Input Substitutability and the Value of the Firm∗

Philipp Hergovich1 and Monika Merz2

1Vienna Graduate School of Economics2Department of Economics, University of Vienna

January 11, 2017

Abstract

How does a firm’s ability to substitute one type of input for another type inthe production of output affect its profit and therefore its market value? Does anincrease in the technical degree of factor substitutability indeed raise firm prof-its when profits amount to rents paid to quasi-fixed factors of production? Toaddress these questions, we develop a firm valuation model that features a CESproduction function with labor and physical capital as main input factors. First,we allow for ad hoc factor adjustment costs. Then we micro-found labor adjust-ment costs by embedding the CES production function into a competitive searchenvironment with capital as in Acemoglu and Shimer (1999) and augmenting itby the possibility that a firm can vary its size by hiring multiple workers. Thissetup endogenously renders efficient wages. It is rich enough to enable us to studythe implications that particular labor market policies or policies that affect theactual input-mix have on a firm’s profit. The model generates a non-monotonicrelationship between the degree of factor substitutability and firm’s profitability.The reason is that there are two opposing forces present in the model. Owningcapital generates a rent for the firm, which is higher if capital is more difficultto substitute, on the other hand too little substitutability limits the productionpossibilities.

Key Words: factor substitutability, quasi-fixed production factor, competitive search,firm value

∗Monika Merz gratefully acknowledges financial support from the Jubiläumsfonds of the AustrianNational Bank, project no. 16253

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1 Motivation

The nature of firms’ output production processes has recently received renewed atten-tion from macroeconomists and labor economists alike. The extent to which factors ofproduction such as labor, or physical capital are substitutes or complements to eachother is of crucial importance for firms’ decisions to adjust their factors of productionin order to secure profits when facing changing relative factor prices, or exogenous pro-ductivity shocks. But even if input factors are technically substitutable, there oftenexist additional obstacles that render actual factor substitution expensive if not impos-sible. Such obstacles include time-to-build for physical capital, (partial) irreversibilityof installed capital, or the fact that it takes time and resources for a firm to attract andhire new workers, or to layoff existing ones.

In this paper, we study the implications that varying degrees of input substitutabil-ity in the output production process have for the market value of firms when labor andcapital are costly to adjust. A firm’s market value corresponds to the present discountedvalue of expected future profits, where profits amount to rents paid to the quasi-fixedfactors of production. We develop a partial equilibrium firm valuation model that fea-tures search frictions in the labor market and irreversible capital. Identical workers areeither employed or unemployed and searching for a job. Firms require capital and laborfor producing output in a competitive environment. Our analytical approach is novelin that it features frictional factor markets – in particular search and matching frictionsin the labor market – together with varying degrees of technical factor substitutability.In order to sensibly study how interactions between these two model features affectfirms’ value, we have to replace the commonly used setup of one firm employing oneworker in exactly one job by one where a firm can hire and employ several workerssimultaneously. We achieve this by assuming firms to inherit a fixed stock of physi-cal capital at the beginning of time while trying to attract workers to operate it bysimultaneously posting vacancies and the associated going wage rate. This effectivelyamounts to a competitive search environment with capital as in Acemoglu and Shimer(1999) augmented by the possibility that a firm can vary its size by hiring more thanone worker. This setup renders efficient wages endogenously.

Our analytical approach seems promising for at least two reasons. First, it usessearch and matching frictions in the labor market that have been shown in variouscontexts to matter empirically and to be essential for understanding labor markets.Second, it integrates those frictions into an equilibrium model featuring optimizingworkers and firms, thereby effectively providing a micro foundation for labor adjustment

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costs. 1 This environment is rich enough to enable us to study the implications thatparticular labor market policies or policies that affect the actual input-mix have on afirm’s market value.

The remainder of this paper is organized as follows. After providing a brief literaturereview, chapter 2 formulates as benchmark a firm valuation model with neoclassicalfactor markets and ad hoc factor adjustment costs. Chapter 3 develops an equilibriumfirm valuation model featuring identical workers who can be employed or unemployedand a competitive search environment with capital and variable firm size. The model iscalibrated and used as a lab for investigating the implications that varying degrees oftechnical factor substitutability have for firms’ value when labor and capital are costlyto adjust. Chapter 4 briefly discusses possible applications. Chapter 5 concludes.

1.1 Related Literature

In the standard neoclassical model, there is no role for the firm. Rather a household candevote the productive assets it owns, more specifically capital and labor, to production,causing disutility of labor and depreciation of capital, in exchange for output goods,which can be consumed or used for next period’s production.

Already Coase (1937) points to the fact, that this view does not capture an essentialfeature of firms, namely that this organizational structure economizes on transactioncosts. Production factors within are worth more to the firm, because it does no longerhave to undergo the process of inquiring the market prices for the inputs and tasks itcurrently needs. This argumentation is striking in a sense that it points to transactionscosts in the market for input factors.

An explicit formulation of these transaction costs can be found in the literatureregarding Tobin’s q, starting with Tobin (1969) and has been applied in the assetpricing and investment literature (for example see Cochrane (1991)), or in our case,determine the value of a firm. This link has been shown by Hayashi (1982). Ananalogous derivation for our problem can be found in the appendix.

The second model used for our analysis allows firms to hire multiple workers, whilea matching friction is present at the labor market, but it is not the first one to doso. Stole and Zwiebel (1996) analyze the problem of the firm, when it hires multipleworkers in a bargaining framework. Each additional worker lowers the marginal productof all previously employed workers thus leading to inefficiencies related to over-hiring,

1Alternatively, we could have micro-founded labor adjustment costs by considering labor contracts,or efficiency wage models.

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in a framework where contracts are not enforceable. This is the case because workerscan renegotiate the wage before production, creating a hold-up problem for the firm.Stole and Zwiebel (1996) carefully develop a bargaining game and characterize the setof contracts, and relates the wages being paid in equilibrium to the wages that wouldarise in a purely neoclassical setup.

We take the following lessons from this: Bargaining solutions are prone to hold-upproblems, especially when we have decreasing returns to scale. These come naturallyfrom the concavity of most production functions (decreasing returns to one factor), orvia capital investments for which the firm wants to hire workers.

We thus adopt a directed search framework à lá Moen (1997), where firms post wagesand vacancies, workers observe these posts and apply optimally. Acemoglu (2001) showsthat this eliminates the holdup problem, which arises when capital is already in placebefore the firm posts vacancies. We hereby move closer to the modelling frameworkproposed by Kaas and Kircher (2015), abstracting from the heterogeneity present intheir model, and we assume a linear vacancy posting cost. We do so to remain inclose vicinity to the standard directed search model. Our focus is not on firm size asin Kaas and Kircher (2015), as we want to study the effects of substitutability amongproduction factors.

Other papers have linked the value of the firm coming from labor, to the equitypremium puzzle (Kuehn et al., 2012) or to various asset pricing anomalities (Shim,2016). None of these papers has explored substitutability among inputs, as the firstpaper abstracts from capital, while the second one keeps the elasticity of substitutionat unity, using a Cobb-Douglas production function.

2 Ad hoc Adjustment Cost Function

Adjustment costs on labor have been studied before in the literature, but mostly witha different objective in mind. The exercises performed were mostly to determine themagnitude of labor adjustment costs, compared to the ones of capital. Examples includethe work by Hamermesh (1993) or Merz and Yashiv (2007). The usual findings are thatthe adjustment costs for labor are smaller than the ones for capital. We will employthis fact when we calibrate the adjustment cost function.

In our attempt to answer the question of how the value of a firm depends on thedegree of substitutability between input factors, we closely follow the Tobin’s q approach(Tobin, 1969).

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The firm’s side

We consider a firm which uses two inputs of production, namely capital and labor,and assume that the production function is of the following form, exhibiting constantelasticity of substitution:

Y (Kt, Lt) = [αLσt + (1− α)Kσt ]

1σ (1)

with

0 < α < 1

σ ∈ (−∞, 1]

This function is homogeneous of degree one in both inputs. This is important forour characterization of the value of the firm later on. The parameter α representsthe capital share, while σ is the degree of substitutability. It’s range from −∞ (beingthe Leontief production function where inputs are perfect complements) to 1 (linearproduction, inputs are perfect substitutes) allows us to consider all scenarios of factorsubstitutability.

The laws of motion for the different input factors are given by,

Kt+1 = (1− δ)Kt + it, 0 < δ < 1

Lt+1 = (1− sL)Lt + ht, 0 < sL < 1

Profits are given by

πt = Y (Kt, Lt)− g(it, Kt, ht, Lt)− wtLt (2)

Notice that the firm does not pay any interest rate or rental costs for capital, as itbuys the capital at a price normalized to one. However it has to pay a wage wt eachperiod to the Lt.

The problem of the firm is now to maximize the sum of future discounted profits

Jt =∞∑t=0

βtπt (3)

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We can write the corresponding Lagrangian as:

L = Et∞∑t=0

βt[Yt(Kt, Lt)− Ltwt − it − g(it, Kt, ht, Lt)

+QKt ((1− δ)Kt + it −Kt+1)

+QLt ((1− sL)Lt + ht − Lt+1)]

(4)

and Derive the following first order conditions.

∂L∂it

:1 + g′(it) = QKt

∂L∂ht

:g′(ht) = QLt

∂L∂Kt+1

:−QKt + β[(Y ′t+1(Kt+1)− g′(Kt+1) + (1− δ)QK

t+1] = 0

∂L∂Lt+1

:−QLt + β[Y ′t+1(Lt+1)− wLt+1 − g′(Lt+1) + (1− sL)QL

t+1] = 0

In Appendix A we show how these expressions can be used to decompose the valueof the firm into:

Jt = Kt+1QKt + Lt+1Q

Lt (5)

following Hayashi (1982). Each of the Q’s are analogous to Tobin’s q, and are givenby the adjustment cost related to a marginal increase in the respective factor.

The household’s side

The household’s side of the economy is purposefully kept simple, because the mainfocus lies on the firm. We assume a representative household, who comprises of a unitmass of workers. It derives utility from consumption and incurs a disutility of workingaccording to the following CRRA utility function:

U(ct, Lt) =c1−γct

1− γc+

(1− Lt)1−γL1− γL

(6)

It supplies labor in exchange for the wage income, and saves in bonds b, i.e.

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bt+1 + ct ≤ wtLt + (1 + rt)bt (7)

The optimality conditions for the household are

(1− Lt)−γLc−γct

= wt

β(1 + rt+1)

(ctct+1

)γc= 1

where β is the household’s intertemporal discount factor.We can solve this model numerically and determine the value of the firm, however

for this, we need a calibration of the model, and choose functional forms.

2.1 Calibration

Following Merz and Yashiv (2007), we adopt adjustment costs of the form:

g(it, Kt, ht, Lt) = [ν1

(itKt

)2

+ ν2

(htLt

)2

]Y (Kt, Lt) (8)

It is important that the adjustment cost function is also homogeneous of degree1, otherwise Hayashi’s Theorem can not be applied. We choose a standard quadraticadjustment cost function instead of using the estimates presented in Merz and Yashiv(2007), because the focus is on the degree of substitutability, not per se about adjust-ment costs.

The parameter values of the model are summarized in the Table number 1. Mostvalues are standard for a yearly calibration.

The depreciation rate δ is 0.1, and the capital share governed by α is 0.3. We assumethat adjustment costs for one unit of labor are half as big as the ones for capital. Thelast two entries are the parameters of the consumer’s utility function.

2.2 Steady State Results

We can now solve the model under our baseline parametrization, and perform experi-ments with respect to the degree of substitutability σ. The results are shown in Table2.

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Variable σ = .4 (Baseline) σ = .3 σ = .5k 2.771 2.033 4.657L 0.451 0.4654 0.420y 0.904 0.779 1.212c 0.626 0.0.5744 0.745w 1.060 1.002 1.187i 0.277 0.203 0.466h 0.023 0.023 0.021J 2.794 2.0528 4.687Qk 1.006 1.008 1.005QL 0.01 0.008 0.014

Table 1: Steady State values with varying σ

We observe that the parameter σ is crucial for the solution to the problem. Com-pared to the baseline scenario, when the inputs become less substitutable capital used inproduction decreases significantly, and more labor is used. However, output producedfalls. The effects are reversed if the degree of substitutability increases, and the inputsbecome better substitutes (third column). Due to the higher capital input, and sincethe firm owns the capital (and thus earns the compensation paid to this factor), we seethat a higher degree of substitutability increases the value of a firm.

Clearly the model with ad hoc adjustment costs is unsatisfying along several dimen-sions. For once, it ignores the fact that we have a satisfying microfoundation, for whyit is costly to adjust labor. For capital, one could argue that time-to-build constraintsare a microfoundation of capital adjustment costs, but for the labor market, there iswide agreement for the existence of search and matching frictions. We want to explorethis approach in the subsequent chapter.

3 A One Shot Model of Directed Search

An important take-away from previous section’s discussion is that a production factorcan only contribute to the value of the firm if there is a friction associated with it. Underthe assumption of neoclassical factor markets, a factor’s marginal product equals thecompensation it receives, so there is no rent to earn. In the model presented above,the friction was an exogenous adjustment cost, which ensured that a production factorwithin the firm was more valuable than the same factor outside of it.

There are several arguments to be made concerning the source of adjustment friction,

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one of which being "time-to-build". This argument is usually made for capital, thatit takes some time to install a new piece of capital into the production process, whichinterferes with the production process, and can inflict output loss. A similar reasoningapplies for training costs and employees. It is twofold there, as training activities areneeded to bring the new employees productivity to the common level, additional to thereal output costs of training.

We want to employ a different friction, and study how this affects the variablesunder considerations, namely a search friction with directed search (wage posting). Thismeans that firm can post vacancies at a wage, and a matching function determines thetransition from applicants to employees. Although this modelling framework in itselfis not novel, we add a new feature to it. In most applications of this model, it isassumed that each vacancy is a firm in itself, which means the production technologyis of Leontief form, where the ratio of factors is fixed, translating into a degree of factorsubstitutability equal to zero.

We abstract from modelling training costs, but rather employ a search and matchingfriction on the labor market. This is done, because it is theoretically well understood,and has been shown to be empirically relevant (see e.g. van den Berg and Ridder(1998)). 2 As our question is about substitutability among input factors we need to gobeyond the usual Leontief style one-firm-one-worker, which is present in the majorityof the search and matching models. We do so by following a proposed extension inAcemoglu and Shimer (1999), where one firm can post multiple vacancies, thus produc-ing with different combinations of capital and labor. In equilibrium however, each firmwill choose the same composition of input factors, because firms are homogeneous. Theframework of directed search helps us avoid the problems associated with decreasingmarginal products of new employees (Stole and Zwiebel, 1996), because all workers di-rect their search to the best existing offer. Again due to homogeneity, we only observeone wage and one number of vacancies being posted.

3.1 The Model

As mentioned above we need to go beyond the one-firm-one-worker relationship and wedo this in the following way.

Firms are born with a given capital stock k, that they can use for production. Theycan post vacancies v at a wage rate w. This wage rate will determine the number of

2There are other frictions one could study like asymmetric information, which leads to contractingproblems or efficiency wages.

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searching workers who apply for each vacancy. The matching function determines howmany of these vacancies are filled. On the other side of the market we have a unit massof workers, all of whom are assumed to be unemployed in the beginning of the period.We analyze a one-shot form of this model, meaning we consider only one period. Theseworkers can apply to one job, and if they get the job with a certain probability, theyreceive the wage associated with the job. Otherwise they receive an unemploymentbenefit b.

Before laying out the formal model, a few words on our notation are in order:

• θ = vudenotes labor market tightness, defined as vacancies in a given submarket

in relation to the number of applicants for these jobs.

• m(u, v) is the matching function which exhibits constant returns to scale

• m(u,v)u

= m(1, θ) = p(θ) job-finding rate for the unemployed worker, m(u,v)v

=

m(θ−1, 1) = q(θ) job-filling rate for the firm

Workers will only apply to a job if the expected value of getting the job is maximalfor them. If we denote the maximal value of a job by U, for all jobs offering wage wwith associated labor market tightness θ the following inequality has to hold:

U ≤ p(θ)w + (1− p(θ))b (9)

Firms know this and take equation (9) with equality as a constraint into considera-tion, when deciding how many vacancies to post. If the inequality was strict, all workerswould apply to this one submarket, driving the labor market tightness down and re-ducing the right hand side. This implies that firms can determine the labor markettightness for the submarket they are operating in by posting the respective wage.

The firm can use the inherited capital along with the labor it hires to produce theoutput good y. It does so via a CES production function, taking capital and labor asinputs.

y(k, h) = (αkσ + (1− α)hσ)1/σ

where α is the factor share and thus between zero and one, while σ ≤ 1 governsthe degree of substitutability. If σ = 1, we get a linear production function and thusperfect factor substitutability, whereas with σ converging to −∞ the isoquants of theproduction function become L-shaped, as in the Leontief production function. 3 For

3This is the case most commonly studied in the labor search literature.

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the other limiting case of σ = 0 the function becomes Cobb-Douglas.The firm thus seeks to hire h workers to maximize profits, given by

π(k, h) = y(k, h(k))− αk − w(k)h(k)− v(k)a (10)

To avoid production without any labor, which is possible when the production func-tion is of CES form, we assume that the firm has to pay a maintenance cost of α for itscapital, which renders profits without hiring any labor equal to zero.

As the labor market exhibits a matching friction firms cannot directly choose l, butcan influence it via posting vacancies v at wage w. As they operate in a submarketwith tightness θ it holds that q(θ)v = l. As capital is fixed, and h is determined bythe number of vacancies and the labor market tightness (and thus the wage), we canrewrite output as

y(vq(θ)) = (αkσ + (1− α)(q(θ)v)σ)1/σ

and profit as

π(v, w, θ) = y(vq(θ))− αk − wq(θ)v − va (11)

Note that all variables in equilibrium depend on the capital stock, which equals afixed parameter in our model.

The firm’s problem consists of maximizing equation (19) taking equation (9) withequality as a constraint.

Depending on how we close the model, there will be profits or no profits, as we willdiscuss below. 4

We write the maximization problem as a Lagrangian:

L(v, w, θ) = y(vq(θ))− αk − wq(θ)v − va− λ(U − p(θ)w − (1− p(θ))b) (12)4Notice that in a dynamic version, existing firms will make profit, which is why the value of the

firm will be different from zero, even if a free-entry-condition is in place.

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∂

∂v:(1− α)

[y(vq(θ)

(vq(θ))

]1−σq(θ)− wq(θ)− a = 0 (13)

∂

∂w:q(θ)v = λp(θ) (14)

∂

∂θ:(1− α)

[y(vq(θ)

(vq(θ))

]1−σvq′(θ)− wvq′(θ) + λp′(θ)(w − b) = 0 (15)

Equation (13), can be rewritten as:

w = (1− α)[y(vq(θ)

(vq(θ))

]1−σ− a

q(θ)(16)

This wage equation has a very important economic intuition. If it is very costly fora firm to find a worker (high a) wages will be below marginal product as the firm wantsto recover the upfront investment cost. Similarly, if there is no wage posting cost, wageswill equal the marginal product of labor, and the matching friction plays no role. Ifit is however very easy to fill a vacancy, which means a high value for q(θ), wages willbe higher. As q(θ) increases, the vacancy posting costs become less important. In thelimiting case of infinite q(θ), the term involving the vacancy posting cost goes to zero.If we could increase capital together with labor (we will call the point where capitalis equal to labor the bliss point) wages would equal marginal product. However, ascapital is fixed, this is only a thought experiment to guide our intuition. To see this,note that

w = (1− α)[(αkσ + (1− α)(vq(θ))σ)1/σ

(vq(θ))

]1−σ− a

q(θ)

w = (1− α)[(α(vq(θ))σ + (1− α)(vq(θ))σ)1/σ

(vq(θ))

]1−σ− a

q(θ)

w = (1− α)− a

q(θ)

limq(θ)→∞

w = (1− α)

Equation (14) relates the two probabilities to the Lagrange multiplier. If we assumea Cobb-Douglas matching function, i.e. m(u, v) = vγu1−γ, then p(u, v) = m(u,v)

u= θγ

and q(u, v) = m(u,v)v

= θγ−1. We get λ = q(θ)p(θ)

v = θγ

θγ−1v = vθ−1. As we are looking atsymmetric equilibria only, where all firms take the same actions in equilibrium, we know

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that the number of vacancies governing labor market tightness θ will be a multiple ofthe vacancies posted by the individual firm. Let vθ = vρ, where ρ is the measure offirms active in the submarket. We then get that λ = u

ρ. The Lagrange multiplier can

be interpreted as the shadow value of relaxing the constraint. Having firms paying ahigher wage, will benefit the whole mass of unemployed who apply to these ρ firms.In equilibrium, as firms post the same wages, the whole unit measure of unemployedapplies to the same firm so, u = 1 for each firm.

The first order condition (15) is a little bit more complicated, but we can substituteout λ and rearrange to get

(1− α)[y(...)

vq(θ)

]1−σv = wv − vθ−1p

′(θ)

q′(θ)(w − b)

(1− α)[y(...)

vq(θ)

]1−σ= w − γ

γ − 1(w − b)

(1− α)[y(...)

vq(θ)

]1−σ= w(1− γ

γ − 1) +

γ

γ − 1b

w =(1− α)

[y(...)vq(θ)

]1−σ+ γ

1−γ b

1 + γ1−γ

= (1− γ)(1− α)[y(...)

vq(θ)

]1−σ+ γb (17)

Where y(...) is the CES output-production function, p′(θ) = γθγ−1 and q′(θ) =

(γ − 1)θγ−2, so p′(θ)q′(θ)

= γγ−1θ.

This wage equation is similar to the familiar expression from bargaining models,where wages are a convex combination of unemployment compensation b and productof the match, where the weighting is done via the bargaining weights. In our modelthese weights depend on the elasticity of the matching function. This follows fromthe efficiency of directed search related to the Hosios condition (Hosios, 1990), whichstates that the random search and bargaining model yields the efficient allocation, whenthe bargaining power is equal to the elasticity in the matching function. However, withwage posting and directed search, Moen (1997) showed that this efficient solution arisesendogenously.

We now have a system of non-linear equations with four unknowns, namely wage,vacancies, labor market tightness and the value for the unemployed U.

To determine U, we simply use the constraint (equation (9)), which has to holdwith equality in equilibrium. Note that as U is defined to be the expected utility of a

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searching worker, it denotes the chance to get a job, at the beginning of the period.We still need to specifiy how we close the model. We consider the following two

options, which are discussed in greater detail on page 974 in Rogerson et al. (2005).

• We can now fix the number of firms at any mass ρ to get ρv = θ and solve theproblem one by one, by simply saying this is how many firms are around andhave capital, necessary to enter the market. In this case, we get positive profits.In order to avoid any deception coming through different normalizations, we willnormalize ρ to 1.

• Alternatively, we can impose free entry, equivalent to a zero profit condition. Inthat case, additionally to equation (19) being equal to zero, enough firms haveto enter such that ρv = θ. This consistency condition implies that firms postvacancies v, and ρ of these firms enter to create the labor market tightness θ.

We provide numerical results for both cases below. Before doing so, we want tostress one is one interesting feature of our model.

3.1.1 An analytic result

Our one shot model is able to provide a micro-foundation for the model with ad hocadjustment costs, presented in section 2. To see this, compare the wage equation derivedfor the matching friction

w =y′(vq(θ))

q(θ)− a

q(θ)

to the wage expression derived in the Ad hoc Adjustment Cost model in steadystate, which was:

−g′(h) + β[Y ′(L)− wL − g′(L) + (1− sL)g′(h)] = 0

where we dropped the time indices as we look at steady state, and g′(h) is theadjustment cost paid for one marginal additional unit of labor.

If we rewrite this equation to express the wage we get

wL = Y ′(L)− (s+1

β)g′(h) (18)

These two expressions are similar in the sense that they both include a wedge be-tween the marginal product and wages, given by how costly it is to adjust the workforce,

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for one additional worker. As our model with matching frictions is one-shot, if we sets = 0 and β = 1 because no matches break apart and there is no discounting, equation(18) states that marginal product minus adjustment costs.

3.2 Calibration

Calibration is important, in the sense that the parameters jointly have to allow for theexistence of a solution. If unemployment benefits b were too high compared to output,which can be produced using capital k, firms couldn’t afford to post a wage to attractany workers.

We already described the functional forms of the matching function to be CobbDouglas, and the production function to be CES. We choose the following parametersas baseline calibration.

Parameter Value Interpretationb 0.2 unemployment benefita 0.05 vacancy posting costk 0.5 capital-endowmentα 0.3 capital-share in CESσ 0.4 substitutability CESγ 0.4 elasticity of the matching function

Table 2: Baseline calibration

As this is a one shot version of the model, the baseline calibration serves only as astarting point for inspecting the mechanism and is not meant to match any empiricalmoments. The main focus of this paper is on the effects of varying the parameter σ,which governs the degree of factor substitutability in output production.

3.3 Results

We can now numerically solve the model with the matching friction and perform com-parative statics analysis. We will discuss the results separately for the two cases wedescribed earlier, the "free entry" and the "fixed number of firms" cases.

It appears that free entry gives rise to smaller firm size, when expressing size interms of employment. This is because more firms enter, posting fewer vacancies, and asthe matching function exhibits effectively decreasing returns to scale (remember, it isconstant returns to scale in both factors, but u is fixed). Notice that in the fixed firms

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Variable Free Entry Fixed No. of Firmsθ 10.07 6.05w 0.5 0.421v 1.999 6.05U 0.956 0.654q(θ) 0.25 0.34p(θ) 2.52 2.053h 0.5 2.053y 0.5 1.4π 0 0.135

Table 3: Simulation Results

case, θ = v holds, as ρ is normalized to unity. This also implies that, vq(θ) = h = p(θ).To see the "excessive entry" in the free entry case, it is instructive to compare v and θ.As all firms behave in the same way, the ratio θ

vgives the measure of firms that operate

in the market denoted with ρ in previous sections. We see that this ratio is around 5.What is interesting to see is the following. In the free entry case, firms reach

their bliss point where k = h. Apparently, the matching friction acts on the expectedwages of the workers, either via jobfinding probabilities, or via the wage itself. It isstraightforward to see why firms would choose to go to their bliss point in productionand overhire, which means hiring more h than k.

First notice that firms earn a rent on their capital, so they can afford to hire"enough" workers. On the other hand, the production function is CRS in both ar-guments, but exhibits decreasing marginal products when increasing only one factorand keeping the other fixed. Therefore the point (0.5,0.5) is the point at which increas-ing labor further would lower the marginal product (as capital is a fixed factor), andthus render the production of the firm inefficient. A new firm could enter, produce atthe efficient point, and cause the firm who overhired to incur losses.

We now examine the results in greater detail separately.

3.3.1 Free Entry

The result that firms go to their blisspoint depends on the assumption that productionwithout labor yields zero profits. However, if firms pay less than αk for their capital,firms would always make profit by not posting any vacancies, and thus infinitely manyfirms would enter the market, producing exclusively with capital.

If however firms have to pay more for their capital, by changing equation (19) to

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π(v, w, θ) = y(vq(θ))− 1.1αk − wq(θ)v − va (19)

We see that the labor hired increases to 0.62, resulting in an increased output of0.58. Firms post more vacancies because they have to make up for the higher capitalpayments, and this drives the measure of firms in the market down.

However, it is interesting that firms reach their blisspoint in our setup, as it meansthat they do choose to produce according to a Leontief production function. However,in this setup we are not able to talk about the degree of substitutability, as the value σcancels out from the relevant equations. To see why, we go back to our CES productionfunction and assume that both factors are used in the same quantities m. We then get

y(m,m) =(αmσ + (1− α)mσ)1/σ = m(y(m,m)

m

)1−σ

=(mm

)σ−1= 1

independently of σWe next turn to the case of a fixed number of firms for further analysis.

3.3.2 Fixed Number of Firms

If we assume that there is a unit mass of firms which is endowed with capital k, we canask the question on how does the outcome of our model change, if we vary the degreeof substitutability between capital and labor.

Variable Fixed Entry σ = .45 Fixed Entry σ = .4 (baseline)θ 6.377 6.05w 0.428 0.421v 6.377 6.05U 0.678 0.654q(θ) 0.329 0.34p(θ) 2.098 2.053h 2.098 2.053y 1.49 1.4π 0.124 0.135

Table 4: Equilibrium outcomes with varying σ

We see that firms want to extend their production, and thus hire more labor. This

17

is because labor and capital are substitutes, and capital is fixed. However, as the degreeof substitutability rises, we see that profits decline, because the fixed factor k becomesless of an issue. This leads to a decline in the rent earned by the firm and resultingin declining profits. If we just look at profits and vary σ between -3 and 1, we get thefollowing relationship in Figure 1.

Figure 1: Profits for different degrees of substitutability

It depicts a non-monotonic relationship between the degree of substitutability be-tween production factors and profits. At first increasing the degree of substitutabilityamong factors leads to higher profits, as the firms become more flexible in their produc-tion. However, after some point the opposing an opposing force dominates this effect.As it becomes easier to substitute between labor and capital, the scarce factor capitalis not as essential in production anymore, thus the rent earned on it declines. This inturn, lowers the market value of the firm. For our calibration, profits are maximized ata value of σ = −1.41. The profit at 0.4 corresponds to the 0.135, depicted in table 5.

A different experiment is illustrated in figure 2, where we analyze how changesin the unemployment benefit change the profits earned by the firms. We obtain a

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Figure 2: Profits for different levels of unemployment benefit

negative relationship between profits and unemployment benefits. This comes, becausea decrease in unemployment benefits makes the workers more willing to work at lowerwages, and relaxes the constraint for the firm (9). Notice that for high enough values ofunemployment benefit, the firm cannot make profits anymore, and thus does not postany vacancies.

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4 Possible Applications

We have developed an equilibrium model that features frictional factor markets and acompetitive market for output goods. The model is populated by optimizing workerswho can be employed or unemployed and searching for a job together with optimizingfirms that devote resources to hiring workers in order to produce output together with itsirreversible stock of capital. This setup can easily form the basis for studying additionalquestions centered at the intersection of the labor market, input substitutability andfirm profits. Our model lends itself to studying the implications for the labor marketand the aggregate economy of the ongoing process of automation and digitization thathas affected the industrialized world. The continuing digitization of output productionhas led to a slow change in the input mix of production factors: Routine and manualtasks are increasingly substituted by machines, whereas cognitive tasks complement theincreased use of machines and – depending on their relative ease of adjustment and theirprice – sometimes even substitute them. Most research in macro and labor economicshas focused on studying the underlying economic causes of this process that has becomeknown as job-polarization. Since our model explicitly allows for factor substitutability,it can help shed light on the consequences that digitization and job-polarization havefor future employment opportunities of different types of labor, the wage structure, orthe distribution of factor income. It can be viewed as an alternative to the task-basedframework in Acemoglu and Restrepo (2016) . It may also serve to better understandthe driving forces underlying the observed downward trend in national labor sharesof income (ILO, 2015). An alternative field of application lies at the intersection ofmacro/labor and finance. Our equilibrium setup lends itself to generating testableimplications on the role of labor and capital in finance. Once we extend it to allow fordynamics and exogenous driving forces such as productivity shocks that may or maynot be input specific, our model creates a link between the variability of a firm’s profitand the degree to which capital and labor are substitutes in the production process. Itcan be used to extend and further explore the hypothesis recently promoted by Shim(2016) that a rise in factor substitutability effectively means that a firm can react moreflexibly to exogenous variations in productivity and therefore exhibit less systematicrisk and a lower expected return on stocks. In our environment where labor and capitalare subject to factor adjustment costs, this hypothesis will have to be modified.

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5 Conclusions

In this paper we have developed an equilibrium model featuring a CES productionfunction with labor and physical capital as main input factors. First, we allow for adhoc factor adjustment costs. Then we micro-found labor adjustment costs by embeddingthe CES production function into a competitive search environment with capital as inAcemoglu and Shimer (1999) and augmenting it by the possibility that a firm canvary its size by hiring multiple workers. This environment endogenously generatesefficient wages. Simulation exercises show that it captures a non-monotonic relationshipbetween the degree of factor substitutability and firm’s profitability. That is becausetwo opposing forces are at work: as input substitutability increases, a firm can reactmore flexibly to exogenous disturbances which tends to increase profits, but at thesame time rents on each input factor decline which tends to reduce profits. The modelpresented can be extended in different directions. If we introduce a dynamic structurein the firm’s and the workers’ decision problems and allow for exogenous disturbances,e.g., productivity shocks and match separation, we can study the implications thatvarying degrees of factor substitutability have for labor turnover, employment, factorprices, and firms’ profits. Alternatively, we can add heterogeneity to the workers’ orthe firm’s side. If we introduce different types of labor that vary in their degree ofsubstitutability with capital, we can study the aggregate implications for the labormarket, factor prices and output of a rise in job-polarization.

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References

Acemoglu, D. (2001). Good Jobs versus Bad Jobs. Journal of Labor Economics 19,1–21.

Acemoglu, D. and P. Restrepo (2016). The Race Between Machine and Man: Implica-tions of Technology for Growth, Factor Shares and Employment. Technical report.

Acemoglu, D. and R. Shimer (1999). Efficient Unemployment Insurance. Journal ofPolitical Economy 107, 893–928.

Coase, R. (1937). The nature of the firm. Economica 4, 386–405.

Cochrane, J. H. (1991). Production-Based Asset Pricing and the Link between StockReturns and Economic Fluctuations. Journal of Finance 46, 209–237.

Hamermesh, D. (1993). Labor Demand. Princeton University Press.

Hayashi, F. (1982). Tobin’s Marginal q and Average q: A Neoclassical Interpretation.Econometrica 50, 213–224.

Hosios, A. J. (1990). On The Efficiency of Matching and Related Models of Search andUnemployment. Review of Economic Studies 57, 279–298.

ILO (2015). The Labour Share in G20 Economies. Report prepared for the g20 em-ployment working group, OECD.

Kaas, L. and P. Kircher (2015). Efficient Firm Dynamics in a Frictional Labor Market.Review of Economic Studies 105, 3030–3060.

Kuehn, L.-A., N. Petrosky-Nadeau, and L. Zhang (2012). An Equilibrium Asset PricingModel with Labor Market Search. Technical report.

Merz, M. and E. Yashiv (2007). Labor and the Market Value of the Firm. AmericanEconomic Review 97, 1419–1431.

Moen, E. R. (1997). Competitive Search Equilibrium. Journal of Political Economy 105,385–411.

Rogerson, R., R. Shimer, and R. Wright (2005). Search-Theoretic Models of the LaborMarket-A Survey. Journal of Economic Literature XLIII, 959–988.

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Shim, K. H. (2016). The Substitutability of Laobr: Implications for the Cross Sectionof Stock Returns. Working Paper .

Stole, L. and J. Zwiebel (1996). Intra-Firm Bargaining under Non-Binding Contracts.Review of Economic Studies 63, 375–410.

Tobin, J. (1969). A General Equilibrium Approach to Monetary Theory. Journal ofMoney, Credit and Banking 1, 15–29.

van den Berg, G. J. and G. Ridder (1998). An Empirical Equilibrium Search Model ofthe Labor Market. Econometrica 66, 1183–1222.

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A Deriving the value of the Firm àlá Hayashi

To obtain the result that the value of the firm can be written in the form of equation5 we proceed according to the following steps.

Multiplying each First order condition with the argument we differentiated the orig-inal problem with, we obtain

0 =− itg′(it) + itQKt

0 =− htg′(ht) + htQLt

Kt+1QKt =Kt+1Et{β[(Y ′t+1(Kt+1)− g′(Kt+1) + (1− δ)QK

t+1]}

Lt+1QLt =Lt+1Et{β[Y ′t+1(Lt+1)− wt+1 − g′(Lt+1) + (1− sL)QL

t+1]}

We now can decompose the value of the firm in what is added by the two productioninputs, and I will demonstrate the case for capital.

We use the first and the fourth equation we just derived and the law of motion forcapital. By inserting the law of motion for the it multiplying QK

t in the first equationwe get

it(1 + g′(it)) = (Kt+1 − (1− δ)Kt)QKt

We then roll it forward by one period and take time t conditional expectation andmultiply with β:

Et[βit+1(1 + g′(it+1))] = Et[β[(Kt+2 − (1− δ)Kt+1)QKt+1]]

We reshuffle it a little bit, to be

Et[β(1− δ)Kt+1QKt+1] = Et[β[(Kt+2Q

Kt+1 − it+1(1 + g′(it+1))]] (20)

We now take the third equation we derived above and multiply out the RHS

Kt+1QKt = Et{β[(Y ′t+1(Kt+1)Kt+1 − g′(Kt+1)Kt+1 + (1− δ)QK

t+1Kt+1]} (21)

We now plug in equation 20 into 21 and get

24

Kt+1QKt = Et{β[(Y ′t+1(Kt+1)Kt+1 − g′(Kt+1)Kt+1 + [(Kt+2Q

Kt+1 − it+1(1 + g′(it+1))]]}

We now define the cash flow to capital to be its contribution to production, minuswhat his needed to sustain this level, which are adjustment costs and investment.

cfKt+1 ≡ (Y ′t+1(Kt+1)Kt+1 − g′(Kt+1)Kt+1 − (1 + g′(it+1))it+1 (22)

Plugging this definition in (A) we get

Kt+1QKt = Etβ{cfKt+1 +Kt+2Q

Kt+1} (23)

which we can again rearrange to be

Etβ{cfKt+1} = Kt+1QKt − Etβ{Kt+2Q

Kt+1} (24)

Because of the constant returns to scale property of the production function andthe adjustment cost function, we can decompose profits into using Euler’s Theorem 5

πt =Y′(Kt)Kt + Y ′(Lt)Lt

− wLt Lt − it− g′(it)it − g′(Kt)Kt − g′(ht)ht − g′(Lt)Lt

We can rewrite this to get

πt = cfKt + cfLt (25)

where

cfKt ≡ Y ′(Kt)Kt − (1 + g′(it))it − g′(Kt)Kt

cfLt ≡ Y ′(Lt)Lt − wLt Lt − g′(ht)ht − g′(Lt)Lt

We now turn to the finance literature where the value of a firm st is given by its5See for example the Appendix in Mas-Colell, Whinston and Green (1993)

25

dividend payments (cashflow) and the value next period. If we price the firm based onfundamentals which implies ruling out bubble solutions, it has to hold that Vt = st.

st = Etβ{cft+1 + st+1} (26)

We can decompose the cft+1 into its two components, and split future st + 1 intothe values coming from each production factor.

st = VKt + VLt= Etβ{VKt+1 + cfKt+1}+ Etβ{VLt+1 + cfLt+1}

We again express the discounted value of future cashflow Etβ{cfKt+1} to be

Etβ{cfKt+1} = VKt − Etβ{VKt+1}

Which gives

VKt − Etβ{VKt+1} = Kt+1QKt − Etβ{Kt+2Q

Kt+1}

This is just a dynamic version of saying:

VKt = Kt+1QKt

Virtually the same analysis leads to the the equations

VLt = Lt+1QLt

As the value of the firm is the sum of future discounted profits, and we decomposedthe profits into values related to the size of each production factor present in the firmmultiplied by its adjustment costs, we have linked the value of a firm to the inputfactors and obtained equation 5.

B Homogeneity of degree one of the Adjustment Cost

Function

g(it, Kt, ht, Lt, ht, Ht) = [ν1

(itKt

)2

+ ν2

(htLt

)2

]Y (Kt, Lt) (27)

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This function is clearly linearly homogeneous, as

g(Ait, AKt, Aht, ALt) = [ν1

(itKt

)2

+ ν2

(htLt

)2

]Y (AKt, ALt)

= [ν1

(itKt

)2

+ ν2

(htLt

)2

]AY (Kt, Lt)

= Ag(it, Kt, ht, Lt)

First notice that A cancels out in the curly brackets, thus remaining only withinthe production function, which has been shown to be homogeneous of degree 1.

This also allows for a convenient way to write the profits.

πt = Y (Kt, Lt)(1− [ν1

(itKt

)2

+ ν2

(htLt

)2

])− wtLt − it (28)

27