Expected Utility Theory and Utility Functions

In the literature, differences in risk preference among individuals or

organizations have been described using expected utility theory and utility functions (von Neumann and Morgenstern, 1944). Expected utility theory is prescriptive: a

decision maker is supposed to choose the policy that maximizes his expected utility over all possible outcomes.

5.3.1 Expected Utility Theory

Using expected utility theory, different risk attitudes are classified into risk-averse, risk-neutral, and risk-seeking, each of which is discussed below.

Representation of Different Risk Attitudes

Consider a hypothetical situation as follow. A variable x can be x1 or x2 with probability p and (1-p). The expected value of this random condition is defined in Equation 5.1.

1 2

[ ] (1 )

E x = px + −p x [5.1]

Let U(x) be a utility function of x, the function that a decision maker tries to maximize.

The expected utility is defined as in Equation 5.2.

1 2

[U( )] U( ) (1 )U( )

E x = p x + −p x [5.2]

A special condition when expected value theory and expected utility theory provide the same result is explained below. This applies when an individual or an organization is risk-neutral. According to expected value theory, E(x) is a linear combination of x1 and x2 as in Equation 5.1, which can be seen on the x axis in Figure 5.1. Figure 5.1 has a straight line AEB, which is a utility function. If U(x) = x, then U(x1) = x1, U(x2) = x2, and E[U(x)] = E[x], as shown on the y axis. Thus, expected utility is identical to expected monetary value.

Consider two lotteries:

ƒ Lottery 1 pays the expected value E x( )= px₁+ −(1 p x) ₂ with certainty.

ƒ Lottery 2 pays either x1 with probability p or x2 with probability (1-p).

If a decision maker has the utility function in Figure 5.1, the individual will be indifferent between the two lotteries and the individual is called risk-neutral. Therefore, a risk-neutral decision maker will make a choice as consistent with expected value theory.

Utility, U(x)

E[U(x)] = E[x]

U(x1) = x1

U(x2) = x2

A

B

E

P

1 - P

x

x1 E[x] x2

P 1 - P

Figure 5.1 Utility Function – Risk-neutral

Next, consider a risk-averse decision maker who has a concave utility function such as the curve ACDB in Figure 5.2. Figure 5.2 describes how E[x] and E[U(x)] can be different and how a decision can be made differently depending on individuals’ risk attitude, in particular the risk-averse case. In the figure, A is the point on the utility function corresponding to U(x1). B is the point on the utility function corresponding to U(x2). On the curve ACDB, the point D corresponds to U[E(x)], the utility of the expected value. Meanwhile, the point C corresponds to E[U(x)], the expected value of the utility.

x

x1 CE(x) E[x] x2

P 1 - P

A

B

C D E

P

1 - P E[U(x)]

U[E(x)]

U(x1) U(x2)

Utility, U(x)

Figure 5.2 Utility Function – Risk-averse

As shown on the y axis in Figure 5.2, U[E(x)] is greater than E[U(x)] as in Equation 5.3. Using Equations 5.1 and 5.2, Equation 5.4 is derived.

U[ ( )] [U( )]E x > E x [5.3]

1 2 1 2

U(px +(1- ) )p x > pU( ) (1- )U( )x + p x [5.4]

Equations 5.3 and 5.4 explain that the utility of the expected value is greater than the expected value of the utility using the concave utility function in Figure 5.2.

Consider again the same lotteries discussed above now for a risk-averse decision maker.

ƒ Lottery 1 pays the expected value E x( )= px₁+ −(1 p x) ₂ with certainty.

ƒ Lottery 2 pays either x₁ with probability p or x2 with probability (1-p).

As described in Equation 5.4, the utility that the risk-averse decision maker will have, if he/she obtains E(x) with certainty, is greater than expected value of the

combination of U(x1) and U(x2) with probability p and (1-p). Therefore, this decision maker will choose lottery 1. The decision maker prefers certainty to uncertainty even though lottery 2 could pay more.

In contrast, for a risk-seeking decision maker, if such an individual exists, the utility that the decision maker will have, if he/she obtains E(x) with certainty, is smaller than expected value of the combination of U(x1) and U(x2) with probability p and (1-p).

Therefore, the decision maker will choose lottery 2. The decision behaviors by a risk-seeking decision maker can be represented by a convex curve, which has the mirror image of the concave curve against in the straight utility function AED in Figure 5.2.

Risk Premium

For a risk-averse decision maker, U[E(x)] > E[U(x)]. The decision maker perceives U[E(x)] more valuable than E[U(x)]. How much more does the decision maker value U[E(x)] than E[U(x)]? The equivalent amount of monetary value when the decision maker chooses Lottery 1 is CE(x), the certainty equivalent of x, which can be calculated using an inverse function, as shown in Figure 5.2. If the difference between E(x) and CE(x) is paid to the risk-averse decision maker, the decision maker would be willing to take the risky choice, lottery 2. In fact, the decision maker becomes

indifferent between the two lotteries. The difference between the expected value and CE(x) is called the risk premium (RP) as in Equation 5.5.

In order to let a risk-averse decision maker choose Lottery 2 instead of Lottery 1, someone needs to pay the risk premium to the risk-averse decision maker. In general, without providing the risk premium or extra benefit, the risk-averse decision maker would not choose a risky choice (lottery 2 in this example).

RP(x) = E[x] – CE(x) [5.5]

Therefore, different risk attitudes can be classified by the value of the risk premium as below:

ƒ RP(x) > 0 for all values of x for a risk-averse individual or organization

ƒ RP(x) = 0 for all values of x for a risk-neutral individual or organization

ƒ RP(x) < 0 for all values of x for a risk-seeking individual or organization

5.3.2 Utility Functions

There are different types of utility functions used in the literature. As explained above, the utility function is a function that a decision maker maximizes in his/her choice. So, a utility function depends on an individual’s risk preference. Note that once U(x) has been defined, making a decision is a purely mathematical operation

(optimization) and no choice is left to the decision maker. Consequently, if U(x) is known, the decision made by the decision maker can be predicted with perfect accuracy.

Frequently used utility functions are exponential functions, power functions, and quadratic functions. Among them, exponential functions (Pratt, 1964) are most

frequently used (Walls and Dyer, 1996), and are discussed below.

The general form of exponential utility functions is found in Equation 5.6.

U( )x = −a e^−cx (0≤ ≤ ∞ x ) [5.6]

Where, a is a constant to make the utility positive (if so desired); and

c is the risk-aversion coefficient. c = 0 indicates risk-neutrality and large values of c indicate greater risk-aversion (greater curvature of the utility function).

The coefficient c is to decide the degree of risk-aversion. The main reason for the frequent use of these exponential functions is their simplicity: the function needs only one parameter, the risk-aversion coefficient c. Figure 5.3 provides a set of

exponential functions with different risk-aversion coefficient values and the constant a = 1, for example. Utility functions of real people would vary.

In Figure 5.3, as the risk-aversion coefficient c increases, the functions have more curvature, which means that U[E(x)] becomes higher than E[U(x)], requiring greater amount of risk premium for a risk-averse decision maker. A more concave curvature in a utility function represents a more risk-averse attitude.

0.0

Figure 5.3 Example of Exponential Utility Functions (Risk-averse Case)