Question

Show that if an individual's utility-of-wealth function is convex then he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence?

Short answer

Answer: The relationship between a convex utility-of-wealth function and an individual's preference for fair gambles over income certainty lies in the expected utility theory. A convex utility function implies that an individual is risk-seeking. According to Jensen's inequality, the expected utility of a fair gamble will be greater than or equal to the utility of the expected wealth for a convex utility function, meaning the individual will prefer the fair gamble over income certainty. However, this risk-taking behavior might not be common, as it can be influenced or limited by factors such as financial constraints, regulatory constraints, cultural and social norms, and limited information and cognitive biases.

Step-by-step solution

Defining a fair gamble

A fair gamble is a situation where the expected value of the gamble is equal to the certain income that the person receives. In other words, the probability of gaining or losing in the gamble multiplied by the respective amounts is the same as the person's income certainty.

Defining expected utility

Expected utility is the sum of the utilities of each possible outcome, each weighted by the probability of that outcome. The expected utility can be represented as E(U) = ∑(p_i * U(w_i)), where p_i is the probability of outcome i, U(w_i) is the utility of the wealth in outcome i, and the summation is over all possible outcomes.

Analyzing a convex utility function and a fair gamble

Assuming the individual has a convex utility-of-wealth function U(w), let's analyze a fair gamble with a potential gain of G and a potential loss of L, both with an equal probability of 0.5. Let w be the initial wealth, then the expected utility of the fair gamble can be written as: E(U) = 0.5 * U(w + G) + 0.5 * U(w - L) Now, since U(w) is a convex function, according to Jensen's inequality, E(U(w)) ≥ U(E(w)), where E(w) is the expected wealth. In other words, the expected utility of the gamble will be greater than or equal to the utility of the expected wealth. Since it's a fair gamble, the expected wealth E(w) is equal to the initial wealth w (because the expected gain and loss cancel each other out). Hence, we have E(U(w)) ≥ U(w), which means the individual will prefer the fair gamble over income certainty.

Discussing the preference for unfair gambles

If the convexity of the utility function is strong (i.e., the individual is very risk-seeking), the individual may also prefer somewhat unfair gambles, where the potential loss is greater than the potential gain. The preference for these gambles depends on the magnitude of the convexity and the degree of unfairness of the gamble.

Discussing risk-taking behavior and limiting factors

While some individuals might exhibit risk-seeking behavior due to a convex utility function, it may not be a common phenomenon. People's risk preferences can be influenced by several factors, such as their financial situations, psychological factors, and past experiences. Moreover, factors that might limit the occurrence of risk-taking behavior include: 1. Regulatory constraints: Policies and regulations often limit the amount of risk that individuals or institutions can undertake. 2. Financial constraints: People with less financial capacity to absorb losses may refrain from taking too much risk. 3. Cultural and social norms: Societal attitudes towards risk-taking can play a role in shaping people's risk preferences. 4. Limited information and cognitive biases: People might not always possess complete information to make decisions, and cognitive biases could lead them to make suboptimal choices. In conclusion, although a convex utility function may imply a preference for fair or somewhat unfair gambles over income certainty, other factors might limit the occurrence of risk-taking behavior, and it might not be a prevalent phenomenon.