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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