Why does John earn a lower salary than Jane? A number of possible reasons come to mind. Jane stayed in school longer; Jane's work is more demanding ...
Frictional Wage Inequality: A Puzzle? Andreas Hornstein (Richmond Fed) Per Krusell (Princeton University and IIES-Stockholm) Gianluca Violante (New York University)
Yale Macro Lunch, November 17th, 2005
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 1/26
What determines wage inequality? Why does John earn a lower salary than Jane? A number of possible reasons come to mind. Jane stayed in school longer; Jane’s work is more demanding; Jane is older or has been with her company longer; She is more highly motivated; John is the victim of discrimination against men; John works in a region where average wage is lower... [Cahuc-Zylberberg, LABOR ECONOMICS, page 246.] • Worker heterogeneity: accumulated skills, innate ability • Compensating differentials: job amenities, risk, location • Discrimination: between demographic groups
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 2/26
What determines wage inequality? Why does John earn a lower salary than Jane? A number of possible reasons come to mind. Jane stayed in school longer; Jane’s work is more demanding; Jane is older or has been with her company longer; She is more highly motivated; John is the victim of discrimination against men; John works in a region where average wage is lower... [Cahuc-Zylberberg, LABOR ECONOMICS, page 246.] • Worker heterogeneity: accumulated skills, innate ability • Compensating differentials: job amenities, risk, location • Discrimination: between demographic groups • Frictions: Jane has been luckier than John in her search!
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 2/26
Related Literature • Estimation of structural search models with the objective to match the wage distribution and labor market transitions • Early attempts: Eckstein and Wolpin (1990), Van den Berg and Ridder (1993, 1994, 1998), Kiefer and Neumann (1994), Bowlus, Kiefer, Neumann (1995) • Most recent efforts: Bontemps, Robin, Van den Berg (1999, 2000), Bowlus and Seitz (2000), Flinn (2005), Moscarini (2005), Postel-Vinay and Robin (2005)
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 3/26
Outline of the talk 1. Make a first attempt at measuring frictional wage dispersion in the data 2. Use the simplest McCall search model to derive a statistic for frictional inequality that is observable and distribution-free 3. Show that the model, appropriately calibrated, is strikingly unable to match the facts 4. Demonstrate robustness of this result to incorporating the search model into the Mortensen-Pissarides model with equilibrium determination of wages 5. Discuss several extensions of the standard model that can, perhaps, better match the facts
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 4/26
Measurement • Data: PSID (1967-1996), heads and spouses 18-55, with positive hours worked, wages not top-coded, hourly wage above 50% of the minimum wage • Method: two-stage procedure to control for: 1) observed and 2) unobserved heterogeneity ◦ Stage 1: Mincerian wage regression on the cross-section of log-hourly wages ◦ Controls: gender, education, experience, race, region, marital status, number of children, union member, disability status, occupation (2 digits), industry (2 digits), urban/rural area of residence, plus interactions ◦ Allow for time-varying coefficients (year by year) ◦ Explanatory power: R2 = 0.43 Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 5/26
Measurement (cont.) • Step 2: Control for unobservable innate ability and unobservable skill accumulation, i.e. ◦ For each i, obtain a time series of wage residuals {²it }t ◦ ∀i, regress ²it = ²¯i + βi expit
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 6/26
Measurement (cont.) • Step 2: Control for unobservable innate ability and unobservable skill accumulation, i.e. ◦ For each i, obtain a time series of wage residuals {²it }t ◦ ∀i, regress ²it = ²¯i + βi expit • Focus on group aged 18-27 to minimize the role of stochastic fluctuations in individual skills • The vector of residuals of this second regression, at time t, is the object of interest
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 6/26
Empirical Findings I 0.9
0.8
Frictional Wage Inequality coefficient of variation Raw data After regression Without fixed individual effects Age 18−27
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1970
1975
1980
Year
1985
1990
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 7/26
Empirical Findings II 6 5.5
Frictional Wage Inequality mean to first percentile ratio Raw data After regression Without fixed individual effects Age 18−27
5 4.5 4 3.5 3 2.5 2 1.5
1970
1975
1980
Year
1985
1990
• ratio of mean wage residual to 1% lowest wage residual is 1.87 Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 8/26
Ideal (?) Measurement • The Bureau of Labor Statistics collects data on the wage distribution by occupation (three-digits) and geographical area (MSA) • Combine two datasets: ◦ Stage 1: Compute inequality statistics by occupation/area ◦ Stage 2: Reduce these statistics by the amount predicted by the two-stage procedure from PSID data, without occupational and geographical controls
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 9/26
McCall search model (continuous time) • Workers are ex-ante equal, infinitely lived, risk-neutral • They discount future at rate r, value leisure/home-production at b • At rate λu , workers encounter wage offers drawn from the exogenous distribution F (w) • Wage level remains constant as long as the match lasts • At rate σ, workers separate from job and re-enter unemployment • No on-the-job-search
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 10/26
Solution of the model • Flow values of employment and unemployment rW (w) = w − σ [W (w) − U ] Z ∞ [W (w) − U ] dF (w) rU = b + λu w∗
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 11/26
Solution of the model • Flow values of employment and unemployment rW (w) = w − σ [W (w) − U ] Z ∞ [W (w) − U ] dF (w) rU = b + λu w∗
• The reservation wage equation is: λu w∗ = rU = b + r+σ
Z
∞
[w − w∗ ] dF (w) w∗
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 11/26
Distribution-free statistic for inequality • Let b = ρw, ¯ with w ¯ = E[w|w ≥ w ∗ ], then: ∗
w
∗
= =
λu [1 − F (w )] ρw ¯+ r+σ λ∗u [w ¯ − w∗ ] ρw ¯+ r+σ
Z
∞
[w − w∗ ] w∗
dF (w) 1 − F (w∗ )
• where λ∗u ≡ λu [1 − F (w ∗ )] and the last step eliminates F (w) which is unobservable
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 12/26
Distribution-free statistic for inequality • Let b = ρw, ¯ with w ¯ = E[w|w ≥ w ∗ ], then: ∗
w
∗
= =
λu [1 − F (w )] ρw ¯+ r+σ λ∗u [w ¯ − w∗ ] ρw ¯+ r+σ
Z
∞
[w − w∗ ] w∗
dF (w) 1 − F (w∗ )
• where λ∗u ≡ λu [1 − F (w ∗ )] and the last step eliminates F (w) which is unobservable • Rearranging, we obtain the Mean-min Ratio (M mR)
M mR =
λ∗ u r+σ λ∗ u r+σ
+1 +ρ
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 12/26
Observations on MmR
M mR =
λ∗ u r+σ λ∗ u r+σ
+1 +ρ
• The higher is “patience” (low value for r + σ) and the higher is the equilibrium contact rate λ∗u , the larger is w ∗ and the lower is frictional wage inequality in equilibrium
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 13/26
Observations on MmR
M mR =
λ∗ u r+σ λ∗ u r+σ
+1 +ρ
• The higher is “patience” (low value for r + σ) and the higher is the equilibrium contact rate λ∗u , the larger is w ∗ and the lower is frictional wage inequality in equilibrium • M mR ∈ (1, 1/ρ], as λ∗u varies in [0, ∞) • Reasonable restrictions for the U.S. economy (monthly basis): λ∗u = 0.21, σ = .01, r = 0.004, ρ = 0.2
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 13/26
Observations on MmR
M mR =
λ∗ u r+σ λ∗ u r+σ
+1 +ρ
• The higher is “patience” (low value for r + σ) and the higher is the equilibrium contact rate λ∗u , the larger is w ∗ and the lower is frictional wage inequality in equilibrium • M mR ∈ (1, 1/ρ], as λ∗u varies in [0, ∞) • Reasonable restrictions for the U.S. economy (monthly basis): λ∗u = 0.21, σ = .01, r = 0.004, ρ = 0.2 • Result: M mR ∼ 1.053
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 13/26
Interpretation and comments • Unemployed workers search longer (turn down jobs) if there is large option value of waiting in the form of a large wage dispersion (or a large spread MmR ww¯∗ )... • ...therefore, a short unemployment duration reveals that the spread is small!! • For typical distribution functions (truncated Log-normal, Gamma, Pareto), our MmR translates into variances of log-wages that are less than 10% of those measured
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 14/26
Interpretation and comments • Unemployed workers search longer (turn down jobs) if there is large option value of waiting in the form of a large wage dispersion (or a large spread MmR ww¯∗ )... • ...therefore, a short unemployment duration reveals that the spread is small!! • For typical distribution functions (truncated Log-normal, Gamma, Pareto), our MmR translates into variances of log-wages that are less than 10% of those measured • Sensitivity with respect to (r, ρ)
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 14/26
Frontier of (r, ρ) pairs Pairs of r and ρ consistent with MmR=1.87
Net value of leisure as a fraction of w (ρ)
1 0
"Reasonable" pairs
−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Monthly interest rate (r)
0.08
0.09
0.1
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 15/26
The Mortensen-Pissarides “chemistry model” • An infinitely elastic supply of ex-ante identical firms • Vacancy-posting cost c, free entry • Each worker-firm pair draws a productivity p according to F (p) • Nash bargaining determines the wage • Matching function with CRS ⇒ contact rates are a function of market tightness θ = v/u
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 16/26
The Mortensen-Pissarides “chemistry model” • An infinitely elastic supply of ex-ante identical firms • Vacancy-posting cost c, free entry • Each worker-firm pair draws a productivity p according to F (p) • Nash bargaining determines the wage • Matching function with CRS ⇒ contact rates are a function of market tightness θ = v/u • Solve it... and get exactly the same expression for the M mR!
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 16/26
Preliminary conclusions • The textbook equilibrium unemployment model generates 6% of the measured frictional wage inequality ⇒ dismal failure • You may say... "This model is designed to explain unemployment, not frictional inequality" • Super-tight link between frictional unemployment and frictional wage dispersion • Reverse our logic: given the observed amount of frictional inequality, the model predicts an expected unemployment duration of 7.7 years!
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 17/26
Ways of generating more frictional wage inequality • Risk-averse workers
• On-the-job search
• Heterogeneity in the value of leisure bi across workers i
• Others?
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 18/26
Risk Aversion • Let preferences be u(c) and w = c, i.e., autarky: upper bound for the effect of risk-aversion • We obtain the reservation wage equation: ∗ λ u [E(u(w)) − u(w ∗ )] ¯ + u (w∗ ) = u(ρw) r+σ
• Consider a second-order Taylor expansion of u(w) around w ¯ 1 2 ¯ (w − w) ¯ + u00 (w) ¯ (w − w) ¯ u (w) ' u (w) ¯ + u0 (w) 2 • Let u(w) = E
µ
w1−γ 1−γ 1−γ
w 1−γ
¶
and take expectation 1−γ
'
1−γ
·
w ¯ var (w) 1 w ¯ + γ (γ − 1) 1−γ 2 1−γ w ¯2
¸
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 19/26
Risk Aversion (cont.) • Substituting, we obtain ³
w ¯ ' ∗ w
1 + 12 γ (γ − 1) cv (w) λ∗ u r+σ
2
+1
´
λ∗ u r+σ
+ ρ1−γ
1 γ−1
• Note, as γ ⇒ 0, we obtain the old expression • Calibration: the only new number we need is cv(w) = 0.23
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 20/26
Numerical results for the model with risk-aversion Mean−min Ratio with Risk Aversion
4.5
ρ=0.2
4
ρ=0.35 ρ=0.5
3.5 3 2.5 2 1.5 1
0
5
10
15
20
γ (relative risk aversion coefficient)
25
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 21/26
On-the-job search • Workers draw wage offers from F (w) at rate λu if unemployed, and at rate λw if employed • When employed, accept offer w 0 if w0 > w
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 22/26
On-the-job search • Workers draw wage offers from F (w) at rate λu if unemployed, and at rate λw if employed • When employed, accept offer w 0 if w0 > w • Solve the model and obtain... M mR =
λ∗ u −λw r+σ ∗ λu −λw r+σ
+1 +ρ
• As λw → λ∗u , M mR → 1/ρ • No reason to turn down any wage above b, since being employed does not hurt chances to find a higher wage later • Potentially, it works: restrictions on λw from observed labor market flows? Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 22/26
Numerical results for OJS model • The average duration of a job in the model is M dur =
σ + λw /2 σ(σ + λw )
• In the data, M dur ∈ [36, 72] months • Express both M mr and M dur as a function of λw • Is there a range of values for λw consistent with both M mr = 1.87 and M dur ∈ [36, 72]?
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 23/26
Mean−min Ratio (MmR)
Numerical results for the model with on the job search 5 4 3 2
Mean job duration (months)
1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Offer arrival rate on the job ( λw )
75 70 65 60 55 50
Offer arrival rate on the job ( λw )
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 24/26
Heterogeneity in value of leisure • The lowest observed wage w ∗ will be the reservation wage of the nerdiest worker (i.e. the one with the smallest value of leisure) • It depends on G(b): suppose we have a small measure of workers with very low b: the mean wage will not be affected
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 25/26
Heterogeneity in value of leisure • The lowest observed wage w ∗ will be the reservation wage of the nerdiest worker (i.e. the one with the smallest value of leisure) • It depends on G(b): suppose we have a small measure of workers with very low b: the mean wage will not be affected • In the data, we control for marital status, number of children and fixed effects which should partly capture heterogeneity in b
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 25/26
Concluding Remarks • Calibrated standard search/matching model has trouble generating much frictional wage inequality • So either frictional wage inequality is mismeasured and actually it is very, very small. . . • . . . or the model needs to be supplemented with something like on-the-job search, risk-aversion, heterogeneity in value of leisure to be able to explain what we “observe” • Relation to Andolfatto-Shimer puzzle: A-S puzzle easier to solve for high values of b
Hornstein-Krusell-Violante, ”Frictional Wage Inequality” – p. 26/26
