Asset Returns, the Business Cycle, and the Labor Market ∗ Burkhard Heera,b, Alfred Maußnerc a Free University of Bolzano-Bozen, School of Economics and Management, 1 piazza universit`a, I-39100 Bolzano-Bozen, Italy, [email protected] b CESifo cCorresponding Author, University of Augsburg, Department of Economics, Univer- sita¨tsstraße 16, 86159 Augsburg, Germany, [email protected] This version: January 31, 2012 JEL classification: G12, C63, E22, E32 Key Words: Equity Premium, Production CAPM, Real-Business Cycle, Labor Market Statistics, Nominal Rigidities Abstract: We review the labor market implications of recent real business cycle and New Keynesian models that successfully replicate the empirical equity premium. We document the fact that all models reviewed in this paper that do not feature either sticky wages or immobile labor between two production sectors as in Boldrin, Christiano, and Fisher (2001) imply a negative correlation of working hours and output that is not observed empirically. Within the class of Neo-Keynesian models, sticky prices alone are demonstrated to be less successful than rigid nominal wages with respect to the modeling of the labor market stylized facts. In addition, monetary shocks in these models are required to be much more volatile than productivity shocks to match statistics from both the asset and labor market. ∗ Wewouldliketothanktwoanonymousrefereesfortheircomments. Allremainingerrorsareours. 1 Introduction Mehra and Prescott (1985) estimate an equity premium of 6.18% p.a. for the US over the period 1889-1979 and demonstrate that a general equilibrium model of an exchange economy is unable to replicate this fact unless the representative consumer is implausibly risk averse. This puzzle seriously challenges business cycle research that rests on representative agent stochastic dynamic general equilibrium (DSGE) models sothatsubstantialefforthasbeenmadetoresolveit. Severalreviewarticles2 document this venture. With respect to models with an exogenously given endowment process modifications of the agents preferences and, more recently, the possibility of rare, but severe crises (Barro (2006)) have been proposed. Kocherlakota (1996) argues that generalized expected utility preferences as proposed by Epstein and Zin (1989) do not resolve the equity premium puzzle in the Mehra and Prescott (1985) data set, while some form of habit formation does. Jermann (1998) demonstrates that habit formation alone is not sufficient to resolve the equity premium puzzle in a production economy. In addition to the consumer being eager to smooth consumption, the adjustment of capital must be costly. However, in his model savings in physical capital is the single vehicle to smooth consumption. Once the consumer is allowed to adjust working hours too, there is second channel to cope with productivity shocks and the equity premium disappears. Subsequent research, thus, has focussed on additional frictions hampering the adjustment of labor. In this paper we consider the ability of these more recent models to resolve the equity premium puzzle while at the same time being consistent with the stylized facts of business cycles. Our main motivation for this venture is the prominent role played by DSGE models in the analysis of monetary policy and our conviction that models suitable for this purpose should be broadly consistent with both asset and labor market stylized facts. Many studies document that these facts are relatively stable both across time and countries.3 For this reason it is more or less a matter of (understandable) taste that we will use German data to gauge the models reviewed below. In Appendix B that is available from the authors upon request, we present the results from redoing the 2See, among others, Abel (1991), Kocherlakota (1996), Campbell (2003), and Cochrane (2008). 3See, among others, Ambler, Clarida, and Zimmermann (2004), Backus and Kehoe (1992), Brand- ner and Neusser (1992), Basu and Taylor (1999), Hodrick and Prescott (1997), and Maußner (1994), for a survey of these facts. 1 analysis presented below with parameter values and benchmark business cycle facts related to the US economy to verify this claim. In our study we consider several ways to introduce frictions in the allocation of labor. Several authors have proposed a habit in leisure which serves as a short-cut to the modeling of either adjustment costs of labor or search frictions in the labor market. Bouakez and Kano (2006) argue that habit formation in leisure fits the US data better with regard to the persistence and propagation of shocks than other standard real- business-cyclemodels, inparticularthoseallowingforlearning-by-doingsuchasChang, Gomes, and Schorfheide (2002). Lettau and Uhlig (2000), however, argue that, with habit formation in leisure, labor input is too smooth over the cycle and output and hours are negatively correlated, which is clearly at odds with the stylized facts of the business cycle. Uhlig (2007) combines habits in consumption and leisure with sticky real wages as proposed by Blanchard and Gal´ı (2005). With a considerable degree of real wage stickiness his model is able to produce a sizable equity premium and a positive correlation of hours and output. In the two sector model of Boldrin, Christiano, and Fisher (2001) (BCF for short), it is not possible to reallocate labor from the consumption goods sector to the investment goodssectoraftertheobservationoftheshock. Accordingly,theequitypremiumresults from variations in the relative price of the two goods rather than from variations in the firm’s value. This model reproduces the positive output-hours correlation found in the data, but fails to predict a positive correlation between the real wage and working hours. Most studies of the equity premium and asset prices are constrained to the analysis of the real economy that is subject to a technology shock. As one of the very few exceptions, De Paoli, Scott and Weeken (2010) examine the behavior of asset prices in a New Keynesian model with sticky prices. They find that the effect of nominal rigidities on the risk premium depends on the nature of the shock. While the risk premium is reduced, if cycles are driven by technology shocks, it increases in the case of monetary shocks. In addition to these models we study a model with sticky nominal wages and a model with both sticky nominal prices and wages. Our results are summarized in Table 1.1. The first column displays the names of the models that we consider in the following sections. The first row presents the 2 Table 1.1 Summary of Results Equity s s /s s /s s /s r r Score Y I Y N Y w Y YN wN premium Data 5.18 1.14 2.28 0.69 1.03 0.40 0.27 Models 2. Real Business Cycle Models Benchmark Exogenous labor 5.18 0.90 2.28 Endogenous labor 0.52 0.51 1.47 1.27 2.08 0.68 0.94 26.43 − − Habit in leisure 5.25 0.65 2.22 0.56 1.53 0.91 0.96 3.52 − − Predetermined hours Firms 0.08 0.86 2.19 0.10 1.71 0.62 0.25 28.18 − − Households 5.23 0.78 2.26 0.37 1.23 0.50 0.73 1.91 − − Sticky real wages 5.58 1.36 2.34 0.59 0.61 0.82 0.38 0.54 Two sector models Stationary growth 4.77 0.95 2.66 0.13 3.28 0.72 0.03 5.88 − Integrated growth 4.71 0.95 1.55 0.08 3.40 0.73 0.08 6.99 − Adjustment costs 4.58 0.92 2.07 0.07 3.29 0.69 0.00 6.04 3. New Keynesian Models Sticky prices 0.43 0.54 1.99 1.06 1.93 0.76 0.94 26.38 − − Sticky wages 5.20 0.98 2.43 1.38 1.14 0.57 0.69 1.47 − Stickypricesandwages 5.05 2.05 2.19 1.40 0.52 0.90 0.66 1.19 Notes: sx:=Standarddeviationoftimeseriesx,wherex∈{Y,I,N,w}andY,I, andN denoteoutput, investment, hours, and the wage, respectively. Empirical as well as model generated time series were HP-filteredwithweight1600. Theempiricalmomentsrelatetopercapitamagnitudes,exceptforthereal wagewhichwasmeasuredashourlyworkercompensation. sx/sy:=standarddeviationofvariablexrelative to standard deviation of output y. rNY:=Cross-correlation of variable hours with output, rwN:=Cross- correlationoftherealwagewithhours. ThecolumnScorepresentsthesumofsquareddifferencesbetween themomentsfromsimulationsofthemodelandthemomentsfromthedata. empirical values in Germany that we aim to match.4 Among the real business cycle modelsconsideredinSection2themodelbyUhlig(2007)comesclosesttotheempirical moments. This model features slowly adjusting external habits in both consumption 4Except for the equity premium, the second moments reported in Table 1.1 are taken from Heer andMaußner(2008), Table1.2,p. 56. TheestimateoftheGermanequitypremiumduring1900-2002 of 5.18 is from Kyriacou, Madsen, and Mase (2004). 3 and leisure and sticky real wages. The two-sector models in the spirit of Boldrin, Christiano, and Fisher (2001), where the reallocation of labor between sectors within the current period is impossible, are less successful in this endeavor. In the class of New Keynesian models with nominal frictions our model with sticky prices and wages performs best. Its score is only slightly worse than the score of the Uhlig (2007) model. We find that sticky prices alone are less important than rigid wages for the modeling of the asset and labor market statistics. In addition, we need a sizeable monetary shock in the nominal models in order replicate empirical regularities. The paper is organized as follows. In the next section we consider real models of the business cycle. We first present the Jermann (1998) model as a benchmark case to which we add one model element after the other. In Subsection 2.2, we show that the equity premium disappears once labor is supplied elastically. In the following subsectionsweconsiderhabitsinconsumptionandworkinghours, hourswhichmustbe determinedbeforetheproductivityshockisrevealedtoeitherthefirmorthehousehold, sticky real wages, and frictions in the allocation of labor between sectors. Section 3 studies models with nominal rigidities. We start with the New Keynesian model of de Paoli, Scott, and Weeken (2010) and show that this model is unable to replicate several labor market statistics. In Sections 3.2 and 3.3, we demonstrate that our model with rigid wages performs much better. All equilibrium conditions and derivations of the individual models are presented in an Appendix that is available from the authors upon request. 2 Real Business Cycle Models 2.1 The Benchmark Model 2.1.1 The Model The first model that we consider is the asset pricing model of Jermann (1998).5 We follow the description of this model in Heer and Maußner (2009). Time is discrete and 5InAppendixA.3weconsiderthetimetoplanmodelofChristianoandTodd(1996)asanalterna- tive to the adjustment costs of capital approach employed by Jermann (1998). The model is also able togeneratetheequitypremiumobservedinthedata, iflaborsupplyisexogenous. AsintheJermann model, the equity premium falls close to zero, if labor supply is endogenous. In a separate paper, we will consider extensions of this model similar to those presented for the Jermann model here. 4 denoted by t. Households. A representative household supplies labor in a fixed amount of N N t ≡ at the real wage w . Besides labor income he receives dividends d per unit of share t t S he holds of the representative firm. The current price of shares in units of the t consumption good is v . His current period utility function u depends on current and t past consumption, C and C , respectively. Given his initial stock of shares S , the t t−1 t households maximizes ∞ (C χCC )1−η 1 E βs t+s − t+s−1 − , η 0, χN [0,1), β (0,1) t 1 η ≥ ∈ ∈ ( ) s=0 − X subject to the sequence of budget constraints v (S S ) w N +d S C . (2.1) t t+1 t t t t t t − ≤ − E The operator denotes mathematical expectations with respect to information as of t period t. The first-order conditions of this problem are: Λ = (C χCC )−η βχCE (C χCC )−η, (2.2a) t t t−1 t t+1 t − − − Λ = βE Λ R , (2.2b) t t t+1 t+1 d +v t t R := , (2.2c) t v t−1 where Λ is the Lagrange multiplier of the budget constraint. t Firms. The representative firm uses labor N and capital K to produce output Y t t t according to the production function Y = Z N1−αKα, α (0,1). (2.3) t t t t ∈ The level of total factor productivity Z is governed by the AR(1)-Process t lnZ = ρZ lnZ +ǫZ, ǫZ N 0,(σZ)2 . (2.4) t t−1 t t ∼ (cid:0) (cid:1) The firm finances part of its investment I from retained earnings RE and issues new t t shares to cover the remaining part: I = v (S S )+RE . (2.5) t t t+1 t t − 5 It distributes the excess of its profits over retained earnings to the household sector: d S = Y w N RE . (2.6) t t t t t t − − Investment increases the firm’s future stock of capital according to: K = Φ(I /K )K +(1 δ)K , δ [0,1], (2.7) t+1 t t t t − ∈ where we parameterize the function Φ as 1−ζ a I 1 t Φ(I /K ) := +a , ζ > 0. (2.8) t t 2 1 ζ K − (cid:18) t(cid:19) The firm’s ex-dividend value at the end of the current period t, V , equals the number t of outstanding stocks S times the current stock price v . This definition implies: t+1 t (2.5) (2.6) V = v S = I +v S RE = I +w N Y +(v +d )S , t t t+1 t t t t t t t t t t t − − (2.2c) = I +w N Y +R V . t t t t t t−1 − Rearranging and taking expectations as of period t, yields Y w N I +V V = E t+1 − t+1 t+1 − t+1 t+1 . t t R (cid:26) t+1 (cid:27) Iterating on this equation using the law of iterated expectations and assuming V E t+s lim = 0 t s→∞ (cid:26)Rt+1Rt+2...Rt+s(cid:27) establishes that the end-of-period value of the firm equals the discounted sum of its future cash flows CF = Y w N I : t+s t+s t+s t+s t+s − − ∞ 1 V = E ̺ CF , ̺ = (2.9) t t t+s t+s t+s R R ...R t+1 t+2 t+s s=1 X The firm’s objective is to maximize its beginning-of-period value, which equals Vbop = t V +CF . Defining ̺ = 1 allows us to write t t t ∞ Vbop = E ̺ CF . (2.10) t t t+s t+s s=0 X The first-order conditions for maximizing (2.10) subject to (2.7) are: w = (1 α)Z N−αKα, (2.11a) t − t t t 1 q = , (2.11b) t Φ′(I /K ) t t q ̺ = E ̺ αZ N1−αKα−1 (I /K )+q Φ(I /K )+1 δ . t t t t+1 t+1 t+1 t+1 − t+1 t+1 t+1 t+1 t+1 − n (cid:2) (cid:3)o(2.11c) 6 In addition, the transversality condition lim E ̺ q K = 0 (2.11d) t t+s t+s t+s+1 s→∞ must hold. Market Equilibrium. Using equations (2.5) and (2.6), the household’s budget con- straint implies the economy’s resource restriction: Y = C +I . (2.12) t t t In equilibrium, the labor market clears at the wage w so that N = 1 for all t. Fur- t t thermore, using (2.2b), ̺ can be replaced by βΛ /Λ so that at any date t the set t+1 t+1 t of equations 1 q = , (2.13a) t Φ′(I /K ) t t Y = Z Kα, (2.13b) t t t Y = C +I , (2.13c) t t t Λ = (C χCC )−η βbE (C χCC )−η, (2.13d) t t t−1 t t+1 t − − − Λ q = βE t+1 αZ Kα−1 (I /K )+q Φ(I /K )+1 δ (2.13e) t t Λt t+1 t+1 − t+1 t+1 t+1 t+1 t+1 − n o K = Φ(I /K )K +(1 δ)K , (cid:2) (cid:3) (2.13f) t+1 t t t t − determines (Y ,C ,I ,K ,Λ ,q ) given (K ,Λ ,q ). t t t t+1 t+1 t+1 t t t Deterministic Stationary Equilibrium. Since our solution strategy rests on a second order approximation of the model, we must consider the stationary equilibrium of the deterministic counterpart of our model that we get, if we put σZ = 0 so that Z equals its unconditional expectation Z = 1 for all t. In this case we can ignore the t expectations operator E . Stationarity implies x = x = x for any variable in our t t+1 t model. As usual, we specify Φ so that adjustment costs play no role in the stationary equilibrium, i.e., Φ(I/K)K = δK and q = Φ′(δ) = 1. This requires that we choose a = δζ, 1 ζδ a = − . 2 1 ζ − 7 These assumptions imply via equation (2.13e) the stationary solution for the stock of capital: 1 1 β(1 δ) α−1 K = − − . (2.14a) αβ (cid:18) (cid:19) Output, investment, consumption, and the stationary solution for Λ are then given by Y = Kα, (2.14b) I = δK, (2.14c) C = Y I, (2.14d) − Λ = C−η(1 χC)−η(1 χCβ). (2.14e) − − 2.1.2 Calibration and the Equity Premium Calibration. We calibrate this and the other models considered here in a two-step procedure. In the first step we choose the parameters for which there is direct or (via the models equilibrium conditions) indirect empirical evidence or that are usually set by researchers to some preferred value. In the second step we set the remaining free parameters so that the respective model best fits certain empirical targets. For the first step we employ seasonally adjusted quarterly data for the West German economy over the period 1975.i through 1989.iv. The parameter settings are taken from Heer and Maußner (2009), Section 6.3.4. Table 2.1 displays the respective values.6 Notice that the wage share in the German data, 1 α = 0.73, is larger than the value − of 0.64 that is often found in comparable studies relying upon US data,7 while the depreciation rate, δ = 0.011, is much smaller and amounts to approximately half the US value. In addition, N = 0.13 is chosen to match the average quarterly fraction of hours spent on work by the typical German household. Notice that many studies set N = 1/3 arguing that the typical worker spends 8 hours per day on the job (see, for example, Hansen (1985)). We consider the typical household to be an average over the total population including children and retired persons rather than consisting of a single worker who is also working on the weekend and does not take any vacation. The discount factor β = 0.994 yields an annual risk free rate in the simulation of the model of about 1 percent. We choose the unobserved parameters χC and ζ to 6For future reference it also presents parameters that will be introduced below. 7See, for example, King, Plosser, and Rebelo (1988) and Plosser (1989). 8 match two statistics: the relative volatility of investment expenditures and the equity premium. The former, measured as the standard deviation of the cyclical component of investment expenditures relative to the standard deviation of the cyclical component of GDP, is 2.28 in our data set. The latter equals 5.18 according to a recent study by Kyriacou, Madsen, and Mase (2004) covering the period 1900-2002 (see footnote 4). The solution of this problem is χC = 0.793 and ζ = 5.53. Table 2.1 Benchmark calibration Preferences β=0.994 η=2 τ=0.20 N=0.13 Production α=0.27 δ=0.011 Stationary Shocks ρZ=0.90 σZ=0.0072 Integrated Shocks lnz¯=0.006 σ =0.0101 lnz Computation of the Equity Premium. The solution of the model are functions gi, i K,Y,C,I,Λ,q , that determine K , Y , C , I , Λ , and q given the current t+1 t t t t t ∈ { } period state variables K , C , and the log of the productivity shock lnZ . t t−1 t In our model the gross risk free rate of return r is given by t Λ t r = . (2.15) t βE Λ t t+1 Since Λ = gΛ(K ,C ,lnZ ) t+1 t+1 t t+1 = gΛ(gK(K ,C ,lnZ ),gC(K ,C ,lnZ ),ρZ lnZ +ǫZ ) t t−1 t t t−1 t t t+1 =: g˜Λ(K ,C ,ρZ lnZ +ǫZ ,) t t−1 t t+1 and ǫZ is normally distributed, the expected value of the Lagrange multiplier equals t+1 ∞ 1 −(ǫZt+1)2 E Λ = g˜Λ(K ,C ,ρZ lnZ +ǫZ ,) e (σZ)2 dǫZ . t t+1 t t−1 t t+1 σZ√2π t+1 Z−∞ We use the quadratic approximation of gΛ at the stationary equilibrium and the Gauss- Hermite 6-point quadrature formula to approximate the integral on the right-hand-side of this equation. 9