Abstract
We consider Giffen goods for the very general setting of an upper semi-continuous utility function. Two not equivalent definitions for this notions are discussed. Concerning this we show that there exists a (well-defined) non-decreasing demand function \({\mathbb{R}}_{++} \rightarrow \mathbb{R}\) that is nowhere strictly increasing. Also we reconsider the relation between Giffen and inferior goods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Indeed: for subsets A and B of \(\mathbb{R}\), A ≤ B means a ≤ b (a ∈ A, b ∈ B) and A < B means a < b (a ∈ A, b ∈ B).
- 2.
But also see Exercise 2.F.13 in [5].
- 3.
‘Nowhere monotone’ means that there is no non-degenerate interval on which the function is monotone.
- 4.
- 5.
We do not discriminate between a set with one element (i.e. a singleton) and the element itself.
- 6.
That is, a subset of \({\mathbb{R}}_{++}\) with \(\#\mathcal{P}\geq 2\).
- 7.
Indeed: suppose good i is not Giffen. This implies that for every \(({\mathbf{p}}_{\hat{\imath }}; m)\) and price section \(\mathcal{P}\) good i is not Giffen on \(\mathcal{P}\) at \(({\mathbf{p}}_{\hat{\imath }}; m)\). In particular for every p i ′ < p i and \(({\mathbf{p}}_{\hat{\imath }}; m)\), \(\breve{{x}}_{i}({p}_{i}\prime ) <\breve{ {x}}_{i}({p}_{i})\) does not hold. So, because \(\#\breve{{x}}_{i}(p) = 1\;(p > 0)\), we have for every p i ′ < p i and \(({\mathbf{p}}_{\hat{\imath }}; m)\) that \(\breve{{x}}_{i}({p}_{i}\prime ) \geq \breve{ {x}}_{i}({p}_{i})\). It follows for every \(({\mathbf{p}}_{\hat{\imath }}; m)\) that good i is ordinary on \({\mathbb{R}}_{++}\) at \(({\mathbf{p}}_{\hat{\imath }}; m)\). Thus good i is ordinary.
- 8.
Here is a proof by contradiction: so suppose u is Giffen. Then there is some price section \(\mathcal{P}\) and budget m such that good 1 is Giffen on \(\mathcal{P}\) at m. Let \({p}_{a},{p}_{b}\,\in \,\mathcal{P}\) with p a < p b . Then \(\breve{{x}}_{1}({p}_{a}) <\breve{ {x}}_{1}({p}_{b})\).
Because of the upper semi-continuity of u the maximum v(p; m) of † p; m is well-defined. Because \(\breve{{x}}_{i}({p}_{b}) \subseteq [0,m/{p}_{b}] \subseteq [0,m/{p}_{a}]\), it follows that, v(p b ; m) < v(p a ; m). Because \(\breve{{x}}_{1}({p}_{a}) <\breve{ {x}}_{1}({p}_{b})\), it follows that \(\breve{{x}}_{1}({p}_{a}) \subseteq [0,m/{p}_{b}]\) and therefore v(p a ; m) < v(p b ; m), a contradiction.
- 9.
See the contribution [2] in this book for an overview of concrete Giffen utility functions with various theoretical properties.
- 10.
That is, a subset of \({\mathbb{R}}_{+}\) with \(\#\mathcal{M}\geq 2\).
- 11.
Here is a proof by contradiction. Suppose u is inferior. Then there is a budget section \(\mathcal{M}\) and price p 1 such that good 1 is inferior on \(\mathcal{M}\) at p 1. Let \({m}_{a},\;{m}_{b} \in \mathcal{M}\) with m a < m b . Then \(\hat{{x}}_{1}({m}_{a}) >\hat{ {x}}_{1}({m}_{b})\). Because \(\hat{{x}}_{1}({m}_{a}) \subseteq [0,{m}_{a}/{p}_{1}] \subseteq [0,{m}_{b}/p]\), it follows that v(p; m a ) < v(p; m b ). Because \(\hat{{x}}_{1}({m}_{b}) <\hat{ {x}}_{1}({m}_{a})\), it follows that \(\hat{{x}}_{1}({m}_{b}) \subseteq [0,{m}_{a}/p]\) and therefore v(p; m b ) < v(p; m a ), a contradiction.
- 12.
Here is a proof by contradiction. Suppose \(\inf \mathcal{M}\leq 0\). Then \(\inf \mathcal{M} = 0\). Take \({m}_{0} \in \mathcal{M}\) with \(\hat{{x}}_{i}({m}_{0}) > 0\). Because \(\inf \mathcal{M} = 0\) and \(\min ({p}_{i}\hat{{x}}_{i}({m}_{0}),{m}_{0}) > 0\), there is \(m \in \mathcal{M}\) with \(m <\min ({p}_{i}\hat{{x}}_{i}({m}_{0}),{m}_{0})\). Now \({p}_{i}\hat{{x}}_{i}({m}_{0}) > m \geq {p}_{i} \cdot \hat{ {x}}_{i}(m) > {p}_{i}\hat{{x}}_{i}({m}_{0})\), a contradiction.
- 13.
Also see Example 8 in the contribution [2] in this book.
- 14.
Note that m′≥ 0.
- 15.
We use for \(\mathbf{x},\mathbf{y} \in {\mathbb{R}}^{2}\) the notation \({I}_{[\mathbf{x},\mathbf{y}]}\; :=\;\{ (1 - \lambda )\mathbf{x} + \lambda \mathbf{y}\;\vert \;0 \leq \lambda \leq 1\}\).
- 16.
Even with the following additional properties: \(\vert F\prime (x)\vert \leq 1\;(x \in \mathbb{R})\), \(\{x \in \mathbb{R}\;\vert \;F\prime (x) = 0\}\) is dense in \(\mathbb{R}\) and F′ is over no interval Riemann-integrable.
- 17.
For if F(a) > F(b), then replace F by − F. And if F(a) = F(b), there is always a constant c with a < c < b with F(a)≠F(c). Now replace F(x) by \(\tilde{F}(x)\; :=\; \left (a\frac{b-c} {b-a} + \frac{c-a} {b-a}x\right )\). Now \(\tilde{F}(a) = F(a)\neq F(c) =\tilde{ F}b\). If \(\tilde{F}(a) >\tilde{ F}(b)\), then replace \(\tilde{F}\) by \(-\tilde{F}\).
References
I. Dasgupta and P. K. Pattanaik. Comparative statics for a consumer with multiple optimum consumption bundles. IZA Discussion Paper 4818, Institute for the Study of Labor, Bonn, 2010.
W. Heijman and P. H. M. von Mouche. A child garden of concrete Giffen utility functions: a theoretical review. 655:69–88, 2011.
Y. Katznelson and K. Stromberg. Everywhere differentiable, nowhere monotone, functions. American Mathematical Monthly, 81:349–353, 1974.
H. Liebhafsky. New thoughts about inferior goods. American Economic Review, 59:931–934, 1969.
A. Mas-Colell, M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, New York, 1995.
A. van Rooy and W. Schikhof. A Second Course on Real Functions. Cambridge University Press, Cambridge, 1982.
P. Sørenson. Giffen demand for several goods. 655:97–104, 2011.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Berlin Heidelberg
About this chapter
Cite this chapter
von Mouche, P., Pijnappel, W. (2012). On the Definitions of Giffen and Inferior Goods. 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_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-21777-7_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21776-0
Online ISBN: 978-3-642-21777-7
eBook Packages: Business and EconomicsEconomics and Finance (R0)