1  Introduction

This paper provides an introduction to the basic analytics of discrete hours labour supply modelling.[1] Discrete hours models are popular in tax policy microsimulation, because it is relatively easy (compared to the continuous models) to incorporate taxation and social security details. To get to the stage where a policy change, such as a change in income taxation rates, can be simulated, several steps are needed. First, a model needs to be specified explaining labour supply behaviour. Second, taxation and social security parameters and individual information on incomes, wages and household composition are needed to calculate net incomes at all possible labour supply levels. Third, the model is estimated using information on individual labour supply, net income at the different labour supply levels and other relevant characteristics. Fourth, once the parameters of the model are estimated, they can be used to predict the effect of policy changes through simulation. In this paper, special attention is given to the three steps of model specification, estimation and microsimulation.

The assumption behind discrete hours labour supply modelling is that utility-maximising individuals choose from a relatively small number of hours levels, rather than being able to vary hours worked continuously. The discrete approach is perhaps more realistic, in that typically only a finite number of part-time or full-time working options are available.[2] It also substantially simplifies the nature of the budget set faced by each individual, who is assumed to face a fixed gross hourly wage. It is assumed that the same set of hours is available to each individual. In the continuous hours context the analysis of choices under piecewise-linear budget lines must deal with the complexities arising from budget sets displaying convex and non-convex ranges, and multiple local equilibria.[3] In practice, the evaluation of the complete range of each individual’s unique budget set is cumbersome, given the complexity of most tax and transfer systems.[4] With discrete hours models it is simply a question of evaluating utility at a small number of points, none of which represents a standard tangency solution.

The advantages of discrete hours modelling are perhaps even stronger in the context of the empirical estimation of individuals’ preference functions. With continuous hours modelling several approaches have been adopted.[5] Often a reasonably flexible labour supply function (relating hours worked to net wage rates, non-wage incomes and a range of individual characteristics) is estimated, and then the utility function is found by appropriate integration methods. Alternatively a supply function is derived from either a direct or (more commonly given the greater flexibility allowed) an indirect utility function. However, considerable problems arise because of, for example, the fact that net wages and hours are jointly determined, and problems exist concerning the determination of virtual non-wage incomes for each linear segment. Indeed, empirical continuous hours models have found it extremely difficult to capture the complexities arising from supply behaviour under piece-wise linear constraints.

Section 2 describes the discrete choice modelling framework. In practice, the determinants of any individual’s behaviour can never be known with certainty. A feature of the discrete hours approach is that the stochastics are introduced at the initial discrete-choice modelling stage in the utility function rather than in the derived labour supply model; measured utility differs from true utility as a result of measurement, optimisation and other errors. This generates a crucially important probability distribution over the set of hours available for work. Section 3 provides a simple numerical example of the way in which such a probability distribution is generated, where the error terms follow a hypothetical discrete distribution. A more detailed and formal examination of the error specification, and its implications for the probability distribution of an individual’s hours worked, is given in section 4. Estimation of the parameters of specified preference functions, using the method of maximum likelihood, is considered in section 5. A numerical example of estimation is given in section 6. Alternative specifications of the model are discussed briefly in section 7. The use of discrete hours labour supply models in behavioural microsimulation is examined in section 8, where a numerical example of a tax reform is presented. Brief conclusions are in section 9.