4  Econometric Specification of a Labour Supply Model

In Section 2 an economic model was introduced that serves as a starting point for the specification of an econometric model. In the following sections, the econometric specification is discussed. First, we examine the implications of nonlinear and non-convex budget sets in Section 4.1. Section 4.2 discusses the utility function and Section 4.3 describes how expected labour supply can be derived using the estimated model.

4.1   Allowing for Nonlinear and Non-convex Budget Sets

Including taxes and benefits for two persons in budget constraints produce highly nonlinear constraints. Looking at the benefit and tax regimes between 1991 and 2001 leads us to expect many kinks in the budget constraint. Since we prefer to keep the representation of taxes and benefits as close to reality as possible, a complex budget constraint cannot be avoided. In the case where one potential worker is considered at a time, the labour supply estimation is already quite complex[15]. The complexity is even greater in the case where income units with two potential workers are analysed, subject to their joint budget constraint.

Restricting the number of possible working hours to a limited set of discrete values, as is done by other authors (for example Van Soest, 1995; Duncan, Giles and MacCrae, 1999; Keane and Moffitt, 1998) facing the same problem, appears an attractive solution.[16] For this limited set of hours, one can calculate the level of utility that each possible combination of hours would generate, according to the specified utility function. An additional (computational) advantage of the discrete approach is that quasi-concavity does not have to be imposed before using maximum likelihood methods to estimate the model, as is necessary in the case of continuous labour supply for some utility functions (see Van Soest, Kapteyn and Kooreman, 1993), but can be checked after estimation.

Instead of being defined on a continuous set of working hours [0,T], in the discrete choice case the budget constraint is defined on a discrete set of points on the interval [0,T]. 0, h₁₁, h₁₂, and etcetera represent the discrete values that labour supply can take. Using these sets, net income x(h₁,h₂) is calculated for all (m+1)×(k+1) combinations of h₁ and h₂ (where m+1 is the number of discrete points for h₁ and k+1 is the number of discrete points for h₂). The static microsimulation model at New Zealand Treasury, TaxMod, can be used to calculate net income at all chosen discrete labour supply points in the different survey years. It is assumed in these calculations that all benefits, for which the household is eligible, are taken up. By increasing the number of different hours in the choice set, the quality of the representation improves. However, the computational load also increases, so a compromise between quality and computational feasibility is necessary. Furthermore, some of the theoretically possible hours ranges may not be observed in the data such as low part-time hours for men, which may mean fewer discrete points are necessary in that range.

Net income x is dependent on labour supply and wage rates of both adults, on non-labour income, on household composition and on eligibility for benefits. Net income for the records originating from the earlier data sets are inflated up to the 2001 level by multiplying the amount by the relevant CPI. In this way, net incomes in the different years are comparable. Wage rates, non-labour income and household composition are considered to be exogenous in this model. The model becomes:

(5)    max U(x,l₁,l₂)

subject to:

(6)    

w₁ and w_{2are the gross wage rates of husband and wife respectively,}

are the sets of discrete points from which values can be chosen for h₁ and h₂,

B(.) is the amount of benefit, for which the household is eligible, given household composition c and income,

τ(.) is the tax function that indicates the amount of tax to be paid.

Adding an error term to the utility function allows for optimization errors made by the household, preventing contributions to the likelihood in any data point from becoming zero. A likelihood function can be formed using the above utility function. Based on the assumption of utility maximization for each household the following can be stated. The contribution of each household to the likelihood function is the probability that its observed hours result in an optimal utility for the household of interest when compared with all other possible choices for hours. This probability looks as follows:

(7)    Pr {U[x((h₁, h₂)_r), (h₁, h₂)_r, ε_r] ≥ U[x((h₁, h₂)_s), (h₁, h₂)_s, ε_s] for all s}

where:

r stands for the combination h₁ and h₂ that is preferred,

s stands for all (k+1)×(m+1) possible combinations that can be made, given the discrete choice sets for hours worked,

ε_r and ε_s represent error terms.

Choosing an Extreme Value specification for the error term in (7) results in a multinomial logit model (see Maddala, 1983). If we can calculate utility levels for each of the possible combinations of leisure and income, and the error terms are specified, then for each possible combination we can calculate the probability of that combination being preferred according to the estimated model:

(8)    

Taking the logarithm of this probability, the log likelihood contribution for couples looks as follows:

(9)    

where:

i indicates the husband’s labour supply;

j indicates the wife’s labour supply;

i’, j’ are the preferred (observed) states of labour supply (combination r in equation 7);

U_ij is the level of utility derived from the state where the husband has labour supply i and the wife has labour supply j.

Expression (8) denotes the probability that the utility in the observed combination of hours is higher than the utility in any other situation. The log likelihood function for all households in the sample is formed by summing all individual contributions. The aim is to choose parameter values for the utility function that maximize the log likelihood function in the observed data points.

For single adult households equation (8) simplifies to:

(10)