https://are.berkeley.edu/courses/ARE251/fall2007/Papers/atkinson70.pdf

On the Measurement of Inequality

OST models of portfolio selection have M been one-period models. I examine the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model whzere his income is generated by returns on assets and these returns or instantaneous "growth rates" are stochastic. P. A. Samuelson has developed a similar model in discrete-time for more general probability distributions in a companion paper [8]. I derive the optimality equations for a multiasset problem when the rate of returns are generated by a Wiener Brownian-motion process. A particular case examined in detail is the two-asset model with constant relative riskaversion or iso-elastic marginal utility. An explicit solution is also found for the case of constant absolute risk-aversion. The general technique employed can be used to examine a wide class of intertemporal economic problems under uncertainty. In addition to the Samuelson paper [8], there is the multi-period analysis of Tobin [9]. Phelps [6] has a model used to determine the optimal consumption rule for a multi-period example where income is partly generated by an asset with an uncertain return. Mirrless [5] has developed a continuous-time optimal consumption model of the neoclassical type with technical progress a random variable.

Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case

THIS PAPER IS A THEORETICAL examination of the stochastic behavior of equilibrium asset prices in a one-good, pure exchange economy with identical consumers. The single good in this economy is (costlessly) produced in a number of different productive units; an asset is a claim to all or part of the output of one of these units. Productivity in each unit fluctuates stochastically through time, so that equilibrium asset prices will fluctuate as well. Our objective will be to understand the relationship between these exogenously determined productivity changes and market determined movements in asset prices. Most of our attention will be focused on the derivation and application of a functional equation in the vector of equilibrium asset prices, which is solved for price as a function of the physical state of the economy. This equation is a generalization of the Martingale property of stochastic price sequences, which serves in practice as the defining characteristic of market "efficiency," as that term is used by Fama [7] and others. The model thus serves as a simple context for examining the conditions under which a price series' failure to possess the Martingale property can be viewed as evidence of non-competitive or "irrational" behavior. The analysis is conducted under the assumption that, in Fama's terms, prices "fully reflect all available information," an hypothesis which Muth [13] had earlier termed "rationality of expectations." As Muth made clear, this hypothesis (like utility maximization) is not "behavioral": it does not describe the way agents think about their environment, how they learn, process information, and so forth. It is rather a property likely to be (approximately) possessed by the outcome of this unspecified process of learning and adapting. One would feel more comfortable, then, with rational expectations equilibria if these equilibria were accompanied by some form of "stability theory" which illuminated the forces which move an economy toward equilibrium. The present paper also offers a convenient context for discussing this issue. The conclusions of this paper with respect to the Martingale property precisely replicate those reached earlier by LeRoy (in [10] and [11]), and not surprisingly, since the economic reasoning in [10] and the present paper is the same. The

/pdf/asset-prices-in-an-exchange-economy-2rw36dol4q.pdf

Asset prices in an exchange economy

A menu of paired lottery choices is structured so that the crossover point to the high-risk lottery can be used to infer the degree of risk aversion With normal laboratory payoffs of several dollars, most subjects are risk averse and few are risk loving Scaling up all payoffs by factors of twenty, fifty, and ninety makes little difference when the high payoffs are hypothetical In contrast, subjects become sharply more risk averse when the high payoffs are actually paid in cash A hybrid "power/expo" utility function with increasing relative and decreasing absolute risk aversion nicely replicates the data patterns over this range of payoffs from several dollars to several hundred dollars

Risk Aversion and Incentive Effects

A menu of paired lottery choices is structured so that the crossover point to the high-risk lottery can be used to infer the degree of risk aversion. With “normal” laboratory payoffs of several dollars, most subjects are risk averse and few are risk loving. Scaling up all payoffs by factors of twenty, fifty, and ninety makes little difference when the high payoffs are hypothetical. In contrast, subjects become sharply more risk averse when the high payoffs are actually paid in cash. A hybrid “power/expo” utility function with increasing relative and decreasing absolute risk aversion nicely replicates the data patterns over this range of payoffs from several dollars to several hundred dollars. Although risk aversion is a fundamental element in standard theories of lottery choice, asset valuation, contracts, and insurance (e.g. Daniel Bernoulli, 1738; John Pratt, 1964; Kenneth Arrow, 1965), experimental research has provided little guidance as to how risk aversion should be modeled. To date, there have been several approaches used to assess the importance and nature of risk aversion. Using lottery choice data from a field experiment, Hans Binswanger (1980) concluded that most farmers exhibit a significant amount of risk aversion that tends to increase as payoffs are increased. Alternatively, risk aversion can be inferred from bidding and pricing tasks. In auctions, overbidding relative to Nash predictions has been attributed to risk aversion by some and to noisy decision-making by others, since the payoff consequences of such overbidding tend to be small (Glenn Harrison, 1989). Vernon Smith and James Walker (1993) assess the effects of noise and decision cost by dramatically scaling up auction payoffs. They find little support for the noise hypothesis, reporting that there is an insignificant increase in overbidding in private value auctions as payoffs are scaled up by factors of 5, 10, and 20. Another way to infer risk aversion is to elicit buying and/or selling prices for simple lotteries. Steven Kachelmeier and Mohamed Shehata (1992) report a significant increase in risk aversion (or, more precisely, a decrease in risk seeking behavior) as the prize value is increased. However, they also obtain dramatically different results depending on whether the choice task involves buying or selling, since subjects tend to put a high selling price on something they “own” and a lower buying price on something they do not, which implies This is analogous to the well-known “willingness to pay/willingness to accept bias.” Asking for a high selling price 1 implies a preference for the risk inherent in the lottery, and offering a low purchase price implies an aversion to the risk in the lottery. Thus the way that the pricing task is framed can alter the implied risk attitudes in a dramatic manner. The issue is whether seemingly inconsistent estimates are due to a problem with the way risk aversion is conceptualized, or to a behavioral bias that is activated by the experimental design. We chose to avoid this possible complication by framing the decisions in terms of choices, not purchases and sales. 3 risk seeking behavior in one case and risk aversion in the other. Independent of the method used to elicit 1 a measure of risk aversion, there is widespread belief (with some theoretical support discussed below) that the degree of risk aversion needed to explain behavior in low-payoff settings would imply absurd levels of risk aversion in high-payoff settings. The upshot of this is that risk aversion effects are controversial and often ignored in the analysis of laboratory data. This general approach has not caused much concern because most theorists are used to bypassing risk aversion issues by assuming that the payoffs for a game are already measured as utilities. The nature of risk aversion (to what extent it exists, and how it depends on the size of the stake) is ultimately an empirical issue, and additional laboratory experiments can produce useful evidence that complements field observations by providing careful controls of probabilities and payoffs. However, even many of those economists who admit that risk aversion may be important have asserted that decision makers should be approximately risk neutral for the low-payoff decisions (involving several dollars) that are typically encountered in the laboratory. The implication, that low laboratory incentives may be somewhat unrealistic and therefore not useful in measuring attitudes toward “real-world” risks, is echoed by Daniel Kahneman and Amos Tversky (1979), who suggest an alternative: Experimental studies typically involve contrived gambles for small stakes, and a large number of repetitions of very similar problems. These features of laboratory gambling complicate the interpretation of the results and restrict their generality. By default, the method of hypothetical choices emerges as the simplest procedure by which a large number of theoretical questions can be investigated. The use of the method relies of the assumption that people often know how they would behave in actual situations of choice, and on the further assumption that the subjects have no special reason to disguise their true preferences. (Kahneman and Tversky, 1979, p. 265) In this paper, we directly address these issues by presenting subjects with simple choice tasks that

/pdf/risk-aversion-and-incentive-effects-3c3wpiipme.pdf

Yrjö Jahnssonin säätiö

Papers

Aspects of the theory of risk-bearing

Frontiers of Economics