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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
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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.
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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.
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Cost Analysis and Estimation 205
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
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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
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Cost Analysis and Estimation 207
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
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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
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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.
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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.
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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
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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:
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Cost Analysis and Estimation 213
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
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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
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Cost Analysis and Estimation 215
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
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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
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Cost Analysis and Estimation 217
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
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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?
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Cost Analysis and Estimation 219
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
= 60,000 cases per month
QNormal = $3 - $5
$110,000
= 55,000 cases per month
QDayton = $3 - $5
$95,000
= 47,500 cases per month
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:
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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
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Cost Analysis and Estimation 221
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:
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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,
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Cost Analysis and Estimation 223
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:
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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:
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Cost Analysis and Estimation 225
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
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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.
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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.