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Distributional effects of true economic indexes
Economics Letters North-Holland
185
16 (1984) 185-189
DISTRIBUTIONAL
EFFECTS
OF TRUE ECONOMIC
INDEXES
Tran VAN HOA University of Melbourne,
Received
30 January
Parkville,
Vict. 3052, Australia
1984
Generalized integrable Working system parameter estimates are used to construct the true differential indexes of (a) the cost of living, (b) the real income, and (c) the marginal prices to study distributional effects of inflation and real income in Australia over the period 197O.IV to 1983.11. The findings provide useful results for disaggregate household analysis and reveal the severe bias of aggregate indexes in sectoral studies of economic welfare.
1. Introduction Although the concept of a constant utility or indifference-defined index was proposed by Konus as early as 1924 to measure the true cost of living (COL), its use in practical situations has not been as widespread as the statistical indexes of Laypeyres and Paasche. This is unfortunate, since the statistical indexes, often necessary in debates on inflation, wage bargaining, tax indexation, current cost accounting, and social security assessment are explicitly void of economic theoretic underpinnings ’ and inherently characterized by over-estimation due to their lack of substitution effects between the various commodities that make up the group index [Theil (19731. Early studies of the COL. and other money metrics of economic welfare in the framework of a Stone-Geary utility function and additive preferences include the work by Van Hoa (1969a, b) for Australia and the United Kingdom. Phlips (1974) obtains a COL for Belgium based on the linear expenditure system. An analysis of the COL t Deaton and Muellbauer (1980) have, however, suggested that the statistical be regarded as first-order approximations to the true indexes.
0165-1765/84/$3.00
0 1984, Elsevier Science Publishers
B.V. (North-Holland)
indexes
may
186
T. Van Hoa / Distrrbutional effects of true indexes
based on the translog utility function was reported by Christensen and Manser (1975) for U.S. meat and produce. A two-way tabulation of the COL based on the PIGLOG cost function was carried out for the United Kingdom by Deaton and Muellbauer (1980). In those studies, the COL depends crucially on the parameters of a system of demand equations whose estimates are not as readily available as the expenditure shares of the various commodities in a household budget. Yet, those expenditure shares are all that are required to derive the statistical indexes. The differential or Divisia approach by Theil (1975,198O) alleviates to some extent the problems associated wiht the measurement of the COL above but its application to disaggregate consuming units within a system framework has hardly been reported. In this paper, we use systems estimates of the generalized Working model [Laitinen et al. (1983) and Van Hoa (1983)] to construct three aggregate indexes of economic welfare and their sub-indexes to study the distributional effects of inflation and income in Australia over the period 197061983. These indexes are (a) the COL, (b) the real income, and (c) the marginal prices. Sub-indexes are characterized by age and by the composition of adults and children in the families. In this context, the bias (if any) of the aggregate indexes and their disaggregate sub-indexes can be seen in two distinct dimensions.
2. The true indexes in differential form Given two price (k X 1) vectors Pi = [Pill, Pz = [ Pi21 for a basket of k commodities and the utility level U, the COL index between P2 and P, given U is generally defined as
where C( U, Pz) and C( U, P,) are the cost functions at the utility-price situations (U, P2) and (U, PI), respectively. At the utility levels U,, U, and the price vector P, the real income (R Y) index between U, and U, given P, is defined similarly as
where C( U,, P,) and C( U,, PI) are the cost functions at the situations (U,, PI) and (U,, P,), respectively. Finally, the marginal price (MP)
T. Van Hoa / Distributional effects of true indexes
187
index at P2 and P, given U is simply the ratio of the two marginal costs of utility at the two situations (U, P2) and (U, PI)or
~p(p,, p,/W = (ac(u,p,)/du)/(
ac(u, p&au).
(3)
The above indexes constitute three measurements of true economic welfare. (1) dates back to Konus (1939) and (2) is an indirect definition of the volume index via the minimum expenditure functions [Theil (1980)]. (3) was first formulated by Frisch (1932) to represent a change in additional expenditures to attain the same level of utility at two price situations. When the cost function is of the LES form, the two indexes (1) and (2) depend crucially on the marginal budget shares and the subsistence parameters. (3) is in this case dependent on the marginal budget shares but independent of the utility level. All three indexes require system parameter estimates of k demand equations. The fact that the calculation of (l)-(3) requires system parameter estimates makes the use of the true indexes unattractive in practical situations. This is particularly true when the generating utility or the cost function is of complex form. Theil (1.975,1980) has, however, demonstrated that (l)-(3) can be represented by indexes in differential form in which the necessary parameters are either expenditure shares [for (1) and (2)J or marginal budget shares [for (3)J. In differential form, the indexes (l)-(3) are written as
COL(P,,
P/U) =
5 yd(log
Pi),
(4)
i=l
RY(Ui, ul/‘f’~>= i Wd(logqi),
(5)
i=l
MP( Pz, Pi/U)
= i
B;d(log
Pi),
(6)
i=l
where, for the i th commodity, w denotes the budget share, 19, the marginal budget share, d(log Pi) = log P,, - log P,,, d(log q,) = log qiz log qil with a1 and q,* being the quantities consumed at two situations. When w = 0, for all i’s, (5) is independent of PI and (4) independent of U.
188
T. Van Hoa / Distributional
effects oj true indexes
3. Empirical evidence
We have computed (4)-(6) to measure various levels of economic welfare over the period 1970-1983 for Australian families disaggregated by age and by the number of children in the households. For convenience, these disaggregated indexes are called age and composition sub-indexes. There are four age groups in our analysis: 15-24, 25-44, 45-64 and over 65 years old. The child composition consists of childless households, households with one or two children, and households with more than two children. Due to a lack of disaggregate time series data, W, adn 6, for each group of households are estimated from the Australian Household Expenditure Survey 1975-76 using a generalized Working model [Laitinen et al. (1983) and Van Hoa (1983)] of eight expenditure equations. These equations represent: (i) household costs and maintenance, (ii) fuel and light, (iii) food, (iv) alcohol and tobacco, (v) clothing and footware, (vi) household equipment and operations, (vii) transport and communications, and (viii) miscellaneous goods and services. The time series data on the individual commodity prices and population-adjusted expenditures are quarterly observations and taken from the various bulletins of the Australian National Accounts and grouped as closely as possible to the survey or cross-section classification. The final results, which are a time series analysis based on cross-section coefficient estimates [see Theil and Finke (1983)], are given in table 1, where all index Table 1 Distributional
effects of inflation
and income
- Australia,
197O.W to 1983.11. a
Attributes
COL
RY
MP
No. households (x 1000)
15-24 years 25-44 years 45-64 years Over 65 years All age groups
127.77 128.31 129.74 132.14 123.74
10.43 7.24 4.31 2.05 6.83
125.20 130.86 130.78 134.50 125.46
344.56 1761.60 1348.67 702.70 4157.53
Childless households Housholds with Q 2 children Households with > 2 children
129.26
6.52
129.22
2118.68
128.59
6.51
130.94
1411.90
129.65
4.81
133.88
626.95
a All numbers for COL, RY and MP are log-changes ( x 100) with the base, period 197O.IV. b The corresponding change of the consumer price index is 125.47.
T. Van Hoa / Distributional effects of true indexes
189
figures represent discrete log-changes (X 100) from the base period of 197O.IV to 1983.11. From the table, we note the variations of the true indexes between aggregate and disaggregate households. More specifically, the aggregate COL is consistently lower than the age and composition sub-indexes. The largest underestimation is found in the elderly age group. The real income aggregate index, however, exceeds all composition and over-45 age sub-indexes but is exceeded by those in the under-45 age groups. The largest overestimation is found again in the elderly age group. Finally, all but one marginal price sub-indexes are greater than the aggregate index. The exception is in the 15-24 age group. The above evidence appears to indicate the bias of aggregate true indexes as measures of sectoral or disaggregate economic welfare. More important to policy evaluation purposes are the findings that all COL sub-indexes in our study reveal the underestimation of the true level of economic welfare as reflected by the general consumer price index (which increases by 125.47 percent in the period under observation) while the COL aggregate index indicates that the reverse is true. References Christensen, L.R. and M.E. Manser, 1975, Cost of living indexes and price indexes for U.S. meat and produce, 1947-1971, in: N.E. Terleckyj, ed., Household production and consumption (National Bureau of Economic Research, New York). Deaton, AS. and J. Muellbauer, 1980, Economics and consumer behaviour (Cambridge University Press, Cambridge, MA). Frisch, R., 1932, New methods of measuring marginal utility (Mohr, Tiibingen). Konus, A.A., 1939, The problem of the true index of the cost of living (originally published in Russian in 1924), Econometrica 7, 10-29. Laitinen, K., H. Theil and T. Raparla, 1983, A generalization of Working’s model, Economics Letters 13, no. 1, 97-100. Phlips, L., 1974, Applied consumption analysis (North-Holland, Amsterdam). Theil, H., 1975, Theory and measurement of consumer demand, Vol. 1 (North-Holland, Amsterdam). Theil, H., 1980, The system-wide approach to microeconomics (University of Chicago Press, Chicago, IL). Theil, H. and R. Finke, 1983, A time series analysis of a demand system based on cross-country coefficient estimates, Mimeo. Van Hoa, Tran, 1969a, Additive preferences and cost of living indexes: An empirical study of the Australian consumer’s welfare, Economic Record 45, 432-440. Van Hoa, Tran, 1969b, Consumer demand and welfare indicators: A dbmparative study for the United Kingdom and Australia, Economica 36, 409-425. Van Hoa, Tran, 1983, The integrability of generalized Working models, Economics Letters 13, no. 1, 101-104.
185
16 (1984) 185-189
DISTRIBUTIONAL
EFFECTS
OF TRUE ECONOMIC
INDEXES
Tran VAN HOA University of Melbourne,
Received
30 January
Parkville,
Vict. 3052, Australia
1984
Generalized integrable Working system parameter estimates are used to construct the true differential indexes of (a) the cost of living, (b) the real income, and (c) the marginal prices to study distributional effects of inflation and real income in Australia over the period 197O.IV to 1983.11. The findings provide useful results for disaggregate household analysis and reveal the severe bias of aggregate indexes in sectoral studies of economic welfare.
1. Introduction Although the concept of a constant utility or indifference-defined index was proposed by Konus as early as 1924 to measure the true cost of living (COL), its use in practical situations has not been as widespread as the statistical indexes of Laypeyres and Paasche. This is unfortunate, since the statistical indexes, often necessary in debates on inflation, wage bargaining, tax indexation, current cost accounting, and social security assessment are explicitly void of economic theoretic underpinnings ’ and inherently characterized by over-estimation due to their lack of substitution effects between the various commodities that make up the group index [Theil (19731. Early studies of the COL. and other money metrics of economic welfare in the framework of a Stone-Geary utility function and additive preferences include the work by Van Hoa (1969a, b) for Australia and the United Kingdom. Phlips (1974) obtains a COL for Belgium based on the linear expenditure system. An analysis of the COL t Deaton and Muellbauer (1980) have, however, suggested that the statistical be regarded as first-order approximations to the true indexes.
0165-1765/84/$3.00
0 1984, Elsevier Science Publishers
B.V. (North-Holland)
indexes
may
186
T. Van Hoa / Distrrbutional effects of true indexes
based on the translog utility function was reported by Christensen and Manser (1975) for U.S. meat and produce. A two-way tabulation of the COL based on the PIGLOG cost function was carried out for the United Kingdom by Deaton and Muellbauer (1980). In those studies, the COL depends crucially on the parameters of a system of demand equations whose estimates are not as readily available as the expenditure shares of the various commodities in a household budget. Yet, those expenditure shares are all that are required to derive the statistical indexes. The differential or Divisia approach by Theil (1975,198O) alleviates to some extent the problems associated wiht the measurement of the COL above but its application to disaggregate consuming units within a system framework has hardly been reported. In this paper, we use systems estimates of the generalized Working model [Laitinen et al. (1983) and Van Hoa (1983)] to construct three aggregate indexes of economic welfare and their sub-indexes to study the distributional effects of inflation and income in Australia over the period 197061983. These indexes are (a) the COL, (b) the real income, and (c) the marginal prices. Sub-indexes are characterized by age and by the composition of adults and children in the families. In this context, the bias (if any) of the aggregate indexes and their disaggregate sub-indexes can be seen in two distinct dimensions.
2. The true indexes in differential form Given two price (k X 1) vectors Pi = [Pill, Pz = [ Pi21 for a basket of k commodities and the utility level U, the COL index between P2 and P, given U is generally defined as
where C( U, Pz) and C( U, P,) are the cost functions at the utility-price situations (U, P2) and (U, PI), respectively. At the utility levels U,, U, and the price vector P, the real income (R Y) index between U, and U, given P, is defined similarly as
where C( U,, P,) and C( U,, PI) are the cost functions at the situations (U,, PI) and (U,, P,), respectively. Finally, the marginal price (MP)
T. Van Hoa / Distributional effects of true indexes
187
index at P2 and P, given U is simply the ratio of the two marginal costs of utility at the two situations (U, P2) and (U, PI)or
~p(p,, p,/W = (ac(u,p,)/du)/(
ac(u, p&au).
(3)
The above indexes constitute three measurements of true economic welfare. (1) dates back to Konus (1939) and (2) is an indirect definition of the volume index via the minimum expenditure functions [Theil (1980)]. (3) was first formulated by Frisch (1932) to represent a change in additional expenditures to attain the same level of utility at two price situations. When the cost function is of the LES form, the two indexes (1) and (2) depend crucially on the marginal budget shares and the subsistence parameters. (3) is in this case dependent on the marginal budget shares but independent of the utility level. All three indexes require system parameter estimates of k demand equations. The fact that the calculation of (l)-(3) requires system parameter estimates makes the use of the true indexes unattractive in practical situations. This is particularly true when the generating utility or the cost function is of complex form. Theil (1.975,1980) has, however, demonstrated that (l)-(3) can be represented by indexes in differential form in which the necessary parameters are either expenditure shares [for (1) and (2)J or marginal budget shares [for (3)J. In differential form, the indexes (l)-(3) are written as
COL(P,,
P/U) =
5 yd(log
Pi),
(4)
i=l
RY(Ui, ul/‘f’~>= i Wd(logqi),
(5)
i=l
MP( Pz, Pi/U)
= i
B;d(log
Pi),
(6)
i=l
where, for the i th commodity, w denotes the budget share, 19, the marginal budget share, d(log Pi) = log P,, - log P,,, d(log q,) = log qiz log qil with a1 and q,* being the quantities consumed at two situations. When w = 0, for all i’s, (5) is independent of PI and (4) independent of U.
188
T. Van Hoa / Distributional
effects oj true indexes
3. Empirical evidence
We have computed (4)-(6) to measure various levels of economic welfare over the period 1970-1983 for Australian families disaggregated by age and by the number of children in the households. For convenience, these disaggregated indexes are called age and composition sub-indexes. There are four age groups in our analysis: 15-24, 25-44, 45-64 and over 65 years old. The child composition consists of childless households, households with one or two children, and households with more than two children. Due to a lack of disaggregate time series data, W, adn 6, for each group of households are estimated from the Australian Household Expenditure Survey 1975-76 using a generalized Working model [Laitinen et al. (1983) and Van Hoa (1983)] of eight expenditure equations. These equations represent: (i) household costs and maintenance, (ii) fuel and light, (iii) food, (iv) alcohol and tobacco, (v) clothing and footware, (vi) household equipment and operations, (vii) transport and communications, and (viii) miscellaneous goods and services. The time series data on the individual commodity prices and population-adjusted expenditures are quarterly observations and taken from the various bulletins of the Australian National Accounts and grouped as closely as possible to the survey or cross-section classification. The final results, which are a time series analysis based on cross-section coefficient estimates [see Theil and Finke (1983)], are given in table 1, where all index Table 1 Distributional
effects of inflation
and income
- Australia,
197O.W to 1983.11. a
Attributes
COL
RY
MP
No. households (x 1000)
15-24 years 25-44 years 45-64 years Over 65 years All age groups
127.77 128.31 129.74 132.14 123.74
10.43 7.24 4.31 2.05 6.83
125.20 130.86 130.78 134.50 125.46
344.56 1761.60 1348.67 702.70 4157.53
Childless households Housholds with Q 2 children Households with > 2 children
129.26
6.52
129.22
2118.68
128.59
6.51
130.94
1411.90
129.65
4.81
133.88
626.95
a All numbers for COL, RY and MP are log-changes ( x 100) with the base, period 197O.IV. b The corresponding change of the consumer price index is 125.47.
T. Van Hoa / Distributional effects of true indexes
189
figures represent discrete log-changes (X 100) from the base period of 197O.IV to 1983.11. From the table, we note the variations of the true indexes between aggregate and disaggregate households. More specifically, the aggregate COL is consistently lower than the age and composition sub-indexes. The largest underestimation is found in the elderly age group. The real income aggregate index, however, exceeds all composition and over-45 age sub-indexes but is exceeded by those in the under-45 age groups. The largest overestimation is found again in the elderly age group. Finally, all but one marginal price sub-indexes are greater than the aggregate index. The exception is in the 15-24 age group. The above evidence appears to indicate the bias of aggregate true indexes as measures of sectoral or disaggregate economic welfare. More important to policy evaluation purposes are the findings that all COL sub-indexes in our study reveal the underestimation of the true level of economic welfare as reflected by the general consumer price index (which increases by 125.47 percent in the period under observation) while the COL aggregate index indicates that the reverse is true. References Christensen, L.R. and M.E. Manser, 1975, Cost of living indexes and price indexes for U.S. meat and produce, 1947-1971, in: N.E. Terleckyj, ed., Household production and consumption (National Bureau of Economic Research, New York). Deaton, AS. and J. Muellbauer, 1980, Economics and consumer behaviour (Cambridge University Press, Cambridge, MA). Frisch, R., 1932, New methods of measuring marginal utility (Mohr, Tiibingen). Konus, A.A., 1939, The problem of the true index of the cost of living (originally published in Russian in 1924), Econometrica 7, 10-29. Laitinen, K., H. Theil and T. Raparla, 1983, A generalization of Working’s model, Economics Letters 13, no. 1, 97-100. Phlips, L., 1974, Applied consumption analysis (North-Holland, Amsterdam). Theil, H., 1975, Theory and measurement of consumer demand, Vol. 1 (North-Holland, Amsterdam). Theil, H., 1980, The system-wide approach to microeconomics (University of Chicago Press, Chicago, IL). Theil, H. and R. Finke, 1983, A time series analysis of a demand system based on cross-country coefficient estimates, Mimeo. Van Hoa, Tran, 1969a, Additive preferences and cost of living indexes: An empirical study of the Australian consumer’s welfare, Economic Record 45, 432-440. Van Hoa, Tran, 1969b, Consumer demand and welfare indicators: A dbmparative study for the United Kingdom and Australia, Economica 36, 409-425. Van Hoa, Tran, 1983, The integrability of generalized Working models, Economics Letters 13, no. 1, 101-104.