Question

Consider an individual whose utility function over income I is U(I), where U is increasing smoothly in I and is concave (in other words, our basic assumptions throughout this chapter). Let I_s = 0 be this person's income if he is sick, let I_H > 0 be his income if he is healthy, let p be his probability of being sick, let E[I] be expected income, and let E[U] be his expected utility when he has no insurance, a Write down algebraic expressions for both E[I] and E[U] in terms of the other parameters of the model. b Consider a full insurance product that guarantees this individual E[I]. Create a diagram in U-I space. Draw the individual's utility curve and the lines representing I_s, I_H, and E[I]. Then draw and label a line segment that corresponds to the utility gain, DeltaU, from buying this insurance product. Draw and label another line segment, M, which corresponds to the consumer surplus from the purchase of insurance (that is, the monetary value of the utility gain from buying insurance), c Derive an algebraic expression for M.

Answer 1

Expected income (E[I]) and expected utility (E[U]) are defined using probabilities and outcomes for income in different health states, and a full insurance policy guarantees a utility gain, with consumer surplus calculated as the monetary value equivalent to this utility gain.

Step-by-step explanation:

The student's question revolves around expected utility theory in the context of insurance and income. In terms of the parameters provided (I_s = 0, I_H > 0, probability p, E[I], E[U]), one can express expected income (E[I]) as E[I] = p * I_s + (1 - p) * I_H, which represents a weighted average of income in healthy and sick states, weighted by the probability of each state occurring. Expected utility (E[U]), without insurance, is then E[U] = p * U(I_s) + (1 - p) * U(I_H), reflecting the utility associated with each income level, again weighted by the probability of being healthy or sick.

For a full insurance product that guarantees the individual an income of E[I], the diagram in U-I space would include the individual's convex utility curve and the lines representing I_s, I_H, and E[I]. The utility gain from buying this insurance product (DeltaU) would be represented by a vertical line segment between the curve at I_H and the curve at E[I]. The consumer surplus from insurance, M, is the monetary equivalent of the utility gain and can be algebraically derived by solving U-1(DeltaU + U(I_s)) = M, where U-1 is the inverse utility function. It represents the difference in income that gives the same utility increase as the insurance.