Total Factor Productivity Computed and Evaluated Using Multi-Step Perturbation

For period t, let qt = f(vt) + τt, where qt denotes measured output quantity, f(⋅) denotes a production function, vt = (v1t, ..., vnt) denotes a vector of n input quantities, τt denotes total factor productivity (TFP), and all variables are in natural-log form. Then, f(vt) = , for 0 < αit < 1 and = 1, is a Cobb-Douglas (CD) 1st-order log-form approximation of a production function. If f(⋅) is approximated as a CD production function, the share parameters, αit, are set to successive two-period-averaged observed inputcost shares, and the observed input quantities are considered optimal or inputcost minimizing, then, qt is the log of Solow-residual TFP (STFP). STFP could be subject to positive or negative input-substitution bias for two reasons. First, the CD production function restricts all input substitutions to one. Second, observed inputs generally differ from optimal inputs, which respond to input-pricesubstitution effects. In this paper, we test the possible inputprice-substitution bias of STFP in capital, labor, energy, materials, and services (KLEMS) inputs data for U.S. manufacturing from 1949 to 2001. (1) Based on maximum likelihood estimation, we determine a best 4th-order approximation of a CES-class production function. The CES class includes not only the standard constant elasticity of input substitution production functions but also includes so called tiered CES production functions, in which prespecified groups of inputs can have their own input-substitution elasticities and input-cost shares are parameterized (i) tightly as constants, (ii) moderately as smooth functions, or (iii) loosely as successive two-period averages. (2) Based on the best or optimal estimated production function, we compute the implied optimal TFP (OTFP) as qt f( ), where f( ) denotes the best estimated production function evaluated at the computed optimal inputs, . (3) For the data in percentagegrowth (%∆) form, we obtain two main conclusions: (i) relative to the average values of %∆STFP and %∆OTFP of about 1%, |%∆STFP %∆OTFP| exceeds about 100% about 30% of the time in the first half of the sample period, so that %∆STFP is frequently significantly biased relative to %∆OTFP; (ii) in the second half of the sample period, %∆STFP %∆OTFP is mostly close to its average of about .1%, so that %∆STFP is not significantly biased relative to %∆OTFP. it n 1 i it v ∑ = α ∑ = α n 1 i it