Decision-Making under Uncertainty – Part I Topic 4
Outline for this Topic Contingent Consumption Utility Function and Probabilities Expected Utility Risk-Averse Agent and Insurance Premium Risk-Lover Agent and Insurance Premium Risk-Neutral Agent Insurance Amount under Risk-Neutral Insurer and Risk-Averse Insuree
Contingent Consumption Under uncertainty, the consumer is concerned with the probability distribution of getting different consumption bundles of goods People have different preferences over probability distributions
Contingent Consumption (cont.) We can think of the different outcomes of some random events as being different states of nature. Then, a contingent consumption plan can be seen as being a specification of what will be consumed in different states of nature (contingent means depending on something, not certain) We will focus on the analysis of choices among contingent monetary outcomes, instead of contingent bundles of goods. Of course, it is not money alone that matters; it is the consumption that money can buy that is the ultimate good being chosen
Utility Function and Probabilities If the consumer has reasonable preferences about “consumption” in different circumstances, then we can use a utility function to describe these preferences We are considering choice under uncertainty, where the value of wealth in one state as compared to another will depend on the probability that these states occur
Utility Function and Probabilities (cont.) Then, the utility function should depend on the probabilities of occurrence of events and outcomes of these events. We will call this utility function expected utility function (or von Neumann-Morgenstern utility function) U(w1, w2, 1, 2) = 1 u(w1) + 2 u(w2) Note: We call a situation when 2 or more mutually exclusive outcomes occur with positive probabilities a “gamble or lottery.” A gamble with the same expected value of wealth as a certain outcome to which it is compared is called fair gamble
Risk–Averse Agent and Insurance Premium A consumer is risk-averse if she/he always prefers a certain outcome to a gamble with the same expected value of wealth u(g) < u(E(g)) Risk-averse behavior corresponds to concave u(w). The diminishing marginal utility of wealth shows the agent risk aversion
Risk–Averse Agent and Insurance Premium (cont.) Maximum risk premium for a risk-averse agent –Find the certain outcome x such that u(x) = u(g) –Maximum premium = (initial wealth endowment) – x –If a fair insurance premium (premium = expected payment by insurance company) is available, the risk-averse agent and the insurance company will always find a mutually beneficial contract (p fair < p maximum)
Risk–Lover Agent and Insurance Premium A consumer is risk-lover if she/he always prefers the fair gamble to the certain outcome u(g) > u(E(g)) Risk-lover behavior corresponds to convex u(w). The increasing marginal utility of wealth reflects the agent risk-loving preferences
Risk–Lover Agent and Insurance Premium (cont.) Maximum risk premium –Find the certain outcome y such that u(y) = u(g) –Maximum premium = (initial wealth endowment) - y –If a fair insurance premium is available, the risk-lover agent and the insurance company will not find a mutually beneficial contract (p fair > p maximum)
Risk-Neutral Agent A consumer is risk-neutral if she/he is indifferent between the fair gamble and the certain outcome u(g) = u(E(g))
Insurance Amount under Risk-Neutral Insurer and Risk-Averse Insuree Assume that the insurer is risk-neutral and the insuree is risk- averse If a fair premium is available, the risk-averse agent will buy a full-insurance policy –Let D = loss under bad event, α = insurance amount. Then, the insuree’s expected utility will be maximized at α* = D