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Declining Equivalence Scales and Cost of Children: Evidence and Implications for Inequality Measurement

Fabrizio Balli and Silvia Tiezzi


This article estimates expenditure-dependent equivalence scales for Italian couples with and without children. Following Donaldson and Pendakur (2006), the generalised absolute equivalence-scale exactness (GAESE) restrictions are incorporated into a translated quadratic almost ideal demand system. We obtain declining-with-expenditure equivalence scales, a pattern that tends to strengthen when the number of children increases. Thus, scale economies in current consumption are lower for families with poor expenditure capacities. We also show that families living in the South bear a substantial additional cost to achieve the same well-being of those living in the North. Finally, we find that ignoring the declining with expenditure pattern may involve a relevant understatement of measured inequality.

JEL classification: D12; D63

Appendix: parameter and elasticity estimates

The TQAI demand system proposed by Lewbel (2003) is derived from the following indirect utility function:


Each scalar entering [7] depends on a distinct group of parameters. Under GAESE, which involves and , these are defined as: (i) (the translog price index), (ii) , (iii) , and (iv) .

As shown by eqs [36], only and are needed to obtain GAESE equivalence scales. In Tables 8 and 9, the estimations of and are calculated at distinct values of the binary demographics and at the average value of continuous variables.

There are no theoretical indications about the sign of the translation terms . However, positive values have a clear-cut interpretation and correspond to subsistence consumption levels. As implied by eq. [7], the negative values actually obtained show the presence of consumption, and therefore positive utility, without a corresponding expenditure. This fact raises the question about the source of such utility. It could be an indication of inter-familiar informal transfers occurring between aged parents and their children’s families. The values of , i.e. the translog price indexes, are all positive, as prescribed by the theory, and significant. All values associated to full sample estimates lie outside the range defined by the corresponding estimates of the two subsamples.

Table 8:

Estimated (€ per month) computed at specific values of demographic dummies and at mean values of other demographics and p.

All demographic dummies are zero, except:RP 40RP 40Full sample
(Reference household)–226.470.18–456.989.97–361.856.84
c1 1–176.470.32–377.391.18–277.056.02
c2 1–190.165.99–328.494.95–248.859.38
c3 1–92.886.82–345.1106.83–179.670.53
e2 1–311.170.93–583.293.75–424.358.80
c1 1; e2 1–261.270.80–503.697.89–339.660.87
c2 1; e2 1–274.874.37–454.794.27–311.467.21
c3 1; e2 1–177.585.72–471.4115.79–242.171.68

Table 9:

Estimated computed at specific values of demographic dummies and at mean values of other demographics and p.

All demographic dummies are zero, except:RP 40RP 40Full sample
log aSElog aSElog aSE
(Reference household)6.920.0687.000.0577.370.079
c1 17.070.0897.100.0838.510.223
c2 17.170.0917.140.0798.550.240
c3 17.030.1197.420.1298.780.301
e2 16.950.0887.050.0736.670.113
c1 1; e2 17.100.1017.150.0897.810.226
c2 1; e2 17.210.1047.200.0927.850.230
c3 1; e2 17.060.1267.480.1508.080.284
nw 16.720.1017.000.0787.350.094
ne 16.770.0946.910.0847.520.107
si 16.930.0957.170.0887.570.109

Table 10 shows expenditure and compensated (Hicksian) own-price elasticities. For each commodity, expenditure and own-price elasticities are reported: for each model, elasticities are calculated at the average value of all variables and, for expenditure, also at the first and the third quartile.

Table 10:

Expenditure and compensated own-price elasticities at the sample mean of all variables; for expenditure, also at the 1st and 3rd quartile.

CommodityExpenditure levelExpenditure elasticitiesCompensated own-price elasticities
RP 40fullRP 40RP 40fullRP 40
(1) Food3rd q.0.640.620.63–0.31–0.38–0.34
1st q.0.670.660.65–0.39–0.49–0.51
(2) Alcohol and tobacco3rd q.0.490.450.31–0.25–0.30–0.14
1st q.0.750.760.71–0.71–0.92–1.21
(3) Clothing3rd q.1.421.221.08–0.85–0.48–0.40
1st q.1.541.321.20–0.47–0.63–0.63
(4) Housing (excluding rent)3rd q.0.520.530.78–1.59–1.42–1.40
1st q.0.480.500.57–1.66–1.68–1.63
(5) Household equipment3rd q.0.940.840.95–2.66–2.02–1.87
1st q.1.100.961.05–2.71–2.48–1.91
(6) Health3rd q.1.762.082.48–1.22–1.80–1.88
1st q.1.401.641.92–1.20–0.87–0.97
(7) Transport3rd q.0.951.151.20–0.59–0.72–0.76
1st q.–0.98–0.95–0.98
(8) Communications3rd q.0.530.550.52–0.59–0.66–0.62
1st q.0.360.400.34–0.68–0.67–0.74
(9) Recreation and culture3rd q.1.160.861.08–0.27–0.23–1.02
1st q.1.391.001.260.16–0.46–0.57
(10) Other goods and services3rd q.1.611.741.40–1.20–1.19–0.86
1st q.1.381.501.32–1.41–1.26–1.11

Note: Significant values – at 95% level – are in bold.

All expenditure elasticities are positive. Necessities include food, alcohol and tobacco, housing, and communications; luxuries include clothing, health, and recreation. Household equipment has an expenditure elasticity close to one. The same for transport, but only for families with younger RP; otherwise it is a luxury. Nearly all compensated own-price elasticities are negative. There is a positive but insignificant value for recreation, when younger household are considered and expenditures are set at the first quartile. Housing, household equipment and health are elastic. In several cases, full sample estimates lie in the range defined by the two corresponding subsample estimates. In other cases, this does not happen, as for expenditure elasticities of household equipments.

The rank of the model allows for complex patterns. For example, alcohol and tobacco become elastic, and health tends to be relatively inelastic, when older RP families with lower income are considered. Also, recreation and culture tend to be relatively elastic when older RP families with a higher expenditure are considered. In general, necessities tend to be more elastic when income is low.

Table 11:

Compensated cross-price elasticities calculated at the sample mean of all variables.

Elasticity rangeaRP 40 s.full s.RP 40 s.
EXY 1E510, E54, E93, E83, E210E54, E83, E510, E93E54, E83, E67Substitutability →
0.5 EXY 1E53, E45, E41, E39, E810, E71E210, E67, E53, E45, E61, E71, E41, E39, E27, E107, E101E93, E61, E71, E210, E76, E45, E94, E39, E41
0 EXY 0.5E67, E27, E105, E61, E87, E107, E710, E101, E57, E35, E38, E64, E110, E94, E102, E76, E17, E14, E49, E46, E108, E72, E75, E610, E78, E104, E410E710, E94, E76, E810, E49, E59, E105, E64, E110, E38, E89, E35, E17, E56, E82, E84, E87, E28, E102, E95, E57, E14, E46, E65, E72, E98, E16, E48, E108, E104, E75, E78, E410E27, E59, E107, E510, E710, E89, E84, E49, E38, E82, E95, E17, E28, E64, E810, E105, E101, E102, E98, E14, E16, E110, E72, E48, E410, E108, E104
–0.5 EXY 0E62, E12, E109, E19, E26, E18, E15, E74, E910, E37, E73, E68, E91, E47, E79E12, E109, E19, E18, E92, E58, E910, E96, E15, E85, E29, E68, E74, E79, E37, E73, E91, E21, E47E12, E32, E18, E15, E58, E68, E79, E85, E23, E52, E37, E69, E73, E74, E25, E96, E97, E47Complementarity →
–1 EXY 0.5E21, E97, E86, E52, E25E52, E97, E86, E25E21, E86, E51, E81
EXY −1E51, E81E81, E51

Notes: [a]

Table 11 shows compensated cross-price elasticities at the average values of all variables. The stronger substitution relationships detected are those between household equipment and housing, recreation and clothing, communication and clothing, housing and food, transport and food, health and transport, household equipment and clothing. On the other hand, the stronger complementarities are those between household equipment and food, communications and food, communications and health, recreation and transport, alcohol/tobacco and household equipment, alcohol/tobacco and food.


Acknowledgements: We thank Pierpaolo Pierani and Pier Luigi Rizzi for comments, suggestions and criticisms. Any errors are our own.


Balli, F., and S. Tiezzi. 2010. “Equivalence Scales, the Cost of Children and Household Consumption Patterns in Italy.” Review of Economics of the Household 8:527–49.10.1007/s11150-009-9068-3Search in Google Scholar

Banks, J., R. Blundell, and A. Lewbel. 1997. “Quadratic Engel Curves and Consumer Demand.” Review of Economics and Statistics 79:527–39.10.1162/003465397557015Search in Google Scholar

Blackorby, C., and D. Donaldson. 1993. “Adult Equivalence Scales and the Economic Implementation of Interpersonal Comparisons of Well-Being.” Social Choice and Welfare 10:335–61.10.1007/BF00182510Search in Google Scholar

Blackorby, C., and D. Donaldson. 1994. “Measuring the Cost of Children: A Theoretical Framework.” In The Measurement of Household Welfare, edited by R. Blundell, I. Preston, and I. Walker, 164–99. Cambridge: Cambridge University Press.10.1017/CBO9780511598968.002Search in Google Scholar

Donaldson, D., and K. Pendakur. 2004. “Equivalent-Expenditure Functions and Expenditure Dependent Equivalence Scales.” Journal of Public Economics 88:175–208.10.1016/S0047-2727(02)00134-2Search in Google Scholar

Donaldson, D., and K. Pendakur. 2006. “The Identification of Fixed Costs from Consumer Behaviour.” Journal of Business and Economic Statistics 24:255–65.10.1198/073500106000000035Search in Google Scholar

ISTAT. 1997–2004. “Indagine sui Consumi delle Famiglie: Manuale d’uso, anno 1997–2004”. ISTAT, Roma.Search in Google Scholar

ISTAT. 2011. “La povertà in Italia nel 2010.” Comunicato stampa, Roma, 15 luglio 2011.Search in Google Scholar

Koulovatianos, C., C. Schroeder, and U. Schmidt. 2005a. “On the Income Dependence of Equivalence Scales.” Journal of Public Economics 89:967–96.10.1016/j.jpubeco.2004.09.005Search in Google Scholar

Koulovatianos, C., C. Schroeder, and U. Schmidt. 2005b. “Properties of Equivalence Scales in Different Countries.” Journal of Economics 86:19–27.10.1007/s00712-005-0141-ySearch in Google Scholar

Lewbel, A. 1989. “Household Equivalence Scales and Welfare Comparisons.” Journal of Public Economics 39:377–91.10.1016/0047-2727(89)90035-2Search in Google Scholar

Lewbel, A. 1997. “Consumer Demand Systems and Households Equivalence Scales.” In Handbook of Applied Econometrics, Vol. 2, edited by M.H. Pesaran, and P. Schmidt. Oxford: Blackwell Publishers.Search in Google Scholar

Lewbel, A. 2003. “A Rational Rank Four Demand System.” Journal of Applied Econometrics 18:127–35. A corrected version is available at: in Google Scholar

Majumder, A., and M. Chakrabarty 2010. “Estimating Equivalence Scales through Engel Curve Analysis.” In Econophysics and Economics of Games, Social Choices and Quantitative Techniques, edited by B. Basu et al. Berlin: Springer.10.1007/978-88-470-1501-2_25Search in Google Scholar

Menon, M., and F. Perali 2010. “Econometric Identification of the Cost of Maintaining a Child (Chapter 10).” In Studies in Applied Welfare Analysis: Papers from the Third ECINEQ Meeting (Research on Economic Inequality, Vol. 18), edited by J.A. Bishop, 219–55. Emerald Ltd.10.1108/S1049-2585(2010)0000018013Search in Google Scholar

Ministero del Lavoro e delle Politiche Sociali (2011), Rapporto ISEE 2010, Roma.Search in Google Scholar

OECD. 2011. “What Are Equivalence Scales?” OECD-Directorate for Employment, Labour and Social Affairs, Paris. in Google Scholar

Perali, F. 2006. Il Costo di Mantenimento di un Bambino. In “Le Dimensioni della Povertà”, a cura di G. Rovati, Carocci Editore.Search in Google Scholar

Shonkweiler, J.S., and S. Yen 1999. “Two-Step Estimation of a Censored System of Equations.” American Journal of Agricultural Economics 81:972–82.10.2307/1244339Search in Google Scholar

  1. 1

    The index of equivalent economic situation (ISEE). In 2009 nearly 30% of Italian residents applied for social benefits awarded on the basis of the ISEE (Ministero del Lavoro e Politiche Sociali 2011).

  2. 2

    Donaldson and Pendakur (2006) handle a wider range of household types (i.e. couples with any number of children) using fewer parameters by imposing a specific functional structure.

  3. 3

    ISTAT delivers monthly price indexes for the provinces, a more disaggregated geographical level than the administrative region. To match with expenditure data, where only the household’s region is known, we take the prices of the province corresponding to the regional capital and consider them as representative of the whole region.

  4. 4

    The reference person (RP) is the holder of the file recorded in the household register handled by each Italian municipality.

  5. 5

    We set rather than estimate the intercept of the translog price index log a(p,z), which is difficult to identify, at the average log total expenditure of the reference household. All parameter estimates are available from the authors upon request.

  6. 6

    These authors computed different values for couples with a child in distinct age ranges (the childless couple is the reference). They obtained the following values: 1.19 for a child between 0 and 5 years old; 1.16 for a child between 6 and 13; 1.18 for a child between 14 and 18.

  7. 7

    Differences between values calculated by these authors at the top and the bottom vigintile of the annual current expenditure distribution are −0.10, −0.13 and −0.15 for households with zero, one and three children, respectively. They consider vigintiles from conditional distributions; in this work deciles are taken from the joint distribution.

  8. 8

    With respect to the older RP sample, in the younger RP subsample couples with three children are less represented and not evenly distributed over time and across geographical areas.

Published Online: 2013-08-06

©2013 by Walter de Gruyter Berlin / Boston

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