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?

A convex utility function, in the context of wealth, represents an individual's preference for taking risks. It has the property that the individual derives more satisfaction from the average utility of two outcomes than from the utility of the average outcome. Mathematically, this can be expressed as follows: For a utility function U(w), if U is convex, U(\frac{w_1 + w_2}{2}) > \frac{U(w_1) + U(w_2)}{2} where w_1 and w_2 are two different wealth levels.

A fair gamble is defined as a situation where the expected monetary value of gain and loss is equal, so there is neither an expected gain nor a loss for the individual participating in the gamble. An example of a fair gamble is a coin toss where you win 10 units of wealth if the coin lands heads and lose 10 units of wealth if the coin lands tails.

With a convex utility function, let's analyze the preference for a fair gamble. Suppose we have two wealth levels, w_1 and w_2, and a fair gamble is offered with an equal probability of ending up with either wealth level (50% chance of getting w_1 and 50% chance of getting w_2). The expected utility from the gamble is: EU_gamble = \frac{1}{2}U(w_1) + \frac{1}{2}U(w_2). Now, let's compare this with the utility from income certainty which is at the average wealth: U(\frac{w_1 + w_2}{2}). As the utility function is convex, we have: U(\frac{w_1 + w_2}{2}) > \frac{U(w_1) + U(w_2)}{2}. But since EU_gamble = \frac{1}{2}U(w_1) + \frac{1}{2}U(w_2), this means that: U(\frac{w_1 + w_2}{2}) > EU_gamble. This demonstrates that an individual with a convex utility function will prefer a fair gamble over income certainty.

An individual with a convex utility function may also be willing to accept somewhat unfair gambles, depending on the exact shape of the utility function and the expected payoff from the gamble. For example, if the potential gains from an unfair gamble are large enough relative to the potential losses, they might outweigh the negative impact of the losses. In this case, the expected utility of the unfair gamble would still be greater than the utility of income certainty.

While it is possible for individuals to have convex utility functions and prefer taking risks, the prevalence of such behavior may be limited by various factors. Some of these factors include: 1. Limited access to wealth: Individuals may be unable to participate in gambles due to limited income or wealth. 2. Cognitive biases and psychological factors: Individuals might overestimate the probability of negative outcomes or be excessively optimistic about the potential gains, thus affecting their decision-making process. 3. Market imperfections and regulation: Legal restrictions or other market imperfections might limit the availability of certain types of gambles. 4. Social and cultural factors: Social norms and cultural values might inhibit risk-taking behavior or influence how individuals perceive risks. Overall, while convex utility functions can help explain some risk-taking behavior, numerous factors might tend to limit its occurrence in reality.