7  Alternative specifications

This section presents a number of alternative specifications of the basic model discussed above. The discussion in this section is meant as an overview only and provides much less detail than the discussion in the previous sections. First, the form of utility functions is examined in subsection 1. Allowance for participation in welfare programmes, often described in terms of the ‘take-up’ of benefits, is examined in subsection 2. Alternative ways in which allowance may be made for individuals’ personal characteristics are discussed in subsection 3. The effect of characteristics of a particular discrete hours point is described in subsection 4. Finally, alternatives to the use of the extreme value distribution are briefly discussed in subsection 5.

7.1  Utility functions

It has been mentioned that the discrete hours approach offers considerable flexibility in the form of utility function that can be used. The linear form used in the numerical example of estimation is obviously highly restrictive. The assumption of constant marginal utilities is implausible and in empirical applications, utility functions usually allow for diminishing marginal utility. A popular extension in applied work is the quadratic utility function:[23]

(30)    

where the marginal utility of income is:

(31)    

An alternative is the translog specification in which the arguments of utility are income and leisure (), rather than income and hours worked:[24]

(32)    

where the marginal utility of income is:

(33)    

Both specifications allow for diminishing returns through the quadratic terms. Thus, if is negative the marginal utility of income decreases with the amount of income. Furthermore the cross-product term allows for complementarity (if is negative or is positive) or substitutability (if is positive or is negative) of income and leisure. For example, the value of income may increase if more leisure time is available, that is extra income may be appreciated less if there is no time for consumption.

Neither the translog nor the quadratic utility function is automatically quasi-concave across the full range of possible parameter values. This is not a problem as long as the optimal parameter values result in a utility function that is quasi-concave in the observed labour supply points. This contrasts with continuous hours labour supply modelling, where the necessary restriction of the parameter space may bias substitution effects upwards and income effects downwards and which is cumbersome in maximum likelihood estimation.[25] In discrete hours labour supply modelling, it is sufficient to check for quasi-concavity after estimation, which is a straightforward check of two necessary conditions.[26]

The quadratic and translog utility function can both be easily extended to allow for households consisting of couples, where both partners simultaneously determine labour supply. This is achieved by assuming that the couple maximises one utility function, which seems a reasonable assumption for households where the members pool their incomes. However, a common criticism of this type of model is that the assumption of one common utility function for the household as a whole is not realistic. Unfortunately, alternatives using bargaining models and other types of non-unitary collective models require detailed data and their own set of assumptions which are needed, for example, to break down consumption into shared and private goods or to construct a sharing rule for income.[27] Such models need to be simplified in other areas. As a result, researchers who focus on tax and benefit policy issues and are interested in incorporating the full detail of tax and benefit systems have mostly chosen unitary utility functions.

The quadratic utility function for a couple can be written as:

(34)    

where the index denotes hours and parameters of the male and the index denotes hours and parameters of the female, and represents joint income. The parameter indicates whether the male’s and female’s labour supply are complements or substitutes.

7.2  Welfare participation

The utility function can be extended through addition of a term for welfare participation, or benefit take-up.[28] The choice between discrete labour supply points is then extended to a choice between discrete labour supply points with and without welfare participation, whenever relevant. In these models, it is expected that disutility is attached to participation in welfare. This disutility could be caused for example by the costs of applying for welfare. These could be pecuniary costs or non-pecuniary costs, such as the time needed to travel to a social security office, or by a psychological effect of being on welfare, where people on welfare feel stigmatised. The latter explanation is more likely to be important when participation in welfare is clearly noticeable to the outside world, such as through payment in shops with Food Stamps in the U.S.

A simple and popular way of adding welfare participation to the utility function is through the addition of a dummy variable for participation.[29] For example, if the person participates and if the individual does not take-up the benefit, even if entitled to it. The coefficient on this variable indicates the disutility associated with participation in welfare; that is, a larger negative value indicates greater disutility. For the quadratic utility function, the specification would therefore be:

(35)    

The participation parameter can be made dependent on individual characteristics in the same way as for the preference for work or income. This is described in the following subsection.

An alternative approach is to estimate an unordered model of moving from one choice to another, where the amount of labour supply and participation in the welfare programme jointly determine choice. In this specification there is no explicit welfare participation parameter, but the gain in utility from a choice with welfare participation compared with a choice without welfare participation can be determined conditional on the income gain associated with the move between these choices and other individual characteristics.[30]