Large Firms
Unemployment Income
The Capitalization Effect
Equilibrium Unemployment Theory
Long-Run Equilibrium and Balanced Growth
Matthias S. Hertweck
University of Basel
March 12, 2012
Matthias S. Hertweck
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Lecture Outline
Large Firms
Unemployment Income
Technological Progress: The Capitalization Effect
Matthias S. Hertweck
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Literature
Recommended Readings:
I Pissarides, C. A. (2000), Equilibrium Unemployment Theory, The MIT
Press, Cambridge, Massachusetts, pp. 67-89.
Optional Readings:
I Faccini, R. & Ortigueira, S. (2010), Labor-market volatility in the
search-and-matching model: The role of investment-specific technology
shocks, Journal of Economic Dynamics and Control 34(8), 15091527.
Matthias S. Hertweck
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Assumptions on the Firms Size
I
firms are large in the sense that they employ many workers
I
they can eliminate all uncertainty about job flows
I
firms are small in the sense that they have no market power
I
they take wages w and labor market tightness θ as given
and, therefore, do not internalize congestion externalities
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Production
I
firms are indexed by i
I
F (Ki , pNi ): CRS production function
I
p: labor-augmenting productivity parameter
I
firm-specific capital stock: Ki
I
firm-specific stock of employment: Ni
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
The Capital Market
I
there is a perfect capital market
I
technological progress is disembodied
I
the price of capital equals the price of output
I
the law of motion of the firm’s capital stock:
K̇i = Ii − δKi
Matthias S. Hertweck
(1)
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Labor Turnover
I
jobs are destroyed at rate λ
I
vacancies are filled at rate q(θ)
I
Vi : number of vacancies posted by firm i
I
the law of motion of the firm’s stock of employment:
Ṅi = q(θ)Vi − λNi
Matthias S. Hertweck
(2)
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
The Net Present Value of the Firm
Z
Πi =
∞
e −rt [F (Ki , pNi ) − wNi − pcVi − Ii ]dt
(3)
0
I
the firm maximizes its NPV, subject to (1) and (2)
I
control variables: investment Ii and vacancies Vi
I
µ: the co-state associated with the state Ki
I
ν: the co-state associated with the state Ni
Matthias S. Hertweck
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
The Firm’s Optimization Problem — Hamiltonian
Hi = Πi + µ [Ii − δKi ] + ν [q(θ)Vi − λNi ]
I
first order conditions:
∂Hi /∂Ii = 0
∂Hi /∂Ki = −µ̇
I
∂Hi /∂Vi = 0
(5)
∂Hi /∂Ni = −ν̇
(6)
Ni (0) = Ni0
(7)
initial conditions:
Ki (0) = Ki0
I
(4)
transversality condition
lim Hi (t) = 0
t→∞
Matthias S. Hertweck
(8)
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
First Order Condition — Investment
I
set the derivative wrt the control Ii equal to zero:
−e −rt + µ = 0
I
(9)
set the derivative wrt the state Ki equal to −µ̇:
e −rt [F1 (Ki , pNi )] − µδ = −µ̇
I
(10)
this yields:
d
e −rt
dt
e −rt [F1 (Ki , pNi ) − δ] = e −rt r
e −rt [F1 (Ki , pNi ) − δ] = −
Matthias S. Hertweck
(11)
(12)
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
First Order Condition — Vacancies
I
set the derivative wrt the control Vi equal to zero:
−e −rt [pc] + νq(θ) = 0
I
set the derivative wrt the state Ni equal to −ν̇:
e −rt [pF2 (Ki , pNi ) − w ] − νλ = −ν̇
I
(13)
(14)
this yields:
e −rt pc
λ = −ν̇
(15)
e −rt [pF2 (Ki , pNi ) − w ] −
q(θ)
pc
pc
−rt
e
pF2 (Ki , pNi ) − w −
λ = e −rt
r (16)
q(θ)
q(θ)
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Steady State Solution
I
Investment/Capital:
F1 (Ki , pNi ) = r + δ
I
(17)
Vacancies/Employment:
pF2 (Ki , pNi ) = w +
Matthias S. Hertweck
r +λ
pc
q(θ)
(18)
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
The Role of Constant Returns to Scale
I
production per worker is given as:
1
F (Ki , pNi ) = F
f (k) =
pNi
Ki
,1
pNi
I
equations (17) and (18) pin down a unique k = Ki /pNi
I
we can express F1 and F2 as functions of k:
F1 (Ki , pNi ) = f 0 (k)
(20)
0
F2 (Ki , pNi ) = f (k) − kf (k)
Matthias S. Hertweck
(19)
(21)
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Cobb-Douglas Example
F (Ki , pNi )
=
=
=
Kiα (pNi )1−α
α
Ki
(pNi )
pNi
k α pNi
α
f (k)
=
k
f 0 (k)
=
αk α−1
Matthias S. Hertweck
F1 (Ki , pNi )
F2 (Ki , pNi )
=
αKiα−1 (pNi )1−α
=
αk α−1
=
(1 − α)Kiα (pNi )−α
=
k α − αk α
=
f (k) − αk α−1 k
= f (k) − kf 0 (k)
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Large Firms
Unemployment Income
The Capitalization Effect
The Job Creation Condition
I
combining (17) and (20) yields:
f 0 (k) = r + δ
I
(22)
combining (18), (21) and (22) yields:
p[f (k) − k(r + δ)] − w −
r +λ
pc = 0
q(θ)
(23)
⇒ equivalent to the small firm model with capital
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
The Wage Equation
I
I
the NPV of the firm is independent of Ni
I
the production function F is of CRS
I
hiring costs are linear in Vi
I
there is a perfect capital market
the firm bargains with each worker separately,
wages of all other workers are taken as given
w = (1 − β)z + βp[f (k) − (r + δ)k + cθ]
(24)
⇒ equivalent to the small firm model with capital
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
The Beveridge Curve
I
all firms post the same number of vacancies (23)
I
we sum over all jobs and vacancies
I
ΣVi
= θuL
(25)
ΣNi
= (1 − u)L
(26)
hence:
Vi =
λ
λNi
⇔u=
q(θ)
λ + θq(θ)
(27)
⇒ equivalent to the small firm model with capital
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Unemployment Income
I
baseline model: increase in p leads to a reduction in u
I
reason: the wage absorbs only the share β:
w = (1 − β)z + βp (1 + cθ)
I
(28)
unemployment declines in the long-run
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Empirical Evidence — Stylized Facts
I
productivity p grows at a variable rate
I
unemployment u fluctuates around a constant rate
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
The Composition of Unemployment Income
I
actual income: unemployment benefits
I
imputed income: value of leisure (depends on wealth)
z = ζr (A + U)
(29)
I
U: human capital, e.g. z might depend on previous wages
I
A: non-human capital, e.g. income from savings
I
ζ: valuation of leisure 0 < ζ < 1
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Case 1: Only Human Capital Income
I
we set A = 0 ⇔ z = ζr U
I
consequently:
rU
rU
z
w
I
β
pcθ
1−β
β
=
pcθ
(1 − ζ)(1 − β)
ζ
β
=
pcθ
1−ζ 1−β
= β f (k) − (r + δ)k +
(30)
= z+
(31)
(32)
1
cθ p
1−ζ
(33)
result: w is directly proportional to p
Matthias S. Hertweck
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Case 2: Human and Non-Human Capital Income
I
I
instead, A > 0 implies:
z
=
w
=
ζ
β
rA +
pcθ
(34)
1−ζ
1−β
(1 − β)ζ
1
r A + β f (k) − (r + δ)k +
cθ p (35)
1−ζ
1−ζ
result: similar to the baseline model
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Interpretation
I
short-run: A is non responsive to shocks
I
long-run: A absorbs shocks like U
I
differences between A and U are important in the short-run
I
but they can be ignored in the long-run
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Choose the Suitable Model
I
baseline model: suitable for short-run studies
⇒ temporary shocks have persistent effects on unemployment
I
Case 1: suitable for long-run analyses
⇒ permanent shocks are fully absorbed: average u is constant
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Unemployment Persistence
I
productivity slowdown in the 70ies
I
result: persistent, but temporary, increase in u
70ies
productivity
slowdown
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Technological Progress: The Capitalization Effect
I
exogenous labor-augmenting technological progress
I
technological progress is disembodied
I
all existing and new jobs benefit
I
no investment is needed
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Assumptions
I
exogenous interest rate r
I
p grows at rate g < r
p(t) = e gt p0
I
(36)
p0 > 0 is some initial productivity level
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Changes in the Optimization Problem
I
equation (9) now reads as:
∂Hi /∂Vi = 0 ⇒ −e −rt [p0 e gt c] + νq(θ) = 0
I
I
we take the derivative wrt time:
p(t)c −rt
ν̇ = −(r − g )
e
q(θ)
(37)
(38)
consequently:
pF2 (Ki , pNi ) − w −
Matthias S. Hertweck
r +λ−g
pc = 0
q(θ)
(39)
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Unemployment Along a Balanced Growth Path
I
the CRS production function implies:
f (k) − kf 0 (k) −
I
I
r +λ−g
w
−
c=0
p
q(θ)
the job creation condition is independent of p:
r +λ−g
βcθ
0
−
c=0
(1 − β) f (k) − kf (k) −
1−ζ
q(θ)
(40)
(41)
u is constant along a balanced growth path
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Impact of Growth on Unemployment
I
we re-write equation (41):
βθq(θ)
q(θ)
−
g = − (1 − β) f (k) − kf 0 (k)
− (r + λ)
c
1−ζ
I
∂g /∂θ > 0: there is a positive relationship between the
growth rate and steady-state market tightness
I
a tighter labor market implies higher vacancies v ,
lower unemployment u, and higher wages w
I
capital and wages grow, but market tightness is constant
along a balanced growth path
Matthias S. Hertweck
(42)
Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Interpretation
I
g reduces the firm’s effective discount rate r − g
I
posting a vacancy entails a cost now,
balanced against future rents
I
future rents are discounted at a lower rate
I
this effect is referred to as “capitalization effect”
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Robustness: Endogenous Interest Rate
I
result may be reversed
I
elasticity of capital supply no longer infinite
I
less capital available per efficiency unit of labor
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Equilibrium Unemployment Theory
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Large Firms
Unemployment Income
The Capitalization Effect
Robustness: Embodied Technological Progress
I
latest technology requires investment
I
endogenous job destruction rises in g
I
higher growth rates lead to higher unemployment,
even if the interest rate is constant
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Equilibrium Unemployment Theory
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