Labor Market Frictions and

**Aggregate** **Employment**

Michael W. L. Elsby Ryan Michaels David Ratner 1

February 15, 2016

Abstract

What are the aggregate employment effects of labor market frictions? Canonical

labor market models share a common theme in exploring the implications of

adjustment frictions. We use this shared microeconomic structure to devise an

empirical diagnostic that allows one to bound the effects of this class of frictions on

the path of aggregate employment. Application of this diagnostic to rich

establishment microdata for the United States suggests that canonical labor market

frictions are unable to explain the majority of observed employment dynamics. This

result can be traced to the failure of canonical models to account for the dynamics of

the firm size distribution observed in establishment microdata.

[Preliminary and incomplete]

JEL codes: E32, J63, J64.

Keywords: Search frictions, adjustment costs, firm dynamics.

1

Elsby: University of Edinburgh (mike.elsby@ed.ac.uk). Michaels: Federal Reserve Bank of Philadelphia

(ryan.michaels@phil.frb.org). Ratner: Federal Reserve Board (david.d.ratner@frb.gov).

We thank seminar participants at Penn State, the 2014 Canon Institute for Global Studies Conference on

Macroeconomic Theory and Policy, ENSAI, Oxford, CIREQ, UBC, Strathclyde, Humboldt, Bonn, Glasgow,

Bristol, and the 2015 Workshop on Search and Matching, Universidad de Chile for helpful comments. All errors are

our own. This research was conducted with restricted access to Bureau of Labor Statistics (BLS) data. The views

expressed here do not necessarily reflect the views of the BLS, the Federal Reserve Bank of Philadelphia, the staff

and members of the Federal Reserve Board, or the Federal Reserve System as a whole.

1

What are the aggregate employment effects of labor market frictions? In this paper, we present a

new approach to this question for a popular class of labor market frictions. This class shares a

common theme in exploring the aggregate implications of (non-convex) adjustment frictions.

Notable examples include canonical models of lumpy adjustment (Caballero, Engel and

Haltiwanger 1997), per-worker hiring and firing costs (Bentolila and Bertola 1990; Hopenhayn

and Rogerson 1993), and search and matching (Diamond 1982; Pissarides 1985; Mortensen and

Pissarides 1994).

Our approach is motivated by two themes. First, the pattern of microeconomic adjustment in

the economy leaves a clear imprint on the dynamics of the firm-size distribution—that is, crosssectional

flows of employment across firms. Canonical frictions impede these flows, and it is

through this microeconomic channel that frictions shape the path of aggregate employment.

Second, many components of the implied firm-size dynamics can be measured, admitting a

clear mapping from theoretical concepts to empirical analogues. We show how establishmentlevel

microdata on employment shed light on these cross-sectional employment dynamics, and

thereby inform an analysis of the aggregate consequences of canonical models of labor market

frictions.

In section 1, we show that conventional models of labor market frictions imply crosssectional

employment dynamics that take on an intuitive and simple structure. The change in the

mass of establishments of a given size n reflects the difference between the inflow of

establishments to size n, and the outflow from that mass. Frictions act to reduce the rates of

adjustment to and from the mass at each point n in the firm-size distribution.

We show how this structure can be used to motivate a diagnostic that has a clear relationship

to the level of aggregate employment that would emerge in an economy without frictions. We

define the flow steady state density of employment as the distribution that would be ground out if

current rates of adjustment to and from each n were held constant. Our diagnostic is defined as

the mean of this distribution, aggregate flow steady state employment. We prove that, under this

class of frictions, the path of aggregate flow steady state employment bounds the path of

aggregate frictionless employment in canonical models of lumpy adjustment, and is

approximately equal to aggregate frictionless employment in models of linear and search and

matching frictions. As a result, the behavior of the diagnostic sheds light on the path of aggregate

employment that would emerge in the absence of frictions.

To see the intuition behind our result, consider the effect on impact of an increase in

aggregate labor productivity. In the absence of frictions, all establishments would wish to hire

more workers. Conditional on adjusting, this increase in desired employment will imply an

increase (decrease) in the mass of large (small) establishments in flow steady state: the flow

steady state density shifts right. In addition, however, the probability of adjustment reacts: Firms

will become more (less) likely to adjustment toward versus away from large (small) employment

levels. This extensive margin effect induces a further rightward shift in the flow steady state

density, overshooting the response of its frictionless counterpart. In this way, the difference

2

etween the paths of aggregate flow steady state and actual employment is an upper bound on

the effect of this class of frictions.

A crucial feature of the diagnostic is that it also can be measured in establishment microdata

on employment. Accordingly, in section 2 we use rich microdata underlying the Quarterly

Census of **Employment** and Wages for the period 1992Q1 through to 2014Q2 to measure the

diagnostic empirically. These microdata report the employment distribution, the inflow of

establishments to each size in the distribution, and the probability of outflow from each size.

These allow one to infer the flow steady state density and thus our diagnostic, aggregate flow

steady state employment.

We find that the empirical properties of the diagnostic resemble those of the empirical path

of actual employment. Specifically, we contrast the paths of (detrended) actual and flow steady

state aggregate employment. Aside from a brief period around the trough of the Great Recession,

the two series are remarkably similar: The median (mean) absolute deviation between the two

series is just 0.5 (0.8) of a log point over the sample period. Viewed through the lens of

canonical models, the inference is that this class of frictions accounts for little of observed

employment dynamics—the path of aggregate employment would have looked similar even in

the absence of these frictions.

1. Labor market frictions and firm size dynamics

Canonical models of labor market frictions share a common theme in exploring the implications

of microeconomic adjustment frictions. Models of discrete, or “lumpy,” adjustment in the

tradition of Caballero, Engel and Haltiwanger (1997) study the effects of fixed costs of

adjustment. Seminal analyses of the effects of firing costs, such as Bentolila and Bertola (1990)

and Hopenhayn and Rogerson (1993), invoke the presence of per-worker, linear costs of

adjustment. Similarly, recent models of search costs that extend the standard Diamond-

Mortensen-Pissarides model to incorporate a notion of firm size have underscored its

interpretation as the presence of a per-worker, linear hiring cost (Acemoglu and Hawkins 2014;

Elsby and Michaels 2013).

In this section we describe the economic channels through which labor market frictions

shape aggregate employment in this canonical class of models. A key observation is that these

frictions affect aggregate employment by impeding the flow of firms across different firm sizes.

We show that canonical models in turn imply a specific structure to these firm size dynamics.

We use this structure to motivate a diagnostic for the aggregate effects of these frictions. A virtue

of this diagnostic that we take up in later sections of this paper is that it can be measured directly

from establishment microdata.

1.1 Fixed costs

A leading model of labor market frictions postulates the presence of a fixed cost of adjusting

employment that is independent of the scale of adjustment. In what follows we review the well-

3

understood distortions of firms’ labor demand policies induced by this friction. More importantly

for our purposes, we use this to infer the (less-widely-understood) implications for firm size

flows, and thereby aggregate employment.

With regard to the structure of labor demand, the key implication of a fixed cost is that

employment will be adjusted only intermittently and, upon adjustment, discretely—hence

“lumpy” adjustment. Thus, labor demand takes the form of a threshold “Ss” policy, illustrated in

Figure 1A:

n ∗ if n ∗ > U(n −1 ),

n = { n −1 if n ∗ ∈ [L(n −1 ), U(n −1 )],

n ∗ if n ∗ < L(n −1 ).

(1)

Here n ∗ is the level of employment that a firm would choose if it adjusted. Under the Ss policy, a

firm’s current employment n is adjusted away from its past level n −1 whenever n ∗ deviates

sufficiently from n −1 , as dictated by the adjustment triggers L(n −1 ) < n −1 < U(n −1 ).

Caballero, Engel and Haltiwanger (1995, 1997) refer to n ∗ as mandated employment. This

can be interpreted as what the firm would choose if the friction were suspended for the current

period. It is, in principle, distinct from frictionless employment, which emerges if the fixed cost

is set to zero now and forever. However, in the Appendix, we verify that for a reasonably

calibrated model within this canonical class, the dynamics of mandated and frictionless

employment are very similar. 2 Henceforth, then, we will interpret n ∗ as frictionless employment

and sometimes use “mandated” and “frictionless” interchangeably. 3

The dynamics of aggregate employment implied by the firm behavior in equation (1) can be

inferred from its implications for firm size flows. Imagine the economy enters the period with a

density of past employment, h −1 (n), and that realizations of idiosyncratic and aggregate shocks

induce a density of mandated employment h ∗ (n). Our strategy is to infer a law of motion for the

current-period density h(n) implied by equation (1). This in turn will imply a path for aggregate

employment in the economy, which we denote by N, since the latter is captured by the mean of

the density, N ≡ ∫ nh(n)dn.

The adjustment policy in Figure 1A suggests a straightforward approach to constructing a

law of motion for the firm-size density h(n) . Consider first the outflow of mass from

employment level n. Among the h −1 (n) mass of firms that enter the period with n workers, only

the fraction whose desired employment n ∗ lies outside the inaction region [L(n), U(n)] will

choose to incur the adjustment cost and leave the mass. Symmetrically, now consider the inflow

of mass to employment level n. Among the h ∗ (n) mass of firms whose desired employment is

equal to n, only the fraction whose inherited employment n −1 lies outside of the inverse inaction

2

This has also been proven analytically for the case of a plausibly small fixed cost (Gertler and Leahy, 2008; Elsby

and Michaels, 2014).

3

To conserve space, Figure A in the Appendix displays impulse responses of all models for the parameterization

that yields an inaction rate in the middle of the range that we explore in our quantitative analysis. This

parameterization still implies considerably more inaction than in the data.

4

egion [U −1 (n), L −1 (n)] will choose to incur the adjustment cost and flow to n. Thus, the change

in the mass at employment level n follows the law of motion

Δh(n) = τ(n)h ∗ (n) − φ(n)h −1 (n), (2)

where τ(n) and φ(n) are respectively the probabilities of adjusting to and from an employment

level n,

τ(n) = Pr(n −1 ∉ [U −1 (n), L −1 (n)]|n ∗ = n) , and

φ(n) = Pr(n ∗ ∉ [L(n), U(n)]|n −1 = n).

Formal derivations of equations (2) and (3) are provided in the appendix.

1.2 An empirical diagnostic

With this theoretical law of motion in hand, our next step is to consider which of its components

can be measured empirically using available data. As we shall see, establishment-level panel

data allow one to observe much of equation (2): One can measure the mass at each employment

level at each point in time, h −1 (n) and h(n); one can also observe the fraction of establishments

at each employment level that adjusts away, φ(n). Crucially, however, we observe only the total

inflow, τ(n)h ∗ (n) , rather than its constituent parts. And, of course, this is precisely the

identification problem that we are attempting to surmount: If we could measure both τ(n) and

h ∗ (n), the latter would allow us to infer a measure of aggregate mandated employment N ∗ ≡

∫ nh ∗ (n)dn. Comparison of N ∗ with the observed path of actual aggregate employment N would

then indicate the wedge between these two induced by the adjustment friction.

Our approach is instead to ask whether we can infer useful information about the path of

desired aggregate employment N ∗ without observing directly the density of mandated

employment h ∗ (n). Our point of departure is to note that, for fixed adjustment rates τ(n) and

φ(n), the law of motion (2) implies that the firm size density will converge to what we shall

refer to as a flow steady state,

ĥ(n) ≡ τ(n)

φ(n) h∗ (n), (4)

ĥ(n) is useful for several reasons. First, it can be measured straightforwardly, since it

requires knowledge only of the total inflow, τ(n)h ∗ (n), and the probability of outflow φ(n),

both of which are observed in establishment panel data.

Second, we argue in what follows that the flow steady state conveys important information

on the path of mandated employment, and thereby on the role of adjustment frictions in shaping

the dynamics of aggregate employment. Specifically, note that the aggregate employment level

implied by the flow steady state, N̂ ≡ ∫ nĥ(n)dn, can be written as

N̂ = N ∗ + cov h ∗ (n, τ(n)

φ(n) ), (5)

(3)

5

where cov h ∗ denotes a covariance taken with respect to the distribution of mandated

employment, h ∗ (n).

Equation (5) reveals that aggregate flow steady-state employment N̂ will bound the path of

aggregate mandated employment N ∗ under a monotonicity condition—namely that firms on

average are more likely to adjust to versus from high (low) employment levels following positive

(negative) innovations to aggregate mandated employment. Under this condition, N̂ will rise

more than N ∗ when the latter rises, and fall more than N ∗ when it falls.

The monotonicity condition that underlies this intuition is closely related to the selection

effect that has been emphasized in the literature on adjustment frictions (Caballero and Engel

2007; Golosov and Lucas 2007). This refers to a property shared by state-dependent models of

adjustment whereby the firms that adjust tend to be those with the greatest desired adjustment.

By the same token, firms in these models also will adjust in the direction of the desired

adjustment.

The forgoing intuition is captured especially clearly in standard models of fixed adjustment

frictions, such as that set out in Caballero and Engel (1999). In this environment, firms face an

isoelastic production function y = pxn α that is subject to idiosyncratic shocks x. Firms thus face

the following decision problem

Π(n −1 , x) ≡ max

n {pxnα − wn − CI[n ≠ n −1 ] + βE[Π(n, x ′ )|x]}, (6)

where p denotes (fixed) aggregate productivity, w the wage, and C the fixed adjustment cost.

Caballero and Engel show that, if idiosyncratic shocks follow a geometric random walk, and

the adjustment cost C is scaled to be proportional to the firm’s frictionless labor costs, the

resultant homogeneity of a firm’s labor demand problem has two tractable implications,

summarized in the following Lemma:

Lemma 1 (Caballero and Engel 1999) Consider the firm’s problem in (6). If (i) ln x′ = ln x +

′ ′

ε x with ε x i.i.d., and (ii) C = Γ ⋅ (αx⁄ w α ) 1⁄ 1−α , then (a) the adjustment triggers in (1) are

linear and time invariant, L(n) = L ⋅ n and U(n) = U ⋅ n for constants L < 1 < U ; and (b)

desired (log) employment adjustments, ln(n ∗ /n −1 ), are independent of initial firm size n −1 .

Proposition 1 uses these properties of the canonical model to formalize the heuristic claim

above that changes in aggregate flow steady state employment bound changes in aggregate

mandated employment. Because of the model’s loglinear structure, the result is most simply

derived in terms of aggregate log mandated employment, which we shall denote by N ∗ , and its

aggregate log flow steady state counterpart N̂ .

Proposition 1 In the canonical model of fixed adjustment costs, relative to a prior constant-N ∗

state, a small change in aggregate log mandated employment, ΔN ∗ , induces on impact a larger

change in aggregate log flow steady state employment, ΔN̂ . That is,

ΔN̂ = A ⋅ ΔN ∗ , where A > 1. (7)

6

Proposition 1 has a number of virtues. First, it holds irrespective of whether adjustment is

symmetric or asymmetric. Second, it holds in the presence of changes in (market) wages w.

Intuitively, the latter simply dampen changes in mandated employment ΔN ∗ . Since the change

in the aggregate flow steady state ΔN̂ is proportional to ΔN ∗ , changes in wages affect both

symmetrically, leaving the result in Proposition 1 unimpaired.

There are also limitations to Proposition 1, however. It relies on the homogeneity of the

canonical model implied by the random walk assumption on shocks. It is also a comparative

statics result, describing the response of the economy to a change in aggregate labor demand,

indexed by p, that is expected to occur with zero probability from the firms’ perspectives. For

these reasons, in the next subsection, we explore the robustness of the bounding result

anticipated above in numerical simulations that relax these assumptions.

1.3 Quantitative illustrations

In what follows, we illustrate the dynamics of fixed costs models that resemble the canonical

model described above, but with two differences. First, we relax the random walk assumption on

idiosyncratic shocks which we allow to follow a geometric AR(1),

ln x′ = ρ x ln x + ε x ′ , where ε x ′ ∼ N(0, σ x 2 ). (8)

Second, we allow for the presence of aggregate productivity shocks, and for their stochastic

process to be known to firms in the model. The evolution of these aggregate shocks also is

assumed to follow a geometric AR(1),

ln p′ = ρ p ln p + ε p ′ , where ε p ′ ∼ N(0, σ p 2 ). (9)

To mirror the timing of the data we use later in the paper, a period is taken to be one quarter.

Based on this, we set the discount factor β to 0.99, consistent with an annual interest rate of

around 4 percent. To parameterize the remainder of the model, we appeal to the empirical

literature that estimates closely related models of firm dynamics. Thus, the returns to scale

parameter α is set to 0.64, as in the estimates of Cooper, Haltiwanger, and Willis (2007, 2015).

The parameters of the idiosyncratic productivity shock process (8) are based on estimates

reported by Abraham and White (2006). Abraham and White’s results imply a quarterly

persistence parameter ρ x of approximately 0.7 and a standard deviation of the idiosyncratic

innovation ε x

′

of around σ x = 0.15. 4

The parameters of the process for aggregate technology in (9) are chosen so that aggregate

frictionless employment in the model exhibits a persistence and volatility comparable to

aggregate employment in U.S. data. This yields ρ p = 0.95 and σ p = 0.0026. Although frictions

augment persistence, and dampen volatility, the intent is for the model environment to resemble

4

The Appendix derives these parameters and contrasts them with other estimates reported in related literature. It

also shows that the implied dynamics of aggregate employment implied by reasonable changes in these parameters

are qualitatively similar to those described here.

7

oadly the U.S. labor market with respect to these unconditional moments. Importantly, the

approach does not build in any persistence in employment conditional on technology.

Finally, with respect to the adjustment cost C , we explore three parameterizations that

successively raise the friction. The first is chosen to replicate the observed fraction of firms that

do not adjust from quarter to quarter. In the data used later in the paper, this inaction rate

averages 52.5 percent. However, the latter inaction rate is implied by a fixed cost equal to 1.3

percent of quarterly revenue, which is at the lower end of available estimates (Bloom 2009;

Cooper, Haltiwanger, and Willis 2007, 2015). For this reason, we also explore fixed costs that

imply quarterly inaction rates of 67 percent and 80 percent, corresponding to adjustment costs of

2.7 percent and 5.2 percent of quarterly revenue, respectively.

We solve the labor demand problem for 250,000 firms via value function iteration on an

integer-valued employment grid, n ∈ {1,2,3 … }. The latter mirrors the integer constraint in the

data, allowing one to construct the flow steady state density ĥ(n) in the simulated data in the

same way as we later implement in the real data.

Figure 2 plots simulated impulse responses of aggregate employment N, together with its

mandated and flow steady state counterparts, N ∗ and N̂ respectively. The bounding result

anticipated in the previous subsection, and in Proposition 1 in particular, is clearly visible in the

model dynamics. For all three parameterizations of the adjustment cost our proposed diagnostic,

flow steady state employment N̂ , responds more aggressively to the aggregate shock than

mandated employment N ∗ . Moreover, the magnitude of the overshooting of N̂ relative to N ∗ is

substantial in the model, responding on impact around twice as much to the impulse.

1.4 Linear costs

Prominent alternative models of labor market frictions appeal instead to linear costs of

adjustment in which the friction is discrete at the margin, and rises with the scale of adjustment.

As noted above, simple models of per-worker firing and hiring costs fit directly into this mold.

But we will also include canonical models of search frictions in this class, since they are models

of time-varying per-worker hiring costs (Elsby and Michaels 2013).

Relative to the fixed costs case examined above, linear frictions alter the structure of both

labor demand and firm size dynamics. Although labor demand will continue to feature

intermittent adjustment a key difference is that, conditional on adjusting, firms will no longer

implement their desired mandated employment n ∗ . Rather, they will reduce the magnitude of

hires and separations, shedding fewer workers when they shrink, and hiring fewer workers when

they expand. Formally, the policy rule for separations, which we shall denote by l(⋅), will differ

from the policy rule used for hiring, denoted u(⋅), inducing the continuous Ss policy illustrated

in Figure 1B,

u −1 (n ∗ ) if n ∗ > u(n −1 ),

n = { n −1 if n ∗ ∈ [l(n −1 ), u(n −1 )],

l −1 (n ∗ ) if n ∗ < l(n −1 ).

(10)

8

The key distinction, that the direction of adjustment must be taken into account in the

presence of linear costs, also leaves its imprint on the law of motion for the firm size distribution.

As before, the labor demand policy in Figure 1B motivates the form of this law of motion. This

reveals that the structure of outflows is qualitatively unchanged—of the h −1 (n) density of firms

currently at employment level n, only those with mandated employment outside the inaction

region [l(n −1 ), u(n −1 )] will adjust away. But inflows are now differentiated by the direction of

adjustment. The inflow of mass adjusting down to n is comprised of firms whose past

employment n −1 is greater than n, and whose mandated employment n ∗ is equal to l(n) < n.

Likewise, the inflow of mass flowing up to n consists of firms with n −1 < n and n ∗ = u(n) >

n.

Piecing this logic together yields the following law of motion for the firm size density,

Δh(n) = τ l (n)h l ∗ (n) + τ u (n)h u ∗ (n) − φ(n)h −1 (n), (11)

where h l ∗ (n) = l ′ (n)h ∗ (l(n)) and h u ∗ (n) = u ′ (n)h ∗ (u(n)) are the densities of employment that

would emerge if all firms adjusted, respectively, according to the separation rule, l(n), and

hiring rule, u(n), and the adjustment probabilities take the form

τ l (n) = Pr(n −1 > n|n ∗ = l(n)) ,

τ u (n) = Pr(n −1 < n|n ∗ = u(n)) , and

φ(n) = Pr(n ∗ ∉ [l(n), u(n)]|n −1 = n).

(12)

Extending the interpretation of the fixed costs case above, τ l (n) is the probability that a firm

adjusts down to n, while τ u (n) is the probability that a firm adjusts up to n.

To construct the flow steady state density for the linear costs case note that, for fixed

adjustment rates τ l (n), τ u (n) and φ(n), the law of motion (11) implies that the firm size density

will converge to

ĥ(n) ≡ τ l(n)

φ(n) h l ∗ (n) + τ u(n)

φ(n) h u ∗ (n). (13)

Once again, the behavior of aggregate flow steady state employment implied by linear

frictions can be formalized most straightforwardly in a canonical linear cost model 5 in which

firms face isoelastic production y = pxn α , and idiosyncratic shocks that follow a geometric

random walk. The key difference is that the adjustment friction is now scaled by the magnitude

of adjustment, so that firms face the decision problem:

Π(n −1 , x) ≡ max

n {pxnα − wn − c + Δn + + c − Δn − + βE[Π(n, x ′ )|x]}. (14)

5

Nickell (1978, 1986) was the first to formalize the linear-cost model in the context of labor demand. Bentolila and

Bertola (1990) introduced uncertainty into Nickell’s continuous-time formulation. Equation (14) can be seen as the

discrete-time analogue to Bentolila and Bertola (though we do not have to assume Gaussian innovations).

9

A simple extension of Caballero and Engel’s (1999) homogeneity results can be used to

show that, if per-worker hiring and firing costs are proportional to wages, the same tractable

properties of Lemma 1 hold for the case of linear frictions:

Lemma 1′ Consider the firm’s problem in (14). If (i) ln x′ = ln x + ε x

′

with ε x

′

i.i.d., and (ii)

c + = γ + w and c − = γ − w, then (a) the adjustment triggers in (10) are linear and time invariant,

l(n) = l ⋅ n and u(n) = u ⋅ n for constants l < 1 < u ; and (b) desired (log) employment

adjustments, ln(n ∗ /n −1 ), are independent of initial firm size n −1 .

As in Proposition 1 above for the fixed costs case, the latter properties allow one to relate the

response of aggregate flow steady state log employment N̂ to the response of aggregate

frictionless log employment N ∗ following a change in aggregate productivity.

Proposition 2 In the canonical model of linear adjustment costs, relative to a prior constant-N ∗

state, a small change in aggregate log mandated employment, ΔN ∗ , induces on impact

approximately the same change in aggregate log flow steady state employment, ΔN̂ . That is,

ΔN̂ ≈ ΔN ∗ . (15)

Proposition 2 shares the same virtues and limitations as Proposition 1: It holds for

asymmetric frictions c + ≠ c − , and in the presence of changes in wages w; but it requires the

homogeneity assumptions in Lemma 1′, and is a comparative statics result.

Proposition 2 also suggests that the flow steady state is again informative with respect to the

path of frictionless employment. A key difference relative to the fixed costs case, however, is

that the response of N̂ no longer bounds that of N ∗ , as in the fixed costs case, but is

approximately equal to it.

The key difference relative to the fixed costs case is that firms adjust only partially toward

their frictionless employment under linear frictions. A rise in aggregate frictionless employment

places more firms on the hiring margin, where employment is set below its frictionless

counterpart, and fewer firms on the separation margin, where employment exceeds its

frictionless level. Both forces serve to attenuate the response of flow steady state aggregate

employment relative to the fixed costs case. Proposition 2 shows that, to a first order, this

attenuation offsets exactly the overshooting of the flow steady state in the fixed costs case.

Figures 3 and 4 show that the result of Proposition 2 is mirrored in numerical simulations of

models that allow the presence of more plausible idiosyncratic shocks x, and fully stochastic

aggregate shocks p . These numerical results are based on the same methods and baseline

parameterization described in section 1.3.

Figure 3 illustrates impulse responses of actual, frictionless and flow steady state aggregate

employment in the presence of symmetric linear frictions where c + = c − . As before, each panel

of Figure 3 successively raises the friction to produce increasingly higher average rates of

inaction in employment adjustment. As foreshadowed by Proposition 2, although the response of

10

actual employment becomes progressively more sluggish as the friction rises, the response of

flow steady state employment lies very close to the frictionless path.

Figure 4 in turn reveals that this result is unimpaired by the presence of asymmetric frictions,

as suggested by Proposition 2. Its first three panels report results for successively higher hiring

costs, c + > 0 and c − = 0; the latter three panels do the same for firing costs, c − > 0 and c + =

0. Strikingly, it is hard to discern differences between the impulse responses for hiring and firing

costs, and between these and the impulse response for the symmetric case in Figure 3.

The message of Figures 3 and 4, then, is that the insight of Proposition 2 is robust to

empirically reasonable parameterizations of canonical models of linear frictions. This reinforces

the message of section 1 that the flow steady state is indeed a useful diagnostic for the path of

frictionless employment, and thereby for the aggregate effects of canonical frictions.

However, Proposition 2 does not allow the adjustment triggers to vary, since these are

independent of ΔN ∗ under the time-invariant linear frictions we have consider thus far. This is a

key distinction with respect to models of search frictions, to which we now turn.

1.5 Search costs

The canonical Diamond-Mortensen-Pissarides (DMP) model of search frictions, in which a

single firm matches with a single worker, can be extended to a setting with “large” firms that

operate a decreasing-returns-to-scale production technology (Acemoglu and Hawkins 2014;

Elsby and Michaels 2013). The presence of search frictions implies two modifications to the

canonical linear cost model studied above.

First, search frictions induce a time-varying per-worker hiring cost. Hiring is mediated

through vacancies, each of which is subject to a flow cost c, and is filled with a probability q that

depends on the aggregate state of the labor market. Under a law of large numbers, the effective

per-worker hiring cost is thus c/q, which varies over time with variation in the vacancy-filling

rate q. The typical firm’s problem therefore takes the form:

Π(n −1 , x) ≡ max

n {pxnα − w(n, x)n − c q Δn+ + βE[Π(n, x ′ )|x]}. (16)

Second, search frictions induce ex post rents to employment relationships over which a firm

and its workers may bargain. In an extension of the bilateral Nash sharing rule invoked in

standard one-worker-one-firm search models, Elsby and Michaels (2013) show that a marginal

surplus-sharing rule proposed by Stole and Zwiebel (1996) implies a wage equation of the form

pxαn α−1

w(n, x) = η

1 − η(1 − α) + (1 − η)ω. (17)

Here η ∈ [0,1] indexes worker bargaining power, and ω is the annuitized value of the threat

point faced by workers. Bruegemann, Gautier and Menzio (2015) show that the marginal

surplus-sharing rule underlying (17) can be derived from an alternating-offers bargaining game

11

etween a firm and its many workers in which the strategic position of each worker in the firm is

symmetric.

As before, we consider a version of the search model with a tractable homogeneity property.

Specifically, we study the case in which the friction, embodied in the vacancy cost, is

proportional to the workers’ outside option ω. 6 The normalized posting cost is γ ≡ c/ω. Under

these assumptions, the Appendix shows that the homogeneity properties of Lemma 1′ continue to

hold, with one exception: although the adjustment triggers remain linear, they no longer are timeinvariant,

for the simple reason that the friction varies with the aggregate state.

The presence of time-varying adjustment triggers modifies Proposition 2 as follows:

Proposition 3 Assume (i) firms are sufficiently patient, β ≈ 1; (ii) frictions are sufficiently

small, γ 2 ≈ 0; and (iii) the distribution of ε x is symmetric. Then, relative to a prior constant-N ∗

state, a small change in aggregate log mandated employment, ΔN ∗ , induces on impact the

following change in aggregate log flow steady state employment,

ΔN̂ ≈ 1 − ε ω

ΔN ∗ , (18)

1 − ε w ∗

where ε ω and ε w ∗ are the elasticities of ω and frictionless wages w ∗ to aggregate productivity p.

Proposition 3 shows that the response of aggregate log flow steady-state employment N̂ still

approximates the response of aggregate log mandated employment N ∗ , under a few additional

restrictions. We argue in what follows that these restrictions are plausible.

The first two restrictions—that firms are patient, and that frictions are sufficiently small—

are quantitative. We address their plausibility by examining results from a numerical model that

does not impose these restrictions. As above, this model sets the discount factor β to match an

annual interest rate of 4 percent, and sets the vacancy cost c to replicate the average probability

of employment adjustment. The numerical results will thus address the extent to which β is close

enough to one, and the friction sufficiently small, for the insight of Proposition 3 to hold.

The third restriction concerns the symmetry of the distribution of idiosyncratic shocks. This

can be justified along two grounds. First, as foreshadowed in the quantitative analyses of

preceding sections, it is conventional to implement shock processes with symmetrically

distributed—typically Normal—innovations. In addition to its being commonplace, it is also

consistent with the observed pattern of employment adjustment, which is close to symmetric (see

Davis and Haltiwanger 1992, and Elsby and Michaels 2013, among others).

These three restrictions aid the proof of Proposition 3, which is based on symmetry. If the

firm is sufficiently patient (β ≈ 1), the cost of hiring in the current period implies an equal cost

of firing in the subsequent period. As a result, one can show that the optimal policy is symmetric,

to a first-order approximation around γ = 0, as long as the driving force ε x is symmetric. In

6

This can be motivated through the presence of a dual labor market in which recruitment is performed by workers

hired in a competitive market, who are paid according to the annuitized value of unemployment ω.

12

terms of the notation of Lemma 1′, the upper and lower adjustment triggers satisfy ln u ≈ − ln l,

and move by approximately the same amount in response to a shift in aggregate productivity.

The final restriction necessary for the change in aggregate log flow steady-state employment

ΔN̂ to approximate its frictionless counterpart ΔN ∗ is that the flexibility of frictionless wages

w ∗ mirrors the flexibility of workers’ outside option in the presence of frictions ω, ε w ∗ = ε ω .

We parameterize these elasticities as follows. In the frictionless case, ε w ∗ is related to the

Frisch elasticity of labor supply. Kimball and Shapiro (2003) use a model of household labor

supply to infer the latter from long-run labor supply responses to wealth changes. 7 We apply

their estimate and set the Frisch labor supply elasticity equal to one. This implies ε w ∗ = 1/(2 −

α) ≈ 0.735 when α is set to equal 0.64.

To calibrate ε ω , we take the model with search frictions as the data-generating process for

observed short-run fluctuations. Once account is taken of biases associated with the shifting

composition of employment over the business cycle, microdata-based estimates suggest that real

wages are about as cyclical as employment (Solon, Barksy, and Parker 1994; Elsby, Solon, and

Shin 2015). Accordingly, we set ε ω to match an elasticity of average real wages with respect to

aggregate employment approximately equal to one. We find that ε ω = 2/3 hits this target. 8

Note that Proposition 3 implies that the response of aggregate flow steady-state employment

should bound that of frictionless employment under this parameterization, since (1 − ε ω )/(1 −

ε w ∗) ≈ 1.25 . Figure 5 shows that this prediction of Proposition 3 is visible in numerical

simulations of the model based on the same methods and baseline parameterization described in

section 1.3—that is, with stationary idiosyncratic shocks x, and fully stochastic aggregate shocks

p. The impulse responses in Figure 5 suggest that flow steady-state employment reacts slightly

more on impact to the aggregate shock than its frictionless counterpart.

2. Empirical implementation

The previous section gave a theoretical rationale for an empirical diagnostic, flow steady state

employment N̂, which provides a bound for the aggregate effects of a canonical class of labor

market frictions. A key virtue of this diagnostic is that it can be measured with access to

establishment panel data on employment. In this section, we apply these insights to a rich source

of microdata from the United States.

2.1 Data

The data we use are taken from the Quarterly Census of **Employment** and Wages (QCEW). The

QCEW is compiled by the Bureau of Labor Statistics (BLS) in concert with State **Employment**

7

An advantage of Kimball and Shapiro’s focus on long-run responses is that they are likely to be less sensitive to

search frictions.

8

The level of ω is chosen to induce an average establishment size of 20, consistent with data from County Business

Patterns. The other new parameter to set is η. This is chosen to be 0.4 to induce a realistic labor share.

13

Security Agencies. The latter collect data from all employers in a state that are subject to the

state’s Unemployment Insurance (UI) laws. Firms file quarterly UI Contribution Reports to the

state agency, which provide payroll counts of employment in each month. These are then

aggregated by the BLS, which defines employment as the total number of workers on the

establishment’s payroll during the pay period that includes the 12 th day of each month. Following

BLS procedure, we define quarterly employment as the level of employment in the third month

of each quarter. 9

From the cross-sectional QCEW data, the BLS constructs the Longitudinal Database of

Establishments (LDE), which we use in what follows. Although data are available for the period

1990Q1 to 2014Q2, we restrict attention to data from 1992Q1 due to difficulty in matching

establishments in the first two years of the sample.

Sample restrictions. The QCEW data are a near-complete census of workers in the United

States, covering approximately 98 percent of employees on non-farm payrolls. The dotted line in

Figure 6 plots the time series of log aggregate employment in private establishments in the full

QCEW sample. Relative to this full sample we apply three further sample restrictions, illustrated

by the successive lines in Figure 6.

First, access for outside researchers to QCEW/LDE microdata is restricted to a subset of

forty states that approve access onsite at the BLS for external research projects. As a result, our

sample excludes data for Florida, Illinois, Massachusetts, Michigan, Mississippi, New

Hampshire, New York, Oregon, Pennsylvania, Wisconsin, and Wyoming.

Second, we restrict our sample to continuing establishments with positive employment in

consecutive quarters. Specifically, we construct a set of overlapping quarter-to-quarter balanced

panels that exclude births and deaths of establishments within the quarter. Note that we do not

balance across quarters, so births in a given panel will appear as incumbents in the subsequent

panel (if they survive). We focus on continuing establishments because the canonical models of

adjustment frictions analyzed above are intended to describe adjustment patterns among

incumbent firms. 10

Our final sample restriction is to exclude establishments with more than 1000 employees in

consecutive quarters. We do this for practical reasons. To measure the flow steady state

employment distribution in equations (4) and (13), and hence the diagnostic suggested by the

theory, we require measures of establishment flows between points in the firm size distribution—

specifically, inflows of mass to each employment level, and the probability of outflow. To

measure the latter with sufficient precision requires sufficient sample sizes at all points in the

9

The count of workers includes all those receiving any pay during the pay period, including part-time workers and

those on paid leave.

10

An additional, more practical reason for focusing on continuers is that the matching of establishments over time is

more subject to error when identifying births and deaths in the QCEW. Among other steps, the latter involves

identification of predecessor and successor establishments to detect spurious births or deaths. By excluding

establishment births and deaths, our sample excludes establishments that require linkage using predecessor or

successor information. The latter is a very small subset of establishments, around 0.1 percent in 2014Q2.

14

distribution. Since establishments with more than 1000 employees comprise a very small fraction

of U.S. establishments—less than 0.1 percent in 2014Q2—sample sizes become impracticably

thin beyond 1000 employees, inducing substantial noise in implied estimates of our diagnostic.

Recall that our goal is to understand whether canonical models of labor market frictions can

account for the dynamics of aggregate employment. A worry, then, is that the forgoing sample

restrictions might alter the dynamics of aggregate employment in our sample relative to the full

sample.

Figure 6 reveals that this is not the case. In terms of levels, the largest loss of sample size

occurs because we are unable to access data for all states, accounting for around 30 percent of

total employment in the United States. The further exclusion of non-continuing establishments

and large establishments accounts, respectively, for around 2 percent and 10 percent of

employment. However, Figure 6 shows that the path of aggregate employment in our sample

resembles, in both trend and cycle, the path of aggregate employment in the full QCEW sample.

The correlation between log aggregate employment in the published QCEW series for all states

and that in our final microdata sample is 0.99.

Measurement. To estimate our diagnostic, we require first an estimate of what we refer to as the

flow steady state distribution of employment, ĥ(n). Rearranging equations (4) and (13), we can

write the latter as

ĥ t (n) = h t−1 (n) + Δh t(n)

φ t (n) , (19)

where t indexes quarters, h t−1 (n) is the previous quarter’s mass of establishments with

employment n, Δh t (n) ≡ h t (n) − h t−1 (n) is the quarterly change in that mass, and φ t (n) is the

fraction of establishments that adjusts away from an employment level of n in quarter t. Thus,

estimation of ĥ t (n) requires only an estimate of the outflow adjustment probability φ t (n), in

addition to measures of the evolution of the firm size distribution h t (n).

The simplest approach to measuring φ t (n) is to use our microdata to compute the fraction of

establishments with n workers in quarter t that reports employment different from n in quarter

t + 1. As alluded to above in motivating our sample restrictions, however, a practical issue that

arises is that sample sizes become small as n gets large, inducing sampling variation in estimates

of φ t (n).

We address this issue by discretizing the employment distribution at large n. An advantage

of the substantial sample sizes in the QCEW/LDE microdata is that we can be relatively

conservative in this regard. In particular, we allow individual bins for each integer employment

level up to 250 workers. In excess of 99 percent of establishments lie in this range, and so sample

sizes in each bin are large, between about 100 and 1.3 million establishments. For establishment

sizes of 250 through 500 workers we use bins of length five, allowing us to maintain sample

sizes above about 80 establishments in each quarter. Further up the distribution, of course,

sample sizes get smaller, so we extend our bin length to ten for employment levels between 500

and 999 workers. In this range, sample sizes are at least 15 establishments in each quarter.

15

Denoting these bins by b, we estimate the firm size mass and the outflow probability as

h t (b) = ∑ I[n it ∈ b]

i

, and φ (b) t = ∑ i I[nit ∉ b|n it−1 ∈ b]

, (20)

∑ I[n it−1 ∈ b]

where i indexes establishments. We use these measures to compute the flow steady state mass in

each bin according to equation (19) as ĥ t (b) = h t−1 (b) + [Δh t (b) ⁄ φ t (b)]. Finally, we compute

aggregate employment and its flow steady state counterpart by taking the inner product of h t and

ĥ t with the midpoints of each bin, denoted m b ,

N t = ∑ m b h t (b)

b

i

, and N̂t = ∑ m b ĥ t (b). (21)

b

2.2 Inferring the aggregate effects of frictions

With this estimate of flow steady state aggregate employment N̂t in hand, we implement in this

section three explorations of its empirical behavior in contrast to the path of actual aggregate

employment N t , and to the predictions of canonical frictions summarized in section 1.

Empirical time series. Our first, and simplest approach is to compare the paths of actual and

flow steady state aggregate employment in the data. Under the class of frictions analyzed in

section 1, the time series of N̂t will bound the path of mandated aggregate employment N t

∗

under

fixed adjustment frictions, and will approximate N t

∗

in the case of linear or search frictions. Thus

a simple comparison of the empirical time paths of N t and N̂t provides, at least, a bound on the

aggregate effects of these canonical frictions.

Figure 7 plots the time series of N t and N̂t derived from application of equation (21) to the

QCEW/LDE microdata. Both series are expressed in log deviations from a quadratic trend. 11

Figure 7 reveals that N̂t is a leading indicator of actual employment N t , and is also more volatile.

Specifically, the standard deviation of N t is 0.025, whereas the standard deviation of N̂t is

0.031.

However, on the whole, the differences between the two series appear pretty modest. The

median (average) absolute difference is just 0.5 (0.8) log points. As Figure 7 indicates, the only

substantial “daylight” between the series emerges in the Great Recession. For instance, in the 5

quarters that bracket the trough (2009q2) of the recession, the average difference between the

series is about 3 log points. However, this difference is shortlived. Since 2010, the two series

have moved in tandem: employment has increased 11.6 log points, whereas steady state has

increased 11.9 log points.

11

We use a quadratic trend rather than the HP filter, because it is consistent with how we detrend the same series to

estimate impulse responses. As Ashley and Verbrugge (2006) have warned, the HP filter is problematic in this

context because it uses future realizations to derive the cyclical component in the present period. As a result, lags of

HP deviations included as regressors in a VAR are not predetermined with respect to the outcome variable.

16

Together these observations give a first suggestion that, even when viewed through the lens

of canonical models of labor market frictions, the aggregate effects of such frictions account for

a modest part of aggregate employment dynamics.

Dynamics of aggregate employment. Our quantitative illustrations in section 1 compared

impulse responses of actual versus flow steady state employment in models with a variety of

adjustment frictions. These emphasized that canonical frictions imply quite distinctive dynamics

of flow steady state employment N̂ . Even when actual employment N moves sluggishly, N̂

jumps on impact of an aggregate shock, and bounds the response of frictionless employment N ∗ .

In this section, we confront these theoretical predictions with the empirical dynamics of our

measures of N and N̂. Our goal is not to use the data to estimate impulse responses of these

variables to identified structural shocks. Instead, we undertake a descriptive analysis of the

persistence properties of aggregate employment. A commonly used gauge for the latter is a

comparison of the dynamics of employment relative to output-per-worker.

We formalize this comparison as follows. Denote log output-per-worker by y t . In a first

stage, we estimate innovations in y t that are unforecastable conditional on lags of y, and lags of

log aggregate employment N. Specifically, we use quarterly data on output-per-worker in the

nonfarm business sector from the BLS Productivity and Costs release and our measure of actual

employment from the QCEW to estimate the following AR(L) specification:

y t = α y + ∑

L

s=1

β s y y t−s

L

+ ∑ γ y s N t−s

s=1

+ δ 1 y t + δ 2 y t 2 + ε t y . (22)

Note that we account for trends in these series using a quadratic time trend.

The estimated residuals from this first-stage regression, ε̂ty , are then used as the innovations

to output-per-worker from which we derive impulse responses of actual and flow steady state

employment in a second stage,

N t = α N + ∑

N̂t = α N̂ + ∑

L−1

β N y

s ε̂t−s

s=0

L−1

s=0

y

β s

N̂ε̂t−s

L

+ ∑ γ N s N t−s

s=1

L

+ δ 1 N t + δ 2 N t 2 + ε t N , and

+ ∑ γ s

N̂N̂t−s + δ 1

N̂t + δ 2

N̂t 2 + ε t

N̂.

s=1

Note that the timing in the lag structure of innovations to output-per-worker permits a

contemporaneous relationship between these innovations and employment, as suggested by the

model-based impulse responses described in section 1.

The estimates from the regressions in equations (22) and (23) allow us to trace out the

dynamic relationship between each measure of log aggregate employment and a one-log-point

innovation in output-per-worker. In practice, we use a lag order of L = 4 in both stages, (22) and

(23), and obtain estimates using the data for the full sample period, 1992Q2 to 2014Q2. 12

(23)

12

The precise peak of the hump shaped impulse responses vary slightly across different lag lengths, but Figure 8 is

representative of results across a number of specifications.

17

Panel A of Figure 8 plots the results. The dynamic response of aggregate employment takes

a familiar shape, rising slowly after the innovation with a peak response of around 1 log point

after five quarters. These hump-shaped dynamics mirror similar results found using different

methods elsewhere in the literature (Blanchard and Diamond 1989; Fujita and Ramey 2007;

Hagedorn and Manovskii 2011). This is one representation of the persistence of aggregate

employment.

As suggested by the time series in Figure 7, the dynamics of the flow steady state diagnostic

N̂ share many of these properties. Although its peak response occurs earlier—after three

quarters—reinforcing the impression of Figure 7 that N̂ is a leading indicator of the path of N, it

exhibits a similar volatility, and a clear hump-shape.

To contrast the empirical dynamics illustrated in Figure 8A with those implied by canonical

models of frictions, we rerun the regressions in equations (22) and (23) using model-generated

data. Specifically, we maintain the baseline parameterization from the quantitative illustrations in

section 1, but with one important difference: We allow for a hybrid of fixed and linear

adjustment frictions, and choose the configuration of these that minimizes the (sum of squares)

distance between the empirical dynamics of actual employment N in Figure 8A and those

implied by the model.

As foreshadowed by the theoretical results of Figures 2 and 3, we find that a pure linear cost

model is best able to generate the substantial empirical persistence of aggregate employment. 13

To achieve this, however, the degree of inaction implied by the model must exceed its empirical

analogue. Each quarter around 85 percent of firms do not adjust employment in the model,

compared to 52 percent in the data. We are thus being generous to the model in allowing it to

violate this moment of the data.

Panel B of Figure 8 reveals that this parameterization of the model is able to generate a

dynamic relationship between employment and output-per-worker that is comparable to the data.

Although the model overstates the impact response, the amplitude and persistence of

employment are similar to their empirical counterparts.

A key result of Figure 8B, however, is that the model-implied dynamics of flow steady state

employment are profoundly different from those seen in the data. Confirming the impression of

the theoretical impulse responses in Figure 3, N̂ jumps in response to innovations in output-perworker

in the model, deviating substantially from the path of actual employment N. In marked

contrast, the empirical dynamics of N̂ in Figure 8A are much more sluggish, bearing a closer

resemblance to the empirical path of actual employment than its model-implied counterpart.

The substantial discrepancy between the implied and observed dynamics of flow steady state

employment is an important failure of canonical models of frictions. Under these models, the

path of flow steady state employment is closely related to the path of frictionless employment.

The gap between the paths of N and N̂ is therefore indicative of the aggregate effects of these

13

We do not pursue the effects of asymmetries in the adjustment costs here: the theoretical results of Propositions 1

and 2, and the quantitative results of Figure 4, suggest that any such asymmetries affect neither the dynamics of

aggregate employment, nor its flow steady state counterpart.

18

canonical frictions. That this gap is relatively small in the data therefore implies that canonical

frictions play a minor role in generating the observed dynamics of aggregate employment.

Dynamics of the firm size distribution. To examine the origins of this failure of canonical

models, recall that the link between our diagnostic flow steady state employment N̂ and

frictionless employment N ∗ is mediated through the behavior of firm size flows—the τs and φs

of equations (4) and (13)—and that canonical frictions have strong predictions regarding the

dynamics of these flows by establishment size.

As we have emphasized, a key benefit of the data is that we can measure aspects of these

flows using the longitudinal dimension of the QCEW microdata—specifically the total inflow to,

and the probability of outflow from, each employment level. Our next exercise, therefore, is to

contrast the dynamics of the firm size distribution in the data to those implied by canonical

models of frictions.

To do this, we first split establishments in the data into three size classes. We choose these

to correspond to the lower quartile (fewer than 15 employees), interquartile range (16 to 170

employees), and upper quartile (171 employees and greater) of establishment sizes. We then

estimate descriptive impulse responses that mirror equations (22) and (23) for the total inflow to,

and probability of outflow from, each size class. 14 As in our previous analysis of the dynamics of

aggregate employment, we repeat these same steps using data simulated from the model

underlying Figure 8B that is calibrated to match as closely as possible the empirical dynamics of

aggregate employment.

Panels C through F of Figure 8 illustrate the results of this exercise. The empirical and

model-implied dynamics share a qualitative property, namely that positive aggregate shocks

render small (large) establishments more (less) likely to adjust away from their current

employment, and induce fewer (more) establishments to adjust to low (high) employment levels.

Aside from this broad qualitative similarity, the quantitative dynamics reveal striking

contrasts. The empirical behavior of firm size flows exhibits an inertia not only in the sense that

their levels are retarded relative to a frictionless environment, but also in the sluggishness of their

responses to aggregate disturbances.

We highlight three manifestations of this general observation. First, note that the empirical

responses of the firm size flows in Figures 8C and 8E are an order of magnitude smaller than

their theoretical counterparts in Figures 8D and 8F. Second, the dynamics of the flows in the data

are much more sluggish than implied by canonical frictions. Firm size dynamics in the model

respond aggressively on impact of the aggregate shock. In the data, the response is mild and

delayed. Third, the empirical dynamics reveal an establishment size gradient in the magnitude of

the response of firm size flows: Flows to and from smaller establishments respond less than their

counterparts for larger establishments.

14

To aggregate within a quartile range, we take a weighted average across establishment sizes, where the weight is

the size’s share of all establishments in the range.

19

The upshot of this exercise is that canonical models of labor market frictions do a poor job

of capturing the empirical dynamics of the firm size distribution. Since the latter is the key

channel through which canonical frictions are supposed to impede aggregate employment

dynamics, this is an important limitation of this class of model.

Matching the time series of aggregate employment. Our final exercise is inspired by an

approach devised by King and Rebelo (1999) and Bachmann (2012). They show that it is

possible to find a sequence of aggregate shocks that generates a path for aggregate modelgenerated

outcomes—in our case employment—that matches an empirical analogue. In what

follows, we use this technique to contrast the time series of flow steady state employment in

model and data when the path of aggregate employment in each is constructed to be the same.

The procedure relies on the ability to summarize the dynamics of aggregate employment

implied by the model using a simple aggregate law of motion. In a related adjustment cost

model, Bachmann shows that an AR(1) specification does an excellent job of summarizing the

dynamics of (log) aggregate employment. We find that the same property holds for our model.

Specifically, we initiate an algorithm with the model underlying Panels B, D and F of Figure

8. Following Bachmann, in a first step we use this model to generate 89 quarters of simulated

data (the same time span as in the data). Second, we estimate via OLS the following AR(1)

process that relates log aggregate employment to its lag and current total factor productivity p t ,

N t = ν̂0 + ν̂1N t−1 + ν̂2p t . (24)

With estimates of equation (24) in hand, it is possible to back out a series for productivity

{p t } that replicates the path of log aggregate employment {N t }. Since the resultant sequence {p t }

may not be consistent with the data generating process assumed, in a final step we re-estimate

the productivity process and re-initialize the model with this updated process. These steps are

repeated until the moments of the implied productivity series are consistent with the

parameterization assumed. In practice, the AR(1) specification in (24) fits the data closely (the

R-squared of the regression is 0.9985), and so the algorithm converges quite quickly, after just a

few iterations. 15

Figure 9 illustrates the results. The model yields an aggregate steady state employment

notably more variable than its empirical counterpart. By the trough of the two NBER-dated

recessions, the model-generated steady state employment has fallen 5 log points more, for

instance. In the wake of the downturns, the model’s steady state employment also recovers more

quickly. In the 6-8 quarters after the Great Recession, for instance, the model’s steady state

employment rises almost 10 log points. Its empirical counterpart increases by less than half that

in that period.

15

The i.i.d. innovations of the implied productivity process have standard deviation 0.0062, which is more than

twice that used in the process that underlies Figures 2-4. But, throughout Figures 2-4, we wanted to hold the

aggregate productivity process fixed, and a standard deviation of 0.0062 would yield much larger fluctuations in

model-generated employment than ever seen in the data.

20

3. Summary and discussion

[To be completed.]

21

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23

Figure 1. Ss policies in the presence of fixed and linear adjustment frictions

A. Fixed costs B. Linear costs

Notes:

24

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Figure 2. Impulse responses of aggregate employment in selected parameterizations: Fixed costs

A. Quarterly inaction rate ≈ 52.5% B. Quarterly inaction rate ≈ 67% C. Quarterly inaction rate ≈ 80%

8%

8%

8%

7%

7%

7%

6%

6%

6%

5%

5%

5%

4%

4%

4%

3%

3%

3%

2%

2%

2%

1%

1%

1%

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

Actual Mandated Flow steady state

Actual Mandated Flow steady state

Actual Mandated Flow steady state

Notes:

25

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Figure 3. Impulse responses of aggregate employment in selected parameterizations: Linear costs

A. Quarterly inaction rate ≈ 52.5% B. Quarterly inaction rate ≈ 67% C. Quarterly inaction rate ≈ 80%

8%

8%

8%

7%

7%

7%

6%

6%

6%

5%

5%

5%

4%

4%

4%

3%

3%

3%

2%

2%

2%

1%

1%

1%

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

Actual Mandated Flow steady state

Actual Mandated Flow steady state

Actual Mandated Flow steady state

Notes:

26

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Figure 4. Impulse responses of aggregate employment in selected parameterizations: Asymmetric linear costs

A. Quarterly inaction rate ≈ 52.5% B. Quarterly inaction rate ≈ 67% C. Quarterly inaction rate ≈ 80%

i. Pure hiring cost

8%

7%

8%

7%

8%

7%

6%

6%

6%

5%

5%

5%

4%

4%

4%

3%

3%

3%

2%

2%

2%

1%

1%

1%

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

Actual Mandated Flow steady state

Actual Mandated Flow steady state

Actual Mandated Flow steady state

ii. Pure firing cost

8%

7%

8%

7%

8%

7%

6%

6%

6%

5%

5%

5%

4%

4%

4%

3%

3%

3%

2%

2%

2%

1%

1%

1%

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

Actual Mandated Flow steady state

Actual Mandated Flow steady state

Actual Mandated Flow steady state

27

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Figure 5. Impulse responses of aggregate employment in selected parameterizations: Search costs

A. Quarterly inaction rate ≈ 52.5% B. Quarterly inaction rate ≈ 67% C. Quarterly inaction rate ≈ 80%

2.00%

2.00%

2.00%

1.50%

1.50%

1.50%

1.00%

1.00%

1.00%

0.50%

0.50%

0.50%

0.00%

0 5 10 15 20

Quarters since shock

0.00%

0 5 10 15 20

Quarters since shock

0.00%

0 5 10 15 20

Quarters since shock

Actual Steady state Frictionless

Actual Steady state Frictionless

Actual Steady state Frictionless

Notes:

28

Figure 6. **Aggregate** employment in the QCEW by sample restriction

Log aggregate employment

18.8

18.6

18.4

18.2

18

17.8

17.6

Published (all states) Microdata (avail. states)

Microdata (continuers) Microdata (sample)

17.4

1992 1996 2000 2004 2008 2012

Notes:

29

Figure 7. Actual and flow steady state log aggregate employment

Log aggregate employment (deviation from quadratic trend)

0.1

0.08

0.06

0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

Actual employment Flow steady state

-0.1

1992 1996 2000 2004 2008 2012

Notes:

30

Deviation since innovation

Deviation since innovation

Deviation since innovation

Deviation since innovation

Deviation since innovation

Deviation since innovation

Figure 8. Descriptive impulse responses of employment and firm size flows: Data versus model

2.5%

A. **Employment**: Data B. **Employment**: Model

2.5%

2.0%

2.0%

1.5%

1.5%

1.0%

1.0%

0.5%

0.5%

0.0%

0 2 4 6 8 10 12

Quarters since innovation

0.0%

0 2 4 6 8 10 12

Quarters since innovation

Actual

Flow steady state

Actual

Flow steady state

0.8%

C. Outflow probability: Data D. Outflow probability: Model

8.0%

0.6%

0.4%

6.0%

4.0%

0.2%

2.0%

0.0%

0.0%

-0.2%

-2.0%

-0.4%

-4.0%

-0.6%

-6.0%

-0.8%

0 2 4 6 8 10 12

Quarters since innovation

-8.0%

0 2 4 6 8 10 12

Quarters since innovation

1

Figure 9. Model-implied time series for flow steady state log aggregate employment

Log aggregate employment (deviation from quadratic trend)

0.15

0.1

0.05

0

-0.05

-0.1 Actual (model = data)

Flow steady state (data)

Flow steady state (model)

-0.15

1992 1996 2000 2004 2008 2012

32

Appendices

A. Laws of motion for the firm size distribution

To derive the laws of motion for the density of employment across firms stated in the main text

for the fixed and linear cost models, we require notation for several distributions. As in the main

text, we denote the densities of employment, lagged employment and mandated employment by

h , h −1 and h ∗ . Conventionally, we will refer to their respective distribution functions by

analogous upper-case letters, H, H −1 and H ∗ . In addition, however, we require notation for the

distributions of mandated employment conditional on lagged employment, which we denote by

H ∗ (ξ|ν) = Pr(n ∗ < ξ|n −1 = ν), and the distribution of lagged employment conditional on

mandated employment, denoted by H(ν|ξ) = Pr(n −1 < ν|n ∗ = ξ). Of course, the latter are

related by Bayes’ rule, h(ν|ξ)h ∗ (ξ) = h ∗ (ξ|ν)h −1 (ν), where lower-case script letters denote

associated density functions. However, we preserve separate notation where it aids clarity.

With this notation in hand, we can use the labor demand policy rules—(1) for the fixed costs

case, (10) for the linear costs case—to construct a law of motion for the distribution function of

actual employment H(n) implied by each type of friction. We then show how these imply the

laws of motion for the density h(n) stated in equations (2) and (11) in the main text.

Fixed costs. Consider a point m in the domain of the employment distribution. We wish to

derive the flows in and out of the mass H(m). To do this, we first derive the flows for a given

lagged employment level n −1 . Then inflows into H(m) are summarized as follows:

1) If m < L(n −1 ), or equivalently n −1 > L −1 (m), then the inflow is equal to H ∗ (m|n −1 ).

2) If m ∈ [L(n −1 ), n −1 ), or equivalently n −1 ∈ (m, L −1 (m)], then the inflow is equal to

H ∗ (L(n −1 )|n −1 ).

Likewise, the outflows from H(m) for a given n −1 can be evaluated as:

3) If m ∈ (n −1 , U(n −1 )], or equivalently n −1 ∈ [U −1 (m), m), then the outflow is equal to

1 − H ∗ (U(n −1 )|n −1 ).

4) If m > U(n −1 ) , or equivalently n −1 < U −1 (m), then the outflows is equal to 1 −

H ∗ (m|n −1 ).

The latter are the flows of mass for a given lagged employment, n −1 . Integrating with

respect to the distribution of lagged employment H −1 (n −1 ) recovers the aggregate flows and

thereby the law of motion for H(m),

ΔH(m) = ∫ H ∗ (m|n −1 )dH −1 (n −1 )

L −1 (m)

m

− ∫ [1 − H ∗ (U(n −1 )|n −1 )]dH −1 (n −1 )

U −1 (m)

U −1 (m)

− ∫ [1 − H ∗ (m|n −1 )]dH −1 (n −1 ).

L −1 (m)

+ ∫ H ∗ (L(n −1 )|n −1 )dH −1 (n −1 )

m

(25)

33

Linear costs. Likewise, one can use the adjustment rule for the linear costs case, (10), to

construct an analogous law of motion for H under this friction. Again, we first fix a given level

of lagged employment, n −1 , and evaluate inflows to, and outflows from, H(m).

These flows are simpler in the linear costs case. Inflows are given by the following case:

1) If m < n −1 , or equivalently n −1 > m, then the inflow is equal to H ∗ (l(m)|n −1 ).

Similarly, outflows are given by:

2) If m > n −1 , or equivalently n −1 < m, then the outflow is equal to 1 − H ∗ (u(m)|n −1 ).

Following the same logic as in the fixed costs case above, the law of motion for H(m) is

thus given by

ΔH(m) = ∫ H ∗ (l(m)|n −1 )dH −1 (n −1 )

m

m

− ∫ [1 − H ∗ (u(m)|n −1 )]dH −1 (n −1 ). (26)

Laws of motion for h(n). Differentiating (25) and (26) with respect to m, cancelling terms, and

ν

using Bayes’ rule to note that ∫ h ∗ (ξ|n −1 )h −1 (n −1 )dn

0

−1 = ∫ h(n −1 |ξ)h ∗ (ξ)dn

0

−1 yields the

simpler laws of motion for the density of employment h(n), equations (2) and (11) in the main

text.

B. Proofs of Propositions

Proof of Lemmas 1 and 1′. Here we provide a proof of both versions of Lemma 1 reported in

the main text. The proof closely follows results provided in Caballero and Engel (1999).

Consider a firm that faces, possibly asymmetric, fixed and linear costs of adjustment. The firm’s

problem is

Π(n −1 , x) ≡ max

n

{pxnα − wn − C + I[n > n −1 ] − C − I[n < n −1 ] − c + Δn + + c − Δn −

(27)

+βE[Π(n, x ′ )|x]},

where {C − , C + } are the fixed costs of adjusting down and up, and {c − , c + } the analogous linear

costs. Frictionless employment solves the first-order condition pxαn ∗α−1 − w = 0.

Note that, since idiosyncratic shocks follow a geometric random walk, ln x ′ = ln x + ε ′ x , so

does frictionless employment, ln n ∗′ = ln n ∗ ′

′

+ ε n ∗ where ε n ∗ = ε ′ x /(1 − α).

Let C −/+ = Γ −/+ wn ∗ and c −/+ = γ −/+ w, define z = n/n ∗ and ζ = n −1 /n ∗ , and conjecture

that Π(n −1 , x) = wn ∗ Π̃(ζ). Under the conjecture, one can write

Π̃(ζ) = max { zα

z α − z − Γ+ I[z > ζ] − Γ − I[z < ζ] − γ + (z − ζ) + + γ − (z − ζ) −

+βE [e ε ′

n ∗ Π̃ (e −ε ′

n ∗ z)]}.

We highlight two important aspects of (28). First, the expectation over the forward value is

no longer conditional, since it is taken over ε ′ n ∗, which is i.i.d. Second, the firm’s problem is

simplified to the choice of a number z = n/n ∗ for each realization of the single state variable

ζ = n −1 /n ∗ .

ν

(28)

34

An Ss policy will thus stipulate that z = ζ for intermediate values of ζ ∈ [1/U, 1/L], and

will set z = 1/u whenever ζ < 1/U , and z = 1/l whenever ζ > 1/L . Mapping back into

employment terms, we have

n ∗ /u if n ∗ > U ⋅ n −1 ,

n = { n −1 if n ∗ ∈ [L ⋅ n −1 , U ⋅ n −1 ],

(29)

n ∗ /l if n ∗ < L ⋅ n −1 .

Noting that the special case of fixed costs implies u = l, while pure linear costs imply l = L <

U = u, establishes part a) of the result.

To establish part b), note that the probability of a desired log employment adjustment of size

less than δ can be written, in general, as

Pr(ln(n ∗′ ′

/n) < δ|n) = Pr(ε n ∗

′

< δ + ln z |n) = ∫ Pr(ε n ∗

< δ + ln z |n, z) dΖ(z|n), (30)

where Ζ(z|n) denotes the distribution function of z given n. In the context of the canonical

′

model, however, (30) simplifies. First, ε n ∗ is independent of n since the former is i.i.d. Second, z

is also independent of n. To see this, note first that if a firm adjusts this period, its choice of z is

uninformed by n—it sets z = 1/u or z = 1/l. If the firm sets n = n −1 but adjusted last period,

then it sets ln z = ln n −1 − ln n ∗ = ln z −1 − ε n ∗ and z −1 is 1/u or 1/l . Thus, z is again

independent of n. More generally, suppose the firm last adjusted T periods ago, that is, n =

n −1 = ⋯ = n −T and z −T = 1/u or 1/l. Then, ln z = ln n −T − ln n ∗ = ln z −T − ∑T−1

. Each

term here is independent of n = n −T . Equation (30) therefore collapses to

which does not depend on n.

Pr(ln(n ∗′ ′

/n) < δ|n) = ∫ Pr(ε n ∗

t=0

ε ∗ n−t

< δ + ln z |z) dΖ(z), (31)

Proof of Proposition 1. Denoting log employment by n, the adjustment rules take the form

L(n) = n − λ and U(n) = n + υ for λ > 0 and υ > 0. The flow steady state density of log

employment is then defined by

1 − H(n + λ|n) + H(n − υ|n)

ĥ(n) ≡

1 − H ∗ (n + υ|n) + H ∗ (n − λ|n) h∗ (n), (32)

where H ∗ (ξ|ν) ≡ Pr(n ∗ < ξ|n −1 = ν) and H(ν|ξ) ≡ Pr(n −1 < ν|n ∗ = ξ). The property of

the canonical model noted in result b) of Lemma 1, that n ∗ − n −1 is independent of n −1 ,

implies that

H ∗ (ξ|ν) = Pr(n ∗ − n −1 < ξ − ν) ≡ H̃ ∗ (ξ − ν). (33)

This implies that the probability of adjusting away from n is independent of n,

1 − H ∗ (n + υ|n) + H ∗ (n − λ|n) = 1 − ∫ h̃∗ (z)dz ≡ φ. (34)

Now consider the probability of adjusting to n. Using Bayes’ rule, we can write this as

υ

−λ

35

n+λ

1 − H(n + λ|n) + H(n − υ|n) = 1 − ∫ h ∗ (n|ν) h −1(ν)

h ∗ (n) dν

Piecing this together, we have

Multiplying both sides by n and integrating yields

N̂ ≡ ∫

∞

−∞

υ

n−υ

n+λ

= 1 − ∫ h̃∗ (n − ν) h −1(ν)

h ∗ (n) dν

n−υ

υ

= 1 − ∫ h̃∗ (z) h −1(n − z)

h ∗ dz.

(n)

−λ

(35)

ĥ(n) = h∗ (n) − ∫ h̃∗ (z)h −1 (n − z)dz

−λ

υ

. (36)

1 − ∫ h̃∗ (z)dz

nĥ(n)dn

= N∗

= N∗

= N∗

φ − 1 φ ∫

−λ

∞

−∞

υ

υ

∫ nh̃∗ (z)h −1 (n − z)dzdn

−λ

∞

φ − 1 φ ∫ h̃∗ (z) ∫ nh −1 (n − z)dn dz

−λ

υ

φ − 1 φ ∫ h̃∗ (z)(N −1 + z)dz

−λ

= N∗

φ − 1 − φ

φ

N −1 − 1 υ

φ ∫ zh̃∗ (z)dz.

−λ

Since there is a constant-N ∗ state prior to the aggregate shock, aggregate log employment is

constant and equal to aggregate flow steady state employment, N −1 = N −2 = N̂−1 . Imposing

this and solving for N̂−1 yields

N̂−1 = N ∗

−1

υ

−∞

(37)

− ∫ zh̃−1

∗ (z)dz. (38)

Now consider a shock to aggregate log mandated employment, ΔN ∗ . On impact this will shift

the mean of the distribution of desired employment adjustments, h̃∗ (⋅), by ΔN ∗ . Starting from

the prior constant-N ∗ state, substitution of (38) into (37) implies

ΔN̂ = ΔN∗

φ

− 1 φ ∫ zΔh̃∗ (z)dz. (39)

−λ

To a first-order approximation around ΔN ∗ = 0,

υ

∫ zΔh̃∗ (z)dz = ∫ zh̃−1

∗ (z − ΔN ∗ )dz − ∫ zh̃−1

∗ (z)dz

−λ

υ

−λ

υ

−λ

≈ − ∫ zh̃−1 (z)dz ΔN ∗

= [1 − φ − υh̃−1

∗ (υ) − λh̃−1

∗ (−λ)]ΔN ∗ .

Since the latter is smaller in magnitude than ΔN ∗ the stated result holds.

−λ

∗ ′

υ

υ

−λ

(40)

36

Proof of Proposition 2. The proof proceeds in the same way as the proof of Proposition 1 above.

The adjustment rules again take the form l(n) = n − λ and u(n) = n + υ for λ > 0 and υ > 0.

The flow steady state density of log employment is then defined by

ĥ(n) ≡ [1 − H(n|n − λ)]h∗ (n − λ) + H(n|n + υ)h ∗ (n + υ)

1 − H ∗ (n + υ|n) + H ∗ . (41)

(n − λ|n)

Since H ∗ (ξ|ν) = Pr(n ∗ − n −1 < ξ − ν) ≡ H̃ ∗ (ξ − ν), the probability of adjusting away from

n is again independent of n,

1 − H ∗ (n + υ|n) + H ∗ (n − λ|n) = 1 − ∫ h̃∗ (z)dz ≡ φ. (42)

Now consider the probabilities of adjusting down and up to n. Using Bayes’ rule, we can write

these as

∞

1 − H(n|n − λ) = ∫ h(ν|n − λ)dν

Piecing this together, we have

ĥ(n) = ∫

H(n|n + υ) = ∫

−λ

−∞

n

∞

υ

−λ

= ∫ h ∗ h −1 (ν)

(n − λ|n)

h ∗ (n − λ) dν

n

∞

= ∫ h̃∗ h −1 (ν)

(n − λ − ν)

h ∗ (n − λ) dν

n

−λ

= ∫ h̃∗ (z) h −1(n − λ − z)

h ∗ dz , and

(n − λ)

−∞

n

−∞

n

h(ν|n + υ)dν

= ∫ h ∗ h −1 (ν)

(n + υ|ν)

h ∗ (n + υ) dν

−∞

n

= ∫ h̃∗ h −1 (ν)

(n + υ − ν)

h ∗ (n + υ) dν

−∞

∞

= ∫ h̃∗ (z) h −1(n + υ − z)

h ∗ dz.

(n + υ)

h̃∗ (z)h −1 (n − λ − z)dz

Multiplying both sides by n and integrating yields

υ

υ

−λ

∞

+ ∫ h̃∗ (z)h −1 (n + υ − z)dz

υ

1 − ∫ h̃∗ (z)dz

(43)

(44)

. (45)

37

−λ

∞

N̂ = 1 φ ∫ h̃∗ (z) ∫ nh −1 (n − λ − z)dn dz + 1 φ ∫ h̃∗ (z) ∫ nh −1 (n + υ − z)dn dz

−∞

−λ

−∞

= 1 φ ∫ h̃∗ (z)(N −1 + λ + z)dz + 1 φ ∫ h̃∗ (z)(N −1 − υ + z)dz

−∞

= N −1 + 1 −λ

∞

φ [λ ∫ h̃∗ (z)dz − υ ∫ h̃∗ (z)dz] + 1 ∞

φ ∫ zh̃∗ (z)dz

−∞

−λ

= N∗

φ − 1 − φ

φ N −1 + 1 φ [λ ∫ h̃∗ (z)dz

−∞

υ

υ

∞

∞

υ

∞

−∞

∞

−∞

− 1 υ

φ ∫ zh̃∗ (z)dz

−λ

− υ ∫ h̃∗ (z)dz] − 1 φ ∫ zh̃∗ (z)dz.

Solving for N̂−1 = N −1 = N −2 in the prior constant-N ∗ state yields

−λ

N̂−1 = N ∗ −1 + λ ∫ h̃−1

∗ (z)dz − υ ∫

−∞

υ

∞

h̃−1

∗

υ

υ

−λ

υ

(46)

(z)dz − ∫ zh̃−1

∗ (z)dz. (47)

Substitution of (47) into (46) implies that a shock to aggregate log mandated employment that

shifts the mean of h̃∗ (⋅) by ΔN ∗ will induce a change in N̂ relative to the prior constant-N ∗

state equal to

ΔN̂ = ΔN∗

φ + 1 φ [λ ∫ Δh̃∗ (z)dz

−∞

To a first-order approximation around ΔN ∗ = 0,

∫

−λ

−∞

∞

−λ

−λ

Δh̃∗ (z)dz = ∫ h̃−1

∗ (z − ΔN ∗ )dz − ∫

−∞

∞

∞

− υ ∫ Δh̃∗ (z)dz

υ

−λ

h̃−1

∗

−∞

∞

∗

(z)dz

∫ Δh̃∗ (z)dz = ∫ h̃−1

∗ (z − ΔN ∗ )dz − ∫ h̃−1 (z)dz

υ

υ

υ

υ

−λ

] − 1 υ

φ ∫ zΔh̃∗ (z)dz

−λ

≈ −h̃−1

∗ (−λ)ΔN ∗ ,

≈ h̃−1

∗ (υ)ΔN ∗ ,

and, as in equation (40) above, ∫ zΔh̃∗ (z)dz ≈ [1 − φ −

−λ

υh̃−1

∗ (υ) − λh̃−1

∗ (−λ)]ΔN ∗ .

Substitution of these into equation (48) yields the stated result, ΔN̂ ≈ ΔN ∗ .

C. Large-firm canonical search and matching model

. (48)

The firm’s problem for this model combines equations (16) and (17) in the main text to obtain:

Π(n −1 , x) ≡ max

n {Apxnα − (1 − η)ωn − c

q(θ) Δn+ + βE[Π(n, x ′ )|x]} ,

1 − η

(50)

where A ≡

1 − η(1 − α) .

To establish Proposition 3 in the main text, it is convenient first to define a notion of quasifrictionless

employment. Lemma 1′′ then extends the homogeneity properties of Lemmas 1 and

1′ to this problem, but with respect to quasi-frictionless employment. Using this homogeneous

problem and a symmetry result summarized in Lemma 2, we are able to prove that the change in

aggregate log flow steady-state employment ΔN̂ is approximately equal to the change in

aggregate log quasi-frictionless employment ΔN ∗ . Proposition 3 follows because ΔN ∗ is simply

related to the change in aggregate log frictionless employment ΔN ∗∗ .

(49)

38

Definitions Quasi-frictionless employment n ∗ solves Apxαn ∗ α−1 ≡ (1 − η)ω ; frictionless

employment n ∗∗ solves pxαn ∗∗ α−1 ≡ w ∗ , where w ∗ denotes the frictionless wage.

Remark The change in aggregate log quasi-frictionless employment ΔN ∗ induced by a change

in aggregate productivity Δ ln p is related to the change in aggregate log frictionless

employment ΔN ∗∗ according to

ΔN ∗ = 1 − ε ω

ΔN ∗∗ , (51)

1 − ε w ∗

where ε ω and ε w ∗ respectively denote the elasticities of the annuitized value of unemployment ω

and the frictionless wage w ∗ to aggregate productivity p.

Lemma 1′′ If (i) ln x′ = ln x + ε x

′

with ε x

′

i.i.d., and (ii) c ∝ ω, then (a) the adjustment triggers

take the form in (10), are linear, l(n) = l ⋅ n and u(n) = u ⋅ n for time-varying l < 1 < u; and

(b) desired (log) employment adjustments, ln(n ∗ /n −1 ), are independent of initial firm size n −1 .

Proof. If vacancy costs are proportional to the annuitized value of unemployment, say c =

γ(1 − η)ω, a conjecture that Π(n −1 , x) = (1 − η)ωn ∗ Π̃(ζ) yields

Π̃(ζ) ≡ max { zα

z α − z − γ

q(θ) (z − ζ)+ + βE [e ε ′

n ∗ Π̃ (e −ε ′

n ∗ z)]}. (52)

Results (a) and (b) follow from the proof to Lemmas 1 and 1′ above.

Lemma 2 If (i) the adjustment triggers are symmetric, − ln l = ln u ≡ μ, and (ii) the distribution

of innovations ε n ∗ is symmetric, E(−ε n ∗) = 1 − E(ε n ∗), then the distribution of desired (log)

employment adjustments ln(n ∗ /n −1 ) is symmetric, H̃ ∗ (−ς) = 1 − H̃ ∗ (ς).

Proof. Note first that the distribution of the desired log change in employment, n ∗ − n −1 ,

conditional on last period’s log gap, z −1 = n −1 − n ∗ −1 , takes the simple form Pr(n ∗ − n −1 <

ς|z −1 ) = E(ς − z −1 ), since ε n ∗ ≡ n ∗ ∗

− n −1 is i.i.d. with distribution function E(⋅). It follows

that the unconditional distribution of n ∗ − n −1 is

μ

μ

H̃ ∗ (ς) = ∫ Pr(n ∗ − n −1 < ς|z −1 ) g(z −1 )dz −1 = ∫ E(ς − z −1 )g(z −1 )dz −1 , (53)

−μ

−μ

where g(z −1 ) is the ergodic density of z −1 . It is simple to verify that E(−ε n ∗) = 1 − E(ε n ∗)

implies H̃ ∗ (ς) = 1 − H̃ ∗ (−ς), provided g(⋅) also is symmetric, which we now establish.

Our strategy is to conjecture that g(⋅) is symmetric and verify that this is implied. Consider

a firm with an initial z −1 = z − ε such that z ∈ (−μ, μ) lies strictly inside the inaction range.

Clearly, this firm migrates to z if it draws ε. Thus, the mass of firms at z this period is given by

g(z) = ∫

z+μ

z−μ

μ

g(z − ε) dE(ε) = ∫ g(y) dE(z − y), (54)

−μ

39

where we have used the change of variable y = z − ε. Under the conjecture that g(y) = g(−y),

one can confirm g(z) = g(−z). To see this, evaluate g(⋅) at −z, use symmetry of E(⋅), a

change of variable ỹ = −y, and standard rules of calculus to obtain

μ

μ

−μ

g(−z) = ∫ g(y) dE(−z − y) = ∫ g(y) dE(z + y) = − ∫ g(−ỹ) dE(z − ỹ)

−μ

−μ

μ

(55)

= ∫ g(−y) dE(z − y).

−μ

Now consider the mass at the lower adjustment barrier, z = −μ. This is comprised of two

parts: first, firms that begin at −μ, draw a negative labor demand shock (ε < 0), and adjust to

remain at −μ; and second, firms that began away from −μ and then migrate there. Thus,

2μ

g(−μ) = E(0)g(−μ) + ∫ g(−μ + ε)dE(−ε) = 1 2μ

0

E(0) ∫ g(−μ + ε)dE(ε) , (56)

0

where the second equality follows from symmetry of E(⋅). A similar argument can be used to

show that the mass at the upper adjustment barrier z = μ satisfies

g(μ) = 1 2μ

E(0) ∫ g(μ − ε)dE(ε) . (57)

0

A conjecture of symmetry g(−μ + ε) = g(μ − ε) is again confirmed, g(−μ) = g(μ). It follows

that g(z) = g(−z) for all z ∈ [−μ, μ], and symmetry of H̃ ∗ (⋅) obtains.

μ

Proof of Proposition 3. The first-order conditions that define the triggers for optimal adjustment

z ∈ {1⁄ u , 1⁄ l} are given by

l 1−α + βD(1⁄ l ; θ) ≡ 1,

u 1−α + βD(1⁄ u ; θ) ≡ 1 + γ

q(θ) , (58)

where D(z; θ) ≡ E [Π̃′ (e −ε ′

n ∗ z)]. The latter satisfies the following recursion

ln(uz)

D(z; θ) = ∫ [e (1−α)ε ′

n ∗ z α−1 − 1 + βD (e −ε ′

n ∗ z; θ)] dE(ε ′ n ∗)

ln(lz)

+ γ

(59)

q(θ) [1 − E(ln(uz))].

We first consider a first-order approximation to the firm’s optimal policies around γ = 0. 16 To

this end, note first that

16

Equation (52) has the form D(z) = C(D, γ)(z), where C is a contraction map on the cross product of the space of

bounded and continuous functions (where D “lives”) and [0, Γ], a closed subinterval of the nonnegative real line

from which γ is drawn. By inspection, this map is continuously differentiable with respect to (w.r.t.) γ ∈ [0, Γ]. It

then follows from Lemma 1 of Albrecht, Holmlund, and Lang (1991) that D is continuously differentiable w.r.t. γ

and satisfies the recursion, D γ (z) = C γ (D, γ)(z) + C D (D γ , D, γ)(z), where C D is the Frechet derivative of C. The

right side of the latter expression defines a(nother) contraction map on a space of bounded and continuous functions.

We have, then, that D γ is bounded and continuous on [0, Γ]. Its calculation in (53) follows.

40

D γ (z; θ) ≈ 1 [1 − E(ln(uz))] + β ∫

q(θ) D

ln(uz)

ln(lz)

γ (e −ε ′

n ∗

z; θ)

dE(ε ′ n ∗)

= 1

(60)

q(θ) [1 − E(ln z)] when γ = 0

Thus we can write D(z; θ) ≈ γ[1 − E(ln z)]/q(θ). Substituting into the first-order conditions

and noting that l = e −λ and u = e υ yields

e −(1−α)λ + β

γ [1 − E(λ)] ≈ 1,

q(θ)

e (1−α)υ + β

γ [1 − E(−υ)] ≈ 1 +

γ

(61)

q(θ) q(θ) .

Next, linearizing the leading terms around λ = 0 and υ = 0, respectively, leads to

−(1 − α)λ + β γ [1 − E(λ)] ≈ 0,

q(θ)

(1 − α)υ + β γ [1 − E(−υ)] ≈

γ

(626)

q(θ) q(θ) .

Imposing β ≈ 1, and E(−ε) = 1 − E(ε) yields λ ≈ υ.

Now return to the relationship between ΔN̂ and ΔN ∗ in equation (48). Time-variation in the

adjustment triggers alters the approximations around small aggregate shocks. Specifically, with

λ ≈ υ ≈ μ, equation (48) becomes

−μ

ΔN̂ ≈ ΔN∗

φ + 1 φ [Δ (μ ∫ h̃∗ (z)dz

−∞

Taking first-order approximations around ΔN ∗ = 0,

−μ

−μ

∞

) − Δ (μ ∫ h̃∗ (z)dz

μ

−μ −1

h̃−1

∗

−∞

)] − 1 μ

φ Δ (∫ zh̃∗ (z)dz

−μ

). (63)

Δ (μ ∫ h̃∗ (z)dz) = μ ∫ h̃−1

∗ (z − ΔN ∗ )dz − μ −1 ∫ (z)dz

−∞

−∞

(64)

≈ [−μ −1 h̃−1

∗ (−μ −1 ) + {H̃−1

∗ (−μ −1 ) − μ −1 h̃−1

∗ ∂μ

(−μ −1 )}

∂ΔN ∗] ΔN∗ ;

similarly,

∞

Δ (μ ∫ h̃∗ (z)dz) = μ ∫ h̃−1

∗ (z − ΔN ∗ )dz − μ −1 ∫

and,

μ

μ

Δ (∫ zh̃∗ (z)dz

−μ

μ

∞

≈ [μ −1 h̃−1

∗

μ

∞

μ −1

h̃−1

∗ (z)dz

(μ −1 ) + {1 − H̃−1

∗ (μ −1 ) − μ −1 h̃−1

∗ ∂μ

(μ −1 )}

∂ΔN ∗] ΔN∗ ;

) = ∫ zh̃−1

∗ (z − ΔN ∗ )dz − ∫

−μ

μ −1

zh̃−1

∗ (z)dz

−μ −1

≈ [1 − φ − μ −1 h̃−1

∗ (μ −1 ) − μ −1 h̃−1

∗ (−μ −1 )

− (μ −1 h̃−1

∗ ∂μ

(−μ −1 )

∂ΔN ∗) + (μ −1h̃−1

∗ ∂μ

(μ −1 )

∂ΔN ∗)] ΔN∗ .

Substituting these approximations back into (63) and cancelling yields

(59)

(65)

41

ΔN̂ ≈ ΔN ∗ + 1 φ {H̃−1

∗ (−μ −1 ) − [1 − H̃−1

∗ ∂μ

(μ −1 )]}

∂ΔN ∗ ΔN∗ . (66)

To complete the proof, note from Lemma 2 that symmetry of the adjustment barriers, and of

E(⋅), implies that H̃ ∗ (⋅) is also symmetric. It follows that H̃−1

∗ (−μ −1 ) − [1 − H̃−1

∗ (μ −1 )] ≈ 0,

and (66) collapses to ΔN̂ ≈ ΔN ∗ . The result then follows from equation (51).

D. Sensitivity analyses

This appendix describes our parameterization of the process for idiosyncratic shocks, contrasts it

with alternatives in the literature, and performs sensitivity analyses on our main results to

reasonable changes in model parameters.

Idiosyncratic shock process. Recall that we specify the process of idiosyncratic productivity as

a geometric AR(1),

ln x t+1 = ρ x ln x t + ε xt+1 , (67)

where ε xt+1 ∼ N(0, σ 2 x ) and t denotes quarters. In what follows, we infer ρ x and σ x using

estimates of the persistence and volatility of idiosyncratic productivity from data of different

frequencies.

Annual data. Abraham and White (2006) use annual plant-level data from the U.S.

manufacturing sector. Their measure of log idiosyncratic productivity would thus map most

naturally to ln ∑4

t=1 x t , the log of the annual sum of quarterly realizations. For tractability,

though, we will interpret the annual data as corresponding to X 4 ≡ ∑4

t=1 ln x t , and flag this

violation of Jensen’s inequality as a caveat to our results.

Abraham and White run least squares regressions of X 4 on X 0 ≡ ∑0

t=−3 ln x t . An average of

the estimated slope coefficients from their weighted and unweighted regressions is

cov(X 4 , X 0 )

= 0.39. (68)

var(X 0 )

Under the assumption that the annual data are generated from the quarterly process (67), one can

relate the latter covariance and variance terms to ρ x and σ x as follows,

and,

cov(X 4 , X 0 ) = cov (∑

4

t=1

0

ln x t , ∑ ln x t )

t=−3

2 (69)

= [ρ x + 2ρ 2 x + 3ρ 3 x + 4ρ 4 x + 3ρ 5 x + 2ρ 6 x + ρ 7 σ x

x ]

1 − ρ2 , x

var(X 0 ) = [4 + 6ρ x + 4ρ 2 x + 2ρ 3 x ]

1 − ρ2 . (70)

x

Solving for the implied quarterly persistence yields ρ x = 0.68. To infer σ x we use Abraham and

White’s estimates to infer that var(X 0 ) = 0.21, which in turn implies that σ x = 0.1034.

Quarterly data. Cooper, Haltiwanger, and Willis (2015) estimate a dynamic labor demand

model using quarterly data. Like us, they specify a geometric AR(1) for idiosyncratic

σ x

2

42

productivity. They estimate a persistence parameter of ρ x = 0.4 and an innovation standard

deviation of σ x = 0.5.

Monthly data. Cooper, Haltiwanger, and Willis (2007) estimate a search and matching

model using monthly data. They specify an idiosyncratic productivity process that follows a

geometric AR(1) at a monthly frequency. Their estimates of the monthly analogues to ρ x and σ x

are ρ x m = 0.395 and σ x m = 0.0449. 17

Under the assumption that quarterly data are generated by the monthly AR(1) estimated by

Cooper, Haltiwanger and Willis, the above logic implies that ρ x can be computed as

ρ x = ρ x m + 2(ρ x m ) 2 + 3(ρ x m ) 3 + 2(ρ x m ) 4 + (ρ x m ) 5

3 + 4ρ x m + 2(ρ x m ) 2 = 0.194. (71)

Likewise, the variance of the innovation to the quarterly process is

σ x 2 = (1 − ρ x 2 ) 3 + 4ρ x m + 2(ρ x m ) 2

1 − (ρ x m ) 2 (σ x m ) 2 = 0.251. (72)

This implies a standard deviation σ x = 0.501.

17

These average Cooper et al.’s results for two specifications of adjustment costs that seem most germane to our

application: (i) a fixed cost to adjust and linear cost to hire, and (ii) a fixed cost to adjust and linear cost to fire.

43

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

E. Additional figures

Figure A. Impulse responses of mandated and frictionless employment (Quarterly inaction rate ≈ 67%)

8%

7%

A. Fixed cost B. Symmetric linear cost

8%

7%

6%

5%

4%

3%

2%

1%

0%

8%

7%

6%

5%

4%

3%

2%

1%

0%

0 5

0 5 10 15 20

10 15 20

Quarters since shock

Quarters since shock

Mandated Frictionless

Mandated Frictionless

C. Pure linear hiring cost D. Pure linear firing cost

8%

7%

6%

6%

5%

5%

4%

4%

3%

3%

2%

2%

1%

1%

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

Mandated

Frictionless

Mandated

Frictionless

44

Deviation from initial steady state

Deviation from initial steady state

Deviation from initial steady state

Figure B. Impulse responses of aggregate employment in selected parameterizations: Hybrid fixed and linear costs

A. Quarterly inaction rate ≈ 52.5% B. Quarterly inaction rate ≈ 67% C. Quarterly inaction rate ≈ 80%

8%

8%

8%

7%

7%

7%

6%

6%

6%

5%

5%

5%

4%

4%

4%

3%

3%

3%

2%

2%

2%

1%

1%

1%

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

0%

0 5 10 15 20

Quarters since shock

Actual Mandated Flow steady state

Actual Mandated Flow steady state

Actual Mandated Flow steady state

Notes:

45