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Estimating utility consistent labour supply functions
Economics Letters 9 (1982) 389-395 North-Holland Publishing Company
389
ESTIMATING UTILITY CONSISTENT LABOUR SUPPLY FUNCTIONS Some Results on Pooled Budget Data * Ranjan
RAY
University
of Manchester,
Manchester
MI3 9PL, UK
Received 4 January 1982
Demand systems AIDS, LES are augmented to include labour supply and then estimated on pooled U.K. budget data allowing for time series and cross section variation in wages. The results point to the danger of constraining the labour supply curve a-priori by use of restrictive utility systems. In addition, hypotheses relating to effects of price/wage movements on composition of ‘full income’ are tested.
1. Introduction The specification and estimation of utility consistent labour supply functions has recently been an area of considerable research. The literature on labour supply has, with very few exceptions, accepted uncritically the view that such curves bend backwards. One such exception is the interesting paper by Barzel and McDonald (1973) who argue convincingly about the danger of constraining the labour supply function a-priori by a restrictive specification of the utility function. Recent studies, notably Abbott and Ashenfelter (1976) and Phlips (1978), have ‘confirmed’ the backward bend of the labour supply curve on time series data. They do so through, for example, use of the Stone-Geary utility function which forces the labour supply curve a-priori * An earlier version of this paper containing the results in greater detail was presented at the 1981 European meeting of the Econometric Society in Amsterdam and is available upon request.
0165-1765/82/0000-0000/$02.75
0 1982 North-Holland
390
R. Ray / Estimuting
utility consistent labour supply functions
to bend backwards beyond a certain point. The use of separable utility systems leads to a considerable distortion of leisure/goods substitution in response to wage movement and contributes to the backward bend by causing the income effect to dominate the substitution effect at a very early stage. An additional reason for the nearly unanimous evidence on the backward bend could have been the general use of annual time series date which, as Killingsworth (1975) suggests, reflects permanent wage changes and exhibits a strong income effect. There has been, as far as I am aware, no empirical study on pooled cross-section data combining time-series variation in prices with timeseries/cross-section variation in wage rate and demographic variables in analysing jointly the leisure/goods allocation mechanism of the household. That is one of the principal features of the present study. This paper also compares alternative augmented functional forms estimated on the same data set. The plan of the paper is as follows. The leisure augmented functional forms for commodity demand systems AIDS and LES are presented in section 2. The data and estimation are briefly discussed in section 3, while the empirical results are presented and analysed in section 4. We end with the concluding note of section 5.
2. Augmented AIDS, augmented LES The leisure augmented cost function, c( u,p, w) shows the minimum ‘full income’ necessary to obtain a specific utility level u at a given price p and wage rate w. The AIDS cost function, due to Deaton and Muellbauer (1980), can be augmented to include leisure I and number of children z and also to take account of wage variation across households and over time (h denotes household),
n,l +
n.l
0th+ f 2 Ix Yk:1WPk l%P, k=l
j=l
7
(2)
R. Ray / Estimating
utility consistent labour supply functions
391
z,,)=logu,(P,w,,~z,)+P,~P,B~w,B~. lOE$h(P7Wh, The ALES commodity form, are given by W;h
=
a, $- 2
Yi,
and leisure demand
logp, + Yi/ log wh
+
equations,
(3)
in budget
share
PI ‘OgCxh/‘h >- eP,zh) i=
l,...,n,
(4)
xh = w,T + _yi is full income, T is maximum where yjj = ( yiT.+ y,3/2. working time, J$ is ‘unearned’ income and P is the leisure augmented AIDS price index. The yij, y,/s. are symmetric and this was enforced in the estimation. The earnings equation, implied by (5) allows for a more general labour supply behaviour than the restrictive forms implied by ALES and other separable systems and widely used in the literature. The ALES cost function is given by c(U,P,W)=~PkYk+“Y,+
( ~PfySW8U.
The limited variation in household size prompted us to ignore the dependence of c or z, while the subscript h has been omitted for clarity. The ALES commodity and leisure demand equations, in budget share form, are given by
w,=[l -P,(l
w,=(l
-S)ly,$
-s)Y,“-s&k$+s. X
Xfl, = 1 and x is, as before, full income. by (8) is quoted widely in the literature.
(8) The earnings
equation,
implied
392
R. Ray / Estimating
utility consistent labour supply functions
3. Data and estimation The data base is the U.K. Family Expenditure Surveys (1968-1978) and involved households whose head belonged to either of 3 professions - Professional & Technical, Clerical and Manual. Data on gross wages and hours of work, obtained from the Department of Employment Gazette, was used to construct series for the net wage, w, which varied between the 3 groups of workers and over the 11 cross sections. A 4-commodity breakdown was used, and the data consisted of 130 observations. The estimation was non-linear FIML and employed the package written by Wymer (1968). (Ye in AAIDS was fixed a priori for computational reasons.
4. Results Table 1 presents the ‘free’ parameter estimates. The y,, estimates imply that a real full income compensated rise in real wage has a positive impact on the full income share of each commodity, the largest being registered for Food. The results generally indicate a considerable amount of leisure/goods substitution in response to wage movements. The insignificance of the 8 estimates under AAIDS justifies the absence of demographic variables in augmented LES. The ALES parameter estimates seem sensible and all the subsistence quantities (vs.) are positive. The log likelihood values are presented in table 2. The first three rows, which form a nested sequence, suggest a decisive rejection of the hypotheses of zero y,,s. and zero yji, y,,s. (as a whole) but a non-rejection of zero y,,s. The ALES performs much worse on likelihood criterion than the most restrictive version of AAIDS. The uncompensated wage elasticities of all the items and labour supply are presented in table 3 and cross-tabulated to facilitate comparison, especially between AAIDS and ALES. The magnitude of the wage elasticities seem considerably higher than in previous studies on pure time series data. Of particular interest and importance are the figures relating to labour supply. A positively sloped curve under least restrictive AAIDS is replaced by a negatively sloped one under ALES. The results appear to confirm our earlier observation about the danger of constraining the labour supply curve u-priori through the use of additively separable utility systems as in ALES.
R. Ray / Estimating
393
utility consistent labour supply functions
I
I
I
I
394
R. Ray / Estimating
Table 2 Log likelihood
values.
utility consistent labour supply functions
Specification
Log likelihood
No. of free parameters
Symmetric AAIDS Symmetric AAIDS with y,, =0 AAIDS with all y s.=O ALES
2470.7 1 2454.81 2448.97 2421.79
19 I5 9 7
Table 3 Wage elasticities
(uncompensated).
Item
Augmented Symmetric
AIDS
Augmented LES Symmetric Y,, =0
Food
All y s.=o
3.613
2.80 1
2.847
2.159
6.010
6.106
4.515
4.332
Fuel and Light
2.383
2.183
2.323
1.552
Durable household goods
7.155
7.558
5.142
5.354
Labour
0.088
1.351
Clothing
and Footwear
supply
-0.335
- 0.338
5. Conclusion The principal interest of this paper has been the estimation of leisure/goods models on a time series of budget surveys allowing for time series and cross section variation of wages. The estimated labour supply function is not as restrictive as those generally used before. The most important of the empirical results is that the celebrated backward bend in labour supply observed in previous studies could not be reproduced within our least restricted framework. This gives reason to believe that such a backward bend could have been in part due to the restrictive utility forms generally employed and in part due to the time series data usually used. The results also point to a greater degree of demand response to wage movements than to price changes.
R. Ray / Estimating
utility consistent labour supply functions
395
References Abbott, M. and 0. Ashenfelter, 1976, Labour supply, commodity demand and the allocation of time, Review of Economic Studies 43, 389-412. Barzel, Y. and R.J. McDonald, 1973, Assets, subsistence and the supply curve of labour, American Economic Review 63, 621-633. Deaton, AS. and J. Muellbauer, 1980, An almost ideal demand system, American Economic Review 70, 312-326. Killingsworth, M.R., 1975, Neoclassical labor supply models: A survey of recent literature on determinants of the supply of time to the labor market, Unpublished paper (Fisk University, Nashville, TN). Phlips, L., 1978, The demand for leisure and money, Econometrica 46, 1025-1043. Wymer, C.R., 1968, Computer program ‘Resimul Manual’, Mimeo., LSE.
Data sources Annual Abstract of Statistics, 1979 (C.S.O., London). Department of Employment Gazette, various years (H.M.S.O., London). Family Expenditure Surveys, 196881978 (Department of Employment, H.M.S.O., London).
389
ESTIMATING UTILITY CONSISTENT LABOUR SUPPLY FUNCTIONS Some Results on Pooled Budget Data * Ranjan
RAY
University
of Manchester,
Manchester
MI3 9PL, UK
Received 4 January 1982
Demand systems AIDS, LES are augmented to include labour supply and then estimated on pooled U.K. budget data allowing for time series and cross section variation in wages. The results point to the danger of constraining the labour supply curve a-priori by use of restrictive utility systems. In addition, hypotheses relating to effects of price/wage movements on composition of ‘full income’ are tested.
1. Introduction The specification and estimation of utility consistent labour supply functions has recently been an area of considerable research. The literature on labour supply has, with very few exceptions, accepted uncritically the view that such curves bend backwards. One such exception is the interesting paper by Barzel and McDonald (1973) who argue convincingly about the danger of constraining the labour supply function a-priori by a restrictive specification of the utility function. Recent studies, notably Abbott and Ashenfelter (1976) and Phlips (1978), have ‘confirmed’ the backward bend of the labour supply curve on time series data. They do so through, for example, use of the Stone-Geary utility function which forces the labour supply curve a-priori * An earlier version of this paper containing the results in greater detail was presented at the 1981 European meeting of the Econometric Society in Amsterdam and is available upon request.
0165-1765/82/0000-0000/$02.75
0 1982 North-Holland
390
R. Ray / Estimuting
utility consistent labour supply functions
to bend backwards beyond a certain point. The use of separable utility systems leads to a considerable distortion of leisure/goods substitution in response to wage movement and contributes to the backward bend by causing the income effect to dominate the substitution effect at a very early stage. An additional reason for the nearly unanimous evidence on the backward bend could have been the general use of annual time series date which, as Killingsworth (1975) suggests, reflects permanent wage changes and exhibits a strong income effect. There has been, as far as I am aware, no empirical study on pooled cross-section data combining time-series variation in prices with timeseries/cross-section variation in wage rate and demographic variables in analysing jointly the leisure/goods allocation mechanism of the household. That is one of the principal features of the present study. This paper also compares alternative augmented functional forms estimated on the same data set. The plan of the paper is as follows. The leisure augmented functional forms for commodity demand systems AIDS and LES are presented in section 2. The data and estimation are briefly discussed in section 3, while the empirical results are presented and analysed in section 4. We end with the concluding note of section 5.
2. Augmented AIDS, augmented LES The leisure augmented cost function, c( u,p, w) shows the minimum ‘full income’ necessary to obtain a specific utility level u at a given price p and wage rate w. The AIDS cost function, due to Deaton and Muellbauer (1980), can be augmented to include leisure I and number of children z and also to take account of wage variation across households and over time (h denotes household),
n,l +
n.l
0th+ f 2 Ix Yk:1WPk l%P, k=l
j=l
7
(2)
R. Ray / Estimating
utility consistent labour supply functions
391
z,,)=logu,(P,w,,~z,)+P,~P,B~w,B~. lOE$h(P7Wh, The ALES commodity form, are given by W;h
=
a, $- 2
Yi,
and leisure demand
logp, + Yi/ log wh
+
equations,
(3)
in budget
share
PI ‘OgCxh/‘h >- eP,zh) i=
l,...,n,
(4)
xh = w,T + _yi is full income, T is maximum where yjj = ( yiT.+ y,3/2. working time, J$ is ‘unearned’ income and P is the leisure augmented AIDS price index. The yij, y,/s. are symmetric and this was enforced in the estimation. The earnings equation, implied by (5) allows for a more general labour supply behaviour than the restrictive forms implied by ALES and other separable systems and widely used in the literature. The ALES cost function is given by c(U,P,W)=~PkYk+“Y,+
( ~PfySW8U.
The limited variation in household size prompted us to ignore the dependence of c or z, while the subscript h has been omitted for clarity. The ALES commodity and leisure demand equations, in budget share form, are given by
w,=[l -P,(l
w,=(l
-S)ly,$
-s)Y,“-s&k$+s. X
Xfl, = 1 and x is, as before, full income. by (8) is quoted widely in the literature.
(8) The earnings
equation,
implied
392
R. Ray / Estimating
utility consistent labour supply functions
3. Data and estimation The data base is the U.K. Family Expenditure Surveys (1968-1978) and involved households whose head belonged to either of 3 professions - Professional & Technical, Clerical and Manual. Data on gross wages and hours of work, obtained from the Department of Employment Gazette, was used to construct series for the net wage, w, which varied between the 3 groups of workers and over the 11 cross sections. A 4-commodity breakdown was used, and the data consisted of 130 observations. The estimation was non-linear FIML and employed the package written by Wymer (1968). (Ye in AAIDS was fixed a priori for computational reasons.
4. Results Table 1 presents the ‘free’ parameter estimates. The y,, estimates imply that a real full income compensated rise in real wage has a positive impact on the full income share of each commodity, the largest being registered for Food. The results generally indicate a considerable amount of leisure/goods substitution in response to wage movements. The insignificance of the 8 estimates under AAIDS justifies the absence of demographic variables in augmented LES. The ALES parameter estimates seem sensible and all the subsistence quantities (vs.) are positive. The log likelihood values are presented in table 2. The first three rows, which form a nested sequence, suggest a decisive rejection of the hypotheses of zero y,,s. and zero yji, y,,s. (as a whole) but a non-rejection of zero y,,s. The ALES performs much worse on likelihood criterion than the most restrictive version of AAIDS. The uncompensated wage elasticities of all the items and labour supply are presented in table 3 and cross-tabulated to facilitate comparison, especially between AAIDS and ALES. The magnitude of the wage elasticities seem considerably higher than in previous studies on pure time series data. Of particular interest and importance are the figures relating to labour supply. A positively sloped curve under least restrictive AAIDS is replaced by a negatively sloped one under ALES. The results appear to confirm our earlier observation about the danger of constraining the labour supply curve u-priori through the use of additively separable utility systems as in ALES.
R. Ray / Estimating
393
utility consistent labour supply functions
I
I
I
I
394
R. Ray / Estimating
Table 2 Log likelihood
values.
utility consistent labour supply functions
Specification
Log likelihood
No. of free parameters
Symmetric AAIDS Symmetric AAIDS with y,, =0 AAIDS with all y s.=O ALES
2470.7 1 2454.81 2448.97 2421.79
19 I5 9 7
Table 3 Wage elasticities
(uncompensated).
Item
Augmented Symmetric
AIDS
Augmented LES Symmetric Y,, =0
Food
All y s.=o
3.613
2.80 1
2.847
2.159
6.010
6.106
4.515
4.332
Fuel and Light
2.383
2.183
2.323
1.552
Durable household goods
7.155
7.558
5.142
5.354
Labour
0.088
1.351
Clothing
and Footwear
supply
-0.335
- 0.338
5. Conclusion The principal interest of this paper has been the estimation of leisure/goods models on a time series of budget surveys allowing for time series and cross section variation of wages. The estimated labour supply function is not as restrictive as those generally used before. The most important of the empirical results is that the celebrated backward bend in labour supply observed in previous studies could not be reproduced within our least restricted framework. This gives reason to believe that such a backward bend could have been in part due to the restrictive utility forms generally employed and in part due to the time series data usually used. The results also point to a greater degree of demand response to wage movements than to price changes.
R. Ray / Estimating
utility consistent labour supply functions
395
References Abbott, M. and 0. Ashenfelter, 1976, Labour supply, commodity demand and the allocation of time, Review of Economic Studies 43, 389-412. Barzel, Y. and R.J. McDonald, 1973, Assets, subsistence and the supply curve of labour, American Economic Review 63, 621-633. Deaton, AS. and J. Muellbauer, 1980, An almost ideal demand system, American Economic Review 70, 312-326. Killingsworth, M.R., 1975, Neoclassical labor supply models: A survey of recent literature on determinants of the supply of time to the labor market, Unpublished paper (Fisk University, Nashville, TN). Phlips, L., 1978, The demand for leisure and money, Econometrica 46, 1025-1043. Wymer, C.R., 1968, Computer program ‘Resimul Manual’, Mimeo., LSE.
Data sources Annual Abstract of Statistics, 1979 (C.S.O., London). Department of Employment Gazette, various years (H.M.S.O., London). Family Expenditure Surveys, 196881978 (Department of Employment, H.M.S.O., London).