Comment on ‘A reevaluation of price elasticities for irrigation water’ by Richard E. Howitt, William D. Watson, and Richard M. Adams

Abstract

The crux of the Howitt et al. [ 1980] article is that the use of quadratic programing (QP) rather than linear programing (LP) in estimating the derived demand for irrigation water will yield more elastic estimates of demand. Their empirical example purports to demonstrate these results, which they first claimed to be conceptually correct on the basis of a mathematical proof. Their assertion was so lacking in intuitive appeal that we were led first to examine the empirical illustration for logical consistency. When the example did not prove credible, we then examined the basic mathematics. While the mathematics itself was correct, interpretation of the crucial equation was not. INTUITIVE CONCEPTSThe practical advantage of QP over LP in estimating the derived demand for water is that instead of assuming product prices constant as the price of water rises (the LP formulation), the QP formulation provides for rising product prices as less of the product is produced for sale. This QP formulation is in conformance with the generally accepted notion of However, intuition suggests that the QP formulation should normally develop a derived demand curve for water that is less elastic than a demand curve developed from a LP formulation, rather than more elastic as Howitt et at. claim. Because product prices are allowed to rise as less water is used in production and output is decreased, the products will be better able to pay for higher-cost water than is projected in the LP formulation, and the QP demand curve will rise more steeply than the LP curve from a common base (see our Figure 1). Beginning at any given point a, representing current reality, a rise in the price of irrigation water should have less of an effect on water use under the QP formulation. One interpretation would be to substitute Py = Py* (a constant) for their market clearing condition (6). However, if output is allowed to vary while the product price is held constant, their theoretical model provides indeterminate results. This difficulty arises from the fact that under a constant product price the homogeneous of degree 1 (constant returns to scale) production function they assume results in undefined maximum profits as well as undefined input and output levels.An alternative interpretation is that the product price is fixed in their market clearing equation. Then, output is also fixed, and