2  The basic model

This section presents the basic model of utility maximisation and discusses the determination of the probability distribution of hours worked. Subsection 1 discusses the discrete choice framework, involving the introduction of a random term reflecting the difference between actual utility and measured utility for an individual. In contrast with a deterministic approach, this gives rise to a probability distribution of hours worked for each individual, as discussed in subsection 2 and more formally in subsection 3. The measurement of labour supply elasticities in this framework is examined in subsection 4.

2.1  Utility maximisation

Consider an individual with a set of measured characteristics, . The individual (who faces a fixed gross wage rate) maximises utility by selecting the number of hours worked, subject to the constraint that only a discrete number of hours levels, are available for work. The level of utility is determined by the amount of leisure and net income. Utility is increasing in both arguments and is bounded by the time and budget constraints. That is, the amount of leisure per week cannot be more than the total amount of time available per week minus the hours of work.[6] Total weekly income is restricted by the available amount of nonlabour and labour income. The latter is the individual’s wage rate multiplied by hours worked (the total available time minus the time spent on leisure). Instead of leisure, hours of work are often used as the argument in the utility function because labour supply is typically the key variable of interest in economics. The individual balances leisure and net income to obtain the highest utility possible, more leisure means less income and vice versa.

The utility associated with each hours level is denoted and is a function of ‘measured’ utility plus an ‘error term’, so that:[7]

(1)

The term arises from factors such as measurement errors concerning the variables in optimisation errors of the individual or the existence of unobserved preference characteristics. Any observation on is of course associated with a set of possible ‘draws’ of the random variables from their respective distributions. Within this framework, there exists a probability distribution over available hours levels that is influenced by the properties of the [8] Without these error terms, the model would be deterministic and knowledge of the form of and the vector would be sufficient to determined the precise utility-maximising choice of hours level.

The issue considered here is how to generate the probability distribution for labour supply, for given assumptions about the distributions

2.2  Probability distributions

The framework, summarised by equation (1), is one in which there is a distribution of utility for each discrete hours level, depending on the distributions of the . Suppose for convenience that there are only three hours points. The three distributions of for , are shown in Figure 1, where in each case increasing utility involves moving upwards along each axis. The choice of any particular hours level is associated with ‘draws’ from these three distributions, where the hours level producing the highest is chosen.

Consider the probability that hours level is chosen, given that the value has been selected from the distribution of . This can only be chosen if it is higher than the values of and selected from their respective distributions. From Figure 1, the probability that is given by the area B. Similarly the probability that is the area C. The joint probability that is chosen, given the selection of , is the probability that and . If the ‘draws’ from the distributions are independent, this probability is the product, BC, of the two areas.[9]

This relates only to one draw, of , from the distribution of . It is necessary to consider the overall probability of being chosen. This is obtained by adding together all the conditional probabilities, for all possible values of .[10] Even for the higher values of , Figure 1 suggests that the conditional probabilities of being selected would in most cases be low. Overall, the probability of producing maximum utility is small.

Figure 1 – Three Hours Levels and Utility Distributions

2.3  A more formal statement

The procedure discussed in the previous subsection is set out more formally here. Consider the hours level, Utility maximisation implies that this hours level is chosen if:

(2)     

Substituting for , using (1), and rearranging, this condition is equivalent to the requirement that:

(3)    

Hence, for any given value of the probability of exceeding all other values is equal to the joint probability that and and so on for all If the various distributions are independent, this joint probability is the product of the separate probabilities, . Therefore, for any given value of the probability that hours level produces maximum utility is equal to:

(4)    

This is the conditional probability, for a given value of . The overall probability is found by aggregating terms like (4) over all possible values of The analysis of this problem is considerably simplified by assuming that the form of the distribution of for each is the same. An example is given in the next section, and this is followed by a more detailed and analytical treatment of the error specification. First, it is necessary to consider the concept of the wage elasticity of labour supply in the discrete context.