10
Quantitative CHAPTER Analyses of
Cost Functions
Learning Objectives
After studying this chapter, a student should be able to:
1. Assess the appropriateness and quality of the data set proposed for use in the
ordinary least squares method of analysis
2. Apply standardized mathematical manipulations to a set of data points and
produce the result of an OLS simple linear regression analysis with benchmark
statistics testing reliability
3. Identify and explain data problems encountered in estimating cost functions
4. Extend the analysis of data sets to include discontinuous linear cost functions
within a relevant range
5. Distinguish between the curvilinear cumulative average time learning model and
incremental unit time learning model
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CHAPTER OVERVIEW
Chapter 10 provides additional tools from those introduced in Chapter 3 for using costvolume-profit analysis so important to planning and control decisions made by managers.
CVP analysis uses costs categorized by behaviour, variable or fixed, in relation to a cost
driver. Recognizing that some costs display behaviour that does not fit either category,
the concepts and tools presented here enable one to sort a mixed or semivariable cost into
component parts of variable and fixed. The process of identifying and measuring costs as
variable or fixed based on past relationships is called cost estimation. Cost estimation is
useful for cost prediction in the decision-making process.
Basic assumptions for cost estimation are carefully described. Approaches to the process
of identifying and measuring costs using a cost function are explained. The analytical
basis of the process of cost estimation is illustrated in step form. Criteria for evaluation of
the relationship between a cost and a level of activity, the cost driver, as developed from
the cost estimation process are presented. One of the more troublesome aspects of cost
estimation, data problems, is described with specific illustrations. Some possible steps to
overcome these problems are included.
Themes carried through the text are integral to the chapter material: accounting systems
that provide useful information to managers for improved decision making, cost/benefit
analysis, behavioural and technological considerations, and different costs for different
purposes.
MyAccountingLab provides students with
diagnostic tests so they can identify areas where they need improvement.
These sample tests generate personalized study plans so students can be
efficient with their study time.
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CHAPTER OUTLINE
I. General issues in estimating cost functions
Learning Objective 1:
Assess the appropriateness and quality of the data set proposed for
use in the ordinary least squares method of analysis
A. Basic assumptions
1. Variations in the level of a single activity explain variations in total costs of a
cost object
2. A linear cost function adequately approximates cost behaviour within the
relevant range
a. Cost function: mathematical expression describing how costs change with
changes in the level of an activity
b. Linear cost function: y = a + bX
Do multiple choice 1.
Assign Exercises 10-16, 10-17, 10-18 or 10-19
Learning Objective 2:
Apply standardized mathematical manipulations to a set of data
points and produce the result of an OLS simple linear regression
analysis with benchmark statistics testing reliability
B. Examples of cost functions
1. Variable cost: y = bX when b is variable cost per unit and X is number of
units
2. Fixed cost: y = a when a is the amount of fixed cost in total
3. Mixed or semivariable cost: y = a + bX
C. Assumptions underlying cost classifications
1. Choice of cost object: cost item may be variable with respect to one cost
object but fixed with respect to another cost object
2. Time span: more costs are variable with longer time span
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3. Relevant range: costs may behave nonlinearly outside relevant range
II. Cause-and-effect criterion in choosing cost drivers
A. Cost estimation: the attempt to measure a past cost relationship between costs
and the level of an activity
B. Cause-and-effect relationship establishes economically plausible relationship
1. Physical relationship between the costs and the level of activity
2. Contractual arrangement
3. Implicit through logic and knowledge of operations
C. Cautionhigh correlation exists in cause-and-effect relationship but high
correlation does not mean a cause-and-effect relationship exists
Learning Objective 3:
Identify and explain data problems encountered in estimating cost
functions
1. Evaluate cost drivers of the estimated cost function
a. Economic plausibility
b. Goodness of fit
c. Slope of regression line: steep slope, stronger relationship; flat slope, weak
relationship
Do multiple choice 6.
Assign Exercise 10-26 and or Problem 10-31
D. Cost drivers and activity-based costing
1. Greater number and variety of cost drivers
2. Cost drivers evaluated using criteria described above
3. Emphasis on long-run relationships between the cost driver and cost
III. Other issues in determining how costs behave
1. Frequently encountered problems and steps to overcome them
a. Differing time periods used for measuring dependent variable and cost
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driver(s) use accrual accounting for better matching
b. Misrepresenting fixed costs as unit costs rather than total costs carefully
distinguish between cost behaviours
c. Missing data design data collection reports that regularly and routinely
obtain the required data; immediately follow up on missing data
d. Extreme values of observations adjust or eliminate unusual observations
before estimating a cost relationship
e. Unstable relationship over time pool data to form a single cost
relationship
f. Effects of inflation remove purely inflationary price effects from data
Do multiple choice 10.
Assign Problem 10-41
IV. Appendix: Regression analysis
Assign Problems 10-37 and 10-38
Learning Objective 4:
Extend the analysis of data sets to include discontinuous linear cost
functions within a relevant range
A. Cost estimation approaches
1. Industrial engineering method
a. Analyzes the relationship between inputs and outputs in physical terms
b. Inputs and outputs may be difficult to specify, may be time-consuming and
very costly
2. Conference method
a. Analyzes costs and their drivers subjectively through assessments gathered
from various departments of an organization
b. Pooling of expert knowledge but dependent upon best judgment of people
providing input
3. Account analysis method
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a. Analyzes using qualitative rather than quantitative approach to classify cost
accounting in the ledger
b. Thorough knowledge of operation necessary; helpful if coupled with
conference method
4. Quantitative analysis methods
a. Analyzes using formal mathematical methods to fit linear cost functions to
past data observations
b. Systematic model approach
Do multiple choice 2.
Assign Exercises 10-21 and 10-22
B. Nonlinearity and cost functions
1. Economies of scale (discounts for large amounts)
2. Step cost functions
Do multiple choice 7.
Learning Objective 5:
Distinguish between the curvilinear cumulative average time learning
model and incremental unit time learning model
3. Learning curves; experience curves
a. Cumulative average-time learning model
b. Incremental unit-time learning model
Do multiple choice 8 and 9.
Assign Exercises 10-27 and 10-28
C. Data collection and adjustment issues
1. Ideal data base
a. Numerous reliably measured observations of the cost driver(s) and cost that
is dependent variable
b. Many values that span a wide range for the cost driver
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CHAPTER QUIZ
1.
A mixed cost function has a constant component of $20,000. If the total cost is
$60,000 and the independent variable has the value 200, what is the value of the
slope coefficient?
a.
2.
b.
$600
c.
$400
d.
$200
[CMA Adapted] Of the following methods, the one that would not be appropriate
for analyzing how a specific cost behaves is
a.
b.
c.
d.
3.
$40,000
the scattergraph method.
linear programming.
the industrial engineering approach.
statistical regression analysis.
When the high-low method is used to estimate a cost function, the variable cost per
unit is found by
a.
b.
c.
d.
performing regression analysis on the associated cost and cost driver data
base.
dividing the difference between the highest and lowest observations of the
cost driver by the difference between costs associated with the highest and
lowest observations of the cost driver.
dividing the difference between costs associated with the highest and lowest
observations of the cost driver by the difference between the highest and
lowest observations of the cost driver.
subtracting the fixed cost per unit from the total cost per unit based on either
the highest or lowest observation of the cost driver.
The following data apply to questions 4 and 5.
Theta Company derived the following cost relationship from a regression analysis of its
monthly manufacturing overhead cost.
y = $80,000 + $12X
where: y = monthly manufacturing overhead cost
X = machine hours
The standard error of estimate of the regression is $6,000. The standard time required to
manufacture one six-unit case of Torys single product is four machine hours. Theta
applies manufacturing overhead to production on the basis of machine hours, and its
normal annual production is 50,000 cases.
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Chapter 10
Chapter 10 Quiz continued
4.
[CMA Adapted] Thetas estimated variable manufacturing overhead cost for a
month in which scheduled production is 10,000 cases would be
a. $480,000.
5.
$320,000.
c.
$160,000.
d.
$80,000.
[CMA Adapted] Thetas predetermined fixed manufacturing overhead rate would
be
a. $4.80/MH.
6.
b.
b.
$4.00/MH.
c.
$3.20/MH.
d.
$1.60/MH.
Three criteria to use in identifying cost drivers from the potentially large set of
independent variables that can be included in a regression model are
a.
b.
independence between independent variables, economic plausibility, and
specification analysis.
c.
goodness of fit, size of the intercept term, and specification analysis.
d.
7.
economic plausibility, goodness of fit, and slope of regression line.
spurious correlation, expense of gathering data, and multicollinearity.
Companies that take advantage of quantity discounts in purchasing their materials
have
a.
linear cost functions.
b.
nonlinear cost functions.
c.
increasing cost functions.
d.
stationary cost functions.
See next page for 8, 9, and 10
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Chapter 10
Chapter 10 Quiz continued
The following data apply to questions 8 and 9.
Sigma Company recently completed and sold an order of 50 units having the following
costs:
Direct materials
$ 1,500
Direct labour (1,000 hours @ $8.50)
8,500
Variable overhead (1,000 hours @ $4.00)
4,000
Fixed overhead
1,400
$15,400
Allocated on the basis of direct labour hours.
Allocated at the rate of 10% of variable cost.
The company has now been requested to prepare a bid for 150 units of the same product.
8.
[CMA Adapted] If an 80 percent learning curve is applicable, Sigmas total cost on
this order would be estimated at
a. $38,500.
9.
$37,950.
c.
$31,790.
d.
$26,400.
[CMA Adapted] If Sigma had experienced a 70 percent learning curve, the bid for
the 150 units would
a.
b.
c.
d.
10.
b.
show a 30 percent reduction in the total direct labour hours required with no
learning curve.
include 1,404 total direct labour hours at $8.50 per hour.
be 10 percent lower than the total bid at an 80 percent learning curve.
include 6.40 direct labour hours per unit at $8.50 per hour.
Common problems encountered in collecting data for cost estimation include all but
which of the following:
a.
b.
c.
d.
missing data
changes in technology
lack of observing extreme values
distortions resulting from inflation
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Chapter 10
CHAPTER QUIZ SOLUTIONS:
1.
[d]
2.
[b]
3.
[c]
4.
[a]
5.
[d]
6.
[a]
7.
[b]
8.
[d]
9.
[d]
10.
[c]
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Chapter 10
WRITING/DISCUSSION EXERCISES
1.
Explain the two assumptions frequently used in cost-behaviour estimation.
Cost-behaviour estimation occurs through the use of mathematical models. What
purpose(s) do models serve?
Mathematical models serve three purposes:
1 generalize a relationship
2 enable prediction based upon a relationship
3 derive values within a relationship
The cost-behaviour estimation models describe how costs y change with changes in
the level of an activity X. The mathematical statement y = a + bX is a general
description of the costs. This statement can be used to predict costs based upon the
level of activity. A manager (through the work of the cost accountant) can evaluate
a situation at a given level of activity by comparing actual costs to derived costs
from use of the model or statement.
2.
Describe linear cost functions and three common ways in which they behave.
How many ways different can a line be drawn within a two dimensional space
(up/down and side-to-side)?
The line could be drawn top to bottom or vice versa, side-to-side, or diagonally.
The diagonal line represents a proportion of the x-axis to the y-axis, a variable
cost. A line side-to-side represents a constant cost (y value) across a changing
volume of activity (x values), a fixed cost. The vertical, top-to-bottom line is not
representative of a cost function as the cost will not vary at one level of activity
unless another variable is involved and that would negate the two-dimensional
aspect. The diagonal could rise to the right or fall to the right.
A diagonal line that falls to the right would be represented by the cost function y =
a bX. From a long-run viewpoint, a negative slope is not economically plausible.
An example of a negative relationship in the short run is that of maintenance costs
and the activity level of miles driven. Decreases in maintenance costs could be
associated with increases in miles driven because
maintenance could be deferred from periods of high demand to periods of low
demand,
some seasons require high maintenance due to adverse weather conditions, or
low times of miles driven arise from vehicles being taken out of operation for
maintenance.
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Over time an obvious relationship exists of higher mileage, higher maintenance.
Anyone familiar with operations would recognize the proper relationship and
adjust the time frame for data accordingly. Maintenance costs would tend to lag
behind miles driven as the cost driver
3.
Recognize various approaches to cost estimation.
When asked what the most important thing science students needed to learn to be
good scientists, a college biology professor involved with the teaching of science
in high schools across the nation replied, They need to learn to observe. Why is
observation an important aspect of cost estimation?
To observe is to notice patterns or relationships. The methods used in cost
estimation result in the development of a relationship between a cost and a level of
activity. All of the methods require someone with knowledge of the activities
associated with the work that incurs the cost be involved at some point in the
process of identifying that connection between the cost incurred and the specific
activity.
4.
Outline six steps in estimating a cost function based on current or past cost
relationships.
What might cause a cost function that was carefully developed using a
combination of acceptable approaches and the six steps to not work?
One of the purposes of a cost function, a mathematical model, is that of prediction.
Cost estimation is obviously prediction. The approaches identified as well as the six
steps use historical data to develop the cost function. If the processes or operations
underlying the historical relationship were to change, a cost function developed
using past data would no longer be useful for predicting. Historical information
must be used with other information to be most effective.
5.
Describe three criteria to evaluate and choose cost drivers
A company invests considerable time and money to develop sophisticated cost
functions that rate high on all evaluative criteria. In the course of using the cost
functions, a manager notes that, in several instances, the actual costs were
different than the predicted costs resulting in lower profits during one quarter of
the year. The question of the value of the cost function is asked. Give some
suggestions on how to answer the manager.
Exercise 10-23 in the text illustrates a situation in which a cost function is used to
reject a job. At some point, actual costs were known and the job would have been
profitable to accept. The opposite situation could occur. A job could be accepted
with the actual costs greater than anticipated, resulting in a loss. Models, regardless
of the level of sophistication, are used to make predictions not to be relied upon for
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actual data, which can only be known after the fact.
Models are generalizations that do not capture every variable.
Models need to be updated to adequately reflect that which they model.
Models make fewer errors when used than would be made if they were not
used. (If this is not the case, the model needs to be redone so that it does work
to provide information resulting in improved decision making.)
The appendix to Chapter 3 highlights the role of events and outcomes as part of
dealing with uncertainty.
6.
Explain and give examples of nonlinear cost functions.
What is the easiest way to check for linearity or nonlinearity with one
independent variable and dependent variable?
A scatter diagram is an easy and effective method of checking for the shape of a
relationship. Plotting observed data is an essential step for developing cost
functions.
7.
Distinguish between the cumulative average time learning model and
incremental unit time learning model.
The two learning curve models, cumulative average-time and incremental unittime, are so similar, why choose one rather than the other?
Though the models are similar, they generate different numbers or information for
a managers use. The one that more accurately approximates the relationship
between costs and labour activity is the better model to choose. Technological and
mathematical advances have enabled the development of more sophisticated
models that can be readily used by many companies. A company should, therefore,
take advantage of the best models available.
8.
Understand data problems encountered in estimating cost functions.
Relate the data problems described in the text with the scorekeeping role
performed by accountants described in Chapter 1.
Accountants who serve in the scorekeeping role are responsible for the reliability of
the reported information. In this regard, accountants are watchdogs for top
management. Management accountants must ensure the integrity of the data used
in the cost control reporting.
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Chapter 10
DEMONSTRATION PROBLEM 1
(Illustration of the high-low method used for cost estimation)
Marie Kahn the president of Sigma Inc. enlisted the assistance of the vice presidents in
charge of sales and of production. Together they constructed two budgetsone
optimistic, one pessimistic. These are shown in columns (1) and (2) below. The actual
results are shown in column (3). The companys cost accountant was perplexed as to how
to present an analysis. Consequently, he produced variances of actual results against both
the optimistic projection and the pessimistic projection, columns (4) and (5), respectively.
(1)
Optimistic
Budget
(2)
Pessimistic
Budget
(3)
(4)
(5)
Actual
Results
Variance
from
Optimistic
Budget
Variance
from
Pessimistic
Budget
________________________________________________________________________
Units sold
200,000
75,000
135,000
65,000
(60,000)
Sales
$2,000,000
$750,000
$1,350,000
$ 650,000 $(600,000)
Direct materials $
Direct labour
Indirect labour
Maintenance
Supplies
Power
Heat
Light
Rent
Insurance
200,000
400,000
106,000
20,000
28,000
160,000
50,000
7,000
80,000
20,000
$ 75,000
150,000
43,500
20,000
15,500
60,000
50,000
4,500
80,000
13,750
$ 140,000
285,000
72,000
22,000
21,000
108,000
53,000
5,900
80,000
17,000
$ (60,000)
(115,000)
(34,000)
2,000
(7,000)
(52,000)
3,000
(1,100)
-0(3,000)
$ 65,000
135,000
28,500
2,000
5,500
48,000
3,000
1,400
-03,250
REQUIRED:
Marie is baffled by the analysis. She has asked you for a more understandable
performance report. In the columns below, in clear and orderly fashion, prepare a new
report. Explain your work to the president. Show supporting computations as needed.
Sales or
Actual
Cost
Revised
Line Item
Results
Function
Budget
Variance
________________________________________________________________________
Units sold
135,000
135,000
0
Sales
$1,350,000
y = $10X
$1,350,000
$0____
Direct materials
$ 140,000
y = $1X
$ 135,000
$ 5,000U
Direct labour
285,000
y = $2X
Indirect labour
72,000
y = $6,000 + $0.50X
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Chapter 10
SOLUTION FOR DEMONSTRATION PROBLEM 1
The problem can be used with Learning Objective 2: Describing linear cost functions and
three common ways in which they behave. The three behaviours are used variable,
fixed, and mixed or semivariable. One can usually spot the three fixed cost behaviours by
looking at the amounts budgeted under each of the two budgets, optimistic and
pessimistic. Looking at units sold and dollar amounts in the budgets fairly quickly
distinguish two of the three pure variable costs. The other four line items must be
divided into variable and fixed components by use of the high-low method of deriving a
cost function.
Sales or Cost
Actual
Function
Revised
Line Item
Results
Budget
Variance
________________________________________________________________________
Units sold
135,000
135,000
0
Sales
$1,350,000
y = $10X
$1,350,000
$0______
Direct materials
$ 140,000
y = $1X
$ 135,000
$ 5,000U
Direct labour
285,000
y = $2X
270,000
15,000U
Indirect labour
72,000
y = $6,000 + $0.50X
73,500
1,500F
Maintenance
22,000
y = $20,000
20,000
2,000U
Supplies
21,000
y = $8,000 + $0.10X
21,500
500F
Power
108,000
y = $0.80X
108,000
-0Heat
53,000
y = $50,000
50,000
3,000U
Light
5,900
y = $3,000 + $0.02X
5,700
200U
Rent
80,000
y = $80,000
80,000
-0y = $10,000 + $0.05X
16,650
250U
Insurance
17,000
$ 803,900
$ 780,450
$23,450U
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Chapter 10
Computations:
Direct mat:$200,000 = a + b200,000 units
$ 75,000 = a + b 75,000 units
$125,000 = b125,000 units
$1 = b
Power:$160,000 = a + b200,000 units
$ 60,000 = a + b 75,000 units
$100,000 = b125,000 units
$0.80 = b
$200,000 = a + $1200,000 units
$160,000 = a + $0.80200,000
-0-= a
y = $0.80X
-0-= a
y = $1X
Direct labour: $250,000 = b125,000 units
$2 = b
-0-= a
y = $2X
Heat: $ -0-= -0-0-= b
$50,000 = a
y = $50,000
Indirect labour: $106,000 = a + b200,000
$ 43,500 = a + b 75,000
$ 62,500 = b125,000
$0.50 = b
$106,000 = a + $100,000
$6,000 = a
y = $6,000 + $0.50X
Light: $7,000 = a + b200,000
$4,500 = a + b 75,000
$2,500 = b125,000
$0.02 = b
$7,000 = a + $4,000
$3,000 = a
y = $3,000 + $0.02X
Maintenance:
y = $20,000
Supplies: $28,000 = a + b200,000
$15,500 = a + b 75,000
$12,500 = b125,000
$0.10 = b
$28,000 = a + $20,000
$8,000 = a
y = $8,000 + $0.10X
Rent: y = $80,000
Insurance: $20,000 = a + b200,000
$13,750 = a + b 75,000
$ 6,250 = b125,000
$0.05 = b
$20,000 = a + $10,000
$10,000 = a
y = $10,000 + $0.05X
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Chapter 10
DEMONSTRATION PROBLEM 2
GIVEN:
The Kappa Company has received a contract to supply 240 units of new
telecommunication equipment. The direct materials costs are $60,000 per unit. The
average direct manufacturing labour costs for each unit (in the first lot of 30 units) was
estimated to be $40,000. Direct manufacturing labour on a per lot basis is subject to a
90% cumulative average-time-learning model. (A 90% learning curve implies q =
0.1520.) Variable manufacturing overhead was estimated to be 60% of direct
manufacturing labour cost. Kappas price includes a markup of 25% on total variable
manufacturing costs.
REQUIRED:
1.
Determine the Kappas cumulative average unit cost of manufacturing labour for
producing the 240 units.
2.
Assume that the Kappa is asked to produce additional telecommunication
equipment beyond the 240 units currently under contract. Kappa anticipates that the
expected average cost incurred to produce the last 120 units is the expected
manufacturing cost per unit for each additional unit after 240 units. Calculate
Kappas average unit cost of manufacturing labour for units beyond the original
240 units.
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Chapter 10
SOLUTION FOR DEMONSTRATION PROBLEM 2
(Illustration of cumulative average time learning model with concept of leveling)
[CMA, adapted]
1.
The learning curve is on a per lot basis. The manufacturing labour cost per unit in
the first lot is $40,000.
(30 units per lot)
Cumulative
Number of
Units
(1)
1
2
4
8
30
60
120
240
Cumulative
Number of Lots
Cumulative Average
Direct Manufacturing
Labour Cost
Cumulative
Per Unit in Each Lot
Total Cost
(2)
(3) = (1) x (2)
$40,000
36,000 (40,000 x .9)
32,400 (36,000 x .9)
29,160 (32,400 x .9)
$1,200,000
2,160,000
3,888,000
6,998,400
Using the formula:
The average cost per unit for 240 units in 8 lots = $40,000 x 8 0.1520 = $29,160.
2.
The average cost per unit for the last round of learning when going from 4 lots to 8
lots (120 units to 240 units) is the change in cumulative total costs divided by 120
units (240 units for 8 lots 120 units for 4 lots):
6,998,400 - $3,888,000 = $25,920
120
This is Coopers expected average cost per unit for each additional unit beyond the
first 240 units.
The learning curve is a leveling type of model. At some point, though the rate of change
continues, the absolute amount of change between lot sizes becomes ever increasingly
smaller. An organization could factor in a leveling off point such as Cooper
Corporation did in this example. This concept does not negate continuous improvement,
however.
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