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Journal of Mathematical Economics 46 (2010) 326–331

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Journal of Mathematical Economics
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Efficient, Pareto-improving processes
Kwan Koo Yun
Economics Department, SUNY at Albany, Albany, NY 12222, United States

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We give two optimization programs fordetermining whether Pareto improving local changes are possible. When they are, the programs compute them. Any procedure generating efficient and Pareto improving changes can be replicated by these programs. The two programs are dual to each other. We apply the programs to Pareto improving exchange processes and to Pareto-improving tax-tariff reforms. © 2009 Elsevier B.V. All rights reserved.Article history: Available online 28 December 2009

JEL classification: C61 D51 H21 Keywords: Pareto improvement Duality Exchange process Piecemeal tax reform

1. Introduction Tax or tariff negotiations often result in piecemeal reforms. Larger reforms may be considered too expensive economically or politically, and failure to reach an agreement results in the status quo. A reasonable requirement ofany agreement is that all participants benefit from it, but when Pareto improving changes are possible, there are usually many of them. An equity criterion may narrow the choices. For example, a large country may argue that it should benefit at least as much as smaller countries. One way to express equity considerations is by specifying the divisions of the total surplus from the change (e.g. theMDP exchange process; Malinvaud, 1972). Alternatively, the divisions of the surplus can be viewed as the outcome of generalized Nash bargaining (Myerson, 1991). In this paper, we formulate two optimization programs reflecting these approaches that characterize efficient piecemeal reforms as their solutions. The programs check whether Pareto improving local changes are possible and generate efficientPareto improving changes whenever they are. We represent the feasible (directions of) local changes by a non-empty, closed convex cone K in Rl and criteria by non-zero vectors {vi }n in Rl . We interpret each vi as the gradient of some criterion function. A local change d in K is Pareto improving i=1 if vi · d > 0, all i. When there is a Pareto improving change, d ∈ K \ {0} is efficient if theredoes not exist d ∈ K of equal size (|d | = |d|) such that vi · d > vi · d, for all i. We assume that the units are chosen in such a way that the norm of a change reflects the cost of it. Denote the standard unit simplex in Rn as ≡ {x ∈ Rn |x ≥ 0, xi = 1}, its relative interior as ◦ , and l by D ≡ {d ∈ Rl |d · d ≤ 1}. Given v in Rl , the unit disk in R K v is the orthogonal projection of v to K.E-mail address: 0304-4068/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2009.12.003

K.K. Yun / Journal of Mathematical Economics 46 (2010) 326–331


Consider: Program 1. Given positive numbers {ci }, i = 1, . . . , n, min
∈ K i i ci vi



Program 2. Given


maxd ˘i (vi · d) i subject to d ∈ K ∩ D, vi · d ≥ 0, i= 1, . . . , n. The two programs share criterion vectors {vi } and the feasible directions K. The positive numbers {ci } specify the first problem and the positive exponents { i } specify the second. The minimum of program 1 or the maximum of program 2 is zero if and only if a Pareto improving change does not exist. A solution to either program gives an efficient, Pareto improving change when aPareto improvement is possible. Program 1 is defined by choosing share ratios of the total surplus. The MDP exchange process (Malinvaud, 1972) is an example of a Pareto improving process that produces prescribed share ratios of the total surplus although it is not efficient in general. Differential equations constructed through program 1 is a special case of the ‘slow solution’ whose properties...
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