From total factor to total resource productivity: an application to agriculture.

The U.S. farm sector has long been recognized as a productivity growth leader. Among the 45 industries studied by Jorgenson, Gollop, and Fraumeni (pp. 192-94), agriculture ranked sixth in terms of productivity growth over the 1947-79 period. In a study updated through 1985, Jorgenson and Gollop

(p. 749) find that the average annual rate of productivity growth in the farm sector was more than three times the average rate in the nonfarm economy. Even over the earlier 1929-66 period, Kendrick (pp. 251-52, 256) finds that agriculture experienced, on average, annual productivity growth more than 50% higher than rates achieved in the private nonfarm economy. Yet, agriculture was not immune to the productivity slowdown of the 1970s. According to Jorgenson, Gollop, and Fraumeni (p. 480), the industry registered negative productivity growth rates in 6 of the 10 years spanning from 1970 to 1979.

Rising energy prices were quickly identified as a likely suspect for the decline in productivity growth, but it was also popular to blame the contemporaneous rise in environmental regulation because major "pesticide cancellations" (withdrawals of chemicals from the market) affected U.S. farmers in the 1970s. Among the most memorable instances was the withdrawal of DDT in 1972. Interestingly, both Jorgenson, Gollop, and Fraumeni (p. 188) and Jorgenson and Gollop (p. 746) identify the productivity slowdown in agriculture beginning in the 1969-73 period.

This article takes the first step toward the ultimate end of empirically evaluating the productivity effects of environmental regulation in agriculture. Conventional productivity measures, even the state-of-the-art total factor productivity (TFP) index, consider only marketable inputs and outputs. As such, TFP measures capture the inputs and input costs associated with mandated abatement efforts but fail to count as output any resulting improvement in environmental quality, mainly improvements in water quality in the case of agriculture. Because the consumption of water resources involves true opportunity costs no less than does the consumption of labor, capital, or material inputs, TFP measures must be viewed as biased barometers of how well society is allocating its scarce resources. The case for broadening TFP to total resource productivity (TRP) is self-evident. The ultimate evaluation of environmental regulation requires it.

The primary objective of this article is to suggest a proper framework for TRP measurement. The proposed model, developed in the first two sections, derives formally from a model of welfare maximization but preserves the "producer-based" orientation of conventional productivity accounting. Application of the TRP model to the U.S. farm sector over the 197293 period is illustrated in the third section using preliminary estimates of marginal abatement cost derived from a data set currently under USDA development and estimates of the value of the marginal disutility of water pollution obtained from the existing literature. Preliminary indications are that increased groundwater pollution from pesticides in the 1972-79 period leads to annual TRP growth rates 0.06 percentage points below TFP growth rates (0.82%). In contrast, reductions in pollution over the 1979-93 period cause annual TRP growth to exceed conventional TFP growth (1.92) by about 0.05 percentage points per year.

The Production Sector

Consider an economy endowed with resources X and technology T. The economy produces conventional outputs, Y, and, as a production by-product, an undesirable output, S. Developing an index of the economy's aggregate output begins by selecting any arbitrary set of nonnegative quantities of outputs Y and S.(1) Given this product set, the economy's aggregate output can be defined as a proportion of all quantities of conventional outputs, Y. The maximum value of aggregate output ([Phi]) then can be expressed as a function of the (Y, S) product set; resources, X; and a time-based technology index, T:

(1) [Phi] = H(Y, S, X, T).

The mix of products in aggregate output is defined by the product vector Y. Their maximum feasible scale, given S, X, and T, is given by [Phi]. The function H is increasing in S, X, and T and decreasing in Y. Ceteris paribus, an increase in S frees resources to produce additional aggregate output; an increase in any [Y.sub.j] consumes resources and therefore reduces aggregate output.

The function H exhibits the usual homogeneity properties; H is homogeneous of degree minus one in Y because, holding S, X, and T constant, any proportional increase in all [Y.sub.j] definitionally generates an equal proportional decrease in [Phi]. In addition, H is assumed to be homogeneous of degree one in X and S. As a result, H is homogeneous of degree zero in Y, S, and X and exhibits constant returns to scale.

The task of the producing sector of the economy is to maximize production given the supplies of primary factors of production X, the sectoral production functions summarized in the technology variable T, existing regulations (if any) on S, and, stated in elasticity form, competitive market equilibrium conditions for inputs X and conventional outputs Y:

(2) [Delta] ln H / [Delta] ln [Y.sub.j] = -[q.sub.j][Y.sub.i]/[Sigma] [q.sub.j][Y.sub.j] (j = 1, . . ., m)

[Delta] ln H / [Delta] ln S = [Rho]S/[Sigma] [q.sub.j][Y.sub.j]

[Delta] ln H / [Delta] ln [X.sub.i] = [w.sub.i][X.sub.i]/[Sigma] [q.sub.i][Y.sub.j] (i = 1, . . ., n),

where [Sigma] [q.sub.j][Y.sub.j] equals the value of aggregate conventional output and p equals the marginal abatement cost of S. The definition of H and the competitive nature of the economy also guarantee that

(3) [Mathematical Expression Omitted].

The former is true by construction. The latter follows from the economic characterization of production. First, abatement of S requires resources X so that [Mathematical Expression Omitted] captures the total cost of producing Y and abating S. Second, given competitive markets, product prices, [q.sub.j], reflect the costs of production and abatement. It follows that [q.sub.j][Y.sub.j] = [summation over j] [w.sub.ij][X.sub.ij] as required by equation (3).

A graphical representation of the production sector is instructive [ILLUSTRATION FOR FIGURE 1 OMITTED]. Consider an economy producing a single conventional output, Y, and an undesirable output, S. The natural reference point or origin for this analysis is Y = S = 0. Because Y is a "good" and S a "bad," the second quadrant provides the appropriate context. Given X and T, the economy can operate efficiently anywhere along its production possibility frontier [G.sup.0] defined between the origin and point [a.sup.0]. Starting [G.sup.0] at the origin posits that (a) there is no costless (input-free) way to produce Y and (b) the production of Y is the only relevant source of the by-product S.(2) Production beyond (to the left of) [a.sup.0] is economically irrelevant. At point [a.sup.0], the economy dedicates all X to the production of Y and none to the abatement of S. It follows that production of conventional output Y and by-product S reach their maximums at [a.sup.0].

The frontier [G.sup.0] has the usual negative slope and smooth curvature, indicating that the marginal cost of producing Y and the marginal abatement cost of reducing S are both increasing in their respective arguments. Note that as the economy approaches [a.sup.0] along [G.sup.0], the marginal abatement cost of S approaches zero.

Increased resource endowments and technical change lead to shifts in the frontier. An increase in X leads to frontiers of the form [G.sup.1], which, like [G.sup.0], begins at the origin but reaches its maximum at a point [a.sup.1] to the left of [a.sup.0], implying that, without a change in technology, added production of Y with zero abatement necessarily implies additional S. In the event of technical change, the frontier again shifts in the northeasterly direction, but the frontier's zero abatement boundary depends on the nature of technical change. If technical improvements are embedded solely in the production of Y, the point of maximum possible Y and zero abatement will occur to the left of [a.sup.0], as is the case for frontier [G.sup.1]. If, however, the process of S abatement is the sole source of technical change, then frontier [G.sup.0] might shift to take the form represented by [G.sup.2], where production of maximum Y (unchanged from [G.sup.0]) with zero abatement leads to a lower level of S. Technical improvements reflecting efficiency gains in both the production of Y and the abatement of S lead to frontiers of the form [G.sup.3].

Productivity Growth

Consider a welfare function for a representative single-consumer economy in which utility is a function of conventional outputs and the undesirable byproduct; that is, U([Phi], S). Substituting H for [Phi] from equation (1), [Omega] is defined as the maximum utility that can be achieved from resources X and technology T:

(4) [Omega] = U[H(Y, S, X, T), S].

It is assumed that, holding S constant, [Delta]U/[Delta]H [greater than] 0 and that, holding H constant, [Delta]U/[Delta]S [less than] 0. The representative consumer maximizes welfare subject to the usual budget constraint

(5) [Mathematical Expression Omitted],

where M is money income. Solving this problem leads to the usual set of first-order conditions for conventional outputs and inputs:

(6) [Delta]U/[Delta]H [Delta]H/[Delta][Y.sub.i] + [Lambda][q.sub.j] = 0 (j = 1, 2, . . ., m)

[Delta]U/[Delta]H [Delta]H/[Delta][X.sub.i] - [Lambda][w.sub.i] = 0 (j = 1, 2, . . ., n),

where [Lambda] is the Lagrange multiplier representing the marginal utility of money income.

Though S has no market price, it has a shadow price [Sigma], which, following the form of conventional first-order conditions for market goods, is defined as follows:

(7) [Sigma] = - ([Delta]U/[Delta]H [Delta]H/[Delta]S + [Delta]U/[Delta]S)/[Lambda].

Because [Lambda] can be expressed equivalently by [Delta]U/[Delta]H [multiplied by] [Delta]H/[Delta]M

(8) [Mathematical Expression Omitted]

and, further, rewriting the first ratio in equation (8) in equivalent logarithmic form and recognizing that [Delta] ln H/[Delta] In M = 1,

(9) [Mathematical Expression Omitted].

Finally, given that [Delta] In H/[Delta] ln S = [Rho]S/M from equations (2) and (5)

(10) [Sigma] = -([Rho] - [Eta]),

where [Rho] ([greater than or equal to] 0) is the marginal abatement cost of S as defined above and [Eta] ([greater than or equal to] 0) is the absolute dollar value of the marginal disutility of S. In short, the shadow price of S is nonzero if and only if the marginal abatement cost of S does not equal the dollar value of disutility associated with S. The magnitude of [Sigma] determines the importance of resource reallocation in the face of market failure, just as the magnitude of the price/marginal cost spread determines the impact of standard market imperfections.

Welfare ([Omega]) in equation (4) is defined in terms of all four arguments: Y, S, X, and T. As such, there are well-defined marginal rates of substitution among the arguments. Consider the conventional definition of productivity growth: the weighted growth in outputs less the weighted growth in inputs. This definition effectively represents a marginal rate of substitution (MRS) equal to one between technology, T, on the one hand and a particular linear combination of outputs and inputs on the other hand. Because any MRS can be represented along a welfare surface holding welfare fixed, the formal specification of TRP can be derived by taking the differential of equation (4) with respect to time while holding welfare constant ([Delta][Omega]/[Delta]T = 0):

(11) 0 = [summation over j] [Delta]U/[Delta]H [Delta]H/[Delta][Y.sub.j] d[Y.sub.j]/dT + [summation over i] [Delta]U/[Delta]H [Delta]H/[Delta][X.sub.i] d[X.sub.i]/dT + ([Delta]U/[Delta]H [Delta]H/[Delta]S + [Delta]U/[Delta]S)dS/dT + [Delta]U/[Delta]H [Delta]H/[Delta]T.

Substituting equilibrium conditions from equations (6) and (7), dividing all terms by M and the marginal utility of income ([Lambda] or its equivalent [Delta]U/[Delta]H [multiplied by] [Delta]H/[Delta]M), and multiplying all terms by well-chosen "ones," yields

(12) [Mathematical Expression Omitted]

Because H is homogeneous of degree one in money income and substituting for [Sigma] from equation (10), equation (12) can be solved for the growth rate of TRP growth [E.sup.TRP]:

(13) [E.sup.TRP] [equivalent to] [Delta] ln H/[Delta]T = [summation over j] [q.sub.j][Y.sub.j]/M d ln [Y.sub.j] / dT - [summation over i] [w.sub.i][X.sub.i]/M d ln [X.sub.i] / dT - ([Eta] - [Rho])S/M d ln S / dT.

The last term in equation (13) addresses the effect of market failure on resource allocation and therefore TRP growth. If the marginal disutility of S exceeds marginal abatement cost, positive growth in S would reduce [E.sup.TRP]. In contrast, a reduction in S would make a positive contribution to [E.sup.TRP]. However, if [Rho] [greater than] [Eta], positive (negative) growth in S would increase (decrease) [E.sup.TRP].

An Application to Agriculture

The modern production techniques that have enabled the U.S. farm sector to enjoy high rates of productivity growth necessarily require the use of pesticides, herbicides, and fungicides. The quality of surface and groundwater sources are clearly affected by the application of these materials. Over time, application practices and chemical types and potency have been modified with the intention of mitigating harm to water quality through chemical runoff and leaching while preserving production levels of farm output. As motivated above, although consistently a productivity leader, the industry's TFP growth slowed at the very time attention began to focus on the sector's impact on water quality. The farm sector becomes a logical candidate for an application of the TRP model.

Applying the TRP formula described in equation (13) requires not only price and quantity data on both conventional animal and crop outputs and labor, capital, and material inputs but also quantity data on the industry's environmental impact (S) and estimates of both the sector's marginal abatement cost ([Rho]) and society's valuation of the marginal disutility of water pollution ([Eta]). The Environmental Indicators and Resource Accounting Branch of the Economic Research Service at USDA has for some time been engaged in projects to develop data that can support, among other research efforts, models such as TRP proposed in this paper. Space constraints prevent a full description of the data. The following overview will have to suffice.

The data are a panel of annual observations for individual states over the period 1972-93. The market-based data on conventional outputs and inputs are formed for each state as Tornquist indexes over detailed output and input accounts. Hundreds of disaggregated farm product categories, capital asset classes, and material goods go into the construction of the output, capital, and material input indexes, respectively. The labor index aggregates over 160 demographically cross-classified labor cohorts for each state. A full description of the underlying data series, sources, and indexing technique is presented in Ball.

The final measure of S will be a state- and year-specific measure of toxicity-adjusted chemicals from both pesticides and fertilizers reaching both surface and groundwater sources. The data will be constructed from county-specific analyses controlling for differing soil conditions, crop and therefore chemical types, application rates per acre, chemical toxicity, and rainfall patterns. That data series is under development. For the purposes of this paper, attention focuses only on pesticides and their effect on ground water. At present, USDA has developed state- and year-specific pesticide acre-treatment (frequency of application) data adjusted for (a) the leaching potential of different applied chemicals and (b) the leaching vulnerability of soil types measured by inches of water that percolate through various soils (for a full description, see Kellogg, Nehring, and Grube). These data are further adjusted with data made available by USDA: a time series of chemical pounds applied per acre treatment (U.S. average) and data on rainfall patterns and "doses" per pound applied. Only the latter requires elaboration. Barnard et al. have defined a chronic health risk "dose" as the quantity of chemical by weight that, if ingested daily over a specified time period, would involve serious health risk to humans.(3) Barnard et al. first compute a dose equivalent for each pesticide, then aggregate over pesticides and states within regions to generate estimates by region of the total change in toxicity and persistence of farm chemicals per pound applied. The resulting measure of S represents total pesticide "doses" generated each year in the farm sector.

Constructing the shadow price [Sigma] for S begins with an estimate of the marginal abatement cost in the farm sector of improving groundwater quality by one dose. Swinand estimates a translog cost function and input cost share equations using the preliminary panel data set described above. Marginal abatement cost ([Rho]) is estimated to equal $0.28. This estimate is adopted in the following illustration.

The shadow price also depends on [Eta], the marginal social value of a unit of clean (dosefree) water required per day for human consumption. Although considerable research has been done attempting to estimate this value, estimates still vary considerably, and consensus within the profession as to the value of these estimates is limited. Two recent survey articles (Boyle, Poe, and Bergstrom, Abdalla) discuss various contingent valuation and avoidance cost studies found in the literature and report a wide range in valuation estimates. From both studies, the estimates of an average household's willingness to pay for clean water range from $56 to $1,154 per year. Dividing by the average 2.7 persons per household and 365 days per year, these estimates convert to $0.06 and $1.17, respectively, per daily allowance of clean (dose-free) water. Limiting attention only to those avoidance cost studies that have been published in peer-reviewed journals, the mean estimate is $428 and converts to $0.43. These three values provide alternative estimates of [Eta] to be used in evaluating [E.sup.TRP].

Both traditional TFP and alternative TRP measures are reported in Table 1. The three TRP measures correspond to high ($1.17), low ($0.06), and mean ($0.43) values for the marginal valuation of a dose-free daily allowance of clean water. All three measures apply the marginal abatement cost measure of $0.28 per dose. Measures of TRP follow directly [TABULAR DATA FOR TABLE 1 OMITTED] from equation (13). The TFP measures ignore the nonmarket by-product, S, and are derived from the conventional TFP growth formula:

(14) [E.sup.TFP] [equivalent to] [summation over j = 1]] [q.sub.j][Y.sub.j]/M d ln [Y.sub.j]/dT - [summation over i] [w.sub.i][X.sub.i]/M d ln [X.sub.i]/dT.

The relationship between corresponding measures of TFP and TRP growth in Table 1 depends, as can be seen from equation (13), on (a) the sign and magnitude of the difference between [Eta] and [Rho]; (b) the sign and magnitude of the growth rate of doses generated, dlnS/dT; and (c) the relative dollar importance of the pollution externality to the market value of agricultural goods, [Sigma]S/M. When [Eta] takes its high or mean value, [Eta] [greater than] [Rho] and reductions (increases) in S lead to an increase (reduction) in productivity growth. At its low value, [Eta] [less than] [Rho], implying (assuming rising marginal abatement cost) that pollution abatement has proceeded to a level where marginal abatement cost exceeds the marginal benefit of pollution reduction.(4) In this case, an increase (decrease) in pollution makes a positive (negative) contribution to productivity growth. Finally, ceteris paribus, the spread between [E.sup.TFP] and [E.sup.TRP] is an increasing function of the weight [Sigma]S/M in equation (13).

The pattern of pollution growth exhibited over the four subperiods examined in the table permits the illustration of the above properties of the TRP index. Water pollution in the 1972-79 period, for example, increased at an annual 4.28% rate, thereby causing TRP growth to fall below conventional TFP growth when [Eta] [greater than] [Rho] and to rise above it when [Eta] takes its low $0.06 value relative to [Rho] = $0.28. The sizable declines in water pollution recorded in each of the three following subperiods (-12.6%, -14.8%, and -29.5% annual rates, respectively) generate precisely the opposite pattern. The growth of TRP exceeds that of TFP when the marginal benefit of abatement exceeds marginal abatement cost, but in the case in which abatement is "excessive" ([Eta] [less than] [Rho]), the reduction in S causes TRP growth to fall below TFP growth.

The magnitudes of the TFP/TRP differences are informative as well. Clearly, the greatest spread occurs when [Sigma] is valued at $0.89 ([Sigma] = [[Eta] - [Rho]] = [$1.17 - $0.28]). The resulting difference between TFP and TRP growth rates equal 0.34, 0.37, 0.23, and 0.19 percentage points per year, respectively, in the 1972-79 through the 1989-93 subperiods. The switch from a positive 4.28% to a negative 12.63% growth rate in water pollution between 1972-79 and 1979-85 further magnifies the differential. Whereas annual TFP growth rates between the two subperiods increased by 1.92 percentage points, the growth rate in TRP (at [Eta] = $1.17) increased by 2.63 percentage points.

The importance of the weight [Sigma]S/M on the growth rate dlnS/dT in equation (13) also can be illustrated in Table 1. For the 1972-79 and 1989-93 subperiods, compare the absolute values of the differences between TFP growth and each measure of TRP growth. Given that the annual rate of growth in water pollution in the latter period (-29.52%) is, in absolute value, seven times its growth rate in the earlier period, one might expect the resulting spread between TFP and TRP growth to be higher in the later period. The opposite turns out to be the case. The difference between TFP growth and each of the three measures of TRP growth in the 1972-79 subperiod is roughly twice each corresponding difference in the 1989-93 subperiod. The reason is that over the full 21 years of the study, water pollution declined at an average 10.6% annual rate. Compounded, this implies that pesticide-related doses (S) reaching groundwater in 1993 equaled only about 10% of doses leached in 1972. Over the same period, the nominal dollar value of agricultural production (M) increased by nearly 135%. As a result, the sevenfold higher growth rate in S in the 1989-93 period has a weight that is only 0.1 of its 1972-79 level.

Comparing the mean-based estimates of TRP growth with TFP growth offers an indication of the expected magnitude of TRP relative to TFP. The Table 1 estimates suggest that increased groundwater pollution from pesticides used in the farm sector in the 197279 period reduced conventionally measured productivity growth by about 0.06 percentage points per year. Annual TRP growth during the period was about 7% below TFP growth. In contrast, declining pollution after 1979 increased productivity growth by an average 0.05 percentage points annually, augmenting TFP growth in the 1979-93 period by nearly 3% per year. Note, however, that these estimates are likely to change, perhaps significantly, once the development of USDA water pollution data is complete and application is extended to both pesticide and fertilizer effects on both groundwater and surface water.

Concluding Comment

The model of TRP growth proposed in this paper has a number of desirable properties. First, while it broadens the notion of TFP growth to include nonmarket goods, it preserves the production orientation of productivity accounting. Total resource productivity measures productivity growth, not welfare growth. Second, the formulation of [E.sup.TRP] in equation (13) provides a natural context within which to evaluate the impact of regulatory policy over time on productivity growth. Regulation affects not only the trend in pollution but the magnitude of the shadow price [Sigma], that is, the spread between [Eta] and [Rho]. Third, the derivation of the shadow price [Sigma] in equation (13) as the spread between [Eta] and [Rho] makes much sense. After all, in the limit, if abatement efforts were such that the marginal social valuation of clean water equaled marginal abatement cost, the resulting zero shadow price would imply that externalities have been properly internalized in the market prices of conventional outputs.

Formal application of the model to the farm sector necessarily must await completion of the environmental data by USDA, but even the preliminary results reported in this paper underscore the importance of careful research into the estimation of society's marginal valuation of clean water. Whether TRP growth exceeds or falls below TFP growth depends importantly on whether the shadow price is positive ([Eta] [greater than] [Rho]) or negative ([Eta] [less than] [Rho]). The present estimates of [Eta] in the literature differ by nearly a dollar. That range needs to be narrowed. Measuring [Eta] is no doubt difficult, but it is critical to the proper evaluation of the effect of changing environmental quality and regulation on productivity growth in agriculture or any other industry.

1 At this stage of the analysis, there is no requirement that the selected output levels Y and S be feasible given X and T. The only requirement is that Y and S be nonnegative.

2 Either or both of these conditions could be relaxed without affecting the analysis that follows. Both are maintained for convenience.

3 Barnard does not define health risk per se but aggregates over several probable maladies, such as increased mortality from carcinogens. This is adequate for the purposes of this paper because, in the context of the consumer choice model, water is considered either clean or dirty.

4 The marginal abatement cost function derived from the second-order cost function estimated by Swinand is found to be increasing in abatement efforts.

References

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Ball, V.E. "Output, Input, and Productivity Measurement in U.S. Agriculture, 1948-79." Amer. J. Agr. Econ. 67(August 1985):475-86.

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Jorgenson, D.W., and F.M. Gollop. "Productivity Growth in U.S. Agriculture: A Postwar Perspective." Amer. J. Agr. Econ. 74(August 1992):745-50.

Jorgenson, D.W., F.M. Gollop, and B.M. Fraumeni. Productivity and U.S. Economic Growth. Cambridge: Harvard University Press, 1987.

Kellogg, R.L., R. Nehring, and A. Grube. "National and Regional Environmental Indicators of Pesticide Leaching and Runoff from Farm Fields," Resource Assessment and Strategic Planning Working Paper, Natural Resources Conservation Service, USDA, forthcoming.

Kendrick, J.W. Postwar Productivity Trends in the United States, 1948-1969. New York: National Bureau of Economic Research, 1973.

Swinand, G.P. "Modeling and Measuring Total Resource Productivity." Working paper, Boston College, 1997.

Frank M. Gollop and Gregory P. Swinand are professor and doctoral candidate, respectively, Department of Economics, Boston College. The authors wish to thank V. Kerry Smith and Robert Weaver for their helpful comments.