The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Workine P a ~ e r8813 DECOMPOSING TFP GROWTH IN THE PRESENCE OF COST INEFFICIENCY, NONCONSTANT RETURNS TO SCALE, AND TECHNOLOGICAL PROGRESS by Paul W. Bauer Paul W. Bauer is an economist at the Federal Reserve Bank of Cleveland. The author would like to thank Michael Bagshaw and C.A. Knox Love11 for helpful comments. John R. Swinton provided valuable research assistance. Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to stimulate discussion and critical comment. The views stated herein are those of the author and not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System. December 1988 ABSTRACT Productivity growth is a major source of economic growth; thus, an understanding of how and why productivity measures change is of great interest to economists and policymakers. This paper explores the relationship between observed total factor productivity (TFP) growth, defined using an index number approach, and examines changes in returns to scale, cost efficiency, and technology. Several decompositions are developed, using alternatively production and cost frontiers. for multiple outputs. The last decomposition developed also allows DECOMPOSING TFP GROWTH IN THE PRESENCE OF COST INEFFICIENCY, NONCONSTANT RETURNS TO SCALE. AND TECHNOLOGICAL PROGRESS I. Introduction Measures of productivity have long enjoyed a great deal of interest among researchers analyzing firm performance and behavior. The observed growth in total factor productivity (TFP) is one of the most widely employed measures of overall productivity. The conventional Divisia index of TFP is defined as1 (1) TFP = - F, where where y is observed output, F is an aggregate measure of observed input usage, w, is the price of the i-th input, xi is the observed use of the i-th input, and C is the observed cost.2 Ohta (1974) and Denny, Fuss, and Waverman (1981), among others, have shown that in the single-product case, with constant returns to scale and cost efficiency, TFP growth equals technological progress. With nonconstant returns to scale and cost efficiency, TFP growth is equal to technological progress plus a term that adjusts for the degree of returns to scale: where f is the production function, x is the vector of inputs, t is a time index, and ecy is the cost elasticity with respect to output. This paper extends the decomposition of observed TFP growth by showing how changes in cost efficiency over time also affect the observed measure of TFP growth. The observed measure of TFP is decomposed into various components roughly stemming from changes in returns to scale, in cost efficiency, and in technological progress. Biased estimates of firm or industry performance will result if changes in cost efficiency are ignored. Furthermore, since these decompositions are derived from an observed quantity, the appropriate decomposition could be included in the estimation of the frontier as an additional equation, thus improving the statistical precision of the estimates by providing additional information and increasing the number of degrees of freedom. In section I1 of this paper, TFP growth is decomposed using a production function approach. Section I11 derives the decomposition using a cost function approach for both the single-product and multiproduct firm. Section IV presents some empirical examples of the use of some of these decompositions, and the conclusion appears in Section V. 11. Production Function Approach Let the production frontier be defined as where y* is the maximum amount of output that can be produced with input vector x at time t. A Farrell-type, output-based measure of technical efficiency can be defined as follows: T P =- J f (x., t) ' where 0 < Tp 5 1. The first TFP decomposition can be derived as follows. First, take the natural log of both sides of (5) and totally differentiate it with respect to time: (6) dlnTp = dt dlny dt C i alnf(x,t) dxi dt axi + alnf(x,t) at - This can be rewritten as where Tp is the time rate of change of technical efficiency and &(x, t) is the time rate of change of technological progress as measured by shifts in the production frontier over time. Next,(7) can be rearranged using the definition of observed TFP in (1): The following substitutions can be made: (9 ci(x,t) = af(x,t) and ax, f(x,' Xi where ci(x,t) is the output elasticity of the i-th input and si is the observed share of the i-th input. This yields the following decomposition: which decomposes observed TFP growth into change in technical efficiency, technological progress, and a term that depends on the degree of the inputspecific returns to scale and cost inefficiency. This decomposition yields the intuitive result that advances in both technological progress and technical efficiency increase observed TFP growth. While the first two terms have straightfornard interpretations, the last term requires further explanation. This term has two informative properties. First, under cost efficiency, this term is equal to the last term in (3) , since cost minimization requires (12) af = 2, £or a11 axi i. Second, when the firm is cost inefficient, this last term is a bundle composed of nonconstant returns to scale and both technical and allocative inefficiency. One can further decompose this term using duality; however, the cost function approach developed in the next section does this in a much more straightforward manner. But first, consider the relation between this decomposition and that of Nishimizu and Page (1982). Nishimizu and Page derived their decomposition as follows. First, they define what might be called the "average" production function, g(x,t), as In contrast to the frontier production function, f(x,t), the observed production function yields what each firm actually produces. They transform (13) by taking the natural logarithm and totally differentiating with respect to time to obtain (14) $(x,t) = y - 1, eg,(x,t) xi, where egi(x,t) is the output elasticity of the i-th input with respect to the "average" production function. Nishimizu and Page then employ an alternative approach to defining TFP. Instead of defining TFP with respect to a Divisia index, they define TFP with respect to the rate of shift in the "average" production function, g(~,t).~ The next step in deriving their decomposition is to rewrite equation (7) (15 9 = '$ + f(x,t) + ri(x,t) i,. Substituting for y in (14) and simplifying yields This is the Nishimizu and Page decomposition; equation (16) separates observed TFP growth into technological progress, change in efficiency, and differences in output elasticities between the frontier and the interior for a firm operating in the interior. While (16) is quite similar in form to (ll), there are two important differences. First, it must be recalled that Nishimizu and Page employ a different definition of TFP than the one employed here. They define it to be the rate of shift in the "average" production function, whereas the decompositions derived here are based on a definition of observed TFP using the Divisia index. The potential advantage of the latter approach is that it creates the possibility of adding another equation to the system to be estimated (in addition to the cost and input share equations) since the left side of equation(11) is observed and the right side of equation (11) is a function of Including the TFP equation in the the parameters to be e~timated.~ regression increases the number of degrees of freedom (since no new parameters are added) and also provides information that is not found in the cost or input demand equations. Second, the use of an "average" production function, g(x,t), may be of use conceptually, given Nishimizu and Page's assumption that firms operating away from the frontier have a good reason for doing so. This is not useful empirically, however, because g(x,t) cannot be estimated simultaneously with the frontier production function unless the reason for the deviation from the frontier is also modeled. Without this type of modeling, the only possible definition of g(x, t) is (17) g(x,t) = f(x,t) - T,. This implies that their "average" production function models not only the frontier production function, but also inefficiency. In other words, it predicts the level of inefficiency--with the same arguments as the frontier production function. The cost function TFP decompositions are now derived. 111. Cost Function Approach The TFP decomposition is first derived in the case of the single-product firm and is then generalized for the multiproduct firm. Let the singleproduct cost frontier be represented by where C* is the efficient cost given (y,w,t). Following Farrell (1957), an overall measure of cost efficiency may be defined as From these input-based measures of technical and allocative efficiency, one can derive (20) E = T . A , which implies (21) E = T + A ,(which will be used later), where T and A are the Farrell measures of technical and allocative efficiency, respectively. The decomposition of TFP growth can now be derived using the cost function approach. Taking the natural logarithm of each side of (19), totally differentiating, and making a few minor substitutions yields t) . ) where ~ ~ ~ ( y , w=, tdlnC(y,w, alny Using the definition of observed TFP in equation (I), equation(22) can be simplified as follows: At this point, note the following: (26) c 2 7ki + WiXi = i c,, and i Substituting (27) into (23) yields Substituting (21) into (28) and making some straightforward substitutions yields the single-product cost function decomposition of observed TFP: This expression decomposes TFP growth into terms related to returns to scale, changes in technical and allocative efficiency, technological progress, and a residual term (which will be discussed below). This decomposition is consistent with expectations; in particular, the expectation that increases in cost efficiency increase observed TFP. The last term clearly reflects the presence of allocative inefficiency. If the firm is allocatively efficient, then si=si(y,w,t), and this term is equal to zero. This term is also equal to zero when input prices change at the same rate, since ~ [ S ~ - S , ( ~t)]=O. ,W, i Some insight into this term can be obtained by noting that in the presence of allocative inefficiency, since the observed input shares, si,are not equal to the efficient input shares, si(y,w,t), the aggregate index of input usage F (used to define observed TFP) does not weight the observed inputs according to the costminimizing input shares. The last term corrects for any bias this may have on observed TFP. A multiproduct version of the decomposition can also be derived. For the multiproduct firm, observed TFP is usually defined as6 (30) TFP = 9P P - I?, where jr = PjYj 1-9 R i. and F 1 WiXi c kip i j where = 9 is a revenue-weighted index of output, F is a cost share index of aggregate input usage, wi is the price of the i-th input, xi is the observed use of the i-th input, and C is the observed cost. Using the same basic steps used in the single-product case above for handling cost inefficiency and in Denny, Fuss, and Waverman(1981) for handling multiple outputs, observed TFP for a multiproduct firm can be shown to be equal to the following: + 1i P c [si-si(y,w,z,t)] wi + (y -y ) , where y = This expression decomposes TFP growth into terms related to ray returns to scale, changes in technical and allocative efficiency, and technological progress. The next-to-last term has the same properties as the last term in equation (25). The last term simply measures any effect that nonmarginal cost pricing may have on the observed measure of TFP. Denny, Fuss, and Waverman have shown that F=yc under marginal cost pricing and proportional markup pricing. These TFP decompositions provide useful conceptual and empirical tools for assigning the observed changes in TFP growth to the various root sources. Note that the cost function approach provides a more complete partitioning of the sources of observed TFP growth than the production approach did. IV. Empirical Application This section illustrates a use of one of the multiproduct TFP decompositions. The example is drawn from the U.S. airline industry, and these results are discussed more fully in Bauer (1988). First, the model that was estimated and the data set that was employed are briefly discussed; then the empirical results and the TFP decomposition are presented. The translog system of cost and input share equations that was estimated is presented below (omitting firm and time subscripts): where y is a vector of outputs, w is a vector of input prices, z is a vector of network characteristics, and t is a time index. The translog functional form was selected on the basis of its being a second-order approximation to any cost function about a point of expansion(here, the sample means) .' Note that the network and time variables were not interacted with input prices in order to reduce the number of parameters to a manageable level and to lessen the effects of multicollinearity. Symmetry and linear homogeneity in input prices impose the following restrictions on the cost system: By construction, lsi(y,w)=l, so that one input share equation must be i dropped before estimation to avoid singularity. Barten (1969) has shown that asymptotically, the parameter estimates are invariant as to which input share equation is dropped. The following distributional assumptions are imposed. The inefficiency term, mode p %t, is assumed to follow a truncated-normal distribution with and underlying variance oU2 such that term, vnt, is assumed to be independent of I+,2 0. The noise xtand to follow a normal distribution with mean zero and variance ov2. The disturbances on the input share equations are assumed to follow a multivariate normal distribution: wnt = (wlnt7 . . . , +-l,nt) ' cv N(a, n) . The likelihood function for this system can be written as8 (35) lnL = - TNM ln(2n) 2 - (TN) - lno" T,N lnlnl - ln[l-F*((-a) (A-~+I)"~)] - (writ-a)' 1 t n n-l (writ-a). Maximum likelihood estimates can be obtained for all the parameters in (35), and these estimates will be asymptotically efficient. A number of specification tests can be performed using likelihood ratio tests similar to those proposed by Stevenson (1980). The data set employed in this paper was constructed by Robin Sickles using the AIMS 41 form that all interstate airlines were required to submit periodically as part of the Civil Aeronautics Board's regulation of the industry. Included are 12 firms and 48 quarters of data from 1970:IQ to 1981:IVQ. The airline industry is considered to produce revenue passenger ton miles (y ) and revenue cargo ton miles (yc) using four inputs: labor (L), P capital (K), energy (E), and materials (M). Labor is an aggregate of 55 separate labor accounts; capital is a combination of flight equipment, ground equipment, and landing fees; energy is the quantity of fuel used converted to BTU equivalents; and materials is an aggregate of 56 different accounts composed mainly of advertising, insurance, commissions, and passenger meals. The network through which airlines supply their outputs has an important influence on the cost of providing that output. The average load factor, zldf, for a given airline in a given time period is the proportion of an airline's capacity that is actually sold in that time period. The average stage length, zStgl, is the average distance of an airline's flights in a given quarter. These two network characteristics are incorporated into the two translog cost models as presented in equation (32). From table 1 it can be seen that all but two of the parameter estimates are statistically significant. The parameters reported here are from a model slightly more restricted than the one developed in section 111. Instead of the more general truncated-normal distribution, the half-normal distribution was assumed, which is equivalent to restricting p=0. This restriction could not be rejected using a t-test based on the results of the more general model. Table 2 reports the results of the TFP decomposition technique. Observed TFP grew on average for all of the firms, although there was a great deal of variation across firms. Much of this increase is the result of technological progress that ran at a rate of 0.274 percent per quarter, as reported earlier. The scale effect was a significant source of TFP gains for the smaller airlines, which were free to grow under the regulatcry reform process, but not for the four largest airlines. The inefficiency effects varied consider'ably from airline to airline, but were generally small. Over time, changes in the airlines' networks have generally boosted productivity. The average load factors and stage lengths of the airlines have risen (although unevenly across airlines), each resulting in increases in observed TFP of about the same order of magnitude as those stemming from technological progress. The biases in the observed measure of TFP as a result of nonrnarginal cost pricing (the output effect) and observed input shares not being equal to the least-cost input shares(the price effect) are found to have a small effect on observed TFP. A "pure" measure of TFP growth could be constructed by summing the scale, cost efficiency, technological change, and network effects. In general, these estimates indicate that the observed measure of TFP is a biased estimate of technological progress, not just because of the scale and output effects (as Denny, Fuss, and Waverman have shown), but also because of the efficiency, network, and input price effects. V. Conclusion Observed TFP growth has been decomposed into scale, change in efficiency, and technological progress effects using both production and cost function approaches for both single-product and multiproduct firms. The production function approach was compared to the decomposition of Nishimizu and Page (1982) and was found to have at least the possible advantage that the observed TFP equation might be added to the system of equations to be estimated. In addition, the decomposition derived here does not depend on the artificial construction of an "average" production function. In this respect, the decomposition proposed here seems to be more firmly based in cost theory and efficiency measurement. The decompositions of TFP developed here will have at least two uses in empirical work. First, there is the potential that the TFP equation could be added to the system of equations to be estimated. Since this equation provides information not contained in the others and increases the number of degrees of freedom, better estimates of technology (as embodied in the production or cost function) and the level of cost efficiency will be obtained. Second, it will also be of use in interpreting and explaining empirical results. For example, TFP growth has been negative in some industries in recent years--a fact that is sometimes difficult to explain in a framework that does not allow for cost inefficiency (see Gollop and Roberts [1981]). Using this decomposition, negative TFP growth could turn out to be a result of declines in cost efficiency, both technical and allocative. Footnotes Variables with a dot over them are defined as follows: i = dlnz dt . See Jorgenson and Griliches (1967), Richter (1966), Hulten (1973), Diewert (1976), and Denny, Fuss, and Waverman (1981), among others, for uses of this definition. Returns to scale can be defined as follows: RTS = 1i ei(x, t) . For a discussion of the various approaches to defining TFP growth, see Diewert (1981). Exactly how to implement this potential advantage both econometrically and practically has not yet been solved. See Denny, Fuss, and Waverman (1981). ' Though the translog functional form is a second-order approximation of the cost function at a point, it is generally only a first-order approximation of the economic measures of technology derived from the cost function. For example, note that the observed input shares are only a linear function of the regressors, being the first derivative of the log of the cost function. Strictly speaking, it is incorrect to model the disturbances in the cost and input share equations as being independent, given the interdependence However, as Schmidt (1984) pointed out, these of alnAJalnwint and %,. terms will tend to be uncorrelated, since both negative and positive deviations from efficient shares raise costs. For a more detailed description of this data set see Sickles (1985). Table 1 MLE Parameter Estimates Parameters Estimate Asymptotic Standard Error *Not statistically significant at the 0.01 level of significance. Source: Author's calculations. Table 2 TFP Decomposition (Average quarterly rate of change, in percent) Airline TFP Scale Effect Output Effect Eff. Technical Price Effect Change Effect Load Factor Stage Length AA AL BR co DL EA F'L NC oz PI UA WA Source: Author's calculations. The key to the carrier abbreviations are as follows: American AA AL USAir Braniff BR Continental CO Delta DL EA Eastern Frontier FL North Central NC Ozark OZ Piedmont PI United UA Western WA References Bauer, Paul W. "TFP Growth, Change in Efficiency, and Technological Progress in the U.S. Airline Industry: 1970 to 1981," Federal Reserve Bank of Cleveland Working- Paoer 8804, June 1988. Denny, Michael, Melvyn Fuss, and Leonard Waverman. "The Measurement and Interpretation of Total Factor Productivity in Regulated Industries, with an Application to Canadian Telecommunications," in Productivitv Measurement in Regulated Industries, T.G. Cowing and R.E. Stevenson, eds., New York: Academic Press, 1981, pp. 179-218. Diewert, W,E. "Exact and Superlative Index Numbers,'' Journal of Econometrics, 4 (May 1976), pp. 115-145. Diewert, W.E. "The Theory of Total Factor Productivity Measurement in Regulated Industries," in Productivitv Measurement in Regulated Industries, T.G. Cowing and R.E. Stevenson, eds., New York: Academic Press 1981, pp. 17-44. Fare, R., S. Grosskopf, and C.A. Knox Lovell. The Measurement of the Efficiency of Production, Boston: Kluwer-NijhoffPublishing Co., 1985. Farrell, M.J. "The Measurement of Productive Efficiency," Journal of the Royal Statistical Societv, General Series A, vol. 120, part 3 (1957), pp. 253-281. Gollop, Frank M., and Mark J. Roberts. "The Sources of Economic Growth in the U.S. Electric Power Industry," in Productivitv Measurement in Re~ulatedIndustries, T.G. Cowing and R. E. Stevenson eds., New York: Academic Press, 1981, pp. 107-143. Hulten, C.R. "Divisia Index Numbers," Econometrica, vol. 41 (November 1973), pp. 1017-1026. Jorgenson, D.W., and 2. Griliches. "The Explanation of Productivity Change," Review of Economic Studies, vol. 34 (July 1967), pp. 249-283. Nishimizu, Mieko, and John M. Page. "Total Factor Productivity Growth, Technological Progress and Technical Efficiency Change: Dimensions of Productivity Change in Yugoslavia, 1965-78," The Economic Journal, vol. 92 (December 1982), pp. 920-936. Ohta, M. "A Note on the Duality Between Production and Cost Functions: Rates of Return to Scale and Rates of Technical Progress," Economic Studies Quarterlv, vol. 25 (December 1974), pp. 63-65. Richter, M.K. "Invariance Axioms and Economic Indexes," Econometrica, vol. 34 (October 1966), pp. 739-755. Sickles, Robin C. "A Nonlinear Multivariate Error Components Analysis of Technology and Specific Factor Productivity Growth with an Application to the U.S. Airlines," Journal of Econometrics, vol. 27, no. 1 (January 1985), pp. 61-78. Stevenson, Rodney E. "Likelihood Functions for Generalized Stochastic Frontier Estimation," Journal of Econometrics, vol. 13, no. 1 (May 1980), pp. 57-66.
@FedFRASER