Christensen and Greene (1976) estimated a generalized Cobb??Douglas cost function of the form ln(C/Pf) = + ln Q + (ln2 Q)/2 +

Christensen and Greene (1976) estimated a generalized Cobb??Douglas cost function of the form ln(C/Pf) = α + β ln Q + γ (ln2 Q)/2 + δk ln(Pk/Pf) + δ1 ln(Pl/Pf) + ε. Pk, Pl and Pf indicate unit prices of capital, labor, and fuel, respectively, Q is output and C is total cost. The purpose of the generalization was to produce a U-shaped average total cost curve. (See Example 7.3 for discussion of Nerlove??s (1963) predecessor to this study) We are interested in the output at which the cost curve reaches its minimum. That is the point at which (?? ln C/?? ln Q) | Q = Q* = 1 or Q* = exp [(1 ?? β)/γ]. The estimated regression model using the Christensen and Greene 1970 data are as follows, where estimated standard errors are given in parentheses: The estimated asymptotic covariance of the estimators of β and γ is ??0.000187067, R2 = 0.991538 and e' e = 2.443509. Using the estimates given above, compute the estimate of this efficient scale. Compute an estimate of the asymptotic standard error for this estimate then form a confidence interval for the estimated efficient scale. The data for this study are given in Table F5.2. Examine the raw data and determine where in the sample the efficient scale lies. That is, how many firms in the sample have reached this scale, and is this scale large in relation to the sizes of firms in the sample?

In(C/Pr) = -7.294 +0.39091 In Q+0.062413(In Q)/2 (0.34427) (0.036988) (0.0051548) +0.07479 In (Pr/Pr) +0.2608 In (Pi/Pf) + e. (0.061645) (0.068109)