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Studies in the Economics of Uncertainty presents some new developments in the economics of uncertainty produced by leading scholars in the field. The contributions to this Festschrift in honor of Professor Josef Hadar of Southern Methodist University cover a broad range of topics centered on the principle of Stochastic Dominance. Topics covered range from theoretical and statistical developments on Stochastic Dominance to new applications of the Stochastic Dominance Theory. The intended audience includes researchers interested in recent developments in tools used for decision-making under uncertainty as well as economists currently applying Stochastic Dominance principles to the analysis of the Theory of Firm, International Trade, and the Theory of Finance.





Stochastic Dominance in Nonlinear Utility Theory

Stochastic dominance has interested mathematicians for more than half a century (Karamata, 1932; Hardy, Littlewood and Polya, 1934; Sherman, 1951; Lehmann, 1955), and its integration into decision theory began nearly forty years ago (Masse and Morlat, 1953; Allais, 1953a, 1953b; Blackwell and Girshick, 1954; Quirk and Saposnik, 1962; Fishburn, 1964). However, it was not until a cluster of important papers on stochastic dominance appeared around 1969–71 (Hadar and Russell, 1969, 1971; Hanoch and Levy, 1969; Whitmore, 1970; Rothschild and Stiglitz, 1970, 1971) that it emerged as a central topic in economic decision theory. Hundreds of papers as well as a survey book (Whitmore and Findlay, 1978) and an extensive research bibliography (Bawa, 1982) testify to its popularity.
P. C. Fishburn

The “Comparative Statics” of the Shackle-Vickers Approach to Decision-Making in Ignorance

The analysis of decision-making under conditions of ignorance developed by Shackle [8] and Vickers [9], [10] begins with the assumption that the decision-maker is able to describe a potential surprise function defined over the subsets of an incomplete2 collection of possible outcomes or states of the world.3 The potential surprise of an outcome-set A is the surprise the decision-maker imagines now that he would experience in the future were A to occur. Let a decision set containing the objects of choice be specified, and suppose that the decision-maker has an ordinal utility function defined over the Cartesian product of the decision set and the incomplete collection of states of the world. For each state of the world, then, the utility function maps objects of choice or elements of the decision set into associated utility values. More importantly, for each object of choice, the utility function also maps states of the world into utility values. Using this latter relation, the potential surprise function can be translated into a potential surprise density (or frequency) function4 defined over utility values. Thus the original potential surprise function, whose domain is a collection of outcome-sets, becomes a potential surprise density function with a domain consisting of utility magnitudes, or real numbers. Although it has a different significance and meaning, the potential surprise density function is often drawn to look something like an inverted probability density function.
Donald W. Katzner

Stochastic Dominance and Transformations of Random Variables

During the past twenty years, the term stochastic dominance (SD) has been used by economists to describe a particular set of rules for ranking random variables. These rules apply to pairs of random variables, and indicate when one is to be ranked higher than the other by specifying a condition which the difference between their cumulative distribution functions (CDF) must satisfy. Various SD ranking procedures have been employed in both empirical and theoretical analysis. First degree stochastic dominance, second degree stochastic dominance, and Rothschild and Stiglitz’ definition of increasing risk are prominent examples.
Jack Meyer

Representative Sets for Stochastic Dominance Rules

Advising someone as to his best course of conduct is always treacherous and trying to steer his economic course can be perilous to personal relationships. If we knew his utility function, economists might say, we could confidently and with courage make optimal selections for anyone. But economic counselors would likely know, at most, only some salient characteristics of the advisee. Depending on what and how much we know about a client’s utility function we could more or less sharply delineate the options which are inferior for him. This idea is at the root of all studies under the topic of “stochastic dominance rules.” Given incomplete information about a person’s utility function the best that we can do is to classify him accordingly to one or more sets of utility functions. He may be one of any number of people whose utility functions are members of some given set. We know no more or less about his utility function than any other in the set. Any choice between two uncertain prospects appropriate to him would likewise be appropriate to anyone else whose utility function is in the same set because this set defines the limits of information about the utility functions we consider. It is equivalent then to ask; “If we know Tom’s utility function can be characterized as thus and so which choice should he make?” or, “If everyone we consider has utility functions which can be characterized as thus and so, which choice should they unanimously make?” The latter question is in the spirit of stochastic dominance research. Primarily, stochastic dominance rules dictate procedures for discovering unanimous orderings of uncer?tain prospects appropriate for utility functions within specified sets.
W. R. Russell, T. K. Seo

Stochastic Dominance for the Class of Completely Monotonic Utility Functions

According to the expected utility axioms, a decision maker with utility function u(x) for wealth x assigns the following subjective value to an uncertain prospect with cumulative distribution function F(x).
$$ E(u;F) = \smallint _0^\infty u(x)dF(x) $$
It is assumed here that wealth level x is positive and that prospect F has moments of all orders.
G. A. Whitmore

Estimation and Testing


The Stochastic Dominance Estimation of Default Probability

The purpose of this paper is two-fold: (1) to present a stochastic dominance technique which can be used to quantify differences in cumulative probability distributions of data, and (2) to demonstrate this technique by quantifying the probability of default as assessed by the bond market. We suggest, then, that the contribution of this paper lies in its introduction of a new methodology which we then use to answer a question in economics and finance.
Mary S. Broske, Haim Levy

Testing for Stochastic Dominance

It is often useful in economic analysis to formulate and test necessary conditions for optimizing behavior; this obviates the need for complete, and perhaps tenuous, maintained hypotheses on the objective function of the decision-maker. For example, the weak axiom of revealed preference can be tested without assuming a parametric family of preferences. For consumers who choose among uncertain alternatives by maximizing von Neumann-Morgenstern utility, it is also useful to identify necessary conditions and to develop methods that allow them to be tested econometrically. One such necessary condition is that an optimal prospect cannot be inferior to another feasible prospect for all increasing utility functions; this condition can be characterized in terms of stochastic dominance between distributions of payoffs.2 Hadar and Russell (1969, 1971, 1974a,b, 1978) introduced the concept of second-degree stochastic dominance for consumers with increasing, risk-averse utility functions; this has become an important tool in the analysis of choice under uncertainty. This paper develops statistical tests for stochastic dominance. Section 2 reviews the concept of stochastic dominance, and makes a few minor extensions to Hadar and Russell’s thorough characterization. Tests for first-degree and second-degree stochastic dominance are discussed in Sections 3 and 4, respectively. Section 5 discusses computation of the test statistics.
Daniel McFadden



Insurance and the Value of Publicly Available Information

A number of recent papers (e.g., Radner (1981, 1985), Townsend (1982), Rubinstein-Yaari (1983), Lambert (1983), Dionne (1983), and Rogerson (1985)) have studied the role of multi-period contracts in situations characterized by asymmetrically informed agents. In the circumstances envisaged by this literature, multi-period contracts perform no useful function when agents are equally well-informed.1 The present paper considers a different set of circumstances and shows that multi-period contracts may be useful even when agents are symmetrically informed.
Marcel Boyer, Georges Dionne, Richard Kihlstrom

Vertical Transactions under Uncertainty

Transactions between entities engaging in successive stages of production often take place outside the spot market. Some producers employ forward contracts to tie down price or quantity; others rely on some form of principal-agent arrangement; still others enter into joint stock ownership or even integrate vertically, thus, placing decision making in the hands of a single authority. These arrangements represent, in varying degrees, the producers’ attempts to bypass the market. In a market-oriented economy, such practices require careful explanation. The purpose of this paper is to study how market uncer­tainties motivate these arrangements and to explain why one of these arrangements or a combination of them is chosen under a given set of market conditions.
J. Horen, S. Y. Wu

Optimal Tariffs and Quotas under Uncertain International Transfer

Developing countries face various sorts of uncertainties. Some of them have been investigated in the literature.1 It is well known that many developing countries receive foreign aids or gifts in one form or another: they are either financial or real. Amounts of such transfers, however, are often not known with certainty in advance, because usable funds in donor countries are not sure owing to uncertain economic conditions such as production or consumption, or uncertain political decision processes in those countries. On the other hand, there exists some time lag between the formulation of trade policies and their implementation in developing countries. In such a case the policy-makers of the developing countries have the problem that they must decide optimal levels of trade policies before the amounts of transfer to receive are known. This paper addresses the issue of the optimal tariffs and quotas when a developing country faces uncertainty about the amount of transfer to receive.
Takao Itagaki

Investment, Capital Structure and Cost of Capital: Revisited

The relevance of capital structure for a firm’s valuation is an unresolved issue (see [10], [7], [16], [4], [8], [17], [9], [15], [13], [14], and others). In a series of papers, Modigliani and Miller conclude that in a world without taxes, capital structure does not matter, but with corporate tax they reach the conclusion that the larger the proportion of debt the larger is the value of the firm (see Modigliani and Miller, [11], [12], [13]). However, Miller [10], in his presidential address has shown that under certain assumptions, with corporate and personal taxes, once again that capital structure does not matter.
Yoram Kroll, Haim Levy

Utility Functions, Interest Rates, and the Demand for Bonds

One riskless investment dominates any other with a lower interest rate in the simplest, most complete way one could hope for. Still, given the risky assets available, a lower riskless interest rate might lead a risk-averse expected-utility maximizer to (a) allocate more of his investment funds to riskless assets, or (b) allocate enough more to obtain greater riskless return including principal, or even (c) allocate enough more to obtain greater riskless income. One easy intuitive explanation is that, at a lower riskless interest rate, greater riskless investment may be needed to guarantee some minimum acceptable income or future wealth. Alternative characterizations of the three possibilities are that, for this individual, (a) demand for riskless future wealth has elasticity less than 1; (b) riskless future wealth is a Giffen good; (c) riskless income is a Giffen good.
John W. Pratt
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