Abstract
Most of the material covered so far deals with a world of certainty. Decision makers know the effects of consumption, employment, output, or pricing choices. This chapter considers decisions under conditions of uncertainty, in which economic actors cannot be sure of outcomes. Risk (we use the words risk and uncertainty interchangeably) relates to the existence of a range of possible outcomes that result from a decision. Specifically, the analysis treats an outcome x as the result of a gamble or lottery. The value of the outcome is random, and the probability of a particular realization of x can be described by a probability density function (or pdf), f(x). The chapter develops a model of risk preferences that relates to the nature of a utility function. The model is applied to ways of measuring differences in risk preferences. Finally, the model is applied to choices related to insurance.
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Notes
- 1.
For simplicity, we do not explore ramifications of the distinction between risk and uncertainty, as developed by Knight [2]. Knight treats risk as a situation in which the effects of decisions are not known, but the decision maker knows the probability distributions that pertain to the possible outcomes. True uncertainty occurs when the decision maker cannot assign probabilities to various possible outcomes. Knight argues that uncertainty but not risk is a source of profits.
- 2.
Determine the expected value when each of the following maximum number of tosses is allowed: 10, 50, 100. Also, determine the maximum amount that you could win in each of the three games and the probability of winning that amount. Consider how much you would pay to play each of these three games.
- 3.
Our analysis is limited to the utility that the person receives from the payoff itself. It does not take into account the pleasure from, or addiction to, the playing of the game. We use the words lottery and gamble interchangeably.
- 4.
- 5.
For an affine transformation, all points that lie on a line initially still lie on a line after the transformation, and ratios of distances are not affected (so that, for example, the midpoint of a line segment remains the midpoint after transformation).
- 6.
Other points on this reflect different odds. At any point, the W value is the expected wealth level, the height of the line segment is the expected utility level, and the height of the solid curve is the utility level that would result if the expected wealth level were a certain level of wealth. Clearly, at every value, the utility level for a certain wealth level exceeds the expected utility level when wealth is not subject to risk. The graph shows, however, that for some probability values, the expected utility exceeds 0.713. We examine the implications of this fact below.
- 7.
Exercise: Given the utility function \(U3(W,R) = {[(W - x)/(y - x)]}^{R}\), with R > 1, how will the expected value of this gamble compared to the certain $25? Note that R here no longer has the same interpretation as above.
- 8.
Exercise: Determine what happens to the difference between the utility of the certain $25 and the expected utility from the lottery as R increases.
- 9.
The constant term here makes the graph look more conventional, with utility going through the origin, but it is not necessary. Remember, expected utility functions are invariant to affine transformations.
- 10.
Exercise: Derive the relative risk aversion measures for U2(W, R) and U3(W, R) and compare them to their absolute risk aversion counterparts.
- 11.
More realistically, a consumer faces a range of possible negative outcomes with different probabilities. We use the two-outcome model for simplicity.
- 12.
Remember units. If this person’s wealth is measured in 1,000 dollar units, then the person is willing to pay $536 “regular” dollars above the actuarially fair premium.
Exercise: Examine the effect of changing R on the consumer’s willingness to pay above the actuarially fair premium.
References
Arrow KJ (1971) The theory of risk aversion. Reprinted in: Essays in the theory of risk bearing. Markham, Chicago
Knight FH (1921) Risk, uncertainty, and profit. Available at http://www.econlib.org/library/Knight/knRUP.html
Nicholson W, Snyder C (2008) Microeconomic theory: basic principles and extensions, 11th edn. South-Western, Mason
Pratt JW (1964) Risk aversion in the small and in the large. Econometrica 32:122–136
Varian HR (1992) Microeconomic analysis, 3rd edn. Norton, New York
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Hammock, M.R., Mixon, J.W. (2013). Uncertainty. In: Microeconomic Theory and Computation. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9417-1_16
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DOI: https://doi.org/10.1007/978-1-4614-9417-1_16
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