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Federal Reserve Bank of Chicago
Risk-Adjusted Capital Allocation
and Misallocation
Joel M. David, Lukas Schmid, and
David Zeke
December 18, 2020
WP 2020-34
https://doi.org/10.21033/wp-2020-34
*
Working papers are not edited, and all opinions and errors are the
responsibility of the author(s). The views expressed do not necessarily
reflect the views of the Federal Reserve Bank of Chicago or the Federal
Reserve System.
Risk-Adjusted Capital Allocation and Misallocation∗
Joel M. David†
Lukas Schmid‡
David Zeke§
FRB Chicago
USC Marshall
USC
December 18, 2020
Abstract
We develop a theory linking “misallocation,” i.e., dispersion in marginal products of
capital (MPK), to macroeconomic risk. Dispersion in MPK depends on (i) heterogeneity
in firm-level risk premia and (ii) the price of risk, and thus is countercyclical. We document
strong empirical support for these predictions. Stock market-based measures of risk premia
imply that risk considerations explain about 30% of observed MPK dispersion among US
firms and rationalize a large persistent component in firm-level MPK. Risk-based MPK
dispersion, although not prima facie inefficient, lowers long-run aggregate productivity by
as much as 6%, suggesting large “productivity costs” of business cycles.
JEL Classifications: D24, D25, E22, E32, G12, O47
Keywords: misallocation, productivity, costs of business cycles, risk premia
∗
We thank Andy Atkeson, David Baqaee, Frederico Belo, Harjoat Bhamra, Vasco Carvalho, Gian Luca
Clementi, Wei Cui, Greg Duffee, Andrea Eisfeldt, Emmanuel Farhi, Brent Glover, Francois Gourio, John Haltiwanger, Nir Jaimovich, Şebnem Kalemli-Özcan, Matthias Kehrig, Pete Klenow, Leonid Kogan, Deborah Lucas,
Ellen McGrattan, Ben Moll, Stijn Van Nieuwerburgh, Ezra Oberfield, Christian Opp, Stavros Panageas, Dimitris Papanikolaou, Adriano Rampini, Diego Restuccia, John Shea, Jules van Binsbergen, Venky Venkateswaran,
Neng Wang, Amir Yaron and many seminar and conference participants for helpful comments and suggestions.
The views expressed here are those of the authors and not necessarily those of the Federal Reserve Bank of
Chicago or the Federal Reserve System.
†
joel.david@chi.frb.org.
‡
lukas@marshall.usc.edu.
§
zeke@usc.edu.
1
Introduction
A large and growing body of work has documented the “misallocation” of resources across firms,
i.e., dispersion in the marginal product of inputs into production, and the resulting adverse
effects on aggregate outcomes, such as productivity and output. Recent studies have found
that even after accounting for a host of leading candidates – for example, adjustment costs,
financial frictions or imperfect information – a substantial portion of observed misallocation
seems to stem from other firm-specific factors, specifically, of a type that are orthogonal to firm
fundamentals and are extremely persistent (if not permanent) to the firm.1 Identifying exactly
what – if any – underlying economic forces lead to this type of distortion has proven puzzling.
In this paper, we propose and quantitatively evaluate just such a theory, linking capital
misallocation to macroeconomic risks. To the best of our knowledge, we are the first to make
the connection between standard notions of the risk-return tradeoff and the resulting dispersion
in the marginal product of capital (MPK). Indeed, our framework provides a natural way to
translate firm-level financial market outcomes into the implications for the allocation of capital
across firms. Further, we are able to quantify the effects of risk considerations – i.e., dispersion in
risk premia and the extent of aggregate volatility (and so aggregate risk) – on macroeconomic
variables, such as aggregate total factor productivity (TFP). Through the marginal product
dispersion they induce, risk premium effects – though not prima facie inefficient – depress the
achieved level of TFP, leading to a previously unexplored “productivity cost” of business cycles
in the spirit of Lucas (1987).2
Our point of departure is a standard neoclassical model of firm investment in the face of
both aggregate and idiosyncratic uncertainty. Firms discount future payoffs using a stochastic
discount factor that is also a function of aggregate conditions. Critically, this setup implies that
firms optimally equalize not necessarily MPK, but rather expected, appropriately discounted,
MPK. With little more structure than this, the framework gives rise to a sharp condition
governing the firm’s expected MPK – firms with higher exposure to aggregate risk require a
higher risk premium on investments, which translates into a higher expected MPK. In fact, the
model implies an asset pricing equation of exactly the same form that is often used to price the
cross-section of stock market returns. The equation simply states that a firm’s expected MPK
should be linked to the exposure of its MPK to aggregate risk (i.e., the firm’s “beta”), and the
“price” of that risk. This firm-specific risk premium appears exactly as what would otherwise
be labeled a persistent distortion or “wedge” in the firm’s investment decision.
We use a combination of firm-level production and stock market data to empirically in1
See, e.g., David and Venkateswaran (2019). We discuss the literature in more detail below.
Our analysis is also reminiscent of the approach in Alvarez and Jermann (2004), who use data on asset
prices to measure the welfare costs of aggregate fluctuations.
2
2
vestigate and verify two key implications of this simple framework: (i) firm-level exposure to
aggregate risk, measured using standard risk factors priced in financial markets, is an important determinant of expected MPK and (ii) MPK dispersion is increasing in the price of risk,
measured using common proxies such as credit spreads and the aggregate price/dividend ratio.
Because the price of risk is countercyclical, this link introduces a countercyclical element into
MPK dispersion as observed in the data. We use the empirical results to perform a back-ofthe-envelope calculation that points to an significant role of risk effects in generating MPK
dispersion. Intuitively, the calculation relies on the facts that (i) dispersion in risk premia –
readily measured using data on expected stock market returns – is large and (ii) regression
estimates yield a sizable elasticity of firm-level MPK to expected returns.
After establishing these empirical results, we interpret them and gauge their magnitudes
through the lens of a quantitative model. To that end, we enrich our theory by explicitly
linking the sources of uncertainty to idiosyncratic and aggregate productivity risk.3 We add
two key elements to this framework: (i) a stochastic discount factor designed to match standard
asset pricing facts and (ii) ex-ante cross-sectional heterogeneity in firm exposure, i.e., beta, with
respect to the aggregate productivity shock. The profitability (e.g., productivity or demand) of
high beta firms is highly sensitive to the realization of aggregate productivity (which captures
the state of the business cycle), low beta firms have low sensitivity, and indeed, the profitability
of firms with negative beta may move countercyclically. The investment side of the model is
analytically tractable and yields sharp characterizations of firm investment decisions and MPK.
This setup is consistent with the key empirical results described above, namely, firm-level
expected MPKs depend on exposures to the aggregate productivity shock (the systemic risk
factor in the economy) and due to the countercyclical nature of the price of risk, the crosssectional dispersion in expected MPK is also countercyclical. Further, we derive an expression
for aggregate TFP, which is a strictly decreasing function of MPK dispersion. By inducing
MPK dispersion, cross-sectional variation in exposure to aggregate risk and a higher price of
risk (which depends on the degree of aggregate volatility) reduce the long-run (average) level
of achieved TFP. Thus, the model provides a novel, quantifiable link between financial market
conditions, i.e., the nature of aggregate risk, and longer-run macroeconomic performance.
The strength of these connections relies on three key parameters – the degree of heterogeneity
in firm-level risk exposures and the magnitude and time-series variation in the price of risk. We
devise an empirical strategy to identify these parameters using salient moments from firm-level
and aggregate stock market data, specifically, (i) the cross-sectional dispersion in expected stock
returns, (ii) the market equity premium and (iii) the market Sharpe ratio. We use a linearized
3
These can also be interpreted as shocks to demand. Later, we show that the environment can be extended
to incorporate multiple risk factors and financial shocks.
3
version of our model to derive analytical expressions for these moments and show that they are
tightly linked to the structural parameters. The latter two pin down the level and volatility of
the price of risk and the first identifies the cross-sectional dispersion in firm-level risk exposures.
Indeed, in some simple cases of our model, the dispersion in expected MPK coming from risk
premium effects is directly proportional to the dispersion in expected stock returns – intuitively,
both of these moments are determined by cross-sectional variation in betas.
Before quantitatively evaluating this mechanism, we add other investment frictions to the
environment, specifically, capital adjustment costs. Although they do not change the main
insights from the simpler model, we uncover an important interaction between these costs and
risk premia – namely, adjustment costs amplify the effects of beta variation on MPK dispersion.
Intuitively, adjustment costs generate a second source of co-movement of firm outcomes with
aggregate conditions – i.e., fluctuations in the value of installed capital – and hence add an
additional component to the risk premium. Because the value of capital turns out to be more
procyclical for high beta firms, the risk premium is in turn higher for these firms. On their
own, adjustment costs have only transitory effects and do not lead to persistent dispersion in
firm-level MPK. However, they can augment the effects of other factors that do, such as the
heterogeneity in risk premia we analyze here.
We apply our methodology to data on US publicly traded firms from Compustat/CRSP. Our
estimates reveal substantial variation in firm-level betas and a sizable price of risk – together,
these imply a significant amount of risk-induced MPK dispersion. For example, our results
suggest risk premium effects can explain as much as 30% of total observed MPK dispersion.
Importantly, this dispersion is largely due to persistent MPK deviations at the firm-level, exactly
of the type that compose a large portion of measured misallocation. Indeed, risk effects can
account for as much as 47% of the permanent component in the data. The implications of
these findings for the long-run level of aggregate TFP are significant – cross-sectional variation
in risk reduces TFP by as much as 6%. Note that this represents a quantitative estimate of
the impact of financial market outcomes on macroeconomic performance and further, a new
connection between the nature of business cycle volatility and long-run outcomes in the spirit
of Lucas (1987). Here, higher aggregate volatility leads to greater aggregate risk, increasing
dispersion in required rates of return and MPK and thus reducing TFP. Our results suggest
these “productivity costs” of business cycles may be substantial.
Our estimates also imply a significant countercyclical element in expected MPK dispersion. For example, the parameterized model produces a correlation between the cross-sectional
variance in expected MPK and the state of the business cycle (measured by the aggregate
productivity shock) of -0.31. To put this number in context, the correlation between MPK dispersion and aggregate productivity in the data is about -0.27. This result provides a risk-based
4
explanation for the puzzling observation, made forcefully by Eisfeldt and Rampini (2006), that
capital reallocation is procyclical, despite the apparently countercyclical gains – due to the
countercyclical nature of the price of risk and high beta of high MPK firms, such reallocation
in downturns would require capital to flow to the riskiest of firms in the riskiest of times.
We pursue two important extensions of our baseline analysis. First, we add a flexible class of
firm-specific “distortions” of the type that have been emphasized in the misallocation literature.
These distortions can be fixed or time-varying and may be correlated or uncorrelated with firmlevel characteristics (including betas) and with the state of the business cycle. We show that
to a first-order approximation, our identification strategy and results are either unaffected by
these distortions or are likely conservative (depending on the exact correlation structure of the
distortions). These findings highlight an important feature of our empirical approach: although
observed misallocation may stem from a wide variety of sources, our approach to measuring risk
premium effects yields a robust estimate of the contribution of this one source alone. Second, we
provide further, direct evidence on the extent of beta dispersion. Rather than relying on stock
market data, we compute firm-level betas using production-side data by estimating time-series
regressions of firm-level productivity on measures of aggregate productivity. The beta is the
coefficient from this regression. This approach yields beta dispersion on par with the dispersion
implied by the cross-section of stock market returns.
Why do firms (within an industry) have different exposure to the business cycle? Although
our analysis does not require us to take a stand on this question, we explore a number of
potential explanations. First, we investigate heterogeneity in production technologies (i.e.,
input elasticities) and markups. We show that these types of heterogeneity can indeed lead
to variation in firm responsiveness/exposure to shocks but at most are likely to account for
about 12% and 6% of the estimated standard deviation of betas, respectively. Although nonnegligible, these results suggest that the majority of beta dispersion stems from other sources.
Next, we show that theories of “trading down” over the business cycle as in, e.g., Jaimovich
et al. (2019), may be a promising explanation. In times of economic expansion, when purchasing
power is high, consumers tend to substitute towards higher quality goods while in downturns
they substitute towards lower quality ones. Thus, higher quality products are more procyclical
and lower quality ones less so. Although systematically quantifying this channel is challenging
(such an analysis would require comprehensive product quality data), we provide evidence from
a single industry – eating places – where we were able to obtain a proxy for quality, namely,
average price. Low-price establishments tend to have lower betas than high-price ones and
further, average price is positively related to betas, expected returns and MPK. This case study
of a single industry helps illustrate the main relationships implied by our theory and suggests
quality differentiation may be an important factor behind differences in firm cyclicality.
5
Related Literature. Our paper relates to several branches of literature. Foremost is the
large body of work investigating resource misallocation, seminal examples of which include
Hsieh and Klenow (2009) and Restuccia and Rogerson (2008). A number of recent papers
have explored the role of financial frictions, for example, Midrigan and Xu (2014), Moll (2014)
and Buera et al. (2011) study collateral constraints and Gilchrist et al. (2013) firm-specific
borrowing costs. Gopinath et al. (2017) and Kehrig and Vincent (2017) study the interaction
of financial frictions and adjustment costs in explaining recent dynamics of misallocation in
Spain and within firms, respectively. We build on this literature by exploring the implications
of a different dimension of financial markets for marginal product dispersion, namely, the riskreturn tradeoff faced by risk-averse agents. The addition of aggregate risk is a key innovation
of our analysis – existing work has typically abstracted from this channel (either by assuming
no aggregate uncertainty or risk-neutral agents). We show that the link between aggregate risk
and observed misallocation is quite tight in the presence of heterogeneous exposures to that
risk.4 Papers studying additional candidates behind marginal product dispersion include Peters
(2016), Edmond et al. (2018) and Haltiwanger et al. (2018) who focus on markup dispersion,
David et al. (2016) information frictions and Asker et al. (2014) capital adjustment costs.5 David
and Venkateswaran (2019) provide an empirical methodology to disentangle various sources of
capital misallocation and establish a large role for highly persistent firm-specific factors. In our
theory, firm-level risk premia manifest themselves as persistent firm-specific “wedges” of exactly
this type.
Kehrig (2015) documents in detail the countercyclical nature of productivity dispersion.
We build on this finding by relating fluctuations in MPK dispersion to time-series variation in
the price of risk. A growing literature, starting with Eisfeldt and Rampini (2006), investigates
the procyclical nature of capital reallocation, which is puzzling since higher cross-sectional
dispersion in MPK in downturns should lead capital to flow to highly productive, high MPK
firms in recessions. Our results bear on that observation by noting that the countercyclicality
of the price of risk, in conjunction with heterogeneity in firm-level risk exposures, goes some
way toward reconciling this puzzle.
Our work also relates to a large literature exploring the link between financial market re4
Gilchrist et al. (2013) find a limited role for firm-specific borrowing costs. Where they focus mainly on
costs of debt, we find a larger role for differences in costs of equity, which is an important source of financing
for the firms in our data (for example, the average leverage ratio in our sample is 0.28). Indeed, in our simple
theory, the Modigliani-Miller theorem holds, i.e., all firms are able to borrow at the common risk-free rate and
thus there is no dispersion in borrowing costs. Yet equity costs – and so total costs of capital – may differ
widely. One contribution of our work is extending the insights in Gilchrist et al. (2013) to a broader notion of
financing costs and showing that the implications for misallocation can be quite different.
5
Many papers study the role of firm-specific distortions, e.g., Bartelsman et al. (2013). Restuccia and
Rogerson (2017), Hopenhayn (2014) and Eisfeldt and Shi (2018) provide excellent overviews of recent work on
capital misallocation/reallocation.
6
turns and the return to capital (“investment returns”). Cochrane (1991), Restoy and Rockinger
(1994) and Balvers et al. (2015) show that stock returns and investment returns are closely
linked (indeed, exactly coincide under constant returns to scale). Recent work builds on this
insight to examine the cross-section of stock returns from the perspective of investment returns, interpreting common risk factors through firms’ investment policies and showing that
investment-based factors are priced in the cross-section of returns, e.g., Zhang (2005), Gomes
et al. (2006), Liu et al. (2009) and Zhang (2017). We examine investment returns and the
marginal product of capital as a joint manifestation of risk premia, most readily measured
through stock returns, and extend this connection to analyze the implications for the allocation
of capital and macroeconomic outcomes, such as aggregate TFP. Binsbergen and Opp (2017)
also investigate the implications of asset market considerations for the real economic decisions
of firms. They propose a framework where distortions in agents’ subjective beliefs lead to “alphas,” i.e., cross-sectional mis-pricings, and real efficiency losses, whereas we focus on the MPK
dispersion induced by heterogeneity in aggregate risk exposures. Our empirical work establishes
a connection between MPK and financial market outcomes and our quantitative work uses a
workhorse macroeconomic model of firm dynamics augmented with risk-sensitive agents and
aggregate risk to evaluate the implications of this connection. One of our key messages shares a
common theme with this line of work – financial market considerations can have sizable effects
on real outcomes by affecting capital allocation decisions.6
2
Motivation
In this section, we lay out a simple version of the standard, frictionless neoclassical theory of
investment to illustrate the main insight of our analysis, namely, the link between firm-level
MPK and risk premia. We use this framework to motivate a number of empirical exercises
exploring this connection and guide a simple back-of-the-envelope calculation that suggests a
significant role for risk in generating MPK dispersion.7 Section 3 enriches this environment
along several dimensions for purposes of our quantitative work.
2.1
MPK and Risk Premia
Firms produce output using capital and labor according to a Cobb-Douglas technology and
face constant (or infinite) elasticity demand curves. Labor is chosen period-by-period in a spot
market at a competitive wage. At the end of each period, firms choose investment in new capital,
6
Relatedly, David et al. (2014) find that risk considerations play an important role in determining the
allocation of capital across countries, i.e., can explain some portion of the “Lucas Paradox.”
7
All derivations for this section are in Appendix A.
7
which becomes available for production in the following period so that Kit+1 = Iit + (1 − δ) Kit ,
where δ is the rate of depreciation. Let Πit = Πit (Xt , Zit , Kit ) denote the operating profits of
the firm – revenues net of labor costs – where Xt and Zit denote aggregate and idiosyncratic
shocks to firm profitability, respectively, and Kit the firm’s stock of capital. Both Xt and Zit
may be vectors, i.e., there may be multiple sources of both idiosyncratic and aggregate risk.
The analysis can accommodate a number of interpretations of the fundamental shocks, for
example, as productivity or demand shifters. With these assumptions, the profit function takes
a Cobb-Douglas form, is homogeneous in K of degree θ < 1 (due to curvature in production
and/or demand) and is proportional to revenues. The marginal product of capital is equal to
Πit
. The payout of the firm in period t is equal to Dit = Πit − Iit .
M P Kit = θ K
it
Firms discount future cash flows using a stochastic discount factor (SDF), Mt+1 , which is
correlated with the aggregate shock(s), Xt . We can write the firm’s problem recursively as
V (Xt , Zit , Kit ) = max Πit (Xt , Zit , Kit ) − Kit+1 + (1 − δ) Kit + Et [Mt+1 V (Xt+1 , Zit+1 , Kit+1 )] ,
Kit+1
(1)
where Et [·] denotes the firm’s conditional expectations. The Euler equation is given by
1 = Et [Mt+1 (M P Kit+1 + 1 − δ)]
∀ i, t .
(2)
MPK dispersion. An immediate implication of expression (2) is that MPK (or even expected
MPK) need not be equated across firms; rather, it is only expected, appropriately discounted
MPK that is equalized. To the extent that firms’ MPK co-move differently with the SDF, their
expected MPK will differ. From here, we can derive the following equation for expected MPK:
Et [M P Kit+1 ] = αt + βit λt .
(3)
Here, αt = rf t +δ is the risk-free user cost of capital, where rf t is the (net) risk-free interest rate,
t+1 ,M P Kit+1 )
captures the elasticity, or exposure, of the firm’s MPK to movements
βit ≡ − covt (M
vart (Mt+1 )
t (Mt+1 )
is the market price of that
in the SDF – i.e., the riskiness of the firm – and λt ≡ var
Et [Mt+1 ]
risk. Expression (3) illustrates the first main insight: expected MPK is not necessarily common
across firms, but rather is a function of the (common) risk-free rate and a firm-specific risk
premium, which depends on the firm’s beta on the SDF – which may vary across firms – and
the market price of risk.
The cross-sectional variance of date t conditional expected MPK is
σE2 t [M P Kit+1 ] = σβ2t λ2t ,
8
(4)
where σβ2t is the cross-sectional variance of time t betas. Expression (4) reveals the second main
insight: the extent to which risk considerations lead to dispersion in expected MPK is increasing
in the price of risk (and in the cross-sectional variation in risk exposures). A key observation
underlying our analysis is that financial market data imply that risk prices are high (e.g., a
large equity premium and observed Sharpe ratios on various investment strategies), suggesting
a potentially important role for differences in risk exposure in leading to MPK dispersion.
Further, given persistence in firm-level betas, the theory implies persistent differences in firmlevel MPK – as observed in the data – driven by dispersion in required rates of return.8
Examples. It is useful to consider a few concrete illustrative examples:
Example 1: no aggregate risk (or risk neutrality). In the case of no aggregate risk, we have
βit = 0 ∀ i, t, i.e., all shocks are idiosyncratic to the firm. Expressions (3) and (4) show
that there will be no dispersion in expected MPK and for each firm, Et [M P Kit+1 ] = rf + δ,
which is simply the riskless user cost of capital (which is constant in the absence of aggregate
shocks). This is the standard result from the stationary models widely used in the misallocation
literature where, without additional frictions, expected MPK should be equalized across firms.9
The same result holds in an environment with aggregate shocks but risk neutral preferences,
which implies Mt+1 is simply a constant (equal to the time discount factor).
Example 2: CRRA preferences. In the case of CRRA utility with coefficient of relative risk
aversion γ, standard approximation techniques give
Et [M P Kit+1 ] = αt +
covt (∆ct+1 , M P Kit+1 )
γvar (∆c ) ,
| t {z t+1}
vart (∆ct+1 )
|
{z
}
λt
βit
where ∆ct+1 denotes log consumption growth. Expected MPK is determined by the covariance
of the firm’s MPK with consumption growth. The price of risk is the product of the coefficient
of relative risk aversion and the conditional volatility of consumption growth.
Example 3: CAPM. In the CAPM, the SDF is linearly related to the return on the aggregate
stock market, i.e., Mt+1 = a − brmt+1 for some constants a and b. Because the market portfolio
8
To see this more clearly, we can take the unconditional expectation of equation (3) to obtain an approximate
2
2 2
2
2
expression for the variance of firms’ mean MPKs as σE[M
P K] ≈ σβ λ , where σβ ≡ σE[βit ] denotes the variance
of unconditional betas and λ ≡ E [λt ] the unconditional expectation of the price of risk. The approximation is
valid as long as cov (βi , cov (βit , λt )) is small. In line with the results in Lewellen and Nagel (2006), we find the
time-series variation in betas to be quite modest. Further, they are persistent (for example, we find that CAPM
betas have an implied one-year autocorrelation of 0.87). In the case of constant betas or if βit is orthogonal to
λt , the expression is exact.
9
With time-to build for capital and uncertainty over upcoming shocks, there may still be dispersion in realized
MPK, but not in expected terms, and so these forces do not lead to persistent firm-level MPK deviations.
9
is itself an asset with β = 1, it is straightforward to derive
Et [M P Kit+1 ] = αt +
covt (rmt+1 , M P Kit+1 )
E [r
− rf t ] ,
| t mt+1
{z
}
vart (rmt+1 )
|
{z
}
λt
βit
i.e., expected MPK is determined by the covariance of the firm’s MPK with the market return,
which is the aggregate risk factor in this environment. The price of risk is equal to the expected
excess return on the market portfolio, i.e., the equity premium.
Extensions. As can be seen from this simple framework, the link between firm-level MPK
and risk is quite general and does not depend on specific assumptions about the SDF. In
Appendix A.2, we provide two additional examples to show that the connection holds under
alternative assumptions on the sources of risk premia. Specifically, we study versions where
firm-specific risk is due to (i) firm-specific distortions to the pricing of risk (which show up
as “alphas” or “mis-pricing”) and (ii) heterogeneity in firm exposures to cyclical fluctuations
in the price of investment goods (in the spirit of, e.g., Kogan and Papanikolaou (2013)). In
both cases, differences in firm-level risk premia leads to MPK dispersion exactly as above, i.e,
an expression analogous to (3) continues to hold. The main message from these excercises is
that the link between MPK and risk is quite robust and does not depend on the precise source
of differences in risk premia (although the normative implications, e.g., whether the resulting
MPK dispersion is efficient or not – and so represents a true “misallocation” – may).
In our quantitative model in Section 3, we add capital adjustment costs, which lead to
endogenous fluctuations in the value of installed capital, i.e., Tobin’s Q, and study the role of
additional factors (e.g., firm-level distortions) that determine expected MPK as emphasized in
the misallocation literature (e.g., Hsieh and Klenow (2009)). In these cases, equations (3) and
(4) do not hold exactly, since there are now additional forces generating MPK dispersion, but
we develop an empirical strategy that accurately captures the portion of this dispersion that
comes from risk effects alone, even in the presence of these other factors.
2.2
Empirical Evidence
In this section, we explore the two key implications of the simple framework laid out thus far:
(i) exposure to aggregate risk – and hence, the firm-specific risk premium – is a determinant of
expected MPK and (ii) MPK dispersion is increasing in the price of risk.
Measuring risk premia. In addition to influencing firms’ capital choices, exposure to aggregate risk affects firms’ stock returns. Indeed, in our model, expected excess stock returns
10
over the risk-free rate (i.e., the predictable component of the excess return) reflects compensation for risk exposure and thus represents a direct measure of the firm-level risk premium. We
can exploit this link to use well-studied measures of firm risk exposures and risk premia from
stock market data to explore the connection with MPK. To motivate this approach, we derive
the following approximate expressions for firm-level expected excess MPK (over the user cost),
e
e
denoted M P Kit+1
, and excess stock market returns, Rit+1
:
e
e
= −covt (mpkit+1 , mt+1 )
≡ log Et M P Kit+1
Empkit+1
e
e
= −ψcovt (mpkit+1 , mt+1 ) ,
≡ log Et Rit+1
Erit+1
(5)
(6)
where ψ is a constant and lowercase denotes natural logs. To a first-order approximation,
expected stock returns and MPK are proportional, since they are jointly determined by the
underlying risk characteristics of the firm.10 Combining, we have
e
Empkit+1
1 e
e
= Erit+1
⇒ var Empkit+1
=
ψ
2
1
e
var Erit+1
,
ψ
(7)
which shows that (in logs) the cross-sectional dispersion of expected MPK is proportional to
the cross-sectional dispersion of expected stock market returns. We can use expression (7) to (i)
verify the link between firm-level MPK and risk premia, measured from expected stock returns
and (ii) calculate a back-of-the-envelope estimate of the MPK dispersion that may stem from
heterogeneous risk premia.
Firm-level MPK and risk. Expression (7) reveals a tight link between expected MPK and
expected stock returns. However, empirically implementing this equation is challenging since
neither expected MPK nor expected returns are directly observable. One approach is to proxy
these variables with their realized values, but this may be problematic: since both variables
respond to the same realizations of shocks, a positive relationship may be mechanical and not
indicative of the relationship between expected values. To overcome these hurdles, we follow
a two-stage instrument variables approach in which we instrument for returns using common
measures of risk exposure. First, in a preliminary step, we estimate time-varying risk exposures
from backwards-looking rolling window regressions of individual firm stock returns on aggregate
risk factors. Then, in the first stage, we estimate period-by-period cross-sectional regressions
of realized stock market returns on the estimated exposures. The predicted values from these
regressions yield measures of expected excess returns, i.e., risk premia, that are driven only
10
The proportionality is exact under a single source of aggregate risk. A slightly modified expression holds
with multiple sources.
11
by exposure to the aggregate risk factors considered.11 Next, we estimate the second stage
regression implied by equation (7) using a panel specification with these predicted values as
instruments. Because the focus in the misallocation literature is generally on within-industry
dispersion in MPK (MPK may vary across industries due to heterogeneity on a number of
additional dimensions), we include a full set of industry-by-year fixed effects. We perform this
procedure using three different sets of aggregate risk factors, taken from three common asset
pricing models: the CAPM, the Fama-French 3 Factor model, and the Hou et al. (2015) q 5
5 factor model. In the CAPM, the aggregate market return is the single source of aggregate
risk. The latter two models add returns on additional portfolios of firms sorted by a number of
different characteristics. We provide further details in Appendix B.1.
In principle, once we have a measure of expected returns, proxying expected MPK with
realized MPK should be innocuous, since the forecast errors in future realizations of MPK
– the difference between the two measures – should be uncorrelated with backwards-looking
data on stock returns. However, for completeness, we also construct a measure of expected
MPK as follows: under our assumptions, realized (log) MPK is given by mpkit = yit − kit (we
suppress constants that play no role), where yit denotes (log) revenue. We assume that firmlevel productivity, equal to ait = yit − θkit , follows an AR(1) process with persistence ρa . Then,
expected MPK is given by Et [mpkit+1 ] = Et [yit+1 ] − kit+1 = ρa ait − (1 − θ) kit+1 . Consistent
with our estimates in Section 4 below, we use ρa = 0.94 and θ = 0.62.12
We implement this approach using firm-level data from the Center for Research in Security
Prices (CRSP) and Compustat.13 We measure firm capital stock, Kit , as the (net of depreciation) value of property, plant and equipment and firm revenue, Yit , as reported sales.14 Full
details of the data sample are in Appendix B.1.
Table 1 reports results from regressions of MPK and expected MPK on lagged expected
stock market returns. The left-hand panel shows that each of the asset pricing models we consider yields a statistically significant relationship between expected returns and future realized
MPK.15 The economic magnitudes are also significant: the estimated coefficients imply that
that a 100 basis point increase in the expected return is associated with a 3-5% increase in
11
Because the estimates of risk exposures use data only through time t, they forecast only the expected
component of realized returns at time t + 1.
12
A simplifying assumption here is that the aggregate and idiosyncratic components of ait have the same
persistence. Our quantitative work in Section (4) suggests this is a reasonable approximation, where we find
the persistences of the two components to be quite close.
13
By using Compustat data, our analysis focuses on large, publicly traded firms for which financial market
data are available. Extending the analysis to private firms would be a valuable exercise, but faces a significant
challenge in deriving accurate measures of risk premia (our alternative approach in Section 4.4 may be one way).
14
Using book assets, a broader notion of firm capital, yields similar results.
15
Standard errors are two-way clustered by firm and year, but do not account for estimation error in measured
risk exposures. In Appendix C, we perform a bootstrapping procedure designed to address this issue.
12
Table 1: MPK and Risk Premia
mpk
Ecapm [ret]
Ef f 3 [ret]
Eq5 [ret]
(1)
4.798∗∗∗
(4.08)
(2)
E[mpk]
(3)
3.284∗∗∗
(6.80)
(4)
5.388∗∗∗
(4.62)
(5)
(6)
4.159∗∗∗
(8.70)
5.016∗∗∗
(8.22)
6.299∗∗∗
(10.71)
Notes: This table reports results from a two-step procedure in which we estimate the elasticity of firm-level MPK to expected excess
stock returns. In the first stage, we instrument for expected returns using measures of firm risk exposures from stock market data.
In the second stage, we run regressions of the form (7) with industry-by-year fixed effects. Standard errors are two-way clustered
by firm and year. t-statistics in parentheses. Significance levels are denoted by: * p < 0.10, ** p < 0.05, *** p < 0.01.
MPK. The right-hand panel reports analogous results using measures of expected, rather than
realized, MPK. The estimated coefficients are similar, and if anything, slightly larger, though
the differences are not statistically significant. Thus, our two-stage approach reveals a significant relationship between firm-level risk premia, instrumented using common measures of risk
exposure from stock market data, and MPK.
The results in Table 1 suggest a simple back-of-the-envelope calculation of the role of risk
premia in driving MPK dispersion. As an example, consider the Fama-French 3 factor model,
perhaps the most widely used framework to study the cross-section of stock returns. Using
this model to estimate expected returns yields a within-industry standard deviation of about
0.127 (details in Appendix B.2). Multiplying this value by the estimated elasticity of MPK
to expected returns in Table 1 (3.3) and squaring yields a predicted cross-sectional variance
of mpk of about 0.17. The total within-industry variance of mpk in the Compustat sample is
about 0.45, implying that variation in risk premia can account for almost 40% of the total. A
similar calculation using expected MPK gives a slightly larger share.
Although suggestive, this calculation ignores a number of factors that may affect the relationship between MPK and returns, e.g., possible nonlinearities, adjustment costs and/or other
investment distortions of the type that have been emphasized in the misallocation literature.
In Sections 3 and 4, we develop a quantitative model and empirical strategy that incorporates
these additional considerations and allows for counterfactual decompositions of the role of risk
effects in driving observed MPK dispersion. The results from the more general model turn out
to be broadly in line with (though slightly smaller than) the simple calculation performed here.
MPK dispersion and the price of risk. Expression (4) illustrates a second key implication
of the simple framework: MPK dispersion is positively related to the price of risk. To explore
13
Table 2: MPK Dispersion and the Price of Risk
σ (mpkit+1 )
Excess Bond Premia
(1)
0.357∗∗∗
(3.08)
σ (Et [mpkit+1 ]))
(2)
(3)
0.191∗∗∗
(2.75)
GZ Spread
174
0.0644
(5)
(6)
0.338∗∗∗
(5.68)
-0.176∗∗∗
(-3.51)
174
0.0660
PD Ratio
Observations
R-squared
(4)
0.633∗∗∗
(6.89)
174
0.0653
174
0.218
174
0.221
-0.228∗∗∗
(-4.09)
174
0.117
Notes: This table reports time-series regressions of four-quarter ahead mpk dispersion on measures of the price of risk. t-statistics
are in parentheses, computed using Newey-West standard errors. Significance levels are denoted by: * p < 0.10, ** p < 0.05, ***
p < 0.01.
this prediction, we estimate time-series regressions of the form
σ (mpkt+1 ) = ψ0 + ψ1 λt + ζt+1 ,
where λt denotes three different proxies for the price of risk: the price/dividend (PD) ratio on the
aggregate stock market and two measures of credit spreads – the Gilchrist and Zakrajsek (2012)
(GZ) spread, a high-information and duration-adjusted measure of the mean credit spread
and the excess bond premium, which measures the portion of the GZ spread not attributable
to default risk.16 Table 2 reports regressions of the within-industry standard deviations of
MPK (left-hand panel) and expected MPK (right-hand panel) on the lagged values of these
measures.17 Each predicts MPK dispersion, and in the direction the theory suggests: the GZ
Spread and excess bond premium predict greater MPK dispersion, while a higher PD ratio
predicts lower dispersion. Because the measures of the price of risk are countercyclical, the
results imply that time-series variation in risk premia induce a countercyclical component in
MPK dispersion, in line with (and potentially in part accounting for) the well known evidence
of countercyclicality documented in Eisfeldt and Rampini (2006) and Kehrig (2015).18
16
We extract the cyclical component of the PD ratio and MPK dispersion using a one-sided Hodrick-Prescott
filter. The credit spread measures do not exhibit significant longer-term trends.
17
Within-industry standard deviations are calculated by first de-meaning mpk (expected mpk) by industryyear. To control for the changing composition of firms, for each quarter, we include only firms that were present
in the previous quarter and calculate changes in the standard deviation for these firms. We then use those
changes to construct a composition-adjusted series that is unaffected by new additions or deletions from the
dataset. Details of this procedure are in Appendix B.1.
18
We report the correlations of these measures with de-trended GDP and TFP in Table 6 in Appendix B.3.
14
Additional exercises. In Appendix C, we explore a number of additional implications of
our framework. We use MPK-sorted portfolios of firms to show that high MPK firms tend to
offer higher expected stock market returns. We verify that firm-level MPK is related to direct
measures of the sensitivity of MPK (rather than stock returns) to aggregate risk, as suggested
by (3). We show that industries with greater MPK dispersion also tend to be those with more
dispersion in expected returns and heterogeneity in firm-level risk exposures.
3
Quantitative Model
In this section, we use a more detailed version of the investment model laid out above to quantitatively investigate the contribution of heterogeneous risk premia to observed MPK dispersion.
The model is kept deliberately simple in order to isolate the role of our basic mechanism, namely
dispersion in exposure to macroeconomic risk. The theory consists of two main building blocks:
(i) a stochastic discount factor, which we directly parameterize to be consistent with salient
patterns in financial markets, i.e., high and countercyclical prices of risk and (ii) a cross-section
of heterogeneous firms, which make optimal investment decisions in the presence of firm-level
and aggregate risk, given the stochastic discount factor. Specifying the stochastic discount
factor exogenously allows us to sidetrack challenges with generating empirically relevant risk
prices in general equilibrium, and focus on gauging the quantitative strength of our mechanism. To hone in on the effects of risk premia, we begin with a simplified version in which we
abstract from additional adjustment frictions. In this case, our framework yields exact closed
form solutions for firm investment decisions. In Section 3.2, we extend the model to include
capital adjustment costs. Our theoretical results there reveal an important amplification effect
of these costs on the impact of risk premia.19
Heterogeneity in risk exposures. The setup is a fleshed-out version of that in Section 2.
We consider a discrete time, infinite-horizon economy. A continuum of firms of fixed measure
one, indexed by i, produce a homogeneous good using capital and labor according to:
Yit = Xtβ̂i Ẑit Kitθ1 Nitθ2 ,
θ1 + θ2 < 1 .
(8)
Firm productivity (in logs) is equal to β̂i xt + ẑit , where xt denotes an aggregate component that is common across firms and β̂i captures the exposure of the productivity
of firm i
¯ 2
to aggregate conditions. We assume that β̂i is distributed as β̂i ∼ N β̂, σβ̂ across firms.
Heterogeneity in this exposure is a key ingredient of our framework – cross-sectional variation
19
We also consider the effects of other investment frictions, e.g., “wedges,” or distortions, in Section 4.3.
15
in β̂i will lead directly to dispersion in expected MPK. The term ẑit denotes a firm-specific,
idiosyncratic component of productivity.20
The two productivity components follow AR(1) processes (in logs):
xt+1 = ρx xt + εt+1 ,
ẑit+1 = ρz ẑit + ε̂it+1 ,
εt+1 ∼ N 0, σε2
2
ε̂it+1 ∼ N 0, σε̂
(9)
.
Thus, there are two sources of uncertainty at the firm level – aggregate uncertainty, with
conditional variance σε2 , and idiosyncratic uncertainty, with variance σ̂ε̃2 .
Stochastic discount factor. In line with the large literature on cross-sectional asset pricing
in production economies, we parameterize directly the pricing kernel without explicitly modeling
the consumer’s problem. In particular, we specify the SDF as
1
log Mt+1 ≡ mt+1 = log ρ − γt εt+1 − γt2 σε2
2
γt = γ0 + γ1 xt ,
(10)
where ρ, γ0 > 0 and γ1 ≤ 0 are constant parameters.21 The SDF is determined by shocks to
aggregate productivity. The conditional volatility of the SDF, σm = γt σε , varies through time as
determined by γt . This formulation allows us to capture in a simple manner a high, time-varying
and countercyclical price of risk as observed in the data (since γ1 < 0, γt is higher following
economic contractions, i.e., when xt is negative). Additionally, directly parameterizing γ0 and
γ1 enables the model to be quantitatively consistent with key moments of asset returns, which
are important for our analysis. The risk free rate is constant and equal to − log ρ. Thus, γ0 and
γ1 only affect the properties of equity returns., easing the interpretation of these parameters.
The maximum attainable Sharpe ratio is equal to the conditional standard deviation of the
SDF, i.e., SRt = γt σε , and the price of risk is equal to the square of the Sharpe ratio, γt2 σε2 .
For simplicity, the setup thus far features (i) a single source of aggregate risk and (ii) a
tight link between financial market conditions (i.e., γt ) and macroeconomic conditions (i.e., xt ).
In Appendix F we extend this framework to (i) include multiple risk factors and (ii) decouple
movements in financial and macroeconomic conditions by including pure financial shocks that
affect the price of risk but otherwise do not impact firm profits/productivity. Similar insights
from the simpler model go through under those extensions.
20
More broadly, expression (8) should be thought of as a revenue-generating function and the “productivity”
components as also capturing demand factors, see, e.g., Section 5.
21
This specification builds closely on those in, for example, Zhang (2005), Gomes and Schmid (2010) and
Jones and Tuzel (2013).
16
Input choices. Firms hire labor period-by-period at a competitive wage, Wt . To keep the
labor market simple, we assume that the equilibrium wage is given by
Wt = Xtω ,
i.e., the wage is a constant elasticity and increasing function of aggregate productivity, where
ω ∈ [0, 1] determines the sensitivity of wages to aggregate conditions.22 Maximizing over the
static labor decision gives operating profits – revenues less labor costs – as
Πit = GXtβi Zit Kitθ ,
θ2
1−θ2
(11)
1
θ1
where G ≡ (1 − θ2 ) θ2 , βi ≡
β̂i − ωθ2 , Zit ≡ Ẑit1−θ2 and θ ≡ 1−θ
. The exposure of
2
firm profits to aggregate conditions is captured by βi , which is a simple transformation of the
underlying exposure of firm productivity to the aggregate component, β̂i , and the sensitivity
1
of wages, ω.23 The idiosyncratic component of productivity is similarly scaled, by 1−θ
. The
2
curvature of the profit function is equal to θ, which depends on the relative elasticities of capital
and labor in production. These scalings reflect the leverage effects of labor liabilities on profits.
From here on, we will primarily work with zit , which has the same persistence as ẑit , i.e., ρz ,
1
1−θ2
2
1
1
ε̂t+1 with variance σε̃2 = 1−θ
σε̂2 . We will also use the fact that
and innovations εit+1 = 1−θ
2
2
2
1
σβ2 = 1−θ
σβ̂2 . Notice that the profit function takes precisely the form assumed in Section
2
2. Thus, the firm’s dynamic investment problem takes the form in expression (1).
Optimal investment. The simplicity of this setting leads to exact analytical expressions for
the firm’s investment decision. Specifically, we show in Appendix D.1 that the firm’s optimal
investment policy is given by:
kit+1 =
1
α̃ + βi ρx xt + ρz zit − βi γt σε2 ,
1−θ
(12)
where α̃ ≡ log θ + log G − α, α ≡ log (rf + δ).24 The firm’s choice of capital is increasing
in xt and zit due to their direct effect on expected future productivity (i.e., βi ρx xt + ρz zit =
Et [βi xt+1 + zit+1 ]), but, ceteris paribus, firms with higher betas choose a lower level of capital.
The magnitude of this effect is larger when γt is large, i.e., in economic downturns. Clearly,
22
This setup follows, for example, Belo et al. (2014) and İmrohoroğlu and Tüzel (2014).
The adjustment term for labor supply, ωθ2 , has a small effect on the mean of the β distribution, but
otherwise does not affect our analysis.
24
More precisely, there are also terms that reflect the variance of shocks. Because these terms are negligible
and play no role in our analysis (they are independent of the risk premium effects we measure), we suppress
them here. The full expressions are given in Appendix D.1.
23
17
with risk neutrality, i.e., γ0 = γ1 = 0, the last term is zero and investment is purely determined
by expected productivity.
For a slightly different intuition, we substitute for γt and write the expression as
kit+1 =
1
α̃ + βi ρx − γ1 σε2 xt + ρz zit − βi γ0 σε2 .
1−θ
(13)
The risk premium affects the capital choice through both the time-varying and constant components of the price of risk: first, a more negative γ1 increases the responsiveness of firms to
aggregate conditions. Intuitively, a high (low) realization of xt has two effects – first, since xt
is persistent, it signals that productivity is likely to be high (low) in the future, increasing (decreasing) investment (this force is captured by the ρx term). Moreover, a high (low) realization
of xt implies a low (high) price of risk, which further increases (decreases) investment. Second,
the constant component of the risk premium, γ0 , adds a firm-specific constant – i.e., a fixed
effect – which leads to permanent dispersion in firm-level capital.
MPK dispersion. By definition, the realized mpk is given by mpkit+1 = log θ + πit+1 − kit+1 .
Substituting for kit+1 ,
mpkit+1 = α + εit+1 + βi εt+1 + βi γt σε2 ,
(14)
and taking conditional expectations,
Empkit+1 ≡ Et [mpkit+1 ] = α + βi γt σε2 ,
(15)
where α is as defined in equation (12) and reflects the risk-free user cost of capital. Expression
(14) shows that dispersion in the realized mpk can stem from uncertainty over the realization
of shocks, as well as the risk premium term, which is persistent at the firm level and depends
on (i) the firm’s exposure to the aggregate shock, βi (and is increasing in βi ), and (ii) the time
t price of risk, which is reflected in the term γt σε2 . Intuitively, firm-level mpk deviations are
composed of both a transitory component due to uncertainty and a persistent component due to
the risk premium. The transitory components are i.i.d. over time and lead to purely temporary
deviations in mpk (even though the underlying productivity processes are autocorrelated); the
risk premium, on the other hand, leads to persistent deviations – firms that are more exposed
to aggregate shocks, and so are riskier, will have persistently high mpk.
Expression (15) hones in on this second force and shows the persistent effects of risk premia
on the conditional expectation of time t+1 mpk, denoted Empk. Indeed, in this simple case, the
ranking of firms’ mpk will be constant in expectation as determined by the risk premium – high
beta firms will have permanently high Empk and low beta firms the opposite. Importantly,
18
the value of Empk will fluctuate with γt , but the ordering across firms will be preserved.
This is the sense that we call this component persistent/permanent. Expression (14) shows
that this ordering will not be preserved in realized mpk – due to the realization of shocks,
the ranking of firms’ mpk will fluctuate, but the firm-specific risk premium adds a persistent
component.25 Because the uncertainty portion of the realized mpk is always additively separable
and is independent of our mechanism, from here on we primarily work with Empk.
Expression (16) presents the cross-sectional variance of Empk:
2
σEmpk
≡ σE2 t [mpkit+1 ] = σβ2 γt σε2
t
2
.
(16)
Cross-sectional variation in Empk depends on the dispersion in beta and the price of risk. Dispersion will be greater when risk prices, reflected by γt σε2 , are high and so will be countercyclical.
The average long-run level of Empk dispersion is given by
2
2 2
2
2
2
2 2
EσEmpk
≡ E σEmpk
=
σ
γ
+
γ
σ
σε
β
0
1
x
t
where σx2 =
σε2
.
1 − ρ2x
(17)
Aggregate outcomes. Appendix D.3 shows that aggregate output can be expressed as
log Yt+1 ≡ yt+1 = at+1 + θ1 kt+1 + θ2 nt+1 ,
where kt+1 denotes the aggregate capital stock, nt+1 aggregate labor and at+1 the level of
aggregate TFP, given by
1 θ1 (1 − θ2 ) 2
at+1 = a∗t+1 −
σ
,
(18)
2 1 − θ1 − θ2 mpk,t+1
2
where σmpk,t+1
is realized mpk dispersion in period t + 1. The term a∗t+1 is the first-best level
of TFP in the absence of any frictions (i.e., where marginal products are equalized). Thus,
aggregate TFP monotonically decreases in the extent of capital “misallocation,” captured by
2
σmpk
. The effect of misallocation on aggregate TFP depends on the overall curvature in the
production function, θ1 + θ2 and the relative shares of capital and labor. The higher is θ1 + θ2 ,
that is, the closer to constant returns to scale, the more severe the losses from mis-allocated
resources. Similarly, fixing the degree of overall returns to scale, for a larger capital share, θ1 ,
a given degree of misallocation has larger effects on aggregate outcomes.
Using equation (16), the conditional expectation of one-period ahead TFP is given by
2
1 θ1 (1 − θ2 ) 2
Et [at+1 ] = Et a∗t+1 −
σβ γt σε2 .
2 1 − θ1 + θ2
25
(19)
With additional adjustment frictions, there will be other factors confounding the relationship between beta
and the realized and expected mpk.
19
The expression shows that risk premium effects unambiguously reduce aggregate TFP and
disproportionately more so in business cycle downturns, since γt is countercyclical. Taking
unconditional expectations gives the effects on the average long-run level of TFP in the economy:
a ≡ E [Et [at+1 ]] = a∗ −
2
1 θ1 (1 − θ2 ) 2 2
σβ γ0 + γ12 σx2 σε2 .
2 1 − θ1 + θ2
(20)
The expression directly links the extent of cross-sectional dispersion in required rates of return
(which are in turn determined by the prices of risk and volatility of aggregate shocks) to the
long-run level of aggregate productivity and gives a natural way to quantify the implications
of these effects. Further, and perhaps more importantly, it uncovers a new connection between
aggregate volatility and long-run economic outcomes, i.e., a “productivity cost” of business
cycles – ceteris paribus, the higher is aggregate volatility (σε2 and σx2 in the expression), the
more depressed will be the average long-run level of TFP (relative to an environment with no
aggregate shocks and/or risk premia).
In Appendix F, we show that our model can be extended to include multiple sources of
aggregate risk and to allow γt to depend on additional factors beyond the state of technology
and so expressions (19) and (20) provide a more general connection between financial conditions
(that may be less than perfectly correlated with the real economy), the cross-sectional allocation
of resources and aggregate TFP.26 Thus, more broadly, these expressions provide one way to
link the rich findings of the literature on cross-sectional asset pricing to real allocations and
macroeconomic outcomes.
3.1
The Cross-Section of Expected Stock Returns and MPK
In this section, we derive a sharp link between a firm’s beta – and so expected mpk – and its
expected stock market return, along the lines developed in Section 2.2. This connection suggests
an empirical strategy to measure the dispersion in beta and so quantify the mpk dispersion that
arises from risk considerations using stock market data. Our key finding is that, to a first-order
approximation, the firm’s expected stock return is a linear (and increasing) function of its beta.27
The implication is that, in the simple model outlined thus far, expected mpk is proportional to
expected stock returns, and thus, the dispersion in expected stock returns puts tight empirical
discipline on the dispersion in betas and so expected mpk arising from risk channels. We use
this connection to provide transparent intuition for our numerical approach in Section 4.
26
Further, we can verify that the additional extensions discussed in Section 2.1, i.e., firm-specific “alphas”
and heterogeneous exposures to investment goods prices have similar implications.
27
It is well known that a first-order approximation may not be sufficient to capture risk premia. In our
quantitative work in Section 4, we work with numerical higher order approximations.
20
We obtain an analytic approximation for expected stock market returns by log-linearizing
around the non-stochastic steady state where Xt = Zt = 1. To a first-order, the (log of the)
expected excess stock return is equal to (derivations in Appendix D.4)
e
e
Erit+1
≡ log Et Rit+1
= ψβi γt σε2 ,
where
ψ=
1
ρ
1
ρ
(21)
+δ−1
1−ρ
.
+ δ (1 − θ) − 1 1 − ρρx + ργ1 σε2
The expected excess return depends on the firm’s beta (indeed, is linear and increasing in beta)
and is increasing in the price of risk. Because the price of risk is countercyclical, risk premia
increase during downturns for all firms and fall during expansions.28 The time t cross-sectional
dispersion in expected excess returns is given by
2
2
2 2
2
σEr
e ≡ σ
e
log Et [Rit+1
t
] = ψ σβ γt σε
2
.
(22)
Similar to our findings for expected mpk, the expression reveals a tight link between beta
dispersion and expected stock return dispersion. Indeed, if firms had identical betas, dispersion
in expected returns would be zero. Moreover, as with expected mpk dispersion, expected stock
return dispersion is increasing in the price of risk and so is countercyclical.
Comparing equations (15) and (21) shows that expected excess returns are proportional to
2
2
expected mpk and equations (16) and (22) show that σEr
e is proportional to σEmpk . Thus,
t
t
the expressions reveal a tight connection between cross-sectional dispersion in expected stock
returns and expected mpk – both are dependent on the variation in betas. Although the exact
proportionality will not hold exactly in the full non-linear solution – or in the presence of other
frictions/distortions to firm investment decisions – we will use this intuition to quantify the role
of risk considerations in generating dispersion in expected mpk.
Specifically, these results suggest an empirical strategy to estimate the three key structural
parameters – γ0 , γ1 and σβ2 – using readily available stock market data. First, it is straightforward to verify that the market index – i.e., a perfectly diversified portfolio with no idiosyncratic
Strictly speaking, these results hold in the approximation so long as 1 − ρρx + ργ1 σε2 > 0. This condition
does not play a role in the numerical solution.
28
21
risk – achieves the maximal Sharpe ratio:29
SRmt = γt σε ,
ESRm ≡ E [SRmt ] = γ0 σε .
(23)
The expression links the market Sharpe ratio to γ0 . Indeed, in this linearized environment, the
mapping is one-to-one (given σε2 ). Next, deriving equation (21) for the market index gives
Ermt+1 = ψ β̄γt σε2 ,
Erm ≡ E [Ermt+1 ] = ψ β̄γ0 σε2 .
(24)
For a given value of γ0 , the equity premium is increasing as γ1 becomes more negative through
its effects on ψ (β̄ denotes the mean beta across firms). Lastly, equation (22) connects dispersion
in beta, σβ2 , to dispersion in expected returns. Together, equations (22), (23) and (24) tightly
link three observable moments of asset market data to the three parameters, γ0 , γ1 and σβ2 .
3.2
Adjustment Costs
In this section, we extend the framework to include capital adjustment costs. Although the
main insights from the previous sections go through, we illustrate an important interaction
between these costs and the effects of risk premia, namely, adjustment costs amplify the impact
of these systematic risk exposures on mpk dispersion.
We assume that capital investment is subject to quadratic adjustment costs, given by
ξ
Φ (Iit , Kit ) =
2
Iit
−δ
Kit
2
Kit .
With these costs, the return on capital is no longer equal to the MPK plus the undepreciated
capital stock, but depends on endogenous fluctuations in the value of installed capital, i.e.,
Tobin’s Q. Although exact analytic solutions are no longer available as in the simpler case, a
first-order approximation yields the return on capital (the investment return) to be30
I
rit+1
= (1 − ρ (1 − δ)) mpkit+1 + ρqit+1 − qit .
(25)
The investment return depends on mpkit+1 , but additionally on qit and qit+1 , where qit ≡
29
The Sharpe ratio for an individual firm is SRit =
s
βi γt σε2
2
2
1−ρρx +ργ1 σε
1−ρρz
, which shows that, due to the
σε̃2 +βi2 σε2
presence of idiosyncratic risk, individual firms do not attain the maximum Sharpe ratio. However, in this linear
environment, the diversified index faces no risk from σε̃2 , so that the expression collapses to (23). Although in the
full numerical solution the market may not exactly attain this value due to the nonlinear effects of idiosyncratic
shocks, the expression highlights that the market Sharpe ratio is informative about γ0 .
30
Throughout this section, we suppress constant terms that play no role.
22
ξ (kit+1 − kit ). As above, the risk premium on capital is equal to (the negative of) the covariance
of the return with the SDF, i.e.,
I
= −covt ((1 − ρ (1 − δ)) mpkit+1 + ρqit+1 , mt+1 ) ,
log Et rit+1
(26)
which shows that adjustment costs add an additional element to the risk premium through
endogenous changes in qit that are correlated with the SDF.
Appendix D.2 derives the log-linearized version of the firm’s optimal investment policy:31
kit+1 = φ1 βi xt + φ2 zit + φ3 kit − φ4 βi γ0 σε2 ,
(27)
where
0 =
φ1 =
φ4
ˆ 2 + ξˆ
(θ − 1) − ξˆ (1 + ρ) φ3 + ρξφ
3
(ρx − γ1 σε2 ) φ3
,
φ2 =
ρz φ3
ξˆ (1 − ρφ3 (ρx − γ1 σε2 ))
ξˆ (1 − ρρz φ3 )
φ3
1
=
,
ˆ
ξ (1 − ρφ3 ) 1 − ρφ3 (ρx − γ1 σε2 )
ξ
and ξˆ ≡ 1−ρ(1−δ)
is a composite parameter that captures the severity of adjustment costs.
Now, the past level of capital affects the new chosen level. The coefficient φ3 captures the
strength of this relationship. It lies between zero and one and is increasing in the adjustment
ˆ It is independent of the risk premium. The other coefficients each have a counterpart
cost, ξ.
in equation (13), but are modified to reflect the influence of adjustment costs. The coefficients
φ1 and φ2 are both decreasing in these costs – intuitively, adjustment costs reduce the firm’s
responsiveness to transitory shocks. Importantly, φ4 is increasing in these costs, showing that
they increase the importance of the firm’s beta in determining its choice of capital.32 The
expression for φ4 also reveals an interaction between adjustment costs and time-varying risk –
the denominator contains the product of φ3 and γ1 , which implies that a more negative γ1 leads
to higher values of φ4 as long as adjustment costs are non-zero. By increasing the value of φ4 ,
this interaction effect strengthens the impact of beta dispersion on Empk dispersion.
From here, we can derive the following expression for conditional expected mpk:
Et [mpkit+1 ] =
1
ˆ t [kit+2 − kit+1 ] − ξˆ (kit+1 − kit ) .
βi γt σε2 + ρξE
1 − ρφ3 (ρx − γ1 σε2 )
31
(28)
As above, we ignore terms reflecting variance adjustments that
are close to zero.
Strictly speaking, this is true so long as 1 − ρφ3 ρx − γ1 σε2 > 0. This condition holds for any reasonable level of adjustment costs, for example, given our estimates of the other parameters, ξ must be less than
approximately 2180.
32
23
Expected mpk depends on both the risk premium and adjustment costs (realized mpk also
depends on the realization of shocks, as above). The last two terms capture the effects of
adjustment costs alone and, conditional on current and expected future capital stocks, are
independent of aggregate risk. The first term captures the risk premium. Without adjustment
costs, φ3 = ξˆ = 0, and the risk premium is identical to expression (13)). The risk premium
is increasing in those costs (i.e., as φ3 gets larger), showing that adjustment costs amplify risk
premium effects. Intuitively, as shown in (26), adjustment costs add an additional source of
co-movement of capital returns with the SDF through fluctuations in firm-level Q. High beta
firms invest more in expansions (both because their expected productivity is high and the cost
of capital is low due to a low price of risk) and so the Q of these firms is negatively correlated
with the SDF (e.g., is more procyclical), inducing a higher risk premium.33
The extended model continues to give rise to mpk deviations that are persistent at the
firm-level. In particular, taking the unconditional expectation of (28) yields expressions for the
persistent component of Empk and its cross-sectional dispersion:
E [Empkit+1 ] =
⇒
2
σE[Empk
it+1 ]
1
βi γ0 σε2
1 − ρφ3 (ρx − γ1 σε2 )
=
1
1 − ρφ3 (ρx − γ1 σε2 )
2
γ0 σε2
(29)
2
σβ2 .
(30)
The risk premium is an essential ingredient for the model to generate persistence in firm-level
mpk; adjustment costs are not sufficient. When γ0 = 0 and hence risk effects are absent, there is
no persistent Empk dispersion, even with adjustment costs (beta dispersion is also necessary).
Thus, on their own, adjustment costs generate dispersion only in the transitory component
of mpk (through their role in expression (28)). However, (29) and (30) show that when the
persistent risk premium component is present, adjustment costs have a further amplification
effect on that component, equal to the fraction in those expressions. How large might this
amplification effect be? Using the parameter values from the next section, which include a
relatively modest level of adjustment costs, the scaling factor from these costs is about 1.75,
which implies a cross-sectional variance in Empk that is scaled up by a factor of three relative
to the case with no adjustment costs. Thus, although they do not change the qualitative
predictions of the model, adjustment costs can have an important quantitative effect on the
results. In contrast, the transitory effects coming from these costs will turn out to be small.
Finally, how do adjustment costs change the relationship between expected mpk, beta and
33
For a related, but slightly different intuition, adjustment costs cause capital to be a long-lived asset and
thus increase the length of the relevant time horizon when considering a capital investment. Because the amount
of risk is increasing in the length of the horizon, the risk premium is naturally larger.
24
expected stock returns? Appendix D.4 shows that to a first-order, expected returns are not
affected by adjustment costs and so the results from Section 3.1 continue to hold.34 Thus, the
arguments made in that section linking the key parameters of the model to moments of asset
returns go through unchanged.
4
Quantitative Analysis
In this section, we use the analytical insights laid out above to numerically quantify the extent
of mpk dispersion arising from risk premia effects.
4.1
Parameterization
We begin by assigning values to the more standard production parameters of our model. Following Atkeson and Kehoe (2005), we set the overall returns to scale in production θ1 + θ2 to
0.85. We assume standard shares for capital and labor of 0.33 and 0.67, respectively, which
gives θ1 = 0.28 and θ2 = 0.57. These values imply θ = 0.65.35 We assume a period length of one
year and accordingly set the rate of depreciation to δ = 0.08. We estimate the adjustment cost
parameter, ξ, in order to match the autocorrelation of investment, denoted corr (∆kt , ∆kt−1 ),
which is 0.38 in our data. Equation (39) in Appendix D.5 provides a closed-form expression for
this moment, which reveals a tight connection with the severity of adjustment frictions.36
To estimate the parameters governing the aggregate shock process, we build a long sample
of Solow residuals for the US economy using data from the Bureau of Economic Analysis on real
GDP and aggregate labor and capital. The construction of this series is standard (details in
Appendix B.4). With these data, we use a standard autoregression to estimate the parameters
ρx and σε2 . This procedure gives values of 0.94 and 0.0247 for the two parameters, respectively.37
34
Although this is only exactly true under our first-order approximation, Table 4 verifies numerically that at
their estimated level, adjustment costs have relatively modest effects on moments of returns.
35
This is close to the values generally used in the literature. For example, Cooper and Haltiwanger (2006)
estimate a value of 0.59 for US manufacturing firms. David and Venkateswaran (2019) use a value of 0.62.
36
The expression also reveals that for ρx close to ρz , which we find in the data, described next, the autocorrelation of within-firm investment is almost invariant to the firm’s beta (indeed, the invariance is exact if
ρx = ρz ). Thus, even with dispersion in betas, we may not see large variation in this moment across firms.
37
The autoregression does not reject the presence of a unit root at standard confidence levels. We have also
worked with the annual TFP series developed by John Fernald, available at:
https://www.frbsf.org/economic-research/indicators-data/total-factor-productivity-tfp/.
These data are only available for the more recent post-war period, but also show that the series is close to
a random walk (i.e., the autocorrelation of growth rates is essentially zero). A potential concern with this
approach is that these series reflect not only the process on exogenous technology, but also the effects of
mpk dispersion itself (since dispersion affects measured aggregate productivity). However, at our estimates,
these effects are small – mpk dispersion primarily impacts the level of aggregate productivity (which does not
affect our estimates of persistence or volatility) but has only a small impact on its time-series properties (we
25
Under our assumptions, firm-level productivity (including the aggregate component) can be
measured directly (up to an additive constant) as yit − θkit . After controlling for the level of
aggregate productivity, a similar autoregression on the residual (firm-specific) component yields
values for ρz and σε̃ of 0.93 and 0.28, respectively.
Turning to the parameters of the SDF, we set ρ = 0.988 to match an average annual risk-free
rate of 1.2%. Following the arguments in Section 3.1, we estimate the values of γ0 and γ1 to
match the post-war (1947-2017) average annual excess return on the market index of 7.7% and
Sharpe ratio of 0.53.38 This strategy is equivalent to matching both the mean and volatility
of market excess returns (the standard deviation is 14.6%). To be comparable to the data,
stock returns in the model need to be adjusted for financial leverage. To do so, we scale the
where D
is the
mean and standard deviation of the model-implied returns by a factor of 1 + D
E
E
debt-to-equity ratio. We follow, e.g., Barro (2006) and assume an average debt-to-equity ratio
of 0.5. Because both the numerator and denominator are scaled by the same constant, the
Sharpe ratio is unaffected. For ease of interpretation, in what follows, we report the properties
of levered returns. To compute the model-implied market return, we must also take a stand
on the mean beta across firms. Assuming that the mean of β̂i (the underlying productivity
beta) is one, and using the value of ω (the sensitivity of wages to aggregate shocks) suggested
by İmrohoroğlu and Tüzel (2014) of 0.20, we can compute the mean beta to be 1.99.39 This
is simply the mean productivity beta adjusted for the leverage effects of labor liabilities. This
procedure yields values of γ0 = 32 and γ1 = −140.
Finally, again following the insights in Section 3.1, we estimate the dispersion in betas to
match the cross-sectional dispersion in expected stock returns. To be consistent with the broad
literature, we use the expected returns predicted from the Fama-French model as computed
in Section 2.2. We de-lever firm-level expected returns following the approach in Bharath and
Shumway (2008) and Gilchrist and Zakrajsek (2012) (details in Appendix B.2). This procedure
yields an estimated average within-industry standard deviation of un-levered expected returns
of 0.127 (we report details and plot the full histogram of the expected return distribution in
Appendix B.2: for example, the mean is about 9%, and the interquartile range is just under
12%; the standard deviation of raw expected returns, i.e., not de-levered or controlling for
industry, is about 0.156).40 Feeding this value into our quantitative model yields an estimate
discuss these different effects in Section 4.2) – suggesting that these series are reasonable approximations to
the exogenous process. Further, we have also constructed an alternative series that is free from this concern
directly from the firm-level data by averaging across the firms in each year. This gives results quite similar to
the baseline, ρx = 0.92 and σε = 0.0245. Details are in Appendix B.4.
38
We calculate these values using annualized monthly excess returns obtained from Kenneth French’s website,
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
39
İmrohoroğlu and Tüzel (2014) estimate this value to match the cyclicality of wages.
40
Our estimates are consistent with those in Lewellen (2015), who reports moments of the expected return
26
Table 3: Parameterization - Summary
Parameter
Production
θ1
θ2
δ
ξ
σβ̂
Stochastic Processes
ρx
σε
ρz
σε̃
ω
Stochastic Discount Factor
ρ
γ0
γ1
Description
Capital share
Labor share
Depreciation rate
Adjustment cost
Std. dev. of risk exposures
Value
0.28
0.57
0.08
0.04
4.80
Persistence of agg. shock
Std. dev. of agg. shock
Persistence of idiosyncratic shock
Std. dev. of idiosyncratic shock
Wage elasticity
0.94
0.0247
0.93
0.28
0.20
Time discount rate
SDF – constant component
SDF – time-varying component
0.988
32
-140
for σβ of 12, and adjusting for the scaling 1 − θ2 gives the dispersion in underlying productivity
betas, σβ̂ , equal to 4.80.41
We parameterize the model using simulated method of moments (details in Appendix E).
Table 3 summarizes our empirical approach/results.
4.2
Risk-Based Dispersion in MPK
Table 4 presents our main quantitative results. We report four variants of the framework.
Column (1) (“Baseline”) corresponds to the full model with time-varying risk and adjustment
costs. Column (2) (“No Risk”) shows results from a version where risk effects are completely
absent (specifically, we set γ0 = γ1 = 0). Column (3) (“Only Risk”) reports the effects of risk
premia alone, without adjustment costs (i.e., ξ = 0). Column (4) (“Constant Risk”) examines
a version with adjustment costs but a constant price of risk (i.e., γ1 = 0). Column (5) (“Only
distribution from a number of predictive models. For example, using monthly data, he finds an annualized
cross-sectional standard deviation of up to 17.5% (Model 3, Panel A, Table 5 of that paper).
41
Although this a significant amount of dispersion, it composes only a modest fraction of overall dispersion in
σ 2
β̂
x2t +
firm-level productivity. To see this, note that the cross-sectional variance of productivity at time t is 1−θ
2
σ2
σz2 , where σz2 = 1−ρε̃ 2 . Plugging in our estimates and assuming, for example, that the economy is 2% above
z
or below trend, gives the first term to be about 8% of the total. It remains relatively modest for reasonable
deviations from trend. Thus, despite firms’ diverse sensitivities to business cycle shocks, our estimates still point
to firm-level idiosyncratic conditions as the dominate factor driving cross-sectional heterogeneity.
27
Table 4: Risk Premia and Misallocation
MPK Implications
2
EσEmpk
2
% of total σmpk
2
σEmpk
2
% of total σmpk
Baseline
(1)
No Risk
(2)
Only Risk
(3)
Constant Risk
(4)
Only Constant Risk
(5)
0.17
37.9%
0.14
0.03
7.4%
0.00
0.05
11.5%
0.05
0.16
35.9%
0.13
0.05
10.4%
0.05
47.3%
0.1%
15.7%
41.9%
15.7%
∆a
2
corr σEmpk
,
x
t
t
0.07
0.01
0.02
0.07
0.02
−0.31
0.01
−0.97
0.45
0.00
Moments
e
Erm
ESRm
corr (∆kt , ∆kt−1 )
0.08
0.53
0.38
0.00
0.00
0.38
0.10
0.61
−0.02
0.05
0.52
0.38
0.06
0.65
−0.03
Constant Risk”) has a constant price of risk and no adjustment costs. Our goal in showing these
different permutations is to understand the role that each element of the model plays in leading
to various patterns in mpk dispersion. In order to interpret the results as decomposition, each
variant holds all the parameters fixed at their estimated values except the one under study.
Long-run effects. The first row of the table shows the average level of mpk dispersion that
arises in each variant of the model.42 The second row shows the percentage of total observed
2
misallocation that this value accounts for. In our sample, overall σmpk
is 0.45. This is the
denominator in that row. Next, we calculate the dispersion stemming from only the permanent
component of firm-level MPK deviations (given by equation (28)), which we report in the third
row of the table. To compute this value in the data, for each firm, we regress the time-series of
its mpk on a firm-level fixed effect. The fixed effect is the permanent component of firm-level
mpk and the residuals transitory components. We then compute the variance of the permanent
2
= 0.30, about two-thirds of the total.43 This is
component, which yields a value of σmpk
the denominator in the fourth row of the table, which displays the model-implied permanent
dispersion as a percentage of the observed permanent component in the data. The next row
quantifies the implications of the estimated dispersion for the long-run level of aggregate TFP.
It reports the gains in the average level of TFP from eliminating the predicted mpk dispersion,
42
With adjustment costs, we do not have analytic expressions for period-by-period Empk dispersion. We
compute these values using simulation and then average over them. Without adjustment costs, we can use
expression (17) directly.
43
Other approaches give a similar breakdown, see, e.g., David and Venkateswaran (2019).
28
denoted ∆a.44 This is essentially an application of expression (18).
Column (1) shows that the full model generates mpk dispersion of about 0.17. This accounts
for about 38% of overall mpk dispersion in the data.45 Of the model-implied dispersion, about
0.14 is permanent in nature, which explains about 47% of the permanent component in the
data. The costs of this dispersion represent a loss in long-run TFP of about 7%. In the full
model, both risk effects and adjustment costs lead to dispersion in Empk. To hone in on the
role of risk alone, column (2) shows the same statistics when we eliminate risk effects and
only adjustment costs are present. Adjustment costs on their own generate relatively modest
2
dispersion in Empk (EσEmpk
= 0.03) and as proved in Section 3.2, do not lead to any dispersion
2
= 0). Thus, risk premia effects are crucial to generating
in the permanent component, (σEmpk
the substantial and persistent dispersion in column (1). Subtracting column (2) from column
(1) captures the contribution of risk effects alone: Empk dispersion of 0.14 (about 30% of
the total in the data), dispersion in the permanent component of 0.14 (47% of the data) and
long-run TFP losses of 6%. These results suggest (i) heterogeneity in risk premia can generate
significant MPK dispersion, particularly when compared to the permanent component in the
data, and (ii) the consequences for measures of aggregate performance such as TFP – i.e., the
“productivity costs” of business cycles – can be substantial.
Column (3) removes adjustment costs to illustrate their amplification of existing risk premia
effects. On their own (i.e., without adjustment costs), risk premia generate mpk dispersion of
2
0.05, which accounts for 11.5% of total σmpk
in the data and explain about 16% of the permanent
component. Thus, although the impact of risk premia remains significant in isolation, they are
less than half of those in column (1), where the amplification from adjustments costs is taken
into account. TFP losses are also smaller, but remain significant, at approximately 2%.
Columns (4) and (5) show that the majority of these effects stems from the presence of a
high persistent component in the price of risk, i.e., γ0 , rather than from the time-variation from
γ1 . Setting γ1 = 0 only modestly reduces the size of these effects in the presence of adjustment
costs (compare columns (1) and (4)) and has a negligible effect on the results without them
(columns (3) vs. (5)). The implication is that time-variation in the price of risk does not add
much to the long-run level of mpk dispersion.
Countercyclical dispersion. The last row in the top panel examines the second main implication of the theory, namely, the countercyclicality of mpk dispersion, which we measure as the
2
correlation of σEmpk
with the state of the business cycle, xt . Column (1) shows that the full
t
44
This calculation does not mean that policies eliminating this source of mpk dispersion would necessarily
be desirable. We merely see this as a useful way to quantify the implications of our findings.
45
The model-implied ratio of the standard deviation of MPK to that in expected stock returns is about 3.25
1
2
( .17
.127 ), which is close to the empirical estimates in Section 2.2.
29
2
model generates significantly countercyclical dispersion in Empk – the correlation of σEmpk
t
with the state of the cycle is -0.31. To put this figure in context, Table 6 in Appendix B.3
2
and the cyclical component of aggregate productivity
shows that the correlation between σmpk
in the data is -0.27. Thus, the model predicts countercyclical dispersion on par with this value.
Column (2) shows that adjustment costs alone do not generate any cyclicality in Empk dispersion. Column (3) shows that as the only factor behind Empk dispersion, the time-varying
risk premium would lead to an almost perfectly negative correlation with the business cycle.
This is a clear implication of equation (16). The additional presence of adjustment costs in the
first column confounds this relationship and leads to a smaller correlation (in absolute value)
that is more in line with the data. Finally, the last two columns illustrate that time-varying
risk is key to generating countercyclical dispersion. Without this element, Empk dispersion
is significantly positive with adjustment costs and without them, is exactly acyclical. Thus,
our findings suggest that the interaction of a countercyclical price of risk with adjustment frictions is crucial in yielding a negative (though far from negative one) correlation between Empk
dispersion and the state of the business cycle.
To highlight the potential implications of the countercyclical Empk dispersion produced
by our model, consider the connection with the empirical findings in Eisfeldt and Rampini
(2006), who show that firm-level dispersion measures tend to be countercyclical, yet most
capital reallocation is procyclical. Our theory can – at least in part – reconcile this observation
due to the countercyclical nature of factor risk prices and the high beta of high MPK firms:
countercyclical reallocation would entail moving capital to the riskiest of firms in the riskiest of
times. Thus, in light of our results, it may not be as surprising that countercyclical dispersion
obtains, even in a completely frictionless environment.46
Moments. In the bottom panel of Table 4, we investigate the role of each element in matching
the target moments. Our full model in column (1) is directly parameterized to match the three
moments, i.e., the equity premium, Sharpe ratio and autocorrelation of investment. Column
(2) shows that without risk aversion, risk premia are essentially zero. Column (3) shows that,
as implied by the approximation in Section 3.2, adjustment costs have a modest effect on the
properties of returns (eliminating them somewhat raises the equity premium and Sharpe ratio).
However, the autocorrelation of investment falls dramatically without them, indeed, becoming
slightly negative (due to the mean-reverting nature of shocks). Thus, some degree of adjustment
costs is crucial for matching this moment. Comparing columns (1) and (4) shows that without
46
The main measure of reallocation in Eisfeldt and Rampini (2006) includes both mergers and acquisitions
(M&A) as well as sales of disassembled capital (sales of property, plant and equipment). Even excluding
M&A, they find the latter is significantly procyclical (correlation with GDP of about 0.4; data from https:
//sites.google.com/site/andrealeisfeldt/home/capital-reallocation-and-liquidity).
30
time-varying risk, the model struggles to match the equity premium, which falls almost by
half, from about 8% to 5%. As implied by expressions (24), (23) and (39), time-varying risk
is tightly linked to average excess returns, but has only modest effects on the average Sharpe
ratio and the autocorrelation of investment. A similar pattern emerges from columns (3) and
(5) – in the absence of adjustment costs, removing time-varying risk significantly reduces the
equity premium but has smaller effects on the other two moments.
In sum, the results in Table 4 show first, heterogeneity in firm-level risk premia leads to
quantitatively important dispersion in mpk, with significant adverse effects on aggregate TFP;
moreover, much of this dispersion is persistent and can account for a significant portion of what
seems to be a puzzling pattern in the data, namely, persistent mpk deviations at the firm-level.
Second, these risk premium effects add a notably countercyclical element to mpk dispersion,
going some way towards reconciling the countercyclical nature of firm-level dispersion measures.
4.3
Other Distortions
Recent work has pointed to a number of additional factors (beyond fundamentals and adjustment frictions) that may affect firms’ investment decisions and lead to mpk dispersion, for
example, financial frictions or policy-induced distortions. Moreover, it has been pointed out
that attempts to identify one of these forces – while abstracting from others – may yield misleading conclusions. This section demonstrates that our strategy of using asset market data is
robust to this critique. In other words, our approach yields accurate estimates of risk premium
effects, even in the presence of other, un-modeled, distortions.
Rather than take a stand on the exact nature of these factors, we follow the broad literature,
e.g., Hsieh and Klenow (2009) and Restuccia and Rogerson (2008), and model these distortions
using a flexible class of “taxes” or “wedges,” which can have a rich correlation structure over
time and with both firm-level characteristics and aggregate conditions (in Section 5 we analyze
two additional sources of measured mpk dispersion, namely, heterogeneity in markups and
production function parameters). Specifically, we introduce a proportional “tax” on firm-level
operating profits, 1 − eτit+1 (so that the firm keeps a portion eτit+1 ), of the form:
τit+1 = −ν1 zit+1 − ν2 xt+1 − ν3 βi xt+1 − ηit+1 .
(31)
The first term captures a component correlated with the firm’s idiosyncratic productivity, where
the strength of the relationship is governed by ν1 . If ν1 > 0, the wedge discourages (encourages)
investment by firms with high (low) idiosyncratic productivity. If ν1 < 0, the opposite is true.
The next two terms capture the correlation of the wedge with the state of the business cycle,
xt . We allow for a component through which all firms are equally distorted by the cyclical
31
portion of the wedge, captured by ν2 , and a component through which high beta firms are
disproportionately affected by the cyclical portion, captured by ν3 . Through this piece, the
wedge can be correlated with firm-level betas. The last term, ηit+1 , captures factors that are
uncorrelated with firm or aggregate conditions. It can be either time-varying or fixed and is
normally distributed with mean zero and variance ση2 . Low (high) values of η spur (reduce)
investment by firms irrespective of their underlying characteristics or the state of the business
cycle.47 David and Venkateswaran (2019) show that a related formulation describes observed
MPK dispersion well (although they do not have beta dispersion or aggregate shocks). We
loosely refer to the wedge as a “distortion,” although we do not take a stand on whether it stems
from efficient factors or not, simply that there are other frictions in the allocation process.
To gain intuition, we analyze each component of the distortion in turn. First, we focus
only on the first and last terms, i.e., we set ν2 = ν3 = 0. In this case, the wedge is purely
idiosyncratic in the cross-section, i.e., it is always mean zero and has no aggregate component.
This formulation is closest to the ones typically used in the literature, which has typically
focused on idiosyncratic distortions with no aggregate shocks. Appendix G derives the following
expressions for expected mpk and its cross-sectional variance:
Empkit+1 = α + ν1 ρz zit + ηit+1 + βi γt σε2 ,
⇒
2
σEmpk
= (ν1 ρz )2 σz2 + ση2 + σβ2 γt σε2
t
2
. (32)
In this case, Empk includes (i) a component that reflects the correlated distortion, ν1 , and
depends on the firm’s expectations of its idiosyncratic productivity (ρz zit ), leading to mpk deviations that are correlated with idiosyncratic productivity, and (ii) a term that reflects the
uncorrelated distortion, η, which leads to mpk deviations that are uncorrelated with productivity. The last term reflects the risk premium. All of these components lead to dispersion in
Empk (dispersion in realized mpk also reflects uncertainty over shocks).
Crucially, expression (32) reveals that the risk premium (and resulting risk-based dispersion)
are unaffected by the presence of these additional distortions. Further, Appendix G proves that
expected stock returns are also unaffected, i.e., equation (21) still holds. The result implies that
the mapping from expected returns to beta is, to a first-order, unaffected by the distortions, as
is the mapping from beta dispersion to its effects on Empk. This leads to an important finding:
even in the richer environment here featuring a common class of mis-allocative distortions,
using stock market data continues to yield accurate estimates of the effects of heterogeneous
risk exposures alone. Clearly, a strategy using mpk dispersion directly does not share this
feature: measuring risk effects alone would be complicated by the presence of other distortions.
47
We have also studied a version where the distortion is size-dependent, i.e., τit+1 = −νk kit+1 . This turns
νk
out to be equivalent to the specification in (31) where ν1 = ν3 = 1−θ+ν
and ν2 = 0.
k
32
Next, we add the components that are correlated with aggregate conditions. First, consider
the case with a common cyclical component, i.e., ν2 6= 0. We can prove a similar result as
with only idiosyncratic wedges – the distortion does not affect the cross-sectional dispersion in
expected stock returns or the risk-related dispersion in Empk.
Finally, consider the case where high beta firms are disproportionately affected by the
aggregate distortion, i.e., ν3 6= 0. If ν3 > 0, the distortion discourages (encourages) investment
by high (low) beta firms in good times and the reverse in bad times (in this sense, it works like a
cyclical productivity-dependent component, since high beta firms are relatively more productive
in good times). There is also an aggregate implication of the wedge: averaging across firms
gives τ̄t+1 = −ν3 β̄xt+1 . If ν3 > 0 (< 0), the tax is pro- (counter) cyclical. Empk is given by:
Empkit+1 = α + ν1 ρz zit + ν3 βi ρx xt + (1 − ν3 ) βi γt σε2 + ηit+1 .
(33)
The second and third terms captures the effects of idiosyncratic and aggregate distortions,
respectively, and are independent of risk. The second to last term captures the risk premium,
which is now scaled by a factor 1 − ν3 . Further, we can show that expected stock returns
are scaled by exactly the same factor. The key implication is that all results from the baseline
version go through, with a reinterpretation of the beta we recover from stock market data: rather
than picking up the true beta alone, stock market returns yield a measure of the distorted beta,
(1 − ν3 ) βi . Since this is the object that also determines the risk premia in MPK, the remainder
of our results stay largely unaffected.48
4.4
Directly Measured Productivity Betas
Our baseline approach to measuring firm-level risk exposures used the link between beta and
expected stock returns laid out in Section 3.1. Here, we use an alternative strategy to estimate
the dispersion in these exposures using only production-side data. In one sense, this approach is
more direct – there is no need to employ firm-level stock market data to measure risk exposures.
On the other hand, computing betas directly from production-side data has its drawbacks –
the data are of a lower frequency (quarterly at best) and the time dimension of the panel is
shorter. Further, it may be difficult to apply this method to firms in developing countries
(where measured misallocation tends to be larger), since most firm-level datasets there have
48
One caveat is that when taking the cross-sectional variance of (33), an additional term arises from the
covariance of the risk premium term with the aggregate distortion term. If the wedge worsens in downturns,
i.e., ν3 < 0, which may be a plausible conjecture, we can prove that our baseline calculations yield a lower
bound on risk premium effects on mpk dispersion. If the wedge is procyclical, i.e., ν3 > 0, we could be at
risk of overstating these effects. However, Appendix G derives an upper bound on this bias at the estimated
parameters and shows that it is quantitatively small.
33
relatively short panels and are at the annual frequency. For those reasons, we view our results
here as an informative check on our baseline findings above.
For each firm, we regress measured productivity growth, i.e., ∆zit + βi ∆xt , on aggregate
productivity growth ∆xt . It is straightforward to verify that the coefficient from this regression
is exactly equal to βi . Using these estimates, we can compute the firm’s underlying productivity
beta, β̂i , and calculate the cross-sectional dispersion in these estimates, σβ̂2 . We have applied
this procedure using three different measures of the aggregate shock: (i) our long sample of
Solow residuals, (ii) the series we construct from firm-level data (both of these are described in
Appendix B.4) and (iii) the Fernald annual TFP series. The results yield values of σβ̂ of 6.4,
4.3 and 5.9, respectively. Recall that our estimate for this value using stock return data was
4.8, which is in line with – and towards the lower end of – the range found here.
4.5
Measurement Concerns
In this section, we address a number of potential measurement-related issues. First, following
the recent literature, e.g., Hsieh and Klenow (2009) and Gopinath et al. (2017), we measure
firm-level capital stocks using reported book values. An alternative approach is to use the
perpetual inventory method along with detailed data on investment flows and investment good
price deflators to construct capital stocks. Although this is in general an important issue for the
firm dynamics/misallocation literatures, our empirical approach allows us to largely avoid this
concern. To see this, notice that our estimation relies on measures of firm-level capital in only
two places: first, to calculate the properties of idiosyncratic shocks, i.e., ρz and σε̃2 , and second,
to calculate the autocorrelation of investment, which largely identifies the extent of adjustment
costs. As shown, for example, in equations (16) and (22), idiosyncratic shocks have no effect
on dispersion in expected mpk or on expected stock returns (the latter to a first-order). In our
framework, idiosyncratic risk, though crucial in explaining firm dynamics, is not priced, and
thus does not affect risk premia.49 David and Venkateswaran (2019) measure firm-level capital
using both approaches and find a larger serial correlation of investment using the perpetual
inventory method. Since our adjustment cost estimate is increasing in the serial correlation,
this approach would likely lead to larger estimates, and, as shown above, further amplify the
risk premium effects we uncover.50 Largely avoiding the use of firm-level capital measures is an
49
We have also verified that idiosyncratic shocks have little effect on our estimates in the full non-linear
model (see Appendix I).
50
If the serial correlation is lower, we can generally think of the results in column 2 of Table 4, where we set
adjustment costs to zero, as a lower bound. Although not part of our estimation, we also use firm-level capital
to calculate total mpk dispersion, e.g., the denominator in the second row of Table 4. David and Venkateswaran
(2019) show that this statistic is very similar under the two measurement approaches (see Tables 2 and 18 in
that paper).
34
important feature of our use of stock market data.51
How about the effects of measurement error? Our use of stock market data is also useful in
this regard – in general, stock market data should be quite precisely measured and so largely
free of this concern. Measurement error in capital may affect our estimate of adjustment
costs, but we can show that this error would likely lead us to a conservative estimate for these
costs. To see this, consider first the case of (classical) measurement error that is iid over time.
This unambiguously reduces the observed serial correlation (i.e., the true one is higher), which
would yield higher adjustment cost estimates. Alternatively, consider the opposite case where
the measurement error is permanent. Then, since we work with the growth rate of capital, our
results would be unaffected. Of course, similarly to mis-measured capital, measurement error
may affect the observed amount of mpk dispersion.
These issues may also be concerns for our estimates of productivity betas in the previous
section, where we used measures of capital to calculate firm-level productivity. However, in Appendix H we show that any potential bias is likely quite small. Loosely speaking, mis-measured
capital introduces error into the dependent variable of the regression, which, under certain conditions, will not affect our estimates (specifically, so long as changes in the measurement error
are uncorrelated with changes in aggregate productivity). In that appendix we also investigate the potential bias in those estimates coming from unobserved heterogeneity in parameters
across firms, i.e., θ, and show that it is quite small.
5
The Sources of Betas
Cross-firm variation in exposure to aggregate shocks, i.e., beta, is an essential ingredient in our
theory. In this section, we investigate some potential sources of this type of heterogeneity –
namely, dispersion in technological parameters (input elasticities in production) and markups
as well as in the sensitivity of demand to business cycle fluctuations. Importantly, we show
that each of these forms of heterogeneity is reflected in our measured betas, so that our main
results on risk premia and mpk dispersion go through unchanged. Our goal here is simply to
gain some further insight into why firms exhibit different sensitivities to aggregate shocks.
Heterogeneous technologies/markups. Firm-level heterogeneity in production function
parameters or markups are potential sources of beta dispersion. Intuitively, both of these forces
lead firms to have different responsiveness and so exposure to aggregate shocks. In Appendix I,
we explore each of these in detail (to allow for markup dispersion, we extend our baseline setup
51
Of course, some of the measured mpk dispersion in the data – i.e., the denominators in rows 2 and 4 of
Table 4 may be coming from mis-measurement of capital.
35
to an environment where firms produce differentiated goods, are monopolistically competitive
and face constant, but potentially heterogeneous, elasticities of demand). First, we show that
a version of our analysis in Section 3 continues to hold in both cases, where the firm’s beta now
also reflects these additional sources of heterogeneity. Second, we calculate how much of the
observed beta dispersion can be attributed to each of these forces. Using dispersion in labor’s
share of revenue as a likely upper bound for technology dispersion, we find it can potentially
account for about 12% of the overall standard deviation of betas from Section 4. Similarly,
using recent estimates of markup dispersion among Compustat firms, we find it can account for
about 6%. Thus, in total, heterogeneity in input elasticities and markups are likely to explain
at most about 18% of measured beta dispersion. Although this is a significant fraction, these
findings also suggest that the majority of beta dispersion seems to arise from other sources.52
Heterogeneous demand sensitivities. A recent literature has pointed out variation in the
response of firm-level demand to the business cycle. For example, Jaimovich et al. (2019)
document a “trading down” phenomenon – during expansions, when purchasing power is high,
households tend to consume higher quality goods and in downturns substitute towards lower
quality ones – Nevo and Wong (2015) show that during the Great Recession, consumers substituted towards cheaper generic products and discount stores and Coibion et al. (2015) show that
during downturns, consumers substitute towards low-price retailers.53 This pattern makes high
quality products more procyclical and lower quality ones less so (or even countercyclical). To
see the implications of those findings for our analysis, consider the following system of demand
and production functions:
µ
Qit = Pit−µ Xtβ̂i Ẑit ,
Yit = Kitθ̂1 Nitθ̂2 .
Here, Xt is interpreted as an aggregate component of demand rather than technology (it is
straightforward to include aggregate technology shocks as well) and Ẑit as idiosyncratic demand.
The firm-specific sensitivity to Xt , β̂i , captures the idea that in expansions, when demand for all
goods is high, consumers substitute towards some goods and away from others. In downturns,
when Xt is low, the opposite pattern holds: consumers substitute away from those same goods.
This is a simple way to capture the “trading down” phenomenon. Firm revenues are given by
Pit Yit = Xtβ̂i Ẑit Kitθ1 Nitθ2 ,
52
Appendix I also investigates potential heterogeneity in the depreciation rate, δ, and the parameters governing idiosyncratic shocks, ρz and σε̃2 , as well as the effects of adjustment costs alone. We find that these forces
are unlikely to account for much of the dispersion in risk premia.
53
A related literature documents a similar “flight from quality” in response to contractionary exchange rate
devaluations. e.g., Burstein et al. (2005), Bems and Di Giovanni (2016) and Chen and Juvenal (2018).
36
where θj = 1 − µ1 θ̂j , j = 1, 2. With this reinterpretation, the expression is exactly equivalent
to (8). In other words, differences in the responsiveness of firm-level demand to the business
cycle may be behind our beta estimates.
Since direct data on quality are hard to come by, systematically quantifying the dispersion
in these “demand betas” is challenging. However, in Appendix I, we examine one industry
where we were able to obtain a proxy for quality, namely average check per person (price) in
SIC 5812, Eating Places (i.e., restaurants). Pricing data are from a number of publicly available
sources, including company SEC filings and investment bank reports. The appendix shows that
higher quality establishments, as proxied by price, have greater exposure to aggregate shocks,
and higher expected stock returns and MPK. Thus, the main message of that study of a single
industry is that differences in the cyclicality of firm-level demand due to quality differences and
“trading down” seems a promising explanation for beta dispersion.
6
Conclusion
In this paper, we have revisited the notion of “misallocation” from the perspective of a risksensitive, or risk-adjusted, version of the stochastic growth model with heterogeneous firms.
The standard optimality condition for investment in this framework suggests that expected
firm-level marginal products should reflect exposure to macroeconomic risks, and their pricing.
To the extent that firms are differentially exposed to these risks, cross-sectional dispersion in
MPK may not only reflect true misallocation, but also risk-adjusted capital allocation. We
provide empirical support for this proposition and demonstrate that a suitably parameterized
model of firm-level investment suggests that, indeed, risk-adjusted capital allocation accounts
for a significant fraction of observed MPK dispersion among US firms. Importantly, much of
this dispersion is persistent in nature, which speaks to the large portion of observed MPK
dispersion that arises from seemingly persistent/permanent sources. Further, our setup leads
to a novel link between aggregate volatility, risk premia and long-run productivity – our results
suggest that there can be substantial “productivity costs” of business cycles.
There are several promising directions for future research. Our framework points to a new
connection between business cycle dynamics and the cross-sectional allocation of inputs. Investigation of this link, for example, a further exploration of the sources of beta variation across
firms, would lead to a better understanding of the underlying causes of observed marginal product dispersion. Much of the misallocation literature examines differences in marginal product
dispersion across countries. A natural next step would be to implement a similar analysis in a
set of developing countries – because those countries typically have high business cycle volatility,
it may be that dispersion in risk premia is larger there. The tractability of our setup allowed
37
us to quantify the effects of financial market considerations, e.g., cross-sectional variation in
required rates of return, on measures of macroeconomic performance, i.e., aggregate TFP. This
link provides a new way to evaluate the implications of the rich set of empirical findings in
cross-sectional asset pricing. For example, pursuing multifactor/financial shock extensions of
our analysis (e.g., along the lines laid out in Appendix F) to incorporate the many risk factors
pointed out in that literature would be fruitful to measure the implications of those factors for
allocative efficiency. Of particular interest would be whether those factors are efficient or not,
e.g., to what extent do capital allocations reflect the “mis-pricing” of assets.
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Appendix: For Online Publication
A
A.1
Motivation
Derivation of equation (3) and examples.
1 = Et [Mt+1 (M P Kit+1 + 1 − δ)]
= Et [Mt+1 ] Et [M P Kit+1 + 1 − δ] + covt (Mt+1 , M P Kit+1 )
The (gross) risk-free rate satisfies Rf t =
1
.
Et [Mt+1 ]
Combining and rearranging yields
Et [M P Kit+1 ] = M P Kf t+1 −
covt (Mt+1 , M P Kit+1 )
Et [Mt+1 ]
= αt + βit λt
where αt , βit and λt are as defined in the text.
No aggregate risk. With no aggregate risk, Mt+1 = ρ ∀ t where ρ is the rate of time discount.
The Euler equation gives
1 = ρ (Et [M P Kit+1 ] + 1 − δ) ∀ i, t
⇒
Et [M P Kit+1 ] =
1
− (1 − δ) = rf + δ
ρ
CAPM. Clearly, −cov (Mt+1 , M P Kit+1 ) = bcov (rmt+1 , M P Kit+1 ) and var (Mt+1 ) = b2 var (rmt+1 ).
Since the market return is an asset, it must satisfy Et [rmt+1 ] = rf t + λbt so that λt = b (Et [rmt+1 ] − rf t ).
Substituting into expression (3) gives the CAPM expression in the text.
CRRA preferences. A log-linear approximation to the SDF around its unconditional mean
gives Mt+1 ≈ E [Mt+1 ] (1 + mt+1 − E [mt+1 ]) and in the case of CRRA utility, mt+1 = −γ∆ct+1
where ∆ct+1 is log consumption growth. Substituting for Mt+1 into expression (3) gives the
CCAPM expression in the text.
A.2
Extensions
Alphas. We model alphas as firm-level distortions in discount rates. Specifically, we assume
the payoffs of firm i are discounted using
M̃t+1 = Mt+1 Tit+1
43
where Tit+1 denotes a firm-specific distortion to the discount factor.
The Euler equation then takes the form
h
i
h
i
1 = (1 − δ) Et M̃t+1 + Et M̃it+1 M P Kit+1
Applying a similar approach as above and, for simplicity, assuming that expected discount
factors are undistorted, we obtain
Et [M P Kit+1 ] = αt + βit λt
covt (M P Kit+1 ,M̃it+1 )
. Thus, even if
where αt and λt are as defined in expression (3) and βit = −
vart (Mt+1 )
all firms have the same dynamic process for MPK, dispersion in expected MPK can arise from
differences in the stochastic processes of the discount factors, M̃it+1 .
Firm-specific investment prices. Here, we allow firms to face different prices of capital,
denoted Qit , so that the cost of new investment in period t is equal to Qit (Kit+1 − (1 − δ) Kit ).
The Euler equation is given by
Qit = Et [Mt+1 (M P Kit+1 + Qit+1 (1 − δ))]
(34)
and rearranging,
Et [M P Kit+1 ] = Qit Rf t − (1 − δ) Et [Qit+1 ] −
covt (M P Kit+1 + (1 − δ) Qit+1 , Mt+1 )
Et [Mt+1 ]
= αit + βit λt
where αit = Qit Rf t − (1 − δ) Et [Qit+1 ] is the (now firm-specific) risk-free cost of capital,
+(1−δ)Qit+1 ,Mt+1 )
t (Mt+1 )
and λt = var
. Thus, even if all firms have the same
βit = − covt (M P Kit+1
vart (Mt+1 )
Et [Mt+1 ]
co-movement of MPK with the SDF, i.e., covt (M P Kit+1 , Mt+1 ) is constant across firms, heterogeneity in the co-movement of Qit with the SDF will lead to differences in risk premia across
firms and hence in expected MPK.
A.3
MPK and Stock Returns
To derive equation (5), use the Euler equation
1 = Et [Mt+1 (M P Kit+1 + 1 − δ)]
= (1 − δ) Et [Mt+1 ] + Et [Mt+1 M P Kit+1 ]
44
Assume that M P K and Mt+1 are jointly log-normal and use the fact that the risk-free rate
satisfies Rf t = Et [M1t+1 ] to obtain
1
Rf t = (1 − δ) + eEt [mpkit+1 ]+ 2 vart (mpkit+1 )+covt (mpkit+1 ,mt+1 )
or, rearranging and suppressing variance terms for simplicity,
e
Empkit+1
≡ Et [mpkit+1 ] − log (rf t + δ) ≈ −covt (mpkit+1 , mt+1 )
To derive equation (6), standard techniques give the risk premium on stocks as
e
Erit+1
≡ Et [rit+1 ] − rf t ≈ −covt (rit+1 , mt+1 )
With a single source of aggregate risk, a first order approximation gives the return as its
expected value plus terms that are linear in the unexpected shocks, i.e.,
rit+1 = Et [rit+1 ] + ψε εit+1 + ψβit εt+1
where ψε and ψ are constants of linearization and βi captures the firm-specific exposure to the
aggregate shock (ψε and εit can either be scalars, or vectors of exposures and realizations of
idiosyncratic shocks). Similarly, mpk satisfies
mpkit+1 = Et [mpkit+1 ] + εit+1 + βit εt+1
Then,
covt (rit+1 , mt+1 ) = ψcovt (mpkit+1 , mt+1 )
Substituting yields
e
≈ −ψcovt (mpkit+1 , mt+1 )
Erit+1
For a detailed derivation of these results, see the approach in Appendix D.1 and D.4. For a
multifactor version, see Appendix F.1.
B
Data
In this appendix, we describe the various data sources used throughout our analysis.
45
B.1
Sources and Series Construction
We obtain firm-level data from COMPUSTAT and CRSP.54 We include firms coded as industrial
firms from 1965-2015. Our time-series regressions and portfolio sorts use data from 1973-2015,
since data on the GZ spread and excess bond (EB) premium begin in 1973 and because there
are relatively few industries with at least 10 firms in a given year pre-1973.55 We further
exclude financial firms by dropping those with COMPUSTAT SIC codes that correspond to
finance, insurance, and real estate (FIRE, SIC codes 6000-6999). We also exclude firms with
missing SIC codes or coded as non-classifiable, as much of our analysis examines within-industry
variables. We measure firm revenue using sales from Compustat (series SALE), and capital
using the depreciated value of plant, property, and equipment (series PPENT). We measure
firm marginal product of capital in logs (up to an additive constant) as the difference between
log revenue and capital, mpkit = yit − kit . Market capitalization is measured as the price times
shares outstanding from CRSP and profitability as the ratio of earnings before interest, taxes,
depreciation, and amortization (EBITDA) divided by book assets (AT). We measure market
leverage as the ratio of book debt to the sum of market capitalization plus book debt, where
book debt is measured as current liabilities (LCT) + 1/2 long term debt (DLTT), following
Gilchrist and Zakrajsek (2012).
We obtain data on aggregate risk factors from the following sources. Data on the FamaFrench factors are from Kenneth French’s website, http://mba.tuck.dartmouth.edu/pages/
faculty/ken.french/, while the Hou et al. (2015) q 5 factors are from http://global-q.org/
factors.html. Updated data on the price/dividend ratio are from Robert J. Shiller’s website,
http://www.econ.yale.edu/~shiller/ and updated measures of the GZ spread and excess
bond premium are from Simon Gilchrist’s website, http://people.bu.edu/sgilchri/.
Computation of betas and expected returns. Here, we describe our procedure to compute betas and expected returns.
We estimate stock market betas by performing time-series regressions of firm-level excess
returns (realized returns from CRSP in excess of the risk-free rate), rite , on aggregate factors,
denoted by the N × 1 vector Ft . For each firm, the specification takes the form
rite = αiτ + βiτ Ft + it
(35)
We estimate these regressions at the quarterly frequency using backwards-looking five-year
rolling windows, i.e., for t ∈ {τ − Nτ + 1, τ − τT + 2, ..., τ }, where βiτ denotes the 1 × N vector
54
Source: CRSP® , Center for Research in Security Prices, Booth School of Business, The University of
Chicago. Used with permission. All rights reserved.
55
The results are qualitatively similar if we use data from the full 1965-2015 sample.
46
of factor loadings and Nτ the length of the window.56 Under the CAPM, the single risk factor
is the aggregate market return. Under the Fama-French 3 factor model, the risk factors are the
market return (MKT), the return on a portfolio that is long in small firms and short in large
ones (SMB) and the return on a portfolio that is long in high book-to-market firms and short
in low ones (HML). Under the Hou et al. (2015) q 5 5 factor model, the risk factors are the
market return, the return on a portfolio that is long in small firms and short in large ones, the
return on a portfolio that is long in low investment firms and short in high investment ones,
the return on a portfolio that is long in high profitability (return on equity) firms and short
in low profitability ones and the return on a portfolio that is long in firms with high expected
1-year ahead investment-to-assets changes and short in firms with low ones.
Next, we estimate the following cross-sectional regression in each period:
rite = αt + λt βit + it
(36)
where λt denotes the 1 × N vector of period t factor risk prices and βit the N × 1 vector of
exposures, estimated as just described. We calculate expected stock returns as αi + λβit , where
βit is as estimated from equation (35), λ is calculated using the estimates from (36), and αi is
P
calculated as αi = T1 Tt=1 (αit + it ) also using the estimates from (36).
Composition-adjusted measures of dispersion. For Table 2, we compute time-series
of the cross-sectional dispersion in MPK. Because Compustat is an unbalanced panel with
significant changes in the composition of firms over time, it is important to ensure that we
measure the variation in dispersion due to changes in firm MPK, rather than additions or
deletions from the dataset (especially since many additions and deletions to the Compustat
data may not be true firm entry or exit). We therefore compute composition-adjusted measures
of the cross-sectional standard deviation in MPK that are only affected by firms who continue
on in the dataset. We use the following procedure:
For each set of adjacent periods, e.g., t and t + 1, we compute the cross-sectional standard
deviation in each time period only for those firms that are present in the data in both periods.
Taking the difference yields the change from time t to t + 1 that is due only to changes in
the common set of firms. Completing this procedure yields a time-series of changes in the
cross-sectional standard deviation of MPK. We then combine this time-series of changes with
the initial value of the standard deviation (across all firms in the initial period) to construct a
synthetic series for the standard deviation that is not affected by the changing composition of
firms in the data.
56
We have also estimated the stock market betas using higher frequency monthly data (and two-year rolling
windows) and obtained similar results.
47
B.2
Expected Return Distribution
Table 5 reports statistics from the cross-sectional distribution of expected returns (E[re ]) and
unlevered expected returns (E[ra ]), which is a measure of expected asset returns, estimated from
the Fama-French model. We de-lever expected returns using an adjustment factor computed
from Black-Scholes following the approach in, e.g., Bharath and Shumway (2008) and Gilchrist
and Zakrajsek (2012). Specifically, we implement an iterative procedure using data on realized
equity volatility, firm debt, and firm market capitalization to compute the implied value of assets
and asset volatility. The Black-Scholes equations imply E[ra ] ≈ M kt.VAcap. Φ(δ1 )E[re ], where VA
is the total firm asset value implied by Black-Scholes as a function of the market capitalization
of equity, book debt, and realized backwards-looking equity volatility and Φ (δ1 ) is the BlackScholes “delta” of equity, as defined in, e.g., Gilchrist and Zakrajsek (2012). We compute the
cap.
Φ(δ1 ) for each firm using daily data and a 21 day backwards-looking
adjustment factor M kt.
VA
window for equity volatility and then calculate a firm-year adjustment factor by averaging
this adjustment factor for each firm-year. Finally, we compute un-levered expected returns for
each firm as the product of its expected equity return multiplied by this factor. To find the
cross-sectional distribution of within-industry expected returns, we de-mean expected returns
by industry-year, keeping industry-years with at least 10 observations. We then add back the
means and report the resulting distribution.57 Figure 1 plots the full cross-sectional distribution
of within-industry expected excess asset and equity returns.
Table 5: The Distribution of Expected Excess Returns
Percentile
10th
E[ra ]
E[re ]
-3.6%
-5.3%
E[ra ]
E[re ]
-3.6%
-4.6%
25th
Mean
75th
Panel A: Not Industry-Adjusted
4.0%
9.8%
17.1%
6.6%
12.1%
20.6%
Panel B: Industry-Adjusted
4.7%
9.8%
16.6%
6.6%
12.1%
20.3%
90th
Std. Dev.
24.7%
28.6%
13.2%
15.6%
23.6%
28.1%
12.7%
15.0%
Notes: This table reports the cross-sectional distributions of un-levered expected excess equity returns, E[ra ], and expected excess
equity returns, E[re ]. Industry adjustment is done by demeaning each measure of expected returns by industry-year. We then add
back the mean returns to these distributions.
B.3
Time-Series Correlations
Table 6 reports contemporaneous correlations between (within-industry) MPK dispersion and
indicators of the price of risk and the business cycle.
57
The results are similar if we compute our cross-sectional statistics within each year or industry-year and
average over the years/industry-years
48
0
0
1
2
2
3
4
4
5
(b) E[re ]
6
(a) E[ra ]
-.5
-.25
0
Histogram
.25
.5
-.5
Kernal Density
-.25
0
Histogram
.25
.5
Kernal Density
Figure 1: Cross-Sectional Distribution of Expected Excess Returns
Notes: This figure displays the cross-sectional distributions of un-levered expected excess equity returns, E[ra ], and expected
excess equity returns, E[re ]. Industry adjustment is done by demeaning each measure of expected returns by industry-year. We
then add back the mean returns to these distributions. The vertical bars denote the histograms of these distributions, while the
solid lines are the results of kernel smoothing regressions with a bandwidth of 0.25.
B.4
Aggregate Productivity Series
Solow residuals. To build a series of Solow residuals, we obtain data on real GDP and
aggregate labor and capital from the Bureau of Economic Analysis. Data on real GDP are
from BEA Table 1.1.3 (“Real Gross Domestic Product”), data on labor are from BEA Table
6.4 (“Full-Time and Part-Time Employees”) and data on the capital stock are from BEA Table
1.2 (“Net Stock of Fixed Assets”). The data are available annually from 1929-2016. With these
data we compute xt = yt − θ1 kt − θ2 nt . We extract a linear time-trend and then estimate the
autoregression in equation (9).
Firm-level series. To construct the alternative series for aggregate productivity from the
firm-level data, we use the following procedure. First, we compute firm-level productivity as
zit + βi xt = yit − θkit . We then average these values across all firms in each year. Because zit is
mean-zero and independent across firms, this yields a scaled measure of aggregate productivity,
β̄xt , where β̄ is the mean beta across firms, which under our assumptions, is approximately
two. We extract a linear time-trend from this series and then estimate the autoregression. The
coefficient from this regression gives ρx . The standard deviations of the residuals gives β̄σε and
after dividing by β̄ gives the true volatility of shocks. Applying this procedure to the set of
Compustat firms over the period 1962-2016 yields values of ρx = 0.92 and σε = .0245.
49
Table 6: Correlations of MPK Dispersion, the Price of Risk and the Business Cycle
MPK Dispersion
PD Ratio
GZ Spread
EB Premium
GDP
TFP
MPK Dispersion
1.00
-0.42
0.39
0.51
-0.53
-0.27
PD Ratio
GZ Spread
EB Premium
GDP
TFP
1.00
-0.51
-0.57
0.46
0.43
1.00
0.68
-0.59
-0.32
1.00
-0.66
-0.44
1.00
0.70
1.00
Notes: This table reports time-series correlations of MPK dispersion, measures of the price of risk and the business cycle. MPK dispersion is measured as the within-industry standard deviation in mpk. The PD ratio is the aggregate stock market price/dividend
ratio. The GZ spread and EB (excess bond) premium are measures of credit spreads. GDP is log GDP and TFP is log TFP. We
extract the cyclical components of GDP, TFP and the PD ratio using a one-sided Hodrick-Prescott filter. All series are described
in more detail in the main text and Appendix B.1. All data are quarterly and are from 1973-2015.
C
Additional Empirical Results
MPK and stock returns. We perform two additional exercises examining the link between
MPK, stock market returns and exposure to aggregate risk. First, we verify that high MPK
firms tend to offer higher expected stock market returns. To do so, we group firms into five bins,
or portfolios, based on their MPK and assess whether the high MPK groups tend exhibit higher
stock market returns than the low MPK groups. We sort firms into five portfolios based on their
year t MPK, where portfolio 1 contains low MPK firms and portfolio 5 high MPK ones. The
portfolios are rebalanced annually. We then compute four versions of the equal-weighted stock
excess stock return to each portfolio: the contemporaneous return, denoted rte , the one-period
e
e
, and the one-period ahead unlevered
, the three-period ahead return, rt+3
ahead return, rt+1
M ktcap
a
a
e
= M ktcap+Debt
, which we calculate using an unlimited liability model, rt+1
return, rt+1
rt+1
.58
We also compute the excess return on a high-minus-low MPK portfolio (MPK-HML), which is
an annually rebalanced portfolio that is long on stocks in the highest MPK portfolio and short
on stocks in the lowest.
Examining firms grouped by MPK helps eliminate firm-specific factors unrelated to MPK
that may affect returns and so allows us to hone in on the predictability of excess returns
by MPK and follows recent practice in empirical finance, which has generally moved from
addressing variation in individual firm returns to returns on portfolios of firms, sorted by
factors that are known to predict returns. In our context, however, this is likely to provide
only a noisy measure of the true relationship between MPK and risk premia. For example, the
main text studies a number of additional reasons why MPK may differ across firms, e.g., the
58
When computing one-period ahead returns, we follow Fama and French (1992) and associate the MPK for
fiscal year t with returns from July of year t + 1 to June of year t + 2. Similar timing holds for three-period
ahead returns. Value-weighted portfolios yield similar magnitudes, though the standard errors are greater
in some specifications since value-weighting the smaller within-industry samples can increase the variance of
portfolio returns. In our log-normal model, equal-weighted dispersion is the key object of interest.
50
realization of unanticipated shocks (equation (14), capital adjustment costs (Section 3.2), other
frictions/distortions (Section 4.3), etc. Each of these forces influence MPK and thus affect the
sorting variable (e.g., the high MPK bin includes firms with a high risk premium, but also firms
with a low risk premium but high realization of the idiosyncratic shock or a large distortion).
The effects of this additional noise is analogous to that of measurement error in the right-hand
side variable of a regression, attenuating the true relationship between the variables. Our twostep strategy in Table 1 is designed in part to address this concern. However, we think it useful
to examine whether MPK is associated with stock returns using this simpler approach, with the
caveat that the quantitative magnitudes should be interpreted with caution and likely represent
a lower bound.
The focus of our analysis (and the misallocation literature more broadly) is on withinindustry variation in MPK and so to control for industry effects, we demean firm-level mpk
by industry-year and sort firms based on this de-meaned measure.59 For completeness we also
present results for total, non industry-adjusted sorts, since the non-adjusted results may be
interesting in their own right (discussed more below) and confirm that the link between MPK
and stock returns holds at various levels of aggregation.
We report within-industry results in Panel A of Table 7. The table reveals a strong relationship between MPK and stock returns – high MPK portfolios tend to earn high excess
returns. The first row shows that the difference in contemporaneous returns between high and
low MPK firms, i.e., the excess return on the MPK-HML portfolio, is over 8% annually. The
second row confirms that this finding does not simply result from the simultaneous response
of stock returns and MPK to the realization of unexpected shocks – one-period ahead excess
returns are in fact predictable by MPK. The predictable spread on the MPK-HML portfolio is
over 2.5% annually. Both the contemporaneous and future MPK-HML spreads are statistically
different from zero at the 99% level. The last two rows of Panel A confirm that the results
continues to hold when examining returns further in the future and thus exhibits persistence
and after de-levering equity returns. Thus, high MPK firms tend to offer high stock returns,
both in a realized and an expected sense, suggesting that MPK differences reflect exposure to
risk factors for which investors demand compensation in the form of a higher rate of return.60
Panel B of Table 7 reports the “total” results not controlling for industry. Comparing the
two panels shows that the relationship between MPK and returns is even stronger when taken
59
There may be heterogeneity across industries on a number of dimensions, for example, in production
function coefficients or industry-level exposure to aggregate shocks.
60
Although there are several measurement differences, the results in Table 7 are related to the “profitability
premium” documented in Novy-Marx (2013) and others, i.e., high profit-to-capital firms earn high excess returns
(both industry and non industry-adjusted). Further, Novy-Marx (2013) finds that the sales-to-assets component
of profitability is the most directly related to higher returns (Appendix A.2 in that paper).
51
unconditionally across industries, suggesting that there is indeed an industry-level component
of excess returns that is predictable by an industry-level component of MPK. Although we do
not explore this finding in more detail, it is reassuring confirmation of the link we are after –
firms in industries with high average MPK tend to offer higher returns (in a predictable sense)
than firms in low MPK industries, suggesting that industry-level exposures to aggregate risk
factors may be important as well. Note that across all the variations reported in Table 7, the
within-industry effects are well over half of the total, implying a key role for the within-industry
component.61
Table 7: Excess Returns on MPK-Sorted Portfolios
Portfolio
rte
e
rt+1
e
rt+3
a
rt+1
rte
e
rt+1
e
rt+3
a
rt+1
Low
6.98
(1.63)
11.10∗∗∗
(2.61)
11.95∗∗∗
(2.99)
6.86∗∗
(2.13)
7.00∗∗
(2.01)
8.60∗∗
(2.48)
9.63∗∗∗
(2.96)
4.64∗
(1.88)
2
3
4
Panel A: Within-Industry
10.59∗∗∗
12.28∗∗∗
(2.52)
(3.05)
(3.30)
11.55∗∗∗
12.71∗∗∗
12.70∗∗∗
(3.35)
(3.75)
(3.50)
12.27∗∗∗
12.04∗∗∗
12.60∗∗∗
(3.71)
(3.75)
(3.60)
7.16∗∗∗
8.04∗∗∗
8.17∗∗∗
(2.94)
(3.37)
(3.15)
Panel B: Total
∗∗
9.08
10.67∗∗∗
12.00∗∗∗
(2.53)
(2.93)
(3.09)
∗∗∗
∗∗∗
12.27
13.48
13.73∗∗∗
(3.47)
(3.80)
(3.62)
12.43∗∗∗
12.69∗∗∗
13.90∗∗∗
(3.69)
(3.71)
(3.81)
7.53∗∗∗
8.69∗∗∗
8.66∗∗∗
(3.07)
(3.53)
(3.26)
8.91∗∗
High
MPK-HML
15.78∗∗∗
(3.73)
13.69∗∗∗
(3.36)
13.82∗∗∗
(3.58)
8.84∗∗∗
(3.04)
8.80∗∗∗
(9.54)
2.59∗∗∗
(2.98)
1.87∗∗
(2.22)
1.97∗∗∗
(2.66)
15.25∗∗∗
(3.71)
13.48∗∗∗
(3.36)
12.99∗∗∗
(3.38)
8.22∗∗∗
(3.02)
8.25∗∗∗
(4.54)
4.87∗∗∗
(2.81)
3.36∗
(1.96)
3.58∗∗∗
(3.05)
Notes: This table reports stock market returns for portfolios sorted by mpk. rte denotes equal-weighted contemporaneous annualized monthly excess stock returns (over the risk-free rate) measured in the year of the portfolio formation from January to December
e
e
of year t. rt+1
denotes the analogous future returns, measured from July of year t + 1 to June of year t + 2 and rt+3
from July of
a
year t + 3 to June of year t + 4. rt+1
denotes equal-weighted unlevered (“asset”) returns from from July of year t + 1 to June of
year t + 2, where we use an unlimited liability model to unlever equity returns. Industry adjustment is done by de-meaning mpk
by industry-year and sorting portfolios on de-meaned mpk, where industries are defined at the 4-digit SIC code level. t-statistics in
parentheses, computed using Newey-West standard errors. Significance levels are denoted by: * p < 0.10, ** p < 0.05, *** p < 0.01.
MPK and measures of risk exposure. Next, we directly relate firm MPK to measures
of risk exposure using the sensitivity of MPK to aggregate risk factors. To do so, we estimate
61
In unreported results, we have verified that the relationship between MPK and returns continues to hold
when we expand the number of portfolios (to 10) and control for size and book-to-market (though they are both
correlated with MPK).
52
regressions of the form
mpkit+1 = ψ0 + ψβ βit + ζit+1 ,
(37)
where βit is a measure of firm i’s MPK exposure to aggregate risk at time t. Although there
are several important measurement concerns in calculating these exposures compared to stock
market-based measures – e.g., they are lower frequency, may be more prone to issues of measurement/sampling error (long enough samples of MPK are only available for a subset of firms)
and require assumptions about the factor structure of aggregate risk in MPK that is less wellexplored than that in stock returns – it can still be useful to examine the relationship of these
exposures to firm-level MPK, keeping in mind these important caveats.62
We calculate measures of MPK exposure using the CAPM and Fama-French models. Specifically, we follow an analogous procedure to (35) and (36), replacing excess stock market returns
on the left-hand side of (35) and (36) with mpkit . The first regression yields measures of βM P K ,
i.e., the exposure of each firm’s MPK to the aggregate risk factors. The second regression
combines these exposures into a single value in the multi-factor Fama-French model using the
coefficients from cross-sectional Fama and MacBeth (1973) regressions, i.e., as
βit,F F = λβit =
X
λx βit,x , x ∈ M KT, HM L, SM B
x
P
where λx = T1 Tt=1 λxt .
We estimate (37) at an annual frequency and lag the right-hand side variable to control
for the simultaneous effect of unexpected shocks on contemporaneous measures of beta and
MPK. We report the results in columns (1)-(2) in Table 8. The estimates imply that firmlevel MPK is significantly related to the sensitivity of MPK to measures of aggregate risk,
i.e., the aggregate market return and the three Fama-French factors.63 In columns (3)-(4),
we estimate analogous regressions with the addition of industry-year fixed effects and a set of
standard firm-level controls, namely, market capitalization, book-to-market ratio, profitability,
and market leverage.64 All of the coefficients remain positive and statistically significant. Thus,
the results help confirm a key implication of expression (3): firm-level risk exposures – measured
using “MPK betas” – are significantly related to firm-level expected MPK.
62
Our two-stage approach in Section 2.2 helps deal with some of these issues.
We report two-way clustered standard errors by firm and industry-year to allow for arbitrary time-series
correlations for a given firm and for correlations across firms within an industry at a particular time. These
standard errors do not account for the error associated with the generated regressors (betas). As in
Guren, McKay, Nakamura, and Steinsson (2018), this requires a bootstrap procedure that clusters only on time
but precludes clustering on other dimensions. In unreported results, we follow Guren, McKay, Nakamura, and
Steinsson (2018) and perform such a bootstrap. The estimates remain significant across almost all specifications.
64
We describe these series in Appendix B.1.
63
53
Table 8: Regressions of MPK on MPK Risk Exposures
βCAP M,M P K
(1)
0.065∗∗∗
(5.46)
βF F,M P K
Observations
F.E.
Controls
79404
No
No
(2)
4.005∗∗∗
(9.49)
78920
No
No
(3)
0.024∗∗∗
(3.80)
72477
Yes
Yes
(4)
1.097∗∗∗
(4.68)
71990
Yes
Yes
Notes: This table reports the results of a panel regression of year-ahead mpk regressed on measures of firm mpk exposure to aggregate risk. Each observation is a firm-year. The dataset contains approximately 10,000 unique firms. F.E. denotes the presence
of industry-year fixed effects. Standard errors are two-way clustered by firm and industry-year. t-statistics in parentheses. Significance levels are denoted by: * p < 0.10, ** p < 0.05, *** p < 0.01.
MPK dispersion and risk premia dispersion. An additional implication of expression
(4) is that across groups of firms or segments of the economy, dispersion in expected MPK
should be positively related to dispersion in risk permia. We investigate this implication using
variation in the dispersion of expected stock market returns and measured risk exposures across
industries. Specifically, for each industry in each year, we compute the standard deviation of
MPK, σ (mpk), expected stock returns, σ (E [r]), and various measures of beta, σ (β). We then
estimate regressions of industry-level MPK dispersion on the dispersion in expected returns and
betas, i.e.,
σ (mpkjt+1 ) = ψ0 + ψ1 σ (xjt ) + ζjt+1 xjt = E [rjt ] , βjt ,
where j denotes industry. To avoid potential simultaneity biases from the realization of shocks,
we lag the independent variables (dispersion in expected returns and betas).
Table 9 reports the results of these regressions and verifies that industries with higher
dispersion in expected stock returns and risk exposures exhibit greater dispersion in MPK.
Column (1) reveals this fact using expected returns calculated from the Fama-French model.
Variation in expected return dispersion predicted by the Fama-French model explains over
20% of the variation in MPK dispersion across industry-years. Column (2) regresses MPK
dispersion on dispersion in each of the three individual factors – variation in the beta on
each factor is significantly related to MPK dispersion. Next, we repeat the exercise using
dispersion in MPK betas (described above) as the right-hand side variables. The results in
column (3) show that industries with greater dispersion in MPK betas (on each of the FamaFrench factors) exhibit greater dispersion in MPK. Columns (4)-(6) add year fixed effects and a
number of controls capturing additional measures of firm heterogeneity within industries – the
standard deviations of profitability, size, book-to-market, and market leverage. Across these
specifications, measures of within-industry heterogeneity in expected returns and aggregate risk
54
exposures remain positive and significant predictors of within-industry dispersion in MPK.65
Table 9: Industry-Level Dispersion in MPK, Expected Stock Returns and Beta
σ(E[r])
(1)
2.71∗∗∗
(30.11)
(2)
(4)
1.20∗∗∗
(9.82)
0.11∗∗∗
(6.48)
0.14∗∗∗
(11.18)
0.14∗∗∗
(13.72)
σ(βM KT )
σ(βHM L )
σ(βSM B )
σ(βCAP M,M P K )
σ(βHM L,M P K )
σ(βSM B,M P K )
Observations
R2
Industries
Year F.E.
Controls
(3)
3203
0.221
157
No
No
3210
0.265
161
No
No
(5)
(6)
0.08∗∗∗
(3.31)
0.10∗∗∗
(5.61)
0.07∗∗∗
(5.77)
0.01∗∗∗
(8.58)
0.06∗∗∗
(7.96)
0.06∗∗∗
(10.38)
2398
0.200
142
No
No
3188
0.261
153
Yes
Yes
3194
0.285
156
Yes
Yes
0.09∗∗∗
(4.08)
0.06∗∗∗
(4.80)
0.06∗∗∗
(5.70)
2380
0.348
138
Yes
Yes
Notes: This table reports a panel regression of the dispersion in mpk within industries on lagged measures of dispersion in risk
exposure within those industries. An observation is an industry-year. E [r] is the expected return computed from the Fama-French
model. β denotes the stock return beta on the Fama-French factors and βM P K the mpk beta on the same factors. t-statistics are
in parentheses. Significance levels are denoted by: * p < 0.10, ** p < 0.05, *** p < 0.01.
Bootstrapped standard errors for Table 1. Table 1 reports standard errors that are twoway clustered by firm and year, but do not account for the estimation error in the measures of
risk exposure used in the first stage, and possibly are limited by the parametric assumptions
imposed by the regression specification. To address this concern, we have performed a custom
bootstrapping procedure that runs a block-bootstrap three ways on our entire procedure, encompassing (i) our rolling-window estimation of firm risk exposures (betas), (ii) the estimation
of firm risk premia from these betas, and (iii) the estimation of the regression of future MPK
on firm risk premia. First, we randomly sample those firms in our dataset (with replacement)
which report sufficient data to compute MPK and appear for long enough to compute measures
of risk exposure. Second, we randomly sample the time-periods to use for our first and second
65
The results are robust to using different asset pricing models to compute betas and expected returns, such
as the CAPM and Hou et al. (2015) investment-CAPM models. The relationship is robust to a variety of
different controls and industry definitions as well. Finally, the results are qualitatively similar when we use the
inter-quartile range instead of the standard deviation as our measure of within-industry dispersion.
55
stage procedures. Third, within each backwards-looking rolling window used to compute the
betas, we randomly sample the time periods of observations used (but use the same time periods for each firm).66 We run this bootstrapping procedure 250 times for each specification.
We compute the standard deviation of these estimates, adjusted for the sample size of the
re-sampled regressions.67 We find that the regression coefficients in Table 1 remain significant,
except for those using the CAPM model. The implied t-statistics for specifications (1)-(6) are,
respectively, 0.36, 2.29, 5.53, 0.36, 2.53, and 6.00.68
D
Baseline Model
This appendix provides detailed derivations for the baseline model and analysis.
D.1
Solution – No Adjustment Costs
The static labor choice solves
max eẑit +β̂i xt Kitθ1 Nitθ2 − Wt Nit
with the associated first order condition
Nit =
θ2 eẑit +β̂i xt Kitθ1
Wt
1
! 1−θ
2
Substituting for the wage with Wt = Xtω and rearranging gives operating profits
Πit = Geβi xt +zit Kitθ
θ2
where G ≡ (1 − θ2 ) θ21−θ2 , βi =
(11) in the text.
1
1−θ2
β̂i − ωθ2 , zit =
66
1
ẑ
1−θ2 it
and θ =
θ1
,
1−θ2
which is equation
This will account for potential correlation of estimation error across firms.
Our random sampling of firms and years leads to, on average, fewer observations than in our baseline
dataset. We adjust the estimated standard deviation for the lower average number of observations.
68
In the case of the CAPM, the bootstrapping algorithm generates a few extreme outliers that lead to the high
standard deviation and low t-statistics. If we were to use the percentiles of the distribution instead, the p-value
would be lower than what the t-statistic implies. These extreme outliers do not occur with the Fama-French or
q 5 models, which are known to be better at matching the cross-sectional distribution of risk premia.
67
56
The first order and envelope conditions associated with (1) give the Euler equation:
θ−1
+1−δ
1 = Et Mt+1 θezit+1 +βi xt+1 GKit+1
θ−1
= (1 − δ) Et [Mt+1 ] + θGKit+1
Et emt+1 +zit+1 +βi xt+1
Substituting for mt+1 and rearranging,
h
i
1 2 2
Et emt+1 +zit+1 +βi xt+1 = Et elog ρ−γt εt+1 − 2 γt σε +zit+1 +βi xt+1
h
i
log ρ+ρz zit +εit+1 +βi ρx xt +(βi −γt )εt+1 − 21 γt2 σε2
= Et e
1
2
1
2 2
2
= elog ρ+ρz zit +βi ρx xt + 2 σε̃ + 2 βi σε −βi γt σε
and
i
h
1 2 2 1 2 2
1 2 2
Et [Mt+1 ] = Et elog ρ−γt εt+1 − 2 γt σε = elog ρ+ 2 γt σε − 2 γt σε = ρ
so that
θ−1
θGKit+1
=
1 − (1 − δ) ρ
1
2
1
2 2
2
elog ρ+ρz zit +βi ρx xt + 2 σε̃ + 2 βi σε −βi γt σε
and rearranging and taking logs,
kit+1
1
=
1−θ
1 2 1 2 2
2
α̃ + σε̃ + βi σε + ρz zit + βi ρx xt − βi γt σε
2
2
where
α̃ = log θ + log G − α
α = − log ρ + log (1 − (1 − δ) ρ) = log (rf + δ)
Ignoring the variance terms gives equation (12).
The realized mpk is given by
mpkit+1 = log θ + πit+1 − kit+1
= log θ + log G + zit+1 + βi xt+1 − (1 − θ) kit+1
= log θ + log G + zit+1 + βi xt+1 − α̃ − ρz zit − βi ρx xt + βi γt σε2
= α + εit+1 + βi εt+1 + βi γt σε2
The time t conditional expected mpk is
Et [mpkit+1 ] = α + βi γt σε2
57
and the time t and mean cross-sectional variances are, respectively,
2
σE2 t [mpkit+1 ] = σβ2 γt σε2
h
i
h
2 i
2
= σβ2 γ02 + γ12 σx2 σε2
E σE2 t [mpkit+1 ] = E σβ2 (γ0 + γ1 xt )2 σε2
D.2
Solution – Adjustment Costs
With capital adjustment costs, the firm’s investment problem takes the form
GXtβi Zt Kitθ − Kit+1 + (1 − δ) Kit − Φ (Iit , Kit )
V (Xt , Zit , Kit ) = max
Kit+1
(38)
+ Et [Mt+1 V (Xt+1 , Zit+1 , Kit+1 )]
Policy function. The first order and envelope conditions associated with (38) give the Euler
equation:
1+ξ
Kit+1
Kit
"
!
2
Kit+2
Kit+2
ξ Kit+2
θ−1
−1
= Et Mt+1 Gθezit+1 +βi xt+1 Kit+1
−1 +ξ
−1
+1−δ−
2 Kit+1
Kit+1
Kit+1
"
!#
2
ξ Kit+2
ξ
θ−1
= Et Mt+1 Gθezit+1 +βi xt+1 Kit+1
+1−δ+
−
2 Kit+1
2
In the non-stochastic steady state,
MP K
Π
P
R
1
θ−1
1 1
1
+δ−1
= GθK
= +δ−1 ⇒ K =
ρ
Gθ ρ
= GK θ ⇒ D = GK θ − δK
ρ
=
D
1−ρ
1
D
= 1+
=
⇒ rf = − log ρ
P
ρ
θ−1
Define the investment return:
I
Rit+1
θ−1
Gθezit+1 +βi xt+1 Kit+1
+ 1 − δ + 2ξ
=
1 + ξ KKit+1
−
1
it
58
Kit+2
Kit+1
2
−
ξ
2
and log-linearizing,
I
= ρGθK θ−1 (zit+1 + βi xt+1 ) + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) kit+1 + ρξkit+2 + ξkit
rit+1
− log ρ − ρGθ (θ − 1) K θ−1 k
where k = log K. Rearranging and suppressing constants yields expression (25).
To derive the investment policy function, conjecture it takes the form
kit+1 = φ0i + φ1 βi xt + φ2 zit + φ3 kit
Then,
kit+2 = φ0i (1 + φ3 ) + φ1 βi (ρx + φ3 ) xt + φ2 (ρz + φ3 ) zit + φ23 kit + φ1 βi εt+1 + φ2 εit+1
Substituting into the investment return,
I
rit+1
=
+
+
+
+
ρGθ (θ − 1) K θ−1 − ξ (1 − ρφ3 ) φ0i − log ρ − ρGθ (θ − 1) K θ−1 k
ρGθK θ−1 ρz + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ2 + ρξ (ρz + φ3 ) φ2 zit
ρGθK θ−1 ρx + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ1 + ρξ (ρx + φ3 ) φ1 βi xt
ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ3 + ρξφ23 + ξ kit
ρGθK θ−1 + ρξφ2 εit+1 + ρGθK θ−1 + ρξφ1 βi εt+1
and
I
rit+1
+ mit+1 =
+
+
+
+
1
1
ρGθ (θ − 1) K θ−1 − ξ (1 − ρφ3 ) φ0i − ρGθ (θ − 1) K θ−1 k − γ02 σε2 − γ12 σε2 x2t
2
2
ρGθK θ−1 ρz + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ2 + ρξ (ρz + φ3 ) φ2 zit
ρGθK θ−1 ρx + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ1 + ρξ (ρx + φ3 ) φ1 βi − γ0 γ1 σε2 xt
ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ3 + ρξφ23 + ξ kit
ρGθK θ−1 + ρξφ2 εit+1 + ρGθK θ−1 + ρξφ1 βi − γ0 − γ1 xt εt+1
59
The Euler equation governing the investment return implies
I
1
I
0 = Et rit+1
+ mt+1 + var rit+1
+ mit+1
2
θ−1
= ρGθ (θ − 1) K
− ξ (1 − ρφ3 ) φ0i − ρGθ (θ − 1) K θ−1 k
+ ρGθK θ−1 ρz + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ2 + ρξ (ρz + φ3 ) φ2 zit
+ ρGθK θ−1 ρx + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ1 + ρξ (ρx + φ3 ) φ1 − ρGθK θ−1 + ρξφ1 γ1 σε2 βi xt
+
ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ3 + ρξφ23 + ξ kit
2
1
ρGθK θ−1 + ρξφ2 σε̃2
+
2
2
1
+
ρGθK θ−1 + ρξφ1 βi2 σε2 − ρGθK θ−1 + ρξφ1 βi γ0 σε2
2
and we can solve for the coefficients from:
0 =
+
+
=
=
=
ρGθ (θ − 1) K θ−1 − ξ (1 − ρφ3 ) φ0i − ρGθ (θ − 1) K θ−1 k
2
1
ρGθK θ−1 + ρξφ2 σε̃2
2
2
1
ρGθK θ−1 + ρξφ1 βi2 σε2 − ρGθK θ−1 + ρξφ1 βi γ0 σε2
2
ρGθK θ−1 ρz + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ2 + ρξ (ρz + φ3 ) φ2
ρGθK θ−1 ρx + ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ1 + ρξ (ρx + φ3 ) φ1 − ρGθK θ−1 + ρξφ1 γ1 σε2
ρGθ (θ − 1) K θ−1 − ξ (1 + ρ) φ3 + ρξφ23 + ξ
Define ξˆ =
ξ
ρGθK θ−1
=
ξ
.
1−ρ(1−δ)
0 =
φ1 =
φ2
Then,
ˆ 2 + ξˆ
(θ − 1) − ξˆ (1 + ρ) φ3 + ρξφ
3
(ρx − γ1 σε2 ) φ3
ξˆ (1 − ρρx φ3 + ργ1 σε2 φ3 )
ρz φ3
=
ˆ
ξ (1 − ρρz φ3 )
φ0i = φ00 − φ01 βi + φ02 βi2
60
where
φ00
φ01
φ02
ρGθ (1 − θ) K θ−1 k + 12 ρGθK θ−1 + ρξφ2
=
ρGθ (1 − θ) K θ−1 + ξ (1 − ρφ3 )
γ0 σε2
φ3
=
ξˆ (1 − ρφ3 ) 1 − ρρx φ3 + ργ1 σε2 φ3
2
σε̃2
2
ρGθK θ−1 ρξφ1 + 21 (ρξφ1 )2 + 21 ρGθK θ−1
=
σε2
ρGθ (1 − θ) K θ−1 + ξ (1 − ρφ3 )
1
Note that φξ̂3 goes to 1−θ
as ξˆ goes to zero and zero as ξˆ goes to infinity. Again ignoring variance
terms, and defining φ4 = γφ001
, the policy function is
σ2
ε
kit+1 = φ1 βi xt + φ2 zit + φ3 kit − φ4 βi γ0 σε2 + constant
which is equation (27) in the text.
Persistent MPK Dispersion. To derive expression (29), take the unconditional expectation
of the policy function to obtain
φ4 βi γ0 σε2
E [kit+1 ] = −
1 − φ3
and thus the unconditional expected mpk as
E [mpkit+1 ] = (θ − 1) E [kit+1 ] + constant
1
(1 − θ) φ3
β γ σ 2 + constant
=
2) i 0 ε
ˆ
1
−
ρφ
(ρ
−
γ
σ
3
x
1 ε
ξ (1 − ρφ3 ) (1 − φ3 )
where we have substituted using the definition of φ4 . Lastly, we can use the definition of φ3 to
show that the first fraction is equal to one and thus,
E [mpkit+1 ] =
D.3
1
βi γ0 σε2 + constant
2
1 − ρφ3 (ρx − γ1 σε )
Aggregation
The first order condition on labor gives
β̂i xt +ẑit
Nit =
θ2 e
Wt
61
Kitθ1
1
! 1−θ
2
and substituting for the wage,
1
1−θ
2
Nit = θ2 e(β̂i −ω)xt +ẑit Kitθ1
Labor market clearing gives:
Z
Nt =
1
1−θ2
Nit di = θ2
e
Z
1
− 1−θ
ωxt
1
e 1−θ2
2
β̂i xt +zit
so that
θ2
1−θ2
θ2
e
2
!θ2
Nt
θ
2 ωx
− 1−θ
t
=
R
1
e 1−θ2
Kitθ di
β̂i xt +zit
Kitθ di
Then,
θ2
Yit = eβ̂i xt +ẑit Kitθ1 Nitθ2 = θ21−θ2 e
1
= R
e 1−θ2
e
β̂i xt +zit
1
β̂ x +zit
1−θ2 i t
Kitθ
θ
2 ωx
− 1−θ
t
2
1
e 1−θ2
β̂i xt +zit
Kitθ
θ
Kitθ di
θ2 Nt 2
By definition,
1
M P Kit = R
β̂i xt +zit
Kitθ−1
θ
θ2 Nt 2
1
β̂ x +z
e 1−θ2 i t it Kitθ di
θe 1−θ2
and rearranging,
Kit =
θe
1
β̂ x +zit
1−θ2 i t
1
! 1−θ
M P Kit
θ2
! 1−θ
Nt
R
e
1
β̂ x +zit
1−θ2 i t
Kitθ di
Capital market clearing gives
Z
Kt =
Nt
1
Kit di = θ 1−θ
θ2
! 1−θ
Z
R
e
1
β̂ x +zit
1−θ2 i t
so that
Kitθ = R
e
1
1
1
β̂i xt + 1−θ
zit
Kitθ di
θ
1
− 1−θ
M P Kit
Kt
1
1
1
− 1−θ
β̂ x +
z
1−θ i t 1−θ it
M P Kit di
1
1
β̂ x + 1 z
1−θ2 1−θ i t 1−θ it
e 1−θ2
1
e 1−θ2 1−θ
62
−
1
M P Kit 1−θ di
and substituting into the expression for Yit ,
e
!θ
e 1−θ2
R
Yit =
R 1
e 1−θ2 β̂i xt +zit
1 β̂ x + 1 z
− 1
1−θ i t 1−θ it M P K 1−θ
it
1
1 β̂ x + 1 z
− 1
t 1−θ it
e 1−θ2 1−θ i
M P Kit 1−θ
1
1
β̂ x +zit
1−θ2 i t
1 β̂ x + 1 z
− 1
1−θ i t 1−θ it M P K 1−θ
it
1
1 β̂ x + 1 z
− 1
t 1−θ it
e 1−θ2 1−θ i
M P Kit 1−θ
Kt
di
θ
1
e 1−θ2
R
θ2 Nt 2
!θ
Kt
di
di
1 β̂ x + 1 z
− θ
1−θ i t 1−θ it M P K 1−θ
it
!θ
1
1 β̂ x + 1 z
− 1
t 1−θ it
e 1−θ2 1−θ i
M P Kit 1−θ di
1
e 1−θ2
R
θ
θ
θ2 Kt 1 Nt 2
=
1 β̂ x + 1 z
− θ
1−θ i t 1−θ it M P K 1−θ di
it
!θ
1 β̂ x + 1 z
1
− 1
t 1−θ it
e 1−θ2 1−θ i
M P Kit 1−θ di
1
R
R
e 1−θ2
Aggregate output is then
Z
Yt =
Yit di = At Ktθ1 Ntθ2
where
1−θ2
1
1
1
− θ
β̂i xt + 1−θ
zit
R e 1−θ
2 1−θ
M P Kit 1−θ di
At =
θ
1
R 1 1 β̂i xt + 1 zit
−
1−θ
e 1−θ2 1−θ
M P Kit 1−θ di
Taking logs,
at
Z
Z
θ
1
1
1
1
1
− 1−θ
− 1−θ
β̂ x + 1 z
β̂ x + 1 z
1−θ2 1−θ i t 1−θ it
1−θ2 1−θ i t 1−θ it
M P Kit di − θ log e
M P Kit di
= (1 − θ2 ) log e
The first expression in braces is equal to
1
1 ¯
θ
1
β̂xt −
m̄pk +
1 − θ 1 − θ2
1−θ
2
−
1
1−θ
2
1
θ
σmpk,β̂i xt +zit
2
(1 − θ) 1 − θ2
63
1
1 − θ2
2
!
x2t σβ̂2
+
σz2
1
+
2
θ
1−θ
2
2
σmpk
and the second to
1 ¯
θ
1
θ
β̂xt −
m̄pk + θ
1 − θ 1 − θ2
1−θ
2
−
1
1−θ
2
1
1 − θ2
2
!
x2t σβ̂2
+
σz2
1
+ θ
2
1
1−θ
2
2
σmpk
θ
1
σmpk,β̂i xt +zit
2
(1 − θ) 1 − θ2
and combining (and using σβ =
gives
1 θ
1 1
2
x2t σβ2 + σz2 −
σmpk
21−θ
21−θ
1
θ
2
= a∗t − (1 − θ2 )
σmpk
2
1−θ
1
θ
(1
−
θ
1
2) 2
= a∗t −
σ
2 1 − θ1 − θ2 mpk
¯
= β̂xt + (1 − θ2 )
at
D.4
1
σ )
1−θ2 β̂
Stock Market Returns
We derive stock market returns in the environment with adjustment costs. This nests the
simpler case without them when ξ = 0.
Dividends are equal to
zit+1 +βi xt+1
Dit+1 = e
θ
Kit+1
ξ
− Kit+2 + (1 − δ) Kit+1 −
2
2
Kit+2
− 1 Kit+1
Kit+1
and log-linearizing,
dit+1
Π
Π
K
Π
K
K
= (zit+1 + βi xt+1 ) + θ + (1 − δ)
kit+1 − kit+2 + log D − θ − δ
k
D
D
D
D
D
D
where k = log K. Substituting for kit+1 and kit+2 from Appendix D.2 and rearranging,
dit+1 = A0i + Ã1 zit + A1 βi xt + Ã2 εit+1 + A2 βi εt+1 + A3 kit
64
where
A0i =
A1 =
Ã1 =
A2 =
Ã2 =
A3 =
K
K
Π
(k − φ0i ) − φ0i φ3
log D − θ − δ
D
D
D
Π
Π K
ρx + θ + (1 − δ − ρx − φ3 ) φ1
D
D D
Π
Π K
ρz + θ + (1 − δ − ρz − φ3 ) φ2
D
D D
Π K
− φ1
D D
Π K
− φ2
D D
Π K
θ + (1 − δ − φ3 ) φ3
D D
By definition, returns are equal to
Rit+1 =
Dit+1 + Pit+1
Pit
and log-linearizing,
rit+1 = ρpit+1 + (1 − ρ) dit+1 − pit − log ρ + (1 − ρ) log
P
D
Conjecture the stock price takes the form
pit = c0i + c1 βi xt + c2 zit + c3 kit
Then,
rit+1
P
= − log ρ + (1 − ρ) log + A0i − c0i + ρc3 φ0i
D
+ (ρρz − 1) c2 + ρc3 φ2 + (1 − ρ) Ã1 zit
+ ((ρρx − 1) c1 + ρc3 φ1 + (1 − ρ) A1 ) βi xt
+ ((ρφ3 − 1) c3 + (1 − ρ) A3 ) kit
+ ρc2 + (1 − ρ) Ã2 εit+1 + (ρc1 + (1 − ρ) A2 ) βi εt+1
and the (log) excess return is the (negative of the) conditional covariance with the SDF:
e
log Et Rit+1
= (ρc1 + (1 − ρ) A2 ) βi γt σε2
65
To solve for the coefficients, use the Euler equation. First,
rit+1 + mit+1
P
1
= (1 − ρ) log + A0i − c0i + ρc3 φ0i − γ02 σε2
D
2
+ (ρρz − 1) c2 + ρc3 φ2 + (1 − ρ) Ã1 zit
+
((ρρx − 1) c1 + ρc3 φ1 + (1 − ρ) A1 ) βi − γ0 γ1 σε2 xt
+ ((ρφ3 − 1) c3 + (1 − ρ) A3 ) kit
1 2 2 2
−
γ1 σε xt
2
+ ρc2 + (1 − ρ) Ã2 εit+1
+ ((ρc1 + (1 − ρ) A2 ) βi − γ0 − γ1 xt ) εt+1
The Euler equation implies
1
0 = Et [rit+1 + mit+1 ] + var (rit+1 + mit+1 )
2
P
1
= (1 − ρ) log + A0i − c0i + ρc3 φ0i + (ρc1 + (1 − ρ) A2 )2 βi2 σε2 − (ρc1 + (1 − ρ) A2 ) βi γ0 σε2
D
2
2
1
ρc2 + (1 − ρ) Ã2 σε̃2
+
2
+ (ρρz − 1) c2 + ρc3 φ2 + (1 − ρ) Ã1 zit
+ (ρρx − 1) c1 + ρc3 φ1 + (1 − ρ) A1 − (ρc1 + (1 − ρ) A2 ) γ1 σε2 βi xt
+ ((ρφ3 − 1) c3 + (1 − ρ) A3 ) kit
and so by undetermined coefficients,
P
1
0 = (1 − ρ) log + A0i − c0i + ρc3 φ0i + (ρc1 + (1 − ρ) A2 )2 βi2 σε2 − (ρc1 + (1 − ρ) A2 ) βi γ0 σε2
D
2
2
1
+
ρc2 + (1 − ρ) Ã2 σε̃2
2
= (ρρz − 1) c2 + ρc3 φ2 + (1 − ρ) Ã1
= (ρρx − 1) c1 + ρc3 φ1 + (1 − ρ) A1 − (ρc1 + (1 − ρ) A2 ) γ1 σε2
= (ρφ3 − 1) c3 + (1 − ρ) A3
66
or
(1 − ρ) A3
1 − ρφ3
ρc3 φ2 + (1 − ρ) Ã1
=
1 − ρρz
ρc3 φ1 + (1 − ρ) (A1 − A2 γ1 σε2 )
=
1 − ρρx + ργ1 σε2
c3 =
c2
c1
Substituting for c1 we can solve for
e ρ2 c3 φ1 + (1 − ρ) (ρA1 + (1 − ρρx ) A2 )
=
βi γt σε2
log Et Rit+1
1 − ρρx + ρσε2 γ1
Solving for
ρA1 + (1 − ρρx ) A2 =
1
ρ
+ δ − 1 − ρθφ1 φ3
1
ρ
+ δ (1 − θ) − 1
ρ2 (1 − ρ) φ1 φ3
= θ
1 − ρφ3
2
ρ c3 φ1
1
ρ
1
ρ
− φ3
+ δ (1 − θ) − 1
substituting into the return equation and simplifying, we obtain
e
log Et Rit+1
= ψβi γt σε2
where
ψ=
1
ρ
1
ρ
+δ−1
1−ρ
+ δ (1 − θ) − 1 1 − ρρx + ργ1 σε2
which is equation (21) in the text.
The Sharpe ratio is the ratio of expected excess returns to the conditional standard deviation
of the return:
ψβi γt σε2
r
SRit =
2
ρc2 + (1 − ρ) Ã2
We can solve for
ρc2 + (1 − ρ) Ã2 =
1
ρ
1
ρ
σε̃2 + ψ 2 βi2 σε2
+δ−1
1−ρ
+ δ (1 − θ) − 1 1 − ρρz
and substituting and rearranging gives the expression in footnote 29.
For a perfectly diversified portfolio (i.e., the integral over individual returns) idiosyncratic
shocks cancel, i.e., σε̃2 = 0 and SRmt = γt σε .
67
D.5
Autocorrelation of Investment
To derive the autocorrelation of investment, define net investment as ∆kit+1 = kit+1 − kit . We
use the following:
cov (∆zit , zit ) = cov ((ρz − 1) zit−1 + εit .ρz zit−1 + εit )
= ρz (ρz − 1) σz2 + σε̃2
1
σε̃2
=
1 + ρz
cov (∆kit , zit ) = cov (∆kit , ρz zit−1 + εit )
= ρz cov (∆kit , zit−1 )
= ρz cov (φ1 βi ∆xt−1 + φ2 ∆zit−1 + φ3 ∆kit−1 , zit−1 )
= ρz (cov (φ2 ∆zit−1 , zit−1 ) + φ3 cov (∆kit−1 , zit−1 ))
1
σ 2 + ρz φ3 cov (∆kit−1 , zit−1 )
= ρz φ 2
1 + ρz ε̃
so that
E [cov (∆kit , zit )] =
ρz
φ2 σε̃2
1 + ρz 1 − φ3 ρz
Next,
cov (∆kit+1 , ∆zit+1 ) = cov (φ1 βi ∆xt + φ2 ∆zit + φ3 ∆kit . (ρz − 1) zit + εit+1 )
= φ2 (ρz − 1) cov (∆zit , zit ) + φ3 (ρz − 1) cov (∆kit , zit )
1
ρz φ3 (ρz − 1) φ2 σε̃2
= φ2 (ρz − 1)
σε̃2 +
1 + ρz
1 + ρz
1 − φ3 ρz
2
ρz − 1 φ2 σε̃
=
1 + ρz 1 − φ3 ρz
Similar steps give
cov (∆kit+1 , ∆xt+1 ) =
68
ρx − 1 φ1 βi σε2
1 + ρx 1 − φ3 ρx
Combining these gives the variance of investment:
2
2
= φ21 βi2 var (∆xt ) + φ22 var (∆zit ) + φ23 σ∆k
σ∆k
+ 2φ1 φ3 βi cov (∆xt , ∆kit ) + 2φ2 φ3 cov (∆zit , ∆kit )
2
2
2
= φ21 βi2
σε2 + φ22
σε̃2 + φ23 σ∆k
1 + ρx
1 + ρz
2 2
2
2φ1 φ3 βi σε ρx − 1 2φ22 φ3 σε̃2 ρz − 1
+
+
1 − φ3 ρx 1 + ρx 1 − φ3 ρz 1 + ρz
2
2
φ3 (ρx − 1)
φ3 (ρz − 1)
2 σε̃
2 2
2 2 σε
1+
+ 2φ2
1+
= φ3 σ∆k + 2φ1 βi
1 + ρx
1 − φ 3 ρx
1 + ρz
1 − φ3 ρz
2
1
1
1
1
=
φ21 βi2 σε2
+ φ22 σε̃2
1 + φ3
1 + ρx 1 − φ3 ρx
1 + ρz 1 − φ3 ρz
Next,
cov (∆kit+1 , ∆kit ) = cov (φ1 βi ∆xt + φ2 ∆zit + φ3 ∆kit , ∆kit )
2
= φ1 βi cov (∆xt , ∆kit ) + φ2 cov (∆zit , ∆kit ) + φ3 σ∆k
1
1
ρx − 1
ρz − 1
2
= φ21 βi2 σε2
+ φ22 σε̃2
+ φ3 σ∆k
1 + ρx 1 − φ3 ρx
1 + ρz 1 − φ3 ρz
and the autocorrelation is:
1
1
2 2 2 ρx −1
2 2 ρz −1
1 + φ3 φ1 βi σε 1+ρx 1−φ3 ρx + φ2 σε̃ 1+ρz 1−φ3 ρz
corr (∆kit+1 , ∆kit ) = φ3 +
1
1
1
1
2 φ21 βi2 σε2 1+ρ
+ φ22 σε̃2 1+ρ
x 1−φ3 ρx
z 1−φ3 ρz
(39)
Notice that this approaches
corr (∆kit+1 , ∆kit ) = φ3 + (1 − φ3 )
ρx − 1
2
as ρz and ρx become close. Further, in the case both shocks follow a random walk, the autocorrelation is simply equal to φ3 .
E
Numerical Procedure
Our numerical approach to parameterize the model is as follows. To accurately capture the
properties of the time-varying risk premium, we solve for returns numerically using a fourthorder approximation in Dynare++. For a given set of the parameters γ0 , γ1 , ξ and σβ2 , we solve
the model for a wide grid of beta-types centered around the mean beta. We use an 11 point
grid ranging from -3 to 7 (the results are not overly sensitive to the width of the grid). We
69
simulate a time series of excess returns for a large number of firms of each type, which results in
a large panel of excess returns. Averaging returns across these firms in each time period yields
a series for the market excess return. We can then compute the mean and standard deviation
(i.e., Sharpe ratio) of the market return.
Next, we compute the expected return
h for each ibeta-type in each time period directly as the
it+1
and then average over the time periods to
conditional expectation Et [Rit+1 ] = Et Dit+1P+P
it
obtain the average expected return for firms of each type. We then use these values to calculate
2
the dispersion in expected returns, σEr
, interpolating for values of β that are not on the grid.
We use a simulated investment series to calculate the autocorrelation of investment.
Finally, we find the set of the four parameters, γ0 , γ1 , σβ2 and ξ that make the simulated
moments consistent with the empirical ones, i.e., (i) market excess return, (ii) market Sharpe
ratio, (iii) cross-sectional dispersion in expected returns and (iv) the autocorrelation of investment. As shown in column (1) of the bottom panel of Table 4, the simulated moments are quite
close to their empirical counterparts.
F
Extensions
Our baseline framework in Section 3 features (i) a single source of aggregate risk and (ii) a tight
connection between financial market conditions and the “real” side of the economy – indeed, the
state of technology determined both the common component of firm-level productivities and the
price of risk simultaneously. In this appendix, we generalize that setup to allow for (i) multiple
risk factors and (ii) more flexible formulations of the determinants of financial conditions.
Although empirically disciplining the additional factors added here may be challenging, we
demonstrate that the same insights from the baseline analysis go through. We also study
versions of the model where heterogeneity in risk premia stem from “alphas” or “mis-pricing” in
addition to betas and from differences across firms in exposure to capital price shocks, rather
than productivity shocks. and show that our main results continue to hold.
F.1
Multifactor Model
There are J aggregate risk factors in the economy. Firms have heterogeneous loadings on these
factors, so that the profit function (in logs) takes the form
πit = βi xt + zit + θkit
70
(40)
where βi is a vector of factor loadings of firm i, e.g., the j-th element of βi is the loading of firm
i profits on factor j, and xt is the vector of factor realizations at time t, i.e.,
0
β1i
β2i
βi =
..
.
0
x1t
x2t
xt =
..
.
βJi
xJt
Each factor, indexed by j, follows an AR(1) process
xjt+1 = ρj xjt + εjt+1 ,
εjt+1 ∼ N
0, σε2j
(41)
where the innovations are potentially correlated across factors. Denote by Σf the covariance
matrix of factor innovations, i.e.,
σε21 σε1 ,ε2 · · · σε1 ,εJ
σε2 ,ε1 σε22 · · · σε2 ,εJ
Σf = .
..
..
.
.
.
.
σεJ ,ε1 σεJ ,ε2 · · · σε2J
The idiosyncratic component of firm productivity follows
zit+1 = ρz zit + εit+1 ,
εit+1 ∼ N 0, σε̃2
(42)
The stochastic discount factor takes the form
1
mt+1 = log ρ − γεt+1 − γΣf γ 0
2
(43)
where γ is a vector of factor exposures, e.g., element γj captures the exposure of the SDF to
the j-th factor, and εt+1 is the vector of innovations in each factor, i.e.,
0
γ1
γ2
γ=
..
.
εt+1
γJ
ε1t+1
ε2t+1
=
..
.
εJt+1
For purposes of illustration, we assume γ is constant through time and there are no adjustment
costs, although these assumptions are easily relaxed. Expressions (40), (41), (42) and (43) are
71
simple extensions of (11), (9) and (10).
Following a similar derivation as Appendix D.1, we can derive the realized mpk:
mpkit+1 = α + εit+1 + βi εt+1 + βi Σf γ 0
where βi and εt+1 denote vectors of factor loadings and shocks. The expected mpk and its
cross-sectional dispersion are given by
Et [mpkit+1 ] = α + βi Σf γ 0 ,
σE2 t [mpk] = γΣ0f Σβ Σf γ 0
where Σβ is the covariance matrix of factor loadings across firms, i.e.,
σβ21 σβ1 ,β2 · · · σβ1 ,βJ
σβ2 ,β1 σβ22 · · · σβ2 ,βJ
Σβ = .
..
..
.
.
.
.
σβJ ,β1 σβJ ,β2 · · · σβ2J
This is the natural analog of expression (16): (i) expected mpk is determined by the firm’s
exposure to (all) the aggregate risk factors in the economy and the risk prices of those factors,
and (ii) mpk dispersion is a function of the dispersion in those exposures across firms as captured
by Σβ .
Similar steps as Appendix D.4 gives the following (approximate) expression for expected
excess stock returns and the cross-sectional dispersion in expected returns:
e
Erit+1
= βi ψΣf γ 0 ,
2
0 0
0
σEr
e = γΣf ψ Σβ ψΣf γ
t
where ψ is a diagonal matrix with
ψjj =
1
ρ
1
ρ
+δ−1
1−ρ
+ (1 − θ) δ − 1 1 − ρρj
where ρj denotes the persistence of factor j. These are the analogs of expressions (21) and (22)
– expected returns depend on factor exposures and the risk prices of those factors. Expected
return dispersion depends on the dispersion in those exposures, here captured by Σβ .
Thus, the same insights from the single factor model go through – dispersion in Empk and
expected returns are both determined by variation in exposures to the set of aggregate factors
and hence, there is a tight relationship between the two. To quantify the impact of these factors
on mpk dispersion, however, we would need to know all the primitives governing the dynamics
72
of the factors, e.g., the vector of persistences ρ and the covariance matrix Σf , and exposures,
i.e., the exposures of the SDF, γ, and the vectors of firm loadings, Σβ . This would likely entail
taking a stand on the nature of each factor, computing their properties from the data and
calibrating/estimating the γ vector and the covariance matrix of firm exposures, Σβ .
F.2
Financial Shocks
Our baseline model tightly linked financial conditions, for example, the price of risk, to macroeconomic conditions, i.e., the state of aggregate technology. However, financial conditions may
not co-move one-for-one with the “real” business cycle. Here, we extend the setup to include
pure financial shocks. The stochastic discount factor takes the form
1
mt+1 = log ρ − γt εt+1 − γt2 σε2
2
γt = γ0 + γf ft ,
(44)
where
εf ∼ N 0, σε2f .
ft+1 = ρf ft + εf ,
In this formulation, ft denotes the time-varying state of financial conditions, which is now
disconnected from the state of aggregate technology. These financial factors may be correlated
with real conditions, xt , but need not be perfectly so. Thus, there is scope for changes in
financial conditions, independent of those in real conditions, to affect the price of risk and
through this channel, the allocation of capital.69 Note the difference between this setup and
the one in Section F.1 – here, the financial factor, ft , does not directly enter the profit function
of the firm, it only affects the price of risk. Thus, it is a shock purely to financial market
conditions. In contrast, the factors considered in Section F.1 directly affected firm profitability.
Keeping the remainder of the environment the same as Section 3, we can derive exactly the
same expressions for expected mpk and its cross-sectional variance, i.e.,
Et [mpkit+1 ] = α + βi γt σε2 ,
σE2 t [mpkit+1 ] = σβ2 γt σε2
2
,
where now γt is a function of financial market conditions. When credit market conditions
tighten (i.e., when ft is small/negative since γf < 0), γt is high and mpk dispersion will rise.
Just as in Section 3, the conditional expectation of one-period ahead TFP is given by
2
1 θ1 (1 − θ2 ) 2
σβ γt σε2
Et [at+1 ] = Et a∗t+1 −
2 1 − θ1 + θ2
69
Our baseline model is the nested case where γf = γ1 and ft and xt are perfectly correlated.
73
(45)
which illustrates the effects of a deterioration in financial conditions on macroeconomic performance – when credit market conditions tighten and risk premia rise (i.e., ft falls), the resulting
increase in mpk dispersion leads to a fall in aggregate TFP.
Finally, the average long-run level of Empk dispersion and aggregate TFP are given by
2
2 2
2
2
2 2
E σEmpkt = σβ γ0 + γf σf σεf ,
2 2
1 θ1 (1 − θ2 ) 2 2
2 2
ā = a −
σ γ + γf σf σεf
,
2 1 − θ1 − θ2 β 0
∗
σε2
where σf2 = 1−ρf2 . The expressions reveal a tight connection between financial conditions and
f
long-run performance of the economy – higher financial volatility (σε2f ), even independent of
the state of the macroeconomy, induces greater persistent MPK dispersion and depresses the
average level of achieved productivity.
G
Other Distortions
With other distortions, the derivations are similar to those in Appendix D.1. The Euler equation
is given by
θ−1
1 = Et Mt+1 θeτit+1 +zit+1 +βi xt+1 GKit+1
+1−δ
θ−1
= (1 − δ) Et [Mt+1 ] + θGKit+1
Et emt+1 +τit+1 +zit+1 +βi xt+1
Idiosyncratic distortions. Substituting for mt+1 and τit+1 and rearranging,
i
h
1 2 2
Et emt+1 +τit+1 +zit+1 +βi xt+1 = Et elog ρ−γt εt+1 − 2 γt σε −ν1 zit+1 −ηit+1 +zit+1 +βi xt+1
i
h
1 2 2
= Et elog ρ+(1−ν)ρz zit +(1−ν1 )εit+1 +βi ρx xt +(βi −γt )εt+1 − 2 γt σε −ηit+1
1
2 2 1 2 2
σε̃ + 2 βi σε −βi γt σε2 −ηit+1
= elog ρ+(1−ν1 )ρz zit +βi ρx xt + 2 (1−ν1 )
so that
θ−1
θGKit+1
=
1 − (1 − δ) ρ
1
2 2 1 2 2
σε̃ + 2 βi σε −βi γt σε2 −ηit+1
elog ρ+(1−ν1 )ρz zit +βi ρx xt + 2 (1−ν1 )
and rearranging and taking logs,
kit+1
1
=
1−θ
1
1 2 2
2 2
2
α̃ + (1 − ν1 ) σε̃ + βi σε + (1 − ν1 ) ρz zit + βi ρx xt − βi γt σε − ηit+1
2
2
where α̃ and α are as defined in Appendix D.1.
74
The realized mpk is given by (ignoring the variance terms)
mpkit+1 = log θ + πit+1 − kit+1
= log θ + log G + zit+1 + βi xt+1 − (1 − θ) kit+1
= log θ + log G + zit+1 + βi xt+1 − α̃ − (1 − ν1 ) ρz zit − βi ρx xt + βi γt σε2 + ηit+1
= α + εit+1 + βi εt+1 + ν1 ρz zit + βi γt σε2 + ηit+1
The conditional expected mpk is
Et [mpkit+1 ] = α + ν1 ρz zit + βi γt σε2 + ηit+1
and the cross-sectional variance is
σE2 t [mpkit+1 ] = (ν1 ρz )2 σz2 + ση2 + γt σε2
2
σβ2
(46)
Deriving stock returns follows closely the steps in Appendix D.4. Dividends are equal to
τit+1 +zit+1 +βi xt+1
Dit+1 = e
θ
Kit+1
ξ
− Kit+2 + (1 − δ) Kit+1 −
2
2
Kit+2
− 1 Kit+1
Kit+1
and log-linearizing,
dit+1
Π
Π
K
K
Π
K
= (τit+1 + zit+1 + βi xt+1 ) + θ + (1 − δ)
kit+1 − kit+2 + log D − θ − δ
k
D
D
D
D
D
D
where k = log K.
Substituting for kit+1 and kit+2 from above,
dit+1 = A0 + Ã1 zit + A1 βi xt + Ã2 εit+1 + A2 βi εt+1 + A3 ηit+1 + A4 ηit+2
75
where
K
α̃
Π
k−
log D − θ − δ
D
D
1−θ
1
Π
K
1
Π
K
+ (1 − δ − ρx )
ρx −
θ + (1 − δ − ρx )
γ1 σε2
1−θ D
D
1−θ
D
D
K
1 − ν1 Π
+ (1 − δ − ρz )
ρz
1−θ D
D
Π
1 K
1 K
−
ρx +
γ1 σε2
D
1
−
θ
D
1
−
θ
D
Π
1 K
−
(1 − ν1 ) ρz
D 1−θD
1
Π
K
−
θ + (1 − δ)
1−θ
D
D
1 K
1−θD
A0 =
A1 =
Ã1 =
A2 =
Ã2 =
A3 =
A4 =
Using the log-linearized return equation,
rit+1 = ρpit+1 + (1 − ρ) dit+1 − pit − log ρ + (1 − ρ) log
P
D
and conjecturing the stock price takes the form
pit = c0i + c1 βi xt + c2 zit + c3 ηit+1
gives
rit+1
P
= − log ρ + (1 − ρ) log + A0 − c0
D
+ (ρρz − 1) c2 + (1 − ρ) Ã1 zit
+ ((ρρx − 1) c1 + (1 − ρ) A1 ) βi xt
+ ρc2 + (1 − ρ) Ã2 εit+1 + (ρc1 + (1 − ρ) A2 ) βi εt+1
+ (ρc3 + (1 − ρ) A4 ) ηit+2 + ((1 − ρ) A3 − c3 ) ηit+1
The (log) excess return is the (negative of the) conditional covariance with the SDF:
e
log Et Rit+1
= (ρc1 + (1 − ρ) A2 ) βi γt σε2
A2 is independent of ν1 and η. Following the same steps as in Appendix D.4, it is easily
verified that c1 is independent of these terms as well. Thus, expected returns are independent
76
of distortions.
Aggregate distortions. Consider the first formulation, i.e.,
τit+1 = −ν1 zit+1 − ν2 xt+1 − ηit+1
Similar steps as above give expression (46). Dispersion in expected stock market returns are
similarly unaffected.
Next, consider the second formulation:
τit+1 = −ν1 zit+1 − ν3 βi xt+1 − ηit+1
In this case, similar steps as above give the conditional expected mpk as
Et [mpkit+1 ] = α + ν1 ρz zit + ν3 βi ρx xt + (1 − ν3 ) βi γt σε2 + ηit+1
and expected excess stock market returns as
e
log Et Rit+1
= (1 − ν3 ) ψβi γt σε2
where ψ is as defined in expression (21). In other words, the risk-premium effect on expected
mpk, as well as expected returns, are both scaled by a factor 1 − ν3 .
The mean level of expected mpk and return dispersion are, respectively,
h
i
= ση2 + (ν1 ρz )2 σz2 + (ν3 ρx )2 σx2 σβ2
2 2
+ (1 − ν3 ) σε2
γ0 + γ12 σx2 σβ2 + 2ν3 (1 − ν3 ) ρx σx2 γ1 σε2 σβ2
h
i
2
2
2 2
2
2 2
E σlog
=
(1
−
ν
)
ψσ
γ
+
γ
σ
σβ
3
e
ε
0
1
x
Et [Rit+1 ]
E
σE2 t [mpkit+1 ]
The last two terms of the first equation capture the mpk effects of risk premia. The last term
there is new and does not have a counterpart in the second equation – in other words, using
dispersion in expected returns would give the second to last term, as usual, but not the last.
If ν3 < 0, it is straightforward to verify that that term is positive (recall that γ1 is negative).
Then, we may be understating risk premium effects. If ν3 > 0, the last terms is negative and
we may be overstating them.
In this latter case, we can obtain an upper bound on the extent of the potential bias as
follows: holding the other parameters fixed, the term is most negative for ν3 = 0.5. Using this
value, along with the estimated values of the other parameters, yields a value of the bias that
77
is at most about 0.03.
H
Robustness – Productivity Betas
In this appendix, we investigate the potential effects of (i) mis-measurement of firm-level capital
and (ii) unobserved heterogeneity in θ on our estimates of productivity betas in Section 4.4.
First, to see the effects of mis-measured capital or measurement error, assume that the measured
capital stock is k̂it = kit + eit , where kit is true capital and eit the mis-measurement. Measured
firm-level productivity growth is then equal to ∆zit +βi ∆xt −θ∆eit . Regressing this on measures
of aggregate productivity, i.e., ∆xt , it is straightforward to see that the estimated β’s would
be unaffected so long as changes in mis-measurement at the firm-level (∆eit ) are uncorrelated
with the business cycle, which may be a reasonable conjecture. Put another way, mis-measured
capital in this analysis leads to measurement errors in the dependent variable, which, under
relatively mild conditions, are innocuous.70
How about unobserved heterogeneity in θ? It turns out this will have small effects as
well. To see this, let θi denote the true firm-specific parameter and θ our assumed common
value. Measured firm-level productivity growth is then equal to ∆zit + βi ∆x − (θ − θi ) ∆kit
it ,∆xt )
, where βi =
and regressing this on aggregate productivity growth gives βi − (θ−θi )cov(∆k
var(∆xt )
β̂i −θ2i ω
1−θ2i
is the effective true β (see Section 5 and Appendix I for a further discussion of this
expression). The second term represents the potential bias, which depends on the covariance
between investment and changes in aggregate productivity. How large is this covariance? As
one example, consider the case with no adjustment costs and a constant price of risk. We can
use the firm’s optimality condition to analytically characterize the covariance, which gives the
θ−θi
it ,∆xt )
βi ρx (1 − ρx ). This term turns out to be negligible.
= 12 1−θ
bias term to be − (θ−θi )cov(∆k
var(∆xt )
i
Intuitively, because of time-to-build, investment in period t + 1 capital is chosen in period
t, before the innovation in productivity is realized. Because of this, the covariance between
changes in capital and contemporaneous productivity is quite small and is only non-zero due
to mean reversion in the AR(1) process (indeed, if productivity follows a random walk or is
iid, i.e., ρx = 1 or ρx = 0, the bias term is zero).71 To verify this result quantitatively, we
have simulated data under the extreme case where heterogeneity in θi is the only source of beta
dispersion (we use the distribution of θ described in Appendix I). As described there, the true
70
Moreover, a non-zero correlation between ∆eit and ∆xt is not itself sufficient to bias the estimates of beta
it ,∆xt )
dispersion. In the case of a non-zero correlation, the regression yields βi − θ cov(∆e
. Thus, if the stochastic
var(∆xt )
process on eit is common across firms, this will add a constant bias to the beta estimates, but will not affect
our estimates of dispersion.
71
Industry-level heterogeneity is a special case where θ varies across industries but not across firms within
an industry.
78
standard deviation of beta is 1.35; the biased estimate is 1.38. Although analytic expressions are
not available in the full model with adjustment costs and time-varying risk, we have simulated
this case as well – the biased estimate remains extremely close to the truth, 1.39. In sum,
because the productivity betas are estimated off of covariances, they are extremely robust to
concerns of both measurement of capital and unobserved parameter heterogeneity.
I
The Sources of Betas
Heterogeneous technologies. With heterogeneity in input elasticities, the production function for firm i is
Yit = Xtβ̂i Ẑit Kitθ1i Nitθ2i
(47)
In this case, we must make a distinction between mpk and the average product of capital,
apk = yit − kit , which is the object we measure in the data. With common parameters, these
are proportional. With parameter heterogeneity, they are not. Following similar steps as in the
baseline analysis, we can derive
apkit+1 = − log θ1i + εit+1 + βi εt+1 + βi γt σε2 + constant
(48)
where
βi ≡
β̂i − θ2i ω
1 − θ2i
In other words, an expression analogous to (14) holds, with two differences: first, variation in
capital elasticities, θ1i , will directly lead to apk dispersion through the first term in (48). Second,
the effective beta is now a combination of the direct sensitivity to the aggregate shock, β̂i , and
the firm-specific labor elasticity, θ2i . Variation in θ2i leads firms to have different exposures
to changes in labor market conditions, captured through the cyclicality of wages, ω. To gain
intuition, consider the extreme case where all heterogeneity in business cycle exposure comes
2i ω
through θ2i , i.e., β̂i = 1 ∀ i. Then, βi = 1−θ
. It is straightforward to show that βi is increasing
1−θ2i
in θ2i as long as ω < 1, i.e., holding all else equal, labor intensive firms are more exposed to
cyclical movements in wages, which in and of itself leads to a higher risk premium.72 Given
this simple reinterpretation of beta, a version of the analysis in Section 3 continues to hold. In
particular, we can derive an expression for expected stock markets returns that is analogous to
(21), but which now also reflects the variation in θ2i – in other words, this type of heterogeneity
should be picked up by our empirical measure of variation in risk premia.
72
Donangelo et al. (2018) explore a related mechanism and provide empirical support for the connection
between “labor leverage” and risk premia. They also find that a necessary condition for this relationship to hold
is that wages are less than perfectly procyclical, i.e., ω must be less than one.
79
How much of the observed beta dispersion can be attributed to variation in production
function parameters? Although precisely pinning down its contribution is challenging, we can
reach one (likely over-) estimate as follows. First, under the (admittedly strong) assumption
that all cross-firm variation in labor’s share of income comes from heterogeneity in θ2i , we have
Wt Nit
= θ2i . This is likely to be an upper bound, since there are many other reasons that
Yit
labor’s share may differ across firms (e.g., labor market frictions or distortions). Donangelo
et al. (2018) (Table XII, Panel C) report a cross-sectional standard deviation of labor’s share
among Compustat firms of 0.186. Using this as an estimate of the dispersion in θ2i , we can
calculate the implied beta dispersion. Specifically, we assume that θ2i is normally distributed
and discretize the distribution on a seven point grid following the method suggested in Kennan
(2006). This yields a range of values for θ2i from 0.31 to 0.84 with standard deviation 0.183.
2i ω
. The standard deviation of the betas is 1.35,
Next, we compute the implied betas as βi = 1−θ
1−θ2i
which represents about 12% of the overall standard deviation of betas in Section 4.73
Heterogeneous markups. As is well known in the literature, the production function in
expression (8) with decreasing returns to scale is isomorphic to a revenue function that arises
with monopolistically competitive firms that produce differentiated products and face constant
elasticity demand functions. Specifically, assume that demand and production for firm i take
the forms
Qit = Pit−µi , Yit = Xtβ̃i Z̃it Kitθ̃1 Nitθ̃2
where µi denotes the (potentially firm-specific) elasticity of demand and θ̃j , j = 1, 2 the technological parameters in the production function, which for this section are assumed to be common
across firms. It is straightforward to derive the following expression for firm revenues:
Pit Yit = Xtβ̂i Ẑit Kitθ1i Nitθ2i
1
µi
1− µ1
1
µi
where β̂i = 1 −
β̃i , Ẑit = Z̃it
and θji = 1 −
θ̃j , j = 1, 2. With these reinterpretations of parameters, this is equivalent to (47) (there, the common price of output
is equal to one). Note that for the case of a common demand elasticity, i.e., µi = µ, the analysis
from Section 3 goes through exactly. With heterogeneity in demand elasticities, the analysis
takes the same form as with technology heterogeneity – variation in technology and markups
show up in the same way. Thus, markup dispersion across firms is an additional candidate for
heterogeneous exposures and, indeed, should be picked up in our measures of firm-level risk
i
73
David and Venkateswaran (2019) investigate technology heterogeneity in detail in a related framework
and provide a sharper upper bound on the extent of this heterogeneity. We have also used their estimate for
Compustat firms and found similar, though somewhat smaller, results.
80
premia. All else equal, firms facing a high demand elasticity (so setting a low markup, which
i
) respond more strongly to shocks and so show greater sensitivity to them.
is equal to µiµ−1
Even with no additional heterogeneity in β̃i , the firm’s beta in the revenue function is given
1
by β̂i = 1 − µ1i = markup
, i.e., is the inverse of the markup. How much of the measured beta
i
dispersion can variation in markups explain? Recent estimates of the within-industry standard
deviation of (log) markups among Compustat firms yield values of about 0.20 (e.g., David
and Venkateswaran (2019)).74 Following a similar approach as in our analysis of technology
heterogeneity, we can compute the resulting dispersion in betas. Specifically, we discretize the
distribution of log markups on a five point grid. The lowest value on the grid implies a markup
less than one, which we set to 1.01. We choose the standard deviation of the distribution
so that the standard deviation of the truncated distribution is 0.20. This yields a range of
markups from 1.01 to 1.63. After optimizing over labor, the implied beta for firm i is given
1−θ̃2 ω
i −θ2i ω
= markup
. We set θ̃2 to a standard value of 0.67 and compute the standard
by βi = β̂1−θ
2i
i −θ̃2
deviation of these betas, which is 0.71. This accounts for about 6% of the overall standard
deviation calculated in Section 4.
Other parameter heterogeneity. We have also examined the potential effects of two other
forms of parameter heterogeneity – in the depreciation rate, δ, and the properties of idiosyncratic
shocks, i.e., their persistence and volatility, ρz and σε̃2 . To a first-order, the latter two parameters
do not enter our estimates of beta dispersion/risk premia anywhere – idiosyncratic shocks, while
extremely important in determining firm dynamics, do not affect covariances and so do not lead
to risk premia. Expression (21) shows that δ does play a role in determining expected stock
returns (through the denominator of ψ, which, with heterogeneity in δ will be firm-specific),
but a numerical simulation suggests these effects are small. For example, allowing δ to range
from 0.04 to 0.16 (so half and double the baseline value) generates a spread in expected returns
of 1.6%, which is modest relative to the extent of expected return dispersion in the data.
For example, Table 5 in Appendix B.2 shows that interquartile range of expected returns is
almost 12%. Halving/doubling ρz and σε̃2 also leads to only limited spreads in expected returns
(1.3% and 2.3%, respectively). These results suggest that unobserved heterogeneity in these
parameters seems unlikely to account for the substantial dispersion in risk premia observed in
the data. Moreover, note that our calculation of productivity betas in Section 4.4 is independent
of these parameters, further emphasizing that the majority of the empirical beta dispersion is
74
The statistics reported in Edmond et al. (2018) imply a roughly similar figure. Haltiwanger et al. (2018)
find the same value using a different empirical method (namely, estimating a variable elasticity of substitution
demand system using detailed data on prices and quantities) on a sample drawn from the Census of Manufactures.
81
unlikely to stem from these parameters.75
Heterogeneous demand sensitivities. SIC 5812 is defined as “Establishments primarily
engaged in the retail sale of prepared food and drinks for on-premise or immediate consumption”
and includes food service establishments ranging from fast food (e.g., McDonalds) to high-end
restaurants (e.g., Ruth’s Chris Steak House). We gathered data (where available) on average
check per person (usually proxied by total check divided by the number of entrees ordered)
from publicly available sources, including company SEC filings and surveys performed by Citi
Research and Morgan Stanley. The data are generally from 2014 to 2015. Matching these prices
to the Compustat data yielded a sample of 20 publicly traded firms in SIC 5812 with data on
prices, betas, expected returns and MPK.
We first extracted the set of all firms in SIC 5812 for which we have sufficient quarterly
observations to compute our measures of risk exposure (20 consecutive quarters are required).
Next we obtained data on average check per person. These data are primarily from surveys
performed by Citi Research and Morgan Stanley, downloaded from https://finance.yahoo.
com/news/much-costs-eat-every-major-201809513.html, dated September 2015. Of the
firms in the Compustat sample, this gave us pricing data for 8 firms: McDonalds (MCD),
Wendy’s (WEN), Sonic (SON), Chipotle (CHP), Cheesecake Factory (CHE), Texas Roadhouse
(TEX), BJ’s Restaurants (BJR) and Red Robin (ROB). We supplemented these data with
figures reported in company 10-K filings with the SEC for the year 2014 for Jack in the Box
(JCK), Panera Bread (PAN), Carrol’s Restaurant Group (the largest Burger King franchisee;
BKG), Chili’s (CHL), Cracker Barrel (CRA), Bob Evans (BOB), Ruth’s Chris Steakhouse
(RUT), Denny’s (DEN), Famous Dave’s (FAM), Kona Grill (KON), Granite City (GRA) and
Darden (DAR). Data on Granite City are from its 2013 10-K filing, where we calculated the
average of the reported range across markets. Darden owns Eddie V’s, Capital Grille, Seasons’s
52, Bahama Breeze, Olive Garden, Longhorn Steakhouse, Fleming’s, Bonefish Grill, Carraba’s
and Outback Steakhouse. It reports an average check for each of these chains separately, which
we combined into a single value using a sales-weighted average. The largest among this group
is Olive Garden. We excluded chains that were confined to a very limited geographic area and
those for which we could not obtain average check data. In total, our sample consists of 20
75
We have also explored the effects of adjustment costs alone by simulating a panel of firms with a common
beta and computing the mean of period-by-period expected return dispersion. We find that adjustment costs on
their own lead to very little dispersion in expected returns (average standard deviation of about 0.015 compared
to 0.127 in the data), suggesting that it is unlikely that our estimates of beta are reflecting the effects of these
costs. We have also verified that this result goes through for larger levels of these costs (e.g., ξ = 3, compared to
0.04 in the baseline). Note that this is in line with our approximate expression for expected returns in equation
(21): that expression show that to a first-order, expected returns are completely independent of adjustment
costs.
82
firms. We computed average betas, expected returns and MPK for these firms over the period
2010-2015.
Figure 2 illustrates the main results from this exercise. The top two panels of the figure
plot average check against CAPM and demand betas, along with the lines of best fit. Both
plots show a strong positive relationship – higher quality restaurants, as proxied by price,
have higher exposure to aggregate shocks, measured using either stock market or operating
data. Firms on the low end include McDonalds (MCD), Burger King (BUR), Wendy’s (WEN),
Sonic (SON), etc., and towards the higher end Kona Grill (KON), Famous Dave’s (FAM) and
Cheesecake Factory (CHE). The highest-price restaurant in the sample is Ruth’s Chris Steak
House (RUT).76 The bottom two panels of the figure go one step further and additionally link
quality to measures of expected returns and MPK. Again, there is a strong positive relationship:
higher quality restaurants – which the top panel shows tend to be those with higher exposure
to aggregate shocks – have higher expected returns and MPK.
Table 10 presents the full set of correlations across average check, betas, expected returns and
MPK for this set of firms (we also add a measure of beta constructed from the Fama-French
factors, which gives similar results). The table shows strong positive correlations between
average check and the various beta measures, as well as between average check and returns
and MPK. Further, the positive correlations between beta, expected returns and MPK show
that high beta and high expected return firms tend to have MPK. However, Figure 2 neatly
summarizes the key message – differences in the responsiveness of firm-level demand to aggregate
conditions due to quality variation and “trading down” seems a promising explanation of beta
dispersion.
Table 10: Correlations – SIC 5812, Eating Places
Ln(avg. check)
CAPM Beta
Demand Beta
FF Beta
Expected Return
Ln(MPK)
Ln(avg. check)
1.00
0.47
0.64
0.61
0.63
0.65
CAPM Beta
Demand Beta
FF Beta
Expected Return
Ln(MPK)
1.00
0.41
0.85
0.42
0.65
1.00
0.47
0.37
0.63
1.00
0.57
0.77
1.00
0.64
1.00
76
Ruth’s Chris is somewhat of an outlier with a price of $76.00 per person, about three times larger than the
next highest. We have verified that omitting Ruth’s Chris does not significantly change the results.
83
4.5
RUT
GRA
2.50
RUT
CHL
4.0
Demand Beta
CAPM Beta
3.5
2.00
FAM
ROB
KON
CHE
DEN
1.50
CHL
BOBCHP GRA
TEX
BJR
PAN
CRA
SON
BUR
WEN
1.00
2.5
CHE
DEN
BOB
CRA
ROB
BUR
WEN JCK PAN
SON
CHP
MCD
FAM
TEX
1.0
2
2.5
3
3.5
4
4.5
1
1.5
2
Ln(avg. check)
2.5
3
3.5
4
4.5
Ln(avg. check)
0.2
RUT
1.5
TEX
DEN
RUT
0.15
CRA
0.1
KON
BJR
DAR
ROB
DEN
CHL CHE
0.05
FAM
WEN
SON
MCD
0
1.5
TEXCHE KON
DAR
ROB GRA
BOB
0.5
BJR
0
GRA
BUR
1
CRA
BOB
CHP
PAN
JCK
-0.05
CHLFAM
1
Ln(MPK)
Expected Stock Return
DAR
BJR
1.5
DAR
MCD
1.5
3.0
2.0
JCK
1
KON
2
2.5
3
3.5
4
4.5
Ln(avg. check)
BUR
WEN JCK
SON
MCD
1
1.5
2
CHP
PAN
2.5
3
3.5
4
4.5
Ln(avg. check)
Figure 2: Average Check, Beta, Expected Returns and MPK in SIC 5812, Eating Places
84
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