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Age and Labor Market Matching
1. Motivation Data Model Results Conclusion References
The Importance of Age in
Labor Market Matching
Christopher Marfisi
Department of Economics
Florida State University
November 2012
2. Motivation Data Model Results Conclusion References
Motivation
• There are empirically robust relationships between age &
wages, and age & unemployment rates
• Wages are “hump-shaped” over the life-cycle
• Unemployment decreases with age
• This paper attempts to model these relationships using a
Mortensen and Pissarides [1994] type model and then
compares the results to data
• The model produces realistic wage profiles and aggregate
unemployment, but the effect of age on wages and
unemployment is much stronger than suggested by data
3. Motivation Data Model Results Conclusion References
Outline
This paper proceeds as follows
1. Data is presented on wages and unemployment rates and
relevant literature is discussed
2. The model of this paper is outlined
3. Results of the model are presented and discussed
4. Criticisms and future work is discussed
4. Motivation Data Model Results Conclusion References
Wages & Age
• “Hump-shape”
• See Thurow [1969], Carroll and Summers [1991], Kotlikoff and
Gokhale [1992], Attanasio et al. [1999], Gourinchas and Parker
[2002]
• Robust to
• education levels
• geography
• sample period
• Popular way to generate this is to feed in “Mincer”
productivity profiles
• Wages, across all ages, are correlated over business cycle
5. Motivation Data Model Results Conclusion References
Quarterly median usual weekly earnings in 2011 $ (CPS)
$0
$100
$200
$300
$400
$500
$600
$700
$800
$900
$1,000
WeeklyEarnings
Year
16-19
20-24
25-34
35-44
45-54
55-64
65+
7. Motivation Data Model Results Conclusion References
Average Weekly Earnings∗
326
426
626
740
768 757
586
$0
$100
$200
$300
$400
$500
$600
$700
$800
$900
16-19 20-24 25-34 35-44 45-54 55-64 65+
WeeklyEarnings
Age Cohort
Average Weekly Earnings 2000-2011
∗
Compare with Slide 35
8. Motivation Data Model Results Conclusion References
Unemployment & Age
• Unemployment decreases with age
• See Dernburg and Strand [1966], Shimer [1998]
• Unemployment rates are highly correlated over the business
cycle
11. Motivation Data Model Results Conclusion References
Unemployment by Age
17.9
10.3
6.9
5.7
5.1
4.6
4.3 4.1 4.0 3.9
3.5
3.1
0
2
4
6
8
10
12
14
16
18
20
16-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-64 65-69 70-74 75+
UnemploymentRate
Age Cohort
12. Motivation Data Model Results Conclusion References
Labor Force Participation by Age
• Previous slide may be misleading because of retirement
48.1
76.1
83.3 83.6 84.1 84.7 83.5
79.1
59.7
24.8
13.9
5.4
0
10
20
30
40
50
60
70
80
90
16-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-64 65-69 70-74 75+
LaborForceParticipation
Age Cohort
13. Motivation Data Model Results Conclusion References
Data Summary
• Wage data is “hump-shaped”
• See Thurow [1969], Carroll and Summers [1991], Attanasio
et al. [1999], Gourinchas and Parker [2002]
• Unemployment decreases with age
• See Dernburg and Strand [1966], Shimer [1998]
• This is true regardless of business cycle conditions as
evidenced by
• high correlations
• similar empirical findings in the literature
14. Motivation Data Model Results Conclusion References
Intuition
• Why use MP to model these phenomenon?
• MP is the workhorse model of unemployment in
macroeconomics
• But, by ignoring age, MP misses important dynamics
• Why would age matter in the MP framework?
• Older workers need to be replaced sooner (which is an implicit
cost)
• This puts downward pressure on wages for older workers
• At the same time, if older workers are more productive there is
an explicit benefit the firm must also weigh
15. Motivation Data Model Results Conclusion References
Environment
• MP environment with two important differences
• Workers only live T periods
• Workers allowed to be risk averse
• Each generation of workers is of measure 1
T
• Workers begin life unemployed with no assets
• The state space of the worker is
X = A × B × E
where A ≡ {1, 2, . . . , T}, B ≡ {bmin, . . . , bmax },
E ≡ { employed, unemployed } with state vector
x = (a, b, ε)
16. Motivation Data Model Results Conclusion References
Environment
• Employment subject to matching friction
• Production F(a) depends only on age
• Wages w(xt) are determined each period by Nash bargaining
and there is no hourly decision
• There is “full” information
• History of unemployment unknown
• Match with (a, b, E)-worker who was unemployed for 1 period
treated same as if unemployed for 50 periods
• History dependence vastly increases state space
• Matches exogenously destroyed with probability δ
• Matches in which a = T always destroyed (worker dies)
• Firing and quits allowed
17. Motivation Data Model Results Conclusion References
Matching
• Given measures of unemployment u and vacancies v,
aggregate matches given by
M(u, v) = Auα
v1−α
• Labor market tightness is defined as
v
u
= θ
• Probability of match for worker
M(u, v)
u
= Aθ1−α
≡ λ
• Probability of match for firm
M(u, v)
v
= Aθ−α
≡ λf
18. Motivation Data Model Results Conclusion References
Worker Problem
• A worker born in period k solves
max
bt+1,it
Ek
T+k
t=k
βt−k
u(ct)
• where u(c) = (c1−σ − 1)/(1 − σ) and
ct =
(1 − τ)w(xt) + (1 + r)bt − bt+1 if εt = employed
z + (1 + r)bt − bt+1 if εt = unemployed
• The worker optimizes by making decisions on savings bt+1
and, if matched, wage acceptance it ∈ { accept, reject }
19. Motivation Data Model Results Conclusion References
Worker Problem
• The value function of the worker is given by
v(xt) =
max(ve
t , vu
t ) if matched
vu
t otherwise
• where
ve
t =
u(ct) + β[δvu
t+1 + (1 − δ)vt+1] if t < T
u[(1 − τ)w(xT ) + (1 + r)bT ] if t = T
vu
t =
u(ct) + β[λvt+1 + (1 − λ)vu
t+1] if t < T
u(z + (1 + r)bT ) if t = T
• Decision rules for savings and wage acceptance are functions
bt+1 =f (xt)
it =g(xt)
20. Motivation Data Model Results Conclusion References
Firm Problem
• Firms are infinitely lived and solve
max
if
t
E0
∞
t=0
βt
J(xt)
• where xt is the state of the worker the firm is matched with at
time t and
J(xt) =
max(JE
t , JU) if matched
JU otherwise
• Firm’s value functions are given by
J(xt) = max{JE
(xt), JU
}
JE
(xt) =
F(a) − w(xt) + β[δJU + (1 − δ)J(xt+1)] if a < T
F(T) − w(xt) + βJU if a = T
JU
= − κ + β λf
A,B
JE
(a, b)
µ(a, b, U)
u
dadb + (1 − λf )JU
21. Motivation Data Model Results Conclusion References
Firm Problem
Free entry implies that JU = 0 so that
0 = −κ + βλf
A,B
JE
(a, b)
µ(a, b, U)
u
dadb
or
λf =
κ
βEµJE (a, b)
22. Motivation Data Model Results Conclusion References
Wages
• Wages determined by Nash bargaining, i.e. w(xt) solves
max
w(xt )
[ve
(xt) − vu
(xt)]φ
[JE
(xt) − JU
]1−φ
• φ is “sharing” parameter and vu, JU are “threat points”
• The solution to the Nash bargain is given by
w(xt) =
F(a) + β(1 − δ)J(xt+1) + 1−φ
φ
vu(xt )−ve(xt )
(1−τ)u (·) if a < T
F(a) + 1−φ
φ
vu(xt )−ve(xt )
(1−τ)u (·) if a = T
• Note that wages are increasing in
• worker productivity F(a)
• discounted expected future value for the firm β(1 − δ)J(xt+1)
• value of unemployment vu
(xt)
23. Motivation Data Model Results Conclusion References
Government
• The government taxes wages at rate τ to pay for
unemployment benefit z
• Total government revenues Z are given by
Z =
A,B
τw(a, b)µ(a, b, E)dadb
• The unemployment benefit is then given by
z =
Z
u
24. Motivation Data Model Results Conclusion References
Distribution
• Define the employment indicator functions
1E (x) =
1 if if (x) = accept = i(x)
0 otherwise
1¬E (x) =
1 if ¬[if (x) = accept = i(x)]
0 otherwise
• Then, the distribution of workers over states µ(xt) evolves as
follows
• Workers begin life unemployed with no assets, so that for
x = (1, b, ε)
µ(1, b, ε) =
1
T if b = bmin, ε = U
0 otherwise
25. Motivation Data Model Results Conclusion References
Distribution
For all other states, µ(x) evolves according to
µ(a + 1, b, E) =(1 − δ)
B
1[b=b (a,ˆb,E)]1E (a + 1, b)µ(a, ˆb, E)dˆb
+ λ
B
1[b=b (a,ˆb,U)]1E (a + 1, b)µ(a, ˆb, U)dˆb
µ(a + 1, b, U) =δ
B
1[b=b (a,ˆb,E)]µ(a, ˆb, E)dˆb
+ (1 − λ)
B
1[b=b (a,ˆb,U)]µ(a, ˆb, U)dˆb
+ (1 − δ)
B
1[b=b (a,ˆb,E)]1¬E (a + 1, b)µ(a, ˆb, E)dˆb
+ λ
B
1[b=b (a,ˆb,U)]1¬E (a + 1, b)µ(a, ˆb, U)dˆb
26. Motivation Data Model Results Conclusion References
Distribution
µ(a + 1, b, E) =(1 − δ)
B
1[b=b (a,ˆb,E)]1E (a + 1, b)µ(a, ˆb, E)dˆb
+ λ
B
1[b=b (a,ˆb,U)]1E (a + 1, b)µ(a, ˆb, U)dˆb
The measure of employed workers aged a + 1 with savings b is the
sum of workers who were:
1. employed at age a, chose savings b, with probability 1 − δ
were not separated and then, along with the firm, chose to
continue employment at age a + 1
2. unemployed at age a, chose savings chose savings b, with
probability λ were matched at age a + 1 and agreed, along
with the firm, to employment.
27. Motivation Data Model Results Conclusion References
Distribution
µ(a + 1, b, U) =δ
B
1[b=b (·)]µ(a, ˆb, E)dˆb + (1 − λ)
B
1[b=b (·)]µ(a, ˆb, U)dˆb
+ (1 − δ)
B
1[b=b (·)]1¬E (a + 1, b)µ(a, ˆb, E)dˆb
+ λ
B
1[b=b (·)]1¬E (a + 1, b)µ(a, ˆb, U)dˆb
The measure of unemployed workers aged a + 1 with savings b is
the sum of workers who were:
1. employed at age a, chose savings b and, with probability δ,
were separated at age a + 1
2. unemployed at age a, chose savings chose savings b and, with
probability 1 − λ, were not matched at age a + 1
3. employed at age a, chose savings b, and with probability 1 − δ
were not separated at age a + 1 but were either fired or quit
4. unemployed at age a, chose savings b, and with probability λ
were matched but that match was not agreed to
28. Motivation Data Model Results Conclusion References
Equilibrium
It is useful, before defining equilibrium, to define
Ξ ={v, J, w, b , i, if
, µ∗
, θ, z, U, r}
and
ξ(x1, . . . , xn) ={y | y ∈ Ξ, x1 = y, . . . , xn = y}
Equilibrium is then the set Ξ such that
• (HH optimization): Given ξ(b , i, v), the functions b and i
are decision rules for v
• (Firm optimization): Given ξ(if , J), the function if is the
decision rule for J
• (Nash bargaining): Given ξ(w), wages w satisfy the Nash
bargaining FOC for all x ∈ X
• (Free entry): Given ξ(θ), the value for θ satisfies JU = 0
29. Motivation Data Model Results Conclusion References
Equilibrium
and
• (Government BC): Given ξ(z), the unemployment benefit z
satisfies the government budget constraint
z =
τ A,B w(a, b)µ(a, b, E)dadb
U
• (Unemployment): Given ξ(U), the measure of unemployed
workers U is given by
U =
A,B
µ(a, b, U)dadb
• (Invariant distribution): Given ξ(µ∗) the distribution µ∗ is
the invariant distribution associated with the law of motion
for µ
• (Savings Market): Given ξ(r), the interest rate r satisfies
A,B,E
b(ˆa, ˆb, ˆε)µ∗
(x)dˆadˆbdˆε = 0
30. Motivation Data Model Results Conclusion References
Production
• How productivity changes over time is a crucial assumption
• If older workers leave matches sooner and are less productive,
then there is little benefit to employing older workers
• Kotlikoff and Gokhale [1992] suggests older workers may in
fact be less productive
• van den Berg and Ridder [1998] suggests that productivity may
be “hump-shaped”
• To see how the model changes based on productivity are
considered to investigate
1. increasing
2. normally distributed
3. uniform
4. decreasing
• In each case, total lifetime production is equivalent
• In future, want to estimate these profiles
31. Motivation Data Model Results Conclusion References
Production
0 20 40 60 80 100 120 140 160
0
2
4
6
8
10
12
14
Age
F(Age)
Increasing
Normal
Uniform
Decreasing
32. Motivation Data Model Results Conclusion References
Parameters
Value Description
φ 0.5 Nash bargain parameter
A 0.3 Match parameter
α 0.5 Elasticity of match function
δ 0.09 Exogenous separation rate
T 160 Life span
σ 1 Worker utility parameter
κ 0.5 Vacancy posting cost
β 0.99 Discount factor
τ 1% Unemployment tax rate
34. Motivation Data Model Results Conclusion References
Wages
• (a) & (c) demonstrate that even if the old have productivity
≥ productivity of young, their wages go down because of
implicit cost
• (d) shows that decreasing productivity, suggested by Kotlikoff
and Gokhale [1992], generates wages unlike the data
35. Motivation Data Model Results Conclusion References
Wage Profiles†
0 20 40 60 80 100 120 140 160
0
1
2
3
4
Age
wavg(Age)
Increasing
Normal
Uniform
Decreasing
†
Compare with Slide 7
37. Motivation Data Model Results Conclusion References
Unemployment Profiles
0 20 40 60 80 100 120 140 160
0
0.2
0.4
0.6
0.8
1
Age
Unemployment
Increasing
Normal
Uniform
Decreasing
Data
Though unemployment rates are similar to data, age effects are
much stronger in model
38. Motivation Data Model Results Conclusion References
Results
Variable
Production Data
(1) (2) (3) (4)
θ 8.01 4.45 9.70 7.88
λ 0.85 0.64 0.93 0.84
λf 0.11 0.14 0.10 0.11
U 9.17% 27.2% 5.68% 9.81% 6.1%
max(w)/Q1(w) 2.71 50.8 1 3.01 1.96
• Q1(w) is the median of the 1st quartile of w and is used to
measure hump
• Model produces reasonable unemployment rates
• Hump’s are not as flat in (1), (2) & (4) but too flat in (3)
compared to data
39. Motivation Data Model Results Conclusion References
Conclusion
• Model shortcomings
• No idiosyncratic risk
• Unrealistic full information about (a, b) between worker & firm
• No history dependence, which may be important for
unemployment
• Age effects are severe relative to data
• Asset heterogeneity impacts wages, but not employment
decisions
• Model accomplishments
• Wages and unemployment rates “qualitatively” similar to data
• Can now be used to analyze effects of policy changes (τ, z) on
w, µ(a, b, U)
40. Motivation Data Model Results Conclusion References
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Evidence. In National Saving and Economic Performance, NBER Chapters, pages 305–348. National Bureau of
Economic Research, Inc, Jan-Jun 1991.
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Value of Workers’ Earnings. The Quarterly Journal of Economics, 107(4):pp. 1215–1242, 1992.
Dale T Mortensen and Christopher A Pissarides. Job Creation and Job Destruction in the Theory of
Unemployment. Review of Economic Studies, 61(3):397–415, July 1994.
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11–61, 1998. ISSN 08893365.
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