ch16Solution manual

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Chapter 16 General Equilibrium Theory Solutions to Review Questions 1. What is the difference between a partial equilibrium analysis and a general equilibrium analysis? When analyzing the determination of prices in a market, under what circumstances would a general equilibrium analysis be more appropriate than a partial equilibrium analysis? A partial equilibrium analysis studies the determination of price and output in a single market, taking as given the prices in all other markets. In general equilibrium analysis, we study the determination of price and output in more than one market at the same time. One would employ a partial equilibrium analysis in situations where the concerns focused on a single market; for example, how does an increase in rainfall affect the price of corn? One would use general equilibrium analysis when one was concerned with how changes in price and output in one market affect the price and output in another market; for example, how does an increase in the price of natural gas affect the price and output for electric furnaces? 2. In a general equilibrium analysis with two substitute goods, X and Y, explain what would happen to the price in market X if the supply of good Y increased (i.e., if the supply curve for good Y shifted to the right). How would your answer differ if X and Y were complements? If the supply of good Y increased, the equilibrium price of good Y would fall. Since X and Y are assumed to be substitutes, when the price of good Y falls relative to good X , the demand for good X will fall, lowering the equilibrium price and quantity for good X . If X and Y are complements, when the price of good Y falls, the demand for good X will increase, increasing the equilibrium price and quantity for good X . 3. What role does consumer utility maximization play in a general equilibrium analysis? What is the role played by firm cost minimization in a general equilibrium analysis? In a general equilibrium, demand for finished products comes from utility maximization by households, while demand for inputs comes from cost minimization by firms. The supply of finished products comes from profit maximization by firms, while the supply of inputs comes from profit maximization by households. 4. What is Walras’ Law? What is its significance? Walras’ Law implies that a general equilibrium analysis will only be able to determine prices in N  1 of the markets being studied. This implies that a general equilibrium determines the prices of all goods and inputs relative to the price of another good or input, rather than determining the absolute levels of all prices.

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Chapter 16 - 1

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

5. What is an economically efficient allocation? How does an economically efficient allocation differ from an inefficient allocation? An allocation of goods and inputs is economically efficient if there is no other feasible allocation of goods and inputs that would make some consumers better off without hurting other consumers. In contrast, an allocation of goods and inputs is economically inefficient if there is an alternative feasible allocation of goods and inputs that would make all consumers better off as compared with the initial allocation. 6. What is exchange efficiency? In an Edgeworth box diagram, how do efficient allocations and inefficient allocations differ? Exchange efficiency occurs when a fixed amount of consumption goods cannot be reallocated among consumers in an economy without making at least some consumers worse off. Efficient allocations in an Edgeworth Box occur at points where the indifference curves for different consumers are tangent. Inefficient allocations occur at points where the indifference curves for different consumers intersect. 7. How does exchange efficiency differ from input efficiency? Could an economy satisfy the conditions for exchange efficiency but not the conditions for input efficiency? Input efficiency occurs when a fixed stock of inputs cannot be reallocated among firms in an economy without reducing the output of at least one of the goods that is produced in the economy. It is quite possible that an economy could enjoy exchange efficiency, where the amount of goods available in the economy is allocated so that no consumer can be made better off without making some other consumer worse off, and not enjoy input efficiency, where the amount of inputs could be reallocated to produce more of all goods. 8. Suppose an economy has just two goods, X and Y. True or False: If the condition of input efficiency prevails, we can increase the production of X without decreasing the production of Y. Explain your answer. False. If an economy has input efficiency, then the inputs cannot be reallocated among firms in an economy without reducing the output of at least one of the goods that is produced in the economy. That is, input efficiency implies that an expansion of output in one industry necessitates a reduction in output in another industry. 9. What is the production possibilities frontier? What is the marginal rate of transformation? How does the marginal rate of transformation relate to the production possibilities frontier? The production possibilities frontier describes the combinations of consumption goods that can be produced in an economy given the economy’s available supply of inputs. The points on the frontier satisfy input efficiency, while the points inside the frontier have input inefficiency.

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 2

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

The marginal rate of transformation is the absolute value of the slope of the production possibilities frontier at some point. This measures the amount of one good the economy must give up in order to gain one additional unit of output for some other good. 10. Explain how consumers in an economy can be made better off if the marginal rate of transformation does not equal consumers’ marginal rates of substitution. If the marginal rate of transformation is not equal to consumer’s marginal rate of substitution, then consumers in the economy can be made better off. As an example, suppose the marginal rate of substitution was 3 and the marginal rate of transformation was 1 for goods x and y . By producing one more unit of x , the economy would need to sacrifice one unit of y . Consumers are willing to sacrifice 3 units of y , however, to get one additional unit of x . Therefore, if the economy produces one more unit of x and one fewer unit of y , consumers would be better off. 11. Explain how the conditions of utility maximization, cost minimization, and profit maximization in competitive markets imply that the allocation arising in a general competitive equilibrium is economically efficient. In a general competitive equilibrium, the economy will satisfy exchange, input, and substitution efficiency. This implies that all consumers are maximizing utility given the prices of goods in the economy and producers are maximizing profit at the point where prices equal marginal costs. P That is, MRS x , y  x Py , px  MCx , and Py  MC y . Together these imply that P MCx MRS x , y  x   MRTx , y Py MC y That is, utility maximization by consumers and profit maximization by producers implies the marginal rate of substitution will equal the marginal rate of transformation. This guarantees substitution efficiency is satisfied at the general competitive equilibrium. In other words, the allocation that arises in a general competitive equilibrium is economically efficient. 12. What is comparative advantage? What is absolute advantage? Which of these two concepts is more important in determining the benefits from free trade? Comparative advantage implies that one country has a lower opportunity cost in the production of some good, expressed in units of some other good forgone, than another country. Absolute advantage implies that one country can produce a product at a lower cost in terms of units of some input, labor for example, than another country. In determining the benefits from free trade, one must compare the opportunity costs to identify the benefits. For example, in a two-good world, while one country may have an absolute advantage in the production of both goods, it is still likely that each country has a comparative advantage in the production of different goods, and through specialization and trade, each country can be made better off.

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 3

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Solutions to Problems 16.1. Consider the markets for butter (B) and margarine (M), where the demand curves are Q = 20 – 2PM + PB and Q = 60 – 6PB + 4 PM and the supply curves are QM = 2PM and QB = 3PB. a) Find the equilibrium prices and quantities for butter and margarine. b) Suppose that an increase in the price of vegetable oil shifts the supply curve of margarine to QM = PM. How does this change affect the equilibrium prices and quantities for butter and margarine? Using words and graphs, explain why a shift in the supply curve for margarine would change the price of butter. a) In equilibrium we must have quantity supplied equal to quantity demanded in both the butter and margarine markets. This implies in equilibrium we will have QMd  QMs QBd  QBs Substituting in the given curves implies 20  2 PM  PB  2 PM 60  6 PB  4 PM  3PB Solving for PB in the first equation and substituting into the second equation implies 60  4 PM  9(4 PM  20) 60  4 PM  36 PM  180 PM  7.5 When PM  7.5 , PB  10 . At these prices, QM  15 and QB  30 . b)

When the supply curve for margarine shifts to QMs  PM , we have 20  2 PM  PB  PM 60  6 PB  4 PM  3PB

Solving the first equation for PB and substituting into the second equation implies 60  4 PM  9(3PM  20) 60  4 PM  27 PM  180 PM  10.43 When PM  10.43 , PB  11.30 . At these prices, QM  10.43 and QB  33.91 . The increase in the price of vegetable oil increases the price of margarine and decreases the quantity of margarine

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 4

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

consumed. As consumers switch to butter, the price of butter rises and the quantity of butter consumed goes up. The price of butter rises when the price of vegetable oil rises because butter and margarine are substitutes. The effects can be seen in the following graphs.

Market for Margarine

Price

40 S'

30 20

S

10

D

0 0

5

10

15

20

25

30

35

Quantity

Price

Market for Butter 25 20 15 10 5 0

S D' D 0

10

20

30

40

50

60

70

Quantity

Because the goods are substitutes, when the supply of margarine shifts inward from S to S’, raising the price of margarine, consumers substitute butter for margarine, shifting demand for butter outward from D to D’. This raises both the equilibrium price and quantity of butter. 16.2. Suppose that the demand curve for new automobiles is given by QA = 20 – 0.7PA – PG where QA and PA are the quantity (millions of vehicles) and average price (thousands of dollars per vehicle), respectively, of automobiles in the United States, and PG is the price of gasoline (dollars per gallon). The supply of automobiles is given by Q5A = 0.3PA. Suppose that the demand and supply curves for gasoline are QdG = 3 – PG and QSG = PG. a) Find the equilibrium prices of gasoline and automobiles.

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 5

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

b) Sketch a graph that shows how an exogenous increase in the supply of gasoline affects the prices of new cars in the United States. a) In equilibrium, the quantity supplied and the quantity demanded for both goods will be equal. This implies QAd  QAs QGd  QGs Substituting in the given curves implies 20  0.7 PA  PG  0.3PA 3  PG  PG Here we have two equations and two unknowns. Solving the second equation for PG yields PG  1.5 . Substituting into the first equation results in 20  0.7 PA  1.5  0.3PA PA  18.5 At these prices, QA  5.55 and QG  1.5 . b) If the supply of gasoline increases, the supply curve for gasoline will shift to the right lowering the equilibrium price of gasoline as seen in the graph below.

Market for Gasoline 4 S

Price

3 2

S'

1

D

0 0

1

2

3

4

Quantity

Because gasoline is a complement good for autos, the reduction in the price of gasoline will increase the demand for autos. This will shift the demand curve to the right, increasing the equilibrium price and quantity for autos as seen in the following graph.

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 6

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Market for Autos 80 S

Price

60 40 20

D'

0 0

5

10

15

D 20

Quantity

16.3. Studies indicate that the supply and demand schedules for ties (t) and jackets (j) in a market are as follows:

The estimates of the schedules are valid only for prices at which quantities are positive. a) Find the equilibrium prices and quantities for ties and jackets. b) Do the demand schedules indicate that jackets and ties are substitute goods, complementary goods, or independent goods in consumption? How do you know? a)

In equilibrium (1) the supply and demand for ties will be equal

410 – 5Pt – 2Pj = –60 + 3PK , and (2) the supply and demand for jackets will be equal 295 – Pt – 3Pj = –120 + 2Pj Solving these two simultaneous equations, we find that Pj = 75 and Pt = 40. Also, using either the demands or supply schedules, we calculate that the equilibrium quantity of jackets is 30, and the equilibrium quantity of ties is 60. b) The demand function for ties shows that a higher price of jackets decreases the demand for ties. Similarly the demand function for jackets shows that a higher price of ties decreases the demand for jackets. Ties and jackets are therefore complements in consumption. 16.4. Suppose that the demand for steel in Japan is given by the equation QdS = 1200 – 4PS + PA + PT, where QS is the quantity of steel purchased (millions of tons per year), PS is the price of steel (yen per ton), PA is the price of aluminum (yen per ton), and PT is the price of

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 7

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

titanium (yen per ton). The supply curve for steel is given by QSS = 4PS. Similarly, the demand and supply curves for aluminum and for titanium are given by QdA = 1200 – 4PA + PS + PT (demand curve for aluminum), QSA = 4PA (supply curve for aluminum), QdT = 1200 – 4PT + PS + PA (demand curve for titanium), and QST = 4PT (supply curve for aluminum). a) Find the equilibrium prices of steel, aluminum, and titanium in Japan. b) Suppose that a strike in the Japanese steel industry shifts the supply curve for steel to QSS = PS. What does this do to the prices of steel, aluminum, and titanium? c) Suppose that growth in the Japanese beer industry, a big buyer of aluminum cans, fuels an increase in the demand for aluminum so that the demand curve for aluminum becomes QdA = 1500 – 4PA + PS + PT. How does this affect the prices of steel, aluminum, and titanium? a) In equilibrium the quantity supplied will equal the quantity demanded in all three markets. Algebraically this implies QSd  QSs QAd  QAs QTd  QTs Substituting in the given curves implies 1200  4 PS  PA  PT  4 PS 1200  4 PA  PS  PT  4 PA 1200  4 PT  PS  PA  4 PT Solving the first equation for PT and substituting into the second equation implies 1200  4 PA  PS  (8 PS  PA  1200)  4 PA 9 PS  9 PA PS  PA Substituting these results into the third equation implies 1200  4(8 PA  PA  1200)  PA  PA  4(8 PA  PA  1200) 10,800  54 PA PA  200 At PA  200 , PS  200 and PT  200 . The equilibrium quantities are QA  800 , QS  800 , and QT  800 . b)

Substituting the new supply curve for steel into the equilibrium condition implies

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Chapter 16 - 8

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

1200  4 PS  PA  PT  PS 1200  4 PA  PS  PT  4 PA 1200  4 PT  PS  PA  4 PT Again solving for PT in the first equation and substituting into the second equation implies 1200  4 PA  PS  (5PS  PA  1200)  4 PA 6 PS  9 PA PS  1.5 PA Substituting these results into the third equation implies 1200  4(5(1.5 PA )  PA  1200)  1.5 PA  PA  4(5(1.5 PA )  PA  1200) 10,800  49.5PA PA  218.18 At PA  218.18 , PS  327.27 and PT  218.18 . At these prices, the equilibrium quantities are QA  872.72 , QS  327.27 , and QT  872.72 . The shift in the supply of steel raises the equilibrium price for all three goods, lowering the equilibrium quantity of steel and raising the equilibrium quantities of aluminum and titanium. This last effect comes as a result of the demand curves for aluminum and titanium increasing in response to the shift in the steel supply curve. c)

Returning to the original equilibrium, this shift in the demand for aluminum implies 1200  4 PS  PA  PT  4 PS 1500  4 PA  PS  PT  4 PA 1200  4 PT  PS  PA  4 PT

Solving the first equation for PT and substituting into the second equation implies 1500  4 PA  PS  (8 PS  PA  1200)  4 PA 9 PS  300  9 PA PS  PA  33.33 Substituting these results into the third equation implies 1200  4(8( PA  33.33)  PA  1200)  ( PA  33.33)  PA  4(8( PA  33.33)  PA  1200) 12,900  54 PA PA  238.89

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 9

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

At PA  238.89 , PS  205.56 and PT  205.56 . At these prices, the equilibrium quantities are QA  955.56 , QS  822.24 , and QT  822.24 . An increase in the demand for aluminum will raise the equilibrium prices and quantities in all three markets. The price and quantity in the steel and aluminum industries increase because as the price of aluminum rises, the demand for steel and titanium increases. 16.5. Consider a simple economy that produces two goods, beer (denoted by x) and quiche (denoted by y), using labor and capital (denoted by L and K, respectively) that are supplied by two types of households, those consisting of wimps (denoted by W) and those consisting of hunks (denoted by H). Each household of hunks supplies 100 units of labor and no units of capital. Each household of wimps supplies 10 units of capital and no units of labor. There are 100 households of each type. Both beer and quiche are produced with technologies exhibiting constant returns to scale. The market supply curves for beer and quiche are

where w denotes the price of labor and r denotes the price of capital. The market demand curves for beer and quiche are given by

where X and Y denote the aggregate quantities of beer and quiche demanded in this economy and IW and IH are the household incomes of wimps and hunks, respectively. Finally, the market demand curves for labor and capital are given by

There are four unknowns in our simple economy: the prices of beer and quiche, Px and Py, and the prices of labor and capital, w and r. Write the four equations that determine the equilibrium values of these unknowns. First, in equilibrium, the quantity supplied of beer and quiche must equal the quantity demanded of beer and quiche. This implies 20 IW  90 I H w1/ 6 r 5/ 6  X 80 IW  10 I H w3/ 4 r1/ 4  Y Now, since each hunk supplies 100 units of labor and no units of capital and each wimp supplies 10 units of capital and no units of labor, Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 10

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

IW ( w, r )  10r I H ( w, r )  100 w Substituting these into the conditions above yields our first two equations: 200r  9000 w X 800r  1000 w  Y

w1/ 6 r 5/ 6  w3/ 4 r1/ 4

Second, in equilibrium, the quantity supplied of labor and capital must equal the quantity demanded of labor and capital. Since there are 100 households of each type, we will have L  100(100)  10, 000 and K  100(10)  1, 000 . Setting these equal to demand yields the third and fourth equations: X 10, 000  6

 r    w

5 X  w 1, 000    6  r

5/ 6

 r  w

1/ 4

Y  w   4  r

3/ 4

3Y   4

1/ 6

16.6. In an economy, there are 40 “white-collar” households, each producing 10 units of capital (and no labor); the income from each unit of capital is r. There are also 50 “bluecollar” households, each producing 20 units of labor (and no capital); the income from each unit of labor is w. Each white-collar household’s demand for energy is XW = 0.8MW/PX, where MW is income in the household. Each white-collar household’s demand for food is YW = 0.2MW/PY. Each blue-collar household’s demand for energy is XB = 0.5MB/PX, where MB is income in the household. Each blue-collar household’s demand for food is YB = 0.5MB/PY. Energy is produced using only capital. Each unit of capital produces one unit of energy, so r is the marginal cost of energy. The supply curve for energy is described by PX = r, where PX is the price of a unit of energy. Food is produced using only labor. Each unit of labor produces one unit of food, so w is the marginal cost of food. The supply curve for labor is described by PY = w, where PY is the price of a unit of food. a) In this economy, show that the amount of labor demanded and supplied will be 1,000 units. Show also that the amount of capital demanded and supplied will be 400 units. b) Write down the supply-equals-demand conditions for the energy and food markets. c) In equilibrium how will the price of a unit of energy compare with the price of a unit of food? d) In equilibrium how will the income of each white-collar family compare with the income of each blue-collar family?

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Chapter 16 - 11

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

a) The total amount of capital produced (all by white collar households together) is (40 households)(10 units/household) = 400 units. The total amount of labor produced (all by blue collar households together) is (50 households)(20 units/household) = 1000 units. b) The income in each white collar household is MW = 10r. The income in each blue collar household is MB = 20w. The aggregate demand for energy will be X = [50(0.5MB) + 40(0.8MW)]/PX = [500w + 320r]/PX The aggregate demand for food will be Y =[50(0.5MB) + 40(0.2MW)]/PY = [500w + 80r]/PY. The supply-equals demand condition in the energy market is r = [500w + 320r]/X = [500w + 320r]/400, or r = 6.25w The supply-equals demand condition in the food market is w = [500w + 80r]/Y= [500w + 80r]/1000, or, as before r = 6.25w (same as above by Walras’ Law) c) PX = r and PY = w. Since r = 6.25w, the price of a unit of energy is 6.25 times as large as the price of a unit of food. d) The income of a blue collar family is MB = 20w. The income of a white collar family is MW = 10r = 10(6.25w) = 62.5w. So a white collar family has an income 3.125 (= 62.5/20) times larger than that of a blue collar family. 16.7. One of the implications of Walras’ Law is that the ratios of prices (rather than the absolute levels of prices) are determined in general equilibrium. In Learning-By-Doing Exercise 16.2, show that price labor will be 25/52 ≈ 0.48 of the price of capital, as illustrated in Figure 16.9. When we equated the supply and demand for energy, we eliminated the price of energy to derive Equation 16.4: w1/3r2/3 , we find that . Call this equation A. Similarly, when we equated the supply and demand for food, we eliminated the price of food to derive Equation 16.5: w1/2r1/2

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Chapter 16 - 12

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

, we find that . Call this equation B. When we substitute Equations A and B into the supply-equals-demand in the labor market (Equation 16.6), we find that 7000 =

+

=

+

=

+

=  0.48.

Thus: 42000w =

16.8. One of the implications of Walras’ Law is that the ratios of prices (rather than the absolute levels of prices) are determined in general equilibrium. In Learning-By-Doing Exercise 16.2, show that the ratio of the price of energy to the price of capital is about 0.79, as illustrated in Figure 16.9. When we equated the supply and demand for energy, we eliminated the price of energy to derive Equation 16.4: w1/3r2/3 , we find that . Call this equation A. Similarly, when we equated the supply and demand for food, we eliminated the price of food to derive Equation 16.5: w1/2r1/2 , we find that . Call this equation B.

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 13

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

When we substitute Equations A and B into the supply-equals-demand in the labor market (Equation 16.6), we find that 7000 =

+

=

+

=

+

=  0.48.

Thus: 42000w =

Finally, from the relationship describing the supply of energy we know that Px = w1/3r2/3 =

r=

r  0.79 r

16.9. One of the implications of Walras’ Law is that the ratios of prices (rather than the absolute levels of prices) are determined in general equilibrium. In Learning-By-Doing Exercise 16.2, show that the ratio of the price of food to the price of capital is about 0.7, as illustrated in Figure 16.9. When we equated the supply and demand for energy, we eliminated the price of energy to derive Equation 16.4: w1/3r2/3 , we find that . Call this equation A. Similarly, when we equated the supply and demand for food, we eliminated the price of food to derive Equation 16.5: w1/2r1/2 , we find that . Call this equation B.

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 14

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

When we substitute Equations A and B into the supply-equals-demand in the labor market (Equation 16.6), we find that 7000 =

+

=

+

=

+

=  0.48.

Thus: 42000w = Finally, from the relationship describing the supply of food we know that PY = w1/2r1/2 =

r=

r  0.7 r

16.10. Two consumers, Josh and Mary, together have 10 apples and 4 oranges. a) Draw the Edgeworth box that shows the set of feasible allocations that are available in this simple economy. b) Suppose Josh has 5 apples and 1 orange, while Mary has 5 apples and 3 oranges. Identify this allocation in the Edgeworth box. c) Suppose Josh and Mary have identical utility functions, and assume that this utility function exhibits positive marginal utilities for both apples and oranges and a diminishing marginal rate of substitution of apples for oranges. Could the allocation in part (b)—5 apples and 1 orange for Josh; 5 apples and 3 oranges for Mary—be economically efficient? a & b) Mary

4

Oranges

3 2 1 0 Josh 0

2

4

6

8

10

Apples

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 15

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

c) To be economically efficient, the two consumers must have identical marginal rates of substitution at the allocation. While we are not given the MRS for each consumer, we are told that each has an identical utility function. This implies that at an efficient allocation where the MRS for each consumer is the same, the ratio of apples to oranges must be the same. Since at the current allocation Josh has a ratio of apples to oranges equal to 5 and Mary has a ratio of 1.67, this allocation cannot be efficient. The contract curve in this case will be a straight line between the origins for each consumer. 16.11. Ted and Joe each consume peaches, x, and plums, y. The consumers have identical utility functions, with Together, they have 10 peaches and 10 plums. Verify whether each of the following allocations is on the contract curve: a) Ted: 8 plums and 9 peaches; Joe: 2 plums and 1 peach. b) Ted: 1 plum and 1 peach; Joe: 9 plums and 9 peaches. c) Ted: 4 plums and 3 peaches; Joe: 6 plums and 7 peaches. d) Ted: 8 plums and 2 peaches; Joe: 2 plums and 8 peaches. To be on the contract curve, an allocation must yield identical marginal rates of substitution for each consumer. a) MRSTed = 80/9 < MRSJoe = 20/1. Not on the contract curve. b) MRSTed = 10/1 = MRSJoe = 90/9. On the contract curve. c) MRSTed = 40/3 > MRSJoe = 60/7. Not on the contract curve. d) MRSTed = 80/2 > MRSJoe = 20/8. Not on the contract curve. 16.12. Two consumers, Ron and David, together own 1,000 baseball cards and 5,000 Pokémon cards. Let xR denote the quantity of baseball cards owned by Ron and yR denote the quantity of Pokémon cards owned by Ron. Similarly, let xD denote the quantity of baseball cards owned by David and yD denote the quantity of Pokémon cards owned by David. Suppose, further, that for Ron, MRSRx,y = yR/xR, while for David, MRSDx,y = yD/2xD. Finally, suppose xR = 800, yR = 800, xD = 200, and yD = 4,200. a) Draw an Edgeworth box that shows the set of feasible allocations in this simple economy. b) Show that the current allocation of cards is not economically efficient. c) Identify a trade of cards between David and Ron that makes both better off. (Note: There are many possible answers to this problem.) a)

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Chapter 16 - 16

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

David

Pokemon Cards

5000 4000 3000 2000 1000 0 Ron

0

200

400

600

800

1000

Baseball Cards

b) To be economically efficient, the MRS for the two consumers must be equal. At this allocation we have y 800 MRS xR, y  R  1 xR 800 y 4200 MRS xD, y  D   10.5 2 xD 2(200) Since MRSD > MRSR, the current allocation is not economically efficient. c) At the current allocation Ron is willing to trade one baseball card for one Pokemon card, and David is willing to trade one baseball card for 10.5 Pokemon cards. If David gives Ron 9 Pokemon cards in exchange for one baseball card, both consumers will be better off. Or, in other words, Ron thinks a baseball card is worth just one Pokemon card while David thinks it is worth 10.5 Pokemon cards. So both will be better off if Ron sells David a baseball card for anything more than one Pokemon card and less than 10.5 Pokemon cards. 16.13. There are two individuals in an economy, Joe and Mary. Each of them is currently consuming positive amounts of two goods, food and clothing. Their preferences are characterized by diminishing marginal rate of substitution of food for clothing. At the current consumption baskets, Joe’s marginal rate of substitution of food for clothing is 2, while Mary’s marginal rate of substitution of food for clothing is 0.5. Do the currently consumed baskets satisfy the condition of exchange efficiency? If not, describe an exchange that would make both of them better off. Since the marginal rates of substitution are not equal for both people, the current consumption baskets do not satisfy exchange efficiency. Joe would be willing to give up 2 units of food to get 1 additional unit of clothing. Mary would be willing to give up 0.5 units of food to get 1 additional unit of clothing; put another way, Mary would be willing to give up 2 units of clothing to get 1 additional unit of food.

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

One exchange that would make both better off would be for Joe to give 1 unit of food to Mary in exchange for 1 unit of clothing. Joe is better off (he would have been willing to give up 2 units of food to get 1 additional unit of clothing). But what about Mary? To get the 1 additional unit of clothing, she would have been willing to give up 2 units of clothing; but with the proposed exchange, she only had to give up 1 unit of clothing. So the proposed exchange also makes her better off. 16.14. Consider an economy that consists of three individuals: Maureen (M), David (D), and Suvarna (S). Two goods are available in the economy, x and y. The marginal rates of substitution for the three consumers are given by MRSMaureenx,y = 2yM/xM, MRSDavidx,y = 2yD/xD and MRSSuvarnax,y = yS/xS. Maureen and David are both consuming twice as much of good x as good y, while Suvarna is consuming equal amounts of goods x and y. Are these consumption patterns economically efficient? To be economically efficient, the marginal rates of substitution for all consumers must be equal. From the given information we know xM  2 yM , xD  2 yD , and xS  yS . Substituting into the marginal rates of substitution we have 2y MRS xMaureen  M 1 ,y 2 yM 2 yD MRS xDavid  1 ,y 2 yD y MRS xSu, yvarna  S  1 yS Thus, each consumer has an identical marginal rate of substitution. This consumption pattern is therefore economically efficient. 16.15. Two firms together employ 100 units of labor and 100 units of capital. Firm 1 employs 20 units of labor and 80 units of capital. Firm 2 employs 80 units of labor and 20 units of capital. The marginal products of the firms are as follows: Firm 1: MP1l = 50, MP1k = 50; Firm 2: MP2l = 10, MP2k= 20. Is this allocation of inputs economically efficient? To satisfy input efficiency, the marginal rates of technical substitution must be equal across firms. Here we have

MRTS

1 l ,k

MPl1 50   1 MPk1 50

MRTS l2,k 

MPl 2 10   0.5 MPk2 20

Thus, the allocation of inputs is not economically efficient.

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

16.16. There are two firms in an economy. Each of them currently employs positive amounts of two inputs, capital and labor. Their technologies are characterized by diminishing marginal rate of technical substitution of labor for capital. At the current operating basket, Firm A’s marginal rate of technical substitution of labor for capital is 3, while Firm B’s marginal rate of technical substitution of labor for capital is 1. Do the current production baskets satisfy the condition of input efficiency? If not, describe an exchange of inputs that would improve efficiency. Since the marginal rates of technical substitution are not equal for both firms, the current production baskets do not satisfy input efficiency. Firm A would be willing to give up 3 units of capital to get one additional unit of labor. Firm B would be willing to give up 1 unit of capital to get 1 additional unit of labor; put another way, Firm B would be willing to give up 1 unit of labor to get 1 additional unit of capital. One exchange that would make both firms better off would be for Firm A to give 2 units of capital to Firm B in exchange for 1 unit of labor. Firm A is better off (it would have been willing to give up 3 units of capital to get 1 additional unit of labor). But what about Firm B? To get the 2 additional units of capital, it would have been willing to give up 2 units of labor; but with the proposed exchange, Firm B only had to give up 1 unit of labor. So the proposed exchange also makes Firm B better off. 16.17. Two firms together employ 10 units of labor (l) and 10 units of capital (k). The marginal rate of technical substitution of each firm is given by: MRTS1lk = k1/l1 and MRTS2lk = 4k2/l2. Which of the following input allocations satisfy the condition of input efficiency? a) Firm 1 uses 5 units of labor, 5 units of capital; Firm 2 uses 5 units of labor, 5 units of capital. b) Firm 1 uses 5 unit of labor, 8 units of capital; Firm 2 uses 5 units of labor; 2 units of capital. c) Firm 1 uses 9 units of labor, 9 units of capital; Firm 2 uses 1 unit of labor; 1 unit of capital. d) Firm 1 uses 2 units of labor; 5 units of capital; Firm 2 uses 8 units of labor; 5 units of capital. To satisfy input efficiency, the marginal rates of technical substitution must be equal across firms. a) MRTS1 = 5/5 = 1 < MRTS2 = 4(5)/5 = 4. The allocation does not satisfy input efficiency. b)

MRTS1 = 8/5 = 1.6 = MRTS2 = 4(2)/5 = 1.6. The allocation satisfies input efficiency.

c)

MRTS1 = 9/9 = 1 < MRTS2 = 4(1)/1 = 4. The allocation does not satisfy input efficiency.

d)

MRTS1 = 5/2 = 2.5 = MRTS2 = 4(5)/8 = 2.5. The allocation satisfies input efficiency.

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

16.18. Two firms together employ 20 units of labor and 12 units of capital. For Firm 1, which uses 5 units of labor and 8 units of capital, the marginal products of labor and capital are MP1l = 20 and MP1k = 40. For Firm 2, which uses 15 units of labor and 4 units of capital, the marginal products are MP2l = 60 and MP2k = 30. a) Draw an Edgeworth box for inputs that shows the allocation of inputs across these two firms. b) Is this allocation of inputs economically efficient? Why or why not? If it is not, identify a reallocation of inputs that would allow both firms to increase their outputs. a) Firm 2

12

Capital

10 8 6 4 2 0 Firm 1 0

5

10

15

20

Labor

b)

To satisfy input efficiency we must have MRTS L1 , K  MRTS L2, K MPL1 MPL2  MPK1 MPK2

Substituting in the given information implies 20 60  40 30 0.5  2 Since the MRTS are not equal, the current allocation of inputs is not economically efficient. At the current allocation, Firm 1 can trade 2 units of labor for 1 unit of capital without changing output. By giving up one unit of labor to receive one unit of capital the firm can increase its output. At the current allocation Firm 2 can trade 2 units of capital for one unit of labor without affecting output. By giving up only one unit of capital in exchange for one unit of labor Firm 2

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

can increase its output. Therefore, by reallocating one unit of capital from Firm 2 to Firm 1 and one unit of labor from Firm 1 to Firm 2, both firms can produce more output. 16.19. Consider an economy that produces two goods: food, x, and clothing, y. Production of both goods is characterized by constant returns to scale. Given current input prices, the marginal cost of producing clothing is $10 per unit, while the marginal cost of producing food is $20 per unit. What is the marginal rate of transformation of x for y? How much clothing must the economy give up in order to get one additional unit of food? In general equilibrium, MRTx,y = MCx / MCy = 20/10 = 2. To get one additional unit of food (x), the economy must sacrifice 2 units of clothing (y). 16.20. An economy consists of two consumers (Julie and Carina), each consuming positive amounts of two goods, food and clothing. Food and clothing are both produced with two inputs, capital and labor, using technologies exhibiting constant returns to scale. The following information is known about the current consumption and production baskets: The marginal cost of producing food is $2, and the price of clothing is $4. The wage rate is 2/3 the rental price of capital, and the marginal product of capital in producing clothing is 3. In a general competitive equilibrium, what must be a) The price of food? b) The marginal rate of transformation of food for clothing? c) The shape of the production possibilities frontier for the economy? d) The marginal product of labor in producing clothing? a) In a competitive equilibrium, price must equal marginal cost for each good. Thus, the price of food must be $2. b) By the same reasoning as in (a) the marginal cost of clothing must be $4. MRTfood,clothing = MCfood / MCclothing = $2/$4 = 0.5 c) Because all production occurs with constant returns to scale, the marginal costs will be constant, and thus the MRTfood,clothing is always 0.5. On a graph with food on the horizontal axis and clothing on the vertical axis, the Production Possibilities Frontier will be a straight line with a slope of -0.5. d)

With input efficiency, w/r = MPL/MPK ; thus, 2/3 = MPL/3. So MPL = 2.

16.21. Consider an economy that uses labor and capital to produce two goods, beer (x) and peanuts ( y), subject to technologies that exhibit constant returns to scale. The marginal cost of a 12-ounce can of beer is $0.50. The marginal cost of a 12-ounce tin of peanuts is $1.00. Currently, the economy is producing 1 million 12-ounce cans of beer and 2 million 12-ounce tins of peanuts. The marginal rates of technical substitution of labor for capital in the beer and peanut industries are the same. Moreover, there are 1 million identical

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

consumers in the economy, each with a marginal rate of substitution of beer for peanuts given by MRSx,y = 3y/x. a) Sketch a graph of the economy’s production possibilities frontier. Identify the economy’s current output on this graph. b) Does the existing allocation satisfy substitution efficiency? Why or why not? a) Because the technologies exhibit constant returns to scale, the production possibilities frontier will be a straight line with slope MC x 0.50 MRTx , y    0.50 MC y 1.00 Here is a graph of the production possibilities frontier for this economy. 3

Current Production

Peanuts

2.5 2 1.5 1 0.5 0 0

1

2

3

4

5

6

Beer

b) To achieve substitution efficiency we must have MRTx , y  MRS x, y . At the current allocation this implies MCx x  MC y 3 y At the current allocation we have 0.50 1  1.00 3(2) 0.50  0.17 Since MRT < MRS, consumer utility would go up if more resources were devoted to beer production (x) and less resources were devoted to peanut production (y). 16.22. The United States and Switzerland both produce automobiles and watches. The labor required to produce a unit of each product is shown in the following table:

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

a) Which country has an absolute advantage in the production of watches? In the production of automobiles? b) Which country has a comparative advantage in the production of watches? In the production of automobiles? a) From the information given in the table, the U.S. has an absolute advantage in the production of watches because the production of one watch takes only 50 hours per watch in the U.S. compared with 60 hours per watch in Switzerland. The U.S. also has an absolute advantage in the production of automobiles since the U.S. spends only 5 hours per auto produced compared with 20 hours per auto in Switzerland. b) In the U.S. the opportunity cost of producing one watch is 10 autos. In Switzerland the opportunity cost of producing one watch is 3 autos. Because the opportunity cost is lower for Switzerland than the U.S., Switzerland has a comparative advantage in the production of watches. In the U.S. the opportunity cost of producing one auto is 1/10 of a watch. In Switzerland, the opportunity cost of producing one auto is 1/3 of a watch. Because the opportunity cost is lower for the U.S., the U.S. has a comparative advantage in the production of autos. 16.23. Brazil and China can produce cotton and soybeans. The labor required to produce a unit of each product is shown in the following table:

a) Which country has an absolute advantage in the production of cotton? In the production of soybeans? b) Which country has a comparative advantage in the production of cotton? In the production of soybeans? a) From the information given in the table, Brazil has an absolute advantage in the production of cotton because it takes only 10 hours of labor per unit of cotton in Brazil compared with 20 hours per unit cotton in China. Brazil also has an absolute advantage in the production of soybeans since it spends only 80 hours of labor per unit of soybeans compared with 100 hours per unit of soybeans in China.

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 23

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

b) In Brazil the opportunity cost of producing one unit of cotton is 8 soybeans. In China the opportunity cost of producing one unit of cotton is 5 soybeans. Because the opportunity cost is lower for China than Brazil, China has a comparative advantage in the production of cotton. In Brazil the opportunity cost of producing one unit soybeans is 1/3 of a unit of cotton. In China, the opportunity cost of producing one unit of soybeans is 1/5 of a unit of cotton. Because the opportunity cost is lower for Brazil than China, Brazil has a comparative advantage in the production of soybeans.

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 16 - 24