¹ Cambridge: Harvard University Press, 1961. Pp. xix, 472. $8.75.

² Charles Schultze of the University of Maryland has most helpfully suggested changes in this section to increase clarity.

³ If curvature is slight, as in segments W₁ and in Figure 4, the Laspeyres ratio, , is a better estimate of the true ratio than is the Paasche, , and the Paasche ratio is a better estimate of the true ratio, , than is the Lespeyres measure . If curvature is marked, as with segments W₂ and , the Laspeyres measure is a better estimate of the true ratio and the Paasche measure is a better estimate of the true ratio .

⁴ When curvature is slight, as with E₁ and in Figure 5, the Laspeyres ratio is a superior estimate of the true change and the Paasche ratio is a better estimate of the true change . When curvature is marked, as with E₂ and , the Laspeyres ratio is a better estimate of the true change while the Paasche ratio is a better estimate of the true change .

⁵ In measuring the direction of bias, the Laspeyres ratio should be considered on the second vector and the Paasche on the first. Whenever the function considered is homogeneous, each measure will give an ambiguous estimate of the direction of bias on the other vector. This is shown in Figure 5, where the Laspeyres estimate on vector 2, is unambiguously an underestimate of the true change in capacity to produce the second mix whether the production possibilities locus is E₁ or E₂, but where the Laspeyres ratio on the first vector, is an underestimate of the true change on vector 1 when curvature is slight but an overestimate when curvature is marked. Similar conclusions may be developed for the Paasche ratio along vector 2.

⁶ A similar result can sometimes be obtained using a welfare standard when the Paasche index exceeds the Laspeyres, but this is atypical, whereas such results will be typical for the estimation of change in production potential whenever Gerschenkron's hypothesis about relative price changes holds true. The only case where the true estimates of efficiency might fall between the Laspeyres and Paasche indices in Bergson's series is when the Paasche exceeds the Laspeyres— and even then bracketing is not assured.

⁷ While adding information about midperiod changes, it is interesting to note that a chain linked index might actually prove an inferior estimate of change for the period as a whole, because there appears to be a reversal in the trend of relative marginal rates of substitution between commodities which were increasing slowly and those which were increasing most rapidly. This is suggested by the fact that the Paasche measure exceeds the Laspeyres in the period 1944 to 1950. (See the charts on pp. 217 and 226.)