دانلود مقاله ISI انگلیسی شماره 1531
عنوان فارسی مقاله

خوردن کیک، استخراج منابع تمام شدنی،صرفه جویی چرخه عمر و بازی های غیراتمی: نظریه های موجودیت برای یک بخشی از مشکلات تخصیص بهینه

کد مقاله سال انتشار مقاله انگلیسی ترجمه فارسی تعداد کلمات
1531 2009 16 صفحه PDF سفارش دهید محاسبه نشده
خرید مقاله
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عنوان انگلیسی
Cake eating, exhaustible resource extraction, life-cycle saving, and non-atomic games: Existence theorems for a class of optimal allocation problems
منبع

Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)

Journal : Journal of Economic Dynamics and Control, Volume 33, Issue 6, June 2009, Pages 1345–1360

کلمات کلیدی
خوردن کیک - استخراج منابع تمام شدنی - صرفه جویی چرخه عمر - بازی های غیراتمی - تخصیص بهینه - موجودیت -
پیش نمایش مقاله
پیش نمایش مقاله خوردن کیک، استخراج منابع تمام شدنی،صرفه جویی چرخه عمر و بازی های غیراتمی: نظریه های موجودیت برای یک بخشی از مشکلات تخصیص بهینه

چکیده انگلیسی

This paper investigates the problem concerning the existence of a solution to a diverse class of optimal allocation problems which include models of cake eating, exhaustible resource extraction, life-cycle saving, and non-atomic games. A new formulation that encompasses all these diverse models is provided. Examples of these models for which a solution does not exist and the causes of the non-existence are studied. Two theorems are provided to tackle the existence problem under different conditions. Several analytical examples with a closed-form solution are offered to illustrate the usefulness of the existence theorems.

مقدمه انگلیسی

This paper studies the problem concerning the existence of a solution to a diverse class of optimal allocation models which have appeared in various forms and found a lot of applications in economics. The optimal allocation models have the following common underlying structure: equation(1) View the MathML sourcemaxc(t)∈Φ∫01α(t)f(t)g(c(t))dt Turn MathJax on subject to equation(2) c(t)⩾0,c(t)⩾0, Turn MathJax on equation(3) S(t)⩾0,S(t)⩾0, Turn MathJax on equation(4) S′(t)=j(t)S(t)+m(t)-f(t)c(t),S′(t)=j(t)S(t)+m(t)-f(t)c(t), Turn MathJax on and equation(5) S(0)=S0,S(0)=S0, Turn MathJax on where c,S,α,j,m,f:[0,1]→Rc,S,α,j,m,f:[0,1]→R, g:R→Rg:R→R, S0∈[0,∞)S0∈[0,∞), ΦΦ denotes the space of piecewise continuous functions, and RR denotes the real line. Model (1), (2), (3), (4) and (5) is an optimal control problem with a control constraint and a state constraint. For convenience, the range of t is taken to be [0,1][0,1], but it can be any bounded subset of the non-negative real line. A wide variety of economic models can be formulated in the form of (1), (2), (3), (4) and (5), e.g., exhaustible resource extraction ( Hotelling, 1931), cake-eating ( Gale, 1967), life-cycle saving ( Yaari, 1965), and non-atomic games ( Aumann and Shapley, 1974). The formulation (1), (2), (3), (4) and (5) provides a new and convenient way to encompass all these diverse models. Karlin (1959, pp. 210–214) was the first to study the existence problem for a special case of (1), (2), (3), (4) and (5) in which j(t)=0j(t)=0, m(t)=0m(t)=0, and f(t)=1f(t)=1 are assumed. In this case, the optimization problem (1), (2), (3), (4) and (5) becomes a calculus of variations problem. Yaari (1964) advances Karlin's (1959) analysis and provides an interesting example to show that the variational problem may not have a solution even when g(·)g(·) is strictly concave and the other functions are smooth and well defined. Yaari's (1964) example is counter-intuitive because it means that there is no optimal way to allocate a given amount of resources to maximize a well-defined objective in a simple and reasonable setting. Perhaps even more puzzling is that a solution to the example will exist if S0S0 is sufficiently small. In other words, there is no optimal way to allocate the endowment S0S0 if it is sufficiently large. Since Yaari's (1964) work, a number of follow-up and refinement studies have appeared in both the economics and mathematics literatures, e.g., Aumann and Perles (1965), Kumar (1969), Abrham (1970), Artstein, 1974 and Artstein, 1980, and Ioffe (2006). These studies offer a variety of sufficient conditions to guarantee the existence of a solution to the variational problem. While previous studies have focused on the special case where j(t)=0j(t)=0, m(t)=0m(t)=0, and f(t)=1f(t)=1, the existence problem for the general model (1), (2), (3), (4) and (5) has not yet been studied in the literature. The objective of this paper is to develop sufficient conditions for the existence of a solution to the general model (1), (2), (3), (4) and (5) under the assumption that α(1)=0α(1)=0. As will be explained later, α(1)=0α(1)=0 is a special but not arduous restriction. When the condition holds, a precise existence result can be obtained. The solution to (1), (2), (3), (4) and (5), if it exists, possesses a distinctive feature which can be utilized to generate a simple sufficient condition that guarantees the existence of a solution to the optimal allocation problem. When applied to the special case where j(t)=0j(t)=0, m(t)=0m(t)=0, and f(t)=1f(t)=1, the sufficient condition is substantially simpler than the existing ones in the literature. The existence results reveal why (1), (2), (3), (4) and (5) may not have a solution and whether the existence problem depends on the presence of j(t)j(t), m(t)m(t), and f(t)f(t). In addition, the analysis provides a complete solution to the puzzle raised by Yaari's (1964) counter-intuitive example. While there are many general existence theorems for optimal control problems in the literature (e.g., Cesari, 1983), the relatively simple and explicit structure of this class of optimal allocation problems commensurately deserves a simple and direct existence result. The plan of the paper is as follows. Section 2 presents the assumptions and three economic examples for (1), (2), (3), (4) and (5). Section 3 investigates the existence problem and provides a series of analytical examples with a closed-form solution to illustrate the usefulness of the existence theorems. Section 4 discusses the role of several major assumptions in the existence results and explores the consequences if the assumptions are relaxed. Section 5 concludes the paper.

نتیجه گیری انگلیسی

This paper investigates the problem concerning the existence of a solution to a diverse class of optimal allocation problems which include models of cake eating, exhaustible resource extraction, life-cycle saving, and non-atomic games. A new formulation that encompasses all these diverse models is provided. Examples of these models for which a solution does not exist and the causes of the non-existence are studied. Two theorems are provided to tackle the existence problem under different conditions. Several analytical examples with a closed-form solution are offered to illustrate the usefulness of the existence theorems. This paper considers only the case where α(1)=0α(1)=0. The existence theorems are built on this pivotal assumption as it provides a special route by which the existence problem can be resolved. While the assumption may appear restrictive, it is not contrived because it does include a meaningful class of optimal allocation models such as Yaari's (1965) life-cycle model of saving and the non-atomic games in Aumann and Shapley (1974). For α(1)>0α(1)>0, existence results are available only for the case where j(t)=0j(t)=0 and m(t)=0m(t)=0, see Ioffe (2006) and the references therein. The existence problem for the general model (1), (2), (3), (4) and (5) where α(1)>0α(1)>0 has not yet been investigated in the literature. Whether the results of this paper can shed light on the existence problem for the general model remains to be studied.

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