University of Turin Frisch Centre - Oslo Slutsky elasticities when the utility is random John K. Dagsvik Statistics Norway and the Frisch Centre, Oslo Marilena Locatelli University of Turin, CHILD, and the Frisch Centre, Oslo Steinar Str ø m University of Turin and Oslo, CHILD, and the Frisch Centre, Oslo

2 Novel features Labor supply and sectoral choice New method to evaluate labor supply elasticites when utility functions and opportunity sets are random to the econometricians Whe calculating the Slutsky elasticities we explicitly take into account that the random part of the utility function depends on the labor supply choice Choice probabilities are random due to random variables in the wage equations. The assumption of IIA is thus avoided. Dagsvik J. and S. Strøm (2006), Sectoral Labour Supply, Choice Restrictions and Functional Form, Journal of Applied Econometrics, 21

3 Indifference curves With a random utility function indifference curves have no meaning Instead we have indifference band and we can derive iso-probability curves Along these curves the probability of A being preferred to B is the same. The iso-probability curve corresponding to 0.5 (when there are two choices) implies that the individual is indifferent between A and B Quandt (1956), Dagsvik and Karlstrom (2006)

4 The model For expository reasons we start with explaining the one sector model and where the opportunity sets are deterministic Next we show how the choice probabilities are modified when the agents can choose to work either in the private or in the public sector. Agent faces a choice set of feasible jobs with job- specific (given) hours of work

5 Notation U= utility h=annual hours of work W= hourly wage rate I= vector of non-labor income( spouse income, capital income,child allowances) C= household disposable income z= indexes jobs and captures other attributes of the jobs than hours of work and wages

6 Notation contiuned B(h)=sets of available jobs with offered hours of work h m(h)=number of jobs with offered hours of work h in the choice set B(h)

7 More notation B(0)={0} m(0)=1 C=f(hW,I) f(.)= household disposable income function

8 Even more notation U=U(C,h,z)=v(C,h) ε(z) v(C,h) is a positive deterministic function ε(z) is a positive random taste shifter capturing unobserved individual characteristics and job attributes ε(z) is assumed to be i.i.d. across agents and jobs and with c.d.f.: P(ε(z)≤x)=exp(1/x), x>0 (Extreme value distribution) Ψ(h,W,I)=v(f(hW,I),h)

9 One sector choice probability We derive the probability that an agent will choose a job with hours of work h within the choice set B(h).

10 The probability that job z will be chosen

11 The probability of choosing any job within the choice set B(h) Weighted and mixed logit:

12 The two sector model: choice probabilities Density of offered hours (g) Availability of jobs (θ) Disposable income taking into account unobseved heterogeneity

13 Some interpretation The average of the choice probabilities over individuals can be interpreteded as the share in the populations working h hours in sector j Multiplying the choice probabilities with hours (measured in 7 catgories in each sector) and summing over hours we get expected labor supply in hours (conditional on working in sector j or in any sector or unconditional). Summing over individuals we get total expected labor supply in the population

14 The model: choice probabilities (cont.) The choice probabilities depend on the random variables in the wage equations {η 1, η 2 }. We assume that log η j, j=1,2, are normally distributed with zero expectations and variances  j. To represent the choice sets when the model is estimated or used in policy simulation, we have to draw from the distribution of  j, j=1,2. What we do is to draw 50x50=2500 from the distribution of {  1,  2} for each individual. By taking the average of  j over these 2500 draws we obtain the average choice probabilities, which can be interpreted as the choice probabilities where the unobserved heterogeneity in the choice sets is integrated out.

15 Empirical specification Utility function Co=1G, A=age, CUx=no of children less than and above 6

16 Wage equations Z1=experience,Z2=experience squared, Z3=education

17 Estimates of the wage equation Human capital variables like experience and education are priced marginally higher in the public sector compared to the private sector. The standard deviation of the error term in the public sector,  1, is estimated to be 0.243, whereas in the private sector the corresponding standard deviation  2 is estimated to be The wage level, as well as the dispersion in wages, is slightly higher in the private sector than in the public sector, whereas observed human capital is priced higher out on the margin in the public sector.

18 Estimates of the job opportunities We would expect that offered hours in the public sector are more concentrated at full-time hours than in the private sector. The unions are stronger with a much higher coverage in the public than in the private sector. We would also expect that there are more jobs available for the higher educated woman in the public sector than in the private sector. These expectations are confirmed by the estimates

19 Data description Data on the labor supply of married women in Norway, used in this study, consists of a merged sample from “ Survey of Income and Wealth, 1994 ”, Statistics Norway (1994) and “ Level of living conditions, 1995 ”, Statistics Norway (1995). Data covers married couples as well as cohabiting couples with common children. The age of the spouses ranges from 25 to 64. None of the spouses are self-employed and none of them are on disability or other type of benefits.

20 Table 1. Estimation results for the parameters of the labor supply probabilities obs Source: J.Dagsvik, and S. Strom, 2005, “Sectoral labor supply, choice restrictions and functional form”,”forthcoming in Journal of Applied Econometrics

21 Estimate of labour supply probailities and hours of work from previous table: We observe that: The deterministic part of the utility function is quasi-concave; The interaction term betwen consumption and leisure is negative and significantly different from zero which means that separability between consumption and leisure is rejected Marginal utilities with respect to consumption and leisure are positive. Marginal utility of leisure declines with age to around 32 years of age and thereafter it increases with age. The number of young and “ old ” children has a similar and positive effect on the marginal utility of leisure. Thus, when the woman is young and has children she has a reduced incentive to take part in work outside the home and when the children have grown up she gradually again gets a weakened incentive to participate in the labor market because of becoming older. The estimates of the opportunity densities imply: - for the higher educated women there are more jobs available in the public than in the private sector - the full-time peak is more distinct in the public than in the private sector

22 Table 2. Observed and predicted aggregates, married women, Norway 1994 Variables Not workingPublic sectorPrivate sector ObservedPredict ed Obser ved Predic ted Observ ed Predicted Choice probabilities Annual hours McFadden’s  2 McFadden’s adjusted ρ

23 Table 3. Choice probabilities and their variation with socio economic variables, married women, Norway Percent.

24 Table 4. Conditional expectations of annual hours and their variation with socio- economic variables, married women, Norway In both sector, vary little across age Drop sharply in both sectors when the households get 2 or more children Expected hours are also considerably lower in the upper deciles compared to the lower deciles

25 Compensated probabilities To calculate the Slutsky elasticities we first have to derive compensated probabilities The compensated probabilities are the probabilities of choice, given that the utility is constant and given that the utility is random These compensated probabilities can be used to derive iso-probabilistic curves. Along a curve the probability of choice is the same

26 Compensated elasticites cont. Let k: sector k, k=0 (not working), k=1 (public sector), k=2 (private sector) r: r=0 (initial state), r=1 (after a 1% increase in wage level for all individuals).

27 Continued Let be the wage level; k=1,2 and r=0,1. Note that when k=h=0, then For k=0, h1=0 For k=1, h={315, 780, 1040, 1560, 1976, 2340, 2600), seven categories, hi, i=2,3,,,8 For k=2, h={315, 780, 1040, 1560, 1976, 2340, 2600), seven categories, hi, i=9,3,,,15 Note that the agent can choose between sector and hours

28 Continued Let I be non-labor income, and Let

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33 Compensated probabilities Let be the probability that the agent chooses (k,h) after the wage increase, given that utility is the same as before the wage increase. Then for all h:

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35 The compensated change Absolute and relative change

36 Integrating out random effects Y in (19) are optimal determined decile limits in the distribution of disposable income

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38 Change in aggregate probabilities for 810 females 1st deciles (indexed 1) :

39 Change in aggregate probabilities for 810 females 2 nd to 9 th deciles, (indexed 2-1)

40 Change in aggregate probabilities for 810 females 10th deciles (indexed 3-2):

41 Aggregate Slutsky elasticities Elasticity of probability of working:

42 Aggregate Slutsky elasticities Elasticity of probability of working in sector j, j=1,2

43 Aggregate Slutsky elasticities Elasticity of unconditional expected hours :

44 Aggregate Slutsky elasticities Elasticity of conditional expected hours :

45 Elasticities of participation 1 st deciles2-9 th deciles10 th decilesAverage all Working Public sector Private sector Elasticities of unconditional expected hours 1 st deciles2-9 th deciles10 th decilesAverage all All sectors Public sector Private sector Elasticities of conditional expected hours 1 st deciles2-9 th deciles10 th decilesAverage all All sectors Public sector Private sector

46 The results The Slutsky elasticities are small, and smaller than traditional deterministic models have produced. Have the costs of taxation been overestimated before? The Slutsky elasticities are highest related to participation; overall and between sector (jobs)

47 U-shaped elasticities The Slutsky elasticities are lowest for the mid- deciles. In the lowest deciles there is a tendency to work longer hours in the private sector. This is what they can do given limited job-opportunities The women in the 10 th deciles move to the private sector- and work less Should tax-cut reforms focus on those with the lowest and highest household incomes? And tax the middle class harder?