A Monte Carlo Study of Alternative Estimates of the Cobb-Douglas Production Function

A Monte Carlo experimeilt is carried out to examine the small sample
properties of ordinary least squares, indirect least squares, Hoch's, and
Klein's estimates of the parameters of the Cobb-Douglas production function.
A perfectly competitive model of firms in a single industry is considered in
nine situations which differ in tlie behavior of the disturbances, the variability
of inputs, and the positio~l of the average firm. In each case 200 samples
of size 20 and 200 samples of size 100 were obtained to approximate the
sampli~lgd istribution of the various estimators.
1 . INTRODUCTIOS
THECONDITIOXS of profit maximization and the specification of the production
function fully determine the equilibrium position of a firm that operates under
conditions of perfect competition in the product market, obtains its inputs at
fixed prices, arid experiences decreasing returns to scale. If all the relationships
hold exactly, all firms in the industry will be producing identical
quantities of output and will be employing identical quantities of inputs,
providing the inputs are freely variable and substitutable.2 I-ariations from
firm to firm will exist if one or illore of the inputs are fixed; in this case, the
profit maximizing quantities of output and of inputs kvill depend on the
amount of the fixed input or inputs in each firm. If, however, the production
function as well as the profit-maximizing dccision ecluations contain stochastic
disturbances, differences in actual outputs and inputs of firms will appear
even in the absence of fixed factors of prodt~ctionI. n this case a solution of
the system of equations for the quantities of variable inputs and of output
of any firm shows that each quantity is a function of all disturbances in the
system. Consequently, the inputs are not independent of the dsturbance
in the production function, and single-equation least-squares estimates of
the production function parameters based on cross-sectional data will be, in
general, biased and inconsistent. This was first noted in a classical article by
Marschak and Andrews [5] in 1944. Alternative methods of estimation have
since been proposed; these include Klein's, Hoch's, and the indirect leastsquares
procedure. FYith the exception of the first, no small sample properties
of the suggested estimators have been derived.