The Consumer's Problem

The economy is populated by consumers with expected lifetime utility,

where c_t is consumption, s_t is the index of status, n_t is employment, and 0<β<1. The functional form for u(c, s) follows Smith (2001):

where η_c≥0, η_s≥0, and γ≥1. The parameter η_s measures the extent to which individuals care about status. Importantly, the individual does not care about status when η_s=0. Using the terminology of Smith (2001), the effective coefficient of relative risk aversion is γ^e=1−(η_c+η_s)(1−γ), the effective elasticity of intertemporal substitution is σ^e=1/[1+(γ−1)η_c], and the effective rate of time preferences is ρ^e=[η_c/(η_c+η_s)][(1−β)/β]. This formulation of preferences, or a slight variant of, is used in Bakshi and Chen (1996), Futagami and Shibata (1998), Gong and Zou (2002), and Luo, Smith, and Zou (2002).

The functional form for the subutility function v(n) is:

where θ≥0 and ζ≥1. We explore the implications of the baseline model for two different versions. These versions are distinguished by their assumed labor market frictions. Version 1 assumes that workers choose current hours worked based on the information set available at the end of last period, as in Boldrin, Christiano, and Fisher (2000). The result is that date t labor supply cannot be adjusted in response to date t state of the world. Version 2 assumes that workers supply labor inelastically, as in Jermann (1998). The result is that labor supply is never adjusted.

Consumers face the budget constraint:

where w_t is the wage rate, a_t and p_t are the quantity and price of the risky asset, d_t is dividends paid by the risky asset, and b_t and q_t are the quantity and price of the risk-free asset.

Finally, we focus our attention on the absolute wealth is status framework, in which status is defined exclusively as the level of financial wealth:

Consumers choose consumption, employment, and asset holdings to maximize expected lifetime utility subject to the budget constraint. For version 1, the problem yields the following first-order conditions:

where λ^c_t is the multiplier associated with the budget constraint (4) and

. The functions u_c(c_t, s_t) and u_s(c_t, s_t) are the derivatives of u(c_t, s_t) with respect to its arguments, and v_n(n_t) is the derivative of v(n_t) with respect to its argument. Also, the risky return is:

The risk-free return is:

Finally, for version 2, the problem yields identical first-order conditions, except that condition (7) is replaced by n_t=N.

The Firm's Problem

The firm wishes to maximize its value. The firm's value is (p_t+d_t)A_t, where A_t=1 is the total amount of shares issued. Using the equity return (10), the firm's value can be expressed as the discounted sum of future dividends:

where

and χ₀=1.

Dividends are:

where y_t is output and x_t is investment. Output is produced using:

where z_t is the level of technology or total factor productivity, k_t is the capital stock, and 0<α<1. Capital accumulates via:

where 0<δ<1. Capital accumulation is costly. The adjustment cost function ϕ(x_t/k_t) is:

where ξ>0. The standard no-adjustment cost version is obtained when ξ→∞. Also, ω₁ and ω₂ ensure that ϕ(δ)=δ and ϕ_δ(δ)=1 in the deterministic steady state, where ϕ_δ(x_t/k_t) is the derivative of ϕ(x_t/k_t) with respect to its argument.

Finally, total factor productivity follows the stochastic process:

where

is the stationary level of technology, ε_zt is a mean zero random variable with variance σ_z², and 0<ρ_z<1.

The firm chooses employment and investment to maximize its value subject to the different constraints. The firm's problem yields the following first-order conditions:

where μ_t is the multiplier associated with the capital accumulation equation (15).

Market Clearing Conditions

The economy is closed by the following market clearing conditions. The equity market clearing condition is:

the bond market clearing condition is:

and the goods market clearing condition is:

The consumers' constraints (2)–(5) and necessary conditions (6)–(9), the firm's constraints (13)–(17) and necessary conditions (18)–(20), and the market clearing conditions (21)–(23) are used to compute the economy's competitive equilibrium allocation. This allocation is then used to compute asset prices.

Simulation Method and Calibration

The asset pricing and business cycle implications of the above economic environment are analyzed using simulations. The simulation method is similar to that in Jermann (1998). The method combines the log-linear approximation of King, Plosser, and Rebelo (2002) to solve for quantities with a log-normal approximation to solve for the risk-free return.

The method works as follows. First, we employ the log-linear approximation to generate 1,000 time-series of 200 periods each for consumption, investment, capital, employment, and output. These series are then used to construct the risky return. As in Christiano, Boldrin, and Fisher (2000), we approximate the risky return as the return on capital:

where:

Note that

when investment is fully equity financed.

The quantity series are also used to construct the risk-free return. To do so, we rewrite condition (9) as

where

,

, and λ^c and λ^s are steady state values. As in Jermann (1998), we approximate the expectational terms as

This method requires numerical values for all parameters. To ease comparison with previous studies, we base our values on Boldrin, Christiano, and Fisher (2000) and Jermann (1998).³

, ρ_z=0.979, and σ_z=0.0072 as in King and Rebelo (2000).

For the remaining parameters, Bakshi and Chen (1996) provide estimates for a variant of the preferences displayed here. Using the notation of Smith (2001), their variant is

. This restricts the parameters to γ=γ_BC, η_c=1, and η_s=−λ_BC/(1−γ_BC). For an assumed value of β=1, their unrestricted estimates of γ_BC vary between 2.27 and 3.08, whereas those of λ_BC vary between 0.75 and 1.27. Following these estimates, we set β=0.99999, γ=2, η_c=1, and η_s=−λ_BC/(1−γ) for λ_BC=1.

We also reproduce the asset return and business cycle implications for two other popular specifications of the production economy. We do so because the popular specifications provide a good comparison benchmark, and to assure that the application of the method yields results that are similar to previous studies.

The first specification is that of a standard RBC model with indivisible labor. For this, we set β=0.99, γ=2, α=0.36, δ=0.021, and ζ=1. We also set θ to ensure that employment n=1 in the steady state. We remove the factor market frictions. That is, we let ξ→∞, ω₁=1, and ω₂=0, and we let labor supply depend on all current information.

The second specification is that of a habit formation model. For this, we retain the two versions of the SOC model, but change preferences to:

where ψ≥0. For the two versions of the habit formation model, we retain the previous calibrations and set ψ=0.90.⁴

Simulation Results

Tables 1 and 2 report financial and business cycle statistics. The tables report historical statistics for United States data as well as statistics for simulated data.

The data statistics come from two sources. The financial statistics are those shown in Boldrin, Christiano, and Fisher (2001) and are estimates taken from Cecchetti, Lam, and Mark (1993).⁵

% and

%. Ratio presents the Sharpe Ratio:

.

The business cycle statistics are constructed from seasonally adjusted quarterly data for the 1964:1 to 2001:4 period. In all cases, the statistics are based on the logarithm of quarterly per capita real variables, where the logarithm of each variable is detrended using the Hodrick-Prescott filter with a smoothing parameter of 1,600. The statistics refer to volatility, correlation, and persistence of different variables. Volatility presents the ratio of the standard deviations of consumption, investment, and employment to the standard deviation of output: σ_c/σ_y=0.80, σ_x/σ_y=2.61, and σ_n/σ_y=0.99. Correlation presents the contemporaneous correlations of consumption, investment, and employment with output: ρ(c, y)=0.96, ρ(x, y)=0.94, and ρ(n, y)=0.80. Finally, persistence presents the first autocorrelation of consumption and output: ρ(c′, c)=0.86 and ρ(y′, y)=0.89.

The RBC model does not replicate the financial statistics but replicates several of the business cycle statistics. The RBC model generates too high a risk-free rate, no sizeable equity premium, and too low a volatility for both the risk-free and risky rates. Similar results are found in Jermann (1998), Lettau (2003), Tallarini (2000), and Boldrin, Christiano, and Fisher (2001). As Boldrin, Christiano, and Fisher (2001) argue, the RBC model does not generate an equity premium because the linearity of capital accumulation and the unrestricted labor adjustments allow consumers to smooth consumption too much. As shown in Lettau (2003), the implied reduction in the variance of the marginal utility of consumption (and, hence, the pricing kernel) raises the risk-free rate. Both factors work to reduce the volatility of the risky rate. In particular, the linearity of capital accumulation eliminates potential capital gains and effectively makes the supply of capital perfectly elastic. Furthermore, the ability of firms to adjust employment reduces the need to respond to productivity shocks by adjusting capital. Thus, the fluctuations in the price of capital are limited, which reduces the volatility of the risky rate and the appearance of an equity premium.

The Habit model replicates some of the financial statistics but fails to replicate the main business cycle facts. The Habit model explains a lower risk-free rate and a sizeable equity premium. Admittedly, version 1 of the Habit model generates too much volatility for both the risk-free and risky rates, and version 2 generates too low a risk-free rate. The Habit model's success comes from a correction of the main failure of the RBC model: the Habit model adds habit formation and restrictions to capital and labor adjustments. Habit formation forces consumers to care about the volatility of consumption. That is, habit formation generates a large volatility of the marginal utility from a small volatility of consumption. In addition, the restrictions to capital and labor adjustments ensure sizeable variations in the price of capital. Similar results are found in Boldrin, Christiano, and Fisher (2000) and Jermann (1998). As its stands, the Habit model does not explain the behavior of employment. Version 1 overpredicts the volatility of employment, and counterfactually predicts that employment is acyclical and that output is negatively autocorrelated. Version 2 eliminates the volatility of employment, but predicts that output is positively autocorrelated.

The baseline SOC model can partially replicate the financial statistics. The model predicts a very low (negative) risk-free rate. The low risk-free rate results from the preferences for status. To see this, the pricing equation of the risk-free rate of the RBC model is:

A comparison between equations (27) and (30) suggests that the SOC model produces a low risk-free rate because the marginal utility of status is positive. That is, the expected future return to financial assets includes not only higher future consumption, but also higher future status. This latter effect reduces the equilibrium risk-free rate, and helps resolve the risk-free rate puzzle. Otherwise, the baseline SOC model produces a reasonable amount of volatility for the risk-free rate but not enough for the risky rate. In addition, the model produces only a minimal equity premium. The equity premium is considerably larger than that produced by the RBC model, but it is an order of magnitude that is too small.

The baseline SOC model fails to replicate the business cycle statistics. The model seriously overpredicts the volatility of consumption and employment, while it underpredicts the volatility of investment. Version 1 overpredicts the volatility of employment, and incorrectly predicts that output and consumption are negatively autocorrelated. Version 2 eliminates the volatility of employment but predicts that output and consumption are positively autocorrelated.

A closer look at the SOC model reveals that its calibration implies a very large steady state investment rate (I/Y), and thus a very large capital output ratio. The investment rate of the baseline SOC model is roughly 68%. For comparison, the investment rate is only 24% in the RBC model and 36% in the Habit model. The large investment rate may be responsible for the negative risk-free rate. That is, the investment rate is large because the subjective discount factor β is large. In turn, the large value of β reduces the risk-free rate.

The large investment rate also may affect the volatility of consumption and investment. In the RBC model, large investment rates are associated with high volatility of consumption and low volatility of investment. This association might well carry over to the SOC model. Admittedly, the low volatility of investment might result from other parts of the calibration. The baseline calibration implies stringent adjustment costs to investment, in which the elasticity of investment with respect to Tobin's q (ξ) is small. In addition, the baseline calibration puts equal weights on consumption and status in preferences. This is important because consumers wish to smooth a composite of consumption and status, where status is linked to the capital stock.

For these reasons, we study an alternative calibration of the SOC model. In this alternative, we lower the subjective discount factor to match the observed risk-free rate of 1.19%. For version 1, this requires lowering the subjective discount factor to β=0.936, whereas, for version 2, it requires β=0.978. In both cases, the result is that the investment rate is 32%. This is higher than in the RBC model but smaller than in the Habit model. We also arbitrarily set the elasticity of investment with respect to Tobin's q to ξ=2.⁶

Tables 1 and 2 report financial and business cycle statistics for the alternative calibration. Under this calibration, the SOC model no longer predicts a higher equity premium than the RBC model, and now underpredicts the volatility of asset returns. The model now predicts that investment is more volatile than output but the volatility of investment is still too low. Version 1 of the SOC model now correctly predicts the volatility of employment but incorrectly predicts that employment is countercyclical.

Status

The baseline SOC model uses the absolute wealth is status framework of Bakshi and Chen (1996). These authors propose two other frameworks. The first alternative framework assumes that “the ratio of one's own wealth to the social index determines status.” In this case, status is defined as relative wealth:

where v_t=p_t−1a_t+q_t−1b_t and V_t is the social wealth index. In what follows, we assume that the social wealth index is V_t=p_t−1A_t+q_t−1B_t, and that every consumer is middle class in equilibrium (s_t=1). The second alternative framework assumes that “self-perception determines happiness.” In this case, although status is defined as relative wealth, the term that enters preferences is:

where κ≥0.

We introduce these definitions of status in version 2 of the SOC model. For the self-perception framework, we restrict 0<κ<1, because the social index is V_t=p_t−1A_t+q_t−1B_t and v_t=V_t in equilibrium. We set κ=0.9, but the results are invariant in that range.

The results appear under Status in Tables 3 and 4. The different status frameworks of the SOC model do not significantly alter the financial nor the business cycle statistics. Note that the SOC model with the self-perception framework generates financial and business cycle statistics that are identical to those generated by the absolute wealth framework. The relative wealth framework generates a larger equity premium than the other status frameworks. The relative wealth framework, however, does not improve the ability of the SOC model to predict the business cycle statistics.

Composite

The baseline SOC model assumes that consumers have preferences over a Cobb-Douglas composite of consumption and status. The composite can be easily generalized. For this, we adopt a constant elasticity of substitution aggregator, such that preferences are modified to:

where 1/(1−e) is the elasticity of substitution. In addition to the Cobb-Douglas case, we study three alternative cases. The separable case sets e = 0 and γ = 1, so that consumption and status are separable and preferences reduce to u(c_t, s_t)=η_cln(c_t)+η_sln(s_t). The complement case sets e = −9, so that consumption and status are complements. The substitute case sets e=0.8, so that consumption and status are substitutes.

The results appear under Composite in Tables 3 and 4. The different form of preferences do not significantly alter the financial nor the business cycle statistics. The complement case produces a higher equity premium, a higher volatility of asset returns, and a higher volatility of investment, but none of these statistics are sufficiently high to match the data.

Capital Adjustment

The baseline SOC model adopts a one-sector production structure with adjustment costs similar to that of Jermann (1998). His structure limits capital adjustment by assuming that it is costly. Boldrin, Christiano, and Fisher (2000, 2001) study two other frameworks that limit capital adjustment. The time-to-plan framework replaces the adjustment cost assumption by a time-to-plan assumption. The two-sector framework uses a two-sector production structure in which capital cannot be reallocated across sectors.

For the time-to-plan framework, we eliminate the adjustment costs, so that k_t+1 = x_t +(1−δ)k_t,

, and

. The time-to-plan assumption requires that firms make investment decisions based only on the information set available at the end of last period. This ensures that E_t−1(μ_t)=1, instead of μ_t=1.

For the two-sector framework, we split the firm's problem in two. The firm in the consumption sector seeks to maximize the discounted sum of its future dividends d_ct=y_ct−q_ctx_ct−w_tn_ct. The firm sells its output c_t=y_ct to consumers, purchases new equipment goods x_ct from the investment sector firm at price q_ct, and hires labor n_ct. The firm produces goods using the production technology

and capital accumulates as k_ct+1=x_ct+(1−δ)k_ct, where k_ct is the capital stock of the firm in the consumption sector. The firm in the investment sector also seeks to maximize the discounted sum of its future dividends d_it=q_ctx_ct−w_tn_it. The firm sells part of its output x_ct to the firm in the consumption sector and hires labor n_it. The firm produces goods using the production technology

and capital accumulates as k_it+1=x_it+(1−δ)k_it, where k_it is the capital stock of the firm in the investment sector and x_t=y_it=x_it+x_ct.

The risky return in the two-sector model is:

The risky returns are

and

, where

and

.

The results appear under Capital Adjustment in Tables 3 and 4. The time-to-plan framework produces a small and negative equity premium, while the two-sector framework produces a larger equity premium. Note, however, that it is still an order of magnitude that is too small. The time-to-plan framework generates much more volatility for investment but still predicts a large volatility for consumption. The two-sector framework produces too large a volatility for both consumption and investment. In addition, investment is countercyclical. Interestingly, the correlation between fluctuations in consumption and investment (or the capital stock) is −0.97, which is consistent with consumers smoothing the composite of consumption and status.

Shocks

Finally, the baseline SOC model assumes that total factor productivity is the only source of risk in the economy. Greenwood, Hercowitz, and Krusell (2000) suggest investment-specific technological change as another source of risk. This source of risk directly affects the price of capital.

We add investment-specific shocks to the baseline SOC model. For this, we modify the capital accumulation equation to:

where a_t is the stochastic investment-specific shock. The investment-specific shock follows the stochastic process:

where ε_at is a mean zero random variable with variance

, and 0<ρ_a<1. The risky return is as in the baseline SOC model but μ_t=1/[a_tϕ_δt] and

.

We calibrate the stochastic process as in Greenwood, Hercowitz, and Krusell (2000):

, ρ_a = 0.64, and σ_z = 0.035. We also assume that productivity shocks and investment-specific shocks are uncorrelated.

The results appear under Shocks in Tables 3 and 4. The addition of investment-specific shocks clearly raises the equity premium and the volatility of asset returns. The model with investment-specific shocks still incorrectly predicts a large volatility of consumption, and understates the extent to which consumption and investment are procyclical.