The State Dependent Effectiveness of Hiring Subsidies

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1290 July 2020 The State Dependent Effectiveness of Hiring Subsidies Sebastian Graves Please cite this paper as: Graves, Sebastian (2020). “The State Dependent Effectiveness of Hiring Subsidies,” International Finance Discussion Papers 1290. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2020.1290. NOTE: International Finance Discussion Papers (IFDPs) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

The State Dependent Effectiveness of Hiring Subsidies (cid:3) Sebastian Graves July 1, 2020 Abstract The responsiveness of job creation to shocks is procyclical, while the responsiveness of job destruction is countercyclical. This new (cid:12)nding can be explained by a heterogeneous-(cid:12)rm model in whichhiring costs lead to lumpyemploymentadjustment. The model predicts that policies that aim to stimulate employment by encouraging job creation, such as hiring subsidies, are signi(cid:12)cantly less effective in recessions: These are times when few (cid:12)rms are near their hiring threshold and many (cid:12)rms are near their (cid:12)ring threshold. Policies that target the job destruction margin, such as employment protection subsidies, are particularly effective at such times. Keywords: Labor market frictions, hiring costs, hiring subsidies, employment stabilization policies, time-varying volatility JEL Classi(cid:12)cations: E24, E32, E63 (cid:3)Federal Reserve Board. Email: sebastian.h.graves@frb.gov. I am grateful to William Gamber, Simon Gilchrist,VirgiliuMidrigan,andseminarparticipantsatNYUforhelpfulcomments. Theviewsexpressedin thispaperaresolelythoseoftheauthorandshouldnotbeinterpretedasre(cid:13)ectingtheviewsoftheBoardof Governors of the Federal Reserve System or any other person associated with the Federal Reserve System.

1 Introduction Aggregate employment growth can be decomposed into the contributions from job creation, the increase in employment coming from expanding or entering establishments, and job destruction, the decrease in employment coming from contracting or exiting establishments1. The main contribution of this paper is to show that the relative contribution of job creation and job destruction to changes in aggregate employment is not constant over the business cycle. Job creation is signi(cid:12)cantly more responsive to aggregate shocks in expansions, while job destruction is more responsive in recessions. This time-varying responsiveness has important implications for the effectiveness of various labor market policies at different stages of the business cycle. I begin by using panel data on job creation and destruction rates at the state-level to show that job creation and destruction exhibit signi(cid:12)cant time-varying responsiveness. Using a predicted employment growth strategy, as inBartik (1991), I showthat the relativecontribution of job creation and job destruction to changes in employment growth varies signi(cid:12)cantly over the business cycle. Consider a shock that raises employment. In states where annual employment growth is 3%, half of the increase in employment comes from higher job creation and half from lower job destruction. However, when annual employment growth is -3%, the contribution of lower job destruction rises to two-thirds and that of higher job creation falls to one-third. An implication of this time-varying responsiveness is that the job creation rate should be more volatile when employment growth is high, and the job destruction rate should be more volatile when employment growth is low. In the state-level data I (cid:12)nd exactly this: the volatility of the job creation rate is roughly 60% higher at the 95th percentile of the employment growth distribution than at the 5th percentile. To understand the causes and implications of this time-varying responsiveness, I study a heterogeneous-(cid:12)rm business cycle model with lumpy employment adjustment. In the model, employment adjustment is lumpy because (cid:12)rms face per-worker hiring costs, while (cid:12)ring workers is costless. Such kinked adjustment costs lead to an inaction region in (cid:12)rms’ policy functions: for a range of productivity levels, (cid:12)rms keep their employment unchanged. This model is capable of generating time-varying responsiveness of job creation and destruc- 1These de(cid:12)nitions were proposed by Davis, Haltiwanger, and Schuh (1998). 1

tion because of movements in the underlying distribution of (cid:12)rms over the business cycle. In an expansion, more (cid:12)rms are either hiring workers or are near their hiring threshold, and fewer (cid:12)rms are (cid:12)ring or near their (cid:12)ring threshold. This makes the job creation rate more responsive to either aggregate shocks or unexpected policy changes than it would be in a recession. The opposite is true for the job destruction rate. The model with lumpy employment adjustment is able to quantitatively match the timevarying responsiveness of job creation and destruction seen in the state-level data. I show that the presence of adjustment frictions is crucial: in a frictionless model, where there is no inaction and all (cid:12)rms are either hiring or (cid:12)ring each period, the responsiveness of job creation and destruction to aggregate shocks is acyclical. I then investigate the aggregate implications of time-varying responsiveness by matching the model to US employment data from 1977 to the present. Due to the sharp decline in employment associated with the COVID-19 pandemic, the model implies that the job creation rate is currently almost entirely unresponsive, while the job destruction rate is almost three times as responsive as usual. In short, job destruction is currently the only relevant margin for the vast majority of (cid:12)rms’ employment decisions. In the (cid:12)nal section I investigate the policy implications of this time-varying responsiveness by estimating the impact of unexpected hiring subsidies or employment protection subsidies2 at different points in time. The effectiveness of these policies is highly state-dependent. The modelimpliesthathiringsubsidies,whichoperateatthejobcreationmargin,aresigni(cid:12)cantly less effective at stimulating employment during recessions. Employment protection subsidies (or (cid:12)ring taxes), which operate at the job destruction margin, are signi(cid:12)cantly more effective than normal at these times. This suggests that providing incentives for (cid:12)rms to retain their existing employees is likely the most effective way to support employment levels during the COVID-19 pandemic. Indeed, the Paycheck Protection Program in the US tries to do exactly this for small businesses. Policies that attempt to stimulate new hiring would likely be entirely ineffective at this time. 2I de(cid:12)ne an employment protection subsidy as a payment made to (cid:12)rms whose employment does not contract. 2

1.1 Literature Review There is a large literature studying models of lumpy employment adjustment. The model in this paper is related to that in Hopenhayn and Rogerson (1993). However, their paper only studies the steady-state implications of adjustment frictions in the form of a (cid:12)ring tax, while I focus on the cyclical implications of lumpy adjustment caused by hiring costs. My model is also related to the more recent multiple-worker search and matching models of Elsby and Michaels (2013) and Fujita and Nakajima (2016). In those papers the adjustment friction takes the form of vacancy posting costs, implying that the cost of hiring a worker is time-varying, as it depends on the probability that a vacancy is (cid:12)lled. In contrast to these search models, in this paper the cost of hiring a worker is constant over time3. In this paper the focus is on the time-varying responsiveness of job creation and destruction rates over the business cycle. The mechanism in this paper is related to that in Foote (1998), which studies the implications of trend employment growth for the relative volatility of job creation and destruction rates. His paper argues that the high relative volatility of job destruction in the manufacturing sector is explained by the fact that the manufacturing sector in the US is in a secular decline, and consequently relatively more (cid:12)rms are close to the job destruction threshold than the job creation threshold. The model in this paper is consistent with the empirical evidence on employment adjustment put forward in Caballero, Engel, and Haltiwanger (1997). They use micro-data from the Longitudinal Research Database (LRD) to characterize the employment adjustment process of manufacturing establishments. They show that employment adjustment is characterized by both frequent inaction and an increasing adjustment hazard: (cid:12)rms respond more to large deviations of employment from their target level than small ones. In Appendix D I show that (cid:12)rms in my model adjust their employment in exactly this fashion. This paper is also related to Bachmann, Caballero, and Engel (2013) and Berger and Vavra (2015). These papers show that aggregate investment and durable consumption are signi(cid:12)cantly less responsive to shocks in recessions. The key difference between the case of employment and either investment or durable consumption is that the establishment-level employment growth distribution is symmetric, implying that the job destruction and job creation margins are equally important for aggregate employment dynamics. Hence, while job creation is less responsive in recessions, job destruction is more responsive. 3Consistent with the evidence cited in Christiano, Eichenbaum, and Trabandt (2016). 3

2 Empirical Evidence of Time-Varying Responsiveness In this section, I provide evidence of the time-varying responsiveness of job creation and destruction using two complementary approaches. First, I use a local labor demand shock approach, following Bartik (1991). I construct a measure of predicted state-level employment growth, based on local industry shares, and national variation in industry employment growth rates. I then investigate the extent to which this predicted employment growth feeds through into changes in job creation or job destruction, and whether this pass-through varies over time. The second approach is based on conditional heteroskedasticity in job creation and destruction rates. Both methods lead to the same conclusions: job creation is around 60% more volatile at the 95th percentile of the state-level employment growth distribution than when it is at the 5th percentile of the distribution. The opposite is true of job destruction. 2.1 Approach 1: Bartik Method In this section, I estimate the response of state-level job creation and destruction to changes in predicted employment growth, constructed following Bartik (1991). I construct predicted employment growth for state i at time t using industry-level employment growth rates in the remainder of the country at time t and weights based on each industry’s employment share in state i in a given base period: K ‚ B (cid:16) φ gN (2.1) i;t i;k;(cid:28) (cid:1)i;k;t k(cid:16)1 where φ is the employment share of industry k in state i in the base year (cid:28) (cid:16) 1990, i;k;(cid:28) and gN is national employment growth in industry k excluding state i in year t. The (cid:1)i;k;t Bartik method predicts high employment growth in a state if the industries that the state has specialized in are growing fast in the rest of the country4. Using this measure of predicted employment growth, I then use the following regression 4This method has become popular as a method of identifying exogenous changes in labor demand at the state-level. For example, Notowidigdo (2011) or Charles, Hurst, and Notowidigdo (2012). Goldsmith- Pinkham, Sorkin, and Swift (2018) provides a thorough discussion of the Bartik approach. 4

speci(cid:12)cation to estimate time-varying responsiveness: ∆Y (cid:16) (cid:11) (cid:0)(cid:13) (cid:0)(cid:12) B (cid:0)(cid:12) B (cid:4)gN (cid:0)(cid:0)1Z (cid:0)ϵ (2.2) i;t i t 0 i;t 1 i;t i;t(cid:1)1 i;t(cid:1)1 i;t where ∆Y is the change in the state’s job creation rate, job destruction rate, or employment i;t growth, gN is lagged state employment growth and Z is a vector of control variables. i;t(cid:1)1 i;t(cid:1)1 The main coefficient of interest is (cid:12) . This shows how the impact of a change in predicted 1 employment growth on the outcome variable is affected by the current cyclical position of the state, measured by local employment growth in the previous year. I estimate these regressions using annual data at the state-level from the Census Bureau’s Business Dynamics Statistics (BDS) database. I construct the Bartik instrument using annual data on state-industry employment from the Quarterly Census of Employment and Wages (QCEW). Appendix B gives more details on the data used, the construction of the Bartik instrument, and the controls used to estimate equation 2.2. Table 1a shows the results. The (cid:12)rst columns shows the effect of the Bartik shock on overall employment growth. As might be expected, on average the change in employment growth predicted by the Bartik measureiscorrect: employmentgrowthmovesone-for-onewiththeBartikshock. Thesecond row shows that there is no evidence of time-varying responsiveness of overall employment ^ ^ growth. The third and fourth rows of the table use the estimated values (cid:12) and (cid:12) to com- 0 1 pare estimates of the response at the mean level of lagged employment growth and when employment growth is one standard deviation higher. The (cid:12)fth row compares the responsiveness at the 5th and 95th percentiles of the state-level employment growth distribution. The responsiveness of employment growth is acyclical. Thenexttwocolumnsshowthattheacyclicalresponsivenessofemploymentmaskssigni(cid:12)cant ^ time-varying responsiveness in job creation and destruction. The estimate of (cid:12) is positive 1 andsigni(cid:12)cantlydifferentfromzeroforbothjobcreationandjobdestruction. Aonestandard deviation increase in employment growth from its mean level increases the responsiveness of thejobcreationratebyjustunder20%, andreducestheresponsivenessofthejobdestruction rate by a similar amount. Looking at the tails of the distribution: job creation is around 60% more responsive at the 95th percentile compared to the 5th percentile of the employment growth distribution. The opposite is true of job destruction. 5

Table 1: Empirical Evidence of Time-Varying Responsiveness (a) Bartik Method ∆gN ∆JC ∆JD i;t i;t i;t B 0.99 0.40 -0.59 i;t (0.20) (0.13) (0.14) B (cid:4)gN -0.003 0.025 0.027 i;t i;t(cid:1)1 (0.017) (0.010) (0.010) (cid:12) ^ (cid:0)(cid:12) ^ g(cid:22)N 0.99 0.45 -0.53 0 1 (cid:12) ^ (cid:0)(cid:12) ^ pg(cid:22)N (cid:0)1SDq 0.98 0.54 -0.44 0 1 logp(cid:27)95q -0.03 0.60 -0.55 (cid:27)5 Observations 1173 1173 1173 R2 0.415 0.305 0.369 (b) Conditional Heteroskedasticity |∆gN| |∆JC | |∆JD | i;t i;t i;t gN 0.015 0.080 -0.101 i;t(cid:1)1 (0.028) (0.013) (0.024) logp(cid:27)95q 0.06 0.63 -0.64 (cid:27)5 Observations 1836 1836 1836 R2 0.17 0.06 0.16 Notes: Panel (a) uses data from 1990-2014. Panel (b) uses data from 1977- 2014. Standard errors clustered by state. The mean and standard deviation of state-level employment growth are 2.1% and 3.3%. The 5th and 95th percentilesofthestate-levelemploymentgrowthdistributionare-3.4%and7.1%. 6

2.2 Approach 2: Conditional Heteroskedasticity An alternative approach to measuring time-varying responsiveness is simply to investigate whether or not the volatility of job creation and destruction is state dependent. If the job creation rate is particularly responsive during expansions, then the volatility of the job creation rate should be higher at such times. The opposite should be true of the job destruction rate. To test this prediction, I regress the absolute value of changes in a variable on its lagged value and the lagged value of employment growth in that state, using the same state-level panel of job creation and destruction rates from the BDS database. |∆Y | (cid:16) (cid:11) (cid:0)(cid:13) (cid:0)(cid:12) |∆Y |(cid:0)(cid:12) gN (cid:0)ϵ (2.3) i;t i t 0 i;t(cid:1)1 1 i;t(cid:1)1 i;t Again, the main parameter of interest is (cid:12) . In this case, (cid:12) measures the extent to which 1 1 the absolute size of changes in a variable are related to lagged employment growth. Table 1b shows the results of estimating this regression for employment growth, job creation, and job destruction. The results mirror those using the Bartik approach: There is no evidence of time-varying volatility of total employment growth, the job creation rate is more volatile when lagged employment growth is high, and the job destruction rate is more volatile when lagged employment growth is low. The magnitude of these effects is also similar to those from the Bartik approach: The (cid:12)nal row of Table 1a shows that the volatility of the job creation rate isaround60%higheratthe95thpercentilecomparedtothe5thpercentileoftheemployment growth distribution, while the opposite is true for the job destruction rate5. Overall, the results from Section 2.1 and Section 2.2 are consistent. When employment growth is high, the job creation rate is much more responsive to shocks than the job destruction rate. The opposite is true when employment growth is low. An explanation for this (cid:12)nding is that in expansions many (cid:12)rms are near a hiring threshold, where they decide to hire extra workers, while in recessions more (cid:12)rms are close to a (cid:12)ring threshold, where they decide to lay off employees. In the next section, I study a model which formalizes this argument. 5I estimate these counterfactuals at the mean value of the (cid:12)xed effects estimated across states, holding the lagged value of volatility at its average level. 7

3 A Model of Lumpy Employment Adjustment In this section, I study a heterogeneous-(cid:12)rm business cycle model, in order to understand the causes and implications of the time-varying responsiveness of job creation and job destruction rates. In the baseline model, (cid:12)rms are subject to linear hiring costs, which leads to infrequent employment adjustment. By comparing this model to one in which employment adjustment is frictionless, I will show that infrequent adjustment is crucial for matching the empirical evidence presented in Section 2. Below I describe the (cid:12)rm problem, then that of the representative household, before de(cid:12)ning an equilibrium and discussing computational issues. 3.1 The Firm’s Problem The economy consists of a continuum of regions, each containing a continuum of (cid:12)rms. The mass of (cid:12)rms and regions is normalized to one. Each (cid:12)rm operates a decreasing returns to scale production function using only labor, n, as an input. Firms are subject to aggregate, regional, and idiosyncratic productivity shocks. The production function is: y (cid:16) Az z n(cid:11) (3.1) r i where A, z , and z denote aggregate, region, and idiosyncratic productivity, respectively. r i All productivity processes are AR(1) in logs. The (cid:12)rm’s idiosyncratic state variables are their employment level, n, and their idiosyncratic and regional productivity, z and z . The i r aggregate state variables are the distribution of (cid:12)rms over their idiosyncratic states, (cid:22), and aggregate productivity, A. I denote the aggregate state by S (cid:16) pA;(cid:22)q. Firm employment is predetermined. After productivity shocks are realized, (cid:12)rms make their employment decision for the next period. Firing workers is costless, but (cid:12)rms are subject to a per-worker hiring cost, (cid:20), paid in units of output. The (cid:12)rm’s problem can be written recursively as: Vpz ;z ;n;Sq (cid:16) maxAz z n(cid:11) (cid:1)wpSqn(cid:1)gpn;n1q(cid:0) E r(cid:3)pS;S1qVpz1;z1;n1;S1qs (3.2) r i n1 r i z r 1;z i 1;A1 r i subject to gpn;n1q (cid:16) (cid:20)pn1 (cid:1)nq1pn1 ą nq 8

(cid:22)1 (cid:16) (cid:0)pA;(cid:22)q logA1 (cid:16) (cid:26) logA(cid:0)(cid:27) ϵ1 A A A logz1 (cid:16) (cid:26) logz (cid:0)(cid:27) ϵ1 r r r r r logz1 (cid:16) (cid:26) logz (cid:0)(cid:27) ϵ1 i i i i i where ϵ1 , ϵ1, and ϵ1 are iid Np0;1q random variables, wpSq is the wage, and (cid:3)pS;S1q is the A r i stochastic discount factor of the representative household, whose problem is outlined in the nextsection. Thepresenceofthelinearhiringcostmeansthatthe(cid:12)rm’soptimalemployment decision is characterized by two thresholds, npz ;z ;Sq and n(cid:22)pz ;z ;Sq. If employment is r i r i below npz ;z ;Sq then the (cid:12)rm raises employment to this threshold in the next period. If r i employment is above n(cid:22)pz ;z ;Sq then the (cid:12)rm reduces its employment to this threshold. If r i employment is between these thresholds then the (cid:12)rm leaves employment unchanged. The thresholds are de(cid:12)ned by following (cid:12)rst-order conditions: E r(cid:3)pS;S1qV pz1;z1;npz ;z ;Sq;S1qs (cid:16) (cid:20) (3.3) z r 1;z i 1;A1 n r i r i E r(cid:3)pS;S1qV pz1;z1;n(cid:22)pz ;z ;Sq;S1qs (cid:16) 0 (3.4) z r 1;z i 1;A1 n r i r i where E r(cid:3)pS;S1qV pz1;z1;n;S1qs is the expected marginal bene(cid:12)t of a worker to the z r 1;z i 1;A1 n r i (cid:12)rm. 3.2 The Household’s Problem Firms are owned by a continuum of identical households. As in Khan and Thomas (2008), it is sufficient to focus on the (cid:12)rst-order conditions of the household’s problem that determines the equilibrium wage and stochastic discount factor. Households have the following preferences6: 1 (cid:2) N1(cid:0)ϕ (cid:10)1(cid:1)(cid:13) UpC;Nq (cid:16) C (cid:1) (3.5) 1(cid:1)(cid:13) 1(cid:0)ϕ 6As in Greenwood, Hercowitz, and Huffman (1988). 9

Consequently, the stochastic discount factor can be written as: (cid:4) (cid:12)(cid:1)(cid:13) CpS1q(cid:1) NpS1q1(cid:0)ϕ (cid:3)pS;S1q (cid:16) (cid:12)(cid:5) 1(cid:0)ϕ (cid:13) (3.6) CpSq(cid:1) NpSq1(cid:0)ϕ 1(cid:0)ϕ The (cid:12)rst-order conditions of the household’s intra-temporal problem de(cid:12)ne the equilibrium wage: U pC;Nq wpSq (cid:16) (cid:1) N (cid:16) NpSqϕ (3.7) U pC;Nq C The choice of preferences, combined with the fact that labor is predetermined in the model, implies that the wage is also predetermined. This simpli(cid:12)es computation of the model substantially, as discussed in Appendix C. 3.3 Equilibrium De(cid:12)nition A recursive competitive equilibrium of the model is a set of functions tV;n1;w;(cid:3);C;N;(cid:0)u such that: 1. Taking w;(cid:3);(cid:0) as given, n1pz ;z ;n;Sq solves the (cid:12)rm’s problem (3.2) and Vpz ;z ;n;Sq r i r i is the associated value function. 2. Taking w as given, household’s labor supply satis(cid:12)es (3.7). (cid:3) is implied by household consumption and labor supply as in (3.6). 3. The goods market clears: ż CpSq (cid:16) rAz z n(cid:11) (cid:1)(cid:20)pn1pz ;z ;n;Sq(cid:1)nq1pn1pz ;z ;n;Sq ą nqsd(cid:22) r i r i r i 4. The labor market clears: ż NpSq (cid:16) nd(cid:22) 5. The evolution of the distribution, (cid:22)1 (cid:16) (cid:0)pA;(cid:22)q is induced by the policy function n1pz ;z ;n;Sq and the exogenous processes for z , z and A. r i r i 10

Table 2: Parameter Values Parameter Baseline Frictionless Hiring cost (cid:20) 0.47 0 Regional shock volatility (cid:27) 0.0045 0.0025 r Idiosyncratic shock volatility (cid:27) 0.10 0.079 i Aggregate shock volatility (cid:27) 0.0049 0.0039 A Regional productivity persistence (cid:26) 0.97 0.97 r Idiosyncratic productivity persistence (cid:26) 0.97 0.97 i Aggregate productivity persistence (cid:26) 0.974 0.984 A Decreasing returns to scale (cid:11) 0.65 0.65 Discount factor (cid:12) 0.99 0.99 Risk aversion (cid:13) 1 1 Elasticity of labor supply 1 2 2 ϕ Disutility of labor supply 0.78 0.73 3.4 Equilibrium Calibration and Computation The model period is one quarter. Table 2 summarizes the parameter values for the baseline and frictionless versions of the model. The key parameters governing employment adjustment in the model are the hiring cost, (cid:20), and the dispersion of idiosyncratic and regional productivity shocks, (cid:27) and (cid:27) . In the baseline model, I set (cid:20) equal to 60% of the quarterly i r wage in steady-state, in line with the evidence provided by Silva and Toledo (2009). This value corresponds broadly to the lower end of estimates of hiring costs in the literature. In both calibrations, I choose (cid:27) to match the standard deviation of annual employment r growth at the state level in the US, equal to 0.012. I choose (cid:27) to match the standard i deviation of annual employment growth among continuing establishments of 0.4 reported in Davis, Haltiwanger, Jarmin, and Miranda (2007). I choose to target only continuing establishments as the model abstracts from (cid:12)rm entry and exit and because entry and exit do not contribute to the volatility of aggregate job creation and destruction rates7. I set (cid:26) (cid:16) (cid:26) (cid:16) 0:97 and choose (cid:26) and (cid:27) to match the persistence and volatility of de-trended r i A A US employment8. I follow Cooper, Haltiwanger, and Willis (2007) in setting the curvature of the production 7See Figure 3 in the Appendix. 8I de-trend quarterly US employment using the HP (cid:12)lter with (cid:21) (cid:16) 105, the parameter used in Shimer (2005) and related papers. 11

function, (cid:11), to 0.65. I set the remaining parameters to conventional values. The discount factor(cid:12) is0.99andIassumethat thehouseholdhaslogpreferences. Isetϕ (cid:16) 0:5, implyinga Frisch elasticity of labor supply of 2. In the Appendix I show that the main results are robust to lower values of the labor supply elasticity or a risk-neutral representative household. I select , the parameter governing the disutility of labor supply, to normalize aggregate employment to 1 in the steady-state of the model. It is not computationally feasible to solve the (cid:12)rm’s problem (3.2), as (cid:22) is an in(cid:12)nite dimensional object. I use the method proposed in Krusell and Smith (1998) and approximate (cid:22) by the (cid:12)rst moment of the employment distribution. Further details of my computational strategy and proof of its accuracy are given in Appendix C. 4 Model Validation: The Importance of Hiring Costs To show the importance of hiring costs in generating time-varying responsiveness, I now replicate the experiments from Section 2 in both versions of the model. As the model does not include industries, it is not possible to exactly replicate the Bartik experiment in the model. However, it is possible to use aggregate employment growth as a measure of predicted (cid:12)rm employment growth, in order to run the same regressions as in Section 2.1. In the model, aggregate employment growth is equal to mean (cid:12)rm employment growth by construction. For the volatility regressions in Section 2.2, it is possible to run the regressions exactly as in the data. Table 3a shows the results of the Bartik regressions for the baseline and frictionless models. The baseline model replicates closely the time-varying responsiveness seen in Table 1a: employment growth does not exhibit time-varying responsiveness, while the responsiveness of job creation is procyclical, and that of job destruction is countercyclical. The (cid:12)nal three rows of the table show that the magnitudes of the time-varying responsiveness are quantitatively very close to those seen in Table 1a. The frictionless model generates almost no time-varying responsiveness. The coefficients on the interaction terms are all close to zero and the (cid:12)nal three rows of the table show that there is no quantitatively signi(cid:12)cant time-varying responsiveness. Without employment adjustment frictions, the relative contribution of job creation and destruction to changes in 12

Table 3: Time-Varying Responsiveness in the Model (a) Bartik Method Baseline Model Frictionless Model ∆gN ∆JC ∆JD ∆gN ∆JC ∆JD i;t i;t i;t i;t i;t i;t B 1.04 0.58 -0.46 0.99 0.54 -0.45 i;t B (cid:4)gN 0.005 0.027 0.022 0.002 0.003 0.000 i;t i;t(cid:1)1 (cid:12) ^ (cid:0)(cid:12) ^ g(cid:22)N 1.04 0.58 -0.46 0.99 0.54 -0.45 0 1 (cid:12) ^ (cid:0)(cid:12) ^ pg(cid:22)N (cid:0)1SDq 1.06 0.67 -0.38 0.99 0.54 -0.45 0 1 logp(cid:27)95q 0.05 0.49 -0.51 0.03 0.05 -0.00 (cid:27)5 (b) Conditional Heteroskedasticity Baseline Model Frictionless Model |∆gN| |∆JC | |∆JD | |∆gN| |∆JC | |∆JD | i;t i;t i;t i;t i;t i;t gN 0.021 0.068 -0.048 0.007 0.012 -0.005 i;t(cid:1)1 logp(cid:27)95q 0.11 0.67 -0.58 0.02 0.06 -0.03 (cid:27)5 Notes: Iruntheregressionsondatageneratedbysimulatingthemodelfor10000 periods. In the model mean employment growth is zero. I use the standard deviation of state employment growth of 3.3% estimated in the data. For the 95th and 5th percentiles of the employment growth distribution I use -5.2% and 5.2% (de-meaning the values of -3.4% and 7.1% from the data). 13

employment growth does not vary over the business cycle. Table 3b shows the results of the volatility regressions for both versions of the model. Once again, thebaselinemodelwithlumpyemploymentadjustmentgeneratestime-varyingvolatility that is almost exactly the same magnitude as seen in the data. In contrast, in the frictionless version of the model the volatility of employment growth, job creation, and job destruction is acyclical. 4.1 What Causes Time-Varying Responsiveness? The previous section showed that the baseline model is able to generate a signi(cid:12)cant degree of time-varying responsiveness of both job creation and destruction rates, whereas the frictionless model fails to do so. Why is this the case? As emphasized by Caballero and Engel (2007), time-varying responsiveness can be decomposed into extensive and intensive margin effects. For example, a positive aggregate productivity shock will increase job creation by increasing the number of (cid:12)rms that increase their employment (the extensive margin) as well as by increasing the job creation of (cid:12)rms who would already have been hiring (the intensive margin). Consequently, the responsiveness of the job creation rate depends on the number of (cid:12)rms already adjusting, as well as the number of (cid:12)rms that are near their hiring threshold. In the baseline model both of these forces contribute to procyclical time-varying responsiveness of the job creation rate and countercyclical time-varying responsiveness of the job destruction rate: in an expansion more (cid:12)rms are either creating jobs or are close to their job creation threshold, while the opposite is true in recessions. Figure 3 shows this in a stylized way in the model by sketching the distribution of (cid:12)rms over the marginal bene(cid:12)t of an extra worker, E r(cid:3)pS;S1qV pz1;z1;n;S1qs. The shaded area in the n i s left tail denotes (cid:12)rms that are (cid:12)ring, while the shaded area on the right denotes (cid:12)rms that are hiring. The unshaded section of the distribution shows (cid:12)rms that keep their employment unchanged. The left panel sketches the distribution in a recession, while the right panel plots the distribution in an expansion. As the distribution shifts over time, it clearly affects both the number of (cid:12)rms creating or destroying jobs, as well as the number that are close to the thresholds. As I will show in Section 5.1, time-varying responsiveness has signi(cid:12)cant implications for the 14

effectiveness of different types of employment stabilization policy at different points in time. Considertheeffectofaone-periodunexpectedhiringsubsidyequalto(cid:28) pernewworker. The effect of this policy in the model is to temporarily lower the hiring cost from (cid:20) to (cid:20)1 (cid:16) (cid:20)(cid:1)(cid:28) for one period. This policy will increase job creation through the intensive and extensive margins described above. Figure 3 predicts that both of these mechanisms will be weaker in a recession than in an expansion. In contrast, policies which aim to stimulate aggregate employment by discouraging job destruction, such as an employment protection subsidy (or (cid:12)ring tax) are likely to be much more potent in a recession than in an expansion. 5 Aggregate Implications The previous section showed that the baseline model is consistent with the cross-sectional evidencefromSection2. Inthissection, Iconsidertheaggregateimplicationsoftime-varying responsiveness. First, I match the model to the US data, and show that the time-varying responsiveness of aggregate job creation and destruction is quantitatively signi(cid:12)cant. I then investigate the implications of this for various policies that are used to support employment during recessions. To match the model to the US data, I (cid:12)nd the particular sequence of aggregate productivity shocks9 such that aggregate employment in the model exactly replicates the path of the cyclical component of US employment from 1977 to the present10. To estimate the degree of time-varying responsiveness of job creation and destruction in the model, I follow Bachmann et al. (2013) in constructing \responsiveness indices", which measure the impact of a one standard deviation aggregate productivity shock on job creation, job destruction, and employment growth at each point in time: RJC (cid:17) JCpexpplogpA q(cid:0)(cid:27) q;(cid:22) q(cid:1)JCpA ;(cid:22) q (5.1) t t A t t t RJD (cid:17) JDpexpplogpA q(cid:0)(cid:27) q;(cid:22) q(cid:1)JDpA ;(cid:22) q (5.2) t t A t t t RN (cid:17) NpexpplogpA q(cid:0)(cid:27) q;(cid:22) q(cid:1)NpA ;(cid:22) q (5.3) t t A t t t 9Bachmann et al. (2013) and Berger and Vavra (2015) use a similar procedure to show time-varying responsiveness of investment and durable consumption. 10I assume that the model is in steady-state in June 1977. In the Appendix, Figure 7 shows the implied path of aggregate productivity in the model and Figures 8 and 9 show that the model exhibits realistic movements in quarterly job creation and destruction rates, as well as the proportion of (cid:12)rms expanding or contracting. 15

Figure 1b plots the responsiveness indices for the baseline model, normalized such that the mean value is equal to one. The baseline model implies a signi(cid:12)cant degree of time-varying responsiveness in aggregate job creation and destruction. The model implies that the job creation rate was almost 40% less responsive during the Great Recession in 2009 than it was in the pre-crisis period. Conversely, job destruction was around 50% more responsive in 2009 than in 2006. Turning to the most recent data, the model implies that the job creation rate is currently almost entirely unresponsive, while the job destruction rate is almost three times as responsive as usual. In short, job destruction is the only relevant margin for the vast majority of (cid:12)rms’ employment decisions in response to the COVID-19 pandemic. Another way of seeing the time-varying responsiveness generated by the baseline model is to plot the response of job creation and destruction to an aggregate shock at different points in time. Figure 2a plots the impulse response function to a positive aggregate productivity shock in the baseline model in the third quarter of 2005 and compares this to the response if the same shock had occurred in the third quarter of 2008. As implied by Figure 1b, during a recession the impact of the shock on job destruction is larger and on job creation is smaller. 5.1 Time-Varying Policy Effectiveness The previous sections have shown that the responsiveness of job creation is procyclical, the responsiveness of job destruction is countercyclical, that this time-varying responsiveness is quantitatively signi(cid:12)cant, and that it is offsetting such that aggregate employment shows little time-varying responsiveness. But if aggregate employment does not exhibit time-varying responsiveness, should macroeconomists care about the implications of lumpy employment adjustment at the microeconomic level? The answer is yes, as the time-varying responsiveness of job creation and destruction has signi(cid:12)cant policy implications. Employment stabilization policies can be categorized into those that aim to encourage job creation, those that aim to discourage job destruction, and those that aim to operate on both margins. The Paycheck Protection Program that the US Small Business Administration (SBA) has initiated in response to the COVID-19 pandemic is an example of a policy that aims to discourage job destruction. The program provides loans to small businesses that will be forgiven \if all employees are kept on the payroll for eight weeks and the money is 16

Figure 1: Time-Varying Responsiveness of Job Creation and Destruction (a) Sketch of Model Distribution 0 (cid:20)1 (cid:20) 0 (cid:20)1 (cid:20) E p(cid:3)1V1q E p(cid:3)1V1q n n Recession Expansion Notes: The distribution sketched is over the expected marginal bene(cid:12)t of an extra worker. When this is above (cid:20) the (cid:12)rm increase its employment. When it is below 0 the (cid:12)rm will (cid:12)re workers. (b) Model-Implied Responsiveness Indices 1.6 1.4 1.2 1.0 0.8 0.6 1980 1985 1990 1995 2000 2005 2010 2015 2020 1 = naeM Job Creation Job Destruction Employment 2.59 in Q2 2020 0.21 in Q2 2020 Notes: Responsiveness indices show the impact on job creation, job destruction and employment of a one SD aggregate productivity shock. The mean response is normalized to one. 17

Figure 2: State Dependence of IRFs and Policy Effectiveness (a) Impulse Response Functions 0.20 0.16 0.12 0.08 0.04 0.00 0 1 2 3 4 5 egnahC tnecreP Job Creation Job Destruction 0.00 Q3 2005 Q3 2008 0.04 0.08 0.12 0.16 0.20 0 1 2 3 4 5 Notes: ImpactonjobcreationandjobdestructionofaoneSDaggregateproductivityshockinanexpansion and in a recession. (b) Time-Varying Policy Effectiveness 1.3 1.2 1.1 1.0 0.9 0.8 1980 1985 1990 1995 2000 2005 2010 2015 2020 1 = naeM Hiring Subsidy Employment Protection Subsidy 3.12 in Q2 2020 0.20 in Q2 2020 Notes: Impact on employment of an unanticipated hiring subsidy or employment protection subsidy equal to 25% of the average quarterly wage. The mean response is normalized to one. 18

used for payroll, rent, mortgage interest, or utilities"11. This policy has similarities with the short-time work schemes that are common in European countries. In such schemes, (cid:12)rms are able to temporarily reduce employee’s working hours, with the government providing income support to these workers12. On the other hand, in previous recessions many employment policies in the US have focused on the job creation margin. For example, the original version of the 2010 Hiring Incentives to Restore Employment (HIRE) Act proposed a $5,000 tax credit for every net new employee hired by small businesses. The New Jobs Tax Credit (NJTC) of 1977-1978 provided a signi(cid:12)cant wage subsidy for (cid:12)rms who increased their employment by more than 2%. To investigate the quantitative impact of time-varying responsiveness for different labor market policies in the model, I consider the impact on aggregate employment of one-period unanticipated policy shocks at each point in time. In particular, I consider the effect on employment of an unexpected hiring subsidy or an unexpected employment protection subsidy equal to 25% of the average quarterly wage13. A hiring subsidy of (cid:28) reduces the cost of increasing employment from (cid:20) to (cid:20)(cid:1)(cid:28). The hiring threshold now satis(cid:12)es: E r(cid:3)pS;S1qV pz1;z1;npz ;z ;Sq;S1qs (cid:16) (cid:20)(cid:1)(cid:28) (5.4) z r 1;z i 1;A1 n r i r i I model an employment protection subsidy as a payment of (cid:28) per worker to any (cid:12)rm that does not decrease their employment level. This changes the (cid:12)ring threshold to : E r(cid:3)pS;S1qV pz1;z1;n(cid:22)pz ;z ;Sq;S1qs (cid:16) (cid:1)(cid:28) (5.5) z r 1;z i 1;A1 n r i r i Figure 2b shows the impact of these policies on aggregate employment at each point in time in the baseline model, with the mean impact normalized to one. As is expected, the impacts of the policies broadly mirror the responsiveness indices shown in Figure 1b. The effectiveness of a hiring subsidy on aggregate employment is signi(cid:12)cantly procyclical, while that of an employment protection subsidy is signi(cid:12)cantly countercyclical. While it is beyond the scope of the model in this paper to study the impact of short-time work schemes, it is likely that they are also particularly effective in recessions, given that they operate on the job destruction margin. 11https://www.sba.gov/funding-programs/loans/coronavirus-relief-options/paycheck-protection-program 12For more detail on such schemes, see Hijzen and Venn (2011). 13The impact of an employment protection subsidy on the (cid:12)ring threshold is exactly equivalent to a (cid:12)ring tax of the same magnitude. 19

6 Conclusion In this paper I have used state-level data to show that job creation and destruction rates exhibit signi(cid:12)cant time-varying responsiveness. The job creation rate is most responsive in expansions, while the job destruction rate is most responsive in recessions. This timevarying responsiveness is quantitatively signi(cid:12)cant: a one standard deviation in state-level employment growth leads to an almost 20% increase in the responsiveness of job creation and a 20% decline in the responsiveness of job destruction. I have shown that a heterogeneous-(cid:12)rm business cycle model with lumpy employment adjustment is capable of explaining this new fact. The job creation rate is more responsive in expansions as these are times when more (cid:12)rms are either already hiring or are near their hiring threshold. The opposite is true for the job destruction rate. The model suggests that the sharp decline in employment induced by the COVID-19 pandemic means that the aggregate job creation rate is currently almost entirely unresponsive, while the job destruction rate is signi(cid:12)cantly more responsive than usual. This implies that providing incentives for (cid:12)rms to retain their existing employees is likely the most effective way to support employment levels during the COVID-19 pandemic. Infuturework, Iplantouseasimilarmodeltostudytheimpactofundertakinglabormarket reforms at different times in the business cycle. The direct effect of removing (cid:12)ring costs is that it is cheaper for (cid:12)rms in the left-tail of the distribution to (cid:12)re workers. The indirect effect is that (cid:12)rms in the right-tail have a larger incentive to hire workers, as they no longer expect to have to pay (cid:12)ring costs if they need to (cid:12)re those workers in the future. My model would suggest that the direct effect is likely to be larger in recessions, and consequently that the short-run impact on employment of removing (cid:12)ring costs may be most negative at these times. 20

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Appendix For Online Publication A Supplementary Figures Figure 3: Job Creation and Destruction Rates: Continuing vs. Entry/Exit 6 5 4 3 2 1 1995 2000 2005 2010 2015 2020 % JC (Continuing) JD (Continuing) JC (Entry) JD (Exit) Figure 4: Fraction of Establishments Adjusting Employment 24.0 22.5 21.0 19.5 18.0 1995 2000 2005 2010 2015 2020 stnemhsilbatsE fo % Expansion Rate Contraction Rate Notes: Data from the BLS Business Employment Dynamics database. 24

Figure 5: State and National Employment Growth 8 4 0 4 8 1995 2000 2005 2010 2015 tnemyolpmE egnahC % Notes: Light-blue lines depict state-level annual employment growth. Black line depicts national annual employment growth. Data is total nonfarm employment from the BLS Current Employment Statistics Database. Figure 6: Cyclical Component of US Employment 4 0 4 8 12 1980 1985 1990 1995 2000 2005 2010 2015 2020 dnerT morf tnemyolpmE fo noitaiveD % Notes: Cyclical component of quarterly US employment de-trended using the Hodrick-Prescott (cid:12)lter with (cid:21)(cid:16)1e5. B Data For Section 2.1 and Section 2.2 I use state-level data on job creation and destruction rates derived from establishment-level data from the US Census Bureau’s Business Dynamics 25

Figure 7: Model-Implied Aggregate Productivity Series 1.04 0.96 0.88 0.80 1980 1985 1990 1995 2000 2005 2010 2015 2020 ytivitcudorP etagerggA Baseline Baseline - Risk Neutral Baseline - High Frictionless Figure 8: Model-Implied Job Creation and Destruction Rates 10.0 7.5 5.0 2.5 0.0 1980 1985 1990 1995 2000 2005 2010 2015 2020 tnemyolpmE fo % Job Creation Rate Job Destruction Rate Statistics (BDS) database. This provides annual data from 1977 to 2014. ForSection2.1, Ialsorequirestate-industryemploymentdata. Thisisavailableonanannual basis from the BLS Quarterly Census of Employment and Wages (QCEW) database from 1990 to 2016. In both cases I use data on the 50 states of the US as well as the District of Columbia. 26

Figure 9: Model-Implied Expansion and Contraction Rates 50 40 30 20 10 0 1980 1985 1990 1995 2000 2005 2010 2015 2020 stnemhsilbatsE fo % Expansion Rate Contraction Rate In Section 5 I use total non-farm payrolls from the BLS14 as my measure of US employment. B.1 Details on Bartik Approach To construct a Bartik measure of predicted employment growth, I use state-industry employment data at the 3-digit NAICS level. I use all 3-digit industries apart from those with the following NAICS codes: 482, 491, 516, 521. This leaves K (cid:16) 88 industries. Bartik predicted employment growth is de(cid:12)ned as: K ‚ B (cid:16) φ g (B.1) i;t i;k;(cid:28) (cid:1)i;k;t k(cid:16)1 φ is the employment share of industry k in state i in the base year (cid:28) (cid:16) 1990. If any i;k;(cid:28) state-industry employment observations are missing for 1990 then I set φ (cid:16) 0 for those i;k;(cid:28) observations. g is national employment growth in industry k excluding state i in year (cid:1)i;k;t t. Goldsmith-Pinkham et al. (2018) point out that it is important to drop state i when calculating national employment growth rates in this stage. 14FRED code: PAYEMS 27

In equation 2.2 the vector of control variables Z contains two lags of the change in the i;t(cid:1)1 job creation and destruction rates, the lagged level of employment growth, and the lagged value of Bartik predicted employment growth. C Computational Method Below I outline the computational algorithms used to solve the baseline and frictionless model. C.1 Baseline Model To solve the (cid:12)rm’s problem, I approximate the expected marginal value function using linear splines. A similar computational procedure is used in Fujita and Nakajima (2016). I follow Khan and Thomas (2008) and re-write the (cid:12)rm’s recursive problem in terms of utils of the representative household. Consequently, the problem can be written: Vpz ;z ;n;Sq (cid:16) maxppSqrAz z n(cid:11) (cid:1)wpSqn(cid:1)(cid:20)pn1 (cid:1)nq1pn1 ą nqs(cid:0)(cid:12)E rVpz1;z1;n1;S1qs r i n1 r i z r 1;z i 1;A1 r i (C.1) s.t. (cid:22)1 (cid:16) (cid:0)pA;(cid:22)q where (cid:2) N1(cid:0) (cid:10)(cid:1)(cid:13) ppSq (cid:17) U pC;Nq (cid:16) C (cid:1) (C.2) C 1(cid:0) The above problem is not computable due to the in(cid:12)nite dimensionality of (cid:22). I use the technique of Krusell and Smith (1998) and approximate (cid:22) by the (cid:12)rst moment of its distribution over employment (equivalent to aggregate employment). I approximate (cid:0) using log-linear forecast equations. The problem which I compute is: Vpz ;z ;n;A;Nq (cid:16) maxppA;NqrAz z n(cid:11) (cid:1)wpNqn(cid:1)(cid:20)pn1 (cid:1)nq1pn1 ą nqs (C.3) r i r i n1 (cid:0)(cid:12)E rVpz1;z1;n1;A1;N1qs z r 1;z i 1;A1 r i s.t. 28

logN1 (cid:16) a (cid:0)b logN (cid:0)c logA N N N logp (cid:16) a (cid:0)b logN (cid:0)c logA p p p The (cid:12)rm’s hiring and (cid:12)ring thresholds are described by the following FOCs: E V pz ;z ;npz ;z ;A;N;pq;A;Nq (cid:16) p(cid:20) (C.4) z1;z1;A1 n r i r i r i E V pz ;z ;n(cid:22)pz ;z ;A;N;pq;A;Nq (cid:16) 0 (C.5) z1;z1;A1 n r i r i r i The (cid:12)rm’s envelope condition for this problem is: V pz ;z ;n;A;Nq (cid:16) ppA;NqrAz z (cid:11)n(cid:11)(cid:1)1 (cid:1)wpNqs n r i r i $ ’ ’ ’ 0 if (cid:12)E rV n pz r 1;z i 1;n;A1;N1qs ă 0 & (cid:0) (cid:12)E rV pz1;z1;n;A1;N1qs if 0 ď (cid:12)E rV pz1;z1;n;A1;N1qs ď ppA;Nq(cid:20) n r i n r i ’ ’ ’ %ppA;Nq(cid:20) if (cid:12)E rV pz1;z1;n;A1;N1qs ą ppA;Nq(cid:20) n r i (C.6) The expected marginal value function, before the realization of z ,z and A, is then: i r Wpz ;z ;n;A;Nq (cid:17) E V pz ;z ;n;A;Nq (C.7) r i z1;z1;A1 n r i r i (cid:16) E rA1z1z1(cid:11)n(cid:11)(cid:1)1(cid:1)w(cid:0)minpmaxr(cid:12)Wpz1;z1;n;A1;Nq;0s;ppA1;Nq(cid:20)qs z1;z1;A1 r i r i r i C.1.1 Equilibrium Algorithm (Baseline Model) 1. Guess an initial forecast rule system: (cid:0)^ (cid:16) ta ;b ;c u i i i i(cid:16)N;p 2. Given the forecast rule system, solve for the expected marginal value function by iterating equation (C.7) until convergence. 3. UsetheexpectedmarginalvaluefunctionalongwiththeFOCs(C.4andC.5)toapproximate the thresholds that describe the (cid:12)rm’s policy function: npz ;z ;A;N;pq and n(cid:22)pz ;z ;A;N;pq. r i r i Note that the (cid:12)rm’s policy can depend on the market-clearing price p. 4. Simulate the model for T periods using the non-stochastic approach of Young (2010), i.e. on a discrete (but dense) grid of points for z , z and n. Each period in the simulation, the r i market-clearing price p must be determined. t 29

5. When the simulation for T periods is complete, discard an initial T(cid:22) periods, and then use the remaining periods to update the forecast rules using OLS regression. If these coefficients (cid:0)~ have converged with (cid:0)^, the algorithm is complete. Otherwise, update (cid:0)^ and return to step 2. C.2 Frictionless Model In the frictionless model the (cid:12)rm’s problem is: Vpz ;z ;n;Sq (cid:16) maxppSqrAz z n(cid:11)(cid:1)wpSqns(cid:0)(cid:12)E rVpz1;z1;n1;S1qs (C.8) r i r i z1;z1;A1 r i n1 r i s.t. (cid:22)1 (cid:16) (cid:0)pA;(cid:22)q where (cid:2) N1(cid:0) (cid:10)(cid:1)(cid:13) ppSq (cid:17) U pC;Nq (cid:16) C (cid:1) (C.9) C 1(cid:0) The (cid:12)rms employment decision for the following period is implied by the following (cid:12)rst-order condition: E V pz ;z ;n;A;Nq (cid:16) 0 (C.10) z1;z1;A1 n r i r i The (cid:12)rm’s envelope condition is: V pz ;z ;n;Sq (cid:16) ppSqrAz z (cid:11)n(cid:11)(cid:1)1(cid:1)wpSqs (C.11) n r i r i Using the previous two equations, the employment policy function is given by: n1pz ;z ;Sq (cid:16) (cid:18) (cid:11)E (cid:18) A1z r 1z i 1ȷȷ 1(cid:1) 1 (cid:11) (C.12) r i z r 1;z i 1;A1 wpS1q Consequently, in the frictionless version of the model there is no need to forecast p in order to (cid:12)nd the (cid:12)rm’s policy functions. This simpli(cid:12)es the algorithm. C.2.1 Equilibrium Algorithm (Frictionless Model) 1. Guess an initial forecast rule system: (cid:0)^ (cid:16) ta ;b ;c u N N N 2. Given the forecast rule system, solve for the (cid:12)rm’s policy functions using equation C.12. 30

3. Simulate the model for T periods using the non-stochastic approach of Young (2010), i.e. on a discrete (but dense) grid of points for z , z and n. r i 4. When the simulation for T periods is complete, discard an initial T(cid:22) periods, and then use the remaining periods to update the forecast rules using OLS regression. If these coefficients (cid:0)~ have converged with (cid:0)^, the algorithm is complete. Otherwise, update (cid:0)^ and return to step 2. C.3 Computational Accuracy Table 4 shows the coefficients of the estimated log-linear forecast rules in the Krusell and Smith (1998) approach in both the baseline and frictionless models. It is clear from these coefficients that thebaselinemodelinducespersistenceinaggregateemployment. Themostbasictestofaccuracyof these forecast equations is their R2. While these are extremely high, they are also a poor measure of accuracy, as pointed out by Den Haan (2010). The basis issue is that one-period ahead forecast errors are a poor way of ensuring that the approximated law of motion for the model is close to the true one. Consequently, I follow Den Haan’s recommendation and simulate the model for a large number of periods (T = 5000)15. I then compare the average and maximum percentage deviation betweenlevelsofpandN impliedbythemodelandthosethatoccurfromiteratingontheestimated forecast rule system. The last four rows of Table 4 show that both mean and maximum percentage errors from the forecast rule system are small. This con(cid:12)rms that the Krusell and Smith (1998) approach provides a very accurate approximation. D Implications of Hiring Costs The top panel of Figure 10 shows the (cid:12)rm’s employment policy function in the steady-state of the model. For each level of idiosyncratic productivity, the (cid:13)at regions of the policy function correspondingtolevelofemploymentthat(cid:12)rmsadjusttoiftheyeitherhireor(cid:12)reworkers. Inthese regions, future employment does not depend on current employment. There is also an intermediate rangeofemploymentlevelswhere(cid:12)rmsleavetheiremploymentunchanged. Inthisareaofthestate space, the policy function is clearly upward sloping in current employment. The bottom panel of Figure 10 shows the distribution of employment gaps and adjustment probabilities implied by the model, where I de(cid:12)ne a (cid:12)rm’s target employment level as the mid-point 15Note, this is not the same sample for which the equilibrium coefficients of the forecast rules were found. 31

Table 4: Accuracy of Equilibrium Forecasting Rules Baseline Frictionless a 0.001 -0.004 N b 0.515 0.000 N c 0.555 1.170 N a 0.365 N/A p b -0.184 N/A p c -1.569 N/A p R2 0.99982 0.99999 N R2 0.99997 N/A p Max Error N(%) 0.17 0.11 Mean Error N (%) 0.04 0.10 Max Error p (%) 0.11 N/A Mean Error p (%) 0.04 N/A Notes: Mean/maximum errors constructed by simulating the model for 5000 periods and comparing p and N series from the model with those from the forecasting rules. between the hiring and (cid:12)ring thresholds for their current levels of idiosyncratic and regional productivity. Firms whose gap is small are unlikely to adjust. As the employment gap gets larger, the adjustment probabilities smoothly increase. This shows that the model is capable of generating employment gaps and adjustment probabilities that are qualitatively similar to those estimated using Longitudinal Research Database (LRD) micro-data by Caballero et al. (1997). E Robustness In this section I show that the time-varying responsiveness of job creation and job destruction is robusttotwoalternativecalibrationsofthemodel: onewitharisk-neutralrepresentativehousehold, and one with a lower aggregate labor supply elasticity. I show that the time-varying responsiveness of aggregate job creation and destruction rates predicted by the model is unaffected by either of these calibration changes. 32

Figure 10: Lumpy Employment Adjustment in the Model (a) Employment Policy Function Notes: Employment policy functions shown in the steady-state of the model, holding regional productivity equal to one. (b) Employment Gaps and Adjustment Probabilities 1.0 0.8 0.6 0.4 0.2 0.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Log Employment Gap ytilibaborP tnemtsujdA 1.0 0.8 0.6 0.4 0.2 0.0 ytisneD Notes: Employment gap is de(cid:12)ned as the deviation between current employment and the mid-point of the hiring and (cid:12)ring thresholds for the current level of productivity. 33

Figure 11: Robustness: Model-Implied Responsiveness Indicies 1.6 1.4 1.2 1.0 0.8 0.6 1980 1985 1990 1995 2000 2005 2010 2015 2020 1 = naeM JC (Risk-Neutral) JC (High ) JD (Risk-Neutral) 2.45-2.51 in Q2 2020 JD (High ) 0.18-0.20 in Q2 2020 E.1 Risk-Neutral Representative Household KhanandThomas(2008)showedthatprocyclicalrealinterestratesingeneralequilibriumhavethe ability to neutralize the time-varying responsiveness of aggregate investment in models of lumpy capital adjustment. The baseline model in this paper uses a standard speci(cid:12)cation of the representative household’s stochastic discount factor, which implies similar movements in real interest rate movements, yet this model generates signi(cid:12)cant time-varying responsiveness of job creation and destruction rates. To understand the impact of general equilibrium effects on the time-varying responsiveness in the model, I redo the exercise of Section 5 in a model where the representative householdisrisk-neutral,i.e. (cid:13) (cid:16) 0,andconsequentlywhererealinterestratesareconstant. Figure 11 shows that the responsiveness indices from this model, which are very similar to those in the baseline model. Why do real interest rate movements have such a limited effect in the case of lumpy labor adjustment? The key reason is that the timing of employment adjustment has little impact on consumptionoftherepresentativehousehold. InthemodelofKhanandThomas(2008), generalequilibrium effects are important because of the consumption smoothing motive of the representative household, which causes large real interest rate movements in the face of consumption volatility. In this modelthe onlyimpactthatemploymentadjustmenthas onconsumptionisthroughthe hiringcost, which is small. 34

E.2 Lower Labor Supply Elasticity In the baseline calibration I use a Frisch labor supply elasticity of 2, a value that is common in the macro literature but higher than micro estimates. In this section I repeat the experiment of Section 5 by generating \responsiveness indices" for the model assuming that the Frisch labor supply elasticity is lowered to 1. The responsiveness indices shown in Figure 11 are almost identical to those in Figure 1b. The only difference between this calibration of the model and the baseline calibration is that aggregate productivity now needs to be more volatile to induce the changes aggregate employment seen in the data: the standard deviation of aggregate productivity shocks is raised from 0.0049 in the baseline model to 0.0073 with the lower labor supply elasticity. This can be seen in Figure 7: the baseline model requires a productivity decline of around 6% to generate the decline in employment seen in the Great Recession. For the model with a low labor supply elasticity (high ϕ), the required decline in productivity is closer to 10%. 35