Estimating and simulating with a random utility random opportunity model of job choice. Presentation and application to Belgium

In general, the ruro model is an economic model of human choice behaviour. Human decision makers are assumed to choose the best element from a set of choice possibilities or opportunities, where ‘best’ is defined in terms of preferences (or, vice versa, preferences are derived from observed choice behaviour as that objective which would be maximised given those choices). Applied to job choice, the set of opportunities is to be thought of as a set of possible activities a particular individual might choose to execute. Some of these activities are rendered available through job offers. These job offers will be indexed by j, where the variable j belongs to an index set, say J. A job offer stipulates an amount of labour time to be supplied when accepting the offer, say h, and pays a wage, w. It is assumed that this wage can be expressed in units of time effectively spent on the job, so that (gross) revenues earned by the job equal the amount of time spent on the job times the wage.

Gross earned labour income is then equal to wh. Gross earned labour income together with some other characteristics, say x_f, among which non-earned gross income (exclusive of transfers), determine the outcome of the gross to net (disposable) income function c = f (w, h;x_f), where c stands for consumption, which in a static model as the present one coincides with disposable income.⁵ That is, saving is considered as part of consumption. The function f converts gross income components into net disposable income, by subtracting taxes to be paid and adding transfers and subsidies. Usually, the generation of disposable income is constructed from raw data on gross income, labour time and other characteristics, by means of a microsimulation model.

Besides time spent on the job and the remuneration, jobs exhibit a number of other characteristics such as degree of responsibility, variation and challenge of the tasks, safety, healthiness, physical effort, stress, relation with colleagues and superiors. Preferences over these non–pecuniary attributes affect job choice.

One might also decline all job offers. Evidently, not executing a formal job does not require any time to be spent on the formal labour market (h = 0), and is assumed not to pay a wage (w = 0). A person who does not work, receives a net transfer (that is after deducting income taxes to be paid from her replacement income) equal to f (0, 0;x_f). In that case, time is spent on executing some of the available non–market opportunities. However, the set of activities⁶ one has alternatively available is not the same for all individuals, neither is the extent to which a particular alternative is available. When living in a small town, attending concerts, theatre or visiting museums is certainly not as easy as for big city dwellers. If you are in a wheel chair, hiking is not an option. Which of the available non–market activities will be chosen, in case no job offer is accepted, again depends on preferences (or, vice versa, what one chooses to do allows to derive something on the shape of preferences that person supposedly has). Non–market alternatives will be indexed by i, belonging to the index set I. The index sets for jobs and non–market alternatives respectively, are disjointI ∩ J = Ø. We will also use the index variable z to indicate an alternative in general, that is either a job or a non–market alternative. So, z ∊ Z := I ∪ J. To be really precise, an index z refers to a set of activities. If this set involves one or more jobs, the index z will belong to J, while it belongs to I otherwise. An alternative including several part time jobs is described by the total number of hours these jobs involve and the hourly wage these pay together (calculated as earnings divided by total hours).

In the model, preferences are defined over the number of hours h spent on jobs (which is zero if one chooses not to accept any job offer), consumption, c, and a set of other attributes that a job or certain non–market activities possess, and that a person might care for. These other attributes are not observed by the researcher.

The observable bundle of characteristics an alternative z ∊ Z exhibits, is denoted by (C(z), H(z)), where C(z) refers to the individually specific net disposable income resulting from executing activities indicated by z, and H(z) to the labour time involved by the set of activities indicated by z. The utility derived from these observable characteristics is denoted by V (C(z), T − H(z); x_V), where x_V are the specific values of a set of preference shifters for the individual under consideration, and T denotes the number of time units available in the period over which labour time h is registered (e.g. 168 hours a week, if labour time is expressed in hours worked per week). It is assumed that the econometrician can derive some evidence on the shape of the function V on the basis of observations on (C(z), T − H(z)) and x_V. So, no individual preference differences apart from those explained by observable characteristics x_V, are allowed for in this part of the utility function, and V is therefore called the systematic part of the utility function, and, hence, of preferences.

Since the other attributes besides disposable income and labour time are not observable, their contribution to utility will be specified as a random term. Thus, the utility generated by these attributes of set of activities z, is denoted by the random variable ε (z). It is assumed that this utility from non–pecuniary attributes, ε (z), enters overall utility of an alternative z in a multiplicatively separable way from the systematic part of the utility function. In order to make sense, this requires both, the systematic part of the utility, and the random term, to be non–negatively valued functions.

In summary, the total utility derived from picking an alternative z ∊ Z, say U (C(z), H(z), z;x_V), equals:

As c = f (w, h;x_f), the systematic part of the utility function, V (c, T – h;x_V), induces a utility function, say Ψ, defined over hours worked on the formal labour market, h, and wage, w:

Consequently,

where W (z) is the wage paid by activity z. More in particular, for someone not accepting any job offer, and choosing an alternative i ∊ I, utility equals:

The domain of the systematic part of the utility function in the wage-hours space, Ψ(·; x_V, x_f), is [0, ∞) × [0, T).

From now on we will drop the conditioning variables x_V and x_f in the systematic part of the utility functions.

Both, jobs and non–market activities, are not equally available to all individuals. This is captured by the notion of intensity with which alternatives are rendered available to a specific individual. The probability to receive a job offer as a civil engineer, for someone who has only completed secondary school, is e.g. zero. Something similar holds for non–market activities: they are not all equally available to all agents. Someone having lost her legs will not be able to run (or not in the same fashion as before), though she might continue to be fond of it.⁷

The intensity with which a job is offered to an individual depends on a number of personal characteristics, such as skills, education, experience, and on the characteristics of the job itself, more specifically, the wage, the labour time regime of the job, and its other attributes. For the sake of simplifying notation we will drop the individual conditioning variables.

In Equation (2), preferences were defined over the continuous set of all possible amounts of time spent on the jobs. In the real world, however, jobs requiring a non–rational number of hours a week, are not available. Be it alone to organise the production process, it might sometimes be required to have a number of people working together during a fixed number of hours. So, in practice, full-time, three-quarter time, half time, one-quarter, or 20% jobs are more densely offered than other labour time regimes. Let g₂(h) be the density with which jobs requiring h hours of labour supply, are rendered available to a specific individual. Similarly, jobs pay different wages. By g₁ (w|h) we denote the density of jobs paying a wage w, within the class of job offers available to a specific individual, that require h hours of work.

Persons do not only care for the wage a job pays, and the number of hours to be worked, but also for other job attributes. From a behavioural theoretic point of view (see Equation 1), these are only important in as far as they yield a specific value for the multiplication factor in the utility function for those alternatives. Two jobs, j₁ and j₂ say, paying the same wage and requiring the same amount of hours, with attributes yielding the same value of the multiplication factor in the utility function for those alternatives, that is ε(j₁) = ε(j₂), are thus, according to the behavioural model of Equation (1), equivalent to each other in the present model, and therefore will be considered as the same opportunity. The variable indicating the value of this multiplication factor is denoted by ν.

The intensity with which job offers yielding a value ν, are arriving to a specific individual is denoted by λ¹ (ν).⁸ We assume the following functional form for λ¹:

where q is a measure for the demand by the market for capacities and characteristics an individual possesses, relative to those capacities which are currently unsolicited by jobs, and which are therefore called non–market opportunities.

The functional form of λ¹ implies that attributes which are particularly disliked (yield a very small value for ν) are excessively abundant, while those that are particularly liked (yielding a very high value of ν), are extremely scarce. This functional form guarantees independence of irrelevant alternatives in the probability of choosing jobs (Dagsvik, 1994).

Now, let ℋ be the set of all possible labour time regimes of jobs offered in the market, and W the set of possible wage offers. Wages can obtain any positive value. Let ℬ := ℬ_h × ℬ_w be the Cartesian product of a measurable subset of labour time regimes ℬ_h ⊆ ℋ, and wage offers ℬ_w = (0, w), for some positive w. The arrival of job offers⁹ to an individual, is modelled by an inhomogeneous spatial Poisson process. Events are job offers that are characterised by a labour time regime h, a wage offer w, and the utility that can be derived from other attributes ν. The intensity parameter of this Poisson process is equal to g2 (h)g1 (w|h)λ¹ (ν). Define next the set of job offers specifying a labour time regime t £ ℬ_h, paying a wage r lower than w (that is r £ ℬ_w), and which will yield a utility level at least equal to u, as U_h,w,u := { (t, r, ν) £ ℬ_h × ℬ_w × ℝ₊|Ψ(r, t) ν ≥ u}, and define:

Let N(ℬ, u) be the number of job offers with a wage r belonging to ℬ_w, the number of hours to be worked in ℬ_h, and yielding a utility level larger than or equal to u. The probability for an individual to be offered n such jobs is under the present assumptions governed by a Poisson distribution, and is thus equal to:

Thus, Λ¹ (U_h,w,u) can be interpreted as the expected number of job offers with labour time regime, wage, and utility level in U_h,w,u.

As the arrival of job offers is modelled as a stochastic process, the utility level obtained from jobs with a wage in ℬ_w, number of hours to be worked in ℬ_h, and other attributes yielding a value for the multiplicative factor equal to ν, is a random variable too, denoted by U_ℬ. The probability that U_ℬ is less than u, denoted by P (U_ℬ < u), equals the probability that the number of available alternatives with utility larger than or equal to u, is zero. From Equation (7) with n = 0, it thus follows that:

Hence, U_ℬ turns out to be Fréchet distributed with location parameter μ = 0, scale parameter

and shape parameter α = 1.¹⁰

The derivation of this distribution is equally valid for any (measurable) subset ℬ of the space of possible working times and wage combinations, job offers might exhibit. More in particular, it holds for the complement of ℬ in the set of all possible working time wage combinations, defined as ${ℬ}^c:={ℬ}_h^c\times {ℬ}_w^c=\{\mathscr{H}\backslash {ℬ}_h\}\times [w,\infty )$. It follows that the random variable U_ℬc, denoting the utility level derivable from possible job offers with working time wage combinations in ℬ^c, is Fréchet distributed with location parameter μ = 0, scale parameter

and shape parameter α = 1.

The distinguishing value of different potential non–market activities, is completely absorbed by the different values of the multiplication factor in the utility function they generate. Indeed, the systematic part of the utility function is for all non–market alternatives equal to Ψ(0, 0). As for jobs with the same wage and required labour input, two non–market activities, i₁ and i₂ say, with attributes yielding the same value of the multiplication factor in the utility function for those alternatives, that is ε(i₁) = ε (i₂), are from a behavioural theoretic point of view equivalent to each other, and will therefore be considered as one and the same opportunity. For the same reason as in the case of job offers, it will be assumed that leisure activities which are particularly disliked, are abundantly available, while those that are intensely desired, are rather difficult to obtain. The intensity with which nonmarket activities yielding a multiplication factor equal to ν, denoted by λ⁰ (ν), are accessible to an individual, is thus assumed to be equal to:

Let ℰ_u be the set of values for ν such that non–market activities yield at least a utility level larger than or equal to u: ℰ_u = {ν ∊ ℝ₊|Ψ(0,0)ν≥ u}. Define Λ⁰(ℰ_U), as:

If one assumes that λ⁰ is the intensity measure of an inhomogeneous spatial Poisson process, then the number of non–market alternatives that yield a utility level of at least u, available to a person, is Poisson distributed (see Appendix A1). Let the number of available non–market activities yielding a utility level of at least u, be denoted by N(Ψ(0,0) ν ≥ u). The probability that N(Ψ(0,0)ν ≥ u) = n, is, according to the Poisson distribution, equal to:

The value of Λ⁰ (ℰ_u) equals the expected number of non–market opportunities available to an individual, yielding a utility level at least equal to u. The higher the value of Λ⁰ (ℰ_u), the more skewed to the right this distribution becomes, that is, the higher the probability that the number of available non–market alternatives yield a utility level of at least u is relatively big.

The probability that all available non–market alternatives to an individual, yield a utility level lower than u, equals the probability that the number of available alternatives with utility larger than or equal to u, is zero:

From the last equation, it can be concluded that the utility that can be derived from the available non–market alternatives, which is a stochastic variable denoted by U₀ = Ψ (0,0) ν, is Fréchet distributed with location parameter μ = 0, scale parameter σ₀ = Ψ(0,0), and shape parameter α = 1. This will prove useful when formulating the likelihood function in the next section (Section 3.1).

Now we turn to the behavioural implications of the model explained in Section 2. From the available job offers and non–market opportunities, a person will choose that alternative she likes most. The probability that this will be an alternative including a job offer with a working time wage combination in the set ℬ, is equal to the probability that U_ℬ > max {U0, U_ℬc}.

As the processes governing the arrival of job offers and non–market opportunities are assumed to be independent, the probability that U_ℬ (the utility from a job offer with a labour time and wage combination in ℬ) is equal to or greater than max {U₀, U_ℬc} is equal to the product of the probability that U_ℬ is greater than or equal to U₀ and the probability that U_ℬ is greater than or equal to U_ℬc. That amounts to:

In a similar fashion it can be derived that the probability to choose a non–market alternative, is equal to the probability that U0 is equal to or greater than U_ℬ∪ℬc, which is equal to:

An additional assumption for identification is the independence of the wage offer distribution from the hours specified by the job offers. That is g1 (w|h) = g1 (w), ∀h ∊ ℋ.

The likelihood that an individual will choose one particular job offer requiring labour time h, and paying a wage w, can thus be obtained from Equation (15):

Similarly, the likelihood an individual’s most preferred non–market alternative is preferred to any of the job offers, equals:

We exclusively concentrated on individual decision makers. The model can be extended to the case of households consisting of couples (with or without children), if one is willing to assume a unitary decision making model. We provide this extension in Appendix A2.

It is worthwhile to compare the likelihood function (15)–(15’) with what is obtained in a random utility function based upon discrete choice of labour time regimes (such as in Van Soest, 1995). In this approach, the wage a person obtains, is, apart from measurement problems (see note 2), a fixed individual characteristic reflecting that person’s productivity. Choice of labour time is free but limited to a discrete set {h_i;k = 1,2,..., K}. Under the assumption that the stochastic parts of the utility functions are Fréchet distributed, the likelihood (probability) to observe an individual choosing a labour time regime hi equals:

The difference with (15)–(15’) is twofold. In the ruro model utilities are weighted with the intensity with which alternatives are rendered available to an individual. Next, the wage is part of the job offer. Consequently the denominator sums over all possible pairs of wages and labour time regimes (w, h), job offers contain, and not only over possible labour time regimes for a given wage.

Some parts of the model are non–parametrically identified. A fuller treatment of this issue is provided in Aaberge, Columbino and Strøm (1999), the working paper version of Dagsvik and Strøm (2006) (See Dagsvik & Strøm, 2004), and Dagsvik and Jia (2016). The main line of argument for identifying the wage offer distribution from preferences runs as follows. Isolate two observationally equivalent groups of individuals in the population, supplying different number of hours, say h₁ and h₂, for the same wage w. The relative proportion of these groups in the population is φ (w, h₁) / φ (w, h₂), which according to the model in Equation (17) reduces to:

Doing this for different levels of wages, allows to identify the function Ψ (w, h) g2 (h). Looking then at persons performing the same number of hours, but accepting different wages, w₁ and w₂ say, gives:

As Ψ (w, h) g2 (h) was already identified in the previous step, it is now possible to identify g1 (w), using the fact that it is a density, and thus that ∫_w∈W g1 (w) d w = 1.

Then, consider an observationally equivalent group of persons in the population. Some of them will be engaged in a formal job, and some of them not. The relative proportion of those groups in the population are:

As Ψ is a utility function, we can normalize the value of Ψ (0,0), which allows to identify q from this equation. In our empirical application, we tried to improve upon the non–parametric identification of q by introducing an exclusion restriction. More specifically, a group specific unemployment rate¹¹ is added as an explanatory variable for q. We assume that this variable does not affect individual preferences, but, obviously, it has some relation with labour demand.

The utility function Ψ (w, h) and the distribution of offered labour time regimes g2 (h) are however not separately non–parametrically identified. One way out is to give a more fundamental justification of the functional form used for preferences. For example, Dagsvik and Røine Hoff (2011) and Dagsvik (2013) give a non–parametric justification of the preferences embodied by the Box-Cox type of utility functions that we will use (see the next section).

Moreover, it can be argued that the occurrence of peaks in the distribution of the number of hours worked, as observed in many datasets, around half time, three quarter time and full time work, are not easily explained by the traditional way preferences are shaped in economics, neither by the kinks in the shape of the budget set caused by different tax structures.

In the present section we present the functional forms of the different components of the model that will be used in the empirical application in Section 5.

At the preference side,

• the systematic part of the log utility function for singles is of the Box-Cox type: ln $V(c,T-h;x_V)={\beta }_c\cdot \left(\frac{c^{\alpha c}-1}{{\alpha }_c}\right)+({\beta }_h^{\text{'}}{x}_V).\left(\frac{{((T-h)/T)}^{\alpha }h-1}{{\alpha }_h}\right)$ , with α_c, α_h < 1. Intensity of preferences for leisure is increasing (decreasing) in an element of x_V, if the associated parameter of βh is positive (negative).¹² The exponents, a_c and ah, determine the curvature of the indifference curves in terms of labour time and consumption (that is, while keeping other attributes affecting preferences fixed). The lower these are, the less substitutable leisure and consumption are;

• for couples, a unitary decision model is assumed, but spouses’ leisure time is considered to be an assignable good. So, the systematic part of preferences is defined over consumption and each spouse’s leisure time. Partner’s time endowments are equal. An interaction term capturing potential complementarities between partners’ leisure time, is added to the utility function:

  (23) $\text{In}V(c,T-h_1,T-h_2\text{};x_V)={\beta }_{c,g}\cdot \left(\frac{c^{\alpha c,g}-1}{{\alpha }_{c,g}}\right)+{\displaystyle {\sum }_{i=1,2}({\beta }_h^{\text{'}}{x}_V).}\left(\frac{{((T-h_i)/T)}^{{\alpha }_h}i-1}{{\alpha }_h{}_{{}_i}}\right)+{\beta }_{h_1,h_2}\cdot {\Pi }_{i=1,2}\left(\frac{{((T-h_i)/T)}^{{\alpha }_h}i-1}{{\alpha }_h{}_{{}_i}}\right),$

  with α_c,g, α_hi < 1 (i = 1,2). The interpretation of the exponents and the ${\beta }_{h_i}^{\text{'}}$ (i = 1,2) remains the same as for singles; in addition, β_h1,h2 > (<)0 means that partners’ leisure are complements (substitutes).

At the opportunity side,

• the log of the intensity of job offers relative to the availability of non–market alternatives is linear in the covariates: ln $q(x_{opp})=n_q^{\text{'}}{x}_{opp}$. The vector x_opp resumes covariates that might affect the job offer intensity, and should contain a constant term. The coefficient associated to this constant term is denoted by η_q,0;

• the wage density g1 (w; x_w) is assumed to be lognormal:

  $g_1(w;x_w)=\frac{1}{w\cdot \sigma \cdot \sqrt{2\pi }}\exp \left(-\frac{1}{2}{\left(\frac{\text{In}w-{\delta }_{g_1}^{\text{'}}{x}_w}{\sigma }\right)}^2\right)$. It is dependent on covariates x_w that might affect the wage offer distribution;

• the distribution of the labour time regimes offered, is piecemeal uniform. There are a number of, say K, peaks, indexed by k = 1,2,..., K, around which the bulk of the job offers’ labour time regimes are concentrated (typically around half time, that is 18.5 to 20.5 hour a week in our application, three quarter time, or 29.5 to 30.5 hours a week, and full time, or 37.5 to 40.5 hours). The lower and upper bound of peak k (k = 1, 2,..., K) are denoted by respectively $\underset{\_}{H_k}$ and $\overline{H_k}$. There is a lower limit, H_mj_n, below which job offers are not considered to belong to the formal labour market (fixed at one hour a week in the application below); and an upper limit of labour time spent on formal jobs, denoted by H_max, and fixed at 70 hours per week in our application. This results in the following density function:

  (24) $\begin{array}{ccc}{g}_2\left(h;x_h\right)& =& \{\begin{array}{ll}{\gamma }_1\hfill & \text{if }h\in [H_{\text{min}}\text{,}{\underset{\_}{H}}_1[,h\in [{\overline{H}}_k\text{,}{\underset{\_}{H}}_{k+1}[,\text{or }h\in [{\overline{H}}_k\text{,}{H}_{\text{max}}[,k\text{ = 1,2, }\mathrm{...}\text{ ,}K-\text{1;}\hfill \\ {\gamma }_1\exp {\gamma }_{k+1}\hfill & \text{if }h\in [{\underset{\_}{H}}_k,{\overline{H}}_k[,k\text{ = 1,2, }\mathrm{...}\text{ ,}K.\hfill \end{array}\end{array}$

  An example of such a distribution function is given in Figure 1. In our application, the only covariate in x_h allowed to affect this function is the sex of the person .

To estimate the parameters governing preferences, the relative intensity of market over non–market alternatives, and the distribution of wage offers and labour time regimes, a likelihood function, say ℒ, is constructed on the basis of Equations (17) and (18’) (for the case of the couples, see Equation (18”) of Appendix A2). The individual contributions of a single to that likelihood function are composed of the likelihood that the observed choice is the most preferred one, reflected in Equations (17), or (18’), depending on whether the observed choice involves participation on the formal labour market executing a job (or set of jobs) requiring h hours of work, and paying a wage w, or whether it is the non–market alternative. In these expressions, the numerator is thus evaluated at the actually observed choice, when constructing the likelihood function. Similarly for couples, the first, second, third, or fourth equation in (15”) (in Appendix A2) applies, dependent on whether both partners, only partner j (j = 1, 2), or none of both actually are engaged in formal jobs.

Figure 1

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In practice, we do not observe the set of wage offers, W, nor the offered labour time regimes, ℋ. Therefore, a set of alternatives in the space of wages and labour time regimes is sampled from a prior density function, say ℙ (w, h). Denote the set of sampled combinations of wage offers and labour time regimes, possibly including the non–market alternative, by D. The observed choice (w^obs, h^obs) is to be always included in the sampled choice set. From the sampling densities ℙ (w, h), the likelihood to sample a set D given that the observed choice equals (w^obs, h^obs), can be constructed.¹³ It is denoted by P (D | (w^obs, h^obs)), and it equals:

Recall that the probability (density) that a job paying a wage w, and requiring a number of h hours to be worked, would be optimal given a choice set C := {0,0} ∪ W } ℋ, was derived in Equations (17), and (18’) if the non–market alternative would be the most preferred option. The unconditional probability to sample a choice set D, denoted by Π (D), can thus be written as:

Using Bayes’ law, the probability (density) to observe an agent choosing a job offer that pays a wage w_i, and requires h_i, hours of labour time from the sampled set D, thus equals:

Using Equations (25) and (26), we can thus reformulate the simulated likelihood to observe someone choosing an alternative (w, h) from a choice set D sampled according to the prior ℙ (w, h), as:

The corresponding expression for choosing the non–market alternative equals:

One further normalisation issue is in order. Note that the constant term of the q (x_opp)–function, exp(η_q,0), occurs in any term of the likelihood where γ1 appears (that is, in those terms of the sum pertaining to a job offer on the formal labour market, (w, h) : w, h > 0), and each time these terms appear as a product. Therefore both, η_q,0 and γ1, cannot be estimated separately. But γ1 is still identified by the identity:

This means in practice that one does not estimate all the parameters in the likelihood (23) (respectively 23’), but rather reduces these equations to:

where:

For the likelihood to choose a non–market alternative, this becomes:

One is able to back out an estimate of η_q0 by using Equation (30). Indeed, subtracting ln $\widehat{{\text{γ}}_{\text{1}}}$ (with $\widehat{{\text{γ}}_{\text{1}}}$, the estimated value of γ₁ from applying Equation (30) using the estimates for 7_k+1, for k = 1,2,..., K) from the estimated constant of ln $(\tilde{q}(x_{opp}))$.

We used the following specification of ℙ (w, h) for constructing a choice set D: wages are sampled from a lognormal distribution with parameters m and ς, labour time is sampled from the uniform distribution on the [H_min, H_max)–interval, and the probability to sample non–market alternatives is the observed inactivity degree in the sample (that is the relative number of persons in the sample being engaged in formal jobs for less than one hour a week), say ${\pi }_0^{\text{obs}}$.¹⁴ That is:

In our implementation, we do not specify a functional form for the disposable income function f (w, h; x_f). For each draw (w, h) from the prior ℙ (w, h), a disposable income f (w, h; x_f) is instead calculated on the basis of the existing Belgian tax rules, as implemented in the microsimulation model eu-romod, using where necessary additional information on non–earned income and relevant household characteristics from the silc.¹⁵

In order to evaluate the fit of the estimates, or the estimated model’s prediction of behavioural reactions to changes in explanatory variables, a simulation method can be used. Thereto, a choice set is to be drawn (possibly capturing changes in the intensity with which certain alternatives become available to certain persons) and then it is determined what an agent’s best choice would be within this simulated choice set, according to the estimated preferences of that person. If simulation is used for evaluating the fit of the estimated model, then the choice set is drawn according to the model estimates (the relative intensity of job offers, the wage offer distribution, and the labour time regime offers), and the simulated choices from that set are to be compared with actual ones, as observed in the data. This is done in Figures 6-11 below (Section 6.1). Next we will use the simulation method also for calculating elasticities, and to evaluate the effects of changes in the education level on labour market participation (see Sections 6.2 and 7).

For example, when simulating to assess the fit, one uses the estimated measure of intensity with which alternatives (w, h) are offered to an agent, that is, using the estimates of the q-function, the estimated wage offer distribution, g1, and the estimated hours distribution, g2, to sample a choice set {(w_s, h_s); s = 1,2,..., S}. Next, one draws for each of the sampled alternatives, (w_s, h_s), a random variable from the Fréchet distribution, say ε (w_s, h_s). Next, it is evaluated which of the drawn alternatives yields highest utility: $\widehat{V}(f(w_s,h_s;x_f),T-h{}_s;x_V)\epsilon (w_s,h_s)$.¹⁶ The alternative (w_r,h_r) thus yielding the highest utility is considered to be the agent’s optimal choice according to the model.

To assess preference estimates we investigate the shape of the estimated indifference curves for labour time and consumption. As jobs contain also other, unobserved, attributes, this approach is only valid under the assumption that these other attributes are kept fixed when considering variations in consumption and labour time. The marginal rate of substitution between consumption and labour time for singles equals:

and for couples it is equal to:

Notice that the covariates influencing preferences affect the marginal rate of substitution only through their influence on $\left({\beta }_{hj}^{\text{'}}{x}_V\right)$. More specifically, as $\left({\beta }_{hj}^{\text{'}}{x}_V\right)$ increases in one of the covariates, the marginal rate of substitution in any point (c, h_j) becomes higher for a person with a larger value on that variable. Hence, a person exhibiting a higher value on that covariate will have relatively steeper indifference curves. That is, she will exhibit a more intense preference for leisure relative to consumption.

We illustrate this in Figure 2 for the education level. For males, both in couples (thin black curves in Figure 2), and as singles (fat black curves in Figure 2), the effect of education is non–monotonous. Both highly and lowly educated (dashed, respectively dotted curves) men have less intense preference for leisure relative to consumption, as compared to their fellows with a middle education level (full lines). This effect is more outspoken, but much less precisely estimated, in the case of singles. Higher educated females have less intense preferences for leisure relative to consumption, both when living in couples (dark grey curves) or as a single (light grey curves).

Figure 2

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Figure 3 represents the estimation of the wage offer distributions. The dotted lines apply to females, the full lines to males. The sex differences are small, as compared to the impact of the other covariates, and not always in the disadvantage of females. The latter is the case for persons with a middle education level.

Figure 3

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Higher education shifts the wage offer distribution to the right: compare the fat black, grey, and light grey curves for the distributions at low experience age (10 years), and the thin black, grey, and light grey curves for the impact of education at high experience age (25 years). Potential experience also shifts the wage offer distribution to the right: compare fat and thin lines of the same color. However, when potential experience would exceed the level of 37 years, for males, and 32 years, for females, the wage offer distribution would start shifting to the left again. For experience levels below these bounds, higher education, which implies less potential experience, has therefore two countervailing effects on the wage distribution. The net effect is usually positive, that is a shift of the wage offer distribution to the right. Whether this would also results into an increased acceptance of jobs, depends on income and substitution effects (in preferences), and cannot be fixed apriori.

Notice that this is a wage offer distribution. Simulated and observed wages of accepted jobs are discussed in Section 6.

Figure 4 represents the distribution of offered labour time regimes.¹⁷ Again, these are not actual labour time regimes nor those chosen according to the model. The most salient observation is that this distribution is different for males and females, the latter receiving more part time, and less full time job offers.

Figure 4

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Figure 5 reports the impact of education on the intensity of suitable job offers (the q–function).¹⁸ As the type specific unemployment rate varies with education level (and age) we report the joint effect of both. More specifically, for each age class we report the intensity of suitable job offers of males (black bars), respectively females (grey bars) of a lowly (dotted bar), highly educated persons (dashed bar), and those with middle education level (fully coloured bars).

Considered per se the education level has a positive effect on the availability of suitable job offers, though the effect of high versus middle education is small for males.

From Table 2 it can be seen that the type specific unemployment rate is decreasing in the education level. Now, for males, a higher type specific unemployment rate increases the intensity of suitable job offers. As a consequence, the already small positive effect of education is slightly attenuated, leading ultimately to a net negative effect of high as compared to middle education level for the middle age classes. For females on the other hand the net effect of high education is outspokenly positive, as both the effect of the lower unemployment rate with education and education per se, push into the same direction.

Notice that Figure 5 also reveals that the availability of suitable job offers is decreasing slightly with age for males, while we don’t see a similar decline for females.

Figure 5

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We now evaluate the fit of the estimated model.

1. Couples

The mean estimated consumption within couples is 3070 € per month. Compare that with the observed mean of 3143 € per month reported in Table 1. The simulated distribution (black curve in Figure 6) slightly overestimates the number households with lower incomes, at the expense of those with modal disposable incomes (compare the black curve with the observed values represented by the gray dashed one).

The fit of the (conditional) distribution of female wages is good (RhS panel of Figure 7): the black (simulated) and grey dashed (observed) curve almost coincide. The simulated wage distribution of the males (black line on the LhS panel of Figure 7) is more populated than the observed one (grey dashed line on the LhS panel of Figure 7) at lower and moderately high wages, at the expense of a smaller occurrence of modal and extremely high wages.

Figure 6

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The number of non–participants in the labour market is overestimated. Compare thereto the full grey (observed) and unfilled black bordered (simulated) left most spike in both panels of Figure 8. Still, the number of cases in which none of both partners work, is underestimated. The estimated peaks reasonably well fit the observed values, except for the three quarter time jobs, the occurrence of which is underestimated by the model, both for male and female partners. The percentage of females having a full time job is also slightly overestimated by the model.

Figure 7

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Figure 8

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2. Singles

Figures 9–11 represent the fit of the model for singles. Consumption of single females is reasonably well approximated (rhs of Figure 9). Mean consumption of males is almost perfectly replicated by the estimates (1585 € per month fitted versus 1588 € per month observed), but the empirical distribution is somewhat less good approximated, with, amongst other things, an underestimation of the lower tail. The latter is also the case for the single females.

Figure 9

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Figure 10

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Similarly, the wage distribution of single females (rhs of Figure 10) is better fitted than that of males (lhs of Figure 10).

Labour market participation of single males (cf. lhs of Figure 11) is overestimated, while that of single females is almost perfectly fitted (rhs of Figure 11). The observed peak for half time jobs is underestimated for males, and that of three quarter time jobs is overestimated for both, males and females. The occurrence of full time jobs for males is overestimated. That of females underestimated.

Figure 11

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Table 4 reports the total reaction in terms of labour time and the effect on participation to the labour market (extensive margin), following a shift of the density of the males’, respectively females’, wage offer distribution to the right by 10% (augmenting the estimated location parameter with ln1.1). Additionally, we report intensive margins (effect on labour supply conditional on participating in the base line, inclusive of the labour market leavers). The variable ‘part in’ gives the percentage of entrants into the labour market, while ‘part out’ represents the percentage of leavers.

Compared to Marshallian elasticities in the literature estimated by static models using micro data, the total elasticity estimates produced here are rather large (Compare e.g. with the figures reported in Tables 6 and 7 of Keane, 2011). However, as far as these total elasticities include the extensive margin, and are calculated as the proportional change in total labour time for the whole sub-sample, these need to be compared with macro elasticities, which are usually much larger, even as compared to the figures obtained here.¹⁹ Still, it should be stressed that the figures reported here are conceptually of a different nature, in that actually obtained wages in the present model are the result of choosing the most attractive job offer according to the persons’ preferences. Therefore, a reaction to an exogenous change in that wage cannot be conceived of in the framework we used. What was, alternatively, done, is to shift the entire distribution of the wages included in the job offers, to the right. This cannot be considered the same as a change in an exogenously given wage. As the ruro model incorporates frictions due to restrictions in the labour market opportunities an agent faces, this might account for the lower values of the elasticities reported here, compared to values obtained for macro figures.

Recent overviews of the model and its applications are provided by Aaberge and Colombino (2014), and Dagsvik, Jia, Kornstad, and Thoresen (2014).

Some contributions do allow for unobserved wage heterogeneity, see e.g. Van Soest, Das and Gong (2002), Löffler et al. (2013), and the second model discussed in Dagsvik and Jia (2016). Van Soest (1995) already incorporated the problem of imperfect observation of wages for non–participants in the extended version of his model. Besides, there is an earlier literature accounting for the fact that wages are non–linear in hours (See for example Moffitt, 1984). However, none of these treats wages as an object of job choice behaviour. In our model, there is no unobserved wage heterogeneity, neither measurement error in wages. The wages in our model are, as in the first class of models of Dagsvik and Jia (2016), part of the job offers available to an individual.

Statistics on Income and Living Conditions (silc) is a survey held yearly in EU-member states and some other countries under supervision of eurostat. Permission to use eu-silc data for Belgium was obtained in the framework of the euromod-project. More in formation on eu-silc can be found on http://ec.europa.eu/eurostat/web/microdata/european-union-statistics-on-income-and-living-conditions. See Section 4 for more information on the data we used from eu-silc.

Currently males’ educational attainment level is lagging behind that of females.

We include the gross wage, and the number of hours worked as separate arguments in that function, as some aspects of the tax system, such as the Belgian work bonus, may depend on the wage, rather than on labour income, wh. We are however aware that this might cause problems for the non–parametric identification of the ruro model.

The word ‘activity’ will be used here in a broad sense, including occupations which are not very ‘active’ such as sleeping and day dreaming. A certain type of agency or control is however presumed, since otherwise it would be difficult to talk about choice behaviour.

In Appendix A1, we provide a brief introduction to the type of stochastic process that describes the degree to which job offers and non–market alternatives become available to an individual, and which is known as an inhomogeneous spatial Poisson process.

Job offer arrivals depend on personal capacities and skills which are subdivided in those apt to execute formal jobs, and those suited for performing leisure activities. Next there may be personal characteristics on the basis of which discrimination in job offers by employers might take place. As indicated before, we omit the impact of these conditioning variables for the sake of notational simplicity.

As mentioned before, a ‘job offer’ is a short hand for ‘an alternative containing at least one job offer’.

In general, the class of Fréchet distributions is defined as: $F(x;\mu ,\sigma ,\alpha ):=\exp \left[-{\left(\frac{x-\mu }{\sigma }\right)}^{-\alpha }\right]$, where μ is a location parameter, σ a scale parameter, and α is a shape parameter.

Ideally one would use the number of vacancies for suitable jobs for certain identifiable groups of persons in the population, but, such information is not really observable.

More details are provided in Section 5.1.

The issue of sampling choice sets for estimating the ruro model is discussed more in detail in McFadden (1978), Ben-Akiva and Lerman (1985), Aaberge, Colombino and Wennemo (2009), Train (2009), and Lemp and Kockelman (2012).

The exact figures are ${\pi }_0^{\text{obs}}=.104,m=2.71,\text{and}\varsigma \text{=}\text{.308}$ for males, and the corresponding numbers for females are .246, 2.63, and .297.

Version F5.5 was used. For more information about euromod, see Sutherland and Figari (2013) and http://www.iser.essex.ac.uk/euromod.

For each draw (w_s, h_s), the disposable income f (w_s, h_s; x_f) is again obtain from the microsimulation model euromod.

Admittedly, this part of the model is not non–parametrically identified (see Section 3.2). So, if one feels more for explaining this peak pattern by preferences, we cannot tell this to be wrong on purely empirical grounds.

We actually report values for q/ (1 + q). In this way the results are normalised to lie between zero and one.

On the controversy about micro versus macro estimates, see amongst others Chetty et al. (2011), Chetty (2012), Fiorito and Zanella (2012), Keane and Rogerson (2012), Jantti et al. (2015), and the references therein.

As midas runs on administrative data that do not include information on education levels, these were imputed, and some basic scenarios were developed to assess their evolution in the future, as new, future generations enter the model. More information on midas can be found in Dekkers et al. (2009).

Education levels were assigned on the basis of a comparison of a random draw from the [0, 1]–uniform distribution for each case, with the cumulative distribution of education levels in the corresponding gender and age class of the individual.