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Prospect Theory and Asset Pricing in an RBC Framework Julia Lendvai and Rafal Raciborski First Draft: June 2006 PRELIMINARY Abstract We construct a fully-ﬂedged production economy model with Kahne- man and Tversky’s Prospect Theory features. The agents’ objective func- tion is a weighted sum of the usual utility over consumption and leisure and the utility over the relative changes of the agents’ wealth. It is also assumed that the agents are more sensitive to wealth losses than to gains. Apart from the changes in the utility, our RBC model is standard. We study prices of diﬀerent assets in our economy. It is demonstrated that under plausible parametrizations of the objective function our model ex- plains all ﬁrst and second moments of the returns on risky assets, short bonds and long bonds. In particular, we are able to match the empirical Sharpe Ratio, the equity premium and the volatilities of the risky asset and the riskless bond. 1 Introduction In his survey on behavior of ﬁnancial asset prices Campbell (1999) documents a host of stylized facts about the stock markets. The average real return on stocks is high and volatile. The returns on Treasury Bills are low and not very volatile. Additionally, real consumption growth is very smooth and it is only mildly correlated with real stock returns. These ﬁndings seem robust both across countries (see Campbell, 1999) as well as in time (Siegel, 1992a,b). Yet, explaining them within the framework of the consumption-based asset pricing model, an approach that has become a standard in the ﬁeld, constitutes a veri- table challenge for the economists. As it is clearly stated by Campbell, there are two fundamental problems arising from the application of this approach, when confronted with the data. First, for what is believed to be a plausible utility parametrization, it is hard to explain the magnitude of the premium of average stock returns over the risk-free rate. This problem, arguably the best known of the ﬁnancial puzzles, was termed by Mehra and Prescott (1985) the ”equity premium puzzle”. Second, it is diﬃcult to reconcile the high volatility of stock returns with the low volatility of the short term real interest rate. This is what Campbell calls the ”stock market volatility puzzle”. 1 For a long time all attempts to explain the aforementioned anomalies had been notoriously unsuccessful (for a list of works trying to challenge the puz- zles see Constantinides et al, 2002; see also the surveys: Kocherlakota, 1996, Campbell, 1999 and Campbell, 2000). It is only relatively recently that the economists have proposed some modiﬁcations to the baseline consumption- based, representative-agent asset pricing model that in the endowment-economy framework seem to be helpful in resolving the asset market puzzles. Among them, the Campbell-Cochrane habit persistence model (1999) is perhaps con- sidered the most successful, as except the historical means and volatilities of the asset returns it is able to match a number of other ﬁnancial statistics. The task of explaining the puzzles in the fully-ﬂedged production economy framework occurs even more demanding. The challenge comes from the fact that the consumers in this framework possess multiple means to alleviate the impact of an unfortunate investment. Consequently, their price for risk is low and thus are the risk premia. An early example of the diﬃculties one encounters when passing from an endowment economy setting to an endogenous-consumption and dividends model was provided by Rouwenhourst (1995). One simple way to match the equity premium in the endowment economy is to assume that the degree of agents’ risk aversion is considerably higher than what was traditionally considered plausible1 (Kandel and Stambaugh, 1991). However Rouwenhourst observed that in the standard real business cycle model very risk-averse agents are able to substantially reduce their consumption stream ﬂuctuations, which provides them with a good insurance against the consumption risk. Hence, an extreme degree of risk-aversion must be assumed in order to obtain non- negligible risk premia. Then, the model also produces counterfactually low volatility of consumption. Also attempts to carry over the success of ’habits’ to production economy models encountered serious problems. Just like in the case of increasing risk aversion, introducing the habit persistence to a standard real business cycle model pushes the agents to smooth excessively their consumption paths, leaving the equity premium unaﬀected. It is only when additional frictions on capital and labor mobility are added to the model that the favorable properties of the habit persistence are restored (Jermann 1998, Boldrin et al 2001). In fact, Boldrin et al argue that their two sector economy model with habit persistence is also able to signiﬁcantly improve the business cycle properties of the standard RBC model. Even so, the RBC model with habits does not seem satisfactory in all di- mensions. As it is shown in Jermann, the long term premium of long bond returns over the risk-free rate is of the same size as the equity premium in this type of models, a clearly counterfactual prediction. Also, the model consider- ably overshoots the risk-free rate volatility (Jermann, 1998; Boldrin et al, 2001). 1 But see Campbell (1999) who argues that it does not resolve the volatility puzzle. The assumption of high risk aversion in these models leads also to what Weil (1989) dubbed the ”risk-free rate puzzle”. For a short discussion of ”what a plausible degree of risk aversion is” see Kocherlakota (1996). 2 In fact both problems had been known already before and seem to be insepa- rably related to the habit persistence. Although in their endowment economy habit persistence model Campbell and Cochrane (1999) were able to engineer the habit stock process in such a way that the risk-free rate becomes virtually constant, they themselves admit that this is a knife-edge result. Moreover, Let- tau and Uhlig (1997) show that in the Campbell-Cochrane environment under some parametrizations consumption bunching is desirable, a certainly disturb- ing ﬁnding. Finally, the empirical evidence for the strength of habit persistence seems mixed (for a short discussion see Guvenen, 2003). In particular, K. Dynan (2000) does not ﬁnd any evidence of habit formation in the food consumption data from the Panel Study on Income Dynamics. This paper attempts to extend the standard endogenous production set- ting in another direction. We construct an RBC model with Kahneman and Tversky’s (1979) Prospect Theory features. Prospect Theory emerges from the experiments on the process of decision making under risk and is recently widely recognized among psychologists as a good descriptive model of such decisions. In their model Kahneman and Tversky posit that people assign value to gains and losses rather than to ﬁnal absolute position of their assets. They also introduce the concept of loss aversion: In their theory the individuals are substantially more sensitive about the losses than the gains they experience. Kahneman and Tversky’s theory is able to explain a set of anomalies observed in the individ- uals’ decision making that contradict the expected utility paradigm. Although originally based on laboratory experiment results on simple gambles, since the date of its publication the theory has gained a wealth of empirical support (see for instance Camerer, 1998 and List, 2004). Evidence consistent with this the- ory was recently found also in the case of the professional asset traders behavior (Coval and Shumway, 2001). Our work is a continuation of a strand of the behavioral ﬁnance literature that seems promising in resolving the ﬁnancial market puzzles. In a partial equi- librium model, Benartzi and Thaler (1995) show that loss-averse agents demand a high average risk premium in order to be willing to invest in stocks. Barberis, Huang and Santos (2001, BHS for future reference) examine whether the same will be true in a general equilibrium, endowment economy framework. Their answer is negative as far as it concerns a standard optimizing-agents model with the loss aversion as the only extra feature. However, once additional mod- iﬁcations based on psychological theory developments are added, their model is shown to explain the mean and volatility of stock returns. Moreover, it re- produces the predictability of the returns on risky assets. In a follow-up paper (2001) Barberis and Huang further develop this approach. The goal of this paper is to check whether Prospect Theory helps explaining the two previously discussed ﬁnancial puzzles also in a fully-ﬂedged production economy framework. To this end we extend the agents’ preferences in the other- wise standard RBC model to account for the utility derived from changes in the value of the ﬁnancial wealth they hold and assume that agents are loss-averse. Contrary to BHS, we do not consider any other modiﬁcations suggested by the psychological evidence. Hence, our setting corresponds to the simplest version 3 of the model considered by BHS. Our methodology is similar in spirit to the rapidly developing literature aiming at reproducing the ﬁnancial facts in models with nontrivial production sectors. In particular, we draw on the works of Jermann (1998) and Boldrin et al (2001). Other important papers in the ﬁeld are Guvenen (2003), Uhlig (2006) and Guvenen and Kuruscu (2006). There is one major diﬀerence be- tween these papers and our approach. The modiﬁcations of the baseline RBC model introduced by these authors (habit persistence in the case of Jermann and Boldrin et al, heterogeneity of agents in the case of Guvenen and Guvenen and Kuruscu and rigidity of wages in the case of Uhlig) in order to reproduce the asset market statistics usually do not leave unaﬀected the real side of the model economies. Therefore, a lot of attention is paid to ensure that the outcome of these modiﬁcations could be reconciled also with the stylized macroeconomic facts2 . This type of problems does not appear in the setting with loss-averse agents. As it will be shown later in the paper, the loss-aversion term that enters the agents’ utility in an additively separable way does not have any eﬀect on the dynamics and the volatility of real variables like consumption, output or investment. Hence, in our setting we are able to fully disentangle the ﬁnancial- and the real side of the model economy. Since there exists reach literature on reproducing macroeconomic facts in the RBC framework (see for instance King and Prescott, 1995 or King and Rebelo, 2000), here we focus solely on matching the asset market statistics. There are six statistics we attempt to ﬁt: The Sharpe Ratio, the equity pre- mium, the long-term bond premium, the real interest rate and the volatilities of the stock returns and the real interest rate returns. When a standard bare-bone RBC model with Prospect Theory features is considered, we are able to match only the Sharpe Ratio, and the short-term real interest rate moments. This is not to be neglected, as ﬁtting the empirical Sharpe Ratio alone has proven dif- ﬁcult even in the endowment economy framework (see Lettau and Uhlig, 2002). In order to ﬁt the other statistics we introduce capital adjustment costs and a moderate degree of ﬁnancial leverage to our model. This version of the RBC model allows us to reproduce with a high precision all asset market statistics of interest. Hence, none of the ﬁnancial puzzles discussed before emerge in our framework. The presence of the Prospect Theory features in the consumers’ utility func- tion is neutral as regarded the real side of the economy. However the intro- duction of the adjustment costs is shown to deteriorate the performance of the model in this dimension. Hence more work is needed in order to reconcile the capital adjustment costs with the business cycle facts. The rest of the paper is organized as follows. In the next section some characteristics of the Prospect Theory are displayed. Then the setting of the model is explained and the ﬁrst order conditions derived. The calibration issues and the numerical solution technique are also presented in that section. We 2 See for instance Lettau and Uhlig (2000) on the problem of consumption oversmoothing in the presence of habit persistence. 4 discuss our results in section 3. The last section concludes. 2 The model In this section we describe our simple RBC model with loss-averse agents. We start with a short discussion of the Prospect Theory. 2.1 Loss aversion The Kahneman and Tversky’s Prospect Theory (1979) is a model of decision making under risk. More precisely, it aims at describing the behavior of an agent facing a choice of risky prospects, or gambles. There are four main char- acteristics that distinguish the Prospect Theory from the more widely applied Expected Utility Theory: 1. The agents assign the value to wealth reductions and increases rather than to the absolute magnitude of the ﬁnal wealth. These reductions (losses) or increases (gains) are measured as compared to some initial reference level. 2. The agents are loss-averse: They are more sensitive to losses than to gains (their objective function has a kink at the reference point and is steeper for losses than for gains). 3. The agents’ objective function is concave for gains and convex for losses. 4. The agents overweigh low probabilities. As it was noted by Campbell (2000), the Theory was tested in experiments necessarily involving only relatively small payoﬀs. Therefore it needs not be readily applicable to decisions with higher risks at stake. For this reason, despite their apparent attractivity for our purposes, we decided to disregard the last two features of the Prospect Theory and focus solely on the loss aversion of the agents. In doing so we closely follow BHS (2001), who also concentrate on loss aversion. The Kahneman and Tversky’s original formulation of the Theory postulates that the individuals care solely about the relative changes of their wealth. As in BHS, we consider a more general setting in which we allow the agents’ objective function to be a weighted sum of the standard utility over consumption stream and leisure and the utility over the relative changes of the wealth (further re- ferred to as a Loss Aversion or LA term). In order to specify the latter we need to deﬁne three concepts (see Campbell 2000): the time horizon, the reference level and the type of wealth the variations of which enter the LA term. We start the description from the last. The issue of specifying the argument of the LA term is closely related to the psychological concept of the narrow framing (Thaler, 1980; Thaler et al, 1997). The standard economical theory predicts that what matters for the agents in 5 a given moment is their total wealth, regardless of its particular partition on diﬀerent types of assets. Contrary to this viewpoint, the psychological evidence suggests that when doing their mental accounting, people tend to focus on each narrowly deﬁned component of their wealth separately: They engage in narrow framing. This controversy has a direct impact on our work. In order to comply with the standard theory we would need to consider the loss aversion over the total wealth. Alternatively, we could deﬁne it over the portfolio of ﬁnancial assets. An extreme position (although not an unreasonable one, see Barberis et al, 2003) is to assume that the agents are exposed to the loss aversion over ﬂuctuations in the value of individual assets they hold. Under this assumption, the investors create separate mental accounts for stocks of every single ﬁrm. Barberis and Huang (2001) demonstrate that this last approach is especially successful in reproducing observed asset market statistics. In order to single out the inﬂuence of the pure loss aversion we do not make a distinction between individual stocks. All ﬁrms in the economy will be assumed identical so that their equities are homogeneous. Hence, we closely follow BHS (2001) who also consider only one type of stock3 . In addition, we consider a second type of risky asset, a long-term bond. We postulate an ad hoc degree of narrow framing by positing that the LA terms are separately deﬁned for the stocks and the long bonds. There are two other types of assets in our framework: one period riskless bonds and one period ﬁrms’ discount bonds. Since both assets guarantee a certain payout next period, we do not apply the Prospect Theory in their case4 . The next question concerns the deﬁnition of the reference level. The most basic formulation (Kahneman and Tversky, 1979) posits that individuals mea- sure their gains and losses in reference to the status quo, which in our case is the value of the initial investement in a risky asset. Let ar denote the number of t shares or long bonds purchased by an agent in t and held until t+1 and Ptr their current price. Then the initial investment value and the status quo reference level is simply ar Ptr . BHS suggest instead that we should take into account the t opportunity cost of the risky investment that is, the wealth one could obtain f from investing ar Ptr in a riskless asset. Let denote Rt the gross return on the t r one period (riskless) discount bond and Rt+1 the stochastic return on the risky asset, both over a t to t + 1 period. Then we assume that the reference level of f r an individual is ar Ptr Rt . Hence, her gain or loss Xt+1 is: t f r r Xt+1 := ar Ptr Rt+1 − ar Ptr Rt+1 t t The form of the loss aversion term will be then deﬁned as: r r r Xt+1 Xt+1 ≥ 0 V (Xt+1 ) = r for r (1) φXt+1 Xt+1 < 0 In order to capture the property of the loss aversion, we set φ > 1. 3 Note that the return on this single stock may be interpreted as a return on the market portfolio. 4 We do not consider a default risk in our model. 6 The last issue to be dealt with is the timing horizon. The question to be asked here is how frequently the agents evaluate their asset positions. Benartzi and Thaler (1995) argue that it is natural to assume that a serious evaluation is carried out once a year. It would then correspond to the frequency the taxes are ﬁled, the fund reports are published and the money managers are most thor- oughly scrutinized (Benartzi and Thaler 1995). Based on this argumentation, BHS calibrate their model on the annual basis. Instead, we decided to work with a quarterly calibration. We do so since the host of the RBC models (including all production economy models listed in the introduction) are using quarters as their basic time unit. Experimentation with yearly calibration does not seem to alter the results of our model. As it was mentioned in the introduction, BHS ﬁnd that loss aversion alone is insuﬃcient to generate high equity premia. Guided by another strand of the psychological research (Thaler and Johnson 1990) BHS relate the intensity with which the agents experience gains and losses from an investment to their prior investment outcomes. So modiﬁed model performs very well in terms of the generated average equity premium. Nonetheless, we do not follow their strategy. Just as in the BHS case, the results of a baseline RBC model with loss averse agents do not come even close to the empirical equity premium. However, we are able to remedy this problem by enriching the model with measures considered standard extensions to the simple RBC setting. These extensions are capital adjustment costs and a moderate degree of the ﬁnancial leverage. As a next step, we pass now to specifying our benchmark RBC setting. 2.2 The economy The economy is populated with an inﬁnite number of identical, inﬁnitely-lived ﬁrms and households. The only good in the economy is produced with a stan- dard Cobb-Douglas production function, subject to random productivity shocks. It may be consumed or serve to replenish the capital stock. Households. The representative household maximizes: ∞ 1−γ 1−χ C (1 − Nt ) max E0 βt t +θ + βBt r V (Xt+1 ) (2) Ct ,Nt ,at t=0 1−γ 1−χ r=s,l As usually, β denotes the subjective time discount factor. The ﬁrst two terms in the brackets are standard in the RBC model: The consumers maximize their utility over consumption Ct and leisure Lt = 1 − Nt , where Nt is the time spent on working. The total amount of time available to households in a single period is normalized to one. The last part of the expression in brackets introduces the r two LA terms V (Xt+1 ). The superscript r takes the value ’s’ for shares and r ’l’ for long bonds. As it is discussed in BHS, V (Xt+1 ) must be properly scaled so that it does not dominate the utility function as the aggregate wealth grows 7 over time. We follow BHS in specifying the scaling factor as: ¯ −γ Bt = B Ct ¯ where Ct is the aggregate per capita consumption in a given period. The choice of this particular exogenous variable is neutral for the main message of the ¯ paper. Without any inﬂuence on our results, instead of using Ct we could apply any exogenous variable whose growth in the steady state is equal to the growth of the economy, for instance the aggregate per-capita wealth or the aggregate capital. The households face a standard budget constraint: Ct + ai Pti = Wt Nt + t ai i i t−1 Pt + Dt (3) i i In a period t they assign a part of their wealth to the current consumption, while the remainder is invested in a vector of assets at = {as , af , al , aq } held t t t t until the period t + 1, where the superscripts i = s, f , l or q stand respectively for shares, risk-free bonds, long-term bonds and ﬁrms’ discount bonds. A given asset i may be purchased in a period t at a price Pti . The payout of an asset i held from period t − 1 to t is denoted by Dt . The agents receive also a labor income Wt Nt , where Wt denotes the date t wage. Firms. Firms maximize the present value of their future dividend ﬂow discounted at the marginal rate of substitution β i ΛΛt of their owners: t+i ∞ Λt+i max Et βi s Dt+i (Kt−1+i , Kt+i , Nt+i ) (4) Kt+i ,Nt+i i=0 Λt where Λt+i C −γ = t+i −γ Λt Ct is the stochastic discount factor of the economy. The dividends are deﬁned as: Dt (Kt−1 , Kt , Nt ) ≡ Dt = Yt − Wt Nt − It + Ptq µKt − µKt−1 s (5) Yt is the current output, It - investment and Kt - end of period capital stock. The constant µ measures the degree of ﬁnancial leverage of the ﬁrms. When µ = 0 the new capital of the ﬁrm is fully ﬁnanced through retained earnings. The maximization is subject to the Cobb-Douglas production function: 1−α α Yt = At Ntd d Kt−1 , (6) d where Ntd and Kt−1 denote respectively labor and capital demand of the ﬁrm. The technology coeﬃcient At is growing at a constant growth rate g in the 8 steady state. The deviation of At from its steady state follows an AR(1) process. ˜ At ˜ Denote At ≡ Ass . It is assumed that At follows: t ˜ ˜ log At = log At−1 + t with < 1 and t i.i.d such that E t = 0, and std( t ) = σε . By this deﬁnition, ˜ ˜ At ∼ Ass elog At = Ass eg+ρ log At−1 + t . = t t−1 It is assumed that the transformation of the investment It into capital Kt is costly. As it is argued in Jermann (1998), assuming positive capital adjustment cost allows the shadow price of installed capital to diverge from the price of an additional unit of capital, which brings the model closer to the reality. In specifying the capital adjustment cost we follow Uhlig (2006). Hence, our capital accumulation equation takes the form: It Kt = (1 − δ) Kt−1 + G Kt−1 (7) Kt−1 with the function G(·) such that 1−1/ξ It a1 It G ≡ + a2 (8) Kt−1 1 − 1/ξ Kt−1 where a1 , a2 are two positive constants. For ξ < ∞ the adjustment cost becomes strictly positive. Returns on assets. The gross return in t + 1 on an asset i is: i i i Pt+1 + Dt+1 Rt+1 = Pti s In the case of the risky asset we will use Rt+1 ≡ Rt+1 . Consider the vector of purchase prices in t: {Pts , Ptf , Ptl , Ptq } = {Pt , Ptf , Ptl , Ptq } One period later, after the dividends have been paid and the coupons stripped, the prices of these assets are: s f l q l {Pt+1 , Pt+1 , Pt+1 , Pt+1 } = {Pt+1 , 0, Pt+1 , 0} while the vector of payouts: f q s l {Dt+1 , Dt+1 , Dt+1 , Dt+1 } = {Dt+1 , 1, Dl , 1} Therefore the risky assets returns are: l l Pt+1 + Dt+1 Pt+1 + Dl {Rt+1 , Rt+1 } = { , } (9) Pt Ptl 9 with Dl > 0 being a constant payoﬀ of the long bond and Dt+1 deﬁned as in (5). The vector of risk-free asset returns is: f q 1 1 {Rt , Rt } = { , } (10) Ptf Ptq Note our convention to use the superscript t for a riskless asset return realized in t + 1. It is to stress the fact that these returns are known in advance. 2.3 The solution Clearing conditions. In the equilibrium the transversality condition holds. Also, all markets clear: • Goods market: Ct + Kt − (1 − δ)Kt−1 = Yt d • Capital market: Kt = Kt ; • Labor market: Nt = Ntd ; • Risk-free bonds market: af = 0; t • Long-term bonds market: al = 0; t • Firms’ bonds market: aq = µKt t • Equity market: as = 1 t First order conditions. Let xs denote the equity premium and xl t+1 t+1 the long bond premium over the risk-free rate: f r xr := Rt+1 − Rt t+1 for r = s, l Deﬁne v(xr ) as: t+1 xr xr ≥ 0 v(xr ) = t+1 for t+1 t+1 φxr t+1 xr < 0 t+1 From the consumers’ problem (2) and (3) and taking into account the clearing conditions we obtain the following set of Euler Equations (EE): −γ f −γ Ct = βRt Et Ct+1 (11) the EE on the riskless asset; −γ q −γ Ct = βRt Et Ct+1 (12) the EE on the ﬁrms’ discount bond; −γ −γ Ct = βEt {Ct+1 [Rt+1 + Bv(xs )]} t+1 (13) 10 the EE on the ﬁrms’ shares; −γ −γ l Ct = βEt {Ct+1 [Rt+1 + Bv(xl )]} t+1 (14) the EE on the long-term bond. The returns are deﬁned as in (9) and (10). Note that the ﬁrst two Euler Equations have a form typical for the standard RBC models. They imply the usual intertemporal trade-oﬀ between consuming goods for one more dollar today or investing this dollar in an asset with return i Rt in order to enjoy a bit higher consumption tomorrow. On the other hand the EEs on shares and long bonds contain an additional term implied by the formulation of our model in the spirit of the Prospect Theory: Consuming a bit less today and investing the proceeds in a risky asset brings a slightly higher expected consumption utility tomorrow, but it also exposes the consumer to a higher degree of the loss-aversion risk. The ﬁrst order condition with respect to labor gives the standard expression: θ(1 − Nt )−χ −γ = Wt (15) Ct Finally from the clearing conditions we have: Ct + It = Yt (16) The maximization problem (4), (6) and (7) of the ﬁrm gives the following stan- dard expressions for the price of capital and labor respectively5 : −γ −γ k Ct = βEt {Ct+1 Rt+1 } (17) where It+1 It G Kt−1 At+1 Nt+1 1−α 1−δ+G Kt It+1 k Rt+1 = α −µ+ − 1−G It µPtq Kt G It+1 Kt Kt−1 Kt (18) and α Kt−1 (1 − α)A1−α t = Wt (19) Nt These, together with the capital accumulation equation (7) and the production function (6) end the description of the main equations of the model. The balanced growth path. As it was already noted in King et al (1988), in a standard RBC model with additively separable consumption and leisure and a positive productivity growth rate g, the existence of the balanced growth path is assured only for the special case γ = 1. This observation is also true in our model. There are several ways to deal with this problem (see Lettau and Uhlig 1997 or Lettau et al 2001), but we simply choose to keep g = 0. This complies with most of the RBC literature. 5 Note that ﬁrms’ managers are supposed not to be loss-averse. Therefore Rk , the return t on capital, is determined by the Euler Equation of the standard form. An interesting question emerges how the economy would be aﬀected, had also the managers were loss-averse. 11 Calibration. The baseline RBC parameter values are set in line with the standard RBC literature (Cooley and Prescott, 1995; King and Rebelo, 2000). We choose γ = 1 so that the elasticity of intertemporal substitution of the consumption good is also 1. The leisure utility is parametrized as follows. The power parameter χ is set equal to 1. Then the parameter θ is picked so that the proportion of time spent by the consumers on working in the steady state is equal to 0.3. We set β equal 0.995. This, combined with the zero productivity growth rate will give the equilibrium risk-free interest rate of around 2%, which is slightly above its true value6 . For the production technology, the parameter governing the persistence of the technology shock ρ = 0.95. Then, the value of the standard deviation of the shock t is set in such a way that std[(1 − α) t ] ≈ 0.0072 as in Lettau et al (2001)7 . The capital share α is set to the standard value of 0.33. In calibrating the functional form of the cost-of-adjustment we closely stick to the parametrization used in Uhlig (2006). The constants a1 and a2 are chosen so that G(δ) = δ and G (δ) = 1, where δ = 0.025 is the quarterly depreciation rate. We focus on two values of the parameter ξ that often appear in the literature: ξ = ∞ (no adjustment cost) and ξ = 0.23. For the loss-aversion parameters φ and B, BHS (2001) use φ = 2.25, a magnitude based on the experimental ﬁndings of Kahneman and Tversky (1979), and B = 2 as their benchmark. Our baseline parametrization of the LA term is identical to theirs. We also provide a sensitivity analysis for these parameters. Finally, the payout of the long term-bond is normalized to 1%, Dl = 0.01, which is paid every period ad inﬁnitum. We allow for diﬀerent values of the ﬁnancial leverage ratio, varying from µ = 0 to µ = 0.5. Details of the solution approximation. The LA term in the consumers’ utility is a kinked function and is hence not continuously diﬀerentiable. The model can therefore not be linearized. Instead, we use a non-linear solution method based on Coleman (1991) and following Davig (2004a and 2004b).8 . See the appendix for details. 3 The results 3.1 The data. The statistics a successful model of ﬁnancial markets should be able to repro- duce are reported in Table 1. We focus on the Sharpe Ratio which, given its 6 The papers on the asset pricing in the RBC framework assume usually a positive produc- tivity growth rate g. In most cases it demands having β very close to 1 in order to match the risk-free interest rate (see for instance Boldrin et al, 2001). The choice of g does not aﬀect other ﬁnancial market statistics we intend to match. 7 See also King and Rebelo (2000) 8 We are grateful to Troy Davig for having sent us sample codes for the solution of a bare-bone RBC model. 12 interpretation of the price for risk, is argued by Uhlig (2006) to be a more re- vealing statistics than a simple equity premium. Its logarithmic unconditional version is deﬁned as: f E[log Rt+1 − log Rt ] SR = σlog r 2 where σlog r = var[log Rt+1 ]. The annualized logarithmic Sharpe Ratio was found by Lettau and Uhlig (2002) to be around 0.27. Uhlig (2006) ﬁnds SR only slightly higher. Other authors consider rather the arithmetic version of the Ratio: E[xs ] t+1 SRa = σr with σr being the volatility of the stock returns. Based on the stock return moments estimated by Cecchetti et al (1993), Boldrin et al (2001) calculated the arithmetic Ratio to be 0.34. A yet larger number is given in Campbell (2004), SRa = 0.46. As it is suggested by Uhlig (2006), a part of the diﬀerence between his and others’ estimates is probably due to diﬀerent deﬁnitions of the Sharpe Ratio, although it is not a full story. In accordance with the tradition initiated by Mehra and Prescott (1985), we attempt also to match the equity premium, which is estimated to be about 7% on the yearly basis. In order to account for the volatility puzzle identiﬁed by Campbell (1999) two other ﬁnancial statistics we want to reproduce are the risk- free rate volatility and the equity return volatility. The latter one is around 19% in the data. There is some kind of controversy regarding the former. According to Cecchetti et al (1993), the standard deviation of the risk-free rate is around 5% in the data. However Campbell 2000 ﬁnds the volatility of the returns on the US Treasury Bills to be 1.76%. Additionally, he argues that half of it could be due to ex-post inﬂation shocks. Finally, we also check how well our model performs in terms of the mean risk-free rate and the mean premium on the long bonds. The risk free premium is about 1%, while the premium on the long bonds over the risk free rate is found in the data to be below 2%. Table 1 Standard asset statistics moment value a SR 0.27b f Ert 1.19 Exs t 6.63 σr f 5.27 σr 19.4 Exl t 1.70c a The estimates from this column are taken from Cecchetti et al (1993) as reported by Boldrin et al (2001). They are based on the U.S. data from the period 1892-1987. b The Sharpe Ratio estimate is taken from Lettau and Uhlig 2002. c The estimate of the premium on the mean long term bond is taken from Jermann (1998). Table 2 reports the main business cycle facts. We do not attempt to match these statistics and they are provided solely for the comparison purposes. We 13 can see that the quarterly volatility of output is around 2%. The consumption is less volatile than the output while the investment is three times as much volatile. The hours worked vary about as much as the output. Finally, the consumption, the investment and the labor are strongly procyclical. Table 2 Standard business cycle statistics moment value a σY 1.81 σC /σY 0.74 σI /σY 2.93 σN /σY 0.99 ρY C 0.88 ρY I 0.80 ρY N 0.88 a The estimates from this column are taken from King and Rebelo (2000). 3.2 Results: The Basic Framework In this section we consider a basic framework with loss-aversion and capital adjustment cost but without leverage. In order to compute the moments of variables of interest we simulated the economy for 10200 periods. The moments reported are based on time series consisting of last 10000 observations. The ﬁrst row of Table 3 presents the statistical asset moments for the sim- plest RBC model. It demonstrates the well-known fact that in the classical RBC framework with moderately risk averse agents, one is unable to reproduce any of the most important ﬁnancial statistics. In particular, the Sharpe Ratio produced by the model appears virtually zero. As it was argued in the intro- duction, even a signiﬁcant increase in the parameter of risk aversion does not resolve the puzzle. The addition of the capital adjustment cost by itself cannot do the job either, see the second row of the table (compare also Jermann, 1998). In contrast, as we will see shortly, the introduction of the loss-aversion terms has the potential to signiﬁcantly improve the performance of the model. We consider ﬁrst a bare-bone RBC model (no adjustment costs nor leverage) with LA term parameters b = 2 and φ = 2.25. This is the baseline parametriza- tion of BHS (2001). The addition of the LA features to the model change the things radically: The Sharpe Ratio becomes 0.25, which matches the estimate given by Uhlig (0.27). Hence, the usefulness of the loss-aversion in explaining the ﬁnancial market puzzles appears hard to be questioned also in the production economy framework. Intuitively, the mechanism that pushes up the Sharpe Ratio in our model may be explained as follows. The agents in this framework care not only about the consumption level, but also about the changes in their risky asset positions. Since they are loss-averse, the feeling of pain they experience in the case of losses is relatively stronger than the satisfaction they achieve in the case of gains. This implies that the risk premium the households demand is high. Hence, for a given amount of risk (as measured by the standard deviation of the risky asset return), the Sharpe Ratio increases. 14 Note that we were able to obtain this result without constraining the labor or the capital mobility in the model. This is in contrast to those production economy models in which the explanation of the ﬁnancial puzzles is based on the habit persistence (Jermann 1998, Boldrin et al 2001), which demand a consid- erable amount of frictions on both, the labor and the consumption-capital good markets in order to match the Ratio. For instance Boldrin et al (2001) report a Sharpe Ratio of 0.002 for their parametrization of a production economy habit formation model without labor and capital frictions. The intuition behind this stark diﬀerence between the two types of models is straightforward. The performance of the habit persistence models relies heavily on the high consumers’ aversion to the consumption risk. They are capable to reproduce easily the high empirical Sharpe Ratio in the endowment economy framework where the consumption volatility, even low, is given exogenously. However, in the models with fully-ﬂedged production economies the degree of consumption volatility depends on the actions taken by consumers. They are able to smooth their consumption paths by varying the labor eﬀort and the rate of capital accumulation and thus establish an insurance against unexpected changes in consumption. It turns out that if at least one of these channels is easily accessible, the consumers are able to substantially reduce their exposure to the consumption risk, even if their risky asset position is high. What follows, they do not deem investing in risky assets truly risky. It is only when the labor-leisure choice and the capital-consumption choice are both considerably constrained that the habit persistence models are able to generate the observed risk premium. The models based on the Prospect Theory do not demand a high degree of aversion to the consumption risk. It is because, as it was pointed out by Campbell, ”they generate risk-averse behavior [...] also from direct aversion to wealth ﬂuctuations” (Campbell 2000, p. 1554). In other words, the consumers exhibit what is called ”ﬁrst-order risk aversion”9 . Clearly, the households cannot easily insure against the risk related to the aversion to the wealth ﬂuctuations. Neither working more nor investing less (and thus consuming more) will alleviate the pain from an unfortunate investment. It is because a loss of wealth is hurtful independently of the level of consumption chosen. In fact the only way to avoid the pain is to refrain from investing in risky assets. Since in the equilibrium all assets must be owned by the households, the implied price for risk must be high. Although this version of the model comes close to matching the empirical Sharpe Ratio, it fails to reproduce other asset markets statistics. In particular it produces only a tiny equity premium (Exs = 0.03%). It is not very hard to t ﬁnd the reason for this failure. The model implies a very low volatility of stock returns (σr = 0.33%) - two orders smaller than the observed volatility. The introduction of the loss-aversion to the model aﬀected the intensity with which the agents experience the exposure to the risky asset return ﬂuctuations, and hence the price for risk, but did not have an impact on the amount of the risk 9 See Epstein and Zin (1990) for a diﬀerent model of ﬁrst-order risk aversion. 15 as measured by the asset return volatility. Hence, in order to match the equity premium we need to ratchet up the standard deviation of the returns. In the fourth row of Table 3 the results of the model with loss-aversion and capital adjustment costs are reported. As expected, the volatility of the stock returns is now much higher, σr = 4.19%. Still, it does not reach the true stock return volatility of almost 20%. Therefore, the implied equity premium still does not come close to the level of 7%, Exs = 1.06%. t We can check the sensitivity of our results to changes in the loss-aversion parameters b and φ (two last rows of Table 3). The choice of b = 100 implies that the consumers’ utility is fully dominated by the LA term. Despite this fact, the increase of the stock return volatility and of the equity premium is only moderate. Setting the parameter φ to 4 makes the losses almost doubly as painful as in the baseline parametrization. This can be measured by the Sharpe Ratio, that has increased almost twice. Yet, this change does not lead either to more realistic stock return moments. On the other hand it helps in matching the premium on the long term bonds over the risk-free rate (the last column of Table 3). Table 3 Financial statistics (no leverage) f model SR Exs t σr Ert σr f Exlt bare-bone RBC 0.00 0.00 0.32 2.03 0.29 0.01 RBC with adj. costs 0.01 0.10 3.80 1.99 0.52 0.07 bare-bone RBC with loss 0.25 0.03 0.33 2.02 0.31 0.18 -aversion (baseline calibration) RBC with adj. costs and loss 0.24 1.06 4.19 2.01 0.49 0.74 -aversion (baseline calibration) RBC with adj. costs and 0.32 1.45 4.31 2.01 0.54 0.96 loss-aversion (b = 100) RBC with adj. costs and 0.43 2.01 4.55 2.00 0.46 1.29 loss-aversion (φ = 4) Baseline calibration: loss-aversion term: b=2, phi=2.25; adjustment costs: ksi=0.23 3.3 Leverage So far we have assumed that the ﬁrms ﬁnance their activity through retained earnings. Yet, in reality a non-negligible part of ﬁrms’ capital is ﬁnanced by borrowing. Allowing for a non-zero ﬁnancial leverage in our model will obviously increase the riskiness of holding stocks. In this subsection we examine whether a plausible degree of ﬁnancial leverage may be helpful in matching the stock return volatility and the equity premium. The ﬁnancial moments generated in the model with leverage are reported in Table 4. The following results stand out. The ﬁrst row of the table shows that a combination of a plausible degree of leverage with capital adjustment costs alone is not suﬃcient to produce a reasonable values of the ﬁnancial statistics. As expected, it does considerably increase the volatility of risky returns, which may be interpreted as an increase of 16 the quantity of risk. However the price of risk in the economy is low. Therefore the equity premium hardly reaches the mere 0.5%. The leverage ratio used was µ = 0.5, which implies that the ﬁrms’ ﬁnance mix is composed in 50% of stocks and in 50% of credits10 . Assuming a much higher leverage ratio would certainly do the job of further ratcheting up the premium. It would not however do the trick with the Sharpe Ratio, which in this version of the model is extremely low (0.02). Clearly, in order to explain both high equity premium and high Sharpe Ratio, the model must be capable of generating not only a non-negligible quantity, but also a high price of risk. This last feature is guaranteed if the agents are loss-averse. Indeed, adding the loss-aversion to the setting with adjustment cost and leverage allows for matching basically all statistics of interest. Take ﬁrst the BHS baseline calibration with b = 2 and φ = 2.25. As we have seen already before, the model with thus calibrated LA terms produces the Sharpe Ratio of about 0.25. Due to the leverage ratio of 0.5, the stock holders bear now twice as much risk as they did before. This translates into the volatility σr = 16.85, a number that is only slightly lower than the empirical equity return volatility of 19.4%. This combination of the high price for risk and the high quantity of risk results in a very reasonable equity premium Exs = 5.16. We ﬁnd an even t better match with the data, if we depart from the BHS baseline calibration of the LA term. Assuming φ = 3 (and decreasing the leverage ratio µ to 0.4) while still keeping b = 2 causes a further increase of the equity premium by 0.5%. Finally, increasing the parameter b also to 3 produces the premium of 6.5%, which almost perfectly matches its empirical value (6.6%). Note that this undeniable success of matching the empirical equity pre- mium did not come at the expense of other statistics of interest. In partic- ular, our approach does not give rise to a too high long bond risk premium or too volatile short-term real interest rate returns, two problems that plague the habit-formation based models (see for instance Boldrin et al, 2001 and Jermann, 1998). The reason why the short-term real interest rate does not vary much in our model is that the expected consumption stream the model produces is not very volatile either. For the value of the power utility parameter we are using (γ = 1) this implies that the volatility of the expected marginal utility and hence also of the expected stochastic discount factor (SDF) is modest. Since the Euler f Equation (11) determining the short-term real interest rate rt is standard and f depends only on this last expectation, the volatility of rt must also be low. It is also intuitive why we did not obtain a too high mean long bond risk premium, despite the fact that the agents in our economy are loss-averse over the returns on these bonds. It is because the introduction of the leveraged ﬁrms’ ownership does not inﬂuence the risk of holding long bonds. Compare the mean levels of the long bond risk premium in the case of the BHS baseline calibration with capital adjustment costs but no leverage (Table 3, row 4) and in 10 Boldrin et al (2001) also apply the leverage ratio of 0.5. Some authors consider slightly higher values of this ratio (see Jermann 1998). However in the light of the empirical research we do not ﬁnd it legitimate to allow for much higher numbers. For instance Masulis’ (1988) market values estimates of the leverage ratio for the US do not surpass 0.44. 17 the case of identical calibration but with µ = 0.5 (Table 4, row 2). In both cases Exl t 0.7511 . For our preferred calibration (b = 3, φ = 3, ξ = 0.23, µ = 0.4, see Table 4, row 5) the mean level of the long bond premium is 1.1%. It is of the same order of magnitude as the empirically observed value Exl = 1.7%. t Our preferred calibration implies the logarithmic Sharpe Ratio of 0.36, which is larger than the number SR = 0.27 given by Lettau and Uhlig (2002). As it was argued earlier, in order to compare our results to other studies, we need to compute the Sharpe Ratio based on the arithmetic returns. For the calibration under discussion (b = 3, φ = 3, ξ = 0.23, µ = 0.4) it gives SRa = 0.42, which is almost the same as the value reported by Campbell (0.46). We conclude that our result for the Sharpe Ratio lies well within the bounds of what is considered a plausible value for this statistics. It is also important to stress that in our model the elasticity of intertemporal substitution is still equal to the inverse of the power utility parameter γ. Hence, the assumption that γ = 1 implies that the consumers do not have a strong desire to smooth their consumption over time. This suppresses the mean value of the short-term real interest rate, which in our model does not diﬀer much from its steady state value. For our calibration (discount factor β = 0.995, productivity f growth g = 0) it gives rt 2%, but could even be lower, should we choose a higher β. Hence the risk-free rate puzzle (Weil 1989) is not an issue in our approach. Table 4 Financial statistics f model SR Exs t σr Ert σr f Exl t RBC with adjustment cost and 0.02 0.46 7.56 1.93 0.51 0.13 leverage, no loss-aversion RBC, baseline calibration 0.24 5.16 16.85 1.99 0.48 0.76 RBC with adj. costs, leverage µ = 0.4 0.31 5.56 15.01 2.01 0.51 1.00 and loss-aversion (φ = 3) RBC with adj. costs, leverage µ = 0.4 0.36 6.48 15.63 2.00 0.48 1.10 and loss-aversion (b = 3, φ = 3) RBC with leverage and loss 0.23 0.05 0.37 2.04 0.29 0.16 -aversion , no adjustment cost RBC with adj. costs ξ = 2, leverage 0.48 2.97 5.91 2.01 0.34 1.07 µ = 0.4 and loss-aversion (b = 4, φ = 4) Baseline calibration: loss-aversion term: b=2, phi=2.25; leverage: mu=0.5; adjustment costs: ksi=0.23 3.4 The macroeconomic performance of the model. As far as the real variables statistics reported in Table 2 are regarded, the bare- bone RBC model reproduces the qualitative characteristics of the true economies (see Table 5, row 1). On the other hand, although the capital adjustment costs may be useful in explaining some other dimensions of the data (for instance the sluggishness of the inﬂation and the output), when combined with the simplest 11 The tiny diﬀerence follows from the fact that the results are based on simulations 18 RBC framework, they lead to an overall deterioration of the model macroe- conomic performance. In order to see it, regard the second row of Table 5, where real variables statistics for the bare-bone model with adjustment-costs (ξ = 0.23) are reported. Clearly, the standard deviation of output is too low, the volatility ratios are reversed, with consumption being the most volatile and the volatility of investment being only a third of that of output, and the cor- relations of consumption and investment with output are too extreme. The model generates also a countercyclical behavior of labor. This last feature is perhaps not of a concern however, as the procyclicality of this variable in the data has been recently hotly debated (compare Uhlig, 2006 and the references given therein). The reason why the capital adjustment costs have adverse eﬀects, when added to a simple RBC model is intuitive. When the agents face adjustment costs, they prefer the stock of capital to adjust only smoothly in response to unexpected changes in the productivity. This implies that the stock of capital does not get a signiﬁcant boost in the times of prosperity. This in turn leads to a lower volatility of output. The fall of the variability of the investment stream is even more pronounced. On the other hand the volatility of consumption con- siderably increases: If not invested, the extra amount of the consumption good produced due to a positive productivity shock will be immediately consumed. And when the shock is negative, the agents are willing to give up an important part of their consumption in order to keep the stock of capital at its stable level. Finally, since an immediate increase of the scale of production is not proﬁtable, a positive productivity development leads the agents to work less hard. In eﬀect, the hours worked behave countercyclically. This situation does not change when we assume the households to be loss- averse. It is because the introduction of the loss-aversion to the model does not aﬀect the dynamics nor the volatilities of variables like consumption, output or investment. In fact, its only eﬀect is to shift the stochastic steady states of these variables. The third and the fourth row of Table 5 report the statistical moments of these variables for the calibrations of the LA term that we found the most successful in resolving the asset market puzzles. Clearly, the reported statistics are virtually identical to the ones coming from the model with only adjustment costs. What happens if we switch oﬀ the adjustment costs? Consider an RBC model with leverage (µ = 0.5) and the baseline loss-aversion (b = 2, φ = 2.25), but without adjustment costs. The real economy statistics for this version of the model are reported in the ﬁfth row of Table 5. Clearly, they do not diﬀer at all from the ones obtained in the simplest RBC framework, and hence are crudely in line with the data. The question now arises whether this version of the model does also generate a considerable equity premium. As it is shown in the ﬁfth row of Table 4, the answer is negative. The reason is that the leverage ratio of 0.5 alone is not enough to reproduce the high volatility of risky asset returns12 . And as we already learned, when the standard deviation of the equity 12 It is worth stressing again that allowing for a higher µ would increase the return volatility, 19 returns is low, adding the loss aversion terms to the utility helps in explaining the empirical Sharpe Ratio, but does not make it easier to match the mean risky asset premia. In particular, setting very high values of the parameters b and φ is of no help: It raises the Sharpe Ratio excessively, but increases the equity premium only marginally. Finally, one might wonder whether setting an intermediate value of the ad- justment cost parameter would not allow to match the empirical statistics in both dimensions. To answer this question, we pick the adjustment cost param- eter ξ = 2 and consider the leverage ratio µ = 0.4 and the Prospect Theory parameters b = 4 and φ = 4. The last row of Table 5 shows that, as far as the real economy statistics are concerned, the chosen value of ξ appears too low. The output volatility σY = 1.01 is too small of order of two as compared to the data (σY = 1.81). Also the volatility ratios σY = 1.45 and σN = 0.12 must σI σY σI be considered too low (in the data σY = 2.93 and σN = 0.99). Yet, as it is σY demonstrated in the last row of Table 4, this value of ξ is still too high to be of much help in matching the asset market statistics. With high values of b and φ chosen we obtain the Sharpe Ratio SR = 0.48. It is much above its reliable estimates. Still, the mean equity premium Exs = 3% is only a half of what we t ﬁnd in the data. As in all previously discussed cases, the origin of the problem is the same: The stock returns are not suﬃciently volatile. Unless we assume an implausible degree of ﬁnancial leverage, the only easy way to increase this volatility is to consider higher capital adjustment costs. Table 5 Real variables statistics σC σI σN model σY σY σY σY ρY C ρY I ρY N bare-bone RBC 1.41 0.29 2.99 0.51 0.90 0.99 0.99 RBC with adj. costs 0.83 1.25 0.33 0.17 1.00 1.00 -1.00 RBC, baseline calibration 0.81 1.26 0.33 0.17 1.00 1.00 -1.00 RBC with adj. costs, leverage µ = 0.4 0.85 1.25 0.33 0.17 1.00 1.00 -1.00 and loss-aversion (b = 3, φ = 3) RBC with leverage and loss 1.37 0.29 3.00 0.51 0.90 0.99 0.99 -aversion , no adjustment cost RBC with adj. costs ξ = 2, leverage 1.01 0.83 1.45 0.12 1.00 1.00 1.00 µ = 0.4 and loss-aversion (b = 4, φ = 4) Baseline calibration: loss-aversion term: b=2, phi=2.25; leverage: mu=0.5; adjustment costs: ksi=0.23 4 Conclusions In this paper we have attempted to provide a resolution to several famous ﬁ- nancial market puzzles. In contrast to the gross of the classic literature on the subject, we have taken up the challenge to work in a fully-ﬂedged general equilibrium model, where the consumption and dividend streams are generated endogenously. In this framework we have shown that the assumption that the and hence also the risk premium. 20 agents act in line with the behavioral Prospect Theory of Kahneman and Tver- sky, and in particular that they are loss-averse over the ﬂuctuations of the value of the risky assets they hold allows to easily match the empirical Sharpe Ratio. Further, when combined with capital adjustment costs and a moderate degree of ﬁnancial leverage, it also helps to exactly reproduce the US economy mean equity premium. The consumers’ elasticity of intertemporal substitution in our model is dis- entangled from the time-varying degree of risk aversion and constant. For the parametrization of the utility function considered in the paper, this elasticity is equal to one. Therefore the agents are not urged to borrow from the future, as it is often the case in other models aiming at explaining the high equity premium. Thus, the risk-free rate puzzle does not emerge. Additionally, our model gener- ates a low volatility of the risk-free rate and a moderate premium on long-term bonds. This is in contrast to the popular habit-formation based models, usually ﬂawed with too high long bond returns and an implausible variability of the short-term real interest rate. The Prospect Theory, with the loss-aversion as one of its central elements, provides us with a mechanism that allows to generate a high price of risk in the economy. Simply put, the agents with the Prospect Theory utility are extremely risk averse on the border between the gains and the losses. Due to this local risk sensitivity, the agents demand high premium for bearing a given amount of risk. When applied to ﬁnancial markets, this implies that the agents expect a high average return on risky assets, as compared to the quantity of risk these assets bear. In absolute terms however, the equity premium the model generates does not need to be large. The reason is that the simple RBC model does not produce high variation in risky returns. The loss-aversion by itself does not provoke a notably higher variation in the risky asset returns either. Therefore, in order to obtain a sizeable equity premium, one has to rely on an additional mechanism that would produce a considerable volatility of stock returns. In our paper we obtain a high standard deviation of the equity returns by incorporating costly capital adjustment to the model. The rationale behind adding the cost-to-adjust to the macroeconomic models is that it generates a time-varying Tobin’s q. It is also widely used as a tool to enhance the propagation properties of the output and inﬂation dynamics. Nevertheless, in a simple RBC framework, it leads to several counterfactual results. The question therefore remains, how to generate a plausible variability of stock returns, without deteriorating the real economy properties of the RBC model. 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[32] Weil, Philippe (1989): ”The Equity Premium Puzzle and the Risk-Free Rate Puzzle”, Journal of Monetary Economics 24, 401-421 6 Technical Appendix. The complete model is described by the equations (6), (7) and (9) to (17). The solution conjectures candidate decision rules for Kt , Nt and It denoted ˜ ˜ ˜ respectively by hk Kt−1 , At , hn Kt−1 , At , and hi Kt−1 , At and conjec- f tured pricing functions for Rt , Pts and Ptl : hr ˜ ˜ Kt−1 , At , hps Kt−1 , At , ˜ hpl Kt−1 , At respectively. The conjectured rules and pricing functions are substituted into the model’s equations. For the solution, we descretize the state space. For any given state (Kt−1 , At ) , the decision rules and pricing functions are taken as given. This allows us to f compute Ptq = 1/Rt ; Yt by the production function (6), Ct by the resource con- straint (16) and G by its deﬁnition (8) for the given state conjecturing agents decisions for capital, labor and investment and the pricing rules. For the expectational terms, we ﬁrst compute the conjectured decisions and f f ˜ ˜ prices for the period t + 1: Kt+1 = hk Kt , At+1 , Rt+1 = hr Kt , At+1 , ˜ ˜ ˜ ˜ ˜S ˜ ˜L Nt+1 = hn Kt , At+1 , Xt+1 = hx Kt , At+1 , Pt+1 = hps Kt , At+1 , Pt+1 = ˜ ˜ q ˜ hpl Kt , At+1 for εt+1 given K and At . From these, we can compute Pt+1 Kt , At+1 , ˜ ˜ ˜ ˜ ˜ ˜ Yt+1 Kt , At+1 , Ct+1 Kt , At+1 , G (t + 1), G (t + 1), Dt+1 Kt , At+1 , Rt+1 Kt , At+1 , x ˜ l ˜ Rt+1 Kt , At+1 , Rt+1 Kt , At+1 again for ˜ εt+1 given K and At . 1 1 f The expectations for Et ˜ eg Ct+1 , Et ˜ eg Ct+1 Rt+1 + B˜(Rt+1 , Rt ) v , x 1 ˜L L f Rt+1 Et ˜ eg Ct+1 Pt+1 +D + v L B˜(Rt+1 , Rt ) and Et e are computed ˜t+1 gC by numerical quadrature. The solution technique treats the conjectured rules as six unknowns. It solves the sytem of the four Euler equations (11, 12, 13, 14), the consumption- leisure trade-oﬀ and the capital accumulation equation for the unknowns in any set of state variables deﬁned over the discrete partition of the state space. This procedure is repeated until the iteration improves the current decision rule at any given state vector by less than some tolerance level which we set to 10−12 . 24

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