Abstract
We demonstrate that a well-behaved utility function can generate Giffen behavior, where “well-behaved” means that its indifference curves are smooth, convex, and closed in a commodity space; the resulting demand function of each good is differentiable with respect to prices and income. Moreover, we show that Giffen behavior is compatible with any level of utility and an arbitrarily low share of income spent on the inferior good. This contrasts sharply with the common view that the Giffen paradox tends to occur when households’ wealth levels are low.
Comment from the editors: this article concerns a reprint from Economic Theory, Volume 41(2), pages 247–267, 2009.
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Notes
- 1.
Considering a general equilibrium model, Nachbar [5] points out that there is a fundamental difficulty in conducting empirical analyses on Giffen goods, since price changes in important goods often result in changes in households’ income. However, in the case of Sørensen’s example, the Giffen effect arises at the point where the substitution effect is zero, and hence, the amount of goods consumed is irrelevant.
- 2.
Using a specific utility function, Vandermeulen [9] also shows that expenditure shares are irrelevant to Giffen behavior because Giffen behavior depends solely on a property of the indifference map, i.e., the slope of indifference curves along vertical lines, and conjectures that a good is predisposed toward a Giffen good when it is a necessity and a household is rapidly satiated with its consumption.
- 3.
This yields another difference between the function we will propose and the other functions: the demand curve based on the former is smooth over the whole domain, while those based on the latter have a kink, where their slope changes from negative to positive discontinuously when the price falls below its threshold value. One can confirm that this discontinuity occurs because when one good turns into a Giffen good, the optimal consumption bundle changes from a corner solution to an interior one, or from an interior one where the first-order conditions are met with equalities to another interior one where they are not.
- 4.
Even if the utility function takes the form of г over \({\mathbb{R}}_{+}^{2}\), we can obtain all the following results except for global non-decreasingness and quasi-concavity of the utility function (Lemma 2).
- 5.
Since the first derivatives u 1 and u 2 are nonnegative and positive, respectively, over \({\mathbb{R}}_{+}^{2}\), the budget constraint necessarily holds with equality.
- 6.
Since \(G({x}_{1}) = \gamma p\big{(}{x}_{1} - \frac{I} {2p}{\big{)}}^{2} + \alpha -\frac{\gamma {I}^{2}} {4p}\), Type I budget constraints correspond to the case of \(I < 2{(\alpha p/\gamma )}^{\frac{1} {2} }\) and vice versa.
- 7.
As shown in the preceding section, for any (p, I), the optimal consumption pair exists in \({\mathbb{R}}_{++}^{2} - {\Lambda }_{0}\); therefore, the value of z that corresponds to the pair must be between 0 and α.
- 8.
Needless to say, from the definition of ζ, the equality \(\zeta (p/\gamma {I}^{2}) = \gamma {\phi }_{1}(p,I){\phi }_{2}(p,I)\) holds.
- 9.
\((\beta - \alpha ) -\sqrt{\beta (\beta - \alpha )/2} < 0\) holds for
$$\begin{array}{rcl} \left [(\beta - \alpha ) -\sqrt{\frac{1} {2}\beta (\beta - \alpha )}\right ] \times \left [(\beta - \alpha ) + \sqrt{\frac{1} {2}\beta (\beta - \alpha )}\right ]& =& {(\beta - \alpha )}^{2} -\frac{\beta (\beta - \alpha )} {2} \\ & =& (\beta - \alpha )\left (\frac{\beta } {2} - \alpha \right ), \\ \end{array}$$which is negative due to the Assumption.
- 10.
The definitions of income expansion path and price offer curve are given in standard textbooks of microeconomic theory. For example, see Varian [10, p. 116–118].
References
Doi, J., Iwasa, K., Shimomura, K.: Indeterminacy in the free-trade world. J Differ Equ Appl 13, 135–149 (2007).
Iwasa, K., Shimomura, K.: A family of utility functions which generate Giffen paradox. J Econ Kwansei Gakuin Univ 60(3), 29–45 (2007).
Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford: Oxford University Press (1995).
Moffatt, P.G.: Is Giffen behavior compatible with the axioms of consumer theory? J Math Econ 37, 259–267 (2002).
Nachbar, J.H.: The last word on Giffen goods? Econ Theory 11, 403–412 (1998).
Nishimura, K., Shimomura, K.: Indeterminacy in a dynamic two-country model. Econ Theory 29, 307–324 (2006).
Sørensen, P.N.: Simple utility functions with Giffen demand. Econ Theory 31, 367–370 (2007).
Spiegel, U.: The case of a Giffen good. J Econ Educ 25, 137–147 (1994).
Vandermeulen, D.C.: Upward sloping demand curve without the Giffen paradox. Am Econ Rev 72, 453–458 (1972).
Varian, H.R.: Microeconomic Analysis, 3rd edn. New York: W.W. Norton & Company (1991).
Weber, C.E.: The case of a Giffen good: Comment. J Econ Educ 28, 36–44 (1997).
Wold, H., Jureen, L.: Demand Analysis. New York: Wiley (1953).
Acknowledgements
Comments from Murray C. Kemp, Binh Tran-Nam, Ngo Van Long, Masao Oda, Noritsugu Nakanishi, and Chiaki Hara have greatly improved the paper. We have also benefited from discussions with Koichi Hamada, Satya Das, Takashi Kamihigashi, Tomoyuki Kamo, Toru Kikuchi, Katsufumi Fukuda, Yu-chin Chen, Fahad Khalil, Takeshi Nakatani, Kazuo Nishimura, Ken-Ichi Shimomura, and Stephen J. Turnovsky. We thank the anonymous referee for his/her helpful comments. Iwasa would like to acknowledge financial support from Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Professor Koji Shimomura passed away on February 24, 2007. This paper was completed after his untimely death.
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Doi, J., Iwasa, K., Shimomura, K. (2012). Giffen Behavior Independent of the Wealth Level. In: Heijman, W., von Mouche, P. (eds) New Insights into the Theory of Giffen Goods. Lecture Notes in Economics and Mathematical Systems, vol 655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21777-7_9
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