1. A household has Cobb-Douglass utility over housinghand other goodsc:u(c,h) =αlnc+ (1−α) lnh.The household’s budget constraint isc+h≤y ,whereyis total income and we are normalizing the prices of housing and other goodsto one. Assume there is no labor supply decision, soyis fixed. You can think ofhascapturing housing quality or expenditure, where higher quality housing is more expensive.(a) Solve for the household’s optimal bundle (c∗,h∗). Your answer will depend on the parameter α.Suppose the government subsidizes housing by reimbursing a household for a fractiontof their spendingh. The household only pays (1−t)hout of their own pocket.(b) Solve for the optimal bundle (c(t),h(t)) as a function of the subsidy rate. How doesit compare to (c∗,h∗)?(c) Graphically illustrate how the housing subsidy affects the household’s chosen bundle.Decompose the income and substitution effects (you may draw multiple graphs).Now suppose that the subsidy is a fixed amountHthat can be spent on housing. If ahousehold spends less thanHon housing, they pay nothing out of their own pocket.If they spend more thanH, they payh−Hand the government pays the rest.(d) Calculate the optimal bundle (c(H),h(H)) as a function ofH. For which values ofHdoesh(H) =H?(e) Graphically, decompose the income and substitution effects of a fixed subsidyHinthe two cases of (i)h(H) =H, and (ii)h(H)> H.(f) In which case is the housing subsidy equivalent to an unrestricted cash transfer?2. You earn $40,000 per year at your current job. With probabilityp, you will lose yourjob next year and earn only $10,000. With probability 1−pyou will keep your job andcontinue to earn $40,000. Your utility function over your incomeyisu(y) =√y1
(a) What is theexpected valueof your income next year? (this is a function ofp)(b) What is yourexpected utilitynext year?(c) What is thecertainty equivalentof your income next year?Suppose a risk-neutral insurance company offers an insurance policy that pays you$30,000 if you lose your job, but nothing otherwise. In return, you pay a premium of$Pin all states of the world.(d) Show that this policy fully insures you against income risk, regardless of the premium.(e) What is theactuarially fairpremium, as a function of your layoff probabilityp?(f) What is the maximum premium you would be willing to pay for this insurance policy(as a function ofp)? We’ll call this yourwillingness-to-payfor full insurance.(g) Compare your willingness-to-pay to the actuarially fair premium forp=12, andp=110. This is thevalueof fully insuring yourself against income risk.(h) For what value ofpis the value of full insurance the largest? How is your answerrelated to the amount of uncertainty you face? (Hint: compare thevarianceof nextyear’s income as a function ofpto the value of insurance)3. Suppose everyone in the U.S. earns $40,000 per year with certainty. However, each personcan get sick. If they get sick, they incur $30,000 in medical expenses, and so their incomenet of medical costs falls to $10,000. If they do not get sick, her health care costs are zero.Suppose that each American receives utilityu(y) =√yfrom their net incomey.So far, this problem is equivalent to the previous one. But suppose there are two types ofpeople. The first type is “healthy,” and has a 10% chance of getting sick. The second typeis “unhealthy,” and has a 50% chance of getting sick. Let (p1,p2) = (0.1,0.5) denote theseprobabilities, using 1 to denote the healthy type.(a) Describe a health insurance contract which fully insures each type of person againsttheir risk of incurring health care costs.(b) For each type, what is the actuarially fair premium for full insurance (P1,P2)? Whatis each type’s willingness-to-pay for full insurance?(c) Suppose instead that utility is given byu(y) = ln(y)How do you answers to parts (a) and (b) change?Now suppose that an insurance company offers an insurance policy that fully covershealth care costs. However, the company cannot distinguish between healthy andunhealthy people. If the company offers the policy at premium $P, both types canbuy it. The company does know the fraction of unhealthy types,λ, in the population.(d) If all Americans buy the insurance policy, what is the actuarially fair premiumP(λ)at which the insurance company makes zero profits? (Note: this is a function ofλ)(e) Suppose all Americans have the utility functionu(y) = ln(y), and a fractionλ=112of the U.S. population is unhealthy. Would unhealthy people be willing to buy insur-ance at the actuarially fair price you found in part (d)? What about unhealthy people?2
For the rest of the problem, we’ll go back to assumingu(y) =√y. It is still the casethatλ=112.
(f) How does your answer to part (e) change if everyone has utility functionu(y) =√y?
(g) Who will buy insurancein equilibrium, assuming the insurance company chargesan actuarially fair premium (makes zero profits)? What is the equilibrium premium?
(h) Suppose the U.S. government passes a law mandating that every individual buy full health insurance coverage. Would this policy be a Pareto improvement over the equilibrium in part (g)? Which type(s) would be better or worse off? Explain.
