Wage Rigidity: A Quantitative Solution to Several Asset Pricing Puzzles

Favilukis, Jack; Lin, Xiaoji

doi:10.1093/rfs/hhv041

Abstract

In standard production models, wage volatility is far too high, and equity volatility is far too low. A simple modification–sticky wages because of infrequent resetting together with a constant elasticity of substitution (CES) production function leads to both smoother wages and higher equity volatility. Further, the model produces several other hard-to-explain features of financial data: high Sharpe ratios, low and smooth interest rates, time-varying equity volatility and premium, a value premium, and a downward-sloping equity term structure. Procyclical, volatile wages are a hedge for firms in standard models; smoother wages act like operating leverage, making profits and dividends riskier.

Received July 30, 2013; accepted July 6, 2015 by Editor Geert Bekaert.

In most production models used for finance, for example, Jermann (1998); Boldrin, Christiano, and Fisher (2001); Kaltenbrunner and Lochstoer (2010), the volatility of excess equity returns is far too low.¹ This low equity volatility is closely related to the equity premium puzzle. Many standard models also fail to match several other important features of financial and accounting data, including conditional variation in the equity volatility and expected return, the value premium, and a downward-sloping equity term structure. A seemingly unrelated characteristic of these models is that wages are too volatile and too highly correlated with output. We show that the failure to match wage dynamics is closely related to the failure to explain financial data. Introducing sticky wages brings the model quantitatively close to the data for these diverse financial phenomena. To our knowledge, we are the first to capture quantitatively such a wide array of financial moments in a reasonably calibrated general equilibrium model.

In the standard frictionless model, wages are equal to the marginal product of labor, which is perfectly correlated with output and fairly volatile. This model fails because wages act as a hedge for shareholders. Profits are roughly equal to output minus wages, thus highly volatile and procyclical wages make profits very smooth. Dividends are roughly equal to profits minus investment; because profits are smooth and investment is procyclical, dividends are countercyclical. The firm appears too safe, and its equity return is too smooth relative to the data. For example, in standard frictionless models, equity return volatility is between 1% and 5%, compared with above 20% in the data.

Infrequent wage resetting causes the average wage paid by firms to be equal to the weighted average of historical spot wages. This makes the average wage smoother than the marginal product of labor, and less correlated with output. A constant elasticity of substitution (CES) production function, through complementarities between labor and capital, also smoothes wages. When wages are smoother than the marginal product of labor, they are less of a hedge for the firm's shareholders. Profits, which are the residual after wages have been paid, are more volatile; dividends are now procyclical. We refer to this effect as labor leverage, and it leads to a more volatile return on equity.

Quantitatively, to get an equity volatility close to the data, our model employs other ingredients, as well; these include decreasing returns to scale, fixed costs, idiosyncratic productivity shocks, and financial leverage. Our best model has an equity volatility of 15.61%, which explains more than three quarters of the equity volatility in the data. To be fair, this improvement is not purely due to wage rigidity. When we decompose the total effect into various ingredients, the two channels we highlight–infrequent wage negotiation and CES production–play the crucial role. Their contributions are similar in size, and both work through a labor leverage channel; depending on the calibration, they increase equity volatility by a factor of 2 to 4. Without the labor leverage channel (i.e., even with the other ingredients mentioned above), the model produces an equity volatility of only 5.3%.

For the labor leverage channel to be quantitatively important, the labor share (payments to employees as a fraction of output) needs to be large, volatile, and countercyclical. Indeed, the labor share in our model behaves like the data and is countercyclical; whereas in standard models, it is constant. Analogously, for the labor leverage channel to be quantitatively important, the total wage bill needs to be (relatively) smooth and not too procyclical. Again, our model's wage bill behaves as in the data: it is smoother than output and is imperfectly correlated with output; whereas in standard models, it is perfectly correlated with output and has the same volatility as output. Labor leverage resulting from sticky wages acts in a similar way to operating (and even financial) leverage. Because equity is the residual, higher leverage implies riskier equity.

The importance of the labor leverage channel is confirmed by a novel empirical finding. We show that industries with high and countercyclical labor share have higher equity volatility and higher capital asset pricing model (CAPM) betas. In fact, labor share can explain 38% of all cross-sectional variation in industry volatility, and 54% of the variation in CAPM beta (Figure 1 and Table 1). When we allow firms in our model to differ in their amount of wage rigidity, we find that firms that are more rigid have more countercyclical labor share, higher volatility, and higher CAPM betas.

Figure 1

Labor market variables and industry return, volatility, and beta

Figure 1 plots the univariate relationships between the average industry return and the average industry labor share (upper left), the average industry return and the covariance of the industry's labor share with aggregate private output (lower left), the industry return volatility and the average industry labor share (upper middle), the industry return volatility and the covariance of the industry's labor share with aggregate private output (lower middle), the industry CAPM beta and the average industry labor share (upper right), and the industry CAPM beta and the covariance of the industry's labor share with aggregate output (lower right).

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Table 1

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Labor market variables explain industry return, volatility, beta

Panel A: Return
Specification	1	2	3
$B_{E [L S]}$	2.57		2.59
	(1.03)		(1.06)
$B_{C o v [L S, G D P]}$		−0.03	0.02
		(0.11)	(0.09)
$R^{2}$	0.07	0.00	0.07

Panel B: Volatility
Specification	1	2	3
$B_{E [L S]}$	20.96		18.70
	(3.06)		(2.66)
$B_{C o v [L S, G D P]}$		−2.79	−2.44
		(2.60)	(2.54)
$R^{2}$	0.23	0.20	0.38

Panel C: CAPM beta
Specification	1	2	3
$B_{E [L S]}$	0.77		0.63
	(2.44)		(2.72)
$B_{C o v [L S, G D P]}$		−0.17	−0.15
		(3.69)	(4.21)
$R^{2}$	0.19	0.42	0.54

Table 1 presents results of cross-sectional regressions where the unconditional average return, volatility, or CAPM beta of an industry are regressed on industry characteristics. The characteristics are the average labor share in an industry and the covariance of an industry's labor share with aggregate private GDP. Specification 1 includes just average labor share, specification 2 includes just the covariance, and specification 3 includes both. T-statistics are in parentheses. Results are for 1929–2000.

Table 1

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Labor market variables explain industry return, volatility, beta

Panel A: Return
Specification	1	2	3
$B_{E [L S]}$	2.57		2.59
	(1.03)		(1.06)
$B_{C o v [L S, G D P]}$		−0.03	0.02
		(0.11)	(0.09)
$R^{2}$	0.07	0.00	0.07

Panel B: Volatility
Specification	1	2	3
$B_{E [L S]}$	20.96		18.70
	(3.06)		(2.66)
$B_{C o v [L S, G D P]}$		−2.79	−2.44
		(2.60)	(2.54)
$R^{2}$	0.23	0.20	0.38

Panel C: CAPM beta
Specification	1	2	3
$B_{E [L S]}$	0.77		0.63
	(2.44)		(2.72)
$B_{C o v [L S, G D P]}$		−0.17	−0.15
		(3.69)	(4.21)
$R^{2}$	0.19	0.42	0.54

Table 1 presents results of cross-sectional regressions where the unconditional average return, volatility, or CAPM beta of an industry are regressed on industry characteristics. The characteristics are the average labor share in an industry and the covariance of an industry's labor share with aggregate private GDP. Specification 1 includes just average labor share, specification 2 includes just the covariance, and specification 3 includes both. T-statistics are in parentheses. Results are for 1929–2000.

Cross-sectional variation in labor leverage has additional asset pricing implications. Low-productivity firms that are saddled with high committed wages relative to output are riskier. These firms are value (low market-to-book) firms. In particular, value firms are loaded with high wages that do not fall as much as output in bad times, which leads to a lower profit-to-labor-expense ratio for value firms and a sizable value premium. Our mechanism in generating value premium is different from that of Zhang (2005), in that we focus on the endogenous operating leverage effect induced by rigid wages, which affects value firms more than it affects growth firms, especially in economic downturns, whereas Zhang (2005) emphasized the real frictions on firms' investment. Moreover, the growth rate shocks to aggregate productivity in our model are essentially the long-run risk shocks as in Bansal and Yaron (2004), whereas the pricing kernel in Zhang (2005) is effectively habit persistence, as in Campbell and Cochrane (1999). Our mechanism is also different from those of Garleanu, Panageas, and Yu (2012); Ai, Croce, and Li (2013); Ai and Kiku (2013), who used large infrequent technological innovations, intangible capital, and growth options, respectively, to explain the value premium.

Similar to the cross-section, labor leverage is not constant through the business cycle. Because wages are smoother than output, leverage associated with wages is higher in recessions than in expansions. Consistent with financial data, this leads to higher equity volatility and a higher expected equity premium during bad times. Favilukis and Lin (2015) explored additional empirical implications of labor leverage and show that consistent with the model in this paper, wage growth forecasts excess stock returns negatively at the aggregate, industry, and state level. This happens because during bad (good) times, wages fall (rise) by less than output leading to an increase (decrease) in leverage and therefore equity risk. These finding echo Santos and Veronesi (2006), who showed that the labor-income-to-consumption ratio is a good predictor of long-horizon stock returns.

Finally, our model is able to reproduce the downward-sloping term structure of equity dividends that van Binsbergen, Brandt, and Koijen (2012) found in the data. The intuition is again due to rigid wages. Wages and output are cointegrated, and thus in the long-run, wages and dividends are expected to be at their normal shares of output. However, in contrast to standard models, in the short-run, wages can be far from their normal share, making profits and dividends highly procyclical. This results in short-term dividends being riskier. Our channel is distinct from that of Belo, Collin-Dufresne, and Goldstein (2015). They showed that a mean-reverting financial leverage policy can also produce a downward-sloping term structure of equity dividends. Our channel is also distinct from that of Ai, Croce, Diercks, and Li (2015), who have earlier showed that most production-based long-run risk (LRR) models imply an upward sloping term structure of equity.

Although the macroeconomic literature on wages and labor is quite substantial (e.g., Pissarides 1979), there has been surprisingly little work done relating labor frictions to finance; notable exceptions are Uhlig (2007); Merz and Yashiv (2007); and Belo, Lin, and Bazdresch (2014).

Our mechanism is similar to the work of Danthine and Donaldson (2002), which showed that the operating leverage effect, caused by exogenous bargaining and risk sharing between households and capital owners, can lead to higher equity volatility. However, the staggered wage contract in our model implies endogenously smoother wages. More importantly, our model is quantitatively much closer to the data; it also generages conditional variation in the time series and the cross-section of expected returns, whereas Danthine and Donaldson (2002) were silent on these hard-to-match moments. Our paper is also complimentary to Gourio (2007), who noted that wages are smoother than output and explored the implications of this for cross-sectional asset pricing. Unlike Gourio (2007), we explore the asset pricing implications of wage rigidity in a dynamic stochastic general equilibrium (DSGE) model.

Petrosky-Nadeau, Zhang, and Kuehn (2013) explored how search frictions affect asset prices in a general equilibrium setting with production. Unlike our model, their channel works mostly through rare events, as in Barro (2006), during which unemployment spikes up. Li and Palomino (2014) studied nominal price and wage rigidity. They found that both types of rigidity increase expected equity returns, but wage rigidities have a larger effect. Berk and Walden (2013) showed that firms write labor contracts to provide insurance for risk-averse workers; similar to our mechanism, this results in riskier equity. Donangelo (2014) studied the link between labor mobility, operating leverage, and asset prices. Gomes, Jermann, and Schmid (2015) investigated the rigidity of nominal debt, which creates long-term leverage that works in a similar way to our labor leverage.

Our paper is also related to the literature on long-run risk (LRR); however, our contribution is to make LRR viable in a production economy. Bansal and Yaron (2004) have shown that the combination of a high intertemporal elasticity of substitution (IES) and a persistent consumption growth rate can deliver a high Sharpe ratio, even with a low risk aversion. Croce (2014); Kaltenbrunner and Lochstoer (2010); and Kung and Schmid (2015) have shown that this can work in a production economy, but the excess volatility and time variation of returns remain unresolved. Our model is similar to all of these models, but it adds infrequent wage resetting. It is important to note that LRR alone cannot produce time-varying excess returns or volatilities. Bansal and Yaron (2004) devoted the second half of their paper to adding an exogenous state variable which controls the volatility of equity returns but that is orthogonal to LRR. Thus, in addition to a high volatility of equity, an important contribution of our work is to showcase a channel for endogenous conditional variation in equity returns and volatilities in a LRR world.

Finally, our paper is related to the extensive literature on wage rigidities and unemployment dynamics. Shimer (2005); Hall (2006); Gertler and Trigari (2009); Pissarides (2009), among others, showed that wage rigidities are crucial to explain U.S. labor market dynamics. For example, Hall wrote, “The incorporation of wage stickiness makes employment realistically sensitive to driving forces.” Our paper differs from these macro-papers in that we study asset pricing implications of staggered wage setting, whereas the models in labor economics fail to match the asset prices observed in the data. This is a problem endemic to most standard models, as observed by Mehra and Prescott (1985).

1. The Model

In the model financial markets are complete; therefore, we consider one representative household who receives labor income, chooses between consumption and saving, and maximizes utility as in Epstein and Zin (1989).

U_{t} = \max {((1 - β) C_{t}^{1 - \frac{1}{ψ}} + β E_{t} {[U_{t + 1}^{1 - θ}]}^{\frac{1 - \frac{1}{ψ}}{1 - θ}})}^{\frac{1}{1 - \frac{1}{ψ}}}

(1)

W_{t + 1} = (W_{t} + N_{t} * {\bar{w}}_{t} - C_{t}) R_{t + 1},

(2)

where the second equation is the budget constraint of the representative household. The household enters period

t

with financial wealth

W_{t}

⁠, earns labor income

N_{t} * {\bar{w}}_{t},

² consumes

C_{t}

⁠, and invests the remainder in a portfolio of financial assets with return

R_{t + 1}

⁠. Risk aversion is given by

θ

⁠, and the IES by

ψ

⁠.

The aggregate labor supply is $N_{t}$ ⁠. Labor and leisure do not enter into the household's utility function. However, we allow $N_{t}$ to be a function of the aggregate stochastic shock. There are several ways to motivate this. For example, this would be the case in a model where all individuals expect to spend 8 hours per day working, searching for jobs, or acquiring new skills, but there is time variation in search frictions or in the fit between employee skills and skills demanded by firms. For example, Andolfatto (1996) showed that time-varying search efficiency is important quantitatively for the standard real business cycle (RBC) model to match the comovement between labor and wages. We allow the labor supply to vary because it is important for the validity of our exercise to match both quantities and prices.

An earlier version of the paper had constant aggregate labor supply; although the asset pricing results were quantitatively similar, the model could not match the behavior of the total wage bill or the labor share.³

1.1 Firms

The interesting frictions in the model are on the firm's side. We assume a large number of firms (indexed by $i$ and differing in idiosyncratic productivity) choose investment and labor to maximize the present value of future dividend payments, where the dividend payments are equal to the firm's output net of investment, wages, operating costs, and adjustment costs. Output is produced from labor and capital. Firms hold beliefs about the discount factor $M_{t + 1}$ ⁠, which is determined in equilibrium.

1.1.1 Technology

The variable $Z_{t}$ is an exogenously specified total factor productivity common to all firms; idiosyncratic productivity of firm $i$ is $Z_{t}^{i}$ ⁠; their calibration is described below.

Firm

i

's output is given by

Y_{t}^{i} = Z_{t}^{i} {(α {(K_{t}^{i})}^{η} + (1 - α) {(Z_{t} N_{t}^{i})}^{η ρ})}^{\frac{1}{η}} .

(3)

Output is produced with CES technology from capital (⁠

K_{t}^{i}

⁠) and labor (⁠

N_{t}^{i}

⁠), where

Z_{t}

is labor augmenting aggregate productivity, and

Z_{t}^{i}

is the firm's idiosyncratic productivity.

ρ

determines the degree of return to scale (constant return to scale if

ρ = 1

⁠);

\frac{1}{1 - η}

is the elasticity of substitution between capital and labor (Cobb-Douglas production if

η = 0

⁠); and

(1 - α) ρ

is related to the share of labor in production.

1.1.2 The wage contract

In standard production models wages are reset each period, and employees receive the marginal product of labor. We assume that any employee's wage will be reset in the current period with probability $1 - μ$ ⁠.⁴ When $μ = 0$ ⁠, our model is identical to models without rigidity: all wages are reset each period, each firm can freely choose the number of its employees, and each firm chooses $N_{t}^{i}$ ⁠, such that its marginal product of labor is equal to the wage. When $μ > 0$ ⁠, we must differentiate between the spot wage (⁠ $w_{t}$ ⁠), which is paid to all employees resetting wages this period; the economy's average wage (⁠ ${\bar{w}}_{t}$ ⁠); and the firm's average wage (⁠ ${\bar{w}}_{t}^{i}$ ⁠).

When a firm hires a new employee in a period with spot wage

w_{t}

⁠, with probability

μ

⁠, it must pay this employee the same wage next period; on average, this employee will keep the same wage for

\frac{1}{1 - μ}

periods. All resetting employees come to the same labor market, and the spot wage is selected to clear markets. The firm chooses its total labor force

N_{t}^{i}

each period. These conditions lead to a natural formulation of the firm's average wage as the weighted average of the previous average wage and the spot wage:

{\bar{w}}_{t}^{i} N_{t}^{i} = w_{t} (N_{t}^{i} - μ N_{t - 1}^{i}) + {\bar{w}}_{t - 1}^{i} μ N_{t - 1}^{i}

(4)

Here

N_{t}^{i} - μ N_{t - 1}^{i}

is the number of new employees the firm hires at the spot wage, and

μ N_{t - 1}^{i}

is the number of tenured employees with average wage

{\bar{w}}_{t - 1}^{i}

⁠.⁵

Note that the rigidity in our model is a real wage rigidity, although our channel could in principle work through nominal rigidities, as well. There is evidence for the importance of both real and nominal rigidities. Micro-level studies of panel data sets comparing actual and notional wage distributions show that nominal wage changes cluster both at zero and at the current inflation rate, with sharp decreases in density to the left of the two mass points. Barwell and Schweitzer (2007); Devicenti, Maida, and Sestito (2007); and Bauer, Goette, and Sunde (2007) found that downward real wage rigidity is substantial in United Kingdom, Germany, and Italy, and that the fraction of real wage cuts prevented by downward real wage rigidity is more than five times greater than the fraction prevented by downward nominal wage rigidity. Dickens and others (2007) found that the relative importance of downward real wage rigidity and downward nominal wage rigidity varies greatly across countries while the incidences of both types of wage rigidity are roughly the same.

1.1.3 Accounting

The equation for profit is

Π (K_{t}^{i}) = Y_{t}^{i} - {\bar{w}}_{t}^{i} N_{t}^{i} - Ψ_{t}^{i} .

(5)

Π (K_{t}^{i})

is profit, given by output less labor and operating costs. Operating costs are defined as

Ψ_{t}^{i} = f * K_{t}

⁠; they depend on aggregate (but not firm specific) capital to allow fixed costs to be integrated with the rest of the economy, given that productivity is non stationary. Labor costs are

{\bar{w}}_{t}^{i} N_{t}^{i}

⁠.

The total dividend paid by the firm is

D_{t}^{i} = Π (K_{t}^{i}) - I_{t}^{i} - Φ (I_{t}^{i}, K_{t}^{i}),

(6)

which is profit less investment and capital adjustment costs. Capital adjustment costs are given by

Φ (I_{t}^{i}, K_{t}^{i}) = ν {(\frac{I_{t}^{i}}{K_{t}^{i}})}^{2} K_{t}^{i}

⁠.

1.1.4 The firm's problem

The firm maximizes the present discounted value of future dividends

V_{t}^{i} = \max_{I_{t}^{i}, N_{t}^{i}} E_{t} [\sum_{j = 0, \infty} M_{t + j} D_{t + j}^{i}],

(7)

subject to the standard capital accumulation equation

K_{t + 1}^{i} = (1 - δ) K_{t}^{i} + I_{t}^{i},

(8)

as well as Equations (3), (4), (5), and (6). Equation 7 can be rewritten as a Bellman equation:

V_{t}^{i} = \max_{I_{t}^{i}, N_{t}^{i}} D_{t} + E_{t} [M_{t + 1} V_{t + 1}^{i}]

⁠.

The firm's optimal investment choice satisfies the familiar Euler equation in the q-theory of investment:

1 + 2 ν \frac{I_{t}^{i}}{K_{t}^{i}} = E_{t} [M_{t + 1} \frac{\partial V_{t + 1}^{i}}{\partial K_{t + 1}^{i}}] .

(9)

The firm's optimal labor choice is given by:

w_{t} = \frac{\partial Y_{t}^{i}}{\partial N_{t}^{i}} + E_{t} [M_{t + 1} \frac{\partial V_{t + 1}^{i}}{\partial N_{t}^{i}}] .

(10)

It is straightforward to see that when

μ = 0

⁠, then

\frac{\partial V_{t + 1}^{i}}{\partial N_{t}^{i}} = 0

and labor is chosen such that the wage is equal to the marginal product of labor. However, when

μ > 0

⁠, the next period's value function depends on this period's wage and labor choice as the firm considers the effect of today's labor choice on future labor expenses.

1.2 Equilibrium

We assume that there exists some underlying set of aggregate state variables $S_{t}$ ⁠, which is sufficient for this problem. Each firm's individual state variables are given by the vector $S_{t}^{i} = [Z_{t}^{i}, K_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}]$ ⁠. Because the household is a representative agent, we are able to avoid explicitly solving the household's maximization problem and simply use the first-order conditions to find $M_{t + 1}$ as an analytic function of consumption or expectations of future consumption. For instance, with CRRA utility, $M_{t + 1} = β {(\frac{C_{t + 1}}{C_{t}})}^{- θ}$ ⁠, whereas for the Epstein-Zin utility function, $M_{t + 1} = β {(\frac{C_{t + 1}}{C_{t}})}^{- \frac{1}{ψ}} {(\frac{U_{t + 1}}{E_{t} {[U_{t + 1}^{1 - θ}]}^{\frac{1}{1 - θ}}})}^{\frac{1}{ψ} - θ}$ ⁠.

Equilibrium consists of the following: It must also be that given these policy functions, all markets clear and the beliefs are consistent with simulated data. Therefore, these beliefs are rational:

Beliefs about the transition function of the aggregate state variable and the shocks, $S_{t + 1} = Γ (S_{t}, Z_{t + 1})$
Beliefs about the realized stochastic discount factor as a function of the aggregate state variable and the realized shocks, $M (S_{t}, Z_{t + 1})$
Beliefs about the aggregate spot wage as a function of the aggregate state variable, $w (S_{t})$
Firm policy functions for labor demand $N_{t}^{i}$ and investment $I_{t}^{i}$ (functions of $S_{t}$ and $S_{t}^{i}$ ⁠)

The firm's policy functions maximize the firm's problem (satisfy Equations 9 and 10) given beliefs about the wages, the discount factor, and the aggregate state variable.
The labor market clears: $\sum N_{t}^{i} = N_{t}$ ⁠. Recall that $N_{t}$ is a function of the exogenous shock, which is part of the aggregate state $S_{t}$ ⁠.
The goods market clears: $C_{t} = \sum (Π_{t}^{i} + Ψ_{t}^{i} + {\bar{w}}_{t}^{i} N_{t}^{i} - I_{t}^{i}) = \sum D_{t}^{i} + {\bar{w}}_{t}^{i} N_{t}^{i} + Φ_{t}^{i} + Ψ_{t}^{i}$ ⁠. Note that here we are assuming that all costs are paid by firms to individuals and are therefore consumed. The results are very similar if all costs are instead wasted.
The beliefs about $M_{t + 1}$ are consistent with goods market clearing through the household's Euler equation.
Beliefs about the transition of the state variables are correct. For instance if aggregate capital is part of the aggregate state vector $S_{t}$ ⁠, then it must be that $K_{t + 1} = (1 - δ) K_{t} + \sum I_{t}^{i}$ ⁠, where $I_{t}^{i}$ is each firm's optimal policy.

2. Calibration

We solve the model at a quarterly frequency using a variation of the Krusell and Smith (1998) algorithm. We discuss the solution method in the Appendix. The model requires us to choose the preference parameters, $β$ (time discount factor), $θ$ (risk aversion), and $ψ$ (IES), as well as the technology parameters, $α$ and $ρ$ (jointly determine labor share in output and degree of return to scale), $\frac{1}{1 - η}$ (elasticity of substitution between labor and capital), $δ$ (depreciation), $f$ (operating cost), and $ν$ (capital adjustment cost). Finally, we must choose our key parameter $μ$ ⁠, which determines the frequency of wage resetting. Additionally, we must choose a process for labor supply, aggregate productivity shocks, and idiosyncratic productivity shocks.

In Table 2 we present parameter choices for four models of interest: (1) a standard model with Cobb-Douglas technology where all wages are reset each period (⁠ $μ = 0$ ⁠), (2) a model with Cobb-Douglas technology where wages are reset once every ten quarters on average (⁠ $μ = 0.9$ ⁠), (3) a model with a calibrated elasticity of substitution between labor and capital but where all wages are reset each period, and (4) a model with a calibrated elasticity of substitution between labor and capital where wages are reset once every ten quarters on average.

Table 2

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Calibration

Parameter	Description	I	II	III	IV
		Cobb-Douglas	Cobb-Douglas	Calibrated CES	Calibrated CES
		Frequent resetting	Infrequent resetting	Frequent resetting	Infrequent resetting
Preferences
$β$	Time preference	0.994	0.994	0.994	0.994
$θ$	Risk aversion	6.5	6.5	6.5	6.5
$ψ$	IES	2.0	2.0	2.0	2.0
Production
$α$	Capital weight in production	0.23	0.23	0.45	0.45
$ρ$	Determines labor and profit shares	0.77	0.77	0.77	0.77
$δ$	Depreciation	0.025	0.025	0.025	0.025
$\frac{1}{1 - η}$	Labor capital elasticity	1.00	1.00	0.50	0.50
$ν$	Adjustment cost	0.3	1.2	3.5	4.5
$f$	Operating cost	0.0160	0.0195	0.0187	0.0198
$μ$	Probability of no resetting	0	0.9	0	0.9

Parameter	Description	I	II	III	IV
		Cobb-Douglas	Cobb-Douglas	Calibrated CES	Calibrated CES
		Frequent resetting	Infrequent resetting	Frequent resetting	Infrequent resetting
Preferences
$β$	Time preference	0.994	0.994	0.994	0.994
$θ$	Risk aversion	6.5	6.5	6.5	6.5
$ψ$	IES	2.0	2.0	2.0	2.0
Production
$α$	Capital weight in production	0.23	0.23	0.45	0.45
$ρ$	Determines labor and profit shares	0.77	0.77	0.77	0.77
$δ$	Depreciation	0.025	0.025	0.025	0.025
$\frac{1}{1 - η}$	Labor capital elasticity	1.00	1.00	0.50	0.50
$ν$	Adjustment cost	0.3	1.2	3.5	4.5
$f$	Operating cost	0.0160	0.0195	0.0187	0.0198
$μ$	Probability of no resetting	0	0.9	0	0.9

All model parameters are listed in this table. Note that most parameters are shared by all models, and only four parameters (⁠ $η$ ⁠, $ν$ ⁠, $f$ ⁠, and $μ$ ⁠) vary across models. The model is solved at a quarterly frequency.

Table 2

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Calibration

Parameter	Description	I	II	III	IV
		Cobb-Douglas	Cobb-Douglas	Calibrated CES	Calibrated CES
		Frequent resetting	Infrequent resetting	Frequent resetting	Infrequent resetting
Preferences
$β$	Time preference	0.994	0.994	0.994	0.994
$θ$	Risk aversion	6.5	6.5	6.5	6.5
$ψ$	IES	2.0	2.0	2.0	2.0
Production
$α$	Capital weight in production	0.23	0.23	0.45	0.45
$ρ$	Determines labor and profit shares	0.77	0.77	0.77	0.77
$δ$	Depreciation	0.025	0.025	0.025	0.025
$\frac{1}{1 - η}$	Labor capital elasticity	1.00	1.00	0.50	0.50
$ν$	Adjustment cost	0.3	1.2	3.5	4.5
$f$	Operating cost	0.0160	0.0195	0.0187	0.0198
$μ$	Probability of no resetting	0	0.9	0	0.9

Parameter	Description	I	II	III	IV
		Cobb-Douglas	Cobb-Douglas	Calibrated CES	Calibrated CES
		Frequent resetting	Infrequent resetting	Frequent resetting	Infrequent resetting
Preferences
$β$	Time preference	0.994	0.994	0.994	0.994
$θ$	Risk aversion	6.5	6.5	6.5	6.5
$ψ$	IES	2.0	2.0	2.0	2.0
Production
$α$	Capital weight in production	0.23	0.23	0.45	0.45
$ρ$	Determines labor and profit shares	0.77	0.77	0.77	0.77
$δ$	Depreciation	0.025	0.025	0.025	0.025
$\frac{1}{1 - η}$	Labor capital elasticity	1.00	1.00	0.50	0.50
$ν$	Adjustment cost	0.3	1.2	3.5	4.5
$f$	Operating cost	0.0160	0.0195	0.0187	0.0198
$μ$	Probability of no resetting	0	0.9	0	0.9

All model parameters are listed in this table. Note that most parameters are shared by all models, and only four parameters (⁠ $η$ ⁠, $ν$ ⁠, $f$ ⁠, and $μ$ ⁠) vary across models. The model is solved at a quarterly frequency.

Preferences: $β$ is set to 0.994 per quarter, and this parameter effects directly the level of the risk-free rate and is also related to the average investment-to-output-ratio. $θ$ is set to 6.5 to get a reasonably high Sharpe ratio, while keeping risk aversion within the range recommended by Mehra and Prescott (1985). $ψ$ (IES) is set to 2 (IES), and this also helps with the Sharpe ratio; its value is consistent with the LRR literature. Bansal and Yaron (2004) showed that values above one are required for the LRR channel to match asset pricing moments because this ensures that the compensation for long-run expected growth risk is positive.

Technology: $δ$ is set to 0.025 to match quarterly depreciation. Our production function has constant elasticity of substitution (CES), which includes Cobb-Douglas production as a special case. $η$ is set to 0, which implies Cobb-Douglas production, or to −1, which matches empirical estimates of the elasticity of substitution between labor and capital. In our model this elasticity is $\frac{1}{1 - η} = 0.5$ ⁠, which is consistent with estimates between 0.4 and 0.6 in a survey article by Chirinko (2008).

For the Cobb-Douglas production function (⁠ $η = 0$ ⁠), the parameters $α$ and $ρ$ have clear interpretations: $α$ is the share of capital in production, and $ρ$ is the curvature of the production function (equivalently the degree of return to scale). Together, $α$ and $ρ$ determine the labor share $(1 - α) ρ$ and the profit share $(1 - α) (1 - ρ)$ ⁠. We set $ρ = 0.77$ and $α = 0.23$ so that labor share is $(1 - α) ρ = 0.59$ ⁠; this matches payments to labor as a share of output (for the private sector) of 0.59 in the data. The profit share is 0.177, which is consistent with the literature. Burnside, Eichenbaum, and Rebelo (1995) estimated it to be between 0.1 and 0.2; Khan and Thomas (2008) used 0.104; and Bachmann, Caballero, and Engel (2013) used 0.18. Additionally, the combination of $δ$ ⁠, $α$ ⁠, $ρ$ ⁠, and $β$ determine the investment-to-output ratio; in the United States. this is between 20% and 25%, and our model matches this.

For the more general CES production function (⁠ $η = - 1$ ⁠), $α$ and $ρ$ are still related to labor share, profit share, and the investment-to-capital-ratio, however, their relationship is no longer characterized by a simple analytic formula. If we were to choose the same parameters as for Cobb-Douglas, then labor share and capital share would be quite different. We set $ρ = 0.77$ and $α = 0.45$ ⁠, and these allow the model with $η = - 1$ to have roughly the same profit share, labor share, and investment-to-output ratio as those of the Cobb-Douglas model, and as the U.S. economy.

Operating cost: $Ψ_{t} = f * K_{t}$ is a fixed cost from the perspective of the firm; however, it depends on the aggregate state of the economy, in particular, on aggregate capital. In each model, we choose $f$ to match the average market-to-book ratio in the economy, which we estimate to be 1.33 (the actual values of $f$ are in Table 2, and details estimating the market-to-book ratio are in the Appendix). Although we think it is realistic for this cost to increase when aggregate capital is higher (during expansions), the results are not sensitive to this assumption. The results are very similar when $Ψ_{t}$ is simply growing at the same rate as the economy is.

Capital adjustment cost: Within each model, we choose the capital adjustment cost $ν$ to match the volatility of aggregate investment. Models with different $μ$ or $η$ may require a different adjustment cost for investment volatility to match the data; therefore, the level of adjustment cost is different across models. Higher adjustment costs always help to increase equity volatility and the value premium, but they decrease aggregate investment volatility. This restriction on matching aggregate investment volatility limits how much work capital adjustment costs can do in helping to match financial moments.

Productivity and labor shocks: For the standard LRR channel (IES > 1) to produce high Sharpe ratios, aggregate productivity must be non-stationary with a stationary growth rate. We define the growth rate of $Z_{t}$ as $g_{t} = l n (\frac{Z_{t}}{Z_{t - 1}})$ ⁠. $g_{t} \in {g_{1}, g_{2}, g_{3}}$ and follows a symmetric Markov chain, where $P r (g_{t + 1} = g_{j} | g_{t} = g_{i}) = π_{i j} \geq 0$ and $\sum_{j = 1}^{3} π_{i j} = 1$ for each $j$ ⁠. The probabilities are chosen so that the quarterly autocorrelation of 0.80, unconditional mean of 0.005, and unconditional standard deviation of 0.019 match roughly the autocorrelation, growth rate, and standard deviation of output.⁶

We assume that aggregate labor supply is perfectly correlated with aggregate productivity so that $N_{t} = N_{i}$ whenever $g_{t} = g_{i}$ ⁠, where $i \in {1, 2, 3}$ ⁠. We set $N_{1} = 0.93$ ⁠, $N_{2} = 1.0$ ⁠, and $N_{3} = 1.07$ to best match the behavior of aggregate employment, as can be seen in Table 3. Of course we are somewhat limited because of the perfect correlation with productivity growth; in principle, labor supply could follow an independent process; however, this would increase computational time.

Table 3

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Macroeconomic moments

Panel A: Data means
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.57	1.00	1.00	0.54
$c$	0.51	0.72	0.53	0.53	0.79	0.50
$i$	3.04	0.07	0.55	2.84	0.27	0.39
$n$	0.78	0.92	0.55	0.73	0.89	0.47
$w$	0.44	0.60	0.36	0.49	0.61	0.32
$w * n$	0.87	0.84	0.33	0.91	0.85	0.19
$\frac{w * n}{y}$	0.56	−0.49	0.36	0.53	−0.43	0.20

Panel A: Data means
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.57	1.00	1.00	0.54
$c$	0.51	0.72	0.53	0.53	0.79	0.50
$i$	3.04	0.07	0.55	2.84	0.27	0.39
$n$	0.78	0.92	0.55	0.73	0.89	0.47
$w$	0.44	0.60	0.36	0.49	0.61	0.32
$w * n$	0.87	0.84	0.33	0.91	0.85	0.19
$\frac{w * n}{y}$	0.56	−0.49	0.36	0.53	−0.43	0.20

Panel B: Data standard errors
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	0.00	0.00	0.11	0.00	0.00	0.10
$c$	0.05	0.09	0.12	0.05	0.08	0.12
$i$	0.26	0.24	0.14	0.45	0.26	0.12
$n$	0.04	0.03	0.11	0.04	0.03	0.11
$w$	0.04	0.08	0.12	0.04	0.09	0.12
$w * n$	0.04	0.06	0.11	0.06	0.04	0.14
$\frac{w * n}{y}$	0.09	0.07	0.15	0.08	0.10	0.20

Panel B: Data standard errors
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	0.00	0.00	0.11	0.00	0.00	0.10
$c$	0.05	0.09	0.12	0.05	0.08	0.12
$i$	0.26	0.24	0.14	0.45	0.26	0.12
$n$	0.04	0.03	0.11	0.04	0.03	0.11
$w$	0.04	0.08	0.12	0.04	0.09	0.12
$w * n$	0.04	0.06	0.11	0.06	0.04	0.14
$\frac{w * n}{y}$	0.09	0.07	0.15	0.08	0.10	0.20

Panel C: Cobb-Douglas, $μ = 0$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.43
$c$	0.54	0.92	0.51	0.65	0.91	0.64
$i$	3.06	0.95	0.46	2.79	0.90	0.28
$n$	0.84	0.56	0.20	0.82	0.55	−0.08
$w$	0.88	0.61	0.57	0.88	0.63	0.62
$w * n$	1.00	1.00	0.46	1.00	1.00	0.43
$\frac{w * n}{y}$	0.00	0.00	0.00	0.00	0.00	0.00

Panel C: Cobb-Douglas, $μ = 0$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.43
$c$	0.54	0.92	0.51	0.65	0.91	0.64
$i$	3.06	0.95	0.46	2.79	0.90	0.28
$n$	0.84	0.56	0.20	0.82	0.55	−0.08
$w$	0.88	0.61	0.57	0.88	0.63	0.62
$w * n$	1.00	1.00	0.46	1.00	1.00	0.43
$\frac{w * n}{y}$	0.00	0.00	0.00	0.00	0.00	0.00

Panel D: Cobb-Douglas, $μ = 0.9$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.47	1.00	1.00	0.43
$c$	0.60	0.95	0.53	0.68	0.94	0.63
$i$	2.95	0.95	0.43	2.70	0.91	0.25
$n$	0.83	0.55	0.20	0.81	0.54	−0.08
$w$	0.47	0.47	0.70	0.60	0.61	0.86
$w * n$	0.77	0.88	0.19	0.89	0.90	0.17
$\frac{w * n}{y}$	0.48	−0.66	0.45	0.43	−0.46	0.28

Panel D: Cobb-Douglas, $μ = 0.9$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.47	1.00	1.00	0.43
$c$	0.60	0.95	0.53	0.68	0.94	0.63
$i$	2.95	0.95	0.43	2.70	0.91	0.25
$n$	0.83	0.55	0.20	0.81	0.54	−0.08
$w$	0.47	0.47	0.70	0.60	0.61	0.86
$w * n$	0.77	0.88	0.19	0.89	0.90	0.17
$\frac{w * n}{y}$	0.48	−0.66	0.45	0.43	−0.46	0.28

Panel E: CES, $μ = 0$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.42
$c$	0.60	0.95	0.55	0.67	0.93	0.66
$i$	3.04	0.95	0.42	2.86	0.91	0.24
$n$	0.83	0.56	0.20	0.81	0.55	−0.08
$w$	0.83	0.35	0.51	0.85	0.40	0.53
$w * n$	0.76	0.99	0.49	0.80	0.99	0.52
$\frac{w * n}{y}$	0.27	−0.93	0.42	0.25	−0.85	0.25

Panel E: CES, $μ = 0$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.42
$c$	0.60	0.95	0.55	0.67	0.93	0.66
$i$	3.04	0.95	0.42	2.86	0.91	0.24
$n$	0.83	0.56	0.20	0.81	0.55	−0.08
$w$	0.83	0.35	0.51	0.85	0.40	0.53
$w * n$	0.76	0.99	0.49	0.80	0.99	0.52
$\frac{w * n}{y}$	0.27	−0.93	0.42	0.25	−0.85	0.25

Panel F: CES, $μ = 0.9$ (baseline model)
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.42
$c$	0.62	0.95	0.55	0.68	0.94	0.64
$i$	3.00	0.95	0.41	2.83	0.91	0.23
$n$	0.83	0.56	0.20	0.82	0.55	−0.08
$w$	0.40	0.38	0.71	0.55	0.54	0.88
$w * n$	0.74	0.83	0.14	0.86	0.87	0.11
$\frac{w * n}{y}$	0.56	−0.68	0.49	0.50	−0.52	0.35

Panel F: CES, $μ = 0.9$ (baseline model)
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.42
$c$	0.62	0.95	0.55	0.68	0.94	0.64
$i$	3.00	0.95	0.41	2.83	0.91	0.23
$n$	0.83	0.56	0.20	0.82	0.55	−0.08
$w$	0.40	0.38	0.71	0.55	0.54	0.88
$w * n$	0.74	0.83	0.14	0.86	0.87	0.11
$\frac{w * n}{y}$	0.56	−0.68	0.49	0.50	−0.52	0.35

In Table 3 we compare annual macroeconomic moments from the data (1929–2013) to several versions of our model. A variable denoted as $x$ is HP-filtered; a variable denoted as $Δ x$ is a growth rate. The variables are GDP (⁠ $y$ ⁠), consumption (⁠ $c$ ⁠), investment (⁠ $i$ ⁠), employment (⁠ $n$ ⁠), average wage (⁠ $w$ ⁠), wage bill (⁠ $w * n$ ⁠), and labor share (⁠ $\frac{w * n}{y}$ ⁠). The labor market variables $w$ ⁠, $w * n$ ⁠, and $\frac{w * n}{y}$ are for the private sector only, and any scaling is by private sector GDP; private sector data is available from 1948 to 2013. In the data $σ (y) = 3.98 %$ and $σ (Δ y) = 5.12 %$ ⁠; productivity shocks are calibrated such that all models (almost exactly) match these numbers. The data moments and bootstrapped standard errors are in Panels A and B; we also present the Cobb-Douglas frictionless model (C), the Cobb-Douglas sticky wage model (D), the CES frictionless model (E), and the CES sticky wage model (F).

Table 3

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Macroeconomic moments

Panel A: Data means
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.57	1.00	1.00	0.54
$c$	0.51	0.72	0.53	0.53	0.79	0.50
$i$	3.04	0.07	0.55	2.84	0.27	0.39
$n$	0.78	0.92	0.55	0.73	0.89	0.47
$w$	0.44	0.60	0.36	0.49	0.61	0.32
$w * n$	0.87	0.84	0.33	0.91	0.85	0.19
$\frac{w * n}{y}$	0.56	−0.49	0.36	0.53	−0.43	0.20

Panel A: Data means
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.57	1.00	1.00	0.54
$c$	0.51	0.72	0.53	0.53	0.79	0.50
$i$	3.04	0.07	0.55	2.84	0.27	0.39
$n$	0.78	0.92	0.55	0.73	0.89	0.47
$w$	0.44	0.60	0.36	0.49	0.61	0.32
$w * n$	0.87	0.84	0.33	0.91	0.85	0.19
$\frac{w * n}{y}$	0.56	−0.49	0.36	0.53	−0.43	0.20

Panel B: Data standard errors
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	0.00	0.00	0.11	0.00	0.00	0.10
$c$	0.05	0.09	0.12	0.05	0.08	0.12
$i$	0.26	0.24	0.14	0.45	0.26	0.12
$n$	0.04	0.03	0.11	0.04	0.03	0.11
$w$	0.04	0.08	0.12	0.04	0.09	0.12
$w * n$	0.04	0.06	0.11	0.06	0.04	0.14
$\frac{w * n}{y}$	0.09	0.07	0.15	0.08	0.10	0.20

Panel B: Data standard errors
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	0.00	0.00	0.11	0.00	0.00	0.10
$c$	0.05	0.09	0.12	0.05	0.08	0.12
$i$	0.26	0.24	0.14	0.45	0.26	0.12
$n$	0.04	0.03	0.11	0.04	0.03	0.11
$w$	0.04	0.08	0.12	0.04	0.09	0.12
$w * n$	0.04	0.06	0.11	0.06	0.04	0.14
$\frac{w * n}{y}$	0.09	0.07	0.15	0.08	0.10	0.20

Panel C: Cobb-Douglas, $μ = 0$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.43
$c$	0.54	0.92	0.51	0.65	0.91	0.64
$i$	3.06	0.95	0.46	2.79	0.90	0.28
$n$	0.84	0.56	0.20	0.82	0.55	−0.08
$w$	0.88	0.61	0.57	0.88	0.63	0.62
$w * n$	1.00	1.00	0.46	1.00	1.00	0.43
$\frac{w * n}{y}$	0.00	0.00	0.00	0.00	0.00	0.00

Panel C: Cobb-Douglas, $μ = 0$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.43
$c$	0.54	0.92	0.51	0.65	0.91	0.64
$i$	3.06	0.95	0.46	2.79	0.90	0.28
$n$	0.84	0.56	0.20	0.82	0.55	−0.08
$w$	0.88	0.61	0.57	0.88	0.63	0.62
$w * n$	1.00	1.00	0.46	1.00	1.00	0.43
$\frac{w * n}{y}$	0.00	0.00	0.00	0.00	0.00	0.00

Panel D: Cobb-Douglas, $μ = 0.9$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.47	1.00	1.00	0.43
$c$	0.60	0.95	0.53	0.68	0.94	0.63
$i$	2.95	0.95	0.43	2.70	0.91	0.25
$n$	0.83	0.55	0.20	0.81	0.54	−0.08
$w$	0.47	0.47	0.70	0.60	0.61	0.86
$w * n$	0.77	0.88	0.19	0.89	0.90	0.17
$\frac{w * n}{y}$	0.48	−0.66	0.45	0.43	−0.46	0.28

Panel D: Cobb-Douglas, $μ = 0.9$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.47	1.00	1.00	0.43
$c$	0.60	0.95	0.53	0.68	0.94	0.63
$i$	2.95	0.95	0.43	2.70	0.91	0.25
$n$	0.83	0.55	0.20	0.81	0.54	−0.08
$w$	0.47	0.47	0.70	0.60	0.61	0.86
$w * n$	0.77	0.88	0.19	0.89	0.90	0.17
$\frac{w * n}{y}$	0.48	−0.66	0.45	0.43	−0.46	0.28

Panel E: CES, $μ = 0$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.42
$c$	0.60	0.95	0.55	0.67	0.93	0.66
$i$	3.04	0.95	0.42	2.86	0.91	0.24
$n$	0.83	0.56	0.20	0.81	0.55	−0.08
$w$	0.83	0.35	0.51	0.85	0.40	0.53
$w * n$	0.76	0.99	0.49	0.80	0.99	0.52
$\frac{w * n}{y}$	0.27	−0.93	0.42	0.25	−0.85	0.25

Panel E: CES, $μ = 0$
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.42
$c$	0.60	0.95	0.55	0.67	0.93	0.66
$i$	3.04	0.95	0.42	2.86	0.91	0.24
$n$	0.83	0.56	0.20	0.81	0.55	−0.08
$w$	0.83	0.35	0.51	0.85	0.40	0.53
$w * n$	0.76	0.99	0.49	0.80	0.99	0.52
$\frac{w * n}{y}$	0.27	−0.93	0.42	0.25	−0.85	0.25

Panel F: CES, $μ = 0.9$ (baseline model)
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.42
$c$	0.62	0.95	0.55	0.68	0.94	0.64
$i$	3.00	0.95	0.41	2.83	0.91	0.23
$n$	0.83	0.56	0.20	0.82	0.55	−0.08
$w$	0.40	0.38	0.71	0.55	0.54	0.88
$w * n$	0.74	0.83	0.14	0.86	0.87	0.11
$\frac{w * n}{y}$	0.56	−0.68	0.49	0.50	−0.52	0.35

Panel F: CES, $μ = 0.9$ (baseline model)
$x$	$\frac{σ (x)}{σ (y)}$	$ρ (x, y)$	AC(⁠ $x$ ⁠)	$\frac{σ (Δ x)}{σ (Δ y)}$	$ρ (Δ x, Δ y)$	AC(⁠ $Δ x$ ⁠)
$y$	1.00	1.00	0.46	1.00	1.00	0.42
$c$	0.62	0.95	0.55	0.68	0.94	0.64
$i$	3.00	0.95	0.41	2.83	0.91	0.23
$n$	0.83	0.56	0.20	0.82	0.55	−0.08
$w$	0.40	0.38	0.71	0.55	0.54	0.88
$w * n$	0.74	0.83	0.14	0.86	0.87	0.11
$\frac{w * n}{y}$	0.56	−0.68	0.49	0.50	−0.52	0.35

In Table 3 we compare annual macroeconomic moments from the data (1929–2013) to several versions of our model. A variable denoted as $x$ is HP-filtered; a variable denoted as $Δ x$ is a growth rate. The variables are GDP (⁠ $y$ ⁠), consumption (⁠ $c$ ⁠), investment (⁠ $i$ ⁠), employment (⁠ $n$ ⁠), average wage (⁠ $w$ ⁠), wage bill (⁠ $w * n$ ⁠), and labor share (⁠ $\frac{w * n}{y}$ ⁠). The labor market variables $w$ ⁠, $w * n$ ⁠, and $\frac{w * n}{y}$ are for the private sector only, and any scaling is by private sector GDP; private sector data is available from 1948 to 2013. In the data $σ (y) = 3.98 %$ and $σ (Δ y) = 5.12 %$ ⁠; productivity shocks are calibrated such that all models (almost exactly) match these numbers. The data moments and bootstrapped standard errors are in Panels A and B; we also present the Cobb-Douglas frictionless model (C), the Cobb-Douglas sticky wage model (D), the CES frictionless model (E), and the CES sticky wage model (F).

Idiosyncratic productivity of firm $i$ is $Z_{t}^{i}$ ⁠. This also follows a three-state Markov chain. $Z_{t}^{i} \in {Z_{1}^{i}, Z_{2}^{i}, Z_{3}^{i}}$ ⁠, where $P r (Z_{t + 1}^{i} = Z_{j} | Z_{t}^{i} = Z_{k}) = π_{k j}^{Z} \geq 0$ ⁠. The parameters of this process are identical for all firms, but the process is independent across firms. Note that unlike aggregate productivity, the level of firm productivity is stationary. We choose parameters so that the quarterly autocorrelation and unconditional standard deviation of $Z_{t}^{i}$ are 0.9 and 0.14, respectively.⁷ We also set the mean of $Z^{i}$ to be 0.25, so that the average capital in our model is roughly the same as in a model solved annually with the same production function.⁸

Frequency of wage resetting: In standard models wages are reset once per period and employees receive the marginal product of labor as compensation. This corresponds to the $μ = 0$ case. However, wages are far too volatile in these models relative to the data. We choose the frequency of resetting to match roughly the volatility of wages in the data. This results in $μ = 0.9$ or an average resetting frequency of 10 quarters, and it implies realistic behavior for the model's labor share and total compensation.

Although $μ$ is chosen to match macroeconomic moments, its value seems consistent with micro evidence on wage rigidity. Given the importance of this parameter for our results, it is useful to add some discussion. We believe that this number is realistic and may even be on the low side (choosing a higher number would strengthen our results). For example, Rich and Tracy (2004) estimated that the majority of labor contracts last between 2 and 5 years, with a mean of 3 years, and they cite several major renewals (e.g., United Auto Workers and United Steel Workers) that are at the top of the range. Anecdotal evidence suggests that assistant professors, investment bankers, and corporate lawyers all wait even longer to be promoted.

Even if explicit contracts are written for a shorter period than our calibration (or not written at all), we believe that 10 quarters is a reasonable estimate of how long the real wage of many employees remains unchanged. Campbell and Kamlani (1997) conducted a survey of 184 firms and found that implicit contracts are an important explanation for wage rigidity of U.S. manufacturing workers, especially of blue-collar workers. For example, if the costs of replacing employees (for employers) and the costs of finding a new job (for employees) are high, the status quo will remain, keeping wages the same without an explicit contract. Another example are workers who receive small raises every year, keeping their real wage constant or growing slowly; indeed, Barwell and Schweitzer (2007) showed that this type of rigidity is quite common. Such workers do not experience major changes to their income until they are promoted, or let go, or move to another job. Hall (1982) estimated an average job duration of 8 years for American workers, Abraham and Farber (1987) estimated similar numbers just for nonunionized workers (presumably unionized workers have even longer durations).⁹

Financial leverage: We define the unlevered return as

R_{t + 1}^{K} = \frac{V_{t + 1}}{V_{t} - D_{t}}

⁠. However, real world firms are financed both by debt and by equity, with equity being the riskier, residual claim. To compare the model's equity return and dividend to their empirical counterparts, we must make an assumption about financial leverage. Note that the Modigliani and Miller (1958) propositions hold in our model; therefore, all assumptions about financial leverage are completely orthogonal to our model's solution. These assumptions only affect the way we report returns and dividends. We assume that financial debt follows the following process:

\begin{array}{l} B_{t} = ρ^{B} B_{t - 1} + (1 - ρ^{B}) B_{t}^{*} \\ B_{t}^{*} = λ (V_{t} - D_{t}), \end{array}

(11)

where the persistence in leverage

ρ^{B} = 0.9

and the target leverage ratio

λ = 0.37

are estimated from the data; in the Appendix we describe this in more detail. All reported returns are returns on equity, or equivalently, levered returns.

3. Results

3.1 The standard model

Next, we discuss the model where production is Cobb-Douglas and wages are reset once per quarter (⁠ $μ = 0$ ⁠). This is a standard real business cycle model with the addition of LRR. This model is most similar to those in Croce (2014) and Kaltenbrunner and Lochstoer (2010). We will refer to this model throughout the text as the standard, frictionless model. Like other RBC models, starting with Prescott (1986), this model can match most standard macroeconomic moments, as shown in Panel C of Table 3.¹⁰ Note that although the model is solved at a quarterly frequency, we aggregate model data and report annual moments.

One important exception to this model's success in reproducing macroeconomic moments is wage behavior (Rows 5, 6, and 7). In this model the wage is equal to the marginal product of labor, which leads to a constant labor share. The result is counterfactual; in the data, the volatility of labor share in the private sector is more than half of the volatility of output. Additionally, in the data, labor share is countercyclical. Labor share is volatile and countercyclical in the data because the wage is relatively smooth; its volatility is less than half that of output. Thus, after a positive (negative) shock to the economy, wages do not rise (fall) by as much as output, resulting in labor share falling (rising). In the standard model, the wage is nearly twice as volatile as in the data. Because labor income comprises such a large fraction of output, this flaw is quantitatively important and is responsible for many of this model's failures at matching financial data.

Unlike the first generation of RBC models, which did a poor job at matching financial moments (the well-known Equity Premium Puzzle), this model can produce a high Sharpe ratio through the LRR channel. Another common shortcoming of production models that were able to produce high equity volatilities has been that they also implied volatile risk-free rates, but this was also resolved by a high IES.

As shown in Tables 3 and 6, wherever the frictionless model is successful–consumption, investment, the risk-free rate, the Sharpe ratio–all other models we solve are successful, as well. However, in addition to not matching labor market variables, the standard frictionless model has several important flaws. Most evident among these is the volatility of equity returns, which is above 20% in the data but only 5.33% in the model (Row 3 in Panel A of Table 6). Because the volatility of equity is so low, the equity premium (which is the Sharpe ratio multiplied by the volatility of equity) is also quite low.

One way to increase the volatility of equity return is by increasing the capital adjustment cost; this channel has been used to explain the value premium, as well. Figure 3 shows the equity volatility in frictionless models, which differ from each other in adjustment cost only. For example, by raising the adjustment cost parameter $ν$ to 15 in the frictionless Cobb-Douglas model, it is possible to increase the unlevered equity volatility and value premium to 10.27% and 2.10%, respectively, and with financial leverage to 16.89% and 5.25%. However, this will lead to investment being less than half as volatile as in the data. We do not take this route here. Instead, adjustment costs are chosen so that both aggregate and firm-level investment volatility in our model match the data.

Another criticism of LRR models, as well as standard real business cycle models, is that they cannot endogenously produce variations in risk premia or equity volatility across the business cycle. As can be seen in Panels B and C of Table 6, in the data (first row) the expected equity premium and the equity volatility during bad times are higher than during good times. The standard model (second row) produces virtually no variation in either of these quantities.

Additionally, the standard model performs poorly when we consider cross-sectional asset pricing. The well-known value premium puzzle is that low market-to-book (value) stocks have higher average returns than high market-to-book (growth) stocks. However, the opposite is true in the standard model: growth stocks have higher average unlevered returns, whereas the value premium measured on levered returns is very small. These results are in Panel A of Table 6 and in Table 8.

Finally, the standard model performs poorly on several important accounting moments, as can be seen in Panels A and B of Table 5. Profit growth volatility is only 57% that of the data (71% for HP-filtered), and profits are too procyclical. Even more problematic, dividends in the data are procyclical, but in this model, they are highly countercyclical.

3.2 Infrequent resetting of wages

In this section we discuss a model that combines two key features: infrequent wage renegotiation and a calibrated CES production function (as opposed to Cobb-Douglas production in the standard model). We refer to this model as our baseline model. The addition of these two features improves on all of the problems with the standard model discussed previously.

Because wages are set infrequently, the average wage is no longer equal to the marginal product of labor but rather to a weighted average of past spot wages, which results in the average wage being smoother than the marginal product of labor (Row 5 in Panels D and F of Table 3). Smoother wages mean that the total wage bill (the average wage multiplied by total labor) is less volatile than output, and no longer perfectly correlated with output (Row 6). Finally, labor share (the total wage bill divided by output) is no longer constant, it is relatively volatile and, crucially, countercyclical as in the data (Row 7). We believe that most models with similar features will have results qualitatively similar to those discussed next. We view infrequent wage setting as one of several mechanisms responsible for the relatively smooth wages in the data.

The parameter controlling sticky wages is $μ$ ⁠, the fraction of employees who renegotiate their wage in a given period. When $μ > 0$ ⁠, there are two relevant wages. The average employee who did not renegotiate receives the average wage from the previous period ${\bar{w}}_{t - 1}^{i}$ ⁠. All resetting employees receive the current spot wage $w_{t}$ ⁠, which clears the labor market. The average wage is a weighted average between last period's average wage and this period's spot wage. Consistent with the empirical findings of Pissarides (2009), in our baseline model, the spot wage is more volatile than the average wage. Despite this, the average wage is still smoother than the marginal product of labor. Table 3 shows that unlike the standard model, models with wage rigidity (Panels D and F) not only do well at replicating the behavior of standard macroeconomic quantities, such as investment and consumption volatility, but also that of wages, the total wage bill, and labor share.

3.2.1 Elasticity of substitution between capital and labor

As discussed above, infrequent renegotiation of wages leads to wages being smoother than in a standard model, which improves the model's performance on various financial moments. A properly calibrated elasticity of substitution between capital and labor, even without other frictions, also makes wages smoother and will, therefore, have a similar effect on financial moments.

Traditionally, production-based models have used the Cobb-Douglas production function (⁠ $η = 0$ ⁠). This implies that the elasticity of substitution between labor and capital (⁠ $\frac{1}{1 - η}$ ⁠) is one. It also implies that (in the absence of additional frictions), the labor share is constant, which is counterfactual. Empirical estimates of this elasticity are below one (⁠ $η < 0$ ⁠). A lower elasticity strengthens complementarity between labor and capital, whereas a higher elasticity makes them substitutes.¹¹

Suppose that wages are frequently reset (⁠

μ = 0

⁠). In this case, the wage is still equal to the marginal product of labor, regardless of the elasticity of substitution between labor and capital. Ignoring the idiosyncratic component, the marginal product of labor is:

\frac{\partial Y_{t}}{\partial N_{t}} = ρ (1 - α) Y_{t}^{1 - η} Z_{t}^{ρ η} N_{t}^{ρ η - 1} .

(12)

To gain an understanding of how

η

affects the wage, we perform a simple experiment. Holding constant

N_{t}

and

K_{t}

while letting

Z_{t}

vary, we compute the volatility of output and of the marginal product of labor. In Table 4 we present a ratio of the two volatilities and their correlation. As the complementarity between capital and labor increases, the marginal product of labor (and equivalently the wage) become less volatile relative to output. Nevertheless, the marginal product of capital and output are nearly perfectly correlated even for extreme values of

η

⁠. Indeed, consistent with this logic, when we compare a Cobb-Douglas, frictionless model to a frictionless model with calibrated elasticity, wage growth volatility falls by 3% and wage bill growth volatility falls by around 20%; results are even stronger for HP-filtered volatility (Table 3, Panels C and E).

Table 4

Open in new tab

Labor capital elasticity and the wage

$η$	$\frac{σ (\frac{\partial Y}{\partial N})}{σ (Y)}$	$\frac{σ (l n (\frac{\partial Y}{\partial N}))}{σ (l n (Y))}$	$σ (\frac{\partial Y}{\partial N}, Y)$
−2.0	0.70	0.94	0.999
−1.0	0.63	0.90	0.999
−0.5	0.60	0.91	1.000
0.0	0.59	1.00	1.000
0.5	0.64	1.26	1.000
1.0	0.77	1.91	1.000
2.0	1.33	6.49	0.999

$η$	$\frac{σ (\frac{\partial Y}{\partial N})}{σ (Y)}$	$\frac{σ (l n (\frac{\partial Y}{\partial N}))}{σ (l n (Y))}$	$σ (\frac{\partial Y}{\partial N}, Y)$
−2.0	0.70	0.94	0.999
−1.0	0.63	0.90	0.999
−0.5	0.60	0.91	1.000
0.0	0.59	1.00	1.000
0.5	0.64	1.26	1.000
1.0	0.77	1.91	1.000
2.0	1.33	6.49	0.999

Here we set $N = 1$ ⁠; $K = 3$ ⁠; $α = 0.23$ ⁠; $ρ = 0.77$ ⁠; and $Z_{t} \sim N (1, 0.1)$ ⁠. We set $K$ roughly equal to average capital in our frictionless model. We set the volatility of $Z_{t}$ higher than in the model only to highlight the effect of $η$ on volatility of wages; qualitatively everything remains the same for lower volatility of $Z_{t}$ ⁠. All other parameters are identical to our calibration. We compute $Y_{t} = {(α K^{ρ} + (1 - α) * {(Z_{t} N)}^{ρ η})}^{\frac{1}{η}}$ and $\frac{\partial Y_{t}}{\partial N} = ρ (1 - α) Y^{1 - η} Z^{ρ η} N^{ρ η - 1}$ and report the ratios of their volatilities and their correlation. Note that $η = 0$ corresponds to Cobb-Douglas production.

Table 4

Open in new tab

Labor capital elasticity and the wage

$η$	$\frac{σ (\frac{\partial Y}{\partial N})}{σ (Y)}$	$\frac{σ (l n (\frac{\partial Y}{\partial N}))}{σ (l n (Y))}$	$σ (\frac{\partial Y}{\partial N}, Y)$
−2.0	0.70	0.94	0.999
−1.0	0.63	0.90	0.999
−0.5	0.60	0.91	1.000
0.0	0.59	1.00	1.000
0.5	0.64	1.26	1.000
1.0	0.77	1.91	1.000
2.0	1.33	6.49	0.999

$η$	$\frac{σ (\frac{\partial Y}{\partial N})}{σ (Y)}$	$\frac{σ (l n (\frac{\partial Y}{\partial N}))}{σ (l n (Y))}$	$σ (\frac{\partial Y}{\partial N}, Y)$
−2.0	0.70	0.94	0.999
−1.0	0.63	0.90	0.999
−0.5	0.60	0.91	1.000
0.0	0.59	1.00	1.000
0.5	0.64	1.26	1.000
1.0	0.77	1.91	1.000
2.0	1.33	6.49	0.999

Here we set $N = 1$ ⁠; $K = 3$ ⁠; $α = 0.23$ ⁠; $ρ = 0.77$ ⁠; and $Z_{t} \sim N (1, 0.1)$ ⁠. We set $K$ roughly equal to average capital in our frictionless model. We set the volatility of $Z_{t}$ higher than in the model only to highlight the effect of $η$ on volatility of wages; qualitatively everything remains the same for lower volatility of $Z_{t}$ ⁠. All other parameters are identical to our calibration. We compute $Y_{t} = {(α K^{ρ} + (1 - α) * {(Z_{t} N)}^{ρ η})}^{\frac{1}{η}}$ and $\frac{\partial Y_{t}}{\partial N} = ρ (1 - α) Y^{1 - η} Z^{ρ η} N^{ρ η - 1}$ and report the ratios of their volatilities and their correlation. Note that $η = 0$ corresponds to Cobb-Douglas production.

3.2.2 Unconditional financial moments

Profits and dividends are presented in Panels A and B of Table 5. As explained in the introduction, the standard model implies profits that are too smooth and dividends that are countercyclical because the wage, which is too volatile and too procyclical, acts as a hedge.

Table 5

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Accounting Moments

Panel A: Profit
	$\frac{σ (π)}{σ (y)}$	$ρ (π, y)$	AC(⁠ $π$ ⁠)	$\frac{σ (Δ π)}{σ (Δ y)}$	$ρ (Δ π, Δ y)$	AC(⁠ $Δ π$ ⁠)
Data	3.01	0.64	0.23	3.27	0.60	−0.07
	(0.47)	(0.06)	(0.11)	(0.55)	(0.07)	(0.16)
C-D $μ = 0$	1.03	0.98	0.46	1.01	0.97	0.40
C-D $μ = 0.9$	2.14	0.92	0.59	1.86	0.88	0.56
CES $μ = 0$	1.78	0.98	0.46	1.66	0.96	0.35
CES $μ = 0.9$	2.57	0.94	0.57	2.27	0.90	0.51

Panel A: Profit
	$\frac{σ (π)}{σ (y)}$	$ρ (π, y)$	AC(⁠ $π$ ⁠)	$\frac{σ (Δ π)}{σ (Δ y)}$	$ρ (Δ π, Δ y)$	AC(⁠ $Δ π$ ⁠)
Data	3.01	0.64	0.23	3.27	0.60	−0.07
	(0.47)	(0.06)	(0.11)	(0.55)	(0.07)	(0.16)
C-D $μ = 0$	1.03	0.98	0.46	1.01	0.97	0.40
C-D $μ = 0.9$	2.14	0.92	0.59	1.86	0.88	0.56
CES $μ = 0$	1.78	0.98	0.46	1.66	0.96	0.35
CES $μ = 0.9$	2.57	0.94	0.57	2.27	0.90	0.51

Panel B: Dividend
	$\frac{σ (d)}{σ (y)}$	$ρ (d, y)$	AC(⁠ $d$ ⁠)	$\frac{σ (\frac{Δ d}{y})}{σ (Δ y)}$	$ρ (\frac{Δ d}{y}, Δ y)$	AC(⁠ $\frac{Δ d}{y}$ ⁠)	$\frac{σ (d^{e})}{σ (y)}$	$ρ (d^{e}, y)$	AC(⁠ $π$ ⁠)	$\frac{σ (\frac{Δ d^{e}}{y})}{σ (Δ y)}$	$ρ (\frac{Δ d^{e}}{y}, Δ y)$	AC(⁠ $\frac{Δ d^{e}}{y}$ ⁠)
Data	4.08	0.26	0.34	0.58	0.38	0.17	9.55	0.39	0.30	0.49	0.40	0.09
	(0.86)	(0.10)	(0.18)	(0.10)	(0.10)	(0.19)	(2.17)	(0.08)	(0.17)	(0.09)	(0.10)	(0.21)
C-D $μ = 0$	6.80	−0.91	0.40	0.34	−0.75	0.18	3.57	−0.65	0.21	0.24	−0.16	0.10
C-D $μ = 0.9$	6.62	−0.01	0.14	0.38	0.01	−0.08	6.84	0.87	0.54	0.43	0.78	0.50
CES $μ = 0$	1.54	−0.80	0.29	0.07	−0.13	0.26	3.33	0.80	0.36	0.30	0.79	0.25
CES $μ = 0.9$	4.52	0.09	0.23	0.43	0.17	0.09	5.64	0.90	0.56	0.55	0.82	0.51

Panel B: Dividend
	$\frac{σ (d)}{σ (y)}$	$ρ (d, y)$	AC(⁠ $d$ ⁠)	$\frac{σ (\frac{Δ d}{y})}{σ (Δ y)}$	$ρ (\frac{Δ d}{y}, Δ y)$	AC(⁠ $\frac{Δ d}{y}$ ⁠)	$\frac{σ (d^{e})}{σ (y)}$	$ρ (d^{e}, y)$	AC(⁠ $π$ ⁠)	$\frac{σ (\frac{Δ d^{e}}{y})}{σ (Δ y)}$	$ρ (\frac{Δ d^{e}}{y}, Δ y)$	AC(⁠ $\frac{Δ d^{e}}{y}$ ⁠)
Data	4.08	0.26	0.34	0.58	0.38	0.17	9.55	0.39	0.30	0.49	0.40	0.09
	(0.86)	(0.10)	(0.18)	(0.10)	(0.10)	(0.19)	(2.17)	(0.08)	(0.17)	(0.09)	(0.10)	(0.21)
C-D $μ = 0$	6.80	−0.91	0.40	0.34	−0.75	0.18	3.57	−0.65	0.21	0.24	−0.16	0.10
C-D $μ = 0.9$	6.62	−0.01	0.14	0.38	0.01	−0.08	6.84	0.87	0.54	0.43	0.78	0.50
CES $μ = 0$	1.54	−0.80	0.29	0.07	−0.13	0.26	3.33	0.80	0.36	0.30	0.79	0.25
CES $μ = 0.9$	4.52	0.09	0.23	0.43	0.17	0.09	5.64	0.90	0.56	0.55	0.82	0.51

In Table 5 we compare annual accounting moments from the data to several versions of our model. The data on profits are for 1950–2012 (Compustat); the data on dividends are 1946–2013 (FOF). Bootstrapped standard errors are in parentheses. A variable denoted as $x$ is HP-filtered; a variable denoted as $Δ x$ is a growth rate. For dividends we consider the change in dividends normalized by output instead of the dividend growth rate because dividends are sometimes very close to zero, resulting in very large growth rates. We report both the dividend to equity (total dividends plus share repurchases), and the dividend paid by the corporate sector (dividend to equity plus net interest).

Table 5

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Accounting Moments

Panel A: Profit
	$\frac{σ (π)}{σ (y)}$	$ρ (π, y)$	AC(⁠ $π$ ⁠)	$\frac{σ (Δ π)}{σ (Δ y)}$	$ρ (Δ π, Δ y)$	AC(⁠ $Δ π$ ⁠)
Data	3.01	0.64	0.23	3.27	0.60	−0.07
	(0.47)	(0.06)	(0.11)	(0.55)	(0.07)	(0.16)
C-D $μ = 0$	1.03	0.98	0.46	1.01	0.97	0.40
C-D $μ = 0.9$	2.14	0.92	0.59	1.86	0.88	0.56
CES $μ = 0$	1.78	0.98	0.46	1.66	0.96	0.35
CES $μ = 0.9$	2.57	0.94	0.57	2.27	0.90	0.51

Panel A: Profit
	$\frac{σ (π)}{σ (y)}$	$ρ (π, y)$	AC(⁠ $π$ ⁠)	$\frac{σ (Δ π)}{σ (Δ y)}$	$ρ (Δ π, Δ y)$	AC(⁠ $Δ π$ ⁠)
Data	3.01	0.64	0.23	3.27	0.60	−0.07
	(0.47)	(0.06)	(0.11)	(0.55)	(0.07)	(0.16)
C-D $μ = 0$	1.03	0.98	0.46	1.01	0.97	0.40
C-D $μ = 0.9$	2.14	0.92	0.59	1.86	0.88	0.56
CES $μ = 0$	1.78	0.98	0.46	1.66	0.96	0.35
CES $μ = 0.9$	2.57	0.94	0.57	2.27	0.90	0.51

Panel B: Dividend
	$\frac{σ (d)}{σ (y)}$	$ρ (d, y)$	AC(⁠ $d$ ⁠)	$\frac{σ (\frac{Δ d}{y})}{σ (Δ y)}$	$ρ (\frac{Δ d}{y}, Δ y)$	AC(⁠ $\frac{Δ d}{y}$ ⁠)	$\frac{σ (d^{e})}{σ (y)}$	$ρ (d^{e}, y)$	AC(⁠ $π$ ⁠)	$\frac{σ (\frac{Δ d^{e}}{y})}{σ (Δ y)}$	$ρ (\frac{Δ d^{e}}{y}, Δ y)$	AC(⁠ $\frac{Δ d^{e}}{y}$ ⁠)
Data	4.08	0.26	0.34	0.58	0.38	0.17	9.55	0.39	0.30	0.49	0.40	0.09
	(0.86)	(0.10)	(0.18)	(0.10)	(0.10)	(0.19)	(2.17)	(0.08)	(0.17)	(0.09)	(0.10)	(0.21)
C-D $μ = 0$	6.80	−0.91	0.40	0.34	−0.75	0.18	3.57	−0.65	0.21	0.24	−0.16	0.10
C-D $μ = 0.9$	6.62	−0.01	0.14	0.38	0.01	−0.08	6.84	0.87	0.54	0.43	0.78	0.50
CES $μ = 0$	1.54	−0.80	0.29	0.07	−0.13	0.26	3.33	0.80	0.36	0.30	0.79	0.25
CES $μ = 0.9$	4.52	0.09	0.23	0.43	0.17	0.09	5.64	0.90	0.56	0.55	0.82	0.51

Panel B: Dividend
	$\frac{σ (d)}{σ (y)}$	$ρ (d, y)$	AC(⁠ $d$ ⁠)	$\frac{σ (\frac{Δ d}{y})}{σ (Δ y)}$	$ρ (\frac{Δ d}{y}, Δ y)$	AC(⁠ $\frac{Δ d}{y}$ ⁠)	$\frac{σ (d^{e})}{σ (y)}$	$ρ (d^{e}, y)$	AC(⁠ $π$ ⁠)	$\frac{σ (\frac{Δ d^{e}}{y})}{σ (Δ y)}$	$ρ (\frac{Δ d^{e}}{y}, Δ y)$	AC(⁠ $\frac{Δ d^{e}}{y}$ ⁠)
Data	4.08	0.26	0.34	0.58	0.38	0.17	9.55	0.39	0.30	0.49	0.40	0.09
	(0.86)	(0.10)	(0.18)	(0.10)	(0.10)	(0.19)	(2.17)	(0.08)	(0.17)	(0.09)	(0.10)	(0.21)
C-D $μ = 0$	6.80	−0.91	0.40	0.34	−0.75	0.18	3.57	−0.65	0.21	0.24	−0.16	0.10
C-D $μ = 0.9$	6.62	−0.01	0.14	0.38	0.01	−0.08	6.84	0.87	0.54	0.43	0.78	0.50
CES $μ = 0$	1.54	−0.80	0.29	0.07	−0.13	0.26	3.33	0.80	0.36	0.30	0.79	0.25
CES $μ = 0.9$	4.52	0.09	0.23	0.43	0.17	0.09	5.64	0.90	0.56	0.55	0.82	0.51

In Table 5 we compare annual accounting moments from the data to several versions of our model. The data on profits are for 1950–2012 (Compustat); the data on dividends are 1946–2013 (FOF). Bootstrapped standard errors are in parentheses. A variable denoted as $x$ is HP-filtered; a variable denoted as $Δ x$ is a growth rate. For dividends we consider the change in dividends normalized by output instead of the dividend growth rate because dividends are sometimes very close to zero, resulting in very large growth rates. We report both the dividend to equity (total dividends plus share repurchases), and the dividend paid by the corporate sector (dividend to equity plus net interest).

When the average wage and the total wage bill become smoother and less correlated with the marginal product of labor, profits become more volatile relative to the standard model. Volatile and procyclical profits lead to procyclical dividends. Complementarity between labor and capital alone (Row 4), and infrequent wage setting alone (Row 5) each increase the volatility of profits (Panel A) and make dividends more procyclical (Panel B). Our baseline model combines the two (Row 6), and here, profit and dividend behavior is quite similar to the data.

The relationship between wages, profits, and dividends is also evident in Figure 2, in which we plot the impulse responses to a positive aggregate productivity shock lasting for four consecutive quarters for a standard frictionless model (solid line), a CES calibrated frictionless model (dot-dashed line), and our baseline model (dashed line). In Panel A, wages respond much more slowly to a productivity shock in our baseline model, needing around 10 years to fully catch up to the standard frictionless model. However, because of smooth wages, the short-term jump in profits (Panel B) in our baseline model is twice that of the standard frictionless model. Finally, dividends (Panel C) actually fall in response to a positive productivity shock in the standard frictionless model, because profit is relatively small and smooth, whereas investment is procyclical. In our baseline model, dividends respond positively to a positive productivity shock. Note that a frictionless model with a calibrated CES improves on the standard model because of smoother wages caused by the complementarity of labor and capital; however, the improvement is far short of our baseline model.

Figure 2

Impulse responses

Figure 2 plots the impulse responses of the aggregate wage, aggregate profit, and aggregate dividend to a positive productivity shock. The positive productivity shock lasts for 1 year (4 consecutive quarters of high growth). The solid line represents the standard frictionless model (Cobb-Douglas production and no wage rigidity); the dot-dashed line represents the frictionless model with a calibrated CES; and the dashed line represents the calibrated CES production with wage rigidity. The x-axis is years after the shock, the y-axis is model quantity at any time relative to time zero.

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Figure 3

Comparative statics of the capital adjustment cost in the frictionless model

In Figure 3 we compare the volatility of the unlevered equity return (upper panel), and the unlevered value premium (lower panel) in a frictionless model as we vary capital adjustment costs. The x-axis plots the volatility of HP-filtered investment relative to HP-filtered output (this is monotonically related to the adjustment cost). We present both Cobb-Douglas technology (solid line) and a calibrated CES (dashed line).

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With smoother wages no longer being as strong of a hedge, equity volatility in our baseline model is closer to the data, as well. Panel A of Table 6 shows that our baseline model (Row 6) has an equity volatility of 13.15%, compared with 20.26% in the data; it is only 5.33% in the standard model. Because the equity volatility and Sharpe ratio are both closer to the data, it is not surprising that our baseline model has an equity premium of 5.57%, compared with only 2.27% in the standard frictionless model. Note that this is all possible without extremely high risk aversion or excessively smooth investment.

Table 6

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Financial Moments

Panel A: Unconditional financial moments
	$E [R^{F}]$	$σ (R^{F})$	$E [R^{E}]$	$σ (R^{E})$	SR	$E [R^{V} - R^{G}]$	$β^{V} - β^{G}$	$E [R_{U L}^{V} - R_{U L}^{G}]$	$β_{U L}^{V} - β_{U L}^{G}$
Data	0.42	3.58	7.68	20.26	0.36	5.37	0.29
	(0.38)	(0.43)	(2.18)	(1.60)	(0.12)	(2.09)	(0.09)
C-D $μ = 0$	1.03	0.90	3.30	5.33	0.46	0.25	0.11	−0.20	−0.09
C-D $μ = 0.9$	0.98	0.89	4.28	9.85	0.37	0.79	0.23	−0.15	−0.04
CES $μ = 0$	1.19	1.02	5.10	9.21	0.45	1.41	0.32	0.20	0.06
CES $μ = 0.9$	1.00	0.99	6.56	13.15	0.44	2.58	0.49	0.51	0.13

Panel A: Unconditional financial moments
	$E [R^{F}]$	$σ (R^{F})$	$E [R^{E}]$	$σ (R^{E})$	SR	$E [R^{V} - R^{G}]$	$β^{V} - β^{G}$	$E [R_{U L}^{V} - R_{U L}^{G}]$	$β_{U L}^{V} - β_{U L}^{G}$
Data	0.42	3.58	7.68	20.26	0.36	5.37	0.29
	(0.38)	(0.43)	(2.18)	(1.60)	(0.12)	(2.09)	(0.09)
C-D $μ = 0$	1.03	0.90	3.30	5.33	0.46	0.25	0.11	−0.20	−0.09
C-D $μ = 0.9$	0.98	0.89	4.28	9.85	0.37	0.79	0.23	−0.15	−0.04
CES $μ = 0$	1.19	1.02	5.10	9.21	0.45	1.41	0.32	0.20	0.06
CES $μ = 0.9$	1.00	0.99	6.56	13.15	0.44	2.58	0.49	0.51	0.13

Panel B: Conditional financial moments
	$E [R^{e x} \| R]$	$E [R^{e x} \| E]$	$σ (R^{e x} \| R)$	$σ (R^{e x} \| E)$	$E [R_{U L}^{e x} \| R]$	$E [R_{U L}^{e x} \| E]$	$σ (R_{U L}^{e x} \| R)$	$σ (R_{U L}^{e x} \| E)$
Data	10.57	8.36	20.19	18.33
	(1.33)	(1.37)	(1.88)	(0.87)
C-D $μ = 0$	2.22	2.34	4.60	4.63	1.43	1.57	2.61	2.69
C-D $μ = 0.9$	3.39	3.25	8.69	8.19	1.97	1.98	5.07	4.94
CES $μ = 0$	3.91	3.82	8.41	7.46	2.27	2.30	4.75	4.42
CES $μ = 0.9$	5.68	5.26	12.65	10.60	3.29	3.14	7.12	6.29

Panel B: Conditional financial moments
	$E [R^{e x} \| R]$	$E [R^{e x} \| E]$	$σ (R^{e x} \| R)$	$σ (R^{e x} \| E)$	$E [R_{U L}^{e x} \| R]$	$E [R_{U L}^{e x} \| E]$	$σ (R_{U L}^{e x} \| R)$	$σ (R_{U L}^{e x} \| E)$
Data	10.57	8.36	20.19	18.33
	(1.33)	(1.37)	(1.88)	(0.87)
C-D $μ = 0$	2.22	2.34	4.60	4.63	1.43	1.57	2.61	2.69
C-D $μ = 0.9$	3.39	3.25	8.69	8.19	1.97	1.98	5.07	4.94
CES $μ = 0$	3.91	3.82	8.41	7.46	2.27	2.30	4.75	4.42
CES $μ = 0.9$	5.68	5.26	12.65	10.60	3.29	3.14	7.12	6.29

Panel C: Predictive regression
	Return					Volatility
	3	12	60	120	3	12	60	120
Data	−0.15	1.80	1.09	0.79	-	1.51	1.11	0.40
	(0.31)	(2.48)	(2.94)	(2.31)	-	(1.72)	(2.49)	(1.09)
C-D $μ = 0$	−2.16	−5.51	−2.76	0.26	-	−3.26	−0.48	−4.84
C-D $μ = 0.9$	−0.79	−2.09	−0.85	0.94	-	−0.21	1.95	−1.87
CES $μ = 0$	−0.91	0.09	0.33	1.89	-	1.14	3.84	−1.11
CES $μ = 0.9$	0.30	0.64	0.83	1.85	-	2.49	4.95	0.15

Panel C: Predictive regression
	Return					Volatility
	3	12	60	120	3	12	60	120
Data	−0.15	1.80	1.09	0.79	-	1.51	1.11	0.40
	(0.31)	(2.48)	(2.94)	(2.31)	-	(1.72)	(2.49)	(1.09)
C-D $μ = 0$	−2.16	−5.51	−2.76	0.26	-	−3.26	−0.48	−4.84
C-D $μ = 0.9$	−0.79	−2.09	−0.85	0.94	-	−0.21	1.95	−1.87
CES $μ = 0$	−0.91	0.09	0.33	1.89	-	1.14	3.84	−1.11
CES $μ = 0.9$	0.30	0.64	0.83	1.85	-	2.49	4.95	0.15

In Table 6 we compare annual financial moments from the data to several versions of our model. The data on returns are 1929–2013 from Ken French's Web site. In Panel A, the value premium is defined as the difference in average returns between firms in the top quintile and bottom quintile of a book-to-market sorting. For simulated data, we also present the unlevered value premium, which is computed for the entire firm. In Panel B, we present the expected excess (both levered- and unlevered-) return and volatility over the next 5 years, conditional on being in a recession or expansion today (the definition of recessions is described in the main text). In Panel C, we present results from long-horizon regressions of equity returns (3, 12, 60, and 120 quarters) on the book-to-market ratio. For the data, in parentheses we present one standard deviation in Panels A and B, and t-statistics in Panel C.

Table 6

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Financial Moments

Panel A: Unconditional financial moments
	$E [R^{F}]$	$σ (R^{F})$	$E [R^{E}]$	$σ (R^{E})$	SR	$E [R^{V} - R^{G}]$	$β^{V} - β^{G}$	$E [R_{U L}^{V} - R_{U L}^{G}]$	$β_{U L}^{V} - β_{U L}^{G}$
Data	0.42	3.58	7.68	20.26	0.36	5.37	0.29
	(0.38)	(0.43)	(2.18)	(1.60)	(0.12)	(2.09)	(0.09)
C-D $μ = 0$	1.03	0.90	3.30	5.33	0.46	0.25	0.11	−0.20	−0.09
C-D $μ = 0.9$	0.98	0.89	4.28	9.85	0.37	0.79	0.23	−0.15	−0.04
CES $μ = 0$	1.19	1.02	5.10	9.21	0.45	1.41	0.32	0.20	0.06
CES $μ = 0.9$	1.00	0.99	6.56	13.15	0.44	2.58	0.49	0.51	0.13

Panel A: Unconditional financial moments
	$E [R^{F}]$	$σ (R^{F})$	$E [R^{E}]$	$σ (R^{E})$	SR	$E [R^{V} - R^{G}]$	$β^{V} - β^{G}$	$E [R_{U L}^{V} - R_{U L}^{G}]$	$β_{U L}^{V} - β_{U L}^{G}$
Data	0.42	3.58	7.68	20.26	0.36	5.37	0.29
	(0.38)	(0.43)	(2.18)	(1.60)	(0.12)	(2.09)	(0.09)
C-D $μ = 0$	1.03	0.90	3.30	5.33	0.46	0.25	0.11	−0.20	−0.09
C-D $μ = 0.9$	0.98	0.89	4.28	9.85	0.37	0.79	0.23	−0.15	−0.04
CES $μ = 0$	1.19	1.02	5.10	9.21	0.45	1.41	0.32	0.20	0.06
CES $μ = 0.9$	1.00	0.99	6.56	13.15	0.44	2.58	0.49	0.51	0.13

Panel B: Conditional financial moments
	$E [R^{e x} \| R]$	$E [R^{e x} \| E]$	$σ (R^{e x} \| R)$	$σ (R^{e x} \| E)$	$E [R_{U L}^{e x} \| R]$	$E [R_{U L}^{e x} \| E]$	$σ (R_{U L}^{e x} \| R)$	$σ (R_{U L}^{e x} \| E)$
Data	10.57	8.36	20.19	18.33
	(1.33)	(1.37)	(1.88)	(0.87)
C-D $μ = 0$	2.22	2.34	4.60	4.63	1.43	1.57	2.61	2.69
C-D $μ = 0.9$	3.39	3.25	8.69	8.19	1.97	1.98	5.07	4.94
CES $μ = 0$	3.91	3.82	8.41	7.46	2.27	2.30	4.75	4.42
CES $μ = 0.9$	5.68	5.26	12.65	10.60	3.29	3.14	7.12	6.29

Panel B: Conditional financial moments
	$E [R^{e x} \| R]$	$E [R^{e x} \| E]$	$σ (R^{e x} \| R)$	$σ (R^{e x} \| E)$	$E [R_{U L}^{e x} \| R]$	$E [R_{U L}^{e x} \| E]$	$σ (R_{U L}^{e x} \| R)$	$σ (R_{U L}^{e x} \| E)$
Data	10.57	8.36	20.19	18.33
	(1.33)	(1.37)	(1.88)	(0.87)
C-D $μ = 0$	2.22	2.34	4.60	4.63	1.43	1.57	2.61	2.69
C-D $μ = 0.9$	3.39	3.25	8.69	8.19	1.97	1.98	5.07	4.94
CES $μ = 0$	3.91	3.82	8.41	7.46	2.27	2.30	4.75	4.42
CES $μ = 0.9$	5.68	5.26	12.65	10.60	3.29	3.14	7.12	6.29

Panel C: Predictive regression
	Return					Volatility
	3	12	60	120	3	12	60	120
Data	−0.15	1.80	1.09	0.79	-	1.51	1.11	0.40
	(0.31)	(2.48)	(2.94)	(2.31)	-	(1.72)	(2.49)	(1.09)
C-D $μ = 0$	−2.16	−5.51	−2.76	0.26	-	−3.26	−0.48	−4.84
C-D $μ = 0.9$	−0.79	−2.09	−0.85	0.94	-	−0.21	1.95	−1.87
CES $μ = 0$	−0.91	0.09	0.33	1.89	-	1.14	3.84	−1.11
CES $μ = 0.9$	0.30	0.64	0.83	1.85	-	2.49	4.95	0.15

Panel C: Predictive regression
	Return					Volatility
	3	12	60	120	3	12	60	120
Data	−0.15	1.80	1.09	0.79	-	1.51	1.11	0.40
	(0.31)	(2.48)	(2.94)	(2.31)	-	(1.72)	(2.49)	(1.09)
C-D $μ = 0$	−2.16	−5.51	−2.76	0.26	-	−3.26	−0.48	−4.84
C-D $μ = 0.9$	−0.79	−2.09	−0.85	0.94	-	−0.21	1.95	−1.87
CES $μ = 0$	−0.91	0.09	0.33	1.89	-	1.14	3.84	−1.11
CES $μ = 0.9$	0.30	0.64	0.83	1.85	-	2.49	4.95	0.15

In Table 6 we compare annual financial moments from the data to several versions of our model. The data on returns are 1929–2013 from Ken French's Web site. In Panel A, the value premium is defined as the difference in average returns between firms in the top quintile and bottom quintile of a book-to-market sorting. For simulated data, we also present the unlevered value premium, which is computed for the entire firm. In Panel B, we present the expected excess (both levered- and unlevered-) return and volatility over the next 5 years, conditional on being in a recession or expansion today (the definition of recessions is described in the main text). In Panel C, we present results from long-horizon regressions of equity returns (3, 12, 60, and 120 quarters) on the book-to-market ratio. For the data, in parentheses we present one standard deviation in Panels A and B, and t-statistics in Panel C.

Our baseline model differs from the standard model in two ways: the CES parameter and infrequent renegotiation. It is useful to explore each of these in isolation: infrequent renegotiation alone raises equity volatility from 5.33% to 9.85%, whereas a CES production function alone raises it to 9.21%. This is because smoother wages lead to more volatile profits and to more procyclical dividends.

Figure 4 displays the quantitative effect on equity volatility for different degrees of wage stickiness. The solid line is for models with Cobb-Douglas production, whereas the dashed line is for calibrated CES production. In our baseline model, renegotiations happen once every 10 quarters, on average; this corresponds to $μ = 0.9$ ⁠. However, the effects on equity volatility are significant even with shorter contract length.

Figure 4

Comparative statics of $μ$

In Figure 4 we compare the volatility of the unlevered equity return as we vary wage rigidity (⁠ $μ$ ⁠) across different models. All models have capital adjustment costs calibrated to match investment volatility and fixed costs calibrated to match the average market-to-book ratio. On the upper panel, we plot volatility (y-axis) against average job duration $\frac{1}{1 - μ}$ ⁠, on the lower panel we plot volatility (y-axis) against the probability of remaining in a job $μ$ ⁠. We present both Cobb-Douglas technology (solid line), and a calibrated CES (dashed line).

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Comparing the frictionless model with a calibrated CES, to the sticky-wage model with a calibrated CES, also makes evident a subtle but important point. Note that the volatility of the total wage bill in both models is similar to the data (Row 6, Panels A, E, F of Table 3). However, as discussed earlier, the sticky-wage model produces much more equity volatility. This difference highlights the importance of not only the volatility of the total wage bill but also its correlation with output: in the frictionless model, the two are perfectly correlated, but in the sticky-wage model, the correlation is less than perfect, as in the data. Using the same logic to go from wages to profits and from profits to dividends, the frictionless, calibrated CES model performs better than the standard frictionless model does, but its profits are still far too smooth, and its dividends are still countercyclical. This is because despite having the right amount of wage bill volatility, the frictionless model still has too much correlation. The lower correlation between output and the total wage bill in the sticky-wage model allows for more procyclical dividends.

3.2.3 Decomposing the quantitative effect

Although our focus is on wage rigidity, relative to previous models in the literature, such as Kaltenbrunner and Lochstoer (2010), we have also added decreasing returns to scale, fixed costs, CES production, and idiosyncratic productivity risk. Thus, what we refer to as our standard model, is still not quite standard. In this section we decompose the total rise in equity volatility into various components, and in Table 7, we present the results.

Table 7

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Decomposing the effect

Column	1	2	3	4	5	6	7	8	9
Production	C-D	C-D	C-D	CES	C-D	CES	CES	C-D	CES
Return to scale (⁠ $ρ$ ⁠)	1.00	0.77	0.77	0.77	0.77	0.77	0.77	0.90	0.90
Fixed cost	No	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Rigidity	No	No	No	No	Yes	Yes	Yes	No	Yes
Idiosyncratic risk	No	No	No	No	No	No	Yes	No	No
$σ (R^{K})$	0.47	2.33	3.32	5.38	5.18	7.18	7.48	1.97	6.94

Column	1	2	3	4	5	6	7	8	9
Production	C-D	C-D	C-D	CES	C-D	CES	CES	C-D	CES
Return to scale (⁠ $ρ$ ⁠)	1.00	0.77	0.77	0.77	0.77	0.77	0.77	0.90	0.90
Fixed cost	No	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Rigidity	No	No	No	No	Yes	Yes	Yes	No	Yes
Idiosyncratic risk	No	No	No	No	No	No	Yes	No	No
$σ (R^{K})$	0.47	2.33	3.32	5.38	5.18	7.18	7.48	1.97	6.94

In Table 7 we present the unlevered equity volatility from several models.

Table 7

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Decomposing the effect

Column	1	2	3	4	5	6	7	8	9
Production	C-D	C-D	C-D	CES	C-D	CES	CES	C-D	CES
Return to scale (⁠ $ρ$ ⁠)	1.00	0.77	0.77	0.77	0.77	0.77	0.77	0.90	0.90
Fixed cost	No	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Rigidity	No	No	No	No	Yes	Yes	Yes	No	Yes
Idiosyncratic risk	No	No	No	No	No	No	Yes	No	No
$σ (R^{K})$	0.47	2.33	3.32	5.38	5.18	7.18	7.48	1.97	6.94

Column	1	2	3	4	5	6	7	8	9
Production	C-D	C-D	C-D	CES	C-D	CES	CES	C-D	CES
Return to scale (⁠ $ρ$ ⁠)	1.00	0.77	0.77	0.77	0.77	0.77	0.77	0.90	0.90
Fixed cost	No	No	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Rigidity	No	No	No	No	Yes	Yes	Yes	No	Yes
Idiosyncratic risk	No	No	No	No	No	No	Yes	No	No
$σ (R^{K})$	0.47	2.33	3.32	5.38	5.18	7.18	7.48	1.97	6.94

In Table 7 we present the unlevered equity volatility from several models.

A useful starting point is Model Long-Run Risk II in Kaltenbrunner and Lochstoer (2010), which is the production economy version of Bansal and Yaron (2004). This model is similar to what we refer to as the standard model in our paper, the quantitatively important differences being constant returns to scale, compared with decreasing returns to scale (⁠ $ρ = 0.77$ ⁠) in our model, and fixed costs in our model. The model in Kaltenbrunner and Lochstoer (2010) has an unlevered equity return volatility of 0.66% per year. For comparison, we solve our model with constant returns to scale (⁠ $ρ = 1$ ⁠), Cobb-Douglas production (⁠ $α = 0.36$ ⁠), no wage rigidity (⁠ $μ = 0$ ⁠), no fixed costs (⁠ $f = 0$ ⁠), and no idiosyncratic productivity risk (⁠ $σ (Z^{i}) = 0$ ⁠). Similar to Kaltenbrunner and Lochstoer (2010), the unlevered equity return volatility is 0.47%.¹² This model is shown in the first column of Table 7.

Next, we set $ρ = 0.77$ to decrease the return to scale, and the volatility of unlevered equity rises to 2.33% in Column 2. Next, we also increase the fixed cost to match the average market-to-book-ratio; this is the model we refer to as the standard model (for example, Panel C of Table 3). In this model (Column 3), the volatility of unlevered equity is 3.32% (financial leverage raises this number to 5.33%, as can be seen in Table 6).

The fixed costs in the production process also effectively constitute a form of operating leverage. However, its quantitative effect on equity volatility and on the equity premium is limited. This can be seen by comparing the standard frictionless model to our baseline model (Table 6). All of the models contain fixed costs; however, the standard model has a small equity volatility and a negative (unlevered) value premium. What matters is not just the presence of operating leverage, but that operating leverage is less procyclical than output. Leverage resulting from wage rigidities achieves this, while fixed costs alone do not.

Next, one at a time, we add CES production (⁠ $η = - 1$ ⁠, Column 4) or wage rigidity (⁠ $μ = 0.9$ ⁠, Column 5); these raise the unlevered equity volatility to 5.38% and 5.18%, respectively. Recall that as we explained earlier, although CES production is not actually a friction, the channel through which it increases equity volatility is very similar to actual wage rigidity. The two channels together lead to an unlevered equity volatility of 7.18% (Column 6). Note that this model is identical to our baseline model, except that there is no idiosyncratic risk. Finally, in Column 7, we present our baseline model, which has an unlevered equity volatility of 7.48%.

Note that without idiosyncratic productivity shocks, the First Welfare Theorem holds and wage rigidity has no effect on macroeconomic quantities. This is because with a representative firm, market clearing will imply that the firm always hires $N_{t}$ workers, and its marginal product of capital is unaffected; labor supply being exogenous is crucial here. Therefore, the Euler equation for investment is unaffected. Wage rigidity affects equity returns only through a shifting of risk from wages to dividends.

The First Welfare Theorem continues to hold for small idiosyncratic shocks, but no longer holds for larger shocks: this can be seen in Table 2, which shows that models with higher $μ$ require higher adjustment costs to match the volatility of investment in the data. This happens because when shocks are large enough, some low-productivity firms may have negative value. If we allowed these firms to continue operating and to maximize their value, then the First Welfare Theorem would still hold. However, consistent with limited liability, we assume that a firm's value cannot be negative. If, given a firm's state, its value is implied to be negative at $t$ ⁠, then we assume the following: the firm shuts down for the current period; it keeps its capital but not its labor so that it produces nothing at $t$ ⁠; its equity value at $t$ is zero, implying a -100% return to $t - 1$ equity holders; the firm reopens at $t + 1$ with the shuttered firm's capital (minus depreciation) but no labor obligations; the firm's ownership passes to $t - 1$ creditors (this determines the return on debt); the firm starts $t + 1$ with no debt. Because firms are forward looking, they are aware that such a shut down is possible and they choose their policy accordingly. In our baseline model, the fraction of shut-down firms is very small (0.05% in an average quarter, and 0.2% in the maximum quarter of our simulation). As shown in Table 7, the difference in equity volatility between the baseline model and a model with no shocks is not large (7.18% versus 7.48% unlevered equity volatility). Thus, although default may be a realistic feature in the data and may help further improve the model's performance, our key results do not require large idiosyncratic shocks or default.

Although the paper focuses on labor market frictions, it may appear that much of the effect is coming from decreasing returns to scale. This is not the quite true. Indeed, in a model with no wage rigidity, decreasing returns to scale (lower $ρ$ ⁠) increases the volatility of equity returns. However, as will be explained next, the effects of wage rigidity (higher $μ$ ⁠) are actually stronger in a model with constant return to scale. In other words, when comparing models, $\frac{\partial σ (R^{E})}{\partial ρ^{- 1}} > 0$ and $\frac{\partial σ (R^{E})}{\partial μ} > 0$ ⁠, however, $\frac{\partial σ (R^{E})}{\partial ρ^{- 1} \partial μ} < 0$ ⁠. This can be seen in Columns 8 and 9 of Table 7, where we solve a model that is closer to constant returns to scale¹³ with and without wage rigidity. Relative to the $ρ = 0.77$ case, the equity volatility in the frictionless model falls by 41% (compare Columns 8 and 3), whereas in the rigidity case, only by 7% (compare Columns 9 and 7).

The reason the effect of rigidity is stronger when returns to scale are closer to constant is that with constant returns to scale, even a small increase in expected profitability induces large increases in scale and leads to large increases in equity values. Compared with a frictionless model, rigid wages lead to larger increases in expected profitability after a positive productivity shock. Although this is true for both constant and decreasing returns to scale, decreasing returns to scale implies that the firm is unable to take full advantage of changes in profitability associated with wage rigidity because being far away from optimal capital stock is costly.

3.2.4 Conditional asset pricing moments

It is well-known that financial moments exhibit conditional variation. The volatility of equity returns tends to be autocorrelated; it is also higher in recessions than expansions. For example, in our sample, volatility was 20.19% following periods of low GDP growth (bottom 33%) and 18.33% following periods of high GDP growth (top 33%). Average excess equity returns are also higher during recessions (10.57%) than during expansions (8.36%). An extensive body of literature has documented that expected returns are predictable, with business cycle-related variables such as the term spread, the default spread, the dividend yield, and the consumption wealth ratio all having predictive power.

In our model, the effect of sticky wages is much like that of financial or operating leverage: the equity return is the residual after other factors are paid. If these other factor payments are fixed or slow to adjust, the equity return is riskier. Labor leverage makes the equity return riskier on average; however, labor leverage is not constant through the business cycle. Because wages adjust slowly, they are relatively high compared with output during bad times, making bad times especially risky. Panel B of Table 6 compares equity volatility and equity premia during bad times and good times. Similar to the data, the model produces higher expected volatility and return during bad times; this is not just due to leverage, because the same pattern exists in unlevered returns. Panel C of Table 6 regresses long-horizon equity return and volatility on the book-to-market ratio. There is a positive relationship in both the data and our model, but a negative relationship in the frictionless model. When wage obligations are high, firms' value is low but expected returns are high because of operating leverage caused by the high wages.

In a companion paper, Favilukis and Lin (2015) explored additional empirical implications of wage stickiness. They showed that consistent with the intuition of this paper, aggregate wage growth forecasts aggregate stock returns negatively. This is because after a negative (positive) productivity shock, output falls (rises) by more than the wage, leading to a rise (fall) in leverage because of labor. Traditional variables known to forecast long-horizon returns (dividend yield, term spread, default spread, and the consumption wealth ratio) do not subsume wage growth when considered simultaneously. Reversing the regression also showcases the importance of wage stickiness: stock returns (which are forward-looking) positively forecast both GDP growth and wage growth; however, they forecast wage growth minus GDP growth negatively, suggesting that wages adjust sluggishly to positive economic news. Finally, they show that wage growth forecasts stock returns negatively at the U.S. state and the industry level, and that states and industries that are more rigid (defined by the inverse of past wage growth volatility) have returns that are more forecastable. These empirical findings are all supportive of the mechanism in our model.

3.2.5 Cross-sectional asset pricing moments

The model also has implications for the cross-section. We split firms into market-to-book quintiles in the model, using the ex-dividend value (⁠ $V_{t}^{i} - D_{t}^{i}$ from Equation 7) as the market value and $K_{t}^{i}$ as the book value similar to Fama and French (1992). Low book-to-market firms are referred to as growth, and high book-to-market firms are value. Compared with other firms, in both the model and data, growth (value) firms invest and hire more (less).

The value premium puzzle is that value firms have higher average returns than those of growth firms; the difference in returns is the value premium. Just as operating leverage induced by sticky-wages varies through time and leads to conditional variation in expected return, operating leverage induced by sticky-wages also varies cross-sectionally and leads to cross-sectional variation in expected return. We present several statistics for each book-to-market portfolio in Panels A (data) and B (baseline model) of Table 8.¹⁴

Table 8

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The value premium

Panel A: Data
	B/M	$\frac{P r o f i t}{l a b o r e x p e n s e}$	$\frac{D e b t}{D e b t + e q u i t y}$	$α$	$β$	$E [R^{i} - R^{f}]$
Growth	0.20	0.70	0.19	−0.55	0.97	6.51
2	0.47	0.62	0.33	0.28	0.93	7.01
3	0.70	0.61	0.42	1.28	1.02	8.69
4	0.97	0.54	0.49	1.91	1.13	10.10
Value	2.85	0.45	0.78	2.73	1.26	11.88
V-G	2.65	−0.25	0.59	3.28	0.29	5.37

Panel A: Data
	B/M	$\frac{P r o f i t}{l a b o r e x p e n s e}$	$\frac{D e b t}{D e b t + e q u i t y}$	$α$	$β$	$E [R^{i} - R^{f}]$
Growth	0.20	0.70	0.19	−0.55	0.97	6.51
2	0.47	0.62	0.33	0.28	0.93	7.01
3	0.70	0.61	0.42	1.28	1.02	8.69
4	0.97	0.54	0.49	1.91	1.13	10.10
Value	2.85	0.45	0.78	2.73	1.26	11.88
V-G	2.65	−0.25	0.59	3.28	0.29	5.37

Panel B: Model
	B/M	$\frac{P r o f i t}{l a b o r e x p e n s e}$	$\frac{D e b t}{D e b t + e q u i t y}$	$α$	$β$	$E [R^{i} - R^{f}]$	$E [R^{i, U L} - R^{f}]$
Growth	0.32	0.61	0.35	0.09	0.88	4.98	3.10
2	0.49	0.49	0.37	0.13	0.95	5.41	3.34
3	0.61	0.44	0.41	0.14	1.01	5.78	3.36
4	0.79	0.41	0.49	0.06	1.18	6.62	3.38
Value	1.79	0.09	0.53	−0.06	1.37	7.56	3.61
V-G	1.47	−0.52	0.18	−0.14	0.49	2.58	0.51

Panel B: Model
	B/M	$\frac{P r o f i t}{l a b o r e x p e n s e}$	$\frac{D e b t}{D e b t + e q u i t y}$	$α$	$β$	$E [R^{i} - R^{f}]$	$E [R^{i, U L} - R^{f}]$
Growth	0.32	0.61	0.35	0.09	0.88	4.98	3.10
2	0.49	0.49	0.37	0.13	0.95	5.41	3.34
3	0.61	0.44	0.41	0.14	1.01	5.78	3.36
4	0.79	0.41	0.49	0.06	1.18	6.62	3.38
Value	1.79	0.09	0.53	−0.06	1.37	7.56	3.61
V-G	1.47	−0.52	0.18	−0.14	0.49	2.58	0.51

In Table 8 we present average book to market of equity, profit-to-labor-expenses ratio, financial leverage, and mean excess return for five portfolios sorted on book to market. The top panel contains data, and the bottom panel shows results from our baseline model, with wage rigidity (⁠ $μ = 0.9$ ⁠) and a calibrated CES (⁠ $\frac{1}{1 - η} = 0.5$ ⁠). For simulated data, we also present the unlevered value premium, which is computed for the entire firm. Profits are defined as Revenue-COGS-SGA from Compustat; labor expenses are employees from Compustat multiplied by the average wage for the firm's industry from NIPA.

Table 8

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The value premium

Panel A: Data
	B/M	$\frac{P r o f i t}{l a b o r e x p e n s e}$	$\frac{D e b t}{D e b t + e q u i t y}$	$α$	$β$	$E [R^{i} - R^{f}]$
Growth	0.20	0.70	0.19	−0.55	0.97	6.51
2	0.47	0.62	0.33	0.28	0.93	7.01
3	0.70	0.61	0.42	1.28	1.02	8.69
4	0.97	0.54	0.49	1.91	1.13	10.10
Value	2.85	0.45	0.78	2.73	1.26	11.88
V-G	2.65	−0.25	0.59	3.28	0.29	5.37

Panel A: Data
	B/M	$\frac{P r o f i t}{l a b o r e x p e n s e}$	$\frac{D e b t}{D e b t + e q u i t y}$	$α$	$β$	$E [R^{i} - R^{f}]$
Growth	0.20	0.70	0.19	−0.55	0.97	6.51
2	0.47	0.62	0.33	0.28	0.93	7.01
3	0.70	0.61	0.42	1.28	1.02	8.69
4	0.97	0.54	0.49	1.91	1.13	10.10
Value	2.85	0.45	0.78	2.73	1.26	11.88
V-G	2.65	−0.25	0.59	3.28	0.29	5.37

Panel B: Model
	B/M	$\frac{P r o f i t}{l a b o r e x p e n s e}$	$\frac{D e b t}{D e b t + e q u i t y}$	$α$	$β$	$E [R^{i} - R^{f}]$	$E [R^{i, U L} - R^{f}]$
Growth	0.32	0.61	0.35	0.09	0.88	4.98	3.10
2	0.49	0.49	0.37	0.13	0.95	5.41	3.34
3	0.61	0.44	0.41	0.14	1.01	5.78	3.36
4	0.79	0.41	0.49	0.06	1.18	6.62	3.38
Value	1.79	0.09	0.53	−0.06	1.37	7.56	3.61
V-G	1.47	−0.52	0.18	−0.14	0.49	2.58	0.51

Panel B: Model
	B/M	$\frac{P r o f i t}{l a b o r e x p e n s e}$	$\frac{D e b t}{D e b t + e q u i t y}$	$α$	$β$	$E [R^{i} - R^{f}]$	$E [R^{i, U L} - R^{f}]$
Growth	0.32	0.61	0.35	0.09	0.88	4.98	3.10
2	0.49	0.49	0.37	0.13	0.95	5.41	3.34
3	0.61	0.44	0.41	0.14	1.01	5.78	3.36
4	0.79	0.41	0.49	0.06	1.18	6.62	3.38
Value	1.79	0.09	0.53	−0.06	1.37	7.56	3.61
V-G	1.47	−0.52	0.18	−0.14	0.49	2.58	0.51

In Table 8 we present average book to market of equity, profit-to-labor-expenses ratio, financial leverage, and mean excess return for five portfolios sorted on book to market. The top panel contains data, and the bottom panel shows results from our baseline model, with wage rigidity (⁠ $μ = 0.9$ ⁠) and a calibrated CES (⁠ $\frac{1}{1 - η} = 0.5$ ⁠). For simulated data, we also present the unlevered value premium, which is computed for the entire firm. Profits are defined as Revenue-COGS-SGA from Compustat; labor expenses are employees from Compustat multiplied by the average wage for the firm's industry from NIPA.

To summarize operating leverage induced by sticky-wages, we compute the profit-to-labor-expenses-ratio for each quintile, as shown in the second column of Table 8. Value firms have much lower profit-to-labor-expenses than growth firms have, that is, value firms are burdened with high labor expenses. The profit-to-labor-expenses ratio is 0.45 for value and 0.70 for growth in the data, compared to 0.09 for value and 0.61 for growth in our baseline model. Value stocks are riskier in this environment because during bad times, wages are relatively high and most firms want to reduce labor. This is especially true for low-productivity firms, which also tend to be value firms.¹⁵ However, reducing labor is costly because it leaves the firm with an even higher average wage (see Equation 4); therefore, low-productivity firms suffer disproportionately during recessions.

In Panel A of Table 6 we report the value premium (column six). Because there are differences in leverage between value and growth, we also report the difference in unlevered return (return on capital) between the two portfolios; this allows us to separate the effect of financial leverage from labor market frictions. This table shows that the standard frictionless model produces a negative unlevered value premium; that is, value stocks have lower average returns than growth stocks have. Leverage ratios are reported in Table 8. Because value stocks have higher leverage than growth stocks do, the levered value premium is positive in the standard model, despite the unlevered value premium being negative. However, in the standard model the levered value premium is only 0.25%, compared with 5.37% in the data.

When acting alone, higher complementarity between labor and capital raises the unlevered value premium slightly, as does wage rigidity. When the two are combined in our baseline model, the unlevered value premium is 0.51%. This is because higher complementarity between labor and capital makes it hard for value firms to use the less costly input to smooth productivity shocks, and wage rigidity increases further the cross-section of risk dispersion through relatively higher wage obligations for value firms.¹⁶

Combining the effects of labor market frictions together with financial leverage results in a 2.58% value premium in our baseline model. The large difference between the unlevered and levered value premium may seem to imply that financial leverage is doing all of the work. However, it is actually the interaction between financial leverage and labor market frictions that matters. For example, despite all models having financial leverage, the levered value premium in the standard frictionless model is eight times smaller than in our baseline model. Further, the difference in leverage between value and growth in the data is even bigger than in our model (Table 8); therefore, the 0.51% unlevered value premium is not necessarily small. Although we are unable to compute the observed unlevered value premium for the full sample, Choi (2013) showed that in 1982–2007, the unlevered (or asset) value premium was less than one third of the realized equity value premium.

Notably Carlson, Fisher, and Giammarino (2004); Zhang (2005); and Cooper (2006) also generated a sizable value premium. However, our model differs from these papers in three important ways. First, theirs are partial equilibrium models in which equilibrium prices are exogenously specified, whereas ours is a full-fledged general equilibrium model. In general equilibrium, prices may dampen the firms' investment demand, making it harder to generate large cross-sectional risk dispersion. In fact, our frictionless, Cobb-Douglas model is the standard investment-based model with capital adjustment costs in general equilibrium, but the (unlevered) value premium in this model is negative. Second, we use general equilibrium implications to guide the strength of the adjustment cost. As argued previously, increasing adjustment costs can revive the value premium even in the standard model (Figure 3); however, it would result in unrealistically smooth investment. Third, the channel through which our model delivers a value premium is quite different. Our model hinges on firms' wage expense being rigid, for which we provide direct evidence from Compustat firms; the models herein employ a combination of high capital adjustment costs and fixed production costs.

Although these results suggest that wage rigidities are important for the value premium, they cannot tell the full story. Note that in our model, the conditional CAPM very nearly holds, and the value premium is due to $β$ only. However, in the data (1929–2013) it is both due to $α$ and $β$ differences;¹⁷ this can be seen in Columns 4 and 5 of Table 8. Many production models are subject to the same criticism. A potential way forward is to add a second aggregate shock. For example, in addition to a standard TFP shock, Belo, Lin, and Bazdresch (2014) introduced a shock to the cost of adjusting the labor force and found that the model produces higher $α$ 's for value firms.

3.2.6 Term structure of the equity premium

van Binsbergen, Brandt, and Koijen (2012) used option prices to compute the price of the dividend on the aggregate stock market at various horizons, which they term dividend strips. Note that the value of the aggregate stock market is the sum of all dividend strips across all maturities. They computed the term structure of these strips and showed that it is downward-sloping. The expected return and expected volatility on short-term strips is higher than those of long-term strips.

van Binsbergen, Brandt, and Koijen (2012) pointed out that several leading models are unable to produce this downward-sloping term structure of equity. In particular, the standard LRR model, first proposed by Bansal and Yaron (2004), has an upward-sloping term structure, because their calibration implies that most of the risk is associated with variations in expectations of long-horizon consumption growth. Because consumption and dividends are cointegrated, long-horizon dividends load on this risk. However, there is a second, short-run shock that matters for short-term dividends but is not highly correlated with the long-run component. Thus, short-horizon dividends are less risky than long-horizon dividends.

In our model, the term structure of equity return and volatility is downward-sloping. This can be seen on the left-hand side of Figure 5, in which we present unlevered expected returns (top) and volatility (bottom). Our baseline model is the dashed line. Note that the criticism of van Binsbergen, Brandt, and Koijen (2012) applies to the standard, frictionless model (solid line), which produces an upward-sloping term structure of equity with an expected return of -6% (unlevered) at the shortest horizons. The frictionless model with a calibrated CES (dot-dashed line) does slightly better: the term structure is still upward-sloping but nearly flat. Both models with wage rigidity (dotted line for Cobb-Douglas and dashed line for CES) produce a downward-sloping term structure even for unlevered returns, with the short end being about 1.7% per year higher than the long end. Adding financial leverage (right side) can raise this difference to nearly 10%.

Figure 5

Term structure of equity

In Figure 5 we plot the term structure of expected return and volatility for dividend strips, calculated as in van Binsbergen, Brandt, and Koijen (2012). The solid line represents the standard frictionless model (Cobb-Douglas production and no wage rigidity); the dotted line represents the Cobb-Douglas model with wage rigidity; the dot-dashed line represents the frictionless model with a calibrated CES; and the dashed line represents the model with a calibrated CES and wage rigidity. We present unlevered returns on the left, and levered on the right.

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The reason our model is able to produce a downward-sloping term structure whereas a standard model cannot is twofold. First, our productivity process is slightly different from Bansal and Yaron (2004). Our productivity shock has the same key feature necessary for LRR–productivity growth is predictable; however, our process is simpler in that there is only one source of risk. Our process is equivalent to a discretized version of the process in Bansal and Yaron (2004), where the short-run and long-run shocks are perfectly correlated. Thus short-run dividends are riskier here compared with a more complicated two-shock process.

However, having this simpler process is not enough. Note that all of our models have the same productivity process, yet the frictionless model still has an upward-sloping equity term structure. This is because the standard model has countercyclical dividends, as discussed previously.¹⁸ Thus, the standard model's failure at matching the properties of aggregate dividends, and equity volatility is closely related to its failure to match a downward-sloping term structure of equity. In our baseline model, the term structure of equity is downward sloping because wage rigidity makes firms riskier in the short-run–a negative productivity shock results in big drops to profits and dividends. However, wages and profits are cointegrated and therefore expected to return to their normal shares of output in the long-run, and thus short-run dividends are riskier than long-run dividends are.

It is useful to compare our results with those of Belo, Collin-Dufresne, and Goldstein (2015). They showed that a mean-reverting financial leverage policy generates a downward-sloping term structure. Although both channels work because of cointegration, as described in the previous paragraph, our channel is distinct from theirs. This can be seen by comparing unlevered (left) to levered (right) returns in Figure 5. Note that our effect works even without financial leverage, although financial leverage certainly makes it stronger. Further, even with a reasonably calibrated financial leverage, we find that the standard frictionless model has a slightly upward-sloping equity term structure (solid line on the right).

3.3 Extensions

In our baseline model, all firms have identical $μ$ and differ only by their history of idiosyncratic productivity shocks. We have solved three realistic extensions of our baseline model. In all three extensions, aggregate rigidity is the same or slightly weaker than it is in our baseline model. However, we allow for heterogeneity in rigidity across firms or time. The first extension allows rigidity to differ across firms, with $μ$ symmetrically distributed. The second extension is like the first, but it allows for an asymmetric distribution. In the third extension, rigidity is identical across firms, but aggregate $μ$ changes through time with higher rigidity in recessions.

Each of the three extensions leaves the macroeconomic models nearly unchanged, but improves quantitatively the performance of the model for financial moments; for example, the equity return volatility improves from 13.15% in our baseline model to 14.45%, 15.61%, and 14.31%, respectively, in the three extensions. The intuition for this is evident in Figure 4, which shows that across models with different rigidity, aggregate return volatility is a convex function of $μ$ ⁠. Because convexity implies that $E [σ (μ)] > σ (E [μ])$ ⁠, a model with heterogenous $μ$ has more equity volatility.

These extensions help the model explain an additional dimension of the data. In models with heterogenous firms, when we group all high versus low $μ$ firms together, the high-rigidity group has higher average returns, higher volatility, higher CAPM beta, and a more countercyclical labor share. As shown in Figure 1 and Table 1, the data exhibit a similar pattern.

4. Conclusion

In this paper we show that introducing wage rigidity into an otherwise standard model can greatly improve the model's ability to match financial data quantitatively. The problem with standard models is that wages are far too volatile and procyclical relative to the data. Wages therefore act as a hedge for the firm's owners, making profits too smooth and dividends countercyclical. Thus, the equity volatility in the data is about four times that of standard models.

In our model, the average wage is smoother because of infrequent wage resetting and to higher complementarity between labor and capital. Therefore, both profit and dividend behavior are similar to the data, and the volatility of equity returns is now 75% that of the data. The same channel brings the model closer to explaining several other unresolved puzzles in financial data. In particular, in our model, as in the data, equity volatility and expected returns are countercyclical; there is a significant value premium, and the term structure of equity is downward-sloping.

One shortcoming of our modeling strategy is that we are unable to endogenize the labor supply decision. It is possible that introducing leisure would give households an additional channel for insurance and would bring down the Sharpe ratio, as in Lettau and Uhlig (2000). Another shortcoming of our model is that financial leverage is exogenously specified. Relaxing this assumption would allow researchers to look for links between labor markets and corporate policy, as well as bond prices.

Although our model improves significantly on the standard model, the volatility of equity and the value premium are still short of the data. Further, the value premium in our model is due to value firms having higher market risk; in the data the value premium is a combination of higher market risk and higher CAPM alpha. One possible way to rectify this is by adding a second aggregate shock. The labor market may be the right place to look for such a shock; for example, the parameters of the production function that control labor share or the workers' bargaining power that determines contract length may be time varying.

Empirical work exploring the relationship between labor market variables and asset returns may shed light on such a shock. There are additional empirical implications as well. Figure 1 gives a flavor of this. Labor markets frictions are not constant. For example, wage rigidity may vary through time, across firms, or across countries; this has implication for expected returns and firm behavior. We leave such exploration for future work.

Appendix

A.1 Data

Stock returns are from the Ken French's web page: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. All stock return data are from 1929 to 2012. Accounting information is from the CRSP/Compustat Merged Annual Industrial Files, the sample is from 1950 to 2012. We exclude from the sample any firm-year observation with missing data or for which total assets or the gross capital stock are either zero or negative. In addition, as standard, we omit firms whose primary SIC classification is between 4900 and 4999 (regulated firms) or between 6000 and 6999 (financial firms). Firm profit is net sales (SALE) minus the sum cost of goods sold (COGS) and selling, general, and administrative expense (XSGA). We aggregate all firms' profit scaled by investment price deflator to compute total real profit. Corporate payout is from the Flow of Funds (FOF); these data are from 1946 to 2012. We define net equity payouts in a similar manner to Jermann and Quadrini (2012); in particular, they are net dividends paid (line 3 from FOF Table F.102) minus the change in the book value of corporate equities (line 35 from FOF Table F.101). The net payout is the net equity payout plus net interest (line 29 from FOF Table F.7).

All remaining data are from National Income and Product Accounts (NIPA), available for 1929–2013. GDP, consumption (nondurable consumption plus services), and investment (fixed private investment plus durable goods) are from NIPA Table 1.1.5. Employment is full-time and part-time employees from NIPA Table 6.4. Although we are calibrating our aggregate economy (output, investment, consumption, and employment) to match the aggregate U.S. economy, for the labor market variables, we target the private sector only. We believe this is appropriate because we are interested in the effect of labor market frictions on asset prices, which are relevant only for the private sector. Therefore, the frictions in our model should match the labor market frictions in the U.S. private sector. The total wage bill is the sum of compensation of employees in private industries and supplements to wages, both in NIPA Table 2.1. The wage is the total wage bill divided by private employees from NIPA Table 6.4. The labor share is the total wage bill divided by private sector GDP, which we define as GDP minus government expenditures from NIPA Table 1.1.5. Because of large temporary movements of people out of the private sector during World War II, we report the labor market variables for the postwar only; consequently, when we present scaled statistics, the scaling variable is computed for the analogous period rather than the full sample.

Finally, exclusively for Table 1 and Figure 1, we compute industry level variables. Industry labor share is constructed as industry employee compensation divided by the sum of industry compensation, profit, and capital consumption (investment). These data come from NIPA Section 6. After matching industries, we are left with 26 industries for the period 1929–2000; a detailed description of these data and the matching procedure is given in Favilukis and Lin (2015).

All nominal variables are scaled by the Consumer Price Index from the BLS.

A.2 Additional notes on calibration

We calibrate the productivity process to match roughly the short-term autocorrelation of output; this implies that the autocorrelation of consumption growth (which is endogenous) is somewhat higher than the data, though still within error bounds. In the Online Appendix, we plot autocorrelations of output and consumption growth at horizons of 1–10 years. At medium term horizons (3–6 years), both output and consumption growth autocorrelations produced by the model are too high, which is common to many LRR models. We have also solved a model with much lower autocorrelation of output. This model is closer to the data at longer horizons but has far too little autocorrelation at shorter horizons; the equity volatility in this model is 11.66% (compared with 13.15% in our baseline model). Note that our process for aggregate productivity is a simple 3-state Markov process, but we believe a more complicated process is necessary to capture both short- and long-term autocorrelation in output and consumption growth.

A.2.1 Financial leverage

The firm's return on capital is $R_{t + 1}^{K} = \frac{V_{t + 1}}{V_{t} - D_{t}}$ ⁠. However, real-world firms are financed both by debt and by equity, with equity being the riskier, residual claim. To compare the model's equity return and dividend to their empirical counterparts, we must make an assumption about financial leverage. Note that the Modigliani and Miller (1958) propositions hold in our model; therefore, all assumptions about financial leverage are completely orthogonal to our model's solution; these assumptions only affect the way we report returns and dividends.

It is standard in the literature to assume that all firms keep leverage constant¹⁹; however, the data leverage is quite sticky. We have solved the model for both constant and sticky leverage. All return moments we report are very similar for both; however, dividends are too volatile if leverage is constant. Because sticky leverage is realistic and it improves the model's performance, we assume that the firm's choice of debt is sticky and follows:

\begin{array}{l} B_{t} = ρ^{B} B_{t - 1} + (1 - ρ^{B}) B_{t}^{*} \\ B_{t}^{*} = λ (V_{t} - D_{t}), \end{array}

(A1)

where

B_{t}

is the market value of corporate debt (⁠

B_{t}

is borrowed at

t

⁠;

B_{t} R_{t + 1}^{B}

is owed at

t + 1

⁠) and

B_{t}^{*}

is the target debt level. We assume that the target debt level is a constant fraction of the firm's ex-dividend value

V_{t} - D_{t}

⁠.

Combining this assumption about the evolution of debt with the Modigliani and Miller (1958) second proposition implies that the equity dividend and the equity return are:

D_{t}^{E} = D_{t} + B_{t} - B_{t - 1} R_{t}^{B}

(A2)

R_{t}^{E} = \frac{V_{t} - B_{t - 1} R_{t}^{B}}{V_{t - 1} - B_{t - 1} - D_{t - 1}},

(A3)

where

D_{t}

is the total payout to all of the firm's financial stake holders (debt and equity) and

R_{t}^{B}

is the (risky) return on debt. Let

R_{t - 1}^{P}

be the return promised at

t - 1

to be repaid at

t

⁠. Its derivation is discussed below. Default occurs at

t

if

V_{t} < B_{t - 1} R_{t - 1}^{P}

⁠, because of limited liability, equity holders can walk away instead of receiving a net return below −100%. When the firm does not default,

R_{t}^{B} = R_{t - 1}^{P}

⁠; when the firm does default, the equity return is −100% and creditors take over the firm, so that

R_{t}^{B} = \frac{V_{t}}{B_{t - 1}} < R_{t - 1}^{P}

⁠. At this stage the firm's debt is reset to zero and then follows Equation A1. The promised payment

R_{t}^{P}

is set such that

1 = E_{t} [R_{t + 1}^{B} M_{t + 1}] = R_{t}^{p} E_{t} [M_{t + 1} | d \neq 1] * (1 - p (d = 1)) + E_{t} [M_{t + 1} \frac{V_{t + 1}}{B_{t}} | d = 1] * p (d = 1),

(A4)

where

d = 1

indicates default and

p (d = 1)

is the probability of default.

If we were to assume that leverage is constant, we would simply set $ρ^{B} = 0$ ⁠. In this case the equity return takes the more familiar form: $R_{t}^{E} = R_{t}^{B} + \frac{1}{1 - λ} (R_{t}^{K} - R_{t}^{B})$ ⁠.

We estimate the target debt to equity ratio to be 0.59, which implies that $λ = 0.37$ ⁠. We find debt to be quite sticky, with annual estimates of $ρ^{B}$ between 0.46 and 0.99, depending on the specification. Details of these estimates follow. We set quarterly $ρ^{B} = 0.9$ in all of our models, and this allows us to roughly match the dividend volatility in the data. As noted earlier, we have also experimented with $ρ^{B} = 0$ ⁠. With this calibration, all of our key results are very similar to $ρ^{B} = 0.9$ ⁠; however, the process for equity dividends is too volatile.

A.2.2. Estimation of market-to-book and debt-to-equity.

Our model is solved without financial leverage; financial leverage (defined by $ρ^{B}$ and $λ$ and described next) is then added once the model is solved, as described in the text. Therefore, the relevant market-to-book ratio to match during the model solution stage is the market-to-book ratio for the entire firm value (the enterprise value). From Compustat we calculate the market-to-book ratio for equity to be 1.64 and the book debt to market equity ratio to be 0.59. Sweeney, Warga, and Winters (1997) found that outside of the Volcker period, aggregate market-to-book values for debt are very close to one. These numbers imply a market-to-book of 1.33 for enterprise value. This also implies that the debt-to-value ratio is $λ = \frac{0.59}{1 + 0.59} = 0.37$ ⁠; we use this in our calibration. An alternative is to estimate $λ = A v g (\frac{B}{B + E})$ ⁠, where $B$ is the value of debt from the flow of funds and $E$ is the total market value of equity from CRSP (both variables are described in more detail in the next section). The method implies $λ = 0.41$ ⁠. The slightly higher leverage would make the results even stronger.

A.2.3. Estimation of $ρ^{B}$

To estimate the dependence of debt issuance on past issuance, we use levels of debt from the flow of funds. In particular, we aim to estimate

B_{t + 1} = ρ^{B} B_{t} + (1 - ρ^{B}) * λ V_{t},

(A5)

where

B

is the market value of corporate debt and

V

is the enterprise value of the corporate sector (equity plus debt). We define

B

to be non-financial credit market instrument liabilities (line 28 from FOF Table L.101).²⁰ We define

V

to be

B

plus total market value of NYSE/AMEX/NASDAQ from CRSP.²¹ Because the values of debt and equity are non-stationary, we cannot directly estimate Equation A5.

We use several different specifications to estimate $ρ^{B}$ ⁠. In particular, specification 1 simply estimates $ρ^{B}$ to be the autocorrelation of HP-filtered $B$ ⁠. In specification 2 we regress $B_{t + 1} = a_{0} + a_{1} B_{t} + a_{2} V_{t}$ where both $B$ and $V$ are HP-filtered; we either define $ρ^{B} = a_{1}$ (specification 2a, $λ$ unrestricted) or $ρ^{B} = 1 - a_{2} / λ$ (specification 2b, $λ = 0.37$ ⁠). However, the HP-filter may cause the estimation to miss out on important low-frequency dependence of debt on past debt. In specifications 3 and 4 we do not HP-filter. In specification 3, we regress $B_{t + 1} / B_{t} = a_{0} + a_{1} V_{t} / B_{t}$ and either define $ρ^{B} = a_{0}$ (specification 3a, $λ$ unrestricted) or $ρ^{B} = 1 - a_{1} / λ$ (specification 3b, $λ = 0.37$ ⁠. Finally, in specification 4 we regress $B_{t + 1} / B_{t} - B_{t} / B_{t - 1} = a_{0} + a_{1} (V_{t} / B_{t} - V_{t - 1} / B_{t - 1})$ and define $ρ^{B} = 1 - a_{1} / λ$ where $λ = 0.37$ ⁠. The different estimates are presented in Online Appendix.

A.3 Solving the model

A.3.1 Making the Model Stationary

Note that the model is not stationary. To solve it numerically, we must first detrend it and rewrite it in terms of stationary quantities. We will describe the detrending procedure in this section, and the actual numerical solution of the stationary model in the next section.

Along the balanced growth path of this economy, output, consumption, investment, and capital will all grow with

Z_{t}^{ρ}

⁠. To see this, consider the planner's problem analogous to our model but with no frictions:

\begin{array}{l} U_{t} = \max {((1 - β) C_{t}^{1 - \frac{1}{ψ}} + β E_{t} {[U_{t + 1}^{1 - θ}]}^{\frac{1 - \frac{1}{ψ}}{1 - θ}})}^{\frac{1}{1 - \frac{1}{ψ}}} \\ C_{t} = {(α K_{t}^{η} + (1 - α) {(Z_{t} N_{t})}^{η ρ})}^{\frac{1}{η}} - I_{t} \\ K_{t + 1} = (1 - δ) K_{t} + I_{t}, \end{array}

(A6)

where

Z_{t}

and

N_{t}

follow the stochastic processes described in the main text. Because

Z_{t}

is nonstationary, so are many of the other relevant quantities in this problem. Define

k_{t} = \frac{K_{t}}{Z_{t}^{ρ}}

⁠,

i_{t} = \frac{I_{t}}{Z_{t}^{ρ}}

⁠,

c_{t} = \frac{C_{t}}{Z_{t}^{ρ}}

⁠, and

u_{t} = \frac{U_{t}}{Z_{t}^{ρ}}

⁠. The model in Equation A6 can be rewritten as:

\begin{array}{l} u_{t} = \max {((1 - β) c_{t}^{1 - \frac{1}{ψ}} + β E_{t} {[{(\frac{Z_{t + 1}}{Z_{t}})}^{ρ (1 - θ)} u_{t + 1}^{1 - θ}]}^{\frac{1 - \frac{1}{ψ}}{1 - θ}})}^{\frac{1}{1 - \frac{1}{ψ}}} \\ c_{t} = {(α k_{t}^{η} + (1 - α) N_{t}^{η ρ})}^{\frac{1}{η}} - i_{t} \\ k_{t + 1} = {(\frac{Z_{t + 1}}{Z_{t}})}^{- ρ} ((1 - δ) k_{t} + i_{t}) . \end{array}

(A7)

Note that now there are no nonstationary inputs.

We will use the same trend variable

Z_{t}^{ρ}

to rewrite our decentralized problem. The firm's problem is:

\begin{matrix} V (Z_{t}^{i}, K_{t}^{i}, N_{t - 1}^{i}, {\bar{W}}_{t - 1}^{i}; Ω_{t}) = & \max_{I_{t}^{i}, N_{t}^{i}} D_{t} + E_{t} [M_{t + 1} V (Z_{t + 1}^{i}, K_{t + 1}^{i}, N_{t}^{i}, {\bar{W}}_{t}^{i}; Ω_{t + 1})] \\ D_{t} = & Z_{t}^{i} {(α {(K_{t}^{i})}^{η} + (1 - α) {(Z_{t} N_{t}^{i})}^{ρ η})}^{\frac{1}{η}} \\ - ({\bar{W}}_{t - 1}^{i} N_{t - 1}^{i} μ + W_{t} (N_{t}^{i} - N_{t - 1}^{i} μ)) - K_{t} f \\ - ν {(\frac{I_{t}^{i}}{K_{t}^{i}} - δ)}^{2} K_{t}^{i} \\ K_{t + 1}^{i} = & (1 - δ) K_{t}^{i} + I_{t}^{i} \\ {\bar{W}}_{t}^{i} = & \frac{{\bar{W}}_{t - 1}^{i} N_{t - 1}^{i} μ + W_{t} (N_{t}^{i} - N_{t - 1}^{i} μ)}{N_{t}^{i}}, \end{matrix}

(A8)

where

D_{t}

is the net payout of the firm,

Z_{t}^{i}

is the idiosyncratic productivity,

K_{t}^{i}

is the firm's individual capital,

N_{t - 1}^{i}

is the firm's employment last period, and

{\bar{W}}_{t - 1}^{i}

is the firm's average wage last period.

Ω_{t}

is a multi-dimensional vector of all relevant aggregate state variables. In theory,

Ω_{t}

may be infinite dimensional as it must carry information about the full distribution of capital across firms. In our approximate numerical solution,

Ω_{t} = {Z_{t}, j_{t}, j_{t - 1}, K_{t}, {\bar{W}}_{t - 1}}

⁠, where

Z_{t}

is aggregate productivity,

j_{t}

is the discrete state of the Markov process governing productivity growth,

K_{t}

is the average capital in the economy, and

{\bar{W}}_{t - 1}

is the average wage at

t - 1

⁠. Subsequently, we will discuss the rationale for including these variables in the aggregate state.

Aggregating across firms, the aggregate quantities in

Ω_{t}

evolve as:

\begin{array}{l} K_{t + 1} = (1 - δ) K_{t} + \sum I_{t}^{i} \\ {\bar{W}}_{t} = \frac{{\bar{W}}_{t - 1} N_{t - 1} μ + W_{t} (N_{t} - N_{t - 1} μ)}{N_{t}} \end{array}

(A9)

The household's problem implies

\begin{array}{l} M_{t + 1} = β {(\frac{C_{t + 1}}{C_{t}})}^{- \frac{1}{ψ}} {(\frac{U_{t + 1}}{E_{t} {[U_{t + 1}^{1 - θ}]}^{\frac{1}{1 - θ}}})}^{\frac{1}{ψ} - θ} \\ C_{t} = \sum (Z_{t}^{i} {(α {(K_{t}^{i})}^{η} + (1 - α) {(Z_{t} N_{t}^{i})}^{ρ η})}^{\frac{1}{η}} - I_{t}^{i}) \\ U_{t} = {((1 - β) C_{t}^{1 - \frac{1}{ψ}} + β E_{t} {[U_{t + 1}^{1 - θ}]}^{\frac{1 - \frac{1}{ψ}}{1 - θ}})}^{\frac{1}{1 - \frac{1}{ψ}}} . \end{array}

(A10)

Define $k_{t}^{i} = \frac{K_{t}^{i}}{Z_{t}^{ρ}}$ ⁠, $k_{t} = \frac{K_{t}}{Z_{t}^{ρ}}$ ⁠, $i_{t}^{i} = \frac{I_{t}^{i}}{Z_{t}^{ρ}}$ ⁠, $w_{t} = \frac{W_{t}}{Z_{t}^{ρ}}$ ⁠, $d_{t} = \frac{D_{t}}{Z_{t}^{ρ}}$ ⁠, ${\bar{w}}_{t}^{i} = \frac{{\bar{W}}_{t}^{i}}{Z_{t + 1}^{ρ}}$ ⁠, and ${\bar{w}}_{t} = \frac{{\bar{W}}_{t}}{Z_{t + 1}^{ρ}}$ (note that the timing of ${\bar{w}}_{t}^{i}$ and ${\bar{w}}_{t}$ differs from the others). Fundamentally, $Z_{t}^{ρ}$ is the right variable for detrending because $d_{t}$ is not a function of $Z_{t}$ (Equation A11). In addition, we define the vector of detrended aggregate state variables $Θ_{t} = {j_{t}, j_{t - 1}, k_{t}, {\bar{w}}_{t - 1}}$ ⁠.

It is straightforward to check that the firm's value function is linear in

Z_{t}^{ρ}

⁠, that is:

V (Z_{t}^{i}, K_{t}^{i}, N_{t - 1}^{i}, {\bar{W}}_{t - 1}^{i}; Ω_{t}) = Z_{t}^{ρ} v (Z_{t}^{i}, k_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}; Θ_{t}),

where

v

does not depend on the nonstationary

Z_{t}

⁠.²² In this case, Equations A8, A9, and A10 can be rewritten without any nonstationary inputs.

We summarize the stationary economy of our model as follows. The firm's problem is given in Equation A11.

\begin{matrix} v (Z_{t}^{i}, k_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}; Θ_{t}) = & \max_{i_{t}^{i}, N_{t}^{i}} d_{t} + E_{t} [{(\frac{Z_{t + 1}}{Z_{t}})}^{ρ} M_{t + 1} v (Z_{t + 1}^{i}, k_{t + 1}^{i}, N_{t}^{i}, {\bar{w}}_{t}^{i}; Θ_{t + 1})] \\ d_{t} = & Z_{t}^{i} {(α {(k_{t}^{i})}^{η} + (1 - α) {(N_{t}^{i})}^{ρ η})}^{\frac{1}{η}} \\ - ({\bar{w}}_{t - 1}^{i} N_{t - 1}^{i} μ + w_{t} (N_{t}^{i} - N_{t - 1}^{i} μ)) - k_{t} f \\ - ν {(\frac{i_{t}^{i}}{k_{t}^{i}} - δ)}^{2} k_{t}^{i} \\ k_{t + 1}^{i} = & ((1 - δ) k_{t}^{i} + i_{t}^{i}) {(\frac{Z_{t + 1}}{Z_{t}})}^{- ρ} \\ {\bar{w}}_{t}^{i} = & (\frac{{\bar{w}}_{t - 1}^{i} N_{t - 1}^{i} μ + (N_{t}^{i} - N_{t - 1}^{i} μ) w_{t}}{N_{t}^{i}}) {(\frac{Z_{t + 1}}{Z_{t}})}^{- ρ} . \end{matrix}

(A11)

The evolution of the aggregate state variables is given in Equation A12.

\begin{array}{l} k_{t + 1} = ((1 - δ) k_{t} + \sum i_{t}^{i}) {(\frac{Z_{t + 1}}{Z_{t}})}^{- ρ} \\ {\bar{w}}_{t} = (\frac{{\bar{w}}_{t - 1} N_{t - 1} μ + (N_{t} - N_{t - 1} μ) w_{t}}{N_{t}}) {(\frac{Z_{t + 1}}{Z_{t}})}^{- ρ} . \end{array}

(A12)

Additionally, the spot wage

w_{t}

must be such that

\sum N_{t}^{i} = N_{t}

⁠.

The household's stationary equilibrium conditions are as follows:

\begin{array}{l} M_{t + 1} = β {(\frac{Z_{t + 1}}{Z_{t}})}^{- ρ θ} {(\frac{c_{t + 1}}{c_{t}})}^{- \frac{1}{ψ}} {(\frac{u_{t + 1}}{E_{t} {[{(\frac{Z_{t + 1}}{Z_{t}})}^{ρ (1 - θ)} u_{t + 1}^{1 - θ}]}^{\frac{1}{1 - θ}}})}^{\frac{1}{ψ} - θ} \\ c_{t} = \sum (Z_{t}^{i} {(α {(k_{t}^{i})}^{η} + (1 - α) {(N_{t}^{i})}^{ρ η})}^{\frac{1}{η}} - \sum i_{t}^{i}) \\ u_{t} = {((1 - β) c_{t}^{1 - \frac{1}{ψ}} + β E_{t} {[{(\frac{Z_{t + 1}}{Z_{t}})}^{ρ (1 - θ)} u_{t + 1}^{1 - θ}]}^{\frac{1 - \frac{1}{ψ}}{1 - θ}})}^{\frac{1}{1 - \frac{1}{ψ}}} . \end{array}

(A13)

A.3.2 Numerical Solution

We will now explain how we numerically solve the economy described by Equations A11, A12, and A13, and justify our choice of the state space. The algorithm is a variation of the algorithm in Krusell and Smith (1998). Generally, there is no proof that an equilibrium exists. This solution method is referred to as an approximate bounded rational equilibrium. It consists of performing two steps and then repeating them until convergence. The first step solves the firm's problem given a particular set of beliefs; the inputs are beliefs and the outputs are policy functions. The second step updates these beliefs from simulating the economy; the inputs are policy functions and the outputs are beliefs. These steps are repeated until the beliefs have converged and are consistent with simulated data. We will refer to the sequence of step one, followed by step two as an iteration.

The problem is solved using Fortran 77, and parallelized using OpenMP. It then runs on eight parallel processors. The full model takes about 4 hours per iteration and requires 50 to 100 iterations to converge. We are appreciative of The Ohio State University high-performance computing center for the computational resources.

Step 1. We begin this step with beliefs about aggregate investment, the spot wage, aggregate consumption, and the stochastic discount factor as functions of the aggregate state. These beliefs are $\sum i_{t}^{i} = ι^{n} (Θ_{t})$ ⁠, $w_{t} = ω^{n} (Θ_{t})$ ⁠, $c_{t} = ζ^{n} (Θ_{t})$ and $M_{t + 1} = M^{n} (Θ_{t}, j_{t + 1})$ ⁠, where $n$ is the number of the iteration. These beliefs, together with Equations A11 and A12 specify a well-defined (and relatively standard) partial equilibrium firm problem. We solve this problem using value function iteration.

For numerical reasons, we find it better to rescale some of the state variables. In particular instead of $({\bar{w}}_{t - 1}^{i}$ ⁠, $N_{t - 1}^{i})$ ⁠, we use $(N_{t - 1}^{i} {\bar{w}}_{t - 1}^{i}$ ⁠, $N_{t - 1}^{i})$ ⁠. This rescaling is innocuous because one can easily go back and forth. It can be shown that when there are no adjustment costs, the firm's problem is linear in $N_{t - 1}^{i} {\bar{w}}_{t - 1}^{i}$ and $N_{t - 1}^{i}$ ⁠, and thus we believe the rescaling leads to a more efficient algorithm. Further, it allows us to check the accuracy of our solution in the special case of $ν = 0$ ⁠. Instead of ${\bar{w}}_{t - 1}$ ⁠, we use ${\bar{w}}_{t - 1}$ ⁠, scaled by the aggregate marginal product of labor. This is done because ${\bar{w}}_{t - 1}$ is highly correlated with $k_{t}$ ⁠, which makes formation of beliefs more difficult without rescaling. Further, when two state variables are correlated, large areas of the grid are left unused, which wastes computing power. The scaling reduces this correlation.

Our grid sizes are 40 for $k_{t}^{i}$ ⁠, 19 for ${\bar{w}}_{t - 1}^{i}$ ⁠, 12 for $N_{t - 1}^{i}$ ⁠, 20 for $k_{t}$ ⁠, and 6 for ${\bar{w}}_{t - 1}$ ⁠. Recall that we also have grids of size 3 for the aggregate Markov chain at $t$ ⁠, the aggregate Markov chain at $t - 1$ ⁠, and the individual Markov chain at $t$ ⁠. We have experimented with grid sizes extensively and set them large enough that our results are not affected by any further increases. It is important to set the grid edges some distance away from where typical variables reside, despite these values being “off-equilibrium.” At the same setting the edges too far away from model equilibrium will require a very large number of grid points, which is numerically infeasible; therefore, some experimentation is in order. We find that the results are more sensitive to the sizes of firm level grids than to the aggregate grids.

Once the value function iteration is complete, we have two policy functions for the firm: investment $i^{i} (Z_{t}^{i}, k_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}; Θ_{t})$ and employment $N^{i} (Z_{t}^{i}, k_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}; Θ_{t})$ ⁠.

Step 2. In this step we use the policy functions to simulate the economy and then use simulated data to update the beliefs. We simulate the economy for 10,000 firms, and a higher number does not affect any of our results. Before beginning the simulation we must specify an initial distribution of idiosyncratic productivity, capital, past wages, and past labor. We simulate the economy for 3,500 periods and throw away the first 500 periods to let the simulation settle into its normal behavior; this also assures that the initial distribution has no effect on our results.

One complication during the simulation is that we must clear the labor market each period. The difficulty is that each firm's choice of labor is a function of the state variables only. The state variables are fixed and known at the beginning of each period. Thus labor is determined at the start of the period. The firms have beliefs about the spot wage as a function of the state; however, before convergence these beliefs may be incorrect, and, therefore, labor demand may not equal labor supply. The actual market clearing spot wage (as opposed to the belief) is undefined because at this stage in the simulation, nothing can change the state variables or firms' labor demand. To deal with this problem we use the following workaround: During the simulation, we assume that each firm's labor demand is $N_{t}^{i} = N^{i} (Z_{t}^{i}, k_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}; Θ_{t}) \frac{ω^{n} (Θ_{t})}{w_{t}}$ ⁠, where $ω^{n} (Θ_{t})$ is the belief about the spot wage used during the value function iteration step. Note that once our algorithm has converged, the belief is consistent with the spot wage. However, before convergence we are able to pick the spot wage in any period so as to clear markets, that is $w_{t} = ω^{n} (Θ_{t}) \frac{\sum N^{i} (Z_{t}^{i}, k_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}; Θ_{t})}{N_{t}}$ ⁠.

Once the simulation is complete, we have a time series for all relevant aggregate variables. We use these time series to update the beliefs. Krusell and Smith (1998) have suggested regressing the relevant variables on the state variables. However we find this problematic because linear regressions imply strange behavior “off-equilibrium,” which leads to problems in the value-function iteration step. Adding higher-ordered terms does not help because it leads to overfitting.

We propose an alternative, nonparametric approach. We will define a belief separately for each grid point in the state space. There are two types of grid points in the state space: those that are near where the simulated data resides (“on-equilibrium”) and those that are not (“off-equilibrium”).

We define a grid point as “on-equilibrium” if there are more than 20 simulated periods in which the root-mean-square distance between the state variables in that period and the grid point is smaller than a fixed bound. We then run a local regression of our variables of interest (consumption, investment, and the spot wage) on the state variables near the grid point. The predicted value of our variable of interest computed at the grid point is then our updated belief at this grid point.

For the remaining “off-equilibrium” grid points, we use root-mean-square distance to find the closest simulated period. We then shift the distribution of capital and past labor from that period. For example, suppose the grid point has capital and past wage $(k_{a}, {\bar{w}}_{a})$ ⁠. Suppose the nearest simulated period has distributions of capital and past wages with average values $(k_{b}, {\bar{w}}_{b})$ ⁠. Then each firm's capital and average wage are shifted by $k_{b} - k_{a}$ and ${\bar{w}}_{b} - {\bar{w}}_{a}$ ⁠. We then take the shifted distribution as an initial distribution and simulate for one period. The result of this simulation is then assigned as the updated belief to this “off-equilibrium” grid point.

There is one additional caveat. It is important to put a weight on old beliefs during updating; without it the procedure may not converge. We have found that the lower the capital adjustment cost, the higher the required weight. For zero adjustment cost, the weight may sometimes need to be as high as 0.998. For our baseline model, the weight we use is 0.85, and likely an even lower weight would have sufficed.²³

The procedure describes how to form updated beliefs for investment, spot wages, and consumption as a function of the state. These updated beliefs are

\sum i_{t}^{i} = ι^{n + 1} (Θ_{t})

⁠,

w_{t} = ω^{n + 1} (Θ_{t})

⁠, and

c_{t} = ζ^{n + 1} (Θ_{t})

⁠. It still remains to update the belief for the stochastic discount factor. Note that with CRRA utility, this would be straightforward:

M^{n + 1} (Θ_{t}, j_{t + 1}) = β {(\frac{Z_{t + 1}}{Z_{t}})}^{- ρ θ} {(\frac{ζ^{n + 1} (Θ_{t + 1})}{ζ^{n + 1} (Θ_{t})})}^{- \frac{1}{ψ}}

⁠. For the more general case, we instead set

M^{n + 1} (Θ_{t}, j_{t + 1}) = β {(\frac{Z_{t + 1}}{Z_{t}})}^{- ρ θ} {(\frac{ζ^{n + 1} (Θ_{t + 1})}{ζ^{n + 1} (Θ_{t})})}^{- \frac{1}{ψ}} {(\frac{u (Θ_{t + 1})}{E_{t} {[{(\frac{Z_{t + 1}}{Z_{t}})}^{ρ (1 - θ)} u {(Θ_{t + 1})}^{1 - θ}]}^{\frac{1}{1 - θ}}})}^{\frac{1}{ψ} - θ},

where

u (Θ_{t})

comes from separately solving the recursion:

u (Θ_{t}) = {((1 - β) ζ^{n + 1} {(Θ_{t})}^{1 - \frac{1}{ψ}} + β E_{t} {[{(\frac{Z_{t + 1}}{Z_{t}})}^{ρ (1 - θ)} u {(Θ_{t + 1})}^{1 - θ}]}^{\frac{1 - \frac{1}{ψ}}{1 - θ}})}^{\frac{1}{1 - \frac{1}{ψ}}} .

This recursion is also solved with value function iteration using the same aggregate grids as the firm's problem. However, it typically takes less than a second because the state space is much smaller (there are no individual firm variables), and because there are no choice variables.

Once steps one and two are complete, we check whether the algorithm has converged; if it has not, we restart step one with updated beliefs. Convergence means that the absolute distance between $ι^{n + 1} (Θ_{t})$ and $ι^{n} (Θ_{t})$ is sufficiently small (same for $ζ^{n + 1} (Θ_{t})$ and $ω^{n + 1} (Θ_{t})$ ⁠).

In addition to confirming that the beliefs have converged, it is standard to perform other checks. This solution method is referred to as an approximate bounded rational equilibrium. It is rational because the beliefs of the firms and agents are exactly equal to the best forecast an econometrician could achieve with in simulated data using the state defined variables. However, it is bounded because the forecast may still not be very good (low $R^{2}$ ⁠) or because there may be additional variables that may either improve the forecast, or whose inclusion may change the equilibrium allocations or prices.

The lowest $R^{2}$ in our forecasting equations is 0.9996; this compares well with other models solved with the Krusell and Smith (1998) approach and leads us to think that the forecast is fairly accurate.²⁴ We have also experimented, one at a time, with additional state variables: the second moment of the capital distribution, the second moment of the past wage distribution, and the second moment of the past labor demand distribution. Unfortunately, because of numerical constraints, we could not add these as continuous state variables but only as a discrete high or low signal. Adding these moments did not affect our results.

Finally, it may be helpful to discuss why we included the variables we did in the state space. Generally, the whole distribution of capital, labor, and wages across firms may be relevant. Krusell and Smith (1998) have suggested summarizing it by the first moment, and possibly higher moments. Our frictionless economy is similar to the economy solved by Krusell and Smith (1998), with the difference being heterogenous households versus heterogenous firms. They found that the first moment of the capital distribution is enough to summarize the state space because it is this that is most relevant for the output and consumption that the economy will produce, as well as the return on capital. Indeed, in our model we find that average capital contributes most to $R^{2}$ ⁠.

In addition, in our model, firms also must form a belief about the spot wage they will face. When $μ = 0$ ⁠, there is a tight (though not exact) relationship between average capital and the spot wage, so average capital is a sufficient state variable. However, when $μ > 0$ we find that average capital alone does not produce a sufficiently high $R^{2}$ ⁠. It is then natural to add the average of last period's wage to the state space, and indeed this results in a high $R^{2}$ ⁠.

The reason why $Θ_{t}$ contains the current state of the exogenous Markov process $j_{t}$ is obvious; however, there is a subtle reason why $j_{t - 1}$ must be included, as well. Because ${\bar{w}}_{t - 1}$ is part of the aggregate state, firms must be able to forecast ${\bar{w}}_{t}$ ⁠. Equation A12 shows that to forecast ${\bar{w}}_{t}$ ⁠, we need both today's aggregate labor supply $N_{t}$ and last period's $N_{t - 1}$ ⁠. Because, by assumption, labor supply is a function of the Markov process, $j_{t - 1}$ is sufficient for knowing $N_{t - 1}$ ⁠. This is also why adding labor in the utility function is a much more difficult problem to solve numerically. With labor in the utility function, last period's labor is endogenous, and we would need to keep track of it as an additional state variable as opposed to just keeping track of $j_{t - 1}$ ⁠. Unlike the discrete Markov state, this is a continuous variable and would require a finer grid.

We would like to thank Frederico Belo, Gian Luca Clementi, John Cochrane, Max Croce, Bob Goldstein, Vito Gala, Francois Gourio, Jesus Fernandez-Villaverde, Lars Kuehn, Erica Li, Dmitry Livdan, Lars Lochstoer, Stavros Panageas, Monika Piazzesi, Lukas Schmid, Rene Stulz, Stijn Van Nieuwerburgh, Neng Wang, Ivo Welch, Amir Yaron, and Lu Zhang for helpful comments. We thank the seminar participants at the Aarhus University, Beijing University, Cheung Kong Graduate School of Business, Goethe University, Higher School of Economics, London School of Economics, Manchester Business School, Ohio State University, Renmin University of China, University of Hong Kong, University of Minnesota, CEPR Asset Pricing Week in Gerzensee 2013, Chicago Booth-Deutsche Bank 2011 Symposium, NBER 2012 AP meetings, SED 2012 meetings, EFA 2012 meetings, WFA 2013 meetings, China International Finance 2013 Conference, Econometric Society 2013 meetings, and AEA 2015 meetings. We also thank the Ohio State University Super computing center. All remaining errors are our own. Supplementary data can be found on The Review of Financial Studies web site.

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¹ In Jermann (1998) and Boldrin, Christiano, and Fisher (2001), stock returns are volatile; however, most of this volatility is due to extremely volatile risk-free rates.

² Because at any point in time, different workers may be earning different wages, we assume that the economy is made up of many identical households, or families, each with many individuals. Within each family all resources are pooled so that each family's average wage is equal to the average wage in the economy: ${\bar{w}}_{t}$ ⁠.

³ Even though by allowing labor supply to vary, we are able to match both prices and quantities, leaving leisure out of the utility function raises an additional concern. Leisure can provide agents with insurance against consumption fluctuations, as in Uhlig (2007) and Lettau and Uhlig (2000), making it more difficult to match the Sharpe ratio. However, Dittmar, Palomino, and Yang (2014) showed that adding leisure does not interfere (and may help) with matching asset pricing moments in a LRR model. We view our contribution as coming from our model's high volatility and do not believe this channel would be diminished even with a lower Sharpe ratio (and equity premium). Nevertheless, fully incorporating leisure remains an important avenue for future work.

⁴ Because this is independent of the length of employment, we only need to keep track of the number of employees and the average wage as state variables; this way of modeling rigidity is similar to Gertler and Trigari (2009).

⁵ It is possible that $N_{t}^{i} < μ N_{t - 1}^{i}$ ⁠, in which case $μ N_{t - 1}^{i}$ cannot be interpreted as tenured employees. In this case we would interpret the total wage bill as including payments to prematurely laid-off employees. Note that the wage bill can be rewritten as ${\bar{w}}_{t}^{i} N_{t}^{i} = {\bar{w}}_{t - 1}^{i} N_{t}^{i} + (μ N_{t - 1}^{i} - N_{t}^{i}) ({\bar{w}}_{t}^{i} - w_{t})$ ⁠. Here the first term on the right is the wage paid to current employees, and the second term represents the payments to prematurely laid off employees. An earlier version of the paper included the constraint $N_{t}^{i} \geq μ N_{t - 1}^{i}$ ⁠; the results were largely similar.

⁶ The actual values are $g_{1} = - 0.022$ ⁠, $g_{2} = 0.005$ ⁠, and $g_{3} = 0.032$ ⁠, and the transition probabilities are $π_{11} = 0.8$ ⁠, $π_{12} = 0.2$ ⁠, $π_{13} = 0$ ⁠, $π_{21} = 0.1$ ⁠, $π_{22} = 0.8$ ⁠, $π_{23} = 0.1$ ⁠, $π_{31} = 0$ ⁠, $π_{32} = 0.2$ ⁠, and $π_{33} = 0.8$ ⁠.

⁷ The actual values are $Z_{L} = 0.20$ ⁠, $Z_{M} = 0.25$ ⁠, and $Z_{H} = 0.30$ ⁠, and the transition probabilities are $π_{11}^{Z} = 0.965$ ⁠, $π_{12}^{Z} = 0.035$ ⁠, $π_{13}^{Z} = 0$ ⁠, $π_{21}^{Z} = 0.0175$ ⁠, $π_{22}^{Z} = 0.965$ ⁠, $π_{23}^{Z} = 0.0175$ ⁠, $π_{31}^{Z} = 0$ ⁠, $π_{32}^{Z} = 0.035$ ⁠, and $π_{33}^{Z} = 0.965$ ⁠. The volatility of idiosyncratic productivity shocks $Z_{t}^{i}$ depends on the model's scale, that is, which real-world production unit (firm, plant) is analogous to the model's production unit. There is no consensus on the right scale to use. For example, the annual autocorrelation and unconditional standard deviation are 0.69 and 0.40 in Zhang (2005); 0.62 and 0.19 in Gomes (2001); 0.86 and 0.04 in Khan and Thomas (2008), whereas Pastor and Veronesi (2003) estimated that the volatility of firm-level profitability rose from 10% per year in the early 1960s to 45% in the late 1990s. We have experimented with various idiosyncratic shocks and find that our aggregate results are only moderately affected by the size of these shocks.

⁸ The Cobb-Douglas production function is insensitive to rescaling; however, scaling does matter for certain quantities in the CES production function.

⁹ An alternative way to think about job length is the separation rate or the probability of a worker separating from her job in any particular period. Hobijn and Sahin (2009); and Shimer (2005) estimated of separation rates for the United States are around 3% per month. If separations were equally likely for all workers, this would imply an average job length of around 2.8 years. Although this is similar to our contract length, it is actually far smaller than the average job length, because a small number of workers frequently transition between jobs, whereas the majority of workers stay in their jobs for a long time.

¹⁰ For both model and data, we report the volatility of all HP-filtered statistics relative to the volatility of HP-filtered GDP, and the volatility of all growth rates relative to the volatility of the growth rate of GDP. We often just refer to “volatility” or “correlation” in the text, without specifying HP-filtered or growth because the point being made is true for both. In the data, the annual volatilities of HP-filtered GDP and of GDP growth are respectively 3.98% and 5.12%. We calibrate the productivity process so that the model counterparts are virtually identical.

¹¹ For output defined as in our model, $\lim_{η \to - \infty} Y_{t} = \min (K_{t}, Z_{t} N_{t})$ making labor and capital perfect compliments and $\lim_{η \to 1} Y_{t} = \max (K_{t}, Z_{t} N_{t})$ making them perfect substitutes.

¹² We believe the difference is due to a different adjustment function in Kaltenbrunner and Lochstoer (2010).

¹³ For technical reasons our algorithm is unable to solve models where $ρ$ is too close to 1. Because the First Welfare Theorem holds, any frictionless model (without idiosyncratic risk) can be solved using a simple planner's problem algorithm. This is how we solve the $ρ = 1$ case. For the frictionless case where $ρ < 1$ ⁠, we can use either the planner's problem algorithm, or our algorithm; we have confirmed that the two lead to identical results.

¹⁴ The spread in the book-to-market-ratio in our model is smaller than in the data in part because our firm-level productivity shock contains only three states (high/medium/low) and is, therefore, too simple. We have experimented with distributions that are more realistic and the spread in the book-to-market-ratio becomes larger.

¹⁵ Applying similar logic to managerial compensation, Lustig, Syverson, and Van Nieuwerburgh (2011) argued that compensation is set in good times, and therefore, firms who have experienced negative shocks after good times are riskier because of relatively high CEO pay.

¹⁶ The model implied unlevered value premium can be bigger if we introduce an adjustment cost on changes in labor. However, this would further complicate the model mechanism, and we choose to leave it out of the baseline model. The results of the model with costly labor hiring are available upon request.

¹⁷ Many value premium studies start in 1963, and in this case the value premium is due to $α$ only.

¹⁸ It is interesting to note that short-term dividend strips in the frictionless model can be quite volatile. However, this does not necessarily imply volatile equity return because short-term dividend strips carry a negative equity premium, whereas long-term dividend strips carry a positive premium, and thus the two hedge each other to some degree.

¹⁹ One example is Boldrin, Christiano, and Fisher (1999), who provided additional discussion.

²⁰ As an alternative, we defined debt to be credit market liabilities minus assets (line 7 from FOF Table L.101) and the estimated $ρ^{B}$ is very similar because the credit assets of non-financial corporations are very small relative to liabilities.

²¹ Because the set of firms in CRSP is not exactly the same firms as the ones for whom debt is defined in the flow of funds, this definition of $V$ is slightly problematic. However, note that this should mostly affect our estimate of $λ$ but not necessarily $ρ^{B}$ ⁠. For this reason we use other sources to estimate $λ$ ⁠, as described in the previous section. Nevertheless, as discussed in the previous section, the two methods imply similar values for $λ$ ⁠. Further, we have redone everything with logs, or by defining $V$ to be just market value, without adding the debt. Resultant estimates of $ρ^{B}$ are very similar.

²² This can be accomplished by induction. If it is true at $t + 1$ ⁠, then $E_{t} [M_{t + 1} V (Z_{t + 1}^{i}, K_{t + 1}^{i}, N_{t}^{i}, {\bar{W}}_{t}^{i}; Ω_{t + 1})] = Z_{t}^{ρ} E_{t} [{(\frac{Z_{t + 1}}{Z_{t}})}^{ρ} M_{t + 1} v (Z_{t}^{i}, k_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}; Θ_{t})]$ ⁠. Additionally, $d_{t}$ is not a function of $Z_{t}$ ⁠, and thus the right-hand side of Equation A8 can be written as $Z_{t}^{ρ} (d_{t} + E_{t} [{(\frac{Z_{t + 1}}{Z_{t}})}^{ρ} M_{t + 1} v (Z_{t}^{i}, k_{t}^{i}, N_{t - 1}^{i}, {\bar{w}}_{t - 1}^{i}; Θ_{t})])$ ⁠, where the term in parentheses does not depend on $Z_{t}$ ⁠. Because $Z_{t}^{ρ}$ does not affect the maximization problem, it can be pulled outside of the $\max$ operator, and because the right-hand side is linear in $Z_{t}^{ρ}$ ⁠, the left-hand side must be, as well.

²³ This is because even if rational equilibria exist, they are only weakly stable in the sense described by Marcet and Sargent (1989).

²⁴ Because we apply a nonparametric approach, we define the $R^{2} = 1 - \frac{\sum {(x_{t} - E [x_{t} | Θ_{t}])}^{2}}{\sum {(x_{t} - E [x_{t}])}^{2}}$ where $E [x_{t}]$ is the unconditional mean and $E [x_{t} | Θ_{t}]$ is our forecast.

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Article Contents

Wage Rigidity: A Quantitative Solution to Several Asset Pricing Puzzles

Abstract

1. The Model

1.1 Firms

1.1.1 Technology

1.1.2 The wage contract

1.1.3 Accounting

1.1.4 The firm's problem

1.2 Equilibrium

2. Calibration

3. Results

3.1 The standard model

3.2 Infrequent resetting of wages

3.2.1 Elasticity of substitution between capital and labor

3.2.2 Unconditional financial moments

3.2.3 Decomposing the quantitative effect

3.2.4 Conditional asset pricing moments

3.2.5 Cross-sectional asset pricing moments

3.2.6 Term structure of the equity premium

3.3 Extensions

4. Conclusion

Appendix

A.1 Data

A.2 Additional notes on calibration

A.2.1 Financial leverage

A.2.2. Estimation of market-to-book and debt-to-equity.

A.2.3. Estimation of ρB

A.3 Solving the model

A.3.1 Making the Model Stationary

A.3.2 Numerical Solution

References

Supplementary data

Citations

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A.2.3. Estimation of $ρ^{B}$