Public Economics/Finance
Econ 131 Spring 2024 Danny Yagan Problem Set 3 DUE DATE: Friday, April 19th by 4pm on Gradescope Student Name: Student ID: • You must submit your solutions using this template. • Although you may work in groups, each student must submit individual sets of solutions. You must note the names other students that you worked with. Write their names here: 1. True/False Statements Determine whether each statement is true, false, or uncertain and explain why. Answers with no explanation will receive no points. (a) There is evidence that government provision of benefits to the poor reduces private charitable provision of benefits to the poor. (b) If a central government wants to help a local government spend more money on education, an $X block grant that the local government can spend on anything will have the same impact as an $X conditional block grant that can be spent only on education. 2 (c) There is good evidence that loan availability (i.e. borrowing constraints) affects rates of college attendance. (d) If having car insurance (i.e., insurance that provides cash payments in the event of an expensive car accident) makes individuals drive more recklessly, that is an example of moral hazard. 3 2. Externalities US drivers consume billions of gallons of gas every week, and with every gallon consumed harmful emissions are released into the atmosphere. The aggregate demand function for gasoline is given by P = 12 − 2QD , where Q is the quantity of gallons of gas per week in billions and P reflects the price per gallon in dollars. The aggregate supply function is given by P = 1 + QS . IMF economists have estimated a substantial external marginal damage of gas consumption, which we’ll round to $1 per gallon. (a) Solve for the equilibrium private market price and quantity that will be generated without any government intervention. (b) What is the socially optimal demand function taking into account externalities? (c) Solve for the socially optimal equilibrium price and quantity. 4 (d) Graph the market for US gasoline with a supply/demand graph. Be sure to label 8 things: PMC, SMC, PMB, SMB curves, the Marginal Damage, the DWL, the private market equilibrium and the socially optimal equilibrium. (Also pay attention to the slopes – they certainly don’t need to be exact, but try to consider who is the more elastic/inelastic side of the market.) (e) Calculate the dead-weight loss from the externality. (f) If the government uses a gas tax to address this externality, what is the optimal tax to offset the externality? 5 (g) Calculate the revenue that would be raised by this tax. (h) Will there be deadweight loss associated with this tax? If yes, how much? If no, why not? 6 3. Disability Insurance Assume that everybody earns a wage of $100. Individuals face a probability q of getting disabled. If they are disabled their wage becomes $0. Individuals can purchase insurance from private firms. It provides them with a benefit b if they get disabled. The price of insurance (the insurance premium) is p. Assume in all questions that the market for insurance is perfectly competitive (firms earn zero profit) so that insurance plans are actuarially fair (unless explicitly stated otherwise). Assume initially that there are two types of people that differ in the probability of getting disabled q and their utility function over consumption c: • Type L: qL = 5% and u(c) = √ • Type H: qH = 35% and u(c) = c √ c There are 50 people of each type. Assume ONLY IN (a) that there is perfect information, i.e. the insurance company knows the type of each consumer and can offer a different plan to each type (i.e. some plan (bL , pL ) to Type L; some plan (bH , pH ) to Type H.) (a) Assume that the firms offer full insurance. Calculate the benefit and price offered to each type, ((bL , pL ); (bH , pH )). 7 Assume for the rest of the problem that firms cannot observe types. (b) Assume that there is only one insurance company (so it may earn a profit) and it offers only one insurance plan. It offers a plan with a benefit of b = 100 for a price of p = 50. Call this insurance plan Plan #1. Which types will choose to buy it? (Hint: Calculate utilities rather than the maximize utilities) (c) Will the insurance company still be able to offer Plan #1 in a perfectly competitive equilibrium? (d) A new firm decides to offer a second plan so that there are now two plans offered in the economy: Plan #1 and the new plan called Plan #2. Plan #2 has benefit b = 2.5. What is the price for Plan #2 if the new firm expects both types to buy it and it sets the price actuarially fairly? 8 (e) At this price for Plan #2, which types will buy Plan #2, if any? Which types will buy Plan #1, if any? Which will buy no plan at all, if any? (Hint: You’ve computed several of the necessary utilities in (b)) (f) Entrepreneurs learn that the new firm was making a profit under Plan #2. Those entrepreneurs start lots of new firms so that any firm offering a plan with benefit b = 2.5 has to set the premium equal to actual average costs. As a result, Plan #2 disappears, but there is a new plan called Plan #3 with b = 2.5 and premium p equal to the plan’s actual average costs. Assume that the types who used to buy Plan #2 (computed in the previous problem) buy Plan #3 and are the only ones who buy it. What premium will Plan #3 have? 9 (g) Only Plan #1 and Plan #3 are offered in the economy. Which types (if any) will buy Plan #1? Which types (if any) will buy Plan #3? Which types (if any) will buy no plan? (h) Let social welfare equal the sum of all individuals’ expected utility. Under which economic scenario is social welfare maximized: the economic scenario of part (a) or the economic scenario of (g)? (Note: You may but definitely do not have to use algebra in your answer.) 10
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