6  A numerical example of estimation

This section illustrates the application of the maximum likelihood method using a simple numerical example involving a linear form of utility function. Although the example has few individuals and a simple utility specification, the general approach is no different in a more realistic example. Utility is assumed to be independent of an individual’s characteristics except for hours worked, wage and other income; appropriate allowance for dependence on characteristics is discussed in section 7. Hence, all individuals have the same utility function with the same parameters, and this takes the form:

(27)    

This does not mean that all individuals are expected to have the same optimal level of hours. Firstly, people with different wage levels have different levels of income at the hours points and so optimal hours are located at different points and, secondly, the error term introduces random differences in utility caused by unobserved factors. In this simple linear form, the marginal utility of net income is constant and equal to and the marginal utility of hours worked is constant and equal to , so the latter coefficient is expected to be negative. For each individual, the chosen hours point is observed and the value of for each discrete labour supply point can be calculated, given information on the gross wage of each individual and knowledge of the tax and transfer system.

Suppose also that there are only three individuals, whose details are shown in Table 3. There are just three hours levels available for work, 0, 20 and 40 hours, corresponding to not working at all, working part time and working full time respectively. The observed gross wage rates, given the observed hours of work for each individual, give the gross income shown in the final column. In this example, the individual with the highest wage rate works longer hours.[20]

Assuming for simplicity that there are no income taxes or benefit payments, the net income is simply equal to gross income. Substitution of these values, along with (27), into the first-order conditions in (22) give:[21]

(28)    

         

(29)    

    

The result is two nonlinear equations in the two unknowns and . Using an iterative solution procedure, as described in section 2, the maximum likelihood estimates were found to be and .[22]

Consider the wage elasticity of labour supply for person 2, defined as in subsection 4 in terms of changes in expected hours. At the observed wage level, the hours and corresponding net incomes (equal to gross incomes since by assumption there are no taxes) in Table 3 are used, with the parameter estimates, to obtain the utilities corresponding to each hours point, by appropriate substitution in . From these, the probabilities of being at each of the labour supply points are given by and expected labour supply is calculated using . In this example . After increasing the wage by 1 per cent, new net incomes and hence new utilities for each discrete hours point are obtained. Using the resulting new probabilities, expected hours are found to be . This implies a very high elasticity of 43.