consumption of each good by each person. c. On the diagram, show the locus of points that are Pareto efficient (the contract curve).

Size: px

Start display at page:

Download "consumption of each good by each person. c. On the diagram, show the locus of points that are Pareto efficient (the contract curve)."

Victor McLaughlin
5 years ago
Views:

1 Spring / I 350, Intermediate Microeconomics / Problem Set 1. uriously, two old friends, rchie and arney, have found themselves as the only two consumers in a little exchange economy which has only two commodities cheese and wine. rchie's initial endowment of these two commodities is (, W ) = (3,), and arney's initial endowment is (, W ) = (1,6). They have identical utility functions of the form U i ( i,w i ) = i *W i, where the subscripts denote the individual, using for rchie and for arney. a. Modify the Edgeworth box below to illustrate this initial allocation, and to illustrate one indifference curve for each of them, passing through this initial allocation. Use blue ink for rchie's indifference curve and red ink for arney's. b. t any Pareto optimal allocation where both consume some of each good, their marginal rates of substitution must be equal. Write an expression that states this condition in terms of the consumption of each good by each person. c. On the diagram, show the locus of points that are Pareto efficient (the contract curve). d. In this example, at any Pareto efficient allocation, where both persons consume both goods, the slope of rchie's indifference curve will be. Therefore, since we know that competitive equilibrium must be Pareto efficient, we know that at a competitive equilibrium, p, p W e. What are rchie's and arney's consumption bundles in a competitive equilibrium?

2 Spring / I 350, Intermediate Microeconomics / Problem Set Wine heese. onsider a pure exchange economy with two agents, and, and two goods, x and y, where the utility functions of the agents are nd the initial endowments of the two goods are U (x,y) = x 0. y 0.6 U (x,y) = x 0.6 y 0. (x,y) = (30, 0) (x,y) = (0, 30) a. Derive and describe the Pareto efficient equilibrium of this economy under these conditions. b. onstruct an Edgeworth box that illustrates your answer in part (a). c. Derive and describe the Pareto efficient equilibrium of this economy if the only change to the above conditions is that s preferences are the same as s: U (x,y) = x 0.6 y 0. U (x,y) = x 0.6 y 0.

3 pples RHIT / Department of Humanities & Social Sciences / K. hrist Spring / I 350, Intermediate Microeconomics / Problem Set 3. harlotte and Wilber are two agents in a two-agent, two commodity pure exchange economy where apples and bananas are the two commodities. harlotte loves apples and hates bananas. Her utility function is ( ), where a is the number of apples she consumes and b in the number of bananas she consumes. Wilber likes both apples and bananas. His utility function is ( ). harlotte has an initial endowment of no apples and 8 bananas. Wilber has an initial endowment of 16 apples and 8 bananas. a. On the graph below, mark the initial endowment and label it E. Use red ink to draw the indifference curve for harlotte that passes through this point. Use blue ink to draw the indifference curve for Wilbur that passes through this point. b. If harlotte hates bananas and Wilber likes them, how many bananas can harlotte be consuming at a Pareto optimal allocation? On the diagram, use black ink to mark the locus of Pareto optimal allocations of apples and bananas between harlotte and Wilbur. c. We know that a competitive equilibrium allocation must be Pareto optimal and the total consumption of each good must equal the total supply, so we know that at a competitive equilibrium, Wilbur must be consuming how many bananas? If Wilbur is consuming this number of bananas, his marginal utility for bananas will be and his marginal utility of apples will be. If apples are the numeraire, then the only price of bananas at which he will want to consume exactly 16 bananas is. In competitive equilibrium for this harlotte-wilbur economy, Wilbur will consume bananas and apples, while harlotte will consume bananas and apples. d. This problem illustrates something about one of the two welfare theorems discussed on pages of Varian s Intermediate Microeconomics. What does it illustrate? 16 W ananas

4 Spring / I 350, Intermediate Microeconomics / Problem Set. arefully modify the diagram below so that it provides a clear representation of a general equilibrium for a two-agent (, ), two-good (x, y) economy with the following characteristics: The economy s PPF is concave and passes through the points (0, 16), (10, 1), (15, 10), (18, 5) and (0, 0). p y /p x = gent s utility function is U ( x, y) = x y. U x, y = x y gent s utility function is ( ) Your modified diagram should include the following: a. The PPF b. marginal rate of transformation line, the slope of which reflects the price ratio, and which is tangent to the PPF. c. contract curve tracing out the Pareto set and reflecting the obb-douglas preferences given above. d. consumption equilibrium with (x 1, x ) = (5, 7) and (x 1, x ) = (10, 3). e. onsumer indifference curves at equilibrium. f. onsumer terms of trade (or budget) line. g. Is the consumption equilibrium fair in Varian s sense of the term (pp )? h. Show mathematically that a different consumption equilibrium, with (x 1, x ) = (3, 5) and (x 1, x ) = (1, 5) is not fair in Varian s sense of the term. 0 x y 5. Suppose you have a two consumer, two-commodity economy (goods x and y), that the economy s PPF is given by y = 0 x 3 /700, and that the consumers have identical utility functions U(x, y) = x 0.5 y 0.5. alculate the general equilibrium. Recall that the general equilibrium must satisfy the condition that MRS = p 1 /p = MRT, and would specify production amounts for x and y, as well as a price ratio. (Hint: If consumers are identical, perhaps you can treat the problem as if there is just one consumer.)

5 Spring / I 350, Intermediate Microeconomics / Problem Set 6. Suppose the utility possibility frontier for two individuals is given by U + U = 00. On the diagram below, plot the frontier. W U U = max U, U a. In order to maximize a Nietzschean social welfare function, ( ) { } on the utility possibility frontier shown in the diagram, one would set = U =. W U U = min U, U U and = b. If instead we use a Rawlsian criterion, ( ) { } function is maximized on the diagram where = c. Suppose that social welfare is given by ( ) would be maximized when =,, U and,, then the social welfare U. W U, U = U U U and =. In this case, social welfare U. 00 U U , y = x + y 7. Roger and Gordon have identical utility functions, ( ) 10 units of y to be divided between them. U x. There are 10 units of x and a. Draw an Edgeworth box showing some of their indifference curves (Roger = blue, Gordon = red) and mark the Pareto Optimal allocations with black ink. What is it about their preferences that generates a peculiar outcome? b. What are the fair allocations in this case?

6 Spring / I 350, Intermediate Microeconomics / Problem Set 8. Suppose that a honey farm is located next to an apple orchard and each acts as a competitive firm. Let the amount of apples produced be measured by and amount of honey produced be measured by H. The cost functions of the two firms are H H / 100 honey is $ and the price of apples is $3. c H and / 100 H c. The price of a. If the firms each operate independently, the equilibrium amount of honey produced will be and the equilibrium amount of apples will be. b. Suppose that the honey and apple firms merge. What would be the profit-maximizing output of honey for the merged firm? What would be the profit-maximizing amount of apples for the merged firm? c. What is the socially efficient output of honey? If the firms stayed separate, how much would honey production have to be subsidized to induce an efficient supply? 9. clothing store and a jewelry store are located side by side in a small shopping mall. The number of customers who come to the shopping mall intending to shop at either store depends on the amount of money that each store spends on advertising per day. Each store also attracts some customers who come to shop at the neighboring store. If the clothing store spends $ x per day on advertising, and the jeweler spends $ per day on advertising, then the total profits per day of the clothing store are xj ( ) ( ) x, xj = 60 + xj x ~ x (Read ~ as minus ), and the total profits of the jewelry store are ( x x ) = ( x ) x ~ x. J, J J J a. If each store believes that the other store s amount of advertising is independent of its own advertising expenditure, what is the equilibrium amount of advertising for each store and what are the profits for each store? b. Suppose that both stores have the same profit functions as in part (a), but are owned by a single firm that chooses the amounts of advertising so as to maximize the sum of the two store profts. Under these conditions, what is the equilibrium amount of advertising for each store and what are the total profits for the owner of the two stores?

7 Spring / I 350, Intermediate Microeconomics / Problem Set 10. small private airport is located next to a large tract of land owned by a housing developer. The developer would like to build houses on this land, but noise from the airport reduces the value of the land (and the houses he would like to build on it). The more planes that fly, the lower is the amount of profits that the developer makes. Let X be the number of planes that fly per day and let Y be the number of houses that the developer builds. Let the airport s profits be = 8X ~ X (Read ~ H = 60Y ~ Y ~ XY as minus ), and let the developer s profits be. Now consider the outcomes under various institutional rules about property rights and bargaining between the owner of the airport and the developer: a. Free to hoose with No argaining. Suppose that no bargains can be struck between the airport and the developer and that each can decide on its own level of activity. What is the number of planes per day that maximizes profits for the airport? Given that the airport will land this number of planes, what is the number of houses that maximize the developer s profits? What are the profits for the airport? What are the profits for the developer? b. Strict Prohibition. Suppose that a local ordinance makes it illegal to land planes at the airport. How many houses will the developer build?. What are the profits for the developer?. ompare this outcome to part (a). c. Lawyer s Paradise. Suppose that a law is passed that makes the airport liable for all damages to the developer s property values. To maximize his net profits, the developer will choose to build how many houses? To maximize its profits, net of damages paid to the developer, the airport will choose to land how many planes? What are the profits for the airport? What are the profits for the developer? ompare this outcome to parts (a) and (b). d. Merger. Suppose that the housing developer purchases the airport. What is the profit function for the new joint entity?. alculate the profit maximizing number of houses, planes, and total profits under this arrangement. e. Dealing. Suppose that the airport and the developer remain independent. If the original situation was one of free to choose, could the developer increase his net profits by paying the airport to cut back one flight per day if the developer has to pay for all of the airport s lost profits? Suppose the developer, realizing this, decides to get the airport to reduce its flights by paying for all lost profits coming from the reduction of flights. To maximize his own net profits, how many flights per day should he pay the airport to eliminate?

Chapter 5: A Closed-Economy One-Period Macroeconomic Model

Chapter 5: A Closed-Economy One-Period Macroeconomic Model Introduce the government. Construct closed-economy one-period macroeconomic model, which has: (i) representative consumer; (ii) representative