Consider an economy with three people and two goods: private good (x) and public good (y). Suppose that marginal rate of transformation is 1. (one unit of private good is needed for the production of one unit of public good). The utility functions of the individuals are given below. U₁ = 2y¹/2 + x₁ U₂ = 4y¹/2 + X₂ U₂ = 6y¹/2 + x₂ What is the optimal level of public good?

Consider an economy with three people and two goods: private good (x) and public good (y). Suppose that marginal rate of transformation is 1. (one unit of private good is needed for the production of one unit of public good). The utility functions of the individuals are given below. U₁ = 2y¹/2 + x₁ U₂ = 4y¹/2 + X₂ U₂ = 6y¹/2 + x₂ What is the optimal level of public good?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter13: General Equilibrium And Welfare

Section: Chapter Questions

Problem 13.5P

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Two individuals A and B have utility functions defined over two goods, a private good 'x') and a public good, 'F. The utility function of an agent 'i' is defined by ui = 2 log xi + log F where F =FA + FB. Each agent has 200 units of private goods x, as his endowment and 1 unit of private good can be transformed into 1 public good F. Answer the following (a) Find the Nash equilibrium values of FA and FB? (b) What is the Pareto optimal level of F? (c) In what condition the Pareto optimal level of F will not depend on the number of private goods? (d) What would be the consumption of x by each agent if they contribute equally towards the public good (F)? (e) Would it change if an agent had a larger endowment of x to begin with?
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Consider two consumers (1; 2), each with income M to allocate between two goods. Good 1 provides 1 unit of consumption to its purchaser and units of consumption to the other consumer. Each consumer i, i = 1; 2, has the utility function is consumption of good 1 and is consumption of good 2. a. Provide an interpretation of α. b. Suppose that good 2 is a private good. Find the Nash equilibrium levels of consumption when both goods have a price of 1. c. By maximizing the sum of utilities, show that the equilibrium is Pareto-ancient if α = 0 but incident for all other values of α. d. Now suppose that good 2 also provides 1 unit of consumption to its purchaser and a, 0 ≤ α ≤ 1, units of consumption to the other consumer. For the same preferences, find the Nash equilibrium and show that it is ancient for all values of α. e. Explain the conclusion in part d.
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Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods x and total number of firemen M, of the form: u(x, M) M = M4 + M². Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to = 2 In x + In M. The total provision of firemen hired, M, is the sum of the number hired by each of the two persons: providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). How many firemen are hired if the government does not intervene?
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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 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.
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