In this exercise, we will formally show the two properties of indifference curves in I_H–I_S space we discussed in Section 9.2. To prove that indifference curves are downwardsloping, we calculate the slope of the indifference curves ^dIS/_dIH directly. Recall that the individual with income I_H in the healthy state and I_S in the sick state, and with probability p of becoming sick, has an expected utility E[U]_p of

a. Take the total derivative of E[U]_p. This will give a formula explaining how changes in I_H and I_S contribute to changes in E[U]_p.

b. Because indifference curves, by definition, connect points with constant utility, set dE[U]_p equal to 0 and then solve for dI_S/dI_H.

c. Using what you know about the signs of p, U′(I_H), and U′(I_S), prove that the sign of dI_S/dI_H is negative.

d. A curve is convex in I_H–I_S space if its second derivative is positive everywhere. Derive the second derivative of the indifference curves by taking the derivative of your expression for dI_S/ dI_H with respect to I_H.

e. Using what you know about the signs of p, U′(I), and U′′(I), prove that your expression for the second derivative is positive everywhere.

Transcribed Image Text:

E[U]=pU(Is)+(1-p)U(IH) (9.3)