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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