7.3  Personal characteristics

Consider again the simple linear utility function:

(36)    

It is straightforward to extend this to make the preference parameters dependent on personal and household characteristics. Characteristics such as education, number and age of children or an individual’s own age are likely to influence the preference for work and income. Including these characteristics in the preference for work parameter, the utility function could be presented as follows:

(37)    

where, say, if the age of the youngest child is 0 to 4, and otherwise. In this case, two extra parameters for the preference for work are included, so the likelihood now depends on four unknown parameters which need to be estimated.

This specification is more flexible than in the numerical example of section 6, where just one preference parameter for work was estimated. For example, individuals with young children are allowed to have different preferences compared with individuals without young children. This approach can be used to estimate the effect of an individual’s characteristics on preferences and may help to explain differences in behaviour between individuals with similar wages but different personal characteristics.

This addition means that the effects on wage elasticities of labour supply (as defined above in terms of expected hours worked) of characteristics like age or household composition can easily be examined. Expected labour supply can be calculated for two individuals who are exactly the same except for the characteristic of interest. There is thus scope for a wide range of elasticities.

The approach reflected in equation (37) does not incorporate unobserved heterogeneity of individuals because allowance is made only for the measured characteristics. This can be overcome by adding unobserved heterogeneity to the preference parameters. Hence the coefficient on is written as:

(38)    

This introduces an additional error term, , which is typically assumed to be normally distributed. This addition complicates the method of estimation somewhat, such that the method of simulated maximum likelihood is required. However, estimation of such models, including correlated error terms in the different preference terms, remains fairly straightforward using this method.[31]

Some authors have chosen an alternative to the extreme value distribution for the random term to be added to measured utilities. This usually complicates estimation and a sign of the larger complexity is that in such cases it has been possible to distinguish only three discrete hours points. This contrasts with around ten hours points for each individual, when using the extreme value distribution.[32] However, the advantage of the alternative approaches is that greater flexibility is allowed in modelling the relationship between the labour supply of two members of a couple or between labour supply and welfare participation.

7.4  Characteristics of hours points

It is often observed that the probability of obtaining a job offer depends on the desired number of hours of work.[33] For example, finding a job of 5 hours per week may be more difficult than finding a 40-hour job. As a result some discrete hours points are not well-represented by the standard labour supply model, which does not allow for demand side restrictions. For example, it is often found that labour supply models overpredict part-time hours of work. Several methods have been used to overcome this lack of fit to the observed labour supply. Some examples of alternative approaches are discussed briefly here.

First, an ad hoc approach of including a penalty parameter for particular hours of work in the utility function has been used to reduce the utility at certain hours points, so that the probability at these hours points is reduced.[34] A second approach involves the inclusion of the probability of a job offer at the different discrete hours points in the model, which can be applied when desired hours of work are known.[35] Third, a parameter measuring the fixed cost of working can be subtracted from net income in a quadratic utility function.[36] This approach is similar to the first approach but is expressed in dollars rather than units of utility. Thus it is intuitively more appealing, although the costs represented by this parameter are both pecuniary and non-pecuniary costs.

In the fourth approach, the number of job offers in an interval associated with the discrete point is directly used to weight the probabilities derived by using the extreme value distribution.[37] A final example is the approach where an adaptation of the multinomial logit model allows for captivity at particular discrete hours points.[38] This increases the probability of observing an individual at particular hours points. It allows some hours points to have a high probability which does not need to depend on an individual’s characteristics; this may, for example, be expected at the standard full-time 40-hours point.

7.5  Alternative error distributions

The use of the extreme value distribution contains an assumption that has, in previous sections, remained implicit. This form assumes that there is no correlation between the error terms of the different hours alternatives. This is usually referred to as the ‘independence of irrelevant alternatives’ property, and means that taking out one of the choices would not affect the odds ratios of the other choices. For example, suppose that individuals can initially choose between 0, 5, 10, 15, ..., 45 and 50 hours of work. Taking out the 10 hours choice, it seems unlikely that the relative probabilities of the other choices would not change. If 10 hours were no longer an option, it seems likely that individuals previously preferring this discrete point would move to the neighbouring labour supply points, thus changing the odds between the choices. An obvious, but unpopular approach is to extend the extreme value distribution with fixed mean and variance to a version where these two parameters are estimated, allowing for correlation between the choices.

A model related to the multinomial logit described in the previous sections is the nested multinomial logit. Hagstrom (1996) showed that this specification allows for correlation between some of the decisions in the model. In his application, correlation between the wife’s labour supply choice and welfare participation within the husband’s labour supply choice is allowed. This relaxes the independence assumption between all alternatives in the standard multinomial logit model, although some structure is still imposed on the covariance matrix. In addition, a distinction is made between choice-specific variables and individual-specific variables, imposing more structure on the way characteristics influence the different choices by individuals.

An alternative to the extreme value distribution is a normal distribution, which would lead to a probit-type model instead of the logit-type model. However, multivariate probit models are difficult to estimate, even for as few as three categories. An additional problem is that it is impossible analytically to determine the limits of integration which indicate which discrete hours point is preferred. With the recent development of simulation techniques combined with more powerful computers, this type of model has become more feasible and some researchers have explored this option. Fraker and Moffitt (1988) estimate labour supply and participation in two welfare programs for female heads of household in a reduced form model. Three levels of labour supply are distinguished. The choice for these levels of income depends on the preference parameter for work, which depends on an individual’s characteristics, and an unobserved factor which is assumed to be normally distributed. No error terms are added directly to the utility function. The model can be estimated because the ranges for the preference parameter where each hours point is optimal can be written down.[39] This only works when the budget constraint is not too nonconvex, which might otherwise make it impossible for part-time work to be optimal in this specification of the model. The problem of finding the limits of integration, which determine which discrete labour supply point and whether welfare participation is chosen, necessitated the reduced form approach by Fraker and Moffitt. A similar specification using a structural approach can be found in Keane and Moffitt (1998), who overcome the problems with the limits of integration by using advanced simulation techniques. With the simulation approach there is no need to determine analytically the limits of integration. However, estimation is cumbersome and time consuming.

Bingley et al. (1995) use an approach where the difference between utility levels is modelled rather that the utility function itself. Under the assumption of normally distributed error terms on the utility function, a multinomial probit model can be derived. They distinguish three discrete points and model the probability of preferring non-participation over part-time employment and the probability of preferring non-participation over full-time employment. That is the distribution of the differences in utility between nonparticipation and part-time employment and between non-participation and full-time employment are modelled. They allow for correlation across the choices. The variance-covariance matrix is normalized by assuming that the variance of the difference between the part-time and full-time error term has a variance of one. When more than three choices are specified, simulation techniques would be needed for the estimation.

Finally, a flexible non-parametric approach was taken by Hoynes (1996) who added unobserved heterogeneity to the preference parameters for labour supply of husband and wife and for welfare participation. This approach uses a discrete factor representation, where sets of different pairs of unobserved heterogeneity for the husband’s and wife’s preferences for work parameter and for the preference for welfare participation () are observed with a probability where and . The flexibility of this approach is appealing, but it adds a large number of additional parameters to be estimated ( in addition to the number of parameters in a multinomial specification). For large , any correlation between the different error terms can be represented by this specification. In addition to this discrete probability distribution which is meant to capture the correlation between the different preference terms, normally distributed independent error terms are added to the preference for welfare participation and the observed hours of work.[40] Although the intuition behind this model is simple, estimation of the model is difficult, particularly for large .