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Social Choice and Welfare

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01-02-2015

Non fixed-price trading rules in single-crossing classical exchange economies

Author: Mridu Prabal Goswami

Published in: Social Choice and Welfare | Issue 2/2015

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Abstract

This paper defines the single-crossing property for two-agent, two-good exchange economies with classical (i.e., continuous, strictly monotonic, and strictly convex) individual preferences. Within this framework and on a rich single-crossing domain, the paper characterizes the family of continuous, strategy-proof and individually rational social choice functions whose range belongs to the interior of the set of feasible allocations. This family is shown to be the class of generalized trading rules. This result highlights the importance of the concavification argument in the characterization of fixed-price trading rules provided by Barberá and Jackson (Econometrica 63:51–87, 1995), an argument that does not hold under single-crossing. The paper also shows how several features of abstract single-crossing domains, such as the existence of an ordering over the set of preference relations, can be derived endogenously in economic environments by exploiting the additional structure of classical preferences.

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There are several variants of this result for two agents, in restricted domains; for example Schummer (1997) for linear preferences, Ju (2003) for classical quasi-linear preferences, and Hashimoto (2008) for Cobb-Douglas preferences.

These rules have also been shown to be salient in other mechanism design problems. For instance, Hagerty and Rogerson (1987) consider a bilateral trading model with quasi-linear utility functions and show that a strategy-proof, individually rational and budget-balanced SCF is an FPT rule.

For \(x,z \in \mathfrak {R}_{+}^{2}\), by \(x>z\) we mean \(x_{k}\ge z_{k}\) for all \(k=1,2\) and \(x_{k}>z_{k}\) for some \(k\).

We realize that the maximum of \(R_{i}\) on \(B\) does not depend on \(R_{j}\). However as we shall see, this notation helps in defining the tie-breaking rule for FPT rules. Later in the paper while describing our results, we will drop \(R_{j}\) from the notation because tie-breaking is not required in our characterization.

To define the median, an order on the piecewise linear graph needs to be fixed. Let \(a\) and \(b\) be two diagonal allocations that are on the piecewise linear graph. Fix an agent \(i\). We say \(a>_{i}b\) if \(a_{i}^{x}>_{i}b_{i}^{x}\) (or \(a_{i}^{y}<_{i}b_{i}^{y}\)). Let \(\big \{a^{1},a^{2},a^{3}\}\) be a set of diagonal allocations. We define \(\text {median}\big \{a^{1},a^{2},a^{3}\big \}\in \big \{a^{1},a^{2},a^{3}\big \}\) to be the median of \(\big \{a^{1},a^{2},a^{3}\big \}\) if \(\big |\big \{a^{j}|a^{j}\ge _{i} \text {median}\big \{a^{1},a^{2},a^{3}\big \}\big \}\big |\ge 2\) and \(\big |\big \{a^{j}|\text {median}\big \{a^{1},a^{2},a^{3}\}\ge _{i} a^{j} \big \}\big |\ge 2\).

Barberá and Jackson (1995)’s result holds for an arbitrary number of agents, but they impose further assumptions on the SCF.

For any \(a,b \in \Delta , \,aP_{i}b\) means \(a_{i}\) is preferred to \(b_{i}\) by agent \(i\) with preferences \(R_i\), while \(aI_{i}b\) means that agent \(i\) is indifferent between \(a_{i}\) and \(b_{i}\) under \(R_i\)

The notation \(R_{i}|_{\{a,b,c\}}\) refers to \(R_{i}\) restricted to \(\{a,b,c\}\).

The usual definition of single-peaked preferences apply on a diagonal set according to this order. For instance, if a diagonal set is a straight line segment and lies in the interior of \(\Delta \), then all the allocations on it can be sustained as tops for some preference ordering from \(\mathbb {D}^{s}\) (this follows from arguments similar to Proposition 4). Also, on both sides of the top, the allocation nearer (note that under the order, distance between two allocations can be defined in the Euclidean sense) to it are preferred to the one which is further.

An ordered set \(S\) is said to have the least upper bound property if every bounded subset of \(S\) has the supremum in \(S\).

See Munkres (2005), p 169.

Note that such an upper bound need not exist if either the indifference curves are not strictly convex or the allocation that is considered is not in the interior of \(\Delta \).

If \(d\) is not an interior allocations in \(\Delta \), then this may not be true. We will discuss about this later.

For any relevant \(\theta _{1}\) the solution given by \((\frac{3\theta _{1}}{4})^{6}\) indeed corresponds to the maximum. To see this note that the maximization problem can be equivalently written as \(\text {Max}\;\theta _{1}\sqrt{x_{1}}-x_{1}^{\frac{2}{3}}\). The first order condition is \(\frac{\theta _{1}}{2\sqrt{x_{1}^{*}}}=\frac{2}{3}\frac{1}{(x_{1}^{*})^{\frac{1}{3}}}\), which can be equivalently written as \(\frac{3\theta _{1}}{4}=(x_{1}^{*})^ {\frac{1}{6}}\). Since \(\frac{\sqrt{x_{1}}}{x_{1}^{\frac{1}{3}}}=x_{1}^{\frac{1}{6}}\) is an increasing function of \(x_{1}, \,\frac{\theta _{1}}{2\sqrt{x_{1}}}>\frac{2}{3}\frac{1}{x_{1}^{\frac{1}{3}}}\) for \(x_{1}<x_{1}^{*}\) and \(\frac{\theta _{1}}{2\sqrt{x_{1}}}<\frac{2}{3}\frac{1}{x_{1}^{\frac{1}{3}}}\) for \(x_{1}>x_{1}^{*}\). Hence, for the relevant \(\theta _{1}\) the family of functions \(\theta _{1}\sqrt{{x_{1}}}+y_{1}\) concavify \(x_{1}^{\frac{2}{3}}+y\) at the consumption bundles along \(B^{'}\). Therefore, for relevant \(\theta _{1}s\) the solution given by \((\frac{3}{4}\theta _{1})^{6}\) indeed correspond to the maximum.

For \(a,b \in \mathfrak {R}^{2}, \,a>b\) means \(a_{k}\ge b_{k} \) for all \(k \in M\) and \(a_{k}> b_{k}\) for at least one \(k\).

Let \(F_{i}^{x}(R_{i},R_{j})\) and \(F_{i}^{y}(R_{i},R_{j})\) denote the allocation of good \(x\) and \(y\) to agent \(i\) according to the SCF \(F\) at the profile \((R_{i},R_{j})\). Consider the situation \(F_{i}^{x}(\bar{R_{i}},\bar{R_{j}})=F_{i}^{x}(\tilde{R_{i}},\tilde{R_{j}})\) and \(F_{i}^{y}(\bar{R_{i}},\bar{R_{j}})<F_{i}^{y}(\tilde{R_{i}},\tilde{R_{j}})\). By Lemma 4 \(F(\bar{R_{i}},R_{j})\in SEQ_{i}(F(\bar{R_{i}},\bar{R_{j}}))\) for all \(R_{j}\succ \bar{R_{j}}\). By continuity and strategy-proofness there exists \(R_{j}\) such that \(\tilde{R_{j}}\succ R_{j}\succ \bar{R_{j}}\) and \(F(\bar{R_{i}},R_{j})\in int\; THQ_{i}(F(\tilde{R_{i}},\tilde{R_{j}}))\). Hence, the positions of \(F(\bar{R_{i}},\bar{R_{j}})\) and \(F(\tilde{R_{i}},\tilde{R_{j}})\) are without the loss of generality.

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Title: Non fixed-price trading rules in single-crossing classical exchange economies
Author: Mridu Prabal Goswami
Publication date: 01-02-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0834-7

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