Abstract
We characterize the family of non-contestable budget-monotone rules for the allocation of objects and money as those obtained by maximizing a maxmin social welfare function among all non-contestable allocations. We provide three additional seemingly independent approaches to construct these rules. We present three applications of this characterization. First, we show that one can “rectify” any non-contestable rule without losing non-contestability. Second, we characterize the preferences that admit, for each budget, a non-contestable allocation satisfying a minimal or maximal individual consumption of money constraint. Third, we study continuity properties of the non-contestable correspondence.
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Notes
This axioms is commonly referred to as “no-envy.” See Velez (2016) for a discussion why “non-contestability” reflects better the normative content of the axiom.
Money-monotonicity and the compensation assumption imply continuity.
This family of rules can be seen as the implementation of the Rawlsian principle of distributive justice (Rawls 1972) constrained by efficiency and no-envy.
Let \(\{x_k\}_{k\in \mathbb {N}}\) be a bounded sequence of real numbers and \(x\in \mathbb {R}\). If all convergent subsequences of \(\{x_k\}_{k\in \mathbb {N}}\) converge to x, then the sequence \(\{x_k\}_{k\in \mathbb {N}}\) is convergent and its limit is x.
The domain of piece-wise linear preferences can be defined without reference to utility representations by interpolating a finite set of ordered indifference sets.
(i) in the Strict Monotonicity Theorem was first stated and proved by Alkan et al. (1991). Our contribution here is to provide a direct proof of it that requires no linear approximation.
The existence of \(z'\equiv (x',\mu ')\in F_{m+\varepsilon /2}\) is the essential step in our proof. In their construction, Alkan et al. (1991) prove both the strict monotonicity theorem and existence of non-contestable allocations from linear approximations. An earlier version of this paper presented a proof of Theorem 2 based on the following corollary of Svensson (1983): for each positive budget there are non-contestable allocations at which each agent consumes a non-negative amount of money whenever each agent is indifferent among all bundles that have zero money.
This can also be easily seen from the characterization of value-maxmin, value-minmax, money-maxmin, and money-minmax allocations by means of the chains of indifferences that connect the agents’ consumptions at the allocation introduced by Alkan (1994) and used to understand incentive properties of non-contestable rules by Velez (2011) Fujinaka and Wakayama (2015) and Andersson et al. (2014a, b).
An example that shows this is a rule f that assigns the same allotment to an agent, say i, for an interval say [l, h] but otherwise the welfare of all agents increases with the aggregate budget. Here \(M=(-\infty ,l)\cup (h,+\infty )\). If g is non-contestable and budget-monotone and coincides with f on M, all agents prefers g(h) to \(g((l+h)/2)\) and \(g((l+h)/2)\) to g(l). Since g coincides with f on M, for agent i, g(h) is no better than f(h) and g(l) is no worse than f(l). Thus, agent i is indifferent between g(h) and g(l). Thus, there is no such a g.
Here we abuse of notation and write \([l_t',h_t']\) when \(l_t'=-\infty \) or \(h_t'=+\infty \).
The correspondence \(m\in \mathbb {R}\mapsto F_m\) may be discontinuous, for an assignment of objects may be sustained in a non-contestable allocation for only a subset of all possible budgets.
It is an open question to determine whether the rules that are minimally manipulable in the sense of Andersson et al. (2014b) are budget-monotone.
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Thanks to William Thomson, two anonymous referees, an associate editor, and the audience at SCW16 for useful comments and discussions.
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Velez, R.A. Sharing an increase of the rent fairly. Soc Choice Welf 48, 59–80 (2017). https://doi.org/10.1007/s00355-016-1018-4
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DOI: https://doi.org/10.1007/s00355-016-1018-4