4.2  Specification of Utility Functions

The form of the utility function used here is a quadratic specification (following Keane and Moffitt, 1998). This specification is simple but flexible in that it allows for the leisure of each person and income (or consumption) to be substitutes or complements. This means the model can represent complex interactions. Furthermore, the quadratic utility function can be expressed as a function of labour supply rather than leisure without the need to choose a value for total endowment of time (T). T is not important in this specification, as it is a constant, which can be incorporated in the parameters to be estimated.

The above advantages make the quadratic utility function a good choice, even though this utility function is not automatically quasi-concave. However, the latter is not a problem in a discrete labour supply model, because two simple conditions, outlined in Van Soest (1995), can be used to establish whether U is quasi-concave at any data point. In the discrete approach taken here, these two conditions can be tested at all data points after estimation of the parameters. In a model with continuous hours of labour supply, these conditions have to be imposed a priori to guarantee coherency, as has been mentioned earlier.

Many earlier models had the problem of overpredicting part-time hours and underpredicting non-participation. An intuitively appealing approach is to include a fixed cost of working parameter in the income variable x to indicate the cost of working versus non-participation (Callan and Van Soest, 1996). As a result of the inclusion in x, this cost of working parameter is measured in dollars per week. The utility derived from leisure and income can be written as:

(11)    

where α.., β., and φ are preference parameters that have to be estimated; and are the fixed cost of working parameters to be estimated for husband and wife, they are zero when the relevant person is not working.

This quadratic utility function has a simple form and heterogeneity of preferences is easy to include. To account for differences in preferences between households, the parameters β, α, and γ can be made dependent on household and individual characteristics. In the first instance, it is assumed that only , , , γ₁ and γ₂ depend on personal and household characteristics (described in Section 3.2). Simple linear specifications are chosen to include the observed heterogeneity in , , , γ₁ and γ₂.

As an alternative to the individual characteristics in the fixed-cost-of-working parameters, we also estimated separate cost parameters for the different labour supply levels. This is equivalent to making the parameter dependent on the number of hours of work. Instead of specifying a single parameter for all hours of work, separate parameters are introduced for categories of hours. Here parameters are specified for 1 to 10 hours of work, 11 to 20 hours of work, 21 to 30 hours and over 30 hours of work. In addition to allowing for the fixed cost of working, this specification also allows for different costs associated with different hours levels, such as for example the cost of part-time work which may be high if few jobs in this category are available, because a larger search effort is then required.

Adding unobserved heterogeneity to the linear equations in personal characteristics for the preference and fixed-cost-of-working parameters, in the form of a normally distributed error term with zero mean and unknown variance, is quite simple, although exact maximisation would involve a likelihood function with multiple integrals. However, Van Soest (1995) outlines an easier method, replacing the expectation of the log likelihood by a simulated mean and optimising an approximate likelihood function instead of the exact likelihood function. It is straightforward to obtain a simulated mean by: drawing error terms from the distribution based on the current parameter estimates for the covariance matrix for each observation in the sample; calculating the log likelihood function based on these draws; and averaging the log likelihood function over a certain number of draws. Van Soest found that 10 draws seemed sufficient, so the estimation of unobserved heterogeneity in this paper is carried out with the same number of draws.

4.3  Expected Labour Supply

Once the model has been estimated, the results can be used to calculate the expected labour supply from the probabilistic outcomes for people with certain known characteristics and under known social security and taxation rules.

To obtain the expected labour supply of the husband, we first calculate the utility U( (h₁,h₂), h₁, h₂) for each possible combination of labour supply for both adults in the household. This is achieved by substituting the estimated parameter values into equation (10) along with the net income for the relevant combination. Once the utility values are known, a simple logit transformation provides the probability of each possible combination occurring according to the estimated model:

(12)    

These probabilities can be used to calculate the expected value of preferred labour supply for the husband by simply aggregating the probabilities over all possible values of h₂ for each value of h₁. In this manner, the marginal probability of h₁ is obtained, which can be used to calculate the expected value of h₁ in the usual way. The formula for this procedure looks as follows:

(13)    

The expected value for the wife’s labour supply can be obtained in a similar way.