Ch 02b Cost Estimation

Cost Behavior:Statistical Estimation

RelevantRange

A straight line closely

approximates a curvilinear

variable cost line within the

relevant range.

A straight line closely

approximates a curvilinear

variable cost line within the

relevant range.

Activity

Economist’sCurvilinear Cost

Function

The Linearity Assumption and the Relevant Range

Accountant’s Straight-Line Approximation (constant

unit variable cost)

Exh.5-4

Activity

stQuestion: How do you interpret the intercept on

the vertical axis?

The Linearity Assumption and the Relevant Range

Exh.5-4

Estimating the cost function: High-low (two point) method Assume: Y = a + b X Data:

X Y Obs. 1 1,000 1,200 Obs. 2 2,000 1,800

.X=2000, Y =1800

.X=1000, Y=1200 X Required: Estimate a linear cost function based on the two given observations.

b = ΔY / ΔX = 600 / 1000 = .6 a = Y – bX = 1,200 – (.6)(1,000) = 600 Y = 600 + .6 X

Assume that a third observation is available as follows: X = 3,000, Y = 2,000. Based on the three available data points, estimate the cost function. Y

. X=3000, Y = 2000

.X=2000, Y =1800

.X=1000, Y=1200 X

Overhead Costs Overhead Costs Quarter OH Costs

Repair-Hours

1 $9,891 248

2 $9,244 248

3 $13,200 480

4 $10,555 284

5 $9,054 200

6 $10,662 380

7 $12,883 568

8 $10,345 344

9 $11,217 448

10 $13,269 544

11 $10,830 340

12 $12,607 412

13 $10,871 384

14 $12,816 404

15 $8,464 212

Scattergraph Scattergraph

Plot of cost and activity levels

A visual representation

Does it look like a relationship exists between repair-hours and overhead costs?

Manufacturing Overhead

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000

$16,000

0 100 200 300 400 500 600

Repair-Hours

3C Overhead

Repair-Hours

High-Low Cost EstimationHigh-Low Cost Estimation

A method to estimate costs based on two cost observations, usually at the highest and lowest activity level.

Choose two data points.

Use the two points to determine the line representing the cost-activity relation.Draw a total cost line.

The highest and lowest activity.

Overhead Costs Overhead Costs Quarter OH

CostsRepair-Hours

1 $9,891 248

2 $9,244 248

3 $13,200 480

4 $10,555 284

5 $9,054 200

6 $10,662 380

7 $12,883 568

8 $10,345 344

9 $11,217 448

10 $13,269 544

11 $10,830 340

12 $12,607 412

13 $10,871 384

14 $12,816 404

15 $8,464 212

Use the high-low method to estimate the cost function.

Repair hours OH$ RH High (obs.7) $12,883 568 Low (obs.5) $ 9,054 200 Difference $ 3,829 368

Slope: 3829 / 368 = $10.40

Intercept: $6,974

RegressionRegression

Statistical procedure to determine the relationship between variables.

High-Low Method

Regression

Uses two data points.Uses all the data points.

3C Overhead

Regression ContinuedRegression Continued

Independent variable:

Dependent variable:

Repair-hours

Overhead costs

Y = a + b X

Y Intercept

Repair-hours

Fixed costs

The relationship between activities and costs

The Regression Equation

Excel output, Simple Linear Regression

Estimate 3C’s overhead with 520 repair hours.

Given the following regression estimates:

Total cost = $6,472 + (12.52) x (RH) = $6,472 + (12.52) x (520) = $12,982 SEE (Y) = $ 678.00 SE (B1) = $ 1.58

(1) Provide 95% confidence intervals around the estimate of total cost, and around the estimated variable overhead per hour of repair time.

(2) Determine the predetermined overhead rate (POHR) at a budgeted

activity level of 520 repair hours. How much of the POHR represents fixed cost applied per repair hour?

(3) Assume that actual repair hours are 500, and actual total costs are

$ 12,000, Determine the amount of over/under applied overhead for the period.

(3) Assume that actual repair hours are 500, and actual total costs are

$ 12,000, Determine the amount of over/under applied overhead for the period.

Multiple RegressionMultiple Regression

Is repair-hours the only activity that drives overhead costs at 3C?

1 $9,891 248 $1,065

2 $9,244 248 $1,452

3 $13,200 480 $3,500

4 $10,555 284 $1,568

5 $9,054 200 $1,544

6 $10,662 380 $1,222

7 $12,883 568 $2,986

8 $10,345 344 $1,841

9 $11,217 448 $1,654

10 $13,269 544 $2,100

11 $10,830 340 $1,245

12 $12,607 412 $2,700

13 $10,871 384 $2,200

14 $12,816 404 $3,110

15 $8,464 212 $ 752

Quarter OH Costs Repair-Hours Parts

Multiple Regression OutputMultiple Regression Output

Compare the variable overhead perrepair hour across both models. Whydo they differ in this manner?

Multiple Regression Output ContinuedMultiple Regression Output Continued

Estimate 3C’s overhead for 520 repair-hours and $3,500 parts costs.= F + V

1= 6,41

6+ 8.6

Total cost = $6,416 + (8.61) x (RH) + (.77) x (PC) = $6,416 + (8.61) x (520) + (.77) x (3,500) = $13,588 SEE (Y) = $ 517.00 SE (B1) = $ 1.71 SE (B2) = $ 0.24

(1) Provide 95% confidence intervals around the estimate of total cost, and around

the estimated variable overheads per hour of repair time and per dollar of parts cost.

(2) Determine the predetermined variable overhead rate (PVOHR) for repair time

and parts cost. How would you determine the fixed overhead to be assigned to production?

(3) Assume that you are preparing a bid on a special customer order. You estimate

that the project will require 100 repair hours and $ 200 in parts costs. Estimate the amount of variable overhead that you would assign to this project based on the results of the simple and the multiple regressions estimated above. Which of your estimates is more likely to be accurate?

Total cost = $6,416 + (8.61) x (RH) + (.77) x (PC) = $6,416 + (8.61) x (520) + (.77) x (3,500) = $13,588 SEE (Y) = $ 517.00 SE (B1) = $ 1.71 SE (B2) = $ 0.24

Data ProblemsData Problems

Discretionary costs

Inflation

Nonlinearity

Outliers

Residuals assumptions

Outliers

OutliersOutliers

Outlier changes the estimated regression line.

regression line without outlier

Regression line with outlier

outlier

Nonlinearity

Dealing with non-linearity in estimating cost functions:

A simple linear function:Y = a + b (X)

Linearity after transformation:

Piecewise linearity

Linearity after transformation:Y = aXb

Piecewise linearity

ln(Y) = ln(a) + b(ln(X))

Piecewise linearity

ln(Y) = ln(a) + b(ln(X))

Piecewise linearity (adding “dummy” variables to Allow for parameter shifts):

Y = a + b(X) + c(D)(X)

Discretionary costs

Impact of discretionary costs such as advertisingand maintenance spending:

Assume that preventative maintenance is performed when the facilities have idle capacity. In addition, maintenance is deferred when the facilities are very intensely used. What would be the empirical relation between activity levels and maintenance costs?

Impact of discretionary costs such as advertisingand maintenance spending:

Assume that preventative maintenance is performed when the facilities have idle capacity. In addition, maintenance is deferred when the facilities are very intensely used. What would be the empirical relation between activity levels and maintenance costs?

Assume that advertising is increased when sales levels are low. What would be the empirical relation between advertising and sales?

Regression residuals:Discuss the importance to cost analysis of the followingAssumptions concerning the residual or “disturbance”Term in the regression model:

1. Normality

2. Homoscedasticity

3. Independence

1. Normality

2. Homoscedasticity

3. Independence

1. Normality

2. Homoscedasticity

3. Independence

Inflation

Inflation (deflation) is an increase (decrease) in the amount of currency required to buy a specific “market basket” of goods and services.

The relevant “market basket” is firm-specific.

Ignoring inflation will bias the parameter estimates

in a manner similar to the impact of omitted variables.

Ignoring inflation will bias the parameter estimates

in a manner similar to the impact of omitted variables.

To avoid this bias, the currency measures should be

adjusted from historical (nominal) dollars to inflation-adjusted (constant) dollars before computing the parameter estimates.

Example of inflation adjustments using the high-low method:

Observation:

TOH (nominal $)

Price index

TOH (constant $)

Low 10,000 $300,000 1.50 High 15,000 $400,000 1.80 Estimating equation using nominal dollars: F = ? V = ? Estimating equation using constant dollars: F = ? V = ?

Observation:

TOH (nominal $)

Price index

TOH (constant $)

Low 10,000 $300,000 1.50 $360,000 High 15,000 $400,000 1.80 $400,000 Estimating equation using nominal dollars: F = ? V = ? Estimating equation using constant dollars: F = ? V = ?

Observation:

TOH (nominal $)

Price index

TOH (constant $)

Low 10,000 $300,000 1.50 $360,000 High 15,000 $400,000 1.80 $400,000 Estimating equation using nominal dollars: V = $20 F = $300,000 – ($20)(10,000); F = $100,000 Estimating equation using constant dollars: V = $8 F = $360,000 – ($8)(10,000); F = $280,000

Data Problems: RecapitulationData Problems: Recapitulation

Discretionary costs

Inflation

Nonlinearity

Outliers

Handout 2 (a): Using Linear Cost

Functions

Handout 2(a) Linear cost functions: Consider each of the following independent cases:

A. Total fixed costs are $80,000 and variable unit costs are $6.00. Determine the following amounts assuming that the firm produces and sells 10,000 units:

(1) Total cost

(2) Average cost

(3) Unit fixed cost

(4) Unit sales price required to earn a gross margin of $24,000

A. Total fixed costs are $80,000 and variable unit costs are $6.00. Determine

the following amounts assuming that the firm produces and sells 10,000 units:

(1) Total cost TC = $80,000 + 6 x 10,000; = $140,000

(2) Average cost AC = $140,000 / 10,000; = $14.00

(3) Unit fixed cost UFC = $80,000 / 10,000; = $8,00

(4) Unit sales price required to earn a gross margin of $24,000

S = AC + ($24,000 / 10,000); = $14.00 + $2.40; = $16.40

B. Average unit costs are $9.00 at an output of 5,000 units, and

$8.00 at an output of 10,000 units. The total cost function is assumed to be linear. Determine the following:

(1) Average unit costs at an output of 4,000 units

(2) Average unit fixed costs at outputs of 4,000, 5,000 and

10,000 units

(3) Sales prices per unit needed to earn a profit of $12,000 at

outputs of 4,000, 5,000 and 10,000 units

B. Average unit costs are $9.00 at an output of 5,000 units, and $8.00 at an output of 10,000 units. The total cost function is assumed to be linear. Determine the following:

(1) Average unit costs at an output of 4,000 units

(2) Average unit fixed costs at outputs of 4,000, 5,000 and 10,000 units

(3) Sales prices per unit needed to earn a profit of $12,000 at outputs of 4,000, 5,000 and

10,000 units

Units Unit cost Total cost 5,000 $9.00 $45,000 10,000 $8.00 $80,000

Slope: ($80,000 – 45,000) / (10,000 – 5,000) = $7.00 Intercept: $45,000 – ($7.00)x(5,000) = $10,000

Average unit costs: Output UFC UVC UTC UGM Price (UTC + UGM) 4.000 $2.50 $7.00 $9.50 $12,000 / 4,000 = $3.00 $12.50 5,000 $2.00 $7.00 $9.00 $12,000 / 5,000 = $2.40 $11.40 10,000 $1.00 $7.00 $8.00 $12,000 / 10,000 = $1.20 $ 9.20

C. Firm X has $60,000 of total fixed cost and variable costs of $4.00 per unit. Firm Y has $90,000 of total fixed cost and variable costs of $3.00 per unit. At what level of output do both firms have the same total cost?

Set TC(X) = TC(Y)

$60,000 + 4Q = $90,000 + 3Q

Q = 30,000

D Firm Z has the following cost function:

Total cost = $240,000 + (40%) x (Total revenue).

Determine the following: (1) Total costs when revenues are $900,000.

(2) Total gross margin when revenues are $900,000

(3) Total contribution margin when revenues are $900,000

(4) Contribution margin percentage

(5) Gross margin percentage when revenues are $900,000

D. Firm Z has the following cost function:

Total cost = $240,000 + (40%) x (Total revenue).

Determine the following: (1) Total costs when revenues are $900,000. TC = $600,000

(2) Total gross margin when revenues are $900,000 TGM = $300,000

(3) Total contribution margin when revenues are $900,000

TCM = (60%) x (Total revenue); = $540,000

(4) Contribution margin percentage CM% = (100% - VC%); =

(5) Gross margin percentage when revenues are $900,000

GM% = TGM / TRev; = $300,000 / $900,000; = 33 1/3%

Handout 2 (b): Interpreting

Regression Estimates

Handout 2(b): Part A: Linear Constructs, Inc. is revamping its overhead costing, based upon statistical analyses of the relation of overhead costs to various activity levels that are possible “drivers” of overhead costs. As a start, the company’s data analysts have estimated the linear relation of overhead costs (TOH) and direct labor hours (DLH) as follows:

TOH = Intercept + B1 x DLH + e = $9,000 + ($12.00) x (DLH) SEE (Y) = $ 500.00 SE (B1) = $ 1.50

Assume that production activity budgeted at 500 direct labor hours. Based upon these parameter estimates and standard errors, address the following questions (for simplicity assume that 95% confidence intervals are measured as the estimated parameter plus and minus two standard errors).

(1) Provide 95% confidence intervals around the estimate of total cost, and around the estimated variable overhead per direct labor hour.

(2) Estimate the predetermined overhead rate (POHR) at a budgeted activity

level of 500 direct labor hours. How much of the POHR represents fixed cost applied per direct labor hour?

(3) Assume that actual repair hours are 600, and actual total costs are $

15,000. Assume the company used the POHR estimated in (2) above, and determine the amount of over/under applied overhead for the period. Is the variance favorable or unfavorable?

(1) Provide 95% confidence intervals around the estimate of total cost, and around the estimated variable overhead per direct labor hour.

TC = $9,000 + $12 x 500dlh; = $15,000 +/- (2)($500) B1 = $12 +/- (2)($1,50)

Shows the interval of the production.

(2) Estimate the predetermined overhead rate (POHR) at a budgeted

activity level of 500 direct labor hours. How much of the POHR represents fixed cost applied per direct labor hour?

Predetermined overhead account take a look at.

(2) Estimate the predetermined overhead rate (POHR) at a budgeted

activity level of 500 direct labor hours. How much of the POHR represents fixed cost applied per direct labor hour?

POHR = $15,000 / 500dlh; = $30 PFOHR = ($9,000 / 500dlh; = $18

(3) Assume that actual repair hours are 600, and actual total costs are $ 15,000. Assume the company used the POHR estimated in (2) above, and determine the amount of over/under applied overhead for the period. Is the variance favorable or unfavorable?

Applied OH = $30 x 600dlh; = $18,000 Variance = $15,000 - $18,000; = $3,000 over-applied (favorable)

Part B: After evaluating the simple regression results described above, the firm’s data analyst decided to incorporate machine time (MT) as an additional potential driver of overhead costs. The estimating equation was modified as follows:

Required: Assume that production activity is budgeted at 500 direct labor hours and 300 machine hours. Based on this expanded model and parameter estimates,

TOH = Intercept + B1 x DLH + B2 x MT + e = $8,000 + ($9.00) x (DLH) + ($3.00) x (MT) SEE (Y) = $ 400.00 SE (B1) = $ 2.50 SE (B2) = $ 1.25

Multicolinarity.

Assume that production activity is budgeted at 500 direct labor hours and 300 machine hours. Based on this expanded model and parameter estimates,

(1) Provide 95% confidence intervals around the estimate of total cost, and around the estimated variable overheads per direct labor hour and per hour of machine time.

Assume that production activity is budgeted at 500 direct labor hours and 300 machine hours. Based on this expanded model and parameter estimates,

(1) Provide 95% confidence intervals around the estimate of total cost, and around the estimated variable overheads per direct labor hour and per hour of machine time.

TC = $8,000 + $9 x 500dlh + $3 x 300mt; = $13,400 +/- (2)($400) B1 = $9 +/- (2)($2.50) B2 = $3 +/- (2)($1.25)

Discuss why the estimated overhead per direct labor hour is lower, and the confidence interval for the overhead rate per labor hour is wider, when machine time is included as an additional “driver” in the cost estimation model.

Machine time is correlated with direct labor hours, and is an omitted variable in the earlier simple regression equation. Including machine time in the equation removes the coefficient bias, but the standard errors of the slope estimates are higher because of multicollinearity between the two independent variables.

(2) Determine the predetermined variable overhead rate (PVOHR) for labor hours and for machine time.

The predetermined variable overhead rates are simply the estimated slope coefficients for these two variables.

Assume that the intercept estimate in the above model is interpreted as budgeted fixed factory overhead. How would you determine the fixed overhead rate to be assigned to production? Would you develop a fixed overhead rate based on labor hours, machine time, or some other measure? Explain your reasoning.

There is no conceptually correct solution to this problem. In practical cases, the estimated intercept might be interpreted as fixed overhead, and arbitrarily divided by the budgeted level of one of the drivers (e.g., labor hours). In fact, the intercept is affected by omitted variables, non-linearity and measurement errors, and any allocation of this amount is arbitrary.

(3) Assume that you are preparing a bid on a special customer order. You

estimate that the project will require 100 direct labor hours and 200 hours of machine time. Estimate the amount of variable overhead that you would assign to this project based on the results of the simple and the multiple regressions estimated above. Which of your estimates is more likely to be accurate?

Based on simple regression: VOH = ($12 x 100dlh); = $1,200 Based on multiple regression: VOH = ($9 x 100dlh) + ($3 x 200MT); = $1,500

Handout 2 (c): Inflation and cost estimation

Impact of inflation on estimation of cost functions:Linear Structures, Inc. has gathered information on total maintenance costs and total direct labor hours incurred each quarter over the past five years (2007 through 2011). The company considers direct labor hours to be a suitable “driver” for its’ maintenance costs. The amounts of maintenance costs incurred in the lowest and highest activity quarters are provided in the table below:

Maintenance Costs and Direct Labor Hours, lowest and highest activity quarters

Total maintenance costs

Total direct labor hours

Lowest quarter Qtr. 1, 2007 $ 4,000,000 500,000 DLH Highest quarter Qtr. 3, 2011 $ 4,500,000 550,000 DLH

Impact of inflation on estimation of cost functions:Linear Structures, Inc. has gathered information on total maintenance costs and total direct labor hours incurred each quarter over the past five years (2007 through 2011). The company considers direct labor hours to be a suitable “driver” for its’ maintenance costs. The amounts of maintenance costs incurred in the lowest and highest activity quarters are provided in the table below:

Required: Assume that maintenance is a “mixed cost” and behaves in a straight-line fashion. Based on the information in the above table, determine the intercept and the slope of the firm’s maintenance cost function, and discuss the plausibility of your estimates: (a) Intercept $_____________________ (b) Slope $______________per DLH

Required: Assume that maintenance is a “mixed cost” and behaves in a straight-line fashion. Based on the information in the above table, determine the intercept and the slope of the firm’s maintenance cost function, and discuss the plausibility of your estimates: Reminder: Solve for the slope first, and then solve for the intercept. (a) Slope $____10.00_____per DLH $500,000 / 50,000DLH = $ 10.00

(b Intercept $___-$1,000,000 $4,000,000 – $10.00 x 500,000DLH = -$1,000,000

Linear Structures, Inc. has gathered information on total maintenance costs and total direct labor hours incurred each quarter over the past five years (2007 through 2011). The company considers direct labor hours to be a suitable “driver” for its’ maintenance costs. The amounts of maintenance costs incurred in the lowest and highest activity quarters are provided in the table below:

In addition to the information given above, you have just been informed that the types of maintenance costs incurred by the company have inflated by a cumulative amount of about ten percent over the past five years (i.e., prices at the end of 2011 are about ten percent higher than they were in early 2007). The price level is expected to be reasonably stable over the next few years. Based on this additional information, re-estimate the intercept and the slope of the firm’s maintenance cost function.

(a) Intercept $_____________________

(b) Slope $______________per DLH

First, inflation-adjust the 2007 overhead costs: $4,000,000 x 110% = $4,400,000; then solve for the slope and the intercept:

(a) Slope $____2.00_____per DLH $100,000 / 50,000DLH = $ 2.00

(b) Intercept $___ $3,400,000 $4,400,000 – $2.00 x 500,000DLH = $3,400,000

Linear Structures, Inc. has gathered information on total maintenance costs and total direct labor hours incurred each quarter over the past five years (2007 through 2011). The company considers direct labor hours to be a suitable “driver” for its’ maintenance costs. The amounts of maintenance costs incurred in the lowest and highest activity quarters are provided in the table below:

Assume instead that the maintenance costs have inflated by about 15 percent over the past five years, and re-estimate the intercept and slope of the firm’s maintenance cost function, and discuss the plausibility of your estimates.

(a) Intercept $_____________________

(b) Slope $______________per DLH

First, inflation-adjust the 2007 overhead costs: $4,000,000 x 115% = $4,600,000; then solve for the slope and the intercept:

(a) Slope $____(2.00)_____per DLH ($100,000) / 50,000DLH = ($ 2.00)

(b) Intercept $___$5,600,000 $4,600,000 – (- $2.00) x 500,000DLH = $5,600,000

Handout 2(d)

Cost behavior and estimation: multiple choice

questions

1. Utility costs at Service, Inc. are a mixture of fixed and variable components. Records indicate that utility costs are an average of $0.40 per hour at an activity level of 9,000 machine hours and $0.25 per hour at an activity level of 18,000 machine hours. Assuming that this activity is within the relevant range, what is the expected total utility cost if the company works 13,000 machine hours? A. $4,225 B. $3,250 C. $4,000 D. $5,200

The total costs are $4,500 at 18,000 hours and $3,600 at 9,000 hours, so the variable cost per MH is $0.10 ($900/9,000MH). The fixed cost is $2,700 ($3,600 – (.10 x 9,000MH)).

At 13,000 MH: $2,700 + .10 x 13,000MH = $4,000

2. Larson Brothers, Inc., used the high-low method to derive its cost formula for electrical power cost. According to the cost formula, the variable cost per unit of activity is $3 per machine-hour. Total electrical power cost at the high level of activity was $7,600 and at the low level of activity was $7,300. If the high level of activity was 1,200 machine hours, then the low level of activity was: A. 1,000 machine hours B. 900 machine hours C. 800 machine hours D. 1,100 machine hours

The cost varied by $300 and the slope is $3, implying that the activity varied by 100MH. If the high level is 1,200MH, the low level is 1,100MH.

3. The following production and average cost data for a month's operations have been supplied by a company that produces a single product.

The total fixed manufacturing cost and variable manufacturing cost per unit are as follows: A. $3,600; $7.50 B. $7,600; $9.90 C. $7,600; $7.50 D. $3,600; $9.90

3. The following production and average cost data for a month's operations have been supplied by a company that produces a single product.

The total fixed manufacturing cost and variable manufacturing cost per unit are as follows: A. $3,600; $7.50 B. $7,600; $9.90 C. $7,600; $7.50 D. $3,600; $9.90

Total manufacturing overhead is $10,000 at 1,000 units and $12,400 at 2,000 units, so the variable overhead cost is $2.40 and the fixed overhead cost is $7,600 ($10,000 - $2.40 x 1,000). Unit variable cost is $9.90 ($7.50 + 2.40).

4. You are applying the scattergraph method and find that the regression line you have drawn passes through a data point with the following coordinates: 7,500 units and $10,000. The regression line passes through the Y axis at the $4,000 point. Which of the following is the cost formula that represents the slope of this line? A. Y=$4,000+$1.25X B. Y=$4,000+$0.80X C. Y=$10,000+$1.33X D. None of the above is true

The intercept is given as $4,000. The change in Y is $6,000 ($10,000 – 4,000) and the change in X is 7,500 units, so the slope is $.80.

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