A Convenient Utility Function with Giffen Behaviour

The paper proposes a simple utility function that can generate Giffen behaviour. The function suggests an alternative direction where Giffen behaviour can be found and also implies a convenient framework for empirical testing. Moreover, because of its simple form, the utility function is well-suited for teaching purposes. 1. Introduction It was not until the third [1, 2] edition of his Principles that Alfred Marshall stated that the law of demand may not always hold. Marshall inserted a new paragraph with the famous “Giffen paradox,” in which he argues that, under subsistence conditions, a rise in the price of a cheap foodstuff (bread) can force poor families to consume more, rather than less of it. (“(A)s Mr Giffen has pointed out, a rise in the price of bread makes so large a drain on the resources of the poorer labouring families and raises so much the marginal utility of money to them, that they are forced to curtail their consumption of meat and the more expensive farinaceous foods: and, bread being still the cheapest food which they can get and will take, they consume more, and not less of it. But such cases are rare; when they are met with they must be treated separately” ([1], page 208; [2], pages 109, 110).) The precise meaning of this paragraph has kept the minds of many economists occupied for more than a century, as has the more general issue of the possibility of an upward sloping segment of the demand curve (see, [3–8]). The discourse has been fuelled by the difficulty experienced in finding convincing empirical evidence of Giffen behaviour. The standard textbook example of the Irish potato, popularized by Paul Samuelson's Economics ([9], page 432), has been discredited (see, [10, 11]). Only recently, Jensen and Miller [12] claimed to have found the first rigorous evidence of Giffen behaviour—rice consumption by very poor Chinese households. Another difficulty has been that, although it was long recognized that the axioms of consumer theory allow for an upward sloping demand curve, concrete utility functions with this property were hard to formulate. Thus, textbooks usually illustrate the Giffen phenomenon by a picture with an indifference map and some arrows indicating a substitution effect that is offset by a positive income effect. This paper deals with the latter problem by proposing a convenient utility function that implies Giffen behaviour in the case of utility maximization under a fixed-income constraint. As far as I know, there are only a few publications with explicit utility functions with Giffen behaviour; a brief overview