Balakrishnan Mgrl Solutions Ch05

CHAPTER 5 COST-VOLUME-PROFIT ANALYSIS SOLUTIONS

Review Questions 5.1

Profit before taxes = [(Price – Unit variable cost) × Sales volume in units] – Fixed costs = Unit contribution margin × Sales volume in units – Fixed costs.

5.2

The contribution margin statement.

5.3

The sales volume at which profit equals zero.

5.4

The sales dollars at which profit equals zero.

5.5

The unit contribution margin divided by price.

5.6

Taxes reduce profit by a certain percentage beyond the breakeven point. Above the breakeven point, the slope of the profit line decreases by taxes paid.

5.7

We can use the CVP relation to estimate profit at each price, quantity combination.

5.8

The amount by which sales exceed breakeven sales. It equals (Sales in units – Breakeven volume)/Sales in Units or, equivalently, (Revenues – Breakeven revenues)/Revenues.

5.9

The percentage change in profit = the percentage change in sales volume (or revenues) × (1/Margin of safety).

5.10

Operating leverage is a measure of risk from having more fixed costs. It equals Fixed costs/Total costs.

5.11

The relative proportion in which a company expects to sell products – e.g., two units of product A for every unit of product B.

5.12

The contribution margin per average unit.

5.13

The contribution margin per average sales dollar.

5.14

It is easier to work with revenues directly and comparing contribution margin ratios across products makes more sense than comparing unit contribution margins.

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5-2 5.15

(1) Revenues increase proportionally with sales volume, (2) variable costs increase proportionally with sales volume, (3) selling prices, unit variable costs, and fixed costs are known with certainty, (4) a single-period analysis, (5) a known and constant product mix, (6) CVP analysis does not always provide the “best” solution to a short-term decision, and (7) the availability of capacity.

Discussion Questions 5.16

Unit contribution margin equals unit selling price less unit variable cost. Assuming that unit contribution margin is positive, unit selling price is a bigger number than unit variable cost, and therefore a 10% increase in unit selling price will increase the unit contribution margin more than a 10% decrease in unit variable cost. To illustrate, let the unit selling price be $50 and the unit variable cost be $30. The unit contribution margin will be $20 (=$50 -$30). A 10% increase in unit selling price will increase the unit contribution margin to $25 (=$55-$20), but a 10% reduction in unit variable cost will increase the unit contribution margin to only $23 (=$50-$27).

5.17

Profit before taxes

=

.15 * Revenues (fact 1)

Profit before taxes

=

.40*Revenues – $200,000 (fact 2)

Setting these equations equal to each other… Revenues = $200,000/.25 = $800,000. 5.18

It is generally advisable to conduct CVP analysis on a cash basis. Non-cash items such as depreciation are not relevant. However, it is not uncommon to see CVP analysis being used in conjunction with accounting profits---which would include depreciation as an expense---rather than net cash flow. Such an analysis can be particularly erroneous for start-ups and growth firms because the magnitude of non-cash items or accruals is likely to be large.

5.19

Yes. Let us consider an example. Suppose the unit contribution margin is $5 and the fixed costs are $200,000. The breakeven quantity then is 40,000 units (=$200,000/$5). Let us say that the fixed costs increase by $100,000 to $300,000 but the unit contribution margin stays at $5. The new breakeven quantity is 60,000 (=$300,000/$5). That is we need an additional volume of 20,000 units to breakeven. We can also calculate this additional volume needed to breakeven by dividing the change in fixed costs by the unit contribution margin (=$100,000/$5).

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5.20

Many countries use a progressive tax structure. That is, the tax rate increases for higher income brackets. However, the CVP analysis is fundamentally the same except that the profit equation is more elaborate. Consider an example where the tax rate is 30% up to $300,000, but increases to 40% beyond $300,000. In this case, we have to modify the computation of profit after taxes as: If profit before taxes is less than or equal to $300,000 then [EQ]Profit after taxes = Profit before taxes × (1 – 30%) If profit before taxes is greater than $300,000 then [EQ]Profit after taxes = $300,000 × (1 – 30%) + (Profit before taxes - $300,000) × (1 – 40%) Thus, for the first $300,000 of profit, the company would pay tax at the rate of 30%; beyond $300,000 the applicable tax rate would be 40%. Keep in mind, however, that having multiple tax brackets has no consequence for the calculation of the breakeven point because, at the breakeven point, the profit is zero and there are no taxes.

5.21

Yes, we can modify the CVP relation to include step costs. With step costs, fixed costs do not stay fixed for all volumes. It stays fixed for a volume range beyond which it increases to the next level. Consider, for example, a company that leases a copier for its needs and pays a monthly rent. The copier has a certain fixed capacity to make copies over a certain time. Till this capacity is reached, the rent does not vary with the volume of copies made. However, once the company’s copying needs exceeds its capacity, another copier may have to be rented and the rent payment increases by a step to include the rent of the next copier. When fixed cost increases in steps, the CVP analysis may have to be repeated a few times to converge to the answer. Think about computing the breakeven point. First, assume that the breakeven point would fall within the first step. With this assumption, we can calculate the breakeven point in the usual manner described in the text. If this calculation yields a breakeven point that is within the volume range over which the fixed cost does not increase to the next step, we are done. Otherwise, we change the fixed cost to the next step value and repeat our breakeven calculation. We repeat this process until we reach a point where the breakeven volume falls within the range of the assumed step fixed cost!

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5.22

Software companies typically have a high proportion of fixed costs in their cost structure (i.e., high operating leverage) because their primary resource is trained software professionals. Most of these professionals are paid fixed salaries and wages that do not vary with the volume of software programs they generate, or the number of software programs a company like Microsoft sells. Relatively speaking, an automobile company such as Ford would have a greater proportion of variable costs in its cost structure, although over time this proportion has decreased because of increased automation.

5.23

The practice of selling the same good at different prices to different customers is called discriminatory pricing. In general the Robinson-Patman Act of 1936 prohibits discriminatory pricing in certain situations (such as, for example, a wholesaler selling the same product to two retailers at different prices with the purpose of influencing their competitive standings). However, in other situations, such differential pricing may well be necessary depending on the nature of the customer or the specific market (because of customer-specific or market-specific costs). We will discuss this aspect later in Chapter 10, when we discuss Customer Profitability Analysis.

5.24

Margin of safety is a “cushion” that the existing level of operations allows managers in dealing with operating risk. The smaller this cushion, the closer is the manager to making a loss. Thus, when demand uncertainty increases unexpectedly, this cushion “protects” managers from incurring losses. In such situations, it gives them some room to offer discounts or promotions to keep up the volume in the short-term in order to preserve profitability.

5.25

As the sales volume increases, the total variable costs increase but fixed costs stay the same. Recall that operating leverage is the ratio of fixed costs to total costs. Because fixed costs stay the same and the total costs increase (because of the increase in variable costs), the operating leverage decreases.

5.26

As demand conditions fluctuate, short-term profits are more sensitive to consequent changes in sales volume for firms with higher operating leverage. This is why operating leverage is viewed as measure of risk. Referring to Exhibit 5.14, the operating leverage of Sierra Plastics increases with the new technology. Notice that its profits (profit before taxes) fluctuate more with the sales volume with the new technology than without the technology.

5.27

In general, divisions of large firms often have very different cost structures and serve different markets. Because the CVP analysis is essentially a tool for shortterm decision making that helps managers in deciding the level of operations, it makes more sense for individual divisional heads to perform CVP analysis at their respective divisions. At the firm level, the effects of these short-term decisions can then be aggregated to determine the overall state of the firm in the short run, and which divisions are contributing in what measure in this respect.

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5-5

5.28

Yes. In general the unit of one product is not necessarily comparable to a unit of another dissimilar product because they require different amounts of resources--such as raw material, labor, machining time, finishing time---to produce. Therefore, we cannot say that the sports car is more profitable to produce because its unit contribution margin is higher than the contribution margin of an entrylevel vehicle. However, with the contribution margin ratio, we can express relative profitability in terms of sales dollars. We can say for instance that for every sales dollar, the sports car contributes twenty cents toward profit (i.e., CMR of 20%), and the entry-level vehicle contributes ten cents toward profit (i.e., CMR of 10%).

5.29

CVP analysis should be considered as a convenient tool to understand the relations between cost, volume and profit. It makes many assumptions as discussed in the text. Let us consider a few assumptions assumption and identify a setting in which it would be violated.



Assumption 1: Revenues increase proportionally with sales volume. This assumption essentially means that price per unit is constant and does not vary with volume. However, it is well known that as you decrease price per unit, you can sell more and vice versa. In other words, price per unit and sales volume are inversely related. When we allow for this possibility, this assumption is violated. Assumption 2: Variable costs increase proportionally with sales volume. In other words, unit variable cost stays the same over the relevant range of operations so that variable costs increase linearly with volume. This assumption will be violated whenever the sales volume goes beyond relevant range (e.g., when firms stretch existing capacity to meet demand). In such cases, variable costs can increase more than proportionately. Assumption 3: Selling prices, unit variable costs, and fixed costs are known with certainty. In the real world, we have to deal with uncertainty all the time. The assumed selling price, and variable/fixed costs may turn out be different from the actual price and costs because of changes in demand conditions or resource availability. Assumption 4: Single-period analysis. Most business relationships extend beyond a single period, and most short-term decisions have longer-term implications. Please refer to a discussion of such implications in Chapter 2. Such implications would result in a violation of this assumption. Assumption 5: Product-mix assumption. With many products, CVP analysis assumes a known and constant product mix. However, in most instance, the product-mix itself has to be decided. Changing the product-mix may be best the way to react to changes in demand for the different products in the mix.

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





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5-6

Exercises 5.30 a. Recall that: Profit before taxes = (unit contribution margin  sales volume in units) – fixed costs. Additionally, Unit contribution margin

= Unit selling price – Unit variable cost. = $3.00 – $1.00 = $2.00 per package.

The problem also informs us that Ajay’s fixed costs for the month = $600. Thus, Ajay’s profit is: Profit before taxes = ($2.00  number of packages sold) – $600. b. Breaking even implies a profit of zero. Thus, we have: $0 = ($2.00  Breakeven volume) – $600, Or, breakeven volume in packages =

$600 = 300 packages. $2.00

c. Substituting Ajay’s target profit of $1,400 into the expression for profit we developed in part [a], we have: $1,400 = $2.00  Required number of packages – $600. OR, Required sales

=

$600  $1,400 . $2.00

= 1,000 packages.

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5-7

5.31

a. A 50% increase changes Ajay’s variable costs from $1.00 per package to $1.00  (1 + .50), or $1.50 per package. Consequently, the revised unit contribution margin = $3.00 – $1.50 = $1.50 per package. Setting target profit to zero in the expression for profit, we obtain: 0 = $1.50  Breakeven volume – $600. OR, Breakeven volume =

$600 = 400 packages. $1.50

Note: If Exercise 5.30 had been assigned, you may notice that the breakeven volume has increased by 100 packages – this occurs because Ajay now makes a lower contribution margin per package wrapped. b. Writing-out Ajay’s profit in detail, we have: Profit before taxes = (Price – Unit variable cost)  number of packages – Fixed costs. Substituting using the given data, we have: $2,400 = (Price – $1.00)  3,000 – $600. OR, Price = $2.00.

c. If revenue were $4,500, Ajay would have wrapped

$4,500 = 1,500 packages. $3.00

This sales volume would yield a total contribution margin of: 1,500 packages  $2 per package = $3,000. Netting out the fixed costs of $600 yields profit of $2,400. Alternatively, we can substitute 1,500 packages into the profit equation to yield: Profit before taxes = ($2.00  1,500) – $600 = $2,400.

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5-8

5.32 CVP Relation and Profit Planning, Contribution Margin Ratio Approach (LO1, LO2). a. As discussed in the text, we can re-write the profit as: Profit before taxes = (Contribution margin ratio  Revenue) – Fixed costs.

For Gina, her contribution margin ratio is

- $.25 ($1.00 ) $1.00

= 0.75.

Additionally, her fixed costs amount to $6,000 per month. Thus, we have: Profit before taxes = (.75  Revenue) – $6,000. b. Using the profit equation from part [a] and setting profit equal to $0, we have: $0 = (.75  Breakeven revenue) – $6,000. Breakeven revenue = $8,000. c. Substituting sales of $10,000 into Gina’s profit calculation yields: Profit before taxes = (0.75  $10,000) – $6,000 = $1,500. 5.33

CVP and Profit Planning, Hercules (LO1, LO2, LO3 a. We know that the profit equation is: Profit before taxes = (unit contribution margin  sales volume in units) – fixed costs. At breakeven profit before taxes = 0. For Hercules, unit contribution margin is revenue – variable costs = $100 - $35 = $65 per member per month. Fixed costs are $40,950 per month. Substituting, we have: 0 = $65 × Breakeven volume - $40,950 Thus, breakeven volume = $40,950/$65 per member = 630 members. b. First, let us gross up the after-tax target to a required pre-tax amount. We know: After tax profit = Pre-tax profit × (1- tax rate).

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5-9 Plugging in the numbers, we have: $11,375 / 0.65 = $17,500 = required pre-tax profit. Using this estimate in the pre-tax profit equation, we have: $17,500 = $65× required volume - $40,950 Required volume = $58,450/$65 per member = 899 members (rounded). c. We know from part [b] that Hercules needs to earn $17,500 before taxes to reach its goal. We calculate the contribution margin ratio as $65/$100 = 0.65 or 65%. We know: Profit before taxes = contribution margin ratio × required revenue – Fixed costs. $17,500 = 0.65 × required revenue - $40,950 Required revenue = $89,900 (rounding down). Of course, we also can calculate this answer from the answer to part [b]: 899 members × $100 per member = $89,900. d. The Margin of safety = [current sales – breakeven sales] / current sales. Using the answer from part [a], we have: [950 - 630] /950 = 33.68 % e. Operating leverage = Fixed cost / Total cost = $40,950 / [40,950 + 950 × $35] = 55.19% 5.34

Contribution Margin, Unit level costs (LO1). First, we need to calculate unit contribution and fixed costs. We have unit contribution = (price – variable costs) = ($800 - $440 - $40) = $320. At a volume of 15,000 units, fixed costs are $110 + $50 = $160 per unit, or $2,400,000 in total. Then, from the profit equation: 0 = unit contribution × breakeven volume – fixed costs 0 = $320 per unit × breakeven volume - $2,400,000 Breakeven volume = $2,400,000 / $320 per unit = 7,500 units.

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5-10

5.35 CVP Relation and solving for unknowns, Contribution Margin Ratio Approach (LO1, LO2). a. Substituting a target profit of $3,600 into the monthly profit equation, we have: $3,600 = (0.75  Required revenue) – $6,000.

OR, Required revenue =

(

$6,000  $3,600 0.75

)

= $12,800 per

month. b. Substituting the data into Gina’s profit equation, we have: $4,000 = (0.75  $15,000) – Fixed costs. Maximum expenditure on fixed costs = $11,250 – $4,000 = $7,250. c. Gina’s new variable cost is $0.25  (1 + 0.5) = $0.375 per $1.00 of sales.

Thus, Gina’s new contribution margin ratio is:

(

$1.00 - $.375 $1.00

)

= 0.625. Substituting this information into the profit calculation and setting profit equal to $0, we have: $0 = (0.625  Breakeven revenue)– $6,000. Breakeven revenue = $9,600. Note: If Exercise 5.32 has also been assigned, notice that Gina’s breakeven revenue increases by $1,600. From a pedagogical standpoint, instructors may wish to point out that exercises 5.30-5.34 are quite similar. The only difference is whether the problem is cast in terms of sales in units or sales in dollars. This framing, though, very much affects how we proceed.

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5-11

5.36 CVP Relation and Profit Planning, Unit Contribution Margin Approach, Taxes (LO1, LO2). a. When taxes are proportional to pre-tax profit, we can write the profit as: Profit after taxes

= Profit before taxes  (1 – Tax rate). = [(Unit contribution margin × Quantity) – Fixed costs]  (1 – Tax rate).

For SpringFresh, Unit contribution margin = $1.50 – $0.50 = $1.00. Annual Fixed cost = $50,000 × 12 = $600,000. The tax rate = .25 Consequently, SpringFresh’s annual after-tax profit is: Profit after taxes = [($1.00  Pounds laundered) – $600,000] × .75. Note: Please be careful to convert monthly fixed costs to an annual amount. b. SpringFresh’s profit before taxes = ($1.00  750,000 pounds) – $600,000 = $150,000. Taxes paid = $150,000  .25 = $37,500. Profit after taxes = $150,000  .75 = $112,500. c. Since profit and taxes = $0 at the breakeven point, we know that the profit expression in part [a] reduces to: $0 = $1.00  Breakeven volume – $600,000. Thus, Breakeven volume = 600,000 pounds.

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5-12

5.37 Profit after taxes = [(Unit contribution margin × Quantity) – Fixed costs]  (1 – Tax rate). $120,000 = [($1.00  Required volume in pounds) – $600,000] × .75. Thus, Required volume = 760,000 pounds. 5.38 CVP Relation in Non-Profits, Contribution Margin Ratio Approach (LO1, LO2). a. In this case, “profit” is the amount left over after paying for fixed costs. We have: $21,000 = (# of attendees × $50) - $15,000 Or, we need 720 attendees. b. Let us first write out the profit equation: $21,000 = (# of attendees × $50) + (# of attendees × 0.50 × $20) - $15,000 Solving, we find # of attendees = 600 Effectively, the cash bar raises the contribution per member to $60, which lowers the required volume. 5.39 CVP and Profit Planning, Contribution Margin Ratio Approach, Taxes (LO1, LO2). a. Recall that using the contribution margin ratio, we can express profit after taxes as: Profit after taxes = [(Contribution margin ratio  Billings) – Fixed costs] × (1 – Tax rate). For Arena, the contribution margin ratio is 1 – 0.30 = 0.70 (70% of billings). Further, Arena’s monthly fixed costs = $14,000 and the tax rate is 35%. Consequently, Arena’s monthly profit is: Profit after taxes = [(.70  Billings) – $14,000] × .65. b. At the breakeven point, profit after taxes = profit before taxes = $0 (i.e., no tax is due because there is no profit). Consequently, we have (since the tax rate is irrelevant at breakeven): Balakrishnan, Managerial Accounting 1e

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5-13

$0 = (0.70  Breakeven revenue) – $14,000. Solving, we find breakeven billings = $20,000. c. Using the profit expression in [a], we have: Profit before taxes = (0.70  $50,000) – $14,000 = $21,000. Profit after taxes = $21,000  .65 = $13,650. d. Again, using the profit expression in (a), we have: $7,280 = [(0.70  Required billings) – $14,000] × .65. Required billings = $36,000. 5.40 CVP Relation, Inferring Cost Structure, Extension to Decision Making (LO2, LO3). a. The contribution margin ratio =

Price - Unit variable cost ( ). Price With Zap’s information, we have the contribution margin ratio =

$22.00 - Unit variable cost ( $22.00 = 0.60. Unit variable cost = $8.80 for each “ZAP” kit. Alternatively, from the formula for contribution margin ratio, notice that Contribution margin ratio = 1 – (Unit variable cost/Price). The latter term is the “variable cost ratio.” Then, Contribution margin ratio = 1 – Variable cost ratio. Applying the data from the problem, we have: 0.60 = 1 – Variable cost ratio, or Variable cost ratio = 40% of sales price That is, unit variable cost = 0.40 × $22 = $8.80 per unit.

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5-14 b. Let us use the profit expression: Profit before taxes = (Contribution margin ratio  Revenue) – Fixed costs. We know that Zap expects to break even at 17,500 “ZAP” kits – thus, Breakeven revenue = 17,500 × $22.00 = $385,000. Additionally, we know that profit = $0 at the breakeven point. Thus, we have: $0 = (0.60  $385,000) – Fixed costs. Solving, we find that fixed costs = $231,000. Alternatively, we could use the unit contribution margin formulation. We can calculate the breakeven point as: $0 = (Unit contribution margin  Breakeven volume) – Fixed costs. From part [a], we know that the unit variable cost = $8.80. Because the selling price = $22.00, we know that the unit contribution margin = $22.00 – $8.80 = $13.20. Thus, we have: $0 = ($13.20  17,500) – Fixed costs. Again, we find that fixed costs = $231,000. c. The free shipping and handling offer reduces Zap’s revenue per “ZAP” kit to $20.00. With the knowledge acquired in parts [a] and [b] (i.e., the variable cost per “ZAP” kit and Zap’s monthly fixed costs, respectively), we can calculate Zap’s breakeven volume as: $0 = (Unit contribution margin  Breakeven volume) – Fixed costs. or, $0 = [($20.00 – $8.80)  Breakeven volume] – $231,000 Breakeven number of kits (Breakeven volume) = 20,625. Consequently, Zap must sell an additional 20,625 – 17,500 = 3,125 kits to break even if the company decides to offer “free” shipping. Note: Instructors may wish to use this problem to emphasize the importance of knowing both the unit contribution margin approach and the contribution margin ratio approach. In part [a], it was necessary to use the contribution margin ratio to arrive at the variable cost per unit. In part [c], however, the unit contribution margin approach more readily accommodates a reduction in the sales price – i.e., it is relatively straightforward to calculate the unit contribution margin. Calculating the new contribution margin ratio is somewhat more involved, although it will lead to Balakrishnan, Managerial Accounting 1e

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5-15 an equivalent answer as the unit contribution margin approach. Moreover, the new contribution margin ratio is (20 – 8.80)/20 = 0.56 or 56%. 5.41 CVP Relation and Decision Making, Pricing based on a Demand Schedule (LO3). Employing the CVP relation, we can compute the profit at alternative prices to determine the price that yields the maximum profit. The following table contains the detailed computations.

Price

Revenue

$32.50 $30.00 $27.50 $25.00 $22.50

$9,750 $10,500 $11,000 $11,250 $11,250

Variable Costs $1,800 $2,100 $2,400 $2,700 $3,000

Fixed Costs

Profit

$3,000 $3,000 $3,000 $3,000 $3,000

$4,950 $5,400 $5,600 $5,550 $5,250

By inspection, we find $27.50 to be the profit-maximizing price. Greg earns $5,600 in profit at this price. Note: Instructors may wish to point out that firms may use a demand function instead of using a demand schedule like the table above. A demand function gives the revenue for every possible price. (For example, based on the data provided, Greg’s demand function is: Quantity = 950 – 20 × Price). In this case, we use calculus or numerical approximation techniques (via a spreadsheet such as Excel’s solver function) to determine the profit-maximizing price. 5.42 CVP relation and Decision Making, Choosing a Cost Structure, Operating Leverage (LO3, LO4). a. One’s first inclination is to compare the profit across the various popcorn machines. However, for the same number of customers, revenue is equal across the three machines. Thus, we can rank order popcorn machines according to their total costs. In other words, the problem can be formulated as a cost minimization problem as the machine that minimizes cost also maximizes profit. We start by assessing the number of patrons at which the small popcorn machine will cost the same as the medium popcorn machine. We have: $6,000 + (0.50  number of patrons) = $12,000 + (0.35  number of patrons).

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5-16 $6,000 = 40,000. .15

The number of patrons at which the cost is the same =

Thus, when a theater expects less than 40,000 moviegoers a year (or approximately 110 per day), it is optimal to rent the small popcorn machine. Comparing the medium and the large popcorn machines, we have: $12,000 + (0.35  number of patrons) = $18,500 + (0.25  number of patrons). $6,500 = 65,000. .10

The number of patrons at which the cost is the same =

Thus, when a theater expects more than 65,000 moviegoers a year (or approximately 178 per day), it is optimal to rent the large popcorn machine. Additionally, the above analysis informs us that when a theater expects between 40,000 and 65,000 moviegoers a year, it is optimal to rent the medium popcorn machine. Thus, we have the following decision rule: Midwest Cinema Theaters Popcorn Machine Rental Model Annual # of Moviegoers (M)

Popcorn Machine Size

M < 40,000

Small

40,000 < M < 65,000

Medium

M > 65,000

Large

Note: Instructors also may wish to graphically represent the tradeoff – this is perhaps best accomplished by asking students to graph, for each size popcorn machine, how popcorn costs (y-axis) varies as a function of the number of patrons (x-axis). This allows students to see where the lines cross – The instructor can then shade the low cost frontier to see how the preferred machine depends on expected volume. b. Operating leverage

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5-17

For each popcorn machine, we have: Fixed Costs

Variable Costs

Small

$6,000

Medium

$12,000

Large

$18,500

$32,500 (= 65,000 × $0.50) $22,750 (= 65,000 × $0.35) $16,250 (= 65,000 × $0.25)

Operating Leverage .1558 .3453 .5323

As discussed in the text, operating leverage frequently is used as a measure of risk – ceteris paribus, the higher the operating leverage, the higher the risk. Thus, while we see that the large popcorn machine is preferred for volumes of 65,000 moviegoers and higher, it also carries the highest risk, a factor that Leticia may wish to consider in her decision. 5.43 CVP Relation and Decision Making, Margin of Safety, Operating Leverage, Cash-Basis Breakeven Analysis (LO3, LO4). a. Margin of safety =

Current sales - Breakeven sales ( ) Current sales We first need to determine Cottage Bakery’s breakeven sales. Using the contribution margin ratio to write the profit equation, we have: Profit before taxes = (Contribution margin ratio  Revenue) – Fixed costs. Because we do not know Cottage Bakery’s fixed costs, we first have to use the model to “back out” fixed costs and then derive breakeven sales. Accordingly, $7,500 = (0.4  $150,000) – Fixed costs Fixed costs = $52,500. Next, setting target profit equal to $0 will allow us to calculate breakeven revenue: $0 = (0.4  Breakeven revenue) – $52,500 Breakeven revenue = $131,250.

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5-18

(

Cottage Bakery’s Margin of safety =

$150,000 - $131,250 $150,000

)

= 12.5%.

In dollars, the margin of safety equals $150,000 – $131,250 = $18,750. b. Operating leverage = Fixed costs/Total costs. We know that the contribution margin (in dollars) = Contribution margin ratio  Revenue. Thus, we have: Contribution margin = 0.4  150,000 = $60,000. Because contribution margin = revenues – variable costs, variable costs can be calculated as $90,000. Given that fixed costs are $52,500, the total costs are $142,500. Consequently, Operating leverage = fixed costs / total costs = $52,500/$142,500 = 0.368 (rounded) c. If 30% of the fixed costs represent non-cash expenses, the cash fixed expenses equal: 0.70  $52,500 = $36,750. We are now in a position to write the cash profit as: Cash profit = (Contribution margin ratio  Revenue) – Cash fixed costs. To calculate the breakeven point, we set cash profit = $0. $0 = (0.40  Cash breakeven revenue) – $36,750. Cash breakeven revenue = $91,875. Notice the large difference between the revenue required to breakeven on a cash basis ($91,875) and the revenue required to breakeven on a non-cash (accrual) basis ($131,250). Moreover, this problem presents a nice opportunity to talk with students about which profit model is more germane to the firm. While cash basis accounting may paint an unusual picture of the organization’s health (due to the slippage between cash-basis accounting and “economic reality”), many organizations (and people) need to ensure that they remain solvent/liquid in the short-term and that cash expenditures do not exceed cash revenues.

Balakrishnan, Managerial Accounting 1e

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5-19

5.44

Multi-Product CVP Analysis, Unit Contribution Margin Approach (LO5). a. Let us employ a weighted unit contribution margin approach to solve the problem. For Mountain Maples, we have: 2,400 total trees sold – 800, or 1/3 are Butterfly, and 1,600, or 2/3, are Moonfire. Thus, we have: Weighted unit contribution margin = $66.67.

= 1/3  $100 + 2/3  $50.

In turn, Mountain Maples’ profit becomes: Profit before taxes = ($66.67  total number of trees sold) – $75,000. At the breakeven point, we have: $0 = ($66.67  Breakeven number of trees) – $75,000. Solving, we find that the total number of trees sold to breakeven = 1,125. Of these, 1,125  1/3 = 375 Butterfly; 1,125  2/3 = 750 Moonfire. b. Using the weighted unit contribution margin approach, we have: $50,000 = ($66.67  total number of trees) – $75,000. The total number of trees = 1,875. Of these,

1,875  1/3 = 625 are Butterfly, and 1,875  2/3 = 1,250 are Moonfire

c. The change in the product mix affects Mountain Maple’s weighted contribution. With the new information, we have: Weighted contribution margin= (.50  $100) + (.50  $50) = $75.00 The weighted contribution margin is higher than in part [a] because the product mix has shifted toward Butterfly, which has the highest contribution margin per tree. Consequently, the total number of trees required to break even will decrease:

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5-20

0 = ($75.00  Breakeven number of trees) – $75,000. Breakeven number of trees = 1,000. Of these,

5.45

1,000  .50 = 500 are Butterfly, and 1,000  .50 = 500 are Moonfire

Multi-Product CVP Analysis, Contribution Margin Ratio Approach (LO5). a. In this setting, we must use the weighted contribution margin ratio approach given the absence of unit-level data. Accordingly,: Profit before taxes = (RevenueN  Contribution margin rationN) + (RevenueU  Contribution margin ratioU) – Fixed costs, The subscripts ‘N’ and ‘U’ stand for new and used. Additionally, we know that $1,500,000/$2,000,000, or 75% of the revenue is from new cars, and $500,000/$2,000,000, or 25% of the revenue is from used cars. Further, we can calculate the contribution margin ratio for each product using the product-level financial data. We have: (Contribution margin ratio)N =

1,500,000  750,000 = .50. 1,500,000

(Contribution margin ratio)U =

500,000  200,000 = .60. 500,000

Thus, the weighted contribution margin ratio = (.50 × .75) + (.60 × .25) = .525. We can now write Select’s profit in terms of the weighted contribution margin ratio and total revenues: Profit before taxes = (.525 × Total revenue) – $840,000. Setting profit equal to $0, we find: Breakeven total revenue = $1,600,000. This translates into $1,600,000  .75 = $1,200,000 in new auto sales and $1,600,000  .25 = $400,000 in used auto sales. Balakrishnan, Managerial Accounting 1e

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5-21

b. To answer this question, we plug in our desired profit in the equation for profit developed in part [a]. We now have: $1,050,000 = (.525  Total revenue) – $840,000. Solving, we find: Total revenue = $3,600,000. This translates into $3,600,000  .75 = $2,700,000 in new auto sales and $3,600,000  .25 = $900,000 in used auto sales. 5.46 Multi-Product Analysis, weighted contribution margin & weighted contribution margin ratio approach, Hercules (LO5). a. We know that individual and family memberships have a 3:1 ratio (900 individual to 300 families). Thus, the weighted contribution margin is: [3 × ($100-$35) + 1 × ($150 -$60)] / 4 = $71.25. Then, plugging into the profit equation, we have: 0 = $71.25 × total memberships - $42,750, Total memberships = $42,750/$71.25 per average membership= 600. At this volume, Hercules has (3/4) × 600 = 450 individual and (1/4) × 600 = 150 family memberships. b. We calculate the contribution margin for individual and family memberships at 0.65 (= [($100-$35)/$100] and 0.60 (= [($150-$60)/$150] respectively. We know that individual and family memberships have a 2:1 ratio in terms of total revenue (Individual revenue is 900 members × $100 per month = $90,000 and family revenue is 300 memberships × $150 per month = $45,000.) Thus, the weighted contribution margin ratio is: [(2/3) × 0.65 + (1/3) × 0.60] = 0.6333 = 63.33%. Please note that we weight the individual contribution margin ratios by their revenue shares. In contrast, we used the share of memberships to weight individual contribution margins in part [a]. Plugging into the profit equation, we have: 0 = 0.63333 × total revenue - $42,750 Balakrishnan, Managerial Accounting 1e

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5-22

Total memberships = $42,750/0.63333 = $67,500. Note: You can verify this answer by using the answer for part [a]. Total revenue realized at the answer to part [a] is 450 × $100 + 150 × $150 = $67,500!

PROBLEMS 5.47 CVP relation, Profit Planning, Unit Contribution Margin Approach, Extensions to Decision Making (LO1, LO2, LO3). a. The general CVP model is: Profit before taxes

= [(Price – Unit variable cost)  Quantity] – Fixed cost. = (Unit contribution margin  Quantity) – Fixed costs.

Plugging in both the membership fee and the fixed and variable cost information for Garnet’s Gym, we have: Profit

= [($500 – $200)  Number of members] – $1,200,000. = ($300  Number of members) – $1,200,000.

b. To calculate the breakeven point in members, we set profit equal to $0 in the expression for profit in [a]: $0 = 300  Breakeven volume – $1,200,000. Number of members required to break even = 4,000. c. Here, we plug in the number of members from the previous year into the profit equation developed in part [a]. Doing so yields: Profit before taxes = ($300  5,000) – $1,200,000 = $300,000. d. This strategy changes the annual membership fee to $500  .90 = $450. In turn, this changes the per-member contribution margin to $450 – $200 = $250. If membership increases to 6,500 because of the discount, then expected profit is: Profit before Taxes = ($250  6,500) – $1,200,000 = $425,000. This action would increase profit by $125,000 (i.e., $425,000 – $300,000) compared to the previous year. Balakrishnan, Managerial Accounting 1e

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5-23

This seems like a good option to increase profit unless there are significant “congestion costs.” The owners probably should think through the soundness of increasing membership by 1,500 persons, or 1,500/5,000 = 30%. It is possible that this will translate to increased waiting time for machines and equipment, or difficulty finding a parking spot. The owners may wish to survey current members to assess their preferences. Such a strategy could backfire if people believe that their “cozy” gym has become too crowded and, perhaps, filled with the “wrong” type of people (e.g., those who are not serious about training). On the other hand, current members may desire more members as this increases the potential for finding dates, friendships, and so on. e. We start with our finding from part [d] that profit is expected to be $425,000 if the owners reduce the membership fee. We set this amount equal to our target profit and solve for the advertising expense. Accordingly, we have: $425,000 = ($300  6,500) – $1,200,000 – Advertising. We find that the maximum advertising expenditure = $325,000. In terms of comparing the options, the owners should assess which option is likely to be more costly, in terms of foregone contribution margin or out-of-pocket expense, and which option is likely to lead to the greatest increase in membership (if this is the desired outcome). f. The two approaches are mathematically equivalent. To see this, we start with the unit contribution margin approach: Profit before taxes = Unit contribution margin  Quantity – Fixed costs. To arrive at the contribution margin ratio approach, we first multiply UCM  Q by P/P, giving us: (UCM  Q) 

P . P

In turn, this expression can be re-written as: UCM  (Q  P). P

This should look familiar – it is equivalent to: CMR  Revenue. Consequently, the two approaches will produce the same answers and, thus, the owners of Garnet’s Gym need not worry – our answers will not change. Balakrishnan, Managerial Accounting 1e

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5-24

5.48 CVP Relation, Profit Planning, Contribution Margin Ratio Approach, Extensions to Decision Making (LO1, LO2, LO3). a. Given the absence of unit-level data, we need to employ the contribution margin ratio approach. Additionally, since we are asked to find the breakeven point, and profit = $0 at the breakeven point, taxes are not relevant. Thus, we have: $0 = (Contribution margin ratio  Breakeven revenue) – Fixed costs. For Precious Stone Jewelry, the contribution margin ratio is:

- 600,000 ($1,000,000 ) $1,000,000

= 40%.

Additionally, Precious Stone Jewelry’s fixed costs = $260,000. Thus, we have: $0 = (0.40  Breakeven revenue) – $260,000. Solving, we find breakeven revenue = $650,000. b. Increasing the selling price by 20% will increase revenues by 20% (because quantity stays the same) and, in turn, increase the contribution margin ratio. First, we have: Revised revenues = 1.20  1,000,000 = $1,200,000. The new contribution margin ratio is:

- 600,000 ($1,200,000 ) $1,200,000

= 50%.

Thus, we have: $0 = (0.50  Breakeven revenue) – $260,000. Solving, we find breakeven revenue = $520,000. Thus, breakeven revenue would decrease by $650,000 – $520,000 = $130,000. c. If variable costs decrease by 20%, new variable costs will be: Variable costs = .80  600,000 = $480,000. Balakrishnan, Managerial Accounting 1e

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5-25 In turn, the new contribution margin ratio becomes:

( $1,000,000 - 480,000 ) $1,000,000

= 52%

Thus, we have: $0 = (0.52  Breakeven revenue) – $260,000. Solving, we find breakeven revenue = $500,000. Thus, breakeven revenue would decrease by $650,000 – $500,000 = $150,000. d. As discussed in part [a], taxes have no effect on the breakeven point. Since taxes are a percentage of profit, any percentage of $0 is always $0. Thus, the tax rate is not relevant. As illustrated in part [e], however, taxes are relevant when the firm earns positive profit.

e. If all changes do take place, we have: Revenues

= $1,000,000  1.20

Variable costs

= $600,000  (1 – .20)

Contribution margin Fixed costs

Profit after taxes

480,000 $720,000

(stay the same)

Profit before taxes Taxes

$1,200,000

260,000 $460,000

.30  $460,000

138,000 $322,000

Thus, profit is expected to increase by $217,000, from $105,000 to $322,000. Both the change in price and unit variable cost increase Precious Stone’s contribution margin and, in turn, profit. The increase in the tax rate diminishes the amount of Profit before taxes that Precious Stone retains – that is, the increase in the tax rate reduces the slope of the profit after taxes line.

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5-26

5.49

CVP Relation and Profit Planning, Solving for Unknowns (LO1, LO2). a. The general model for the city’s snow removal costs looks like (notice that there is no need for a revenue, or profit, component as the problem deals strictly with cost): Total costs = Fixed costs + (Variable cost per snowfall  Number of snowfalls). We are provided with total costs and snowfall data for two separate years: $300,000 = Fixed costs + Variable cost per snowfall  20. $228,000 = Fixed costs + Variable cost per snowfall  12. We now have two equations with two unknowns – accordingly, we can subtract the second equation from the first equation. Doing so yields: $72,000 = Variable cost per snowfall  8. OR, variable cost per snowfall = $9,000. Plugging this back into either year we find that fixed cost = $120,000. Thus, the city’s snow removal costs can be calculated as: Total Snow Removal Costs = $120,000 + ($9,000  Number of major snowfalls). b. For this question, it is simply a matter of plugging in the anticipated number of snowfalls into the expression for snow removal costs developed in part [a]. We have: Expected Snow Removal Costs = $120,000 + ($9,000  26) = $354,000. Thus, the city should request $54,000 more than it spent this year! This problem, which links nicely back to Chapter 3, is useful for showing students how CVP can be used for planning/budgeting purposes. This problem, though, deals only with the C(ost) and the V(olume) portions of the model.

5.50 Building a CVP Relation that Incorporates Taxes and Bonus Payments using a Contribution Margin Ratio Approach (LO1, LO2). The first step is to write Diamond Jubilee’s after-tax profit as follows: Profit after taxes = (Total wagers – Winnings – Variable costs – Bonus – Fixed costs)  (1 – Tax rate). Balakrishnan, Managerial Accounting 1e

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5-27 Because we are not provided with data regarding how much was wagered in total, we will need to work with the contribution margin ratio approach. The model is more involved because the bonus is a function of pre-tax income. We begin by modeling pre-tax and pre-bonus profit: Pre-tax and pre-bonus profit = (Contribution margin ratio  Total wagers) – Fixed costs. Here, contribution margin ratio is the contribution margin ratio without the bonus; it is $1.00 – $0.82 – $0.08 = $0.10 or 10%. Further, fixed costs equal $27,500. Thus, we have: Pre-tax and pre-bonus profit = (.10  Total wagers) – $27,500. We are now in a position to add the bonus payment to the model as the manager’s bonus equals 5% of the pre-tax and pre-bonus profit. After subtracting the bonus, which is 5% of the pre-tax and pre-bonus profit, we have pre-tax (but after bonus) profit as: Pre-tax profit = [(.10  Total wagers) – $27,500]  (1 – .05). We can next add taxes to the model. With a tax rate of 25%, we have: After-tax profit = {[(.10  Total wagers) – $27,500]  (1 – .05)}  (1 – .25). At this point, we need only substitute the desired monthly after-tax, after-bonus profit of $28,500. Thus, $28,500 = {[(.10  Total wagers) – $27,500]  (1 – .05)}  (1 – .25). Solving, we find that the required monthly level of total wagers (gross gambling revenue) = $675,000. This problem demonstrates how the “basic” CVP relation can be expanded to incorporate additional short-term profit considerations such as bonuses and taxes. 5.51 CVP Relation and Profit Planning, Choosing a Cost Structure (LO1, LO2, LO3). a. To calculate breakeven revenue, we start with the profit calculation using the contribution-margin ratio: Profit = (Contribution margin ratio  Revenue) – Fixed costs. Setting profit = $0,

Balakrishnan, Managerial Accounting 1e

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5-28 Breakeven revenue = Fixed costs/Contribution margin ratio. Given the information in the problem for each cost structure, we have: Monthly breakeven revenue under current cost structure: Fixed costs $36,000 per month Contribution margin ratio 40% Breakeven Revenue $36,000/0.40 = $90,000

Monthly breakeven revenue under new cost structure: Fixed costs $60,000 per month Contribution margin ratio 60% Breakeven revenue $60,000/0.60 = $100,000

Thus, Cecelia’s breakeven revenue will increase by $10,000 if she acquires the new machines. Even though her contribution margin ratio increases (which, ceteris paribus, pushes the breakeven point down) by acquiring the new machines, the substantial increase in fixed costs drives the breakeven point up. b. As in part [a], the profit is: Profit = (Contribution margin ratio  Revenue) – Fixed costs. We now plug in the various parameters to determine profit under each cost structure. Current cost structure: Profit at $95,000 in revenue = (0.40  $95,000) – $36,000 = $2,000. Profit at $150,000 in revenue = (0.40  $150,000) – $36,000 = $24,000. New cost structure: Profit at $95,000 in revenue = (0.60  $95,000) – $60,000 = ($3,000). Profit at $150,000 in revenue = (0.60  $150,000) – $60,000 = $30,000. Thus, Cecelia prefers her current cost structure if monthly revenues are expected to be $95,000, and she prefers to acquire the new machines if revenues are expected to be $150,000. c. We can find this point of indifference, or crossover point, by equating the profit equation under the two cost structures and solving for revenue. Let the required revenue level be $R. The profit at $R is:

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5-29 Current cost structure: New cost structure:

(0.40  $R) – $36,000. (0.60  $R) – $60,000.

Equating the two profit equations, we have: (0.40  $R) – $36,000 = (0.60  $R) – $60,000. $R = ($60,000 – $36,000)/(0.60 – 0.40) = $120,000. With $120,000 of revenue, Cecelia makes the same profit of $12,000 with either cost structure. Also notice that for sales greater than $120,000, Cecelia prefers to acquire the new machines whereas for sales less than $120,000 Cecelia prefers her current cost structure. That is, the slope of the profit line is higher under the new cost structure than the old cost structure. Instructors may wish to graph the two cost structures to illustrate this point to students. 5.52 CVP Relation and Decision Making, Pricing Based on a Demand Schedule (LO3). a. The following table provides the profit computations (and comparisons) if fixed costs were $1,500,000 per year and variable costs were $1 per copy. Introductory Price Users (copies sold) Year 1 Price per copy Variable cost per copy Contribution margin per copy Total contribution margin Fixed costs Profit before taxes

$25 $15 $1 $1 $24 $14 $1,800,000 $2,100,000 $1,500,000 $1,500,000 $300,000 $600,000

$5 $1 $4 $1,200,000 $1,500,000 $(300,000)

Year 2 Price per copy Variable cost per copy Contribution margin per copy Total contribution margin Fixed costs Profit before taxes

$25 $25 $1 $1 $24 $24 $1,800,000 $3,600,000 $1,500,000 $1,500,000 $300,000 $2,100,000

$25 $1 $24 $7,200,000 $1,500,000 $5,700,000

$600,000 $2,700,000

$5,400,000

Year 1 + Year 2 Profit

$25/copy 75,000

$15/copy 150,000

$5/copy 300,000

The table clearly shows that Innova Solutions maximizes two-year profit by setting an introductory price of $5 per copy (this would be true for just about any discount rate).

Balakrishnan, Managerial Accounting 1e

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5-30 b. The following table provides the profit computations (and comparisons) if fixed costs were $200,000 per year and variable costs were $15 per copy. Introductory Price Users (copies sold) Year 1 Price per copy Variable cost per copy Contribution margin per copy Total contribution margin Fixed costs Profit before Taxes Year 2 Price per copy Variable cost per copy Contribution margin per copy Total contribution margin Fixed costs Profit before Taxes Year 1 + Year 2 Profit

$25/copy 75,000

$15/copy 150,000

$5/copy 300,000

$25 $15 $5 $15 $15 $15 $10 $0 ($10) $750,000 $0 ($3,000,000) $200,000 $200,000 $200,000 $550,000 ($200,000) ($3,200,000) $25 $25 $15 $15 $10 $10 $750,000 $1,500,000 $200,000 $200,000 $550,000 $1,300,000

$25 $15 $10 $3,000,000 $200,000 $2,800,000

$1,100,000 $1,100,000

($400,000)

Here, we see that Innova Solutions’ optimal pricing strategy changes – the table shows that Innova Solutions probably is best off by setting an introductory price of $25 per copy (inter temporal considerations also would lead Innova Solutions to go with $25 rather than $15). c. In an industry where an “installed base” of customers is important (as in software or video games), firms often sacrifice today’s profit to build market share and tomorrow’s profit. The key idea is that that the firm can increase profitability in future years by taking advantage of consumers’ switching (transaction) costs. The efficacy of this strategy depends on the current period tradeoff between demand and price. If demand increases sufficiently with a price drop, then the strategy of going for a low introductory price can generate significantly more profit in the long-run. However, as the tables show, the required increase in demand increases as variable costs increase. For Innova, “low-balling” maximized profit when the variable cost was low but not when the variable cost was high. Thus, we often find such a pricing strategy being followed only by those firms with low variable costs (equivalently, high contribution margin ratios) as a percentage of price. While such a strategy may work for software or video games, it is unlikely to work for auto manufacturers.

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5-31 By using Excel, it is relatively straightforward for firms to model the relationship between price and resulting demand. Moreover, firms often spend considerable effort and resources in constructing sophisticated models that capture the economic forces of supply and demand. 5.53

CVP Relation and Margin of Safety (LO4). a. This is a non-standard problem in the sense that there is no variable cost. That said, we can view the commission as Brenda’s contribution margin ratio. With this interpretation, the profit is: Profit = (Contribution margin ratio  Transactions in dollars) – Fixed costs. To calculate monthly breakeven transactions, we set profit = $0 and fixed costs equal to $18,000. This yields: $0 = (0.03  Breakeven Transactions) – $18,000. Thus, the volume of monthly transactions required to breakeven = $600,000. b. Margin of safety =

( current sales - breakeven sales ) current sales

=

.

- $600,000 ($1,000,000 ) $1,000,000

.

= 40%. Thus, Brenda’s transaction volume could decrease by 40%, or $400,000 before she incurs a loss in a month. c. Using the setup from part [b], we find Brenda’s margin of safety for a transaction volume of $1,200,000 to be:

Margin of safety =

- $600,000 ($1,200,000 ) $1,200,000

=

50%.

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5-32 Similarly, for a transaction volume of $1,600,000, we find:

$1,600,000 - $600,000 Margin of safety = ( ) $1,600,000

=

62.50%.

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5-33

d. First, we notice that margin of safety only makes sense at a given sales volume. As sales change, the margin of safety also changes. Second, we notice that that margin of safety increases as transaction volume increases, reflecting the additional sales over the breakeven volume of sales (and, hence, a larger “cushion”). Finally, we notice that the relationship between sales and margin of safety is non-linear. In Brenda’s case, 20% and 60% increases in transaction volume lead to 25% and 56.25% increases in her margin of safety — i.e., 25% = (50-40)/40; 56.25 = (62.50-40)/40. 5.54 CVP Relation and Decision Making, Operating Leverage, Margin of Safety (LO3, LO4). a. We have enough information to construct condensed income statements for each proposal: Revenues Variable Costs* Contribution Margin** Fixed Costs Profit before Taxes

Proposal 1 $2,750,000 1,100,000 $1,650,000 1,500,000 $150,000

Proposal 2 $2,750,000 1,925,000 $825,000 675,000 $150,000

* = (1 – .60)  $2,750,000; (1 – .30)  $2,750,000. ** can also be calculated as: .60  $2,750,000; .30  $2,750,000. Thus, expected profit is equal under both proposals. Turning to operating leverage, we have: Operating leverage = Fixed costs/Total costs. The condensed income statements provide us with all of the information necessary to compute operating leverage: Operating leverage (proposal 1) = $1,500,000/$2,600,000 = 0.577 (rounded). Operating leverage (proposal 2) = $675,000/$2,600,000 = 0.260 (rounded). The margin of safety = (current sales – breakeven sales)/current sales. Thus, we need to calculate the breakeven revenue under each proposal. To do so, we calculate profit using the contribution margin ratio. We have: $0 = Contribution margin ratio  Breakeven revenue – Fixed costs; Breakeven revenue = Fixed costs/Contribution margin ratio. Thus:

Balakrishnan, Managerial Accounting 1e

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5-34 Breakeven revenue (proposal 1) = $1,500,000/.6 = $2,500,000. Breakeven revenue (proposal 2) = $675,000/.3 = $2,250,000. We are now in a position to calculate the margin of safety: Margin of safety (proposal 1) = ($2,750,000 – $2,500,000)/$2,750,000 = .0909 (rounded). Margin of safety (proposal 2) = ($2,750,000 – $2,250,000)/$2,750,000 = .1818 (rounded). Now that we have performed all of the requisite calculations, we are in a position to think about which proposal the bank is likely to support. This is a difficult question. First, notice that expected profit is identical under each alternative. Thus, we need to consider the proposals on some other dimension – the upside potential and/or the downside risk. In terms of the upside, each additional dollar of revenue generated under proposal #1 contributes $.60 to profit, whereas under proposal 2, each additional dollar of revenue generated contributes only $.30 to profit. Thus, proposal 1 has higher upside potential than proposal 2. Both operating leverage and margin of safety provide us with some measure of business risk and, hence, the downside. Operating leverage is a measure of the extent of fixed costs in the business – notice that the first proposal has much higher operating leverage. This is because the first proposal has significantly higher fixed costs than the second proposal. The margin of safety provides us the amount by which sales revenue could decrease before Dan is in the red. Notice that the second proposal has a higher margin of safety than the first proposal – thus, there is more of a cushion in the second proposal (as reflected by the lower break-even point). Thus, proposal 1 has more downside risk than proposal 2. We really need to get inside the banker’s head and think about his/her objective, which likely is “will I get my money back?” Given this objective, a prudent (and probably risk-averse) banker is likely to push the less risky proposal #2. This conclusion is not, however, a fait accompli and this question can generate interesting discussions about risk preferences and decision-making under uncertainty.

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5-35

b. To calculate profit, operating leverage, and margin of safety, we can repeat our approach from part [a]. We have:

Revenues Variable Costs* Contribution Margin** Fixed Costs Profit before Taxes

Proposal 1 $4,500,000 1,800,000 $2,700,000 1,500,000 $1,200,000

Proposal 2 $4,500,000 3,150,000 $1,350,000 675,000 $675,000

* = (1 – .60)  $4,500,000; (1 – .30)  $4,500,000. ** = can also be calculated as: .60  $4,500,000; .30  $4,500,000. Turning to operating leverage, we have: Operating leverage (proposal 1) = $1,500,000/$3,300,000 = .455 (rounded). Operating leverage (proposal 2) = $675,000/$3,825,000 = .176 (rounded). For margin of safety, we have: Margin of safety (proposal 1) = ($4,500,000 – $2,500,000)/$4,500,000 = .4444 (rounded). Margin of safety (proposal 2) = ($4,500,000 – $2,250,000)/$4,500,000 = .50.

Notice that as sales increase, proposal 1 becomes more attractive relative to proposal 2. First, expected profit is significantly higher ($525,000) under proposal 1 than proposal 2. Second, the upside potential of proposal 1 is higher than proposal 2 – as discussed in part [a], each additional $ of revenue contributes $0.60 to profit under proposal 1, but only $0.30 under proposal 2. Finally, compared to part [a], the differences in operating leverage and, particularly, the margin of safety between the proposals are smaller – proposal 1 is only slightly riskier than proposal 2. Given the profit difference and minimal risk, a prudent lender would likely push proposal 1 in this setting.

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5-36

5.55

Multi-Product CVP Analysis (LO5). a. The profit is: Profit before Taxes = [(Price – unit variable cost)  Quantity] – Fixed costs. Substituting for Campus Bagels’ specific information, we have: Profit = [($1.00 – $0.40)  number of bagels sold] – $100,000. Given the 250,000 bagels sold in the previous year, profit was: Profit before taxes = $0.60  250,000 – $100,000. Profit before taxes = $50,000. b. Campus Bagels will now have two different products, bagels and bagel sandwiches. Accordingly, the profit is: Profit = [Quantitybagels  (1.00 – .40)] + [Quantitybagel sandwiches  (Pricebagel sandwiches – 1.25)] – $125,000. Notice that fixed costs have increased by $25,000 and, in order to determine net income, Campus Bagels needs to know the price of bagel sandwiches as well as the quantity of bagel sandwiches to be sold. One might also argue that the quantity of bagels that will now be sold is ambiguous. More generally, whether Campus Bagels will continue to sell 250,000 bagels depends on whether bagels and bagel sandwiches are complements or substitutes. One could argue the case either way – on the one hand, some customers might switch from regular bagels to the “more refined” bagel sandwich. Additionally, some current customers might already convert the bagels they buy into bagel sandwiches. On the other hand, the products may complement each other. Strengthening the product line by adding bagel sandwiches may attract new customers to Campus Bagels who, in turn, not only buy bagel sandwiches but also are drawn to regular bagels. c. It is important to note that the problem setup is somewhat atypical in that it is problematic to specify a quantity without knowing a price – it probably is useful to discuss this issue in class. That said, if we wish to increase profit by $50,000 then we have:

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5-37 $100,000 = [250,000  (1.00 – .40)] + [25,000  (Pricebagel sandwiches – 1.25)] – $125,000. Solving, we find Price= $4.25. d. This scenario is more typical as the demand for one product affects the demand for the other and vice-versa. Now, in addition to the incremental variable and fixed costs associated with introducing bagel sandwiches, there is an opportunity cost associated with selling bagel sandwiches – the lost contribution margin on bagels. We can calculate this opportunity cost directly and add it to the cost of bagel sandwiches or, perhaps more intuitively, we can use the model from part [b], which has this opportunity cost “built in.” In other words, notice that we can directly plug the new quantities into the model, or: $100,000 = [225,000  (1.00 – .40)] + [25,000  (Pricebagel sandwiches – 1.25)] – $125,000. Solving, we find Price = $4.85. Here, we see that relationships among products can be captured in our CVP formulation. At a more general level, we could explicitly model this relationship (e.g., Quantitybagels = 250,000 – Quantitybagel sandwiches). Moreover, when multiple products exist we need to consider both complementary and substitute relationships. Such relationships yield positive or negative externalities and, in turn, reduce or increase profitability. CVP models can help sort-out these issues. 5.56 Multi-Product CVP and Fixed Cost Allocations (LO5). a.

The following table provides the required computations.

Traceable fixed costs Allocated fixed cost Total Contribution margin ratio Breakeven revenue

Retail 175,000 100,000 275,000 66.67% $412,500

Institutional 80,000 100,000 180,000 40% $450,000

At this volume, Jan breaks even for the entire company as well. After all, his total fixed costs are $175,000 + $80,000 + $200,000 = $455,000. Then, at the computed volumes, he generates a contribution of $412,500 × 0.6667 + $450,000 × 0.4 = $455,000. The firm also breaks even at the total level of $862,500.

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5-38 b.

Jan’s weighted contribution margin ratio is $480,000/$900,000 = 53.33%. With this estimate, we can calculate breakeven revenue as $455,000 / 0.53333 = $853,125 Or, $426,562.50 each in retail and institutional sales.

c.

The answers in parts a and b differ because we “fix” different items. In part (b), we fixed the sales mix to be 50% from each segment. With this assumption, the breakeven sales are $853,125. However, sales mix is not fixed in part (a). Rather, we have “fixed” the allocation to be $100K to each segment. Thus, the final answer has a sales mix that is NOT 50-50 across the segments. Moreover, the fixed cost also is not allocated in proportion to the sales mix at breakeven! In general, it makes more sense to fix the sales mix as in part (b) in multi-product CVP analysis. d. In general, we perform multi-product CVP for the entire firm. This is particularly appropriate when the products are similar (e.g., car dealership), are substitutes and share considerable fixed cost. However, we also encounter situations in which products are distinct with few common fixed costs. In this case (e.g., divisions of General Electric or John Deere), it makes sense to compute a division level break even.

5.57 Multi-Product CVP Analysis (LO5). a. Determining the product mix is the first step in computing the breakeven point in a multi-product setting. In Kim’s case, we have the following per 10 customers:

Price Unit variable cost Unit contribution margin # of orders Total contribution margin Total Price:

Sandwich $4.00 $1.25 $2.75 5 $13.75 $20.00

Soup $3.00 $1.00 $2.00

Salad $3.00 $0.75 $2.25

Drink $1.00 $0.25 $0.75

7 4 $14.0 $9.00 0 $21.0 $12.00 0

6 $4.50 $6.00

Thus, we can calculate Kim’s weighted contribution margin ratio as $41.25/$59 = 69.91%. We are now in a position to write down Kim’s profit and compute her breakeven

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5-39 sales. We have: Profit = (0.6991 × revenue) – $4,950. Setting profit = $0, we find: Breakeven revenue = $4,950/0.6991 = $7,080. Because 10 customers represent $59 in revenue, Kim needs to serve $7,080/$59 = 120 sets of customers = 1,200 customers per month to break even. At this volume, Kim will serve 600 sandwiches (120  5), 840 bowls of soup (120  7), 480 salads (120  4), and 720 bottles of water or cans of soda (120  6).

b. The “free” drink offer creates inter-relations among the products. Perhaps the easiest way to deal with such complications is to modify the profit equation to include both paid and free drinks. We can then re-compute the contribution margin and selling price for a ‘bundled’ product. The following table provides the detailed computations:

Price Unit variable cost Unit contribution margin # of orders Total contribution margin Total Price:

Sandwich $4.00 $1.25 $2.75 5 $13.75 $20.00

Soup $3.00 $1.00 $2.00

Salad $3.00 $0.75 $2.25

7 7 $14.0 $15.75 0 $21.0 $21.00 0

Paid Drink $1.00 $0.25 $0.75

Free Drink $0.00 $0.25 -$0.25

3 $2.25

3 -$.75

$3.00

$0.00

Thus, the price for the product mix is $65.00, and its contribution margin is $45.00. In turn, Kim’s profit and breakeven sales are: Profit = ($45  Quantity) – $4,950. Setting profit = $0, we find: Breakeven number of units = $4,950/$45 = 110. Because each bundle represents 10 customers, Kim needs to serve 10  110 = 1,100 customers per month to breakeven. At this volume, Kim will serve 550 sandwiches (110  5), 770 bowls of soup (110  7), 770 salads (110  7), and 660 bottles of water or cans of soda (110  6). Kim also will generate 110  $65 =

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5-40 $7,150 in revenue (i.e., her breakeven point in sales dollars is the breakeven number of bundles multiplied by the revenue per bundle).

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5-41

Thus, Kim needs to serve fewer customers) because the contribution margin for every ten customers has increased. Notice that Kim’s required revenue, however, has changed very little because her bundle selling price has increased by $6 compared to part [a] – this can be calculated as (3  $3 per salad) – (3  $1 per water/soda) = $6. 5.58 Multi-Product CVP Analysis, Weighted Contribution Margin Ratio Approach (LO5). a. We start by setting a price for a new textbook – this will allow us to calculate the price of a used textbook and all of the associated variable costs. Let’s say Pricenew = $100. If Pricenew = $100, then Unit variable costnew = (.75  $100) + (.05  $100) = $80. In turn, the contribution margin ratio on a new book =

($100$100- $80)

= 20%.

For Priceused we have .75  $100 = $75. The unit variable costused is: (.25  $100) + (.05  $100)* = $30. * Note that the variable selling costs (in $ per book, not %) are the same between new and used textbooks, or $5. Accordingly, the contribution margin ratio on a used book =

(

$75 - $30 $75

)

= 60%.

If 40% of University Bookstore’s revenues come from used books and 60% come from new books, we can write University Bookstore’s profit model as: Profit = (.40 × Total revenues × .60) + (.60 × Total revenues × .20) – $360,000. That is, the weighted contribution margin ratio = (.40 × .60) + (.60 × .20) = .36. Setting profit equal to $0, we solve for the breakeven revenue as: $0 = (.36 × Breakeven revenue) – $360,000. Thus, Breakeven revenue = $1,000,000.

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5-42 We can pick any number for the selling price of a new book because we employed a contribution margin ratio formulation, where everything was expressed as a % of selling price. Thus, breakeven revenue is invariant to the selling price we choose (encourage students to try it!). Moreover, University Bookstore needs to generate $1,000,000 in revenue to break even – the price per book affects only the number of textbooks sold at the break-even point, not the required revenue. b. The important factor to consider here is the contribution margin in dollars ($), not the contribution margin ratio (because the selling prices are different). The contribution margin on a new book = $100 – $80 = $20. The contribution margin on a used book = $75 – $30 = $45. Thus, the University Bookstore makes $25 more per book on used books versus new books. Therefore, it strictly prefers to sell used textbooks rather than new textbooks. This explains why bookstores exert significant effort to acquire as many used books as possible – the profit is higher! Note: This relation holds regardless of the number students pick for the selling price of a new book. For any selling price, x, University Bookstore receives .20x in contribution margin (x – .75x – .05x); On a used book, University bookstore receives .45x (.75x – .25x – .05x). Of course, for any positive selling price, .45x strictly exceeds .20x. c. To provide for an equal contribution margin in $, University Bookstore would set the price of a used textbook such that (based on our assumed numbers): $20 = price (used) – $30. OR, price (used) = $50. Thus, the price of a used textbook would be 50/100 = 50% of the price of a new textbook. Notice that the contribution margin ratio on a used textbook is now 20/50 = 40%. Moreover, when considering the “bottom line” we need to consider the absolute contribution margin, not necessarily the contribution margin ratios (unless, of course, the selling prices are equal). 5.59 Multi-Product CVP Analysis, How Best to Spend Advertising Dollars (LO3, LO5). a. Tornado’s profit for the most recent year can be calculated as follows:

Revenues* Variable costs**

F1 $3,750,000 $1,875,000

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F3 F5 $3,000,000 $4,000,000 $1,650,000 $2,400,000

Total $10,750,000 $5,925,000

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5-43 Contribution margin Fixed costs Profit

$1,875,000

$1,350,000 $1,600,000

$4,825,000 $3,860,000 $965,000

* = quantity sold  selling price per unit ** = quantity sold  variable cost per unit b. There are at least two ways to answer this question. The longer and more tedious way is to convert the increase (decrease) in sales to units for each of the three alternatives – i.e., assume that Tornado focuses its advertising campaign on the F1, F3, or F5 market. We can then multiply the sales quantities by the appropriate contribution margin to compute the net increase in margin under each alternative. Naturally, we select the option with the highest margin. The second, and more straightforward, approach is to use contribution margin ratios – which deal with dollars directly. Using the given data, we have: Selling price per unit Variable cost per unit Contribution margin ratio*

F1 $150 $75 .50

F3 $200 $110 .45

F5 $400 $240 .40

* = (unit selling price – unit variable cost)/ unit selling price That is, an increase of $1 in sales yields the greatest contribution if it is from the F1 market. Consequently, it makes most sense for Tornado to focus its campaign on the F1 market. The net effect on profit will be: Gain from F1 market $300,000 Loss from F3 market ($27,000) Loss from F5 market ($24,000) Increase in fixed costs ($150,000) Net increase in profit$99,000

.50  $600,000 .45  $60,000 .40  $60,000 given

The skeptical student may wish to construct tables assuming the advertising campaign were focused on the F3 or F5 market to verify that the strategy of focusing on these markets indeed leads to lower profit than focusing on the F1 market. This problem is useful in highlighting the difference between unit contribution margins and contribution margin ratios. The unit contribution margin calculates the absolute profit, and the contribution margin ratio calculates profitability. Thus, we see that although the F5 has the largest contribution margin and, thus, on a per

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5-44 unit basis contributes the most to absolute profit, it has the lowest profitability per $1 of sales. c. Management is making a number of assumptions. First, they are assuming that the advertising campaign will work and lead to a substantial increase sales for the targeted vacuum cleaner. Second, by assuming that the increase in revenue will be constant at $600,000 regardless of the vacuum cleaner chosen, management is assuming that the “demand kick” in units will be smallest for the F5 and largest for the F1 (the F3 will be in the middle). Specifically, estimated demand for each vacuum cleaner, should it be selected is: F1: $600,000/150 = 4,000 units. F3: $600,000/200 = 3,000 units. F5: $600,000/400 = 1,500 units. These proportions are roughly comparable to the current sales mix. Moreover, management is assuming that the market is “thinner” for the F1 and “thicker” for the F5 – this assumption makes some sense since, ceteris paribus, as price increases we expect quantity demanded (in aggregate) to decrease, particularly when there are substitute products available. Finally, management is assuming that the advertising campaign will cannibalize existing sales – in other words, if the advertising campaign were targeted toward the F3 market, some consumers who would have purchased the F1 or the F5 would, instead, purchase the F3. Again, the constant loss in revenue assumption stipulates that fewer F5 customers will defect than F1 customers, and so on. Moreover, this assumption reflects that while the vacuum cleaners are, to some extent, substitutes, the elasticity of demand likely differs across the products. 5.60 (Advanced) CVP Analysis – A Critical Evaluation: Non-linear Cost Function, Linear Approximation, and Decision Making (LO3, LO6). a. The following graph depicts the relation between the number of families, Q, and Jackrabbit Trails’ weekly total costs for 0  Q  20.

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5-45

Students may recognize that the shape of this cost function is similar to the cost functions they studied in economics (students may also remember that the cubic cost function closely approximates any cost function). The non-linearity reflects economies/diseconomies of scale – i.e., average cost decreases up to a point, but then increases. Such non-linearities are necessary to have a well-defined profit maximization problem – with the typical linear setup if price exceeds variable cost then the firm would push production up to arbitrarily high levels and make unboundedly large profits. b. The linear cost model is developed using the endpoints of Jackrabbit Trails’ relevant range, Q = 4 and Q = 16. The total cost at each of these points is: Total cost(4) = 1,000 + (300  4) – [20  (4)2] + 43 = $1,944. Total cost(16) = 1,000 + (300  16) – [20  (16)2] + 163 = $4,776. We are now in a position to find the slope (unit variable cost) and intercept (Fixed cost): Slope (unit variable cost) = ($4,776 – $1,944)/(16-4) = $236. Thus, $1,944 = intercept + (4  236); or intercept = $1,000. Jackrabbit Trails linear cost function is: TC = $1,000 + ($236 × Quantity). The following graph depicts both the linear and non-linear cost functions:

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5-46

The fit looks reasonably good, and we can see that the deviations between the linear and non-linear models are not very large (except, perhaps, for Q = 19, 20). Moreover, the following table delineates the total cost under each model and the corresponding difference in total cost:

Quantity 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Total Costs Non-Linear Model Linear Model 2 1,000 + 300Q – 20Q + Q3 1,000 + 236Q $1,000 $1,000 $1,281 $1,236 $1,528 $1,472 $1,747 $1,708 $1,944 $1,944 $2,125 $2,180 $2,296 $2,416 $2,463 $2,652 $2,632 $2,888 $2,809 $3,124 $3,000 $3,360 $3,211 $3,596 $3,448 $3,832 $3,717 $4,068 $4,024 $4,304 $4,375 $4,540 $4,776 $4,776 $5,233 $5,012 $5,752 $5,248

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Difference $0 $45 $56 $39 $0 -$55 -$120 -$189 -$256 -$315 -$360 -$385 -$384 -$351 -$280 -$165 $0 $221 $504

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5-47 19 20 AVG:

$6,339 $7,000 $3,367

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$5,484 $5,720 $3,360

$855 $1,280 $7

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5-48 c. The following table shows the profit for 0 to 20 families under each cost structure (the quantity, price combination that would be chosen under each structure and the resulting [expected] profit is in bold). Profit Quantity

Price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 AVG:

1200 1150 1100 1050 1000 950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 700

Non-Linear 1,000 + 300Q – 20Q2 + Q3 -$1,000 -$131 $672 $1,403 $2,056 $2,625 $3,104 $3,487 $3,768 $3,941 $4,000 $3,939 $3,752 $3,433 $2,976 $2,375 $1,624 $717 -$352 -$1,589 -$3,000 $1,800

Linear 1,000 + 236Q -$1,000 -$86 $728 $1,442 $2,056 $2,570 $2,984 $3,298 $3,512 $3,626 $3,640 $3,554 $3,368 $3,082 $2,696 $2,210 $1,624 $938 $152 -$734 -$1,720 $1,807

Notice that Jackrabbit Trails would choose Price = $700 and Quantity = 10 families per week. This yields an expected profit of $4,000 under the non-linear cost model and $3,640 under the linear cost model. One might be tempted to conclude that the cost of using the linear model = $4,000 – $3,640 = $360, but this would not be appropriate because costs will behave in their “true” fashion once the pricing decision is made and, thus, yield identical profit as long as the decision being made is the same (which it is). We could, of course, push the problem around somewhat to show that linear approximation of non-linear relations leads to errors in decision making. The robustness of the linear model, though, is quite remarkable. In this problem, for example, one would have to fit a rather poor linear relation to lead to an error in pricing – e.g., use 0 and 20 families to compute the linear relation.

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5-49 There are multiple benefits/reasons for using a linear model. First, organizations often do not know their true cost curve and a linear setup is likely to be as good an approximation as any other setup. Second, pragmatic considerations lead to linear approximation – non-linear expressions are overbearing/overwhelming – there are diminishing returns to keeping track of detail. Moreover, linear approximation is one of the building blocks of accounting systems. In the particular problem at hand, Jackrabbit Trails likely sacrifices little (if anything) by using a linear approximation of its cost curve. 5.61 (Advanced) CVP Analysis – A Critical Evaluation: Non-linear CVP Relation, Pricing (LO3, LO6). a. The profit is: Profit before taxes = [(Price – unit variable cost)  Quantity] – Fixed cost. The above equation assumes that the quantity sold is independent of price. This is not a good assumption where markets are not perfect and the firm can influence demand via its pricing strategy. As students probably recall from their microeconomics course, we can represent the relation between selling price and demand by constructing a demand curve. Substituting the relation between quantity and price into the profit equation, we get: Profit = {(Price – unit variable cost)  [32,500 – (10  Price)]} – Fixed cost. Substituting the variable cost and fixed cost information gives us: Profit = {(Price – 750)  [32,500 – (10  Price)]} – 14,000,000. Re-arranging terms, we have: Profit = 40,000Price – 10(Price)2 – $38,375,000. b. Solving the revised profit equation for price is a bit more complicated than before. We can solve for this by using calculus or numerically (e.g., using Excel). Using calculus, we find the profit maximizing quantity by differentiating the profit equation with respect to price, setting the derivative equal to zero, and solving for price. Thus, we have: 0 = 40,000 – (20 ×Price) or, Price = $2,000 per unit.

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5-50 Students trained in calculus can readily verify that the associated second order condition (obtained by differentiating the first derivative with respect to P) is negative, assuring us that the above equation yields the profit-maximizing price. (i.e., the profit function is strictly concave). Note that the fixed costs of $14,000,000 plays no role in setting the profitmaximizing price. This is because fixed costs do not change as the price changes. In the short-term, maximizing contribution margin is the same as maximizing profit. We also can solve for the profit-maximizing price by numerically approximation. Consider the following table (with price ranging from $1,000 to $3,000 in increments of $100). Price $1,000 $1,100 $1,200 $1,300 $1,400 $1,500 $1,600 $1,700 $1,800 $1,900 $2,000 $2,100 $2,200 $2,300 $2,400 $2,500 $2,600 $2,700 $2,800 $2,900 $3,000

Profit ($8,375,000) ($6,475,000) ($4,775,000) ($3,275,000) ($1,975,000) ($875,000) $25,000 $725,000 $1,225,000 $1,525,000 $1,625,000 $1,525,000 $1,225,000 $725,000 $25,000 ($875,000) ($1,975,000) ($3,275,000) ($4,775,000) ($6,475,000) ($8,375,000)

By inspection, we see that $2,000 is the profit-maximizing price within this set of choices. By construction, this also turns out to be the profit-maximizing price. We could also use the “solver” function in Excel to find the optimal price. As a final point, instructors may wish to point out to students the “cost of getting it wrong” – notice that if Mr. Park sets price in some “willy-nilly” fashion, his business could lose over $8,000,000 rather than earn a profit of $1,625,000. This can really bring home the point regarding the importance of constructing an accurate profit model. Balakrishnan, Managerial Accounting 1e

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5-51

MINI-CASES 5.62

Made to Order Caps. a. The profit is: Profit before taxes= [(Price – unit variable cost)  Quantity] – Fixed cost. Based on the information provided, we have the following: Profit = ($20.00 price – $4 cost of cap – $2 royalty per cap – $2.50 supplies per cap – [.02  .75  20] expected credit card fees per cap) × Quantity – ($250 for leaflets and brochures + $100 for phone lines + $1,970 for space and equipment + $1,040* for additional help) * = (76 hours the shop is open per week – 50 hours worked by Jessica each week)  $10 per hour for temporary help × 4 weeks per month. This reduces to: Profit = [$11.2  Quantity] – $3,360. b. The breakeven point can be obtained by setting profit in the model from part [a] above equal to $0. Thus, we have: $0 = ($11.20  Breakeven volume) – $3,360. Solving, we obtain Breakeven volume = 300 caps. At a planned price of $20 per cap, this translates to 300  $20 = $6,000 in revenue per month. Thus, in order to breakeven on a monthly basis, Jessica needs to sell approximately 10 caps per day. c. If Jessica sold 1,000 caps per month, her expected profit would be: Profit = ($11.20  1,000) – $3,360 = $7,840 (not bad!) If Jessica wished to earn $4,032 per month, she would need to sell: $4,032 = ($11.2  Q) – $3,360, or Q = 660 caps, roughly 22 caps per day.

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5-52 d. First, notice that Jessica’s research informs her that at a price of $20, demand is expected to be 300 caps per month – as we calculated in part [b], this is Jessica’s breakeven point. Things don’t look good, but perhaps Jessica can do better by exploiting her market research and newfound knowledge of the demand schedule. Let us expand the table to compute Jessica’s expected profit at alternate prices. Note that expected profit in the table is based on the profit equation we developed in part [a]. Specifically, we plug price and quantity into the following equation to arrive at expected profit. Importantly, as the selling price and quantity change, so too do the credit card company charges, which equal 2% of the selling price. Profit = {[Price – 4 – 2 – 2.50 – (.02  .75  Price)] × Quantity} – $3,360. OR, Profit = [(.985Price – 8.50) × Quantity] – $3,360. Price per cap $20 $25 $28 $30 $32 $34

Demand 300 250 220 200 180 160

Expected Profit $0.00 $671.25 $837.60 $850.00 $783.60 $638.40

Jessica maximizes her profit if she sets her price at $30 per cap as her expected profit is highest ($850.00) at this price. Relative to setting the price at $20 per cap, the $30 price results in a lower volume (200 caps versus 300 caps). However, the higher contribution margin per cap ($21.05 rather than $11.20) offsets this. Note (can be skipped without loss of continuity) : Students with a background in economics will notice that we can substitute the demand schedule with a demand function. In this case, the relation between demand and price is Quantity = 500 – (10 × Price). We can modify the expression for profit to incorporate this new piece of information. Thus, we have: Profit= {[500 – (10  Price)]  (.985  Price – 8.5)} – $3,360. This simplifies to: Profit = 492.50(Price) – 9.85(Price)2 – 4,250 + 85P – 3,360. Or, the relation between Jessica’s profit and price is: Balakrishnan, Managerial Accounting 1e

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5-53

Profit = 577.50(Price) – 9.85(Price)2 – 7,610. We can solve for the best P using calculus or Excel’s solver function. The exact answer is P = $29.31 and expected profit is $854.63 e. Taxes affect profit in a relatively straightforward fashion. We can take the profit model developed in part [d] and add the tax variable – doing so yields: Profit after taxes = Profit before taxes – taxes paid. Here, because taxes are proportional to income, taxes paid = (tax rate  pre-tax profit). Adding this column yields: Price per cap $20 $25 $28 $30 $32 $34

Demand 300 250 220 200 180 160

Expected Profit $0.00 $671.25 $837.60 $850.00 $783.60 $638.40

After-tax profit $0.00 $503.44 $628.02 $637.50 $587.70 $478.80

The optimal price continues to be $30 per cap. Notice that the optimal price does not change – this occurs because (in this example), taxes are a linear function of pre-tax profit. Thus, Jessica still wishes to maximize pre-tax profit which, in turn, will also maximize after-tax profit. This aspect of the problem allows instructors to discuss the relation between pre-tax income and tax rates, including extending the discussion to non-linear relations between pre-tax and after-tax income. Taxes will, of course, reduce the amount of Jessica’s profit. In our example, Jessica will now earn an after-tax monthly profit of: $850  .75 = $637.50. Note (Can skip without loss of continuity): In the calculus approach, incorporating taxes yields Profit after taxes = (577.50(Price) – 9.85(Price)2 – 7,610)  (1 – tax rate). Since t = .25, we have: Profit after taxes = (577.50(Price) – 9.85(Price)2 – 7,610)  (.75). Balakrishnan, Managerial Accounting 1e

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5-54

Profit after taxes = 433.125(Price) – 7.3875(Price)2 – 5707.50. Again, we can solve for the best P using calculus or Excel’s solver function. The exact answer is Price = $29.31 (which is exactly what we arrived at earlier). f. While Jessica’s venture has intuitive appeal and our original calculations seemed to indicate that the venture might be a “go” (after all, selling 22 hats a day does not seem like a lot) the numbers simply do not add up. Given the totality of the costs involved and Jessica’s market research, it appears that (at best) Jessica will earn a very modest profit and will be unlikely to maintain a reasonable lifestyle with this business. We can see, though, how modest amounts add up – for example, if Jessica were to “manage” 10 kiosks in various malls around the state, she could earn a reasonable sum of money – this is precisely what franchisers seek to do. 5.63 Rick’s English Hut. a. Currently, Rick’s is generating $60,000 in sales. For alcohol and food, this translates to: Alcohol Sales: $60,000  .55 = $33,000, or $33,000/$4 = 8,250 “alcohol units.” Food Sales: $60,000  .45 = $27,000, or $27,000/$5 = 5,400 “food units.” Monthly profit can then be calculated as: [8,250  ($4 – $2)] + [5,400  ($5 – $4)] – $10,950 = $10,950. Because the proportions are defined in terms of revenue, it may be easier to solve the problem using a contribution margin ratio approach. The contribution margin ratio for food = (5 – 4)/5 = .20, and the contribution margin ration for alcohol is (4 – 2)/4 = .50. Thus, the profit can be calculated as: Profit = ($33,000  0.50) + ($27,000  0.20) – $10,950 = $10,950. To determine breakeven sales, we need to calculate a weighted contribution margin ratio. Since 45% of revenue is from food and 55% is from alcohol, we have: Weighted contribution margin ratio = (0.45  0.20) + (0.55  0.50) = 0.365 or 36.5%. Balakrishnan, Managerial Accounting 1e

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5-55

The profit is then: Profit = Weighted contribution margin ratio  Total Revenue – Fixed cost. At the breakeven point, we have: $0 = 0.365  Breakeven revenue – $10,950. Breakeven revenue = $30,000. Note: It is tempting to calculate a weighted unit contribution margin as (0.45  $1) + (0.55  $2) = $1.55 – this is not appropriate, though, because the weights are the ratio of revenue and not the ratio of the number of units sold. For weighted unit contribution margin, we use unit contribution margins; the weights must also be the ratio of units. If students employ this approach, the weighted unit contribution margin = $1.60. For weighted contribution margin ratio, we use the contribution margin ratio (i.e., contribution per revenue dollar); thus, the weights must be the ratio of revenue. Students can go awry because they combine the revenue weights with unit contribution margins. b. Under this option, the proprietors of Rick’s have decided to change the licensing status from a restaurant to a bar. In terms of calculating profit, revenue stays the same but Rick’s fixed costs have changed. Accordingly, we have: Profit = ($60,000  0.365) – $10,950 – $850 – $318 = $9,782. Rick’s monthly profit has decreased by $1,168, which is the exact amount by which fixed costs have increased. For the breakeven point, the weighted contribution margin ratio stays the same; the fixed costs, however, increase by $1,168 to $12,118: Thus, Breakeven revenue = $12,118/0.365 = $33,200. As would be expected, the breakeven point in revenue has increased. c. Under the second option, Rick’s plans on closing early so that alcohol sales equal the current level of food sales (the revenues from both products are equal). With the new sales mix, the Weighted contribution margin ratio is: Weighted contribution margin ratio (option 2) = 0.5  0.2 + 0.5  0.5 = 0.35. Balakrishnan, Managerial Accounting 1e

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5-56

Additionally, total revenues = $27,000 + $27,000 = $54,000 and fixed costs have decreased by $450 to $10,500. Thus, we have: Profit = $54,000  0.35 – $10,500 = $8,400. This action has reduced Rick’s profit by $2,550 and, at this point, Rick’s would prefer option 1 over option 2. We also have: $0 = 0.35  Breakeven revenue – $10,500. Breakeven revenue = $10,500/0.35 = $30,000. Notice that, compared to part [a] Rick’s breakeven point in revenue has stayed the same even though total fixed costs have decreased. This occurs because the salesmix has shifted to a higher proportion of food items, which is the lower contribution margin product. Notice also that compared to option 1, the breakeven revenue is lower under option 2 (although profit is higher under option 1). The increase in fixed costs under option 1 requires Rick’s to sell more to breakeven – however, it does not restrict the percentage of alcohol sales, so the upside potential is higher. If Rick’s can maintain their current level of sales, they likely would pursue option 1 and bar status. d. Under this option, Rick’s is planning to offer a brunch to make up for the revenue shortfall from current food sales. From part [a], we know that the difference in revenue is $6,000; thus, 6,000/4 = 1,500 brunches need to be sold on the weekends. The brunches also have a negative contribution of -0.08 or a contribution margin ratio of -0.02 = ($4 – $4.08)/$4. The mix of the revenue has also changed. The new weighted contribution margin ratio is: Weighted contribution margin ratio = (27/66)  0.2 + (33/66)  0.5 + (6/66)  (0.02) = 0.33. With this Weighted contribution margin ratio, Rick’s profit is: Profit = 0.33  $66,000 – $10,950 – $105 = $10,725. Notice that this turns out to be Rick’s best option – profit is only $225 lower than the most recent month.

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5-57 For the breakeven point, we have: Breakeven revenue = Fixed cost/ Weighted contribution margin ratio = $11,055/0.33 = $33,500. Notice that compared to option 1, the breakeven revenue is highest under option 3, although it is still considerably below the current sales level of $60,000 (and anticipated sales level of $66,000). Profit also is $943 higher than option 1 and $2,325 higher than option 2. All in all, Rick’s would be hard-pressed not to select option 3. e. Rick’s has learned the value and importance of profit planning at the portfolio (aggregate or business) level rather than at the individual product level. What in isolation appears to be a poor strategy, selling a product at a loss, turns out to be the best strategy for the business as a whole. In essence, when companies offer multiple products or services they need to consider the relationships/externalities among these products and services. The profitability of the business as a whole is of utmost concern, not the profitability of individual products and services. We frequently see such behavior by restaurants and bars – an establishment offers very cheap food/appetizers (e.g., happy hours) to stimulate demand for higher margin alcohol sales. In a similar vein, we also observe restaurants offering “kids eat free” nights. Casinos in Las Vegas or Atlantic City perhaps provide the prototypical example of this behavior. Casinos routinely offer “loss leaders” such as low-price meals (e.g., buffets for $2 or $3), free drinks, and heavily discounted hotel rooms in order to attract and retain customers on the more highly profitable gambling activities. Casino owners make such concessions and offer “comps” to increase profit on their primary product line. Yet another example is banks – banks routinely offer free checking in an attempt to get customers to keep their savings in the bank and/or purchase higher margin home or auto loans. Finally, supermarkets frequently offer loss leaders – they advertise specials on milk, bread, eggs, and the like. Such items are commonly purchased goods and attract customers to the store. Once inside the store, supermarkets hope that customers do the remainder of their shopping there, buying fruits and vegetables, meats, chips, soda, and so on. In all the examples, the fundamental point stays the same – it is important to do profit planning and such planning is appropriately done at the “portfolio” level. In short, the case makes a critical point – we need to think about the interdependencies among the products and services being offered.

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5-58 5.64

Constructing and Interpreting CVP Graphs (LO1, LO2). a. The “vanilla” cost graph would have units (in this case, cups of yogurt) on the x-axis and dollars (total costs) on the y-axis. With regard to the total cost line, the intercept, $3,000, represents Yin-Yang’s monthly fixed costs and the slope, $2, represents Yin-Yang’s variable cost per cup of yogurt. The following graph depicts Yin-Yang’s total costs as a function of the number of cups of yogurt sold: $10,000 Slope = unit variable costs = $2.00

Dolla rs ($)

$8,000 $6,000 $4,000 $2,000

2, 50 0

2, 00 0

1, 50 0

1, 00 0

0

$0

50 0

Intercept = Fixed Costs = $3,000

Cups of Yogurt

b. We can readily add Yin-Yang’s revenue line to our previously-constructed cost graph. The revenue line will start at the origin (after all, if we sell 0 cups we have $0 in revenue) with a slope of $4, representing the price per cup of yogurt. The point where the revenue line crosses the total cost line is the breakeven point. We can drop a line down to the x-axis to read off the required unit sales. We can drop a line to the y-axis to determine the required revenue. If the revenue line is above the total cost line, Yin-Yang will earn a profit. Conversely, if the revenue line is below the total cost line, Yin-Yang will incur a loss. All of these relationships are depicted on the following graph:

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5-59

$10,000 Profit Area

$8,000

Dolla rs ($)

Loss Area

$6,000 $4,000

2, 50 0

2, 00 0

1, 00 0

50 0

0

$0

1, 50 0

Breakeven Point = 1,500 Cups of Yogurt

$2,000

Cups of Yogurt c. By subtracting total costs from revenue, we obtain Yin-Yang’s profit graph. This gives us the standard profit equation: Profit before taxes = (Price – unit variable cost)  Quantity – Fixed costs. For Yin-Yang, we have: Profit before taxes = $2  Number of cups of yogurt sold – $3,000. This line is depicted on the following graph.

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5-60

$4,000

Slope = unit contribution margin = $2 Breakeven point = 1,500 Cups of Yogurt

2, 50 0

2, 00 0

1, 50 0

1, 00 0

$0

50 0

Loss Area

0

Profi t ($)

$2,000

Profit Area

-$2,000 Intercept = - Fixed Costs = -$3,000

-$4,000

Cups of Yogurt The point where the profit line crosses the y-axis represents the net profit when zero cups of yogurt are sold. That is, revenues, variable costs, and contribution margin are all zero. Thus, profit at zero cups of yogurt = –fixed costs = –$3,000. The slope of the profit line = the unit contribution margin; in Yin-Yang’s case, this is $4 - $2 = $2. The point where the profit line crosses the x-axis represents the breakeven point, or the point at which profit = $0. The profit and loss areas can be identified by the profit line’s position relative to the x-axis. If the profit line is above the x-axis, Yin-Yang earns a positive profit. Conversely, if the profit line is below the x-axis, Yin-Yang will incur a loss. d. Yin-Yang’s revised profit graph is presented below. Notice the kink in the profit line because of the tax. The dotted line represents the original, untaxed profit, and the solid line represents the after tax profit. The two lines are the same below the breakeven point because no tax is due if the firm makes a loss. The after-tax profit line is below the pre-tax profit line when the firm makes a profit. The slope of the after tax line is smaller by the tax paid.

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5-61

e. It is possible to construct a “revised” CVP graph without using data on units, as indicated below. In this case, we represent Yin-Yang’s profit as: Profit before taxes = (.50  Revenue) – $3,000. Notice that we are plotting a net contribution line in place of the revenue and total cost lines. The slope (or the steepness) of the contribution line is determined by the contribution margin ratio, which in this case is .50. Moreover, the higher the contribution margin ratio, the steeper the contribution line. Note: Instructors may wish to mention to students that the contribution line will always be less than the 45ْ line because the contribution per sales dollar cannot exceed one!

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5-62

$3,000

-$1,500

10 ,0 00

8, 00 0

6, 00 0

4, 00 0

$0

2, 00 0

Loss Area Revenue < $6,000

0

Profi t ($)

$1,500

Breakeven revenue = $6,000 = (1,500  4)

Profit Area Revenue > $6,000

-$3,000

Revenue ($) 5.65 CVP Analysis with Alternative Cost Structures, Demand Uncertainty, and Risk (Advanced, LO3, LO4, LO6). a. We have the following data regarding each technology:

Selling price per unit Variable cost per unit Unit contribution margin Fixed costs

Labor-Intensive Technology $75 $50 $25

Capital-Intensive Technology $75 $25 $50

$500,000

$2,500,000

We are now in a position to use the profit equation and solve for the breakeven point under each technology. The profit is: Profit = (Unit contribution margin  Quantity) – Fixed costs. We know the contribution margin and level of fixed costs for each technology. Additionally, we know that profit = $0 at the breakeven point. Thus: Breakeven volume (labor intensive) = $500,000/$25 = 20,000 fishing rods. Breakeven volume (capital intensive) = $2,500,000/$50 = 50,000 fishing rods. b. We can use the profit for each technology to compute profit at the two sales levels:

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5-63 Profit (Labor-Intensive) = ($25  Sales in units) – $500,000. Profit (Capital-Intensive) = ($50  Sales in units) – $2,500,000. Plugging-in sales of 40,000 and 90,000 units to each equation yields:

Sales = 40,000 units Sales = 90,000 units

Technology Labor-intensive Capital-intensive  = $500,000  = ($500,000)  = $1,750,000  = $2,000,000

Thus, at sales of 40,000 units Sally prefers the labor-intensive technology and at sales of 90,000 units Sally prefers the capital-intensive technology. We can find the indifference point by setting the two profit equations equal to each other:

($25  Sales in units) – $500,000 = ($50  Sales in units) – $2,500,000.

We find that sales in units at which profit is equal between the options is: 80,000. c. We present a graph of the profit line for each technology below:

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5-64

The graph illustrates that the unit contribution margin is higher under the capitalintensive technology, but so are the fixed costs. The net effect on breakeven sales is ambiguous – in this particular case, breakeven sales increases under the capital intensive technology as indicated in part [a]. Moreover, we see that for sales < 80,000 units Sally strictly prefers the labor-intensive technology, and for sales > 80,000 Sally strictly prefers the capital-intensive technology. d. Using the probabilities given and the profit #’s calculated in part [b], we can calculate expected profit as follows: Expected profit (labor-intensive) =.5  $500,000 + .5  $1,750,000 = $1,125,000. Expected profit (capital-intensive) = .5  -$500,000 + .5  $2,000,000 = $750,000. Thus, expected profit is higher under the labor-intensive technology. e. The range of profit is: $1,750,000 – $500,000 = $1,250,000 under the laborintensive technology. The corresponding range for the capital-intensive technology is: $2,000,000 – ($500,000) = $2,500,000. The capital-intensive technology has the greater range – it is twice as great as the labor-intensive technology. The profit is much more variable with the capitalintensive technology because each unit contributes a great deal more to profit than under the labor-intensive technology. Looking at the graph from part [c], this difference manifests itself via a steeper slope (unit contribution margin) under the capital-intensive technology. In terms of a choice of technology, the labor-intensive technology really seems to be the preferred choice. Almost over the entire range, the labor-intensive technology has the greater profit and, additionally, has much lower variability in profit, implying less risk. 5.66 CVP Relation and Profit Planning, Taxes, Ethics (LO1, LO2, Advanced). a. NOTE TO INSTRUCTOR: Many human rights organizations have rightfully condemned the bidi industry for its extensive exploitation of labor and, in particular, child labor. Bidi workers frequently are afflicted with lung disorders and skin infections because of the hazardous nature of their job where they are constantly exposed to fine tobacco dust. Additionally, bidi workers typically are deprived of even minimum health benefits and education. Finally, like cigarettes, smoking a bidi poses significant health hazards to the user. Instructors may wish to use this problem to raise students’ social and health awareness.

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5-65 To calculate the number of packs of bidis Ganesh must sell, we start with the profit equation: Profit before taxes = (Unit contribution margin × Quantity) – Fixed costs. Additionally, Unit contribution margin = Price – unit variable cost. = 3.00 – .75 – .25 = 2.00 Rupees per pack. Finally, fixed costs = 1,075,000 Rupees and target profit = 750,000 Rupees. With these pieces of information, we have: 750,000 = (2.00  required sales volume in units) – 1,075,000. OR, required sales volume in units =

1,825,000 = 912,500 packs. 2.00

b. We need to modify Ganesh’s CVP model to incorporate income taxes. We start with the CVP model that includes taxes: Profit after taxes = [(Unit contribution margin  Quantity) – Fixed costs] × (1 – Tax rate). With a 40% tax rate, we have: 750,000 = [(2.00  Required sales volume in units) – 1,075,000]  (1 – . 40). OR, required sales volume in units = 1,162,500 packs. We see that Ganesh now needs to sell 1,162,500 – 912,500 = 250,000 more packs of bidis to earn a take-home profit of 750,000 Rupees per month. An alternative method is to convert the after-tax profit to a pre-tax profit target. Using the equation after-tax profit = (1 – tax rate)  pre-tax profit, we have: 750,000 = (1 – .40)  Pre-tax profit Pre-tax profit = 1,250,000 Rupees. We can then substitute this amount into the standard CVP model: 1,250,000 = (2.00  Required sales volume in units) – 1,075,000.

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5-66

Again, we find that the number of packs = 1,162,500. We currently have: c. Profit after taxes = [(2.00  Required sales volume in units) – 1,075,000] × (1 – .40). The value-added tax can be viewed as an additional variable cost per pack of bidis of: (3.00 selling price – .75 materials cost)  .20 = 0.45 Rupees per pack; Thus, Ganesh’s contribution margin is now 2.00 – 0.45 = 1.55 Rupees. In turn, if Ganesh still desires an after-tax profit of 750,000 Rupees, we have: 750,000 = [(1.55  Required sales volume in units) – 1,075,000] × (1 – .40). OR, Required sales volume in units = 1,500,000 packs. Again, we see that taxes place additional demands on Ganesh’s business. Compared to part [b], the firm needs to sell 1,500,000 – 1,162,500 = 337,500 more packs because of the VAT. Compared to part [a], the firm needs to sell 1,500,000 – 912,500 = 587,500 more packs because of both taxes. Finally, notice that the value-added tax is deducted when calculating pre-tax profit. d. After part [c], Ganesh’s CVP model is now: 750,000 = [(1.55  Required sales volume in units) – 1,075,000] × (1 – .40). The excise tax further reduces Ganesh’s contribution margin per pack of bidis – it is now 1.55 – .05 = 1.50 Rupees. Accordingly, the new model is: 750,000 = [(1.50  Required sales volume in units) – 1,075,000] × (1 – .40). Required sales = 1,550,000 packs of bidis (notice again that the excise tax is deducted from pre-tax profit). We see that this tax further increases the amount that must be sold to maintain a desired take-home profit. Collectively, all three taxes require Ganesh to sell 1,550,000 – 912,500 = 637,500 more packs of bidis to earn a take-home profit of 750,000 Rupees per month. e. Both the value-added tax (VAT) and the excise tax are treated as variable costs. Ganesh will incur these costs regardless of his profit level (as long as the firm continues to produce bidis). The excise tax varies as a function of the number of packs of bidis sold – it is a constant 0.05 Rupees per pack, and, Balakrishnan, Managerial Accounting 1e

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5-67 therefore, behaves like a variable cost. The VAT also varies with the packs of bidis sold, but the amount of the tax depends on the ultimate selling price and materials cost. Thus, this tax is a function of these two variables. To reduce the VAT, notice that Ganesh has an incentive to “inflate” his materials cost numbers. Doing so reduces the added “value” and thus reduces the VAT payable. The income tax varies with pre-tax profit – it varies with all cost and revenue variables that affect profit. It is, however, typically structured in steps – no tax on negative or $0 in profit, with the percentages increasing as income increases. Thus, we see that taxes alter firm’s CVP model in significant ways – the exact nature of the change depends on the type and structure of the tax. With regard to a sales tax (or tax on revenue), this tax is borne by consumers. As such, firms ignore it in the CVP model (because sales taxes are explicitly collected from customers). This does not, however, mean that sales taxes do not affect a firm’s profit – sales taxes might indirectly affect profit by affecting consumer demand – it is a cost consumers will consider in their purchasing/consumption decisions. (Instructors may wish to link this to the significant sales tax levied on tobacco products – states increase the costs to consumers to discourage smoking. These are often termed “sin taxes.”)

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