Abstract
The purpose of this study is to investigate a relationship between egalitarian equivalence and envy-freeness in a continuum-agent economy, where tastes vary continuously across individuals. Under efficiency, the two criteria of equity are not compatible, except in the knife-edge case. In particular, when individual utility functions are restricted to the class of Cobb–Douglas-type functions, there exists an efficient, egalitarian-equivalent, and envy-free allocation if and only if all individuals have the same taste over commodities.
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Notes
Pazner and Schmeidler (1978) prove the existence of an egalitarian-equivalent and efficient allocation. For the incompatibility between efficiency and egalitarian equivalence in a social-preference framework, see Tadenuma (2005). Chun et al. (2014) combine egalitarian equivalence with strategy-proofness.
Postlewaite’s argument is incorporated in the work of Daniel (1978, p. 563).
That is, we consider the following Cobb–Douglas utility function with three goods: \(u(x_1,x_2, x_3,t)=\alpha (t) \log x_1 + \beta (t) \log x_2 +(1-\alpha (t)-\beta (t))\log x_3\), where \(\alpha : T \rightarrow (0,1)\) and \(\beta : T \rightarrow (0,1)\) are continuous and differentiable on T. Then Eq. (1) becomes \(\alpha '(t)\log \frac{f_1(t)}{f_3(t)}+\beta '(t)\log \frac{f_2(t)}{f_3(t)}=\alpha '(t)\log \frac{z_1}{z_3}+\beta '(t)\log \frac{z_2}{z_3}\). If \(\alpha '(t)=\beta '(t)\), then we can proceed with the same argument as in the case with two goods. However, under the general case with \(\alpha '(t) \ne \beta '(t)\), our previous argument for two goods is not robust, although the restriction given by (1) is still very restrictive.
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Acknowledgements
I am grateful to Tomoki Inoue, Katsuhito Iwai, Kazuya Kamiya, Tomohiko Kawamori, Toyotaka Sakai, and two anonymous referees of this journal for their many helpful conversations and suggestions. I also thank GATE L-SE for hospitality. This work was financially supported by JSPS KAKENHI (18K01501). This project was also supported by the Postdoctoral Fellowship for Research Abroad of JSPS.
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Cato, S. Compatibility of egalitarian equivalence and envy-freeness in a continuum-agent economy. Econ Theory Bull 8, 97–103 (2020). https://doi.org/10.1007/s40505-019-00168-2
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DOI: https://doi.org/10.1007/s40505-019-00168-2