RDP 1999-02: Reservation Wages and the Duration of Unemployment 4. A Structural Model of Unemployment Duration

Having described the data source and the key variables, we now turn to developing an empirical model from the underlying job-search theory. In Section 4.1 we start by looking at the individual job seeker who receives job offers which arrive randomly at a predetermined rate, with wage offers being drawn randomly from a predetermined wage distribution (the wage-offer distribution). From the basic model we derive the distribution of incomplete unemployment durations facing an individual when they first become unemployed.

In Section 4.2, we discuss the assumptions which are required to incorporate individual characteristics which are likely to affect the duration of unemployment. We also discuss the individual characteristics which we include in the estimation. In Section 4.3, we combine the results of Sections 4.1 and 4.2 to derive an expression for the expected log of incomplete duration for an individual. By assuming that the distribution of individuals who flow into unemployment is the same at each point in time, we can then justify estimating this expression across individuals as a cross-section regression.

4.1 Deriving the Distribution of Incomplete Duration for an Individual

Job seekers in the standard job-search framework set a reservation wage which is the minimum wage that they would be willing to accept. The reservation wage is determined by equating the expected benefits of accepting an offer with the expected costs of further search and the opportunity cost of forgoing potentially better offers. The reservation wage will be a function of things such as the costs of search, the rate at which job offers arrive and the distribution of wage offers. If these determinants do not vary with the duration of unemployment, the reservation wage and the probability of leaving unemployment will remain constant, and the model is said to be stationary.

The starting point for our analysis is to consider the hazard rate. This is the instantaneous probability of exiting unemployment for an individual, equal to the product of the probability that a job offer arrives and the probability that the individual will accept the job offer, i.e. that the wage offer lies above th e individual's reservation wage. More formally, the hazard rate for individual i can be written as:^[11]

If the hazard rate is constant, the distribution of completed unemployment spells, T, facing a newly unemployed individual, i, will be exponential:

However, the data we use from the SEUP are from currently unemployed individuals. Therefore, we are interested in the distribution of incomplete spells of unemployment rather than of completed spells. The probability of observing an individual with an incomplete duration of unemployment of length t, is the probability that the individual did not leave unemployment earlier. This probability is given by 1−G_i(t) for individual i, where G_i(t) is the distribution function corresponding to the density function g_i(t). The expression 1−G_i(t) is also known as the survivor function. The distribution of incomplete spells for an individual will, therefore, be the normalised survivor function:^[12]

The final expression in Equation 3 can be obtained by integrating Equation 2 over T to obtain G(t) and substituting this into Equation 3.

4.2 Incorporating Individual Characteristics

In order to transform the theory into an estimable model, it is necessary to add more structure. The first step is to make an assumption about the wage-offer distribution. One of the most common and tractable assumptions is that wage offers are drawn from a Pareto distribution. This means that the probability of a given wage offer exceeding the reservation wage can be expressed as:

The second step is to assume that the probability of receiving and accepting a job not accounted for by the reservation wage, is an exponential function of the individual's characteristics, X_i:

The explanatory variables, represented by X_i in Equation 5, are included to capture those factors which affect the probability of receiving and accepting job offers, given the reservation wage. The job-offer arrival rate will be affected by the attractiveness of the individual to the employer. Variables which will capture this effect include educational attainment, previous occupation and reasons for leaving your last job. These variables are also likely to capture the important elements of the wage-offer distribution facing an individual. Personal characteristics such as gender, age and English language proficiency may also be important. These variables may also affect the degree of search intensity, which will in turn affect the job-offer arrival rate. Other variables which could be important for explaining search intensity include housing costs and eligibility for benefits which capture an individual's financial capacity to continue job search.

In general these variables are self-explanatory and Appendix A provides more detailed definitions. Some variables, however, have required more construction due to the design and availability of information from the survey. Eligibility for unemployment benefits is likely to be an important explanator of labour-market outcomes and has occupied a large amount of space in the job-search literature. While the SEUP collects unit record data on episodes of income support, including the value of unemployment benefits received, these data are not publicly available due to confidentiality restrictions. We have derived a proxy variable for unemployment benefit eligibility based on answers to questions concerning the main sources of income.

If the respondent was unemployed at the time of the third interview and on 1 September 1996, we classify them as eligible for unemployment benefits if social security was their main source of weekly income in the week prior to the interview. This accounts for around 80 per cent of the people that we classify as unemployment benefit recipients. If they were unemployed on 1 September 1996 but not in the week before the third interview, they are classified as eligible for unemployment benefits if their main source of income in the last financial year was social security.

In general, it is important to control for the fact that different people live in different places, because local labour-market conditions and wage rates are likely to be important. For Australia, it would be obvious to control for the state in which the individual lives as well as the section of state which describes whether the respondent lives in a rural area, a country town, a capital city, or a non-capital city. However, for confidentiality reasons, the Australian Bureau of Statistics (ABS) does not release information about the state of residence.

Instead, we control for the state of the local labour market using an index of socioeconomic disadvantage compiled by the ABS. This index measures the extent to which the local area displays characteristics such as a high proportion of low income families, low average education levels and high unemployment rates.^[13] A higher score in the index of socioeconomic disadvantage suggests that the area is less disadvantaged. The SEUP provides information about the decile of each individual's local area ranked by the index of socioeconomic disadvantage. Gregory and Hunter (1995) have shown that socioeconomic disadvantage and unemployment rates are highly correlated across local areas.

4.3 An Empirical Model

The objective of Section 4 is to derive an empirical model which relates the duration of unemployment to the reservation wage and other explanatory variables. We can derive an expression for the expected log of incomplete unemployment duration for a given individual as follows:

where c is Euler's constant ≈0.577.

Therefore,

In equilibrium, if the flow into unemployment is the same over time, we can treat Equation 6 as a regression model where the individuals used for estimation are a cross-section of unemployed people with incomplete durations at a particular point in time. To the extent that the assumptions made above are acceptable, the parameters of this model can be directly related to the underlying theory, and this model can be regarded as a structural model. If the assumptions which have been made are not acceptable, Equation 6 is still a valid reduced-form regression for which theory has only provided guidance as to the types of variables to include.

Footnotes

The notation is intended to be consistent with Lancaster (1985), although the derivations are more closely aligned with Jones (1988). [11]

This result can be more formally derived using the standard results from renewal theory, see Lancaster (1992). [12]

The local area used by the ABS for this index is the Collection District (CD) which is a small area defined by the ABS for statistical collection purposes. [13]