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 beforecan put it into familiar frameworks
Roy Framework Think about home production There are two jobs Work in labor market, receive W Work at home and produce H People who are relatively more productive in the market will work People who are relatively more productive at home will stay home Work if W >H This is about it
Equalizing Differentials Framework Rather than abstracting from Roy, lets just add on to it Assume that people may prefer to work either at home or at work Let P be an indicator variable indicating that you participate in the labor force Let C be consumption Define utility of individual i as u(C, P) = log(C) − δi P 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 function C : consumption h : hours of work w : wage T : time Y : nonlabor income
Workers maximize u (C, h) subject to C ≤ wh + Y solving the first order conditions and assuming you aren’t at a corner gives: −uh (C, h) =w uc (C, h)
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 Y adjusted to keep the utility constant
Slutsky Equation Let 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 ∗ )) ∂Hm (w, Y (w, u ∗ )) ∂Y (w, u ∗ ) + ∂w ∂Y ∂w ∂Hm (w, Y (w, u ∗ )) ∂Hm (w, Y (w, u ∗ )) − h ∂w ∂Y
= =
and w ∂Hm (w, Y ) h ∂w Ku
w ∂Hh (w, u ∗ ) Y ∂Hm (w, Y (w, u ∗ )) h w + h h ∂w h∂Y Y h ∂ log (Hm (w, Y )) hw = Kc + ∂Y Y =
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 to At+1 = (At + Bt + wt ht − ct ) (1 + rt+1 ) where Bt is nonlabor income (case when t = T is analoguous)
Lets look at all of the first order conditions (assuming not at corner):
uc (ct , ht ) = λt − uh (ct , ht ) = λt wt 0 λt E βVt+1 (At+1 ) = (1 + rt+1 ) 0 Vt (At ) = λt Simplifying, −uh (ct , ht ) uc (ct , ht ) λt
= wt = 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 ) + Qt0 β + εt Think of this as a parametric approximation of labor supply models above There are three separate issues which one must worry about in estimation of these models
Which elasticity are we estimating? This is pretty clear from the theory, it depends what covariates are included in Qt . A standard specification would be log(ht ) = α log (wt ) + θ log(Yt ) + Xt0 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 would have data on them)
Measurement error in wt Measurement error in dependent variables is always problematic This case is even worse We typically measure wages as annual earnings/annual hours Consider measurement error in hours et = log (ht ) + vt log h so et log w
et = log(Et ) − log h = log(Et ) − log (ht ) − vt = log(wt ) − vt
Thus et = α log (wt ) + θ log(Yt ) + Xt0 β + εt + vt log h e t + θ log(Yt ) + Xt0 β + εt + (1 + α) vt = α log w e t is correlated with (1 + α) vt Clearly log w This can be a really serious bias I am really worried whenever my regressor is a function of the dependent 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 lower wages To deal with this one needs an instrument for wt Examples: Age, Local labor market variation, tax changes
An Estimable Dynamic Specification Lets 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 X t=0
hitη citγ β ait − bit γ η t
with the lifetime budget constraint T T X X ct wit hit ≤ Rt Rt t=0
t=0
Lets look at the first order condition for hit : β t bit hitη−1 = λ∗i
wit Rt
where λ∗i is the lagrange muliplier on the full budget constraint so
log(hit ) =
1 η−1
log (λ∗i ) + log (wit ) − log Rt β t − log (bit )
Now notice that since λit = ait citγ−1 = 1
log(hit ) =
η−1
=
η−1
1
λ∗i β t Rt
log (λ∗i ) + log (wit ) − log Rt β t − log (bit )
[log (λit ) + log (wit ) − log (bit )]
is the Frisch labor supply function so
1 η−1
is the Frisch elasticity
Assume further that 0
bit = eXit δ+θi +uit Then we can write log (hit ) = µi + α log(wit ) + ρt + Xit0 δ ∗ + uit∗ where µi
=
log λ∗i + θi η−1
1
α =
η−1
δ∗ =
η−1
uit∗ =
uit η−1
ρt
δ
= − log Rt β 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 ) + ∆Xit0 δ ∗ + ∆ρt + ∆uit∗
Assumption on error term is different and perhaps more reasonable. Wages may be correlated with θi need instead that ∆ log(wit ) is uncorrelated with ∆uit∗ Still need to instrument to deal with measurement error
Outline
Participation
Continuous Hours
Empirical Implementation
Estimates
PSID Data
Started in 1968 with about 4800 households Longitudinal followed annually Follows individuals Follows kids after they have left the house Lots of labor market data Food Consumption A number of other things as well
Macurdy Estimation
Macurdy estimates model using Panel Study of Income Dynamics He uses panel data of first differences Instruments with education, age, year dummies, family background variables
Estimating with Uncertainty Now we will add uncertainty. I will follow Altonji, JPE, 1986. He does two different things. The first extends the Macurdy model 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 is orthogonal to information at time t.
Plugging this into the labor supply equation above
log(hit+1 ) − log(hit ) 1 0 δ + θi + uit+1 = η−1 log (λit+1 ) + log (wit+1 ) + ρt+1 − Xit+1 −
1 log (λit ) + log (wit ) + ρt − Xit0 δ + θi + uit
η−1
1 = η−1 − log(β (1 + rt )) + vit+1 + ∆ log (wit ) + ∆ρt − ∆Xit0 δ + ∆uit+1 As long as we have instruments that are orthogonal to vit+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 error only) 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 ) = ait citγ−1 0
a
a
Assume now that that ait = eXit δa +θi +ui . Now plug this into the labor supply equation: 1
log(hit ) =
η−1
=
η−1
[log (λit ) + log (wit ) − log(bit )]
1
(γ − 1) log (cit ) + log (wit ) + Xit0 δ ∗ + θi∗ + uit∗
where log(ait ) − log(bit ) = Xit0 δ ∗ + θi∗ + uit∗ .
Thus we can just estimate this by IV if we can get good instruments for log(wit ) and log(cit ) Altonji uses an individual-specific permanent compontent of the wage (using alternative measure) as main instrument for log(cit ) as well as alternative wage measure