- Email: [email protected]
Aggregating goods and pollutants
JOURNAL
OF ENWRONMENTAL
ECONOMICS
AND MANAGEMENT
13 245-254 (1986)
Aggregating Goods and Pollutants* ROBERT E. KOHN Department
of Economics,
Box 102, Southern Illinois Universiv Edwarakville. Illinois 62026
at Edwardrville,
Received July 29,1981; revised March 19,1984 and May 23,1985 The mathematical assumptions underlying the aggregation of goods and of pollutants in a class of environmental models are given some possible economic interpretations. The procedures for aggregation are illustrated graphically and with numerical examples. This paper was motivated by the definitive article of Forster (J. Environ. Econ. Manage. 8, 118-133 (1981)) and is intended to complement that article. 0 1986 Academic Press, Inc.
1. INTRODUCTION
Economists often characterize the trade-off of goods for environmental quality with 2-dimensional models that feature a single aggregate good and a single pollutant. (See e.g., Laudadio [6], Oron and Pines [7], and Kohn [4, p. 1471.)Such a model is illustrated in Fig. 1. There is a production possibility frontier, F, which is concave to the pollution axis because total quantities of inputs are fixed, and as pollution is abated to zero, larger and larger quantities of these inputs must be diverted from production to abatement to achieve unit reductions in the pollution level. The indifference curves, U’ < U2 < U3, characterize the preferences of the community. They are concave to the goods axis because, as the pollution level increases, more and more goods are required to compensate for unit increases in pollution in maintaining the given levels of utility. In this model, the optimal combination of goods and pollution is denoted by the point of tangency between the curves, F and I/*. The assumptions that underlie the aggregation of goods and pollutants in these 2dimensional models are often overlooked or taken for granted. At least this was the case until Forster (11 published the definitive article on aggregation in environmental economics. This article has motivated economists such as myself to reexamine their own 2dimensional models and, in the process, discover all of the implicit assumptions that should have been recognized and stated. In this paper, the procedures for aggregating goods and pollutants are illustrated graphically and with numerical examples. Some economic interpretations of the underlying assumptions are also given. Although the analysis here is not as general as Forster’s, many of his results are of such importance as to merit an alternative rendition. Graphical analysis is limited by the fact that the simplest model underlying Fig. 1 is 4-dimensional, consisting of two goods and two pollutants. It is convenient to begin with the case of two goods and a single pollutant, and then move to the case *I am grateful for extensive comments from the anonymous referees. 245 0095~06%/86 $3.00 Copyright Q 1986 by Academic Press. Inc All rights of reproduction in any form reserved
246
ROBERT E. KOHN POLLUTION,
0
P
GOODS,G
FIG. 1. Production possibility frontier and community indifference curves.
of two pollutants and an aggregate good. The analysis is simplified by assuming that, as in Fig. 1 above (1) the production possibility frontier between any good and any pollutant is concave to the axis for the pollutant and (2) indifference curves between any good and any pollutant are concave to the axis for the good. In the case of any two goods or any two pollutants, more generality is allowed. It is assumed (3) that the production possibility frontier between any pair of goods is either linear or concave to the origin and for any pair of pollutants either linear or convex to the origin, and (4) that indifference curves between any pair of goods or pollutants are either linear or have curvature opposite to that in the respective production possibility frontier. (The special case of synergism examined in [3] in which indifference curves between pollutants bend in the same direction as the corresponding frontier is ignored in this paper.) 2. HICKS AGGREGATION OF GOODS POLLUTANT MODEL
IN A SINGLE
It was Hicks [2] who first noted that a set of goods may be aggregated and treated as a single good as long as their relative prices remain constant. As Forster [l, pp. 129-301 has remarked, the assumption of fixed relative prices is extreme in an environmental model in which different goods have different emission rates and different costs of abatement. In such a model the movement along the production possibility frontier to less goods and less pollution almost certainly alters relative prices. This interaction between relative prices and abatement is assumed away in the “pure abatement model” in Kohn [4, pp. 14-161. In that model, it is only the production of intermediate goods that causes pollution, and each of these intermediate goods is used in the production of final goods in some fixed proportion to the single and fully employed input. Unit production costs for each good are constant and therefore relative marginal costs of final goods do not change with shifts in consumption nor with changes in the levels of abatement. The “pure abatement model,” although it is artificial, is somewhat more sophisticated than the alternative assumption that ratios of emissions and abatement costs to pure production costs are the same for every final good. The production possibility frontier for a model such as the “pure abatement model,” in the case of two goods, x and y, and a rising marginal cost of abatement
AGGREGATING
GOODS AND POLLUTANTS
POLLUTANT I
m
GOOD
&OD
247
x
y
FIG. 2. Production possibility frontier.
of pollutant m, is illustrated in Fig. 2. The surface is concave to the pollutant axis but linear in the horizontal plane. Assuming perfect competition, the ratio of relative prices, p,/p,, is equal to the slope, dy/dx. For any level of pollution, say m,, all combinations of x0 and yO along the line tt (which is on the surface of the frontier and is m, units above the xy plane), such as (x’,, y& mo) and (xi, yz, m,), have the same market value, HO = p,x6 + pYy& The combination (Ha, m,), becomes a point on a 2dimensional production possibility frontier such as F in Fig. 1. An indifference surface for the case of two goods and one pollutant is illustrated in Fig. 3. It is convex to the pollutant axis and concave to the goods plane. For any level of pollution, say ml, there is a corresponding indifference curve such as ii. The indifference surface can be collapsed from three to two dimensions, as follows. For the prices, p, and pv, whose ratio equals the slope, &/dx, in Fig. 2, and the
POLLUTANT
/
GOOD
m
Y
FIG. 3. Indifference surface
248
ROBERT
E. KOHN
II z
II +
II E
II II Ii +t AS
,
2 + ? N + Y II
AGGREGATING
II II E=
II e
II v
II II AS
GOODS AND POLLUTANTS
II E
II E
,’ + E II 4
F 7 b
II 01
249
250
ROBERT
E. KOHN
pollutant level, m,, construct a horizontal plane at rnt and, on that plane, the budget constraint with slope dy/du equal to pJp,,, that is tangent to the intersection of the m, plane and the indifference surface. Call that point (LX,*,y;, ml). The value of the aggregate good, which is Hi = p,xf + p&, together with m,, define the point (H,, ml) on a 2dimensional indifference curve such as those in Fig. 1. The entire curve is derived by finding all combinations (Hi, m,) for the given indifference surface. For every 3dimensional surface such as that in Fig. 3, there is a 2-dimensional indifference curve based on a given pair of relative prices. An analogous example of Hicksian aggregation is illustrated by Model 1 in Table I. The optimal solution may be obtained by directly solving the three-variable model, or alternatively, by deriving the Hicksian aggregate, H = 8x + y, from the transformation function. (According to that function, dy/dx = - 8, and the competitive prices of goods x and y are 8 and 1, respectively.) In competitive equilibrium, the marginal rate of substitution of good y for good x, which is -y/x, equals the marginal rate of transformation which is -8 (this is the condition for a tangency such as that in Fig. 3) and therefore, y = 8x. It follows, by substitution, that x = H/16 and y = H/2. Substituting these values of x and y into the transformation and utility functions yields the collapsed functions in Table I. After solving the collapsed system for H* and m *, the aggregate may be disaggregated according to the above relationships. All that is required for Hicksian aggregation are fixed relative prices, not fixed absolute prices. If both prices change by the same factor, then the value of H changes by that same factor. As an argument in a utility function. H is meaningful only if the consumption of goods is measured in constant dollars.
3. PHYSICAL
AGGREGATION POLLUTANT
OF GOODS IN A SINGLE MODEL
There is a special case of that shown in Figs. 2 and 3 in which the marginal rate of substitution in consumption between the two goods is independent of the level of pollution. In that case, any vertical plane through the pollutant axis, such as OABC in Fig. 4, would intersect the indifference surface along a curve such as WI, which has the property that the slope, dy/dx, at any point on that curve is a constant. In this case there is separability in preferences between goods on the one hand and pollution on the other, and the aggregate good need not be defined in constant dollar units, but can be defined in physical units of its components. The collapsed indifference curve, uu, is conveniently viewed in the diagonal plane, OABC (in which case the aggregate good, G, equals (x2 + y*)l/*, or is projected onto the xm plane (in which case G equals x), or is projected onto the ym plane (in which case G equals y). The appropriate cutting plane depends on the optimal ratio, x/y, which is derived from the condition for equality of the marginal rates of substitution and transformation between the two goods. A like cutting plane in Fig. 2 would trace the corresponding 2dimensional production possibility frontier. There is separability in production and preferences in Model 1 in the table and therefore the goods in that system can be aggregated in physical units as well as monetary units. By using the relationships G = (x2 + y *)I’* and y = 8x, the quantities x and y are expressed in units of the physical composite good, G, for substitution into the transformation and utility functions.
AGGREGATING
GOODS AND POLLUTANTS POLLUTANT
251
m
CL I ’ \
GOOD
y
FIG. 4.
“A
Indifference surface.
In Fig. 2 the transformation frontiers between goods, such as tt, are linear, and in Fig. 3 the indifference curves, such as ii, are convex to the origin. If the transformation curves were concave to the origin and the indifference curves were straight lines, there could still be Hicksian aggregation (because outputs would adjust until the ratio of marginal costs equalled the fiied ratio of prices) as well as physical aggregation of the goods. This case is illustrated with Model 2 in Table I, using H = x + 2y for the Hicksian aggregate and, for a slightly different perspective, G = x for the physical aggregate. There is competitive market equilibrium when the marginal rate of substitution in consumption, dy/dx = - i, equals the marginal rate of transformation, - x/4y, in which case x = 2 y. In Models 1 and 2 in Table I there is separability in both the transformation and the utility functions. If there is linearity and separability in the transformation function there need not be separability in the utility function to have Hicksian aggregation. This is illustrated with Model 3 in Table I. The Hicksian aggregate, H = 12x + y, is derived from the transformation function. As before, the utility obtained from good x is independent of the pollution level. However, the utility from good y decreases at an increasing rate as pollution increases. The marginal rate of substitution of good y for good x depends upon m and hence (see Forster [l, p. 122]), goods and pollutant are not separable in the utility function. It is almost counterintuitive that the goods can be aggregated in such a case. They can because the composition of the aggregate is chosen along with m. In this model the condition for competitive market equilibrium, y = 12x + 8, includes a constant. In contrast, Forster [l, p. 1291 notes that there cannot be nonprice aggregation unless there is separability. In such a case, which is characterized by Model 3, the vertical cutting plane corresponding to OABC in Fig. 4 does not go through the origin and the slope, &/dx, along its intersection with the indifference curve, is not a constant. If, instead, there is concavity in the transformation relationship but not separability, as in Model 4, there cannot be Hicksian or physical aggregation, even though there is linearity and separability in the utility function. The transformation function in Model 4 characterizes the well-known case (see, e.g., Kohn [5]) in which
252
ROBERT E. KOHN
pollution generated by industry y reduces the productivity of industry x. The solution of Model 4 must be obtained directly from the 3-variable model. A final case in which there is separability but not linearity in either the transformation or the utility function will be considered later in the paper. 4. HICKSIAN
AGGREGATION
OF POLLUTANTS
The transformation curve between any pair of pollutants is convex to the origin if the pollution axes begin at zero. (This is an empirical observation reported in [3,4].) As long as the marginal rate of transformation between goods is independent of the marginal rate of transformation between pollutants (as in the “pure abatement” model [4]), the production possibility frontier can be constructed for two pollutants and an aggregate good, as in Fig. 5. This surface is convex to the aggregate good axis.
If it is assumed that consumers’ marginal rate of substitution between any pair of pollutants is inversely proportional to linear toxicity weights, as in Kohn [4, pp. 144-1491, and that these weights are independent of the combination of goods consumed, an indifference surface, concave to the aggregate good axis, can be characterized by that in Fig. 6. The,slope, dn/dm, of indifference lines, such as ii, is equal to the toxicity weight of pollutant m divided by the toxicity weight of pollutant n. This is equivalent to the case noted by Forster [l, p, 1301 in which Hicksian aggregation of pollutants is possible if the relative Pigouvian tax structure would be constant. Such a case is illustrated in Model 5. For generality, there are three goods in this model instead of two. The Hicksian aggregate of the two goods, H = x + 2y + 8z, is based upon the transformation function whereas the Hicksian aggregate of the pollutants, P = m + 2n, is derived from the utility function. From the condition for competitive market equilibrium, it follows that x = 2y = 82. From the condition for an efficient combination of pollutants, which is the equality of the marginal rate of transformation, dn/dm = -n’/2/(6m)‘/2, and the marginal rate of substitution in consumption, - i, it follows that 3m = 2n. The above
AGGREGATE GOOD.
/
POLLUTANT
FIG.
n
5. Production possibility frontier.
AGGREGATING
253
GOODS AND POLLUTANTS AGGREGATE GOOD, Gor H
/
POLLUTANT
n
FIG. 6. Indifference surface.
relationships permit x, y, t, n, and m to be expressed in units of the respective aggregates, for substitution into the transformation and utility functions. 5. PHYSICAL
AGGREGATION
OF POLLUTANTS
If the marginal rate of transformation between the two pollutants in Fig. 5 is independent of the quantity of goods produced, there will be some vertical cutting plane through the vertical axis, such as ORST in Fig. 6, whose intersection, II, with the indifference surface is a locus of points (m, n), the ratio of which satisfies the optimality condition for equality of the marginal rates of substitution and transformation between the pollutants. Along this curve, II, the marginal rate of substitution between pollutants will be constant. The indifference curve may be transcribed directly from the plane ORST (in which case the aggregate physical pollutant, Q, equals (m* + n*)l/*), or projected onto the mG plane (in which case Q is measured in units of m), or projected onto the nG plane (in which case Q is measured in units of n). In a second interpretation of Model 5, the two pollutants are physically aggregated by letting Q = n. An alternative case, in which the curvatures are reversed, is characterized by Model 6. In both Models 5 and 6 there can be either Hicksian or physical aggregation of the goods and of the pollutants. In Model 6, Hicksian aggregation of the pollutants is based on the transformation function whereas in Model 5 it is based on the utility function. 6. LINEARITY,
SEPARABILITY,
AND CONCLUDING
REMARKS
In all of the above examples, there is linearity in either the transformation or the utility function. Linearity is necessary if the aggregate good is measured in constant dollars or if the aggregate pollutant is a weighted summation. If both aggregates are measured in physical units, linearity is not required. This is illustrated by Model 7,
254
ROBERT E. KOHN
in which all of the transformation and indifference relationships are nonlinear. The crucial condition for aggregation, that is satisfied in Model 7, is separability. Although six of the seven models in Table I permit aggregation, it should be emphasized that these six examples are not at all realistic. The departure from reality is most flagrant in the transformation functions. In general, the technology of abatement differs from industry to industry, as does the effect of pollution on productivity and the potential of firms to avoid or mitigate the impact of pollution. It is very unlikely that the marginal rate of transformation between pollutants would be independent of the level of output of every good, or that the marginal rate of transformation between goods would be independent of every pollution level. This bears out Forster’s [l, p. 1321 pessimism regarding the “severe limitations for aggregation in environmental economics,” in which case the economy that is characterized by the one-good, one-pollutant model is’what Forster [l, p. 1181 calls a “mythical economy.” But to some extent this is true of most economic models, and it is not necessarily a criticism if such models, as Forster [l] notes, enable “the probanalyst.. . to discover some basic insights” into the “environmental-economic lem.” REFERENCES 1. B. A. Forstcr, Separability, functional structures and aggregation for a class of models in environmental economics, J. Environ. &on. Manage. 8, 118-133 (1981). 2. J. R. Hicks, “Value and Capital,” p. 33, Oxford Univ. Press, London (1946). 3. R. E. Kahn, Optimal air quality standards, Econometrica 39, 983-995 (1971). 4. R. E. Kahn, “A Linear Programming Model for Air Pollution Control,” MIT Press, Cambridge, Mass. (1978). 5. R. E. Kobn, A general equilibrium analysis of the optimal number of firms in a polluting industry, Canad. J. Econ. 18,347-354 (1985). 6. L. Laudadio, On the dynamics of air pollution: A correct interpretation, Canad. J. Econ. 4,563-571 (1971). 7. Y. Oron and D. Pines, The effect of efficient pricing of air pollution on intraurban land-use patterns, Environ. Plann. Ser. A 7, 293-299 (1975).
OF ENWRONMENTAL
ECONOMICS
AND MANAGEMENT
13 245-254 (1986)
Aggregating Goods and Pollutants* ROBERT E. KOHN Department
of Economics,
Box 102, Southern Illinois Universiv Edwarakville. Illinois 62026
at Edwardrville,
Received July 29,1981; revised March 19,1984 and May 23,1985 The mathematical assumptions underlying the aggregation of goods and of pollutants in a class of environmental models are given some possible economic interpretations. The procedures for aggregation are illustrated graphically and with numerical examples. This paper was motivated by the definitive article of Forster (J. Environ. Econ. Manage. 8, 118-133 (1981)) and is intended to complement that article. 0 1986 Academic Press, Inc.
1. INTRODUCTION
Economists often characterize the trade-off of goods for environmental quality with 2-dimensional models that feature a single aggregate good and a single pollutant. (See e.g., Laudadio [6], Oron and Pines [7], and Kohn [4, p. 1471.)Such a model is illustrated in Fig. 1. There is a production possibility frontier, F, which is concave to the pollution axis because total quantities of inputs are fixed, and as pollution is abated to zero, larger and larger quantities of these inputs must be diverted from production to abatement to achieve unit reductions in the pollution level. The indifference curves, U’ < U2 < U3, characterize the preferences of the community. They are concave to the goods axis because, as the pollution level increases, more and more goods are required to compensate for unit increases in pollution in maintaining the given levels of utility. In this model, the optimal combination of goods and pollution is denoted by the point of tangency between the curves, F and I/*. The assumptions that underlie the aggregation of goods and pollutants in these 2dimensional models are often overlooked or taken for granted. At least this was the case until Forster (11 published the definitive article on aggregation in environmental economics. This article has motivated economists such as myself to reexamine their own 2dimensional models and, in the process, discover all of the implicit assumptions that should have been recognized and stated. In this paper, the procedures for aggregating goods and pollutants are illustrated graphically and with numerical examples. Some economic interpretations of the underlying assumptions are also given. Although the analysis here is not as general as Forster’s, many of his results are of such importance as to merit an alternative rendition. Graphical analysis is limited by the fact that the simplest model underlying Fig. 1 is 4-dimensional, consisting of two goods and two pollutants. It is convenient to begin with the case of two goods and a single pollutant, and then move to the case *I am grateful for extensive comments from the anonymous referees. 245 0095~06%/86 $3.00 Copyright Q 1986 by Academic Press. Inc All rights of reproduction in any form reserved
246
ROBERT E. KOHN POLLUTION,
0
P
GOODS,G
FIG. 1. Production possibility frontier and community indifference curves.
of two pollutants and an aggregate good. The analysis is simplified by assuming that, as in Fig. 1 above (1) the production possibility frontier between any good and any pollutant is concave to the axis for the pollutant and (2) indifference curves between any good and any pollutant are concave to the axis for the good. In the case of any two goods or any two pollutants, more generality is allowed. It is assumed (3) that the production possibility frontier between any pair of goods is either linear or concave to the origin and for any pair of pollutants either linear or convex to the origin, and (4) that indifference curves between any pair of goods or pollutants are either linear or have curvature opposite to that in the respective production possibility frontier. (The special case of synergism examined in [3] in which indifference curves between pollutants bend in the same direction as the corresponding frontier is ignored in this paper.) 2. HICKS AGGREGATION OF GOODS POLLUTANT MODEL
IN A SINGLE
It was Hicks [2] who first noted that a set of goods may be aggregated and treated as a single good as long as their relative prices remain constant. As Forster [l, pp. 129-301 has remarked, the assumption of fixed relative prices is extreme in an environmental model in which different goods have different emission rates and different costs of abatement. In such a model the movement along the production possibility frontier to less goods and less pollution almost certainly alters relative prices. This interaction between relative prices and abatement is assumed away in the “pure abatement model” in Kohn [4, pp. 14-161. In that model, it is only the production of intermediate goods that causes pollution, and each of these intermediate goods is used in the production of final goods in some fixed proportion to the single and fully employed input. Unit production costs for each good are constant and therefore relative marginal costs of final goods do not change with shifts in consumption nor with changes in the levels of abatement. The “pure abatement model,” although it is artificial, is somewhat more sophisticated than the alternative assumption that ratios of emissions and abatement costs to pure production costs are the same for every final good. The production possibility frontier for a model such as the “pure abatement model,” in the case of two goods, x and y, and a rising marginal cost of abatement
AGGREGATING
GOODS AND POLLUTANTS
POLLUTANT I
m
GOOD
&OD
247
x
y
FIG. 2. Production possibility frontier.
of pollutant m, is illustrated in Fig. 2. The surface is concave to the pollutant axis but linear in the horizontal plane. Assuming perfect competition, the ratio of relative prices, p,/p,, is equal to the slope, dy/dx. For any level of pollution, say m,, all combinations of x0 and yO along the line tt (which is on the surface of the frontier and is m, units above the xy plane), such as (x’,, y& mo) and (xi, yz, m,), have the same market value, HO = p,x6 + pYy& The combination (Ha, m,), becomes a point on a 2dimensional production possibility frontier such as F in Fig. 1. An indifference surface for the case of two goods and one pollutant is illustrated in Fig. 3. It is convex to the pollutant axis and concave to the goods plane. For any level of pollution, say ml, there is a corresponding indifference curve such as ii. The indifference surface can be collapsed from three to two dimensions, as follows. For the prices, p, and pv, whose ratio equals the slope, &/dx, in Fig. 2, and the
POLLUTANT
/
GOOD
m
Y
FIG. 3. Indifference surface
248
ROBERT
E. KOHN
II z
II +
II E
II II Ii +t AS
,
2 + ? N + Y II
AGGREGATING
II II E=
II e
II v
II II AS
GOODS AND POLLUTANTS
II E
II E
,’ + E II 4
F 7 b
II 01
249
250
ROBERT
E. KOHN
pollutant level, m,, construct a horizontal plane at rnt and, on that plane, the budget constraint with slope dy/du equal to pJp,,, that is tangent to the intersection of the m, plane and the indifference surface. Call that point (LX,*,y;, ml). The value of the aggregate good, which is Hi = p,xf + p&, together with m,, define the point (H,, ml) on a 2dimensional indifference curve such as those in Fig. 1. The entire curve is derived by finding all combinations (Hi, m,) for the given indifference surface. For every 3dimensional surface such as that in Fig. 3, there is a 2-dimensional indifference curve based on a given pair of relative prices. An analogous example of Hicksian aggregation is illustrated by Model 1 in Table I. The optimal solution may be obtained by directly solving the three-variable model, or alternatively, by deriving the Hicksian aggregate, H = 8x + y, from the transformation function. (According to that function, dy/dx = - 8, and the competitive prices of goods x and y are 8 and 1, respectively.) In competitive equilibrium, the marginal rate of substitution of good y for good x, which is -y/x, equals the marginal rate of transformation which is -8 (this is the condition for a tangency such as that in Fig. 3) and therefore, y = 8x. It follows, by substitution, that x = H/16 and y = H/2. Substituting these values of x and y into the transformation and utility functions yields the collapsed functions in Table I. After solving the collapsed system for H* and m *, the aggregate may be disaggregated according to the above relationships. All that is required for Hicksian aggregation are fixed relative prices, not fixed absolute prices. If both prices change by the same factor, then the value of H changes by that same factor. As an argument in a utility function. H is meaningful only if the consumption of goods is measured in constant dollars.
3. PHYSICAL
AGGREGATION POLLUTANT
OF GOODS IN A SINGLE MODEL
There is a special case of that shown in Figs. 2 and 3 in which the marginal rate of substitution in consumption between the two goods is independent of the level of pollution. In that case, any vertical plane through the pollutant axis, such as OABC in Fig. 4, would intersect the indifference surface along a curve such as WI, which has the property that the slope, dy/dx, at any point on that curve is a constant. In this case there is separability in preferences between goods on the one hand and pollution on the other, and the aggregate good need not be defined in constant dollar units, but can be defined in physical units of its components. The collapsed indifference curve, uu, is conveniently viewed in the diagonal plane, OABC (in which case the aggregate good, G, equals (x2 + y*)l/*, or is projected onto the xm plane (in which case G equals x), or is projected onto the ym plane (in which case G equals y). The appropriate cutting plane depends on the optimal ratio, x/y, which is derived from the condition for equality of the marginal rates of substitution and transformation between the two goods. A like cutting plane in Fig. 2 would trace the corresponding 2dimensional production possibility frontier. There is separability in production and preferences in Model 1 in the table and therefore the goods in that system can be aggregated in physical units as well as monetary units. By using the relationships G = (x2 + y *)I’* and y = 8x, the quantities x and y are expressed in units of the physical composite good, G, for substitution into the transformation and utility functions.
AGGREGATING
GOODS AND POLLUTANTS POLLUTANT
251
m
CL I ’ \
GOOD
y
FIG. 4.
“A
Indifference surface.
In Fig. 2 the transformation frontiers between goods, such as tt, are linear, and in Fig. 3 the indifference curves, such as ii, are convex to the origin. If the transformation curves were concave to the origin and the indifference curves were straight lines, there could still be Hicksian aggregation (because outputs would adjust until the ratio of marginal costs equalled the fiied ratio of prices) as well as physical aggregation of the goods. This case is illustrated with Model 2 in Table I, using H = x + 2y for the Hicksian aggregate and, for a slightly different perspective, G = x for the physical aggregate. There is competitive market equilibrium when the marginal rate of substitution in consumption, dy/dx = - i, equals the marginal rate of transformation, - x/4y, in which case x = 2 y. In Models 1 and 2 in Table I there is separability in both the transformation and the utility functions. If there is linearity and separability in the transformation function there need not be separability in the utility function to have Hicksian aggregation. This is illustrated with Model 3 in Table I. The Hicksian aggregate, H = 12x + y, is derived from the transformation function. As before, the utility obtained from good x is independent of the pollution level. However, the utility from good y decreases at an increasing rate as pollution increases. The marginal rate of substitution of good y for good x depends upon m and hence (see Forster [l, p. 122]), goods and pollutant are not separable in the utility function. It is almost counterintuitive that the goods can be aggregated in such a case. They can because the composition of the aggregate is chosen along with m. In this model the condition for competitive market equilibrium, y = 12x + 8, includes a constant. In contrast, Forster [l, p. 1291 notes that there cannot be nonprice aggregation unless there is separability. In such a case, which is characterized by Model 3, the vertical cutting plane corresponding to OABC in Fig. 4 does not go through the origin and the slope, &/dx, along its intersection with the indifference curve, is not a constant. If, instead, there is concavity in the transformation relationship but not separability, as in Model 4, there cannot be Hicksian or physical aggregation, even though there is linearity and separability in the utility function. The transformation function in Model 4 characterizes the well-known case (see, e.g., Kohn [5]) in which
252
ROBERT E. KOHN
pollution generated by industry y reduces the productivity of industry x. The solution of Model 4 must be obtained directly from the 3-variable model. A final case in which there is separability but not linearity in either the transformation or the utility function will be considered later in the paper. 4. HICKSIAN
AGGREGATION
OF POLLUTANTS
The transformation curve between any pair of pollutants is convex to the origin if the pollution axes begin at zero. (This is an empirical observation reported in [3,4].) As long as the marginal rate of transformation between goods is independent of the marginal rate of transformation between pollutants (as in the “pure abatement” model [4]), the production possibility frontier can be constructed for two pollutants and an aggregate good, as in Fig. 5. This surface is convex to the aggregate good axis.
If it is assumed that consumers’ marginal rate of substitution between any pair of pollutants is inversely proportional to linear toxicity weights, as in Kohn [4, pp. 144-1491, and that these weights are independent of the combination of goods consumed, an indifference surface, concave to the aggregate good axis, can be characterized by that in Fig. 6. The,slope, dn/dm, of indifference lines, such as ii, is equal to the toxicity weight of pollutant m divided by the toxicity weight of pollutant n. This is equivalent to the case noted by Forster [l, p, 1301 in which Hicksian aggregation of pollutants is possible if the relative Pigouvian tax structure would be constant. Such a case is illustrated in Model 5. For generality, there are three goods in this model instead of two. The Hicksian aggregate of the two goods, H = x + 2y + 8z, is based upon the transformation function whereas the Hicksian aggregate of the pollutants, P = m + 2n, is derived from the utility function. From the condition for competitive market equilibrium, it follows that x = 2y = 82. From the condition for an efficient combination of pollutants, which is the equality of the marginal rate of transformation, dn/dm = -n’/2/(6m)‘/2, and the marginal rate of substitution in consumption, - i, it follows that 3m = 2n. The above
AGGREGATE GOOD.
/
POLLUTANT
FIG.
n
5. Production possibility frontier.
AGGREGATING
253
GOODS AND POLLUTANTS AGGREGATE GOOD, Gor H
/
POLLUTANT
n
FIG. 6. Indifference surface.
relationships permit x, y, t, n, and m to be expressed in units of the respective aggregates, for substitution into the transformation and utility functions. 5. PHYSICAL
AGGREGATION
OF POLLUTANTS
If the marginal rate of transformation between the two pollutants in Fig. 5 is independent of the quantity of goods produced, there will be some vertical cutting plane through the vertical axis, such as ORST in Fig. 6, whose intersection, II, with the indifference surface is a locus of points (m, n), the ratio of which satisfies the optimality condition for equality of the marginal rates of substitution and transformation between the pollutants. Along this curve, II, the marginal rate of substitution between pollutants will be constant. The indifference curve may be transcribed directly from the plane ORST (in which case the aggregate physical pollutant, Q, equals (m* + n*)l/*), or projected onto the mG plane (in which case Q is measured in units of m), or projected onto the nG plane (in which case Q is measured in units of n). In a second interpretation of Model 5, the two pollutants are physically aggregated by letting Q = n. An alternative case, in which the curvatures are reversed, is characterized by Model 6. In both Models 5 and 6 there can be either Hicksian or physical aggregation of the goods and of the pollutants. In Model 6, Hicksian aggregation of the pollutants is based on the transformation function whereas in Model 5 it is based on the utility function. 6. LINEARITY,
SEPARABILITY,
AND CONCLUDING
REMARKS
In all of the above examples, there is linearity in either the transformation or the utility function. Linearity is necessary if the aggregate good is measured in constant dollars or if the aggregate pollutant is a weighted summation. If both aggregates are measured in physical units, linearity is not required. This is illustrated by Model 7,
254
ROBERT E. KOHN
in which all of the transformation and indifference relationships are nonlinear. The crucial condition for aggregation, that is satisfied in Model 7, is separability. Although six of the seven models in Table I permit aggregation, it should be emphasized that these six examples are not at all realistic. The departure from reality is most flagrant in the transformation functions. In general, the technology of abatement differs from industry to industry, as does the effect of pollution on productivity and the potential of firms to avoid or mitigate the impact of pollution. It is very unlikely that the marginal rate of transformation between pollutants would be independent of the level of output of every good, or that the marginal rate of transformation between goods would be independent of every pollution level. This bears out Forster’s [l, p. 1321 pessimism regarding the “severe limitations for aggregation in environmental economics,” in which case the economy that is characterized by the one-good, one-pollutant model is’what Forster [l, p. 1181 calls a “mythical economy.” But to some extent this is true of most economic models, and it is not necessarily a criticism if such models, as Forster [l] notes, enable “the probanalyst.. . to discover some basic insights” into the “environmental-economic lem.” REFERENCES 1. B. A. Forstcr, Separability, functional structures and aggregation for a class of models in environmental economics, J. Environ. &on. Manage. 8, 118-133 (1981). 2. J. R. Hicks, “Value and Capital,” p. 33, Oxford Univ. Press, London (1946). 3. R. E. Kahn, Optimal air quality standards, Econometrica 39, 983-995 (1971). 4. R. E. Kahn, “A Linear Programming Model for Air Pollution Control,” MIT Press, Cambridge, Mass. (1978). 5. R. E. Kobn, A general equilibrium analysis of the optimal number of firms in a polluting industry, Canad. J. Econ. 18,347-354 (1985). 6. L. Laudadio, On the dynamics of air pollution: A correct interpretation, Canad. J. Econ. 4,563-571 (1971). 7. Y. Oron and D. Pines, The effect of efficient pricing of air pollution on intraurban land-use patterns, Environ. Plann. Ser. A 7, 293-299 (1975).