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- 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?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 750Consider 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.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.)Joe and Martin, two roommates, have access to music in digital and analog formats. Each are initially endowed with 8 GB of high-resolution digital music and 4 vinyl LP's (analog) per month (there are a total of 16 GB's of music in digital and 8LP's in this case). Joe's preferences for music in digital and analog formats are represented by the utility function u,(D, A) = 2D + 2A, where D is the amount of digital music in GB's and A is the numberofLP's. Mạartin's utility function is um(D,A)=D+4A. 1. If Joe convinces Martin to exchange music at terms which would leave Martin at his initial utility level (i.e., without making him worse-off), what is the set of (D, A) that would make Joe better off. Show this on the Edgeworth box that you drew in part (a). What is the final allocation of music under these terms? What is the exchange rate (LP's per GB of digital music)? 2. On the Edgeworth box, show the set of exchange possibilities for Joe and Martin which would make at least one of them…
- Joe and Martin, two roommates, have access to music in digital and analog formats. Each are initially endowed with 8 GB of high-resolution digital music and 4 vinyl LP's (analog) per month (there are a total of 16 GB's of music in digital and 8LP's in this case). Joe's preferences for music in digital and analog formats are represented by the utility function u, (D, A) = 2D + 2A, where D is the amount of digital music in GB's and A is the numberofLP's. Martin's utility function is um(D,A)=D+4A. 1. Draw an Edgeworth box to show the initial endowments and the indifference curves (going through the initial endowments) for Joe and Martin. What are their utility levels from the initial endowments? 2. Calculate the marginal rates of substitution for Joe and Martin. Provide an economic interpretation. 3. Do the initial endowments represent a Pareto efficient allocation? Discuss. 4. If Joe convinces Martin to exchange music at terms which would leave Martin at his initial utility level (i.e.,…Suppose X = R k + for some k ≥ 2, and we define x = (x1 , …, xk ) ≥ = (y1 , …, yk ) if x ≥ y; that is, if for each i = 1, …, k, xi ≥ yi . (This is known as the Pareto ordering on R k + ; it plays an important role in the context of social choice theory in Chapter 8.) (a) Show that ≥ is transitive but not complete. (b) Characterize ≥ Is asymmetric? Is≥ Negatively transitive? Prove your assertions. (c) Characterize ~ defined from ≥ in the usual fashion; that is, x ~ y if x ≥ y and y x. Is ~ reflexive? Symmetric? Transitive? Prove your assertions.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.
- 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.Problem 2. There are two consumers with the following utility functions: x² + √√√x/²2₂ u² (x1, x²) u? (r, rå) x ²2. Total endowment is e = (1,1). = = = (a) Calculate the set of Pareto optimal allocations (a convincing figure is enough). (b) Assume that the planner wants to implement some Pareto efficient allocation (x¹, x²) ((x1, x²), (x², x²)) where x½ ‡ 0, as an equilibrium with transfers. Prove that any price vector such that 0 < p1 < 2√√x½ can support that allocation (normalizing p2 = : 1). (c) What happens if the planner wants to implement precisely the allocation ((1,0), (0, 1))?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?