Equilibrium Unemployment Theory

Transcription

1 Equilibrium Unemployment Theory Model Modifications Matthias S. Hertweck University of Basel March 26, 2012 Matthias S. Hertweck Equilibrium Unemployment Theory 1/38

2 Lecture Outline The Volatility Puzzle Matthias S. Hertweck Equilibrium Unemployment Theory 2/38

3 Literature Recommended Readings: Hagedorn M. and Manovskii I. (2008), The cyclical behavior of equilibrium unemployment and vacancies revisited, American Economic Review 98(4), Hall, R. (2005), Employment fluctuations with equilibrium wage stickiness, American Economic Review 95(1), Gertler M. and Trigari, A. (2009), Unemployment fluctuations with staggered Nash wage bargaining, Journal of Political Economy 117(1), Optional Readings: Costain, J. S. and Reiter, M. (2008), Business cycles, unemployment insurance, and the calibration of matching models, Journal of Economic Dynamics and Control 32(4), Fujita, S. (2004), Vacancy persistence, Working Paper No , Federal Reserve Bank of Philadelphia. Matthias S. Hertweck Equilibrium Unemployment Theory 3/38

4 How to Solve the Volatility Puzzle? Nash bargaining makes the real wage very elastic incumbent workers benefit profits increase only little, firms have almost no incentive to open new vacancies (Shimer, 2005) the wage (price) absorbs the shock, not the quantity solution: real wage rigidity Matthias S. Hertweck Equilibrium Unemployment Theory 4/38

5 so far, the real wage rate was present only implicitly: { } c q(θ t ) = βe c t α(y t+1 z) (1 α)cθ t+1 + (1 λ) q(θ t+1 ) (1) we make it explicit: w t = (1 α) [y t + cθ t ] + αz (2) } c c = βe t {y t+1 w t+1 + (1 λ) (3) q(θ t ) q(θ t+1 ) now we have 5 equations and 5 variables Matthias S. Hertweck Equilibrium Unemployment Theory 5/38

6 Implementation in Dynare 1. we have to declare the variable ww 2. we have to declare the parameter WW 3. we have to compute the steady state value of the real wage rate WW 4. Dynare main file: replace (1) by (2) and (3): 0 = -ww*ww + (1-alfa)*yy*YY + (1-alfa)*c*(VV/UU)*(vv-uu(-1)); 0 = - (cv/(betha*mm))*(vv-mm) + YY*yy(+1) - WW*ww(+1)... + (cv/mm)*(1-lambda)*(vv(+1)-mm(+1)); 5. initial guess ww = 0; Matthias S. Hertweck Equilibrium Unemployment Theory 6/38

7 The Behavior of the Real Wage Rate the graph confirms the conclusion of Shimer (2005) the real wage absorbs most of the shock firms have little incentives to open new vacancies Matthias S. Hertweck Equilibrium Unemployment Theory 7/38

8 the Barro (1977) Critique W W* C B A S' S 0 L* D L without matching frictions, a rigid wage leads to inefficient separations in response to a negative shock labor input L falls from A to C (instead of B) there is only one equilibrium wage Matthias S. Hertweck Equilibrium Unemployment Theory 8/38

9 and the Matching Model matching frictions establish a mutual surplus reservation wages: J t = 0 wt h c = y t + (1 λ) q(θ t ) ( ) 1 α c W t U t = 0 wt l = z (1 λ q(θ t )θ t ) α q(θ t ) (4) (5) if the continuation value c/q(θ t ) of the match is large, the reservation wage of the worker w l t might be negative Matthias S. Hertweck Equilibrium Unemployment Theory 9/38

10 The Bargaining Set individual rationality requires that the real wage rate is always within the bargaining set there are infinitely many equilibrium wage candidates e.g. reducing α, brings w t closer to w h t Matthias S. Hertweck Equilibrium Unemployment Theory 10/38

11 Exogenous infrequent wage bargaining only a fraction (1 ω) of firm-worker pairs is able to re-bargain the real wage rate in period t (Calvo, 1983): ŵ t = ωŵ t 1 + (1 ω)ŵt N (6) otherwise, the real wage rate remains constant new job matches face the same bargaining probability consequently, the aggregate real wage rate behaves sluggish Matthias S. Hertweck Equilibrium Unemployment Theory 11/38

12 Implementation in Dynare declare a new variable: wn the Nash wage wn follows the same rule as before but only the share (1 ω) of the matches is able to bargain 0 = -ww + omega*ww(-1) + (1-omega)*wn; 0 = -wn*ww + (1-alfa)*yy*YY + (1-alfa)*c*(VV/UU)*(vv-uu(-1)); Matthias S. Hertweck Equilibrium Unemployment Theory 12/38

13 Staggered Wage Contracts (Calvo, 1983) we choose ω = 0.85, corresponding to an average wage contract duration of (1 0.85) 1 = 6.7 months the average wage follows a hump-shaped pattern the Nash wage is more elastic than without Calvo staggering, since the movements in labor market tightness θ t are larger Matthias S. Hertweck Equilibrium Unemployment Theory 13/38

14 Endogenous : do not change any equation of the model choose a different calibration target: elasticity of the real wage wrt. labor productivity (0.45) raise α from 0.28 to raise z from 0.4 to Bargaining Power: empirical evidence is scarce mostly, the Hosios (1990) rule is applied: α = η if α is large, firms profit more from a given shock Matthias S. Hertweck Equilibrium Unemployment Theory 14/38

15 Calibration Flow Value of Unemployment: interpretation: pecuniary benefits b and leisure surplus l alternative interpretation: workers try to signal high quality by rejecting wage offers close to b if z is large, the mutual surplus is smaller if profits are small, a given technology shock causes larger percentage deviations (leverage effect) thus, we observe large fluctuations of unemployment and vacancies and small movements of the real wage rate Matthias S. Hertweck Equilibrium Unemployment Theory 15/38

16 Comparison: Shimer, H&M, Calvo Real Wage Unemployment the Calvo wage converges in 10 months to the flexible wage the impulse response function of the H&M wage remains persistently below the flexible wage H&M is more effective, as job creation is a forward-looking activity (the average job duration is 2.5 years) Matthias S. Hertweck Equilibrium Unemployment Theory 16/38

17 Can Match the Data? Y /N V U Θ M/U σ(x )/σ(y /N) Data (0.02) (relative) s.e. Model (0.02) ρ(x, Y /N) Data comovement Model ρ(x t, X t 1 ) Data autocorrelation Model labor market volatility rises significantly raising the matching elasticity η from 0.28 to 0.54 will dampen σ(v ) = 17.1 and further increase σ(u) = 15.6 internal propagation remains weak Matthias S. Hertweck Equilibrium Unemployment Theory 17/38

18 The Beveridge Curve the H&M model matches the data well H&M Original H&M with η = 0.54 if η = 0.54, the model fits the slope of the Beveridge curve (Mortensen & Nagypál, 2007) Matthias S. Hertweck Equilibrium Unemployment Theory 18/38

19 The Matching Elasticity 1. regression of θ t q(θ t ) on θ t yields η = slope of the Beveridge curve implies η = 0.54 ad 1 Brügemann (2008) argues that the estimated value of η rises from 0.28 to 0.40 if the job offer arrival rate and not the job finding probability is taken as regressand ad 2 consistent with this result, the same author shows that a matching model with η = 0.40 and endogenous job destruction replicates the slope of the Beveridge curve Matthias S. Hertweck Equilibrium Unemployment Theory 19/38

20 Job Offer Arrival Rate the probability that an unemployed job searcher receives n job offers within one month follows a Poisson distribution: P(n) = e ftt (f t t) n n! (7) determinant: the job offer arrival rate f t = 0.61 reference: Shimer (2007) Matthias S. Hertweck Equilibrium Unemployment Theory 20/38

21 Number of Job Offers PDF the probability of receiving no single offer equals 55% Matthias S. Hertweck Equilibrium Unemployment Theory 21/38

22 Number of Job Offers CDF the probability of receiving at least one offer equals 45% Matthias S. Hertweck Equilibrium Unemployment Theory 22/38

23 Job Finding Probability the monthly job finding probability is given as: F t = 1 e ftt = 0.45 (8) it follows an exponential distribution Matthias S. Hertweck Equilibrium Unemployment Theory 23/38

24 Job Finding Probability PDF determines the expected waiting time the probability of waiting 5 months is much smaller than the probability of waiting 1 month Matthias S. Hertweck Equilibrium Unemployment Theory 24/38

25 Job Finding Probability CDF the offer arrival rate f t is unbounded the job finding probability F t = 1 e ftt is bounded Matthias S. Hertweck Equilibrium Unemployment Theory 25/38

26 : H&M target the cyclical behavior of the real wage rate therefore, they re-calibrate α and z H&M show that, given an appropriate behavior of the real wage rate, the job matching model can replicate the cyclical volatility of unemployment and vacancies however, how robust are the underlying assumptions? Matthias S. Hertweck Equilibrium Unemployment Theory 26/38

27 The Impact of Benefits on the Unemployment Rate b l {}}{ { }} { value of non-market activity z = an increase in pecuniary benefits by 0.01 units raises the steady-state unemployment rate from 6.87% to 7.37% in other words the semi-elasticity of unemployment wrt. the value of non-market activity in the H&M model is equal to: SE U,z = log(u) z = 7.3 (9) in contrast, Costain & Reiter (2008) estimate: SE U,z [2, 3] Matthias S. Hertweck Equilibrium Unemployment Theory 27/38

28 The Observation of Fujita (2004) the shape of the impulse responses is counter-factual model: vacancies spike and fall back immediately data: vacancies show a distinct hump-shape suggestion: frictional vacancy creation Matthias S. Hertweck Equilibrium Unemployment Theory 28/38

29 The Impulse Response of Vacancies (Fujita, 2004) Matthias S. Hertweck Equilibrium Unemployment Theory 29/38

30 The Hiring Cost Function Standard: Linear Vacancy Posting Costs: ψ(v t ) = cv t Gertler & Trigari (2009): Convex Labor Adjustment Costs: ( mt ) 2 n t 1 ψ(m t, n t 1 ) = c 2 n t 1 evidence using macro data: Merz and Yashiv (AER, 2007) hiring costs are now much less elastic and less persistent Matthias S. Hertweck Equilibrium Unemployment Theory 30/38

31 Firm s FOC with Convex Labor Adjustment Costs cx t = βe t [ y t+1 + c 2 x 2 t w t+1 + (1 λ)cx t+1 ] (10) gross hiring rate (x t = m t /n t 1 ): weakly volatile Firm s FOC with Linear Vacancy Posting Costs c q(θ t ) = βe t [ ] c y t+1 w t+1 + (1 λ) q(θ t+1 ) (11) market tightness (θ t ): very volatile and persistent Matthias S. Hertweck Equilibrium Unemployment Theory 31/38

32 Firm s FOC with Convex Labor Adjustment Costs cx t = βe t [ y t+1 + c 2 x 2 t w t+1 + (1 λ)cx t+1 ] (10) gross hiring rate (x t = m t /n t 1 ): weakly volatile Firm s FOC with Linear Vacancy Posting Costs c q(θ t ) = βe t [ ] c y t+1 w t+1 + (1 λ) q(θ t+1 ) (11) market tightness (θ t ): very volatile and persistent Matthias S. Hertweck Equilibrium Unemployment Theory 31/38

33 Wage Determination Nash bargaining implies following wage rate: [ w t = (1 α) y t + cx t replaces c/q(θ t ) m t 1 n t 1 cx t ] + αz (12) Matthias S. Hertweck Equilibrium Unemployment Theory 32/38

34 Vacancy Impulse Response Matthias S. Hertweck Equilibrium Unemployment Theory 33/38

35 Quantitative Analysis Convex Adjustment Costs firms hire gradually: matches are smooth vacancy posting per se is costless given the matching function: m t = χv η t u 1 η t 1, firms have to post more and more vacancies in order to compensate for the decline in u t 1 in addition, expected marginal matching costs cx t depend negatively on the current level of employment n t 1 Matthias S. Hertweck Equilibrium Unemployment Theory 34/38

36 Quantitative Analysis Linear Vacancy Costs firms anticipate the persistent and long-lasting rise in expected marginal matching costs the surge in c/q(θ t ) is due to the fact that expected marginal hiring costs depend positively on market tightness θ t firms post vacancies v t immediately only one period later, vacancies fall back half way Matthias S. Hertweck Equilibrium Unemployment Theory 35/38

37 Impact of Convex Labor Adjustment Costs Y /N V U Θ M/U σ(x )/σ(y /N) Data (0.02) (relative) s.e. Model (0.02) ρ(x, Y /N) Data comovement Model ρ(x t, X t 1 ) Data autocorrelation Model labor market persistence rises significantly internal propagation improves Matthias S. Hertweck Equilibrium Unemployment Theory 36/38

38 Summary the job matching model is able to replicate the qualitative movements of the labor market over the business cycle the calibrated version of H&M (2008) is able to replicate the cyclical volatility of vacancies and unemployment convex labor adjustment costs improve propagation considering that the model has only 4 equations, it is extremely successful Matthias S. Hertweck Equilibrium Unemployment Theory 37/38

39 Possible Extensions endogenous job destruction on-the-job search flexible hours per worker households optimizing between consumption and capital accumulation Matthias S. Hertweck Equilibrium Unemployment Theory 38/38