question as attached
2. (36 points) An individual earns $500 per week, but faces the risk of getting sick. If she remains healthy, she spends all of her income on non-medical goods. If she gets sick, she incurs $400 of medical expenses, and only has $100 left. for non-medical expenses. She becomes sick with probability 33 each week. Her utility from non- medical Spending, y, is dened on a weekly basis by nub@- 3; Getting sick does not affect the individual's utility directly; it simply lowers the income she has available for non-medical Spending. The individual cannot change her probability of getting sick, or her medical spending if she becomes sick. She also cannot save or borrow; she must spend all of her income each week. (a) (12 points) For this individual, calculate the following as a function of p: i. Her expected utility. (Note: utility is negative, but increasing in y] ii. Her expected income. iii. The certainty equivalent of her uncertain income. iv. Her willmgness-to-pay for being fully insured against the risk of incurring medical expenses. In other words, nd the highest the premium P that the individual would pay every week for an insurance contract which pays her $400 if she gets sick that week. Denote this function 6(p) v. Her expected medical expenses. Denote this function C(p). Now suppose there are many individuals who all have the same utility function given above. However, these individuals have different levels of health risk: 0 20% of the population is \"low-cost," with sickness probability ,oL : . o The remaining 80% are \"high-cost," with sickness probability 1);; = %. There is a competitive insurance market where many rms sell inSurance con- tracts to consumers. The insurers' only costs are the payments they make to sick individuals. An individual's own health risk is private information, so insurance companies must charge the same premium to everyone who buys a given insurance contract. (b) (12 points} Suppose insurance companies can only offer full insurance. This means every insurance contract must pay $400 if the insured individual gets sick. Firms only choose the premium P. i. Compute 9(p) and C (p) for high-cost and low-cost individuals. ii. Calculate the expected cost per customer for inSurers if all individuals pur- chase insurance. iii. In equilibrium, who buys insurance? What is the equilibrium premium? To receive full credit, you must show that your answer satises the necessary conditions for a competitive equilibrium. (Next page...)
