6 Risk and Risk Aversion 179     The evaluation of this game is called the "St. Petersburg Paradox." Although the expected payoff is infinite, participants obviously will be willing to purchase tickets to play the game only at a finite, and possibly quite modest, entry fee. Bernoulli resolved the paradox by noting that investors do not assign the same value per dollar to all payoffs. Specifically, the greater their wealth, the less their "appreciation" for each extra dollar. We can make this insight mathematically precise by assigning a welfare or utility value to any level of investor wealth. Our utility function should increase as wealth is higher, but each extra dollar of wealth should increase utility by progressively smaller amounts.4 (Modern economists would say that investors exhibit "decreasing mar- ginal utility" from an additional payoff dollar.) One particular function that assigns a sub- jective value to the investor from a payoff of $R, which has a smaller value per dollar the greater the payoff, is the function ln(R) where ln is the natural logarithm function. If this function measures utility values of wealth, the subjective utility value of the game is indeed finite, equal to .693.5 The certain wealth level necessary to yield this utility value is $2.00, because ln(2.00) .693. Hence the certainty equivalent value of the risky payoff is $2.00, which is the maximum amount that this investor will pay to play the game. Von Neumann and Morgenstern adapted this approach to investment theory in a com- plete axiomatic system in 1946. Avoiding unnecessary technical detail, we restrict our- selves here to an intuitive exposition of the rationale for risk aversion. Imagine two individuals who are identical twins, except that one of them is less fortu- nate than the other. Peter has only $1,000 to his name while Paul has a net worth of $200,000. How many hours of work would each twin be willing to offer to earn one extra dollar? It is likely that Peter (the poor twin) has more essential uses for the extra money than does Paul. Therefore, Peter will offer more hours. In other words, Peter derives a greater personal welfare or assigns a greater "utility" value to the 1,001st dollar than Paul does to the 200,001st. Figure 6B.1 depicts graphically the relationship between the wealth and the utility value of wealth that is consistent with this notion of decreasing marginal utility. Individuals have different rates of decrease in their marginal utility of wealth. What is constant is the principle that the per-dollar increment to utility decreases with wealth. Functions that exhibit the property of decreasing per-unit value as the number of units grows are called concave. A simple example is the log function, familiar from high school mathematics. Of course, a log function will not fit all investors, but it is consistent with the risk aversion that we assume for all investors. Now consider the following simple prospect:   p 1⁄2