# Amazon’s fulfillment center in Salem can sort and pack boxes. The boxes can be produced using only robots (K), workers (L), or a combination of the two. K and L are perfect substitutes for one another. Assume Amazon has TC = \$100, K costs \$10/robot and L costs \$20/workers. How much K & L will Amazon use? Draw isoquants and an isocost line as part of your answer.

1. If the law of diminishing returns did not apply, how would that change the total production of a product like wheat?

2. Given the cost information listed in the table below:

a. calculate total cost and then, on graph paper, plot the total fixed, total variable, and total cost schedules.

b. is this cost information for the short or long run; how can you tell?

c. using the costs provided in the table, calculate average fixed, average variable, average total, and marginal cost.

d. graph your answers to part c. (do not put them on the same graph used for part a).

e. at approximately what level of output does marginal cost begin to increase?

f. why does marginal cost begin to increase at the output level you identified in part e?

Q

TFC (\$)

TVC (\$)

TC (\$)

0

120

0

1

120

60

2

120

80

3

120

90

4

120

105

5

120

140

6

120

210

1. Suppose a firm’s technology is described by a production function where the quantity produced is equal to the square root of capital times labor (that is, Q = √(L●K)

a. Using graph paper, draw the isoquant for three different levels of output: 2, 4, and 6 units (hint: you will need to find different combinations of L and K that will make Q equal these output values. For example, both L=1, K=4 and L=2, K=2 make Q=2, so both of these combinations of L and K are on the Q=2 isoquant.)

b. Suppose the price of L and K is \$3/hour. On your graph, show isocost lines corresponding to total costs of \$12, \$24, and \$36. Using the isoquants and isocost lines, locate three points on the expansion path and draw the expansion path.

c. Does this production function exhibit increasing, decreasing, or constant returns to scale?

d. Using the three points you found on the expansion path, calculate the firm’s long run total and average costs at each of those points. Summarize your calculations in a table and in a graph.

e. Suppose amount of capital the firm uses in the short run is fixed at 4 units. On your isoquant/isocost diagram illustrate the three points corresponding to output levels of 2, 4, and 6. Calculate the firm’s short run total and average costs and draw them in the same graph with the long run costs.

f. Why do the short run and long run average cost curves have different shapes?

1. Suppose a firm’s technology is described by a production function where the quantity produced is equal to the square root of capital times labor (that is, Q = √(L●K)

a. Using graph paper, draw the isoquant for output = 8 units.

b. Suppose the firm has \$128 to spend on its variable inputs and the price of L and K is \$8. On the same graph on which you drew the isoquant, draw the isocost line. What is the optimal combination of labor and capital?

c. Suppose the price of labor increases to \$16 and the price of capital decreases to \$4. draw the new isocost line and determine the new optimal combination of labor and capital.

d. Repeat c) for price of labor at \$4 and capital at \$16.

1. Fill in the cost information missing in the table below

Q

FC

VC

TC

MC

AFC

AVC

ATC

1

428.5

2

340

67.5

248

3

68.5

4

240

5

22.5

6

616

46

7

623.5

40.5

8

78.5

9

292.5

70.28

10

300

640

30

1. Suppose a firm can spend up to \$1000 acquiring two variable inputs, labor (L) and capital (K). The price of each unit of labor is \$25. The price of each unit of capital is \$100.

a. draw the firm’s isocost line.

b. suppose the firm had \$2000; draw the new isocost line.

c. suppose the firm had \$500; draw the new isocost line.

1. What is does the term “marginal rate of technical substitution” refer to? Illustrate.

2. Why is a firm’s marginal cost unaffected by an increase in fixed costs?

3. This table illustrates a __________ production function because capital and labor _________.

Units of Capital, K

Units of Labor, L

1

2

3

4

1

50.0

66.7

75.5

80.0

2

66.7

100.0

120.0

133.3

3

75.0

120.0

150.0

171.4

4

80.0

133.3

171.4

200.0

1. Suppose the wage rate is \$25 per hour and the rent on capital is \$50 per hour. The equation for the isocost line is:

2. Suppose a firm is producing 2,475 units of output by hiring 50 workers (W = \$20 per hour) and 25 units of capital (R = \$10 per hour). The marginal product of labor and marginal product of capital are 40 and 25, respectively. Is the firm minimizing the cost of producing 2,475 units of output? If not, how should the firm change its usage of L and K?

3. Suppose a firm’s short-run production function is given by Q = 16L0.8. What is the marginal product of the fourth worker?

4. Amazon’s fulfillment center in Salem can sort and pack boxes. The boxes can be produced using only robots (K), workers (L), or a combination of the two. K and L are perfect substitutes for one another. Assume Amazon has TC = \$100, K costs \$10/robot and L costs \$20/workers. How much K & L will Amazon use? Draw isoquants and an isocost line as part of your answer.

5. On separate graphs, draw a SR total product curve, a SR marginal product curve, a SR total cost curve, and a SR marginal cost curve. On the graphs show what economic concept they have in common.

6. What is the marginal-average rule?

7. The firm’s long-run total cost is given by LTC = 100Q – 10Q2 + (1/3)Q3, and long-run marginal cost is given by LMC = 100 – 20Q + Q2. At what output level does the firm have economies of scale?

8. Producing 200 units of good Y and 100 units of good X in the same factory costs the firm \$50,000. In contrast, producing 200 units of good Y in one factory and 100 units of good X in another factory costs the firm \$75,000. So if the firm produces the two goods together, it achieves:

9. The total cost of producing 7 units of output is: image1.png

10. image2.png The firm’s average total cost curve appears in which of the following panels? image3.png

11. Suppose a firm’s total cost is given by TC = 100 + 4Q + 2Q2. Write the expression or equation for TFC, TVC, ATC, AFC, AVC, and MC.