Intermediate Microeconomics

Professor Yongmin Chen

Topic 11. Risk and Information

Expected Values

When a person is uncertain about the outcomes of an action or a choice but he can assign probabilities to these outcomes, we say the person faces a decision (choice) problem involving risk.

Suppose your company offers you the following incentive plan: if the company’s profit goes up by at least 10% this year, you will receive a bonus of $10,000; and otherwise you receive zero bonus. Suppose also that you believe that, with a good effort of yours, there is 50% chances that the company’s profits will grow at least 10% this year. What is your expected value of participating in your company’s incentive plan? $10,000x0.5 + $0x0.5 = $5,000.

In general, suppose there are n possible outcomes, and the payoffs associated with these outcomes are x₁ , x₂ , ..., x_n . Suppose you assign probabilities p₁ , p₂ , ..., p_n to these outcomes. Then the expected value is V = x₁p₁ + x₂ p₂ + ... + x_n p_n.

Exampl. Suppose you are given an option to buy 1000 shares of a company’s stocks at $20 a share one year from now. If the company’s stock price will go up to $30 a share one year from now with probability 0.2, and will remain at $20 a share with probability 0.8, what is expected value of having the option?

Sometimes, it makes sense for a person to try to maximize the expected value when making alternative choices. But that is not always how people behave. Suppose you face two choices: Choice A: having 2 million dollars for sure. Choice B: having 5 million dollars or zero with 50/50 chances. Most people would probably choose A, although B promises higher expected values. To rationalize such behavior, we need a theory called expected utility maximization.

Expected Utility

Utilities are real numbers assigned to different monetary values (or wealth). They measure the levels of satisfaction one obtains from having certain monetary values (or wealth). If x is monetary values, then U(x) is the utility function.

Example. A numerical example of an utility function.

The shape of a person’s utility function reflects his attitude toward risk. A person is risk averse if he prefers a more certain outcome than a less certain outcome if both outcomes have the same expected monetary value. A person who is risk averse has an utility function that is concave (explained in class). A person is risk-seeking if he prefers a less certain outcome to a more certain outcome if both outcomes have the same expected monetary values. A risk-seeker’s utility function is convex.

Example. Examples of utility functions: U(x) = x²; U(x) = log (x + 1).

Example. Why do people buy insurance? How much is a person willing to pay for insurance?

It is possible that a person is risk averse over some values and is risk seeking over some other values.