Measurement and Interpretation of Productivity (1979) / Chapter Skim
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A Note on the Form of the Production Function and Productivity
A Note on the Form of the Production Function and Productivity
Pages 309-317
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From page 309... ...
For productivity measurement, other factors like disequilibrium, measurement errors in inputs and outputs, aggregation 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.
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, 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.
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The purpose of the present note is to examine how differences in the functional forms for aggregator functions affect productivity measurement. The following discussion is based on Diewert (19761.
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p P~'Xo Q = X~'Po `4' The Paasche price and quantity indices are defined by (these use final period weights)
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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)
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L = private domestic labor input quantity index. PL = private domestic labor input price index.
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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
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. 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)
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From page 317... ...
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.
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