Classical Labor Supply: Micro and Macro Elasticities ECON 34430: - - PowerPoint PPT Presentation

Classical Labor Supply: Micro and Macro Elasticities

ECON 34430: Topics in Labor Markets

• T. Lamadon (U of Chicago)

Winter 2016

Agenda

1 Prescott (2004)

• Macro model with labor supply and taxes
• Look at cross-country taxes and labor supply

2 Rogerson and Wallenius (2008)

• Provides a model with extensive and intensive labor supply

decisions

3 Chetty, Guren, Manoli and Weber

• Calibrates Rogerson and Wallenius (2008)
• simulates quasi-experimental results

Prescott 2004 - Why Do Americans Work So Much More Than Europeans?

Intro - rules of the game

1 write a simple macro model of labor supply and taxation 2 calibrate the model with identical parameters for all countries 3 apply countries specific tax codes 4 how much does this explain of labor supply differences?

The model 1

• stand-in household with preference

E[

∞

• t=0

βt log ct + α log(100 − ht)

• ]

(1)

• law of motion for capital stock

kt+1 = (1 − δ)kt + xt

• stand-in firm with market clearing

yt = ct + xt + gt ≤ Aitkθ

t h1−θ t

where gt is public spending

The model 2

• the budget constraint for the household is given by:

(1 − τc)ct + (1 + τx)xt = (1 − τh)wtht + (1 − τk)(rt − δ)kt + δkt + Tt

• rt is rental price of capital
• τx, τc, τh, τk are taxes on consumption, investment, labor and

capital income and define τ = (τh + τc)/(1 + τc)

• Tt is a lump sum transfer

Equilibrium relationships

• marginal rate of substitution between leisure and consumption

α/(1 − h) 1/c = (1 − τ)w

• wage and marginal product of labor

w = (1 − θ)kθh−θ

• which we combine to get

hit = 1 − θ 1 − θ + cit

yit α 1−τit

Equilibrium relationships

• the following expression captures most of the trade-offs:

hit = 1 − θ 1 − θ + cit

yit α 1−τit

• 1 − τ affects the relative price of between consumption in

leisure within a period

• c/y which is directly impacting x and as such the capital

stock, reflects the inter-temporal decision

• bottom line is that this expression links h to c, y, α, τ

Estimating tax rates 1

• define Indirect Tax on consumption as a function of total IT

and C, I : ITc =

• 2/3
• priv cons exp

+1/3 · C C + I

• IT
• this captures that most IT falls on consumption (value added,

sales) but some falls on capital investment (sales tax on equipment, property tax on office building)

• and consumption and output as

c = C + G − Gmil − ITc y = GRP − IT where G is public consumption, Gmil is military

Estimating tax rates 2

• consumption tax rate is given by

τc = ITc C − ITc

• value for the social security tax is

τss = Social Security Taxes (1 − θ)(GDP − IT) where the denominator is labor income when labor is paid marginal product

Estimating tax rates 3

• the average income tax is given by:

¯ τinc = Direct Taxes GDP − IT − Depreciation

• the marginal income tax is set to:

τh = τss + 1.6¯ τinc

• finally we need to parametrize as follows, from what I

understood:

• θ = 0.32 using wage equation?
• α = 1.54 to match the average value of h ?

Labor supply, actual and predicted

Overview of results 1/2

1 surprisingly close! given everything else is ignored 2 in Germany, France and Italy, low participation is explained by

high taxes

• when European and US tax rate were similar, labor supply was

comparable

• US vs France/Germany differences can be explained by

differences in tax rates

Overview of results 2/2

1 a second interesting point is the evolution of labor supply in

the US

• despite tax rates remaining similar, participation went up
• Prescott argues that marginal tax rates of moving from one

wage earner to 2 in household was much lower in 93-96 and in 70-74. And that increased participation was mostly among married women.

2 counterfactual calculations give:

• in France, reducing from 0.6 to 0.4 lifetime consumption would

go up by 19%

• in the US, reducing from 0.4 to 0.3 lifetime consumption

would go up by 7%

What is the value of the labor elasticity?

• Chetty, uses participation data and the numbers from the

paper to get both extensive and intensive margin from this model

• He reports 0.25 for the Hicksian extensive and around 0.33 for

the intensive (this is in the high end of the micro values)

• I took the values from the table and regressed log hours on

log net-of-tax rate and found 0.69981 for the elasticity of aggregate hour.

• Chetty reports an average extensive of 0.25 and Keane gives

an average Hicksian of 0.31 which would give around 0.56 total response. This is not so far.

Lower α

• What if I pick α to match the micro Hicksian elasticity of

0.56?

• Using R I found that it requires α = 0.55 instead of 1.54
• this might generate much weaker responses to taxes
• The main disagreement according to Chetty is on the Frisch

extensive elasticity

RogersonWallenius 2008 - Micro and macro elasticities in life cycle model with taxes

Intro

• the paper argues that one problem is to not consider separatly

extensive and intensive margins

• it builds on Prescott and introduces both a choice on amount
• f time per period and share of life spent working
• using the model they compare predicted micro and macro

elasticities

• and they look at the effect of changing marginal tax rate

The model 1

• continuous time overlapping generation model
• life of an individual is normalized to 1
• at each instant t, individual are endowed with 1 unit of time
• denote by a the age of the agent, preferences are:

1 U (c(a), 1 − h(a))da

• agents choose consumption and work hours paths c(a), h(a)
• no discounting, zero interest rate steady state
• The government taxes labor income at rate τ and

redistributes it as a uniform lump-sum.

The model 2

• labor is the only factor of production, output is given by

Y (t) = L(t).

• L(t) is the input of labor services
• agents hours is mapped into labor services according to:

l = e(a) · g(h)

• e(a) captures variation in life cycle productivity
• provides a driving force for life-cycle employment decision
• assumed to be single peaked
• g(h) captures potential fixed cost of working
• g(h) = max 0, h − ¯

h

• the convexity implies that it could be optimal to randomize

some agents to work full time and other to not work.

• hourly wage rate for part time will be lower

Equilibrium 1

• time zero markets for labor and consumption w(t), p(t)
• market are competitive, production is linear so w(t) = p(t)
• the presence of markets allows for agent to implicitly trade

between period at interest rate p(t)/p(t′)

• the authors focus on zero interest rate steady state equlibrium,

p(t) is constant, so is w(t). They can be both normalized to 1

Equilibrium 2

• a new born optimization problem is given by:

max

c(a),h(a)

1 U (c(a), 1 − h(a)da s.t. 1 c(a)da = 1 e(a)g(h(a))da

• first consider e(a) to be constant
• case 1: h(a) > 0∀a then h is constant
• case 2: h(a) > 0 only in some places then only fraction is

pinned down, not locations of hours worked

• this could be the case to deal with convexities in g

General case

1 the paper shows that h∗(a) has a reservation property:

∃e∗ : h∗(a) > 0 ⇔ e(a) > e∗

• this removes the indeterminacy of the location of work over

the life-cycle

• the assumption that e(a) is single peaked will mean that there

will be a unique starting and stopping age for working

2 the paper also shows that for amount of hours worked we

have that e(a1) ≥ e(a2) ⇒ h∗(a1) ≥ h∗(a2)

3 both property will generate life-cycle participation and hours

Calibration

• for the quantitative section, the model is calibrated in the

following way:

• U (C, 1 − h) = log(c) − α h1+γ

1+γ

• g(h) = max{0, h − ¯

h}

• e(a) = e0 − e1|.5 − a|
• for different values of γ, pick α, ¯

h, e1 to match:

• λ fraction of life spent unemployed
• hmax peak hour of work over the life cycle
• variation in hourly earnings over the life cycle
• model is calibrated with tax of 0.3

Matching wages in the data and in the model

• remember w(t) = 1!
• wage is earnings per hour of work
• if g was linear then we would get e(.5)/e(amax)
• define wh(a) = e(a)g(h(a))/h(a)
• then the targeted wage ratio is

wh(.5) wh(amax) = e(.5)g(h(.5))/h(.5) e(amax)g(h(amax))/h(amax)

• choose e1 to get a wage ratio of 2

Generated micro elasticities

• the author generate data from the calibrated model
• they then run the following regression:

log(ht) = b0 + b1 log(wh

t ) + ǫt

• this should reflect the estimated micro Hicks elasticity
• the non-linearity of g implies that the measured elasticity is

very different from 1/γ (remember here η = 0)

Changing the tax transfer

• using calibrated model, tax is changed from 0.3 to 0.5
• H is aggregate hours, λ is fraction of life being employed,

hmax is peak hour

1 aggreate hours goes down 20% 2 the change in aggregate hours is unaffected by changes in the

γ parameters, and as such by the estimated micro-elasticities

3 shift in γ affects the break down of the change in total hours

between λ and hmax

Conclusion

• a simple model with extensive and intensive margins
• no clean link between γ and estimated micro elasticity
• no effect of γ or estimated micro-elasticities on how taxes

affect aggregate hours

• yet this model would have serious problem replicating

cross-country numbers put forward by Prescott

Chetty paper

Overview

• using the same model as Rogerson, Wallenius (2008)
• calibrates in the same way Rogerson, Wallenius (2008) did
• take γ = 2 instead of a range
• pick α, ¯

h, e1 to match (λ, hmax, ¯ w/w)

• simulate quasi-experiments:

1 EITC reforms 2 SSP program 3 Iceland tax holiday

Tax holiday in Iceland

• average tax rates were 14.5%, 0 and 8.0% in 86,87 and 88
• the response is much stronger in the model
• incentive to work is very high, many do switch within that

given year

SSP experiment

• control group faces a 74.3% tax rate when switching into work
• treatment group only faces a 16.7% tax rate for 3 years
• the model responds too much to these tax incentives

EITC Expansion

• tax rate changed from 50.8% in 1992 to 43.6% in 1996
• the response here seems adequate
• this is a permanent shock, so Hicksian elasticity matters, not

Frisch

Recap on Micro Vs Macro

• Intensive Frisch elasticity is low in many estimated micro

studies, which suggests that all intensive elasticities are low

• see review surveys (Keane: two groups, Saez: basicaly 0,

Chetty: )

• Macro Representative agent seems to require high Frisch

elasticity of aggregate hours

• Prescott (2004) uses around 2.0 to explain cross-country

analysis

• reconciliations ?
• Keane suggests that intensive micro is actually not so low

(include HC)

• Heckman, Chetty point to differences between extensive and

intensive elasticities

• Rogerson Wallenius show that micro and macro might not be

directly linked

• Chetty points out that disagreement mostly in Frisch

Chang and Kim (2006)

Chang and Kim (2006): Introduction

• the model differs from RW in the following way:
• infinite lived agent
• indivisible labor
• incomplete markets (only saving)
• individual heterogeneity
• household decision
• the model is used to
• estimate equivalent micro elasticities
• simulate business cycle moments

Chang and Kim (2006): Model

• Each family is a pair of male and female with preferences:

max E0

• βtu(ct, hmt, hft)

u(ct, hmt, hf t) = 2 log ct 2 − Bm h1+1/γ

mt

1 + 1/γ − Bf h1+1/γ

ft

1 + 1/γ

• workers are heterogeneous in productivity xt
• independent between husband and wife
• evolves according to Markov process πf

x and πm x

• hours are either 0 or ¯

h

• labor earnings are wtxtht
• capital markets are incomplete
• trade claim on physical capital at rate rt, depreciation δ
• only source of insurance
• borrowing constraint on asset at ≥ a

Chang and Kim (2006): Model 2

• This gives the following budget constraints for the household:

ct = wt(xmthmt + xfthft) + (1 + rt)at − at+1 at+1 ≥ a

• Firms produce output according to a Cobb-Douglas:

Yt = F(Lt, Kt, λt) = λtLα

t K 1−α t

• λt follows a Markov process πλ

Chang and Kim (2006): Recursive Equilibrium 1

• Call µ(a, xm, xf ) the distribution over agents

Vℓmℓf (a, xm, xj ; λ, µ) = max

a′≥a u(c, ℓm, ℓh) + βE[max ℓ′

m,ℓ′ f

V ′

ℓ′

m,ℓ′ f | xm, xf , λ]

c = wt(xmthmt + xfthft) + (1 + rt)at − at+1 µ′ = T(λ, µ)

• where T is the transition matrix implied by the decisions
• maxℓ′

m,ℓ′ f V ′

ℓ′

m,ℓ′ f is the labor supply decision

Chang and Kim (2006): Recursive Equilibrium 2

• Equilbrium is characterized by values functions, consumption,

asset and labor supply decisions, as well as K(λ, µ), L(λ, µ), w(λ, µ) ,r(λ, µ)

• such that

1 individual solves the Bellman equation 2 firm maximize profits 3 goods market clear 4 factor markets clear 5 T is defined by individual decisions

• Of course, solving this brute-force is not possible since µ

enters the state space

Chang and Kim (2006): Calibration 1

• parametrize log x ′ = ρx log x + ǫx,

ǫx ∼ N (0, σ2

x), then

log wi

t = ρx log wi t−1 + (log wt − ρx log wt−1) + ǫi x,t

• estimate ρx and σx using this equation for Model I
• use time dummies for wt
• using a selection equation di

t = Z i t b + ui t with ui t ∼ N (0, σ2 u)

for participation including (age, education, age, marital status,...)

• note that this seems in no-way consistent with the

participation decision inside the model!

• for Model II they use the residual of the wage regression in the

cross-section, in which case x might reflect better the transitory component

Chang and Kim (2006): Calibration 2

• no selection is rejected at 1%

Chang and Kim (2006): Calibration 3

• γ = 0.4 is inline with micro estimates
• Bm Bf are set to match employment rates
• πλ is calibrated as a 2-point Markov to match TFP variation

Chang and Kim (2006): Calibration 2

• no selection is rejected at 1%

Results - Steady states

Chang and Kim (2006)

• Bm Bf were set to match employment rates for males females
• turns out to match also household outcomes

Results - Steady states

Chang and Kim (2006)

• Bm Bf were set to match employment rates for males females
• turns out to match also household outcomes

Results: Steady states

Chang and Kim (2006)

• Heterogeneity is important for the distribution of reservation

wages

• model matches well both earnings and wealth
• earnings should be captured by estimated ρx, σx

Results: Reservation wages

Chang and Kim (2006)

• Reservation wage dictates labor participation
• A function of asset and spouse productivities
• earnings should be captured by estimated ρx, σx

Results: Implied elasticity

Chang and Kim (2006)

• Elasticities keeping wealth constant
• Bigger than micro-values, but not too far

Results: Fluctuations

Chang and Kim (2006)

• compute solution using Krussel-Smith like method
• compare Model I and II to representative agent

u(c, h) = log c − B h1+1/γ 1 + 1/γ

• consider γ = 04, 1, 2, 4
• finally, compute Frisch elasticity using simulated panel
• ˆ

h = γ(ˆ w − ˆ c)

• using individual panel
• using aggregate hours

Results: Fluctuations

Chang and Kim (2006)

• calibrated heterogenous agent seems to correspond to γ = 2
• why the scale of the variance is not matched using πλ?
• independently of scale, σ(N )/σ(Y ) is also half of the data

Results: Fluctuations

Chang and Kim (2007)

• calibrated using a = 2 instead of 4, and different πλ, men only
• making individual more borrowed constraint, increases the

response to shocks

• this illustrates that of course there are several free parameters

Results: Fluctuations

Chang and Kim (2006)

• Similar to Rogerson and Wallenius, aggregate elasticities are

larger

Results - Link to Frisch parameter?

Chang and Kim (2006)

• a final word on the Frisch elasticity
• the structural parameter was set to 0.4
• but note that it is irrelevant in this model
• there are 2 constant terms −Bm

¯ h1+1/γ 1+1/γ and −Bf ¯ h1+1/γ 1+1/γ and

Bm and Bf are calibrated to match shares in the population

Conclusion

Chang and Kim (2006)

• individual heterogeneity, and liquidity constraints can help

match micro elasticities together with aggregate hours elasticities

• representative household needs γ = 2 to achieve similar result
• of course model is not very disciplined by the data
• we now turn to Attanasio and al [WP]

Attanasio, Lewell, Low and Sanchez-Marcos (2015) Aggregating Elasticities: Intensive and Extensive Margins of Female Labor Supply

Introduction

Attanasio et Al (2015)

• looks at female labor supply
• takes husband as exogenous
• allows for extensive and intensive
• measures heterogenous elasticities
• measures elasticities over the business cycle
• estimates on Consumer Expenditure Survey
• considers the implication for macro-elasticities

Model 1

Attanasio et Al (2015)

• female labor supply solves

max Et

T

• j=0

βj u(ct+j , lt+j , Pt+j ; zt+j , ζt+j , ξt+j ) subject to the budget constraint At+1 = Rt+1

• At + (wf

t (H − lt) − F(at)Pt + wm t ¯

h

• Pt is indicator of labor force participation
• ζt+j , ξt+j are taste shifters
• zt is a vector of observables
• at is age of the youngest child, F(a) is associated cost

Model 2

Attanasio et Al (2015)

• male always work at age:

ln wm

t

= ln wm

0 + αm 1 t + αm 2 t2 + νm t

• female wages are given by :

ln wf

t = ln wf 0 + ln hf t + νf t

• where human capital hf

t is given by

ln hf

t = αf 1t + αf 2t2

• the paper investigates both when accumulation depends on

participation or on hours.

Model 3

Attanasio et Al (2015)

• both male and female face permanent shocks:

vt = vt−1 + ξt ξt = (ξf

t , ξm t ) ∼ N (µξ, σxi2)

µξ = (−1 2σ2

ξf , −1

2σ2

ξm)

σξ =

• σ2

ξf

ρξf ξm ρξf ξm σ2

ξm

• the covariance is left unrestricted
• the shocks have a negative drift equal to the variance

Model Preference specifications

Attanasio et Al (2015)

• pref over consumption and leisure:

u(ct, lt) = M 1−γ

t

1 − γ exp(πzt + φPt + ζt)

• where

Mt = αt(c1−φ

t

− 1) 1 − φ + (1 − αt)(l1−θ

t

− 1) 1 − θ

• αt = (1 + exp(ψzt + ξt))−1

Intra period decision

Attanasio et Al (2015)

• FOCs give us

wt = ult uct = 1 − αt αt l−θ

t

c−φ

t

• where by taking logs we get

ln wt = ψzt − θ ln lt + φ ln ct + ξt

• which gives a way to extract parameters when individuals are

supplying labor

• this gives direct estimates of Hicks and Marshall elasticities (
• nly φ and θ are needed)

Between period decision, Euler equation

Attanasio et Al (2015)

• we first write the Euler equation away from corner solutions

β(1 + Rt+1uct+1) = uctǫt+1 where E[ǫt+1|It] = 1

• we then take the log of the marginal utility of consumption:

ln uh

ct = −γ ln M h t + ln αh t − φ ln ch t + φPh t + πz h t + ζh t

• this together with the Euler can be used to estimate the Frish

intensive elasticity

Estimation strategy

Attanasio et Al (2015)

The estimation is composed of 3 steps

1 use the intra-temporal expression

• control for selection by specifying a participation equation

(which they claim is consistent with the model)

• deal with endogeneity between w and ξ by using instrumental

variable

2 use the Euler equation at a group level

• also requires instrumental variable
• can make use of the previous estimates

3 solve the model numerically and match moment to recover

extensive margin decisions

Results: MRS estimates

Attanasio et Al (2015)

Results: Euler estimates

Attanasio et Al (2015)

Exogenous parameters

Attanasio et Al (2015)

Results: Moments matching

Attanasio et Al (2015)

Results: Life cycle

Attanasio et Al (2015)

Intensive elasticities at quintile of consumption

Attanasio et Al (2015)

• consumption elasticity seems to pick in the middle
• part of the variation in Frisch is due to level of hours worked

Extensive elasticities

Attanasio et Al (2015)

• Macro combines extensive and intensive
• Younger women are more elastic
• decompose further into maternity state (does not seem to

affect as much as age)

Elasticities in boom and recessions

Attanasio et Al (2015)

Conclusion

Attanasio et Al (2015)

• substantive heterogeneity in elasticities, values relatively larger

in the literature

• different macro shocks will lead to different averaging over

these elasticities as different people will response differently

• macro elasticities are likely to show path dependence

eventhough, they might not at the individual level conditional

• n state.

Chetty survey

Attanasio et Al (2015)

Chetty survey

Attanasio et Al (2015)

References