Evaluating Alternative Representations of the Choice Set In

Models of Labour Supply

Rolf Aaberge, Ugo Colombino and Tom Wennemo

Workshop on Discrete Choice Labour Supply Models 8-9 December 2005, Galway

Purpose

To evaluate alternative methods of representing the choice set in models of labour supply

Background

• Accounting for complicated (non linear, non convex etc.) constraints in modelling behaviour

• Heckman 1974• Hausman 1979, Moffit 1986 etc.• The ”Hausman Approach”• Problems with the ”Hausman Approach”

The ”Hausman Approach”

• Choice is characterized by K.T. conditions for constrained utility maximization

• Complicated and computationally cumbersome with non-convex budget constraints and more than one decision maker or more than two goods

• The problems above contributed to the popularity of a different approach

The “Random Utility” Approach

• McFadden 1974• Dagsvik 1994• Van Soest 1995• Aaberge, Dagsvik and Strøm 1995

The “Random Utility” Approach

• Any opportunity set – however complicated - can be represented by a set of points hi, Ci

• Specify a random utility v(hi, Ci)+i with i i.i.d. Type I Extreme Value

• Then the p.d.f. of choosing (hi, Ci) is

exp(vi)/kexp(vk)

Alternative ways of implementing the “Random Utility” Approach

• Fixed vs Sampled Opportunities

• Number of Opportunities

• Uniform vs Non-Uniform Availability of Opportunities

Fixed opportunities vs Sampled opportunities

• Potentially too many opportunities (e.g. 0, 1,..., 8760 hours of work for each person in the household)

• So, choose a subset of opportunities, but how?

• And, how many?• Most common practice: choose a subset

of fixed opportunities (e.g. 0, 100, 200 etc.)

• More appropriate procedure: sample a subset of opportunities according to some probabilistic rule (McFadden 1978)

Fixed opportunities vs Sampled opportunities (cont.)

• Sampling appears as a more appropriate procedure, however it’s more time consuming

• In the labour supply literature, the fixed opportunities procedure is by far the most commonly used one

• Very little analysis and no systematic testing has been done of the empirical implications of the different procedures (Ben-Akiva and Lerman 1985)

Availability of alternatives:uniform vs non-uniform density

• Some opportunities are more easily accessible than others

• The accessibility of opportunities might vary also across individuals

• How do we account for that?

Availability of alternatives:uniform vs non-uniform density

(cont.)

• Most contributions to the labour supply literature do not account at all for heterogenous accessibility

• Some studies simply assume that only a few opportunities are available, in the same way for everybody (e.g. non participation, part-time, full-time)

• Some other studies assume a non uniform density of opportunities, in general differing across individuals

Accounting for non-uniform density of opportunities

Let p(h,C) be the density in the choice set of opportunities of type (h,C).

The p.d.f. of a choice (h,C) can be written as follows:

,

exp( ( , )) ( , )exp( ( , )) ( , )

x y

v h C p h Cv x y p x y

The simulation exercise

• Generate a sample based on a “true” model

• Estimate various models with different choice-set representations

• Evaluate the estimation and simulation performance of the models

The “true” model

• As the “true” model we take a model estimated on a sample of married/cohabitating females from the Norwegian 1995 Survey of Level of Living.

• It belongs to a class of various models we estimated for Norway, Sweden and Italy.

• Using the parameters estimated for married females, we generate a sample of 10000 observations.

The estimated models

16 models defined according to:• Generation of opportunities: fixed vs

sampled• Number of opportunities: 6 vs 24• Availability of part-time and full-time

jobs: uniform vs non-uniform• Availability of jobs: uniform vs non-

uniform

The estimated models (cont.)

• The “true” model assumes that choice sets vary across individuals and that availability of opportunities is not uniform

• Therefore it is expected that the models accounting for non uniform availability of opportunities perform better

• The issue we want to address is “how much better” and under what circumstances.

Evaluating the models

210

1

( )

kj jk

j j

y yz

y

210

1

( )

kj jk

j j

y yz

y

210

1

( )

kj jk

j j

y yz

y

210

1

( )= relative squared error of model

= prediction of the "true" model for the j-th income decile

= prediction of model for the j-th income decile

In this exercise

kj j

kj j

j

kj

y yz k

y

where

y

y k

we predict income.

Evaluating the models (cont.)

210

1

( )

kj jk

j j

y yz

y

210

1

( )

kj jk

j j

y yz

y

210

1

( )

kj jk

j j

y yz

y

0 1 1 2 2 3 3 4 4 34 3 4

1

2

3

4

( * )

= 1*(sampled alternatives)

= 1*(24 alternatives)

= 1*(non uniform jobs-availability )

= 1*(nonuniform part-time and full-time jobs-availabil

k k k k k k k

k

k

k

k

z x x x x x x

where

x

x

x

x

ty

Contributions to the prediction error:prediction of net income under the

current tax regime

Basic model

Sampled opportunities

24 opportunities

Non-uniformJobs availability

Non-uniformPartime and Fulltime availability

Interaction

Estimate .021 -.002 .007 -.009 -.002 -.003

Standard deviation

.008 .004 .004 .006 .006 .008

Even the basic model performs very well. More complicated representations of the choice set do not significantly contribute to a better fit

Contributions to the prediction error:prediction of net income after a flat tax

reform Basic model

Sampled opportunities

24 opportunities

Non uniformJobs -availability

Non uniformPartime and Fulltime -availability

Interaction

Estimate .128 -.019 -.008 -.029 -.039 .036

Standard deviation

.006 .005 .005 .007 .007 .010

Sampled opportunities, a larger choice set and accounting for heterogeneous availability of opportunities significantly contribute to reduce the prediction error

Evaluating the models: Relative

computation time (estimation)

210

1

( )

kj jk

j j

y yz

y

210

1

( )

kj jk

j j

y yz

y

6 opportunities 24 opportunities

Fixed opportunities

1 4.62

Sampled opportunities

6.70 8.46

Conclusions

• Random Utility models fit very well observed values, even the basic versions with fixed opportunities, and a very small choice set

• When it comes to predicting policy effects, models using sampled opportunities and accounting for heterogeneous availability of opportunities perform significantly better (at the price of larger computational costs)

Heckman J. (1974): Effects of Child-Care Programs on Women’s Work Effort, Journal of Political Economy 82, 2-II, 136-163.

Hausman, J.A. (1979): "The Econometrics of Labour Supply on Convex Budget Sets", Economic Letters, 3.

Moffitt, Robert (1986): "The Econometrics of Piecewise-Linear Budget Constraints: A Survey

and Exposition of the Maximum Likelihood Method," Journal of Business & Economic

Statistics, American Statistical Association, vol. 4(3), pages 317-28.

McFadden, D. (1974): "Conditional Logit Analysis of Qualitative Choice Behavior", in P. Zarembka (ed.), Frontiers in Econometrics, Academic Press, New York.

Dagsvik, J.K. (1994): "Discrete and Continuous Choice, Max-stable Processes and Independence from Irrelevant Attributes", Econometrica, 62, 1179-1205.

van Soest, A. (1995): "Structural Models of Family Labor Supply: A Discrete Choice Approach”, Journal of Human Resources, 30, 63-88.

Aaberge, R.., J.K. Dagsvik and S. Strøm (1995): "Labor Supply Responses and Welfare Effects of Tax Reforms", Scandinavian Journal of Economics, 97, 4, 635-659.

McFadden, D. (1978): "Modelling the Choice of Residential Location" in A. Karlquist, L. Lundquist, F. Snickard and J.J. Weilbull (eds.): Spatial Interaction Theory and Planning Models, Amsterdam, North-Holland.

Ben-Akiva, M., and Lerman, S.R. (1985): Discrete choice analysis, (MIT Press, Cambridge).

References

• Aaberge, R.., J.K. Dagsvik and S. Strøm (1995): "Labor Supply Responses and Welfare Effects of Tax Reforms", Scandinavian Journal of Economics, 97, 4, 635-659.

• Aaberge, R., U. Colombino and S. Strøm (1999): “Labor Supply in Italy: An Empirical Analysis of Joint Household Decisions, with Taxes and Quantity Constraints”, Journal of Applied Econometrics, 14, 403-422.

• Aaberge, R., U. Colombino and S. Strøm (2000): “Labour supply responses and welfare effects from replacing current tax rules by a flat tax: empirical evidence from Italy, Norway and Sweden”, Journal of Population Economics, 13, 595-621.

• Aaberge, R., U. Colombino and S. Strøm (2004): "Do More Equal Slices Shrink the Cake? An Empirical Investigation of Tax-Transfer Reform Proposals in Italy“, Journal of Population Economics, 17