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A Note on the Form of the Production Function and Productivity G. S. MADDALA University of Florida INTRODUCTION The purpose of the present note is to investigate whether measures of multifactor productivity differ significantly with alternative functional forms for production functions. Every input index is based on an im- plicit assumption of a functional form for the underlying production function that describes output as a function of the several inputs. The conclusion of the paper is that, within the limited class of functions considered here (viz. Cobb-Douglas, generalized Leontief, homogeneous translog, and homogeneous quadratic) differences in the functional form produce negligible differences in measures of multi-factor productivity. The intuitive explanation of these results is that the different func- tional forms suggested in the literature differ in their elasticities of sub- stitution (which depends on the second derivatives of the production function) whereas for productivity measurement, all we are concerned with is first derivatives. For productivity measurement, other factors like disequilibrium, measurement errors in inputs and outputs, aggrega- tion problems, etc., are more important than functional forms of the production. Of course, the class of production functions considered here is very limited, but more complicated functional forms result in index number formulae for inputs and outputs that are complicated and rarely used. The conclusion of the paper might have been more obvious to some, but it was not obvious to everyone in the Panel and hence might be worth reporting in this note. 309

310 ANALYSIS PAPERS Consider first the case of a single output Y and two inputs, labor L and capital K. Let the values of these three variables be period 0: Ye, Lo, No period 1: Ye, Lit, Kit. Comparing Yin with Ye, we know by what percentage output has in- creased or decreased. But to find what happened to inputs as a whole, we have to weigh LJLo and KJKo suitably. To get these weights, we have to know how the inputs labor and capital interact with each other to produce the output Y. This interaction is described by the "produc- tion function." For instance, if we say that the production function has the Cobb-Douglas form, we write Y = CL~K'-~. (1) Over a short period of time we assume that ~ remains constant. Then it is changes in C that measure changes in total productivity. Ye = ILK Yo = CoLo'tKo~-a Then Y~/Y0 is the output index, . {L 1 ~ (K15 ~LoJ ~KoJ (2) is the input index, and C~/Co is the productivity index. The functional form for the production function (1), implies the con- struction of the input index (2), which is a geometric mean of the labor input index and the capital input index. Different functional forms for the production function imply different schemes for weighting the input ratios in the construction of the input index. We will discuss these later. To construct the index (2), we need the value of the parameter or. To get this, we make use of the prices of the inputs (on the basis of the im- plicit assumption that these prices are proportional to the marginal pro- ductivities of the respective inputs). Let the prices of labor and capital be, respectively, Pot, pop in period 0 and Pit, pick in period 1. In either period it is possible to produce the given output with several other com- binations of inputs. We have to assume that, given the prices of inputs in that period, the particular combination of inputs was chosen so as to

A Note on the Form of the Production Function and Productivity 311 maximize profits, i.e., to produce Ye, the combination (Lo, Ko) was the most optimal given the prices Pot and pod. Similarly, to produce Ye, the combination ELI, Kit was the most optimal, given the prices Pit, and piK Given this assumption, we get the value ~ as the share of labor in total cost, i.e., 5 LPt LP ~ + KP K If this share is constant between period O and period 1, then there is no problem. If it is not, then there is an empirical question of whether to use the value in period O or period 1 or an average of the two values. Of course, drastic changes in this share suggest either that ~ is not constant over time (structural change) or that the input prices do not reflect marginal products (disequilibrium), of course, assuming that the func- tional form (1) is correct. Thus, what we need to construct the input index is (1) a functional form for the production function, (2) data on input prices, and (3) profit maximization (or cost minimization) hypothesis. The latter two will enable us to get estimates of the parameters of the production function. Since the theory is the same for the construction of an output index from several outputs, we will not go into those details here. We will assume in the following discussion and calculations that there is only one output and several inputs. Diewert (1976) uses a more general term, aggregator functions, instead of production functions. The purpose of the present note is to examine how differences in the functional forms for aggregator functions affect productivity measure- ment. The following discussion is based on Diewert (19761. The construction of the output index (if there are several outputs) and the input index (if there are several inputs) is basically an index number problem. As everyone knows, the Laspeyres, Paasche, and Fisher's Ideal index numbers have been in common usage in many other areas. The same type of index number formulae can be used in pro- ductivity measurement. These index numbers are defined as follows: Let Po be the vector of prices in period 0. Let Pi be the vector of prices in period 1. Let Xo be the vector of quantities in period 0. Let X~ be the vector of quantities in period 1. Let P be a price index and Q a quantity index. P and Q both represent some functions of Pa, Pi, X~ X. Irving Fisher's weak factor reversal test requires that pQ P~'X, total expenditures in period 1 PO'Xo total expenditures in period O .. (3)

312 PAPERS Fisher argued that this is a desirable requirement for all index numbers. The Laspeyres price and quantity indices are defined by (these use base- period weights) p P~'Xo Q = X~'Po `4' The Paasche price and quantity indices are defined by (these use final period weights) PP = P''X~ QP = X 'P · (5) Fisher's ideal price and quantity indices are defined by imp p 'I,' Q = ~Q~Q~/2 (6) Clearly p,Q,= P,,X,, Po Xo and thus Fisher's ideal index numbers satisfy condition (3~. Diewert (1976) shows that Fisher's ideal index numbers are consistent with the aggregator function ,. .. F(x) = (X'AX)~'2 = ([ ~ aijxixj)~2. (7) For the Cobb-Douglas aggregator function the quantity index is the geometric index: Q8= n(X,i) i-l Xoi (8) where sj is the share of the ith item in total cost in period 0. For the homogeneous translog the quantity index is i=, Xo! (9) where so is the ith share of cost in period 1 and so is the ith share of cost in period 0. The corresponding price index is

A Note on the Form of the Production Function and Productivity 313 P° ( ) i=! Po . (10) Now POLO ~ p~'xl/pO'xO. Thus, the indexes PO and QO do not satisfy Fisher's weak factor reversal test in (31. So, corresponding to PO, define the implied quantity index QO that satisfies condition (34. Similarly, corresponding to QO, define the implied price index PO that satisfies condition (3), i.e., Po Qo = Po Qo = , PO XO The pair PO, Qo was advocated by Kloek (19671. The pair PO, QO was used by Christensen and Jorgenson (1969, 1970) in order to measure U.S. real input and output. Jorgenson and Griliches (1972) have also ~ used PO, QO in the context of productivity measurement. They use these indexes for both inputs and outputs. Diewert (1976) derives the index numbers implied by a very general functional form for aggregator functions. This is known as the quadratic mean of r function. It is defined as n n F(x) = [ ~ ~ai~xir/2xjr/2]1/r. i=1 ,i=1 (11) For r = 1 this is the generalized Leontief function suggested by Diewert (19711. If a,j = 0, for i ~ j, we get the constant elasticity of substitution (CES) function. As r-0, this function tends to the homogeneous trans- log function considered by Christensen, Jorgenson, and Lau, for which the appropriate index numbers are the Divisia index numbers defined in (9) and (10~. For r = 2 this function is the homogeneous quadratic function (7) for which Diewert (1976) shows that the appropriate index numbers are Fisher's ideal index numbers. Diewert (1976) shows that for the quadratic mean of r function, the quantity index is Qr = n X r/2 n Xo; r/2 [it=! (Xoi) Nl(-l (Eli) ~ For r-2, we get Q2 (,Xo~p~/X~ rpl ~(QL QP (Fisher's ideal index number). (12)

314 TABLE 1 Data PAPERS Year X L K PI PK 1929 189.8 173.3 87.8 0.324 0.533 1930 172.1 165.4 87.8 0.311 0.437 1931 159.1 158.2 84.0 0.273 0.402 1932 135.6 141.7 78.3 0.236 0.312 1933 132.0 141.6 76.6 0.219 0.318 1934 141.8 148.0 76.0 0.238 0.326 1935 153.9 154.4 77.7 0.248 0.396 1936 171.5 163.5 79.1 0.263 0.423 1937 183.0 172.0 80.0 0.285 0.447 1938 173.2 161.5 77.6 0.281 0.411 1939 188.5 168.6 81.4 0.290 0.442 1940 205.5 176.5 87.0 0.300 0.465 1941 236.0 192.4 96.2 0.337 0.528 1942 257.8 205.1 104.4 0.398 0.589 1943 277.5 210.1 110.0 0.459 0.656 1944 291.1 208.8 107.8 0.494 0.686 1945 284.5 202.1 102.1 0.511 0.702 1946 274.0 213.4 97.2 0.540 0.774 1947 279.9 233.6 105.9 0.594 0.805 1948 297.6 228.2 113.0 0.639 0.828 1949 297.7 221.3 1 14.9 0.647 0.805 1950 328.9 228.8 124.1 0.683 0.908 1951 351.4 239.0 134.5 0.742 0.965 1952 360.4 241.7 139.7 0.782 0.959 1953 378.9 245.2 147.4 0.827 0.932 1954 375.8 237.4 148.9 0.846 0.955 1955 406.7 245.9 158.6 0.880 0.996 1956 416.3 251.6 167.1 0.930 0.971 1957 422.8 251.5 171.9 0.978 0.983 1958 418.4 245.1 173.1 1.000 1.000 1959 445.7 254.9 182.5 1.042 1.028 1960 457.3 259.6 189.0 1.074 1.024 1961 466.3 258.1 194.1 1.103 1.043 1962 495.3 264.6 202.3 1.144 1.091 1963 515.5 268.5 205.4 1.180 1.139 1964 544.1 275.4 215.9 1.229 1.158 1965 579.2 285.3 225.0 1.271 1.2~5 1966 615.6 297.4 236.2 1.335 1.285 1967 631.1 305.0 247.9 1.387 1.245 All figures are constant prices of 1958. X = gross private domestic product quantity index. L = private domestic labor input quantity index. PL = private domestic labor input price index. K = private domestic capital input quantity index. PK = private domestic capital input price index. SOURCE: Christensen and Jorgenson (1970). The indexes in 1967 relative to 1929 (=100) are X = 332.5, L = 176.0, K = 282.3, Pr = 428.1, PK = 233.6.

A Note on the Form of the Production Function and Productivity 315 TABLE 2 Comparison of Productivity Indexes: Weights Changing Every Year Homogeneous Generalized Homogeneous Year Translog Leontief Quadratic 1930 93.071 93.071 93.071 1931 89.946 89.946 89.946 1932 84.131 84.130 84.130 1933 82.709 82.709 82.708 1934 86.916 86.915 86.91S 1935 91.210 91.210 91.210 1936 97.669 97.669 97.668 1937 100.758 100.758 100.757 1938 100.193 100.193 100.192 1939 104.243 104.243 104.242 1940 107.597 107.597 107.596 1941 112.652 112.651 112.651 1942 114.545 114.544 114.543 1943 118.919 118.918 118.917 1944 126.268 126.268 126.267 1945 128.647 128.646 128.645 1946 122.332 122.331 122.331 1947 117.450 117.449 117.449 1948 120.248 120.248 120.247 1949 121.759 121.758 121.758 1950 127.824 127.824 127.823 1951 128.728 128.727 128.726 1952 129.099 129.098 129.097 1953 131.655 131.654 131.653 1954 132.545 132.545 132.543 1955 136.877 136.876 136.875 1956 135.276 135.276 135.274 1957 135.841 135.840 135.839 1958 136.096 136.095 136.094 1959 138.616 138.615 138.613 1960 138.690 138.689 138.688 1961 140.351 140.350 140.349 1962 144.416 144.416 144.414 1963 148.087 148.086 148.084 1964 150.810 150.809 150.807 1965 154.572 154.571 154.568 1966 157.121 157.120 157.118 1967 155.519 155.519 155.516

316 PAPERS TABLE 3 Sensitivity of Productivity Indexes to Changes in Weighting Weights Generalized Homogeneous Changing Leontief Quadratic Every year 155.519 155.516 Every 4 years 153.812 153.806 Every 5 years 153.800 153.793 The table presents the productivity indexes in 1967 under different weighting schemes. QL, QP, and Qua are defined in (4), (5), and (61. Similarly, the price · - nc .ex IS p = ~ P i S n Poi r/2 1/r r [(i=1 (pOi) )I( l-l(pli) )] . Since PrQr ~ P,'X~/P~'XO except for r = 2, it is only Fisher's ideal index numbers that satisfy the factor reversal test (3~.1 To examine the sensitivity of productivity indexes to the specification of the functional forms for the aggregator functions, we used Diewert's general functional form (11) and the associated index (12) for different values of r. Though we did the calculations for several values of r, we will report the results for only three values: r = 0, the homogeneous translog function; r = 1, the generalized Loentief function; r = 2, the homogeneous quadratic function. Table 1 lists the basic data taken from Christensen and Jorgenson (19701. Table 2 gives the productivity indexes for the three functional forms mentioned earlier, when the weights are changed every year. Table 3 shows the sensitivity of results when the weights are changed every 4 years and every 5 years. Of course, one could always pair Qr with Pr or Pr with Qr, where Pr and Qr are defined in a manner similar to Po and Qo earlier.

A Note on the Form of the Production Function and Productivity 317 Christensen and Jorgenson suggest the translog production function and the associated Divisia index numbers. Diewert (1976) argues strongly in favor of the Fisher's ideal index numbers, which are consistent with the homogeneous quadratic function. What the results presented in this note suggest is that from the point of view of productivity measurement it does not make much difference which functional form is used. This is so with yearly changing weights but is also true if weights change every 4 or 5 years. The choice of a particular functional form can, however, still be advocated on the basis of other considerations. REFERENCES Christensen, L. R., and Jorgenson, D. W. (1969) The measurement of U.S. real capital input, 1929-1967. Review of Income and Wealth 15:293-320. Christensen, L. R., and Jorgenson, D. W. (1970) U.S. real product and real factor input, 1929-1967. Review of Income and Wealth 16: 19-50. Diewert, W. E. (1971) An application of the Shepard duality theorem: a generalized Leontief production function. Journal of Political Economy 79:481-507. Diewert, W. E. (1976) Exact and superlative index numbers. Journal of Econometrics 4:115-145. Jorgenson, D. W., and Griliches, Z. (1972) Issues in growth accounting: a reply to Edward F. Denison. Survey of Current Business (Part II) 55(5):65-94. Kloek, T. (1967) On Quadratic Approximations of Cost of Living and Real Income Index Numbers Report 6710. Econometric Institute, Rotterdam.