Abstract
Let a consumer consume two goods, and let good 1 be a Giffen good. Then a well-known necessary condition for such behaviour is that good 1 is an inferior good. This paper shows that an additional necessary condition for such behaviour is that good 1 is a gross substitute for good 2, and that good 2 is a gross complement to good 1 (strong asymmetric gross substitutability). It is argued that identifying strong asymmetric gross substitutability as an additional necessary condition gives better insight into Giffen behaviour, both on an analytical level and an intuitive level.
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Notes
- 1.
Specifically, by Hicks’ Composite Commodity Theorem (for a proof, see Carter [2]), all other goods than good 1 can be treated as a composite good as long as the relative prices of all these other goods do not change. A change in the price of good 2 is then interpreted as a price change that leaves the relative prices of all other goods than good 1 intact.
- 2.
By the fact that the Hicksian demand for good 1 is homogeneous of degree zero, it follows from Euler’s formula that \({p}_{1}(\partial {h}_{1}/\partial {p}_{1}) + {p}_{2}(\partial {h}_{1}/\partial {p}_{2}) = 0\). We know that ∂h 1 ∕ ∂p 1 ≤ 0, so that it follows that ∂h 1 ∕ ∂p 2 ≥ 0.
- 3.
There are also utility functions that lie on the edge of each of these sets, e.g. preferences that are neither characterised by strong asymmetric gross substitutability, nor by weak symmetric gross substitutability. These and other preferences on the edges of the different sets in Fig. 4 have a specific algebraic form, and are treated in [3].
- 4.
For ε < 1, u is not defined in (0, x 2). This problem may be overcome by taking g = e u instead of u, with g(0, x 2) = 0.
- 5.
References
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De Jaegher, K. (2012). Giffen Behaviour and Strong Asymmetric Gross Substitutability. 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_5
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DOI: https://doi.org/10.1007/978-3-642-21777-7_5
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