Exam 35: Public Goods

Remember Bonnie and Clyde from your workbook? Suppose that their total profits are 160H, where H is the number of hours they work per year. Their utility functions are, respectively, U_B(C_B, H) = C_B - 0.04H² and U_C(C_C, H) = C_C - 0.01H², where C_B and C_C are their private goods consumptions and H is the number of hours they work per year. If they find a Pareto optimal choice of hours of work and income distribution, the number of hours they work per year must be

A small coffee company roasts coffee beans in its shop. The unroasted beans cost the company $2 per pound. The marginal cost of roasting coffee beans is $ per pound when q pounds are roasted. The smell of roasting beans imposes costs on the company's neighbors. The total amount that neighbors would be willing to pay to have the shop stop roasting altogether is 5q², where q is the number of pounds being roasted. The company sells its output in a competitive market at $4.50 per pound. What is the socially efficient amount of coffee for the company to roast?

A quiet town in Kansas has 2,000 people, all of whom have the same preferences. There is one private good and one public good. Each person I in town has utility U(x i, y) = x i + y .5, where x i is private good for person i and y is the amount of public good that the town provides. If the private good costs $1 per unit and the public good costs $10 per unit, then the Pareto optimal amount of public good for the town to provide is

Anton, Bertha, and Cecilia all consume crackers and music. Crackers are a pure private good and music is a pure public good. Their utility functions are U A (c A, m) = c A m, U B (c B, m) = c B m, and U B (c C, m) = 2c C m, where c A is Antons cracker consumption, c B is Berthas cracker consumption, c C is Cecilias cracker consumption, and where m is the amount of music jointly consumed by all three of them. Music is measured in hours. Crackers cost $1 each. Music costs $10 an hour. Antons wealth is $30, Berthas wealth is $50, and Cecilias wealth is $20. What is the efficient amount of music for them to consume?

An otherwise charming island is inhabited by two religious groups who hate each other. The island is presided over by a benevolent monarch who is extremely concerned about envy between groups. He chooses the distribution of income on the island so as to maximize the social welfare function, W(x, y) = min{2x - y,2y - x}, where x is the utility of the average member of group X and y is the utility of the average member of group Y. a. If the monarch can accomplish any distribution of utility such that x + 3y = 24, diagram the utility possibilities frontier and the monarchs isowelfare lines. b. What income distribution maximizes W? c. Show that an equal increase in both groups income will always please the monarch. d. If the initial incomes are equal, when do increases in both groups utility reduce W?

One of the problems with the Clarke tax mechanism is that when it is used, people have an incentive to lie about their preferences.

Lucy's utility function is 2X_L + G and Melvin's utility function is X_MG, where G is their expenditures on the public goods they share in their apartment and where X_L and X_M are their respective private consumption expenditures. The total amount they have to spend on private goods and public goods is $33,000. They agree on a Pareto optimal pattern of expenditures in which the amount that is spent on Lucy's private consumption is $9,000. How much do they spent on public goods?

Bob and Ray are thinking of buying a sofa. Bob's utility function is U_B(S, M_B) = (1 + S)M_B and Ray's utility function is U_R(S, M_R) = (3 + S)M_R, where S = 0 if they don't get the sofa and S = 1 if they do and where M_B and M_R are the amounts of money they have respectively to spend on their private consumptions. Bob has a total of $800 to spend on the sofa and other stuff. Ray has a total of $1,600 to spend on the sofa and other stuff. The maximum amount that they could pay for the sofa and still arrange to both be better off than without it is

Bob and Ray are thinking of buying a sofa. Bob's utility function is U_B(S, M_B) = (1 + S)M_B and Ray's utility function is U_R(S, M_R) = (3 + S)M_R, where S = 0 if they don't get the sofa and S = 1 if they do and where M_B and M_R are the amounts of money they have respectively to spend on their private consumptions. Bob has a total of $1,200 to spend on the sofa and other stuff. Ray has a total of $1,600 to spend on the sofa and other stuff. The maximum amount that they could pay for the sofa and still arrange to both be better off than without it is

If preferences are single peaked, then pairwise majority voting among alternative options will not lead to voting cycles.

Nadia Comaneci and Mr. X have preferences defined over pizza, p, and trampolines, t. They have identical utility functions, U(p, t) = . Each pizza costs $1 and each trampoline costs $1,000. Nadia and Mr. X like to share, and indeed trampolines are a public good for them. Pizza, however, is a private good. We don't know their exact incomes, but we do know that each of them earns at least $10,000.

To say that preferences are single peaked means that everybody either prefers more public goods to less or everybody prefers less public goods to more.

A Pareto optimal amount of public goods is shown on a graph (with quantities of public goods on the x axis) by the point at which the horizontal sum of the marginal rate of substitution curves meets the marginal cost curve.

Lucy's utility function is 2X_L + G and Melvin's utility function is X_MG, where G is their expenditures on the public goods they share in their apartment and where X_L and X_M are their respective private consumption expenditures. The total amount they have to spend on private goods and public goods is $28,000. They agree on a Pareto optimal pattern of expenditures in which the amount that is spent on Lucy's private consumption is $10,000. How much do they spent on public goods?

Which of the following is the best example of a public good as defined in your text?

Just north of the town of Muskrat, Ontario, is the town of Brass Monkey, population 6,400. Brass Monkey, like Muskrat, has a single public good, the town skating rink, and a single private good, Labatt Ale. Everyone's utility function is U_i(X_i, Y) = X_i - , where X_i is the number of bottles of ale consumed by i and Y is the size of the skating rink in square meters. The price of ale is $1 per bottle. The cost of the skating rink to the city is $4 per square meter. Everyone has an income of at least $5,000. What is the Pareto efficient size for the town skating rink?

Remember Bonnie and Clyde from your workbook? Suppose that their total profits are 112H, where H is the number of hours they work per year. Their utility functions are, respectively, U_B(C_B, H) = C_B - 0.02H² and U_C(C_C, H) = C_C - 0.02H², where C_B and C_C are their private goods consumptions and H is the number of hours they work per year. If they find a Pareto optimal choice of hours of work and income distribution, the number of hours they work per year must be

Remember Bonnie and Clyde from your workbook? Suppose that their total profits are 40H, where H is the number of hours they work per year. Their utility functions are, respectively, U_B(C_B, H) = C_B - 0.02H² and U_C(C_C, H) = C_C - 0.03H², where C_B and C_C are their private goods consumptions and H is the number of hours they work per year. If they find a Pareto optimal choice of hours of work and income distribution, the number of hours they work per year must be

Lucy's utility function is 2X_L + G and Melvin's utility function is X_MG, where G is their expenditures on the public goods they share in their apartment and where X_L and X_M are their respective private consumption expenditures. The total amount they have to spend on private goods and public goods is $30,000. They agree on a Pareto optimal pattern of expenditures in which the amount that is spent on Lucy's private consumption is $9,000. How much do they spent on public goods?

The Sons of Knute had a hunting lodge up on Loon Lake that burned down last winter. They plan to rebuild it this summer and are trying to decide how large the new lodge should be. The organization has 50 members. The marginal rate of substitution of each of them between square feet of hunting lodge and money for other goods is 1.2 - 0.0004y, where y is the size of the hunting lodge in square feet. What is the efficient size for the new hunting lodge?