(Asset) Pricing the Business Cycle

Eran Yashiv∗Tel Aviv University, CfM (London School of Economics), and CEPR

January 4, 2019

Abstract

Asset values of labor and capital govern firms’ hiring and investment decisions.The paper studies these relations and shows how asset values serve to price thebusiness cycle.

The model is taken to aggregate U.S data using structural estimation, derivingtime series for the unobserved asset values. Local projections methodology is usedto study cyclical behavior.

Findings include pro-cyclical capital and labor values engendering pro-cyclicalvacancy creation and investment. Labor asset values dominate responses, includ-ing capital investment decisions, and are stronger in recessions.

Asset values may serve as instruments in many Macro contexts. The predictivecontent for economic activity inherent in them is implied by the model, which fo-cuses on firms’ forward-looking optimal behavior. I show empirically that indeedthey have a predictive role in forecasting U.S. cyclical fluctuations, especially so forlabor asset values.

Key words: Labor value, capital value, optimal investment, optimal hiring, busi-ness cycles, local projections, predictors.

JEL codes: E22, E23, E24, E32.

∗The Eitan Berglas School of Economics, Tel Aviv University, Tel Aviv 6997801, Israel. Email:yashiv@tauex.tau.ac.il; e.yashiv@lse.ac.uk.

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(Asset) Pricing the Business Cycle1

1 Introduction

When firms hire workers and invest in capital, they take into account current costs andthe expected, present discounted values of labor and of capital. Hence these activitiesare essentially investment activities. The asset values in question are unobserved. Thispaper estimates them using aggregate U.S. data and characterizes their behavior overthe business cycle. I study the response of investment and hiring to the estimated assetvalues and the changing sensitivities to these values over the cycle. I then examine thepredictive role of asset values.

I rely on the existence of frictions in investment and in hiring to formulate the rel-evant dynamic optimization problem, with emphasis on joint optimality. This is anasset-pricing type of analysis, with three relevant shadow values expressing firm, cap-ital, and labor values, which are inter-related. The business cycle is manifested in thecyclical fluctuations of GDP and its main input factors – employment and capital; theafore-going asset values are the “asset prices” of these quantities, hence the title of thispaper.

While using Tobin’s q terminology, the analysis does not rely on financial marketdata or firm price data.2 The idea here is to structurally estimate the shadow values,using data on real variables, such as GDP, wages, relative prices, hiring, and invest-ment. The resulting time series of asset values allow for the study of the followingempirical questions: does the model fit U.S. aggregate data and how does it fare rela-tive to the performance of standard models in this context? What do we learn about themagnitude and cyclicality of the afore-cited asset values? How do investment, vacancycreation, and asset values react to shocks and how do they co-move? Looking at crosseffects, how do capital (labor) values affect vacancy creation (investment)? Do laborand capital values forecast future activity?

Subsequent to estimation, I use a local projections methodology, proposed by Jordà(2005) and by Daly et al (2018), to study the cyclical behavior of the estimated assetvalues as well as of investment, hiring, and output. This analysis looks both at uncon-ditional moments and at moments conditioned by three technology-related shocks –TFP, investment-specific, and matching. It shows that pro-cyclical asset values governpro-cyclical investment and hiring. It documents the changing elasticities of the deci-sion variables with respect to asset values over the cycle. It turns out that labor assetvalues dominate responses, including capital investment decisions, and that they arestronger in recessions.

Asset values may serve as instruments and predictors in a variety of Macro contexts.

1 I am grateful to Larry Christiano and Giuseppe Moscarini for very useful discussions and suggestions, to JohnFernald and Oscar Jordà for helpful correspondence, and to Darin Waisman, Alon Rieger, Oriel Nofekh, and AndreyPerlin for excellent research assistance. I thank Nadav Ben Zeev, John Fernald, and Nicolas Groshenny for provisionof their shocks series. Any errors are my own.

2See Merz and Yashiv (2007) for work using such data in a related context.

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The predictive content for economic activity inherent in them is implied by the modelwhich focuses on firms’ forward-looking optimal behavior. Firms make their hiringand investment decisions as a function of these asset values. Through forecasting re-gressions analysis, I show that they indeed have a predictive role in forecasting U.S.cyclical fluctuations. This is true mostly for labor asset values, and, to a more limitedextent, also to capital values.

The paper proceeds as follows. Section 2 presents the literature. Section 3 delineatesthe model and the relations to be examined empirically. Section 4 presents structuralestimation, deriving time series for shadow asset values. Section 5 discusses the im-plications of the estimates for firms’ optimal behavior. Section 6 presents the cyclicalanalysis and Section 7 the forecasting regressions. Section 8 sums up the key resultsand concludes. Formal derivations and data details are relegated to appendices.

2 Literature

I relate the analysis of the paper to the literature.

2.1 Early Literature

The formulation to be used here was initially suggested by an important early literatureon adjustment costs of factor inputs, studying the accumulation of all factors of produc-tion. Key papers include Lucas (1967) and Mortensen (1973), who derived firm optimalbehavior with convex adjustment costs for n factors of production. Treadway (1971)considered (p.878) “the marginal internal cost of investment (− f ·

x) arising from the cur-

rent product “lost” due to the expansion activity of the firm.” Lucas and Prescott (1971)embedded these convex adjustment costs in stochastic industry equilibrium. Nadiriand Rosen (1969) considered interrelated factor demand functions for labor and capitalwith adjustment costs.

Related literature are the seminal papers introducing the concept of q – papersby Tobin (1969) and Tobin and Brainard (1968). Tobin (1981) defined the variable qK

as “the ratio of market valuation of capital goods to normal replacement cost” (p.21)and posited that investment is a function of qK. He noted that “the deviations of qK

from 1 represent real costs of adjustment, including positive or negative rents, incurredby investing firms in changing the size of their installed capital.” (p.22) Note that thecurrent paper does not use such market data. A formalization of the q concept withinthe latter set of models was offered by Hayashi (1982).

Over the years much empirical work was undertaken. Prominent examples includeNadiri and Rosen (1969), Abel (1980), Summers (1981), Shapiro (1986), and Hall (2004).Chirinko (1993) and Smith (2008) offer reviews.

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2.2 Modelling Hiring and Investment Costs

The current paper places emphasis on hiring frictions of various kinds, investment fric-tions, and their interactions. Faccini and Yashiv (2018) offer a review of the relevantliterature on these costs.

Recent micro studies have looked at hiring costs in their various forms in detail; seeBlatter et al (2016) and Mühlemann and Leiser (2018). Manning (2011) offers a review ofearlier studies. The emerging picture is that of convex costs placed mainly on training,with a more limited role for vacancy costs.

In the model I make a distinction between hiring from other employment (job to jobmovements) and hiring from non-employment. A micro study of a large hospital sys-tem by Bartel, Beaulieu, Phibbs, and Stone (2014) shows such distinction is warranted.They find that the arrival of a new nurse is associated with lowered productivity, butthat this effect is significant only if the nurse is hired externally.

A recent theoretical and empirical literature has given foundations to investment-hiring costs interaction terms, which I use below. This new literature looks at the con-nections between investment in capital, the hiring of workers, and organizational andmanagement changes. A general discussion and overview of this line of research isoffered by Ichniowsky and Shaw (2013) and by Lazear and Oyer (2013). One specificexample is provided by Bartel, Ichniowski and Shaw (2007), who study the effects ofnew information technologies (IT) on productivity. They use data on plants in onenarrowly defined industry, valve manufacturing. Their empirical analysis reveals, in-ter alia, that adoption of new IT-enhanced capital equipment coincides with increasesin the skill requirements of machine operators, notably technical and problem-solvingskills, and with the adoption of new human resource practices to support these skills.They show how investment in capital equipment has a variety of effects on hiring andon training.

In the empirical analysis below I use a convex cost function. While non-convexitieswere found to be significant at the micro level (plant, establishment, or firm), a numberof papers have given empirical support for the use of a convex function in the aggre-gate, showing that such a formulation is appropriate at the macroeconomic level. Thus,Thomas (2002) and Kahn and Thomas (2008, see in particular their discussion on pages417-421) study a dynamic, stochastic, general equilibrium model with nonconvex cap-ital adjustment costs. One key idea which emerges from their analysis is that there aresmoothing effects that result from equilibrium price changes.

2.3 Previous Work

In previous work I have used a similar framework but explored different issues. InMerz and Yashiv (2007) the focus was on production-based asset pricing. We inves-tigated the links between the financial and labor markets, using a production-basedmodel for firms’ market value following Cochrane (1991).3 We inserted into the model

3See also the more recent contributions of Cochrane (2007, 2017).

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labor and capital adjustment costs, with the adjustment costs for labor interacting withthose for capital. We showed that this framework can account well for U.S. stock prices.In Yashiv (2016) I decomposed the future determinants of capital and job values. Ishowed that future returns play a dominant role in determining capital and job values.

The one point of limited overlap of the current paper with the two cited papers isstructural estimation of the firms’ optimality equations. In the current paper the sampleperiod is updated and enlarged and the specification estimated is significantly wider,i.e., it nests the previous ones as special cases.

3 The Model

The model formulates optimal hiring and investment decisions in the presence of fric-tions. The model is a partial equilibrium model, intended to avoid potential misspecifi-cations in other parts of the macroeconomy. I include a discussion of important specialcases, which are prevalent in the capital and labor literatures.

3.1 Firm Optimization

Firms use physical capital (kt) and labor (nt) as inputs in order to produce output goodsyt according to a constant-returns-to-scale production function f with TFP denoted byzt:

yt = f (zt,nt, kt), (1)

TFP follows the process:

ln zt = κ1 + ln zt−1 + εft (2)

where ε ft is a shock and κ1 a parameter.Once a new worker is hired, the firm pays a per-period wage wt. Firms make invest-

ment (it) and vacancy (vt) decisions, subject to frictions, spelled out below. I representthese costs by a function g[it, kt, vt, nt] which is convex in the firm’s decision variables(it, vt) and exhibits constant returns-to-scale, allowing hiring costs and investment coststo interact.

In every period t, the capital stock depreciates at the rate δt and is augmented bynew investment it. Similarly, workers separate at the rate ψt and the employment stockis augmented by new hires qtvt = ht. The laws of motion are:

kt+1 = (1− δt)kt + it, 0 ≤ δt ≤ 1. (3)

nt+1 = (1− ψt)nt + qtvt, 0 ≤ ψt ≤ 1 (4)The vacancy filling rate qt reflects labor market conditions and embodies a matching

shock µt following the AR1 process:

ln µt = κ2 + ρµ ln µt−1 + εµt (5)

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where εµt is a matching shock and κ2, ρµ are parameters.The representative firm chooses sequences of it and vt in order to maximize its prof-

its as follows:

max{it+j,vt+j}

Et∞

∑j=0

(j

∏i=0

ρt+i

)(1− τt+j)

(f (zt+j,nt+j, kt+j)− g

(it+j, kt+j, vt+j, nt+j

)−wt+jnt+j −

(1− χt+j − τt+jDt+j

)p̃It+j it+j

)(6)

subject to the constraints (3) and (4), where τt is the corporate income tax rate, χt the in-vestment tax credit, Dt the present discounted value of capital depreciation allowances,p̃It the real pre-tax price of investment goods, and ρt+j is a time-varying discount factor.In line with the investment technology literature, p̃It is driven by an unanticipated ISTshock as follows:

p̃It ≡1

Θt(7)

ln Θt = κ3 + ρΘ ln Θt−1 + εIt

where εIt is the shock and κ3, ρΘ are parameters.The firm takes the paths of the variables zt, qt, wt, ψt, p̃

It , δt, τt and ρt as given. The

Lagrange multipliers associated with the two constraints are denoted QKt and QNt , re-

spectively. I shall use the term capital value for the former, and labor value for thelatter.

The first-order conditions for dynamic optimality can be written as follows:4

QKt = (1− τt)(

git + pIt

)= Et

[ρt+1 (1− τt+1)

[fkt+1 − gkt+1

+(1− δt+1)(git+1 + pIt+1)

]](8)

QNt = (1− τt)gvtqt= Et

[ρt+1 (1− τt+1)

[fnt+1 − gnt+1 − wt+1+(1− ψt+1)

gvt+1qt+1

]](9)

The capital value (QKt ) is the present value of expected marginal productivities, ad-justed for taxes and depreciation; the labor value (QNt ) is the present value of the profitflows from the marginal worker adjusted for taxes and separation rates.

3.2 Investment and Hiring Costs

The costs function g, captures the frictions in the hiring and investment processes. Hir-ing costs include costs of advertising, screening and testing, matching frictions, train-ing costs, and more. Thus, they pertain to vacancy posting, actual hires from non-employment, and hires from employment (job to job movements). Investment involves

4where I use the real after-tax price of investment goods, given by:

pIt+j =1− χt+j − τt+jDt+j

1− τt+jp̃It+j

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capital installation costs, implementation costs, learning the use of new equipment, andimplementing new organizational structures within the firm.5 In sum, g captures all thefrictions involved in getting workers to work and capital to operate in production.

3.2.1 Hiring and Separation Flows6

Firms hire from non-employment (h1t ) and from other firms (h2t ). Each period, the

worker’s effective units of labor (normally 1 per person) depreciate to 0, in the cur-rent firm, with some exogenous probability ψt. Thus, the match suffers an irreversibleidiosyncratic shock that makes it no longer viable. The worker may be reallocated, witha probability of ψ2t , to a new firm where his/her productivity is (temporarily) restoredto 1. Those who are not reallocated join unemployment with probability ψ1t = ψt − ψ2t .So the fraction ψ2t that enters job to job flows depends on the endogenous hiring flow h

2t .

The firm decides how many vacancies vt to open and, given job filling rates (q1t , q2t ), will

get to hire from the pre-existing non-employed and from the pool of dissolved matches.Employment dynamics are thus given by:

nt+1 = (1− ψ1t − ψ2t )nt + h1t + h2t (10)= (1− ψt)nt + ht, 0 ≤ ψt ≤ 1

h2t = ψ2t nt

The job-filling rates satisfy:

q1t =h1tvt

; q2t =h2tvt

; qt = q1t + q2t

3.2.2 Functional Form of the Costs Function

The parametric form I use is the following, generalized convex function.

g(·) =

e1η1( itkt )

η1

+ e2η2

[(1−λ1−λ2)vt+λ1q1t vt+λ2q2t vt

nt

]η2+ e31η31

(itkt

q1t vtnt

)η31+ e32η32

(itkt

q2t vtnt

)η32 f (zt, nt, kt). (11)

This is a convex function in the rates of activity – investment ( itkt ) and recruiting

( (1−λ1−λ2)vt+λ1h1t+λ2h

2t

nt ). The function is linearly homogenous in its arguments i, k, v, n.The parameters el , l = 1, 2, 31, 32 express scale, and the parameters η1, η2, η31, η32 ex-press the convexity of the costs function. λ1 is the weight in the cost function assigned to

hiring from non-employment ( q1t vtnt ), λ2 is the weight assigned to hiring from other firms

5See Alexopoulos and Tombe (2012).6I am indebted to Giuseppe Moscarini for very useful suggestions to this sub-section.

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( q2t vtnt ), and (1−λ1−λ2) is the weight assigned to vacancy (

vtnt ) costs. The weights λ1 and

λ2 are thus related to the training and production disruption aspects, while the comple-

mentary weight is related to the vacancy creation aspect. The terms e31η31

(itkt

q1t vtnt

)η31and

e32η32

(itkt

q2t vtnt

)η32express the interaction of investment and hiring costs. They allow for a

different interaction for hires from non-employment (h1t ) and from other firms (h2t ). The

function used postulates that costs are proportional to output.This functional formulation may be justified as follows (drawing on Garibaldi and

Moen (2009)): suppose each worker i makes a recruiting and training effort hi; as this isto be modelled as a convex function, it is optimal to spread out the efforts equally acrossworkers so hi = hn ; formulating the costs as a function of these efforts and putting them

in terms of output per worker, one gets c(

hn

)fn ; as n workers do it, the aggregate cost

function is given by c(

hn

)f .

3.3 Important Special Cases

Beyond the general model spelled out above, I examine important special cases, widely-used in the capital and labor literatures.

a. Relying on the seminal contributions of Tobin (1969) and Hayashi (1982), thisapproach assumes no adjustment costs for the other factor of production. In the currentcase, this is convex costs of investment in capital (labor), with no hiring (investment)costs. Typically quadratic costs are posited. Hence in the former case this has e2 = e31 =e32 = 0 and η1 = 2 and in the latter case e1 = e31 = e32 = 0 and η2 = 2.

b. The standard search and matching model – see Pissarides (2000) for an overview –does not consider investment when formulating costs and refers to linear vacancy costs.The specification is thus e2qt

ftnt vt whereby the cost is proportional to labor productivity

ftnt and depends on the average duration of the vacancy

1qt . In terms of the above model

it has e1 = e31 = e32 = 0, λ1 = λ2 = 0 and η2 = 1.

3.4 Asset Values

I formulate the relevant asset values inherent in the analysis. Appendix A shows thefull derivation, yielding (in stationary terms, divided by GDP):

Qtft=

kt+1kt

QKtftkt

+nt+1nt

QNtftnt

(12)

where aggregate firm value is denoted by Qt.The terms on the RHS of (12) are given by the following expressions, in terms of

output per unit of input (using equations (8) –(9)) :

8

QKtftkt

= (1− τt)(

gitftkt

+pItftkt

)(13)

QNtftnt

= (1− τt)gvtqtftnt

(14)

One can further sub-divide the capital value QKt

ftkt

into a term based on the price of

investment pIt and a term relating to the g function costs, as follows:

Q̃Ktftkt

= (1− τt)gitftkt

(15)

QPtftkt

= (1− τt)pItftkt

(16)

4 Estimating Optimal Behavior and Deriving Asset Values

The first stage of the empirical work is to determine whether the model fits the data,how does it fare relative to standard specifications, and to derive time series for thedifferent asset values (formulated in equations (12) – (16)), which are unobserved. Thisis done through structural estimation of the afore-going optimality equations of thefirm.

4.1 Data and Methodology

The data are quarterly and pertain to the aggregate private sector of the U.S. economy.For a large part of the empirical work reported below the sample period is 1994-2016.The start date of 1994 is due to the lack of availability of job to job worker flows (h2t )data prior to that. For another part of the empirical work, the sample covers 1976-2016.The 1976 start is due to the availability of credible monthly CPS data, from which thegross hiring flows from non-employment (h1t ) series is derived. This longer sampleperiod covers five NBER-dated recessions, and both sample periods include the GreatRecession (2007-2009) and its aftermath. Appendix B elaborates on sources and on dataconstruction.

I structurally estimate the firms’ first-order conditions – equation (8) and equation(9) – jointly, using Hansen’s (1982) generalized method of moments (GMM). In whatfollows I outline the methodology and the alternative specifications used. Details areprovided in Appendix C.

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For the production function I use a standard Cobb-Douglas formulation, with aproductivity shock exp(zt):

f (zt,nt, kt) = exp(zt)ntαk1−αt , 0 < α < 1. (17)

Replacing expected values in these equations by actual ones and expectational er-rors (jk,nt ), the estimation equations are given by (estimation is undertaken after divid-ing the investment equation by ftkt and the vacancy equation by

ftnt to induce stationar-

ity):

(1− τt)(

git + pIt

)= ρt+1 (1− τt+1)

[fkt+1 − gkt+1

+(1− δt+1)(git+1 + pIt+1)

]+ jkt (18)

(1− τt)gvtqt= ρt+1 (1− τt+1)

[fnt+1 − gnt+1 − wt+1+(1− ψt+1)

gvt+1qt+1

]+ jnt (19)

The moment conditions estimated are those obtained under rational expectationsi.e., E(Zt ⊗ jt) = 0 where Zt is the vector of instruments. The instrument set includes8 lags of the key variables – the hiring rate ( htnt ) and the investment rate (

itkt ) for both

equations; the rate of growth of output per unit of capital ( ftkt ) and the depreciation rate(δt) for equation (18); and the labor share ( wtntft ) and rate of separation (ψt) for equation(19). I report the J-statistic χ2 test of the over-identifying restrictions.

4.2 Estimation Results

Table 1 presents GMM estimates of equations (18) and (19).

Table 1

Consider panel a. Row (a) estimates all eleven parameters. Eight of these are notprecisely estimated, but suggest a quadratic g function with linear interactions, providefor a very reasonable estimate of the production function, and place the most weightin hiring costs on those associated with the h1t gross flows from non-employment. TheJ-statistic result does not reject the null hypothesis.

Row (b) restricts four of the parameters of the costs function to the point estimatespresented in row (a), yielding a quadratic function (η1 = η2 = 2) with linear inter-actions (η31 = η32 = 1). Here the seven free parameters are precisely estimated, theestimate of α is around the conventional estimate of 0.66, and the J-statistic again has ahigh p-value. The point estimates are close to those of the unrestricted row (a).

Row (c) takes up the same specification as row (b) but ignores job to job flows, i.e.,sets λ2 = e32 = 0 and h2t = ψ

2t = 0. This allows for the use of a longer data sample

– 1976:1-2016:4, with 168 quarterly observations. It, too, yields a J-statistic with a highp-value, and is, for the most part, precisely estimated. Evidently the estimates are notthe same as those of row (b), but there is considerable affinity.

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The three rows yield similar results in terms of the implied costs reported below.The main take-aways from these estimates are: the costs function features quadraticcosts with linear interactions; the latter feature negative coefficients (e31, e32 < 0), im-plying complementarity between hiring and investment; and the bigger weight of re-cruitment costs is assigned to actual hires from non-employment, i.e. λ1 at around 0.65.This is consistent with the results in Yashiv (2000) with Israeli data.

Now consider panel b relating to the standard specifications in the literature. Row(a) follows the standard Tobin’s q for capital model. It has quadratic investment costs,with no role for labor. There is no rejection of the model, but this specification implieshigh marginal investment costs, discussed below. This is reminiscent of the results inmuch of the literature on Tobin’q models for investment

Row (b) posits the same “standard Tobin’s q model” but this time for labor, ignoringcapital. Most parameters are imprecisely estimated and the J statistic rejects the null.

Row (c) reports the results of the standard (Pissarides-type) search and matchingmodel formulation with linear vacancy costs and no other arguments. The J statisticimplies rejection and the estimates imply high total costs, as discussed below.

Hence standard specifications are rejected or deemed implausible. In the next sec-tions, I take row (b) in panel a as the preferred estimates.

4.3 The Estimated Frictions and Asset Values

Table 2 shows the mean and volatility of the estimated costs series derived from theGMM estimates. These also reflect asset values, as delineated in the table.

Table 2

Total costs. Total costs out of GDP ( gtft ) are estimated to be 3.2-3.3% across rows (a)-(c)of panel a with little variation. In terms of panel b, the two Tobin’s q specifications withone factor only indicate 1.1% for the capital case and 2.5% for the labor case; jointlythis is somewhat higher than the panel a estimates. The standard search and matchingmodel estimates, however, imply more than double the costs, 6.8% of GDP, with a bigincrease in their volatility.

Marginal investment costs. These are expressed in terms of the percentage out of

the marginal capital unit price,gitpIt

, and are equal to the ratio of capital values Q̃Kt

QPt. The

results of rows (a) – (c) in panel a point to 2.4%-3.4%, i.e. for every dollar spent on themarginal unit of capital, these costs add 2.4–3.4 cents. These results correspond to thosepapers in the investment q-literature which reported low costs. The one relevant resultin panel b, row (a), Tobin’s q for capital, yields a much larger estimate, 7.5%, reflectinga long-running problem with this widely-used specification.

Marginal hiring costs. These are expressed in terms equivalent to quarterly wages,

gvtqtwt , and are equal to the tax adjusted labor value to wage ratio,

QNt(1−τt)

wt . The results ofrows (a) – (c) in panel a point to the equivalent of 50%-65% of quarterly wages, or theequivalent of 6.4 to 8.4 weeks of wages, for marginal costs.

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The conclusions from this discussion are that hiring and investment costs in rows(a)-(c) of panel a are very moderate. Hence, the analysis below does not rely on exces-sive or implausible costs, an issue that plagued the relevant literatures for decades.

4.4 What Do We Learn?

The afore-going analysis has shown that the model fits U.S. data well, in a periodof over four decades, including the decade of the Great Recession and its aftermath.Prevalent models, in the Tobin’s q and search and matching traditions, do not fit, asthey ignore cross effects between investment and hiring frictions. The interaction esti-mates imply complementarity in investment and hiring activities. The fit relies on verymoderate estimates of the frictions. Time series for the shadow asset values are derivedand used extensively below.

One finding relates to a recurrent question in research on business cycles with laborfrictions (see, for example, Christiano, Trabandt, and Walentin (2011 pp. 2038-2039),Christiano, Eichenbaum, and Trabandt (2016, p.1547), and Faccini and Yashiv (2018,Sections 2 and 5.3)) – what is the share of vacancy costs vs post-match costs (training,for example) in firms hiring costs? The importance of this question stems from thefact that vacancy costs relate to labor market conditions, while training costs relate tointernal firm conditions. The results here indicate that the latter are by far dominant:λ1+ λ2 is estimated at about 0.85, so vacancy costs are only about 15% of total hiringcosts. I will present further analysis of the role of labor market conditions by looking atthe effects of job filling rates (q1t , q

2t ).

5 Optimal Firm Behavior

Given the afore-going estimates, I explicitly formulate the firms’ optimal investmentand vacancies decisions.

The decision rules implied by equations (18) and (19) are as follows, using the pre-ferred parameter estimates (Appendix D shows the full derivation).

Optimal investment is given by:

itkt=

1e1e2Λ2t −Ω2t

[e2Λ2t

(Q̃Kt

(1− τt) ftkt

)− qtΩt

QNt(1− τt) ftnt

](20)

where:

Λt ≡ (1− λ1 − λ2) + λ1q1t + λ2q2t > 0Ωt ≡ e31q1t + e32q2t < 0

Equation (20) implies that the investment rate is unmabiguously a positive functionof asset values and of the corporate tax rate. The intuition for the former result, is thatas the present value of the marginal investment unit or of the marginal hire rise, thefirm invests more. The latter result is explained by the fact that corporate taxes are

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reduced by costs being expensed, so, ceteris paribus, without changes in future rates,the current tax rate rise gives an incentive to invest more now.7

The effects of the two job filling rates (q1t , q2t ), which reflect two sets of labor mar-

ket conditions,8 are ambiguous: on the one hand they operate to raise investment andvacancy rates as the expected present value on the RHS of equation (20) rises. On the

other hand the operate to lower them, as costs on hiring flows ( q1t vtnt ,

q2t vtnt ) rise, expressed

through the e1e2Λ2t term.Optimal vacancy creation is given by:

vtnt=

1e1e2Λ2t −Ω2t

[e1qt

QNt(1− τt) ftnt

−ΩtQ̃Kt

(1− τt) ftkt

](21)

Likewise, the vacancy rate is unmabiguously a positive function of the asset valuesand of the corporate tax rate, for reasons similar to the ones presented above.

Table 3 quantifies these relations by presenting means and standard deviations ofthe elasticities of investment and vacancy rates with respect to their determinants.

Table 3

Tax-adjusted labor asset values ( QNt

(1−τt) ftnt) have stronger effects than net capital val-

ues ( Q̃Kt

(1−τt) ftkt) on both decision variables – elasticities of 0.78 and 0.90 as opposed to 0.22

and 0.10 with respect to the investment rate and the vacancy rate, respectively.The effect of the job filling rate from non-employment (q1t ) is positive on average

with respect to investment and negative with respect to vacancies, as the present valueeffect dominates the costs effect for the former and is dominated for the latter. The ratefrom other employment (q2t ) has a positive effect on both.

The current corporate tax rate has a positive effect, ceteris paribus, which is muchstronger than that of the asset value, of which it is a part. Investment-tax elasticity ismuch higher than vacancy-tax elasticity.

6 The Cyclical Behavior of Hiring, Investment, and Asset Val-ues

In this section two main questions are examined: what are the cyclical patterns of firmsbehavior in terms of investing in inputs (capital and labor), of asset values, and of jobfilling rates, which embody labor market conditions? How does the sensitivity of thefirms’ decisions to asset values and to labor market conditions change over the cycle? Inpart of the analysis I use three relevant shock series, which generate cyclical behavior.

7Note, too, that τt includes χt the investment tax credit, and Dt the present discounted value of capitaldepreciation allowances,

8The rate q1t pertains to worker flows from non-employment while the rate q2t pertains to worker flows

from other employment.

13

6.1 The Shock Series

Asset values are the expected present values of future marginal productivites, adjustedfor separation or deprecation, wages, and taxes. The natural drivers of these productiv-ities are technology-related shocks. I explore three such shocks for the aggregate U.S.economy, taken from the authors of the following papers, for the period 1994-2016.

The TFP shock series is taken from an online data base described in Fernald (2014).In terms of equation (2) above, this is utilization-adjusted ε ft .

The unanticipated IST shock series are taken from Ben Zeev and Khan (2015). BenZeev (2018) shows in his equations (2) and (3) the relevant stochastic specification, akinto equation (7) above.

The matching shocks series was generated by Furlnaetto and Groshenny (2016 a,b).They derive the series within a medium-scale DSGE model. Their model features aCobb Douglas matching function, with a matching technology shock process akin toequation (5) above.

6.2 Methodology

I study cyclical behavior in two stages: in the first, I use local projections (LP) methodsto analyze the IRFs of all variables in response to each shock. In the second, I study thelinear relations between the various variables on the one hand and GDP on the otherhand, conditional on the local projections analysis. The analysis is based on Jordà (2005)and Daly, Fernald, Jordà, and Nechio (2018).

The local projection is given by the equation:

st+h = csih + ξ itλsih + RtΓ

sih + e

st+h (22)

where:

st+h ∈{

ft+h, x1,t+h...xJ,t+h}

xit ∈{

itkt

,vtnt

, q1t , q2t ,

h1tnt

,h2tnt

,QNt

ftnt

,QPt

ftkt

,Q̃Kt

ftkt

,Qtft

, ηt

}

ηt ∈

∂

itkt

∂PKt

PKtitkt

,∂

itkt

∂PNt

PNtitkt

,∂

itkt

∂q1t

q1titkt

,∂

itkt

∂q2t

q2titkt

,

∂vtnt

∂PKt

PKtvtnt

,∂

vtnt

∂PNt

PNtvtnt

,∂

vtnt

∂q1t

q1tvtnt

,∂

vtnt

∂q2t

q2tvtnt

PKt ≡Q̃Kt

(1− τt) ftkt, PNt ≡

QNt(1− τt) ftnt

ξ it ∈{

εft , ε

It , ε

µt

}Equation (22) can be understood as follows:On the LHS st+h is a predicted variable at horizon h; I examine 19 such variables

– GDP ( ft), the decision variables ( itkt ,vtnt ), the job filling rates (q

1t , q

2t ), the hiring rates

( h1t

nt ,h2tnt ), four asset values (

QNtftnt

, QPt

ftkt

, Q̃Kt

ftkt

, Qtft ) and eight elasticities of the decisions variables

14

(ηt). All variables are logged and HP filtered. The forecasting horizon is denoted h andset to 15 quarters.

On the RHS each regression has a constant (csih) and an error term (est+h). The afore-

cited shocks are denoted ξ it; each regression features one of the three shocks delineatedabove. Rt is a vector of control variables. The estimated coefficients are λsih and Γ

sih ,

respectively.I tried alternative formulations for Rt which have yielded similar results. The re-

sults below are based on estimation of equation (22), using for controls two lags of the

shock and one lag of the asset values, QNt−1

ft−1nt−1

, QPt−1

ft−1kt−1

, Q̃Kt−1

ft−1kt−1

. The latter essentially capture the

information known to the firm at the time of its decision on hiring and investment.I then use the results of estimation for the following cyclical analysis. Consider first

an OLS regression of the form:

xjt = aj + bj ft + ejt (23)

The estimated coefficient bj is an indicator of the cyclicality of variable xjt with re-spect to GDP ft. This is an estimate unconditioned by shocks, and does not take intoaccount any dynamics. Daly et al (2018) suggest to use the afore-going LP estimatesof λsih to estimate a shock-conditional bj using Classical Minimum Distance (CMD) asfollows:

b̂ij = (λ̂f ′

i Mλ̂fi )−1(λ̂

f ′

i Mλ̂ji) (24)

with variance of b̂ij given by:

vij = (λ̂f ′

i Mλ̂fi )−1 (25)

and where:

M =(

Ωji)−1

(26)

λ̂fi is the h × 1 vector of LP coefficients of shock i on f , λ̂

ji is the h × 1 vector of LP

coefficients of shock i on variable j, and Ωji a diagonal matrix with variance estimates

of λ̂ji from equation (22).The intuition here is to find the b̂ij which makes the IRF of f (to a shock i) as close

as possible to the IRF of variable j (to a shock i).

6.3 Cyclical Co-Movement

Table 4 reports the point estimates and standard errors of bj from equation (23) and bijfrom equation (24). A positive (negative) coefficient indicates pro-(counter-) cyclicality.

Table 4

15

Without conditioning on any shock, the cyclical patterns are as follows: the deci-sion variables are pro-cyclical; job filling rates are counter-cyclical. Hiring rates are theproduct of the pro-cyclical vacancy rate and the counter-cyclical job filling rates. Hiringrates from non-employment are counter-cyclical, while hiring rates from other employ-ment are pro-cyclical. The part of capital asset values reflecting the price of investment

( QPt

ftkt

) is counter-cyclical. Net capital asset values ( Q̃Kt

(1−τt) ftkt, net of the price of capital) and

labor asset values ( QNt

(1−τt) ftnt) are pro-cyclical.

The elasticity of investment and vacancy rates with respect to net capital asset val-ues is pro-cyclical, while the elasticity related to the value of labor is counter-cyclical.These elasticities are correlated −1, by the model’s equations. In recessions, firms’ de-cisions are more tightly related to the value of labor and less tightly to the adjustmentcost part of the value of capital.

As to job filling rates, the effect of q1t is pro-cyclical for both investment and vacan-cies. Noting, from Table 3, that the former effect is positive and the latter negative,this means that in recessions labor market conditions pertaining to flows from non-employment have relatively weak effects on investment, and stronger effects on va-cancies. The effect of q2t is counter-cyclical for investment and a-cyclical for vacancies.Noting, from Table 3, that both effects are positive, this means that in recessions labormarket conditions pertaining to flows from other employment have relatively strongeffects on investment.

The elasticity with respect to the corporate tax is counter-cyclical, i.e., the effects oftaxes increase in recessions.

There are two notable findings here: (i) the labor value elasticity rises in recessions,i.e., labor values become more important in recessions; and, as noted above, (ii) themean labor value elasticity is much higher than the one for capital asset values, forboth hiring and investment. These results point to the importance of labor values evenfor capital investment decisions, all the more so in recessions.

Conditioning on the shocks, the results stay qualitatively the same, and in mostcases are weaker quantitatively. Some point estimates are imprecisely estimated, possi-bly because of the relatively low number of degrees of freedom. This means that acrossall variables (including asset values), the patterns of co-movement with GDP, followingeach of the three technology-related shocks, are the same as the co-movement with GDPseen in the data independently of shocks. This may be suggestive of the fundamentalrole played by these shocks in generating the data.

7 Asset Values as Instruments and Predictors

Asset values encapsulate the firms expectations about the future. Using the modelwhich features forward-looking optimal firm behavior, they thus theoretically have pre-dictive power for cyclical fluctuations. Is this so empirically, and how can it be used?

Stock and Watson (2003) offer a review and analysis of the role of asset prices in

16

forecasting output and inflation. They depart from the observation that

“Because asset prices are forward-looking, they constitute a class of po-tentially useful predictors of inflation and output growth. The premise thatinterest rates and asset prices contain useful information about future eco-nomic developments embodies foundational concepts of macroeconomics.”(p.788)

But their review of empirical analyses leads to some negative conclusions, includingthe statement whereby:

“the variables with the clearest theoretical justification for use as predic-tors often have scant empirical predictive content...” (p.801)

In terms of methodology, recent contributions, such as Stock and Waston (2016),emphasize FAVAR and SVAR methods in Macroeconomics. They note the use of dy-namic factor models in forecasting, with relatively more success than that implied bythe quote above. In the current context, labor and capital asset values are potential pre-dictors. The empirical questions are: how useful are they, and can they offer marginalpredictive value beyond predictive factors?. In particular, can they serve as instrumentswhen estimating dynamic causal effects?

The following forecasting regression offers some evidence.

ln yt+h = α+ β0 ln yt + β1 ln yt−1 + γ′Ft + et+h (27)

where ln yt+h is the detrended log of an activity variable at horizon h, and Ft is a vectorof forecasting variables; α, β0, β1 and the vector γ are parameters to be estimated.

Table 5 shows six such regressions, each for three different horizons. The predictedvariable is one of the following: the log of GDP, the log of vacancy rates, and the logof investment rates, once in first differences and once detrended with a linear-quadraictrend. The predictors are time t and t − 1 values of the dependent variable (not re-ported), a real activity factor based on the FRED data set, as computed by McCrackenand Ng (2016), which has been shown to have predictive power for real activity in the

U.S. economy,9 and the two asset values QKt

ftkt

and QNt

ftnt

in logs

Table 5

The table shows reasonable forecasts, which usually deteriorate as the horizon getslonger.10 The McCracken and Ng activity factor is often significant at short horizons but

9McCracken and Ng (2016) offer detailed expalnations and present plots and statistics. Essentially thefactor used here is their Factor 1, interpreted as a real activity/employment factor. The factor has beencomputed on FRED data with 134 series for the U.S. economy.

10Though for first differenced variables, forecasts are better at 8 quarters than at 4 quarters.

17

not so at longer ones. Capital asset values, essentially the Tobin’s Q of this paper, havesignificant predictive power only in the minority of cases and usually at the longerhorizons. By contrast, labor values often have predictive power, over and above thepredictive power of the lagged dependent variable and the McCracken and Ng factor.This is especially true for investment and vacancy creation decisions.

8 Summary and Conclusions

The following summarizes the key findings.The decision variables – investment and vacancies - react positively to both capital

and labor asset values. Standard specifications, which do not feature cross effects be-tween capital (labor) value and vacancy creation (investment) are omitting importantvariables and do not fit the data. The effects of labor asset values are stronger on bothhiring and investment and are stronger in recessions. This cross-effect of labor valuesneeds to be emphasized as it is not taken into account in the literature.

Two effects of variables, subsumed in asset values, on the decision variables, arenoteworthy:

(i) Job filling rates, reflecting labor market conditions, affect both expected presentvalues and costs (of hiring and investment) and thus have an ambiguous effect ex-ante. But the empirical analysis shows that they have a positive effect on investment incapital.

(ii) The effects of corporate taxation operate to incentivize both investment and va-cancies and are stronger in recessions.

Labor asset values, and to a lesser extent capital values, may be used as predictorsand as instruments given that they encapsulate firms expectations about the future.

In sum: asset values, while not observed, can be estimated. They play a role in ex-plaining, pricing, and forecasting cyclical aggregate U.S. investment, hiring and output.

18

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22

Tabl

e1

GM

MEs

tim

atio

nR

esul

ts

a.Pr

efer

red

Spec

ifica

tion

ssp

ecifi

cati

onη

1η

2η

31η

32e 1

e 2e 3

1e 3

2λ

1λ

2α

J(χ

2 )a

allf

ree

1.99

1.98

1.02

1.00

757

−12

−8

0.64

0.20

0.67

62(1

.69)

(0.8

6)(1

.06)

(1.4

5)(3

97)

(10)

(60)

(63)

(0.1

0)(0

.14)

(0.0

2)(0

.44)

bη

sre

stri

cted

22

11

818

−11

−10

0.65

0.21

0.66

65−

−−

−(8)

(2)

(2)

(3)

(0.0

6)(0

.07)

(0.0

1)(0

.49)

cη

sre

stri

cted

22

1−

802.

4−

12−

0.34

−0.

6481

e 32=

0;λ

2=

0−

−−

−(1

5)(1

.2)

(4)

(0.2

4)(0

.01)

(0.1

1)19

76-2

016

b.St

anda

rdSp

ecifi

cati

ons

spec

ifica

tion

η1

η2

η31

η32

e 1e 2

e 31

e 32

λ1

λ2

αJ(χ

2 )a

Tobi

n’s

Qfo

rK

2−

−−

31.7

00

0−

−0.

7047

−(1

2.9)

−−

−(0

.00)

(0.1

4)

bTo

bin’

sQ

for

N−

2−

−−

2.72

00

0.44

0.11

0.67

58−−

(6.7

6)−

−(0

.47)

(0.8

6)(0

.03)

(0.0

01)

cSt

anda

rdM

atch

ing

Mod

el−

1−

−−

0.34

00

00

0.65

49(0

.08)

−−

−−

(0.0

1)(0

.02)

Not

es:

1.Th

eta

bles

repo

rtG

MM

poin

tes

tim

ates

wit

hst

anda

rder

rors

inpa

rent

hese

s.Si

gnifi

cant

esti

mat

esar

ebo

lded

.Th

eJ-

stat

isti

cis

repo

rted

wit

hp

valu

ein

pare

nthe

ses.

2.Th

esa

mpl

epe

riod

is19

94:1

–20

16:4

unle

ssno

ted

othe

rwis

e.In

allr

ows,

exce

pt(a

)ofP

anel

a,th

eη

sar

ere

stri

cted

asin

dica

ted.

23

Table 2Moments of the Estimated Frictions and of the Asset Values

a. Preferred Specifications

specification gtft

gi,tftktpItftkt

=Q̃KtQPt

gv,tqtwt =

QNt(1−τt)

wt

a all free mean 0.032 0.027 0.50std. 0.004 0.013 0.05

b ηs restricted mean 0.033 0.024 0.551994-2016 sample std. 0.004 0.014 0.06

c ηs restricted; e32 = λ2 = 0 mean 0.032 0.034 0.651976-2016 sample std 0.015 0.024 0.34

b. Standard Specifications

specification gtft

gi,tftktPItftkt

=Q̃Kt

(1−τt)pIt

gv,tqtwt =

QNt(1−τt)

wt

a Tobin’s Q for K mean 0.011 0.075 −std. 0.002 0.008 −

b Tobin’s Q for N mean 0.025 − 0.47std. 0.005 − 0.08

c Standard Matching Model mean 0.068 − 0.64std 0.011 − 0.10

Notes:The tables report means and standard deviations of time series of the variables listed

in the top rows. The series were generated using the corresponding estimates fromTable 1.

24

Table 3Elasticities of the Decision Variables ( itkt ,

vtnt )

with Respect to Their Determinants

elasticity itktvtnt

mean (std) mean (std)

asset value effects ∂·∂

Q̃Kt(1−τt)

ftkt

Q̃Kt(1−τt)

ftkt· 0.22 0.10

(0.11) (0.05)

∂·∂

QNt(1−τt)

ftnt

QNt(1−τt)

ftnt· 0.78 0.90

(0.11) (0.05)

job-filling rates effects ∂·∂q1t

q1t· 0.10 −0.58

(0.07) (0.07)

∂·∂q2t

q2t· 0.60 0.39

(0.13) (0.07)

corporate tax effects ∂·∂τtτt· 2.24 1.31

(0.23) (0.11)

Notes:1. Point · indicates itkt ,

vtnt

2. The table uses estimation results of row (b) in Table 1a.3. See Appendix D for definitions and derivations of the elasticities.

25

Table 4Cyclicality of the Key Variables

a. Decisions Variables, Job Filling Rates, Hiring Rates; Asset Values

static dynamicTFP shock IST shock Matching shock

ik 2.16

∗∗∗ 0.80∗ 1.61∗∗∗ 0.99∗∗

(0.14) (0.42) (0.32) (0.37)vtnt 4.33

∗∗∗ 3.92∗∗∗ 1.50∗∗ 2.59∗∗∗

(0.24) (1.11) (0.58) (0.85)

q1t −4.61∗∗∗ −3.82∗∗∗ −2.11∗∗ −2.86∗∗(0.26) (0.83) (0.76) (1.00)

q2t −2.84∗∗∗ −3.47∗∗ −0.78NS −1.15NS(0.26) (1.26) (0.55) (0.95)

h1n −0.28∗∗ −0.02NS −0.46NS −0.75NS

(0.12) (0.53) (0.31) (0.45)h2n 1.50

∗∗∗ 0.80NS 0.94∗ 1.16∗∗

(0.17) (1.03) (0.53) (0.46)

static dynamicTFP shock IST shock Matching shock

QNtftnt

1.21∗∗∗ 2.32NS −0.32NS 0.12NS

(0.28) (1.82) (0.58) (0.90)Q̃Kt

ftkt

20.87∗∗∗ −5.02NS 14.33∗∗ 13.28NS

(3.28) (11.35) (6.19) (12.36)QPt

ftkt

−0.80∗∗∗ −0.46NS −0.53∗ −0.45NS

(0.09) (0.32) (0.26) (0.34)Qtft −0.36

∗∗∗ −0.32NS −0.39∗ −0.29NS(0.10) (0.31) (0.22) (0.32)

26

b. Elasticitiesstatic dynamic

TFP shock IST shock Matching shock

∂itkt

∂Q̃Kt

(1−τt)ftkt

Q̃Kt(1−τt)

ftkt

itkt

2.67∗∗∗ 0.59NS 2.23∗∗∗ 1.64∗∗

(0.26) (0.90) (0.74) (0.76)

∂itkt

∂QNt

(1−τt)ftnt

QNt(1−τt)

ftnt

itkt

−2.67∗∗∗ −0.59NS −2.23∗∗∗ −1.64∗∗

(0.26) (0.90) (0.74) (0.76)∂

itkt

∂q1t

q1titkt

1.16∗∗∗ 1.35∗∗ 0.29NS 0.40NS

(0.11) (0.61) (0.21) (0.41)∂

itkt

∂q2t

q2titkt

−0.85∗∗∗ −0.89NS −0.28NS −0.37NS

(0.14) (0.56) (0.41) (0.46)∂

itkt

∂τtτtitkt

−7.00∗∗∗ −4.68∗∗∗ −3.69∗∗ −5.21∗∗∗

(0.34) (0.94) (1.42) (1.45)

static dynamicTFP shock IST shock Matching shock

∂vtnt

∂Q̃Kt

(1−τt)ftkt

Q̃Kt(1−τt)

ftkt

vtnt

1.29∗∗∗ 0.20NS 1.19∗∗∗ 0.87∗∗

(0.14) (0.48) (0.38) (0.39)

∂vtnt

∂QNt

(1−τt)ftnt

QNt(1−τt)

ftnt

vtnt

−1.29∗∗∗ −0.20NS −1.19∗∗∗ −0.87∗∗

(0.14) (0.48) (0.38) (0.39)∂

vtnt

∂q1t

q1tvtnt

1.84∗∗∗ 1.62∗∗∗ 0.74∗∗ 1.01∗∗

(0.12) (0.45) (0.28) (0.44)∂

vtnt

∂q2t

q2tvtnt

0.00NS −0.60NS 0.26NS 0.16NS

(0.07) (0.34) (0.17) (0.22)∂

vtnt

∂τtτtvtnt

−2.83∗∗∗ −2.60∗∗ −0.79NS −1.97∗∗∗

(0.21) (1.06) (0.61) (0.66)

27

Notes:1. ∗ ∗ ∗/ ∗ ∗/ ∗ /NS denote significance at 1%, 5%, 10% or not significant, respec-

tively.2. Static columns report point estimates and standard errors in parentheses for bj in

equation (23): xjt = aj + bj ft + ejt.3. Dynamic columns report point estimates and standard errors in parentheses for

bij in equation (24):b̂ij = (λ̂f ′

i Mλ̂fi )−1(λ̂

f ′

i Mλ̂ji).

4. All variables are logged and HP-filtered.

28

Table 5Forecasting Regressions

GDP M ln ft+h ln ft+h detrendedh 1 4 8 1 4 8

MN factor 1 −0.04∗∗∗ −0.004 −0.01∗ −0.03∗∗∗ −0.02∗∗ 0.02(0.00) (0.69) (0.07) (0.01) (0.01) (0.02)

ln QKt

ftkt

0.07 −0.01 0.05∗∗ 0.04 −0.34∗∗ −0.76∗∗∗

(0.11) (0.90) (0.04) (0.05) (0.15) (0.18)

ln QNt

ftnt

−0.04∗∗∗ −0.01 −0.09∗∗ −0.02 0.09 0.04

(0.01) (0.82) (0.02) (0.02) (0.07) (0.06)

R2 0.50 −0.06 0.12 0.99 0.96 0.93

Vacancy Rate M ln( v

n

)t+h ln

( vn

)t+hdetrended

h 1 4 8 1 4 8MN factor 1 −0.11∗∗∗ 0.02 0.01 −0.11∗∗ −0.13∗∗ 0.004

(0.00) (0.18) (0.50) (0.02) (0.05) (0.067)

ln QKt

ftkt

0.42∗∗ 0.06 0.19 0.19 −0.18 −0.87

(0.03) (0.78) (0.12) (0.17) (0.47) (0.69)

ln QNt

ftnt

−0.21∗∗ −0.08 −0.37∗∗∗ −0.005 0.98∗∗∗ 0.89∗∗∗

(0.02) (0.49) (0.01) (0.09) (0.35) (0.30)

R2 0.37 0.02 0.22 0.95 0.76 0.57

Investment Rate M ln( i

k

)t+h ln

( ik

)t+hdetrended

h 1 4 8 1 4 8MN factor 1 −0.03∗∗∗ −0.02∗ −0.01 −0.03∗∗∗ −0.06∗∗ 0.03

(0.00) (0.07) (0.24) (0.01) (0.02) (0.03)

ln QKt

ftkt

0.08∗ 0.10 0.15∗ 0.06 0.04 −0.62∗∗

(0.10) (0.42) (0.06) (0.04) (0.19) (0.30)

ln QNt

ftnt

0.04 0.04 −0.16∗∗ 0.08∗∗∗ 0.41∗∗∗ 0.54∗∗∗

(0.15) (0.50) (0.02) (0.02) (0.09) (0.11)

R2 0.53 0.03 0.19 0.97 0.84 0.67

29

Notes:1. HAC standard errors were computed using the Bartlett kernel and Newey-West

with fixed bandwidth = 4.0000.2. P-values are reported in parentheses. . ∗ ∗ ∗/ ∗ ∗/ ∗ /NS denote significance at

1%, 5%, 10%, respectively.3. Regressions included the dependent variable at lags 0 and −1, which are not

reported.4. Detrended regressions include a linear quadratic time trend.

30

9 Appendix A: Derivation of Aggregate Q

I derive a general Qt which is a function of QKt and QNt . The following derivations are

based on Hayashi (1982).

9.1 Firm Profits, Cash Flows, and Values

Define firm profits πt :

πt = [ f (kt, nt)− g (it, kt, vt, nt)]− wtnt . (28)Define cash flow payments to firm owners as c ft equal to profits after tax minus pur-chases of investment goods plus investment tax credits and depreciation allowances fornew investment goods:

c ft = (1− τt)πt − (1− χt − τtDt) p̃It it (29)= (1− τt)

(f (kt, nt)− g (it, kt, vt, nt)− wtnt − pIt it

)The representative firm’s value in period t, Qt, is defined as follows:

Qt = Et[ρt+1 (Qt+1 + c ft+1)

]. (30)

This can be split into capital ϑkt and labor values ϑnt as follows:

Qt = ϑkt + ϑnt = Et

[ρt+1

(ϑkt+1 + c f

kt+1

)]+ Et

[ρt+1 (ϑ

nt+1 + c f

nt+1)

], (31)

Using the constant returns-to-scale properties of the production function f and of thecost function, g, and equation (29), decompose the stream of maximized cash flow pay-ments as follows:

c ft = (1− τt)(

fkt kt + fnt nt − wtnt − pIt it − gkt kt − git it − gvt vt − gnt nt)

= (1− τt)[(

fkt kt − pIt it − gkt kt − git it)+ ( fnt nt − wtnt − gvt vt − gnt nt)

]≡ c f kt + c f nt . (32)

9.2 Optimality Equations and Asset Values

Multiply throughout the FOC with respect to investment by it, the FOC with respectto capital by kt+1, the FOC with respect to vacancies by vt, and the one with respect toemployment by nt+1 to get

(1− τt)(

pIt + git)

it = itQKt (33)

(1− τt) gvt vt = vtqtQNt (34)kt+1QKt = kt+1Et

{ρt+1[(1− τt+1)

(fkt+1 − gkt+1

)+ (1− δt+1)QKt+1]

}(35)

nt+1QNt = nt+1Et{

ρt+1

[(1− τt+1) ( fnt+1 − gnt+1 − wt+1) + (1− ψt+1)QNt+1

]}(36)

31

9.2.1 Capital

Insert the law of motion for capital into equation (33), roll forward all expressions oneperiod, multiply both sides by ρt+1 and take conditional expectations on both sides:

Et[ρt+1 (1− τt+1)

(pIt+1 + git+1

)it+1

]= Et

{ρt+1 [kt+2 − (1− δt+1)kt+1]QKt+1

}. (37)

Rearranging:

Et[ρt+1(1− δt+1)

(kt+1QKt+1

)]= Et

{ρt+1

[(kt+2QKt+1 − (1− τt+1)

(pIt+1 + git+1

)it+1

)]}(38)

Combining equations (32), (33), (35), and (38) yields:

kt+1QKt = Et(

ρt+1

(c f kt+1 + kt+2Q

Kt+1

))(39)

Rearranging:Et(

ρt+1c fkt+1

)= kt+1QKt − Et

(ρt+1kt+2Q

Kt+1

). (40)

It follows from the definition of the firm’s market value in equation (31) that

ϑkt − Et(

ρt+1ϑkt+1

)= Et

(ρt+1c f

kt+1

). (41)

Thus,ϑkt − Et

(ρt+1ϑ

kt+1

)= kt+1QKt − Et

(ρt+1kt+2Q

Kt+1

), (42)

which impliesϑkt = kt+1Q

Kt . (43)

9.2.2 Labor

Derive a similar expression for the case of labor. Inserting the law of motion for laborinto equation (34), multiplying both sides by ρt+1, rolling forward all expressions byone period, taking conditional expectations, and combining with equations (32) and(36) get

Et(ρt+1c f

nt+1)= nt+1QNt − Et

(ρt+1nt+2Q

Nt+1

). (44)

The definition of the firm’s value in equation (31) implies that

ϑnt − Et(ρt+1ϑ

nt+1)= Et

(ρt+1c f

nt+1)

. (45)

Thus,ϑnt − Et

(ρt+1ϑ

nt+1)= nt+1QNt − Et

(ρt+1nt+2Q

Nt+1

). (46)

This implies the following expression for the asset value of employment:

ϑnt = nt+1QNt . (47)

32

9.2.3 Aggregation

Hence, the total value of a firm, Qt, equals:

Qt = ϑkt + ϑnt = kt+1Q

Kt + nt+1Q

Nt . (48)

where the components are defined in equations (35) and (36), respectively.

33

10 Appendix B: The Data

10.1 Sample Statistics

Table B1 presents key sample statistics.

Table B1

Descriptive Sample StatisticsQuarterly, U.S. data

a. 1976:1-2016:4 (n = 168)

Variable fk τik δ

wnf

h1n ψ

1 ρ

Mean 0.14 0.37 0.024 0.02 0.62 0.126 0.125 0.99Standard Deviation 0.01 0.05 0.003 0.003 0.03 0.010 0.011 0.005

b. 1994:1-2016:4 (n = 92)Variable fk τ

ik δ

wnf

hn =

h1+h2n ψ = ψ

1 + ψ2 ρ

Mean 0.15 0.34 0.026 0.02 0.61 0.177 0.176 0.99Standard Deviation 0.01 0.005 0.002 0.002 0.03 0.012 0.012 0.004

10.2 Sources and Definitions

variable definition and sourceGDP f gross value added of NFCB; NIPA accounts, table 1.14, line 41GDP deflator p f price per unit of gross value added of NFCB; NIPA table 1.15, line 1wage share wnf numerator: compensation of employees in NFCB; see note 7

discount factor ρ based on non-durable consumption growth; NIPA Table 2.3.3, see note 1employment n employment in nonfinancial corporate business sector; CPS, see note 2hiring h gross hires; CPS, see note 2separation rate ψ gross separations divided by employment; CPS; see note 3vacancies v adjusted Help Wanted Index; Conference Board; see note 4investment i gross investment in NFCB sector; BEA and Fed Flow of Funds; see note 5capital stock k stock of private nonresidential fixed assets in NFCB sector;

BEA and Fed Flow of Funds; see note 5depreciation δ depreciation of the capital stock; BEA and Fed Flow of Funds; see note 5price of capital goods pI real price of new capital goods; NIPA and U.S. tax foundation; see note 6

34

The longest sample period is 1976:1-2016:4; all data are quarterly.

Notes:1. The discount rate and the discount factorThe discount rate is based on a DSGE-type model with logarithmic utility U(ct) =

ln ct. Define the discount factor as ρt ≡ 11+rtIn this model:

U′(ct) = U′(ct+1) · (1+ rt) (49)Hence:

ρt =ct

ct+1(50)

where c is non-durable consumption (goods and services) and 5% of durable consump-tion.

2. EmploymentAs a measure of employment in the nonfinancial corporate business sector (n) I

take wage and salary workers in non-agricultural industries (series ID LNS12032187)less government workers (series ID LNS12032188), less self-employed workers (seriesID LNS12032192). All series originate from CPS databases. I do not subtract workersin private households (the unadjusted series ID LNU02032190) from the above due tolack of sufficient data on this variable.

3. Hiring and Separation RatesThe aggregate flow from non-employment – unemployment (U) and out of the labor

force (O) – to employment is to be denoted OE+UE and the separation rate ψt is rateof the flow in the opposite direction, EU+ EO. Worker flows within employment – i.e.,job to job flows – are to be denoted EE.

I denote h1

n =OE+UE

E andh2n =

EEE the rates of flows from non-employment and

from other employment, respectively. The total flows rate is hn =h1n +

h2n .

Separation rates are given by ψ = ψ1 + ψ2, ψ1 = EO+EUE and ψ2 = EEE =

h2n .

Employment dynamics now satisfies:

nt+1 = (1− ψ1t − ψ2t )nt + h1t + h2t (51)= (1− ψt)nt + ht, 0 ≤ ψt ≤ 1

h2t = ψ2t

To calculate hiring and separation rates for the whole economy I use the following:a. The h1t and ψ

1t flows. I compute the flows between E (employment), U (unemploy-

ment) and O (not-in-the-labor-force) that correspond to the E,U,O stocks published by

35

the CPS. The methodology of adjusting flows to stocks is taken from BLS, and is pre-sented in Frazis et al (2005).11The data till 1990:Q1 were kindly provided by Ofer Corn-feld. The data from 1990:Q2 onwards were taken from the CPS.12 Employment is thequarterly average of the original seasonally adjusted total employment series from BLS(LNS12000000).

b. The h2t and ψ2t flows. The data on EE, available only from 1994:Q1 onward,

were computed by multiplying the percentage of people moving from one employerto another using Fallick and Fleischman (2004)’s13data by the NSA population seriesLNU00000000, taken from the CPS, completing several missing observations and per-forming seasonal adjustment.

4. VacanciesI use the vacancies series based on the Conference Board Composite Help-Wanted

Index that takes into account both printed and web job advertisements, as computedby Barnichon (2012).14 The updated series is available at

https://sites.google.com/site/regisbarnichon/research/publications.This index was multiplied by a constant to adjust its mean to the mean of the JOLTS

vacancies series over the overlapping sample period (2001:Q1–2013:Q4). As this seriesis based on indices, in estimation I estimate a scaling parameter.

5. Investment, capital and depreciationQuarterly series for real investment flow it, real capital stock kt , and depreciation

rates δt were constructed using the following series:

• The quantity index for net stock of fixed assets in NFCB (FAA table 4.2, line 37,BEA) as well as the 2009 current-cost net stock of fixed assets (FAA table 4.1, line37, BEA).

• The chain-type quantity index for depreciation from FAA table 4.5, line 37. Thecurrent-cost depreciation estimates (and specifically the 2009 estimate) are givenin FAA table 4.4, line 37.

• Historic-cost quarterly investment in private non-residential fixed assets by NFCBsector, the Flow of Funds accounts, atabs files, series FA105013005).

The methodology is explained in Yashiv (2016).15

6. Real price of new capital goods

11Frazis, Harley J., Edwin L. Robison, Thomas D. Evans and Martha A. Duff, 2005. Estimating GrossFlows Consistent with Stocks in the CPS, Monthly Labor Review, September, 3-9.

12See http://www.bls.gov/cps/cps_flows.htm13Fallick and Fleischman, 2004. “Employer-to-Employer Flows in the U.S. Labor Market: The Complete

Picture of Gross Worker Flows,” FEDS #2004-34.14Barnichon, Regis, 2012. “Vacancy Posting, Job Separation and Unemployment Fluctuations,” Journal

of Economic Dynamics and Control 36, 315-330.15See Yashiv, Eran, 2016. “Capital Values and Job Values,” Review of Economic Dynamics 19, 190-209.

36

In order to compute the real price of new capital goods, pI , I use the price indicesfor output and for investment goods.

Investment in NFCB Inv consists of equipment Eq and structures St as well as in-tellectual property, which I do not include. I define the time-t price-indices for goodj = Eq, St as p̃jt. The data are taken from NIPA table 1.1.4, lines 10, 11.

I take from http://www.federalreserve.gov/econresdata/frbus/us-models-package.htmthe following tax -related rates:

a. The parameter τ – the statutory corporate income tax rate as reported by the U.S.Tax Foundation.

b. The investment tax credit on equipment and public utility structures, to be de-noted ITC.

c. The percentage of the cost of equipment that cannot be depreciated if the firmtakes the investment tax credit, denoted χ.

d. The present discounted value of capital depreciation allowances, denoted ZPDESt

and ZPDEEq.I then apply the following equations:

pEq = p̃Eq(1− τEq

)pSt = p̃St (1− τSt) ,

1− τSt =(1− τ ZPDESt

)1− τ

1− τEq = 1− ITC− τZPDEEq (1− χITC)

1− τ

Subsequently I compute their change between t− 1 and t (denoted by ∆pjt) :

∆pInvtpInvt−1

= ωt∆pEqtpEqt−1

+ (1−ωt)∆pSttpStt−1

where

ωt =

(nominal expenditure share of Eq in Inv)t−1+ (nominal expenditure share of Eq in Inv)t

2.

The weights ωt are calculated from the NIPA table 1.1.5, lines 9,11.I divide the series by the price index for output, p ft , to obtain the real price of new

capital goods, pI .As all of these prices are indices, in estimation I estimate a scaling parameter ea.7. Labor shareNIPA table 1.14, line 20 (compensation of employees in NFCB) divided by line 17 in

the same table (gross value added in NFCB).

37

11 Appendix C: GMM Estimation of the FOC

This Appendix elaborates on the GMM estimation discussed in Section 4.

11.1 The Cost Function and its Derivatives

g(·) =

e1η1( itkt )

η1

+ e2η2

[(1−λ1−λ2)vt+λ1q1t vt+λ2q2t vt

nt

]η2+ e31η31

(itkt

q1t vtnt

)η31+ e32η32

(itkt

q2t vtnt

)η32 f (zt, nt, kt). (52)

gitftkt

=

[e1( itkt )

η1−1

+e31(

q1t vtnt

)η31( itkt )

η31−1 + e32(

q2t vtnt

)η32( itkt )

η32−1

](53)

gvtftnt

=

e2[(1−λ1−λ2)vt+λ1q1t vt+λ2q2t vt

nt

]η2−1 [(1− λ1 − λ2) + λ1q1t + λ2q2t

]+e31q1t

(itkt

)η31 ( q1t vtnt

)η31−1+e32q2t

(itkt

)η32 ( q2t vtnt

)η32−1 (54)

gktftkt

= −[

e1(itkt)η1 + e31

(q1t vtnt

itkt

)η31+ e32

(q2t vtnt

itkt

)η32](55)

+(1− α)

e1η1( itkt )

η1

+ e2η2

[(1−λ1−λ2)vt+λ1q1t vt+λ2q2t vt

nt

]η2+ e31η31

(itkt

q1t vtnt

)η31+ e32η32

(itkt

q2vtnt

)η32

gntftnt

= −

e2 [ (1−λ1−λ2)vt+λ1q1t vt+λ2q2t vtnt ]η2+e31

(q1t vtnt

itkt

)η31+ e32

(q2t vtnt

itkt

)η32 (56)

+α

e1η1( itkt )

η1

+ e2η2

[(1−λ1−λ2)vt+λ1q1t vt+λ2q2t vt

nt

]η2+ e31η31

(itkt

q1t vtnt

)η31+ e32η32

(itkt

q2t vtnt

)η32

38

11.2 The Estimating Equations

11.2.1 The General Specifications

Replacing expected variables by actual ones and a rational expectations forecast error,the estimating equations are:

(1− τt)(

git + pIt

)= ρt+1 (1− τt+1)

[fkt+1 − gkt+1

+(1− δt+1)(git+1 + pIt+1)

]+ jkt (57)

I estimate this equation after dividing throughout by ftkt .

(1− τt)gvtqt= ρt+1 (1− τt+1)

[fnt+1 − gnt+1 − wt+1+(1− ψt+1)

gvt+1qt+1

]+ jnt (58)

I estimate this equation after dividing throughout by ftnt .As explained in the text, estimation pertains to α, e1, e2, e31, e32, η1, η2, η31, η32, λ1, λ2.

11.2.2 Tobin’s Q Approach

This approach ignores the other factor of production (i.e., assumes no adjustment costsfor it). For the investment in capital equation e2 = e31 = e32 = 0 and η1 = 2 and onlyequation (57) is estimated. For the vacancy creation equation e1 = e31 = e32 = 0 andη2 = 2 and only equation (58) is estimated.

11.2.3 The Standard Search and Matching Model

In this case e1 = e31 = e32 = 0, η2 = 1, λ1 = λ2 = 0 and there is only the hiring equationgiven by:

(1− τt)e2qt=

ρt+1 (1− τt+1) ft+1nt+1ftnt

α− wt+1ft+1nt+1

+ (1− ψt+1)e2

qt+1

+ jt (59)This is estimated for e2 and α.

Instruments The instrument set consists of 8 lags of the following variables – the hir-ing rate ( hn ) and the investment rate (

ik ) for both equations; the rate of growth of output

per unit of capital ( fk ) and the depreciation rate (δ) for the investment equation; and thelabor share ( wnf ) and rate of separation (ψ) for the vacancies equation.

39

12 Appendix D: Solving for the Decision Variables

12.1 Solution for the Investment Rate and Vacancy Rate

Use the FOC, the estimates of Table 1, and the derivatives of the cost function g inAppendix C to solve for the decision variables as follows:

QKtftkt

= (1− τt)([

e1(itkt) + e31

(q1t vtnt

)+ e32

(q2t vtnt

)]+

pItftkt

)(60)

itkt

=1e1

QKt

ftkt

(1− τt)− p

It

ftkt

−[

e31

(q1t vtnt

)+ e32

(q2t vtnt

)]

QNtftnt

=

(1− τt)

e2 vtnt [(1− λ1 − λ2) + λ1q1t + λ2q2t ]2+e31q1t

(itkt

)+ e32q2t

(itkt

) qt

(61)

vtnt

=1

e2[(1− λ1 − λ2) + λ1q1t + λ2q2t

]2 qt

QNtftnt

(1− τt)− e31q1t

(itkt

)− e32q2t

(itkt

)Denote:

Λt ≡ (1− λ1 − λ2) + λ1q1t + λ2q2t (62)Ωt = e31q1t + e32q

2t

Thus:

itkt=

1e1

QKt

ftkt

(1− τt)− p

It

ftkt

−Ωtvtnt

(63)

vtnt=

1e2Λ2t

qtQNt

ftnt

(1− τt)−Ωt

itkt

(64)Therefore:

40

itkt

=1e1

QKt

ftkt

(1− τt)− p

It

ftkt

−Ωt

1e2Λ2t qt

QNtftnt

(1− τt)−Ωt

itkt

=1

e1e2Λ2t −Ω2t

[e2Λ2t

(Q̃Kt

(1− τt) ftkt

)− qtΩt

QNt(1− τt) ftnt

](65)

where I have used

QKtftkt

= (1− τt)(

gitftkt

+pItftkt

)gitftkt

=QKt

(1− τt) ftkt− p

It

ftkt

=Q̃Kt

(1− τt) ftktSimilarly for vacancy creation:

vtnt

=1

e2Λ2t

qtQNt

ftnt

(1− τt)−Ωt

1e1

QKtftkt

(1− τt)− p

It

ftkt

−Ωtvtnt

=1

e1e2Λ2t −Ω2t

[e1qt

QNt(1− τt) ftnt

−ΩtQ̃Kt

(1− τt) ftkt

](66)

12.2 Sensitivities

12.2.1 Investment

itkt=

1e1e2Λ2t −Ω2t

[e2Λ2t

(Q̃Kt

(1− τt) ftkt

)− qtΩt

QNt(1− τt) ftnt

](67)

itkt=

e2[(1− λ1 − λ2) + λ1q1t + λ2q2t

]2 PKt − (q1t + q2t ) [e31q1t + e32q2t ] PNte1e2

[(1− λ1 − λ2) + λ1q1t + λ2q2t

]2 − [e31q1t + e32q2t ]2 (68)where:

41

PKt ≡Q̃Kt

(1− τt) ftkt> 0

PNt ≡QNt

(1− τt) ftnt> 0

∂ itkt

∂Q̃Kt

(1−τt) ftkt

=e2Λ2t

e1e2Λ2t −Ω2t> 0 (69)

∂ itkt

∂QNt

(1−τt) ftnt

=−qtΩt

e1e2Λ2t −Ω2t> 0 (70)

∂ itkt∂q1t

=2e2λ1ΛtPKt − PNt

[2e31q1t + q

2t (e31 + e32)

]e1e2Λ2t −Ω2t

−[e2Λ2t P

Kt −ΩtqtPNt

][2(e1e2λ1Λt − e31Ωt)][

e1e2Λ2t −Ω2t]2 (71)

∂ itkt∂q2t

=2e2λ2ΛtPKt − PNt

[2e32q2t + q

1t (e31 + e32)

]e1e2Λ2t −Ω2t

−[e2Λ2t P

Kt −ΩtqtPNt

][2(e1e2λ2Λt − e32Ωt)][

e1e2Λ2t −Ω2t]2 (72)

The estimates of Table 1 indicate that e1e2Λ2t −Ω2t > 0 and that Ωt < 0.Hence itkt is a positive function of

Q̃Kt(1−τt) ftkt

and QNt

(1−τt) ftnt.

12.2.2 Vacancy Creation

vtnt=

1e1e2Λ2t −Ω2t

[e1qt

QNt(1− τt) ftnt

−ΩtQ̃Kt

(1− τt) ftkt

](73)

vtnt=

[e1(q1t + q

2t)

PNt −[e31q1t + e32q

2t]

PKt]

e1e2[(1− λ1 − λ2) + λ1q1t + λ2q2t

]2 − [e31q1t + e32q2t ]2 (74)

42

∂ vtnt

∂Q̃Kt

(1−τt) ftkt

=−Ωt

e1e2Λ2t −Ω2t> 0 (75)

∂ vtnt

∂QNt

(1−τt) ftnt

=e1qt

e1e2Λ2t −Ω2t> 0 (76)

∂ vtnt∂q1t

=e1PNt − e31PKte1e2Λ2t −Ω2t

−[e1qtPNt −ΩtPKt

][2e1e2λ1Λt − 2e31Ωt][

e1e2Λ2t −Ω2t]2 (77)

∂ vtnt∂q2t

=e1PNt − e32PKte1e2Λ2t −Ω2t

−[e1qtPNt −ΩtPKt

][2e1e2λ2Λt − 2e32Ωt][

e1e2Λ2t −Ω2t]2 (78)

Hence vtnt is a positive function ofQ̃Kt

(1−τt) ftktand Q

Nt

(1−τt) ftnt.

43