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Equilibrium Capital Investment and Asset Returns in
Oligopolistic Industries∗
Hitesh Doshi
University of Houston
Praveen Kumar
University of Houston
This version: August 28, 2019
Abstract
Oligopolistic industries are ubiquitous. We analyze a multi-good general equilibrium production-
based asset pricing model with a competitive and an oligopolistic sector that is consistent with
important asset and product markets phenomena at the industry level. Even under the “classi-
cal”assumptions of power utility preferences without habit persistence, Markov shock structure,
and reasonable risk aversion, the model matches the industry equity risk premia and Sharpe
ratios of concentrated US manufacturing industries for 1958-2011. The interaction of aggregate
and sectoral productivity shocks raises the volatility of the stochastic discount factor and thus
helps also explain the observed market Sharpe ratio. Modeling the effects of product market
power on capital investment and input choice improves the fit of the model – with respect to
both product and asset markets variables – compared to a benchmark competitive industry.
Keywords: Asset pricing, oligopoly, capital investment, price-cost margin, equity risk pre-
mium, Sharpe ratio
JEL classification codes: G12, D25, D43
∗We thank Franklin Allen, Stephen Arbogast, Cristina Arellano, Tim Bresnahan, Tom George, Nils Gottfries, KrisJacobs, Ravi Jagannathan, Sang Seo, Ken Singleton, Vijay Yerramilli, and participants in the 2019 Summer Meetingsof the Econometric Society (Seattle) for helpful comments and discussions. Send correspondence to: Praveen Kumar,C.T. Bauer College of Business, 4750 Calhoun Road, Houston, TX 77204; e-mail: [email protected].
1 Introduction
Oligopolistic industries are ubiquitous. Indeed, in light of long-standing theoretical and empirical
literatures on oligopolies, it hardly requires much justification for considering effects of oligopolistic
market structures on asset returns. In particular, the empirical heterogeneity of expected equity
risk premia (ERP) at the industry level is well known (Fama and French (1997)), and presumably
reflects, at least partly, differences in market structure across industries. While the production-
based asset pricing literature mostly assumes competitive firm behavior, the role of product market
power has recently attracted interest (Van Binsbergen (2016), Garlappi and Song (2017)). But
dynamic investment and asset-pricing implications of oligopolistic firm behavior in production-based
general equilibrium asset-pricing models (with multiple consumption goods) appear unexplored.1
In this paper, we argue that incorporating the strategic capital investment and input choices of
oligopolistic firms is important in explaining salient asset markets phenomena at the industry level.
Intuitively, firms with product market power strategically consider the effects of investment and
other input choices on product prices. Hence, the profit-risk exposure of such firms to aggregate
and sectoral productivity shocks will differ from competitive firms and be reflected in their –
presumably distinct – ERP and Sharpe ratio. Moreover, in a general equilibrium context, it
is empirically plausible that oligopolistic industries may also raise the volatility of the stochastic
discount factor (SDF) and its covariance with asset returns, thereby increasing more generally the
ERP and raising the upper bound on Sharpe ratios (Hansen and Jagannathan (1991)).
We build on this intuition and theoretically and empirically examine product and security
market dynamics in an infinite-horizon, two-sector general equilibrium model in an economy with
two consumption goods. One of the goods is “produced” in a large competitive sector through
an exogenous Markov process (similar to Lucas (1978)). The second good is produced by an
oligopolistic sector. The competitive good can be used for consumption or utilized for productive
inputs – capital and materials – by the oligopolistic sector, which is also exposed to sector-
specific Markov productivity shocks. The representative consumer has power utility preferences
defined over a consumption index (a là Dixit and Stiglitz (1977)). Firms in the oligopolistic sector
1Garlappi and Song (2017) consider a two-sector model with a competitive finished good sector and an intermediategood sector with monopolistic competition. Van Binsbergen (2016) studies cross-sectional asset pricing implicationsof a general equilibrium model with imperfect competition with habit formation, but does not consider investment.Another strand of the literature in real options considers optimal investment exercise by oligopolistic firms in apartial equilibrium setting, taking as given the industry demand function and the pricing kernel for asset valuation(Grenadier (2002)).
1
are subject to Bertrand (1883) price competition and dynamically choose investment and material
inputs in a dynamic stationary Cournot (1838) equilibrium to maximize the expected marginal
utility of real dividends of the representative consumer.2 The distinction between aggregate and
sectoral shocks is emphasized by the real business cycle literature (Long and Plosser (1987), Forni
and Reichelin (1998)). Indeed, Foerster et al. (2011) document the increasing role sectoral shocks
since the 1980s.
Conceptually, our framework makes two main points with respect to asset pricing. First, as
mentioned already, in a multi-good model the interaction of aggregate and sectoral shocks can
raise the volatility of the SDF and covariance of asset returns with the SDF, thereby raising the
ERP and the maximal Sharpe ratio. To see this, recall that with power utility (and relative risk
aversion coeffi cient γ), the SDF is proportional to (Ct+1/Ct)−γ . As is well known, in a single
consumption good model, the percent variability in the SDF and the covariance of asset returns
with consumption growth are restricted by the low volatility of the real consumption growth rate
in the data. In particular, in a log-normal framework, implausibly high values of γ are needed to
explain observed market ERP and Sharpe ratios (Campbell (2000), Lettau and Uhlig (2002)). But
in our two good model, the real consumption is Ct = Zt/Pt, where Zt is effectively aggregate output
and Pt is an aggregate price-index that depends on the product prices of the goods. Hence, the
volatility of the SDF and the covariance between asset returns and the SDF now also depend on
the variance-covariance matrix of aggregate and sectoral shocks that drive aggregate income and
industry product price.
Second, incorporation of the strategic price effect in productive input choice by oligopolistic
firms tends to reduce their (shareholder payoff) risk exposure to aggregate and sectoral shocks.
To explicate, at the margin, higher investment today lowers current dividends for two reasons: It
reduces net profits for a given product price and lowers the equilibrium price by reducing the supply
of the competitive good (since it is also used for productive inputs). While the former effect holds
for competitive firms, the latter arises for oligopolistic firms whose production choices may have
general equilibrium effects. And firms with market power also choose material inputs taking into
account the general equilibrium effects on product prices.
Now consider a positive industry productivity shock. Ceteris paribus, this raises current in-
2Goods in the model are non-storable so Bertrand pricing arises naturally. In essence, firms in the oligopoly playan infinite-horizon version of the Kreps and Scheinkman (1983) (KS) game. In the two-period game, KS showed thatwith capacity pre-commitment and Bertrand pricing the unique first period equilibrium capacity strategies are thosegiven in a Cournot (1838) equilibrium.
2
dustry output and lowers general equilibrium product price (by raising the relative supply of the
good). Because investment and material input demand are also negatively related to equilibrium
price, the strategic price effect ceteris paribus lowers optimal investment, thereby reducing the neg-
ative impact of sectoral shocks on current dividends. Conversely, a positive aggregate shock would
generate higher industry investment relative to the competitive equilibrium, again ameliorating the
positive impact of extraordinary aggregate output shocks on industry price. Thus, there will be
a “smoothing”of investment, inputs, and output in oligopolies relative to benchmark competitive
firms, resulting in relatively lower ERP and volatility of excess returns.
We analyze the model empirically through calibrated, numerical simulations of the log-linearized
approximation of the equilibrium.3 We use aggregate data from the Bureau of Economic Affairs
(BEA) and industry data from the NBER-CES Manufacturing Industry Database (1958-2011).
We classify concentrated oligopolistic industries – where there is likely to be significant product
market power – as those where more than 70% of output is generated by the largest four firms.
This classification is done using the 1997 industry population analyses reports by the US Census
Bureau at the six-digit North American Industry Classification System (NAICS) levels. We have 31
concentrated industries whose combined output is about 11% of the aggregate output. We compare
the model simulations with the benchmark competitive industry.
For the simulations, we use the data to calibrate the variance-covariance matrix of aggregate
and industry productive shocks, along with the mean values of aggregate output and productivity
(obtained from NBER-CES). The representative consumer’s utility weights of the two goods and
the elasticity of substitution (ES) between them are internally calibrated to fit the observed average
price-cost ratio in the oligopolistic industries. The steady state values of the endogenous variables,
that is, capital stock and material inputs, are derived from the equilibrium conditions of the model.
We undertake the simulations for two levels of industry concentration, namely, when the number of
firms is 4 or 8; these assumptions are reasonable given our methodology for classifying oligopolistic
countries.
Under the “classical” assumptions of power utility without habit persistence, a Markov (or
3Because of the endogenous industry demand function and asset pricing kernel, the general equilibrium consump-tion, dividends and asset returns are not conditionally lognormal, despite the classical assumptions of the log-normalframework. Pohl et al (2018) show that log-linear approximations of long run risk models with multiple highly per-sistent processes can generate errors. We do not have long run risks in our model. The two exogenous productivityshocks are driven by Markov processes that are together not highly persistent (in the range indicated as problematicby Pohl et al. (2018)).
3
AR(1)) shock structure, and assuming a risk aversion of 10, the (unconditional) expected ERP
generated by the model is over 5% and is quite close to the data for both levels of industry con-
centration.4 The ERP in the competitive benchmark industry is over 6% and hence is significantly
higher than that in the model and the data. Meanwhile, the model implied aggregate ERP is about
3.5%, which is lower than in the data, but significantly higher than that generated by single con-
sumption good asset pricing models with comparable assumptions. The model also matches very
closely the volatility of the industry equity risk premium (about 18%) and hence also the observed
industry Sharpe ratio of 0.28. Consistent with the intuition given above, the market Shape ratio is
0.37, slightly exceeding that in the data (0.35). We also verify the theoretical prediction that the
volatility of returns in oligopolies is lower than that in competitive benchmark industries.
Turning to endogenous product market variables, the model fits reasonably well the investment,
material inputs, and output volatilities in the data. Consistent with the theoretical predictions, the
volatilities of inputs and output are lower in oligopolies compared with competitive benchmarks.
Moreover, the cyclical properties of equilibrium investment and material inputs use (with respect to
aggregate and industry shocks) are qualitatively consistent with the data. In particular, the model
generates procylical investment and material input with respect to aggregate shocks, which supports
the view that short-run marginal costs are procyclical (Bils (1987), Rotemberg and Woodford
(1991)). We also find that the cyclical properties of productive inputs with respect to aggregate
shocks are significantly different from those with respect to sectoral shocks, which is consistent with
the distinction highlighted in the business cycle literature.
Overall, our analysis indicates that modeling industry market structures in multi-good general
equilibrium models may help explain important product and asset markets phenomena. Our results
complement Garlappi and Song (2017) who find that flexible capital utilization and high market
power in monopolistically competitive industries with persistent technological growth and recursive
preferences (with early resolution of uncertainty) can explain the observed market ERP.5 We find
that considering oligopolistic firms in a two consumption goods model can explain the industry
level ERP and Sharpe ratio with power utility and Markov shocks. Our study is also related to,
but distinct from, some other strands of the literature. The “New Keynesian”models (see, for
4As is widely discussed on the literature, there is no consensus on the appropriate parameterization of the RRA.However, the literature (Mehra and Prescott (1985)) considers a reasonable upper bound on RRA to be about 10,and this value is used by many asset pricing models (see, for e.g,.Bansal and Yaron (2004))
5Garlappi and Song (2017) provide a very useful summary of production-based asset models with investment-specific technology shocks.
4
example, Gali (2008)) develop dynamic, stochastic equilibrium models to examine the effects of
imperfect competition on the macroeconomy; these models typically abstract from asset pricing
implications, however. Opp et al. (2014) examine price-cost markups in general equilibrium model
but do not focus on equity risk premia. There is also a literature on the impact of market power
on risk and dynamics with firm heterogeneity in partial equilibrium (for example, Carlson et al.
(2014)) with exogenous industry demand and SDF.
In the rest of the paper, Section 2 describes the basic model, which is analytically characterized
in Section 3. Section 4 analyzes the equity premia and maximal Sharpe ratios in the log-linear
approximation of the model. Section 5 undertakes empirical tests using calibrated simulations.
Section 6 concludes. Proofs of results and computation details are provided in the Appendix.
2 The Model
2.1 Firms and Industry Structure
There are two sectors in the economy, specializing in the production of non-storable goods x and
y. We will identify these as sectors x and y, respectively, and use capital letters to denote their
outputs. For simplicity, output in sector x is modeled as an exogenous stochastic process Xt∞t=0
that is sold competitively. This good also serves as the numeraire and its price (px) is normalized
to unity each period. It is convenient to consider a representative firm that sells Xt at unit price
each period. Finally, good x can be either consumed or used to facilitate production in the other
sector that is described next.
The second sector is an oligopoly with N firms (labeled i = 1, ..., N), who produce an identical
good y. All firms utilize an identical production technology that stochastically converts their
beginning-of-the-period capital (Kit) and material input chosen during the period (Hit) to output
according to
Yit = θt(Kit)ψK (Hit)
ψH , i = 1, ...N. (1)
Here, θt represents the stochastically evolving industry-wide productivity level and ψK , ψH are
the output elasticities of capital and material inputs, respectively, such that ψK + ψH ≤ 1. The
industry output at t is given by Yt =∑N
i=1 Yit. Firms use x for their material input. For notational
parsimony, it is assumed that x is directly converted to material input so that Hit also represents
the total material cost of production at t.
5
To introduce general equilibrium effects of sectoral investment in a tractable way, we assume
that production in sector y uses x (or the numeraire good) for capital input or investment. There
is a cost of converting x to investment, however. Letting Iit denote the investment by firm i at t,
the investment cost function is6
A(Iit,Kit) = Iit + 0.5υ
(IitKit
)2
Kit. (2)
This formalizes the notion of sector-specific costs of converting the numeriare good to investment.
Conditional on Iit, the firms capital accumulation process (over the set of non-negative reals)
is given by
Kit+1 = (1− δ)Kit + Iit,Ki0 = Ki0, (3)
where δ is the per-period depreciation rate (that is common for all firms in the sector) and the
initial capital stocks are pre-specified.
The output in the competitive sector and the productivity levels in the imperfectly competitive
sector evolve according to correlated and persistent lognormal processes,
logXt = ρx logXt−1 + εxt ; log θt = ρθ log θt−1 + εθt . (4)
Here, for j ∈ X, θ, 0 ≤ ρj ≤ 1 are the autocorrelation parameters and, conditional on (Xt−1, θt−1),
εjt are bivariate normal mean zero variables with the variance-covariance matrix Λ = [λij ].
All firms in the model are unlevered and publicly owned, with their equity being traded in
frictionless security markets. The number of shares outstanding at t is denoted by Qxt and Qyit, i =
1, ...N. Because the revenue of the sector x firm at t is Xt, and there are no investment costs, its
dividend payout is Dxt = Xt. Given the “Lucas tree”structure of this sector, we fix the number of
outstanding shares to unity without loss of generality (that is, Qxt ≡ 1).
Meanwhile, the dividends of firms in sector y are
Dyit = pyt Yit −Hit −A(Iit,Kit), i = 1, ..., N. (5)
Dividends can be negative, financed by equity issuance. In the absence of taxes and transactions
6This investment cost function can also be interpreted through the well known capital adjustment costs (see Abeland Eberly (1994)). Here, the quadratic parameterization conforms to strictly convex adjustment costs (Summers(1981), Cooper and Haltiwanger (1996)) and is useful for interior optima used in the numerical simulations.
6
costs, negative dividends are equivalent to the market value of new equity share issuance.
2.2 Consumers
There is a continuum of identical consumers in the economy. The representative consumer-investor
(CI) maximizes the expected discounted time-additive utility of random consumption streams of
the two goods subject to period-by-period budget constraints. In addition to investing in the stocks
issued by firms, the CI has access every period to a (one-period) risk-free security (f) that pays a
unit of the numeraire good next period. The number of the risk-free security is also fixed at unity.
The profile of securities outstanding at t is thus Qt = (Qxt = 1, Qy1, .., QyN , Q
ft = 1).
Thus, in each period t, the representative consumer chooses the consumption vector ct = (cxt , cyt )
taking as given product prices pt = (1, pyt ). The portfolio of asset holdings at the beginning of the
period is denoted qt = (qxt , qy1 , .., q
yN , q
ft ). Along with consumption, the CI simultaneously chooses
the new asset holdings qt+1, taking as given the corresponding asset prices St = (Sxt , Sy1 , .., S
yN , S
ft ).
For simplicity, there is no other endowment or labor income. Hence, the CI is subject to a wealth
constraint determined by the dividend payouts Dt = (Xt, Dy1 , .., D
yN , 1). More precisely, let Zt be
the wealth net of new asset purchases during the period – that is, the consumers disposable income
available for consumption. Then, the representative CI’s optimization problem is
maxE0
[ ∞∑t=0
βtC1−γt − 1
1− γ
], γ ≥ 0, β < 1, (6)
s.t., pt · ct ≤ qt · (Dt + St)− qt+1 · St ≡ Zt, ct ≥ 0. (7)
In (6), γ determines the representative CI’s degree of risk aversion; β is the subjective discount
factor; and Ct ≡ C(ct) is an aggregated consumption index with constant elasticity of substitution
(CES) between the consumption of the two goods:
C(ct) =[(1− φ)(cxt )(σ−1)/σ + φ(cyt )
(σ−1)/σ]σ/(σ−1)
. (8)
Here, σ > 1 is the ES and 0 < φ < 1 is a pre-specified consumption weight for good y.
Because preferences are strictly increasing, the budget constraint (7) will be binding in any
optimum and hence Zt also represents the total consumption expenditure at t. Because the number
of assets outstanding is fixed at unity for each type, it follows from (7) that, as long as the asset
markets clear, the disposable income is Zt = Dxt +
∑Ni=1D
yit + 1.
7
The optimal consumption demand functions derived from the optimization problem (6)-(7) are
multiplicatively separable in Zt and pt (see Appendix A)
cj∗t (pt, Zt) =ZtPt
[Ptφ
j
pjt
]σ, j = x, y, (9)
where pxt = 1, φx ≡ (1− φ), φy ≡ φ, and Pt ≡ P (pt) is the aggregate price index
P (pt) =[(1− φ)σ + (φ)σ(pyt )
1−σ]1/(1−σ). (10)
It follows from (9)-(10) that, at the optimum, the aggregate real consumption C∗t = C(c∗t ) satisfies
the consistency condition C∗t = ZtPt.
2.3 Capital Investment and Asset Returns
Firms (in sector y) choose the time-profile of their investments Ii = (Ii0, ...) and input choices
Hi = (Hi0, ...) to maximize the present discounted value of real dividends (Dyi1P1, ...). In general,
there will not exist complete contingent markets in this model; hence, the discount rate is given by
the representative consumers marginal utility of real consumption (Brock (1982), Horvath (2000)).
Using (6), the definition of Dyit, and Ct = Zt
Pt, the investment problem for firm i is7
maxIi,Hi≥0
E0
[ ∞∑t=0
βt(ZtPt
)−γ (pyt Yit −Hit −A(Iit,Kit)
Pt
)], s.t., (1)—(3). (11)
Note that investment is reversible and hence unconstrained and if the optimal Dyit < 0, then
Dyit = Syit(Q
yit+1 −Q
yit).
The equilibrium asset price vector can be derived from the representative CI’s optimal portfolio
condition (see Appendix A), namely,
St = Et
[β
(PtPt+1
)(C∗t+1
C∗t
)−γ(Dt+1 + St+1)
]. (12)
In the usual fashion, the pricing kernel (or the SDF) for future equity payoffs is defined in terms of
the intertemporal marginal rate of substitution of real consumption (IMRS). Noting that Ct = ZtPt,
7Note that the optimization problem in (11) implies flexible investment, that is, absence of capital irreversibility.This assumption is made for the sake of parameter parsimony and ease of numerical computations. Of course, thechoice of Ii is constrained by the capital stock remaining non-negative and the initial stock K′i0.
8
the SDF (or pricing kernel) is given byMt+1 ≡ β(Zt+1Zt
)−γ (Pt+1Pt
)γ−1. Equation (12) thus becomes
St = Et [Mt+1(Dt+1 + St+1)] . And in terms of the gross returns Rjt+1 = (Djt+1 + Sjt+1)/Sjt (with
Rft+1 = 1/Sft ), the asset market equilibrium condition can be written in the standard way as
1 = Et [Mt+1Rt+1] . (13)
(Here, 1 is the four-dimensional unit column vector and Rt = (Rxt , Ry1, .., R
yN , R
ft )).
2.4 Equilibrium
An equilibrium specifies the profiles of the representative CI’s consumption and portfolio choices
c∗t ,q∗t ∞t=0 (where q
∗0 is pre-specified as 1); investment and input choice profiles for the firms
in sector y, I∗it, H∗it∞t=0 , i = 1, ..., N ; and product and asset price profiles p∗t ,S∗t
∞t=0 . These
equilibrium quantities generate the profile of disposable income Z∗t ∞t=0 , so that in equilibrium:
1. The representative CI’s consumption and portfolio choices solve the constrained optimization
problem given in (6)-(7);
2. The asset prices satisfy (12) (or (13)) and clear the security markets, that is, q∗t = Qt;
3. For each firm i, j = 1, ...N, in sector y, the investment and input choice policies I∗it, H∗it∞t=0
are optimal with respect to (11), given the investment and material input strategies of rival
firmsI∗jt, H
∗jt
∞t=0
;
4. The product prices p∗t clear the goods markets:
cx∗t (p∗t , Z∗t ) +
N∑i=1
[A(I∗it,Kit) +H∗it)] = Xt (14)
cy∗t (1, py∗t , Z∗t ) = Yt (15)
The requirements of optimal consumption and portfolio policies of consumers and the market
clearing conditions in the product and asset markets are standard. The novel aspect of the equi-
librium here – with respect to the production-based asset pricing literature – is the requirement
of optimal investment policies of oligopolistic firms and the implications of these policies for the
equilibrium product and asset prices. In particular, the equilibrium investment condition has a
9
dynamic Cournot flavor with respect to investment, but the pricing competition in each period
(given in (14)-(15)) is Bertrand where both firms sell their entire current output in the market.
As is typical in the oligopoly literature, We focus on a symmetric equilibrium where both firms
adopt the same investment strategy; that is, along the equilibrium path I∗it = I∗t (so that the
equilibrium industry investment is I∗t = NI∗t and∑N
i=1[A(I∗it,Kit) +H∗it] = N [A(I∗t ,Kt) +H∗t ].8 It
follows from (1)-(3) that both firms also have symmetric capital stocks investments Kt and output
Yt and hence the industry output is Yt = NYt. The (nominal) dividends per firm are
Dyt = pyt Yt −H∗t −A(I∗t ,Kt) (16)
and the industry dividends are Dyt = NDy
t . Furthermore, we will assume realistically that con-
sumption weight of y (that is, φ) is suffi ciently small so that the effects of investment by firms in
sector y on the aggregate consumption C and the aggregate price index P are of order small. This
implies that firms take the pricing kernel (M) as a given, which is a reasonable assumption. This
convention also allows one to treat Xt as a proxy for aggregate output, which will be useful for the
empirical interpretation of the results.
3 Equilibrium Characterization
3.1 Investment and Product Prices
It is convenient to define the vector of state variable at each t as Γt = (Kt, Xt, θt), which completely
determines all the endogenous quantities in the model along with equilibrium prices p∗t (Γt) and
S∗t (Γt). We will focus on a stationary equilibrium where the firms’ equilibrium investment and
material input strategy is a time-invariant function of the state Γt; hence, the equilibrium price
functions in the product and financial markets are also stationary.
Because of imperfect competition, the investment decision by the firms takes into account its
effect on the product price (py). In particular, investment has two effects on (nominal) dividends,
8Kreps and Scheinkman (1983) show that in the two stage game, with capacity choices chosen in first stage andprice competition in the second stage, there is a unique pure strategy symmetric equilibrium with identical capacitychoices in the first stage and identical prices in the second stage. Of course, in an infinite horizon model there is apossibility of multiple non-stationary equilibria. In particular, the folk theorem of dynamic oligopoly implies thatvarious levels of collusion are possible for suffi ciently high discount factors of shareholders through subgame perfectthreats of “punishment phases” of high investments and low prices. However, the stationary equilibrium allowsequilibrium computations around the steady state (see below) that are useful for the analysis. Similarly, while afinite-horizon model may yield uniqueness, it will lead to a non-stationary equilibrium.
10
Dyt (see (16)), at the margin. It decreases dividends, holding the price fixed. Moreover, higher
investment ceteris paribus lowers the consumption of x (from the materials balance condition (14)),
which affects the equilibrium py through the market clearing condition (15). These effects are seen
more clearly in a formal depiction of the equilibrium.
In the standard way, one uses the consumer optimum conditions (9)-(10), along with the market
clearing conditions (14)-(15), to obtain py∗t . Using this in the Euler and optimality conditions for
investment and material inputs, yields the following characterization of the equilibrium investment,
material input choice, and product price. For notational ease, we will write η ≡ φ/(1−φ), the partial
derivatives of the investment cost function as AI(I,K) ≡ 1 + υ(I/K), AK(I,K) ≡ −0.5υ(I/K)2,
and the net market supply of good x as W ∗t ≡ Xt −N [A(I∗t ,Kt) +H∗t ])
Theorem 1 The equilibrium price of the good produced in the oligopolistic sector is
py∗t =
(W ∗tNY ∗t
)1/σ
η. (17)
Furthermore,
∂Dy∗t
∂It= −AI(I∗t ,Kt)
[1 +
(py∗t )1−σησ
Nσ
], (18)
∂Dy∗t
∂Kt= py∗t
∂Yt∂Kt
[1− 1
Nσ
]−AK(I∗t ,Kt). (19)
And the optimal interior investment I∗t and material input choice H∗t in the oligopolistic sector
satisfy, respectively,
−∂Dy∗t
∂It= Et
[M∗t+1
(∂Dy∗
t+1
∂Kt+1− (1− δ)
∂Dy∗t+1
∂It+1
)], (20)
1 +(py∗t )1−σησ
Nσ= py∗t
∂Yt∂Ht
[1− 1
Nσ
]. (21)
In a general equilibrium, the relative price of y (in terms of the numeraire), py, should be
decreasing with the supply of y relative to that of x. And the sensitivity of py to this relative
supply should be increasing (in algebraic terms) with the ES. Furthermore, ceteris paribus, py
should be positively related to the weight of good y in the consumer’s utility function, φ. These
properties are satisfied by the equilibrium price function (17) since the supply of goods y and x
are Y ∗t and W∗t , respectively. Furthermore, η is increasing with the consumer’s utility weight of y,
11
namely, φ.
Next, the Euler condition for optimal investment (20) uses the equilibrium price function (17)
and has a ready interpretation. The left hand side is the marginal cost of current investment,
which is specified in (18). There is a direct marginal cost, given by the first term in (18), because
of the one-to-one reduction in the dividend for a given increase in investment costs. This is familiar
from the existing literature (see, e.g., Love (2003)). But, because of market power, each firm also
takes into account the negative effect of its investment on price (as discussed above), taking as
given the investment of rival firms. The second term in (18), thus, represents the effect of strategic
considerations on investment cost by firms with market power.
The right hand side (RHS) of (20) represents the discounted expected marginal value of current
investment. The first term in the RHS represents the discounted expected marginal effect of higher
capital on dividends next period, which is recursively given by (19). The first term in (19) indicates
that market power reduces the firms gain from investment because of its negative impact on next
period’s price (through the marginal productivity of capital). Hence, ceteris paribus, optimal
investment is lower in the oligopoly equilibrium relative to a competitive industry. The second
term in (19) is the effect of higher capital, at the margin, on next period’s investment costs (that is,
−AK(I∗t+1,Kt+1)). Finally, the second term in the RHS of (20) captures the discounted expected
marginal value of higher capital stock (namely, Kt+1) for τ ≥ t+ 2.9
Equation (21) represents the optimal material input demand condition. The left hand side is
the total marginal cost that now includes the negative effect of higher input demand of the firm
on the industry price. Note the symmetry between the marginal cost of material inputs here and
the marginal cost of investment (in the left hand side of) Equation (20) – the differences are only
due to the investment cost function A(I,K). The right hand side of (21) is the marginal revenue
product of inputs. Again, there is (partial) symmetry in the marginal gains from material inputs
and the expected marginal from investment in (20). The latter is lower, the higher is the ES or the
higher is the net supply of x in equilibrium.
To help build intuition on the effect of market power on equilibrium product market outcomes,
it is useful to compare with a benchmark when sector y is competitive: when all firms in this
sector take prices as given and equate them to the marginal cost. Thus, firms choose investment
9Similar Euler equations are derived in the literature in the presence of financing constraints by Whited (1992),Bond and Meghir (1994), and Gilchrist and Himmelberg (1998). However, these papers do not consider the strategiceffect of investment on general equilibrium product prices. On the other hand, we do not consider internal financingconstraints.
12
to maximize the optimization problem (11) but do not consider the effect of investment on current
and future prices, and they choose material input till the marginal cost equals the price. We will
denote the equilibrium endogenous quantities in the competitive industry by (pyt , It, Ht).
Corollary 1 In a symmetric equilibrium, the output price, optimal investment, and material input
demand in a competitive industry in sector y satisfy
pyt =
(Wt
NYt
)1/σ
η =
(∂Yt∂Ht
)−1
, (22)
AI(It,Kt) = Et[Mt+1
(pyt+1
∂Yt+1
∂Kt+1−AK(It+1,Kt+1) − (1− δ)AI(It+1,Kt+1)
)]. (23)
Finally, the asset market equilibrium is given by (13).
Equation (22) reflects the competitive equilibrium pricing condition where prices clear the mar-
kets and industry price equals the marginal cost. And (23) is the Euler condition with respect
to investment. Propositions 1 and 2 suggest that product market power will also affect the sec-
ond moments of equilibrium investment and material input choice. Specifically, oligopolistic firms
strategically incorporate the effects of investment and input use on prices, which tends to “smooth
out” the effects of aggregate and industry shocks on optimal factor demands and hence output.
The intuition for this has been provided in the Introduction.
Finally, we comment on whether the equilibrium price-marginal cost markup or ratio is pro-
cyclical or countercyclical, since this issue attracts much attention (as noted above). In this model,
the marginal cost is with respect to material inputs. Standard cost function construction yields the
marginal cost of material inputs as(∂Yt∂Ht
)−1. Hence, the price-marginal cost ratio is pmcrt ≡ pyt ∂Yt∂Ht
.
But it follows from (21) that in equilibrium pmcrt ∝ (py∗t )1−σ. Since σ > 1, it follows that pmcrt
and py∗t have the opposite sign in terms of their cyclical properties. Now the equilibrium price
(17) is positively related to X if and only if 1 > N∂[A(I∗t ,Kt)+H
∗t ]
∂X ]; that is, if the marginal effect of
aggregate output on the industry investment and material input costs does not exceed one. Thus,
theoretically pmcr can be procylical or countercyclical depending on whether the marginal invest-
ment and input costs are suffi ciently procyclical relative to the marginal income effect on price. If
firms are competitive, and price equals marginal cost, then the cyclicality of pmcr is determined by
whether marginal cost is procyclical or countercyclical (Bils (1987)). However, with market power,
the cyclical properties of pmcr depend on whether marginal cost is suffi ciently procyclical relative
13
to product price. In a similar fashion, the cyclicality of pmcr with respect to industry productivity
shock θ is also theoretically ambiguous.
3.2 Asset Returns
The equilibrium investment and product prices, given by (20)-(17), determine the time-path of
firms’capital stocks K∗t (through the law of motion (3)) and dividends, Dy∗t , conditional on the
realizations of Xt and θt. These dividends, along with Xt and the unit payout from the riskless
security, then determine the disposable income of the representative consumer Z∗t = Xt+NDy∗t +1.
And, given py∗t , the aggregate price index P∗t is determined by (10). These quantities then determine
the optimal consumption vector (cx∗t , cy∗t ) and the aggregate consumption index C∗t , according to
(9)-(10) and (8), respectively. Finally, with the knowledge of the equilibrium investment and
product pricing rules and, conditional on the state Γt, the representative CI forms expectations on
the pricing kernel M∗t+1 = β(Z∗t+1Z∗t
)−γ (P ∗t+1P ∗t
)γ−1, which determines the equilibrium asset prices
according to (12).
Note that the SDF in this model is more complex compared with the benchmark single-good
consumption CAPM, that is,(C∗t+1C∗t
)−γ. Here, the SDF is the product of the growth of aggregate
income (raised to the power −γ) and the growth of the aggregate price index (raised to the power
(γ − 1)). As seen in (10) and Proposition 1, Pt+1Ptis affected by both the aggregate shock (Xt) and
the industry shock (θt), along with the ES (σ), the production function parameters, and the market
structure (N). It is useful to examine the implications of this on the equity risk-premia and the
maximal Sharpe ratio.
4 Equity Risk-Premia and Maximal Sharpe Ratio
Even though X and θ are conditionally lognormal, the dividend Dy∗ and the pricing kernel M∗ are
not lognormal in equilibrium. Note that (from (17))
Dy∗t = N−1/ση (W ∗t )1/σ [θt(K
∗it)ψK (H∗it)
ψH ]−1/σ −H∗t −A(I∗t ,Kt), (24)
which is generally not lognormally distributed (conditional on Γt). It follows that income Z∗ and
the aggregate price index P ∗ are also not conditionally lognormal, and hence neither is the (pricing
kernel) M∗. It follows that income Z∗ and the aggregate price index P ∗ are also not conditionally
14
lognormal, and hence neither is the (pricing kernel)M∗. This complicates substantially the analysis
of the equilibrium. We follow the standard approach (Woodford (1986), Christiano (1988, 2002))
and analyze the equilibrium by computing its log-linearized approximation around the steady state
where (i) the production in sector x and the technology levels in sector y are non-stochastic with
Xt = E[X](≡ X) and θt = E[θ] (≡ θ) (for each t) and (ii) the equilibrium quantities in sector y are
time-invariant. The details of the computation procedure outlined below are given in Appendix B.
For tractability, we will therefore compute the equilibrium by log-linearizing the investment
optimality conditions around the steady state with Taylor series expansions. The log-linear frame-
work also allows a clean conceptual comparison of the asset pricing implications of model with the
existing asset pricing literature with single good models.
4.1 Log-Linear Approximations
4.1.1 Real Variables
Note that for any time index τ , we can write the firm’s investment Iτ as the first-order forward
equation in capital stocks,
Iτ = Kτ+1 − (1− δ)Kτ . (25)
Hence, by replacing It, It+1 with the appropriate forward equation we can represent the system
of equilibrium conditions (17)—(21) in terms of the vector of costate and state variables Ωt =
(Kt+2,Kt+1,Kt, Ht+1, Ht, Xt+1, Xt, θt+1, θt). In particular, the equilibrium investment condition
(20) can be written in terms of Ωt as Et[ΦI(Ωt)] = 0 where
ΦI(Ωt) ≡ −Z−γt P γ−1t
∂Dyt
∂It+ β
[Z−γt+1P
γ−1t+1
(∂Dy
t+1
∂Kt+1− (1− δ)
∂Dyt+1
∂It+1
)], (26)
(and It+1 = Kt+2 − (1 − δ)Kt+1 etc.). Similarly the optimality condition for material inputs (21)
can be written Et[ΦH(Ωt)] = 0 where
ΦH(Ωt) ≡ −(
1 +(pyt )
1−σησ
Nσ
)+ pytψHθt+1(Kt+1)ψK (Ht+1)ψH−1
[1− 1
Nσ
].
And (26) and (27) use
pyt (Ωt) =
(Xt −N(A(It,Kt), Ht)
Nθt(Kt)ψK (Ht)ψH
)1/σ
η. (27)
Then one solves for the equilibrium policies by using (27) and log-linearizing ΦI(Ωt), ΦH(Ωt)
15
around their steady state values of Ω (denoted Ω) with a first-order Taylor Series expansion. Using
the standard notation, the log deviation around the steady state quantity for any variable wt is
denoted by wt ≡ ln(wtw ) ' wt−ww (for small deviation) (and the log-deviation form of Ω will be
labeled Ω). Then let πt = [Kt+1 Ht]. The solution to the log-linearized version of the model takes
the form
πt = V πt−1 + UXXt + Uθθt, (28)
where the square matrix V = [vjz], j = K,H, z = 1, 2, and the vectors UX , Uθ (with elements ujX
and ujθ, j = K,H) are determined by the solution to log-linearized versions of the Euler conditions.
4.1.2 Financial Asset Returns
Denoting the logarithms of variables by small letters, the equilibrium asset return condition (13)
can be written as
1 = Et[exp
(mt+1 + rjt+1
)], j ∈ x, y, f. (29)
In this model, the log of the pricing kernel is
mt+1 = −γgz,t+1 + (γ − 1)gp,t+1, (30)
where gz,t+1 and gp,t+1 are the log changes in the aggregate income ln(Zt+1)− ln(Zt) and the price
index ln(Pt+1) − ln(Pt), respectively, between t and t + 1. Hence, the log of the pricing kernel is
driven by the aggregate shock Xt+1 the industry productivity shock θt+1. But since the aggregate
income and the price index depend on the equilibrium product price, investment, and material
inputs, the pricing kernel is also affected by the ES σ, the production parameters (ψK , ψH , δ) and
the number of firms N. The influence of the number of active firms (or the industry concentration)
on the pricing kernel is of particular interest.
As noted above, mt+1 and equity returns rjt+1 (j = x, y) are not generally conditionally joint
normal. Sincemt+1 and rjt+1 are also functions of Ωt, we take the first-order Taylor series approxima-
tion around their steady state values. It is important to note that the log-linearized approximations
of the pricing kernel and the equity returns will be derived around the steady state general equi-
librium in the real economy and, hence, will also be functions of the state and costate variables,
namely, Ω. Nevertheless, the resultant approximations are joint normal and hence the expected
equity risk premia on the stocks in the two sectors (x and y) can be computed in the standard
16
fashion.
In the steady state, M = β. Hence, log-linearization of the pricing kernel gives mt+1 ' log β +
mt+1, for
mt+1 = a · Ωt + ωmxεXt+1 + ωmθε
θt+1, (31)
where the coeffi cient vector a is determined by taking the first-order Taylor approximation of
mt+1(Ωt+1) around the steady state Ω. Note that the coeffi cient for the shocks ωm = (ωmX , ωmθ)
are time-invariant because shocks to the logXt and log θt process are additive (see (4)) with a
stationary variance-covariance matrix Λ. In fact, we can use the equilibrium solution (28) to express
a ·Ωt in (31) in terms of the log-deviation form of the state variable vector Γt = (Kt, Xt, θt), namely,
a·Γt(see Appendix B). Thenmt+1 is conditionally normal with the mean (log β+a·Γt) and variance
ωmΛωm. It follows immediately that the equilibrium risk-free rate is
rft+1 = −(log β + a · Γt)−ωmΛωm
2. (32)
To calculate the equilibrium equity returns, we utilize the standard log-linearization of re-
turns in the literature (Campbell and Shiller (1988)). In the situation at hand, the steady state
dividend-price ratio for equities (in both sectors) is Dj
Sj= 1−β
β and log-linearization yields the return
approximation (see Appendix B)
rjt+1 ' −[β log β + (1− β) log (1− β)] + βξjt+1 − ξjt + gjd,t+1, (33)
where gjd,t+1 ≡ djt+1 − djt is the log growth rate of dividends between t and t + 1 and ξjt is the log
stock price-dividend ratio (that is, log(Sjt /Djt )) at t of equity j = x, y. But here (unlike Campbell
and Shiller (1988)) the evolution of the log price-dividend ratio and dividend growth is determined
by the general equilibrium. As noted above, the equilibrium dividends in sector y (Dy∗) will not
generally be conditionally lognormal. (Of course, dxt = logXt, and is conditionally normal.) But
gjd,t+1 will be a function of the costate and state variables. Hence, following a similar approach to
above, log-linearization yields
gjd,t+1 ' b·Γt + ωydxεXt+1 + ωymθε
θt+1, (34)
and it follows from (29) that log-linearization of ξjt takes the form ξj
t ' ej0 + ej ·Γt. The coeffi cients
17
of gjd,t+1 and ξj
t are computed through the equilibrium condition (29). Inserting these relationships
in (33) yields
rjt+1 = vj0 − νj · Γt + ωjrxεxt+1 + ωjrθε
θt+1, j = x, y, (35)
With these linearized relationships in hand, mt+1 and rjt+1 are jointly normal, conditional on
Γt. Hence, using the properties of exponential functions of joint normal variables, we obtain in the
usual fashion (for j = x, y) :
Et[rjt+1 − rft+1] = −Covt(mt+1, r
jt+1)−
Var(rjt+1)
2(36)
= −ωmΛωjr2
− ωjrΛω
jr
2, (37)
where ωjr = (ωjrx, ωjrθ) and ωm = (ωmX , ωmθ) (as defined earlier). We note that the equilibrium ex
ante equity risk premia are time-invariant, conditional on Γt. Analogously, the conditional volatility
of the ERP and hence the Sharpe Ratio, are also time-invariant. In sum, the time-invariance of
the conditional ERP and the Sharpe Ratio arises here because of the stationary equilibrium and
the assumption of additive output shocks (in sector x) and technology shocks (in sector y) with
time-invariant moments.
In a log-linear framework, with the joint (conditional) log-normality of mt and rjt , one can
compare the equilibrium equity premia and Sharpe ratios of the model at hand with the standard
single good asset pricing models that have been extensively studied with similar distributional
assumptions. In the standard way, by using (30) in (36) we can also write the equilibrium equity
premium as
Et[rjt+1 − rft+1] = γCovt(gz,t+1, r
jt+1)− (γ − 1)Covt(gp,t+1, r
jt+1)−
Var(rjt+1)
2, j = x, y. (38)
Equation (38) indicates that the risk premium is positively related to the covariance of the asset
return with log change in aggregate income Z, and negatively related (for γ > 1) to the covariance of
asset return with the log change in the aggregate price index P. In terms of empirical magnitudes,
gz,t+1 would be largely driven by shocks to the aggregate output (x), which is similar to single
good models. Nevertheless, the percentage change in industry dividends gyd,t+1 would also affect
gz,t+1 (as long as the industry is not infinitesimal compared with the aggregate output). Since
industry productivity shocks have a first order impact on gyd,t+1, it follows that the θt process will
18
also influence the first term. Turning to the second term, from the definition of P (see (10)), gp,t+1
is determined by log changes in the industry price, which is driven by shocks to both aggregate
output and industry productivity. Thus, compared to single good models, the industry equity risk
premium and its volatility in this model are determined by the second moments of the growth rates
of total aggregate dividends (Z) and the price index (P ). The latter depends on the industry price
that is driven by investment along with the aggregate and industry shocks.
We can also derive the Hansen-Jagannathan (1991) upper bound on the Sharpe ratios for assets
in the model. Using the fact that Rf = 1/E[m] is close to 1, we have
SRmax =
√Var(m)
E[m]'√γ2Var(gz) + (γ − 1)2Var(gp)− 2γ(γ − 1)Cov(gz, gp). (39)
Hence, the maximal Sharpe ratio depends on the variance of (non-linear functions of) log changes
in X and θ and the covariance between them. In comparison, the maximal Sharpe ratio in the
consumption CAPM is approximately γVol(gC). As is well known, the low variability in per capita
consumption growth in the data restricts SRmax to be quite low for the reasonable range of risk
aversion. However, (39) indicates that, depending on the volatility of gθ and the covariance of
aggregate output and industry productivity shocks, the maximal Sharpe ratio can be relatively
high even if γ is restricted to acceptable levels.
5 Empirical Tests
We now turn to the empirical tests of the model. We analyze the model empirically through
calibrated, numerical simulations of the log-linearized approximation of the equilibrium.
5.1 Empirical Measures and Data
For empirical tests of the model, we need industry data on capital, investment, material inputs,
sales, and productivity. We take these data from the NBER-CES manufacturing database. The
latest data available are for 1958-2011.
The estimation of the model also requires information about oligopolistic industries. We em-
pirically identify the oligopolistic sector in the model by defining oligopolies – that is, industries
where at least some firms have market power – as those where more than seventy percent of the
output is generated by the largest four firms. The data on the proportion of industry output ac-
19
counted for by the largest four and eight firms are available through the US Census Bureau – at
the six-digit NAICS level – for 1997, 2002, and 2007. Hence, we have to hold concentration at the
1997 levels for years prior to 1997. To maintain a uniform classification of industries for the entire
sample period, we therefore use the industry concentration data from 1997.10 We have a total of
473 unique six-digit industries. We require 20 firms in a given industry, which drops the number of
industries to 456. Of these, 31 industries (6.8% of the total) satisfy our definition of oligopolies –
that is, where the top 4 firms generate more than 70% of the output. Table 1 provides the industry
codes and names of these oligopolistic industries.
We measure the output (Y ) of oligopolies as the sum of the output of all industries in the
oligopoly sector based on the procedure annunciated earlier. The output is measured as the value
of shipments obtained from the NBER-CES database. As noted above, this database also provides
information about the investment (I), material costs (H), and capital (K). We measure the output
of the non-oligopolistic “aggregate”sector (X) as the difference between the aggregate output of all
sectors – obtained from the US Bureau of Economic Affairs (BEA) – and the combined output
of the oligopolies. For all of these quantities, the data also provides information about the relevant
price deflator in 1997 dollars. We use these deflators to convert the values in real terms.
We also require the growth in the aggregate income (Z). We proxy for the growth in Zt using
consumption growth. The data on consumption growth are obtained from the Federal Reserve
Bank of St. Louis. We use the consumption price deflator to convert the data in real 1997 dollars.
We also use the consumption price deflator to adjust the returns data.
Finally, to compute the financial variables of the model – namely, the ERP and its volatility
(and hence Sharpe ratio) of the oligopoly and aggregate sectors – we use annual CRSP value-
weighted returns and the annual risk-free rate obtained from Kenneth French’s website. We compute
the sectoral financial variables as follows. We first map the 1997 NAICS codes to 1987 Standard
Industry Classification (SIC) codes. We then use four-digit SIC codes to compute the portfolio
returns. Following the standard procedure in the literature, we compute the value-weighted index
monthly returns of all firms in all industries classified as oligopolies. Using these returns, we obtain
the annualized ERP, annualized equity premium volatility, and the Sharpe ratio for the oligopolistic
sector (y). In a similar fashion, we obtain the financial variables for the aggregate sector (x) using
10Changing the industry identification in 2002 and 2007 leads to significant in-sample data “discontinuities” andtherefore muddles inference. Namely, are the time-variations in results due to changes in the competitive environmentof the oligopolistic sector or are they due to changes in the composition of the sector?.
20
the annual CRSP value-weighted index returns as the proxy.
5.2 Parameterization
We compute the log-linearized version of the model (described in Section 4.1 and Appendix B)
above by calibrating the parameters with data described above. These parameters are summarized
in Table 2. We now explain the parameterization choice and then discuss the results.
For the production sector parameterization, the estimates from (40) provide the values for the
production parameters ψK and ψH (see Table 2). The elements of the variance-covariance matrix of
the shocks Xt and θt are obtained from the data.11 The values for the autocorrelation coeffi cients
ρj , j = X, θ are also estimated from the data using the first-order autocorrelations.
It is well known that because different types of capital – equipment, structures, and intellectual
property – depreciate at different rates, estimating the empirical depreciation rates is challenging.
The literature notes that depreciation rates have been trending upwards because of the increased use
of computer equipment and software since this lowers the useful life of capital stock (Oliner (1989)).
Moreover, the depreciation rates on such equipment themselves have been rising. For example,
Gomme and Rupert (2007) mention that the annual depreciation rates of computer equipment
have risen from 15% in 1960-1980 to 40% in 1990s. And they give estimates for depreciation rates
of software in the range of 50%. Meanwhile, Epstein and Denny (1980) estimate the depreciation
rate of physical capital (in the first part of our sample-period) to be about 13%. We use an annual
depreciation rate of 25%. Untabulated results show that the results are quite robust to variations
in the value of depreciation rate parameter (δ).
There is a wide range of estimates available in the literature regarding the capital investment (or
adjustment) cost parameter υ. Using US plant level data, Cooper and Haltiwanger (2006) report
υ = 0.125, when estimating a strictly convex adjustment cost function, as used in our model. Hence,
we use this parameter value for our simulations.
Turning to the consumption side of the model, the discount rate β is set to (1.03)−1 = 0.97,
which implies a three percent annual discount, which is consistent with the literature (Horvath
(1999)). The elasticity of substitution σ and the consumption weight of the oligopolistic sector φ,
being parameters of unobservable utility function of the representative consumer, are calibrated
internally by matching the average price-marginal cost ratio for the oligopolistic industry in the
11The covariance between percent variability in X and θ is very low in the data (about 0.0004). We hence set it tozero in the simulations.
21
data, which in our model is pmcr ≡ py
mc , mc =(∂Yt∂Ht
)−1. For estimating pmcr, we use an empirical
model based on standard cost minimization conditions for material inputs. These conditions relate
the output elasticity of inputs to its expenditure share in the total sales. If the firm is a price-taker
in inputs, as is assumed in our model, then it well known (see, for example, De Loecker et al.
(2018)) that pmcr = ξYH
(pY YH
)−1, where ξYH = ∂ lnY
∂ lnH , is the output elasticity of H. We then run
the regression
lnY = βK lnK + βH lnH + ln θ + ε. (40)
where ε is an error term. Hence, ξYH = βH .
Finally, there is still no consensus on the appropriate parameterization of the relative risk
aversion (RRA) coeffi cient γ. However, the literature (Mehra and Prescott (1985), Bansal and
Yaron (2004)) considers a reasonable upper bound on RRA to be about 10. We take γ = 10 for
our calculations, which facilitates comparison with some of the existing literature that is based on
consumption good asset pricing models.
5.3 Results
We now present and discuss the results of the equilibrium path computations for endogenous prod-
uct market and financial variables. These results are based on 5000 replications of the equilibrium
paths of a 54 year model economy (1958-2011).
5.3.1 Product Market Variables
Table 3 shows the equilibrium computations for product market variables for N = 4 and 8. These
levels of industry concentration are natural given the definition of oligopolies in our empirical
analysis. The steady state values of the firm-level choice variables, namely, the capital stock (K)
and material inputs (H) are computed from the steady state analogs of the optimality conditions
given in Propositions 1 and 2. The ES σ and consumption share φ, conditional on the number
of firms (N), are calibrated to match the observed mean pmcr. Meanwhile, the volatilities of the
aggregate and sectoral productivity shocks are chosen to match the data. As seen in Table 2, the
volatility of the aggregate output and industry productivity shocks in the simulations match data,
as does the mean pmcr. And the endogenous pmcr of the competitive benchmark is set at 1.
Because the model does not have a growth component in productivity and computations are in
terms of deviation from the steady state, we report the second moments of the control variables.
22
Equilibrium investment and material input volatilities from the model are lower than those in
the competitive benchmarks. This is consistent with the intuition from the model (made explicit
in Section 3) that the strategic price effects of market power on optimal factor demands tend to
smooth out the effects of aggregate output and industry productivity shocks. Consequently, the
output volatility is also lower for oligopolies relative to the competitive benchmarks. Relative to the
data, the model tends to generate excess volatility with respect to investment, but lower volatility
with respect to material inputs and output. Hence, the fit of the model (with an oligopolistic
sector) is better for the volatility of investment compared with the competitive benchmark, but the
latter is a better fit for the volatility of material inputs.
We turn now to the correlations of equilibrium industry investment and inputs to the shocks
(which are independent of N by definition). Consistent with the data, the model generates pro-
cyclical investment and material input demands with respect to aggregate output and industry
productivity. However, the model overstates the correlation of percent variability of capital in-
vestment with respect to both the percent changes in aggregate and sectoral productivity shocks,
especially the latter. This could be because the assumptions of strictly convex production technol-
ogy and capital adjustment costs result in smooth optimal investment demand functions, distinct
from the “lumpy”investment behavior that is observed in the data (Doms and Dunne (1998), Ca-
ballero and Engel (1999)). Meanwhile, our theoretical framework indicates that input choices with
product market power will be more sensitive to productivity shocks – aggregate and sectoral –
relative to competitive firms because oligopolistic firms attempt to offset the effects of these shocks
on product prices. This prediction is confirmed by the simulations in Table 3.
Meanwhile, the model understates somewhat the correlation of percent changes in material
inputs with respect to the percent variability in aggregate shocks, but matches quite well the
correlation with percent variability in sectoral shocks. We note that the procyclical material input
demand implies procyclical marginal costs (due to the strict concavity of the production function).
Hence, the results here are consistent with the view in the literature that short-run marginal costs
overall are procyclical (Bils (1987)).
Finally, we note that the fit of the model is not uniformly superior in terms of the number
of oligopolistic firms (N = 4 or N = 8). In fact, each level of assumed industry concentrations
provides a superior fit for about one-half of the number of endogenous variables.
23
5.3.2 Asset Markets Variables
In Table 4, we present the equilibrium computations with respect to unconditional expected indus-
try and market equity risk-premia (ERP) and their volatilities; this allows us to relate the results
from the model to the industry (and market) Sharpe ratios. The ERP generated by the model of
over 5% are quite close to the data for both levels of industry concentration, with the lower concen-
tration being a slightly better fit. The ERP in the competitive benchmark industry is over 6% and
hence is significantly higher than that in the oligopoly and the data. We will discuss further below
the effects of industry structure on the ERP. Finally, the aggregate ERP is about 3.5%, which is
lower than in the data. Not surprisingly, perhaps, the aggregate ERP is unaffected by the industry
market structure of sector y – that is, oligopoly or competitive – since it represents only about
10% of the total output of the manufacturing industry sample.
We reiterate that our model makes the “classical”assumptions on consumer preferences, pro-
duction risks, and absence of security market frictions. In particular, there is no ‘habit formation’
in consumer preferences, which is a common feature of single good (or aggregate) production-based
asset pricing models (Jermann (1998), Boldrin, Christiano and Fisher (2001)). As a benchmark,
Jermann (1998) reports an aggregate ERP of 0.7% without habit formation, but including capital
adjustment costs (as is also the case in our model). And, as another benchmark, Bansal and Yaron
(2004) obtain an (aggregate) ERP of about 1.2% when γ = 10, the intertemporal elasticity of
substitution (IES) is 0.5, and there are non-fluctuating (or homoscedastic) long-run risks. In our
model, the IES is the inverse of the RRA because of the time-additive utility structure and, hence,
equals 0.1. That is, the model is able to generate a substantially higher aggregate ERP even when
the IES < 1, which is of substantial interest since there is no consensus on whether the IES is above
or below 1 in the literature.
In sum, the aggregate ERP from the model is substantially higher than comparable single-
good asset pricing models in the literature. We also note that we obtain industry ERP of over 5%
independent of the industry market structure, that is, with or without product market power. Hence,
we conclude that the interaction of the covariance between aggregate and sectoral productivity risk
– as made explicit in (38) – contributes to the higher ERP of assets.
We note, next, that the unconditional volatility of the industry risk premium from the model
is also very close to the data (for both levels of industry concentration). But, as in the case of
the ERP, this volatility is higher than the data for the competitive industry benchmark. That is,
24
product market power lowers both the mean and volatility of the equity risk premium. And while
the aggregate volatility of the risk premium from the model is somewhat lower than in the data, it
is much higher than reported by single-good production-based models with analogous assumptions
(Jermann (1998)).
Because the model matches well both the mean and volatility of the industry equity risk pre-
mium, it also matches the observed Sharpe ratio at the industry level. In fact, the match is almost
exact for N = 4. However, the Sharpe ratios are lower than the data for the competitive bench-
marks. Meanwhile, the Sharpe ratio for the market is also close to – and in fact somewhat higher
– than in the data. That is, the fit of the market Sharpe ratio from the model is better than the
fit with respect to the expectation or volatility of the market risk premium.
As mentioned at the outset of the paper, the maximal Sharpe ratio in the classical single-
good consumption asset pricing model (CCAPM) is restricted by the relatively low volatility of
per capita consumption growth in the data. Section 4.1.2 discussed the potential of this model
to raise the maximal Sharpe ratio because the volatility of the SDF is determined by the joint
second moments of the aggregate and industry productivity shocks. The market Sharpe ratio of
0.37 quantifies this effect. To illustrate, using the average post-war annual volatility of consumption
growth of about 1% in the data (Stock and Watson (2002)), in a CCAPM world a Sharpe ratio of
0.37 would require γ to be over 35. But we obtain this with γ = 10 implying that the volatility of
mt = −γgz,t + (γ − 1)gp,t is about 3.7%.
A corollary of the relatively high volatility of the SDF is that the equilibrium unconditional
risk-free rate should also be lower than in the benchmark single good consumption model. And
this is what we observe in Table 4. The model-generated risk-free rate is 2.6%, which is actually
lower than in the data. In contrast, the benchmark models tend to generate equilibrium interest
rates that are higher relative to the data (Weil (1989)).
We return now to the negative effect of product market power on the mean and volatility of
the risk premium that is apparent in Table 4. Average profitability – represented by the average
price-cost ratio (pmcr) – in oligopolistic industries is endogenously higher than in the competitive
benchmarks because firms with market power take into account the effects of investment and
input choices on product prices. Quantitatively, this is reflected in the average pmcr of 1.08 in
oligopolies compared with 1 in the competitive benchmark. Hence, returns in competitive firms are
more exposed to variations in aggregate and sectoral risk (or productivity shocks) compared with
25
oligopolies, which results in higher moments of the risk premium. Effectively, competitive industry
firms are the high book-to-market equity firms in this model and earn higher returns because of
greater exposure to non-diversifiable risk.
6 Summary and Conclusions
The canonical single consumption-good, competitive models (with classical assumptions on con-
sumer preferences and aggregate shocks) lead to a number of empirical “puzzles” that are widely
analyzed. In particular, these models require implausibly high values of RRA to be consistent with
the observed market expected equity premium (ERP) and Sharpe ratios. But do these empirical
puzzles persist at the industry level under the classical assumptions in multi-good models with
empirically prevalent oligopolistic market structures? This question is addressed in a two-sector,
two goods, general equilibrium model with a large competitive sector (the “aggregate”) and a
smaller oligopolistic sector (the “industry”). To allow us to focus on the explanatory power of
the multi-good economy and oligopolistic market structure, we adopt the classical assumptions of
power utility and productivity shocks driven by log-normal Markov processes.
Conceptually, in a multi-good model the interaction of aggregate and sectoral shocks can raise
the volatility of the SDF and covariance of asset returns with the SDF, thereby raising the expected
equity premium and the maximal Sharpe ratio. Furthermore, oligopolistic firms take into account
the effects of capital investment and inputs on current and future product prices, unlike competitive
firms. This has significant ramifications for product and financial market outcomes. In particular,
the strategic price effect “smooths out”the effects of aggregate and industry productivity shocks on
the firm’s factor demands and hence reduces the shareholders’payoff risk exposure to these shocks.
Calibrated simulations of equilibrium paths based on aggregate and concentrated manufacturing
industry data (1958-2011) – the proxies, respectively, for the competitive and oligopolistic sectors
– show that the model is able to fit well the ERP and Sharpe ratio at the industry level even
under the classical assumptions and a risk aversion of 10 (that is considered a reasonable upper
bound in the literature). The oligopolistic model performs better in terms of explaining these asset
markets phenomena relative to the competitive industry benchmark. The volatilities of investment,
material inputs, and output are also a better fit to the data compared with the competitive in-
dustry benchmark. We conclude that modeling industry market structures in multi-good general
equilibrium models may help explain important product and asset markets phenomena.
26
Appendix A: Proofs
A.1 Derivation of Optimal Consumption and Portfolio Policies
Since the objective function is strictly increasing and concave, the optimal consumption and portfolio
policies can be characterized through a two-step process, where optimal consumption ct is determined as
a function of available consumption expenditure Zt, and the optimal portfolio is then determined taking
as given the optimal consumption policy. Using the dynamic programming principle (DP), at any t, the
representative consumers optimization problem (6)-(7 can be written as
maxct,qt+1
Et
[ ∞∑τ=t
βτ−tC1−γτ − 1
1− γ
]+ χt [Zt − pt · ct] . (A1)
Here, χt is the Lagrange multiplier for the budget constraint (7). Since preferences are strictly increasing,
the budget constraint is binding and χt > 0. Next, using the definition of aggregate consumption (8), the
first order optimality conditions for cjt , j = x, y, can be written
(Ct)1−γσσ (cjt )
− 1σφj = χtp
jt , (A12)
where pxt = 1, φx ≡ (1− φ), φy ≡ φ. It follows from (A12) that
pjtcjt = χ−σt (pjt )
1−σ(Ct)−(1−γσ)(φj)σ (A13)
Then recognizing that Zt = pt · ct, and using (A13), and the definition of the aggregate price index Pt (see
(10)) allows one to solve for the Lagrange multiplier as
χt =
(ZtPt
)− 1σ
P−1t (Ct)
1−γσσ . (A14)
Substituting this in (A12) and rearranging terms then gives the optimal consumption functions given in (9).
Next, for any τ ≥ t, let Uτ ≡ βτ−t C∗1−γτ −11−γ denote the indirect period utility function with the optimal
consumption functions given in (9). The envelope theorem then yields χτ = ∂Uτ∂Zτ
. Using the fact that
Zτ = qτ · (Dτ + Sτ )− qτ+1 · Sτ then yields the optimality conditions for qt+1
χtSt = Et[βχt+1(Dt+1 + St+1)
]. (A15)
But using C∗t = ZtPtand substituting in (A14) gives χt =(C∗t )
−γP−1t . Since this holds for any τ , inserting
27
in (A15) yields Eq. (12).
A.2 Proofs of Propositions 1-2
Proof of Theorem 1: Substituting the optimal consumption functions (9) in the market clearing conditions
(14)-(15) in a symmetric equilibrium yield
ZtPt
[Pt(1− φ)]σ = Xt −N∑i=1
[A(I∗it,Kit) +H∗it)] (A21)
ZtPt
[Ptφ
pyt
]σ= Yt (A22)
Dividing (A21) by (A22) and rearranging terms yields py∗t given in (17) for a symmetric equilibrium. Next,
the constrained optimization problem for firm i is (for Ii = (Ii0, ...),Hi = (Hi0, ...)),
maxIi,Hi
E0
[ ∞∑t=0
βt(ZtPt
)−γ (py∗t Yit −Hit −A(Iit,Kit)
Pt
)], s.t., (1)—(3), Hi ≥ 0. (A23)
Substituting the constraints in the objective function, and using the assumption that firms take the pricing
kernel as exogenous, the dynamic programming (DP) principle implies that along any equilibrium path, at
any t and conditional on the state Γt = (Kt, Xt,θt), Kt = (K1t, ...,KNt), if the firm takes as given the
rival firmsinvestment profile Ijτ , Hjττ≥t (j = 1, ...i−1, i+1, ..N) and its own future optimal investment
I∗iττ≥t+1, then the firms indirect value function is given by
Πit(Γt) = maxIit,Hit≥0
βt(ZtPt
)−γ (py∗t Yit −Hit −A(Iit,Kit)
Pt
)+ βt+1Et [Πit+1(Γt+1)] , (A24)
where, for τ ≥ t, py∗τ is given in (17) and Yiτ = θτ (Kiτ )α(Hiτ )ψ, and Dy∗iτ = py∗τ Yiτ −Hiτ −A(Iiτ ,Kiτ ).
Then the optimal (interior) investment and material input path satisfies the following system of equations
0 = −∂Πit(Γt)
∂Kit+ βt
(ZtPt
)−γ ( 1
Pt
)∂Dy∗
it
∂Kit+ βt+1(1− δ)∂Et [Πit+1(Γt+1)]
∂Kit+1, (A25)
0 = −βt(ZtPt
)−γ ( 1
Pt
)∂Dy∗
it
∂Iit+ βt+1∂Et [Πit+1(Γt+1)]
∂Iit, (A26a)
0 =∂(py∗t Yit)
∂Hit− 1. (A26b)
Furthermore, in a symmetric equilibrium withKiτ = Kτ , Yiτ = Yτ , and Iiτ = Iτ , Hiτ = Hτ for i = 1, ...N
28
and each τ = 1, 2... Hence,∑N
j=1,i 6=j [A(Iiτ ,Kiτ ) +Hiτ )] = (N − 1)[A(Iτ ,Kτ ) +Hτ ]. Now let
Wiτ ≡ Xτ − (N − 1)(A(Iτ ,Kτ ) +Hτ )− (A(Iiτ ,Kτ ) +Hiτ ). (A27)
Note that in a symmetric equilibrium, Wτ = Wiτ , i = 1, ...N ,(WiτYτ
)−1= (py∗τ )−σησ
N and
py∗τ Yiτ = η(Wiτ )1/σ((N − 1)Yτ + Yiτ )−1σ Yiτ . (A28)
Therefore, recognizing that ∂Yiτ∂Iiτ= 0, we have in a symmetric equilibrium,
∂(py∗t Yit)
∂Iit= σ−1η(NYt)
−1/σ (Wt)1/σ
(Wit
Yit
)−1(∂Wit
∂Iit
)
= −(py∗t )1−σAI(Iit,Kt)ησ
Nσ.
Hence, in any symmetric equilibrium, for I∗it = I∗t ,
∂Dy∗it
∂Iit= −
[AI(I
∗t ,Kt)
(1 +
(py∗t )1−σησ
Nσ
)]. (A29)
Furthermore, ∂Kt+1∂It= 1 and hence ∂Et[Πt+1(Γt+1)]
∂It= ∂Et[Πt+1(Γt+1)]
∂Kt+1. Then (A25) and (A26a) together
imply that the Euler condition characterizing the equilibrium investment path is given by
∂Dy∗it
∂Iit= Et
[M∗t+1
(∂Dy∗
t+1
∂Kt+1− (1− δ)
∂Dy∗t+1
∂It+1
),
](A30)
where in (A30), we have used iterated expectations and recursively substituted the optimality condition for
I∗t+1 (using A(26a)),
−βt+1
(Zt+1
Pt+1
)−γ ( 1
Pt+1
)∂Dy∗
t+1
∂It+1= βt+2∂Et+1 [Πt+2(Γt+2)]
∂Kt+2.
Now, using the envelope theorem (that sets the indirect effects of ∂Kt+1 on the optimally chosen I∗t+1 and
H∗t+1 to zero), (A28) implies that in a symmetric equilibrium with Yit+1 = Yt+1, we have
∂Dy∗t+1
∂Kt+1=
∂Yit+1
∂Kt+1η(Wit+1)1/σ(NYt+1)−1/σ
[1− Yt+1
σNYt+1
]−AK(I∗t+1,Kt+1)
= py∗t+1
∂Yt+1
∂Kt+1
[Nσ − 1
Nσ
]−AK(I∗t+1,Kt+1). (A31)
29
(A29)-(A31) then together characterize the equilibrium path for investment in a symmetric equilibrium.
Finally, to determine H∗it, using (A28), we have
∂(py∗t Yt)
∂Hit= σ−1ηW
1σt (NYt)
− 1σ
(∂Wit
∂Hit
)(Wit
Yt
)−1
+∂Yit∂Hit
η(Wit)1/σ(NYt)
−1/σ
[1− Yt+1
σNYt+1
]= −(py∗t )1−σησ
Nσ+∂Yit∂Hit
py∗t
[Nσ − 1
Nσ
]. (A32)
Inserting this in (A26b) and rearranging terms gives,
1+(py∗t )1−σησ
Nσ=∂Yit∂Hit
py∗t
[Nσ − 1
Nσ
].
Proof of Corollary 1: This follows straightforwardly from Proposition 1 and noting that for the competitive
firm
∂Dyt
∂Iit=
∂(pyt Yit −A(Iit,Kt)−Hit)
∂Iit= −AI(Iit,Kt), (A33)
∂Dyt+1
∂Iit= pyt+1θt+1
∂Yit+1
∂Kt+1−AK(I∗it+1,Kt+1), (A34)
∂(pyt Yit −A(Iit,Kt)−Hit)
∂Hit= −1. (A35)
And the equilibrium product price function follows from the market clearing conditions (14)-(15).
Appendix B: Equilibrium Computations
B.1 Capital Investment and Material Input Policies
Log-linearization allows one to write ΦI(Ωt)
ΦI(Ωt) ' αK1Kt+2 + αK2Kt+1 + αK3Kt + αK4Ht+1 + αK5Ht +
αK6Xt+1 + αK7Xt + αK8θt+1 + αK9θt, (B1)
where αK1 = K ∂ΦI∂Kt+2
, ..., αK9 = θ ∂ΦI∂θt
. In particular, the steady state endogenous variables (K, H)
are derived from specializing the optimality conditions in Theorem 1 to the steady state with Kt = K,
30
It = δK (≡ I), and Ht = H. In a similar fashion, we have
ΦH(Ωt) ' αH1Kt+1 + αH2Kt + αH3Ht + αH4Xt + αH5θt, (B2)
where αH1 = K ∂ΦH∂Kt+1
, ..., αH5 = θ ∂ΦI∂θt
. Then the linearized Euler condition for πt = [Kt+1 Ht] is
Et[ς0πt+1 + ς1πt + ς2πt−1 + νX0Xt+1 + νX1Xt + νθ0θt+1 + νθ1θt
]= 0, (B3)
where ς0 =
αK1 αK4
0 0
, ς1 =
αK2 αK5
αH1 αH3
, ς2 =
αK3 0
αH2 0
, νX0 =
αK6
0
,νX1 =
αK7
αH4
, νθ0 =
αK8
0
, νθ1 =
αK9
αH5
. Therefore, if πt = V πt−1 + UXXt + Uθθt,
then the Euler condition (B3) imposes the restriction,
ς0V2 + ς1V + ς2I = 0, (B4)
ρX(νX0 + ς0UX) + ((ς1 + ς0V )UX + νX1) = 0, (B5)
ρθ(νθ0 + ς0Uθ) + (ς1Uθ + νθ1) = 0. (B6)
(where I is the identity matrix). Writing V =
VK1 VK2
VH1 VH2
, a solution to (B4) is found by VK2 =
VH2 = 0 and VK1, VH1 that satisfy
αK1(VK1)2 + αK2VK1 + αK4VH1VK1 + αK3 = 0, (B7)
αH1VK1 + αH3VH1 = 0. (B8)
The condition for saddlepoint stability requires that there should be one non-explosive (that is, with modulus
less than 1) and two explosive roots of (B4). The non-explosive root, say V ∗K1, is chosen.
Given V, the elements of UX = [UXK UXH ] and Uθ = [UθK UθH ] are then derived from (B5)-(B6).
B.2 Financial Asset Returns
Given the pricing kernel Mt+1 ≡ β(Zt+1Zt
)−γ (Pt+1Pt
)γ−1,mt+1 = logMt+1 is
mt+1 = log β − γgz,t+1 + (γ − 1)gp,t+1, (B10)
31
where gz,t+1 ≡ ln(Zt+1) − ln(Zt) and gp,t+1 ≡ ln(Pt+1) − ln(Pt). Now, from the definitions of the
aggregate income Zt and price index Pt , it follows that mt+1 is a function of the state and costate vector
Ωt. Then using the relation mt+1 = β exp(mt+1), the first order Taylor series expansion of mt+1 around
steady state values gives
mt+1 = ϕm1Kt + ϕm2Kt+1 + ϕm3Kt+2 + ϕm4Ht + ϕm5Ht+1 + (B11)
ϕm6Xt + ϕm7Xt+1 + ϕm8θt + ϕm9θt+1,
where ϕm1 = K ∂mt+1∂Kt
, ϕm2 = K ∂mt+1∂Kt+1
, ..., ϕm9 = θ ∂mt+1∂θt+1(when these derivatives are evaluated at the
steady state). But using the relation πt = V πt−1 + UXXt + Uθθt (where V, UX and Uθ have been
determined as specified previously) and the facts that θt+1 = ρθθt + εθt+1 and Xt+1 = ρxXt + εxt+1 in
(B11) yields the following coeffi cients for mt+1 ' a · Γt + ωymxεXt+1 + ωymθεθt+1
a1 = ϕm1 + ϕm2vK1 + ϕm3(vK1)2 + ϕm4vH1 + ϕm5vH1vK1,
a2 = ϕm6 + uHX(ϕm4 + ϕm5vH1) + uKX(ϕm3vK1 + ϕm2) + ρx(ϕm7 + uHXϕm5 + uKXϕm3),
a3 = ϕm8 + uHθ(ϕm4 + ϕm5vH1) + uKθ(ϕm3vK1 + ϕm2) + ρθ(ϕm9 + uHθϕm5 + uKθϕm3),
ωmx = ϕm7 + ϕm3uKX + ϕm5uHX ,
ωmθ = ϕm9 + ϕm3uKθ + ϕm5uHθ. (B12)
Then, following a procedure similar to that for approximating mt+1, the first order Taylor series expan-
sion of gydt+1 yields the appropriate coeffi cients b, ωdx, and ωdθ, so that gyd,t+1 = b·Γt+ω
ydxε
Xt+1 +ωymθε
θt+1.
Next, the log equity return (for j = x, y) can be written
rjt+1 = sjt+1 − sjt + log(1 + exp(djt+1 − s
jt+1)). (B13)
Treating this as a function of djt+1 − sjt+1, taking the first order Taylor approximation around the steady
state rj = − log β, recalling that dj − sj = log(1− β)− log β, and adding and subtracting djt , yields the
approximation rjt+1 ' − log β + rjt+1, where
rjt+1 = βξj
t+1 − ξj
t + gyd,t+1. (B14)
Furthermore, letting ξj
t ' ej0 + ej · Γt, the coeffi cients ejn (n = 0, 1, 2, 3) are determined as follows. In
32
log-linearized form, the Equilibrium return condition (29) can be written
1 = Et[exp
(κj0 + κj1Kt + κj2Xt + κj3θt + κj4ε
xt+1 + κj5ε
θt+1
)], (B15)
where κjn (n = 0, 1, 2, 3) are linear functions of the coeffi cients of V,UX , Uθ, an, bn (when j = y), and ejn.
Now let j = y. Collecting together the coeffi cients for Kt from the first-order Taylor series expansions of
mt+1 and ryt+1 around the steady state, one gets
κy1 = a1 + ey1(βvK1 − 1) + b1. (B16)
But since (B15) must hold for all realizations of Γt, κj1 = 0 (n = 1, 2, 3). Hence, from (B16), it follows that
ey1 = b1+a1(1−βvK1) . In a similar fashion, we can compute,
κy2 = βuXKey1 + ey2(ρxβ − 1) + b2 + a2. (B17)
Since ey1 is already determined from (B16), it follows that ey2 =βuXKe
y1+b2+a2
(1−ρxβ) , and following an analogous
computation, ey3 =βuXθe
y1+b3+a3
(1−ρθβ) . Finally, κy0 is obtained as follows. Since ejn (n = 1, 2, 3) are chosen to
set κjn = 0 (n = 1, 2, 3), the equilibrium condition (B15) must satisfy
1 = Et[exp
(κy0 + κy4ε
xt+1 + κy5ε
θt+1
)]. (B18)
Then, by collecting the appropriate terms, we can compute
κy0 = (1− β)[log β − log (1− β)] + (β − 1)ey0 (B19)
κy4 = ωmx + ωdx + βey2;κy5 = ωmθ + ωdθ + βey3. (B20)
Now let ry ≡ m + ry denote logarithm of the discounted return MRy. Hence, using the foregoing,
Var(ry) ≡ (κy4)2λ2x + (κy5)2λ2
θ + 2λxθκy4κ
y5. Then, exploiting the bivariate normality of (εXt+1, ε
θt+1) and
taking the logarithm of both sides of (B18) gives κy0 + 0.5 Var(ry) = 0, which implies that
ey0 = log
(β
1− β
)+
0.5 Var(ry)1− β . (B21)
33
Using (33) and collecting together the relevant terms from above, one can write
ryt+1 = vy0 + vy1Kt + vy2Xt + vy3 θt + ωyrxεxt+1 + ωyrθε
θt+1, (B22)
with the following coeffi cients:
vy0 = −[β log β + (1− β) log (1− β)] + ey0(β − 1) = − log β − 0.5 Var(ry);
vy1 = ey1(βvK1 − 1) + b1 = −a1; vy2 = βey1a2 + ey2(ρxβ − 1) + b2 = −a2;
vy3 = βey1a3 + ey3(ρθβ − 1) + a3 = −a3;ωyrx = βey2 + b2; ωyrθ = βey3 + b3. (B23)
Turning to security x,let
rxt+1 = vx0 + vx1 Kt + vx2 Xt + vx3 θt + ωxrxεxt+1 + ωxrθε
θt+1.
Note that the coeffi cients exn (n = 0, 1, 2, 3) are similarly obtained, except that in this case the log of
dividends is directly obtained as dxt = logXt ≡ xt. Note that xt+1 − xt = Xt+1 − Xt (by subtracting
log X from both xt+1 and xt). Then repeating the foregoing procedure (allowing for the difference in the
log dividend growth) leads to the following:
ex1 =b1
(1− βvK1); ex2 =
βuXKex1 + b2 + (ρx − 1)
(1− ρxβ); ex3 =
βuXθex1 + b3
(1− ρθβ). (B24)
Since the equilibrium condition (B15) must hold given (B24) (for j = x),
κx0 = (1− β)[log β − log (1− β)] + (β − 1)ex0 ,
κx4 = ωmx + 1 + βex2 ;κx5 = ωmθ + βex3 , (B25)
Then, since Var(rx) ≡ (κx4)2λ2x + (κx5)2λ2
θ + 2λxθκx4κ
x5), ex0 = log
(β
1−β
)+ 0.5 Var(rx)
1−β . Hence, rxt+1 has
the following coeffi cients:
vx0 = − log β − 0.5 Var(rx); vx1 = −a1;
vx2 = −− a2; vx3 = −− a3;ωxrx = βex2 + 1; ωxrθ = βex3 . (B26)
References
34
Abel, A., and J. Eberly, 1994, A unified model of investment under uncertainty, American Economic Review
84, 1364-1389.
Aghion, P., E. Farhi, and E. Kharroubi, 2015, Liquidity and growth: The role of countercyclical interest
rates, Working Paper No. 489, Bank of International Settlements.
Bartelsman, E., and W. Gray, 1996, The NBER manufacturing productivity database, Technical Working
Paper 205, National Bureau of Economic Research.
Basu, S., 1996, Procyclical productivity: increasing returns or cyclical utilization, Quarterly Journal of
Economics 111, 719-752.
Beaudry, P., and A. Guay, 1996, What do interest rates reveal about the functioning of real business cycle
models?, Journal of Economic Dynamics and Control 20, 1661-1682.
Bertrand, J., Theorie mathématique de la richesse sociale, Journal des Savants 67, 499—508.
Bils, M., 1987, The cyclical behavior of marginal cost and price, American Economic Review 77, 838—55.
Boldrin, M., L. Christiano, and J. M. Fisher, 2001, Habit persistence, asset returns, and the business cycle,
American Economic Review 91, 141-166.
Bond, S., and C. Meghir, 1994, Dynamic investment and the firms investment policy, Review of Economic
Studies 61, 197-222.
Brock, W., 1982, Asset pricing in production economies, In The Economics of Information and Uncertainty
(J. McCall, ed.), University of Chicago Press.
Caballero, R.., and R.Engel, 1999, Explaining investment dynamics in U.S. manufacturing: A Generalized
(S, s) Approach, Econometrica 67, 783-826.
Campbell, J., 2000, Asset pricing at the millenium, Journal of Finance 55, 1515-1567.
Campbell, J., and R. Shiller, 1988, The dividend-price ratio and expectations of future dividends and discount
factors, Review of Financial Studies 1, 195-228.
Carlson, M., Fisher, A., Giammarino, R., and Dockner, E., 2014, Leaders, followers, and risk dynamics in
industry equilibrium, Journal of Financial and Quantitative Analysis 49, 321-349..
Chan, R., 2010, Financial constraints, working capital and the dynamic behavior of the firm, Working Paper,
World Bank, Washington D.C.
Christiano, L., 1988, Why does inventory investment fluctuate so much, Journal of Monetary Economics 21,
247-280.
Christiano, L., 2002, Solving general equilibrium models by a method of undetermined coeffi cients, Compu-
tational Economics, 20, 21-25.
35
Cochrane, J., 1991, Production-based asset pricing and the link between stock returns and economic fluctu-
ations, Journal of Finance 46, 209—237.
Cooper, R., and J. Haltiwanger, 2006, On the nature of capital adjustment costs, Review of Economic Studies
73, 611-633.
Cournot, A., 1838, Recherches sur les principles mathematiques de la theorie des richesses, Paris.
Doms, M., and T. Dunne, 1998, Capital adjustment patterns in manufacturing plants, Review of Economic
Dynamics 1, 409-430.
De Loecker, J., J. Eeckhout, and G. Unger, 2018, The rise of market power and the macroeconomic impli-
cations, Working paper, NBER.
Dixit, A., and J. Stiglitz, 1977, Monopolistic competition and optimum product diversity, American Eco-
nomic Review 67, 297-308.
Fama, E., and K. French, 1993, Common risk factors in the returns on stocks and bonds, Journal of Financial
Economics 33, 3—56.
Fama, E., and K. French, 1997, Industry costs of equity, Journal of Financial Economics 43, 153—193.
Foerster, A., P. Sarte, and M. Watson, 2011, Sectoral versus aggregate shocks: A structural factor analysis
of industrial production, Journal of Political Economy 119, 1-38.
Forni, M., and L. Reichlin, 1998, Let’s get real: A dynamic factor analytical approach to disaggregated
business cycle, Review of Economic Studies 65, 453—74.
Gali, J., 2008, Monetary policy, inflation, and the business Cycle: An introduction to the new Keynesian
framework, Princeton University Press.
Garlappi, L. and Z. Song, 2017, Capital utilization, market power, and the pricing of investment shocks,
Journal of Financial Economics 125, 447—470.
Gilchrist, S., and C. Himmelberg, 1998, Investment: fundamentals and finance, NBER Macroeconomics
Annual 13, 223-262.
Gomme, P., and P. Rupert, 2007, Theory, measurement and calibration of macroeconomic models, Journal
of Monetary Economics 54, 460-497.
Grenadier, S., 2002, Option exercise games: An application to equilibrium investment strategies of firms,
Review of Financial Studies 15, 691-721.
Hansen, L., and R. Jagannathan, 1991, Implications of security market data for models of dynamic economies,
Journal of Political Economy 99, 225-262.
Horvath, M., 2000, Sectoral shocks and aggregate fluctuations, Journal of Monetary Economics 45, 69-106.
36
Hou, K., and D. Robinson, 2006, Industry concentration and average stock returns, Journal of Finance 61,
1927-1956.
Irvine, P., and J. Pontiff, 2008, Idiosyncratic return volatility, cash flows, and product market competition,
Review of Financial Studies 22, 1149-1177.
Jermann, U., 1998. Asset pricing in production economies, Journal of Monetary Economics 41, 257—275.
Kreps, D., and J. Scheinkman, 1983, Quantity precommitment and Bertrand competition yields Cournot
outcomes, Rand Journal of Economics 14, 326—377.
Kydland, F., and E. Prescott, 1982, Time to build and aggregate fluctuations, Econometrica, 50, 1345-1370.
Li, H., P. McCarthy, and A. Urmanbetova, 2004, Industry consolidation and price-cost margins: evidence
from the pulp and paper industry, Working paper, Georgia Institute of Technology.
Long, J., and C. Plosser, 1987, Sectoral vs. aggregate shocks in the business cycle, American Economic
Review 77, 333-336.
Love, I., 2003, Financial development and financing constraints: International evidence from the structural
investment model, Review of Financial Studies 16, 765-791.
Lucas, R., 1978, Asset prices in an exchange economy, Econometrica 46, 1429-1445.
Mehra, R., and E. Prescott, 1985, The equity premium: A puzzle, Journal of Monetary Economics 15,
145-161.
Oliner, S., 1989, The formation of private business capital: trends, recent development, and measurement
issues, Federal Reserve Bulletin 75, 771-783.
Opp, M., C. Parlour, and J. Walden, 2014, Markup cycles, dynamic misallocation, and amplification, Journal
of Economic Theory 154, 126-161.
Rotemberg, J., and M. Woodford, 1991, Markups and the business cycle, In NBER Macroeconomics Annual
1991 (O. Blanchard and S. Fischer, eds.), 63—129, MIT Press.
Shiller, R., 1982, Consumption, asset markets and macroeconomic fluctuations, Carnegie-Rochester Confer-
ence on Public Policy 17, 203-238.
Spence, M., 1976, Product Selection, fixed costs, and monopolistic competition, Review of Economic Studies
43, 217-235.
Stock, J., and M. Watson, 2002, Has the business cycle changed and why?, National Bureau of Economic
Research Macroeconomics Annual 17, 159-218.
Summers, L., 1981, Taxation and capital investment: A q-theory approach, Brookings Papers on Economic
Activity 1, 67-140.
37
Van Binsbergen, J., 2016, Good-specific habit formation and the cross-Section of expected Returns, Journal
of Finance 71, 1699-1732.
Weil, P., 1989, The equity premium puzzle and the risk-free rate puzzle, Journal of Monetary Economics 24,
401-421.
Whited, T., 1992, Debt, liquidity constraints, and corporate investment: Evidence from panel data, Journal
of Finance 47, 1425-1459.
Woodford, M., 1986, Stationary sunspot equilibria, the case of small fluctuations around a deterministic
steady state, Working Paper, University of Chicago.
38
Table 1. List of Oligopolistic Industries
This table lists the 6-digit NAICS Codes and names of industries in our sample of oligopolies, that is,
industries where the largest 4 firms account for more than 70% of industry output in 1997.
NAICS Code Industry Name
311221 Wet corn milling
311222 Soybean processing
311230 Breakfast cereal mfg.
311320 Beans
311919 Other snack food mfg.
311930 Flavoring syrup & concentrate mfg.
312120 Breweries
316212 House slipper mfg.
321213 Engineered wood member
325181 Alkalies & chlorine mfg.
325191 Gum & wood chemical mfg.
325312 Phosphatic fertilizer mfg.
326192 Resilient floor covering mfg.
326211 Tire mfg (except retreading).
331528 Other nonferrous foundries (except die-casting).
332992 Small arms ammunition mfg.
332995 Other ordnance & accessories mfg.
333315 Photographic & photocopying equipment mfg.
333611 Turbine & turbine generator set unit mfg.
335110 Electric lamp bulb & part mfg.
335222 Household refrigerator & home freezer mfg.
335912 Primary battery mfg.
336111 Automobile mfg.
336112 Light truck & utility vehicle mfg.
336120 Heavy duty truck mfg.
39
336391 Motor vehicle air-conditioning mfg.
336411 Aircraft mfg.
336412 Aircraft engine & engine parts mfg.
336419 Auxiliary equip mfg.
336992 Military armored vehicle, tank and tank component mfg.
339995 Burial casket mfg.
40
Table 2. Parameter Assumptions
This table displays the baseline parameterization of the model. The notation is as in the text, but the various
parameters are defined for convenience.
Global Parameters
β 0.97 Discount factor
δ 0.25 Depreciation rate
γ 10 Relative risk aversion
υ 0.125 Capital adjustment cost
1958-2011
X ($ billion) 2715.1 Mean X
λ0.5X × 100 4.2 Annual volatility of εX
ρX 0.99 Autocorrelation coeffi cient of X
ψK 0.43 Output elasticity of capital
ψH 0.63 Output Elasticity of material inputs
θ 1.20 Mean θ
λ0.5θ × 100 1.9 Volatility of εθ
λXθ 0.0 Covariance of shocks
ρθ 0.96 Autocorrelation coeffi cient of θ
41
Table 3. Oligopolistic Manufacturing Industries: Product Market Variables
This table presents salient statistics on equilibrium capital investment, material input de-
mand, output and price-cost margins for two different levels of industry concentration (N = 4
and 8) using the NBER-CES sample of manufacturing industries (1958-2011). The inter-
nally calibrated values for the elasticity of substitution (σ) are 4 and 3.7 for N = 4 and 8,
respectively, while the consumption weight of the good produced by the oligopolistic sector
in the utility function (φ) is set at 0.5. The other parameters of the model are specified in
Table 2. The statistics are derived from numerical simulations involving 5000 replications of
the equilibrium paths of a 54-year model economy. For any variable w, gw denotes the log
change in adjacent periods. The p-values of the correlations are given in the parentheses.
Data Model (N=4) Benchmark (N=4) Model (N=8) Benchmark (N=8)
Vol(εX) 4.21% 4.26% 4.26% 4.26% 4.26%
Vol(εθ) 1.87% 1.90% 1.90% 1.90% 1.90%
Mean(pmcr) 1.08 1.07 1.00 1.08 1.00
Vol(gI) 17.69% 23.11% 39.15% 22.59% 37.18%
Vol(gH) 7.85% 4.45% 4.94% 4.29% 4.71%
Vol(gY ) 7.12% 5.40% 6.10% 5.20% 5.85%
Corr(gI, gX) 0.37 0.55 (0.0) 0.47 0.58 (0.0) 0.50
Corr(gI , gθ) 0.13 0.75 (0.0) 0.66 0.71 (0.0) 0.62
Corr(gH , gX) 0.72 0.49 (0.0) 0.44 0.54 (0.0) 0.49
Corr(gH , gθ) 0.62 0.60 (0.0) 0.54 0.59 (0.0) 0.54
42
Table 4. Equilibrium Asset Markets Variables
This table presents salient statistics on equilibrium asset markets variables for the industry (y) – at two
different levels of industry concentration (N = 4 and 8) – and the market (x) using the NBER-CES
sample of manufacturing industries (1958-2011). The calibrations for σ and φ are given in Table 3, while the
other parameters are specified in Table 2. The statistics are derived from numerical simulations involving
5000 replications of the equilibrium paths of a 54-year model economy.
Data Model (N = 4) Benchmark (N=4) Model (N = 8) Benchmark (N=8)
E(ry − rf ) 5.09% 5.46% 6.03% 5.35% 6.13%
E(rx − rf ) 5.55% 3.47% 3.47% 3.47% 3.47%
Volu(ry − rf ) 18.28% 18.77% 23.70% 17.68% 23.19%
Volu(rx − rf ) 15.69% 9.45% 9.45% 9.45% 9.45%E(ry−rf )
Volu(ry−rf ) 0.28 0.29 0.25 0.30 0.26E(rx−rf )
Volu(rx−rf ) 0.35 0.37 0.37 0.37 0.37
E(rf ) 5.14% 2.60% 2.60% 2.60% 2.60%
43