Review the Powerpoint Slides on Costs and Then Try the Following Problems

PIMS821

Spring 2012

Mike Conlin

Read Chapter 5, page 175 (start at the section entitled “The Cost Function”) thru page 186. Skip the section entitled “Algebraic Forms of Cost Functions”.

Review the powerpoint slides on costs and then try the following problems.

COST PROBLEMS TO BE WORKED ON BEFORE April 28th

(Solutions are at the end of the problem set)

1. You are the manager at Brueger bagels and must decide whether to open on Sunday. Your lease of the space costs you $6,000 a month. (Assume there are 30 days in the month.) If you open on Sunday, you would sell 1,000 bagels at a price of $.75 per bagel and 500 cups of coffee at a price of $1 per cup. The material (flour, onions, raisins, etc.) for each bagels costs you $.20 and the material for the coffee costs you $.30 per cup. To stay open on Sunday, you require 5 workers. Each worker costs you $100 per day. Assume the utilities used on a Sunday costs you $50. Should you open on Sunday? Show your calculations.

2. Read the article below entitled “Big Salt Producer Says Winter Storms Are Mixed Blessing” that appeared in the Wall Street Journal. Briefly discuss (two sentences and a graph will do) what it implies about the shape of Akzo’s short run cost curves. (Again, the idea is to relate it to the material discussed in class.)

AMSTERDAM -- The winter storms that battered parts of the U.S. have proved a mixed blessing for AkzoNV, the big Dutch chemical company. Akzo is the largest producer of salt products in the U.S., including road salts. And with all the snow and ice this winter, salt has been selling like mad. But at the same time the extreme weather has made it harder for Akzo to do business. "Yes, we're going to be more profitable because of the storms, but costs have also gone up," says Catherine Bolton, a spokeswoman for Akzo Salt in the U.S. Akzo doesn't break out division sales, but analysts say the U.S. salt operations are substantial. The U.S. accounts for about one-quarter of Akzo's annual worldwide sales of more than 16 billion guilders ($8.25 billion). Those sales totals exclude the company's recent acquisition of Nobel Industrier AB of Sweden. Ms. Bolton says Akzo has shipped a third more road de-icing salt than usual so far this year, a total of 3.5 million tons of rock salt since Jan. 1. "It has been an unprecedented winter, with record snowfall since Dec 15. We've had storm after storm after storm," she notes. But, she adds, "it costs more to transport the salt. And while we make more profit, we can't produce the salt as economically." To replenish depleted customer stockpiles, Akzo has been hustling to move salt "to the market as quickly as needed. It's been like fighting a war," Ms. Bolton says. To offset the problems of moving rock salt de-icers by road and rail, Akzo already has shipped a "couple thousand tons" of sea salt-based deicers via boats from its Caribbean unit at Bonaire. Akzo has three rock salt mines in the U.S., with the main mine in Retsof, N.Y. Akzo's U.S. salt operations got a big boost in 1987, when the company acquired Diamond Crystal Salt. Akzo is scheduled to report its 1993 net profit next week, with analysts' estimates roughly between 525 million and 600 million guilders after extraordinary charges of more than 100 million guilders. That would be down from 1992's net profit of 646 million guilders on sales of 16.9 billion guilders.

3. Below is a partially completed cost table for Wilson tennis racquets. Complete the table assuming that Wilson’s total fixed costs is $60.

At what output does Wilson begin to experience diminishing marginal returns? Explain.

4. The following is a partially completed cost table for Spartan Chassis’s production of mobile homes. Complete the table (for realism sake, assume the figures below are in thousand of dollars).

Number of

Mobile HomesTFC TVC TCMCAFC AVC ATC

0

1 50

8

220

3 30

4 84

16

5

6 15

50

7

5. The average variable cost (AVC) for a firm if it produces 30 units of a good is $15 and the firm’s total fixed cost (TFC) is $280. Suppose the marginal cost (MC) associated with the 30th unit is $30 and the marginal cost associated with the 29th unit is $28. Based on this information, what is the average total cost (ATC) for the firm if it produces 28 units?

6. There are 30 students in a class. The average grade for 29 of the students is 87. The grade of the remaining student is 84. Given this student’s grade, what is the average grade of the 30-student class? Explain how this relates to the relationship between marginal cost (MC), average total cost (ATC) and average variable cost (AVC).

7. Suppose Tub-A-Lub is a plastic swimming pool manufacturer and Tub-A-Lub can sell each swimming pool for $110 and has marginal costs indicated in the table below.

Quantity Marginal Cost

0

90

1

70

2

60

3

80

4

95

5

105

6

115

7

125

8

a) Suppose Tub-A-Lub’s total fixed cost is $150. How many swimming pools does Tub-A-Lub produce in the short run? What is Tub-A-Lub’s profits in the short run? Show calculations. What does Tub-A-Lub do in the long run?

b) Suppose Tub-A-Lub’s total fixed cost is $200. How many swimming pools does Tub-A-Lub produce in the short run? What is Tub-A-Lub’s profits in the short run? Show calculations. What does Tub-A-Lub do in the long run?

8. The graph below depicts the marginal cost (MC) and average variable cost (AVC) of a firm.

a) Suppose the firm maximizes profits by producing an output of 9. At this output, the firm’s total revenue is $146 and its profits are $20. If this is the case, what is the firm’s total fixed cost (TFC)? Show calculations.

b) If the firm’s total revenue increases from 100 to 110 by producing an output of 4 compared to an output of 3, how will the firm’s profits change from producing an output of 4 compared to an output of 3? Show calculations.

9. The graph below depicts the cost curves for a particular firm.

Suppose the firm is currently producing an output of 11 units and making total revenue of $200.

a)What are the firm’s total fixed costs? Show calculations.

b)What are the firm’s profits? Show calculations.

c) Suppose the firm is able to sell two additional units of output to a particular consumer for a price of $32 (or $16 for each of the additional units). Assume that the revenue from the first 11 units remains at $200 even if the firm does sell these two additional units of output. How much will the firm’s profits change if they do sell these two additional units? Show calculations.

SOLUTIONS

1. You are the manager at Brueger bagels and must decide whether to open on Sunday. Your lease of the space costs you $6,000 a month. (Assume there are 30 days in the month.) If you open on Sunday, you would sell 1,000 bagels at a price of $.75 per bagel and 500 cups of coffee at a price of $1 per cup. The material (flour, onions, raisins, etc.) for each bagel costs you $.20 and the material for the coffee costs you $.30 per cup. To stay open on Sunday, you require 5 workers. Each worker costs you $100 per day. Assume the utilities used on a Sunday costs you $50. Should you open on Sunday? Show your calculations.

The additional revenue (i.e. marginal or incremental benefit associated with opening) obtained from opening on Sunday is 1000(.75)+500(1)= $1250. The additional cost(i.e. marginal or incremental cost associated with opening) is 1000(.2)+500(.3) +5(100)+50=900. Because the additional revenue is greater than the additional cost, you should open on Sunday. Note that the cost of the lease does not impact your decision because you have to pay the lease whether or not you open on Sunday.

2. Read the article below entitled “Big Salt Producer Says Winter Storms Are Mixed Blessing” that appeared in the Wall Street Journal. Briefly discuss (two sentences and a graph will do) what it implies about the shape of Akzo’s short run cost curves. (Again, the idea is to relate it to the material discussed in class.)

AMSTERDAM -- The winter storms that battered parts of the U.S. have proved a mixed blessing for AkzoNV, the big Dutch chemical company. Akzo is the largest producer of salt products in the U.S., including road salts. And with all the snow and ice this winter, salt has been selling like mad. But at the same time the extreme weather has made it harder for Akzo to do business. "Yes, we're going to be more profitable because of the storms, but costs have also gone up," says Catherine Bolton, a spokeswoman for Akzo Salt in the U.S. Akzo doesn't break out division sales, but analysts say the U.S. salt operations are substantial. The U.S. accounts for about one-quarter of Akzo's annual worldwide sales of more than 16 billion guilders ($8.25 billion). Those sales totals exclude the company's recent acquisition of Nobel Industrier AB of Sweden. Ms. Bolton says Akzo has shipped a third more road de-icing salt than usual so far this year, a total of 3.5 million tons of rock salt since Jan. 1. "It has been an unprecedented winter, with record snowfall since Dec 15. We've had storm after storm after storm," she notes. But, she adds, "it costs more to transport the salt. And while we make more profit, we can't produce the salt as economically." To replenish depleted customer stockpiles, Akzo has been hustling to move salt "to the market as quickly as needed. It's been like fighting a war," Ms. Bolton says. To offset the problems of moving rock salt de-icers by road and rail, Akzo already has shipped a "couple thousand tons" of sea salt-based deicers via boats from its Caribbean unit at Bonaire. Akzo has three rock salt mines in the U.S., with the main mine in Retsof, N.Y. Akzo's U.S. salt operations got a big boost in 1987, when the company acquired Diamond Crystal Salt. Akzo is scheduled to report its 1993 net profit next week, with analysts' estimates roughly between 525 million and 600 million guilders after extraordinary charges of more than 100 million guilders. That would be down from 1992's net profit of 646 million guilders on sales of 16.9 billion guilders.

The article states that costs have gone up as the Akzo Salt increases it output. (The article appears to be referring to total costs.) This is not very interesting and doesn’t tell us much about Akzo’s short run cost curves. The article also states that “it costs more to transport the salt” and “to offset the problems of moving rock salt de-icers by road and rail, Azko already has shipped a couple thousand tons of sea salt-based deicers via boat”. This suggests that the marginal cost of transporting salt increases with output; as Akzo has increased output to 3.5 millions tons of rock salt this year.[This is somewhat of a subjective questions so there are numerous correct answers.]

MC

3.5 millionQ

3. Below is a partially completed cost table for Wilson tennis racquets. Complete the table assuming that Wilson’s total fixed costs is $60.

At what output does Wilson begin to experience diminishing marginal returns? Explain.

Wilson first experiences diminishing marginal returns when they increase output from 3 to 4 tennis racquets. This is when marginal cost begins to increase.

4. The following is a partially completed cost table for Spartan Chassis’s production of mobile homes. Complete the table (for realism sake, assume the figures below are in thousand of dollars).

Number of

Mobile HomesTFC TVC TCMCAFC AVC ATC

040 0 40 - - -

10

140 10 5040 10 50

8

240 18 5820 9 29

12

340 30 7013.3 10 23.3

14

440 44 8410 11 21

16

540 60 100 8 12 20

30

640 90 1306.7 15 21.7

50

740 140 1805.7 20 25.7

5. The average variable cost (AVC) for a firm if it produces 30 units of a good is $15 and the firm’s total fixed cost (TFC) is $280. Suppose the marginal cost (MC) associated with the 30th unit is $30 and the marginal cost associated with the 29th unit is $28. Based on this information, what is the average total cost (ATC) for the firm if it produces 28 units?

Total Variable Cost at an output of 30 is AVC*Q=15*30=450. If Total Fixed Cost is 280, then Total Cost at an output of 30 is 450+280=730. Because the MC associated with the 30th unit is $30, Total Cost must increase $30 if 30 units are produced compared to 29 units. Therefore, the Total Cost of producing an output of 29 is 730-30=700. Because the MC associated with the 29th unit is $28, Total Cost must increase $28 if 29 units are produced compared to 28 units. Therefore, the Total Cost (TC) of producing an output of 28 is 700-28=672 and the ATC at an output of 28 is TC/Q=672/28=24. Notice that ATC increases from 24 at an output of 28 to 24.14 at an output of 29. The reason is that the MC of $28 associated with the 29th unit is greater than the ATC at an output of 28 (which is 24). The same logic holds when output increases from 29 units to 30 units.

6. There are 30 students in a class. The average grade for 29 of the students is 87. The grade of the remaining student is 84. Given this student’s grade, what is the average grade of the 30-student class? Explain how this relates to the relationship between marginal cost (MC), average total cost (AVC) and average variable cost (AVC).

The total number of points for the 29 students is 29x87=2523. The total number of points for all 30 students is 2523+84=2607. Therefore, the average grade of the 30 students is 2607/30=86.9. The average went from 87 to 86.9 when the 30th person was included (the 30th person is the “marginal” person) because the 30th person’s grade of 84 is below the average of 87.

This relates to the MC, ATC and AVC because if the MC is less than the ATC, then the ATC is decreasing. If the MC is greater than the ATC, then the ATC is increasing. The same argument applies for the AVC.

7. Suppose Tub-A-Lub is a plastic swimming pool manufacturer and Tub-A-Lub can sell each swimming pool for $110 and has marginal costs indicated in the table below.

Quantity Marginal CostTVC

0 0

90

1 90

70

2 160

60

3 220

80

4 300

95

5 395

105

6 500

115

7 615

125

8 740

a) Suppose Tub-A-Lub’s total fixed cost is $150. How many swimming pools does Tub-A-Lub produce in the short run? What is Tub-A-Lub’s profits in the short run? Show calculations. What does Tub-A-Lub do in the long run?

The marginal revenue of a pool is $110. In the short run, Tub-A-Lub will produce 6 pools (they will produce the sixth because MC<MR and not produce the seventh because MC>MR).

TR=110*6=660 , TVC=90+70+60+80+95+105=500 , TFC=150

Profits=TR-TC=TR-TVC-TFC=660-500-150=+10

In the long run, Tub-A-Lub stays in business and continues to produce 6 because profits are positive assuming price and costs do not change and assuming Tub-A-Lub cannot achieve profits of more than $10 by changing the size of their plant. [I do not expect you to provide this precise answer for the long run because we have not really gone into detail about this. However, we will and I want you to have thought about it.]

b) Suppose Tub-A-Lub’s total fixed cost is $200. How swimming pools does Tub-A-Lub produce in the short run? What is Tub-A-Lub’s profits in the short run? Show calculations. What does Tub-A-Lub do in the long run?

The marginal revenue of a pool is still $110 and the change in fixed costs from $150 to $200 does not change marginal costs. Therefore, Tub-A-Lub will still produce 6 pools in the short run assuming that the fixed costs are all sunk. If Tub-A-Lub can produce .

TR=110*6=660 , TVC=90+70+60+80+95+105=500 , TFC=200

Profits=660-500-200=-40

In the long run, Tub-A-Lubeither exits the industry because profits are negative (assuming price and costs do not change)or changes the size of their plant if that allows them to achieve positive economic profits..[I do not expect you to provide this precise answer for the long run because we have not really gone into detail about this. However, we will and I want you to have thought about it.]

8. The graph below depicts the marginal cost (MC) and average variable cost (AVC) of a firm.

a) Suppose the firm maximizes profits by producing an output of 9. At this output, the firm’s total revenue is $146 and its profits are $20. If this is the case, what is the firm’s total fixed cost (TFC)? Show calculations.

If the firm’s total revenue is $146 and its profits are $20 at an output of 9, then its total cost at an output of 9 is 146-20=126. Therefore, the firm’s ATC is 14 (TC/Q=126/9=14) at an output of 9. The firm’s AFC is ATC-AVC=14-10=4 at an output of 9. The firm’s TFC must then be AFC(q)=4(9)=36.

b) If the firm’s total revenue increases from 100 to 110 by producing an output of 4 compared to an output of 3, how will the firm’s profits change from producing an output of 4 compared to an output of 3? Show calculations.

If revenue increases by 110-100=10 and total costs increase by 6 (just the marginal cost of the 4th unit of output) from producing the 4th unit of output, profits must increase by 10-6=4.

9. The graph below depicts the cost curves for a particular firm.

Suppose the firm is currently producing an output of 11 units and making total revenue of $200.

a)What are the firm’s total fixed costs? Show calculations.

Select a q- say q=7. (You can select any q and obtain the correct answer.) AFC=ATC-AVC=10-6=4 at q=7. So TFC=AFC*q=4*7=28.

b)What are the firm’s profits? Show calculations.

TR-TC=TR-ATC*q=200-11*12=68.

c) Suppose the firm is able to sell two additional units of output to a particular consumer for a price of $32 (or $16 for each of the additional units). Assume that the revenue from the first 11 units remains at $200 even if the firm does sell these two additional units of output. How much will the firm’s profits change if they do sell these two additional units? Show calculations.

TR increases by 32. MC of 12th unit is 21 and MC of 13th unit is 23. So TC increases by 44. Therefore, profits change by 32-44=-12.