Q1. Suppose in a community 10 people each have a Marginal Benefit from streetlights equal to MB=5 - 2Q, and five people each have a Marginal Benefit for streetlights of MB=7 - Q, where Q is the quantity of streetlights. The marginal cost of a streetlight is MC = 3.
a. Sketch the MSB curve for the society.
b. What is the socially optimal number of streetlights (it's OK here to have a fractional number of streetlights).
Q2. Arnold and Betty have identical marginal benefit formulas for a public good. The MBs are given by:
MBA = α(YA + qB) - βQ*
MBB = α(YB + qB) - βQ*
Where, Q* = qA + qB
α and β are constant positive numbers, α < 1. YA, YB, are incomes of Arnold and Betty. qA, qB are purchases of the public good by Arnold and Betty.
The marginal cost of a unit of the public good is constant: MC=C.
a. As in lecture, determine the best response functions for Arnold and Betty.
Suppose now that α=.5, β=5, YA = YB = 200, and MC=C.
b. What is the strategic equilibrium quantity of the public good that gets produced?
c. What is the socially optimal quantity of the public good that gets produced?
d. Claim: as the slope of the MB curve gets steeper, the difference between the strategic equilibrium and the socially optimal quantity increases. Agree or disagree. How do the formulas help you answer the question? In words briefly explain the economics of this result.
Q3. The US Navy is building an aircraft carrier. Price: $12 billion. There are about 300 million people in the US. Suppose half the population has a marginal benefit of this aircraft carrier equal to $100 each. The other half has a marginal benefit of the aircraft carrier equal to -$30 each. (That is, they'd be willing to pay up to $30 to NOT build the carrier.) Should the aircraft carrier be built?
4. The US government funds basic research at universities through agencies such as the National Science Foundation or the National Institute of Health. How is our discussion of public goods is relevant to such funding? Write a mini-essay of approximately 300 words in which you clearly describe your thoughts.
5. Lewis and Martin are successful comedians who get utility (U) from consumption (C). They each have the same utility function U(C) = ln(C), where ln is the natural logarithm. (This function, like the square root function used in the lecture, also demonstrates decreasing marginal utility of consumption. You can find the natural logarithm on any scientific calculator or google can calculate it too.) They each have income of $1000. As is common with celebrities, occasionally they get sued for a variety of transgressions. The costs of going to court are $500. There is a 10% chance that Lewis goes to court, and 25% that Martin goes to court. There is an insurance product called "liability insurance" that can protect individuals against lawsuits.
a. Calculate the expected consumption and expected utility for each.
b. Calculate the risk premium and actuarially fair insurance premium for full insurance for each. That is, an insurance policy that replaces 100% of the lost consumption.
c. Suppose the insurance company is unable to distinguish between the riskiness of Lewis and Martin. What would the actuarially fair premium be if the government required them to buy the insurance? (Liability insurance is often required for certain enterprises.)
d. If Lewis and Martin were not required to buy insurance, what would happen to the market? Be specific about who gets insurance and how much it could cost.
6. Suppose a society contains two individuals. Joe, who smokes, and Tanya, who does not. They each have the same utility function U(C) = ln(C). If they are healthy, they will each get to consume their income of $15,000. If they need medical attention, they will have to spend $10,000, leaving them $5,000 for consumption. Smokers have a 12% chance of needing medical attention, and nonsmokers have a 2% chance.
An insurance company is willing to insure Joe and Tanya. The twist here is that the insurance company offers two different kinds of policies. One policy is called the "low deductible," (L) for which the insurance company will pay any medical costs over $3,000. The other is a "high deductible," (H) for which the insurance company will pay any medical costs over $8000.
a. What is the actuarially fair premium for each type of policy for Joe and Tanya?
b. If the insurance company can determine who smokes and who does not, and they charge the actuarially fair prices to each, what policy will Joe select? Tanya? (Think carefully about calculating expected utilities for each under the different policies.)
c. Now, suppose that the insurer cannot determine who smokes and who doesn't. The insurer sets prices for each product. The price of L is $840 and the price of H is $40. (Why did I choose these numbers?) What will Joe and Tanya choose to do? Will adverse selection push Tanya out of the market? [Hint: No.] Calculate the total expected utility for our society under this outcome.
d. What has happened here? What does the second policy option accomplish?
e. Suppose the government were to intervene and provide full insurance at a single price and charge everyone the same actuarially fair amount. How would the total social utility compare to that of part c? (Ignore any moral hazard or other unintended consequences.).