Consider an economy with two consumers and two goods. One of these goods, Y, is a public good, and the other good, X, is a private good. The consumers' preferences can be represented by u1(X1, y) = Xıy and u2(x2, y) = X2y², where x; is the consumption level of good X by consumer i, and y is the consumption level of the public good Y. The public good can be produced with the production function f(x) = 2x. Consumer 1 has 10 units of X and consumer 2 has 20 units of X initially. a) Find the Pareto efficient amount of Y in this economy. b) Find the market outcome (voluntary contribution, private provision) equilibrium level of the public good in this economy. Is it efficient?

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…

3. Amherst, MA, has 1,000 people. Citizens of Amherst consume one private good and one public good, the town skating rink. Although they may differ in other respects, citizens have the same utility function, which is given by u(x,G) = -100/G, where x; is the amount of the private good consumed by citizen i and G is the size of the town skating rink, measured in square meters. The price of the private good is normalized at 1 1 and the production function of the public good is given by G =z, where 10 1000 1000 z=w, is the amount of the private good used in producing the public good. (1) Write out the equation implied by the Samuelson condition, i.e.,the efficiency condition when there is a public good. (2) Show that this equation uniquely determines the efficient rink size for Muskrat. What is that size?

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.

1. Consider two consumers who each choose how to split their income, m, between a private good, x, and a public good. The amount of public good consumer į buys is denoted by g, and the total amount of public good is G. The price of each good is 1 and the consumer's utility functions are as below: u, = 2 log(x,)+log(G) uz = log(x,)+ log(G) a) By referring to the utility functions, explain why goodG satisfies the definition of a public good. b) Find each individual's best-response function. c) Find the Nash Equilibrium amount of public goods purchased. d) Use the Samuelson rule to show that the level of public goods purchased in the Nash Equilibrium is inefficient. Explain your answer.

Just north of the town of Muskrat, Ontario, is the town of Brass Monkey, population 6900. 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 U i = X i - 81 Y where x; is the number of bottles of ale consumed by i and Y is 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 $6 per square meter. Everyone has an income of at least $5,000. What is the Pareto efficient size for the town skating rink?

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 is 1 and the price of the public good is 10. Each person has a total income of 1,000O. Find the Pareto efficient amount of G in this economy.

Draw the following situations as a graph format: 1. The graph will have the price (P) on the y-axis and quantity (Q) on the x-axis. The demand or marginal private benefit curve (PMB) will be the same as the social marginal benefit (SMB) curve and it is sloping downwards. The PMC will be lower than the SMC and upward sloping. The social optimal is where SMC and PMB meet and at this point, the P is higher and Q is lower than the P where PMC and PMB meet. Since there is no restriction, the firm produces where PMC and PMB meet. 2. The graph will have the price (P) on the y-axis and quantity (Q) on the x-axis. The Deadweight Loss (DWL) is captured by the triangular area whose height is the gap between the social output and the firm’s output and base is the external cost (point on SMC where the firm is producing minus the point where PMC=demand).

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Consider the market illustrated in the figure to the right. Supply curve S₁ represents the private cost of production and demand curve D₁ represents the private benefit from consumption. Suppose production of this good creates a negative externality. Show how the externality affects the market. 1.) Use the line drawing tool to draw either a new supply (S₂) or demand (D₂) curve incorporating the negative externality in production. Properly label this line. 2.) Use the point drawing tool to indicate the market equilibrium price and quantity. Label this point 'Market equilibrium'. 3.) Use the point drawing tool to indicate the efficient equilibrium price and quantity. Label this point 'Efficient equilibrium'. Carefully follow the instructions above, and only draw the required objects. Price Quantity S₁ D₁