Given a Cobb-Douglas production function where 0.5 Find the equation for the isoquant when Q 2,000

Is it possible that Will and David have different marginal productivity functions but the same marginal rate of technical substitution functions?Explain.Will and David

4/9/2018Homework 4 (Chapter 6)-Maria Bernedo20/3223.24.ID: Concept Question 4.1Given a Cobb-Douglas production function where= = 0.5:Q = KL0.50.5Find the equation for the isoquant when Q = 2,000.A.K =2,000L0.52B.K =2,000L20.5C.K =22,0000.5D.L =2,000K0.52The marginal rate of technical substitution can be written as the ratio of the marginal productivities of the two inputs:MRTS = −= −MPLMPKqLqKFind the marginal rate of technical substitution whenQ = 2,000 and L = 100.MRTS =(enter your response as an integer and remember to include the sign of this value).LID: Concept Question 4.2Consider the following production function:.+ 4L−Lq = 7LK213Given the following expressions for the marginal productivity of each input:and+ 8L − LMPL= 7K2MP= 7LAssuming capital is plotted on the vertical axis and labor is plotted on the horizointal axis, determine the value of the marginal rate of technical substitution when K =and L =.(Round your answer up to two decimal places and include the proper sign.)20MRTS =3K10.

4/9/2018Homework 4 (Chapter 6)-Maria Bernedo21/3225.ID: Text Question 5.3Show in a diagram that a production function can have constant returns to scale.Assume the firm currently produces at point a, using 1 worker and 1 unit ofcapital on isoquant q1.=Using the three-point curved line drawing tool, graph an isoquant for 2 units ofoutput.Label the curve 'q2.'Carefully follow the instructions above, and only draw the required object.45L, Workers per dayK, Units of capital per dayq = 1=0123450123a

4/9/2018Homework 4 (Chapter 6)-Maria Bernedo22/32

Producer Behavior Chapter 6 67

*1. Consider the production function presented in the table below:

Capital (K )

Labor (L)

0123456

1 100 200 300 400 500 600

2 200 400 600 800 1,000 1,

3 300 600 900 1,200 1,500 1,

4 400 800 1,200 1,600 2,000 2,

5 500 1,000 1,500 2,000 2,500 3,

6 600 1,200 1,800 2,400 3,000 3,

a. If the fi rm decides to employ 6 units of capital and 1 worker, what is its output? b. What other combinations of capital and labor could be used to produce the same level of output you found in (a)? c. Plot the combinations you determined in (a) and (b) on a graph, with labor on the horizontal axis and capital on the vertical axis. Connect the dots to form the production isoquant corresponding to 600 units of output.

1. a. The fi rm’s output is 600. b. Output of 600 can also be achieved with either 3 units of capital and 2 units of labor, 2 units of capital and 3 units of labor, or 1 unit of capital and 6 units of labor. c.

Labor

1

2

5

6

4 3

134625

Capital

0

Q = 600

1. The table to the right represents the production function for Hawg Wild, a small catering company specializing in barbecued pork. The numbers in the cells represent the number of customers that can be served with various combi- nations of labor and capital. a. Is this production function a short-run or long-run production function? How can you tell? b. Suppose that Hawg Wild employs 5 units of capital and 2 workers. How many diners will be served? c. Suppose that Hawg Wild employs 5 units of capital and 2 workers, but that the owner, Billy Porcine, is consid- ering adding his nephew to the payroll. What will the marginal product of Billy’s nephew be? d. Notice that when Hawg Wild uses 1 unit of capital, the marginal product of the fi fth unit of labor is 16. But when Hawg Wild uses 5 units of capital, the mar- ginal product of the fi fth unit of labor is 43. Does this production function violate the law of diminishing marginal product of labor? Why or why not?

Capital (K )

Labor (L)

0123456

1 100 132 155 174 190 205

2 152 200 235 264 289 310

3 193 255 300 337 368 396

4 230 303 357 400 437 470

5 263 347 408 457 500 538

6 293 387 455 510 558 600

Producer Behavior

67

68 Part 2 Consumption and Production

e. Suppose that Hawg Wild employs 5 units of capital and 2 workers, but that the owner, Billy Porcine, is considering adding another meat smoker to the kitchen (which will raise the amount of capital input to 6 units). What will the marginal product of the smoker be? f. Hawg Wild employs 5 units of capital and 2 workers. Billy is considering the choice between hiring another worker or buying another smoker. If smokers cost $8 and workers $12, then at the margin, what is the most cost- effective choice for Billy to make?

1. a. This is a long-run production function. In the short-run production function, capital is assumed to be fi xed. b. 347 diners. c. The marginal product of Billy’s nephew will be 408 – 347 = 61. d. The production function does not violate the law of diminishing marginal product of labor. In particular, the mar- ginal product of labor decreases as L increases, holding the level of capital fi xed. e. The marginal product of the smoker will be 387 – 347 = 40. f. The marginal product of an additional labor is 61 and the marginal product of an additional smoker is 40. Thus, given the cost of the smoker and the worker, Billy would marginally rather hire an additional worker. Since MPL_w = _ 61 12

= 5 is greater than _MPKr = _ 40

= 5, Hawg Wild will get more additional production per dollar when it spends that dollar on labor as opposed to capital.

1. Complete the table below:

Labor Input

Total Product

Marginal Product

Average Product

00 — —

170

2 135

363

451

557

6 324

1. Jerusha, a woodworker, builds coffee tables using both labor (L) and tools (capital, or K ). Her production function for coffee tables is a Cobb–Douglas production function: Q = 4 K .5 L.. a. Can Jerusha build any coffee tables without tools? b. Can Jerusha completely mechanize coffee table production? c. Jerusha currently has 16 tools, and in the short run can neither acquire more tools nor sell existing tools. Her woodshop is capable of holding up to 49 employees. What is Jerusha’s short-run production function?

Labor Input

Total Product

Marginal Product

Average Product

00 ——

1 70 70 70

2 135 65 67.

3 189 54 63

4 240 51 60

5 285 45 57

6 324 39 54

70 Part 2 Consumption and Production

1. Abel, Baker, and Charlie all run competing bakeries, where each makes loaves of bread. a. At Abel’s bakery, the marginal product of labor is 15 and the average product of labor is 12. Would Abel’s average product increase or decrease if he hired another worker? b. At Baker’s bakery, the marginal product of labor is 7 and the average product of labor is 12. Would Baker’s aver- age product increase or decrease if he hired another worker? c. At Charlie’s bakery, the MP L is –12. Does this mean her average product must also be negative? d. Based on your answers to (a), (b), and (c), can you generalize the nature of the relationship between the average and marginal products of labor?

2. a. Average product would increase, because marginal product exceeds average product at this quantity. b. Average product would decrease, because marginal product is below average product at this quantity. c. No. A negative marginal product means that average product is declining. Yet average product can still be positive. d. Average product will increase when marginal product is greater than average product; it will decrease when marginal product is less. When marginal product is negative, each additional worker causes output to decline; yet average product will still be positive as long as some output is produced.

3. Suppose that a fi rm’s production function is given by Q = K 0 L 0 , where MP K = 0 K – 0 L 0 and M P L = 0 K 0 L – 0.. a. As L increases, what happens to the marginal product of labor? b. As K increases, what happens to the marginal product of labor? c. Why would the MP L change as K changes? d. What happens to the marginal product of capital as K increases? As L increases?

4. a.

MP L = 0 ( _K

From the above, as L increases, the marginal product of labor decreases. b. Using the formula above, as K increases, the marginal product of labor increases. c. As capital increases, labor becomes more productive. d.

MP K = 0 ( _L

Marginal product of capital decreases as K increases, and increases as L increases.

1. Fetzer valves can be made in either China or the United States, but because labor in the United States is more skilled, on aver- age, than labor in China, the production technologies differ. Consider the two production isoquants in the fi gure. Each rep- resents either the production technology for the United States or for China. Based on the MRTS, which production isoquant is more likely to represent the United States and which represents China? Explain.

2. Both countries use exactly the same labor/capital combination at the point where their isoquants cross. Assume you decrease capital by a fi xed amount. The steeper isoquant indicates that in order to remain on the isoquant, Country A can use less labor than country B. Thus, country A is the United States and country B is China.

*8. Consider the production functions given below: a. Suppose that the production function faced by a 30-weight ball bearing producer is given by Q = 4 K 0 L 0 , where M P K = 2 K – 0 L 0 and M P L = 2 K 0 L – 0. Do both labor and capital display diminishing marginal products? Find the marginal rate of technical substitution for this production function. (Hint: The MRTS = M P L /M P K .) Does this production function display a diminishing marginal rate of substitution?

Capital (K)

Labor (L)

Q = 1, (Country A)

Q = 1, (Country B)

Producer Behavior Chapter 6 71

b. Suppose that the production function faced by a 40-weight ball bearing producer is given by Q = 4 KL, where MP K = 4 L and MP L = 4 K. Do both labor and capital display diminishing marginal products? Find the marginal rate of technical substitution for this production function. Does this production function display a diminishing marginal rate of substitution? c. Compare your answers to (a) and (b). Must labor and capital display diminishing marginal products in order for the MRTS to diminish?

1. a. The marginal product of capital will decrease as capital increases, and the marginal product of labor will decrease as labor increases. Thus, they both exhibit diminishing marginal products. The MRTS is

(2 K 0 L –0 )/(2 K –0 L 0 ) = K/L

As labor increases, this value will diminish. Therefore, the marginal rate of substitution is diminishing. b. For a given amount of labor, the marginal product of capital is constant. For a given amount of capital, the mar- ginal product of labor is constant. Hence, neither input exhibits diminishing marginal returns. The MRTS is

4 K/ 4 L = K/L.

As labor increases, the MRTS will diminish. c. In both (a) and (b), the production function displays a diminishing MRTS. Yet in (b), both inputs were charac- terized by a constant marginal product. So a diminishing rate of substitution can be found even when marginal product is not diminishing.

*9. Suppose that Manny, Jack, and Moe can hire workers for $12 per hour, or can rent capital for $7 per hour. a. Write an expression for Manny, Jack, and Moe’s total cost as a function of how many workers they hire and how much capital they employ. b. Assume that Manny, Jack, and Moe wish to hold their total costs to exactly $100. Use your answer from (a) to fi nd the equation for an isocost line corresponding to exactly $100 of costs. Rearrange your equation to isolate capital. c. Graph the equation for the isocost line, putting labor on the horizontal axis and capital on the vertical axis. d. What is the vertical intercept of the line you drew? The horizontal intercept? What does each represent? e. What is the slope of the line you drew? What does it represent? f. Suppose that bargaining with the local labor union raises wages. Manny, Jack, and Moe must now pay $14 per hour. What happens to the isocost line corresponding to $100 of expenditure? Explain. Show the new isocost line on your graph.

1. a. The cost function is

$12 × L + $7 × K

b.

$100 = $12 × L + $7 × K

K = _ 100 7

• _ 12 7

c.

100 Labor (L) 12

Capital (K)

0

Slope = 

100 7

12 7

d. The vertical intercept indicates the quantity of capital that can be rented with $100 if no labor is hired. The hori- zontal intercept indicates the quantity of labor that can be hired with $100 if no capital is rented. e. The slope is the (negative) ratio of the price of labor (wage) and the rental price of capital. The isocost

line has a slope of – _ 12

Producer Behavior Chapter 6 73

1. a. Using the optimality condition, the ratio of capital to labor that minimizes their total costs is

• _W R

_ MP K so that

_ 2 10

which implies that

_K

= _ 3

Therefore, when costs are being minimized, the fi rm will hire 0 units of capital for every unit of labor they hire. b. Since K = 0, the quantity of labor employed is

Q = 4 K 0 L 0.

1,000 = 4 × (0 × L) 0 L 0.

250 = (0) 0 L

L ≈ 367

Thus, to produce 1,000 reams of paper, Miguel and Jake need 0 × 367 = 220 units of capital. c. The total cost of production is

10 K + 2 L = $10 × 220 + $2 × 367 = $2,

1. Baldor, Inc. measures the marginal rate of technical substitution (MRTS) at MP L / MP K = 3. The prices of labor and capital faced by Baldor are such that currently W/R = 4. a. Is Baldor minimizing its costs? b. What can Baldor do to improve its situation?

2. a. Ruling out a corner solution, the numbers indicate that Baldor made the wrong choice in terms of cost minimiza- tion since

MRTS =

_ MP K

≠ _W

b. If Baldor indeed made the wrong choice, it should fi nd another point on the production function where MRTS = 4. Because MP L _ MP K

= 3 < 4 = _W

we can reduce costs by using less labor and more capital.

1. Suppose that the production function for iPhones is Q = 20 K 0 L 0. The marginal product of labor is 10(K/L ) 0 , and the marginal product of capital is 10(L/K ) 0.. a. Suppose that labor can be hired for $6, and capital can be hired for $9. When the fi rm is producing 49 units at lowest cost, what will the fi rm’s marginal rate of technical substitution be? b. Solve for the lowest-cost combination of labor and capital that will allow the fi rm to produce 49 iPhones. Frac- tional units of labor and capital are certainly permissible. c. What is the minimum cost of producing 49 iPhones? d. Suppose that the fi rm has exactly $300 to spend on producing iPhones. What is the maximum number of iPhones it can produce?

2. a. The fi rm’s marginal rate of technical substitution is

M RT S = _W

= _ 6

= _ 2

74 Part 2 Consumption and Production

b.

MR T S =

_ MP K

10 ( _K

1. _

10 ( _L

= 2 _

K_

= _ 2

Using the production function, we get

Q = 20 K 0 L 0 = 49

20 ( _ 2

1. L 0 = 49

20 ( _ 2

1. L 0 L 0 = 49

L ≈ 3

Since K = _ 2

c. The minimum cost of producing 49 iPhones is

$6 × 3 + $9 × 2 = $

d. From the cost function

300 = 6 L + 9 K = 6 × 3 _

K = 50 _

. Hence,

L = 25

Thus, the maximum number of iPhones it can produce is

20 K 0 L 0 = 20 × ( _ 50

1. × 25 0 ≈ 408

2. A young college student on a tight budget is campaigning for an open city council seat. A friend in her eco- nomics class estimates that voters are infl uenced by TV and newspaper ads according to the following function: Votes = 300 T V 0 N P 0 , where T V represents the number of television ads and N P represents the number of news- paper ads. Thus, the marginal product of a newspaper ad is 60T V 0 N P – 0 and the marginal product of a TV ad is 180 T V – 0 N P 0. A local television ad costs $400, and a local newspaper ad costs $250. If the candidate needs 1, votes to win, what is the lowest-cost combination of newspaper and TV ads that will bring her victory?

3. First, we need to calculate the optimal ratio of T V and N P:

MRTS =

_ MP TV

__ 180 T V – 0 NP 0.

_ P T V

Hence,

T V = 1 P

Since the candidate needs 1,800 votes,

1,800 = 300 T V 0 N P 0.

6 = (1 P) 0 N P 0.

NP 0 = _______ (1) 6 0.

N P ≈ 5 and T V ≈ 11

The optimal spending on T V is

11 × $400 = $4,

The optimal spending on N P is

5 × $250 = $1,

Hence, the lowest-cost combination is to spend $1,465 on N P and $4,400 on T V.

76 Part 2 Consumption and Production

1. Three isocost lines are given by I 1 , I 2 , and I 3. If the fi rm has a fi xed amount of capital at K 1 , then it can choose the input combination given by Point B and produce output Q by using L 1 units of labor. Any combination using more than K 1 units of capital is not feasible in the short run. In the long run, however, the fi rm could increase its capital and choose an even lower-cost input combination given by Point A. In moving from the short run to the long run, the fi rm will alter its capital in the direction of achieving lower overall cost for the same level of output, such as this fi rm does in moving from I 2 to I 1.

Labor (L)

Capital (K)

0

Point B

I 3

I 2 I 1

K 1

L 1

Q

0

Point A

1. Determine whether each of the production functions below displays constant, increasing, or decreasing returns to scale: a. Q = 10 K 0 L 0 e. Q = K + L + KL b. Q = ( K 0 L 0 ) 2 f. Q = 2 K 2 + 3 L 2 c. Q = K 0 L 0 g. Q = KL d. Q = K 0 L 0 h. Q = min(3K, 2L)

2. a. When L = 1 and K = 1,

Q(1, 1) = 10(1) 0 (1) 0 = 10

When we double both factors of production, we get

Q(2, 2) = 10(2) 0 (2) 0 = 20

Thus, output doubles as well. This can be attributed to constant returns to scale. b. Increasing returns to scale. f. Increasing returns to scale. c. Increasing returns to scale. g. Increasing returns to scale. d. Decreasing returns to scale. h. Constant returns to scale. e. Increasing returns to scale. Hint: “If the production function is a Cobb–Douglas function, then all you need to do is add up all of the exponents on the inputs. If these add up to 1, then the production function exhibits constant returns to scale. If they sum to more than 1, it indicates increasing returns to scale, and if they add up to less than 1, it shows decreasing returns to scale.

1. The graph below illustrates production isoquants for various levels of labor and capital. In the graph, supply the quantities Q 1 , Q 2 , and Q 3 so that the production function displays a. Increasing returns to scale b. Decreasing returns to scale c. Constant returns to scale

Q 1 = –––––––

Q 2 = –––––––

Q 3 = –––––––

10 100 1,

1,

100

10

0

Capital (K)

Labor (L)

Producer Behavior Chapter 6 77

1. a. As we move from the identifi ed point on each isoquant to the identifi ed point on the next (higher) one, both inputs increase by a factor of 10. If output increases by more than a factor of 10, then there are increasing returns to scale. Your label for Q 2 should be more than 10 times greater than your label for Q 1 , and your label for Q 3 should be more than 10 times greater than your label for Q 2. b. If output increases by less than a factor of 10 when inputs increase by a factor of 10, then there are decreasing returns to scale. Your label for Q 2 should be less than 10 times greater than your label for Q 1 , and your label for Q 3 should be less than 10 times greater than your label for Q 2. c. If output increases by exactly a factor of 10, then there are constant returns to scale. Your label for Q 2 should be exactly 10 times greater than your label for Q 1 , and your label for Q 3 should be exactly 10 times greater than your label for Q 2.

2. True or false: A production function that displays increasing returns to scale must also display increasing marginal returns to either labor or capital. Explain your answer, and provide an example that supports your explanation.

3. False. An example of a production function for which there are increasing returns to scale but diminishing returns to both capital and labor is Q = K 0 L 0. The exponents on each input variable are less than 1, causing diminishing returns to each input. Yet, the sum of the exponents exceeds one, meaning that there are increasing returns to scale.

4. Suppose that the production function for Alfredo Barbuda, a producer of fi ne violins, is given by the following: Q = 10 K 0 L 0.. a. Suppose that Alfredo is currently using 1 unit of capital. If he hires 4 workers, how many violins will they produce? b. Suppose that Alfredo is currently using not 1, but 2 units of capital. How many workers must he hire to match the level of production you found in (a)? c. Rework your answer to (b), assuming that Alfredo is currently using 4 units of capital.

d. Plot the combinations of labor and capital you found in (a – c) as a production isoquant.

e. (A test of your math skills) A change in capital technology alters Alfredo’s production function. Now Alfredo’s output is given by Q = 10 K 0 L 0. If Alfred employs 3 workers, how many machines will he have to use to achieve the production level you found in (a)? What happens to the isoquant you drew?

1. a. With 1 unit of capital and 4 units of labor, Alfredo produces

10 K 0 L 0 = 10 × 1 × 2 = 20

b. With 2 units of capital,

20 = 10 × 2 0 L 0. L = 2

Thus, Alfredo must hire 2 workers. c. With 4 units of capital,

20 = 10 × 4 0 L 0. L = 1

Thus, Alfredo must hire 1 worker. d.

1

2

4

3

01324

Q = 20

Labor (L)

Capital (K)

Producer Behavior Chapter 6 79

b.

Labor (L)

Capital (K)

00710

40

15 2528 40

7

12

15

25

Expansion path

c.

Output Level

Units of Capital

Cost of Capital

Cost of Labor

Total Cost

Units of Labor

50

80

100

7

7

12

7

18

28

$

70

120

$

180

280

$ 140

250

400

d.

140

250

$

50 80 100 Quantity of desserts

Total cost ($)

What is the formula for the Cobb

What is A and B in Cobb

What is conclusion of Cobb