Consider an economy with a private good and public good. The economyconsists of two consumers whose utility functions are1u₁(x₁, y) = 1⁄² ln x₁ + 1⁄lny and u₂(x, y) = ln x₂ + ²/² ln y23The endowment of consumer 1 is w = 2, and endowment of consumer 21is w.2= 3, both in units of private goods. The production of public gooduses the linearTechnology y = z.a. Find the voluntary contributions equilibrium.b. Find Lindahl equilibrium

Given the utility functions for 2 consumers: u1(x1,y)=12lnx1+12lnyu2(x2,y)=13lnx2+23lny The…

Consider an economy with a private good and public good. The economy consists of two consumers whose utility functions are In u₁(x, y) = ln x₁ + 1⁄lny and u₂(x, y) = /²/ ln x₂ + ²/²lny ½⁄/ The endowment of consumer 1 is w = 1 2, and endowment of consumer 2 is w = W₂ 3, both in units of private goods. The production of public good uses the linear Technology y = z. a. Find the voluntary contributions equilibrium. b. Find Lindahl equilibrium

Consider an economy with a private good and public good. The economy consists of two consumers whose utility functions are In u₁(x, y) = ln x₁ + 1⁄lny and u₂(x, y) = /²/ ln x₂ + ²/²lny ½⁄/ The endowment of consumer 1 is w = 1 2, and endowment of consumer 2 is w = W₂ 3, both in units of private goods. The production of public good uses the linear Technology y = z. a. Find the voluntary contributions equilibrium. b. Find Lindahl equilibrium

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter19: Externalities And Public Goods

Section: Chapter Questions

Problem 19.7P

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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.
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.
. There are three people who consume public and private goods. As usual, the public good is x, and ¡ represents person i's consumption of the private good. The prices of both the public good and the rivate good are $1 per unit. The initial endowments of the private good are (w1, w2, w3) = (10, 10, 10). 'he three people have the following utility functions: u1 (x, Y1) = In x + Y1, u2 (x, Y2) = 2 ln x + Y2, u3 (x, y3) = 3 ln x + y3- (i) Assume that the public good is purchased privately, and that person 3 is the first to go to the market and buy the public good. Assume he does not act strategically-he ignores persons 1 and 2 when he buys x, and thinks only of his own utility maximisation problem. What is the outcome? How much of the public good does person 3 buy? How much do persons 1 and 2 buy?
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).
Consider an economy with a private good and public good. The economy consists of two consumers whose utility functions are: u₁(x, y) = — ln x₁ + 1⁄lny and u₂(x, y) = ln x₂ + y / The endowment of consumer 1 is w = 2, and endowment of consumer 2 is 1 W₂ 3, both in units of private goods. The production of public good uses the linear Technology y = z. = Lindahl equilibrium sets different price p, of public good for each consumer. Suppose price for private good is 1 to solve. Find x and y* for consumer 1, and x and y* for consumer 2, then find the Lindahl equilibrium.
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Given these two individual demand curves, what is the equation for the market demand curve for a private good? i. P=200 - 10Q ii. P =200 - (20/3)Q Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. P= 200 - 4Q b P=200 - 8.33Q P= 100 - 8.33Q d P= 100 - 1.67Q
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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 are n individuals in the economy with utility functions U₁(x,y) = α₁ = a.lny + x, where y is the public good, x, is the private good consumed by individual i 1,..., n i 1 2 n and 0 0. Find the optimal provision of the public good as well as how much is provided under voluntary contributions.