3. Amherst, MA, has 1,000 people. Citizens of Amherst consume one private good andone public good, the town skating rink. Although they may differ in other respects,citizens have the same utility function, which is given by u(x,G) = -100/G, wherex; is the amount of the private good consumed by citizen i and G is the size of the townskating rink, measured in square meters. The price of the private good is normalized at 11and the production function of the public good is given by G =z, where1010001000z=w, is the amount of the private good used in producing the public good.(1) Write out the equation implied by the Samuelson condition, i.e.,the efficiencycondition when there is a public good.(2) Show that this equation uniquely determines the efficient rink size for Muskrat. Whatis that size?

1. Samuelson condition means that the MRS of public and private goods is equal to the ratio of the…

Answered: 3. Amherst, MA, has 1,000 people.…

Principles of Economics, 7th Edition (MindTap Course List)

7th Edition

ISBN:9781285165875

Author:N. Gregory Mankiw

Publisher:N. Gregory Mankiw

Chapter11: Public Goods And Common Resources

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Problem 2PA

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Just north of the town of Muskrat, Ontario, is the town of Brass Monkey, population 6900. 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 - 81 Y where x; 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 $6 per square meter. Everyone has an income of at least $5,000. What is the Pareto efficient size for the town skating rink?
Consider an economy with n consumers, one private good and one public good. We think of the private good as "all other goods," and measure it in terms of money. There aren individuals in the economy with utility functions Ui(xi , y) = ai In y + xi , where y is the public good, xi is the private good consumed by individual i = 1, . .., n, and 0 0. Find the optimal provision of the public good as well as how much is provided under voluntary contributions.
Just north of the town of Muskrat, Ontario is the town of Brass Monkey, population 4,500. 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 UXX₁, XY) = X; - 81/Y, where X; is the number of bottles of ale consumed by i and Yis 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 $5 per square meter. Everyone has an income of at least $5,000. What is the Pareto efficient size for the town skating rink? 270 square meters. 140 square meters. 195 square meters. 390 square meters. 545 square meters.
Consider an economy with n consumers, one privategood and one public good. We think of the private good as “all othergoods,” and measure it in terms of money. There are n individuals inthe economy with utility functions Ui(xi, y) = αi ln y + xi, where y is thepublic good, xi is the private good consumed by individual i = 1, . . . , n,and 0 < α1 < α2 < · · · < αn are parameters. Initial endowment of privategood of each consumer is given by wi. We donote the aggregate endowment 1of the private good by w =Pi wi. The cost of producing the public good is c(y) = cy for some c > 0. Find the optimal provision of the public good as well as how much is provided under voluntary contributions.
Categorize each of the following five goods as a Private Good, Common Good, Club Good, or Public Good. Descriptions of each good will serve as hints. Remember, our textbook outlines only two types of goods, see the notes and lecture video for the other two types. a) An NPR (radio) signal - which is available to everyone with a radio, whether or not they contribute, the signal does not diminish no matter how many people tune in. [Select] b) A new Ford truck this can only be purchased by one person, you can not drive it off the lot without paying for it. [Select] c) A subscription to Disney+ - you can not have a subscription without paying for it, purchasing a subscription does not lower the total number of subscriptions available. [ Select] d) A fireworks show on the 4th of July - the fireworks are visible to everyone in the town, whether or not they contributed anything to the display, one person viewing the fireworks does not make them less visible for anyone else. [Select] e) A…
3. Consider an economy with two individuals, A and B, with quasi-linear preferences over a private good x; with i = A, B, and over a public good, G, represented by the utility function U (x₁, G) = x₁ + InG. Each of them has income mi with i = A, B that can be spent on the private good x₁ or contributed to the public good, G. Each individual's contribution is g₁. One unit of individual contribution is used to produce one unit of the public good, G. The social planner has a Rawlsian social welfare function given by: W(UA, UB) = min{5UA, 5UB}. a) Write the budget constraint for each individual, A and B. b) Derive the Lagrangean corresponding to the optimization problem of individual B. c) Calculate the first order conditions. d) Derive the best response function for individual B, 9B. e) Calculate the demand functions for x, (the private good) and G (the public good).
1. Consider two consumers who each choose how to split their income, m, between a private good, x, and a public good. The amount of public good consumer į buys is denoted by g, and the total amount of public good is G. The price of each good is 1 and the consumer's utility functions are as below: u, = 2 log(x,)+log(G) uz = log(x,)+ log(G) a) By referring to the utility functions, explain why goodG satisfies the definition of a public good. b) Find each individual's best-response function. c) Find the Nash Equilibrium amount of public goods purchased. d) Use the Samuelson rule to show that the level of public goods purchased in the Nash Equilibrium is inefficient. Explain your answer.
Consider the model of public goods in the last section of this chapter. (a) Suppose that preferences over private consumption C and public goods G are such that these two goods are perfect substitutes; that is, the marginal rate of substitution of public goods for private goods is a constant b > 0 . Determine the optimal quantity of public goods that the government should provide, and interpret your results. Make sure you show all of the relevant cases. What happens when b changes, or when q changes? (b) Repeat part (a), except with perfect complements preferences; that is, for the case where the representative consumer always wishes to consume private consumption goods and public goods in fixed proportions, or C = aG, with a > 0.
Suppose there are two consumers A and B, and two goods x and y. A's utility function has the following form UA = XA - (XB)1. Which of the following is true? Consumer A's consumption of good x exhibits a negative externality. Consumer A's consumption of good x exhibits a positive externality. Consumer B's consumption of good x exhibits a negative externality. Consumer B's consumption of good x exhibits a positive externality.
Sylvester and Rachel live together on an island with two goods, food and shelter. Food is a private good and shelter is a public good. The utility function of Sylvester is U= Fs + 2(Zs + Zr) where Fs is the amount of food he buys, Zs is the amount of shelter he buys and Zr is the amount of shelter Rachel buys. Similarly, the utility function of Rachel is U= Fr + Zs + Zr where Fris the amount of food she buys. Suppose the price of food is 10 and both Sylvester and Rachel have 100 to spend. Suppose also that it is not possible for Sylvester and Rachel to reach any kind of agreement. a. Determine the quantity of shelter they will buy and the efficient quantity of shelter if the price of shelter is 5? (Enter your answer in whole number.) Quantity of Shelter: 120 units Efficient b. What is the quantity of shelter they will buy if the price of the shelter is 15? (Round your answer to 2 decimal places (e.g., 32.16).) Quantity of Shelter: 30 units Not Efficient c. What is the quantity of…
Consider the market for CD players, illustrated in the figure to the right. Suppose there are network externalities in this market such that the quantity of a good demanded grows in response to the growth of purchases by other individuals (as indicated by the demand curve "Demand" in the figure). Suppose that the price is initially $110 where the quantity demanded is 90 (thousand CD players per month). If the price of CD players falls to $50, demand will increase to thousand CD players per month. (Enter your response using an integer.) of this increase, thousand units of the 90 thousand-unit increase is the pure price effect and thousand units of the increase is the bandwagon effect. The bandwagon effect causes the demand for CD players to be more otherwise be the case (without network externalities). ▼than would 200- 180 160 Demand 140 120- 100- 80- 60- 40- 20- 0+ 0 Deo 20 D150 D80 P120 P180 40 60 80 100 120 140 160 180 200 220 CD Players (thousands per month) Q Next
10-) Suppose 10 people each have the demand Q = 20 – 4P for streetlights and 5 people have the demand Q = 18 – 2P for streetlights. The cost of building each streetlight is 3. If it is possible to purchase a fractional number of streetlights, how many streetlights are socially optimal?

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