Abstract
We present a theoretical review of the literature on concrete utility functions for Giffen and inferior goods within the context of the utility maximisation problem under a budget restriction and provide new functional forms. The presentation is organised around the specific properties such utility functions have. These properties include strict increasingness, quasi-concavity, and the applicability of Gossen’s second law.
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Notes
- 1.
An exception is homogeneity.
- 2.
For example in the labour-leisure-model with a Leontief utility function u(l, x) = min(l, x) and budget restriction wl + px ≤ wT, the demand function for leisure is strictly increasing.
- 3.
See the contribution [14] in this book for a specific problem for the case of n goods.
- 4.
In the case the domain is a subset of \({\mathbb{R}}_{+}^{2}\) it is easier to provide Giffen utility functions with reasonable properties. (Also see [3].)
- 5.
Doi et al. [2] is reprinted in this book.
- 6.
- 7.
Also see the contribution [5] in this book for other relationships for Giffen goods.
- 8.
- 9.
The interpretation of (4) is as follows: it only is present for \(({x}_{1},{x}_{2}) \in {\mathbb{R}}_{++}^{2}\) in which u is partially differentiable.
- 10.
See for example [1].
- 11.
In this context, one could wish to add here ‘and that this system easily can be solved by hand’, because if it is difficult to solve the system of equations, the utility function is not really appropriate for text book purposes.
- 12.
Note that for small m a maximiser may be not unique, as can be seen from Fig. 1 by looking to its south west corner.
- 13.
In all our figures the x 1 is on the abscissa and x 2 on the ordinate.
- 14.
Please note the scaling-down on the abscissa.
- 15.
Therefore in [15] also a ‘repaired’ version of u 1 T was used to improve the properties of the indifference sets. The repair consisted on modifying u 1 T for x 1 > 50. But this only improved these properties a little bit (the modified function still is not quasi-concave) and lead to a piece-wise definition, which made this function more complicated and thereby destroyed the partially differentiability of the utility function (on \({\mathbb{R}}_{++}^{2}\)).
- 16.
Even if it can with other parameter values, then it is correct to say that this problem first was solved in [15].
- 17.
But not necessarily interior for positive budget.
- 18.
Even demand functions that are not log-convex seem to be rare.
- 19.
But not smooth.
- 20.
Furthermore it may be worthwhile to note that in Example 2 the maximisers are in the good region D (as the proof of the example shows).
- 21.
In fact in [6] the correctness of the claim that the utility function is inferior is not proved. There just is referred to a book with a not correct analysis: it is stated that the formula for \(\hat{\mathbf{x}}(m)\) in Example 8 for the case \(m > 2{p}_{2}\sqrt{\alpha }\) and t(m) > 0 also is correct in case t(m) ≤ 0. However, for \(\alpha = \frac{1} {10},\;{p}_{1} = 1,\;{p}_{2} = 1,\;m = \frac{21} {10}{p}_{2}\sqrt{a}\), one has t(m) < 0 which (now referring to our proof in the appendix) implies \(U(m/{p}_{1}) > U({y}_{-}(m)/{p}_{1})\) from which it follows that (0, m ∕ p 1) is the unique maximiser instead of \(({y}_{-}(m)/{p}_{1},{y}_{+}(m)/{p}_{2})\).
- 22.
Here is a proof: let f be the function \({c}_{1}^{-}- {c}_{2}\). Suppose m is a zero of f. Then \({(\alpha /(m + 1) - \alpha /2)}^{2} ={ (\sqrt{{\alpha }^{2 } /{(m + 1)}^{2 } - 2} -\sqrt{({\alpha }^{2 } - 8m)/2}\;)}^{2}\). Evaluating this equation, next simplifying and then squaring the result after appropriate rearranging, leads to \(2{m}^{4} - ({\alpha }^{2} + 4){m}^{2} + 2m{\alpha }^{2} - {\alpha }^{2} + 2 = 0\). Because \(2{m}^{4} - ({\alpha }^{2} + 4){m}^{2} + 2m{\alpha }^{2} - {\alpha }^{2} + 2 = 2{(m - 1)}^{2}(m - \alpha /\sqrt{2} - 1)(m + \alpha /\sqrt{2} + 1)\), we can conclude that the zeros of f are 1 and \(\alpha /\sqrt{2} - 1\). Noting that \( f\prime (1) = (\sqrt{{\alpha }^{2 } - 8} - a) < 0\) and that f is continuous, it follows that f < 0 on the interval \((1,\alpha /\sqrt{2} - 1)\), as desired.
It would be interesting to understand the deeper meaning why the inequality c 1 − (m) < c 2(m), that in fact came up in a natural way, holds.
- 23.
A straightforward calculation shows that
$$t\prime \,=\,\frac{1} {4} \frac{\left (\sqrt{{m}^{2 } \,-\, 4{p}_{2 }^{2 }\alpha }\left (m\,+\,\sqrt{{m}^{2 } \,-\, 4{p}_{2 }^{2 }\alpha }\right )\right ){e}^{\frac{{\left (m+\sqrt{{m}^{2 } -4{p}_{2 }^{2 }\alpha }\right )}^{2}} {8{p}_{2}^{2}} } \left (m\,+\,\sqrt{{m}^{2 } \,-\, 4{p}_{2 }^{2 }\alpha }\right ){\left (\frac{m-\sqrt{{m}^{2 } -4{p}_{2 }^{2 }\alpha }} {2m} \right )}^{\alpha }} {m{p}_{2}^{2}\sqrt{{m}^{2 } - 4{p}_{2 }^{2 }\alpha }}.$$This implies that t′ > 0 and thus that t is strictly increasing. Therefore for m 0 even each \({m}_{0} > 2{p}_{2}\sqrt{\alpha }\) with t(m 0) > 0 can be taken.
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Heijman, W., von Mouche, P. (2012). A Child Garden of Concrete Giffen Utility Functions: A Theoretical Review. 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_6
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