This document discusses methods for estimating cost functions and determining the optimal scale of operations. It covers: 1. Short-run and long-run cost functions which can be estimated statistically using regression analysis and engineering cost techniques. 2. Factors that influence costs such as output mix, input prices, and technology. Controling for these is important to isolate the cost-output relationship. 3. Break-even analysis and contribution margin which are used to determine output levels required to cover fixed costs and earn profits. The relevance of different costs depends on whether alternatives are avoidable. 4. Operating leverage and business risk which depend on a firm's fixed costs, sales variability, and degree of operating leverage.

1. CHAPTER 9
ESTIMATING COST FUNCTIONS
To make optimal pricing and production decisions, the firm must have knowledge of the shape and
characteristics of its short-run cost function.
Cost function is a schedule,graph,or mathematical relationshipshowing the total,average,or marginal
cost of producing various quantities of output.
To decide whether to accept or refuse an order offered at some particular price, the firm must identify
exactly what variable cost and direct fixed costs the order entails.
Long-run cost function is associated with the longer-term planning period in which all the inputs to the
production process are variable and no restrictions are placed on the amount of an input that can be
employed in the production process.
Economiccost isrepresentedbythevalueof opportunitiesforgone,whereasaccountingcostismeasured
by the outlays that are incurred.
If extractioncostsof the companybeingstudiedare low,thesetwocostmethodswilldiverge because the
equilibriummarketprice isalwaysdeterminedbythe considerablyhighercostof the marginal producer.
Directcosts exclude all overheadandanyotherfixedcostthat must be allocated(so-calledindirectfixed
costs).
Depreciation can be dividedinto two components: time depreciation representsthe decline in value of
an asset associated with the passage of time, and use depreciation represents the decline in value
associated with use.
 onlyuse depreciationvarieswiththe rate of output,onlyuse depreciationisrelevantin
determining the shape of the cost-output relationship.
Capital asset values are often stated in terms of historical costs. In periods of rapidly increasing price
levels, this approach will tend to understate true economic depreciation costs.
Cost is a function of other factors, such as output mix, the length of production runs, employee
absenteeism and turnover, production methods, input costs, and managerial efficiency.
To isolate the cost-output relationship itself, one must control for these other influences by:
• Deflatingor detrendingthe cost data. Wheneverwage ratesorraw material priceschange significantly
overthe periodof analysis,onecandeflatethe costdatatoreflectthese changesinfactorprices.Provided
suitable price indicesare availableorcanbe constructed,costsincurredatdifferentpointsintime canbe
restatedas inflation-adjustedreal costs. (Two assumptionsare implicitinthisapproach: No substitution
takes place between the inputs as prices change, and changes in the output level have no influence on
the pricesof the inputs.Formore automatedplantsthat incorporate onlymaintenance personnel,plant
engineers,andmaterial supplies,theseassumptionsfitthe realityof the productionprocessquitewell.)

2. • Using multiple regression analysis. Suppose a firm believes that costs should decline gradually over
time asaresultof innovativeworkersuggestions.One waytoincorporatethiseffectintothecostequation
would be to include a time trend t as an additional explanatory variable: C = f (Q, t)
The Form of the Empirical Cost-Output Relationship
Total cost function in the short run (SRTC)- S-shaped curve that can be represented by a cubic
relationship:
SRTC = a + bQ + cQ2 + dQ3
U-shaped marginal and average cost functions then can be derived from this relationship.
MC = d(SRTC)/ dQ = b+ 2cQ + 3dQ2
The average total cost function is
ATC = SRTC/Q = a/Q + b + cQ + dQ2
.If the resultsof a regressionanalysisindicate thatthe cubicterm(Q3 ) is notstatisticallysignificant,then
short-run total cost can be represented by a quadratic relationship:
SRTC = a + bQ + cQ2
In thisquadraticcase,total costsincrease at an increasingrate throughoutthe typical operatingrange of
output levels. The associated marginal and average cost functions are:
MC = d(SRTC)/ dQ = b+ 2cQ
ATC = SRTC/Q = a/Q + b + cQ
This quadratic total cost relationship implies that marginal costs increase linearly as the output level is
increased.
Statistical Estimation of Short-Run Cost Functions
Short-run cost functions have been estimated for firms in a large number of different industries—for
example, food processing, furniture, railways, gas, coal, electricity, hosiery, steel, and cement.
Statistical Estimation of Long-Run Cost Functions
Long-run costs can be estimatedovera substantial periodof time in a single plant(time-seriesdata) or
with multiple plants operating at different rates of output (cross-sectional data).

Cross-sectional data assumes that each firm is using its fixed plant and equipment and
variable inputs to accomplish minLRAC productionfor that plant size along the envelop
of SRAC curves.

Time-seriesdataassumes thatinputprices,the productiontechnology,andthe products
offered for sale remain unchanged.
o cross-sectional data are more prevalent in estimating long-run cost functions

3. Determining the Optimal Scale of an Operation
Economies of scope
- Economiesthatexistwheneverthe costof producingtwo(ormore) productsjointlybyone plantor
firm is less than the cost of producing these products separately by different plants or firms.
-Occur whenever inputs can be shared in the production of different products.
For example, in the airline industry, the cost of transporting both passengers and freight on a single
airplane islessthanthe cost of using two airplanes to transport passengers and freight separately.
Engineering cost techniques
- A method of estimating cost functions by deriving the least-cost combination of labor, capital
equipment,andraw materialsrequiredtoproduce variouslevelsof output,usingonlyindustrial
engineering information
-
provide analternativewaytoestimate long-runcostfunctionswithoutusingaccountingcostdata.
Using production data, the engineering approach attempts to determine the least-cost
combinationof labor,capital equipment,andraw materialsrequiredtoproduce variouslevelsof
output.
Advantages
-It is generally much easier with the engineering approach to hold constant such factors as input
prices,productmix,and productefficiency,allowingone toisolate the effectsoncostsof changesin
output.
-Use of the engineering method avoids some of the cost-allocation and depreciation problems
encountered when using accounting data.
Survivor technique
- involvesclassifyingthe firmsin an industryby size and calculatingthe share of industryoutput
coming from each size class over time.11 If the share decreases over time,then thissize class is
presumedtobe relativelyinefficientandto have higheraverage costs.Conversely,anincreasing
share indicates that the size class is relatively efficient and has lower average costs.
The rationale for this approach isthat competitionwill tendtoeliminate those firmswhose size
is relatively inefficient, allowing only those size firms with lower average costs to survive.
Limitation
Because the technique does not use actual cost data in the analysis, it cannot assess the
magnitude of the cost differentials between firms of varying size and efficiency.

4. Break-even analysis
- The study of the interrelationships among a firm’s sales, costs, and operating profit at various
anticipated output levels.
- based on the revenue-output and cost-output functions of microeconomic theory.
- Total revenue is equal to the number of units of output sold multiplied by the price per unit.
- The difference between total revenue and total cost at any level of output representsthe total
profit that will be obtained.
- Basic linear break-even chart shows the constant selling price per unit and a constant variable
cost per unit yield the linear TR and TC functions
- Operatingprofitisequal tothe difference betweentotal revenues(TR) andtotal (operating)costs
(TC).
- break-even point occurs where the total revenue and the total cost functions intersect.
- If a firm’soutputlevel is belowthisbreak-evenpoint(i.e.,if TR< TC),it incurs operatinglosses.If
the firm’s output level is above this break-even point (if TR > TC), it realizes operating profits.
Algebraic Method
To determine the firm’sbreak-evenpointalgebraically,one mustsetthe total revenue andtotal
(operating) costfunctionsequal toeachotherandsolve theresultingequationforthe break-evenvolume.
Total revenue is equal to the selling price per unit times the output quantity:
TR = P × Q
Total (operating) costisequal to fixedplusvariable costs,where the variable costis the product
of the variable cost per unit times the output quantity:
TC = F + (V × Q)
Setting the total revenue and total cost expressions equal to each other and substituting the
break-even output Qb for Q results in
TR = TC or
PQb
= F + VQb
Difference between the selling price per unit and the variable cost per unit, P – V, is referred to as the
contribution margin- It measures how much each unit of output contributes to meeting fixed costs and
operating profits.
Thus, the break-even output is equal to the fixed cost divided by the contribution margin.
Doing a Break-Even versus a Contribution Analysis
Break-even analysis- assumes that all types of costs except the incremental variable cost of additional
unit sales are avoidable and asks the question of whether sufficient unit sales are available at the
contribution margin to cover all these relevant costs. If so, they allow the firm to earn a net profit.
Normally,these questionsarise atentryand exitdecisionpointswhere afirmcan avoidessentiallyall its
costs if the firm decides to stay out or get out of a business.

5. Contributionanalysis- Itcalculateswhethersufficientgrossoperatingprofitsresultfromthe incremental
salesattributable tothe advertisingcampaign,the new product,orthe promotiontooffsetthe proposed
increase infixedcost. It assumesthat many fixedcostsremainunavoidable andare therefore irrelevant
to the decision (indirect fixed costs), while other fixed costs will be newly committed as a result of the
decision(directfixedcosts) andtherefore could be avoided by refusing to go ahead with the proposal.
Some Limitations of Break-Even and Contribution Analysis
Composition of Operating Costs- In doing break-even analysis, one assumesthat costs can be classified
as eitherfixedorvariable.Infact,some costsare partlyfixedandpartlyvariable.Furthermore,some fixed
costs increase in a stepwise manner as output is increased; they are semi-variable.
For example,machinery maintenance is scheduled after 10 hours or 10 days or 10 weeks of use. These
direct fixed costs must be considered variable if a batch production decision entails this much use.
Multiple Products- The break-evenmodel alsoassumesthatafirmisproducingandsellingeitherasingle
product or a constant mix of different products. In many cases the product mix changes over time, and
problems can arise in allocating fixed costs among the various products.
UncertaintyStill- anotherassumptionof break-evenanalysisisthatthe sellingpriceandvariable costper
unit,aswell asfixedcosts,are knownateachlevel of output.Inpractice,these parametersare subjectto
uncertainty. Thus, the usefulness of the results of break-even analysis depends on the accuracy of the
estimates of the future selling price and variable cost.
Inconsistency of Planning Horizon- break-even analysis is normally performed for a planning period of
one year or less; however, the benefits received from some costs may not be realized until subsequent
periods.Forexample,researchanddevelopmentcostsincurredduringaspecificperiodmaynotresultin
newproductsforseveral years.Forbreak-evenanalysistobe adependable decision-makingtool,afirm’s
operatingcostsmustbe matchedwithresultingrevenues for the planning period under consideration.
Operating Leverage
Operatingleverage involvesthe use of assets thathave fixedcosts.Forexample depreciation. A firmuses
operating leverage in the hope of earning returns in excess of the fixed costs of the assets, thereby
increasing the returns to the owners of the firm.
Degree of operating leverage (DOL)- Is the percentage change in a firm’s earnings before interest and
taxes (EBIT) resulting from a given percentage change in sales or output.
DOL at Q = Percentage change in EBIT/ Percentage change in Sales
The degree of operatingleverageisanalogoustothe elasticityof demandconcept(e.g.,price andincome
elasticities)becauseitrelatespercentagechangesinone variable(EBIT) topercentage changesinanother
variable (sales).
Business Risk
refers to the inherent variability or uncertainty of a firm’s EBIT or earnings before interestand
taxes.

6. Factors:
 DOL- The greater a firm’s DOL, the larger the change in EBIT will be for a givenchange in sales.
Thus, all other things being equal, the higher a firm’s DOL, the greater the degree of business
risk.
 Variabilityor uncertainty of sales- A firm with high fixedcosts and stable saleswill have a high
DOL, but it will also have stable EBIT and, therefore, low business risk. Public utilities and
pipelinetransportationcompaniesare examplesof firmshavingthese operating characteristics.
 Uncertainty concerning selling prices and variable costs- A firm having a low DOL can still have
high business risk if selling prices and variable costs are subject to considerable variabilityover
time.
Break-Even Analysis and Risk Assessment
Two most important decisions companies make:
1. Optimal pricing decisions (Pricing techniques: target pricing techniques, cost plus, mark-up and
etc)
2. Production decision (e.g volume, product type, scheduling, special orders)
Issues:
Common potential issues:
1. Measures depreciation- e.g. time component or usage component
2. Measuring Variable costs-
3. Valuation of capital assets (historical vs Current Market Value)
1. Inflation
2. Price levels
3. Rate of obsolescence- the equipment is fully depreciated
Controlling for other variables
-It is very important to isolate cost-output relationship (better predictive indicator)
Consideration for the ff:
1. Deflating or detrending cost date (normalization)
Harmonization- applyingtwodifferentpolicies appliedtoaparticularaccountingelement
2. Using multiple regression analysis (multi-factor models)
 Adding other factors to the model makes it more accurate
3. Economies of scope
 Synergy
The form of the Empirical cost-output relationship
The short run total cost- S curve because of the effectsof economiesof scale and diseconomies of scale

7. If regression analysis results to statistically not significant, you can use quadratic total cost function
Regression is not significant when, the P value is less than 5.
Marginal cost and average total cost- is U curve
Statistical Estimation of short-run cost functions
Example: 1. Johnston (multi-product food processing) 2. Electricity Generation
Excel is one of the most basic software we use in practice.
Statistical Estimation of long-run cost functions
Usually 10-30 years to project
Multiple scenarios usually includes sophisticated assumptions (heroic) and variables
Heroic assumptions also called as hypothetical assumptions
Not much used in short-run decision making- serves as a guide only
Determining the optimal scale of an operation
-Economies of scale
-size of the market
-Degree of variable costs and fixed costs (ex retail and utility generator)
-Inutilitygenerators,theyare heavilyinvestedinfixedcosts, angilahangshape isnegative curve
or downward sloping. The more they manufacture, the unit cost also drives down
Theyjustneedto sell ordistribute asmuchas possible,sothattheycan achieve the lowest cost
-Objective in retail which is heavily invested in variable cost, is to achieve optimum level.
Economies of scale vs economies of scope
Economies of scope
-production of one good reduces the cost of producing another related good. It occurs when
producingawidervarietyof goodsorservices ismore costeffectiveforafirmthanproducingless
of a variety or producing each good independently.
- Arise whenunitcosts are lowerwhena businessproducesa WIDER RANGE OF PRODUCT rather
than specialize in just one or a few products.
- Ex: A company wants to increase its product line and remodels its manufacturing building to
produce a variety of electronic devices such as laptops, tablets and phones. Since the cost of

8. operating the manufacturing bldg. is spread out across a variety of products, the average total
cost of production decreases (this is called the synergy effect). Because of that, your marginal
cost and your average cost is expected to decline.
Economies of scale
- Arise when unit costs fall as output rises
Engineering cost techniques
- Focuses more on the production data
- Common objective: to attempt to determine the least cost combination (combination of labor,
capital, equipment and raw materials required to produce various levels of output.
DISADVANTAGE
- They are costly to determine and sometime the cost outweigh the benefit.
ADVANTAGE
- Theyare much easiertoholdconstantsasfactorsas inputprices,productmix andprocesschoice
efficiently, allowing one to isolate cost changes in output.
The survivor technique
- Hypothesized that a plant or efficient size would survive in competition with other plants.
Break even analysis
- The calculation and examination of the margin of safety for an entity based on the revenues
collected and associated costs.
- Analyzing different price levels relating to various levels of demand a business uses break-even
analysistodetermine whatlevelof salesare necessarytocoverthe company’s total fixed costs.
- A demand-side analysis would give a seller significant insight regarding selling capabilities.
- Contribution margin is equals to fixed cost
- When your profit is positive, then you are operating at safety level, but once you drop down at
that safety level, you are already in the loss spectrum.
Relevant Range- in which the variable and fixed remains true
Algebraic Method- relationship between the revenue, fixed costs, and variable costs
BEP= Fixed costs/ Price per unit - Variable cost per unit (also known as Contribution
Margin)
With the BEP, a business is able to determine the price of the product, how many need to be sold,
identify and potentially reduce excessive fixed costs allowing the business to achieve profit.

9. Limitations of Break even analysis
1. Multiple products- okaylanguntaif dili interchangeable ormuag substitute andisakaproductto
the other but if there are multiple products which are compliment with each other, and the
demand goes with each other, this makes your BEP not useful.
2. Uncertainty- dependentonVCandFC. If these uncertaintiesaffectthe FCandVC, there mightbe
complications in our break even analysis.
3. Inconsistency of planning horizon- The normal planning horizonin using BEP is usually 1 year. If
the historical WA NAHUMAN
BREAK EVEN VS CONTRIBUTION ANALYSIS
-BEA assumes all types of costs (except VC) are avoidable.
-CA goes beyond fixed and variable costs and determine the nature of costs in terms of control
(avoidable or not avoidable)
-CA focuses on the incremental benefit of an alternative, that is:
LIMITATIONS OF CA
- Stepwise Operating Cost (semi-variable)- costs which increases in a certain production level.
o Regression ( y= a+bx)
o High low
Operating Leverage
- The use of fixed cost in an effort to increase expected returns.
- Measure of risk especially operating risks for businesses.
Degree of operating leverage- percentage change in earnings before interests in taxes divided by
percentage change in sales.
- A multiple thatmeasureshowmuchthe operatingincome of a companywill change inresponse
to a change insales.Companieswithlarge proportionof fixedcoststovariable costshave higher
levels of operating leverage.

10. -
DOL= (SALES- VARIABLE COSTS)/ PROFIT
Inherent business risk
Inherent business risk = variability of EBIT
The more your earnings becomes volatile, the higher the business risk. Using fixed costs makes your
volatility higher.
HIGHER DEGREE OF OPERATING LEVERAGE= HIGHER BUSINESS RISK