“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

The literature on supermodular optimization and games is surveyed from the perspective of potential users in economics. This methodology provides a new approach for comparative statics based only on critical assumptions, and allows a general analysis of games with strategic complementarities. The results are presented in a simplified yet rigourous manner, without reference to lattice theory, for the special case of onedimensional parameter and actions sets, with the emphasis being on wide accessibility. Detailed applications are presented for well-known models of consumer behavior, monopoly pass-through, Bertrand and Cournot competition, strategic R&D, search and matching. Wherever appropriate, useful tricks for applications and comparative comments are inserted.

/pdf/supermodularity-and-complementarity-in-economics-an-5ac8x76tgh.pdf

Supermodularity and Complementarity in Economics: An Elementary Survey

The literature on supermodular optimization and games is surveyed from the perspective of potential users in economics. This methodology provides a new approach for comparative statics based only on critical assumptions, and allows a general analysis of games with strategic complementarities. The results are presented in a simplified yet rigourous manner, without reference to lattice theory, for the special case of one-dimensional parameter and actions sets, with the emphasis being on wide accessibility. Detailed applications are presented for well-known models of consumer behavior, monopoly pass-through, Bertrand and Cournot competition, strategic R&D, search, and matching. Wherever appropriate, useful tricks for applications and comparative comments are inserted.

/pdf/supermodularity-and-complementarity-in-economics-an-2e6e0d1ab7.pdf

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Recursive Methods In Economic Dynamics

In this paper, we provide a set of sufficient conditions under which recursive competitive equilibrium exist and are unique for a large class of distorted dynamic equilibrium models with capital and elastic labor supply. We develop a monotone map approach to the problem. The class of economies for which we are able to obtain our existence result is apparently considerably larger than those considered in previous work. Additionally unlike previous work, we are able to also prove that this equilibrium is unique. We conclude by applying the new results to some important examples of monetary economies often used in applied work.

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

https://isiarticles.com/bundles/Article/pre/pdf/23306.pdf

Externalities, Market Power, and Resource Extraction*

The literature on experimentation and learning typically imposes a special dynamic structure: The only connection between periods is the updating of beliefs. Hence, both the present action and present signal realization only affect the future by changing the distribution of future beliefs. In many dynamic problems, however, either the present action or the present signal realization may enter future payoffs directly and not just through future beliefs: This is called signal dependence. We analyze optimal experimentation, under signal dependence, and show that experimentation may reduce information. We also provide sufficient conditions on the primitives for information-increasing experimentation.

Optimal Experimentation in Signal‐Dependent Decision Problems*

The paper introduces trade into dynamic models with externalities and capital accumulation, and evaluates the efficiency of the Cournot–Nash equilibrium. It considers mixed economies characterized by a blend of strategic and nonstrategic sectors. Also, there are two sources of interdependence: the existence of production externalities and the endogenous determination of market prices. It is shown that policy coordination is not needed when preferences are the same. In this case, the production externalities are internalized, so that an inefficient solution becomes the efficient integrated world equilibrium due to trade.

Manjira Datta

Papers

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

Existence and Uniqueness of Equilibrium in Distorted Dynamic Economies with Capital and Labor

Externalities, Market Power, and Resource Extraction*

Optimal Experimentation in Signal‐Dependent Decision Problems*

Dynamic Externalities and Policy Coordination