Model and econometric issues

I apply a static neo-classical structural labor supply model with single decision-makers. The physician’s utility depends on income, leisure and other characteristics of the jobs. The utility maximization problem is solved by discretizing the nonlinear budget set and choosing the optimal job type, hours of work and income combination from a finite set of alternatives. The approach presented here assumes that agents choose among “job packages”, or more

specifically - combinations of jobs, each being defined by a main job and an extra job with specific choices of hours. Examples of other applications of this framework includes Aaberge et al (1995) and van Soest (1995).

A “job package” is described by i, the choice of main job (and the matching extra job), the hours H in the main job, and hours _ij h in the extra job. The individual specific wage rate per _ik

hour in the main job W H depends on hours worked. The wage _ij( _ij) w in the extra job is _ik

independent of hours. In addition there are other job characteristics that may affect

preferences and hence choices. As an example we may think of specific skills involved in the job, patient mix or shift work. I let the i represent these factors in the set-up.

Since preferences are unknown to the analyst, I will assume a random utility model. The utility depends on consumption C, hours in the main job H, hours in the side job h and other characteristics i. ( , , , ) ( , , ) ( , , , ) U C H h i =v C H h +ε C H h i (1) where ( ( ) ) ijk ij ij ij ik ik C C= H W H +h w , H =H_ij, h h= _ik

Cijk is consumption in the job-package, with practice type i, with specific hours of work Hij in

the main job and hik in the extra job. εijk is a stochastic term with an iid extreme value

distribution with an expected mean of 0 and a variance of _{σ π}2 2_{/ 6. The random term ε}

ijk

captures the fact that attributes other than income and hours not observed here affect labor supply, e.g. type of job, shift work etc. The last element in the random term represents other characteristics of both jobs in the job combination, as the choice of an extra job is fixed when the main job is chosen.

Wij(Hij) is a piecewise linear wage relation in main job i capturing the agreed terms of

overtime compensation. This is particularly important when analyzing the labor supply of hospital consultants, as they have a relatively moderate regular wage rate, but a complicated package of different compensations for extended working hours and night shifts2. In the private practice, physicians face the same costs, reimbursements and fees for the marginal patient as for the first. This is only an approximation as fixed costs like office rent and

medical equipment are significant for some specialties. The earnings in the main job and extra job are expressed as

Rij= Wij (Hij) Hij (2)

2 _{A hospital consultant has a basic 37.5 hours working week, but shift work reduces this to 35.5 hours per week.}

Most physicians have agreed to a contract of extended working hours with 2.5 hours per week. This is paid with a regular wage rate, but compensated for with an additional transfer of NOK 19,900 per year. For the interval from 38 to 40.5 hours per week they are compensated with 50 percent extra per hour on top of their regular wage. This rises to 100% for the next five hours, whereas shift plans with more than 45.5 hours per week

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ik ik ik

r =w h (3)

The consumption or more correctly the disposable income corresponding to the choice i, j, k is given by the budget constraint

Cijk= f(Rij+r )+I _ik (4)

The f(.) function represents the net-of-tax income, which is a compound of earnings in the main job and earnings in the extra job. I is family income other than the physician’s own earnings (capital income after tax, spouses income after tax, transfers). A non trivial assumption made is that the spouse’s hours of work are exogenous as there is reason to believe that the spouse’s choice of working hours will correlate either negatively, e.g. if one of the parents must look after the children, or positively as they prefer spending their leisure time together.

Let B be the opportunity set, i.e. it contains all the feasible “job-packages” available to the individual. We exclude non-market opportunities from B as the share of physicians not participating in the labor market is negligible3. Thus for all physicians Hij >0, but h_ik ≥ . The 0

physicians do not differ with regard to the number of available job sectors or practice types, as I have chosen four practice categories that should be feasible to all physicians4. Note that for the same physician, wage rates may differ across jobs, and that the wage rates vary with hours worked at hospitals and in primary care. Having access to their employment contracts, we are able to derive the compensation schemes for extended hours.

The physicians have a choice of Hij ={18, 22, 28, 35.5, 37.5, 40.5, 45.5, 50, 55} hours per

week in the main job. In addition to a main job, the model gives them the possibility for hik =

{0, 6, 12, 18, 24} hours per week in the extra job. As stated above I assume that the

3 _{See Aaberge , Colombino, Strøm & Wennemo (1998) for an example including non-market opportunities.} 4 _{There are of course differences in choice sets related to specialties and geographic regions, but the broad}

categories of job types applied here should not be too limiting. The data restricts the number of job types we are able to model. E.g. we cannot separate income from a municipal casualty clinic or a private practice.

physicians chose the same type of extra job, given their main job. E.g. if the main job is as a hospital consultant, the extra job is in a private practice, the most common type of extra job observed for each practice type.

In traditional labor supply offered wages are determined by human capital characteristics, and offered hours are uniformly distributed. However, in real life wages may vary across sectors for observationally identical workers, and jobs with a specific number of hours may be more available in the market than other jobs, e.g. “full-time” jobs. I introduce an opportunity density where I assume that offered hours are uniformly distributed except for full-time hours and for private practice jobs. This density is assumed to reflect that offered hours, except for full-time workload, is equally available in the market. It also corrects for the fact that if the physicians choose to work in a private practice, the hours available in the market will be less regulated (or not at all) relative to jobs in the public sector. Hours in the side job are

uniformly distributed.

Since hours of work and consumption are given when the job package is given, the

physician’s choice problem is a discrete one, namely to find the job that maximizes utility. As already mentioned, the analyst does not observe preferences and neither does he observe all details of the job-packages available in the market. The problem solved by the physician looks like this:

( ,i H hmaxij,ik) [ ijk, ij ik, ]

U C H +h i (5)

s.t.

(H h W H_ij, _ik, _ij( _ij),w i_ik, )∈ B. (6)

Let (P H h be the probability that the physician will choose a “job package” with H_{ijk ij}, _ik) ij

hours of work in the main job and hik hours of work in the side job. When the random error

terms are iid extreme value distributed, the probability can be expressed as

{ } ( , ) Pr( max ) ijk ij ik ijk rts P H h U U ∈ = = (7)

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I follow the modeling explained in Aaberge, Colombino and Strøm (1999) and get

exp( / ) ( ) ( , ) exp( / ) ( ) ijk ij ijk ij ik rst rs r s t V g H P H h V g H σ σ =

∑∑∑

Due to the assumption of extreme value distributed utilities it follows readily that the choice probabilities are multinominal logits. By setting (.)g =1 in (8) we get the standard

multinominal logit.

The analyst has incomplete knowledge or information about variables entering the choice set

B, and one way to take account of this incomplete knowledge is to specify probability

distributions for these variables. The (.)g function is a probability density that enters the choice probabilities due to job-specific offered hours available in the market. The

interpretation of the “opportunity density extended” version of the standard multinominal logit given in (8), is that the attractiveness of a choice measured by exp( / )V_i σ is weighted by a function saying how available this choice is in the market. For more details about this methodology I refer to Aaberge, Colombino and Strøm (1999).

Next we have

1 2

( _ij) exp( _{ij ij})

g H = ν K +ν L (9)

where (g H is the marginal probability density of offered hours. We will assume that offered _ij) hours are uniformly distributed except for full-time hours. This density is assumed to reflect that offered hours, except for full-time workloads, are equally available in the market. K =1 _ij

if the main job is a full-time job (35.5 hours per week or more), and K =0 otherwise. _ij L =1 if _ij

the main job is private, and L =0 otherwise. The latter captures the fact that if the main job is _ij

private, the hours available in the market will be less regulated (or not at all) relative to jobs in the public sector.

It should be noted that the offered wages depend on hours worked; that is W_ij =W H_ij( _ij). This expression also enters the deterministic part of the utility function through disposable income

ijk

C . The reason why I am able to identify V_ijk /σ is because I use detailed institutional

information to derive how offered wages W vary with hours worked. Given this institutional _ij

information, wage equations are estimated to capture how human capital characteristics and sector-specific constants affect expected wages.

The deterministic part of the preferences is represented by the following “Box-Cox” type utility function, 6 (10 ) 1 ((8760 ) / 8760) 1 ( ) ijk ij ik ijk C H h V X λ γ α β λ γ − _{− − − −} = + (10) where 0 ( )X _qX q_q, 1,..,7 β =β +β =

See for instance, Heckman and MaCurdy (1980), and Aaberge, Dagsvik and Strøm (1995) for empirical analyses applying this specification. An advantage of using this specification is that it is flexible enough to yield both negative (backward-bending labor supply curve) and

positive wage elasticities. 8,760 is the total number of annual hours, while α , λ , γ and the

s

β′ are unknown parameters. For the utility function to be quasi-concave, we require λ <1 and γ <1. Note that if λ→0 and γ → , the utility function converges to a log-linear 0 function. An alternative is to represent the utility function with a polynomial like van Soest (1995).

The characteristics are: X1= Age of the physician, X2= Number of children below six years of

age, X3=1, if the spouse is not working, =0 otherwise, X4=1, if the individual is from Norway;

=0 otherwise, X5=1, if female, =0 otherwise. X6=1, if the physician is a specialist in surgery,

internal medicine or laboratory medicine, =0 otherwise.

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