Consider an economy with 1,000 people and two goods, one private and one public. The people all have the same utility function Ui(Xi, G) = Xi – 100/G , where Xi is the quantity of the private good consumed by person i and G is the quantity of the public good. The price of the private good
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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 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.)Consider an economy consisting of three people – A, B, and C. Person i (where i is A, B, or C) has the utility function where z is the quantity of public good provided and ci is person i’s consumption of private goods. Person i’s income yi is divided between consumption of the private good and his contribution to the financing of the public good. Person i’s contribution is the product of his “tax price” ti and the quantity of the public good, so Person A’s income is 90, person B’s income is 120 and person C’s income is 150. Twenty units of private good must be given up to produce each unit of public good. Find the Lindahl equilibrium
- Consider an economy with a private good and public good. The economy consists of two consumers whose utility functions are: u1(x1,y)=1/2 ln x1 + 1/2 ln y and u2(x2,y)=1/3 ln x2 + 2/3 ln y (better written equations attached) The endowment of consumer 1 is w1=2, and the endowment of consumer 2 is w2=3, both in units of private goods. The production of public good uses the linear Technology y=z. Find Lindahl equilibrium.Consider a two-good economy with one private and one public good. There are four consumers in this economy who contribute to public good provision. The price of the private good is $1 and the demand function of the public good for each consumer is as follows: p1 = 20 - 2G, p2 = 30-(5/10)G, p3 = 100-(G/2) and p4 = 60-G where G is the number of units of the public good and p^i is the price of each unit for consumer i in dollars. The cost of providing one extra unit of the public good is (alpha). (a) With 170 < alpha < 210, what is the optimal level of provision of the public good? Show the optimal point on a graph. (b) Is there a possibility that the public good is not supplied at all? Why? (c) If the public good is not supplied at all, what is the size of the deadweight loss due to this market failure? (d) If at least one person contributes, for what values of the public good will be supplied?Suppose the demand for standard sized bottled water in the US is Qd=120-30.5P where Qd is monthly quantity demanded in millions and P is the price per bottle in dollars and cents. If the marginal private cost (MPC) of producing the bottled water is one dollar, calculate the market equilibrium quantity. Explain what a constant marginal cost implies. Does that mean the total opportunity cost of producing bottled water is unrelated to how many are produced? Let’s assume that the marginal private benefit (MPB) of bottled water equals the marginal social benefit (MSB). Explain what that means. At the equilibrium calculated in part A, what do you know about buyers’ willingness to pay in each transaction?
- Finn is in charge of decorations for an upcoming festival, and he is planning to decorate withclovers (C) and flags (F). Suppose his preferences over decorations can be represented by theutility function U(C, F) = C^(3/4)F^(1/4) For this problem, assume C and F are infinitely divisible so you don’t need to worry aboutrestricting to whole-number answers.(a) Write Finn’s budget constraint as a function of the prices PC, PF , and his budget I.(b) Write Finn’s constrained optimization problem in Lagrangian form and derive the threefirst order conditions.(c) Use two of the first order conditions to show that Finn’s marginal rate of substitution(MRS) equals the marginal rate of transformation (MRT) at the optimum. (Note: Youdo not need to solve the constrained optimization any more than this.)Consider an economy with a private good and public good. The economyconsists of two consumers whose utility functions are:u1(x1, y) = 1/2 ln x1 +1/2 ln y and u2(x2, y) = 1/3 ln x2 +2/3 ln y.The endowment of consumer 1 is w1 = 2, and endowment of consumer 2 is w2 = 3, both in units of private goods. The production of public good uses the linear technology y = z. 1. Find Lindahl equilibrium.Chris and Dana live in an exchange economy with two goods: good Q and good R. Chris starts off with an endowment of 6 units of Q and 10 units of R. Dana starts off with an endowment of 8 units of Q and 8 units of R. Suppose that the price of good R is pR=1 and the price of good Q is pQ=2. a )At these prices, does the market clear? Yes or no? Explain your answer. b) What relationship must hold between the consumption of each agent and the price of the two goods at the market clearing equilibrium? Write the equation
- A local government is considering building new public hiking trails in town.Suppose that the trails are public goods. There are two types of people in a smalltown economy, 1000 avid hikers and 1000 non-hikers, who have typical individualmarginal benefit curves: MB_H = 350 − 5Q and MB_N = 240 − 6Q for the hikersand the non-hikers respectively, where Q is the miles of trails. On the same graph,draw the marginal benefit curves for each type of person and MB of the overalleconomy. If the marginal cost providing this public good is $27000 per mile whatis the optimal level of the public good?Now, consider the incentive facing Beth individually. The following table looks similar to the previous one, but this time, it is partially completed with the individual benefit data for Beth. As shown previously, if both Beth and Eleanor contribute to a public good, Beth receives a benefit of $480. On the other hand, if Eleanor contributes to the building mural and Beth does not, Beth receives a benefit of $640. Complete the right-hand column of the following table, which shows the individual benefits of Beth. Hint: You are not required to consider the benefit of Eleanor. Eleanor contribute doesn't contribute Contribute $480 ??? Beth Doesn't contribute $640 ??? If Eleanor decides to contribute to the building mural, Beth would maximize her benefit by…1. Write down an exchange economy model with two individuals and two goods, in whichthere is no consumption externality. State and explain the key assumptions of the model.Using the model, answer the following questions:(a) Explain carefully how the concept of Pareto efficiency differs from that of socialwelfare.(b) State the Second Theorem of Welfare Economics and explain carefully why it holds.What is the importance of the assumption of convex preferences to this theorem?(c) Explain the concept of the utility possibility frontier. What is the role of the utilitypossibility frontier in the social welfare maximization problem of a welfarist society?
