Labor Supply

Christopher Taber

February 17, 2010

Outline

Participation

Continuous Hours

Empirical Implementation

Estimates

Outline

Participation

Continuous Hours

Empirical Implementation

Estimates

Participation

Lets first just think about the participation decision

Do I work or not?

That is really no different than what we have done before-

can put it into familiar frameworks

Roy Framework

Think about home production

There are two jobs

Work in labor market, receive WWork at home and produce H

People who are relatively more productive in the market willwork

People who are relatively more productive at home will stayhome

Work ifW > H

This is about it

Equalizing Differentials FrameworkRather than abstracting from Roy, lets just add on to it

Assume that people may prefer to work either at home or atwork

Let P be an indicator variable indicating that you participate inthe labor force

Let C be consumption

Define utility of individual i as

u(C,P) = log(C)− δiP

Thus this individual chooses to work if

log (Wi/Hi) > δi

Again this is it-this is the theory

Econometric Implementation

This is just a generalized Roy model

Identification issues we talked about all carry over to this case.

Outline

Participation

Continuous Hours

Empirical Implementation

Estimates

Continuous hours decisions

I will follow Blundell and Macurdy here

First consider a static model

Let

u : utility functionC : consumptionh : hours of workw : wageT : timeY : nonlabor income

Workers maximizeu (C,h)

subject toC ≤ wh + Y

solving the first order conditions and assuming you aren’t at acorner gives:

−uh (C,h)

uc (C,h)= w

Marshallian Elasticity

from this we can solve for the Marshallian demand function:

h = Hm (w ,Y )

The uncompensated (Marshallian) elasticity is defined as:

Ku =∂ log Hm (w ,Y )

∂ log(w)

Hicksian Elasticity

The other important concept is the compensated elasticity.

Let Hh be the hicksian labor supply term defined as

h = Hh (w ,u)

The compensated (Hicksian) elasticity is defined as

Kc =∂ log(Hh (w ,u))

∂ log(w)

The describes how much labor I would supply at wage w if Yadjusted to keep the utility constant

Slutsky EquationLet Y (w ,u) denote the amount that income would change(basically the expenditure function)

then for a given u∗

Hh (w ,u∗) = Hm (w ,Y (w ,u∗))

so

∂Hh (w ,u∗)∂w

=∂Hm (w ,Y (w ,u∗))

∂w+∂Hm (w ,Y (w ,u∗))

∂Y∂Y (w ,u∗)

∂w

=∂Hm (w ,Y (w ,u∗))

∂w− ∂Hm (w ,Y (w ,u∗))

∂Yh

and

wh∂Hm (w ,Y )

∂w=

wh∂Hh (w ,u∗)

∂w+

Y∂Hm (w ,Y (w ,u∗))

h∂YhY

hwh

Ku = Kc +∂ log (Hm (w ,Y ))

∂YhwY

The Slutzky equation

Dynamics

Lets think about a model with full certainty

Write down the model using the Bellman’s Equation:

For t < T

Vt (At ) = max u(ct ,ht ) + βE [Vt+1 (At+1)]

subject toAt+1

(1 + rt+1)= (At + Bt + wtht − ct )

where Bt is nonlabor income

(case when t = T is analoguous)

Lets look at all of the first order conditions (assuming not atcorner):

uc(ct ,ht ) = λt

− uh(ct ,ht ) = λtwt

E[βV ′t+1 (At+1)

]=

λt

(1 + rt+1)

V ′t (At ) = λt

Simplifying,

−uh(ct ,ht )

uc(ct ,ht )= wt

λt = E [(1 + rt )βλt+1]

Can solve for Frisch demand functions

ct = Cf (wt , λt )

ht = Hf (wt , λt )

Ki =∂ log(HF (wt , λt ))

∂ log(wt )

As long as leisure is a normal good

Ki > Kc > Ku

Outline

Participation

Continuous Hours

Empirical Implementation

Estimates

Empirical Implementation

We often write something like

log(ht ) = α log (wt ) + Q′tβ + εt

Think of this as a parametric approximation of labor supplymodels above

There are three separate issues which one must worry about inestimation of these models

Which elasticity are we estimating?

This is pretty clear from the theory, it depends what covariatesare included in Qt .

A standard specification would be

log(ht ) = α log (wt ) + θ log(Yt ) + X ′t β + εt

where Yt represents income

Xt is other variables that may affect tastes

In this case α is the uncompensated (Marshallian) elasticity

Alternatives are also clear (but harder to see how we wouldhave data on them)

Measurement error in wt

Measurement error in dependent variables is alwaysproblematic

This case is even worse

We typically measure wages as annual earnings/annual hours

Consider measurement error in hours

log(

h̃t

)= log (ht ) + vt

so

log(w̃t)

= log(Et )− log(

h̃t

)= log(Et )− log (ht )− vt

= log(wt )− vt

Thus

log(

h̃t

)= α log (wt ) + θ log(Yt ) + X ′t β + εt + vt

= α log(w̃t)

+ θ log(Yt ) + X ′t β + εt + (1 + α) vt

Clearly log(w̃t)

is correlated with (1 + α) vt

This can be a really serious bias

I am really worried whenever my regressor is a function of thedependent variable

Correlation between wt (or Yt) and εt

In the model εt represents something like “tastes for leisure”

We may well believe that people who are lazy would have lowerwages

To deal with this one needs an instrument for wt

Examples: Age, Local labor market variation, tax changes

An Estimable Dynamic SpecificationLets take a simple version of the model (based on e.g.Macurdy, JPE 1981)

He uses continuous time-but I will use discrete time

Take no uncertainty

Assume that utility is

T∑t=0

βt(

aitcγ

itγ− bit

hηitη

)with the lifetime budget constraint

T∑t=0

ct

Rt≤

T∑t=0

withit

Rt

Lets look at the first order condition for hit :

βtbithη−1it = λ∗i

wit

Rt

where λ∗i is the lagrange muliplier on the full budget constraintso

log(hit ) =1

η−1

[log (λ∗i ) + log (wit )− log

(Rtβ

t)− log (bit )]

Now notice that since

λit = aitcγ−1it =

λ∗iβtRt

log(hit ) =1

η−1

[log (λ∗i ) + log (wit )− log

(Rtβ

t)− log (bit )]

=1

η−1 [log (λit ) + log (wit )− log (bit )]

is the Frisch labor supply function so 1η−1 is the Frisch elasticity

Assume further that

bit = eX ′itδ+θi+uit

Then we can write

log (hit ) = µi + α log(wit ) + ρt + X ′itδ∗ + u∗it

where

µi =log(λ∗i)

+ θiη−1

α =1

η−1

δ∗ =δ

η−1

u∗it =uitη−1

ρt = − log(Rtβ

t)

Note that this is a standard fixed effect model:

We can first difference to get rid of µi (and thus λi and θi )

∆ log (hit ) = α∆ log(wit ) + ∆X ′itδ∗ + ∆ρt + ∆u∗it

Assumption on error term is different and perhaps morereasonable.

Wages may be correlated with θi

need instead that ∆ log(wit ) is uncorrelated with ∆u∗itStill need to instrument to deal with measurement error

Outline

Participation

Continuous Hours

Empirical Implementation

Estimates

PSID Data

Started in 1968 with about 4800 householdsLongitudinal followed annuallyFollows individualsFollows kids after they have left the houseLots of labor market dataFood ConsumptionA number of other things as well

Macurdy Estimation

Macurdy estimates model using Panel Study of IncomeDynamics

He uses panel data of first differences

Instruments with education, age, year dummies, familybackground variables

Estimating with Uncertainty

Now we will add uncertainty.

I will follow Altonji, JPE, 1986.

He does two different things. The first extends the Macurdymodel to deal with uncertainty.

Recall from above that λt = Et [(1 + rt )βλt+1]

Altonji considers the following model:

log(λit+1) = − log(β (1 + rt )) + log(λit ) + vit+1

He assumes that the first order approximation that vit+1 isorthogonal to information at time t .

Plugging this into the labor supply equation above

log(hit+1)− log(hit )

=1

η−1

[log (λit+1) + log (wit+1) + ρt+1 − X ′it+1δ + θi + uit+1

]− 1

η−1

[log (λit ) + log (wit ) + ρt − X ′itδ + θi + uit

]=

1η−1

[− log(β (1 + rt )) + vit+1 + ∆ log (wit ) + ∆ρt −∆X ′itδ + ∆uit+1

]As long as we have instruments that are orthogonal tovit+1,Macurdy’s procedure will work.

The first instruments Altonji uses a different measure of∆ log (wit ) that is contained in the PSID (for measurement erroronly)

This assumes that the wage is known one period in advance.

He next uses stuff measured prior to period t

Altonji’s second approach makes use of the consumption data.

Recall from the dynamic model above

λit = uc(cit ,hit )

= aitcγ−1it

Assume now that that ait = eX ′itδa+θa

i +uai .

Now plug this into the labor supply equation:

log(hit ) =1

η−1 [log (λit ) + log (wit )− log(bit )]

=1

η−1

[(γ − 1) log (cit ) + log (wit ) + X ′itδ

∗ + θ∗i + u∗it]

where log(ait )− log(bit ) = X ′itδ∗ + θ∗i + u∗it .

Thus we can just estimate this by IV if we can get goodinstruments for log(wit ) and log(cit )

Altonji uses an individual-specific permanent compontent of thewage (using alternative measure) as main instrument forlog(cit ) as well as alternative wage measure