Technological Innovation: Winners and Losers

Transcription

1 Technological Innovation: Winners and Losers Leonid Kogan Dimitris Papanikolaou Noah Stoffman September 19, 2012 Abstract We analyze the effect of innovation on asset prices in a tractable, general equilibrium framework with heterogeneous households and firms. Innovation is weakly correlated to future aggregate consumption growth, yet innovation risk carries a significant premium. In particular, technological improvements embodied in new capital benefit workers, while displacing existing firms and their shareholders. This displacement process is uneven: newer generations of shareholders benefit at the expense of existing cohorts; and firms well positioned to take advantage of these opportunities benefit at the expense of firms unable to do so. Under standard preference parameters, the risk premium associated with innovation is negative. Our model delivers i) realistic moments for the equity premium, risk-free rate and value premium; ii) the failure of the CAPM and CCAPM; iii) return comovement of firms with similar characteristics and the success of empirical factor models; iv) low correlation between labor income and dividends; v) lower risk premium to human capital than financial wealth; vi) higher return exposure for value firms to consumption of shareholders than growth firms. We derive and test the predictions of our model regarding the cross-section of firms and households using a direct measure of innovation. Our findings support the model s predictions. We thank Carola Frydman, Lars Hansen, Camelia Kuhnen, Martin Lettau, Monika Piazzesi Amit Seru, Martin Schneider, and the seminar participants in CITE, Northwestern University for valuable discussions. Dimitris Papanikolaou thanks the Zell Center for Risk and the Jerome Kenney Fund for financial support. Leonid Kogan thanks J.P. Morgan for financial support. MIT Sloan School of Management and NBER, lkogan@mit.edu Kellogg School of Management and NBER, d-papanikolaou@kellogg.northwestern.edu Kelley School of Business, nstoffma@indiana.edu

2 Introduction The history of technological innovation is a story of displacement. New technologies emerge that render old capital and processes obsolete. New technologies are typically embodied in new vintages of capital. Hence the process of adoption is not costless. For instance, the invention of the automobile by Karl Benz in 1885 required investment in new types of capital, such as paved highways. Hence, resources needed to be diverted in the short run into investment, in order for the economy to benefit in the long-run. However, not every economic agent benefitted similarly from the automobile. Railroad firms, which in the late 19th century accounted for 50% of the market capitalization of of all NYSE-listed firms, are displaced as the primary mode of transport. 1 Railroads, in turn, had displaced earlier forms of transportation, such as canals and waterways (see e.g. Meyer, 1979). In this paper we analyze how embodied technological innovation shocks are priced by the market and how such shocks affect the cross-section of stock returns. To understand how innovation risk is priced, we must consider how innovation affects the marginal investor in financial markets. As we show empirically, the relation between innovation and aggregate consumption growth is nonmonotone and relatively weak. Yet, there is strong evidence that exposure to innovation shocks is priced in the cross-section of asset returns. These patterns are difficult to reconcile with the standard complete-market equilibrium setting, in which consumption risk exposures of all agents are aligned thanks to perfect risk sharing and the stochastic discount factor is a function of aggregate consumption. We therefore consider an economy in which households have naturally heterogeneous background exposure to innovation shocks. Because of the lack of perfect risk sharing, innovation induces wealth reallocation among households, driving a wedge between the consumption risk of the marginal investor in financial markets and aggregate consumption risk. We develop a general equilibrium model with heterogenous firms and households with two types of aggregate technological growth. The first type of technological growth is disembodied, affecting equally all vintages of capital goods. The second type is embodied in new vintages 1 Flink (1990) writes: The triumph of the private passenger car over rail transportation in the United States was meteoric. Passenger miles traveled by automobile were only 25 percent of rail passenger miles in 1922 but were twice as great as rail passenger miles by 1925, four times as great by

3 of capital goods. This embodied shock leads to displacement in the cross-section of firms. Households do not fully share the risk of innovation shocks and therefore are also exposed to displacement. We emphasize two economically significant channels for the propagation of innovation shocks among households. First, households in our model have finite lives and each new cohort of households brings with it embodied technological advances. Currently present cohorts of households do not capture directly all of the profits generated by the future waves of innovation. Yet, innovations reduce the profitability of older vintages of capital owned collectively by the current cohorts of agents. Intergenerational risk sharing is lacking in our model, since households cannot trade with future cohorts prior to their arrival in the economy, and therefore periods of significant innovation result in wealth transfer from the current set of households to the future cohorts. Second, in our framework innovation shocks affect the value of physical capital and labor differently. A positive innovation shock effectively reduces the replacement cost of older vintages of physical capital while increasing demand for labor, which is used to create new, more productive vintages of capital goods. This creates a natural need for risk sharing between the owners of the physical capital stock (the inventors in our model) and the owners of labor (the workers ). Motivated by the well-documented empirical fact that large fractions of households derive most of their wealth from labor income and have low rates of financial market participation, we assume that the workers in our model do not participate in financial markets. Thus, there is a break-down of risk sharing between the owners of physical capital and workers. As a result, aggregate innovation shocks not only shift the shares of capital and labor in the flow of aggregate profits, but also lead to wealth reallocation between the owners of capital and workers. 2 Thus, asset holders in our model are more exposed to innovation risk than suggested by the aggregate or age-cohort-level consumption data. In summary, positive innovation shocks improve the investment opportunity set in the economy, but because of incomplete risk sharing, they lower the consumption share of the existing stockholders, leading to an increase in their the marginal utility. As a result, innovation shocks carry a negative price of risk in equilibrium. 2 This property of the model is in line with recently reported empirical evidence on the dynamics of income shares of financial and human capital in Lustig and Nieuwerburgh (2008), Lustig, Nieuwerburgh, and Verdelhan (2008), and Lettau and Ludvigson (2011). 2

4 Embodied technology shocks have a heterogenous impact on the cross-section of firms. Improvements in the frontier level of technology benefit firms able to capture a larger share of rent from the new inventions relative to firms that are heavily invested in older vintages of capital. Due to this heterogenous exposure to innovation shocks, stock returns of firms with similar access to growth opportunities are highly correlated with each other, above and beyond of what is implied by their exposures to the market returns. Specifically, stock returns in our model have a two-factor structure with one of the factors mimicking the displacive innovation shock. Because innovation shocks earn a negative price of risk in equilibrium, firms with more growth opportunities and thus higher exposure to embodied innovation shocks earn lower average returns. Observable firm characteristics, such as valuation ratios or past investment rates, are correlated with their growth opportunities. This endogenous correlation leads to realistic patterns of return comovement among firms with similar characteristics, and the cross-sectional relations between such characteristics and firms average stock returns. We calibrate our model to match several basic moments of real economic variables and asset returns, including the mean and volatility of the aggregate consumption growth rate, the equity premium, and the risk-free rate. Our model generates realistic differences in average returns between firms with high- and low- market-to-book ratios or investment rates, as well as return comovement among firms with similar characteristics. Further, our model replicates the failure of the CAPM and the Consumption CAPM in pricing the cross-section of stock returns, since neither the market portfolio nor aggregate consumption is a sufficient statistic for the marginal utility of market participants. We concentrate our empirical analysis on the properties of the model directly linked to its main economic mechanism displacement in the cross-section of households and firms generated by embodied innovation shocks. In our empirical tests, we use the measure of embodied technology shocks constructed in Kogan, Papanikolaou, Seru, and Stoffman (2012), which aggregates the stock market reaction to patent announcements. We construct the Kogan et al. (2012) in simulated data from the model, and show that their measure is a function of the frontier level of technology in our theoretical setting. We find empirical support for the model s prediction that innovation shocks generate displacement in the cross-section of households. Innovation shocks are correlated with cohort 3

5 effects in consumption of households in a manner predicted by the model. In particular, the level of technological innovation during the year when household heads enter the economy is associated with higher lifetime consumption, while innovation shocks following the cohort s entry tend to lower its consumption level relative to the rest of the economy. Moreover, as in the model, higher innovation predicts lower consumption growth of stockholders relative to non-stockholders. We relate the measure of innovation to firm output and stock returns. Consistent with the results of Kogan et al. (2012), we find that an increase in innovation of the firm s competitors is associated with its lower subsequent output growth. We find that firms with richer growth opportunities, measured either using the market-to-book ratio or past investment, are less exposed to displacement by their competitors. Such firms also have a relatively high return exposure to innovation shocks. We confirm empirically that innovation shocks earn a negative price of risk. A stochastic discount factor including innovation shocks prices a cross-section of book-to-market and investment-rate portfolios with low pricing errors. The point estimates of the market price of innovation risk are negative, statistically significant, and close to the estimates implied by the calibrated model. Our model replicates two additional stylized facts documented in the consumption-based asset pricing literature. First, the return differential between value and growth firms in the model has a relatively high exposure to the consumption growth of stockholders but only a weak exposure to the aggregate consumption growth. Second, the growth rate of labor income and dividends is weakly correlated. These two patterns are closely connected in our setting. Embodied innovation shocks raise equilibrium wages while reducing dividends on existing firms, which is the reason for the low correlation between the growth of dividends and labor income. As a result of limited risk sharing, this drives a wedge between innovation risk exposures of stockholders and non-stockholders: stockholders are relatively more vulnerable to displacement risk, as are firms with poor growth opportunities. We should emphasize that we study a particular form of technological innovation, specifically innovation that is embodied in new vintages of intermediate goods. Accordingly, our empirical measure of embodied shocks relies on patent data, since innovation that is embodied 4

6 in new products is more easily patentable (see for example Comin, 2008, for a discussion on patentable innovation). The type of innovation that we study could be related to other forms of innovation, such as skill-biased technical change, but the two need not be positively related. For instance, the first industrial revolution, a technological change embodied in new forms of capital the factory system led to the displacement of skilled artisans by unskilled workers, who specialized in a limited number of tasks (see e.g. Sokoloff, 1984, 1986; Atack, 1987; Goldin and Katz, 1998). Further, skill-biased technical change need not be related to firms growth opportunities in the same manner as the embodied technical change we consider in this paper. Nevertheless, we use the terms innovation and capital-embodied change interchangeably in this paper. Our work is related to asset pricing models with production (for a recent review of this literature, see Kogan and Papanikolaou, 2012a). Papers in this literature construct structural theoretical models with heterogenous firms and analyze the economic source of cross-sectional differences in firms systematic risk, with a particular focus on understanding the origins of average return differences among value and growth firms. Most of these models are in partial equilibrium (e.g., Berk, Green, and Naik, 1999; Carlson, Fisher, and Giammarino, 2004; Zhang, 2005; Kogan and Papanikolaou, 2011), with an exogenously specified pricing kernel. Some of these papers develop general equilibrium models (e.g. Gomes, Kogan, and Zhang (2003)), yet most of them feature a single aggregate shock, implying that the market portfolio conditionally spans the value factor. In contrast, our model features two aggregate risk factors, one of them being driven by embodied technology shocks. Using a measure of embodied technical change, we provide direct evidence for the model mechanism rather than relying only on indirect model implications Our work is related to the voluminous literature on embodied technology shocks (e.g., Cooley, Greenwood, and Yorukoglu, 1997; Greenwood, Hercowitz, and Krusell, 1997; Christiano and Fisher, 2003; Fisher, 2006; Justiniano, Primiceri, and Tambalotti, 2010). Technology is typically assumed to be embodied in new capital goods new projects in our setting. Several empirical studies document substantial vintage effects in the productivity of plants (see Foster, Haltiwanger, and Krizan, 2001, for a survey of the micro productivity literature). For instance, Jensen, McGuckin, and Stiroh (2001) find that the 1992 cohort of new plants 5

7 was 50% more productive than the 1967 cohort in its entry year, controlling for industry-wide factors and input differences. Further, our paper is related to work that explores the effect of technological innovation on asset returns (e.g., Greenwood and Jovanovic, 1999; Hobijn and Jovanovic, 2001; Laitner and Stolyarov, 2003; Kung and Schmid, 2011; Garleanu, Panageas, and Yu, 2012). The focus of this literature is on exploring the effects of innovation on the aggregate stock market. We contribute to this literature by explicitly considering the effects heterogeneity in both firms and households in terms of their exposure to embodied technology shocks. The closest related work is Papanikolaou (2011), Garleanu, Kogan, and Panageas (2012) and Kogan and Papanikolaou (2011, 2012b). Papanikolaou (2011) demonstrates that in a general equilibrium model, capital-embodied technology shocks are positively correlated with the stochastic discount factor when the elasticity of intertemporal substitution is less than or equal to the reciprocal of risk aversion. However, the price of embodied shocks in his model is too small relative to the data. We generalize the model in Papanikolaou (2011), allowing for both firm and household heterogeneity and imperfect risk sharing among households. Our model delivers quantitatively more plausible estimates of the risk premium associated with innovation, as well as more testable predictions. Our model shares some of the features in Garleanu et al. (2012), namely intergenerational displacement risk and technological improvements embodied in new types of intermediate goods. We embed these features into a model with capital accumulation, limited market participation, and a richer, more realistic cross-section of firms. In addition, we construct an explicit empirical measure of innovation shocks and use it to directly test the empirical implications of our model s mechanism. Last, our work is related to Kogan and Papanikolaou (2011, 2012b), who analyze the effect of capital-embodied technical progress in partial equilibrium. The general equilibrium model in this paper helps understand the economic mechanism for pricing of such innovation shocks, and provides further insights into how these shocks impact the economy. The rest of the paper is organized as follows. In Section 1 we build a general equilibrium model with heterogenous agents and heterogenous firms. In Section 2 we describe the competitive equilibrium. In Section 3 we describe the calibration of the model and discuss its quantitative predictions. In Section 4 we test the model s predictions in the data. In 6

8 Section 5 we connect our paper to findings in the consumption-based literature. Section 6 concludes. 1 Model setup In this section we formulate a general equilibrium model that links innovation to asset prices. 1.1 Firms and technology There are three production sectors in the model: a sector producing intermediate consumption goods; a sector that aggregates these intermediate goods into the final consumption good; and a sector producing investment goods. Firms in the last two sectors make zero profits due to competition and constant returns to scale, hence we explicitly model only the intermediategood firms. Intermediate-good firms Production in the intermediate sector takes place in the form of projects indexed by j. Projects are introduced into the economies by the new cohorts of inventors, who lack the ability to implement them on their own sell the blueprints to the projects to existing intermediate-good firms. There is a continuum of infinitely lived firms. Each firm owns a finite number of projects, which they have acquired from inventors. We index projects by j and firms by f. We denote the set of projects owned by firm f by J f, and the set of all active projects in the economy by J t. The set of firms F is fixed. 3 Active projects Projects are differentiated from each other by three characteristics: a) their scale k j chosen irreversibly at their inception; b) the level of frontier technology at the time of project creation s; and c) the time-varying level of project-specific productivity, u jt. A project j created at 3 While we do not explicitly model entry and exit of firms, firms occasionally have zero projects, thus temporarily exiting the market, whereas new entrants can be viewed as a firm that begins operating its first project. 7

9 time s produces a flow of output at time t equal to y jt = u jt e ξs k α j, (1) where α (0, 1); ξ denotes the level of frontier technology at the time the project is implemented and u is a project-specific shock that follows a mean-reverting process. In particular, the random process governing project output evolves according to: du jt = θ u (1 u jt ) dt + σ u ujt dz jt, (2) All projects created at time t are affected by the embodied shock ξ: dξ t = µ ξ dt + σ ξ db ξt. (3) The embodied shock ξ captures the level of frontier technology in implementing new projects. In contrast to the disembodied shock x, an improvement in ξ affects only the output of new projects. In most respects, the disembodied shock ξ is formally equivalent to investmentspecific technological change. All new projects implemented at time t start at the long-run average level of idiosyncratic productivity, u jt = 1. Thus, all projects managed by the same firm are ex-ante identical in terms of productivity, but differ ex-post due to the project-specific shocks. Last, active projects expire independently at a Poisson rate δ. Firm investment opportunities new projects There is a continuum of firms in the intermediate goods sector, that own and operate projects. Firms are differentiated by their ability to attract inventors, and hence initiate new projects. We denote by N ft the Poisson count process that denotes the number of projects the firm has acquired. The probability that the firm acquires a new project, dn t = 1, is firm-specific and equal to λ ft = λ f λ ft. (4) The likelihood that the firm acquires a new project λ ft is composed of two parts. The first part λ f captures the long-run likelihood of firm f receiving new projects, and is constant 8

10 over time. The second component, λ ft is time-varying, following a two-state, continuous time Markov process with transition probability matrix S between time t and t + dt given by ( ) 1 µl dt µ S = L dt. (5) µ H dt 1 µ H dt We label the two states as {λ H, λ L }, with λ H > λ L. Thus, at any point in time, a firm can be either in the high-growth (λ ft = λ f λ H ) or in the low-growth state (λ ft = λ f λ L ), and µ H dt and µ L dt denote the instantaneous probability of switching to each state respectively. We impose that E[ λ f,t ] = 1, which implies the parameter restriction 1 = λ L + µ H µ H + µ L (λ H λ L ). (6) Market participants observe a long history of the economy, hence we assume they know the long-run mean of firm-specific project arrival rate λ f. However, they do not observe whether the firm is currently in the high-growth or low-growth phase. Hence, λ ft is an unobservable, latent process. The market updates its beliefs about the current state of the firm based on project arrivals. Last, our specification implies that the aggregate rate of project creation E[λ ft ] = λ is constant. Implementing new projects Implementing new projects requires new capital k purchased at the equilibrium market price q. Once a project is acquired, the firm chooses its scale of production k j to maximize the value of the project. A firm s choice of project scale is irreversible. It cannot liquidate existing projects and recover their original costs. Capital-good firms Firms in the capital-good sector use labor to produce productive inputs (e.g., physical capital) needed to implement new projects in the intermediate-good sector. Firms produce capital using labor, I t = e xt L It. (7) The labor augmenting productivity shock x evolves according to dx t = µ x dt + σ x db xt. (8) 9

11 Final-good firms There is a continuum of firms that take use the intermediate good to produce the final consumption good. They do so using a constant returns to scale technology C t = Y φ t (e xt L Ct ) 1 φ, (9) where Y t is the total output of the intermediate good, L C the amount of labor allocated to the final consumption goods sector, and x t is the labor augmenting productivity shock defined in (8). 1.2 Households In this economy there are two types of households, each with a unit mass: hand-to-mouth workers who supply labor; and inventors who also supply labor, but more importantly, supply ideas for new projects. Both types of households have finite lives: they die stochastically at a rate µ, and are replaced by a household of the same type. Inventors have no bequest motive and have access to a market for state-contingent life insurance contracts. Hence, each household is able to perfectly share its mortality risk with other households of the same cohort. Inventors At the time of entry into the economy, each inventor is endowed with a measure λ/µ of ideas for new projects. Inventors are endowed with no other resources, and lack the ability to implement these project ideas on their own. Hence, they sell these projects to existing firms. Thus, at each instant there is a total measure λ/µ µ dt = λ dt of new projects created. Inventors and firms bargain over the surplus created by new projects. We assume that each inventor captures a share η of the value of each project. Inventors have access to complete financial markets, including an annuity market. After they sell their project, inventors invest their proceeds in financial markets. Inventors maximize a utility index J t over sequences of consumption C s. Inventor s utility takes a recursive form J t = E t t f(c s, C s, J s )ds, (10) 10

12 where f(c, C, J) ρ + µ 1 θ 1 ( C ( ) ) h 1 θ 1 C C ((1 γ)j) γ θ 1 1 γ (1 γ) J. (11) Our preference specification nests Keeping up with the Joneses and non-separability across time (see e.g. Epstein and Zin, 1989; Duffie and Epstein, 1992). Household preferences depend on own consumption C, but also on the consumption of the household relative to the aggregate, with the parameter h capturing the weight on the latter. In this formulation, ρ is the time-preference parameter, γ is the coefficient of relative risk aversion, and θ is the elasticity of intertemporal substitution (EIS) defined over the composite good C(C/ C) h.the fact that households face an exponentially distributed time of death leads to an increase in the effective rate of time discounting by µ. Inventors are only endowed with projects upon entry, and cannot subsequently innovate. This assumption implies that each new successive generation of inventors can potentially displace older cohorts. Workers Workers inelastically supply 1 unit of labor that can that can be freely allocated between producing consumption or investment goods L I + L C = 1. (12) Workers are hand-to-mouth; they do not have access to financial markets and consume their labor income every period. Allowing the workers to save in a riskless storage technology has no effect on the prices of financial assets, since they do not participate. Allowing workers to save results in a smoother aggregate consumption process, as they use their savings to buffer the shocks to their labor income. 2 Competitive equilibrium Here, we describe the competitive equilibrium of our model. To obtain intuition about the behavior of the solution, we first examine the first order conditions to the problem. Then, 11

13 we briefly describe the solution to the fixed point problem. We relegate the details of the derivation to Appendix A. 2.1 Firm optimality conditions First, we describe the firm s first order conditions. Demand for intermediate goods Consumption firms use the intermediate good and labor services as input. In particular, they purchase the intermediate good Y at a price p Y and hire labor L C at a wage w to maximize their value. Their first order condition with respect to their demand for intermediate goods yields an expression for the equilibrium price p Y consumption good φ Y φ 1 t (e xt L Ct ) 1 φ = p Y t. of the intermediate good in terms of the Market clearing implies that the total output of the intermediate good, Y t, equals the sum of the output of the individual projects: Y t = F y f,t = K t. (13) The variable K represents the effective capital stock in the economy, K t e ξ j kj α dj. (14) j J t adjusted for the productivity of each vintage captured by ξ at the time the project is created and for decreasing returns to scale. Demand for capital Intermediate good firms choose the scale of investment k j to maximize the net present value of the project. Hence the optimal level k j depends on the market value of existing projects. Suppose that the equilibrium price of a new project equals P t e ξt k α, where P t is a function of only the aggregate state of the economy. Firms choose k to maximize the market value of the new project, minus its implementation cost max NP V = P t e ξt k α q t k. (15) k 12

14 The optimal scale of investment is a function of the ratio of the market value of capital to its purchase cost which in our formulation depends only on aggregate variables. 4 ( α e kt ξ t ) 1 1 α P t =, (16) q t The unit cost of implementing projects q t is determined by the market clearing condition (7) I t = kft dn ft = λ kt. (17) F Aggregation is simplified by the fact that the optimal scale of all projects created at time t is the same. In particular, the resources allocated per project equal the total amount of investment divided by the number of new projects received. Combining equations (16) and (17) yields the equilibrium price of the investment good in terms of the consumption good: ( ) 1 α λ q t = αe ξt P t. (18) e xt L It Demand for labor Labor is used to produce either the final consumption good, or the capital needed to implement new projects. The first order condition of the firms producing the final consumption good with respect to labor input links their labor choice L C to the competitive wage w t (1 φ) K φ t e (1 φ)xt L φ Ct = w t. (19) The profit maximization in the investment-goods sector implies that Combining the labor market clearing condition e xt q t = w t. (20) L Ct + L It = 1, (21) along with (19) and (18) leads to ( ) 1 α λ (1 φ) K φ t e (1 φ)xt (1 L It ) φ = α e α xt+ξt P t. (22) L I 4 Allowing for ex-ante cross-sectional differences in project profitability is straightforward. 13

15 Equation (22) determines the allocation of resources between producing consumption and investment goods, and is an important part of the model mechanism. All else equal, an increase in the frontier technology ξ increases the demand for new investment goods. As a result, the economy reallocates resources away from producing consumption goods towards producing investment goods. 2.2 Household optimization Here, we describe the household s optimality conditions. Inventors Upon entry, inventors sell the blueprints to their projects to firms and use the proceeds to invest in financial markets. A new inventor entering at time t acquires a share of total financial wealth W t equal to where NP V t b tt = η λnp Vt, (23) µ W t is the maximand in (15), η is the share of the project value captured by the inventor, and W t is total financial wealth in the economy, which equals the market value of all firms. As new inventors acquire shares in financial wealth, they displace existing inventors. The share of total financial wealth W held at time t by an inventor born at time s < t equals ( t ) b ts = b ss exp µ(t s) µ b uu du. (24) Inventors share total financial wealth W with all cohorts of inventors born subsequently. In addition, since agents insure the risk of death with other members of the same cohort, surviving agents experience an increase in the growth rate of per-capital wealth equal to probability of death µ. Existing inventors are identical up to their level of wealth and have homothetic preferences. As a result, they all have the same consumption to wealth ratio s C ts b ts = C ts b ts, (25) 14

16 for all cohorts s s t. We guess and subsequently verify that the value function of an inventor born in time s is given by where F t is a function of the aggregate state only. J ts = 1 1 γ b1 γ ts F t, (26) All existing agents at time t share the same growth rate of consumption going forward, as they share risk in financial markets. To see this, note that from equations (24)-(25) the consumption inequality between cohorts born at s and s depends on the history of the economy between s and s, but not on time t. As a result, all existing inventors are identical, differing only in the level of financial wealth. Hence, aggregation is simplified as the homotheticity of preferences implies that all agents participating in financial markets have the same marginal rate of substitution ( π s ) s = exp f J (C u, π C fc (C s, u, J u ) du C s, J s ) t f C (C t, C t, J t ), (27) t where J is the utility index defined recursively in equation (10), and f is the preference aggregator defined in equation (11). Workers Workers inelastically supply one unit of labor and face no investment decisions. Every period, they consume an amount equal to their labor proceeds 2.3 Asset prices C W t = w t. (28) The last step in characterizing the competitive equilibrium requires computing the components of financial wealth. Since firms producing capital goods and the final consumption good have constant returns to scale technologies and no adjustment costs, they make zero profits in equilibrium. Hence, we only focus on the sector producing intermediate goods. The value of a firm in the intermediate sector consists of the value of assets in place and the value of growth opportunities V ft = V AP ft + P V GO ft. (29) 15

17 Below, we compute the two components of firm value separately. Value of Assets in Place A single project produces a flow of the intermediate good, whose value in terms of consumption is p Y,t. The value (in consumption units) of an existing project with productivity level u jt equals E t [ t e δ s π ] s p Y,s u j,s e ξ j kj α ds =e ξ j kj α π t [ P t + P ] t (u j,t 1), (30) where P t and P t are functions of the aggregate state of the economy verifying our conjecture above. The value of assets in place is equal to the value of all active projects j J t in the economy V AP t e ξ j kj α j J t where K is defined in equation (14). Value of Growth Opportunities [ P t + P ] t (u j,t 1) dj = P t K t, (31) The present value of growth opportunities is equal to the present value of rents from future projects that accrue to the firm π s P V GO ft =(1 η)e t λ fs NP Vs ds t π t =λ f (1 η) [ ( )] Γ L t + p ft Γ H t Γ L t, (32) where 1 η represents the fraction of the value of new projects captured by firms; p ft denotes the probability the firm is in the high-growth state ( λ ft = λ H ); and Γ L t and Γ H t determine the value of a firm in the low- and high-growth phase, respectively. The market s belief that the firm is in the high-growth state p f evolves according to the following stochastic process: ( ) ( ) µh λ f λ H dp ft = (µ L + µ H ) p ft dt + p ft 1 µ L + µ H ˆλ ft dm ft, (33) 16

18 where ) ˆλ ft = λ f (λ L + p ft (λ H λ L ) (34) is the market s current assessment of the firm s likelihood of acquiring new project ideas. The drift term of equation (33) captures the drift of p ft towards its stationary mean. The stochastic component of (33) depends on the martingale M t, which captures the updating of market beliefs based on observing the arrival of projects dm ft = dn ft ˆλ ft dt. (35) Here, N ft is a poisson count process with dn ft = 1 if the firm invests at time t and zero otherwise. See Liptser and Shiriaev (1977) for a textbook treatment of filtering results for point processes. Aggregating (32) across firms, the total value of growth opportunities in the economy is equal to Value of Stock Market [ P V GO t = λ(1 η) Γ L t + µ H ( ) ] Γ H µ L + µ t Γ L t (36) H The total value of the stock market is equal to the sum of the value of existing assets (31) plus the value of growth opportunities (36) W t = V AP t + P V GO t. (37) The value of financial wealth also corresponds to the total wealth of inventors, which enters the denominator of the displacement effect Solution to the fixed-point problem Definition 1 (Competitive Equilibrium) The competitive equilibrium is a sequence of quantities {C S t, C W t, Y t, L Ct, L It }; prices {p Y t, q t, w t }; firm investment decisions {k t }; and beliefs {ˆλ ft } such that given the sequence of stochastic shocks {x t, ξ t, u jt, N ft }, i) shareholders choose consumption and savings plans to maximize their utility (10); ii) intermediate-good firms maximize their value according to (15); iii) Final-good and investment-good maximize 17

19 profits; iv) the labor market (12) clears; v) the market for capital clears (17); vi) the market for consumption clears Ct S + Ct W = C t ; vii) the resource constraints (7)-(9) are satisfied; and viii) market participants rationally update their beliefs about λ ft using all available information. To obtain the competitive equilibrium of the model, we solve the fixed-point problem as follows. First, we solve the firms optimization problem taking prices as given. Second, we solve the innovators and workers optimization problem. Given the households consumption process we obtain the investor s marginal rate of substitution, or stochastic discount factor. Then we solve for equilibrium asset prices, which feed back into the investor s consumption problem through the cohort displacement effect. Last, we solve for the value of individual firms. To conserve space, we relegate the details of the solution to the appendix. The state of the economy can be characterized by the state vector Z t = [χ t, ω t ]. The first state variable χ is difference-stationary and captures the stochastic trend in the economy: χ (1 φ)x + φ ln K, (38) where x is the disembodied shock (8) and K is the effective capital stock (14), which grows at the average rate of new project creation λ times the size of new projects times the vintage shock ξ, and depreciates at rate δ. Using the investment-good market clearing condition (17), K evolves as dk t = ( λ e ω t ( LIt λ ) α δ) K t dt. (39) The second state variable ω captures the real investment opportunities in the economy, which depend on the deviation of the effective capital stock K from an optimal level determined by the two aggregate shocks x and ξ ω α x t + ξ ln K. (40) The state variable ω is mean-reverting and captures transitory fluctuations along the stochastic growth path. The following variables are stationary and depend only on ω: the optimal allocation of labor across sectors L I and L C ; the consumption share of workers C w / C; the labor-output ratio; the rate of displacement of existing shareholders b. The instantaneous growth rate of the quality-adjusted capital stock, dk t /K t, is also a function of ω t. We 18

20 therefore interpret shocks to ω t as shocks to the investment opportunity set in this economy: the latter are affected both by the embodied innovation shocks dξ t and the disembodied productivity shocks dx t. 3 Market price of innovation risk In this section we calibrate our model. We then examine how innovation shocks are priced in equilibrium, and analyze their effect on the key properties of aggregate and firm-level stock returns in the model. 3.1 Numerical solution and calibration The competitive equilibrium is described by a system of five nested ODEs in the unknown functions f, g, g, ν, ν defined in the Appendix, and a functional equation in L I. We solve for the equilibrium using finite differences on a grid with 2,000 points. Parameter choice The model has a total of 18 parameters. We choose these parameters to approximately match a set of moments. Table 1 shows our choice of parameters. Table 2 displays the moments generated by the model, and we mark moments targeted in calibration with a star. We choose the returns to scale parameter at the project level α to approximately match the correlation between investment rate and Tobin s Q. We choose a depreciation rate of δ in line with typical calibrations of RBC models. We choose the share of capital in the production of final goods φ to match the average level of the labor share. We choose the average rate of acquisition of new projects, λ to match the average investment-to-capital ratio in the economy. We choose the parameters governing the acquisition of new blueprints, λ H, µ L and µ H, to match the persistence, the dispersion and the lumpiness in firm investment rate. Regarding the parameters of the two technology processes x and ξ, we choose the mean growth rates µ x and µ ξ to match the growth rate of the economy. We choose the volatilities of the disembodied shock σ x and the embodied shock σ ξ to match the volatility of 19

21 shareholder consumption and investment growth, respectively. We select the parameters of the idiosyncratic shock, σ u and θ u, to match persistence and dispersion in firm output-capital ratios. For our preference parameters we choose a low value of time preference ρ, based on typical calibrations. We select the coefficient of risk aversion γ and the elasticity of intertemporal substitution θ to match the level of the premium of financial wealth and the volatility of the risk free rate. We choose the preference weight on relative consumption h = 1 following Garleanu et al. (2012), so that households attach equal weights to own and relative consumption. We select the bargaining parameter η between innovators and firms to match the volatility of cohort effects. We choose the probability of death µ = 0.025, so that the average length of adult life is 1/µ = 40 years. We create returns to equity by levering financial wealth by 2. Simulation We simulate the model at a weekly frequency dt = 1/52 and then aggregate the data to form annual observations. We simulate 1,000 model histories of 3,000 firms and 120 years each. We drop the first third of each history to eliminate the impact of initial conditions. When we compare the output of the model to our empirical analysis, we report the median parameter estimate across simulations. 3.2 Model properties In our discussion of the properties of asset prices in the model, we first consider the mechanism for how innovation risk is priced. Then we discuss cross-sectional differences in exposure to innovation risk among firms, and resulting differences in expected stock returns. Equilibrium price of innovation shocks Equilibrium risk premia are determined by the covariance of returns with the equilibrium stochastic discount factor π t, which is given by the gradient of the utility function of the stock holders in the model. In particular, the form of the stochastic discount factor in (27) and (11) implies dπ t π t = [ ] dt θ 1 ( ( dcts dcts + h (1 θ) d C )) ts γ θ 1 dj ts, C ts C ts C ts 1 γ J ts 20

22 where C ts and J ts denote the equilibrium consumption rate and the value function of a stockholder household of any older cohort s, s < t. The investor s marginal value of consumption depends on their own consumption growth; their consumption growth C relative to the aggregate C, due to relative consumption concerns parameterized by h; and their growth in continuation utility J. The price of innovation shocks is determined by these three effects, which we analyze separately below. First, an embodied technological innovation has no immediate effect of productivity of the capital stock in the economy because it only affects productivity of new vintages of capital. Therefore, the immediate effect of innovation on the aggregate output of consumption good is negative due to a shift of productive inputs (labor) from the intermediate-good sector to the investment-good sector, as we see in panel a of Figure 1. Second, the increased demand for labor services resulting from innovation has an additional effect of raising real wages, which benefits workers and hurts the owners of capital, shifting income from capital to labor as shown in panel b. Because the workers do not share risks with the stockholders in our model, the result of the two effects above is that consumption of stockholders declines in response to an embodied technological advance, more so than aggregate consumption, as we see in panels c and d respectively. Further, innovation shocks affect the continuation value function of stockholders. As we see in Panel e, the value function J of asset holders is negatively exposed to the innovation shock. This negative effect arises due to the combination of two additional channels. First, a positive innovation shock accelerates the rate at which new cohorts of inventors enter the economy and reduce profitability of the old capital stock owned by the stockholders. This displacement effect lowers the continuation utility of stockholders and is captured by the term b tt. As we see in Panel f, the magnitude of this displacement effect is increasing in the level of embodied technology. Second, embodied innovations have a positive long-term effect on aggregate output and consumption. Depending on the model parameters and the current state of the economy, this may imply a net positive or a net negative effect of innovation on the aggregate market value of growth opportunities, which are owned by the stockholders and affect their utility level. The combination of these three mechanisms above result in a negative risk premium for 21

23 the innovation shock. To summarize these effects, we write the stochastic discount factor as a function of the two technology shocks x and ξ [ = r ft dt γ (1 φ) + α dπ t π t ( θ 1 l (ω t ) l(ω t ) γ θ 1 γ 1 ( θ 1 l (ω t ) f (ω t ) f(ω t ) l(ω t ) γ )] θ 1 f (ω t ) σ x dbt x γ 1 f(ω t ) ) σ ξ db ξ t. (41) where f(ω) captures the dependence of the value function of stockholders on the embodied shock, and l(ω) is the function of consumption share of stockholders. Both of these functions are defined in the Appendix A. Stock market returns and the risk-free rate Table 2 shows the moments implied by the model. In addition to the moments we target, the model generates realistic moments for aggregate quantities. In line with the data, our model delivers a higher volatility of shareholder consumption growth and a positive correlation between investment and consumption growth. In addition, aggregate payout to capital owners dividends, interest payments and repurchases minus new issuance are volatile and weakly correlated with consumption and labor income. Our model generates a high equity premium in line with an empirical estimate, and volatile equity returns, although the volatility is one-third smaller than the empirical number. The choice of a relatively high EIS ensures that our model generates a stable risk-free rate. The level of the risk-free rate is somewhat higher than the post-war average, but lower than the average level in the long sample in Campbell and Cochrane (1999). The relatively high level of the risk-free rate is the result of finite lives (see e.g. Blanchard, 1985). Thus, our model performs at least as well as most general equilibrium models with production in matching the moments of the market portfolio and risk-free rate (e.g., Jermann, 1998; Boldrin, Christiano, and Fisher, 2001; Kaltenbrunner and Lochstoer, 2010). Innovation shocks interacted with limited risk-sharing are an important source of the high equity premium in our model. Existing asset holders bear most of the displacement risk innovation. The correlation between the aggregate stock market returns and consumption growth of stockholders is 63%, more than twice the correlation of market returns with consumption growth of non-stockholders (25%). Hence, the standard consumption-capm 22

24 does not hold in our model, and to evaluate the effect of innovation shocks on asset risk premia we must focus on the risk exposure of asset holders. Cross-section of stock returns Our model features a rich equilibrium cross-section of firms. Firms differ endogenously in their exposure to the disembodied productivity shock x and the embodied shock ξ. Since both technology shocks are priced in equilibrium, heterogeneous risk exposures lead to differences in risk premia across firms. We first describe the economic mechanism that generates heterogeneity in risk exposures in the model. We then relate firms stock return risk to firm characteristics correlated with cross-sectional differences in growth opportunities among firms, such as past investment rates and firms Tobin s Q, or equity market-to-book ratio. We show that our model reproduces the empirical relations between these firm characteristics and risk premia. Moreover, we show that the conditional CAPM model misprices the portfolios of stocks sorted on growth-related characteristics in our model similarly to the corresponding empirical patterns. Applying Ito s lemma to the value of the firm (29) yields dv ft V ft =[ ] dt + (1 φ) σ x db x t + B ft ( σ ξ db ξ t + α σ x db x t ). (42) Differences in firms systematic risk is captured by differences in the state variable ω capturing the current real investment opportunity set ( ) ( ) B ft ζ ν(ω) + ζ ν(ω) A v ft V AP ft + ζ 1 + A v ft V g(ω) + ζ g(ω) A g ft ft 1 + A g ft where and ζ ν (ω) = ln ζ g (ω) = ln ( ( ) 1 l(ω t ) θ 1 f(ω t ) γ θ 1 γ 1 ν(ω)), ζ ν (ω) = ln ( ( ) 1 l(ω t ) θ 1 f(ω t ) γ θ 1 γ 1 g(ω)), ζ g (ω) = ln P V GO ft V ft. (43) ( ν(ω) ν(ω) ( g(ω) g(ω) ), ), (44) A v ft = ν(ω t) j J ft e ξ j kj α (u j,t 1), (45) ν(ω t ) j J ft ε ξ j k α j A g ft = ( p ft µ H µ L + µ H 23 ) (λ H λ L ) g(ω t) g(ω t ). (46)

25 The first stochastic term in (42), (1 φ) σ x dbt x, is identical across firms, and is driven ) solely by the disembodied productivity shocks. The second term, B ft (σ ξ db ξ t + α σ x dbt x, represents unanticipated changes in aggregate investment opportunities, and is driven by both by the labor-augmenting and the capital-embodied productivity shocks. The second term differs across firms and depends on the composition of firms assets, in particular, on the richness of firms growth opportunities captured by V AP ft /V ft. A firm being a portfolio of assets in place and growth opportunities, its risk exposure is a weighted average of their corresponding risk exposures. The weights are determined by the fraction of firm market value derived from growth opportunities. Furthermore, risk exposure of assets in place is heterogeneous across firms, and depends on their idiosyncratic productivity. This dependence arises because idiosyncratic productivity shocks are transient in nature, and the timing of firms cash flows matters for the risk of its assets in place. Because of the quasi-linear functional form we assume for the idiosyncratic productivity shocks in (2), the effect of cash flow timing can be fully summarized by a linear function of two terms common across firms, ζ ν (ω) and ζ ν (ω), and a firm-specific loading A v ft /(1 + Av ft ). The term ζ ν (ω) equals the risk exposure of a project with productivity equal to u = 1. The function ζ ν (ω) adjusts the risk of assets in place for transient productivity shocks. The firm-specific loading on this function is a linear function of the deviations of idiosyncratic project productivity levels from their average values, as shown in (45). Similarly, risk exposure of the firms growth opportunities depends on the term structure of the expected arrival of new investment opportunities. Again, the simplicity of the setup reduces this dependence to a linear function of two terms common across firms, ζ g (ω) and ζ g (ω), and the firm-specific loading A g ft /(1 + Ag ft ). The first term, ζ g(ω), reflects the risk exposure of a firm that is comprised entirely of growth opportunities and the market s beliefs estimate of its growth opportunities is at its average level, p ft = µ H /(µ H + µ L ). The function ζ g (ω) modifies the risk of firm s growth opportunities to reflect time-variation in the project arrival rate. It is therefore scaled by the firm-specific term A g ft /(1 + Ag ft ), where Ag ft which is proportional to the current estimate of the project arrival rate, as shown in (46). We plot the components of individual firm exposures the ζ i (ω) functions in Figure 3. The left panel plots risk exposures of assets in place and growth opportunities assuming 24