Risk avoidance and risk taking under uncertainty: a graphical analysis.(Statistical data)

1. Introduction

Market insurance (risk avoidance) and "gambling" (risk taking) are economic activities concerned with choices under uncertain environments. It is known that yon Neumann and Mogenstern's (1944) theory of expected utility maximization and Arrow (1963) and Pratt's (1964) measures of risk aversion have been widely adopted to examine the economics of choices involving risk. Because the utility function of income under uncertainty is unique up to an affine transformation in preference ordering, Arrow (1984) indicates that

 
   [A]ll the intuitive feelings which lead to the 
   assumption of diminishing marginal utility 
   are irrelevant, and we are free to assume that 
   marginal utility is increasing so that the existence 
   of gambling can be explained with the 
   theory. (p. 28) 

In explaining the coexisting phenomena of insurance and gambling discussed by Friedman and Savage (1948), Arrow (1984) further remarks that

 
   Insurance is rational if the utility function 
   has a decreasing derivative over the interval 
   between the two incomes possible (decreasing 
   on the average but not necessarily everywhere), 
   while gambling is rational if the utility 
   has a predominantly increasing derivative 
   over the interval between the possible outcomes. 
   In view of the structure of gambles 
   and insurance ..., this requires that the utility 
   function have an initial segment where 
   marginal utility is decreasing, followed by a 
   segment where it is increasing. (pp. 28-29) 

Instead of analyzing the behavior of risk-lovers--agents with increasing marginal utility of wealth/income, this paper focuses its analysis on the behavior of risk averters. We wish to examine the following two questions. Under what conditions will a utility-maximizing individual with diminishing marginal utility of income choose to undertake risky activities? Will risk-averse individuals with different income positions engage in gambling activities at the same time?

Based on the state-preference framework of Arrow (1964, 1965) and Ehrlich and Becker (1972), we examine changes in optimizing behavior from risk avoidance to risk taking for risk-averse individuals. We focus the analysis on changes in decisionmaking under uncertainty for an individual at different income positions and for individuals facing different economic opportunities. Moreover, we pay particular attention to factors that influence changes in optimal demand for insurance or gambling. These factors include the degree of risk aversion in preferences, the actuarial fairness/unfairness of market insurance terms, and an individual's subjective evaluations of incomes in different states of nature.

In the analysis, we adopt a pedagogical graphical approach to characterize explicitly variations in optimal decisions in response to changes in economic environments. The graphical approach serves as a very useful alternative to a more complicated analytical approach. Moreover, graphical techniques are important pedagogically to allow for a visualization of equilibrium concepts under uncertainty. The paper graphically demonstrates the familiar result that if the insurance premium is larger than the certainty equivalent premium, risk-averse individuals will not buy insurance. Several other interesting findings are presented as follows. First, the coexistence of insurance and gambling for an individual at different income positions may result from a sufficiently strong degree of decreasing risk aversion as income endowment increases. Second, an individual whose preferences exhibit constant absolute risk aversion purchases less and less market insurance and eventually becomes a risk taker when his endowed incomes in "good" and "bad" states are decreasing to critically low levels. Third, with no change in potential losses, an individual whose preferences exhibit constant relative risk aversion would purchase less and less market insurance and eventually become a risk taker when his endowed incomes in good and bad states are increasing to critically high levels.

The economic rationale for behavioral changes under uncertain situations is straightforward. A risk-averse individual may choose to switch from risk avoidance to risk taking when his subjective evaluation of the bad-state income in terms of the good-state income that he is willing to give up differs from what has to be given up in the marketplace for insurance. Consequently, it is rational for an individual to purchase market insurance at one income position, but become a "risk taker" at another income position. It is also rational for both low-and high-income risk-averse individuals to engage in risk-taking activities (i.e., demand for "gambling") at the same time. These results are consistent with the observations that risk averters may become risk takers if existing economic opportunities are sufficiently favorable. Thus, predictions about changes in attitudes toward risk cannot be made independently of available economic environments or opportunities.

The remainder of the analysis is organized as follows. In Section 2, we discuss the traditional two-state-preference approach to insurance and use it as an analytical framework for the subsequent analysis. In Section 3, we examine the effect of changes in income endowment on behavioral change from risk avoidance to risk taking. Section 4 summarizes and concludes.

2. The Traditional Framework of Two-State Preferences

To analyze risk avoidance and risk taking, we use the state-preference framework originally developed by …