University of Turin

Frisch Centre - Oslo

Slutsky elasticities when the utility is Slutsky elasticities when the utility is randomrandom

2.3.2008 2.3.2008 John K. Dagsvik

Statistics Norway and the Frisch Centre, Oslo

Marilena LocatelliUniversity of Turin, CHILD, and the Frisch Centre, Oslo

Steinar StrømUniversity of Turin and Oslo, CHILD, and the Frisch Centre, Oslo

2

Novel featuresNovel featuresLabor 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 curvesIndifference 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 modelThe 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

NotationNotation

• 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 contiunedNotation 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)

h 0

m(h)

m(h)g(h)

thus

m(h) g(h)

7

More notationMore notation

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

8

Even more notationEven 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 One sector choice probabilityprobability

• 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 The probability that job z will be chosenbe chosen

( )

( )

, , , ,, , ( ) max max , , ( )

, , ( ) , ,

x k B x

x xk B x

h W I h W IP h W I z x W I k

x W I m x x W I

11

The probability of choosing The probability of choosing any job within the choice set any job within the choice set

B(h)B(h)• Weighted and mixed

logit:

( )( )

( )0

0,

; , , , , ( ) max max , , ( )

, , ( ) , ,

, , ; , 0,0, ( ) , ,

( ) , ,.

0,0, ( ) , ,

x k B xz B h

z B hx x

x x D

h W I P H h W I P h W I z x W I k

h W I m h h W I

m x W I x W I I m x x W I

g h h W I

I g x x W I

12

The two sector model: choice The two sector model: choice probabilitiesprobabilities

Density of offered hours (g)Availability of jobs (θ)

Disposable income taking into account unobseved heterogeneity

2

1 0,

0 2

1 0,

, , ( ); ,

0,0, , , ( )

for 0, 1,2 ,and

0,0,0; ,

0,0, , , ( )

for 0

j j j

j

k k kk x x D

k k kk x x D

h W I g hh W I

I x W I g x

h j

IW I

I x W I g x

h

13

Some interpretationSome 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 The model: choice probabilities (cont.)(cont.)

The choice probabilities depend on the random variables in the wage equations {η1, η2}. We assume that log ηj, 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.

j

15

Empirical specificationEmpirical specification

• Utility function• Co=1G, A=age, CUx=no of children less than

and above 6

31

31

400 2

2 4 5 6 7 81 3

4

0 09

1 3

1[10 ( )] 1log ( , ) log (log ) 6 717

[ ]10 1 1

L LC Cv C h A A CO C

C C L L

0 1 3640,L L h

16

Wage equationsWage equations

• Z1=experience,Z2=experience squared, Z3=education

*1 0 1 1 2 2 3 3 1exp log ji ji i j j i j i j i j jiW w Z Z Z

17

Estimates of the wage Estimates of the wage equationequation

• 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 0.274.

• 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 Estimates of the job opportunitiesopportunities

• 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 descriptionData 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 Table 1. Estimation results for the parameters of the labor supply probabilities - 810 obs- 810 obs

Variables Parameters Estimate t-values

Consumption:

Exponent 0.64 7.6

Scale 10-4 1.77 4.2

Subsistence level C0 in NOK per year 60 000Leisure:

Exponent -0.53 -2.1

Constant 111.66 3.2

Log age -63.61 -3.2

(log age)2 9.20 3.3

# children 0-6 1.27 4.0

# children 7-17 0.97 4.1

Consumption and Leisure, interaction -0.12 -2.7Subsistence level of leisure in hours per year

5120

Constant, public sector (sector 1) f11 -4.20 -4.7

Constant, private sector (sector 2) f21 1.14 1.0

Education, public sector (sector 1) f12 0.22 2.9

Education, private sector (sector 2) f22 -0.34 -3.3

Full-time peak, public sector (sector 1) 1.58 11.8

Full-time peak, private sector (sector 2) 1.06 7.4

Part-time peak, public Sector 0.68 4.4

Part-time peak, private Sector 0.80 5.2

Public sector 0.059 18.6

Private sector 0.075 17Log likelihood -1760.9

Uniformly distributed offered hours with part-time and fulltime peaks

Opportunity density of Offered hours, gk2(h), k=1,2

Heterogeneity in wages

Preferences:

The parameters b1 and b2; j j1 j2logb f f S

12 Full 12 0log g h g h

22 Full 22 0log g h g h

12 Part 12 0log g h g h

22 Part 22 0log g h g h

2122

11

2

3

4

5

6

7

8

9

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 Estimate of labour supply probailities and hours of work from previous table: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

VariablesNot working Public sector Private sector

Observed Predicted

Observed

Predicted

Observed

Predicted

Choice probabilities 0.080 0.079 0.492 0.483 0.428 0.438

Annual hours 0 0 1641 1585 1570 1632

McFadden’s 2

•McFadden’s adjusted ρ2

0.211•0.195

23

Table 3. Choice probabilities and their variation with socio economic Table 3. Choice probabilities and their variation with socio economic variables, married women, Norway 1994. Percent.variables, married women, Norway 1994. Percent.

Variable

Not working Public sector Private sector

Age range:

25-34 10.45 47.32 42.23

35-44 7.75 49.05 43.20

43-64 6.80 44.71 48.49

Number of children

0 4.89 46.02 49.09

1 6.18 48.88 44.94

2 10.09 46.76 43.15

more than 2 16.79 47.03 36.18

Woman education

low (≤9 years) 9.71 27.54 62.74

Intermediate (10-13years) 9.05 43.42 47.52

High (15-17 years) 4.42 73.27 22.31

Non-labor income

1 decile 2.89 45.40 51.71

2-9 deciles 8.04 47.03 44.92

10 decile 15.29 48.71 36.00

1994

24

Table 4. Conditional expectations of annual hours and their variation with Table 4. Conditional expectations of annual hours and their variation with socio-economic variables, married women, Norway 1994.socio-economic variables, married women, Norway 1994.

Public sector Private sector Variables

Age range:

25-34 1530 1576

35-44 1571 1631

43-64 1598 1608

Number of children:

0 1689 1694

1 1627 1662

2 1490 1530

More than 2 1310 1363

Education:

Low, 9 years 1535 1531

Intermediate, 10-14 years 1552 1604

High, 15-17 years 1607 1768

Non-labor income:

1decile 1690 1706

2-9 deciles 1567 1602

10 decile 1457 1502

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 probabilitiesCompensated 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 Compensated elasticites cont.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).

r rk k k(1) U (h) v (h, y) (h) ;r 0,1; k 0,1,2

27

ContinuedContinued

• 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

rkW

r0W 0

28

ContinuedContinued• Let I be non-labor income,

and• Let

0 1k k

0 1k k k, ,(2) v (h,W I) v (h,W y (h)) ; k 0,1,2

ky (h) be determined by

29

3

1

13

rk

3

4 r0kr

2k1

25 74 6 8

4 r0k

91 3

k

, y)(3) (h,W , y)

(1 h /3640) 1

, y 1

10 (C(h,W C ) 1log v

( logA (logA) C06 C7,17)

10 (C(h,W ) C ) (1 h /3640) 1

y (hy

)

I

30

rk 0k

r

r r rk k k

3rk ik i k k

i 1

,

( )

(4) C(h,W y) W h T(W h) y

(5) W R exp( Z log )

1for r 0(6) R

1.01for r 1

1k

k k

0 0 1 1k k k k k k k k k k

0 0k k k k k k k

Let ,

, I

that is

, W

I I

y (h, ) be the result of (2), then :

y (h, ) W ( )h T(W ( )h) W ( )h T(W ( )h)

y (h, I) 0.01W ( )h T( ( )h) T(W ( )h) I

I

(7)

31

00 1

0 1

1

1

1 31

j j 0 1 0 1 4j k j k j j k k0 0 1

1 2jk 21 2

k k

1k k

4 1o2 9 k k

3

y (h , )

y (h , )

, ,(9) ,

, ,

,

(h ) (h ) v (h , )dv (h , y)10Q (h ,h ; , y ) ;for( j,h ) (k,h )

K(I y, )

where

(10) dv (h , y)

(1 h / 3640) 110 (C(h ,W ( ), y) C )

Ig g

1 1

0 1i i

1 1k k

1 22

0 1i i i i i i i i

i 1 h 0

v ,

, ,

)

y

11

(h , )dy

I , ) max(v(0,0,I),v(0,0, y))

max( (h) v (h,W ( ),I), (h) v (h,W ( ), y)

( ) y

g g

K(

32

j j k k

j j k k jk

(12)If y (h, ) y (h, ) then ok

If y (h, ) y (h, ) then probability Q 0

0

1 0k k k k k k

1 2kk 0 01 2k k k

,y

,kg (h) v (h,W ( ), y (h, y ))

Q (h,h, , ,K(y , y (h , y ), , )

(13) )

k 1 2

0 1i i

20 1

i i i i i i i i k ii 1 h 0

, ,I , )

)

y max(v(0,0,I),v(0,0, y))

max( (h) v (h,W ( ),I), (h) v (h,W ( ), y (h, , I)

(14) K(

g g

33

Compensated probabilitiesCompensated 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:

*k (h)

34

1 2kk*

1 2 1 2k jkxj k

*0 1 2 1 2 00 1 2j0

xj 0

, ,I Q (h,h, , , I); for k 1,2(h, , ) Q (x,h, , I)

(15a) (0, , ,I) Q (x,0, , ,I) Q (0,0, , ,I); for k 0

(15)

35

The compensated changeThe compensated change

• Absolute and relative change

*1 2 1 2k k, ,I(16) (h, , ) (h, , I)

*1 2 1 2k k

1 2k

, I , I

, I

(h, , ) (h, , )(17)

(h, , )

36

Integrating out random Integrating out random effectseffects

• Y in (19) are optimal determined decile limits in the distribution of disposable income

1 2

** 0 1 2 0 1 20

0 1 2

, ,

,

(0, , I) (0, , I)1(18) (0,I, y) k(0,I, y)2500 (0, , I)

where

1 if C(0,I) y(19) k(0,I, y)

0 otherwise

%

37

1 2

*1 2 1 2j j*

j j1 2j

jj

(h, , , I) (h, , , I)1(20) (h,I, y) k(h, , I, y)2500 (h, , , I)

where

1if C(h, , I) y(21) k(h, , I, y)

0 otherwise

%

38

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

810

1n 1

1 1

*jn

1 I810(22) P (h, j, y ) ;for j 0,1,2;when j 0, then h 0

0.1

(h, ,y ) %

39

Change in aggregate Change in aggregate probabilities for 810 femalesprobabilities for 810 females

• 2nd to 9th deciles, (indexed 2-1)

810* *

2 1jn jnn 1

2 1 2 1,

1 (h,I, y ) (h,I, y )810(23) P (h, j, y y ) ;for j 0,1,2;when j 0, then h 0

0.8

% %

40

Change in aggregate Change in aggregate probabilities for 810 femalesprobabilities for 810 females

• 10th deciles (indexed 3-2): 2

810* *

3jn jnn 1

3 2 3 1

1

2

3

1 (h,I, ) (h,I, y )810(24) P (h, j, , y ) ;for j 0,1,2;when j 0, then h 0

0.1

whereNOK 254 000

NOK 382 200

NOK 900 000

yy

yyy

% %

41

Aggregate Slutsky Aggregate Slutsky elasticitieselasticities

• Elasticity of probability of working:

2

j

j 11 2

*

* *

(d)F( j,d)

(d) (d)(25) F(d) ;d 1,2 9,10

42

Aggregate Slutsky Aggregate Slutsky elasticitieselasticities

• Elasticity of probability of working in sector j, j=1,2

j

d d *x 0

*(x, j,d); j 1,2

(d)(26) ( j,d) P (x, j, y ) ;d 1,2 9,10F

43

Aggregate Slutsky Aggregate Slutsky elasticitieselasticities

• Elasticity of unconditional expected hours :

u2u

2j 1 x 0

j 1 x 0

All

F (d, j)F (d) x (x, j,d) ;d 1,2 9,10

x (x, j,d)(27)

d du x 0

x 0

xP (x, j, y ) (x, j,d)F ( j,d) ;d 1,2 9,10; j 1, 2

x (x, j,d)(28)

44

Aggregate Slutsky Aggregate Slutsky elasticitieselasticities

• Elasticity of conditional expected hours :

c u

All

F (d) F (d) F(d)(29) ;d 1,2 9,10

c u

For j 1,2

( ( ((30) F d, j) F d, j) F d, j) 1,2 9,10

45

Elasticities of participation

1st deciles 2-9th deciles 10th deciles Average all

Working 0.0910 0.0178 0.2840 0.0517

Public sector 0.0902 0.0156 0.2163 0.0431

Private sector 0.0916 0.0202 0.3914 0.0645

Elasticities of unconditional expected hours

1st deciles 2-9th deciles 10th deciles Average all

All sectors 0.0927 0.0137 0.2126 0.0415

Public sector 0.0785 0.0097 0.1424 0.0299

Private sector 0.1044 0.0177 0.32112 0.0567

Elasticities of conditional expected hours

1st deciles 2-9th deciles 10th deciles Average all

All sectors 0.0016 -0.0041 -0.0714 -0.0103

Public sector -0.0120 -0.0059 -0.0738 -0.0133

Private sector 0.0128 -0.0025 -0.0702 -0.0077

46

The resultsThe 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 elasticitiesU-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 10th 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?