Appendix for "Productivity and Unemployment over the Business Cycle" Regis Barnichon October 03, 2009

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1 Appendix for "Productivy and Unemployment over the Business Cycle" Regis Barnichon October 3, 29 A. Data Raw statistics The data on labor productivy, unemployment, employment and hours per worker are taken from the U.S. Bureau of Labor Statistics (BLS). Labor productivy is measured as real average output per hour in the non-farm business sector and unemployment is the quarterly average of the monthly unemployment rate series constructed by the BLS from the Current Population Survey. The employment series is the number of employed workers (in millions) in the nonfarm business sector. Hours per worker is derived from subtracting (log) employment from (log) total hours in the non-farm business sector (the results in the paper are robust to using total private average weekly hours from the Current Employment Survey (BLS)). The vacancy series is the compose Conference Board help advertising index (see Barnichon, 29). All series are expressed as deviations from an HP- lter wh smoothing parameter 6, and the conclusions from the paper are independent of the smoothing parameter. The construction of the TFP series is relatively standard and extends the series constructed by Beaudry and Portier (26). Using nonfarm private business sector data from the BLS, I retrieve two annual series: labor share and capal services, and I interpolate the capal services series a quarterly series, assuming constant growth whin the quarters of the same year. Output and total hours are quarterly and seasonally adjusted nonfarm business measures, from 948Q to 28Q4 (also from the BLS). I then construct a measure of (log) TFP as T F P t = log(y t =H s h t K s h t ), where s h is the average level of the labor share over the period.

2 VAR evidence I use quarterly data taken from the U.S. Bureau of Labor Statistics (BLS) covering the period 948:Q to 28:Q4. Labor productivy x t is measured as real average output per hour in the non-farm business sector, and unemployment u t is the quarterly average of the monthly unemployment rate series constructed by the BLS from the Current Population Survey. Following Fernald (27), I allow for two breaks in x t, 973:Q and 997:Q. The next section describes the details of the estimation method. A2. Estimation of technology and non-technology shocks I am interested in estimating the system x t u t C B A = "a t " d t C A = C(L)" t () where x t is labor productivy de ned as output per hours, u t unemployment, C(L) an invertible matrix polynomial and " t the vector of structural orthogonal innovations comprised of " a t technology shocks and " d t non-technology shocks. I use the estimation method of Shapiro and Watson (988) and Francis and Ramey (23) to allow for time-varying variance of the structural innovations. Whout loss of generaly, () can be wrten x t = u t = px px xx;j x t j + ~ xu;j u t j + " a t j= px uu;j u t j + j= j= j= px ux;j x t j + " a t + " m t As discussed in Shapiro and Watson (988), imposing the long run restriction that only technology shocks have a permanent e ect on x t is equivalent to restricting the variable u t to enter 2

3 the rst equation in di erences. Consequently, the system reduces to x t = u t = px Xp xx;j x t j + xu;j u t j + " a t (2) j= px uu;j u t j + j= j= j= px ux;j x t j + " a t + " m t (3) Since u t j is correlated wh " a t, equation (2) must be estimated wh instrumental variables. I use lags to p = 4 of x t and u t as instruments. The residual from this IV regression is the estimated technology shock ^" a t. The second equation can be identi ed by OLS but using ^" a t in place of " a t. Finally to allow for time-varying variance of the structural innovations (or more generally heteroskedasticy), I follow Francis and Ramey (23) and estimate both equations jointly using GMM. That way, I can estimate the variance-covariance matrix of the estimates and generate the standard error bands for the impulse response functions. The error bands are derived by generating random vectors from a multivariate normal distribution wh mean equal to the coe cient estimates and variance-covariance matrix equal to the estimated one, and then calculating the impulse response functions. A3. Robustness of the empirical ndings VAR evidence A number of researchers question the robustness of Gali s (999) ndings, and my approach may su er from similar criques. In this section, I discuss the robustness of my results in light of their ndings. Christiano, Eichenbaum and Vigfusson (23) argue that the negative response of labor input to a technology shock may be the result of a mistreatment of labor input in the empirical model. Depending on the lter applied to hours, the response of hours can change sign. In the paper, I follow Fernald (27), remove the low-frequency movements in productivy by allowing for two breaks in x t, 973:Q and 997:Q, and my results do 3

4 not depend on the method used to detrend unemployment. Chang and Hong (26) question Gali s (999) use of output per hour as a measure of productivy. They argue that, because output per hour, unlike TFP, is in uenced by permanent shifts in input mix (e.g. shocks a ecting permanently the capal-labor ratio), Gali (999) mislabels changes in input mix as technology shocks and does not properly identify the response of total hours worked to technology shocks. In Figure, I reproduce my VAR exercise using TFP instead of output per hour. Encouragingly, the impulse responses look very similar to the ones using output per hour, and technology shocks increase unemployment temporarily. Finally, Chang and Hong (26) and Holly and Petrella (28) show that while technology shocks may decrease hours worked at the aggregate level, this nding does not necessarily hold at the industry level. Unfortunately, the focus on unemployment does not allow me to test the validy of my results in this dimension. I also estimate a higher dimensional (4 variable) VAR wh the (logged) job nding probabily and the (logged) employment ex probabily as addional variables. Figure 2 plots the impulse responses to a posive technology shock. The responses of unemployment and output per hour are similar to the ones obtained from a bivariate VAR. The job nding probabily declines signi cantly on impact and after two quarters displays a similar behavior to that of unemployment. The employment ex probabily increases on impact before reverting quickly to s long run value. However, the inial response is only marginally signi cant. The volatily drop in demand shocks There is reassuring evidence (see Galí and Rabanal, 24) that technology shocks are correctly identi ed by long run restrictions but, since I emphasize the role played by aggregate demand shocks, I also look at the Romer and Romer (24) monetary shocks and verify that they The results are available upon request. Fernald (27) showed that the presence of a low-frequency correlation between labor productivy growth and unemployment, while unrelated to cyclical phenomena, could signi cantly distort the estimates of short run responses obtained wh long run restrictions. An alternative proposed by Fernald (27) would be to separately analyze subsamples wh no breaks in technology growth. In a robustness check, I restrict the sample period to where there is no clear trend break. Results remain very similar. 4

5 experienced a volatily drop similar to the one used in the simulation. Those shocks are identi ed wh a di erent method, but we can see in Figure 3 that, notwhstanding the large volatily increase in the late 7s, their volatily in 975 is twice as high as that in 99, a volatily drop similar to the one used in the simulation. A4. Derivation of the model Hours/e ort decision and procyclical productivy When a rm and a worker meet, I assume that both parties negotiate the hours/e ort decision by choosing the optimal allocation. More precisely, they solve h min h + h e t + h t e +e t (4) h ;e + h + e subject to satisfying demand A t n h e = yd at date t. The rst-order condions imply that +e e ort per hour is a function of hours per worker e = e h where e = h +e e h +e e posive constant. Thus, changes in hours can proxy for changes in e ort, and I can rewre the rm production function as is a y = y A t n h (5) wh y = e and = + h + e. Wage bargaining As is usual in the search lerature, rms and workers bargain individually about the real wage and spl the surplus in shares determined by an exogenous bargaining weight (see e.g. Krause and Lubik, 27 and Trigari, 29). On the rm s side, the surplus J i (w ) obtained from a marginal worker equals his marginal 5

6 contribution to pro ts so J i (w ) = = P P t y w n + E t+ ( )J i (w + w + E t+ ( )J i (w + ) (6) wh w the wage, t the marginal utily of consumption and t+ = t+ t the stochastic discount factor. In a context of monopolistic competion and infrequent price adjustment, once the rm has set a price, s revenue is independent of n. Therefore, the contribution of the marginal worker to ow pro ts is given, not by the marginal revenue product of the worker n ) P y P = ) P @n "Yt w = h = )), but by the marginal reduction in the w ). If the worker walked away from the job, and given the impossibily of hiring a replacement immediately, the rm would need to increase the number of hours of (and therefore the wage payments to) all other workers in order to meet s demand. A vacancy is lled wh probabily q( t ) and remains open otherwise. Wh c t the cost of keeping a vacancy open at date t, the value V i (w ) of posting a vacancy in terms of current consumption is given by V i (w ) = c t + E t t+ [q( t )J i (w + ) + ( q( t ))V i (w + )] (7) Note that the rm will post vacancies as long as the value of a vacancy is greater than zero. In equilibrium, V i (w ) = so that c t q( t ) = E t t+ [J i (w + )]: (8) Turning to the worker s problem, denote W i (w ) and U t the value of being respectively employed and unemployed in uns of consumption goods. The worker s asset value of being 6

7 matched to rm i is W i (w ) = w t h h + h e + h t e +e + h + e + E t t+ [( )W i (w + ) + U t ] (9) and the value of being unemployed U t is U t = b t + E t t+ Z t q( t ) v jt W j (w + )dj + ( t q( t ))U t+ v t () wh b t the value of home production or unemployment bene ts. A worker receives wage payments minus the disutily of labor, and has a probabily of becoming unemployed next period. When unemployed, a worker receives b t, has a probabily t q( t ) v jt v t period wh rm j and a probabily t q( t ) to remain unemployed. to nd a job next The equilibrium wage w satis es w =argmax w (W i (w ) U t ) (J i (w )) so that the surplus-sharing rule implies W i (w ) U t = J i(w ): () Denoting the worker s surplus S = W i (w ) U t, I can wre E t S + = E t t+ [ Z = E t t+ [ = t q( t ) v jt S jt+ dj + ( v t Z )E t t+ S + t q( t ) v jt v t J j (w + )dj + ( )J i (w + )] c t q( t ) ( tq( t )) wh (8) (2) Rewring () using (6), (9), () and (2), the equilibrium wage satis es w b h t t + h h + h t + h e t + e e +e t + c t q( ( t) tq( t )) = w + ( ) ct q( t) 7

8 or after rearranging, w = + c t t + ( ) b t + g(h t; e t ) : t While the wage equation (3) is a weighted average of both parties surpluses and is similar to other bargained wages derived in e.g. Krause and Lubik (27) or Trigari (29), the rm s surplus is not given by the marginal product of labor. Indeed, once the rm has chosen s price, is demand constrained and a marginal worker will not increase the rm s revenue. Instead, the rst term of (3) is the change in the wage bill caused by substuting the intensive margin (hours and e ort) wh the extensive one (employment). A solution to (3) is given by w = c t t + ( )b t + ( ){ h+ h (4) t wh { = + h +e h (+ h )e (+ h).. Deriving the rst-order condions Given the aggregate price level, rm i will choose a sequence of price fp g and vacancies fv g to maximize the expected present discounted value of future pro ts subject to the demand constraint, the Calvo price setting rule, the hours/e ort choice and the law of motion for employment. Formally, the rm maximizes s value E t X j j u (C t+j ) u (C t ) Pi;t+j yi;t+j d n i;t+j w i;t+j c t+j v i;t+j P t+j 8

9 subject to 8 >< >: y d = ( P i;t P t ) " Y t y = y A t n h n + = ( )n + q( t )v w = c t t + ( )b t + ( ){ h+ h t The optimal vacancy posting condion takes the form c t q( t ) = E t t+ + + c t+ ( ) q( t+ ) (5) wh, the shadow value of a marginal worker, given w = w h = w + ( ){ + h h + h t : Using the wage equation, I can rewre the marginal worker s value as = c t ( ) b + h + ( ) t t {h + h Y t : (7) The level of hours per worker drives the rm s incentives to post vacancies. Wh < + h, the longer hours are, the larger is the wage bill reduction obtained wh an extra worker. As hours increase because of a higher demand for the rm s products, the worker s marginal value increases, and the rm post more vacancies to increase employment. 2 Wh Calvo-type price setting, a rm i resetting s price at date t will satisfy the standard Calvo price setting condion: E t X j= j j P P t+j s +j Y t+j P " t+j = (8) 2 Note that this mechanism is di erent from the one at play in models wh a retail sector and a wholesale sector as in Trigari (24) and Walsh (24). In those models, hiring rms are not demand constrained and the contribution of an addional worker is given by the marginal product of labor minus the wage bill. 9

10 where the optimal mark-up is = " " and the rm s real marginal cost s = + h ( ){ Y t h + h : (9) A t The rm will choose a price P that is, in expected terms, a constant mark-up over s real marginal cost for the expected lifetime of the price. Finally, the household rst-order condions for consumption and money holding take the usual form C t = E t ( + i t ) Pt P t+ C t+ and Mt P t = C t i t +i t : (Non-stationary) Equilibrium In this non-stationary model economy, I rescale the non-stationary variables wh the technology index A t : Denoting rescaled variables wh lower-case letters, the frictionless economy is described by the following system wh 5 equations and 5 unknowns, y, h, e and n : y = Yt A t e = e (h ) h +e = y n h c = q( ( ( )) ) + = c h ( )b + ( ) = + h ( ){y h + h n = q( ) + q( ) {h + h y where y, e and { are posive constants de ned previously.

11 Log-linearization and the New-Keynesian Phillips curve Log-linearizing the vacancy posting condion equation around the (zero-in ation) steady state, I get c q( t = E t )^ ^ + + c( ) q( ) ^t+ (2) where the value of a marginal worker ^ is given by ^ = c^ t + n ( + h = c^ t + n ( + h )^h (2) ) (^y ^n ) (22) where ^ = ln( =At ), ^ t = ln t, ^h = ln h h ; ^n = ln n n, and ^y = ln Yt=At y. To derive the New-Keynesian Phillips curve, I log-linearize the price-setting condion around the zero in ation equilibrium. However, because employment is a state variable and because of rms ex-post heterogeney, the derivation is not straightforward. I follow Woodford s (24) similar treatment of endogenous capal in a New-Keynesian model wh Calvo price rigidy. In my case, employment is the state variable and plays the role of capal in Woodford s model. Log-linearizing the price-setting condion gives me X () k ^Ei t [~p +k ^s +k ] = (23) k= wh ^s +k = ^n +k + + h (^y +k ^n +k ) ^y +k + ^y t+k (24) The notation ^E i t denotes an expectation condional on the state of the world at date t but integrating only over future states in which rm i has not reset s price since period t: ~p log P P t is the rm s relative price. Denoting log prices by lower-case letters and p the optimal (log) price for rm i at t, the demand curve for rm i at date t + can be wrten ^y + = ^y t+ "(p p t+ ) if cannot

12 reset s price at t + and ^y + = ^y t+ "(p + p t+ ) if can reset s price. Averaging across all rms, I get Z 2 ^y + di = ^y t+ " 4( Z Z p di p t+ ) + ( )( 3 p +di p t+ ) 5 = ^y t+ " (p t p t+ ) + ( )(p t+ p t+ ) (25) Z where p t+ = p +di is the average price chosen by all price setters at date t +. Wh Calvo price-setting, I can wre p t+ = ( )pt+ " + p t " " or p = ( ) t+ p t+ " + pt p t+ " : Log-linearizing around the zero-in ation equilibrium gives (p t+ p t ) = ( )(p t+ p t+ ) Z and combining wh (25) gives ^y + di = ^y t+. Further, ^n di = ^n t. Averaging (24) across all rms, I can rewre the real marginal cost as Z + h ^s +k = ^s t+k + ( "~p +k ~n +k ) (26) where ~n +k = n +k n t+k is the relative employment of rm i. Using that ^E i t ~p +k = p E t p t+k and (26) in (24) yields + " +h = ( ) p X () k ^Ei t h^s t+k + k= + " +h p t+k i (27) +h ~n +k 2

13 Moreover, subtracting (2) from s average, I get ~n + = E t (^y + ^y t+ ) (28) = "E t (p p t+ ) + ( )(p + p t+ = " ~p "( )(p + p t+) since p t+ = p t + ( )p t+. The rm s pricing decision depends on s employment level and the economy s aggregate state. But to a rst order, the log-linearized equations are linear so that the di erence between p and p t, the average price chosen by all price setters, is independent from the economy s aggregate state and depends only on the relative level of employment n n t = ~n. So as in Woodford (24), I guess that the rm s pricing decision takes the form p p t = ~n (29) wh a constant to be determined. Hence, (28) becomes ~n + = " "( ) ~p = f()~p Since this was shown for any t >, I also get ~n +k = f()~p +k, 8k > so that I can rewre (27) as p = ( wh = ) X () k ^Ei t ^s t+k + k= + " +h + h + " +h f(). p t+k + h ( ) ~n (3) Subtracting (3) from s average, I obtain + (p p h t ) = ( ) ~n : (3) 3

14 This equation is of the conjectured form (29) if and only if satis es = + " +h ( ) + h +h : (32) f() Averaging (3) and using t = (p t p t ), I obtain the New-Keynesian Phillips curve t = :^s t + E t t+ (33) wh = ( )( ) : 3 Finally, log-linearizing the household s rst-order condions and denoting ^m t = ln Mt=PtA t (M=P ) the log-deviation of real rescaled money from s constant value in the zero-in ation equilibrium, I get ^y t = E t^y t+ (^{ t E t t+ ) and ^m t = ^y t i^{ t wh ^{ t = ln + +i. The log-linearized law of motion for employment can be wrten ^n t+ = ( q())^n t + n n ( )q()^ t. A5. Embodied Technology and creative destruction In this section, I discuss an alternative interpretation of the empirical relationship between labor productivy and unemployment studied in Section 2 of the paper. In this alternative, one could ignore aggregate demand altogether and emphasize instead the Schumpeterian aspect of technological progress. Indeed, Michelacci and Lopez-Salido (27) and Canova, Michelacci and Lopez-Salido (27) argue that a technology shock wh a permanent impact on productivy may increase unemployment through creative destruction. The introduction of new technologies brings about a simultaneous increase in the destruction of technologically obsolete jobs which prompts a contractionary period during which employment temporarily falls. In that framework, the sign ip of in the mid 8s could be due to an acceleration of creative destruction spawned by the Information Technology (IT) revolution. Put di erently, technology 3 In independent work, Thomas (29) shows that the New-Keynesian Phillips curve (3) is atter than the ( )( ) Phillips curve of a standard New-Keynesian model (for which = ). He shows quantatively that such real rigidies strongly increase the persistence of in ation compared to standard New-Keynesian models. 4

15 was disembodied before 984 but became more embodied wh the IT revolution, and there is no need to appeal to aggregate demand to explain the sign swch. However, I see a number of arguments suggesting that creative destruction is not the most plausible explanation. First, if technical progress had become more embodied, this should have appeared in the empirical impulse responses. But as we saw, technology shocks had a quantatively smaller negative impact on unemployment after 984; exactly the oppose of what more creative destruction would imply but consistent wh an improvement in the conduct of monetary policy. Second, in a world wh creative destruction, when productivy increases, unemployment goes up temporarily because rms destroy old and less productive jobs. If technology had become more embodied after 984, movements in the separation rate should contribute to a larger fraction of unemployment uctuations. However, Shimer (27) nds that the proportion of unemployment uctuations accounted for by variations in the separation rate actually decreased from 2% to only 5% after Moreover, wh embodied technology, rms need to post vacancies to create new matches wh the latest level of technology. Hence, an acceleration of creative destruction in the mid 8s could have caused the correlation between productivy and unemployment to become posive, but could not have caused the correlation between productivy and vacancies to become negative. In contrast, an explanation emphasizing the role of aggregate demand shocks is consistent wh large movements in both correlations: when productivy increases because of a technology shock, aggregate demand does not adjust immediately, rms post fewer vacancies, and unemployment goes up. Looking again at Figure 2 of the paper, the -year rolling correlation of labor productivy and vacancies displays a sign swch similar to and favors the latter explanation. Finally, is important to distinguish between embodiment in new jobs and embodiment in new capal. For example, technology may be embodied in capal but disembodied in jobs. 5 4 The job- nding rate accounting for the residual. 5 An example given by Pissarides and Vallanti (25) is the one of a secretary using Microsoft Windows. A 5

16 In order to explain the sign swch of wh creative destruction, technological progress needs to be embodied in new jobs. Studying the impact of productivy growth on unemployment, Pissarides and Vallanti (25) nd that technology embodied in jobs and creative destruction play no role in the dynamics of unemployment. new version of Windows may require a more powerful computer but the secretary keeps the job and only needs to learn how to use the new version: technology is embodied in new capal but not in new jobs. 6

17 References [] Beaudry, P., and F. Portier. Stock Prices, News, and Economic Fluctuations, American Economic Review, 96(4): , 26. [2] Barnichon, R. Vacancy Posting, Job Separation and Unemployment Fluctuations, Mimeo, 29. [3] Canova, F., C. Michelacci and D. López-Salido, Schumpeterian Technology Shocks, mimeo CEMFI, 27. [4] Chang, Y. and J. Hong. Do Technological Improvements in the Manufacturing Sector Raise or Lower Employment?, American Economic Review, 96(), , 26. [5] Christiano, L., M. Eichenbaum and R. Vigfusson. What Happens After A Technology Shock? Federal Reserve Board International Finance Discussion Paper 768, 23. [6] Fernald, J. Trend Breaks, long run Restrictions, and the Contractionary E ects of Technology Shocks, Working Paper, 25. [7] Francis, N. and V. Ramey, The Source of Historical Fluctuations: An Analysis using long run Restrictions, NBER International Seminar on Macroeconomics, 24. [8] Holly, S. and I. Petrella. Factor Demand Linkages and the Business Cycle: Interpreting Aggregate Fluctuations as Sectoral Fluctuations, Working Paper 28. [9] Krause, M and T. Lubik. The (Ir)relevance of Real Wage Rigidy in the New Keynesian Model wh Search Frictions, Journal of Monetary Economics, 54 pp , 27. [] López-Salido, D. and C. Michelacci Technology Shocks and Job Flows, Review of Economic Studies, 74, 27. [] Pissarides, C. and G. Vallanti. The Impact of TFP Growth on Steady-State Unemployment, International Economic Review, Forthcoming, 25. 7

18 [2] Shapiro, M. and M.Watson. Sources of Business Cycle Fluctuations, NBER Macroeconomics Annual, pp. 48, 988. [3] Shimer, R. Reassessing the Ins and Outs of Unemployment, NBER Working Paper No. 342, 27. [4] Trigari, A. Equilibrium Unemployment, Job Flows and In ation Dynamics, Journal of Money, Cred and Banking, 29. [5] Woodford, M. In ation and Output Dynamics wh Firm-Speci c Investment, Working Paper, 24. 8

19 TFP Unemployment TFP Unemployment Figure : Impulse response functions to technology and non-technology shocks. Productivy is measured wh TFP unadjusted for capacy utilization. Dashed lines represent the 95% con dence interval Output per Hour Unemployment Job Finding Probabily Employment Ex Probabily Figure 2: Impulse response functions to a technology shock in a 4 variables VAR over Dashed lines represent the 95% con dence interval. 9

20 Figure 3: 5-year rolling standard-deviation of Romer and Romer monetary shocks. 969:Q- 996:Q4. 2