attained as the number of sectors in the economy increases. When firms have heterogeneous constant returns to scale technologies we find that an increase in common ownership leads to markets that are more concentrated.


Introduction
Oligopoly is widespread and allegedly on the rise. Many industries are characterized by oligopolistic conditions-including, but not limited to, the digital ones dominated by GAFAM: Google (now Alphabet), Apple, Facebook, Amazon, and Microsoft. These firms, as well as others, have influence in the aggregate economy. 1 Yet oligopoly is seldom considered by macroeconomic models, which focus on monopolistic competition because of its analytical tractability. A typical limitation of monopolistic competition models is that they have no role for market concentration to play in conditioning competition because the summary statistic for competition is the elasticity of substitution. In the field of international trade, a few papers consider oligopoly-but with a continuum of sectors and hence with negligible firms in relation to the economy (Neary, 2003a,b;Atkeson and Burstein, 2008). Furthermore, all these papers assume that firms maximize profits even with ownership structures which induce a departure from profit maximization.
In this paper we build a tractable general equilibrium model of oligopoly allowing for ownership diversification, characterize its equilibrium and comparative statics properties, and then use it to analyze the effect of competition policies. Our contribution is mostly methodological, although we have applied the multisector version of our model elsewhere to explain the evolution of macroeconomic magnitudes in a calibration exercise Vives, 2018, 2019a). We adopt this approach in light of (a) the increasing concentration in the US economy with respect to both product and labor markets and (b) the increasing extent of common ownership due to the increase in institutional investment-especially in index funds (thus, for almost 90% of S&P 500 firms, the largest proportion of shares is held by the "Big 3" asset managers: BlackRock, Vanguard, and State Street). These trends have raised concerns of increased market power and markups Azar, Schmalz, and Tecu, 2018;De Loecker et al., 2020) as well as calls for antitrust action and regulation of common ownership, topics that are hotly debated (see e.g. Elhauge, 2016;Posner, Scott Morton, and Weyl, forth.).
The difficulties of incorporating oligopoly into a general equilibrium framework have hindered the modeling of market power in macroeconomics and international trade. The reason is that there is no simple objective for the firm when firms are not price takers. 2 In a general equilibrium, moreover, firms with pricing power will affect not only their own respective profits but also the wealth of consumers and therefore demand (these feedback effects are sometimes referred to as "Ford effects"). Firms that are large relative to factor markets also have to take into account their impact on factor prices. Gabszewicz and Vial (1972) proposed the Cournot-Walras equilibrium concept assuming firms maximize profit in general equilibrium oligopoly but then equilibrium depends on the choice of numéraire. 3 This problem 1 In 2019, GAFAM accounted for nearly 15% of US market capitalization. An extreme example is provided by Samsung and Hyundai, which are large relative to Korea's economy (Gabaix, 2011). In the United States, General Motors and Walmartdespite never employing more than 1% of the country's workforce-often figure prominently in local labor markets.
2 With price-taking firms, a firm's shareholders agree unanimously that the objective of the firm should be to maximize its own profits. This result is known as the Fisher separation theorem (DeAngelo, 1981), which Hart (1979) extends to incomplete markets. In fact, Arrow's impossibility result on preference aggregation was derived precisely when attempting to generalize the theory of the firm with multiple owners (see Arrow, 1984). 3 When firms have market power, the outcome of their optimization depends on what price is taken as the numéraire since has been sidestepped by assuming that there is only one good (an outside good or numéraire;that owners of the firm care about see e.g. Mas-Colell, 1982) or that firms are small relative to the economy-be it in monopolistic competition (Hart, 1983) or sector oligopoly (Neary, 2003a).
Furthermore, a question arises as to what is the objective of the firm when there is overlapping ownership due to owners' diversification. If a firms' shareholders have holdings in competing firms, they would benefit from high prices through their effect not only on their own profits, but also on the profits of rival firms, as well as internalizing other externalities between firms (Gordon, 1990;Hansen and Lott, 1996). Rotemberg (1984) proposes a parsimonious model in which the firm's manager maximizes a weighted average of shareholders' utilities and thus internalizes inter-firm externalities. 4 We build a model of oligopoly under general equilibrium, allowing firms to be large in relation to the economy, and then examine the effect of oligopoly on macroeconomic performance. The ownership structure allows investors to diversify both intra-and inter-industry. We assume that firms maximize a weighted average of shareholder utilities in Cournot-Walras equilibrium. The weights in a firm's objective function are given by the influence or "control weight" of each shareholder. This approach solves the numéraire problem because indirect utilities depend only on relative prices and not on the choice of numéraire. Firms are assumed to make strategic decisions that account for the effect of their actions on prices and wages. When making decisions about hiring, for instance, a firm realizes that increasing employment could result in upward pricing pressure on real wages-reducing not only the firm's own profits but also the profits of all other firms in its shareholders' portfolios.
We develop first a base model with one sector to present our equilibrium concept and comparative static results and then we extend it to a multi-sector economy suitable for calibration. The multi-sector model is parsimonious and identifies the key parameters driving equilibrium: elasticity of substitution across industries, elasticity of the labor supply, together with the market concentration of each industry, and the ownership structure (i.e., extent of diversification) of investors.
Our approach may shed light on some leading questions. How do output, labor demand, prices, and wages depend on market concentration and the degree of common ownership? To what extent are markups in product markets, and markdowns in the labor market, affected by how much the firm internalizes other firms' profits? Can common ownership be pro-competitive in a general equilibrium framework? How do common ownership effects change when the number of industries increases? In the presence of ownership diversification, is the monopolistically competitive limit (as described by by changing the numéraire the profit function is generally not a monotone transformation of the original one (see Ginsburgh, 1994). 4 The maximization of the objective function "weighted average of shareholder utilities" depends on the cardinal properties of shareholders' preferences (violating Arrow's ordinal postulate). However, it can be microfounded using a purely ordinal model-provided shareholder preferences are random from the perspective of the managers who run the firms (Azar, , 2017Brito, Osório, Ribeiro, and Vasconcelos, 2018).  and Brito, Osório, Ribeiro, and Vasconcelos (2018) show that, in a probabilistic voting setting where two managers compete for shareholder votes by developing strategic reputations, the firm's objective will be to maximize a weighted average of shareholder utilities without any coordination of the shareholders. It is worth noting that the Big 3 together have stakes that average close to 20% of each publicly traded company in the United States (Fichtner, Heemskerk, and Garcia-Bernardo, 2017). This gives them enough voting power to be pivotal often. Moreover, Aggarwal, Dahiya, and Prabhala (2019) show that shareholder dissent hurts directors and that director elections matter because of career concerns. In particular, these authors show that increasing the votes withheld by only 10% leads to a 24% increase in the likelihood of director turnover. Dixit and Stiglitz, 1977) attained when firms become small relative to the market?-and, more generally, how is that limit affected by ownership structure? Is traditional antitrust policy a complement or rather a substitute with respect to controlling common ownership when the aim is boosting employment? 5 In the base model that we develop, there is one good in addition to leisure; also, the model assumes oligopoly in the product market and oligopsony in the labor market. Firms compete by setting their labor demands à la Cournot and thus have market power. There is a continuum of risk-neutral owners, who each have a proportion of their respective shares invested in one firm and have the balance invested in the market portfolio (say, an index fund). This formulation is numéraire-free and allows us to characterize the equilibrium. The extent to which firms internalize rival firms' profits depends on market concentration and investor diversification. We demonstrate the existence and uniqueness of equilibrium, and then characterize its comparative statics properties, while assuming that labor supply is upward sloping (and allowing for some economies of scale in production). The results establish that, in our model of a one-sector economy, the markdown of real wages with respect to the marginal product of labor is driven by the common ownership-modified Herfindahl-Hirschman index (HHI) for the labor market and also by labor supply elasticity (but not by product market power, since ownership is proportional to consumption). We perform comparative statics on the equilibrium (employment and real wages) with respect to market concentration and degree of common ownership, and we develop an example featuring Cobb-Douglas firms and consumers with additively separable isoelastic preferences. We find that increased market concentration-due either to fewer firms or to more diversification (common ownership)-depresses the economy by reducing employment, output, real wages, and the labor share (if one assumes non-increasing returns to scale). When firms have different constant returns to scale (CRS) technologies, an increase in common ownership leads to a more concentrated market (as measured by the HHI) because more efficient firms then gain market share at the expense of weaker rivals. Furthermore, the minimal relative productivity for the least productive firm to be viable is increasing in the extent of common ownership.
We extend our base model to allow for multiple sectors, and for differentiated products across sectors, with constant elasticity of substitution (CES) aggregators. The firms supplying each industry's product are finite in number and engage in Cournot competition. We allow here for investors to diversify both in an intra-industry fund and in an economy-wide index fund. In this extension, a firm deciding whether to marginally increase its employment must consider the effect of that increase on three relative prices: (i) the increase would reduce the relative price of the firm's own products, (ii) it would boost real wages, and (iii) it would increase the relative price of products in other industries (i.e., because overall consumption would increase). This third effect, referred to as inter-sector pecuniary externality, is internalized only when there is common ownership involving the firm and firms in other industries. In this case, the markdown of real wages relative to the marginal product of labor increases with the modified HHI values for the labor market and product markets but decreases with the pecuniary externality (weighted by the extent of competitor profit internalization due to common ownership). We find that common ownership always has an anti-competitive effect when increasing intra-industry diversification but that it can have a pro-competitive effect when increasing economy-wide diversification if the elasticity of labor supply is high in relation to the elasticity of substitution among product varieties. In this case, the relative impact of profit internalization on the level of market power in product markets is higher than in the labor market. It is worth remarking that when the elasticity of labor supply is high enough, an increase in economy-wide common ownership always has a pro-competitive effect, no matter how many sectors the economy has.
We then consider the limiting case when the number of sectors tends to infinity. This formulation allows us to check for whether-and, if so, under what circumstances-the monopolistically competitive market of Dixit and Stiglitz (1977) or the oligopolistic ones of Atkeson and Burstein (2008) and Neary (2003a,b) are attained, in the presence of common ownership, when firms become small relative to the market; it also enables a determination of how ownership structure affects that competitive limit.
We find that with incomplete asymptotic diversification, as the number of sectors N in the economy grows, the monopolistically competitive limit is attained if there is either one firm per sector or full intra-industry common ownership. If full diversification is attained at least as fast as 1/ √ N, then profit internalization is positive in the limit and the Dixit-Stiglitz limit is not attained. We obtain that the limit degree of profit internalization is increasing in market concentration and in how rapidly diversification is achieved. The limit markdown may increase or decrease with profit internalization.
Competition policy in the one-sector economy can foster employment and increase real wages by reducing market concentration (with non-increasing returns) and/or the level of diversification (common ownership), which serve as complementary tools. When there are multiple sectors, it is optimal for worker-consumers to have full diversification (common ownership) economy-wide but no extra diversification intra-industry-that is, when the elasticity of substitution in product markets is low relative to the elasticity of labor supply. In this case, competition policy should seek to alter only intra-industry ownership structure.
The rest of our paper proceeds as follows. Section 2 describes some further connections with the literature. Section 3 develops a one-sector model of general equilibrium oligopoly with labor as the only factor of production; this is where we derive comparative statics results with respect to the effects of market concentration on employment, wages, and the labor share. In Section 4 we extend the model to allow for multiple sectors with differentiated products, and we then derive results that characterize the limit economy as the number of sectors approaches infinity. We also offer some illustrative calibrations of the model. Section 5 discusses the implications for competition policies, and we conclude in Section 6 with a summary and suggestions for further research. Appendix A provides more detail about the case of increasing returns in production, and the proofs of most results are given in Appendix B.

Connections with the literature 2.1 Theory
Our paper is related to four strands of the literature. The first is the general equilibrium with oligopoly à la the Cournot models of Gabszewicz and Vial (1972), Novshek and Sonnenschein (1978), and Mas-Colell (1982), where the proposed Cournot-Walras equilibrium assumes that firms maximize profits.
Here we assume instead that a firm's manager maximizes a weighted average of shareholder utilities and also consider an ownership structure that allows for common ownership.
The second strand encompasses the macroeconomic models with Keynesian features that have incorporated market power. A precursor of those models is the work by one of Keynes's contemporaries, Michal Kalecki, on the macroeconomic effects of market power in a two-class economy (Kalecki, 1938(Kalecki, , 1954. The most closely related papers are perhaps Hart (1982) andd'Aspremont, Ferreira, andGérard-Varet (1990). 6 Hart's (1982) work differs from ours in assuming that firms are small relative to the overall economy and have separate owners. Unions have the labor market power in his model and so equilibrium real wages are higher than the marginal product of labor; in our model's equilibrium, real wages are lower than that marginal product.
In d' Aspremont, Ferreira, and Gérard-Varet (1990), firms are large relative to the economy; however, it is still assumed that firms maximize profits in terms of an arbitrary numéraire and that they compete in prices while taking wages as given with an inelastic labor supply. We consider instead the more realistic case of an elastic labor supply, which yields a positive equilibrium real wage even when market power reduces employment to below the competitive level. Our approach differs from theirs also in that we derive measures of market concentration, discuss competition policy in a general equilibrium, and consider effects on the labor share. 7 Furthermore, instead of assuming the existence of consumerworker-owners (as is typical in the literature), we follow Kalecki (1954) and distinguish between two groups: worker-consumers and owner-consumers. Our model has a Kaleckian flavor also in relating product market power to the labor share, since in Kalecki (1938), the labor share is determined by the economy's average Lerner index.
The third strand of this literature focuses on international trade models with oligopolistic firms.
Neary (2003a) considers a continuum of industries with Cournot competition in each industry, taking the marginal utility of wealth (instead of the wage) as given. Workers supply labor inelastically and firms maximize profits. Neary finds a negative relationship between the labor share and market concentration.
Our work differs in that firms are large relative to the economy, and therefore have market power in both product and labor markets, and in considering the effects of firms' ownership structure. Neary (2003a) also assumes a perfectly inelastic labor supply, so that changes in market power can affect neither employment nor output in equilibrium. In contrast, we allow for an increasing labor supply function 6 See Silvestre (1993) for a survey of the market power foundations of macroeconomic policy. 7 Gabaix (2011) also considers firms that are large in relation to the economy but with no strategic interaction among them; his aim is to demonstrate how microeconomic shocks to large firms can create meaningful aggregate fluctuations. Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2012) pursue a similar goal but assume that firms are price takers. and examine more potential effects of competition policy. Atkeson and Burstein (2008) also consider a continuum of sectors with Cournot competition in each industry. These authors assume that goods produced in a country within a sector are better substitutes than across sectors. The aim of the paper is to reproduce stylized facts regarding international relative prices.
It is worth noting that in both Atkeson and Burstein (2008) and Neary (2003a), as well as in Dixit and Stiglitz (1977), there is a representative household that owns a market portfolio in all the firms. And yet, the firms are assumed to maximize their own profits even though no shareholder would actually want this. Thus, there is a tension between the assumed ownership structure and the profit maximization assumption. The results in Section 4.3, under our assumptions with two classes of agents, in which we consider the limit as the number of sectors N tends to infinity, make this tension clear. Specifically, with full asymptotic diversification as N tends to infinity, we obtain the results associated to Dixit-Stiglitz or Neary only when there is no rivals' profit internalization in the limit, and this happens when full diversification is attained very slowly (more slowly than 1/ √ N).
The fourth strand relates to ownership structure and oligopoly in partial equilibrium. In our model managers internalize the control of the firm by the different owners as in Rotemberg (1984) and O 'Brien and Salop (2000), but ours is not a model of the stakeholder corporation as in Magill, Quinzii, and Rochet (2015) since managers only internalize the welfare of owners. The fact that overlapping ownership may relax competition was observed by Rubinstein and Yaari (1983) and explored by Reynolds and Snapp (1986), and Bresnahan and Salop (1986). Since overlapping ownership may internalize externalities between firms, it may have ambiguous welfare effects. Indeed, overlapping ownership may increase market power and raise margins yet simultaneously internalize technological spillovers and increase productivity (López and Vives, 2019); see He and Huang (2017) for compatible evidence and Geng, Hau, and Lai (2016) for how vertical common ownership links may improve the internalization of patent complementarities. Here we will show how common ownership can have pro-competitive effects in a multi-sector economy. 8

Empirics
Our approach may speak about macro trends in the economy in relation to the effects of the evolution of institutional investment and common ownership patterns, product and labor market concentration, markups and the declining labor share, the consequences for competition and investment, and the implications for policy.
The world of dispersed ownership described by Berle and Means (1932) no longer exists in the United States. The rise in institutional stock ownership over the past 35 years has been formidable.
Pension, mutual, and exchange-traded funds now own the lion's share of publicly traded US firms. The asset management industry is concentrated around the three largest managers (BlackRock, Vanguard, and State Street), and there has been a shift from active to passive investors (who are more diversified).
This evolution of the asset management industry has transformed the ownership structure of firms. In any industry today, large firms are likely to have common shareholders with significant shares (Azar, Schmalz, and Tecu, 2018). 9 Before surveying the evidence on these macroeconomic trends, let us examine what evidence there is on how common ownership might affect the incentives of managers. Common owners in an industry may have the ability and incentive to influence management. Indeed, both voice and exit can strengthen with common ownership (Edmans, Levit, and Reilly, 2019), and not pushing for aggressiveness in management contracts is a mechanism by which common owners can relax competition (Antón, Ederer, Gine, and Schmalz, 2018). Note also that, even if a fund follows a passive strategy and even if a good part of the increase in common ownership is due to the rise of passive funds, we cannot assume that the fund is a passive owner (Appel, Gormley, and B.Keim, 2016). In fact, large passive funds tend to exhibit a more "disciplinarian" attitude toward management (Bolton, Li, Ravina, and Rosenthal, 2019)-and institutional common owners not only internalize governance externalities but also are more likely to vote against management (He, Huang, and Zhao, 2019). 10 There are, however, countervailing agency problems: Bebchuk and Hirst (forth.) point out that index fund managers may not have incentives to monitor management (for evidence that index funds are less likely to vote against management than are active funds, see Brav, Jiang, Li, and Pinnington, 2019;Heath, Macciocchi, Michaely, and Ringgenberg, 2019). Schmidt and Fahlenbrach (2017) show that increased passive ownership impedes high-cost governance activities and increases agency costs. 11 In short, the link between increased passive diversification and relaxed competition may stem either from the internalization by managers of the common owners' interests or from increased agency costs that allow managers to slack. 12 Recent empirical research has renewed interest in the issue of aggregate market power and its consequences for macroeconomic outcomes. Grullon, Larkin, and Michaely (2019) claim that concentration has increased in more than 75% of US industries over the past two decades and also that firms in industries with larger increases in product market concentration have enjoyed higher profit margins and positive abnormal stock returns-suggesting that market power is the driver of these outcomes. 13 De Loecker et al. (2020) document, for the US economy, a large increase in markups (in excess of the increased overhead) and in economic profits since 1955. These authors attribute those increases to a re-allocation of market share: from low-productivity, low-markup, high-labor share firms to highproductivity, high-markup, low-labor share firms (in line with the results reported in Autor,Dorn,Katz,9 Minority cross-ownership is also common and has anti-competitive effects (Dietzenbacher, Smid, and Volkerink, 2000;Brito, Osório, Ribeiro, and Vasconcelos, 2018;Nain and Wang, 2018). 10 Furthermore, portfolio managers have incentives to increase, even marginally, the value of firms in their portfolio because doing so increases management fees (Lewellen and Lewellen, 2018). Jahnke's (2019) field research, based on 50 interviews with large-asset managers, supports the view that they have considerable incentives to engage in corporate governance activities for the purpose of increasing portfolio values. This finding is consistent with the views expressed by large-asset managers themselves in their "corporate stewardship" reports (e.g., BlackRock, 2019). 11 Hansen and Lott (1996) observe that higher agency costs may be associated with more managerial discretion when managers internalize externalities through portfolio value maximization. 12 Yet when managers hold shares in their firm, agency costs could mitigate the anti-competitive effects of common ownership (see Azar, 2020). 13 Autor, Dorn, Katz, Patterson, and Van Reenen (forth.) state that, for the period 1982-2012, "according to all measures of sales concentration, industries have become more concentrated on average." Patterson, and Van Reenen, forth. and Kehrig and Vincent, 2018). Autor, Dorn, Katz, Patterson, and Van Reenen (forth.) posit that globalization and technological change lead to concentration and to the rise of what they call "superstar" firms, which have high profits and a low labor share. As the importance of superstar firms rises (with the increase in concentration), the aggregate labor share falls. 14 We find that increased common ownership generates a re-allocation of market share from low-productivity, low-markup, high-labor share firms to high-productivity, high-markup, low-labor share firms. There is also substantial evidence that large firms have market power not just in product markets but also in labor markets. 15 Furthermore, there are claims also of increasing labor market concentration (Benmelech, Bergman, and Kim, forth.).
In addition to increases in concentration as traditionally measured, recent research has shown that: (i) increased overlapping ownership of firms by financial institutions (and by funds in particular)what we refer to as common ownership-has led to substantial increases in effective (i.e., augmented by common ownership) concentration indices in the airline and banking industries; and (ii) this greater concentration is associated with higher prices (Azar, Schmalz, and Tecu, 2018). Gutiérrez and Philippon (2017b) suggest that the increase in index and quasi-index fund ownership has played a role in the decline of aggregate investment. 16 Summers (2016) and Stiglitz (2017) link increases in market power to the potential secular stagnation of developed economies, and Boller and Morton (2020) use an event study of inclusion in the S&P 500 index to conclude that common ownership increases profits.
Some of the recent empirical papers develop theoretical frameworks that link changes in market power to the labor share (Eggertsson, Robbins, and Wold, 2018;Barkai, forth.) and to investment and interest rates (Brun and González, 2017;Gutiérrez and Philippon, 2017a;Eggertsson, Robbins, and Wold, 2018). The models developed by Brun and González (2017), Gutiérrez and Philippon (2017a), Eggertsson, Robbins, and Wold (2018), and Barkai (forth.) are based on a monopolistic competition framework with markups determined exogenously by the parameter reflecting the elasticity of substitution among products. In all cases, only product market power is considered and the firms are assumed to have no market power in labor or capital markets. Our theoretical framework differs from these because we explicitly model oligopoly and strategic interaction between firms in general equilibrium, which enables the study of how competition policy affects the macro economy. The concern about market power in both product and labor markets is a subject of policy debate; for example, the Council of Economic Advisers produced two reports (CEA, 2016a,b) on the issue of market power. Increased common own- 14 Blonigen and Pierce (2016) attribute the US increase in markups to increased merger activity. Barkai (forth.) documents declining labor and capital shares in the US economy over the past 30 years, an outcome that is consistent with an increase in markups. Acemoglu and Restrepo (2019), summarizing a body of work, argue that automation always reduces the labor share in industry value added and that it will tend also to reduce the economy's overall labor share. For example, Acemoglu and Restrepo (2018) report that the labor share declines more in industries (e.g., manufacturing) that are more amenable to automation. 15 A thriving literature in labor economics has established that individual firms face labor supply curves that are imperfectly elastic, which is indicative of substantial labor market power (Falch, 2010;Ransom and Sims, 2010;Staiger, Spetz, and Phibbs, 2010;Matsudaira, 2013;Azar, Marinescu, and Steinbaum, 2020). 16 There is an empirical debate on the validity and robustness of these results, since the Modified HHI is endogenous (see Gramlich and Grundl, 2017;Kennedy, O"Brien, Song, and Waehrer, 2017;O'Brien and Waehrer, 2017;Dennis, Gerardi, and Schenone, 2019). Backus, Conlon, and Sinkinson (2018) adopt a structural approach in their study of the cereal industry and find large potential (but not actual) implied effects of common ownership relative to mergers. ership has also raised antitrust concerns (Baker, 2016;Elhauge, 2016) and led to some bold proposals for remedies (Posner, Scott Morton, and Weyl, forth.;Scott Morton and Hovenkamp, 2018) as well as calls for caution (Rock and Rubinfeld, 2017).
There is an empirical debate about the trends in concentration and markups. Indeed, Rossi-Hansberg, Sarte, and Trachter (2018) find diverging trends for aggregate (increasing) and (decreasing) concentration. Rinz (2018) and Berger, Herkenhoff, and Mongey (2019) find also that local labor market concentration has gone down. Traina (2018) and Karabarbounis and Neiman (2019) find flat markups when accounting for indirect costs of production. Increases in concentration are modest overall in both product and labor markets and/or on too broadly defined industries to generate severe product market power problems (e.g, HHIs remain below antitrust thresholds in relevant product and geographic markets, e.g. Shapiro, 2018).
The question, then, is how to reconcile the evolution of concentration in relevant markets with evidence on the evolution of margins, increasing corporate profits, and decreased labor share. According to the monopolistic competition model, margins increase when products become less differentiated. It is however not plausible that large changes in product differentiation happen in short spans of time. We provide an alternative framework in which market concentration and ownership structure both have a role to play.

One-sector economy with large firms
In this section we first describe the model in detail. We then characterize the equilibrium and comparative statics properties with homogeneous and heterogeneous firms before offering a constant elasticity example. We conclude with a summary and by describing an extension that allows for investment.

Model setup
We consider an economy with (a) a finite number of firms, each of them large relative to the economy as a whole, and (b) an infinite number (a continuum) of people, each of them infinitesimal relative to the economy as a whole. There are two types of people, workers and owners, and both types consume the good produced by firms. Workers obtain income to pay for their consumption by offering their time to a firm in exchange for wages. The owners do not work for the firms; an owner's income derives instead from ownership of the firm's shares, which entitles the owner to control the firm and to a share of its profits. There is a unit mass of workers and a unit mass of owners, and we use I W and I O to denote (respectively) the set of workers and the set of owners. There are a total of J firms in the economy.
There are two goods: a consumer good, with price p; and leisure, with price w. Each worker has a time endowment of T hours but owns no other assets. Workers have preferences over consumption and leisure-as represented by the utility function U(C i , L i ), where C i is worker i's level of consumption and L i is i's labor supply. We assume that the utility function is twice continuously differentiable and satisfies U C > 0, U L < 0, U CC < 0, U LL < 0, and U CL ≤ 0. 17 The last of these expressions implies that the marginal utility of consumption is decreasing in labor supply.
The owners hold all of the firms' shares. We assume that the owners are divided uniformly into J groups, one per firm, with owners in group j owning 1 − φ + φ/J of firm j and owning φ/J of the other firms; here φ ∈ [0, 1]. Thus φ can be interpreted as representing the level of portfolio diversification, or (quasi-)indexation, in the economy. 18 If we use π k to denote the profits of firm k, then the financial wealth of owner i in group j is given by Total financial wealth is equal to ∑ J k=1 π k , the sum of the profits of all firms. The owners obtain utility from consumption only, and for simplicity we assume that their utility function is U O (C i ) = C i . A firm produces using only labor as a resource, and it has a twice continuously differentiable production function F(L) with F > 0 and F(0) ≥ 0. We allow for both F ≤ 0 and F > 0. We use L j to denote the amount of labor employed by firm j. Firm j's profits are π j = pF(L j ) − wL j .
We assume that firm j's objective function is to maximize a weighted average of the (indirect) utilities of its owners, where the weights are proportional to the number of shares. In other words, we suppose that ownership confers control in proportion to the shares owned. 19 In this simple case, because shareholders do not work and there is only one consumption good, their indirect utility (as a function of prices, wages, and their wealth level) is V O (p, w; W i ) = W i /p. Therefore, the objective function of the firm's manager is Brien and Salop (2000) for other possibilities that allow for cash flow and control rights to differ.
After some algebra we obtain that, for firms' managers, the objective function simplifies to maximizing (in terms of the consumption good) the sum of own profits and the profits of other firms-discounted by a coefficient λ. Formally, we have π j p We interpret λ as the weight-due to common ownership-that each firm's objective function assigns to the profits of other firms relative to its own profits. This term was called the coefficient of "effective sympathy" between firms by Edgeworth (1881) and also by Cyert and DeGroot (1973). The weight λ increases with φ, or the level of portfolio diversification in the economy, and also with market concentration 1/J. We remark that λ = 0 if φ = 0 and λ = 1 if φ = 1, so all firms behave "as one" when portfolios are fully diversified.
Next we define our concept of equilibrium.

Equilibrium concept
An imperfectly competitive equilibrium with shareholder representation consists of (a) a price function that assigns consumption good prices to the production plans of firms, (b) an allocation of consumption goods, and (c) a set of production plans for firms such that the following statements hold.
(1) The prices and allocation of consumption goods are a competitive equilibrium relative to the production plans of firms.
(2) Production plans constitute a Cournot-Nash equilibrium when the objective function of each firm is a weighted average of shareholders' indirect utilities.
It follows then that if a price function, an allocation of consumption goods, and a set of production plans for firms is an imperfectly competitive equilibrium with shareholder representation, then also a scalar multiple of prices will be an equilibrium with the same allocation of goods and production. The reason is that (a) the indirect utility function is homogeneous of degree 0 in prices and income; and (b) if a consumption and production allocation satisfies both (1) and (2) with the original price function, then it will continue to do so when prices are scaled.
We start by defining a competitive equilibrium relative to the firms' production plans-in the particular model of this section, a Walrasian equilibrium conditional on the quantities of output announced by the firms. To simplify notation, we proxy firm j's production plan by the quantity L j of labor demanded, implicitly setting the planned production quantity equal to F(L j ).
Definition 1 (Competitive equilibrium relative to production plans). A competitive equilibrium relative to (L 1 , . . . , L J ) is a price system and allocation [{w, p}; (ii) Labor supply equals labor demand by the firms: i∈I W L i di = ∑ J j=1 L j .
(iii) Total consumption equals total production: i∈I A price function W(L) and P(L) assigns prices {w, p} to each labor (production) plan vector L ≡ (L 1 , . . . , L J ), such that for any L, A given firm makes employment and production plans conditional on the price function, which captures how the firm expects prices will react to its plans as well as its expectations regarding the employment and production plans of other firms. The economy is in equilibrium when every firm's employment and production plans coincide with the expectations of all other firms.
Definition 2 (Cournot-Walras equilibrium with shareholder representation). A Cournot-Walras equilibrium with shareholder representation is a price function (W(·), P(·)), an allocation ( , and a set of production plans L * such that: (ii) the production plan vector L * is a pure-strategy Nash equilibrium of a game in which players are the J firms, the strategy space of firm j is [0, T], and the firm's payoff function is Here p = P(L), w = W(L), and π j = pF(L j ) − wL j for j = 1, . . . , J.
Note that the objective function of firm j depends only on the real wage ω = w/p, which is invariant to any normalization of prices.

Characterization of equilibrium
Given firms' production plans, we derive the real wage-under a competitive equilibrium-by assuming that workers maximize their utility U(C i , L i ) subject to the budget constraint C i ≤ ωL i . This constraint is always binding because utility increases with consumption but decreases with labor. Substituting the budget constraint into the utility function of the representative worker yields the following (equivalent) maximization problem: Our assumptions on the utility function guarantee that the second-order condition holds. Hence the first-order condition for an interior solution implicitly defines a labor supply function h(ω) for worker i such that labor supply is given by L i = min{h(ω), T}; this coincides with aggregate (average) labor supply, which is i∈I L i di. Let η denote the elasticity of labor supply. We assume that preferences are such that h(·) is increasing. 20 This assumption is consistent with a wide range of empirical studies showing that the elasticity of labor supply with respect to wages is positive. A meta-analysis of such studies based on different methodologies (Chetty, Guren, Manoli, and Weber, 2011) concludes that the long-run elasticity of aggregate hours worked with respect to the real wage is about 0.59. We assume that the range of the labor supply function is [0, T], which-when combined with the preceding maintained assumptionguarantees the existence of an increasing inverse labor supply function h −1 that assigns a real wage to every possible labor supply level on [0, T]. In a competitive equilibrium relative to the vector of labor demands by the firms, labor demand has to equal labor supply: Any competitive equilibrium relative to firms' production plans L must satisfy either In what follows we shall use the price function that assigns ω = h −1 (T) when L = T. Given that the relative price depends only on L, we can define (with only minor abuse of notation) the competitive equilibrium real-wage function ω(L) = h −1 (L).

Cournot-Walras equilibrium: Existence and characterization
Here we identify the conditions under which symmetric equilibria exist. We also offer a characterization that relates the markdown of wages (relative to the marginal product of labor) to the economy's level of market concentration.
The objective of firm j's manager is to choose an L j that maximizes the following expression: We start by noting that firm j's best response depends only on the aggregate response of its rivals, the elasticity of the inverse labor supply's slope. Then a sufficient condition for the game (among firms) to be of the "strategic substitutes" variety is that E ω < 1. In this case, one firm's increase in labor demand is met by reductions in labor demand by the other firms and so there is an equilibrium (Vives, 1999, Thm. 2.7). Furthermore, if F ≤ 0 and E ω < 1 then the objective of the firm is 20 We can obtain the slope of h by taking the derivative with respect to the real wage in the first-order condition. This procedure yields The implication here is that the competitive equilibrium real wage as a function of (L 1 , . . . , L J ) depends on firms' individual labor demands only through their effect on aggregate labor demand L. strictly concave and the slope of its best response to a rival's change in labor demand is greater than −1. In that event, the equilibrium is unique (per Vives, 1999, Thm. 2.8).
Proposition 1. Let E ω < 1. Then the game among firms is one of strategic substitutes and an equilibrium exists. Moreover, if returns are non-increasing (i.e., if F ≤ 0) then the equilibrium is unique, symmetric, and locally stable under continuous adjustment (unless F = 0 and λ = 1). In an interior symmetric equilibrium, if the total employment level L * ∈ (0, T) then the following statements hold.
(a) The markdown of the real wage ω * is given by where H = (1 + λ(J − 1))/J is the labor market-modified HHI and where H and µ are each increasing in φ.
(b) Both L * and ω * are increasing in J and decreasing in φ.
Remark. To ensure a unique equilibrium, it is enough that −F (L j ) + (1 − λ)ω (L) > 0 if the secondorder condition holds. In this case we may have a unique (and symmetric) equilibrium with moderately increasing returns. Note that F < 0 is required if the condition is to hold for all λ. Furthermore, it is possible to show that together the inequalities −F + (1 − λ)ω > 0 and ω > 0 are enough to ensure that a symmetric equilibrium exists and, in addition, that there are no asymmetric equilibria. And if also F ≤ 0 and E ω < 2 when evaluated at a candidate symmetric equilibrium, then the symmetric equilibrium is unique for any λ (and is stable provided that λ < 1). These relaxed conditions allow for strategies that are strategic complements.
Remark. If F = 0 (constant returns) and if λ = 1 (φ = 1, firm cartel), then there is a unique symmetric equilibrium and also multiple asymmetric equilibria, with each firm employing an arbitrary amount between zero and the monopoly level of employment and with the total employment by firms equal to that under monopoly. The reason is that the shareholders in this case are indifferent over which firm engages in the actual production.
Remark. The market power friction at a symmetric equilibrium can also be expressed in terms of the markup of product prices over the effective marginal cost of labor (mc ≡ w/F (L/JN)), rather than in terms of the markdown The Lerner-type misalignment of the marginal product of labor and the real wage (i.e., the markdown µ of real wages) is equal to the modified HHI divided by the elasticity η of labor supply. The question then arises of why there is no effect of the concentration and/or the residual demand elasticity in the product market. In other words: why does there seem to be no effect of product market power? The reason is that, when there is a single good, this effect (equal to product market modified HHI divided by demand elasticity) is exactly compensated by the effect of owners internalizing their consumption-that is, since they are also consumers of the product that the oligopolistic firms produce. Owners use firm profits only for purchasing the good. 22

Additively separable isoelastic preferences and Cobb-Douglas production
We now consider a special case of the model, one in which consumer-workers have separable isoelastic preferences over consumption and leisure: where σ ∈ (0, 1) and χ, ξ > 0. The elasticity of labor supply is η = (1 − σ)/(ξ + σ) > 0, and the equilibrium real wage in the competitive equilibrium-given firms' aggregate labor demand-can be written as The production function is F(L j ) = AL α j , where A > 0, 0 < α ≤ 1, and returns are non-increasing. The objective function of each firm is strictly concave, and so Proposition 1 applies. It is easily checked that total employment under the unique symmetric equilibrium is . Figure 1 illustrates that an increase in common ownership-that is, an increase in φ or a decrease in the number of firms-reduces equilibrium employment and real wages. With increasing returns to scale, however, reducing the number of firms involves a trade-off between market power and efficiency.
In that case, a decline in the number of firms can increase real wages under some conditions. The symmetric equilibrium is locally stable if α − 1 < (1 − λ)(Jη) −1 (1 + H/η) −1 , which means that a range of increasing returns may be allowed provided that an equilibrium exists. If α > 1, then neither the inequality −F + (1 − λ)ω > 0 nor the payoff global concavity condition need hold. In Appendix B we characterize the case where α ∈ (1, 2) and η ≤ 1 and then give a necessary and sufficient condition for an interior symmetric equilibrium to exist when returns are increasing. Under that condition, L * is decreasing in φ; yet it may either increase or decrease with J depending on whether the effect on the markdown or the economies of scale prevail.

Heterogenous firms
When firms have access to different constant returns to scale technologies (CRS), we confirm the results in Proposition 1 and establish a positive association between common ownership and the dispersion of market shares.
Proposition 2. Let E ω < 1 and firms have potentially different CRS technologies with F j (L j ) = A j L j , A j > 0, j = 1, . . . , J. Then an equilibrium exists and is unique with λ < 1. In an interior equilibrium with L * ∈ (0, T), the following statements hold.
(a) The markdown of real wages for firm j is given by where s * j ≡ L * j /L * and where the weighted average markdown is here H = HHI(1 − λ) + λ is the modified HHI, and both H andμ are increasing in φ.
(b) The total employment level L * and the real wage ω * are each decreasing in φ.
(d) If technologies are heterogeneous, then: (a) both the HHI and the minimal relative productivity for the least productive firm to be viable (i.e., A min /Ā, whereĀ = ∑ J j=1 A j /J) are increasing in φ; and (b) only the most productive firm is active when φ → 1.
Thus, under firm heterogeneity, an increase in common ownership as measured by φ (and λ) leads endogenously to an increase in the Herfindahl-Hirschman index. This follows because a firm's market share s j increases (resp. decreases) when λ increases if j has above-average (resp. below-average) productivity. The implication is that, when common ownership increases, the variance of s j increases and so the HHI increases as well. The effect of common ownership is similar to the behavior of a multi-plant monopolist who shifts production towards the more efficient plants. 23 Common ownership thus generates a re-allocation of market share from low-productivity, low-markup, high-labor share firms to highproductivity, high-markup, low-labor share firms. As stated in Section 2.2, a pattern of re-allocation from low-to high-markup US firms in recent decades is documented by De Loecker et al. (2020), and Autor, Dorn, Katz, Patterson, and Van Reenen (forth.) and Kehrig and Vincent (2018) both find evidence of a re-allocation from high-labor share to low-labor share firms.
Remark. At a Cournot interior equilibrium with constant marginal costs, total output does not depend on the distribution of costs (e.g., Bergstrom and Varian, 1985). Here, too, we have that total employment depends only on average productivityĀ = ∑ J j=1 A j /J and not on its variance. However, a technological change that induces a discrete increase in the dispersion of productivities, large enough to induce the exit of inefficient firms from the market, does affect equilibrium employment. In addition, an increase in common ownership may reinforce this effect. Indeed, we can show that the minimal relative productivity for the least productive firm to be viable (A min /Ā) is given by 24 Moreover, if it is not profitable for the jth least productive firm to produce a positive amount in equilibrium, then it is also not profitable to produce with λ > λ. As φ → 1, only the most productive firm survives and behaves like a monopsonist.

Summary and investment extension
So far we have shown that the simple model developed in this section can help make sense of some recent macroeconomic stylized facts-including persistently low output, employment, and wages in the presence of high corporate profits and financial wealth-as a response to a permanent increase in effective concentration (due either to common ownership or to a reduced number of competitors). Because we have yet to incorporate investment decisions into the model, there is no real interest rate and so we have nothing to say about how it is affected. Even so, the model can be extended to include saving, capital, investment, and the real interest rate. In Azar and Vives (2019a) we present a model with workers, owners and savers and show that-for investors who are not fully diversified-either a fall in the number J of firms or a rise in φ, the common ownership parameter, will lead to an equilibrium with lower levels of capital stock, employment, real interest rate, real wages, output, and labor share of income. Under certain (reasonable) conditions, the changes just described will lead also to a declining capital share.
When firms are large relative to the economy, an increase in market power implies that firms have an incentive to reduce both their employment and investment below the competitive level; this follows because, even though such firms sacrifice in terms of output, they benefit from lower wages and lower interest rates on every unit of labor and capital that they employ. The effect described here is present only when firms' shareholders perceive that they can affect the economy's equilibrium level of real wages and 23 It follows that increases in common ownership will raise the relative incentives of the more efficient firms to invest in cost reduction-that is, since they will end up producing more (see the model in López and Vives, 2019). 24 To see that the threshold is increasing in λ we use Lemma 1 in Appendix B, which shows that in equilibrium ∂η ∂λ + 1 > 0. real interest rates by changing their production plans. Thus, when oligopolistic firms have market power over the economy as a whole, their owners can extract rents from both workers and savers. 25

Multiple sectors
In this section we extend the model to multiple sectors in a Cobb-Douglas isoelastic environment. We characterize the equilibrium, uncover new and richer comparative statics results, and proceed to analyze large markets and convergence to the monopolistic competition outcome as the number of sectors grows large. We end the section with a note on calibration of the model.

Model setup
Consider an economy with N sectors, each offering a different consumer product. We assume that both the mass of workers and the mass of owners are equal to N. So as we scale the economy by increasing the number of sectors, the number of people in the economy scales proportionally. The utility function of worker i is as in the additively separable isoelastic model: ; here c ni is the consumption of worker i in sector n, and θ > 1 is the elasticity of substitution indicating a preference for variety. 26 For each product, there are J firms that can produce it using labor as input. The profits of firm j in sector n are given by π nj = p n F(L nj ) − wL nj ; here, the production function is F(L nj ) = AL α nj with A > 0 and α > 0. The ownership structure is similar to the single-sector case, except that now (i) there are J × N groups of shareholders and (ii) shareholders can diversify both in an industry fund and in a economy-wide fund. Group nj owns a fraction 1 − φ − φ ≥ 0 in firm nj directly, an industry index fund with a fraction φ/J in every firm in sector n, and an economy-wide index fund with a fraction φ/N J in every firm.
The owners' utility is simply their consumption C i of the composite good. Solving the owners' utility maximization problem yields the indirect utility function of shareholder i (i.e., V(P, w; W i ) = W i /P) when prices are {p n } N n=1 , the level of wages is w, shareholder wealth is W i , and P ≡ 1 is the price index. 25 Our model does not account for the possibility of inter-firm technological spillovers due to investment. López and Vives (2019) show that, if spillovers are high enough, then increased common ownership may boost R&D investment as well. 26 The form of C i is the one used by Allen and Arkolakis (2016). The weight (1/N) 1/θ in C i implies that, as N grows, the indirect utility derived from C i does not grow unboundedly and is consistent with a continuum formulation for the sectors (replacing the summation with an integral) of unit mass. More precisely: if the equilibrium is symmetric then, regardless of N, the level of consumption C i is equal to the consumer's income divided by the price.
The objective function of the manager of firm j in sector n is to choose the firm's level of employment, L nj , that maximizes a weighted average of shareholder (indirect) utilities. By re-arranging coefficients so that the coefficient for own profits equals 1, we obtain the objective function π nj P own profits +λ intra ∑ k =j π nk P industry n profits, other firms where the lambdas are a function of (φ, φ, J, N).
Thus the firm accounts for the effects of its actions not only on same-sector rivals but also on firms in other sectors. Note that the manager's objective function depends on N + 1 relative prices-that is, on w/P in addition to {p n /P} N n=1 for N > 1. We can show that the Edgeworth sympathy coefficient for other firms in the same sector as the focal firm is and that the Edgeworth sympathy coefficient for firms in other sectors is given by Observe that λ intra is no less than λ inter . This follows because the former sums the profit weights of both the industry fund and the economy-wide fund. We can show (see Lemma 2 in Appendix B) that λ intra and λ inter are always in [0, 1], increasing in φ and φ, and-for φ > 0 and φ + φ < 1-decreasing in N and J.
When φ + φ = 1, we have λ intra = 1 and λ inter = (1 − φ 2 )/[1 + φ 2 (N − 1)]; as a result, if agents are fully invested in the two index funds then λ intra = 1 regardless of the share in each fund. In contrast, the sympathy λ inter for firms in other sectors decreases as shares are moved from the economy index fund to the own-industry index fund φ. 27 Indeed, if everything is invested in the industry fund then φ = 1, λ intra = 1, and λ inter = 0. If there is no economy-wide index fund, then φ = 0, λ inter = 0, and λ intra = 28 Finally, if everything is invested in the economy-wide index fund then φ = 1 and λ intra = λ inter = 1.

Cournot-Walras equilibrium with N sectors
We start by characterizing the competitive equilibrium in terms of relative prices w/P and of {p n /P} N n=1 , given the production plans of the J firms operating in the N sectors: L ≡ {L 1 , . . . , L J }, where L j ≡ (L 1j , . . . , L Nj ). Then we characterize the equilibrium in the plans of the firms.

Relative prices in a competitive equilibrium given firms' production plans
Because the function that aggregates the consumption of all sectors is homothetic, workers face a twostage budgeting problem. First, workers choose their consumption across sectors (conditional on their aggregate level of consumption) to minimize expenditures; second, they choose labor supply L i and consumption level C i to maximize their utility U(C i , L i ) subject to the budget constraint PC i = wL i , where P is the price index.
We can therefore write the first-stage problem as The solution to this problem yields the standard demand of consumer i for each product n conditional on aggregate consumption C i : It follows from homotheticity that, for every consumer, total expenditures equal the price index multiplied by their respective level of consumption: In the second stage, the first-order condition for an interior solution is given by Since workers are homogeneous, it follows that total labor supply i∈I L i di is simply N times the individual labor supply L i ; moreover, because total labor demand L must equal total labor supply, equation (4.2.2) implicitly defines the equilibrium real wage (now relative to the price of the composite good) as a function ω(L) of the firms' total employment plans. We retain the assumptions for increasing labor supply that ensure ω > 0. Then ω(L) = χ 1/(1−σ) (L/N) 1/η , where again η = (1 − σ)/(ξ + σ) is the elasticity of labor supply. Shareholders maximize their aggregate consumption level conditional on their income. Their consumer demands, conditional on their respective levels of consumption, are identical to those of workers.
Adding up the demands across owners and workers, we obtain In a competitive equilibrium, consumption demand must equal the sum of all firms' production of each product: Using equation (4.2.1) and integrating across consumers, we have that c n = 1 N p n P −θ C. So given firms' production plans, the following equality holds in a competitive equilibrium: The elasticity of the relative price of sector n, p n /P, with respect to the aggregate production c n of the sector for given production in the other sectors (c m for m = n), when evaluated at a symmetric equilibrium, is −(1 − 1/N)/θ. Its absolute value is decreasing in the elasticity of substitution of the varieties (θ) and increasing in the number of sectors (N). Increasing c n has a direct negative impact on p n /P of −1/θ for a given C, and an indirect positive impact on p n /P by increasing aggregate real income C, yielding 1/(θN). When there is only one sector (N = 1) there is obviously no impact on the relative price. Furthermore, the overall effect increases with the number N of sectors because then the indirect effect is weaker.
We can now use equations (4.2.3) and (4.2.4) to obtain an expression for ρ n ≡ p n /P in a competitive equilibrium conditional on firms' production plans L: Observe that-unlike the previous case of a real-wage function, where the dependence was only through total employment plans-relative prices under a competitive equilibrium depend directly on the employment plans of each individual firm.
Proposition 3. Given the production plans L ≡ {L mj } of firms with aggregate labor demand L, the competitive equilibrium is given by the real wage ω(L) and the relative prices of the N sectors: ρ n (L) for n = 1, . . . , N. If firm j in sector n expands its employment plans, then ω increases; in addition, ρ n decreases (∂ρ n /∂L nj < 0) while ρ m , m = n, increases (∂ρ m /∂L nj > 0).
An increase in employment by a firm in sector n increases the relative supply of the consumption good of that sector relative to other sectors, thereby reducing the relative price of the focal sector's good. Since this increased employment increases overall supply of the aggregate consumption good while leaving supply of the other sectors unchanged, the relative prices of goods in those other sectors increase.

Cournot-Walras equilibrium
The optimization problem of firm j in sector n is given by where π nj /P = ρ n F(L nj ) − ω(L)L nj . The first-order condition for the firm is When a firm in a given sector considers hiring an additional worker, it faces the following tradeoffs. On the one hand, expanding employment increases profits by the value of the marginal product of labor (VMPL), which the shareholders can consume after paying the new workers the real wage. On the other hand, expanding employment will increase real wages for all workers because the labor supply is upward sloping. So when there is common ownership, the owners will take into account the wage effect not just for the firm that expans employment (or just for the firms in the same industry) but for all firms in all industries. Furthermore, expanding employment will increase output in the firm's sector and thereby reduce that sector's relative prices; as before, owners internalize that reduction not just for the firm itself but for all firms in the sector in which they have common ownership. Finally, expanding output in the firm's sector decreases consumption in all the other sectors and thus increases their relative prices; the owners of the firm, if they have common ownership involving other sectors, internalize these increased relative prices as a positive pecuniary externality. However, we will show that the own-sector negative price effect always dominates the effect of increased demand in other sectors.
As we establish in Appendix B, a firm's objective function is strictly concave if α ≤ 1. We therefore have the following existence and characterization result. 29 Proposition 4. Consider a multi-sector economy with additive separable isoelastic preferences and a Cobb-Douglas production function under non-increasing returns to scale (α ≤ 1). There exists a unique symmetric equilibrium, and equilibrium employment is given by The equilibrium markdown of real wages is where H labor ≡ (1 + λ intra (J − 1) + λ inter (N − 1)J)/N J is the labor market-modified HHI and H product ≡ (1 + λ intra (J − 1))/J is the product market-modified HHI for each sector.
Remark. Simulations reveal that µ * may be non-monotone in φ also if φ > 0. In fact, we can show that if η is large enough then µ * is decreasing in φ for φ > 0 small and increasing in φ for JN large. Furthermore, µ * is either increasing or decreasing in N.
In the multiple-industry case we find that the equilibrium real wage, employment, and output are analogous-as a function of the markdown-to those in the single-industry case. The only difference is that the markdown is now more complicated owing to the existence of multiple sectors and of product differentiation across firms in different sectors. An important result that contrasts with the singlesector case is that employment, output, and the real wage may all increase with diversification using the economy-wide fund φ.
Perfect substitutes. As the elasticity of substitution (θ) tends to infinity, the products of the different sectors become close to perfect substitutes; then the equilibrium is just as in the one-industry case but with JN firms instead of J firms. This outcome should not be surprising given that, in the case of perfect substitutes, all firms produce the same good and so-for all intents and purposes-there is but a single industry in the economy.
The two wedges of the markdown. The markdown of wages below the marginal product of labor can be viewed as consisting of two "wedges", one reflecting labor market power and one reflecting product market power. In particular, the labor market wedge is 1 + H labor /η. The markdown is increasing in H labor /η, which reflects the level of labor market power (and so decreases with JN and η). The product market wedge is (H product − λ inter )(1 − 1/N)/θ. This wedge has two components: the first is H product (1 − 1/N)/θ, reflecting the level of market power in the firm's sector; the second is λ inter (1 − 1/N)/θ, reflecting the inter-sectoral externality (note that the latter diminishes as products become more substitutable and θ increases). The markdown is increasing in the first component of the product market wedge, and decreasing in the second component. 30 From the previous paragraph it follows that µ * is positively associated with λ intra because so also are both the labor and product wedges-that is, since H labor and H product are increasing in λ intra . However, µ * may be positively or negatively associated with λ inter because, when λ inter > 0, we must account for the effect of expanding employment (by firm j in sector n) on the profits of other firms. Expanding employment in one sector benefits firms in other sectors by increasing the relative prices in those sectors (pecuniary externality) via the increase in overall consumption generated by firm nj's expanded employment plans. The result is that H product is then reduced by λ inter (note that H product ≥ λ inter always). If an increase in λ inter increases the labor market wedge more than it reduces the product market wedge, then µ * is decreasing in λ inter ; the converse of this statement holds as well.
Case with no industry fund. When φ = 0, we have λ intra = λ inter = λ; then the net effect of an increase in λ (due to an increase in φ) will be to diminish the product market wedge. To see this, note that (H product − λ)(1 − 1/N)/θ = (1 − λ)(1 − 1/N)/(θ J). In the limit, when φ = 1 and λ = 1, we have a cartel or a monopoly and the two product market effects cancel each other out exactly. 31 It is worth noting that µ * may either increase or decrease with portfolio diversification φ depending on whether labor market effects or rather product market effects prevail. The markdown will be decreasing in φ when the increase in the labor market wedge (due to the higher φ) is more than compensated by the lower product market wedge (due to the pro-competitive inter-sectoral pecuniary externality)-in other words, when the effect of profit internalization on the level of market power in product markets is higher than it is in the labor market. This happens when the elasticity of substitution θ is small in relation to the elasticity of labor supply η. When η → ∞, common ownership always has a pro-competitive effect. If N is large, then the anti-competitive effect of common ownership prevails provided that η < 1. This outcome follows because then θ/(1 + η) > 1/2 and Under the parameter configurations for the elasticities considered in Azar and Vives (2019a), θ = 3 and with a conservative η = 0.6, we have that θ 1+η > 1 2 . In consequence, the anticompetitive effect will prevail for N large. 30 Recall that, when evaluated at a symmetric equilibrium, the (absolute value of the) elasticity of "inverse demand" p n /P with respect to c n is (1 − 1/N)/θ; this explains why H product (1 − 1/N)/θ is the indicator of product market power (note that this indicator decreases with J and θ but increases with N). 31 When portfolios are perfectly diversified (φ = 1), the economy can be viewed as consisting of a single large firm that produces the composite good. Since the owner-consumers own shares in each of the components of the composite good in the same proportion and since they use profits only to purchase that good, these owner-consumers are to the same extent shareholders and consumers of the composite good. So just as in the single-sector economy, the effects cancel out exactly. The N = 1 case is the one-sector model developed in Section 3. Here λ = 1 can be understood in similar terms except that, in this case, there is an aggregate good C.

Large economies
Most of the literature on oligopoly in general equilibrium considers the case of an infinite number of sectors such that each sector, and therefore each firm, is small relative to the economy. Monopolistic competition can be viewed as a special case of a model with infinite sectors in which there is only one firm per industry. Here we consider what happens when the number of sectors, N, tends to infinity.
Our aim is to identify the conditions under which the monopolistically competitive limit is obtained (as in 1977monopolistic). We consider the following cases where owners: (i) hold fully diversified portfolios, (ii) are not diversified (iii) are fully diversified only intra-industry for N large, and (iv) are fully diversified for N large.

Case 1: Full diversification
When all the owner-consumers hold market portfolios, λ intra(N) = λ inter(N) = 1. This means that we have a sequence of economies with an increasing number of sectors and firms but in which the equilibrium outcome remains the same. The product market wedge disappears because the owner-consumers fully internalize the effect of firms' decisions on themselves as consumers. As with the one-sector model, the labor market wedge remains at the monopsony level-here, because owner-consumers still have an incentive to reduce the real wages of worker-consumers. We remark that, if the model had a representative agent rather than owner-consumers and worker-consumers, then the labor market wedge would also disappear and the equilibrium would be efficient.

Case 2: No diversification
Consider now the case in which owner-consumers hold shares in only one firm. In this case, as the number of sectors tends to infinity, the labor market wedge disappears as the number of firms interacting in the labor market goes to infinity (this result would not hold if the labor markets were segmented, for example, by industry). With J > 1 firms, the limit economy is equivalent to that of Neary (2003b): a continuum of sectors, no labor market power, and a homogeneous-goods Cournot equilibrium in each sector (if goods were heterogeneous within sector, then the limit economy would be equivalent to that of Atkeson and Burstein, 2008). In the case of J = 1, the limit economy in this case is equivalent to that of the Dixit and Stiglitz (1977) monopolistic competition model.
One must bear in mind, however, that obtaining these economies as a limit in the model requires heterogeneous agents: owner-consumers and worker-consumers; also, within the owner-consumers, there must be different groups with each group having ownership in just one firm. If, as in Neary (2003b), Atkeson and Burstein (2008), and Dixit and Stiglitz (1977), we assumed a representative agent (a) there would be fully diversified owner-worker-consumers; and (b) the equilibrium of the economy at each point in the sequence of economies would be efficient, with price equal to marginal cost. Even though the models in these papers assume profit maximization, no shareholder would actually want the firms to maximize profits. This tension was discussed in Section 2.

Case 3: Only intra-industry asymptotic diversification
When φ = 0, oligopsony power vanishes in the limit because (again) the number of firms competing in the labor market goes to infinity and there is no inter-industry internalization effect (λ inter(N) = 0). Thus, the limit economy is equivalent to that of Neary (2003b) but with horizontal, within-industry common ownership. In this case, for any N we have the same formula as for the one-sector model except with φ N instead of φ: Here the markdown increases with φ when J > 1 (in this formula, H product(∞) refers to the limit product market MHHI, which is 1/J + λ intra(∞) (1 − 1/J)). If φ = 1, then H product(∞) = 1 and µ * ∞ = 1/(θ − 1). Recall that the market power friction at a symmetric equilibrium can also be expressed in terms of the markup of product prices over the effective marginal cost of labor (mc ≡ w/F (L/JN)), rather than in terms of the markdown. We have thatμ * → 1/θ (the monopolistic competition markup of Dixit and Stiglitz, 1977) when there is essentially one firm per sector (either J = 1 or λ intra(∞) = 1; e.g., φ → 1). 32

Case 4: Full asymptotic diversification
We consider now the case with full asymptotic diversification (φ N → 1). For simplicity, we assume no industry fund: We start by observing that, if φ N → φ < 1, then λ N → 0. This is so because, as the number of sectors in the economy increases: for a shareholder in group nj, the fraction held in each of the other firms (when φ is constant) is φ/(N J), which goes to zero, while the fraction 1 − φ + φ/(N J) held in firm nj does not. In this case, then, the equilibrium of the limit economy is like the one in Case 2 (no diversification); hence it is equivalent to Neary (2003b) when J > 1 and to Dixit and Stiglitz (1977) when J = 1.
Consider now the case when φ N → 1, or, equivalently, 1 − φ N → 0. In that case, the limit lambdas can take values between zero and one, depending on the speed of convergence. In particular, to have 32 Similar results are obtained with some economy-wide diversification. Suppose φ < 1 and φ > 0 are fixed; then, as N → ∞, we have that λ inter → 0 (and oligopsony power vanishes, since H labor → λ inter(∞) = 0) but that where γ ≡ (1 − φ)/ φ > 0 is the ratio of undiversified investment to investment in the industry fund. The parameter γ ranges from 1 to infinity: γ = 1 when 1 − φ = φ (e.g., as when φ = 1); and γ → ∞ as φ → 0. If γ = 1 then λ intra(∞) = 1, and if γ → ∞ then λ intra(∞) = 0. λ N → λ ∈ (0, 1], we need the sequence 1 − φ N to approach zero (full diversification) at least as rapidly as 1/ √ N (i.e., √ N(1 − φ N ) → k for k ∈ [0, ∞)). If the convergence rate is faster than 1/ √ N with k = 0, then [above]the limiting λ is always equal to 1, and the equilibrium in the limit economy is the same as in Case 1 (full diversification). If the convergence rate is slower than than 1/ √ N then the limiting λ is equal to zero, and the equilibrium in the limit economy is the same as in Case 2 (no diversification).
For sequences 1 − φ N with convergence rates equal to 1/ √ N, the value of λ in the limit is determined by k, the constant of convergence: The impact of λ ∞ on the markdown depends, as before, on whether (or not) its effect on the labor market wedge effect dominates its effect on the product market wedge. The labor market wedge effect dominates the product market wedge effect if and only if the elasticity η of labor supply is lower than θ J − 1. These results are summarized in our next proposition.

Proposition 5. Consider a sequence of economies
-which is increasing in concentration 1/J and in the speed of convergence of φ N → 1 as measured by the constant 1/k.

The limit markdown is µ *
That is to say: if full diversification is attained at least as fast as 1/ √ N as the economy grows large, then profit internalization is (a) positive in the limit and (b) increasing both in concentration and in how rapidly diversification is achieved. The limit markdown increases with profit internalization if and only if θ/(1 + η) > 1/J. Only when λ ∞ = 0 do we obtain the markdown associated with the Dixit-Stiglitz or Neary µ * ∞ = 1/(Jθ − 1). When λ ∞ > 0, however, we obtain a different limit. In this case, if J → ∞ then there is no product market power and so the markdown λ ∞ /η (i) is due only to labor market power and (ii) increases with λ ∞ . When η → ∞, the labor market is competitive and the markdown is decreasing in λ ∞ . Finally, if λ ∞ = 1 then we obtain the monopsony solution µ * ∞ = 1/η.

Calibration
The model is parsimonious enough that it can be calibrated with only a few parameters. In the US economy and under our maintained assumption of proportional control, the weights that managers put on rivals' profits (i.e., the lambdas) have increased dramatically over the past decades. In the United States, for example, the 1,500 largest firms (by market capitalization) nearly doubled their calibrated average intra-industry lambdas: from about 0.41 in 1985 to about 0.72 in 2017 (see Figure 2). We adjust these lambdas downward in our calibration to account for privately held firms (which we assume have no common ownership) representing 58.7% of sales in the economy (Asker, Farre-Mensa, and Ljungqvist, 2014  The model has been extended in Azar and Vives (2019a) to include savings and capital, and shown able to reproduce macroeconomic trends such as the secular decline in the US economy's labor share, and also approximate the decline in the capital share. The key to their approximation is using the evolution of effective (i.e., including the influence of common ownership) concentration in product and labor markets, thereby combining market power in product and labor markets with the evolution of common ownership. 34 With this we do not claim that common ownership is the cause of the evolution of markups and markdowns and of the decline of labor and capital shares, but only that it has the potential to explain it.
The question arises as to what explains observed increases in the lambdas. Banal-Estañol, Seldeslachts, and Vives (2018)  This need not lead necessarily to a higher degree of internalization of rivals' profits, since passive investors could (in principle) exert less control than active ones. However, passive shareholders are more diversified, and the shift toward passive investors does help explain (statistically) the increase in profit internalization. The authors also report, for a cross-section of industries, a positive association between increases in the intra-industry lambda and increasing markups. Atkeson and Burstein (2008) also calibrate a model of oligopoly in general equilibrium. Our calibrations are similar along some dimensions but differ along others. These authors calibrate higher product 34 We do not need to assume symmetric firms for the simulation since we can input the modified HH I for an asymmetric market structure. Indeed, an industry with a very uneven distribution of firms' market shares may have a high HH I even with a large number of firms. market power parameters, 35 but the two quantitative models differ substantially since we consider labor market power and common ownership whereas they do not. 36 We have already mentioned that, although firms in their model are under full common ownership (by the representative household that owns all firms), those firms are still assumed to maximize profits. In our calibration, common ownership is only partial, but it is taken into account by the firms, reducing the effective number of firms to 3.2 in the product market and to 3.3 in the labor market in 2017 (the corresponding numbers for 1985 are 4.5 and 4.2). In addition, common ownership in our model implies a pro-competitive internalization of the inter-sectoral externality. 37 Overall, these differences imply that in our model the (p − c)/p markup (including the labor and product market wedges) increases from 33% to 38% over the  period-as compared with the markup of 29% calculated by Atkeson and Burstein (2008) (even though their product market markup is much higher than our product market wedge). 38

Competition policy
In this section, we show how equilibrium outcomes in oligopolistic economies are suboptimal from a social welfare perspective before considering the potentially beneficial effects of competition policies.
Our model is static and should therefore be interpreted as capturing only long-run phenomena. In this model, then, the low levels of output and employment are of a long-run nature and so could be affected by fiscal policy but not by monetary policy. 39 Competition policy (broadly understood to encompass regulation) can influence aggregate outcomes by directly affecting product and labor market concentration-that is, by affecting the number of firms and also the extent of their ownership overlap. 40 We illustrate the analysis with the one-sector model Cobb-Douglas isoelastic specification. We explore in turn the social planner allocation (first best) and competition policy (second best); we then conclude with some remarks on the multi-sector model. 35 For the inter-sectoral elasticity of substitution we set θ = 3 based on estimates by Hobijn and Nechio (2015); in contrast, Atkeson and Burstein (2008) use a value of 1.01 "to keep sectoral expenditure shares roughly constant." Hence their calibration implies a much lower market-level elasticity of demand and thus the potential of far more market power in the product market. Since we assume that goods within a sector are homogeneous, our calibration of the intra-sector elasticity of substitution parameter is that it is infinite, while they calibrate it to 10, which is a large number, but still implying some differentiation and therefore more product market power than in our model. Their assumptions imply an effective number of firms equal to 6.7 in the product market, whereas we calculate that number to decline from 6.1 to 4.6 over the period 1985-2017. 36 Atkeson and Burstein (2008) assume that firms are price takers in the labor market, whereas we assume that they have market power in the labor market. In particular, for the calibration we assume that labor markets are segmented by industry and that (based on estimates by Chetty, Guren, Manoli, and Weber, 2011) the market-level elasticity of labor supply is 0.59. Our labor market HHI increases from 1,798 to 1,965 over 1985-2017 (i.e., the effective number of firms declines from 5.6 to 5). 37 The internalization of the externality does not disappear when the number of firms tends to infinity when firms are owned by a representative household owning the market portfolio, and therefore it still exists in a model with a continuum of firms. 38 Finally, we remark that their model assumes constant returns in labor whereas we (a) assume decreasing returns and (b) calibrate the associated function parameter so that our calibrated model's labor share matches the BLS labor share in the year 1985. 39 The effects of government employment policies are examined in Azar and Vives (2019b). 40 We do not consider here conduct regulation to limit markdowns and markups under a free entry constraint (see, e.g. Vives, 1999, Sec. 6). Note, however, that conduct regulation is approximated here by controlling common ownership because of its direct link with margins.

Social planner's solution in the one-sector model
Here we characterize-in the one-sector, Cobb-Douglas, additively separable, isoelastic model-the allocation that would be chosen by a benevolent social planner who maximizes a weighted sum of the utilities of all owner-consumers with weight κ ∈ [0, 1] and of all worker-consumers with weight 1 − κ. 41 We assume that the social planner can choose the allocation of labor and consumption as well as the number of firms (with access to a large number J max ). Let (C, L) be the consumption and labor supply of a representative worker, and let C O be the consumption of a representative owner; then the social planner's problem is constrained by C + C O ≤ J A(L/J) α = AL α (1/J) α−1 . This constraint will always hold, since otherwise it would be possible to increase welfare by increasing workers' consumption until the constraint binds. Hence the problem can be rewritten as This problem can be solved in two steps. First we choose the welfare-maximizing C and L conditional on the number J of firms that are used (symmetrically) in production. Second, we maximize over J to obtain the optimal number of firms from the social planner's perspective.
The first-order conditions (which are sufficient under non-increasing returns to scale) for the first maximization problem ensure that, in an interior solution, (i) the marginal utility C −σ of workers' consumption is equal to κ/(1 − κ) multiplied by the owners' marginal utility of consumption (which is constant and equals 1) and (ii) C −σ is equal also to the marginal disutility from working divided by the marginal product of labor: χL ξ /(Aα(L/J) α−1 ). 42 This condition cannot hold in an oligopsonistic equilibrium because the markdown of wages relative to the marginal product of labor is positive, which introduces a wedge between the marginal product of labor and the real wage.
How many firms will the social planner choose to involve in the production process? If there are decreasing returns to scale, then social benefits are increasing in J and so the optimal choice is J max .
With constant returns to scale, the number of firms in operation is irrelevant. Under increasing returns to scale, the social planner would choose to produce using only one firm; however, the planner would still set-contra the monopsonistic outcome-the marginal product of labor equal to the marginal rate of substitution between consumption and labor. 43 Thus, from the viewpoint of a social planner, there is no Williamson trade-off because the planner can set the "shadow" markdown to zero and still benefit fully from the economies of scale due to producing with only one firm. Next we address the second-41 One can interpret κ as determining the welfare standard used by society. Thus κ = 0 represents the case of a " workerconsumer welfare standard" in which owners' utilities are assigned zero weight; this case is analogous-in our general equilibrium oligopoly model-to that of the usual partial equilibrium consumer welfare standard. The case κ = 1/2 corresponds to a "total welfare standard" in which all agents' utilities are equally weighted. 42 However, it is possible-for sufficiently low values of κ-for there to be a corner solution such that all the output is assigned to the workers and the consumption of the owners is zero; that is, C = AL α and C O = 0. 43 With increasing returns to scale, and α < 1 + ξ, the objective of the social planner is convex in L below a threshold, and concave in L above that threshold. This guarantees that the optimal L is strictly positive (however, just like in the nonincreasing returns case, there can be a corner solution for the consumption of the workers and the owners, that is C = AL α and C O = 0). If α > 1 + ξ, in some cases there could be a corner solution with L = 0. best allocation, under which the planner can affect the oligopoly equilibrium only by controlling the variables J and φ.

Competition policy
The models developed so far illustrate how the level of competition in the economy has macroeconomic consequences, from which it seems reasonable to conclude that competition policy may stimulate the economy by boosting output and inducing a more egalitarian distribution of income. We showed that if returns to scale are non-increasing then employment, output, real wages, and the labor share all decrease under higher market concentration and more common ownership.
In the one-sector case, the equilibrium modified HHI (H) is the same for the product and labor markets and is proportional to the markdown of wages relative to the marginal product of labor in the economy. In the multi-sector case, the markdown is a function of both the within-industry and the economy-wide modified HHIs, of which the latter is most relevant for the labor market. (In practice, labor markets are segmented and so the labor market modified HHI would differ from the economywide one; however, the insight would be similar.)

Worker-consumer welfare
We can view the competition policy in our model as setting a policy environment that affects-in a symmetric equilibrium-the number of firms per industry and/or the extent of common ownership. We start by showing that 1 − φ and J are complements as policy tools. Then common ownership mitigates the effect of "traditional" competition policy on employment because increasing the number of firms has less of an effect on concentration when firms' shareholders are more similar. Proposition 6. Let α < 1 + 1/η and let L * be a symmetric equilibrium. Then reducing common ownership (increasing 1 − φ) and reducing concentration (increasing J) are complements as policy tools for increasing equilibrium employment.
The proposition follows because it can be shown that for J > 1, η < ∞, and ∂λ ∂(1−φ) < 0. We remark that this proposition holds under decreasing returns and also in our increasing returns example (see Appendix A) with η ≤ 1 and α ∈ (1, 2).
Under either constant or decreasing returns to scale, it is always welfare-increasing for workerconsumers if the planner's policy reduces diversification (common ownership) and increases the number of firms-although the latter claim need not apply under increasing returns. Under non-increasing returns, the result follows because L * increases with both 1 − φ and J, equilibrium real wages increase with employment, and worker-consumer utility increases with real wages. Under increasing returns, however, there is a trade-off between market power and efficiency; in this scenario, the optimal number of firms (from the perspective of worker-consumer welfare) is limited. 44 In short: if returns to scale are increasing, then a decrease in the equilibrium markdown does not always translate into an increase in worker-consumer welfare. When returns are non-increasing, however, competition policy can lead to equilibria that are arbitrarily close to the social planner's as J max becomes large. This is because the markdown then becomes arbitrarily close to zero.
Entry. Until now we have assumed that the number of firms is fixed. We could consider an extension of the model whereby a large number of groups of potential owners can create new firms by paying a fixed cost. Once a new firm is created, its shares can be traded on the stock market. If we assume that the group creating the firm must retain a fraction 1 − φ of the firm's shares yet can also exchange up to φ of their shares for shares in the index, then we can easily re-create our model as a post-entry stage during which entry decisions depend on the entrant's expected profitability. In this world, common ownership will tend to magnify the excess entry results that hold in a Cournot market (see, e.g., Vives, 1999) although, according to some preliminary results, it will lead to decreased output and depressed wages as φ increases, only punctuated by upward jumps when a new firm enters.

Positive weight on owner-consumer welfare
The polar case of κ = 1, when the social planner maximizes the utility of the owner-consumers only, can easily be seen to imply-if we assume η ≤ 1-that setting φ = 1 will result in a completely concentrated economy in terms of the modified HHI, while choosing the number of firms to produce as efficiently as possible, which implies setting J = J max in the case of decreasing returns, J = 1 in the case of increasing returns, and any J ∈ {1, . . . , J max } in the case of constant returns. For intermediate values of κ, there is no simple analytic solution to the problem of choosing a competition policy that maximizes social welfare.
Yet we do know that, as κ increases, owner-consumer welfare increases while worker-consumer welfare declines; the implication is that equilibrium employment and wages are both lower when κ is higher. Azar and Vives (2018) simulate the optimal policy as a function of κ; they find that φ = 1 and ownerconsumers' welfare (resp., weakly increases, and employment and worker welfare weakly decrease, with κ. 45

Heterogeneous firms
Suppose firms have heterogeneous CRS technologies. Then, by Proposition 2(b), it is optimal to set φ = 0 if the aim is to maximize employment. Now suppose that the least efficient firm exits the market; then average productivity of the remaining firms will increase but total output and hired labor may 44 One can easily check that, for α ∈ (1, 2) and η ≤ 1, the total employment level L * increases with 1 − φ and peaks for J (when considered as a continuous variable) at η −1 (2 − α)/(α − 1). If J > η −1 (2 − α)/(α − 1), then α − 1 > (η J) −1 (1 + (η J) −1 ) −1 and the equilibrium would be unstable (see Appendix A). 45 With increasing returns to scale it easy to generate examples where it is optimal-even from the worker-consumers' standpoint, κ = 0-if some market power is allowed so as to exploit economies of scale. Typically, the number of firms declines as κ increases.
decline. This is what happens with a constant elasticity of labor supply. 46 Although removing the least productive firm reduces worker welfare, total welfare (including both worker-consumer and ownerconsumer welfare) can either increase or decrease.
As an example, consider an economy with two firms and parameters σ = 1/3, ξ = 1/3, χ = 1, and A 1 = 1. Suppose the common ownership parameter is φ = 3/4 (yielding λ = 0.5172) and that the social welfare function parameter κ is 1/2. In that case, if A 2 = 0.8 (i.e., if firm 2 is 80% as productive as firm 1) then removing firm 2 increases total welfare; whereas if A 2 = 0.9 then removing firm 2 reduces total welfare. If φ = 1/8 (such that λ = 0.1064), then removing firm 2 increases total welfare if A 2 = 0.6 but reduces it if A 2 = 0.9. Dropping one firm will be the outcome of a merger to monopoly, which owners will always favor despite the possibility of its reducing total welfare.

Competition policy with multiple sectors
In the one-sector case with the worker-consumer welfare standard (κ = 0), it is always efficient to force completely separate ownership of firms, regardless of how many firms there are, because there are no efficiencies associated with common ownership. In the multi-sector case, however, common ownership is associated with internalization of demand effects in other sectors; this means that-depending on the elasticity of substitution, the elasticity of labor supply, and the number of firms per industry-workerconsumers could be better-off under complete indexation of the economy. In any case, if maximizing employment is the goal, then it is better to set the intra-industry index fund ownership to zero (i.e., φ = 0), and, if returns to scale are decreasing, produce with the maximum number J max of firms. Along these lines, what follows can be viewed as a corollary of our previous results.
In short, with N sectors and non-increasing returns to scale, employment, real wages, and the welfare of worker-consumers are maximized when J = J max ,φ = 0, and when φ = 0 (resp., φ = 1) if θ(J max − 1/N) > (1 + η)(1 − 1/N) (resp., if inequality is reversed). So if the product market wedge effect dominates the labor market wedge effect (i.e., low θ and high η), then allowing full economy-wide common ownership increases equilibrium employment. Conversely, if the labor market wedge effect dominates the product market wedge effect then the optimal policy, as in the one-sector case, is no common ownership.
For large economies, the following analogous result holds. There exists anN such that, for economies with N >N, maximizing employment requires that the planner: (i) set φ = 0 and J = J max ; and (ii) set

Conclusion
We have provided a tractable model of oligopoly in general equilibrium that accommodates the influence of ownership structure. By assuming that managers maximize a weighted sum of utilities of 46 Proof available on request. 47 Even under Neary's (2003b) assumption of no common ownership, competition policy has an effect when firms across all sectors employ the same CRS technology. This result follows because, in our model, the supply of labor is elastic (and so changes in the real wage affect both employment and output) and there are two types of agents. If our model included only worker-owner-consumers, then the representative agent would always choose the optimal level of employment.
shareholders in a firm, we identify a numéraire-free Cournot-Walras equilibrium and characterize it.
In our model, firms' employment decisions affect prices in both product and factor markets. We find that a higher effective market concentration, which accounts for portfolio diversification and common ownership, increases markups and reduces both real wages and employment. Furthermore, when firms have heterogenous CRS technologies an increase in common ownership tilts the scales in favor of more efficient (superstar) firms and raises market concentration. When there are multiple industries, common ownership can have a positive or negative effect on the equilibrium markup: the sign of the effect depends on the relative magnitudes of the elasticities of product substitution and of labor supply. We find also that the monopolistically competitive limit (as in, e.g., Dixit and Stiglitz, 1977) or the oligopolistic one (Neary, 2003a,b;Atkeson and Burstein, 2008) may or may not be attained as the number of sectors in the economy grows large depending on the parallel evolution of diversification.
Competition policy can increase employment and improve welfare. In the one-sector economy we find that controlling common ownership and reducing concentration are complements in terms of fostering employment. With multiple sectors, to foster employment traditional competition policy on market concentration is adequate. However, common ownership can have a positive or negative effect on employment. Although its effect is negative in the intra-industry case, it could be positive in the case of economy-wide common ownership. Some caveats to our results follow from considering vertical relations between firms, and possibly different patterns of consumption between owners and workers. For example, vertical relations imply that products of one sector may serve as inputs for another sector. Then common ownership may lead to partial internalization of double marginalization and decrease markups. 48 In general, our results indicate a need to go beyond traditional partial equilibrium analyses of competition policy, where consumer surplus is king. However, traditional competition policy (e.g., lowering market concentration) remains a valid approach-as is limiting intra-industry ownership. That said, policy regarding economy-wide common ownership requires a more nuanced approach.
The models presented here are extremely stylized. We do not consider asymmetries in ownership structure across firms. Because the ownership structure is exogenous, with a separation between owners and workers, we consider neither the benefits of diversification in an uncertain world nor the effects

A Increasing returns to scale
If α > 1, then neither the inequality −F + (1 − λ)ω > 0 nor the payoff global concavity condition need hold. We characterize the situation where α ∈ (1, 2) and η ≤ 1. Then, with respect to L j , firm j's objective function has a convex region below a certain threshold and a concave region above that threshold. Hence we conclude that there are no more than two candidate maxima for L j , when given the other firms' decisions, at a symmetric equilibrium: L j = 0; and the critical point in the concave region (if there is any). We identify (after some work) the following necessary and sufficient condition for the candidate interior solution to be a symmetric equilibrium: For small λ, if an equilibrium exists then it is stable. Here L * is decreasing in φ, but it may either increase or decrease with J: Increasing the number of firms has two effects on a symmetric equilibrium with increasing returns to scale: a positive effect from fewer markdowns, and a negative effect from reduced economies of scale.
Thus a merger between two firms (decreasing J) would involve a so-called Williamson trade-off between higher market power and the efficiencies stemming from a larger scale of production. In our example, a merger would increase equilibrium employment if α were high enough to dominate the markdown effect.
A higher MHHI (the H in our formulation) makes it more difficult for the scale effect to dominate.
Yet for a given H, a higher internalization λ makes it easier for that effect to dominate because if λ is high enough then firms will act jointly irrespective of their total number J. In fact, if they act fully as one firm (λ = 1) then the condition is always fulfilled. Thus reducing J improves scale but does not affect the markdown because it is already at the monopoly level. It is easy to generate examples where, under increasing returns, there are multiple equilibria and some firms do not produce.
The first derivative ∂ζ/∂L j is given by F − ω − ω L j + λ ∑ k =j L k , so the best response of firm j depends only on ∑ k =j L k . The cross-derivative ∂ 2 ζ/∂L j ∂L m equals where s j ≡ L j /L and s −j ≡ ∑ k =j L k /L. If E ω ≡ −ω L/ω < 1, then the cross-derivative is negative because s j + λs −j ≤ 1 and In this case, Theorem 2.7 of Vives (1999) guarantees the existence of an equilibrium. The second derivative ∂ 2 ζ/(∂L j ) 2 equals F − 2ω − L j + λ ∑ k =j L k ω , and it is negative provided that F ≤ 0 also. Let L −j ≡ ∑ k =j L k and let R(L −j ) denote the best response of firm j. Then If the second-order condition holds, then R > −1 whenever −F + (1 − λ)ω > 0 and, indeed, whenever F ≤ 0 (except if F = 0 and λ = 1). When R > −1, Theorem 2.8 in Vives (1999) guarantees that the equilibrium is unique.
(a) From the first-order condition we have that, in a symmetric equilibrium, s j = 1/J for every j and The derivative of the markdown with respect to λ is given by The term F (L/J)ω(L) 1 J ∂L ∂λ is nonnegative if ∂L ∂λ < 0, and −F (L/J)ω (L) ∂L ∂λ is positive if ∂L ∂λ < 0. We show in part (b) of this proof that ∂L ∂λ < 0. (b) The symmetric equilibrium is given by the fixed point of L −j /(J − 1) = R(L −j ). Total employment is L = L −j + R(L −j ), which is increasing in L −j because R > −1. Furthermore, R is decreasing in λ because the objective function's first derivative is decreasing in λ. This implies that L −j -and hence also that L and ω(L) are decreasing in λ (and in φ). We have in addition that L −j is increasing in J since R < 0 and since R is itself increasing in J (i.e., because R is decreasing in λ and λ is decreasing in J).
Therefore, in equilibrium, L and ω(L) increase with J. given that returns to scale are non-increasing, F(L/J) − (L/J)F (L/J) ≥ 0. 50 Since employment is decreasing in φ, that implies the labor share is decreasing in φ as well.
The following lemma will be useful in the proof of Proposition 2.
Lemma 1. Suppose E ω < 1 and that firms have (possibly heterogeneous) CRS production functions. Then, in equilibrium, ∂η ∂λ + 1 > 0. Proof. We calculate the derivative ∂η ∂λ in two parts: as the product of ∂η ∂ log L and ∂ log L ∂λ . We find that where the last inequality holds because both η and (1 − E ω ) are positive.
To obtain an expression for ∂ log L ∂λ , we take a simple average of the first-order conditions of the firms and then differentiate with respect to λ: the absolute value of this expression is less than 1. This fact, when combined with the inequality ∂η ∂ log L < 1, implies that ∂η ∂λ > −1; therefore, ∂η ∂λ + 1 > 0. Proof of Proposition 2. As in the proof of Proposition 1, our analysis establishes the existence of a unique equilibrium when λ < 1. This claim follows directly because the slope of firm j's best response is given by the same expression as before just letting F = 0.
(a) From the first-order condition for firm j we obtain that its markdown is given by Taking an average weighted by market shares now yields Proposition 2's expression for the weighted average markdown: Solving for s j yields 50 If F(x) is increasing and concave for x ≥ 0 with F(0) ≥ 0, then F(x)/x ≥ F (x).
whereĀ ≡ ∑ J j=1 A j /J. We show that the HHI increases with φ whenever there is variation in firms' productivities, which is equivalent to showing that the HHI increases with λ. We find that Note that the last factor is positive, because the weighted average of the productivities is larger than the unweighted average: productivities. The first factor in the last line of (B.1) is positive as long as (a) there is dispersion in the productivities and (b) 1 + ∂η ∂λ > 0, which Lemma 1 establishes while assuming that E ω < 1. Therefore, an increase in λ increases the Herfindahl-Hirschman index.
The derivative of the modified HHI with respect to λ is ∂H ∂λ To show thatμ is increasing in λ, we rewrite it as since 1 + ∂η ∂λ > 0 and ∂L ∂λ < 0. (b) That L * decreases with λ was shown in part (a). Because labor supply is increasing, ω(L * ) also decreases with λ.
(c) The labor share in this case is As shown in part (a),μ is increasing in λ and so the labor share must be decreasing in λ.
(d) We showed in part (a) that HHI increases with λ when technologies differ. Now we show that the minimal productivity for a firm to be viable is increasing in λ. Assume, without loss of generality, that the firms are sorted by productivity: A 1 ≥ A 2 ≥ · · · ≥ A J . From the firms' first-order conditions, it follows that the condition for all firms to produce a positive amount in equilibrium is that The implication is that, in order for the least productive firm to produce in equilibrium, its productivity relative to average productivity (A J /Ā) must exceed the threshold (η+λ)/(1−λ) (η+λ)/(1−λ)+1/J . This threshold is increasing in λ (assuming that E ω < 1). To see that the threshold is increasing in λ, note that its derivative with respect to λ is (η+λ)+(1−λ)(∂η/∂λ+1) J[(η+λ)/(1−λ)+1/J] 2 (1−λ) 2 , which is positive when E ω < 1 because (by Lemma 1) ∂η ∂λ + 1 > 0. More generally, a necessary condition for the jth least productive firm to produce a positive amount in equilibrium is A k /j is the average of the productivities of the most efficient firms 1, . . . , j. Therefore, if it is not profitable for firm j to produce in equilibrium with λ, then neither is it profitable to produce in equilibrium with λ > λ.
The following lemma establishes the comparative statics properties of λ intra and λ inter with respect to the common ownership parameters φ, φ, N, and J.
Proof. Using the expressions for λ intra and λ inter from Section 4.1, we proceed by establishing these four claims in turn.
(i) The sign of the derivative of λ intra with respect to φ is given by In this expression: the first term is always nonnegative (and positive if 1 − φ − φ > 0); the second term is always nonnegative (and positive if 1 − φ − φ > 0, and either φ > 0 or φ > 0). Hence the derivative is positive in the interior of φ's domain, from which it follows that λ intra increases with φ.
The sign of the derivative of λ inter with respect to φ is given by sgn We have that therefore,the derivative is positive in the interior of φ's domain and so λ intra increases with φ.
and the derivative is positive in the interior of φ's domain. We therefore conclude that λ inter is increasing in φ.
(ii) The sign of the derivative of λ intra with respect to N is given by As a result, if φ > 0 and 1 − φ − φ > 0 then λ intra is decreasing in N.
With respect to λ inter , the term that multiplies N in the denominator (i.e., is positive for φ < 1. The numerator of λ inter is positive for φ > 0, so λ inter decreases with N for φ ∈ (0, 1).
Hence the denominators of both λ intra and λ inter are increasing in J as long as (a) 1 − φ − φ > 0 (we have shown already that, if 1 − φ − φ = 0, then λ intra and λ inter do not depend on J) and (b) given this condition λ intra and λ inter are decreasing in J.
(iv) Since [2(1 − φ) − φ] ≥ 0 with equality for φ = 1, it is immediate that the minimum value λ intra or λ inter can assume is 0. We have shown that λ intra and λ inter are either decreasing or constant in N. Thus they attain their maxima when N = 1, for which .
Proof of Proposition 3. The derivative of the relative price of a firm's own sector with respect to employment can be written as follows: Referring to Section 4.2.2, multiplying and dividing by L in the wage effect term, by c n in the ownindustry relative price effect term, and by c m in the other industry relative price terms, and using equation (B.3), the first-order condition for firm nj simplifies to: ρ n F (L nj ) − ω(L) − ω (L)L[s L nj + λ intra s L n,−j + λ inter (1 − s L nj − s L n,−j )] + ∂ρ n ∂L nj c n [s nj + λ intra (1 − s nj ) − λ inter ] = 0; here s nj ≡ F(L nj )/c n is the share of firm j in the total production of sector n, s L nj ≡ L nj /L, and s L n,−j ≡ ∑ k =j L nk /L. For the objective function of firm j in sector n, the second derivative is ∂ρ n ∂L nj F (L nj ) + ρ n F (L nj ) − 2ω (L) − ω (L) L nj + λ intra ∑ k =j Here ∂ 2 ρ n (∂L nj ) 2 = ∂ρ n ∂L nj ∂ρ n ∂L nj 1 ρ n 1 + (θ − 1) p n c n /PC 1 − p n /c n PC Replacing the latter in our expression (B.4) for the objective function's second derivative and then regrouping terms, we obtain The first row of this expression is negative because ∂ρ n ∂L nj is negative, F is positive, and the term in braces is positive since 1 θ 1 − p n c n PC + 1 − 1 θ p n c n PC < 1. The term in the second row is clearly negative. The third row's first term is nonpositive but its second term is nonnegative. Yet we can combine them to write ∂ρ n ∂L nj c n F (L nj ) F (L nj ) − θ 1 − p n c n /PC which is the product of three nonpositive factors (rendering the entire expression nonpositive). The fourth row is strictly negative because, with the constant elasticity functional form of utility, it is equal to The term 2 + 1 η − 1 [s L nj + λ intra s L n,−j + λ inter 1 − s L nj − s L n,−j )] is greater than 1 and is also multiplying a negative factor − ω L 1 η , so the second-order condition's fourth row is negative.
The objective function of each firm is thus globally strictly concave; therefore, any solution to the system of equations implied by the first-order conditions is an equilibrium. So in order to find the symmetric equilibria, we first simplify the first-order condition of firm nj when it is evaluated at a symmetric equilibrium-using c n = c for all n and p n = p for all n-and then note that c n /C = c/C = 1/N in the symmetric case.
In a symmetric equilibrium, the marginal product of labor is equal to F (L/JN). Using this equality and substituting c n /C = c m /C = 1/N in our expression (B.2) for the change in the relative price of the firm's industry when the firm expands employment plans, we can simplify it to Dividing the first-order condition by the real wage and then substituting the derivatives of the relative price that we just calculated yields In a symmetric equilibrium, the employment share of firm j in sector n is L nj /L = 1/JN for all sectors n and all firms j within that sector-that is, since the employment shares of all firms are the same. Similarly, the product market share of firm j in sector n is F(L nj )/c = 1/J. Plugging these into the previous equation implies that We can now express this in terms of MHHI values for the labor market and product markets as follows: Here H labor is the modified HHI for the labor market, which equals (1 + λ intra (J − 1) + λ inter (N − 1)J)/N J, and H product is the modified HHI for the product market of one industry, which equals 1/J + λ intra (1 − 1/J). We can obtain a closed-form solution for the constant-elasticity labor supply and Cobb-Douglas production function case. In that case, the equilibrium total employment level becomes where 1 + µ * = 1 + H labor /η 1 − (1/θ)(H product − λ inter )(1 − 1/N) .
We next prove four claims as follows. The equilibrium markdown of real wages µ * is: (1) increasing in φ; (2) decreasing in J if φ + φ < 1 but constant as a function of J if φ + φ = 1; (3) decreasing in the elasticity of labor supply η, and (4) decreasing in θ, the elasticity of substitution among goods by consumers, if φ < 1-but constant as a function of θ otherwise.
(1) According to Lemma 2, both λ intra and λ inter are increasing in φ and so likewise is H labor . We also The sign of this expression is given by which is positive for (1 − φ − φ) > 0 and so (H product − λ inter ) is increasing in φ. Furthermore, (H product − λ inter ) ≤ 1 (with equality when φ = 1) and so-in the fraction of our expression for µ * > 0-the numerator is increasing and the denominator is decreasing in φ; therefore, µ * increases with φ.

Consider now
which is decreasing in J as long as 1 − φ − φ > 0; otherwise, it is constant in J. We conclude that: (a) if 1 − φ − φ > 0, then the numerator and denominator in the fraction of our expression for µ * are (respectively) decreasing and increasing in J; and (b) if φ + φ = 1 then those two components are each constant as a function of J. So if 1 − φ − φ > 0 then the equilibrium markdown decreases with J; otherwise, it is unaffected by J.