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: pkumar@uh.edu.

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