THE NATURE OF COSTS

P 2-1: Solution to Darien Industries (10 minutes)

[Relevant costs and benefits]

Current cafeteria income

Sales $12,000

Variable costs (40% × 12,000) (4,800)

Fixed costs (4,700)

Operating income $2,500

Vending machine income

Sales (12,000 × 1.4) $16,800

Darien’s share of sales

(.16 × $16,800) 2,688

Increase in operating income $ 188

P 2-2: Negative Opportunity Costs (10 minutes)

[Opportunity cost]

Yes, when the most valuable alternative to a decision is a net cash outflow that would have occurred is now eliminated. The opportunity cost of that decision is negative (an opportunity benefit). For example, suppose you own a house with an in-ground swimming pool you no longer use or want. To dig up the pool and fill in the hole costs $3,000. You sell the house instead and the new owner wants the pool. By selling the house, you avoid removing the pool and you save $3,000. The decision to sell the house includes an opportunity benefit (a negative opportunity cost) of $3,000.

P 2-3: Solution to NPR (10 minutes)

[Opportunity cost of radio listeners]

The quoted passage ignores the opportunity cost of listeners’ having to forego normal programming for on-air pledges. While such fundraising campaigns may have a low out-of-pocket cost to NPR, if they were to consider the listeners’ opportunity cost, such campaigns may be quite costly.

P 2-4: Solution to Silky Smooth Lotions (15 minutes)

[Break even with multiple products]

Given that current production and sales are: 2,000, 4,000, and 1,000 cases of 4, 8, and 12 ounce bottles, construct of lotion bundle to consist of 2 cases of 4 ounce bottles, 4 cases of 8 ounce bottles, and 1 case of 12 ounce bottles. The following table calculates the break-even number of lotion bundles to break even and hence the number of cases of each of the three products required to break even.

Per Case

4 ounce

8 ounce

12 ounce

Bundle

Price

$36.00

$66.00

$72.00

Variable cost

$13.00

$24.50

$27.00

Contribution margin

$23.00

$41.50

$45.00

Current production

2000

4000

1000

Cases per bundle

Contribution margin per bundle

$46.00

$166.00

$45.00

$257.00

Fixed costs

$771,000

Number of bundles to break even

3000

Number of cases to break even

6000

12000

3000

P 2-5: Solution to J. P. Max Department Stores (15 minutes)

[Opportunity cost of retail space]

Home Appliances
Televisions
Profits after fixed cost allocations
$64,000
$82,000
Allocated fixed costs
7,000
8,400
Profits before fixed cost allocations
71,000
90,400
Lease Payments
72,000
86,400
Forgone Profits
– $1,000
$ 4,000

We would rent out the Home Appliance department, as lease rental receipts are more than the profits in the Home Appliance Department. On the other hand, profits generated by the Television Department are more than the lease rentals if leased out, so we continue running the TV Department. However, neither is being charged inventory holding costs, which could easily change the decision.

Also, one should examine externalities. What kind of merchandise is being sold in the leased store and will this increase or decrease overall traffic and hence sales in the other departments?

P 2-6: Solution to Vintage Cellars (15 minutes)

[Average versus marginal cost]

a. The following tabulates total, marginal and average cost.

Quantity

Average Cost

Total Cost

Marginal Cost

$12,000

10,000

20,000

$8,000

8,600

25,800

5,800

7,700

30,800

5,000

7,100

35,500

4,700

7,100

42,600

7,100

7,350

51,450

8,850

7,850

62,800

11,350

8,600

77,400

14,600

9,600

96,000

18,600

b. Marginal cost intersects average cost at minimum average cost (MC=AC=$7,100). Or, at between 5 and 6 units AC = MC = $7,100.

c. At four units, the opportunity cost of producing and selling one more unit is $4,700. At four units, total cost is $30,800. At five units, total cost rises to $35,500. The incremental cost (i.e., the opportunity cost) of producing the fifth unit is $4,700.

d. Vintage Cellars maximizes profits ($) by producing and selling seven units.

Quantity

Average Cost

Total

Cost

Total

Revenue

Profit

$12,000

$9,000

-$3,000

10,000

20,000

18,000

-2,000

8,600

25,800

27,000

1,200

7,700

30,800

36,000

5,200

7,100

35,500

45,000

9,500

7,100

42,600

54,000

11,400

7,350

51,450

63,000

11,550

7,850

62,800

72,000

9,200

8,600

77,400

81,000

3,600

9,600

96,000

90,000

-6,000

P 2-7: Solution to ETB (15 minutes)

[Minimizing average cost does not maximize profits]

a. The following table calculates that the average cost of the iPad bamboo case is minimized by producing 4,500 cases per month.

Monthly Production and Sales
Production (units)

3,000

3,500

4,500

5,000

Total cost

$162,100

$163,000

$167,500

$195,000

Average cost

$54.03

$46.57

$37.22

$39.00

b. The following table calculates net income of the four production (sales) levels.

Monthly Production and Sales
Production (units)
3,000
3,500
4,500
5,000

Revenue
$195,000
$227,500
$292,500
$325,000
Total cost
162,100
163,000
167,500
195,000
Net income
$32,900
$64,500
$125,000
$130,000

Based on the above analysis, the profit maximizing production (sales) level is to manufacture and sell 5,000 iPad cases a month. Selecting the output level that minimizes average cost (4,500 cases) does not maximize profits.

P 2-8: Solution to Taylor Chemicals (15 minutes)

[Relation between average, marginal, and total cost]

a. Marginal cost is the cost of the next unit. So, producing two cases costs an additional $400, whereas to go from producing two cases to producing three cases costs an additional $325, and so forth. So, to compute the total cost of producing say five cases you sum the marginal costs of 1, 2, …, 5 cases and add the fixed costs ($500 + $400 + $325 + $275 + $325 + $1000 = $2825). The following table computes average and total cost given fixed cost and marginal cost.

Quantity

Marginal

Cost

Fixed

Cost

Total

Cost

Average

Cost

$500

$1000

$1500

$1500.00

400

1000

1900

950.00

325

1000

2225

741.67

275

1000

2500

625.00

325

1000

2825

565.00

400

1000

3225

537.50

500

1000

3725

532.14

625

1000

4350

543.75

775

1000

5125

569.44

950

1000

6075

607.50

b. Average cost is minimized when seven cases are produced. At seven cases, average cost is $532.14.

c. Marginal cost always intersects average cost at minimum average cost. If marginal cost is above average cost, average cost is increasing. Likewise, when marginal cost is below average cost, average cost is falling. When marginal cost equals average cost, average cost is neither rising nor falling. This only occurs when average cost is at its lowest level (or at its maximum).

P 2-9: Solution to Emrich Processing (15 minutes)

[Negative opportunity costs]

Opportunity costs are usually positive. In this case, opportunity costs are negative (opportunity benefits) because the firm can avoid disposal costs if they accept the rush job.

The original $1,000 price paid for GX-100 is a sunk cost. The opportunity cost of GX-100 is -$400. That is, Emrich will increase its cash flows by $400 by accepting the rush order because it will avoid having to dispose of the remaining GX-100 by paying Environ the $400 disposal fee.

How to price the special order is another question. Just because the $400 disposal fee was built into the previous job does not mean it is irrelevant in pricing this job. Clearly, one factor to consider in pricing this job is the reservation price of the customer proposing the rush order. The $400 disposal fee enters the pricing decision in the following way: Emrich should be prepared to pay up to $399 less any out-of-pocket costs to get this contract.

P 2-10: Solution to Verdi Opera or Madonna? (15 minutes)

[Opportunity cost of attending a Madonna concert]

If you attend the Verdi opera, you forego the $200 in benefits (i.e., your willingness to pay) you would have received from going to see Madonna. You also save the $160 (the costs) you would have paid to see Madonna. Since an avoided benefit is a cost and an avoided cost is a benefit, the opportunity cost of attending the opera (the value you forego by not attending the Madonna concert) is $40 – i.e., the net benefit foregone. Your willingness to pay $30 for the Verdi opera is unrelated to the costs and benefits of foregoing the Madonna concert.

P 2-11: Solution to Dod Electronics (15 minutes)

[Estimating marginal cost from average cost]

Dod should accept Xtron’s offer. The marginal cost to produce the 10,000 chips is unknown. But since management is convinced that average cost is falling, this means that marginal cost is less than average cost. The only way that average cost of $35 can fall is if marginal cost is less than $35. Since Xtron is willing to pay $38 per chip, Dod should make at least $30,000 on this special order (10,000 x $3). This assumes (i) that average cost continues to fall for the next 10,000 units (i.e., it assumes that at, say 61,000 units, average cost does not start to increase), and (ii) there are no other costs of taking this special order.
Dod can’t make a decision based on the information. Since average cost is increasing, we know that marginal cost is greater than $35 per unit. But we don’t know how much larger. If marginal cost at the 60,001th unit is $35.01, average cost is increasing and if marginal cost of the 70,000th unit is less than $38, then DOD should accept the special order. But if marginal cost at the 60,001th unit is $38.01, the special order should be rejected.

P 2-12: Solution to Napoli Pizzeria (15 minutes)

[Break-even analysis]

The break-even number of servings per month is:

($300 – $75) ÷ ($3 – $1)

= ($225) ÷ ($2)

= 112.5 servings

b. To generate $1,000 after taxes Gino needs to sell 881.73 servings of espresso/cappuccino.

Profits after tax = [Revenues – Expenses] x (1– 0.35)

$1,000 = [$3N + $75 – $1N – $300] x (1– 0.35)

$1,000 = [$2N – $225] x .65

$1,000 ÷ .65 = $2N – $225

$1,538.46 = $2N – $225

$2N = $1,763.46

N = 881.73

P 2-13: Solution to JLT Systems (20 minutes)

[Cost-volume-profit analysis]

Since we know that average cost is $2,700 at 200 unit sales, then Total Cost (TC) divided by 200 is $2,700. Also, since JLT has a linear cost curve, we can write, TC=FC+VxQ where FC is fixed cost, V is variable cost per unit, and Q is quantity sold and installed. Given FC = $400,000, then:
TC/Q = (FC+VxQ)/Q = AC
($400,000 + 200 V) / 200 = $2,700
$400,000 + 200 V = $540,000
200 V = $140,000
V = $700
Given the total cost curve from part a, a tax rate of 40%, and a $2,000 selling price, and an after-tax profit target of $18,000, we can write:

($2000 Q – $400,000 – $700 Q) x (1- 40%) = $18,000

1300 Q -400,000 = 18,000 / .60 = 30,000

1300 Q = 430,000

Q = 330.8

In other words, to make an after-tax profit of $18,000, JLT must have 330.8 sales and installs per month.

The same solution is obtained if you set marginal revenue (where MR is 2600 – 4Q) equal to marginal cost (700), and again solve for Q, or

Q = 475

4Q = 1900

First derivative: 2600 – 4Q -700 = 0

Total Profit = (2600 – 2Q) Q -400,000 -700 Q
- The simplest (and fastest way) to solve for the profit maximizing quantity given the demand curve is to write the profit equation, take the first derivative, set it to zero, and solve for Q.
- 2600 – 4Q = 700
- Q = 475
The more laborious solution technique is to use a spreadsheet and identify the profit maximizing price quantity combination.
- As before, we again observe that 475 sales and installs maximize profits.
Miles per year = 1,250 × 12 = 15,000 miles per year

M = $100/.08 = 1,250 miles per month

$100 = M(.16 – .08)

b.

Miles per year = 1,666.66 × 12 = 20,000

M = $100/.06 = 1,666.66 miles per month

$100 = M(.12 – .06)

a. The $1,500 upfront payment is irrelevant since it applies to both alternatives. To find the break-even mileage, M, set the monthly cost of both vehicles equal:

  [Break-even analysis]

P 2-18: Solution to Affording a Hybrid (20 minutes)

F = $192,000

F = 24,000 x 20 – 288,000

Substituting P = $20 back into eq. (1) from above yields:

P = $20

408,000 =30,000 P – 24,000 P + 288,000

48,000 =30,000 P – 360,000 – (24,000 P – 288,000)

Substituting in eq. (1) from above yields:

48,000 =30,000 P – 360,000 – F

33,600 = 0.70 (30,000 P – 30,000 x 12 – F)

33,600 = (1 – 0.30) (30,000 P – 30,000 V – F)

Where T = tax rate = 0.30

Profits after tax = (1-T) (P Q – V Q – Fixed Cost)

From the after tax data we can write down the following equation:

F = 24,000 P – 288,000 (1)

24,000 = F / (P – 12)

Substituting the data into this equation yields

V = variable cost per unit

where: P = price per unit

Break-even Q = Fixed Costs / (P – V)

The formula for the break-even quantity is

[Finding unknown quantities in cost-volume-profit analysis]

P 2-17: Solution to Stahl Inc. (25 minutes)

(iii) One of the simplifying assumptions made early in the problem was that the sale of the special display items did not affect the unit sales of competitive items in the store. Suppose that some of the Texcan oil sales came at the expense of other oil sales in the store. Discuss how this would alter the analysis.

(ii) This problem also illustrates that retail stores track contribution margins and volumes very closely in deciding which items to stock and where to display them.

(i) This problem introduces the concept of the opportunity cost of retail shelf space. With the proliferation of consumer products, supermarkets’ valuable scarce commodity is shelf space. Consumers often learn about a product for the first time by seeing it on the grocery shelf. To induce the store to stock an item, food companies often give the store a number of free cases. Such a giveaway compensates the store for allocating scarce shelf space to the item.

Additional discussion points raised

With 50 free units of car wax, it is now profitable to replace the oil display area with the car wax. The opportunity cost of replacing the oil display is its forgone contribution ($350), whereas the benefits provided by the car wax are $445.

Contribution $445

× expected volume    750 300

Contribution margin $0.40

less: Unit cost $2.50

Selling price $2.90

Contribution from remaining 750 units:

Contribution from 50 free units (50 × $2.90) $145

b. With 50 free units of car wax, Armadillo’s contribution is:

Clearly, since the Armadillo car wax yields a lower contribution margin than all three of the existing planned promotions, management should not change their planned promotions and should reject the Armadillo offer.

Contribution $ 320

× expected volume 800

Contribution margin $0.40

less: Unit cost $2.50

Selling price $2.90

Texcan oil is the promotion yielding the lowest contribution and therefore is the one Armadillo must beat out. The contribution of Armadillo car wax is:

Planned Promotion Displays
For Next Week

End-of-
Aisle

Front
Door

Cash
Register

Item

Texcan Oil

Wiper blades

Floor mats

Projected volume (week)

5,000

200

70

Sales price

69¢/can

$9.99

$22.99

Unit cost

62¢

$7.99

$17.49

Contribution margin

7¢

$2.00

$5.50

Contribution
(margin × volume)

$350

$400

$385

a. The question involves computing the opportunity cost of the special promotions being considered. If the car wax is substituted, what is the forgone profit from the dropped promotion? And which special promotion is dropped? Answering this question involves calculating the contribution of each planned promotion. The opportunity cost of dropping a planned promotion is its forgone contribution: (retail price less unit cost) × volume. The table below calculates the expected contribution of each of the three planned promotions.

[Opportunity cost of retail display space]

P 2-16: Solution to Home Auto Parts (20 minutes)

Now, ticket sales in the first two weeks need only be about 25 percent higher than in weeks three and four to replace “Paris” with “I Do.”

Q1 > 1.245Q2

2.65Q1 > 3.3Q2

p1 > p2 if

p2 = 1.30Q2 + 2Q2 – 2,000

p1 = .65Q1 + 2Q1 – 2,000

c. With average concession profits of $2 per ticket sold,

b. Taxes of 30 percent do not affect the answer in part (a).

In other words, they should keep “Paris” for four weeks unless they expect ticket sales in weeks one and two of “I Do” to be twice the expected ticket sales in weeks three and four of “Paris.”

Q1 > 2Q2.

.65Q1 – 2,000 > 1.3Q2 – 2,000, or

p1 > p2, or

“I Do” should replace “Paris” if

p2 = .2(6.5Q2) – $2,000

p1 = .1(6.5Q1) – $2,000

Let Q1 be the number of tickets sold in the first two weeks, and Q2 be the number of tickets sold in weeks three and four. Then, profits in the first two weeks, p1, and in weeks three and four, p2, are:

a. Both movies are expected to have the same ticket sales in weeks one and two, and lower sales in weeks three and four.

[Break-even analysis for an operating decision]

P 2-15: Solution to American Cinema (20 minutes)

Alternatively, suppose price is $3, variable cost is $3, and fixed cost is $50. Contribution margin in this case is zero. Doubling output from 100 to 200 causes average cost to fall from $3.50 ([100 × $3 + $50]÷100) to $3.25 ([200 × $3 + $50]÷200), but profits are still zero.

Profits will increase with volume even if the firm has no fixed costs, as long as price is greater than variable costs. Suppose price is $3 and variable cost is $1. If there are no fixed costs, profits increase $2 for every unit produced. Now suppose fixed cost is $50. Volume increases from 100 units to 101 units. Profits increase from $150 ($2 ×100 – $50) to $152 ($2 × 101 – $50). The change in profits ($2) is the contribution margin. It is true that average unit cost declines from $1.50 ([100 × $1 + $50]÷100) to $1.495 ([101 × $1 + $50]÷101). However, this has nothing to do with the increase in profits. The increase in profits is due solely to the fact that the contribution margin is positive.

Notice that average fixed costs per unit (FC÷Q) falls as Q increases, but with more volume, you have more fixed cost per unit such that (FC÷Q) × Q = FC. That is, the decline in average fixed cost per unit is exactly offset by having more units.

= Q(P – VC) – (FC ÷ Q)Q

Profits = P × Q – (FC – VC × Q) = Q(P – VC) – FC

b. Write the equation for firm profits:

a. False.

[Cost-volume-profit]

P 2-14: Solution to Volume and Profits (15 minutes)