Decomposing growth in revenues and costs into price, quantity and total factor productivity contributions

Abstract

This article employs the superlative Fisher and Törnqvist indexes for exact decomposition of growth in nominal revenues and costs. The findings confirm the well-known result that these indexes very closely approximate each other, implying that the mathematically simpler and computationally easier Törnqvist is the more practicable index. Moreover, this article's nominal growth decomposition yields all the results from the more common real growth decomposition and is also more informative for policy purposes. Application to the US agricultural sector during 1948–2001 shows that of the 3.31% average annual growth in revenues, TFP growth contributed 1.90 percentage points (pct. pts.); growth in output prices added 1.43 pct. pts.; while growth in input quantities contributed – 0.02 pct. pts. (i.e. fewer inputs). Therefore, real output growth (or revenue growth less output price growth) was 1.88 pct. pts., revealing that TFP's 1.90 pct. pts. growth contribution was fully responsible for real output growth with fewer inputs. Since revenues measure incomes, these results suggest that policy should focus more on measures to foster TFP growth than on specific price or quantity instruments to enhance income growth.

Notes

¹The factor reversal property defined above and the time reversal property—that the product of a price (or quantity) index when time moves from s to t and the same price (or quantity) index when time moves from t to s equals 1—together make an index ‘ideal’ according to Fisher (1922).

²Diewert (1976) used ‘superlative’ for an index that is exact in that it can be derived algebraically from a flexible behavioural equation, i.e. the equation provides a second-order approximation to an arbitrary twice continuously differentiable linearly homogeneous aggregator function (e.g. a consumer's expenditure function or a producer's cost function). By this criterion, Diewert showed that the Fisher and Törnqvist indexes are both superlative, except that the aggregator function for the Fisher index is ‘quadratic’ while that for the Törnqvist index is ‘translog’.

³The Diewert and Morrison (1986) decomposition is also presented in Diewert and Nakamura (2003).

⁴Our decompositions of growth in revenues and costs are presented later in Tables 1 and 2. These decompositions cover the entire US agricultural sector during the period 1948 to 2001.

⁵The identity (≡) is used above for definitions and the equality (=) is used for results of algebraic operations.

⁶Balk (2004) showed that the Fisher quantity indexhas both additive and multiplicative decompositions. The difference is that the additive decomposition yields the arithmetic growth ratewhile the multiplicative decomposition yields the logarithmic growth rate. Dumagan (2002) derived the exact additive decomposition—that Balk (2004) recognized as a rediscovery of van IJzeren's (1952) decomposition—and showed that the Fisher arithmetic growth rateand the Törnqvist logarithmic growth rate mathematically approximate each other. This present article provides (in a later section) a new analytic basis for the closeness between the logarithmic growth ratesandto solidify the earlier finding that the Fisher and Törnqvist indexes approximate each other.

⁷Since we are assuming constant returns to scale and competitive profit maximization, technical change and TFP are the same.

⁸In general, if x represents the index level or the relative change from s to t, where t = s + 1, then ln(x) implies the growth equation, x = exp[ln(x)]. By being the exponent of the base of natural logarithms, ln(x) is, by definition, the logarithmic growth rate.

⁹The Törnqvist TFP indexes in Equation 23 are exact algebraic derivations from a hypothesized flexible translog functional form for the theoretical productivity index. For details of the derivation, see Diewert and Morrison and Kohli mentioned earlier in this article.

¹⁰The inequalities in Equation 44 may be visualized as follows. The arithmetic percent growth of the Laspeyres quantity index, for example, is defined by the straight line, y =  × 100. The logarithmic percent growth is defined by the curved line, y* = ln] × 100. The two lines are tangent at y = y* = 0 when  = 1, but the straight line lies above the curved line for all values of other than 1, i.e. y > y* when  ≠ 1.

¹¹In Equations 46 and 47, if the difference between the two terms in the left-hand side is positive then in the right-hand side their ratio is greater than 1 so that the logarithm of this ratio is positive. Conversely, a negative difference in the left-hand side yields in the right-hand side a ratio less than 1 that has a negative logarithm.