Agricultural productivity revisited

American Journal of Agricultural Economics, Nov, 1997 by V. Eldon Ball, Jean-Christophe Bureau, Richard Nehring, Agapi Somwaru

The U.S. Department of Agriculture (USDA) has long been concerned with sectoral productivity growth. An early innovator, it was for more than two decades the sole government agency regularly to compile and publish total factor productivity indexes.

While recognized for taking the lead in compiling total factor productivity indexes, a number of economists have suggested ways in which the quality of the information available for productivity research could be improved. In 1960, Griliches articulated the need for several modifications in data collection and inference used in the USDA productivity series. Others, including Christensen; Penson, Hughes, and Nelson; an American Agricultural Economics Association (AAEA) task force on productivity measurement (USDA 1980); Ball; and Shumway, have reinforced Griliches' basic tenets and have proposed further changes. Until now, few of the recommendations have affected either the data collection or procedures used by USDA in the official productivity data series.

The purpose of this paper is to describe procedures currently being implemented by the USDA to measure agricultural productivity. Output is defined as gross production leaving the farm as opposed to real value added. Inputs are not limited to capital and labor but include intermediate inputs as well. We derive index numbers of gross output, capital, labor, and intermediate inputs. These data are used to construct indexes of total factor productivity. We then compare the contributions of input growth and productivity growth to economic growth. The important role of productivity growth in agriculture becomes immediately apparent.

Index Number Procedures

A productivity index is generally defined as an output index divided by an input index. The question facing the analyst is which functional form for an index number should be used. To address this question, we relate known functional forms for index numbers to functional forms for the underlying aggregator function.

Two important examples of quantity indexes are the Laspeyres and Paasche indexes. The Laspeyres quantity index [Q.sub.L] is defined as

(1) [Q.sub.L]([p.sup.0], [p.sup.1], [x.sup.0], [x.sup.1]) [equivalent to] [p.sup.0] [multiplied by] [x.sup.1]/[p.sup.0] [multiplied by] [x.sup.0]

where [p.sup.t] [greater than] 0 is a vector of prices in period t, t = 0, 1, and x [greater than] 0 is the corresponding vector of quantities. The Paasche quantity index [Q.sub.P] is defined as

(2) [Q.sub.P]([p.sup.0], [p.sup.1], [x.sup.0], [x.sup.1]) [equivalent to] [p.sup.1] [multiplied by] [x.sup.1]/[p.sup.1] [multiplied by] [x.sup.0].

Fisher suggested the geometric mean of [Q.sub.L] and [Q.sub.P]

(3) [Q.sub.F]([p.sup.0], [p.sup.1], [x.sup.0], [x.sup.1]) [equivalent to] [[[p.sup.0] [multiplied by] [x.sup.1]/[p.sup.0] [multiplied by] [x.sup.0] [p.sup.1] [multiplied by] [x.sup.1]/[p.sup.1] [multiplied by] [x.sup.0]].sup.1/2].

Let f(x) define an aggregator function and suppose that [x.sup.t] [greater than] [0.sup.N] is the solution to

(4) [Mathematical Expression Omitted]

where f(x) [equivalent to] [(x[prime]Ax).sup.1/2] and A = [[a.sub.ij]] is a symmetric matrix of coefficients. Then Konus shows that

(5) f([x.sup.1])/f([x.sup.0]) = [Q.sub.F]([p.sup.0], [p.sup.1], [x.sup.0], [x.sup.1]).

If a quantity index Q([p.sup.0], [p.sup.1], [x.sup.0], [x.sup.1]) and a functional form for the aggregator function f satisfy (5), then the quantity index Q is said to be exact for the aggregator function f. Konus has shown that the Laspeyres quantity index is exact for a fixed coefficient aggregator function while the Paasche quantity index is exact for a linear aggregator function. Afriat, Diewert (1976), Pollak, and Samuelson and Swamy provide other examples of exact index numbers. However, among the exact index numbers discussed, only the Fisher quantity index corresponds to a functional form for f that can provide a second-order approximation to an arbitrary twice-differentiable linearly homogeneous function.(1) These results provide an economic justification for our use of the Fisher index in the empirical applications that follow.

Gross Output

The development of a measure of output begins with disaggregated data for physical quantities and market prices of crops and livestock. These data were compiled by the Economic Research Service's Resource Economics Division. The quantity data exclude production that is used on the farm as input.

For purposes of productivity measurement, output includes the quantities of commodities sold off the farm (including unredeemed Commodity Credit Corporation loans) plus net additions to inventory and quantities consumed as part of final demand in farm households during the calendar year.

Prices corresponding to each disaggregated output reflect the value of that output to the producer; that is, subsidies are added and indirect taxes are subtracted from market values. Prices received by farmers, as reported in Agricultural Prices, include an allowance for net Commodity Credit Corporation loans and purchases by the government valued at the average loan rate. However, direct payments under federal commodity programs are not reflected in the data. To prices for wool, mohair, and program crops, we add government payments per unit of production; dairy assessments are subtracted from receipts for milk.