Sudhir Shah - Academia.edu (original) (raw)

Papers by Sudhir Shah

Journal of Mathematical Economics, Oct 1, 2017

The riskiness of random processes is compared by (a) employing a decision-theoretic equivalence b... more The riskiness of random processes is compared by (a) employing a decision-theoretic equivalence between processes and lotteries on pathspaces to identify the riskiness of the former with that of the latter, and (b) using the theory of comparative riskiness of lotteries over vector spaces to compare the riskiness of lotteries on a given path-space. We derive the equivalence used in step (a) and contribute a new criterion to the theory applied in step (b). The new criterion, involving a generalized form of second order stochastic dominance, is shown to be valid by establishing its equivalence to the standard decision-theoretic criterion. We demonstrate its tractability via diverse economic applications featuring risk embodied in random processes.

Working Papers, Sep 1, 2007

We present theorems that establish dualities, i.e., bijections, between specified sets of direct ... more We present theorems that establish dualities, i.e., bijections, between specified sets of direct utility functions, indirect utility functions and expenditure functions. The substantive properties characterizing the specified set of direct utility functions are strong monotonicity, upper semicontinuity and quasi-concavity. Our results are strictly intermediate between two classes of analogous results in the literature. We also provide applications that use all the three classes of duality results.

We consider organizations with a single principal and many agents who interact in an environment ... more We consider organizations with a single principal and many agents who interact in an environment with the following features: (a) Nature imperfectly informs the principal via a state-contingent signal, but not the agents, about the state of the world, (b) the principal selectively shares this information with the agents, thereby endogenously endowing them with private information that is coarser than his own, (c) the principal assigns action spaces to the agents, and (d) an agent's control over the choice from his assigned action space is inalienable. Designing an organization involves specifying (c) and specifying an information dissemination system for implementing (b). Searching for an optimal design involves (1) deriving optimal performance from each design, and (2) comparing designs on the basis of their best performances. Our existence results show the feasibility of performing Step (1) in a large class of cases.

We study an incomplete information, non-cooperative model of the determination of national emissi... more We study an incomplete information, non-cooperative model of the determination of national emission endowments under a Kyoto type protocol with heterogeneous nations. The model generates a link between national types and equilibrium national emission caps. We analyze this link to (a) derive the type-contingent ordering of emission allocations, (b) study the effects of growth on emission allocations, and (c) study the strategies that nations can use to manipulate the emission allocation process. Synthesizing these results allows us to derive the implications of national heterogeneity and asymmetry of economic power in the capping process.

SSRN Electronic Journal, 2008

Given a set of possible vector outcomes and the set of lotteries over it, we define sets of (a) v... more Given a set of possible vector outcomes and the set of lotteries over it, we define sets of (a) von Neumann-Morgenstern representations of preferences over the lotteries, (b) mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) mappings that yield the risk premia corresponding to a lottery, (d) mappings that yield the acceptance set of lotteries corresponding to an outcome, and (e) vector-valued functions that yield generalized Arrow-Pratt coefficients corresponding to an outcome. Our main results establish bijections between these sets of mappings for very general specifications of outcome spaces, lotteries and preferences. As corollaries of these results, we derive analogous dual representations of risk averse preferences. Some applications to financial theory illustrate the potential uses of our results. Finally, we provide criteria for comparing the risk aversion of preferences in terms of the dual representations.

Pratt (1964) and Yaari (1969) contain the classical results pertaining to the equivalence of vari... more Pratt (1964) and Yaari (1969) contain the classical results pertaining to the equivalence of various notions of comparative risk aversion of von Neumann-Morgenstern utilities in the setting with real-valued outcomes. Some of these results have been extended to the setting with outcomes in < n . We obtain ana-logues of the classical results in the setting with outcomes in ordered topological vector spaces when differentiability is not required, and in the setting with out-comes in ordered Hilbert spaces when differentiability is required, as is the case when we work with a vector-valued generalized notion of an Arrow-Pratt coeffi-cient.

We consider a decision-making environment with an outcome space that is a convex and compact subs... more We consider a decision-making environment with an outcome space that is a convex and compact subset of a vector space belonging to a general class of such spaces. Given this outcome space, we de¯ne gen- eral classes of (a) risk averse von Neumann-Morgenstern utility func- tions de¯ned over the outcome space, (b) multi-valued mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) multi-valued mappings that yield the risk premia corresponding to a lottery, and (d) multi-valued mappings that yield the acceptance set of lotteries corresponding to an outcome. Our duality results establish that the usual mappings that generate (b), (c) and (d) from (a) are bi- jective. We apply these results to the problem of computing the value of ¯nancial assets to a risk averse decision-maker and show that this value will always be less than the arbitrage-free valuation.

Journal of Economic Dynamics and Control, 2005

This paper studies an incompletely informed regulator's problem of inducing a firm producing dura... more This paper studies an incompletely informed regulator's problem of inducing a firm producing durable pollution to adopt a socially optimal pollution storage technology. We construct a sparse, yet flexible, theoretical model that can be applied directly to concrete situations as it is stated explicitly in terms of statistical parameters. Next, we show the existence of an optimal regulatory contract and examine its qualitative features for one of the many regulatory problems suggested by the general model. Our model extends the standard inventory model by making the firm's storage technology a strategic choice induced by the regulatory contract. Moreover, by providing a structural model of the firm, our model generates useful information that has to be assumed in an abstract regulatory model.

A Non-cooperative Theory of Quantity--rationing International Transfrontier Pollution

Journal of Mathematical Economics, Oct 1, 2017

The riskiness of random processes is compared by (a) employing a decision-theoretic equivalence b... more The riskiness of random processes is compared by (a) employing a decision-theoretic equivalence between processes and lotteries on pathspaces to identify the riskiness of the former with that of the latter, and (b) using the theory of comparative riskiness of lotteries over vector spaces to compare the riskiness of lotteries on a given path-space. We derive the equivalence used in step (a) and contribute a new criterion to the theory applied in step (b). The new criterion, involving a generalized form of second order stochastic dominance, is shown to be valid by establishing its equivalence to the standard decision-theoretic criterion. We demonstrate its tractability via diverse economic applications featuring risk embodied in random processes.

Working Papers, Sep 1, 2007

We present theorems that establish dualities, i.e., bijections, between specified sets of direct ... more We present theorems that establish dualities, i.e., bijections, between specified sets of direct utility functions, indirect utility functions and expenditure functions. The substantive properties characterizing the specified set of direct utility functions are strong monotonicity, upper semicontinuity and quasi-concavity. Our results are strictly intermediate between two classes of analogous results in the literature. We also provide applications that use all the three classes of duality results.

We consider organizations with a single principal and many agents who interact in an environment ... more We consider organizations with a single principal and many agents who interact in an environment with the following features: (a) Nature imperfectly informs the principal via a state-contingent signal, but not the agents, about the state of the world, (b) the principal selectively shares this information with the agents, thereby endogenously endowing them with private information that is coarser than his own, (c) the principal assigns action spaces to the agents, and (d) an agent's control over the choice from his assigned action space is inalienable. Designing an organization involves specifying (c) and specifying an information dissemination system for implementing (b). Searching for an optimal design involves (1) deriving optimal performance from each design, and (2) comparing designs on the basis of their best performances. Our existence results show the feasibility of performing Step (1) in a large class of cases.

We study an incomplete information, non-cooperative model of the determination of national emissi... more We study an incomplete information, non-cooperative model of the determination of national emission endowments under a Kyoto type protocol with heterogeneous nations. The model generates a link between national types and equilibrium national emission caps. We analyze this link to (a) derive the type-contingent ordering of emission allocations, (b) study the effects of growth on emission allocations, and (c) study the strategies that nations can use to manipulate the emission allocation process. Synthesizing these results allows us to derive the implications of national heterogeneity and asymmetry of economic power in the capping process.

SSRN Electronic Journal, 2008

Given a set of possible vector outcomes and the set of lotteries over it, we define sets of (a) v... more Given a set of possible vector outcomes and the set of lotteries over it, we define sets of (a) von Neumann-Morgenstern representations of preferences over the lotteries, (b) mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) mappings that yield the risk premia corresponding to a lottery, (d) mappings that yield the acceptance set of lotteries corresponding to an outcome, and (e) vector-valued functions that yield generalized Arrow-Pratt coefficients corresponding to an outcome. Our main results establish bijections between these sets of mappings for very general specifications of outcome spaces, lotteries and preferences. As corollaries of these results, we derive analogous dual representations of risk averse preferences. Some applications to financial theory illustrate the potential uses of our results. Finally, we provide criteria for comparing the risk aversion of preferences in terms of the dual representations.

Pratt (1964) and Yaari (1969) contain the classical results pertaining to the equivalence of vari... more Pratt (1964) and Yaari (1969) contain the classical results pertaining to the equivalence of various notions of comparative risk aversion of von Neumann-Morgenstern utilities in the setting with real-valued outcomes. Some of these results have been extended to the setting with outcomes in < n . We obtain ana-logues of the classical results in the setting with outcomes in ordered topological vector spaces when differentiability is not required, and in the setting with out-comes in ordered Hilbert spaces when differentiability is required, as is the case when we work with a vector-valued generalized notion of an Arrow-Pratt coeffi-cient.

We consider a decision-making environment with an outcome space that is a convex and compact subs... more We consider a decision-making environment with an outcome space that is a convex and compact subset of a vector space belonging to a general class of such spaces. Given this outcome space, we de¯ne gen- eral classes of (a) risk averse von Neumann-Morgenstern utility func- tions de¯ned over the outcome space, (b) multi-valued mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) multi-valued mappings that yield the risk premia corresponding to a lottery, and (d) multi-valued mappings that yield the acceptance set of lotteries corresponding to an outcome. Our duality results establish that the usual mappings that generate (b), (c) and (d) from (a) are bi- jective. We apply these results to the problem of computing the value of ¯nancial assets to a risk averse decision-maker and show that this value will always be less than the arbitrage-free valuation.

Journal of Economic Dynamics and Control, 2005

This paper studies an incompletely informed regulator's problem of inducing a firm producing dura... more This paper studies an incompletely informed regulator's problem of inducing a firm producing durable pollution to adopt a socially optimal pollution storage technology. We construct a sparse, yet flexible, theoretical model that can be applied directly to concrete situations as it is stated explicitly in terms of statistical parameters. Next, we show the existence of an optimal regulatory contract and examine its qualitative features for one of the many regulatory problems suggested by the general model. Our model extends the standard inventory model by making the firm's storage technology a strategic choice induced by the regulatory contract. Moreover, by providing a structural model of the firm, our model generates useful information that has to be assumed in an abstract regulatory model.

A Non-cooperative Theory of Quantity--rationing International Transfrontier Pollution