Dynamic efficiency measurement: theory and application.

The kernel estimation procedure generates point estimates of the marginal cost of adjustment for each quasi-fixed factor and for each farm in each year. (13) The value of these estimates is nearly zero for all quasi-fixed factors and for all farms in all years, implying small changes in the initial stock of the quasi-fixed factors has no impact on the behavioral value function. The standard errors associated with these estimates are approximately zero implying there is little variability in these estimated values. (14)

The lower bound on the dynamic cost efficiency in equation (24) requires information on the value of the dynamic undercost, rW(.,[V.sub.o]). The procedure to generate the dynamic undercost is similar to the one used to compute the dynamic overcost. Consider the dynamic undercost in equation (7) and the Kuhn-Tucker conditions in equation (8). The second condition in equation (8) indicates that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be related with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Recognizing this relation, the Kuhn-Tucker conditions in equation (8) become

(43) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Kuhn-Tucker conditions in equation (43) can also be written as the LCP in equation (40). Similarly, the LCP can be restated as the quadratic programming problem in equation (41). The solution obtained by solving equation (41) provides the optimal variable input and gross investment vectors solving the minimization problem in equation (7), the value of the dynamic undercost function and the underlying shadow values of the quasi-fixed factors.

Empirical Efficiency Results

The efficiency measures are empirically illustrated using this balanced panel data set of Pennsylvania dairy operators during the period 1987-92. Efficiency scores are generated for each Pennsylvania farm operator in each year. Due to space restrictions, efficiency levels are not reported for each farm. The efficiency results by farm are available in Silva and Stefanou (2007).

Short-Run Efficiency

The empirical results indicate that upper and lower bounds on technical (allocative) efficiency convey similar information on the technical (allocative) performance of Pennsylvania farm operators in the use of variable inputs (tables 1-4). The median of the upper and lower bounds on both efficiency levels is equal to one in each year which combined with the excess of kurtosis indicate a unimodal distribution with a concentration of mass in the upper tail of the distribution. For illustrative purposes, figures 3 and 4 depict the cross-farm frequency distributions of upper and lower bounds on technical and allocative efficiency in the first and last years. As expected, technical and allocative efficiency scores cannot be approximated by a normal distribution: normality is rejected in all years at the 1% significance level. (15)

[FIGURES 3-4 OMITTED]

Similarity of efficiency results provided by the two bounds is confirmed by the Smirnov two-tailed test conducted on upper and lower bounds on each efficiency level. The statistical test results indicate that empirical distribution functions of the lower and upper bounds on each efficiency level are identical in all years. (16) The p-value of the two-tailed test is greater than 0.20 for all years in the case of technical efficiency. Considering the allocative efficiency, the p-value is greater than 0.20 in all years, except in 1988 the value is approximately equal to 0.20.

Additionally, the Page test is performed on the lower and upper bounds on each efficiency level to detect whether there is no difference in the average efficiency over the six-year period or the average efficiency is increasing or decreasing over time. (17) Three patterns of efficiency performance can be identified for dairy operators as a whole from this test: (i) stable performance, meaning there is no difference in the average efficiency, (ii) progressive performance, meaning that the average efficiency is increasing over time and (iii) regressive performance, meaning that the average efficiency tends to become smaller as time goes on. Results of the Page test indicate a stable (technical and allocative) efficiency performance over the six-year period. When (ii) [(iii)] is the alternative hypothesis postulated for technical efficiency, the p-value is 0.571 [0.429] for the upper bound and 0.135 [0.865] for the lower bound. When the alternative hypothesis postulates the average allocative efficiency is increasing (decreasing) over the six-year period, the p-value is 0.884 (0.116) for the upper bound and 0.11 (0.879) for the lower bound.

Although the cross-farm frequency distributions of the upper and lower bounds on technical and allocative efficiency scores have similar features (e.g., skewed to the right), the concentration of mass in the upper tail of the distribution is more accentuated in the case of technical efficiency scores. This suggests that Pennsylvania dairy operators reveal more difficulty in choosing the optimal proportion of variable inputs given market input prices than exploring their production potential.

Long-Run Efficiency

Contrarily to short-run results, the long-run empirical results on the upper and lower bounds on technical (allocative) efficiency provide substantially different information on the technical (allocative) performance of dairy operators (tables 5-8). There is strong evidence that empirical distribution functions of the upper and lower bounds on each efficiency level are not identical. The Smirnov two-tailed and one-tailed tests are conducted for both bounds on each efficiency level in each year and results find the p-value is 0.01 (0.005) for the two-tailed (one-tailed) test in all years.

Statistical information on upper and lower bounds on the technical efficiency of all inputs are presented in tables 5 and 6. In both cases, the median is less than one in all years and technical efficiency levels cannot be approximated by a normal distribution. The frequency distribution of the upper bound on technical efficiency scores by year is negatively skewed toward the lower values while there is a moderate concentration of mass in the lower tail of the distribution of the lower bound. For illustrative purposes, figure 5 depicts the cross-farm frequency distributions of the upper and lower bounds on technical efficiency in 1987 and 1992.

[FIGURE 5 OMITTED]

Upper and lower bounds on technical efficiency also diverge with respect to the efficiency performance over time. The Page test indicates no difference in the average of the upper bound on technical efficiency over the six-year period, suggesting Pennsylvania dairy operators as a whole reveal a stable technical efficiency performance over time. The p-value is 0.899 (0.123) when the alternative hypothesis postulates an increasing (decreasing) average efficiency over time. The average of the lower bound on technical efficiency is increasing over time (the null hypothesis is rejected at the 0.1% significance level), indicating dairy operators as a whole are improving in terms of technical efficiency. Despite this divergence, the stable technical efficiency performance and the progressive performance inferred from upper and lower bounds, respectively, may indicate that the two bounds on the technical efficiency of all inputs tend to converge over time.

Tables 7 and 8 report statistical information on upper and lower bounds on the allocative efficiency of all inputs. The median of the lower bound is less than one in all years and allocative efficiency scores can be approximated by a normal distribution in all years, except in 1990 and 1992 (table 7). In 1990 and 1992, normality is rejected at the 3% and 1% significance level, respectively. The distribution of allocative efficiency scores in 1990 and 1992 is slightly skewed to the right. Contrasting with the lower bound, the distribution of the upper bound on allocative efficiency scores cannot be approximated by a normal distribution. Normality is rejected at [alpha] = 1% in all years. The median is equal or close to the maximum value across the years and the distribution of the upper bound is negatively skewed toward the lower values (table 8). As a matter of illustration, figure 6 depicts the frequency distribution of the two bounds on allocative efficiency in the first and last years.

[FIGURE 6 OMITTED]

Additionally, there is some evidence that upper and lower bounds on allocative efficiency differ with respect to the efficiency performance over time. Results from the Page test performed on the upper bound indicate the average allocative efficiency is decreasing over time (the null hypothesis is rejected at the 0.1% significance level in one of the tests). In the case of the lower bound, there is some evidence of no difference in the average efficiency over the six-year period (we fail to reject the null hypothesis at the 5% significance level, although rejection occurs at the 10% level). Nevertheless, the regressive allocative performance inferred from the upper bound and the stable allocative performance indicated by the lower bound suggest that these bounds tend to converge over time.