notes on labor supply and unemployment...the slutsky equation implies: "m = "h-s 1 -s...
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Notes onLabor Supply and Unemployment
Vu T. Chau
Harvard University
May 7, 2019
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Outline
Survey of Labor Supply Elasticities
DMP Model of Unemployment
Shimer Puzzle and Debate on Wage Rigidity
• Hagedorn - Manovskii (2008) calibration
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Labor Supply
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Labor Supply Elasticities I
Classify elasticities by:• Static / “steady state” vs. dynamic
• Static: Hicksian (substitution), Marshallian (substitution +
incsome)• Dynamic: Frisch (intertemporal substitution)
• Extensive vs. intensive
• Intensive: change in aggregate hours due to currently working
people• Extensive: change in aggregate hours due to people moving
between work and non-work.
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Static Labor Supply
Static model of labor supply:
V (w , y) ≡ max u(c , n)
s.t. c = wn + y
Marshallian (uncompensated) demand c(w , y) and n(w , y).
Dual cost minimization problem gives Hicksian (compensated)
demand c(w , u) and n(w , u).
Define Marshallian and Hicksian elasticity of substitution:
εM ≡ ∂ ln n(w , y)∂ lnw
, εI ≡ ∂ ln n(w , u)∂ lnw
, εI ≡ ∂ ln n(w , y)∂ ln y
The Slutsky equation implies:
εM = εH −s
1 − sεI (1)
where s ≡ wn/(wn + y) is the labor income share. Proof
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Dynamic Labor Supply and Frisch Elasticity I
Question: suppose relative wage between today and future
change. How much is the change in labor supply today vs the
future?
Dynamic problem:
max E0
[ ∞∑t=0
βtu(ct , nt)
]s.t. ct + at+1 = wtnt + (1 + rt)at
Perturbate the intratemporal FOC, holding marginal utility
constant at uc,t = λ :
un(ct , nt) = −uc(ct , nt)wt = λwt (2)
dun(ct , nt) = unndnt + uncdct = λdwt
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Dynamic Labor Supply and Frisch Elasticity II
Assume that utility is separable between consumption and
leisure, so unc = 0.
The Frish elasticity is given by:
εF ≡ d ln ntd lnwt
=un,t
unn,tnt
Specific utility:
u(ct , nt) =c1−γt − 1
1 − γ− ζ
n1+ 1
ϕt
1 + 1ϕ
(3)
Frisch elasticity is then constant:
εF =un,t
unn,tnt= ϕ
• Generally, εF depends on level of hours worked.• ϕ can be interpreted exactly as the intertemporal elasticity of
substitution for labor.7 / 29
Dynamic Labor Supply and Frisch Elasticity III
If utility is Cobb-Douglas:
u(ct , nt) = ln ct + α ln(T − nt)
the Frisch elasticity is
εF =un,t
unn,tnt=
T − ntnt
Prescott (2004): nt/T ≈ 0.25, so εF ≈ 3.
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Substitution: Hicks vs. Frisch Elasticity
Both the Hicksian elasticity (εH) and the Frisch elasticity (εF )
measures substitution effect.
Crucial difference: εH measures substitution effect in steadystates, while εF intertemporal substitution.
• εH appropriate in context of, say, cross-country difference in
labor supply due to tax differences.• εF appropriate to calibrate for business cycle variation.
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Labor Supply Elasticities II
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Labor Supply Elasticities III
Micro estimates (from quasi-experimental studies) tend to be
small.
Macro estimates of Hicksian elasticities (from cross-country
tax variation) are also small, agreeing with micro estimates
(macro: 0.50 vs. micro: 0.59)
In data: aggregate hours are very volatile.
• Macro models match this moment by introducing large Frisch
elasticity (2.84)• Stark contrast with micro Frisch elasticity (0.82).
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Diamond - Mortensen -Pissarides (DMP)
Model of Unemployment
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DMP: Motivation
Bulk (5/6) of aggregate hours fluctuation are due to extensive
margin.
Standard macro model hard to tractably differentiate
intensive/extensive margin.
Unemployment is important and rather volatile, so need to
study it.
Search model provides nice, tractable way to studyunemployment.
• And other labor issues: heterogeneity, on the job search,
laborpolicy, etc.• Framework extended to other fields, e.g. money search.
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DMP Model
Workers:• Normalize labor force = 1 (ignore inactive people)• ut ∈ [0, 1] unemployed people, looking for and find jobs at rate
ft (endogenous). 1 − ut employed people earn wage wt , but
potentially lose job (separation) at rate st (exogenous):
Ut = zt + βEt [ftWt+1 + (1 − ft)Ut+1]
Wt = wt + βEt [(1 − st)Wt+1 + stUt+1]
Firms:• Unit mass of firms, posting vacancies, hiring, and producing.• Each employed worker (filled position) brings revenue pt to the
firm, but may separate workers at rate st . Each vacancy costs
κt to post, and firms find employees at rate qt (endogenous):
Jt = pt − wt + βEt [(1 − st)Jt+1 + stΞt+1]
Ξt = −κt + βEt [(1 − qt)Ξt+1 + qtJt+1] (4)
• There is free entry to posting vacancies.
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Matching Function
Matching takes place on centralized market
Number of matches: mt = m(ut , vt)
• u and v are num. of unemployed people and num. of
vacancies.• Reduced-form assumption.• Commonly assumed to be CRS.
Define:
• Market tightness: θt ≡ vtut
.• Prob. of finding a job: ft(θt) ≡ mt
ut= m(1, θt).
• Prob of filling a vacancy: qt(θt) ≡ mt
vt= m(θ−1
t , 1).• Note: ft(θt) = qt(θt) · θt . f ′(θ) > 0, q ′(θ) < 0.
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Wage Determination I
Total surplus of an employment:
St ≡Wt − Ut + Jt − Ξt
(in equilibrium, Ξt = 0).
Nash bargaining picks wage that maximize:
maxwt
(Wt − Ut)µ(Jt − Ξt)
1−µ
s.t. (Wt − Ut) + (Jt − Ξt) = St
µ: bargaining power of workers.
Note that St is independent of wage.
Solution:Wt − Ut
µ=
Jt − Ξt1 − µ
= St
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Wage Determination II
Wage that supports this solution:
wt = µ(pt + κtθt) + (1 − µ) zt
• µ = 0: firms extracts 100% rent, and worker earns only outside
option (below MPL pt)• µ = 1: workers extract 100% rent (MPL pt plus the vacancy
cost that the firm can avoid next period κtθt)
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DMP Equilibrium Conditions
1 Free entry to posting vacancies implies Ξt = 0 for all t. (4)
implies:
EtJt+1 =κt
βqt2 Nash bargaining, which splits total surplus from an
employment / filled vacancy, determines wage:
wt = (1 − µ)zt + µpt + µκtθt
3 Plug wt → Jt to find job-creation condition:
κt
βqt(θt)= Et
[(1 − µ)(pt+1 − zt+1) − µκt+1θt+1 + (1 − st+1)
κt+1
βqt+1
](5)
4 Finally, law of motion for ut :
ut+1 = (1 − ft(θt))ut + st(1 − ut) (6)
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DMP Steady State
Job-creation condition (5) pins down market tightness in
steady state:
sκ
βq(θ)+ µκθ = (1 − µ)(p − z) (7)
In v − u space, this is a straight line through the origin.
Law of motion for u in steady state gives an inverse
relationship between u and v (Beveridge curve):
u =s
s + f (v/u)
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DMP Steady State
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Comparative Statics: Example
Example: what is the effect of higher worker’s bargaining
power (µ) on equilibrium outcomes?
η does not shift the Beveridge curve.
Consider the job-creation condition:
Γ(θ, µ;Λ) ≡ sκ
βq(θ)+ µκθ− (1 − µ)(p − z) = 0
where Λ ≡ (s, κ, p, z , Φ, α) is the vector containing remaining
parameters.
Implicit Differentiation Theorem:
dθ
dµ= −
Γµ
Γθ
Γµ = κθ+p−z > 0, Γθ = −sκ
βq(θ)2q ′(θ)+µκ > 0 (q ′ < 0)
Thus: dθ/dµ < 0, i.e. the job creation line becomes less
steep. So: u ↑, v ↓, θ ↓, w ↑.21 / 29
Efficiency
Is the search equilibrium efficient?
• This is beyond inefficiencies arising from the search frictions.
The question is: would the social planner do anything
differently when facing the same matching function?
When someone decides to search for jobs, it:
• makes it easier for firms to fill a vacancy (thick-market
externality).• makes it harder for another job-seeker to find a job (congestion
externality).
Two effects exactly cancel out when µ = α, and market is
efficient (Hosios condition).
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Can search model match data? Shimer (2005)
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Can basic search model match data?
Shimer’s Puzzle: basic search model cannot match volatilityof u, v , θ in equilibrium.
• Model generates volatility an order of magnitude lower than
that observed in data.• In contrast, σ(w) and corr(w , p) too high in model compared
to data.
Problem in Shimer (2005): εw ,p too high and εθ,p too low.
• Recall:
wt = (1 − µ)zt + µ (pt + κtθt)
wt − zt = µ(pt − zt + κtθt)
• When p − z increases by 1% and suppose θ increases by 1%.
Given µ ≈ 0.72 and κ ≈ 0.21, w − z goes up by almost 1%.• Large increase in w soaks up benefit of higher p, dampening
firms’ incentive to post vacancies.
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Proposed solutions to Shimer (2005) Puzzle
Hall (2005)’s solution: assume rigid wages, perhaps due tosocial norms.
• But this is not an inefficient outcome, since any wage
remaining in bargaining set satisfies efficiency.
Hagedorn and Manovskii (2008)’s solution: assume z ≈ p, so
profit change at a larger rate.
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Hagedorn and Manovskii (2008) I
Hagedorn and Manovskii (2008) shows that:
εθ,p = B · p
p − z
where B ≈ [1, 2] for reasonable calibration and any µ ∈ [0, 1].
In data: εθ,p ≈ 20. Shimer calibration: z ≈ 0.4p,
εθ,p ≤ 2 · 11−0.4 ≈ 3.s
To generate volatility: Hagedorn and Manovskii (2008) sets
z = 0.95p.
Intuition: p − w responds little to p, so set high z so that
w ≥ z also close to p. Smaller level of profits implies larger
percentage change in profits following a productivity shocks.
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Hagedorn and Manovskii (2008) II
What about wage? Recall:
wt = (1 − µ)zt + µ (pt + κtθt)
wt − zt = µ(pt − zt + κtθt)
How to reconciles large εθ,p ≈ 20 with small εw ,p ≈ 0.45?
• i.e. how to avoid the Shimer (2005) problem that wage moves
by too much?
Traditional calibration: set µ = α to satisfy Hosios condition.
Hagedorn and Manovskii (2008) instead calibrates µ to match
εw ,p ≈ 0.45. Find µ ≈ 0.052 (much smaller than literature).
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Extra Slides
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Deriving equation (1)
Slutsky equation:
∂n(w , y)
∂w=
∂n(w , u)
∂w︸ ︷︷ ︸substitution effect
−n(w , y)∂n(w , y)
∂y︸ ︷︷ ︸income effect
Multiply by w/n
∂n(w , y)
∂w
w
n︸ ︷︷ ︸εM
=∂n(w , u)
∂w
w
n︸ ︷︷ ︸εH
−wn
y
∂n(w , y)
∂y
y
n︸ ︷︷ ︸εI
Let s ≡ wn/(wn + y). Then wn/y = s/(1 − s).
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