The two curves in Fig. 9.5 exemplify the graphs of quadratic functions, which may be expressed generally in the form v = ax2 + bx +c ia^ 0)

From our discussion of the second derivative, we can now derive a convenient way of determining whether a given quadratic function will have a strictly convex (U-shaped) or a strictly concave (inverse U-shaped) graph.

Since the second derivative of the quadratic function cited is d2y/dx2 = 2a, this derivative will always have the same algebraic sign as the coefficient a. Recalling that a positive second derivative implies a strictly convex curve, we can infer that a positive coefficient a in the preceding quadratic function gives rise to a U-shaped graph. In contrast, a negative coefficient a leads to a strictly concave curve, shaped like an inverted U,

As intimated at the end of Sec. 9.2, the relative extremum of this function will also prove to be its absolute extremum, because in a quadratic function there can be found only a single valley or peak, evident in a U or inverted U* respectively.

The most common application of the concept of marginal utility is to the context of goods consumption. But in another useful application, we consider the marginal utility of income, or more to the point of the present discussion, the payoff to a betting game, and use this concept to distinguish between different individuals' attitudes toward risk.

Consider the game where, for a fixed sum of money paid in advance (the cost of the game), you can throw a die and collect $10 if an odd number shows up, or $20 if the number is even. In view of the equal probability of the two outcomes, the mathematically expected value ofpayoff is

FIGURE 9.6

FIGURE 9.6

The game is deemed a fair game, or fair bet> if the cost of the game is exactly $ 15. Despite its fairness, playing such a game still involves a risk, for even though the probability distribution of the two possible outcomes is known, the actual result of any individual play is not. Hence, people who are "risk-averse" would consistently decline to play such a game. On the other hand, there are "risk-loving57 or "risk-preferring" people who would welcome fair games, or even games with odds set against them (i.e., with the cost of the game exceeding the expected value of payoff).

The explanation for such diverse attitudes toward risk is easily found in the differing utility functions people possess. Assume that a potential player has the strictly concave utility function U = U{x) depicted in Fig. 9.6a, where x denotes the payoff, with J7(0) = 0, £/'(*) > 0 (positive marginal utility of income or payoff), and U"(x) < 0 (diminishing marginal utility) for all x. The economic decision facing this person involves the choice between two courses of action: First, by not playing the game, the person saves the $ 15 cost of the game EV) and thus enjoys the utility level f/($15), measured by the height of point A on the curve. Second, by playing, the person has a .5 probability of receiving $10 and thus enjoying C/($10) (see point M), plus a .5 probability of receiving $20 and thus enjoying f/($20) (see point N). The expected utility from playing is, therefore, equal to

which, being the average of the height of M and that of W, is measured by the height of point B, the midpoint on the line segment MN. Since, by the defining property of a strictly concave utility function line segment MN must lie below arc MN, point B must be lower than point A; that is, EU, the expected utility from playing, falls short of the utility of the cost of the game, and the game should be avoided. For this reason, a strictly concave utility function is associated with risk-averse behavior.

For a risk-loving person, the decision process is analogous, but the opposite choice will be made, because now the relevant utility function is a strictly convex one. In Fig. 9.6b,

C/(S15), the utility of keeping the $15 by not playing the game, is shown by point A' on the curve, and EU, the expected utility from playing, is given by Bfz the midpoint on the line segment MrNf. But this time line segment M!Ne lies above arc Af jV', and point B' is above point A'. Thus there definitely is a positive incentive to play the game. In contrast to the situation in Fig. 9.6a, we can thus associate a strictly convex utility function with risk-loving behavior.

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