Age and Labor Market Matching

Motivation Data Model Results Conclusion References
The Importance of Age in
Labor Market Matching
Christopher Marﬁsi
Department of Economics
Florida State University
November 2012

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

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

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

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+

Wage Correlations
16-19 20-24 25-34 35-44 45-54 55-64 65plus
1 0.9058 0.8771 0.8555 0.8524 0.8675 0.7937 16-19
1 0.9385 0.9318 0.9271 0.9293 0.8637 20-24
1 0.9774 0.9774 0.9750 0.9224 25-34
1 0.9863 0.9881 0.9499 35-44
1 0.9827 0.9553 45-54
1 0.9432 55-64
1 65plus
• Wages move together

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

Unemployment & Age
• See Dernburg and Strand [1966], Shimer [1998]
• Unemployment rates are highly correlated over the business
cycle

Quarterly unemployment rates from CPS
0
5
10
15
20
25
30
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010
UnemploymentRate
Year
16-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-64
65-69
70-74
75+

Unemployment Rate Correlations
16-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-64 65-69
1 0.96 0.93 0.93 0.92 0.94 0.93 0.94 0.94 0.77 16-19
1 0.98 0.97 0.96 0.96 0.95 0.95 0.95 0.73 20-24
1 0.98 0.97 0.96 0.95 0.96 0.95 0.69 25-29
1 0.98 0.97 0.96 0.97 0.96 0.71 30-34
1 0.97 0.96 0.97 0.96 0.74 35-39
1 0.97 0.97 0.97 0.81 40-44
1 0.97 0.98 0.79 45-49
1 0.97 0.78 50-54
1 0.81 55-64
1 65-69
• Unemployment rates move together
• Similar to Shimer [1998]

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

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

Data Summary
• Wage data is “hump-shaped”
• See Thurow [1969], Carroll and Summers [1991], Attanasio
et al. [1999], Gourinchas and Parker [2002]
• 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

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

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)

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

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

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 }

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)

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

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)

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)

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

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

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

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.

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

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

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(â, ˆb, ˆε)µ∗
(x)dâdˆbdˆε = 0

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

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

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

Figure: Wages
0
0.5
1
1.5
2
2.5
3
0
50
100
150
200
0
1
2
3
4
5
6
7
SavingsAge
Wage
(a) Increasing
0
0.5
1
1.5
2
2.5
3
0
50
100
150
200
0
2
4
6
8
10
SavingsAge
Wage
(b) Normal
0
0.5
1
1.5
2
2.5
3
0
50
100
150
200
1.5
2
2.5
3
3.5
4
SavingsAge
Wage
(c) Uniform
0
0.5
1
1.5
2
2.5
3
0
50
100
150
200
0
1
2
3
4
5
6
7
SavingsAge
Wage
(d) Decreasing

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

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

Figure: Worker Employment Decisions, 1 = accept
0
0.5
1
1.5
2
2.5
3
0
50
100
150
200
1
1.2
1.4
1.6
1.8
2
SavingsAge
EmploymentChoice
(a) Increasing
0
0.5
1
1.5
2
2.5
3
0
50
100
150
200
1
1.2
1.4
1.6
1.8
2
SavingsAge
EmploymentChoice
(b) Normal
0
0.5
1
1.5
2
2.5
3
0
50
100
150
200
0
0.5
1
1.5
2
EmploymentChoice
SavingsAge
(c) Uniform
0
0.5
1
1.5
2
2.5
3
0
50
100
150
200
1
1.2
1.4
1.6
1.8
2
SavingsAge
EmploymentChoice
(d) Decreasing

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

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

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)

Orazio P. Attanasio, James Banks, Costas Meghir, and Guglielmo Weber. Humps and Bumps in Lifetime
Consumption. Journal of Business & Economic Statistics, 17(1):pp. 22–35, 1999.
Christopher D. Carroll and Lawrence H. Summers. Consumption Growth Parallels Income Growth: Some New
Evidence. In National Saving and Economic Performance, NBER Chapters, pages 305–348. National Bureau of
Economic Research, Inc, Jan-Jun 1991.
Thomas Dernburg and Kenneth Strand. Hidden Unemployment 1953-62: A Quantitative Analysis by Age and Sex.
The American Economic Review, 56(1/2):pp. 71–95, 1966. ISSN 00028282.
Pierre-Olivier Gourinchas and Jonathan A. Parker. Consumption over the Life Cycle. Econometrica, 70(1):pp.
47–89, 2002.
Laurence J. Kotlikoﬀ and Jagadeesh Gokhale. Estimating a Firm’s Age-Productivity Proﬁle Using the Present
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.
Robert Shimer. Why Is the U.S. Unemployment Rate so Much Lower? NBER Macroeconomics Annual, 13:pp.
11–61, 1998. ISSN 08893365.
Lester C. Thurow. The Optimum Lifetime Distribution of Consumption Expenditures. The American Economic
Review, 59(3):pp. 324–330, 1969.
Gerard J. van den Berg and Geert Ridder. An empirical equilibrium search model of the labor market.
Econometrica, 66(5):1183–1221, 1998.

Age and Labor Market Matching

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