Labor Relations, Endogenous Dividends and the Equilibrium Term Structure of Equity

Transcription

1 Labor Relations, Endogenous Dividends and the Equilibrium Term Structure of Equity Roberto Marfè Swiss Finance Institute and University of Lausanne Preliminary version Comments welcome ABSTRACT Leading asset pricing models are inconsistent with the recent empirical findings which document downward sloping term structures of equity risk and premia. This paper shows that a simple general equilibrium model can accommodate the stylized facts about dividend strips as long as dividend distributions endogenously obtain from an explicit model of labor relations. Unlike standard Walrasian models but in line with the empirical evidence, wages do not correspond to the marginal product of labor but incorporate an income insurance from shareholders to workers. Such a distributional risk provides a rationale to the countercyclical labor share and the high riskiness of owning capital. Fluctuations in the degree of income insurance that workers can exploit within the firm over the business cycle lead to short-term risk of equity returns. Therefore, the model captures simultaneously the negative slope of the term structure of equity and dividends as well as traditional asset pricing facts, such as the high equity premium and the excess volatility and their endogenous time-variation. Keywords: asset pricing term structure of equity dividend strips distributional risk income insurance equilibrium JEL Classification: D53 E24 E32 G12 I would thank Philippe Bacchetta, Ravi Bansal, Vincent Bogousslavsky, Andrea Buraschi, Jérôme Detemple (Gerzensee discussant), Pierre Collin-Dufresne, Stefano Colonnello, Marco Della Seta, Bernard Dumas, Campbell Harvey, Loriano Mancini, Michael Rockinger, Lukas Schmid and the participants at the 12th Swiss Doctoral Workshop in Finance for helpful comments. All errors remain my only responsibility. Part of this research was conducted when the author was visiting Duke University. Financial support by the NCCR FINRISK of the Swiss National Science Foundation and by the Associazione per la Facoltà di Economia dell Università di Torino is gratefully acknowledged. Usual disclaimer applies. Postal address: Swiss Finance Institute, Unil-Dorigny, Extranef Building 21. CH-115 Lausanne, Switzerland. Roberto.Marfe@unil.ch. Webpage:

2 I. Introduction Recent empirical evidence, as in van Binsbergen, Brandt, and Koijen (212, hereafter B12a) and van Binsbergen, Hueskes, Koijen, and Vrugt (212, hereafter B12b), questions the short-term implications of leading asset pricing models such as Campbell and Cochrane (1999) and Bansal and Yaron (24). They provide a long-term explanation of traditional puzzles like the equity premium and the excess volatility but are inconsistent with the highly risky returns of dividend strips in the short-term and the downward sloping term structures of growth rates and volatility of both dividends and dividend strips returns. What is the missing ingredient of asset pricing models? What are the macroeconomic determinants of such a short-term risk, priced at equilibrium? This paper, for the first time, highlights the crucial role of labor relations in the determination of the equilibrium term-structures of both dividends and equity returns. Consistent with Favilukis and Lin (212b), I show that the failure to model volatility and cyclicality of wages is closely linked to the failure to match financial data in most of real business cycle and production-based models. 1 This paper considers a simple risk-sharing model with potentially time-varying bargaining power among workers and shareholders in order to endogenously determine wages and dividend distributions and, hence, to reconcile traditional asset pricing facts with the recent empirical evidence about dividend strips. Intuitively, standard Walrasian models imply counterfactual highly pro-cyclical and volatile wages and, hence, lead to smooth and counter-cyclical dividends, which act as an hedge for the firms. Instead, as pointed out in Boldrin and Horvath (1995), real wages do not correspond to the actual labor marginal productivity but include an insurance component, which does not obtain in the Walrasian framework. The ideas that distributional risk among workers and shareholders is at the heart of labor relations and, also, that the very role of the firm is that of insurance provider have a long tradition since Knight (1921) as well as Baily (1974) and Azariadis (1978). They suggest that, on the one hand, labor s claim on output is partially fixed in advance and, hence, shareholders bear most of aggregate uncertainty and, on the other hand, in exchange of income insurance, shareholders gain flexibility in labor supply. Therefore, by removing the Walrasian market clearing mechanism from labor market with an explicit model of labor relations makes possible to capture the countercyclical dynamics of the labor share and to enhance the riskiness of owning capital. Indeed, income insurance to workers within the firm endogenously leads to a strong desire for the shareholders to smooth consumption intertemporally. The latter essentially substitutes for exogenous approaches such as habit formation or some form of market incompleteness. In spirit of Danthine and Donaldson (1992, 22), I model labor relations through a simple contract rule which qualitatively captures the implications of contract arrangements such as in Boldrin and Horvath (1995) and Gomme and Greenwood (1995) that produces an income insurance to workers as long as those are more risk averse than shareholders, in line with the empirical literature (see Moskowitz and Vissing-Jørgensen (22)). Similar to other equilibrium models, for the sake of exposition and in order to make unambiguous the equilibrium implications of labor relations, production is modelled in reduced form and limited market participation is assumed such that workers do not own capital. Such an extreme scenario prevents some forms of consumption smoothing to shareholders which can hide the role of labor relations in the endogenous determination of wage payments and dividend distributions. 1 Leading production-based asset pricing models, such as Jermann (1998), Boldrin, Christiano, and Fisher (21) and Kaltenbrunner and Lochstoer (21), on the one hand, do not capture the smooth dynamics of wages and, on the other hand, underestimate the riskiness of excess returns on equity. 1

3 Distributional risk due to labor relations increases the riskiness of dividends even if the latter are not riskier than the firm s operational cash-flows in the long-run. In line with Danthine and Donaldson (22) and Danthine, Donaldson, and Siconolfi (26), such a distributional risk and potentially fluctuations in the bargaining power among workers and shareholders are first order determinants of financial data. In particular, uncertainty about the riskiness of dividends commanded by the risk sharing model leads to short-term risk: shareholders are subject to fluctuations around the steady-state degree of income insurance that workers are able to exploit within the firm. Therefore, dividends are risky at short horizons and feature downward sloping term structures of both growth rates and volatility. This result is consistent with the empirical work by Guiso, Pistaferri, and Schivardi (25) which documents an economically significant income insurance from employers to employees, mostly related to temporary, instead of permanent, shocks to the firm s productivity. Moreover, in line with the real data, labor relations induce counter-cyclical dynamics to the labor share and, hence, reduce the hedge component of dividends which is a counterfactual implication of Walrasian models. Similar results could obtain by modelling wage rigidity with infrequent resettling as in Favilukis and Lin (212b): wage payments are smoother than in the Walrasian framework since they are a weighted average of spot wages and previous arrangements. 2,3 Given the endogenous stream of dividends, equilibrium returns on equity depend on the marginal rate of substitution of market participants. Under limited market participation, shareholders act as a representative agent and their elasticity of intertemporal substitution is crucial to the equilibrium timing of risk. As long as the intertemporal substitution effect dominates the wealth effect, the term structures of premia and volatility on equity returns can feature negative slopes. I characterize how the volatility of dividend strips returns decomposes in its fundamental risk, transient risk and forward-looking risk and how the latter governs both riskiness and compensation for equity over the time-horizon. Such result allows to reconcile the recent empirical evidence about dividend strips with traditional asset pricing facts such as high equity premium and excess volatility. In particular, the model can account for a good fit of such moments even under time-separable power utility and reasonable parameters. Furthermore, counter-cyclical premia and Sharpe ratio as well as Garch effects in return volatility obtain endogenously. More sophisticated preferences, like habit formation or recursive utility, can help to capture simultaneously additional stylized facts such as return predictability and cross-sectional properties. In particular, tractable solutions are derived when shareholders feature recursive utility: differently from long-run risk models, early resolution of uncertainty is not at odds with downward sloping term structures of equity and a non-monotone relation with the equity premium obtains. Hence, the empirical evidence about dividend strips not only represents a new stylized fact but, at least in the present framework, provides additional guidance in the understanding of the true nature of the equity premium and the role of investors preferences. The equilibrium explanation of the term structure of equity and, hence, the understanding of the findings by B12a and B12b is currently under investigation and few attempts have provided some insights. Lettau and Wachter (211) and Santos and Veronesi (21) consider the role of cash-flow duration and return cross- 2 This paper differs from Favilukis and Lin (212b) because, first, I focus on the term structure of equity and dividends whereas Favilukis and Lin (212b) look at traditional asset pricing facts; second, I propose a simpler and more parsimonious model to recover the minimal ingredients needed to reconcile short-term and long-term asset pricing patterns in general equilibrium. Moreover, even if my model is silent about production, Danthine, Donaldson, and Siconolfi (26) have shown that the risk sharing model improves the asset pricing implications of a standard real business cycle model without deteriorating standard business cycle predictions. 3 Further recent works about the effect of labor relations on traditional asset pricing implications are Gourio (28), Lochstoer and Bhamra (29), Favilukis and Lin (212a) and Kuehn, Petrosky-Nadeau, and Zhang (212). 2

4 sectional implications. Croce, Lettau, and Ludvigson (27) and Marfè (213) study the term structure of equity in the long-run risks model. Belo, Collin-Dufresne, and Goldstein (212) emphasize the role of financial leverage on the term structure of dividends. Ai, Croce, Diercks, and Li (212) relate the term structure of equity with production under heterogeneous investment risk. Similar to Ai, Croce, Diercks, and Li (212), this paper studies dividends as a direct outcome of the firm s decisions, whereas previous contributions assume exogenous cash-flows. 4 However, whereas Ai, Croce, Diercks, and Li (212) focus on production and investment, my approach is both alternative and complementary and highlights the role of labor relations. The latter are not considered by Ai, Croce, Diercks, and Li (212) which does not capture the counter-cyclical dynamics of labor share and, hence, the short-term risk of dividends due to the income insurance that workers exploit within the firm. Moreover, Ai, Croce, Diercks, and Li (212) focus on the term structure of the first moment of equity returns, while this paper also captures the downward sloping term structure of equity risk, which is likely the most important finding by B12a and B12b since it lies at the heart of the true nature of the equity premium. This paper is also closely related to Belo, Collin-Dufresne, and Goldstein (212): the latter shows in partial equilibrium that exogenous fluctuations in financial leverage can reallocate the riskiness of dividends towards the short horizon. Therefore, labor relations, financial leverage and investment decisions are three channels which can contribute to the equilibrium explanation of the timing of equity risk and their joint analysis could provide a deeper understanding of the topic. However, such three channels can have a different effect depending on the context: for instance, financial leverage can be very important for some firms, but at the aggregate level small fluctuations in the labor-share are likely more relevant than leverage adjustments since the cost of labor is about two thirds of the representative firm s output while the service of debt is about only one or two percent. For this reason the emphasis on labor relations seems to be particularly important from a general equilibrium perspective. This paper is also related to the literature about preference heterogeneity. Under complete markets, Chan and Kogan (22) show how preference heterogeneity leads to counter-cyclical price of risk as a result of endogenous time-variations in the aggregate relative risk aversion. 5 Exactly the same endogenous counter-cyclical dynamics obtains in this paper but it is due to variations in dividend s volatility instead of risk aversion. Such a counter-cyclical mechanism is due to the income insurance from shareholders to workers and increases with their preference heterogeneity. Instead, Guvenen (29) generates a similar result through two groups of agents heterogeneous in risk preferences: both are allowed to trade in bonds but only less risk averse agents are shareholders. Then, bonds induce higher risk to shareholders consumption and, hence, play a role similar to the distributional risk driven by labor relations in the present paper. The paper is organized as follows. Section II describes the ingredients of the model: workers and shareholders problems, the firm s decisions and labor relations. Section III and IV derive the equilibrium pricing kernel, cash flows and asset prices as well as their term structures. Asset pricing results are presented in Section V. Alternative specifications of the model are highlighted in Section VI. Section VII concludes. 4 Even if they do not analyse explicitly the term structure of equity, also Longstaff and Piazzesi (24) and Santos and Veronesi (26) propose exogenous cash-flows dynamics compatible with downward sloping term structures. 5 Therefore, Chan and Kogan (22) s model can be interpreted as a micro-founded version of nonlinear habit models such as Campbell and Cochrane (1999). See Xiouros and Zapatero (21) for the empirical shortcomings of such an approach. 3

5 II. The Economy A. Agents The economy consists of two classes of agents, workers (w) and shareholders (s). Both are equipped with power utility: U (C,γ) = 1 1 γ C1 γ, (1) such that workers utility is denoted by U(C w ) = U (C w,γ w ) and shareholders utility is V (C s ) = U (C s,γ s ), where C w and C s stand for the consumption level of the two groups. Risk attitudes differ as follows. Assumption 1 Preference heterogeneity. shareholders: γ w > γ s. Workers feature larger constant relative risk aversion than It will be shown that preference heterogeneity is crucial to both the endogenous riskiness of dividend distributions as well as the equilibrium pricing of risk. In order to emphasize the workers need for income smoothing in the context of their employment relationship, I make the following assumption. Assumption 2 Limited market participation. Workers do not participate in the financial markets and, hence, inelastically supply labor and consume their wage. Such an assumption provides a simplistic and extreme representation of workers and their permanent relation with the firm. However, it allows to point out a contract where inelasticity of labor supply obtains in exchange of wages potentially less variable than workers marginal productivity. Consequently, workers solve the following problem: sup C w,t,n w,t E s.t. C w,t W t n w,t, n w,t 1, e βt U(C w,t )dt, (2) where E t is the conditional expectation operator under full, public and rational expectations, β is the subjective discount factor, W t is the time t wage and n w,t is the labor supply. The problem has simple solution: C w,t = W t and n w,t = 1. Shareholders act as rentiers: they consume dividends and trade securities in the financial markets. Under Assumption 2, shareholders own all traded securities and also do not supply labor services. Accordingly, they face the following problem: sup C s,t,x s,t E e βt V (C s,t )dt, (3) s.t. dx s,t = X s,t (r(t) + x s,t (µ P (t) r(t)))dt C s,t dt + X s,t x s,t (σ P,y (t)db y,t + σ P,λ (t)db λ,t ), C s,t D t x s,t, x s,t 1. 4

6 where X s,t is the shareholders wealth process and x s,t is the wealth proportion invested in the claim on the flow of firm s dividends D t. For simplicity, the shareholders subjective discount factor coincides with that of the workers. The risk-free rate r(t), the stock expected return µ P (t) and the volatilities σ P,y (t) and σ P,λ (t) have to be determined in equilibrium. The Brownian shocks B y,t and B λ,t are defined later. Under standard concavity and differentiability conditions, the above problem leads to the standard pricing equation where the shareholders marginal utility evaluated at optimal consumption is the valid state-price density: with C s,t = D t and x s,t = 1. ξ t,u = e β(u t) V (D u ) V, u t, (4) (D t ) B. The Firm There exists a representative firm which behaves competitively and lives forever. The firm activity is modelled in a simplified way: it is financed by equity only and the investment decision is modelled in reduced form through a stochastic operational cash flows process. Such a simplistic modelling choice is made for the sake of tractability and to emphasize the role of labor relations, as the unique channel for income smoothing. A cash-flows process embeds the firm decisions and operations: Y t = A t F(K t, N t ) I t G(K t, N t, I t ) =Ŷ t I t G(K t, N t, I t ) =Ȳ exp(y t ), (5) where Ȳ >, the first line is the usual profit identity and the third line captures the reduced form approach with d y t = ηy t dt + σdb y,t. (6) Aggregate risk is then described by y t, which is a stationary, mean-reverting and homoscedastic process. The cash-flows process also determines the resources to be distributed to the workers and the shareholders: Y t = W t + D t. (7) The resource constraint in Eq. (7) captures the endogenous nature of wages and dividends which is determined by labor relations, given the firm s cash-flows. C. Labor Relations The standard Walrasian Cobb-Douglas economy leads to constant capital and labor shares consistent with workers paid at their marginal productivity: W t = (1 α)y t, with α (,1). (8) 5

7 The opposite extreme scenario would be the case where workers are promised perfect income smoothing: wages are constant and equal to the unconditional expected marginal productivity: W t = W = E[(1 α)y t ]. (9) Here, I postulate that workers and shareholders agree in advance on an optimal risk sharing rule. Namely, a contract C is designed to effect the optimal risk sharing between workers and shareholders: wages smoother (riskier) than output obtain at the cost of riskier (smoother) dividends. A simple and intuitive approach is as follows. The two agent types arrange an agreement such that their marginal utilities are proportional to each other. Assumption 3 Bargaining power. The contingent wage payments and dividend distributions are governed by the following sharing rule: and, hence, under Assumption 2 in equilibrium, { C = W t : U (C w,t ) V (C s,t ) = Λ t,w t + D t = Y t } t, (1) U (W t ) V (D t ) = Λ t = Λexp(λ t ), (11) where Λ > and dλ t = φλ t dt + νdb λ,t. (12) The factor Λ t is intended to reflect the relative bargaining power of the two parties and determines simultaneously the average consumption shares as well as their relative variability. The time-varying nature of Λ t represents the inability of workers and shareholders to commit once and for all for the degree of bargaining power. I assume that the factor Λ t captures small and persistent variations due to the social and political environment and, hence, taken as exogenous with respect to the two agent types. I refer henceforth to λ t as the social risk, which captures the exogenous uncertainty in the distributional risk among workers and shareholders and generates the low frequency movements in the wage share. Similarly to y t, also λ t is a stationary, mean-reverting and homoscedastic process. Since a larger (smaller) λ t gives a smaller (larger) share of aggregate income to workers, social risk can be interpreted as a source of idiosyncratic income variation. The latter affects also the shareholders consumption variability and the equilibrium state-price density through the risk sharing rule of Eq. (11). The interpretation of Assumption 3 is then that observed variations in income shares reflect the timevarying bargaining power of workers and shareholders in labor negotiations. Accordingly, the calibration of the exogenous process Λ t can be guided by the objective of replicating the time-series properties of these income shares. This approach allows for correlation among the Brownian shocks: d B y,t,b λ,t = ρdt. The risk sharing rule of Assumption 3 implies a degree of income insurance from shareholders to workers which is consistent with the priority of wage payments and the time-variation of the wage share over the business cycle. Therefore, a distributional risk hangs over the shareholders and magnifies the risk properties of the residual payments, e.g. dividend distributions. Such a risk is sometimes called operational leverage and acts similarly to financial leverage: however, at least at the aggregate level, the former is supposed to be more significant for the riskiness of equity returns since wages account for about two thirds of GDP, whereas interests on corporate bonds account for about 2% only. 6

8 A simple and alternative representation of the workers and shareholders problem is as follows. Assume a benevolent central planner who maximizes a weighted sum of the two agent types utilities under the aggregate resource constraint and with the distinguishing feature that the welfare weights are exogenously stochastic: such that Λ s,t /Λ w,t = Λ t. sup W t +D t =Y t E ( Λs,t V (D t ) + Λ w,t U(W t ) ) dt, (13) Assumption 1 is crucial to the economic meaning of the risk sharing rule of Eq. (11). Indeed, under constant bargaining power (ν = ), its insurance content can only arise under preference heterogeneity: as long as shareholders are less risk averse than workers, the endogenous distributional risk smooths wages and enhances the risk of owning capital. The opposite relation holds if shareholders are more risk averse than workers, whereas no distributional risk obtains if the two agent types have the same risk preferences. In particular, the income insurance that workers achieve within the firm is proportional to the degree of preference heterogeneity: the larger the distance in preferences h = γ w /γ s > 1, the smoother the wage payments relative to the dividend distributions. Figure 1 illustrates a simple example. Insert Figure 1 about here. The upper panel shows wages and dividends under the choice of model s parameters in Section V.A: h > 1 guarantees a smoothing effect on wages at the cost of risky dividends. Indeed, the former are a concave function of the latter. The opposite holds for h < 1 (lower panel), whereas a linear relation obtains in absence of preference heterogeneity (h = 1, middle panel). These two scenarios are computed by changing Λ in order to recover the same steady state dividend and labor shares (black lines). The shadow area represents the range of variation due to uncertainty in the bargaining power (λ t ): it is worth noting that, even in absence of preference heterogeneity, workers can exploit income insurance within the firm as long as aggregate and social risk are positively correlated (ρ > ). Indeed, in good states (high y t ) equilibrium wages are more likely to lie between the blue and the green line (high λ t ) then between the red and blue line (low λ t ). The opposite holds in bad states (low y t ). C.1. Wage Rigidity The contract defined by Eq. (11) accounts for the income insurance the workers can exploit within the firm. A similar smoothing effect can obtain by including in the model wage rigidity. Consistent with Favilukis and Lin (212b), infrequent or incomplete adjustments of wages lead to stable wage payments and reduces their cyclicality. Let define the following model of wage dynamics. I set the standard Walrasian Cobbs-Douglas wage as Wt = (1 α)y t, whereas the true wage payments is denoted by W t. Consider the following dynamics: d logw t = R g(w t,w t )dt + (1 R)d logw t, (14) where W t is a target wage which satisfies a sharing rule as in Eq. (11): U (W t ) V (D t ) = Λ t, 7

9 R {,1} is an indicator function which denotes wage rigidity when it equals one, and g(x, z) is an adjustment function which captures the degree of wage rigidity such as g(x, z) = κ(log x log z). This model of wages covers various cases. When h 1 income insurance within the firm (with social risk ν ) takes place as in the previous section if R = or under wage rigidity if R = 1. In the latter case, wages are locally deterministic and adjust toward a target wage at a speed of reversion κ: when h = 1,ν = such a target is the Walrasian contract Wt. When both wage rigidity and income insurance are turned off (R =, h = 1,ν = ), wage dynamics reduces to the standard walrasian case and then d logw t = d logw t = d logy t. Both income insurance and wage rigidity lead to smooth wages and countercyclical labor-share. Therefore, they enhance the riskiness of owning capital and are useful to generate short-term equity risk and, hence, downward sloping term-structures of dividends and equity. For the sake of exposition and analytic tractability, hereafter I consider the model without wage rigidity and I focus on the equilibrium timing of risk. However, wage rigidity is both a complementary and alternative source of distributional risk to income insurance within the firm. Favilukis and Lin (212b) and Danthine and Donaldson (22) have shown that both channels can improve the traditional asset pricing implications of otherwise standard real business cycle models, whereas this paper shows that labor relations is also crucial to the equilibrium determination of the new empirical evidence about the term-structure of equity risk and premia. III. The Equilibrium The equilibrium for the economy described in Section II is a pair D t and W t which jointly satisfies Eq. (4), (7) and (11). Equivalently, dividend distributions are optimal to shareholders given the risk sharing rule, which defines wage payments, and the resources constraint. A. State price density From the shareholders perspective, that is the point of view of market participants, ξ t,u in Eq. (4) denotes the state price density in the economy and, then, the price at time t of an arbitrary payoff stream {F u, u (t, )} is given by E t [ t ξ t,u F u du]. Proposition 1 The equilibrium state price density ξ,t satisfies: Y t = (Λ t e βt ξ,t ) 1/γ w + (e βt ξ,t ) 1/γ s. (15) Although the state price density equals the marginal utility of shareholders, it depends on the risk attitudes of both agent types. Indeed, the endogenous determination of wages and dividends due to the income insurance from shareholders to workers (h > 1) is proportional to the degree of heterogeneity in risk preferences: the larger the distance in the risk aversion of the agents, the smoother the wages and the riskier the dividends. 8

10 A second consequence of the risk sharing rule between the agents is the dependence of the state price density on the social risk: time-variations of the bargaining power in wage negotiations lead to temporary variations in the equilibrium discount rates since they alter the workers ability to smooth consumption within the firm and, in turn, the riskiness of owning capital. Under preference heterogeneity (h 1) and CRRA form of preferences, the undiscounted state price density (i.e. exp(βt)ξ,t ) is stationary only if the state variables are stationary too. The specification of aggregate and social risks respectively in Eq. (6) and (12) preserves then the stationarity of the equilibrium. Instead, an integrated cash-flows process is incompatible with a stationary equilibrium under heterogeneous risk attitudes and CRRA (as well as recursive) utility. As long as the degree of preference heterogeneity is an integer, h N, the clearing condition in Eq. (15) can be interpreted as a polynomial of degree h in z = ξ 1/γ w,t : Y t = (Λ t e βt ) 1/γ w z + (e βt ) h/γ w z h, (16) and admits analytical solutions for h {1, 2, 3, 4} (see Wang (1996)). doubles that of shareholders the state price density is given by: Namely, when workers risk aversion ξ,t = e βt 2 γ w Λ t ( 1 + 4Y t Λ 2/γ w t 1) γw. (17) Unfortunately, as it will be shown, the degree of preference heterogeneity required to capture simultaneously a high equity premium and the downward-sloping term structure of equity risk and premia does not allows for an analytical solution to ξ,t. However, on the one hand, the general properties of the equilibrium state price density can be easily characterized and, on the other hand, even in the simple case of h = 2 in Eq. (17) closed form solutions to asset prices do not obtain. Corollary 1 The equilibrium state price density has partial derivatives satisfying: y ξ,t < and second order partial derivatives satisfying: and λ ξ,t <, (18) 2 y 2 ξ,t >, 2 λ 2 ξ,t > and 2 y λ ξ,t >. (19) The state price density is decreasing in both aggregate and social risks. Therefore, the state variables y t and λ t are interpreted by the market participants as procyclical variables. Indeed, an increase in y t implies a persistent increase in the available resources from the firm s production and, hence, a decrease in the discount rates. Instead, a decrease in λ t implies a persistent increase in the workers bargaining power and therefore a larger income insurance cost to the shareholders. Such a cost magnifies the riskiness of dividends distributions and, hence, increases the discount rates. The convexity implied by the second-order derivatives is a result of preference heterogeneity: the larger the distance in risk attitudes between workers and shareholders, the larger the curvature of the state price density. Such a result is an endogenous effect of the distributional risk induced by labor relations. The logarithm of the state price density is a nonlinear function of the states and, therefore, is endogenously heteroscedastic even if 9

11 the state variables are homoscedastic as in Eq. (6) and (12). The time-variation in the instantaneous volatility of the state price density is an equilibrium result of the combined effect of labor relations and preference heterogeneity, although the limited market participation assumed for workers. The nonlinearity of the state price density leads to the following endogenous dynamics of the risk-free rate and of the prices of risk. Proposition 2 The equilibrium state price density has dynamics given by dξ,t = r(t)ξ,t dt θ y (t)ξ,t db y,t θ λ (t)ξ,t db λ,t, (2) where the instantaneous risk-free rate satisfies r(t) = β + ( yξ,t )ηy t + ( λ ξ,t )φλ t 1 2 ( yyξ,t )σ ( λλξ,t )ν 2 ( yλ ξ,t )σνρ ξ,t, (21) and the instantaneous prices of aggregate and social risk are given respectively by θ y (t) = σ yξ,t ξ,t = σy t 1 γ w W t + h, (22) γ w D t and with h = γ w /γ s, θ λ (t) = ν λξ,t ν =, (23) ξ,t 1 + hd t /W t yy ξ,t = γ ( wy t ((h + γw )hd t + (1 + γ w )W t )Y t (hd t +W t ) 2) ξ,t (hd t +W t ) 3, (24) λλ ξ,t = D ( tw t h 2 D t + (h + γ w )hw t + γ w D 3 ) t W2 t ξ,t γ w (hd t +W t ) 3, (25) yλ ξ,t = Y ( ) tw t (h + γw 1)hD t + γ w W t (hd t +W t ) 3. (26) ξ,t The endogenous determination of dividends results in the pricing at equilibrium of both aggregate and social risks. The price of aggregate risk in Eq. (22) has the following interpretation. It is proportional to the inverse of weighted average of the agents risk tolerances with weights given by the corresponding consumption shares. Such a quantity equals the aggregate risk aversion of a pure exchange economy, without limited market participation, with two types of agents, whose risk aversion coefficients are given by those of workers and shareholders. 6 Preference heterogeneity (h 1) leads to endogenous variations in the price of aggregate risk: θ y (t) increases as long as the consumption distribution shifts from the lowest risk averse agent to the most risk averse one and vice-versa. Under Assumption 1 (i.e. h > 1), the consumption share of shareholders is high (low) when an additional unit of consumption is cheap (costly) and, therefore, the price of aggregate risk is low in good times and high in bad times. Such a countercyclical dynamics obtains as an endogenous result even if the state price density equals the classical marginal utility of CRRA agents: V (D t ). 6 Such a framework is investigated by Chan and Kogan (22) under complete markets and closely captures the reduced-form approach of (nonlinear) habit models such as Campbell and Cochrane (1999). 1

12 The price of social risk in Eq. (23) has the following rationale. Fluctuations in the bargaining power of workers in wage negotiations lead to time-variations in the degree of income insurance that workers can exploit within the firm. Dividend distributions are subject to such social risk and, in turn, the marginal utility of shareholders is affected too. Preference heterogeneity (h 1) makes the pricing of social risk time-varying since it leads to time-variations in the relative weight of the consumption share between the two types of agents. Under Assumption 1, the price of social risk is decreasing in the ratio of dividends to wages: this obtains because when consumption distribution shifts towards shareholders (workers) a persistent decrease (increase) in the riskiness of dividend distributions is expected and, then, the volatility of shareholders marginal utility reduces (increases) too. The ratio of dividends to wages is high (low) when an additional unit of consumption is cheap and, hence, the price of social risk has countercyclical dynamics from the perspective of shareholders. Both the prices of risk θ y (t) and θ λ (t) are endogenously time-varying and their variability increases with the degree of preference heterogeneity h: the latter induces convexity in the logarithm of the state price density not through variations of the aggregate risk aversion (as in Campbell and Cochrane (1999) or Chan and Kogan (22)), but through the endogenous heteroscedasticity of dividends, due to the reallocation of the firm s resources between workers and shareholders over the state of the economy. B. Endogenous dividends and wages Dividends and wages evolve endogenously at equilibrium. Their dynamics is characterized by the resource constraint and the contract rule, which lead to D t =(Y t D t ) h Λ h/γ w t, (27) W t =(Y t W t ) 1/h Λ 1/γ w t. (28) Both D t and W t are stationary functions of the states y t and λ t, but unfortunately we cannot solve analytically for their levels. However, under Assumption 1 workers benefit from an implied income insurance provided by shareholders, who receive volatile dividends. The growth rates of dividends and wages can be written as dh t = µ H (t)h t dt + σ H,y (t)h t db y,t + σ H,λ (t)h t db λ,t, with H t = {D t,w t } and µ H (t) = Ht 1 ( y H t ηy t λ H t φλ t y,yh t σ λ,λh t ν 2 + y,λ H t ρσν ), σ H,y (t) = H 1 t y H t σ, σ H,λ (t) = H 1 t λ H t ν. The next Lemmas characterize the moments of these growth rates with respect to the state-variables. Lemma 1 Under Assumption 1, dividends and wages satisfy the following conditions: y D t D t >, y W t W t >, y,y D t D t >, y,y W t W t >, λ D t D t >, λ W t W t <, λ,λ D t D t, λ,λ W t W t, y,λ D t D t >, y,λ W t W t <. (29) 11

13 Aggregate risk positively affects the levels of both dividends and wages since it increases the total consumption. Instead, social risk has the opposite effects on dividends and wages since it alters the bargaining power of workers and shareholders in the determination of dividends distributions and wage payments. Lemma 2 The instantaneous volatilities of dividends are given by: σh σ D,y (t) =, (3) 1 + (h 1)D t /Y t ν σ D,λ (t) = γ w (1/h + D t /W t ), (31) and relate with the instantaneous volatilities of wages as follows: Instantaneous volatilities have the following limits: σ W,y (t) = 1 h σ D,y(t), (32) σ W,λ (t) = D t W t σ D,λ (t). (33) if D t Y t { σd,y (t) hσ, σ W,y (t) σ, σ D,λ (t) νh γ w, σ W,λ (t), and if D t Y t 1 { σd,y (t) σ, σ W,y (t) σ h, σ D,λ (t), σ W,λ (t) ν γ w. The exposure of dividends to aggregate risk, σ D,y (t), is increasing in preference heterogeneity (h) and decreasing in the dividend-share under Assumption 1 (i.e. h > 1). An high dividend-share implies a lower volatility because in these states of the world the income insurance implied in the contract rule is cheap relative to the states of the world where D t /Y t is low. This mechanism is similar to that of an economy with preference heterogeneity but without limited market participation: in good times wealth shifts from high risk averse agents towards low risk averse agents (i.e. from workers to shareholders), the aggregate risk aversion decreases too and so does the price of risk. The opposite holds in bad times. Here, labor relations lead to a similar countercyclical dynamics of the aggregate price of risk but this mechanism obtains through the endogenous dynamics of dividends instead of the aggregate risk aversion. Similarly to the price of aggregate risk, σ D,y (t) can be rewritten as σ relative to the weighted average of the wage- and dividend-share with weights respectively 1/h and one. The exposure of dividends to social risk, σ D,λ (t), is increasing in preference heterogeneity (h) and decreasing in the relative risk aversion of workers as well as in the ratio of dividends relative to wages. When D t /W t is high, time-variation in the bargaining power in labor relations has a limited effect on the riskiness of dividend distributions, since, when the dividend-share is high, the cost of the income insurance for shareholders is cheap relatively to the states of the world in which dividends are low. The second part of Lemma 2 describes the smoothing effect of labor relations on the riskiness of both dividends and wages growth rates. Under Assumption 1, aggregate risk leads to a lower instantaneous volatility for wages than for dividends. Such an effect, in Eq. (32), is proportional to the degree of preference heterogeneity h: the larger the relative risk aversion of workers relative to that of shareholders, the larger the income insurance that the former can exploit within the firm and the larger the risk of owning capital. The instantaneous volatility induced by social risk, in Eq. (33), has the opposite sign for dividends and wages growth rates. The relative difference in the magnitude of these volatilities is proportional to the ratio 12

14 of the levels of dividends and wages: the riskiness of time-varying bargaining power in labor relations which falls back on workers and shareholders is inversely proportional to their shares of total consumption. Eq. (33) can be rewritten to disentangle the role of risk aversion from that of preference heterogeneity: σ W,λ (t) = 1 h σ D,λ(t) ν = ν ( ) 1 1. γ w γ w 1 + hd t /W t On the one hand, preference heterogeneity smooths the volatility of wages relative to that of dividends if h > 1, and, on the other hand, the higher the risk aversion of workers the lower the magnitude of the volatility induced by social risk. However, in real data the average share of dividends is significantly lower than that of wages (D t /W t is about less than 1% at the aggregate level) and, hence, Eq. (33) leads most of the times to an endogenous smoothing effect of the riskiness of wages, due to labor relations. The third part of Lemma 2 shows the limit behavior of instantaneous volatilities of dividends and wages. Under Assumption 1, dividends expositions to aggregate and social risks are counter-cyclical in the sense that they decrease (in magnitude) with the labor share and the opposite holds if h < 1. Instead, wages expositions to aggregate and social risks are respectively counter-cyclical and pro-cyclical. The range of variation is fully captured by the degree of preference heterogeneity, h. From the perspective of shareholders, in the limit good state (D t /Y t 1) that is when workers do not exploit income insurance within the firm the dividends exposition to aggregate risk reduces to σ D,y σ and the exposition to social risk disappears σ D,λ. Viceversa dividends expositions are larger in the limit bad state (D t /Y t ). Instead, wages feature minimal volatility when the labor share tends to one: the exposition to aggregate risk is reduced by the degree of preference heterogeneity σ W,y σ/h and the whole uncertainty about bargaining power falls on shareholders σ W,λ. Vice-versa wages volatility increases in magnitude as long as the the labor share reduces. The focus now turns on the term structure of cash-flows and in particular on the slope of expected growth rates and volatility with respect to the time horizon. Unfortunately, the system in Eq. (27)-(28) does not allow for analytic solutions. However, accurate approximations can be easily computed. Recall that both dividend and wage levels as well as the (undiscounted) state-price density are functions of the states y t and λ t only. Let consider the following approximation: ξ p,t D q t M D,t(p, q) = e pβt m j= a j(p, q)e b j(p,q)y t +c j (p,q)λ t. (34) Coefficients {a j, b j, c j } and the integer m can be set to minimize the approximation error over the state-space {y t,λ t }, which are stationary state-variables. Under reasonable model parameters, the approximation is extremely accurate for m small. Since the state-variables belong to the affine class and the approximation function is a weighted sum of exponential affine functionals, expectations at any horizon can be easily computed. In order to preserve both analytic tractability and the economic meaning of primitive parameters, an alternative approximation can be computed after linearizing the logarithm of the dividend process: log D t log D + ( Y D) Ȳ D y t + ( Λ D) Ȳ Λ λ t = d + d y y t + d λ λ t, (35) 13

15 where the steady state level d = log D is an arbitrary fraction of Ȳ (with W = Ȳ D), characterized by the constants Ȳ and Λ and Eq. (27) at the steady state (y t = λ t = ), and the constants d y and d λ satisfy: d y = hȳ W + h D, d 1 λ = γ w /h + γ w D/ W. (36) Notice that under the log-linearization, as usual, the dividend process is still increasing in the state-variables but inherits their homoscedasticity and, hence, the prices of risk are no more time-varying. 7 The main quantity of interest is the Laplace transform of the logarithm of dividends and other cash-flows. Proposition 3 The Laplace transform of the logarithm of dividends has the following approximation: D t (τ, q) = E t [D q t+τ ] m j= a j(, q)e A j(,q,τ)+b j (,q,τ)y t +C j (,q,τ)λ t, (37) where q and model parameters are such that the expectation exists finite, the approximation make use of Eq. (34) and A j,b j and C j are deterministic functions of time derived in the Appendix. A similar expectation obtains with the log-linearization of Eq. (35). With this result in hand, the term structures of expected dividend growth and volatility are computed as g D (t,τ) = 1 ( ) τ log Dt (τ,1) D t (,1) and σ D (t,τ) = ( ) 1 τ log Dt (τ,2) D t (τ,1) 2. (38) Similar expectations W t (τ, q) and Y t (τ, q) and term-structures {g W (t,τ),σ W (t,τ)} and {g Y (t,τ),σ Y (t,τ)} can be computed respectively for wages and total consumption by use of approximations similar to that of Eq. (34). IV. Equilibrium Asset Prices A. Equilibrium dividend strips Given the state-price density, the next proposition establishes the equilibrium price of the market dividend strip. Proposition 4 The equilibrium price of the market dividend strip with maturity τ is given by P t,τ = E t [ ξt,t+τ D t+τ ] ˆξ 1,t m j= a j(1,1)e A j(1,1,τ)+b j (1,1,τ)y t +C j (1,1,τ)λ t (39) where model parameters are such that the expectation exists finite, the approximation makes use of Eq. (34), ˆξ,t = ξ,t e βt and A j,b j and C j are deterministic functions of time derived in the Appendix. A similar price formula obtains with the log-linearization of Eq. (35). The price of the dividend strip (and its ratio with the current dividend value) is a stationary function of the aggregate risk and social risk factors and the maturity. The price is a weighted sum of exponential affine functionals of the factors, therefore the approximation of Eq. (34) does not rule out state-dependence from the return moments. Hence, the dividend strip s returns are endogenously heteroscedastic and its premium and volatility are time-varying even if y t and λ t are homoscedastic. 7 The prices of risk are given by: θ y (t) = σ γ w (1+(h 1) δ) and θ ν(1 δ) λ(t) = 1+(h 1) δ with δ = D/Ȳ. 14

16 Proposition 5 The instantaneous volatility and premium on the dividend strip with maturity τ are given by σ P (t,τ) = σ P,y (t,τ) 2 + σ P,λ (t,τ) 2 + 2ρσ P,y (t,τ)σ P,λ (t,τ), (4) (µ P r)(t,τ) =θ y (t) ( σ P,y (t,τ) + ρσ P,λ (t,τ) ) + θ λ (t) ( ρσ P,y (t,τ) + σ P,λ (t,τ) ), (41) where σ P,y (t,τ) = θ y (t) + σ ydcf(t,τ) DCF(t,τ) = σ + θ y (t) + Ψ y (t,τ), σ P,λ (t,τ) = θ λ (t) + ν λdcf(t,τ) DCF(t,τ) = ν + θ λ (t) + Ψ λ (t,τ), (42) with DCF(t,τ) = ˆξ,t P t,τ implicitly defined in Eq. (39). The return on the dividend strip depends on both the shocks to aggregate and social risk through respectively the expositions σ P,y and σ P,λ. These terms depend on the state-variables y t and λ t as well as on the maturity τ and have the following decomposition. Volatilities are indeed the sum of three terms: the instantaneous volatility of fundamentals (σ and ν), the transient risk (which is exactly equal to the prices of risk θ y (t) and θ λ (t), since the utility of shareholders is time-separable) and the forward-looking risk (Ψ y (t,τ) and Ψ λ (t,τ)). The latter obtains from the semi-elasticity of discounted cash-flows (DCF(t,τ)) to the state-variables and is the only term responsible for the slope of the term-structures of the dividend strip s premium and volatility. The premium is time-varying since both the state-price density and the cash-flows are endogenously heteroscedastic. 8 The former is heteroscedastic under preference heterogeneity (h 1), whereas the latter feature state-dependent volatility because of both preference heterogeneity and social risk (ν > ). Consequently, both the priced factors covary with the return process and lead to equity compensations. 9 premium is as follows. The cyclicality of the On the one hand, the prices of risk are decreasing with the state-variables under Assumption 1, and, on the other hand, the return volatilities can have both decreasing and increasing components. Under Assumption 1, transient risk is decreasing with the states, whereas forward-looking risk can be increasing or decreasing as long as respectively the wealth effect dominates the intertemporal substitution effect (h < γ w ) or vice-versa (h > γ w ). Even in the former case, under reasonable parameters, transient risk more than offset forward-looking risk in most of the states and, hence, both the premium and the volatility are countercyclical, but the dynamics of the premium is more pronounced, in line with the empirical evidence. The nonlinearity of Eq. (27) and (28) does not allow to study analytically the slope of the term structures of premium and volatility of the dividend strips. However, numerical evaluation of Eq. (4) and (41) as a function of maturity τ determines the behavior of such term structures over the state-space { y, λ}. Under the baseline calibration of Section V.A, the sign of the slopes of σ P (t,τ) and (µ P r)(t,τ) depends only on the forward-looking components of return volatility. Namely, we have sign( τ σ P (t,τ)) = sign( τ (µ P r)(t,τ)) = sign ( x E t τ [ E t [ D 1 γs t+τ D 1 γs t+τ ] ] ), (43) with x = {y,λ} and ρ (or small enough in magnitude). Intuitively, dividends D t+τ are increasing with the state-variables and, since those are mean-reverting processes, the semi-elasticity of dividends with respect 8 These two channels of premia s time-variation are usually separately and exogenously considered respectively in habit and longrun risk models. 9 More formally, since shareholders have power utility, both aggregate and social risk contribute to the only compensation for shareholders consumption risk. 15

17 to the states changes negatively with the maturity τ. When we look at discounted dividends D 1 γ s t+τ such a negative relation requires that shareholders are less risk averse than the logarithmic investor, that is γ s < 1, or, equivalently, h > γ w : the intertemporal substitution effect should dominate the wealth effect. Under reasonable parameters, γ s smaller enough than unity is sufficient to generate downward sloping term structures of both premium and volatility of the dividend strip returns, in line with the empirical evidence. Such a result holds true under the log-linearization of Eq. (35) as described by the following proposition. Proposition 6 Under the log-linearized dividend process of Eq. (35), the slope of the term-structures of equity risk and premia satisfy ( τ σ 2 P (t,τ) = 2 (γ h 2 w h) d 2 y ησ2 (γ w (e ητ 1) + h)e 2ητ + d 2 λ φν2 (γ w (e φτ 1) + h)e 2φτ( ) + d y d λ σνρ(γ w (η(e φτ 1) + φ(e ητ 1)) + h(η + φ))e (η+φ)τ /2, (44) τ (µ P r)(t,τ) = γ w h 2 (γ w h) ( d y ησ(d y σ + d λ νρ)e ητ + d λ φν(d λ ν + d y σ + ρ)e φτ). (45) Hence, when ρ the sign of the term-structures of equity is given by sign( τ σ 2 P (t,τ)) = sign( τ(µ P r)(t,τ)) = sign(γ w h) = sign(γ s 1). (46) Shareholders preferences govern the equilibrium timing of risk that is the term-structure of equity despite both the risk attitudes of workers and the degree of preference heterogeneity. Instead, workers risk aversion and preference heterogeneity matter in the determination of the equilibrium amount of risk that is the endogenous dynamics of the prices of risk and the endogenous riskiness of dividends. B. Equilibrium market asset Given the state-price density, the next proposition establishes the equilibrium price of the market asset. Proposition 7 The equilibrium price of the market asset is given by P t = E t [ t ] ξ t,u D u du ˆξ 1 m,t j= a j(1,1) t e A j(1,1,u t)+b j (1,1,u t)y t +C j (1,1,u t)λ t du (47) where model parameters are such that the expectation exists finite, the approximation makes use of Eq. (34), ˆξ,t = ξ,t e βt and A j,b j and C j are deterministic functions of time derived in the Appendix. A similar price formula obtains with the log-linearization of Eq. (35). Notice that the market price is given by the time integral of the dividend strip price over the infinite horizon: P t = P t,τdτ. The market price (and its ratio with the current dividend value) is a stationary function of the aggregate risk and social risk factors. The price is a weighted sum of exponential affine functionals of the factors up to time integration. Therefore the approximation of Eq. (34) does not rule out state-dependence from the return moments. Hence, the market returns inherit from those of the dividend strip the endogenous time-variation of the premium and volatility, even if y t and λ t are homoscedastic. 16

18 Proposition 8 The instantaneous volatility and premium on the market asset are given by σ P (t) = σ P,y (t) 2 + σ P,λ (t) 2 + 2ρσ P,y (t)σ P,λ (t), (48) (µ P r)(t) =θ y (t) ( σ P,y (t) + ρσ P,λ (t) ) + θ λ (t) ( ρσ P,y (t) + σ P,λ (t) ), (49) where σ P,y (t) = θ y (t) + σ ydcf(t) DCF(t) = σ + θ y (t) + Ψ y (t), σ P,λ (t) = θ λ (t) + ν λdcf(t) DCF(t) = ν + θ λ (t) + Ψ λ (t), (5) with DCF(t) = ˆξ,t P t implicitly defined in Eq. (47). The moments of the market returns have a derivation similar to that of dividend strip s returns. Both the exposure to aggregate and social risk depends on both the state-variables and the premium is given by the product of such exposures with the prices of risk. The return volatilities are given by the sum of three components that can be interpreted respectively as fundamental risk (σ and ν), transient risk (θ y (t) and θ λ (t)) and forward-looking risk (Ψ y (t) and Ψ λ (t)). The latter differs from that of dividend strips since it accounts for the semi-elasticity of the whole stream of discounted dividends on the infinite horizon with respect to the state-variables. The sum of the transient and forward-looking components of return volatility results in the so-called excess-volatility of market return over fundamentals. It is worth noting that it obtains even if i) the term structure of dividends volatility converges in the long-run to that of total consumption, which is smooth and homoscedastic; ii) market participants, that is shareholders, feature constant relative risk aversion. The endogenous heteroscedasticity of the state-price density and of the dividends, due to labor relations and preference heterogeneity, leads to non-trivial Garch effects in the market volatility governed by both the rates of reversion in the state variables (κ and φ) as well as in the degree of heterogeneity in risk aversion (h), since it determines how time-varying bargaining power among workers and shareholders translates (state-by-state) into the riskiness of dividends. Under Assumption 1, the prices of risk are counter-cyclical (i.e. they are higher when an additional unit of shareholders consumption is more valuable and vice-versa) and so does the equity premium. Indeed, the forward-looking components of return volatility are counter-cyclical, if the intertemporal substitution effect dominates the wealth effect (γ s < 1). In the opposite scenario they are pro-cyclical but do not offset the endogenous variation in the prices of risk under reasonable parameters. Therefore, the model endogenously generates the main properties of market returns, such as excessvolatility, Garch effects and countercyclical premia. C. Equilibrium bond and equity yields Given the state-price density, the next proposition establishes the equilibrium price of the non-defaultable zero-coupon bonds. Proposition 9 The equilibrium price of the zero-coupon bond with maturity τ is given by B t,τ = E t [ ξt,τ ] ˆξ 1,t m j= a j(1,)e A j(1,,τ)+b j (1,,τ)y t +C j (1,,τ)λ t (51) 17

19 where model parameters are such that the expectation exists finite, the approximation makes use of Eq. (34), ˆξ,t = ξ,t e βt and A j,b j and C j are deterministic functions of time derived in the Appendix. A similar price formula obtains with the log-linearization of Eq. (35). The bond price is a stationary function of the aggregate risk and social risk factors and the maturity. The price is a weighted sum of exponential affine functionals of the factors. Therefore the approximation of Eq. (34) does not rule out state-dependence from the bond dynamics. Hence, the bond s yield and its volatility feature endogenous time-variation, even if y t and λ t are homoscedastic. Namely, the real yield is nonlinear in the state-variables: ε(t,τ) = 1 τ ( log m j= a j(1,)e A j(1,,τ)+b j (1,,τ)y t +C j (1,,τ)λ t logξ,t ). (52) Such nonlinearities usually ruled out by most of term-structure models are due to labor relations and preference heterogeneity. The conditional volatility of the real yield is then a stationary function of both the states and the maturity. A generic bond, which pays coupons over a finite or infinite horizon T, has price simply given by B t = T (τ)b t,τdτ, where (τ) denotes an indicator function for coupon payments. Armed with these results, the focus turns on the equity yields as introduced by B12b. The model s equity yield is defined as p(t,τ) = 1 ( ) Pt,τ τ log = 1 ( log ) m D t τ j= a j(1,1)e A j(1,1,τ)+b j (1,1,τ)y t +C j (1,1,τ)λ t logξ,t D t and, hence, is a stationary function of the states and the maturity only. Moreover, it can be decomposed as follows: (53) p(t,τ) = ε(t,τ) g D (t,τ) + ϱ(t,τ). (54) The equity yield is given by the difference among the yield on the non-defaultable bond, ε(t, τ), and the dividend expected growth, g D (t,τ), plus a premium, ϱ(t,τ). The latter is implicitly defined by Eq. (38)-(52)-(53) and is a state-dependent function of y t, λ t and the maturity. V. Asset Pricing Results A. Model Calibration Model calibration is made by choosing parameters about exogenous uncertainty in the usual range of values in the literature and by choosing preference parameters to provide a good fit of cash-flows properties. Moments of equity returns obtain as an equilibrium result. Total consumption, Y t, is smooth and its logarithm features constant instantaneous volatility of σ = 3.5% and speed of reversion towards the mean of η =.5. The scale factor Ȳ is standardized to one for simplicity. The social risk factor, Λ t, is somewhat not observable but is calibrated to match the main properties of the dividend share (or equivalently labor share) in spirit of Danthine and Donaldson (22) and Danthine, Donaldson, and Siconolfi (26). 1 Social risk is smooth and persistent and positively correlated with aggregate risk: ν = 13%,φ =.5 and ρ =.5. The scale factor, Λ is set to fix the steady state (y t = λ t = ) dividend share 1 With the notation of Danthine and Donaldson (22), Λ t corresponds to µ 1 t. 18

20 at about 1%, given the preference parameters, in line with Longstaff and Piazzesi (24) and Santos and Veronesi (26). Preference parameters are set as follows. The relative risk aversion of shareholders is γ s =.92 and allows to match the yearly volatility of dividends growth rates as in Belo, Collin-Dufresne, and Goldstein (212): σ D,1 = 14.5%. The excess of risk of dividend distributions over total consumption is due to labor relations and obtains under large enough preference heterogeneity. In accord with Assumption 1 (i.e. h > 1), I set h = 4.7 such that the implied workers relative risk aversion is γ w = Since γ s < 1, downward term structures for dividends and equity returns can obtain. The subjective time discount rate (common to both workers and shareholders) is set to 2.4% to match the unconditional risk-free rate. Model s parameters and implied cashflows moments are summarized respectively in Table I and II. Insert Table I and II about here. B. The term structure of cash flows Under Assumption 1, labor relations lead to smooth wage payments and highly risky dividend distributions. In particular, the latter do not feature the counter-factual counter-cyclical dynamics implied by Walrasian models. Under the choice of model s parameters commented above, on the one hand, wages feature a yearly volatility of growth rates about σ W,1 = 1.8% and, hence, lower than that of total consumption (σ Y,1 = 2.8%). On the other hand, dividends match the empirical volatility of about 14.5%. The income insurance that workers exploit within the firm endogenously produces the counter-cyclical dynamics of the labor share. Although its simplicity, the model predicts realistic correlations D t /Y t has correlation with changes in Y t of about.3 and, hence, W t /Y t has correlation about minus.3 and, more importantly, their sign is correct. Such a qualitative result fails to obtain in most of production-based models and is crucial to the modelling of the term structure of both cash-flows and equity returns. Table II shows how dividend and labor shares move with aggregate risk by changing the degree of preference heterogeneity, h: workers more risk averse than shareholders and/or high social risk are the needed ingredient to capture the correct sign in the data. 11 Figure 2 shows simulated paths of the model implied cash-flows and dividend share. Insert Figure 2 about here. The smoothing effect due to labor relations can be observed by comparing the instantaneous volatilities of both wages and dividends for various levels of the degree of preference heterogeneity h. The upper panels of Figure 3 show how the exposition to aggregate risk σ D,y (t) in Eq. (3) (σ W,y (t) in Eq. (32)) of dividends (wages) increases (decreases) with h despite the current level of the dividend share. Similarly, the effect of h on the expositions to social risk (σ D,λ (t) and σ W,λ (t) in Eq. (31) and (33)) is depicted in the lower panels of Figure 3 as a function of D t /Y t. Insert Figure 3 about here. The term structures of growth rates and volatilities of total consumption, dividends and wages are shown in Figure 4. Since total consumption Y t is defined as a stationary process, we focus on the term structures of risk (right panels: σ H,τ for H = {Y,W, D}). The slope of the term structures reflects the distributional risk in 11 Notice that when h < 1, D t /Y t moves negatively with changes in Y t if ν is small enough. 19