Where decisions take place in world of certainty, consumers know for sure the utility they will receive given a choice of goods, but when this certainty is removed and a series of different outcomes may occur, then individuals will react differently, given their attitudes towards risk. Firms know for sure the profit they will receive from a chosen set of inputs, this does not describe the real world, technological uncertainty, market uncertainty, and many other issues cannot be addressed without considering uncertainty, e.g. stock market, insurance, futures markets. When people have to make decisions in the presence of uncertainty rational decision making still exists. The standard tools for analyzing rational choice can be modified to accommodate uncertainty. A person in an uncertain environment chooses among contingent commodities, whose value depends on the eventual outcome or state of the world. As with ordinary commodities, people have preferences for contingent commodities that can be represented by an indifference map. The slope of the budget constraint between two contingent commodities depends on the payoff associated with each state of the world. The curvature of the indifference curve depends on whether the individual is risk averse, risk loving or risk neutral. A risk-averse person will not accept an actuarially fair bet. Risk-averse people purchase insurance in order to spread consumption more evenly across states of the world. When risk-averse people are allowed to purchase fair insurance, they will insure themselves fully in the sense that their consumption is the same in every state of the world. The amount of insurance demanded depends on the premium and on the probability that the insurable event will occur. People with von Neumann-Morgenstern (1944) utility functions, in which the probability of each state of the world is multiplied by the utility associates with that state of the world, seek to maximise the expected value of their utility. The assumption of expected utility maximisation, together with decision trees, can be used to break up complicated decisions into simple components that can be readily solved. By comparing the expected utility of each option, the individual can determine their optimal strategy. An individual’s attitude towards risk, previously described as being either risk loving, risk hating or risk neutral, and the extent to which they fit into these categories will vary, as some people will be more risk loving than others, which could also be described as their risk aversion preference. The Arrow-Pratt coefficient for risk aversion for a utility function is given by r(w)=u'(w)/u”(w). This is a measure of the curvature of the utility function and measures the marginal willingness to pay for a mall change in the absolute risk. The measurement can be understood in relation to the concept of an acceptance set, A(w), which is the set of all gambles than an individual will accept given their current wealth, w. There are implications of changing absolute and relative risk aversion, with the most straightforward implications of increasing or decreasing absolute or relative risk aversion, and the ones that motivate a focus on these concepts, occurring in the context of forming a portfolio with one risky asset and one risk-free asset, Arrow (1971), Pratt (1964). If the person experiences an increase in wealth, they will choose to increase, keep unchanged, or decrease, the number of dollars of the risky asset held in the portfolio if absolute risk aversion is decreasing, constant, or increasing. Economists, generally, avoid using utility functions, because of the unrealistic behavioural implication. Also, if the person experiences an increase in wealth, they will choose to increase, or keep unchanged, or decrease, the fraction of the portfolio held in the risky asset, depending on their relative risk aversion. As a theory of individual behaviour, the expected utility model shares many of the underlying assumptions of standard consumer theory, yet, the expected utility theory comes under criticism by Rabin & Thaler (2001). They argue that expected utility theory is inadequate to explain risk aversion and hence should be discarded as a theory of choice under risk and uncertainty. Watt (2002) addresses this argument stating that all the exercises in Rabin & Thaler (2001) demonstrate only that an unrealistically high degree of risk aversion produces preposterous results. For a person with a high level of wealth to turn down a bet for moderate stakes that has a positive expected value will require either an unreasonably high level of risk aversion, or some other unusual peculiarity in the utility function. Under standard models of risk aversion, their large-scale bets will not be rejected and neither will their moderate-scale bets. Expected utility theory certainly faces problems in explaining certain empirical evidence, as do other competing theories. But in this case, it reveals a useful truth, that risk-averse, wealth-loving people should be willing to accept certain moderate bets with positive expected value, even though at first glance, the bets may not appear attractive to them. Expected utility theory is normative as some people believe that the empirical evidence does not remove the expected utility hypothesis from being a normative theory. It reflects how people ought to behave in order to maximise their well-being under uncertainty. Indeed, many people correct their decisions once their ‘error’ is pointed out to them. Others think that the theory is just plain wrong. Each axiom is open to scrutiny in this regard, such as the independence axiom, which is not always believed to be intuitive. Considering the lottery as a whole, rather than assume independence between the components, may be a solution to this. People try and avoid disappointment in making their decisions, and so a prospect with a small probability of receiving £0 might be enough to deter someone. There are two different issues that are often discussed of the expected utility theory, firstly, the technical, and secondly, the normative. The technical issue is that this theory is analytically convenient, in the sense that it is pervasive in economics. Whereas the normative issue that is discussed by many economists is that expected utility may provide a valuable guide to action, as people often find it hard to think systematically about risky alternatives. Also there is the issue of the Allais Paradox (1953), which is an example of choice behaviour that can be explained under anticipated utility theory. In the paradox, an individual is asked to choose between two gambles. Gamble A, where an individual has a 100% chance of receiving 1 million or Gamble B, where the same person has a 10% chance of 5million, an 89% chance of 1 million, and a 1% chance of nothing. An individual must pick one of these gambles, and then consider the following two gambles. Gamble C, with an 11% chance of 1million, and an 89% chance of nothing. Gamble D, with a 10% chance of 5million, and a 90% chance of nothing. Again, the individual must pick one of these two preferred gambles. Many people prefer A to B and D to C. However, these choices violate the expected utility axioms. The evidence can be interpreted in light of the understanding of the expected utility theory as being positive or normative. If normative, it is evidence of irrational behaviour. If positive, it is a damning indictment of the theory. Various arguments have been posited defending the theory, firstly, that it is a normative theory, so if mistakes are highlighted, people will adapt their choices, also the theory is an approximation, which is a useful predictive tool. Expected utility is a theory of aggregate behaviour and the Allais paradox is an optical illusion. It has also been argued that experiments do not reflect real choices made by individuals. Machina (1982) has also developed a theory of choice under risk that allows for violations of the independence axiom. Machina (1982) proves that the basic results of expected utility theory do not depend on the independence axiom, but may be derived from the much weaker assumption of smoothness of preferences over alternative probability distributions. Unlike anticipated utility theory, Machina’s (1982) theory does not employ a utility function that maps outcomes into the real line. The theory has no separation between outcomes and probabilities in the evaluation function. There are alternative theories, such as Kahneman & Tversky’s (1979) prospect-theory, who formulate that uncertain outcomes are defined relative to a reference point, which is typically current wealth. Outcomes are interpreted as gains and losses. Risky outcomes are referred to as prospects and the decision maker is assumed to choose among alternative prospects by choosing the one with the highest value. The value of a prospect is expressed in terms of two scales, first, a decision weight function, Aâ‚¬, which associates with each probability, p, giving Aâ‚¬(p) reflecting the impact of p. Aâ‚¬(p) is not a probability p in evaluating a prospect. The value function assigns to each outcome x a number v(x), which encodes the decision maker’s subjective value of outcome. Kahneman & Tversky’s (1979) formulation focuses on simple prospects which have at most two non-zero outcomes. The theory can be extended to more complicated prospects, but this poses certain difficulties as it can violate dominance, and hence transitivity, among prospects with more than two outcomes. Potential violations may occur due to the fact that the decision weights in prospect theory are derived by applying the decisiotrn weighting function to individual probabilities rather than to the entire probability density of outcome (Quiggan, 1982). For example, in the market for insurance, assuming the probability of the insured risk is 1%, the potential loss is £1,000 and the premium is £15. In order to apply prospect theory, it is first necessary to set a reference point, such as current wealth. Setting the frame to the current wealth, the decision would be to either pay £15, which gives the prospect theory-utility of u(-15), or a lottery with outcomes £0, with a probability of 99% or a 1% chance of A¢Ë†’£1,000, which yields the prospect-theory utility of w(0.01)x u(-1000) + w(0.99)x v(0). These expressions can be computed numerically. For typical value and weighting functions, the former expression could be larger due to the convexity of v in losses, and hence the insurance looks unattractive. Setting the frame to A¢Ë†’£1,000, both alternatives are set in gains. The concavity of the value function in gains can then lead to a preference for buying the insurance. In this example a strong overweighting of small probabilities can also undo the effect of the convexity of v in losses, the potential outcome of losing £1,000 is over-weighted. The interplay of overweighting of small probabilities and concavity-convexity of the value function leads to the so-called fourfold pattern of risk attitudes, such that risk-averse behaviour in gains involving moderate probabilities and of small probability losses; risk-seeking behaviour in losses involving moderate probabilities and of small probability gains. In the case of subjective probability theory, Savage (1954), argues that even if states of the world are not associated with recognisable, objective probabilities, consistency, such as restrictions on preferences among gambles still imply that decision makers behave as if utilities were assigned to outcomes, probabilities were attached to states of nature, and decisions were made by taking expected utilities. This rationalisation of the decision maker’s behaviour with an expected utility function can be seen uniquely, up to a positive linear transformation for the utility function. Ththeory is basically an extension and generalisation of the expected utility theory. The Ellsberg (1961) paradox concerns subjective probability theory. You are told that an urn contains 300 balls. One hundred of the balls are red and 200 are either blue or green. In gamble A, receive £1,000 if the ball is red. In gamble B, receive £1,000 if the ball is blue. An individual chooses which of these gambles they prefer, and then must consider the following two gambles. Firstly gamble C, where an individual will receive £1,000 if the ball is not red or gamble D, where the individual will receive £1,000 if the ball is not blue. It is common for people to strictly prefer A to B and C to D. But these preferences violate standard subjective probability theory. To see why, let R be the event that the ball is red, and A¬R be the event that the ball is not red, and define B and B’ accordingly. By ordinary rules of probability, p(R)=1-p(A¬R) p(B)=1-p(A¬B) Normalize u(0)=0 for convenience. Then if A is preferred to B, we must have p(R)u(1000)>p(B)u(1000), from which it follows that p(R)>p(B). If C is preferred to D, we must have p(A¬R)u(1000)> p(A¬B)u(1000) from which it follows that p(A¬R) > p(A¬B) However, it is clear that the above expressions are inconsistent. The Ellsberg (1961) paradox seems to be due to the fact that people think that betting for or against R is “safer” than betting for or against “blue.” Opinions differ about the importance of the Allais (1953) paradox and the Ellsberg (1961) paradox. Some economists think that these anomalies require new models to describe people’s behaviour. Others think that these paradoxes are akin to optical illusions. Another theory of interest in choice under uncertainty is the state dependent theory. This theory discusses the idea that when there are only monetary outcomes from lotteries then a complete description of the outcome of a pound gamble should include not only the amount of money available in each outcome but also the prevailing prices in each outcome. State dependent utility could also be described as being the preferences among the goods under consideration depending on the state of nature under which they become available. An example from Varian (1992), of state dependent utility function, looks at health insurance, where the value of a unit of currency may depend on one’s health. Varian (1992, p.190) asked the question “how much would a million dollars be worth to you if you were in a coma?”, and stated utility as a function of health and of money. Quiggan’s (1982) anticipated utility theory maintains properties of dominance and transitivity but employs a weakened version of the independence axiom. The model is consistent with a considerable range of choice behaviour that violates von Neumann-Morgenstern expected utility theory (1944). It also is free of the violations of dominance that can occur under prospect theory. Risk attitude under anticipated utility theory, discussed in Hilton (1988), follow Pratt (1964) and Arrow’s (1971) analysis of risk attitude under von Neumann-Morgenstern (1944) expected utility theory and characterize a decision maker’s attitude toward the risk inherent in a prospect by the decision maker’s risk premium for the prospect. Hilton (1988) also tests risk attitude under prospect theory, but slightly modifies the perspective on prospect theory from the theory’s original statement by Kahneman & Tversky (1979), required by two features of prospect theory. First there is the problem of dominance violations, which Kahneman & Tversky (1979) state that stochastically dominated alternatives are eliminated in the editing phase of the theory, acknowledging that such a procedure raises the problem of intransitivity. Bell (1982), Fishburn (1982) and Loomes & Sugden’s (1983) regret theory generalizes Savage’s (1954) mini-max regret approach. Choice is modelled as the minimising of a function of the regret vector, defined as the difference between the outcome yielded by a given choice and the best outcome that could have been achieved in that state of nature. A decision maker’s preference function is defined over pairs of prospects. It is possible that prospect A is preferred to B, B preferred to C, C preferred to A. Regret theory is another aspect of the rational consumer and the choice under uncertainty. The important comparison is between what is, or the prize you win, and what might have been, or what you could have won. In the Allais (1953) paradox, in the first instance choosing A and receiving £0, when the alternative was £1000 with certainty, would generate considerable regret in some people, in fact there is regret with probability 1. However, winning £0 in option C when there was only a 0.11 probability of winning something in option D, means that there is only an 11% chance of regret in choosing option C. This could rationalise the choice of B over A and then C over D. Regret theory is quite complicated as it implies that what is chosen is judged in relation to what is not chosen. Unlike other mental relationships such as fear and disappointment, which emerge from the comparison of components of a particular prospect, regret is defined across prospects. Hargreaves et al. (1994) explain that preferences need not be transitive, and this could lead to preference reversals. An example of this would be when choosing between X and Y, where X involves the chance of 1/3 of large regret if a green ball is pulled from the urn. If one chooses Y there is a 2/3 chance of regret, but it is less intense. This can lead to intransitive cycles in pair-wise comparisons, such that Y preferred to X, Z preferred to Y and X preferred to Z. So, this can be considered to be quite problematic. In conclusion, it is difficult to determine an accurate view of an individual’s choice under uncertainty, due to the variety of different theories that have been proposed and the limitation of some of the assumptions made by these theories. Although some of the theories discussed are merely an extension of other theories, also discussed, they do allow for some of the assumptions to be ignored. It can also be argued that this is not entirely an issue for economics, but that psychology also plays a role in a person’s decision making, as well as the value of the potential gain or loss, either in a monetary or non-monetary sense, in establishing a realistic outlook on an individual’s behaviour.
References & Biblography
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