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Harberger versus Marshall
123
Economics Letters North-Holland
25 (1987) 123-126
HARBERGER Approximating
VERSUS MARSHALL General Equilibrium Welfare Changes
V. Kerry SMITH
*
North Carolinu State Unioersit.y, Rnleigh. NC 17695, USA
Received
30 July 1987
This paper illustrates how computable general equilibrium models can be used to evaluate the limits of partial equilibrium analysis. The example used for the analysis involves welfare measurement for economy-wide exogenous shocks. Comparison of Harberger’s general equilibrium approximation to consumer surplus and the Marshallian partial equilibrium measures indicates that there are cases where the latter offers a reasonably accurate measure of welfare change in a general equilibrium setting.
1. Introduction Benefit-cost analysis was originally intended to evaluate small public projects or policies with limited scope. It is not surprising therefore that the methods proposed for measuring welfare changes associated with these policies were designed for situations involving changes in a single commodity’s price or output for a subset of the households comprising the aggregate economy. ’ Subsequent analysis has developed in two traditions: (I) the U.S. school, continuing the partial equilibrium focus, with its efforts devoted primarily to estimating the values of non-marketed commodities [see Freeman (1985)], and (2) the British school, adopting a general equilibrium framework, but focusing on measuring the appropriate values for marketed commodities in distorted economies [see Dreze and Stern (1987)]. The performance of partial equilibrium welfare measures as approximations to large scale economy-wide changes has not occupied the attention of either tradition. * Yet, there is increasing evidence that these types of problems must be evaluated on a routine basis. In the United States, for example, the Reagan Administration has required a benefit-cost evaluation of all major regulations (i.e., those imposing in excess of $100 million in costs annually on the economy). Large scale national and international problems such as acid rain or climate change promise to stretch the credibility of the partial equilibrium assumptions even more. The purpose of this paper is to propose a new use for computable general equilibrium (CGE) models - as the equivalent of economic laboratories to evaluate the limits of partial equilibrium * Partial support for this research was provided by NSF grants ATM8217307 and ATM8317619. and by the Woods Hole Oceanographic Institute. ’ See ch. 2 of Krutilla and Eckstein (1958) for an overview of the conditions assumed to be required in a classic early description of the theoretical rationale for benefit-cost analysis. ’ This is not to suggest that the issue underlying general equilibrium welfare measures have not been discussed in the literature. See ch. 9 and Appendix D of Just, Hueth and Schmitz (1982) for a discussion of the issues involved.
0165-1765/87/$3.50
0 1987. Elsevier Science Publishers
B.V. (North-Holland)
V.K. Smith / Approximatrng
124
general equilibrium
welfare changes
methods - and to use this method to compare the Harberger (1971) approximate general equilibrium welfare measure with the Marshallian partial equilibrium consumer surplus for large scale, exogenous shocks to a small closed economy. The results of this analysis confirm and extend economic intuition for these types of problems. Exogenous shocks to an economy that have direct effects confined to a limited number of sectors and that lead to small or consistent indirect effects across sectors so that the relative prices for these commodities do not change (or change in the same direction and approximate magnitude), can be evaluated with reasonable accuracy with the Marshallian welfare measures. Indeed, the Harberger general equilibrium approximation is not uniformly superior to these measures. The comparative performance of these two types of approximations depends on both the nature of each household’s preferences and on the type of exogenous shock. Aggregation over households tends to ‘smooth’ errors observed at the individual level and does tend to support the Harberger method.
2. Measuring
welfare changes and the CGE model
This analysis is confined to two monetary measures of the welfare changes arising from the general equilibrium price changes resulting from exogenous changes to a simple economy. The first of these is the Harberger (1971) general equilibrium approximation of consumer surplus given in eq. (I): H” = t
P,‘AX,‘+
; 5 AP,AX,‘>
(1)
1=1
I=1
where AX = X’” _ X0” I I I )
AP,=P,‘-
Py,
X,‘, X,’ = the quantities of the ith final consumption good for the s th household before (0) and after (1) the price change, P,‘, P,” = the prices for the ith final consumption good in equilibrium before (0) and after (1) the exogenous shock. The second is the Marshallian consumer surplus based on a single sector partial equilibrium demand function with the assumption that demand can be described using a constant elasticity s denotes the household) as in eq. (2): 3 function (i.e. X,S= afP,‘; where the superscript (P;)‘+“:_
(p;)l+p:].
(2)
A constant elasticity demand function is not the correct demand function underlying a CES utility function. However, it is a reasonably good approximation [see Sato (1972)]. To implement the function we used the own price elasticity implied for the CES function for each household at the base case solution: e,’ = - 1 +(I - a,)(1 - k,‘,),
where
in consumption, = s th household’s elasticity of substitution k :o = budget share at initial (base case) prices for the ith commodity
0,
a^: is estimated so that the predictions from the constant for the ith commodity for the sth household.
elasticity
by the sth household.
demand
function
correspond
to the base solution
demand
V. K. Smith / Approxtmatmg
general
equilibrium
125
welfare changes
Since the objective of this analysis is to illustrate a methodological use for CGE models in evaluating partial equilibrium modeling strategies, our CGE model corresponds to the well-documented, two-sector model described in Shoven and Whalley (1984). This model includes two final goods and two inputs. CES functions are used to describe production and the two households’ utility functions. 4 Our parameterization corresponds to the one reported in their paper. Two types of exogenous shocks were used in the evaluation of these approximate welfare measures _ increases in the unit costs of production and proportionate reductions in the endowment of one input. The Hicksian expenditure function is used to define the correct welfare measure in each case. ’ The calculations for all measures assume the correct general equilibrium price and quantity changes with are known. Thus, the primary sources of error for the Harberger method are associated and ignores indirect approximation while that for the Marshallian measure includes approximations effects. 3. Results Table 1 reports the findings for five scenarios. The first three involve changes in sector two’s unit costs while the last two introduce proportionate reductions in the total capital stock. Across the top Table 1 A comparative Welfare
(b)
of approximate
measures
True Hicksian
(a)
evaluation
Individual Rich Poor
welfare
measures. Unit cost scenarios 5%
10%
0.916 2.103
1.852 4.338
3.019
Endowment 20%
scenarios
10%
50%
3.892 9.443
1.086 2.078
9.970 20.605
6.190
13.335
3.164
30.575
0.958 2.084
2.064 4.184
4.452 8.831
1.610 2.021
6.931 18.026
3.042
6.248
13.283
3.631
24.957
0.904 2.067
1.828 4.216
3.773 8.853
0.715 1.632
6.336 15.218
GE measure
household
Aggregate
Harherger
(a)
(b)
Individual Rich Poor Aggregate
Marshallran
(a)
household
consumer
surplus
Individual Rich Poor
household
(b)
Aggregate
(over households
2.971
6.044
12.626
2.347
21.554
(c)
Aggregate
(over goods and households)
2.971
5.994
12.451
3.111
26.422
In the solution of the model we replace the production functions with the cost-minimizing input requirements per unit of output for a range of relative prices corresponding to Shaven and Whalley’s (1984) reported solution and variations in equal increments above and below it. This approach was originally proposed by Scarf (1973) as one means to speed solutions. With a model this simple, it is not really necessary, but was used to facilitate computations on a PC. As Shaven and Whalley (1984) document for the CES specification, Hicksian consumer surplus measures can be expressed directly in terms of the values of utility functions realized in each solution. The expenditure function was used here because partial equilibrium Hicksian surplus measures were also calculated to gauge the importance of the specification error introduced by a constant elasticity of demand specification for consumer demand.
126
V.K. Smith / Approxmating
general equilibrium
welfarechanges
of the table we report the correct Hicksian welfare measures for Shoven and Whalley’s individual households, using their designation of rich and poor, as well as the aggregate of these measures. Comparing each row with the respective Harberger and Marshallian measures, the Marshallian single sector measure is superior to the Harberger index for all unit costs scenarios for the rich household, while the opposite ranking is generally maintained for the poor household. Moreover, the improvement in the Harberger performance in these cases is sufficient to make it superior for the aggregate measure (i.e., across households). However, in these scenarios, the differences between the two measures are not pronounced. Even at fairly large unit cost increases in the second sector (i.e., the one used for the Marshallian measure), both approximations perform quite well. This record is not maintained with the endowment based scenarios. Much larger errors are encountered, especially for the household whose income is affected by the capital loss (i.e., the rich household). Here with largest reduction, there is about a 30 percent error in the Harberger and a 36 percent error in the Marshallian measures. Aggregation smooths errors in this case, because the endowment change has less effect on the poor household. In general, the Harberger measure is superior in this case, but not superior in the aggregate to Marshallian measures aggregated over households and commodities.
4. Implications Experiments of this type illustrate how it is possible to evaluate partial equilibrium methods in a general equilibrium framework. This is especially important when analytical results to characterize the performance of these approximations are not available. While these conclusions are specific to the model used, the CGE framework offers a clear opportunity for a more complete evaluation of general equilibrium welfare measures. They suggest that partial equilibrium welfare measures can be used to approximate welfare changes for single sector changes, even if the price changes involved are fairly large, provided the indirect effects are small or they move in the same direction. When this is not the case, the Harberger measure that more explicitly accounts for the indirect effects has a clear advantage over the single sector Marshallian measure.
References Dreze, Jean and Nicholas Stem, 1987, The theory of cost benefit analysis, in: A. Auerbach and M. Feldstein. eds., Handbook of public economics, Vol. 2 (North-Holland, Amsterdam) forthcoming. Freeman, A. Myrick, III, 1985, Methods for assessing the benefits of environmental programs, in: A.V. Kneese and J.L. Sweeney, eds., Handbook of natural resource and energy economics, Vol. I (North-Holland, Amsterdam). Harberger, Arnold C., 1971, Three basic postulates for applied welfare economics: An interpretive essay, Journal of Economic Literature 5, Sept., 785-797. Just, Richard E., Darrell L. Hueth and Andrew Schmitz, 1982, Applied welfare economics and public policy (Prentice Hall, Englewood Cliffs, NJ). Krutilla, John V. and Otto Eckstein. 1958. Multiple purpose river development: Studies in applied economic analysis (Johns Hopkins University, Baltimore, MD). Sato, Kazuo, 1972, Additive utility functions with double log consumer demand functions, Journal of Political Economy 80, Jan./Feb., 103-124. Scarf, Herbert, 1973, The computation of economic equilibria (Yale University, New Haven, CT). Shaven, John B. and John Whalley, 1984, Applied general equilibrium models of taxation and international trade, Journal of Economic Literature 22, Sept., 1007-1051.
Economics Letters North-Holland
25 (1987) 123-126
HARBERGER Approximating
VERSUS MARSHALL General Equilibrium Welfare Changes
V. Kerry SMITH
*
North Carolinu State Unioersit.y, Rnleigh. NC 17695, USA
Received
30 July 1987
This paper illustrates how computable general equilibrium models can be used to evaluate the limits of partial equilibrium analysis. The example used for the analysis involves welfare measurement for economy-wide exogenous shocks. Comparison of Harberger’s general equilibrium approximation to consumer surplus and the Marshallian partial equilibrium measures indicates that there are cases where the latter offers a reasonably accurate measure of welfare change in a general equilibrium setting.
1. Introduction Benefit-cost analysis was originally intended to evaluate small public projects or policies with limited scope. It is not surprising therefore that the methods proposed for measuring welfare changes associated with these policies were designed for situations involving changes in a single commodity’s price or output for a subset of the households comprising the aggregate economy. ’ Subsequent analysis has developed in two traditions: (I) the U.S. school, continuing the partial equilibrium focus, with its efforts devoted primarily to estimating the values of non-marketed commodities [see Freeman (1985)], and (2) the British school, adopting a general equilibrium framework, but focusing on measuring the appropriate values for marketed commodities in distorted economies [see Dreze and Stern (1987)]. The performance of partial equilibrium welfare measures as approximations to large scale economy-wide changes has not occupied the attention of either tradition. * Yet, there is increasing evidence that these types of problems must be evaluated on a routine basis. In the United States, for example, the Reagan Administration has required a benefit-cost evaluation of all major regulations (i.e., those imposing in excess of $100 million in costs annually on the economy). Large scale national and international problems such as acid rain or climate change promise to stretch the credibility of the partial equilibrium assumptions even more. The purpose of this paper is to propose a new use for computable general equilibrium (CGE) models - as the equivalent of economic laboratories to evaluate the limits of partial equilibrium * Partial support for this research was provided by NSF grants ATM8217307 and ATM8317619. and by the Woods Hole Oceanographic Institute. ’ See ch. 2 of Krutilla and Eckstein (1958) for an overview of the conditions assumed to be required in a classic early description of the theoretical rationale for benefit-cost analysis. ’ This is not to suggest that the issue underlying general equilibrium welfare measures have not been discussed in the literature. See ch. 9 and Appendix D of Just, Hueth and Schmitz (1982) for a discussion of the issues involved.
0165-1765/87/$3.50
0 1987. Elsevier Science Publishers
B.V. (North-Holland)
V.K. Smith / Approximatrng
124
general equilibrium
welfare changes
methods - and to use this method to compare the Harberger (1971) approximate general equilibrium welfare measure with the Marshallian partial equilibrium consumer surplus for large scale, exogenous shocks to a small closed economy. The results of this analysis confirm and extend economic intuition for these types of problems. Exogenous shocks to an economy that have direct effects confined to a limited number of sectors and that lead to small or consistent indirect effects across sectors so that the relative prices for these commodities do not change (or change in the same direction and approximate magnitude), can be evaluated with reasonable accuracy with the Marshallian welfare measures. Indeed, the Harberger general equilibrium approximation is not uniformly superior to these measures. The comparative performance of these two types of approximations depends on both the nature of each household’s preferences and on the type of exogenous shock. Aggregation over households tends to ‘smooth’ errors observed at the individual level and does tend to support the Harberger method.
2. Measuring
welfare changes and the CGE model
This analysis is confined to two monetary measures of the welfare changes arising from the general equilibrium price changes resulting from exogenous changes to a simple economy. The first of these is the Harberger (1971) general equilibrium approximation of consumer surplus given in eq. (I): H” = t
P,‘AX,‘+
; 5 AP,AX,‘>
(1)
1=1
I=1
where AX = X’” _ X0” I I I )
AP,=P,‘-
Py,
X,‘, X,’ = the quantities of the ith final consumption good for the s th household before (0) and after (1) the price change, P,‘, P,” = the prices for the ith final consumption good in equilibrium before (0) and after (1) the exogenous shock. The second is the Marshallian consumer surplus based on a single sector partial equilibrium demand function with the assumption that demand can be described using a constant elasticity s denotes the household) as in eq. (2): 3 function (i.e. X,S= afP,‘; where the superscript (P;)‘+“:_
(p;)l+p:].
(2)
A constant elasticity demand function is not the correct demand function underlying a CES utility function. However, it is a reasonably good approximation [see Sato (1972)]. To implement the function we used the own price elasticity implied for the CES function for each household at the base case solution: e,’ = - 1 +(I - a,)(1 - k,‘,),
where
in consumption, = s th household’s elasticity of substitution k :o = budget share at initial (base case) prices for the ith commodity
0,
a^: is estimated so that the predictions from the constant for the ith commodity for the sth household.
elasticity
by the sth household.
demand
function
correspond
to the base solution
demand
V. K. Smith / Approxtmatmg
general
equilibrium
125
welfare changes
Since the objective of this analysis is to illustrate a methodological use for CGE models in evaluating partial equilibrium modeling strategies, our CGE model corresponds to the well-documented, two-sector model described in Shoven and Whalley (1984). This model includes two final goods and two inputs. CES functions are used to describe production and the two households’ utility functions. 4 Our parameterization corresponds to the one reported in their paper. Two types of exogenous shocks were used in the evaluation of these approximate welfare measures _ increases in the unit costs of production and proportionate reductions in the endowment of one input. The Hicksian expenditure function is used to define the correct welfare measure in each case. ’ The calculations for all measures assume the correct general equilibrium price and quantity changes with are known. Thus, the primary sources of error for the Harberger method are associated and ignores indirect approximation while that for the Marshallian measure includes approximations effects. 3. Results Table 1 reports the findings for five scenarios. The first three involve changes in sector two’s unit costs while the last two introduce proportionate reductions in the total capital stock. Across the top Table 1 A comparative Welfare
(b)
of approximate
measures
True Hicksian
(a)
evaluation
Individual Rich Poor
welfare
measures. Unit cost scenarios 5%
10%
0.916 2.103
1.852 4.338
3.019
Endowment 20%
scenarios
10%
50%
3.892 9.443
1.086 2.078
9.970 20.605
6.190
13.335
3.164
30.575
0.958 2.084
2.064 4.184
4.452 8.831
1.610 2.021
6.931 18.026
3.042
6.248
13.283
3.631
24.957
0.904 2.067
1.828 4.216
3.773 8.853
0.715 1.632
6.336 15.218
GE measure
household
Aggregate
Harherger
(a)
(b)
Individual Rich Poor Aggregate
Marshallran
(a)
household
consumer
surplus
Individual Rich Poor
household
(b)
Aggregate
(over households
2.971
6.044
12.626
2.347
21.554
(c)
Aggregate
(over goods and households)
2.971
5.994
12.451
3.111
26.422
In the solution of the model we replace the production functions with the cost-minimizing input requirements per unit of output for a range of relative prices corresponding to Shaven and Whalley’s (1984) reported solution and variations in equal increments above and below it. This approach was originally proposed by Scarf (1973) as one means to speed solutions. With a model this simple, it is not really necessary, but was used to facilitate computations on a PC. As Shaven and Whalley (1984) document for the CES specification, Hicksian consumer surplus measures can be expressed directly in terms of the values of utility functions realized in each solution. The expenditure function was used here because partial equilibrium Hicksian surplus measures were also calculated to gauge the importance of the specification error introduced by a constant elasticity of demand specification for consumer demand.
126
V.K. Smith / Approxmating
general equilibrium
welfarechanges
of the table we report the correct Hicksian welfare measures for Shoven and Whalley’s individual households, using their designation of rich and poor, as well as the aggregate of these measures. Comparing each row with the respective Harberger and Marshallian measures, the Marshallian single sector measure is superior to the Harberger index for all unit costs scenarios for the rich household, while the opposite ranking is generally maintained for the poor household. Moreover, the improvement in the Harberger performance in these cases is sufficient to make it superior for the aggregate measure (i.e., across households). However, in these scenarios, the differences between the two measures are not pronounced. Even at fairly large unit cost increases in the second sector (i.e., the one used for the Marshallian measure), both approximations perform quite well. This record is not maintained with the endowment based scenarios. Much larger errors are encountered, especially for the household whose income is affected by the capital loss (i.e., the rich household). Here with largest reduction, there is about a 30 percent error in the Harberger and a 36 percent error in the Marshallian measures. Aggregation smooths errors in this case, because the endowment change has less effect on the poor household. In general, the Harberger measure is superior in this case, but not superior in the aggregate to Marshallian measures aggregated over households and commodities.
4. Implications Experiments of this type illustrate how it is possible to evaluate partial equilibrium methods in a general equilibrium framework. This is especially important when analytical results to characterize the performance of these approximations are not available. While these conclusions are specific to the model used, the CGE framework offers a clear opportunity for a more complete evaluation of general equilibrium welfare measures. They suggest that partial equilibrium welfare measures can be used to approximate welfare changes for single sector changes, even if the price changes involved are fairly large, provided the indirect effects are small or they move in the same direction. When this is not the case, the Harberger measure that more explicitly accounts for the indirect effects has a clear advantage over the single sector Marshallian measure.
References Dreze, Jean and Nicholas Stem, 1987, The theory of cost benefit analysis, in: A. Auerbach and M. Feldstein. eds., Handbook of public economics, Vol. 2 (North-Holland, Amsterdam) forthcoming. Freeman, A. Myrick, III, 1985, Methods for assessing the benefits of environmental programs, in: A.V. Kneese and J.L. Sweeney, eds., Handbook of natural resource and energy economics, Vol. I (North-Holland, Amsterdam). Harberger, Arnold C., 1971, Three basic postulates for applied welfare economics: An interpretive essay, Journal of Economic Literature 5, Sept., 785-797. Just, Richard E., Darrell L. Hueth and Andrew Schmitz, 1982, Applied welfare economics and public policy (Prentice Hall, Englewood Cliffs, NJ). Krutilla, John V. and Otto Eckstein. 1958. Multiple purpose river development: Studies in applied economic analysis (Johns Hopkins University, Baltimore, MD). Sato, Kazuo, 1972, Additive utility functions with double log consumer demand functions, Journal of Political Economy 80, Jan./Feb., 103-124. Scarf, Herbert, 1973, The computation of economic equilibria (Yale University, New Haven, CT). Shaven, John B. and John Whalley, 1984, Applied general equilibrium models of taxation and international trade, Journal of Economic Literature 22, Sept., 1007-1051.