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 stockholder reporting reporting purpose purposes? s?
Q8.1
ANSWER
Theoretically, it would be preferable to use current costs for income tax calculations and stockho stockholde lderr repor reportin ting. g. On a practi practical cal level level,, howev however er,, this this would would be nearl nearly y impossible. impossible. Estimation of current current cost, based upon current market values, would would be a difficul difficultt task with a great great deal of room for subjectivi subjectivity. ty. This could could result in many arbitrary cost designations, and the "policing" of tax returns would become a much more formidable formidable task. task. On a practical basis, the use use of historical historical costs for for tax and stockholder reporting purposes has obvious advantages over the theoretically superior current costs. Q8.2
Assume Assume that two years ago, ago, you purchased a new Jeep Wrangler Wrangler SE 4WD with a soft top for $16,500 using five-year five-year interest-free interest-free financing. financing. Today, Today, the remaining loan balance balance is $9,900 $9,900 and your your Jeep has a tradetrade-in in value value of $9,500. $9,500. What What is your opportunity cost of continuing to drive the Jeep? Discuss the financing risk exposure exposure of the lender.
Q8.2
ANSWER
The opportunity cost of continuing continuing to drive the Jeep Jeep is $9,500. If you sell sell the Jeep, $9,500 $9,500 can be generated generated to pay down your remaining remaining loan balance. It is the current current cost or replacement value of your current vehicle. vehicle. It is the relevant relevant economic cost of conti continuin nuing g to drive drive the Jeep. Jeep. Hist Histori orical cal cost cost of $16,50 $16,500 0 and the remaini remaining ng loan loan balance of $9,900 are irrelevant for decision-making purposes. 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 borrower default. default. Aggres Aggressive sive interest-f interest-free ree financing financing offered offered by the major automakers has the potential to create big debt collection problems p roblems in the future. Q8.3
Southwest Airlines offers four flights per weekday from Cleveland, Ohio to Tucson, Arizona. Arizona. If adding a fifth flight per weekday weekday would cost $15,000 per flight, or $110 per available available seat, calculate calculate the incrementa incrementall costs borne by Southwest Southwest following following a decision decision to go ahead with a fifth flight per day for a minimal minimal 60-flight trial period. period. What is the marginal marginal cost? In this case, is incremental incremental cost or marginal marginal cost relevant for decision decision making making purpose purposes? s?
190
Q8.3
Chapter 8
ANSWER
Marginal costs are the cost effect effect of one-unit one-unit changes in output. Incremental cost is the cost effect effect associated associated with a given given managerial decision. decision. Increment Incremental al costs 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 $900,000 (= $15,000 $15,000 × 60). The marginal marginal cost per passenge passengerr is only $110. $110. In this this case, case, the incre incremen mental tal cost cost of $90 $900,0 0,000 00 is the relev relevant ant cost cost for decision decision making 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 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. Explain.
Q8.4
ANSWER
Any cost that is invariant across decision decision alternatives is a sunk cost. cost. Sunk costs are irrelevant for current decision-making purposes and should not enter into decision analysis. Before the lease agreement agreement has been signed, all costs costs are variable, and the incre incremen mental tal cost cost per mon month th of site site rental rental is $5,200 $5,200 per mon month. th. Afte Afterr the lease 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 relation between between production production functions and cost cost functions? functions? Be sure to include in your discussion the effect of competitive conditions in input factor markets.
Q8.5
ANSWER There There is a direct direct relation between between production production and cost cost functions. functions. A cost function function is determined by combining a given production function with the related price functions for the inputs actually employed employed in production. If inputs 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). 8.4). With With imperfect imperfect competition competition in input markets, markets, the relation becomes becomes somewhat somewhat more complex. complex. In all cases, cost/prod cost/productio uction n relations relations can be employed 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 ε Q = ΔQ/Q ÷ ΔX/X (X represents all inputs), whereas the definition of point cost elasticity is ε C = ΔC/C ÷ ΔQ/Q. Explain why εQ > 1 indicates increasing returns to scale, whereas ε C < 1 indicates economies of scale.
Cost Analysis and Estimation
Q8.6
191
ANSWER
The definition of output elasticity is ε Q = ΔQ/Q ÷ ΔX/X (X represents all inputs), whereas the definition of cost elasticity is ε C = ΔC/C ÷ ΔQ/Q. Therefore: If
Then
Implies
εQ > 1
ΔQ/Q > ΔX/X
Risi Rising ng Q/X rati ratio, o, fall fallin ing g AC. AC.
εC < 1
ΔC/C < ΔQ/Q
Fall Fallin ing g C/Q C/Q rati ratio, o, fall fallin ing g AC. AC.
This means that εQ > 1 and εC < 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 material costs. However However,, the controller 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 incr increa easi sing ng inpu inputt pric prices es is quite quite feas feasib ible le.. It stems stems from from an incr increa ease se in input input productivity that could result from any number of causes. One obvious possibility possibility would be the introduction of new capital equipment, either replacement or expansion, into the production system. 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 insurance picks up most most of their health- care costs. costs. Companies Companies complain complain that this gives workers little incentive to help control medical insurance costs, and those costs are spinning spinning out of control. control. Can you suggest suggest ways of giving workers workers better better incentives to control employer medical insurance costs?
Q8.8
ANSWER
In hopes of slowing the growth growth in medical costs, some companies are moving towards “consumer “consumer driven” medical coverage coverage that gives employees employees a financial financial stake stake in what they pay for medical medical care. Such plans plans feature feature high deductibles deductibles of as much as $500 per year for prescriptions prescriptions and $1,000 per year for all other medical costs. To help pay these these costs costs,, some some compani companies es deposi depositt $30 $300 0 to $1,80 $1,800 0 per year in an “emplo “employee yee benefits account.” At some firms, like like Whole Foods Market, Inc., 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 n ext year. The Whole Foods Foods plan, which workers workers themselves themselves chose over two competing competing plans after a series of votes last summer, has no premiums at all for many workers. workers.
192
Chapter 8
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 deductible is reached, the plan operates more like a traditional traditional one, picking picking up 80% of most medical expenses. expenses. The hope is that once the money feels as though 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 tend to be a substantial share of total costs. When fixed fixed costs are large, the degree of operating operating leverage also tends to be high, and an d 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 operating strategies strategies of average average cost minimi minimization zation and profit profit maximization maximization always always lead to identical levels of output?
Q8.10
ANSWER
No, operating strategies 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. output. Average Average cost minimization minimization is an appropriate strategy strategy when managers managers wish wish to produce a target level of output output in an optimal optimal or least-cos least-costt fashion. 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 SELF-TEST PROBLEMS AND SOLUTIONS SOLUTIONS ST8.1
Suppose Modern Merchand Merchandise, ise, Inc., makes makes and markets markets do-itdo-itLearning Learning Curves Curves . Suppose yourself yourself hardware, hardware, housewares housewares,, and industrial industrial products. products. The company's company's new Aperture Aperture Miniblind Miniblind is winning winning customers customers by virtue of of its high quality and quick quick order turnaround turnaround time. The product product also benefits because because its price point bridges the gap
Cost Analysis and Estimation
193
betwee between n readyready-ma made de vinyl blinds blinds and their high-pric high-priced ed custom custom counterp counterpart. art. In addition, the company's expanding product line is sure to benefit from cross-selling across different lines. Given the success success of the Aperture Aperture Miniblind product, Modern Merchandise Merchandise plans to open a new production facility near Beaufort, Beaufort, South Carolina. Carolina. Based Based on information information provided provided by its chief financial officer, officer, the company company estimates estimates fixed costs costs for for this product product of of $50,000 $50,000 per year year and and average average variable variable costs costs of: AVC = $0.5 + $0.0025Q, $0.0025Q, where AVC is average variable cost (in dollars) and Q is output. A.
Estimate Estimate total cost and average average total cost for the projected projected first-year first-year volume of 20,000 units.
B.
An increase increase in worker worker productivity productivity because because of greater greater experience experience or learning learning during during the course course of the year result resulted ed in a substan substantial tial cost cost saving saving for the company. Estimate the effect of learning on average total cost if actual secondsecond year total total cost cost was $848,000 $848,000 at an actual actual volume volume of 20,000 20,000 units. units.
ST8.1
SOLUTION
A.
The total variable cost c ost function for the first year is: TVC = AVC × Q = ($0.5 ($0.5 + $0. $0.00 0025Q 25Q)Q )Q = $0.5 $0.5Q Q + $0. $0.00 0025 25Q Q2 At a volume of 20,000 units, estimated total cost is: TC
= TFC + TVC = $50 $50,00 ,000 0 + $0.5 $0.5Q Q + $0. $0.00 0025Q 25Q2 = $50 $50,000 ,000 + $0.5(20 $0.5(20,000 ,000)) + $0.0025(2 $0.0025(20,00 0,000 02) = $1,060 ,060,0 ,000 00
Estimated average cost is: AC = TC/Q = $1,0 $1,060 60,0 ,000 00/2 /20, 0,00 000 0
194
Chapter 8
= $53 pe per ca 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 $848,0 ,000 00//20,0 20,000 00 = $42.40 per per case ase Therefore, greater experience or learning has resulted in an average cost saving of $10.60 per case since: Lear Learni ning ng effe effect ct = Actu Actual al AC AC - Esti Estima mate ted d AC AC = $42.40 - $53 = -$10 $10.60 .60 per per cas case Alternatively, Learning ra rate
=
AC2 1 × 100 AC 1
=
1 - $42.40 × 100 $53
= 20% ST8.2
Minimum Efficient Efficient Scale Estimation. Estimation. Assume Kanata Corporation is a leading manufacturer of telecommunications telecommunications equipment based in Ontario, Canada. Its main product product is micromicro-proce processor ssor controlled controlled telephone telephone switching switching equipment, equipment, called automa automatic tic private private branch branch exchan exchanges ges (PA (PABX BXs), s), capable capable of handlin handling g 8 to 3,000 3,000 telephone telephone extensions. extensions. Severe Severe price cutting throughout throughout the PABX industry industry continues continues to put pressu pressure re on sales sales and margins. margins. To better better compet competee against against increas increasingl inglyy aggressive rivals, the company is contemplating the construction of a new production facility capable capable of producing producing 1.5 million million units per year. Kanata's Kanata's in-house in-house engineering estimate of the total cost function for the new facility is:
TC = $3,0 $3,00 00 + $1,0 $1,000 00Q Q + $0.0 $0.003 03Q Q2 , MC =
∂ TC/ ∂ Q = $1,000 + $0.006Q
Cost Analysis and Estimation
195
where TC = Total Costs in thousands of dollars, Q = Output in thousands of units, and MC = Marginal Costs in thousands of dollars. A.
Estimate Estimate minimu minimum m efficient efficient scale scale in this industry. industry.
B.
In light of current current PAB PABX X demand of 30 million million units per year, year, how would you evaluate the future potential for competition in the industry?
ST8.2
SOLUTION SOLUTION
A.
Minimum Minimum efficient efficient scale is reached when average average costs costs are first first minimized. This occurs at the point where MC = AC. Average Co Costs
= AC = TC/Q = ($3,0 ($3,000 00 + $1,0 $1,000Q 00Q + $0.00 $0.003Q 3Q2)/Q =
$3,000 Q
+ $1,000 + $0.003Q
Therefore, MC
= AC
$1,000 + $0. $0.006Q
=
0.003Q
=
3,000 Q
$3,000 Q
+ $1,000 + $0.003Q
3,000 Q
2
= 0.003
Q2
= 1,000,000
Q
= 1,000(000) or or 1 million
[ Note 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 εC = 1.
196
Chapter 8
=
2
= 1
($1,000 + $0.006 Q) Q ($3,000 + $1,000 Q + $0.003 Q ) 1,000Q + 0.006Q2 0.003Q2 Q2 QMES B.
∆ TC Q × ∆ Q TC
εC
= 3,000 3,000 + 1,000 1,000Q Q + 0.00 0.003Q 3Q2 = 3,000 = 1,000,000 = 1,000( 1,000(000 000)) or 1 mill million ion
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. Potent Potential ial Numbe Numberr of of Effi Efficie cient nt Compet Competito itors rs
= =
Market Size MES Size 30,000,000 1,000,000
= 30 Thus, there is the potential for n = 30 efficiently sized competitors and, therefore, vigorous competition in Kanata's industry. PROBLEMS PROBLEMS AND SOLUTIONS P8.1
Cost Relations Relations . Determine Determine whethe whetherr each of the the following following is true or or false. Explain Explain why. A.
Average Average cost equals margin marginal al cost at at the minimu minimum m efficient efficient scale scale of plant. plant.
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 εC > 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.
Cost Analysis and Estimation E.
197
An increase in average variable cost always increases the degree of operating operating leverage for firms making a positive net profit.
P8.1
SOLUTION SOLUTION
A.
True. The point of minimum minimum average cost identifies identifies the minimum efficient efficient scale scale of plant. By definition, average average and marginal costs are equal at this point.
B.
False. False. The breake breakeve ven n activi activity ty level level is where where Q = TFC/ TFC/(P (P - AVC). AVC). As averag averagee variable cost (AVC) increases, this ratio and the breakeven activity level will also increase.
C.
True. When εC > 1, the percentage change in cost exceeds a given percentage change in output. This describes describes a situation of increasing average costs and diseconomies diseconomies of scale.
D.
True. When long-run long-run average costs are declining, 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)/(Q(P - AVC) AVC) TFC). TFC). Therefore, when 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
Indicat atee wheth whether er each each of the follo followi wing ng involv involves es an upwa upward rd or Cost Curves Curves. Indic downward shift in the long-run average cost curve or, instead, involves a leftward or rightward rightward movement movement along a given curve. curve. Also indicate indicate whether each will have an increasing, decreasing, or uncertain effect on the level of average cost. A.
A rise in wage wage rates. rates.
B.
A decline decline in output. output.
C.
An energy-s energy-saving aving technica technicall change. change.
D.
A fall in interest interest rates. rates.
E.
An increase increase in learning learning or experie experience. nce.
P8.2
SOLUTION 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. LRAC.
198
Chapter 8
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 Incremental Cost Cost . South Park Software, Inc. produces innovative interior decorating decorating software software that it sells to design design studios, home home furnishing furnishing stores, stores, and so on. The yearly 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
Unit pr profit before tax
-$175 $ 75
Managemen Managementt is evaluating evaluating the possibility possibility of using the Internet Internet to sell its software software dire direct ctly ly to cons consum umer erss at a price price of $300 $300 per per unit. unit. Alth Althou ough gh no added added capi capital tal invest investme ment nt is requir required, ed, additio additional nal shippin shipping g and handlin handling g costs costs are estima estimated ted as follows: follows: Direct Direct labor
$30 per per unit
Variable overhead
$5 per unit
Variable selling expenses
$2 per unit
Fixed Fixed selling selling expenses expenses
$20,000 $20,000 per year year
Calcula Calculate te the incremen incremental tal profit profit that that South South Park Park would would earn earn by custom customizin izing g its instruments and marketing them directly to end users. P8.3
SOLUTION SOLUTION
Cost Analysis and Estimation
199
This problem should should be answered by using using incremental profit analysis. The analysis deals only with the incremental revenues and costs associated with the decision to engage in further processing. processing. Incremental revenue per unit ($300 - $250) Incremental variable cost per unit ($30 + $5 + $2) Incremental profit contribution per unit
$ 50 -$37 $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 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 summer break. break. This is an alternative to summer summer employment employment with a local firm, where where they would each earn $6,000 $6,000 over the three-month three-month summer summer period. period. A fully equippe equipped d facilit facilityy can be leased leased at a cost of $8,000 $8,000 for the summer. summer. Additio Additional nal projected projected costs are $1,000 for insurance insurance and $3.20 per unit for materials materials and supplies. supplies. Their fruit smoothie smoothiess would would be priced priced at $5 per unit. 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 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 SOLUTION
A.
The accounting cost function is: Total Fixed Variable Accounting = TCA = leasing plus + materials plus Cost insurance costs supplies costs = $8,000 + $1,000 + $3.2 Q = $9,000 + $3.2 Q
200 B.
Chapter 8
The economic cost function is: Total Economic Summer employment = + TCA Cost opportunit y cost = 3($6,000) + $9,000 + $3.2 Q = $27,000 + $3.2 Q
C.
The economic breakeven point is reached when: TFC
QBE = =
P - AVC $27,000 $5 - $3.20
= 15,000 units P8.5
Angelica ca Pickles Pickles is manager manager of a Quick Quick Copy franch franchise ise in Profit Contribution Contribution . Angeli White Plains, Plains, New York. Pickles Pickles projects that by reducing reducing copy charges from 5¢ to 4¢ each, Quick Copy's $600-per-week profit contribution will increase by one-third. A.
If average average variable variable costs are 2¢ per copy, calculate calculate Quick Quick Copy's Copy's projected projected increase in volume.
B.
What is Pickles' estimate of the arc price elasticity of demand for copies?
P8.5
SOLUTION SOLUTION
A.
The initial, or before-price reduction, copy volume can be calculated using the profit contribution formula. Profi rofitt cont contrribut ibutio ion n $600 Q
= (P - AVC)Q1 = ($0.05 - $0.02)Q1 = 20,000
After the price reduction, a profit contribution of $800 (=1.33 × 600) requires an output level of 40,000 units since: Prof Profit it Cont Contri ribu buti tion on = (P - AVC AVC)Q )Q2 $800 = ($0.04 - $0.02 .02)Q2
EP =
=
Q2 - Q1 P 2 - P1
×
P 2 + P1 Q2 + Q1
40,000 - 20,000 $0.04 - $0.05
×
$0.04 + $0.05 40,000 + 20,000
Cost Analysis and Estimation
201
= - 3 (Elastic)
Q2 = 40,000 Therefore, Pickles' Pickles' projected increase in volume is: Projected increase = Q2 - Q1 = 40,0 40,000 00 - 20,0 20,000 00 = 20,00 20,000 0 copie copiess per per wee week k B.
Given the large magnitude of this price reduction, use of the arc price elasticity formula is appropriate.
P8.6
Textbook publishers publishers evaluate market market size, the degree degree Cost-Volume-Profit Cost-Volume-Profit Analysis . Textbook of competition, competition, expected expected revenues, revenues, and costs for each prospective 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 breake breakeven ven point point based based upon upon standa standard rd assum assumptio ptions, ns, they they may may conside considerr cuttin cutting g production production costs by reducing reducing the number number of illustrations, 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, process, consider the following data: Cost Category Fixed Costs Copyediting an and ot other ed editorial co costs Illustrations Typesetting Total fixed costs
Variable Costs Printing, binding and paper Bookstore discounts Sales staff commissions Author royalties General and administrative costs Total variable costs per copy List price per copy
Dollar Amount
$15,750 32,750 51,500 $100,000
$22.50 25.00 8.25 10.00 26.25 $92.00 $100.00
Fixed costs costs of $100,000 can be estimated estimated quite accurately. accurately. Variable Variable costs are linear and set by contract. contract. List prices prices are variable, variable, but competition competition keeps prices prices within a narrow range. Variable costs for the proposed proposed book are $92 a copy, and the expected
202
Chapter 8 wholesale price is $100. This means means that each copy sold provides the publisher with an $8 profit contribution. A.
Estimate Estimate the volume volume necessary necessary to to reach reach a breakeve breakeven n level of of output.
B.
How many many textbooks would would have to be sold to generate a profit contribution contribution of $20,000?
C.
Calc Calcul ulat atee the the econ econom omic ic prof profit it cont contri ribu buti tion on or loss loss resu result ltin ing g from from the the acceptance acceptance of a book club offer to buy 3,000 copies directly from the publisher publisher at a price of $77 per per copy. Should the offer be accepted?
P8.6
SOLUTION SOLUTION
A.
Applying the breakeven formula, the breakeven sales volume is 12,500 units, calculated as Q=
$100,000 $8
= 12,500 units . B.
To fi find the the nu number of of co copie pies one one mus mustt se sell to to ear earn n a $20 $20,00 ,000 pro profit, si simply ad add the the $20,000 profit requirement to the book's fixed costs, and then divide this total amount by the profit contribution per unit. The sales volume required in this case is 15,000 books, found found as:
Q= =
Fixed Costs + Profit Requiremen t Profit Contributi on $100,000 + $20,000
$8 = 15,000 units . C.
Becaus causee fix fixeed co costs sts do do not not var vary wi with re respect pect to cha chang ngees in in the the num numbe berr of of boo book ks sol sold, d, they should be ignored. Variable costs per copy are $92, but note that $25 of this cost represent representss bookstore bookstore discounts discounts.. Because Because the 3,000 copies are being sold directly directly to the club, this cost cost will not be incurred. incurred. Hence, the relevant relevant variable cost is only $67 (= $92 - $25). $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. $30,000. Assuming that that these 3,000 copies would 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.
Cost Analysis and Estimation P8.7
203
Pow Power Broke rokers rs,, Inc. Inc. (PB (PBI), I), a disc discou ount nt brok broker erag agee firm firm,, is Cost Elasticity Elasticity . contem contemplat plating ing opening opening a new regiona regionall office office in Provi Providen dence, ce, Rhode Rhode Island Island.. An accounting accounting cost analysis of monthly 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 $0.006Q where where AV AVC C is averag averagee variab variable le costs costs (in dollars) dollars) and Q is output output me measu asured red by number of stock and bond trades. A typical typical stock or bond trade results results in $100 gross commission commission income, income, with PBI paying paying 35% of this this amount amount to its its sales sales represent representatives. atives. A.
Estimate Estimate the trade volume necessary necessary for PBI to reach reach a target target return of $7,500 per month month for for a typical typical office. office.
B.
Estimate Estimate and interpret interpret the elasticity elasticity of cost with respect respect to output at the trade volume found in part A.
P8.7
SOLUTION 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: Q= Q= Q=
Total fixed costs + Target return P - AVC $4,500 + $7,500 (1 - 0.35)($100 ) - $59 + $0.006 Q 12,000 6 + 0.006 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,
204
Chapter 8
- b ± b2 - 4 ac Q= 2a - 6 ± 62 - 4(0.006)(- 12,000) = 2(0.006) = =
- 6 ± 324 0.012 - 6 ± 18
0.012 = 1,000 or - 2,000 trades per month . 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, ε c
=
∆ TC/TC ∆ TC Q = × ∆ Q/Q ∆ Q TC
where TC = Fixed xed Cos Costts + Varia ariabl blee Cos Costts = $4,500 $4,500 + ($5 ($59 9 - $0.00 $0.006Q 6Q)Q )Q = $4,5 $4,500 00 + $59 $59Q Q - $0. $0.00 006Q 6Q2 At Q = 1,000, TC = $4,5 $4,500 00 + $59 $59(1 (1,0 ,000 00)) - $0.0 $0.006 06(1 (1,0 ,000 002) = $4,500 $4,500 + $59,0 $59,000 00 - $6,00 $6,000 0 = $57,500 Therefore, at Q = 1,000 εc =
∆ TC Q × ∆ Q TC
= ($59 ($59 - $0.01 $0.012Q 2Q)) × Q/T Q/TC C = ($59 ($59Q Q - $0.0 $0.012 12Q Q2)/TC
Cost Analysis and Estimation
205
= $59(1,0 $59(1,000) 00) - $0.012( $0.012(1,00 1,000 02)/57,500 = 0.82 Since εC = 0.82 < 1, 1, economies of scale are suggested. suggested. A 1% increase in output leads to a 0.82% increase in costs, and average costs will will fall as output expands. P8.8
Appa Appala lach chia ia Beve Bevera rage ge Comp Compan any, y, Inc. Inc. is cons conside iderin ring g Multiplant Multiplant Operation Operation . alternative alternative proposal proposalss for expansio expansion n into the Midwest. Midwest. Alternative Alternative 1: Construct Construct a single plant in Indianapolis, Indianapolis, Indiana, with a monthly production production capacity of 300,000 cases, a monthly fixed cost of $262,500, and a variable cost of $3.25 per case. Alternative Alternative 2: Construct Construct three three plants, one each each in Muncie, Muncie, Indiana; Indiana; Normal, Normal, Illinois; Illinois; and Dayton, Ohio, with capacities of 120,000, 100,000, and 80,000, respectively, and monthly monthly fixed costs of $120,000, $110,000, $110,000, and $95,000 each. Variable Variable costs would be only $3 per case case because because of lower lower distribu distribution tion costs. costs. To achieve achieve these these cost cost savings, savings, sales from each smaller plant would be limited limited to demand demand within its home state. The total estimated estimated monthly monthly sales volume volume of 200,000 200,000 cases cases in these three Midwestern Midwestern states states is distributed distributed as follows: follows: 80,000 cases cases in Indiana, 70,000 70,000 cases in Illinois, Illinois, and 50,000 50,000 cases cases in Ohio. Ohio. A.
Assuming Assuming a wholesale wholesale price of $5 per case, case, calculate calculate the breakeven breakeven output quantities for each alternative.
B.
At a wholesale wholesale price of $5 per case in all states, states, and assuming assuming sales sales at the projected projected levels, which which alternative expansion expansion scheme scheme provides provides Appalachia Appalachia with the highest profit per month?
C.
If sales increase to production capacities, capacities, which alternative alternative would prove to be more profitable?
P8.8
SOLUTION SOLUTION
A.
The breakeven output quantity for the single plant alternative is: Q =
=
TFC P - AVC $262,500 $5 - $3.25
= 150, 150,00 000 0 cases cases per per mont month h The breakeven output quantities for the multiple plant alternative is:
206
Chapter 8
QMuncie =
$120,000 $5 - $3
= 60,0 60,000 00 cas cases es per per mon month th Q Normal = Normal
$110,000 $5 - $3
= 55,00 55,000 0 case casess per per mon month th QDayton =
$95,000 $5 - $3
= 47,5 47,500 00 cas cases es per per mon month th Thus, the firm-level firm-level breakeven quantity for the multiple plant plan t alternative is: Q = 60,000 + 55,000 + 47,500 = 162, 162,50 500 0 case casess per per month month provided that demand was distributed among the states in amounts equal to the breakeven quantities for each each individual plant. B.
Single plant alternative:
Cost Analysis and Estimation
207
π
= TR - TC = P × Q - TFC - AVC AVC × Q = $5(200, $5(200,000) 000) - $26 $262,5 2,500 00 - $3.2 $3.25(2 5(200,0 00,000) 00) = $87,500
Multiple plant alternative: π
= TR - TC = P Η Q - TFCM - TFC N - TFCD - AVC Η Q = $5(200 $5(200,000) ,000) - $120,0 $120,000 00 - $110,000 $110,000 - $95,000 $95,000 - $3(200,0 $3(200,000) 00) = $75,000
Management would prefer the single plant alternative because of its greater profitability. C.
Single plant at full capacity: π
= TR - TC = P × Q - TFC TFC - AVC × Q = $5(300,0 $5(300,000) 00) - $262,50 $262,500 0 - $3.2 $3.25(3 5(300,0 00,000) 00) = $262,500
Multiple plants at full capacity: π = TR - TC = P × Q - TFCM - TFC N - TFCD - AVC × Q = $5(300,0 $5(300,000) 00) - $120,0 $120,000 00 - $110,00 $110,000 0 - $95,000 $95,000 - $3(300,0 $3(300,000) 00) = $275,000 At peak capacity, capacity, management would prefer the multiple multiple plant option option because because of its greater profitability.
208 P8.9
Chapter 8 Thomas Winery plans to open a new production production facility in Learning Learning Curves. The St. Thomas the Napa Valley Valley of California. Based Based on information information provided by the accounting accounting department, the company estimates fixed costs of $250,000 per year and average variable costs of AVC = $10 + $0.01Q $0.01Q where AVC is average variable cost (in dollars) and Q is output measured in cases of output per year. A.
Estimate Estimate total cost and average average total cost for the coming coming year at a projected projected volume of 4,000 cases.
B.
An increase increase in worker worker productivity productivity because because of greater greater experience experience or learning learning during during the course course of the year resulted resulted in a substan substantial tial cost saving saving for the company. company. Estimate Estimate the effect effect of learning on average average total cost if actual total cost was $522,500 at an actual volume of 5,000 cases.
P8.9
SOLUTION SOLUTION
A.
The total variable cost function for the coming year is: TVC
= AVC × Q = ($10 ($10 + $0 $0.01Q .01Q)Q )Q = $10 $10Q + $0.0 $0.01 1Q2
At a volume of 4,000 units, estimated total cost is: TC
= TFC + TVC = $250 $250,0 ,000 00 + $10Q $10Q + $0. $0.01 01Q Q2 = $250,000 $250,000 + $10( $10(4,000 4,000)) + $0.01(4 $0.01(4,000 ,0002) = $450,000
Estimated average cost is: AC
= TC/Q = $450 $450,0 ,000 00/4 /4,0 ,000 00 = $112 $112.5 .50 0 per per case case
Cost Analysis and Estimation
B.
209
Without learning, estimated total cost and average total cost at a volume of 5,000 cases are: TC
= $250 $250,0 ,000 00 + $10 $10(5 (5,0 ,000 00)) + $0.0 $0.01( 1(5 5,000 ,0002) = $550,000
AC
= TC/Q = $550 $550,0 ,000 00/5 /5,0 ,000 00 = $110 pe per ca 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 $522,5 ,500 00/5 /5,0 ,000 00 = $104 $104.5 .50 0 per per cas casee Therefore, greater experience or learning has resulted in an average cost saving of $5.50 per case since: Lear Learni ning ng Effe Effect ct
= Actu Actual al AC - Est Estimat imated ed AC = $10 $104.50 4.50 - $110 $110 = ($5. ($5.50 50)) per per case case
Alternatively, Learning Rate
=
AC2 1 × 100 AC1
=
1 - $104.50 × 100 $110.00
= 5%
210 P8.10
Chapter 8 Untouc Untouchab hable le Packag Packagee Servic Servicee (UPS) (UPS) offers offers Degree Degree of Operating Operating Leverage. Leverage. overni overnight ght packag packagee delive delivery ry to Canad Canadian ian busines businesss custom customers ers.. UP UPS S has recently recently decide decided d to expand expand its faciliti facilities es to better better satisfy satisfy current current and projec projected ted demand demand.. Current volume totals two million packages per week at a price of $12 each, and average average variable costs costs are constant constant at all output levels. levels. Fixed costs costs are $3 million per week, week, and profit contribution contribution averages averages one-third one-third of revenues revenues on each delivery. delivery. After completion completion of the expansion project, fixed costs will double, 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 Assuming that volume remains remains at two million packages packages per week, calculate the change in the degree of operating leverage that is due to expansion.
C.
Again assuming assuming that volume volume remains remains at two million million packages packages per week, what what is the effect of expansion on weekly profit?
P8.10
SOLUTION SOLUTION
A.
Average variable costs are $8 since: (P - AVC)Q P - AVC
= 1/3P(Q) = 1/3P
AVC
= 2/3($12)
AVC
= $8
Therefore, the breakeven levels of output before and after expansion are: Before Expansion: QB = =
TFC P - AVC $3,000,000
$12 - $8 = 750,000 packages After Expansion:
Cost Analysis and Estimation
211
QA = =
TFC P - AVC $6,000,000
$12 - $6 = 1,000,000 packages The change in the weekly breakeven output level due to expansion is: Change in Breakeven
= QA - QB = 1,000,000 - 750,000 = 250,000 packages per month
B.
The degrees of operating leverage before and after expansion are: Before Expansion: DOLB = =
Q(P - AVC) Q(P - AVC) - TFC 2,000,000( $12 - $8) 2,000,000( $12 - $8) - $3,000,000
= 1.6 After Expansion: DOLA = =
Q(P - AVC) Q(P - AVC) - TFC 2,000,000( $12 - $6) 2,000,000( $12 - $6) - $6,000,000
=2 The change in degree of operating leverage due to expansion is: Change in DOL
= DOLA - DOLB = 2 - 1.6
C.
= 0.4 Profits before before and after expansion are:
212
Chapter 8
Before Expansion: πB = (P - AVC AVC)Q )Q - TFC TFC = ($12 ($12 - $8)2,000 $8)2,000,000 ,000 - $3,000,000 $3,000,000 = $5 milli illio on After Expansion: πA = (P - AVC AVC)Q )Q - TFC TFC = ($12 ($12 - $6)2,00 $6)2,000,000 0,000 - $6,000,00 $6,000,000 0 = $6 milli illio on The change in profits due to expansion is: Change in π = πA - πB = $6,000 $6,000,000 ,000 - $5,00 $5,000,0 0,000 00 = $1 mill millio ion n per per wee week k
Cost Analysis and Estimation
213
CASE STUDY FOR CHAPTER 8 Estimating Estimating Hospitalization Hospitalization Costs Costs for Regional Regional Hospitals
Cost estimation and cost containment are an important concern for a wide range of for-profit and not-for-pro not-for-profit fit organizations organizations offering health-care health-care services. services. For such organizations, organizations, the accurate accurate meas me asur urem emen entt of cost costss per per patie patient nt day day (a me meas asur uree of outpu output) t) is nece necess ssar aryy for for effe effecti ctive ve manageme management. nt. Similarly, Similarly, such cost estimates estimates are of significant interest interest to public officials at the federal, federal, state, and local government government levels. For example, many many state Medicaid Medicaid reimburse reimbursement ment programs programs base base their their payment payment rates rates on historical historical account accounting ing measures measures of average average costs costs per unit unit of service. service. However However,, these historical historical average average costs may or may not be relevant relevant for hospital hospital management decisions. During periods of substantial excess capacity, the overhead component of average costs may 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. usage. From From a public-polic public-policyy persp perspect ective ive,, a furthe furtherr potentia potentiall proble problem m arises arises when hospital 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 average publicl publicly-s y-supp upporte orted d Medica Medicaid id patient patient actuall actuallyy costs costs more more than than that that receiv received ed by non-Medicaid patients, Medicaid reimbursement based on average costs would be inequitable to providers providers and could create create access access barriers barriers for for Medicaid Medicaid patients. patients. As an alternative alternative to accounting accounting cost estimation estimation methods, methods, one might might consider consider using engineering engineering techniques techniques to estimate nursing nursing costs. For example, example, the labor cost cost of each type of service service could be estimated estimated as the product product of an approximatio approximation n of the time required to perform perform each service times times the estimated wage rate rate per unit of time. Multiplying this figure by an estimate of the frequency frequency of service service gives an engineering engineering estimate estimate of the cost of the service. service. A possible limitat limitation ion to the accuracy accuracy of this this enginee engineering ring cost-est cost- estima imation tion method method is that that treatm treatment ent of a variety variety of illnesses illnesses often requires requires a combination combination of nursing services. services. To the extent that multiple services services can be provided provided simultaneously, simultaneously, the engineering engineering technique will tend to overstate overstate actual costs unless the effect of service "packaging" is allowed for. Cost Cost estima estimation tion is also also possib possible le by me means ans of a carefu carefully lly designe designed d regres regression-base sion-based d 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. period. With sufficient sufficient variability variability in cost and service service levels over time, useful estimates estimates of variable labor costs become possible for each type of service and for each patient category (Medicaid, (Medicaid, non-Medicaid, etc.). etc.). An important important advantage of a regressionregression-based based approach approach is that it explicitly allows for the effect of service packaging on variable costs. For example, if shots and and woun wound-dr d-dres essi sing ng serv servic ices es are are typi typica call llyy prov provid ided ed toge togeth ther er,, this this will ill be refl reflec ecte ted d in the the regression-based estimates of variable costs per unit. Long-run costs per nursing nursing facility can be estimated estimated using either crosscross- section section or time-series time- series methods. methods. By relating total facility costs to the service service levels provided by a number of hosp hospita itals, ls, nurs nursing ing home homes, s, or out-pati out-patien entt care care facili facilitie tiess duri during ng a spec specifi ificc perio period, d, usefu useful l cros cross-sec s-sectio tion n estim estimat ates es of total total serv servic icee cost costss are are poss possib ible. le. If case case mixe mixess were were to vary vary
214
Chapter 8
dramatically according to type of facility, then the type of facility would have to be explicitly accounted accounted for in the regression regression model analyzed. analyzed. Similarly, Similarly, if patient mix or serviceservice- provider provider efficiency is expected to depend, at least in part, on the for-profit or not-for-profit organization status of the care facility, facility, the regression regression model model must also recognize recognize this factor. factor. These These factors plus price-level adjustments adjustments for inflation would be accounted accounted for in a time-series approach approach to nursing cost estimation. To illustrate a regression-based approach to nursing cost estimation, consider a hypothetical anal analys ysis is of varia variable ble nurs nursing ing costs costs cond conduc ucted ted by the the South Southea east st Asso Associ ciati ation on of Hosp Hospita ital l Administra Administrators tors (SAHA) (SAHA).. Using Using confidential confidential data provided provided by 40 40 regional regional hospitals, hospitals, SAHA SAHA studied studied the relation between nursing costs per patient day and four typical categories of nursing services. These These annual data appear in Table 8.2. The four categories categories of nursing services services studied studied include shots, intravenou intravenouss (IV) therapy, therapy, pulse taking and monitoring, monitoring, and wound dressing. dressing. Each service service is measured in terms of frequency per patient day. An output of 1.50 in the shots service category means means that, on average, patients patients received one and one-half shots per day. day. Similarly, 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 categories of nursing nursing services, services, the not-for- profit or for-profit status of each hospital hospital is also indicated. indicated. Using Using a "dummy" "dummy" (or binary) binary) variable variable approach, the profit status variable equals 1 for the 8 for-profit hospitals included in the study and zero for the remaining remaining 32 not- for-profit h Cost estimation results for nursing costs per patient day derived using a regression-based approach are shown in Table 8.3. A.
Interpret Interpret the coefficient coefficient of determi determination nation (R2 ) estimate estimated d for the nursing nursing cost function. function.
B.
Describe Describe the economi economicc and statistical statistical significance significance of each each estimated estimated coefficient coefficient in the nursing cost function.
C.
Average Average nursing nursing costs for the eight for-profit hospitals hospitals in the sample sample are only $318.52 per patient day, or $33.07 per patient day less than the $351.59 average cost experienced experienced by the 32 not-for- profit hospitals. hospitals. How can can this fact be reconciled 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-funde publicly-funded d nursing nursing cost reimbu reimbursem rsement ent system systems? s?
CASE STUDY SOLUTION A.
Cost estimation estimation results results provided indicate that R2 = 76.81 76.81%, %, meaning meaning that that 76.8 76.81 1 percent of the total variation in nursing costs per patient day can be explained by the five factors factors studied. studied. For cross-se cross-sectio ctional nal analysis, analysis, such a level level of cost explanation explanation is often quite adequate for gaining useful insight concerning cost characteristics.
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
Cost Analysis and Estimation
215
vari variab able le indi indicat cates es an aver averag agee nurs nursin ing g labo laborr cost cost of roug roughl hly y $72. $72.76 76 per per shot shot.. Similarly, IV therapy results in $215.68 in nursing costs per patient day, pulse taking and monitoring monitoring costs $36.24, $36.24, and wound dressing dressing costs costs $156.04 $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 service categories. categories. This sugges suggests ts that the greater greater efficiency 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 output differences, 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 captur captured ed by the four four servic servicee catego categorie ries, s, for-pr for-profi ofitt organi organizat zations ions appear appear to have 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 Despite obvious obvious limitations, limitations, such such a regressi regression-bas on-based ed approach approach can provide provide useful 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, Illinois, West West Virgi irgini nia, a, Ohio, hio, and and Maryl arylan and d have have impl implem emen ente ted d Medic edicai aid d nurs nursin ing g home home reimburse reimbursement ment system systemss based based on this concept, concept, and several several other states states have similar case-mix reimbursement systems under development.
