Hirschey 12E Chapter 8

Chapter 8 COST ANALYSIS AND ESTIMATION

QUESTIONS AND ANSWERS Q8.1 What advantages or disadvantages do you see in using current costs for tax and

stockholder reporting purposes? Q8.1 ANSWER

Theoretically, it would be preferable to use current costs for income tax calculations and stockholder reporting. On a practical level, however, this would be nearly impossible. Estimation of current cost, based upon current market values, would be a difficult task with a great deal of room for subjectivity. This could result in many arbitrary cost designations, and the "policing" of tax returns would become a much more formidable task. On a practical basis, the use of historical costs for tax and stockholder reporting purposes has obvious advantages over the theoretically superior current costs.

Q8.2 Assume that two years ago, you purchased a new Jeep Wrangler SE 4WD with a soft top

for $16,500 using five-year interest-free financing. Today, the remaining loan balance is $9,900 and your Jeep has a trade-in value of $9,500. What is your opportunity cost of continuing to drive the Jeep? Discuss the financing risk exposure of the lender.

Q8.2 ANSWER

The opportunity cost of continuing to drive the Jeep is $9,500. If you sell the Jeep, $9,500 can be generated to pay down your remaining loan balance. It is the current cost or replacement value of your current vehicle. It is the relevant economic cost of continuing to drive the Jeep. Historical cost of $16,500 and the remaining loan balance of $9,900 are irrelevant for decision-making purposes. With a current market value of only $9,500 against a remaining loan balance of $9,900, the lender faces the risk of borrower default. Aggressive interest-free financing offered by the major automakers has the potential to create big debt collection problems in the future.

Q8.3 Southwest Airlines offers four flights per weekday from Cleveland, Ohio to Tucson,

Arizona. If adding a fifth flight per weekday would cost $15,000 per flight, or $110 per available seat, calculate the incremental costs borne by Southwest following a decision to go ahead with a fifth flight per day for a minimal 60-flight trial period. What is the marginal cost? In this case, is incremental cost or marginal cost relevant for decision making purposes?

Q8.3 ANSWER

Cost Analysis and Estimation 203

Marginal costs are the cost effect of one-unit changes in output. Incremental cost is the cost effect associated with a given managerial decision. Incremental costs may also relate to output changes, but the output change involved is that of a relevant block or increment of service. In this instance, the incremental cost associated with a decision to go ahead with a fifth flight per day for a minimal 60-flight trial period is $900,000 (= $15,000 60). The marginal cost per passenger is only $110. In this case, the incremental cost of $900,000 is the relevant cost for decision making purposes. With expected revenues in excess of $900,000, Southwest should go ahead with the planned expansion.

Q8.4 Suppose the Big Enchilada restaurant has been offered a binding one-year lease

agreement on an attractive site for $5,200 per month. Before the lease agreement has been signed, what is the incremental cost per month of site rental? After the lease agreement has been signed, what is the incremental cost per month of site rental? Explain.

Q8.4 ANSWER

Any cost that is invariant across decision alternatives is a sunk cost. Sunk costs are irrelevant for current decision-making purposes and should not enter into decision analysis. Before the lease agreement has been signed, all costs are variable, and the incremental cost per month of site rental is $5,200 per month. After the lease agreement has been signed, lease costs are sunk, and the incremental cost per month of site rental is $0.

Q8.5 What is the relation between production functions and cost functions? Be sure to

include in your discussion the effect of competitive conditions in input factor markets. Q8.5 ANSWER

There is a direct relation between production and cost functions. A cost function is determined by combining a given production function with the related price functions for the inputs actually employed in production. If inputs are purchased in competitive markets so that their prices are constant irrespective of how many are purchased, the relation between production and cost functions is straightforward (see Figures 8.2, 8.3, and 8.4). With imperfect competition in input markets, the relation becomes somewhat more complex. In all cases, cost/production relations can be employed either to minimize total costs subject to an output constraint or to maximize output subject to a total cost or budget constraint.

Q8.6 The definition of point output elasticity is Q = Q/Q X/X (X represents all inputs),

whereas the definition of point cost elasticity is C = C/C Q/Q. Explain why Q > 1 indicates increasing returns to scale, whereas C < 1 indicates economies of scale.

204 Chapter 8 Q8.6 ANSWER

The definition of output elasticity is Q = Q/Q X/X (X represents all inputs), whereas the definition of cost elasticity is C = C/C Q/Q. Therefore:

If

Then

Implies

Q > 1 Q/Q > X/X

Rising Q/X ratio, falling AC.

C < 1 C/C < Q/Q

Falling C/Q ratio, falling AC.

This means that Q > 1 and C < 1 are both consistent with falling average costs. Q8.7 The president of a small firm has been complaining to the controller about rising labor

and material costs. However, the controller notes that average costs have not increased during the past year. Is this possible?

Q8.7 ANSWER

Yes, the phenomenon of constant (or even decreasing) average costs coupled with increasing input prices is quite feasible. It stems from an increase in input productivity that could result from any number of causes. One obvious possibility would be the introduction of new capital equipment, either replacement or expansion, into the production system.

Q8.8 With traditional medical insurance plans, workers pay a premium that is taken out of

each paycheck and must meet an annual deductible of a few hundred dollars. After that, insurance picks up most of their health-care costs. Companies complain that this gives workers little incentive to help control medical insurance costs, and those costs are spinning out of control. Can you suggest ways of giving workers better incentives to control employer medical insurance costs?

Q8.8 ANSWER

In hopes of slowing the growth in medical costs, some companies are moving towards consumer driven medical coverage that gives employees a financial stake in what they pay for medical care. Such plans feature high deductibles of as much as $500 per year for prescriptions and $1,000 per year for all other medical costs. To help pay these costs, some companies deposit $300 to $1,800 per year in an employee benefits account. At some firms, like Whole Foods Market, Inc., medical claims fell 13% in the first year such a plan was adopted, and about 90% of employees had money left over to use next year.


The Whole Foods plan, which workers themselves chose over two competing plans after a series of votes last summer, has no premiums at all for many workers. But the deductible is a relatively hefty $1,500. Whole Foods each year puts money into an account for each worker to use for health-care expenses. If employees don't spend their money in one year, they get to carry it over to future years. After the deductible is reached, the plan operates more like a traditional one, picking up 80% of most medical expenses. The hope is that once the money feels as though it belongs to them, people won't get an MRI when an X-ray (or an ice pack) might do. Already at Whole Foods, the plan is inducing the company's butchers, bakers and baggers to take responsibility for cutting costs by buying generic drugs, asking for fee waivers on lab tests and other procedures, and keeping a closer eye on what doctors charge for their services.

The plans have one big drawback: People with chronic conditions can take a big hit, since they have little choice about how often they go to the doctor. Some critics fear that the plans will discourage people from getting the care they need. (See: Ron Lieber, New Way to Curb Medical Costs: Make Employees Feel the Sting, The Wall Street Journal Online, June 23, 2004. (http://online.wsj.com)

Q8.9 Will firms in industries in which high levels of output are necessary for minimum

efficient scale tend to have substantial degrees of operating leverage? Q8.9 ANSWER

Yes, in industries where the minimum efficient scale is large, long-run average costs tend to decrease rapidly as output increases. Fixed costs tend to be a substantial share of total costs. When fixed costs are large, the degree of operating leverage also tends to be high, and firms with high levels of output necessary for minimum efficient scale will tend to have a substantial degree of operating leverage.

Q8.10 Do operating strategies of average cost minimization and profit maximization always

lead to identical levels of output? Q8.10 ANSWER

No, operating strategies of average cost minimization and profit maximization lead to identical rates of input combination, but do not typically lead to identical levels of total output. Average cost minimization is an appropriate strategy when managers wish to produce a target level of output in an optimal or least-cost fashion. On the other hand, profit maximization implies production of an optimal level of output, as revealed by product demand, in an optimal or least-cost fashion.

SELF-TEST PROBLEMS AND SOLUTIONS

206 Chapter 8 ST8.1 Learning Curves. Suppose Modern Merchandise, Inc., makes and markets do-it-

yourself hardware, housewares, and industrial products. The company's new Aperture Miniblind is winning customers by virtue of its high quality and quick order turnaround time. The product also benefits because its price point bridges the gap between ready-made vinyl blinds and their high-priced custom counterpart. In addition, the company's expanding product line is sure to benefit from cross-selling across different lines. Given the success of the Aperture Miniblind product, Modern Merchandise plans to open a new production facility near Beaufort, South Carolina. Based on information provided by its chief financial officer, the company estimates fixed costs for this product of $50,000 per year and average variable costs of:

AVC = $0.5 + $0.0025Q,

where AVC is average variable cost (in dollars) and Q is output.

A. Estimate total cost and average total cost for the projected first-year volume of 20,000 units.

B. An increase in worker productivity because of greater experience or learning

during the course of the year resulted in a substantial cost saving for the company. Estimate the effect of learning on average total cost if actual second-year total cost was $848,000 at an actual volume of 20,000 units.

ST8.1 SOLUTION A. The total variable cost function for the first year is: TVC = AVC Q = ($0.5 + $0.0025Q)Q = $0.5Q + $0.0025Q2

At a volume of 20,000 units, estimated total cost is: TC = TFC + TVC = $50,000 + $0.5Q + $0.0025Q2 = $50,000 + $0.5(20,000) + $0.0025(20,0002) = $1,060,000


Estimated average cost is: AC = TC/Q = $1,060,000/20,000 = $53 per case B. If actual total costs were $848,000 at a volume of 20,000 units, actual average total costs

were: AC = TC/Q = $848,000/20,000 = $42.40 per case

Therefore, greater experience or learning has resulted in an average cost saving of $10.60 per case since:

Learning effect = Actual AC - Estimated AC = $42.40 - $53 = -$10.60 per case

Alternatively,

Learning rate = 100 ACAC - 1

1

2

= 100 $53

$42.40 - 1

= 20% ST8.2 Minimum Efficient Scale Estimation. Assume Kanata Corporation is a leading

manufacturer of telecommunications equipment based in Ontario, Canada. Its main product is micro-processor controlled telephone switching equipment, called automatic private branch exchanges (PABXs), capable of handling 8 to 3,000 telephone extensions. Severe price cutting throughout the PABX industry continues to put pressure on sales and margins. To better compete against increasingly aggressive

208 Chapter 8

rivals, the company is contemplating the construction of a new production facility capable of producing 1.5 million units per year. Kanata's in-house engineering estimate of the total cost function for the new facility is:

TC = $3,000 + $1,000Q + $0.003Q2, MC = TC/ Q = $1,000 + $0.006Q

where TC = Total Costs in thousands of dollars, Q = Output in thousands of units, and MC = Marginal Costs in thousands of dollars.

A. Estimate minimum efficient scale in this industry.

B. In light of current PABX demand of 30 million units per year, how would you

evaluate the future potential for competition in the industry? ST8.2 SOLUTION A. Minimum efficient scale is reached when average costs are first minimized. This occurs

at the point where MC = AC. Average Costs = AC = TC/Q = ($3,000 + $1,000Q + $0.003Q2)/Q

= Q

$3,000 + $1,000 + $0.003Q

Therefore, MC = AC

$1,000 + $0.006Q = Q

$3,000 + $1,000 + $0.003Q

0.003Q = Q

3,000

Q

3,0002 = 0.003

Q2 = 1,000,000

Cost Analysis and Estimation 209 Q = 1,000(000) or 1 million

[Note: AC is rising for Q > 1,000(000)].

Alternatively, MES can be calculated using the point cost elasticity formula, since MES is reached when C = 1.

C = TCQ

QTC

)Q$0.003 + Q$1,000 + ($3,000

QQ)$0.006 + ($1,0002 = 1

1,000Q + 0.006Q2 = 3,000 + 1,000Q + 0.003Q2 0.003Q2 = 3,000 Q2 = 1,000,000 QMES = 1,000(000) or 1 million B. With a minimum efficient scale of 1 million units and total industry sales of 30 million

units, up to 30 efficiently sized competitors are possible in Kanata's market.

Potential Number of Efficient Competitors = SizeMESSizeMarket

= 1,000,00030,000,000

= 30

Thus, there is the potential for n = 30 efficiently sized competitors and, therefore, vigorous competition in Kanata's industry.

PROBLEMS AND SOLUTIONS P8.1 Cost Relations. Determine whether each of the following is true or false. Explain why.

A. Average cost equals marginal cost at the minimum efficient scale of plant.

210 Chapter 8

B. When total fixed cost and price are held constant, an increase in average variable cost will typically cause a reduction in the breakeven activity level.

C. If C > 1, diseconomies of scale and increasing average costs are indicated.

D. When long-run average cost is decreasing, it can pay to operate larger plants with

some excess capacity rather than smaller plants at their peak efficiency.

E. An increase in average variable cost always increases the degree of operating leverage for firms making a positive net profit.

P8.1 SOLUTION A. True. The point of minimum average cost identifies the minimum efficient scale of

plant. By definition, average and marginal costs are equal at this point. B. False. The breakeven activity level is where Q = TFC/(P - AVC). As average variable

cost (AVC) increases, this ratio and the breakeven activity level will also increase. C. True. When C > 1, the percentage change in cost exceeds a given percentage change in

output. This describes a situation of increasing average costs and diseconomies of scale. D. True. When long-run average costs are declining, it can pay to operate larger plants

with some excess capacity rather than smaller plants at their peak efficiency. E. False. The degree of operating leverage is defined DOL = Q(P - AVC)/(Q(P - AVC) -

TFC). Therefore, when total fixed costs are zero, DOL is a constant and an increase in average variable cost will have no effect on DOL.

P8.2 Cost Curves. Indicate whether each of the following involves an upward or downward

shift in the long-run average cost curve or, instead, involves a leftward or rightward movement along a given curve. Also indicate whether each will have an increasing, decreasing, or uncertain effect on the level of average cost.

A. A rise in wage rates.

B. A decline in output.

C. An energy-saving technical change.

D. A fall in interest rates.

E. An increase in learning or experience.

Cost Analysis and Estimation 211 P8.2 SOLUTION A. A rise in wage rates will cause an upward shift in the LRAC curve and increase LRAC. B. A decline in output will be reflected in a leftward movement along a given LRAC curve

and involve an uncertain effect on the level of LRAC. C. Energy-saving technical change will cause a downward shift in the LRAC curve and

decrease LRAC. D. A fall in interest rates gives rise to a downward shift in the LRAC curve and a decrease

in LRAC. E. Learning, like any beneficial technical change or innovation, will cause a downward

shift in the LRAC curve and decrease LRAC. P8.3 Incremental Cost. South Park Software, Inc. produces innovative interior decorating

software that it sells to design studios, home furnishing stores, and so on. The yearly volume of output is 15,000 units. Selling price and costs per unit are as follows:

Selling Price

$250

Costs:

Direct material

$40

Direct labor

60

Variable overhead

30

Variable selling expenses

25

Fixed selling expenses

20

-$175

Unit profit before tax

$ 75

Management is evaluating the possibility of using the Internet to sell its software directly to consumers at a price of $300 per unit. Although no added capital investment is required, additional shipping and handling costs are estimated as follows:

Direct labor

$30 per unit

Variable overhead $5 per unit

Variable selling expenses $2 per unit

Fixed selling expenses $20,000 per year

212 Chapter 8

Calculate the incremental profit that South Park would earn by customizing its instruments and marketing them directly to end users.

P8.3 SOLUTION

This problem should be answered by using incremental profit analysis. The analysis deals only with the incremental revenues and costs associated with the decision to engage in further processing.

Incremental revenue per unit ($300 - $250)

$50

Incremental variable cost per unit ($30 + $5 + $2)

-$37 Incremental profit contribution per unit

$13

Yearly output volume in units

15,000 Incremental variable profit per year

$195,000

Incremental fixed cost per year

-$20,000 Yearly incremental profit

$175,000

Since the incremental profit is positive, the decision to engage in further processing would be more profitable than continuing the present operating policy.

P8.4 Accounting and Economic Costs. Three graduate business students are considering

operating a fruit smoothie stand in the Harbor Springs, Michigan, resort area during their summer break. This is an alternative to summer employment with a local firm, where they would each earn $6,000 over the three-month summer period. A fully equipped facility can be leased at a cost of $8,000 for the summer. Additional projected costs are $1,000 for insurance and $3.20 per unit for materials and supplies. Their fruit smoothies would be priced at $5 per unit.

A. What is the accounting cost function for this business?

B. What is the economic cost function for this business?

C. What is the economic breakeven number of units for this operation? (Assume a $5

price and ignore interest costs associated with the timing of lease payments.) P8.4 SOLUTION A. The accounting cost function is:


Q$3.2 + $9,000 =

Q$3.2 + $1,000 + $8,000 =

costs suppliesplus materials

Variable +

costs insuranceplus leasing

Fixed = TC =

CostAccounting

TotalA

B. The economic cost function is:

Q$3.2 + $27,000 =

Q$3.2 + $9,000 + 3($6,000) =

TC + costy opportunitemploymentSummer = Cost

Economic TotalA

C. The economic breakeven point is reached when:

units 15,000 =

$3.20 - $5$27,000 =

AVC - P

TFC = QBE

P8.5 Profit Contribution. Angelica Pickles is manager of a Quick Copy franchise in White

Plains, New York. Pickles projects that by reducing copy charges from 5 to 4 each, Quick Copy's $600-per-week profit contribution will increase by one-third.

A. If average variable costs are 2 per copy, calculate Quick Copy's projected

increase in volume.

B. What is Pickles' estimate of the arc price elasticity of demand for copies? P8.5 SOLUTION A. The initial, or before-price reduction, copy volume can be calculated using the profit

contribution formula. Profit contribution = (P - AVC)Q1

214 Chapter 8 $600 = ($0.05 - $0.02)Q1 Q = 20,000

After the price reduction, a profit contribution of $800 (=1.33 600) requires an output level of 40,000 units since:

Profit Contribution = (P - AVC)Q2 $800 = ($0.04 - $0.02)Q2 Q2 = 40,000

Therefore, Pickles' projected increase in volume is: Projected increase = Q2 - Q1 = 40,000 - 20,000 = 20,000 copies per week B. Given the large magnitude of this price reduction, use of the arc price elasticity formula

is appropriate.

(Elastic) 3- =

20,000 + 40,000$0.05 + $0.04

$0.05 - $0.0420,000 - 40,000 =

Q + QP + P

P - PQ - Q

= E12

12

12

12P

P8.6 Cost-Volume-Profit Analysis. Textbook publishers evaluate market size, the degree of

competition, expected revenues, and costs for each prospective new title. With these data in mind, they estimate the probability that a given book will reach or exceed the breakeven point. If the publisher estimates that a book will not exceed the breakeven point based upon standard assumptions, they may consider cutting production costs by reducing the number of illustrations, doing only light copy editing, using a lower grade of paper, or negotiating with the author to reduce the royalty rate. To illustrate the process, consider the following data:

Cost Category Dollar AmountFixed Costs


Copyediting and other editorial costs $15,750Illustrations 32,750Typesetting 51,500Total fixed costs $100,000

Variable Costs Printing, binding and paper $22.50Bookstore discounts 25.00Sales staff commissions 8.25Author royalties 10.00General and administrative costs 26.25Total variable costs per copy $92.00

List price per copy $100.00

Fixed costs of $100,000 can be estimated quite accurately. Variable costs are linear and set by contract. List prices are variable, but competition keeps prices within a narrow range. Variable costs for the proposed book are $92 a copy, and the expected wholesale price is $100. This means that each copy sold provides the publisher with an $8 profit contribution.

A. Estimate the volume necessary to reach a breakeven level of output.

B. How many textbooks would have to be sold to generate a profit contribution of

$20,000?

C. Calculate the economic profit contribution or loss resulting from the acceptance of a book club offer to buy 3,000 copies directly from the publisher at a price of $77 per copy. Should the offer be accepted?

P8.6 SOLUTION A. Applying the breakeven formula, the breakeven sales volume is 12,500 units, calculated

as

.units 12,500 =

$8$100,000 = Q

B. To find the number of copies one must sell to earn a $20,000 profit, simply add the

$20,000 profit requirement to the book's fixed costs, and then divide this total amount by

216 Chapter 8

the profit contribution per unit. The sales volume required in this case is 15,000 books, found as:

.units 15,000 = $8

$20,000 + $100,000 =

onContributiProfit tRequiremenProfit + Costs Fixed = Q

C. Because fixed costs do not vary with respect to changes in the number of books sold,

they should be ignored. Variable costs per copy are $92, but note that $25 of this cost represents bookstore discounts. Because the 3,000 copies are being sold directly to the club, this cost will not be incurred. Hence, the relevant variable cost is only $67 (= $92 - $25). Profit contribution per book sold to the book club is $10 (= $77 - $67), and $10 times the 3,000 copies sold indicates that the order will result in a total profit contribution of $30,000. Assuming that these 3,000 copies would not have been sold through normal sales channels, the $30,000 profit contribution indicates the increase in profits to the publisher from accepting this order.

P8.7 Cost Elasticity. Power Brokers, Inc. (PBI), a discount brokerage firm, is contemplating

opening a new regional office in Providence, Rhode Island. An accounting cost analysis of monthly operating costs at a dozen of its regional outlets reveals average fixed costs of $4,500 per month and average variable costs of

AVC = $59 - $0.006Q

where AVC is average variable costs (in dollars) and Q is output measured by number of stock and bond trades.

A typical stock or bond trade results in $100 gross commission income, with PBI

paying 35% of this amount to its sales representatives.

A. Estimate the trade volume necessary for PBI to reach a target return of $7,500 per month for a typical office.

B. Estimate and interpret the elasticity of cost with respect to output at the trade

volume found in part A. P8.7 SOLUTION A. To earn a target return of $7,500 per month, Power Brokers must generate sufficient

revenues to cover both fixed costs and the target return, or $4,500 + $7,500 = $12,000


per month. The trade volume necessary to reach a target return of $7,500 per month can be calculated as:

Q0.006 + 612,000 = Q

Q$0.006 + $59 - )0.35)($100 - (1$7,500 + $4,500 = Q

AVC - PreturnTarget + costs fixed Total = Q

0.006Q2 + 6Q - 12,000 = 0

which can be solved using the quadratic formula where a = 0.006, b = 6 and c = -12,000,

.monthper trades2,000-or 1,000 = 0.012

18 6- =

0.012324 6- =

2(0.006)12,000)4(0.006)(- - 6 6- =

a2ac4 - b b- = Q

2

2

Since -2,000 is an infeasible negative output, an activity level of 1,000 trades per month would allow Power Brokers to meet its target return.

B. By definition,

TCQ

QTC =

Q/QTC/TC = c

where

TC = Fixed Costs + Variable Costs = $4,500 + ($59 - $0.006Q)Q = $4,500 + $59Q - $0.006Q2

218 Chapter 8

At Q = 1,000, TC = $4,500 + $59(1,000) - $0.006(1,0002) = $4,500 + $59,000 - $6,000 = $57,500

Therefore, at Q = 1,000

c = TCQ

QTC

= ($59 - $0.012Q) Q/TC = ($59Q - $0.012Q2)/TC = $59(1,000) - $0.012(1,0002)/57,500 = 0.82

Since C = 0.82 < 1, economies of scale are suggested. A 1% increase in output leads to a 0.82% increase in costs, and average costs will fall as output expands.

P8.8 Multiplant Operation. Appalachia Beverage Company, Inc. is considering alternative

proposals for expansion into the Midwest. Alternative 1: Construct a single plant in Indianapolis, Indiana, with a monthly production capacity of 300,000 cases, a monthly fixed cost of $262,500, and a variable cost of $3.25 per case. Alternative 2: Construct three plants, one each in Muncie, Indiana; Normal, Illinois; and Dayton, Ohio, with capacities of 120,000, 100,000, and 80,000, respectively, and monthly fixed costs of $120,000, $110,000, and $95,000 each. Variable costs would be only $3 per case because of lower distribution costs. To achieve these cost savings, sales from each smaller plant would be limited to demand within its home state. The total estimated monthly sales volume of 200,000 cases in these three Midwestern states is distributed as follows: 80,000 cases in Indiana, 70,000 cases in Illinois, and 50,000 cases in Ohio.

A. Assuming a wholesale price of $5 per case, calculate the breakeven output

quantities for each alternative.

B. At a wholesale price of $5 per case in all states, and assuming sales at the projected levels, which alternative expansion scheme provides Appalachia with the highest profit per month?


C. If sales increase to production capacities, which alternative would prove to be more profitable?

P8.8 SOLUTION A. The breakeven output quantity for the single plant alternative is:

Q = AVC-P

TFC

= $3.25 - $5

$262,500

= 150,000 cases per month

The breakeven output quantities for the multiple plant alternative is:

QMuncie = $3 - $5

$120,000


QNormal = $3 - $5

$110,000


QDayton = $3 - $5

$95,000


Thus, the firm-level breakeven quantity for the multiple plant alternative is: Q = 60,000 + 55,000 + 47,500 = 162,500 cases per month

provided that demand was distributed among the states in amounts equal to the breakeven quantities for each individual plant.

B. Single plant alternative:

220 Chapter 8 = TR - TC = P Q - TFC - AVC Q = $5(200,000) - $262,500 - $3.25(200,000) = $87,500

Multiple plant alternative:

= TR - TC = P Q - TFCM - TFCN - TFCD - AVC Q = $5(200,000) - $120,000 - $110,000 - $95,000 - $3(200,000) = $75,000

Management would prefer the single plant alternative because of its greater profitability.

C. Single plant at full capacity: = TR - TC = P Q - TFC - AVC Q = $5(300,000) - $262,500 - $3.25(300,000) = $262,500

Multiple plants at full capacity: = TR - TC = P Q - TFCM - TFCN - TFCD - AVC Q = $5(300,000) - $120,000 - $110,000 - $95,000 - $3(300,000) = $275,000


At peak capacity, management would prefer the multiple plant option because of its greater profitability.

P8.9 Learning Curves. The St. Thomas Winery plans to open a new production facility in the

Napa Valley of California. Based on information provided by the accounting department, the company estimates fixed costs of $250,000 per year and average variable costs of

AVC = $10 + $0.01Q

where AVC is average variable cost (in dollars) and Q is output measured in cases of output per year.

A. Estimate total cost and average total cost for the coming year at a projected

volume of 4,000 cases.

B. An increase in worker productivity because of greater experience or learning during the course of the year resulted in a substantial cost saving for the company. Estimate the effect of learning on average total cost if actual total cost was $522,500 at an actual volume of 5,000 cases.

P8.9 SOLUTION A. The total variable cost function for the coming year is: TVC = AVC Q = ($10 + $0.01Q)Q = $10Q + $0.01Q2

At a volume of 4,000 units, estimated total cost is: TC = TFC + TVC = $250,000 + $10Q + $0.01Q2 = $250,000 + $10(4,000) + $0.01(4,0002) = $450,000

Estimated average cost is:

222 Chapter 8 AC = TC/Q = $450,000/4,000 = $112.50 per case B. Without learning, estimated total cost and average total cost at a volume of 5,000 cases

are: TC = $250,000 + $10(5,000) + $0.01(5,0002) = $550,000 AC = TC/Q = $550,000/5,000 = $110 per case

Since estimated average cost without learning falls between 4,000 and 5,000 units (see part A), the company is operating in a range of economies of scale.

If actual total costs were $522,500 at a volume of 5,000 cases, actual average total costs were:

AC = TC/Q = $522,500/5,000 = $104.50 per case

Therefore, greater experience or learning has resulted in an average cost saving of $5.50 per case since:

Learning Effect = Actual AC - Estimated AC = $104.50 - $110 = ($5.50) per case

Alternatively,


Learning Rate = 100 ACAC - 1

1

2

= 100 $110.00$104.50 - 1

= 5% P8.10 Degree of Operating Leverage. Untouchable Package Service (UPS) offers overnight

package delivery to Canadian business customers. UPS has recently decided to expand its facilities to better satisfy current and projected demand. Current volume totals two million packages per week at a price of $12 each, and average variable costs are constant at all output levels. Fixed costs are $3 million per week, and profit contribution averages one-third of revenues on each delivery. After completion of the expansion project, fixed costs will double, but variable costs will decline by 25%.

A. Calculate the change in UPS's weekly breakeven output level that is due to

expansion.

B. Assuming that volume remains at two million packages per week, calculate the change in the degree of operating leverage that is due to expansion.

C. Again assuming that volume remains at two million packages per week, what is

the effect of expansion on weekly profit? P8.10 SOLUTION A. Average variable costs are $8 since: (P - AVC)Q = 1/3P(Q) P - AVC = 1/3P AVC = 2/3($12) AVC = $8

Therefore, the breakeven levels of output before and after expansion are: Before Expansion:

224 Chapter 8

packages 750,000 = $8 - $12

$3,000,000 =

AVC - PTFC = QB

After Expansion:

packages 1,000,000 = $6 - $12

$6,000,000 =

AVC - PTFC = QA

The change in the weekly breakeven output level due to expansion is:

monthper packages 250,000 = 750,000 - 1,000,000 =

Q - Q = Breakeven

in ChangeBA

B. The degrees of operating leverage before and after expansion are: Before Expansion:

1.6 = $3,000,000 - $8) - $122,000,000(

$8) - $122,000,000( =

TFC - AVC) - Q(PAVC) - Q(P = DOLB

After Expansion:


2 = $6,000,000 - $6) - $122,000,000(

$6) - $122,000,000( =

TFC - AVC) - Q(PAVC) - Q(P = DOLA

The change in degree of operating leverage due to expansion is:

0.4 = 1.6 - 2 =

DOL - DOL = DOLin Change

BA

C. Profits before and after expansion are:

Before Expansion: B = (P - AVC)Q - TFC = ($12 - $8)2,000,000 - $3,000,000 = $5 million After Expansion: A = (P - AVC)Q - TFC = ($12 - $6)2,000,000 - $6,000,000 = $6 million

The change in profits due to expansion is: Change in = A - B = $6,000,000 - $5,000,000 = $1 million per week

226 Chapter 8 CASE STUDY FOR CHAPTER 8 Estimating Hospitalization Costs for Regional Hospitals Cost estimation and cost containment are an important concern for a wide range of for-profit and not-for-profit organizations offering health-care services. For such organizations, the accurate measurement of costs per patient day (a measure of output) is necessary for effective management. Similarly, such cost estimates are of significant interest to public officials at the federal, state, and local government levels. For example, many state Medicaid reimbursement programs base their payment rates on historical accounting measures of average costs per unit of service. However, these historical average costs may or may not be relevant for hospital management decisions. During periods of substantial excess capacity, the overhead component of average costs may become irrelevant. When the facilities are fully used and facility expansion becomes necessary to increase services, then all costs, including overhead, are relevant. As a result, historical average costs provide a useful basis for planning purposes only if appropriate assumptions can be made about the relative length of periods of peak versus off-peak facility usage. From a public-policy perspective, a further potential problem arises when hospital expense reimbursement programs are based on average costs per day, because the care needs and nursing costs of various patient groups can vary widely. For example, if the care received by the average publicly-supported Medicaid patient actually costs more than that received by non-Medicaid patients, Medicaid reimbursement based on average costs would be inequitable to providers and could create access barriers for Medicaid patients.

As an alternative to accounting cost estimation methods, one might consider using engineering techniques to estimate nursing costs. For example, the labor cost of each type of service could be estimated as the product of an approximation of the time required to perform each service times the estimated wage rate per unit of time. Multiplying this figure by an estimate of the frequency of service gives an engineering estimate of the cost of the service. A possible limitation to the accuracy of this engineering cost-estimation method is that treatment of a variety of illnesses often requires a combination of nursing services. To the extent that multiple services can be provided simultaneously, the engineering technique will tend to overstate actual costs unless the effect of service "packaging" is allowed for.

Cost estimation is also possible by means of a carefully designed regression-based approach using variable cost and service data collected at the ward, unit, or facility level. Weekly labor costs for registered nurses (RNs), licensed practical nurses (LPNs), and nursing aides might be related to a variety of patient services performed during a given measurement period. With sufficient variability in cost and service levels over time, useful estimates of variable labor costs become possible for each type of service and for each patient category (Medicaid, non-Medicaid, etc.). An important advantage of a regression-based approach is that it explicitly allows for the effect of service packaging on variable costs. For example, if shots and wound-dressing services are typically provided together, this will be reflected in the regression-based estimates of variable costs per unit.

Long-run costs per nursing facility can be estimated using either cross-section or time-series methods. By relating total facility costs to the service levels provided by a number of hospitals,

Cost Analysis and Estimation 227 nursing homes, or out-patient care facilities during a specific period, useful cross-section estimates of total service costs are possible. If case mixes were to vary dramatically according to type of facility, then the type of facility would have to be explicitly accounted for in the regression model analyzed. Similarly, if patient mix or service-provider efficiency is expected to depend, at least in part, on the for-profit or not-for-profit organization status of the care facility, the regression model must also recognize this factor. These factors plus price-level adjustments for inflation would be accounted for in a time-series approach to nursing cost estimation.

To illustrate a regression-based approach to nursing cost estimation, consider a hypothetical analysis of variable nursing costs conducted by the Southeast Association of Hospital Administrators (SAHA). Using confidential data provided by 40 regional hospitals, SAHA studied the relation between nursing costs per patient day and four typical categories of nursing services. These annual data appear in Table 8.2. The four categories of nursing services studied include shots, intravenous (IV) therapy, pulse taking and monitoring, and wound dressing. Each service is measured in terms of frequency per patient day. An output of 1.50 in the shots service category means that, on average, patients received one and one-half shots per day. Similarly, a value of 0.75 in the IV service category means that on average, patients received 0.75 units of IV therapy per day, and so on. In addition to four categories of nursing services, the not-for-profit or for-profit status of each hospital is also indicated. Using a "dummy" (or binary) variable approach, the profit status variable equals 1 for the 8 for-profit hospitals included in the study and zero for the remaining 32 not-for-profit hospitals.

Cost estimation results for nursing costs per patient day derived using a regression-based approach are shown in Table 8.3. A. Interpret the coefficient of determination (R2) estimated for the nursing cost function. B. Describe the economic and statistical significance of each estimated coefficient in the

nursing cost function. C. Average nursing costs for the eight for-profit hospitals in the sample are only $318.52

per patient day, or $33.07 per patient day less than the $351.59 average cost experienced by the 32 not-for-profit hospitals. How can this fact be reconciled with the estimated coefficient of -39.156 for the for-profit status variable?

D. Would such an approach for nursing cost estimation have practical relevance for

publicly-funded nursing cost reimbursement systems? CASE STUDY SOLUTION A. Cost estimation results provided indicate that R2 = 76.81%, meaning that 76.81 percent

of the total variation in nursing costs per patient day can be explained by the five factors studied. For cross-sectional analysis, such a level of cost explanation is often quite adequate for gaining useful insight concerning cost characteristics.

228 Chapter 8 B. Each individual coefficient estimate is statistically significant at the 99% confidence

level, with the exception of the Shots coefficient estimate, which is significant at the 95% level. In terms of economic interpretation, the 72.765 coefficient for the shots variable indicates an average nursing labor cost of roughly $72.76 per shot. Similarly, IV therapy results in $215.68 in nursing costs per patient day, pulse taking and monitoring costs $36.24, and wound dressing costs $156.04 per unit. Each of these four services appears to have a clear impact on nursing costs per patient day. Interestingly, a coefficient of -39.156 for the profit-status variable indicates that, on average, nursing costs per patient day are roughly $39.16 lower in for-profit than in not-for-profit hospitals after accounting for differences in patient mix as captured by the four service categories. This suggests that the greater efficiency or operating philosophy of for-profit hospitals may be responsible for a substantial portion of the lower nursing costs these hospitals enjoy.

C. By considering differences in the nursing services provided, along with the for-profit or

not-for-profit status of each hospital, it is possible to learn whether average cost differences are related to differences in patient mix or, perhaps, to other factors, such as efficiency or operating philosophy. After accounting for the influences associated with variation in assorted output categories, the organization design variable appears relevant. Before accounting for output differences, for-profit organizations appear to have nursing costs per patient day that are roughly $33.07 less than the average reported by not-for-profit hospitals. After accounting for differences in patient mix as captured by the four service categories, for-profit organizations appear to have nursing costs per patient day that are roughly $39.16 less than the average reported by not-for-profit hospitals. This is an interesting finding, but additional analysis would be necessary to determine if this effect is due to operating advantages of for-profit hospitals or instead due to subtle differences in geographic location, patient mix, and so on.

D. Despite obvious limitations, such a regression-based approach can provide useful

measures of costs for both private and public decision making. In practice, nursing care cost estimation and cost reimbursement methodologies that reflect the care needs of patients can be based on a manageable number of services. In fact, Illinois, West Virginia, Ohio, and Maryland have implemented Medicaid nursing home reimbursement systems based on this concept, and several other states have similar case-mix reimbursement systems under development.

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