General Equilibrium. Additional exercises

Transcription

1 General Equilibrium Additional exercises Microeconomics 2 - Róbert Veszteg 1. (Varian: 29.2.) Consider a small exchange economy with two consumers, Astrid and Birger, and two commodities, herring and cheese. Astrid s initial endowment is 4 units of herring and 1 unit of cheese. Birger s initial endowment has no herring and 7 units of cheese. Astrid s utility function is U(H A, C A ) = H A C A. Birger is a more inflexible person. His utility function is U(H B, C B ) = min{h B, C B }. (a) Draw an Edgeworth box, showing the initial allocation and sketching in a few indifference curves. (b) Find the locus of Pareto optimal allocations. 2. (Varian: 1.2.) Suppose the utility possibility frontier for two individuals is given by U A + 2U B = 200. Find the utility pairs, (U A, U B ), that maximize the following social welfare functions. (a) Nietzschean social welfare function : W (U A, U B ) = max{u A, U B }. (b) Rawlsian social welfare function: W (U A, U B ) = min{u A, U B }. (c) Cobb-Douglas social welfare function: W (U A, U B ) = U 1 2 A U 1 2 B.. (Varian: 1.6.)Roger and Gordon have identical utility functions, U(x, y) = x 2 + y 2. There are 10 units of x and 10 units of y to be divided between them. (a) Draw an Edgeworth box showing some of their indifference curves and mark the Pareto efficient allocations. (b) What are the fair allocations in this case? 4. (Varian: 1.8.) Romeo loves Juliet and Juliet loves Romeo. Besides love, they consume only one good, spaghetti. Romeo likes spaghetti, but he also likes Juliet to be happy and he knows that spaghetti makes her happy. Juliet likes spaghetti, but she also likes Romeo to be happy and she knows that speghetti makes Romeo happy. Romeo s utility function is U R (S R, S J ) = S α R S1 α J and Juliet s utility function is U J (S J, S R ) = SJ αs1 α R, where S R and S J are the amount of spaghetti for Romeo and the amount of spaghetti for Juliet respectively. There is a total of 24 units of spaghetti to be divided between Romeo and Juliet. 1

2 (a) Suppose that α = 2. If Romeo got to allocate the 24 units of spaghetti exactly as he wanted to, how much would he give himself? How much would he give Juliet? (b) If Juliet got to allocate the 24 units of spaghetti exactly as she wanted to, how much would she give herself? How much would she give Romeo? (c) What are the Pareto efficient allocations? (d) When we had to allocate two goods between two people, we drew an Edgeworth box with indifference curves in ti. When we have just one good to allocate between two people, all we need is an Edgeworth line and instead of indifference curves, we will just have indifference dots. Draw an Edgeworth line and let the distance from left to right denote spaghetti for Romeo and the distance from right to left denote spaghetti for Juliet. Show Romeo s favorite point and Juliet s favorite point. (e) Suppose that α = 1. If Romeo got to allocate the spaghetti, how much would he choose for himself? If Juliet got to allocate the spaghetti, how much would she choose for herself? (f) Draw an Edgeworth line and show the two people s favorite points and the locus of Pareto optimal points. (g) When α = 1, at the Pareto optimal allocations what do Romeo and Juliet disagree about? 5. (Varian: 1.9.) Hatfield and McCoy hate each other but love corn whiskey. Because they hate each other to be happy, each wants the other to have less whiskey. Hatfield s utility function is U H (W H, W M ) = W H WM 2 and McCoy s utility function is U M (W M, W H ) = W M WH 2, where W M is McCoy s daily whiskey consumption and W H is Hatfield s daily whiskey consumption (both measured in quarts). There are 4 quarts of whiskey to be allocated. (a) If McCoy got to allocate all of the whiskey, how would he allocate it? If Hatfield got to allocate all of the whiskey, how would he allocate it? (b) If each of them gets 2 quarts of whiskey, what will the utility of each of them be? If a bear spilled 2 quarts of their whiskey and they divided the remaining 2 quarts equally between them, what would the utility of each of them be? If it is possible to throw away some of the whiskey, is it Pareto optimal for them each to consume 2 quarts of whiskey? (c) If it is possible to trow away some whiskey and they must consume equal amounts of whiskey, how much should they throw away? 6. Robinson Crusoe has decided that he will spend exactly 8 hours a day gathering food. He can either spend this time gathering coconuts or catching fish. He can catch 1 fish per hour and he can gather 2 coconuts per hour. 2

3 (a) Write an equation for the line segment that is Robinson s production possibilities frontier. Plot this production possibilities frontier between fish and coconuts on a graph. (b) Robinson s utility function is U(F, C) = F C, where F is his daily fish consumption and C is his daily coconut consumption. Sketch the indifference curve that gives Robinson a utility of 8 on your graph. How many fish will Robinson choose to catch per day? How many coconuts will he collect? (c) Suppose Robinson is not isolated on an island in the Pacific, but is retired and lives next to a grocery store where he can buy either fish or coconuts. If fish cost $1 per fish, how much would coconuts have to cost in order that he would choose to consume twice as many coconuts as fish? (d) Suppose that a social planner decided that he wanted Robinson to consume 4 fish and 8 coconuts per day. He could do this by setting the price of fish equal to $1. How much will coconuts have to cost? How much money should the social planner give Robinson as daily income? (e) Back on his island, Robinson has little else to do, so he pretends that he is running a competitive firm that produces fish and coconuts. He wonders, What would the price have to be to make me do just what I am actually doing? Let s assume that fish are the numeraire and have a price of $1. And let s pretend that I have acces to a competitive labor market where I can hire as much labor as I want at some given wage. There is a constant returns to scale technology. Fill in the spaces in Robinson s argument. An hour s labor produces one fish or 2 coconuts. At wages above $... per hour, I wouldn t produce any fish at all, because it would cost me more than $1 to produce a fish. At wages below $... per hour, I would want to produce infinitely many fish since I would make a profit on every one. So the only possible wage rate that would make me choose to produce a positive finite amount of fish is $... per hour. Now what would the price of coconuts have to be to induce me to produce a positive number of coconuts. At the wage rate I just found, the cost of producing a coconut is $... At this price and only at this price, would I be willing to produce a finite positive number of coconuts. 7. Tip and Spot finally got into college. Tip can write term papers at the rate of 10 pages per hour and solve workbook problems at the rate of per hour. Spot can write term papers at the rate of 6 pages per hour and solve workbook problems at the rate of 2 per hour. (a) Which of these two has comparative advantage in solving workbook problems? (b) Tip and Spot each work 6 hours a day. They decide to work together and to produce a combination of term papers and workbook problems that lies on their joint production possibilities frontier. Plot on a graph their joint production possibilities frontier.

4 (c) How do these two divide work if they produce less than 60 pages of term papers? (d) And if they produce more than 60 pages of term papers? 8. The Isle of Veritas has made it illegal to trade with the outside world. Only two commodities are consumed on this island, milk and wheat. On the north side of the island are 40 farms. Each of these farms can produce any combination of nonnegative amounts of milk and wheat that satisfies the equation m = 60 6w. On the south side of the island are 60 farms. Each of these farms can produce any combination of non-negative amounts of milk and wheat that satisfies the equation m = 40 2w. The economy is in competitive equilibrium and 1 unit of wheat exchanges for 4 units of milk. (a) Plot the production possibilities set for a typical farmer from the north side of the island. Given the equilibrium prices, will this farmer specialize in wheat, specialize in milk, or produce both goods? (b) Now plot the production possibilities set for a typical farmer from the south side of the island. Given the equilibrium prices, will this farmer specialize in wheat, specialize in milk, or produce both goods? (c) Draw on your graphs the budget that farmers face in their role as consumers if they make the optimal choice of what to produce. (d) Suppose that peaceful Viking traders discover Veritas and offer to exchange either wheat for milk or milk for wheat at an exchange rate of 1 unit of wheat for units of milk. If the Isle of Veritas allows free trade with the Vikings, then this will be the new price ratio on the island. At this new price ratio, would either type of farmer change his output? Draw the budget that each farmer faces in their role as a consumer in this new situation. (e) The council of elders of Veritas will meet to vote on whether to accept the Viking offer. The elders from the north end of the island get 40 votes and the elders from the south end get 60 votes. Assuming that everyone votes in the selfish interest of his end of the island, how will the northerners vote? How will the southerners vote? How is it that you can make a definite answer to the last two questions without knowing anything about the farmers consumption preferences? (f) Suppose that instead of offering to make exchanges at the rate of 1 unit of wheat for units of milk, the Vikings had offered to trade at the price of 1 unit of wheat for 1 unit of milk and vice versa. Would either type of farmer change his output? How will the northerners vote now? Show will the southerners vote now? Explain why it is that your answer to one of the last two questions has to be it depends. 4

5 9. Norton and Ralph have a utility possibility frontier that is given by the following equation, U R + UN 2 = 100, where R and N signify Ralph and Norton respectively. (a) Plot the utility possibility frontier for Norton and Ralph on a graph. (b) Both Ralph and Norton believe that the ideal allocation is given by maximizing an appropriate social welfare function. Ralph thinks that U R = 75, U N = 5 is the best distribution of welfare, and presents the maximization solution to a weighted-sum-of-the-utilities social welfare function that confirms this observation. What was Ralph s social welfare function? (c) Norton, on the other hand, believes that U R = 19, U N = 9 is the best distribution. What is the social welfare function Norton presents? Exam Exercises 10. Hatfield and McCoy burn with hatred for each other. They both consume corn whiskey. Hatfield s utility function is U H = W H W 2 8 M and McCoy s utility is U M = W M W 2 8 H, where W H is Hatfield s whisky consumption and W M is McCoy s whisky consumption, measured in gallons. The sheriff has a total of 28 units of confiscated whisky that he could give back to them. For some reason, the sheriff wants them both to be as happy as possible, and he wants to treat them equally. The sheriff should give them each (a) 14 gallons. (b) 4 gallons and spill 20 gallons in the creek. (c) 2 gallons and spill 24 gallons in the creek. (d) 8 gallons and spill the rest in the creek. (e) 1 gallons and spill the rest in the creek. 11. A little exchange economy contains just two consumers, named Ken and Barbie, and two commodities, quiche and wine. Ken s initial endowment is 4 units of quiche and 2 units of wine. Barbie s initial endowment is 2 unit of quiche and units of wine. We write Ken s utility function as, U(Q K, W K ) = Q K W K and Barbie s utility function as U(Q B, W B ) = Q B + W B, where Q K and W K are the amounts of quiche and wine for Ken and Q B and W B are amounts of quiche and wine for Barbie (a) Draw an Edgeworth box to illustrate this situation. Put quiche on the horizontal axis and wine on the vertical axis. Locate the initial endowment point and call it W. Draw some indifference curves for both consumers. (b) Find the contract curve in your Edgeworth box. Write a mathematical expression that defines the contract curve in this exercise. (c) Find the competitive equilibrium both mathematically and graphically. That is, find the relative price, p Q p W, that defines the competitive equilibrium. 5

6 (d) What is Ken s consumption bundle in competitive equilibrium? How about Barbie s? 12. Al and Bill are the only workers in a small factory which makes geegaws and doodads. Al can make geegaws per hour or 15 doodads per hour. Bill can make 2 geegaws per hour or 6 doodads per hour. Assuming that neither of them finds one task more odious than the other, (a) Al has comparative advantage in producing geegaws, and Bill has comparative advantage in producing doodads. (b) Bill has comparative advantage in producing geegaws, and Al has comparative advantage in producing doodads. (c) Al has comparative advantage in producing both geegaws and doodads. (d) Bill has comparative advantage in producing both geegaws and doodads. (e) Both persons have comparative advantage in producing doodads. 1. A little exchange economy contains just two consumers, named Ken and Barbie, and two commodities, quiche and wine. Ken s initial endowment is units of quiche and 2 units of wine. Barbie s initial endowment is 2 unit of quiche and units of wine. Ken and Barbie have identical utility functions. We write Ken s utility function as, U(Q K, W K ) = Q K W K and Barbie s utility function as U(Q B, W B ) = min(q B ; W B ), where Q K and W K are the amounts of quiche and wine for Ken and Q B and W B are amounts of quiche and wine for Barbie (a) Draw an Edgeworth box to illustrate this situation. Put quiche on the horizontal axis and wine on the vertical axis. Locate the initial endowment point and call it W. Draw some indifference curves for both consumers. (b) Find the contract curve in your Edgeworth box. Write a mathematical expression that defines the contract curve in this exercise. (c) Find the competitive equilibrium both mathematically and graphically. That is, find the relative price, p Q p W, that defines the competitive equilibrium. (d) What is Ken s consumption bundle in competitive equilibrium? How about Barbie s? 14. A little exchange economy contains just two consumers, named Ken and Barbie, and two commodities, quiche and wine. Ken s initial endowment is 4 units of quiche and 2 units of wine. Barbie s initial endowment is 4 unit of quiche and units of wine. We write Ken s utility function as, U(Q K, W K ) = Q 1 2 K W 1 2 K and Barbie s utility function as U(Q B, W B ) = min(q B ; W B ), where Q K and W K are the amounts of quiche and wine for Ken and Q B and W B are amounts of quiche and wine for Barbie 6

7 (a) Draw an Edgeworth box to illustrate this situation. Put quiche on the horizontal axis and wine on the vertical axis. Locate the initial endowment point and call it W. Draw some indifference curves for both consumers. (b) Find the contract curve in your Edgeworth box. Write a mathematical expression that defines the contract curve in this exercise. (c) Find the competitive equilibrium both mathematically and graphically. That is, find the relative price, p Q p W, that defines the competitive equilibrium. (d) What is Ken s consumption bundle in competitive equilibrium? How about Barbie s? 15. Paula and Robert live on an island and are not able to exchange goods with other people. They look all day for bananas (B) and coconuts (C). At the end of the day they always end up with the following quantities of the two goods: Paula gathers 2 bananas and 1 coconut, Robert gathers 1 banana and 2 coconuts. They have the same utility function over these two goods: U(B, C) = BC. (a) Represent this small economy with an Edgeworth box. Carefully name the axes and plot some indifference curves for both consumers. (b) Derive a mathematical equation for the contract curve of this economy, and mark the set of Pareto efficient allocations in the Edgeworth box. (c) Let us suppose that the price of coconuts, p C, is equal to 1. What is the price of bananas, p B, in the competitive equilibrium? (d) Find the consumption bundles in the competitive equilibrium. 16. Mutt and Jeff have 8 cups of milk and 10 cups of juice to divide between themselves. Each has the same utility function given by u(m, j) = min{m, j}, where m is the amount of milk and j is the amount of juice that each has. That is, each of them believes that the two liquids are complementary goods. (a) Represent this situation with an Edgeworth box. Plot some of each consumer s indifference curves. (b) Show the geometric location of the Pareto efficient allocations in your graph. (c) Find the allocations in which every consumer as satisfied with his own consumption bundle as would be with the other s. (Envy-free allocations.) 7