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 + x3 What is the optimal level of public good?
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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.Q1. Let X be a public good and Y be a private good. Two individuals, A and B, derive utility from the consumption of X and Y according to the following quasi-linear preferences: U"(X,Y)=aX +Y and U* (X,Y)= BX +Y, where a + B and both a and ß are greater than one. If the marginal rate of transformation between X and Y is C, find X.Suppose that two individuals, Jon and David, form a community and would like to construct a communal fort that would protect them from attacks. They consume both good X, a private good, and the protection from the fort, P. One unit of good X costs 1 unit while one unit of P costs 2 units, so the budget constraint for each is given by: Xi + 2Pi = 100. Both Jon and David have an income of 100 and a utility function of the form: U = log(Xi) + 2log(Pj + Pd) (a) How much protection, P, will be privately provided? What is optimal consumption of X, the private good? (b) What are the socially optimal amounts of protection, P, and consumption, X, of the private good? How do the socially optimal amounts compare to that privately provided? Explain why.
- An economy consists of two individuals, A and B and two goods, a private good, which is called good 1, and a public good, which is called good 2. Each individual i = A, B is endowed with 24 units of time, which can be used to produce an amount of a private good at a labour cost of a units of time, and an amount of a public good at a labour cost of 2x units of time. Individual i's utility function is given by u(xi, x2) = (xi)¹/2 (g)¹/2, for both i = A, B and where g denotes the aggregate provision of the public good, i.e. g = x2 + x2. (a) Derive the allocation that maximises utilitarian welfare (i.e. the sum of both individ- ual's utility functions). 3 (b) Suppose both individuals decide on provision of feasible bundles (x,x) and (x,x) simultaneously and have full information about the other individual's utility function. Derive the Nash equilibrium of the noncooperative game. (c) Explain concisely (max three sentences) why there is usually under-provision of the public good (you can…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?Consider the model of a rational consumer that cares about consumption of private goods and consumption of broadcast public television (a public good). Suppose that the total level of broadcast public television provided through voluntary contributions is 10 hours of programming. Then the government decides to raise money through a tax and provide 10 hours of programming to the public. What would we predict about crowd-out of voluntary contributions to broadcast public television when government does this? How would the answer change if consumers get warm glow utility from donating to broadcast public television in addition to utility from the public television itself? (Be specific.)
- 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.Bonnie and Clyde are working together. Suppose that their total profits are 170H, where H is the number of hours they work per year. Their utility functions are, respectively, Ug(CB, H) = CB-0.01H² and UdCG H) = Cc- 0.04H², where Cg and Care their private goods consumptions and His 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 is 1,700 2,400 850 1,600 750Sylvester 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 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.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 B's consumption of good x exhibits a negative externality. Consumer A's consumption of good x exhibits a negative externality. Consumer B's consumption of good x exhibits a positive externality. Consumer A's consumption of good x exhibits a positive externality.Assume your apartment complex decides to build new open access common area for its residents. Each individual has value for using the common area represented by the following value function: V= 40-0.5Q where Q represents the number of individuals using the common area. The marginal value function is: MV= 40-Q Assume each individual’s opportunity cost of time is $8 [e.g. MC=8] A) How many people use the common are at the social optimum? B) Calculate the net value (NV) at this social optimum. C) What characteristic of the good must be changed in order to avoid over use of the common area?