Chapter 10
Financial Analysis

Pricing for Profit

Internal financial considerations and external market considerations are, at most companies, antagonistic forces in pricing decisions. Financial managers allocate costs to determine how high prices must be to achieve profit objectives. Marketing and sales staff analyze buyers to determine how low prices must be to achieve sales objectives. The pricing decisions that result are politically charged compromises, not thoughtful implementations of a coherent strategy. Although common, such pricing policies are neither necessary nor desirable. An effective pricing decision should involve an optimal blending of, not a compromise between, internal financial constraints and external market conditions.

Unfortunately, few managers have any idea how to facilitate such a cross-functional blending of these two legitimate concerns. From traditional cost accounting, they learn to take sales goals as “given” before allocating costs, thus precluding the ability to incorporate market forces into pricing decisions. From marketing, they are told that effective pricing should be entirely “customer driven,” which ignores costs except as a minimum constraint below which the sale would become unprofitable. Perhaps along the way, these managers study economics and learn that, in theory, optimal pricing is a blending of cost and demand considerations. In practice, however, they find the economist’s assumption of a known demand curve hopelessly unrealistic.

Consequently, pricing at most companies remains trapped between cost- and customer-driven procedures that are inherently incompatible. This chapter suggests how managers can break this tactical pricing deadlock and infuse strategic balance into pricing decisions. Many marketers argue that costs should play no role in market-based pricing. This is clearly wrong. Without perfect segmentation (the ability to negotiate independently a unique price for every customer), pricers must make trade-offs between charging higher margins to fewer customers and lower margins to more customers. Once the true cost and contribution of a sale are understood, managers can appropriately integrate costs into what is otherwise a market-driven approach to pricing strategy.

This chapter describes a simple, logically intuitive procedure for quantitatively evaluating the potential profitability of a price change. First, managers develop a baseline, or standard of comparison, to measure the effects of a price change. For example, they might compare the effects of a pending price change with the product’s current level of profitability, or with a budgeted level of profitability, or perhaps with a hypothetical scenario that management is particularly interested in exploring. Second, they calculate an incremental “break-even” for the price change to determine under what market conditions the change will prove profitable. Marketing managers must then determine whether they can actually meet those conditions.

The key to integrating costs and quantitatively assessing the consequences of a price change is the incremental break-even analysis. Although similar in form to the common break-evens that managers use to evaluate investments, incremental break-even analysis for pricing is quite different in practice. Rather than evaluating the product’s overall profitability, which depends on many factors other than price, incremental break-even analysis focuses on the incremental profitability of price changes. Consequently, managers start from a baseline reflecting current or projected sales and profitability at the current price. Then they ask whether a change in price could improve the situation. More precisely, they ask:

• How much would the sales volume have to increase to profit from a price reduction?
• How much could the sales volume decline before a price increase becomes unprofitable? Answers to these questions depend on the product’s contribution margin.

The sample problems in this chapter introduce the four equations involved in performing such an analysis and illustrate how to use them. They are based on the experience of Westside Manufacturing, a small company manufacturing pillows for sale through specialty bedding and dry cleaning stores. Although the examples are, for simplicity, based on a small manufacturing business, the equations are equally applicable for analyzing any size or type of business that cannot negotiate a unique price for each customer.¹ If customers can be somewhat segmented for pricing, the formulas apply to pricing within a segment.

Following are Westside Manufacturing’s income and costs for a typical month:

Westside is considering a 5 percent price cut, which, it believes, would make it more competitive with alternative suppliers, enabling it to further increase its sales. Management believes that the company would need to incur no additional fixed costs as a result of this pricing decision. How much would sales have to increase for this company to profit from a 5 percent cut in price?

Break-Even Sales Analysis: The Basic Case

To answer Westside’s question, we calculate the break-even sales change. This, for a price cut, is the minimum increase in sales volume necessary for the price cut to produce an increase in contribution relative to the baseline. Fortunately, making this calculation is simple, as will be shown shortly. First, however, it may be more intuitive to illustrate the analysis graphically (see Exhibit 10-1).

In this exhibit, it is easy to visualize the financial trade-offs involved in the proposed price change. Before the price change, Westside receives a price of $10 per unit and sells 4,000 units, resulting in total revenues of $40,000 (the total area of boxes a and b). From this Westside pays variable costs of $5.50 per unit, for a total of $22,000 (box b). Therefore, before the price change, total contribution is $40,000 minus $22,000, or $18,000 (box a). In order for the proposed price cut to be profitable, contribution after the price cut must exceed $18,000.

After the 5 percent price reduction, Westside receives a price of only $9.50 per unit, or $0.50 less contribution per unit. Since it normally sells 4,000 units, Westside would expect to lose $2,000 in total contribution (box c) on sales that it could have made at a higher price. This is called the price effect. Fortunately, the price cut can be expected to increase sales volume.

The contribution earned from that increased volume, the volume effect (box e), is unknown. The price reduction will be profitable, however, when the volume effect (the area of box e) exceeds the price effect (the area of box c). That is, in order for the price change to be profitable, the gain in contribution resulting from the change in sales volume must be greater than the loss in contribution resulting from the change in price. The purpose of break-even analysis is to calculate the minimum sales volume necessary for the volume effect (box e) to balance the price effect (box c). When sales exceed that amount, the price cut is profitable.

So, how do we determine the break-even sales change? We know that the lost contribution due to the price effect (box c) is $2,000, which means that the gain in contribution due to the volume effect (box e) must be at least $2,000 for the price cut to be profitable. Since each new unit sold following the price cut results in $4 in contribution ($9.50 - $5.50 = $4), Westside must sell at least an additional 500 units ($2,000 divided by $4 per unit) to make the price cut profitable.

The minimum percent change in sales volume necessary to maintain at least the same contribution following a price change can be directly calculated by using the following simple formula (see Appendix 9A for derivation):

In this equation, the price change and contribution margin may be stated in dollars, percents, or decimals (as long as their use is consistent). The result of this equation is a decimal ratio that, when multiplied by 100, is the percent change in unit sales necessary to maintain the same level of contribution after the price change. The minus sign in the numerator indicates a trade-off between price and volume: Price cuts increase the volume and price increases reduce the volume necessary to achieve any particular level of profitability. The larger the price change—or the smaller the contribution margin—the greater the volume change necessary to generate at least as much contribution as before.

Assume for the moment that there are no incremental fixed costs in implementing Westside’s proposed 5 percent price cut. For convenience, we make our calculations in dollars (rather than in percents or decimals). Using the contribution margin equation (see Chapter 8), we derive the following:

Given this, we can easily calculate the break-even sales change as follows:

Thus, the price cut is profitable only if sales volume increases more than 12.5 percent. Relative to its current level of sales volume, Westside would have to sell at least 500 units more to maintain the same level of profitability it had prior to the price cut, as shown below:

If the actual increase in sales volume exceeds the break-even sales change, the price cut will be profitable. If the actual increase in sales volume falls short of the break-even sales change, the price change will be unprofitable. Assuming that Westside’s goal is to increase its current profits, management should initiate the price reduction only if it believes that sales will increase by more than 12.5 percent, or 500 units, as a result.

If Westside’s sales increase as a result of the price change by more than the break-even amount—say, by an additional 550 units—Westside will realize a gain in profit contribution. If, however, Westside sells only an additional 450 units as a result of the price cut, it will suffer a loss in contribution. Once we have the break-even sales change and the profit contribution, calculating the precise change in contribution associated with any change in volume is quite simple: It is simply the difference between the actual sales volume and the break-even sales volume, times the new contribution margin (calculated after the price change). For Westside’s 550-unit and 450-unit volume changes, the change in contribution equals the following:

The $4 in these equations is the new contribution margin ($9.50 - $5.50). Alternatively, you might have noticed that the denominator of the percent break-even formula is also the new contribution margin.

We have illustrated break-even analysis using Westside’s proposed 5 percent price cut. The logic is exactly the same for a price increase. Since a price increase results in a gain in unit contribution, Westside can “absorb” some reduction in sales volume and still increase its profitability. How much of a reduction in sales volume can Westside tolerate before the price increase becomes unprofitable? The answer is this: until the loss in contribution due to reduced sales volume is exactly offset by the gain in contribution due to the price increase. As an exercise, calculate how much sales Westside could afford to lose before a 5 percent price increase becomes unprofitable.

It is important to note that the calculation resulting from the break-even sales change formula is expressed as the percent change in unit volume required to break even, not the percent change in monetary sales (for example, the percent change in dollar sales) required to break even. In the case of a price cut, the percent break-even sales change in units necessary to justify the price cut is larger than the percent break-even sales change in sales dollars because the price is now lower.

To convert from the percent break-even sales change in units to the percent break-even sales change in dollars, you can apply the following simple conversion formula:

For example, for Westside’s proposed 5 percent price cut above, the percent break-even sales change in unit volume terms was 12.5 percent. What is the corresponding percent break-even sales change in dollar sales terms? The answer is calculated as follows:

Thus, to break even on the proposed 5 percent price cut, Westside would have to increase its total dollar sales by 6.88 percent, which is exactly equivalent to a 12.5 percent increase in unit volume.

Break-Even Sales Incorporating a Change in Variable Costs

Thus far, we have dealt only with price changes that involve no changes in unit variable costs or in fixed costs. Often, however, price changes are made as part of a marketing plan involving cost changes as well. A price increase may be made along with product improvements that increase variable costs, or a price cut might be made to push the product with lower variable selling costs. Expenditures that represent fixed costs might also change along with a price change. We need to consider these two types of incremental costs when calculating the price–volume trade-off necessary for making pricing decisions profitable. We begin this section by integrating changes in variable cost into the financial analysis. In the next section, we do the same with changes in fixed costs.

Fortunately, dealing with a change in variable cost involves only a simple generalization of the break-even sales change formula already introduced. To illustrate, we return to Westside Manufacturing’s proposed 5 percent price cut. Suppose that Westside’s price cut is accompanied by a reduction in variable cost of $0.22 per pillow, resulting from Westside’s decision to use a new synthetic filler to replace the goose feathers it currently uses. Variable costs are $5.50 before the price change and $5.28 after the price change. By how much would sales volume have to increase to ensure that the proposed price cut is profitable?

When variable costs change along with the price change, managers simply need to subtract the cost change from the price change before doing the break-even sales change calculation. Unlike the case of a simple price change, managers must state the terms on the right-hand side of the equation in currency units (dollars, euros, yen, and so forth) rather than in percentage changes:

where Δ indicates “change in,” P = price, and C = cost. Note that when the change in variable cost ($ΔC) is zero, this equation is identical to the break-even formula previously presented. Note also that the term ($ΔP - $ΔC) is the change in the contribution margin and that the denominator (the original contribution margin plus the change) is the new contribution margin. Thus, the general form of the break-even pricing equation is simply written as follows:

For Westside, the next step in using this equation to evaluate the proposed price change is to calculate the change in contribution margin. Recall that the change in price is $9.50 - $10 or -$0.50. The change in variable costs is -$0.22. Thus, the change in contribution can be calculated as follows:

Previous calculations illustrated that the contribution margin before the price change is $4.50. We can, therefore, calculate the break-even sales change as follows:

In units, the break-even sales change is 0.066 x 4,000 units, or 265 units.

Given management’s projection of a $0.22 reduction in variable costs, the price cut can be profitable only if management believes that sales volume will increase by more than 6.6 percent, or 265 units. Note that this increase is substantially less than the required sales increase (12.5 percent) calculated before assuming a reduction in variable cost. Why does a variable cost reduction lower the necessary break-even sales change? Because it increases the contribution margin earned on each sale, making it possible to recover the contribution lost due to the price effect with less additional volume. This relationship is illustrated graphically for Westside Manufacturing in Exhibit 10-2. Westside can realize a gain in contribution due to the change in variable costs (box f), in addition to a gain in contribution due to any increase in sales volume.

Break-Even Sales with Incremental Fixed Costs

Although most fixed costs do not impact the incremental profitability of a pricing decision (because they do not change), some pricing decisions necessarily involve changes in fixed costs, even though these costs do not otherwise change with small changes in volume. The management of a discount airline considering whether to reposition as a higher-priced business travelers’ airline would probably choose to refurbish its lounges and planes. A regulated utility would need to cover the fixed cost of regulatory hearings to gain approval for a higher price. A fast-food restaurant would need to advertise its promotionally priced “special-value” meals to potential customers. These are incremental fixed costs, necessary for the success of a new pricing strategy but unrelated to the sales volume actually gained at those prices. Recall also that semifixed costs remain fixed only within certain ranges of sales. If a price change causes sales to move outside that range, the level of semifixed costs increases or decreases. Such changes in fixed and semifixed costs need to be covered for a price change to be justified, since without the price change these incremental costs can be avoided.

Fortunately, calculating the sales volume necessary to cover an incremental fixed cost is already a familiar exercise for many managers evaluating investments independent of price changes. For example, suppose a product manager is evaluating a $150,000 fixed expenditure to redesign a product’s packaging. The product’s unit price is $10, and unit variable costs total $5. How many units must be sold for the firm to recover the $150,000 incremental investment? The answer, as found in most managerial economics texts, is given by the following equation:

Remembering that the $CM equals price variable cost, the break-even sales volume for this example is:

How can the manager do break-even analysis for a change in pricing strategy that involves both a price change and a change in fixed cost? She simply adds the calculations for (a) the break-even sales change for a price change and (b) the break-even sales volume for the related fixed investment.

The break-even sales change for a price change with incremental fixed costs is the basic break-even sales change plus the sales change necessary to cover the incremental fixed costs. Since we normally analyze the break-even for a price change as a percent and the break-even for an investment in units, we need to multiply or divide by initial unit sales to make them consistent. Consequently, the unit break-even sales change with a change in fixed costs is as follows:

The calculation for the percent break-even sales change is as follows:

In both cases, if the “$ change in fixed costs” is zero, we have the break-even sales change equation for a simple price change.

To illustrate the equations for a price cut, return again to the pricing decision faced by Westside Manufacturing. Westside is considering a 5 percent price cut. We already calculated that it could profit if sales increase by more than 12.5 percent. Now suppose that Westside cannot increase its output without incurring additional semifixed costs. At the company’s current rate of sales—4,000 units per month—it is fully utilizing the capacity of the equipment at its four workstations. To increase capacity enough to handle 12.5 percent more sales, the company must install equipment for another workstation, at a monthly cost of $800. The new station raises plant capacity by 1,000 units beyond the current capacity of 4,000 units. What is the minimum sales increase required to justify a 5 percent price reduction, given that it involves an $800 increase in monthly fixed costs? The answer is determined as follows:

The company could profit from a 5 percent price reduction if sales increased by more than 700 units (17.5 percent), which is less than the 1,000 units of added capacity provided by the new workstation. Whether a prudent manager should actually implement such a price decrease depends on other factors as well: How likely is it that sales will increase substantially more than the break-even minimum, thus adding to profit? How likely is it that sales will increase by less, thus reducing profit? How soon could the decision be reversed, if at all, if sales do not increase adequately?

Even if management considers it likely that orders will increase by more than the break-even quantity, it should hesitate before making the decision. If orders increase by significantly less than the break-even minimum, this company could lose substantially, especially if the cost of the new workstation is largely sunk once the expenditure has been made. On the other hand, if orders increase by significantly more, the most the company could increase its sales without bearing the semifixed cost for further expansion is 25 percent, or 1,000 units. Consequently, management must be quite confident of a large sales increase before implementing the 5 percent price reduction.

Consider, however, if the company has already invested in the additional capacity and if the semifixed costs are already sunk. The monthly cost of the fifth workstation is then entirely irrelevant to pricing, since that cost would have to be borne whether or not the capacity is used. Thus, the decision to cut price rests entirely on management’s judgment of whether the price cut will stimulate unit sales by more than 12.5 percent. If the actual sales increase is more than 12.5 percent but less than 17.5 percent, management will regret having invested in the fifth workstation. Given that this cost can no longer be avoided, however, the most profitable course of action is to price low enough to use the station, even though that price will not fully cover its cost.

Break-Even Sales Analysis for Reactive Pricing

So far we have restricted our discussion to proactive price changes, where the firm contemplates initiating a price change ahead of its competitors. The goal of such a change is to enhance profitability. Often, however, a company initiates reactive price changes when it is confronted with a competitor’s price change that will impact the former’s sales unless it responds. The key uncertainty involved in analyzing a reactive price change is the sales loss the company will suffer if it fails to meet a competitor’s price cut, or the sales gain the company will achieve if it fails to follow a competitor’s price increase. Is the potential sales loss sufficient to justify cutting price to protect sales volume? Or is the potential sales gain enough to justify forgoing the opportunity for a cooperative price increase? A slightly different form of the break-even sales formula is used to analyze such situations.

To calculate the break-even sales changes for a reactive price change, we need to address the following key questions: (1) What is the minimum potential sales loss that justifies meeting a lower competitive price? (2) What is the minimum potential sales gain that justifies not following a competitive price increase? The basic formula for these calculations is this:

To illustrate, suppose that Westside’s principal competitor, Eastside, has just reduced its prices by 15 percent. If Westside’s customers are highly loyal, it probably would not pay for Westside to match this cut. If, on the other hand, customers are quite price sensitive, Westside may have to match this price cut to minimize the damage. What is the minimum potential loss in sales volume that justifies meeting Eastside’s price cut? The answer (calculated in percentage terms) is as follows:²

Thus, if Westside expects sales volume to fall by more than 33 percent as a result of Eastside’s new price, it would be less damaging to Westside’s profitability to match the price cut than to lose sales. On the other hand, if Westside expects that sales volume will fall by less than 33 percent, it would be less damaging to Westside’s profitability to let Eastside take the sales than it would be to cut price to meet this challenge.

This analysis has focused on minimizing losses in the face of a competitor’s proactive price reduction. However, the procedure for analysis is the same when a competitor suddenly raises its prices. Suppose, for example, that Eastside raises its price by 15 percent. Westside might be tempted to match Eastside’s price increase. If, however, Westside does not respond to Eastside’s new price, Westside will likely gain additional sales volume as Eastside’s customers switch to Westside. How much of a gain in sales volume must be realized in order for no price reaction to be more profitable than a reactive price increase? The answer is similarly found using the break-even sales change formula with a reactive price change. If Westside is confident that sales volume will increase by more than 33.3 percent if it does not react, a nonreactive price policy would be more profitable. If Westside’s management does not expect sales volume to increase by 33.3 percent, a reactive price increase would be more profitable.

Of course, the competitive analysis we have done is, by itself, overly simplistic. Eastside might be tempted to attack Westside’s other markets if Westside does not respond to Eastside’s price cut. And Westside’s not matching Eastside’s price increase might force Eastside to roll back its prices. These long-run strategic concerns might outweigh the short-term profit implications of a decision to react. In order to make such a judgment, however, the company must first determine the short-term profit implications. Sometimes long-term competitive strategies are not worth the short-term cost.

Calculating Potential Financial Implications

To grasp fully the potential impact of a price change, especially when the decision involves incremental changes in fixed costs, it is useful to calculate the profit impact for a range of potential sales changes and to summarize them with a break-even table and chart. Doing so is relatively simple after having calculated the basic break-even sales change. Using this calculation, one can then simulate what-if scenarios that include different levels of actual sales volume following the price change.

The top half of Exhibit 10-3 is a summary of the basic break-even sales change analysis for Westside’s 5 percent price cut, with one column summarizing the level of contribution before the price change (the column labeled “Baseline”) and one column summarizing the contribution after the price change (the column labeled “Proposed Price Change”). The bottom half of Exhibit 10-3 summarizes nine what-if scenarios showing the profitability associated with changes in sales volume ranging from 0 to 40 percent given incremental semifixed costs of $800 per 1,000 units. Columns 1 and 2 show the actual change in volume for each scenario. Columns 3 through 5 calculate the change in profit that results from each change in sales.

To illustrate how these break-even sales-change scenarios are calculated, let us focus for a moment on scenario 6, where actual sales volume is projected to increase 20 percent. A 20 percent change in actual sales volume is equivalent to an 800-unit change in actual sales volume, since 800 units is 20 percent of the baseline sales volume of 4,000 units. How does this increase in sales translate into changes in profitability? Column 3 shows that a 20 percent (or an 800-unit) increase in sales volume results in a change in contribution after the price change of $1,200. This is calculated by taking the difference between the actual unit sales change (800 units) and the break-even sales change shown in the top half of Exhibit 10-3 (500 units) and multiplying by the new contribution margin after the price change ($4). However, the calculations made in column 3 do not take into account the incremental fixed costs required to implement the price change (shown in column 4). Column 5 shows the change in profit after subtracting the change in fixed costs from the incremental contribution generated. Where there is inadequate incremental contribution to cover the incremental fixed costs, as in scenarios 1 through 4, the change in profit is negative. Scenario 5 illustrates the break-even sales change. Scenarios 6 through 9 are all profitable scenarios since they result in greater profit after the price change than before.

The interrelationships among contribution, incremental fixed costs, and the sales change that results from a price change are often easier to comprehend with a graph. Exhibit 10-4 illustrates the relationships among the data in Exhibit 10-3. Appendix 10B (at the end of this chapter) explains how to produce break-even graphs, which are especially useful in comprehending the implications of price changes when many fixed costs become incremental at different sales volumes.

Break-Even Sales Curves

So far we have discussed break-even sales analysis in terms of a single change in price and its resultant break-even sales change. In the example above, Westside Manufacturing considered a 5 percent price reduction, which we calculated would require a 17.5 percent increase in sales volume to achieve enough incremental contribution to cover the incremental fixed cost. However, what if the company wants to consider a range of potential price changes? How can we use break-even sales analysis to consider alternative price changes simultaneously? The answer is by charting a break-even sales curve, which summarizes the results of a series of break-even sales analyses for different price changes.

Constructing break-even sales curves requires doing a series of what-if analyses, similar to the simulated scenarios discussed in the last section. Exhibits 10-5 and 10-6 show numerically and graphically a break-even sales curve for Westside Manufacturing, with simulated scenarios of price changes ranging from 125 percent to 220 percent. Note in Exhibit 10-6 that the vertical axis shows different price levels for the product, and the horizontal axis shows a volume level associated with each price level. Each point on the curve represents the sales volume necessary to achieve as much profit after the price change as would be earned at the baseline price. For example, Westside’s baseline price is $10 per unit, and baseline sales volume is 4,000 units. If, however, Westside cuts the price by 15 percent to $8.50, its sales volume would have to increase 70 percent to 6,800 units to achieve the same profitability. Conversely, if Westside increases its price by 15 percent to $11.50, its sales volume could decrease 25 percent to 3,000 units and still allow equal profitability.

The break-even sales curve is a simple, yet powerful tool for synthesizing and evaluating the dynamics behind the profitability of potential price changes. It presents succinctly and visually the dividing line that separates profitable price decisions from unprofitable ones. Profitable price decisions are those that result in sales volumes in the area to the right of the curve. Unprofitable price decisions are those that result in sales volumes in the area to the left of the curve. What is the logic behind this? Recall the previous discussion of what happens before and after a price change. The break-even sales curve represents those sales volume levels associated with their respective levels of price, where the company will make just as much net contribution after the price change as it made before the price change. If the company’s sales volume after the price change is greater than the break-even sales volume (that is, actual sales volume is to the right of the curve), the price change will add to profitability. If the company’s sales volume after the price change is less than the break-even sales volume (that is, the area to the left of the curve), the price change will be unprofitable. For example, for Westside a price of $8.50 requires a sales volume of at least 6,800 units to achieve a net gain in profitability. If, after reducing its price to $8.50, management believes it will sell more than 6,800 units (a point to the right of the curve), then a decision to implement a price of $8.50 per unit would be profitable.

The break-even sales curve also clearly illustrates the relationship between the break-even approach to pricing and the economic concept of price elasticity. Note that the break-even sales curve looks suspiciously like the traditional downward-sloping demand curve in economic theory, in which different levels of price (on the vertical axis) are associated with different levels of quantity demanded (on the horizontal axis). On a traditional demand curve, the slope between any two points on the curve determines the elasticity of demand, a measure of price sensitivity expressed as the percent change in quantity demanded for a given percent change in price. An economist who knew the shape of such a curve could calculate the profit-maximizing price.

Unfortunately, few firms use economic theory to set price because of the unrealistic expectation that they first have to know their demand curve, or at least the demand elasticity around the current price level. To overcome this shortcoming, we have addressed the problem in reverse order. Rather than asking, “What is the firm’s demand elasticity?” we ask, instead, “What is the minimum demand elasticity required?” to justify a particular pricing decision. Break-even sales analysis calculates the minimum or maximum demand elasticity required to profit from a particular pricing decision. The break-even sales curve illustrates a set of minimum elasticities necessary to make a price cut profitable, or the maximum elasticity tolerable to make a price increase profitable. One is then led to ask whether the level of price sensitivity in the market is greater or less than the level of price sensitivity required by the firm’s cost and margin structure.

This relationship between the break-even sales curve and the demand curve is illustrated in Exhibits 10-7 and 10-8, where hypothetical demand curves are shown with Westside’s break-even sales curve. If demand is more elastic, as in Exhibit 10-7, price reductions relative to the baseline price result in gains in profitability, and price increases result in losses in profitability. If demand is less elastic, as in Exhibit 10-8, price increases relative to the baseline price result in gains in profitability, and price reductions result in losses in profitability. Although few, if any, managers actually know the demand curve for their product, we have encountered many who can comfortably make judgments about whether it is more or less elastic than is required by the break-even sales curve. Moreover, although we have not found any market research technique that can estimate a demand curve with great precision, we have seen many (described in Chapter 12 on price and value measurement) that could enable management to confidently accept or reject a particular break-even sales level as achievable.

Watching Your Baseline

In the preceding examples, the level of baseline sales from which we calculated break-even sales changes was assumed to be the current level. For simplicity, we assume a static market. In many cases, however, sales grow or decline even if price remains constant. As a result, the baseline for calculating break-even sales changes is not necessarily the current level of sales. Rather, it is the level that would occur if no price change were made.

Consider, for example, a company in a high-growth industry with current sales of 2,000 units on which it earns a contribution margin of 55 percent. If the company does not change its price, management expects that sales will increase by 20 percent (the projected growth of total industry sales) to 2,400 units. However, management is considering a 5 percent price cut in an attempt to increase the company’s market share. The price cut would be accompanied by an advertising campaign intended to heighten consumer awareness of the change. The campaign would take time to design, delaying implementation of the price change until next year. The initial sales level for the constant contribution analysis, therefore, would be the projected sales in the future, or 2,400 units. Consequently, the break-even sales change would be calculated as follows:

or

If the current sales level is used in the calculation, the unit break-even sales change is calculated as 200 units, understating the change required by 40 units.

Covering Nonincremental Fixed and Sunk Costs

By this point, one might be wondering about the nonincremental fixed and sunk costs that have been ignored when analyzing pricing decisions. A company’s goal must surely be to cover all of its costs, including all fixed and sunk costs, or it will soon go bankrupt. This concern is justified and is central to pricing for profit, but it is misguided when applied to justify higher prices.

Note that the goal in calculating a contribution margin and in using it to evaluate price changes and differentials is to set prices to maximize a product’s profit contribution. Profit contribution, you will recall, is the income remaining after all incremental, avoidable costs have been covered. It is money available to cover nonincremental fixed and sunk costs and to contribute to profit. When managers consider only the incremental, avoidable costs in making pricing decisions, they are not saying that other costs are unimportant. They simply realize that the level of those costs is irrelevant to decisions about which price will generate the most money to cover them. Since nonincremental fixed and sunk costs do not change with a pricing decision, they do not affect the relative profitability of one price versus an alternative. Consequently, consideration of them simply clouds the issue of which price level will generate the most profit to cover them.

All costs are important to profitability since they all, regardless of how they are classified, have to be covered before profits are earned. At some point, all costs must be considered. What distinguishes value-based pricing from cost-driven pricing is when they are considered. A major reason that this approach to pricing is more profitable than cost-driven pricing is that it encourages managers to think about costs when they can still do something about them. Every cost is incremental and avoidable at some time. For example, even the cost of product development and design, although it is fixed and sunk by the time the first unit is sold, is incremental and avoidable before the design process begins. The same is true for other costs. The key to profitable pricing is to recognize that customers in the marketplace, not costs, determine what a product can sell for. Consequently, before incurring any costs, managers need to estimate what customers can be convinced to pay for an intended product. Then they decide what costs they can profitably incur, given the expected revenue.

Of course, no one has perfect foresight. Managers must make decisions to incur costs without knowing for certain how the market will respond. When their expectations are accurate, the market rewards them with sales at the prices they expected, enabling them to cover all costs and to earn a profit. When they overestimate a product’s value, profit contribution may prove inadequate to cover all the costs incurred. In that case, a good manager seeks to minimize the loss. This can be done only by maximizing profit contribution (revenue minus incremental, avoidable costs). Shortsighted efforts to build nonincremental fixed and sunk costs into a price that will justify past mistakes will only reduce volume further, making the losses worse.

Summary

The profitability of pricing decisions depends largely on the product’s cost structure and contribution margin and on market sensitivity to changes in price. In Chapter 9, we discuss the importance of identifying the costs that are most relevant to the profitability of a pricing decision, namely, incremental and avoidable costs. Having identified the right costs, one must also understand how to use them. The most important reason to identify costs correctly is to be able to calculate an accurate contribution margin. An accurate contribution margin enables management to determine the amount by which sales must increase following a price cut, or by how little they may decline following a price increase, to make the price change profitable. Understanding how changes in sales will affect a product’s profitability is the first step in pricing the product effectively.

It is, however, just the first step. Next, one must learn how to judge the likely impact of a price change on sales, which requires understanding how buyers are likely to perceive a price change and how competitors are likely to react to it. We consider these subjects in the next two chapters on competition and value measurement.

Appendix 10A Derivation of the Break-Even Formula

A price change can either increase or reduce a company’s profits, depending on how it affects sales. The break-even formula is a simple way to discover at what point the change in sales becomes large enough to make a price reduction profitable, or a price increase unprofitable.

Exhibit 10A-1 illustrates the break-even problem. At the initial price P, a company can sell the quantity Q. Its total revenue is P times Q, which graphically is the area of the rectangle bordered by the lines 0P and 0Q. If C is the product’s variable cost, then the total profit contribution earned at price P is (P - C)Q. Total profit contribution is shown graphically as the rectangle left after subtracting the variable cost rectangle (OC, OQ) from the revenue rectangle (OP, OQ).

If this company reduces its price from P to P, its profits will change. First, it will lose an amount equal to the change in price, ΔP, times the amount that it could sell without the price change, Q. Graphically, that loss is the rectangle labeled A. Somewhat offsetting that loss, however, the company will enjoy a gain from the additional sales it can make because of the lower price. The amount of the gain is the profit that the company will earn from each additional sale, P' - C, times the change in sales, ΔQ. Graphically, that gain is the rectangle labeled B. Whether or not the price reduction is profitable depends on whether or not rectangle B is greater than rectangle A, and that depends on the size of ΔQ.

The logic of a price increase is similar. If P were the initial price and Q the initial quantity, then the profitability of a price increase to P would again depend on the size of ΔQ. If ΔQ were small, rectangle A, the gain on sales made at the higher price, would exceed rectangle B, the loss on sales that would not be made because of the higher price. However, ΔQ might be large enough to make B larger than A, in which case the price increase would be unprofitable.

To calculate the formula for the break-even ΔQ (at which the gain from a price reduction just outweighs the loss or the loss from a price increase just outweighs the gain), we need to state the problem algebraically. Before the price change, the profit earned was (P - C)Q. After the change, the profit was (P' C)Q'. Noting, however, that P' = P + ΔP (we write, “+ ΔP” since ΔP is a negative number) and that Q' = Q + ΔQ, we can write the profit after the price change as (P + ΔP - C)(Q + ΔQ). Since our goal is to find the ΔQ at which profits would be just equal before and after the price change, we can begin by setting those profits equal algebraically:

Multiplying this equation through yields

We can simplify this equation by subtracting PQ and adding CQ to both sides to obtain

Note that all the remaining terms in the equation contain the “change sign” Δ. This is because only the changes are relevant for evaluating a price change. If we solve this equation for ΔQ, we obtain the new equation

which, in words, is

To express the right side in percentages, multiply the right side by

Appendix 10B Break-Even Analysis of Price Changes

Break-even analysis is a common tool in managerial accounting, particularly useful for evaluating potential investments. Unfortunately, the traditional forms of break-even analysis appropriate for evaluating investment decisions are often misleading when applied to pricing decisions. Individual investments (Should the company buy a new computer? Should it develop a new product? Should it field a sales force to enter a new market?) can often be evaluated apart from other investments. Consequently, it is appropriate to use traditional break-even analysis, which compares the total revenue from the investment with its total cost.

Usually, however, one cannot set a price for each individual sale independent of other sales. To gain an additional sale by charging one customer a lower price normally requires charging other customers, at least others in that same market segment, the lower price as well. Consequently, it is usually misleading to evaluate the profitability of an additional sale by comparing only the price earned from that sale to the cost of that sale. To comprehend the profit implications of a price change, one must compare the change in revenue from all sales with the change in costs.

The need to focus attention on the changes in revenues and costs rather than on their totals requires a different kind of break-even analysis for pricing decisions. Where traditional break-even analysis of investments deals with total revenue and all costs, break-even analysis of pricing decisions deals with the change in revenue in excess of variable cost (the dollar contribution margin) and with the change in incremental fixed costs. In the body of this chapter, you learned a number of formulas for break-even analysis of pricing decisions and saw how to use them in the Westside Manufacturing example. In this appendix, you will learn how to use those equations to develop break-even graphs and to analyze more complex pricing problems involving multiple sources of fixed costs that become incremental at different quantities.

Developing a Break-Even Chart

A break-even chart, such as Exhibit 10-4 is useful in determining the possible effects of a price change. It plots both the change in the dollar contribution margin and the changes in relevant costs, enabling the pricing analyst to see the change in net profit that a change in sales volume would generate. To develop such a chart, it is useful to begin by preparing a table, such as Exhibit 10-5, organizing all relevant data in a concise form.

As an illustration, let us examine the case of PQR Industries. PQR manufactures and markets home video equipment. One of the most popular items in the company’s product line is a digital video recorder with current sales of 4,000 units at $250 each. Sales have been growing rapidly and are expected to reach 4,800 units in the next year if the price remains unchanged. Variable costs are $112.50 per unit, resulting in the following percent contribution margin:

Despite its projected growth in sales at the current price, PQR is considering a 5 percent price cut to remain competitive and retain its share in this rapidly growing market. Since the cut would be implemented in the next year, the initial sales level, or baseline, is next year’s projected sales (4,800 units). Calculation of the break-even sales change is as follows:

Production capacity is currently limited to 5,000 units but can be increased by purchasing equipment that costs $15,000 for each additional 1,000 units of capacity. The break-even sales change, considering this change in fixed costs, is:

Note that the price cut brings the price down to $237.50, resulting in a new dollar contribution margin of $125 per unit.

Since the actual sales change that would result from the price cut is unknown, a break-even table and chart should be prepared to show the profitability of the price change at various possible sales changes.

Exhibit 10B-1 shows a break-even table for PQR’s proposed 5 percent price cut. The first two columns show the

EXHIBIT 10B-1 Break-Even Table for PQR Industries’ Proposed 5% Price Cut

potential levels of sales changes. Column 3 shows the change in total contribution margin that would result at each level using the change in profit formula. In the case of a 5 percent change in sales, the result would be:

Subtracting the change in fixed costs shown in column 4 from column 3 results in column 5, the change in profit contribution. Alternatively, we could have generated column 5 more directly by calculating the break-even sales change including the change in fixed costs and substituting that number in the change in profit equation (see Exhibit 10B-1 and Exhibit 10B-2).

When plotted on a graph, the data from this table form a break-even chart (Exhibit 10B-2). The horizontal axis represents the change in unit sales and the vertical axis represents dollars of change. The line labeled “change in fixed costs” shows the increase in costs due to added capacity, as taken from column 4 of the table. The data in column 3 were used to plot the “change in contribution margin” line. The distance between the two lines represents the change in profit contribution (column 5). At the points where the “change in contribution margin” line is above the “change in fixed costs” line, the change in profit contribution (and net profit) is positive. The price cut would be profitable if sales changed by those amounts.

Break-Even Analysis with More than One Incremental Fixed Cost

To this point, we have always assumed that a company has only one fixed cost that changes with a price change. Frequently, however, a company will have several semifixed costs that change at different levels of volume. This makes analysis of a price change more complicated and the use of break-even analysis more essential to the management of that complexity.

Let us return to PQR Industries. Because of the cost of adding new equipment, management investigated alternative methods of increasing production. It was determined that the addition of one machine operator could delay purchase of new equipment until sales exceeded 5,400 units (or 600 units more than the baseline initial sales). Although labor costs are normally variable with production, machine operators are skilled laborers who, according to union rules, can only be hired as full-time employees working only at their specialties. The result is that a machine operator’s salary is a semifixed cost. It was also discovered that the union contract required one skilled worker to be added for each 1,000 units of increased production. Finally, the plant engineer informed management that there was space for only one additional piece of equipment. If more equipment were purchased, more space would have to be rented, at a cost of $105,000 per year. The situation is summarized as follows:

Semifixed costs:

Due to the complexity of these costs, there is more than one break-even sales change and a single calculation is not sufficient. For example, any increase in sales will require the hiring of a machine operator. The break-even sales change for a 5 percent price cut becomes:

If, however, the total sales exceed 5,400 units and equipment must be purchased, a new calculation is required as follows:

If more space would have to be rented, still another break-even calculation would be required.

It seems obvious that when there are multiple sources of incremental fixed costs, analysis via calculation of break-even sales changes could become both tedious and confusing. A break-even table and chart are usually essential to the clear sorting out of these problems. Organizing the data into a table (Exhibit 10B-3) and

plotting it on a graph (Exhibit 10B-4) make the options much clearer. At changes in sales of between 540 and 600 units, the price change is slightly profitable. Once the change in sales exceeds 600 units, however, fixed costs must rise again due to the need for more equipment, and profits will become negative and will not return to positive again until the change in sales exceeds 660 units, causing the change in contribution to rise above the change in fixed costs.

Note that 12.5 percent, or 600 units, is the maximum sales change possible before additional costs must be incurred. To determine whether the investment in additional equipment is worthwhile, management must decide whether the possibility that sales will grow enough to achieve a profit contribution of more than $7,500 (that is, by more than 15 percent) after the equipment is purchased would be enough to justify forgoing the more certain profit to be gained from reaching only a 12.5 percent increase in sales.

This type of situation arises whenever there is a change in fixed costs, most dramatically when the second machine is bought and space must be rented. It seems unlikely that sales growth due to the price change would be sufficient to justify renting more space. In fact, sales would have to increase by more than 53 percent before such an increase in fixed costs would produce a positive net profit. Moreover, the investment would not be justified if the profit that could be earned by not meeting the entire sales gain were higher.

Break-Even Graphs

The calculations for determining break-even sales changes for a price increase are the same as those for a price cut. For price increases, however, sales volumes decline rather than increase. Consequently, the direction of the horizontal axis measures declines in sales volumes rather than increases.

To illustrate, let us consider again the case of PQR Industries. In addition to digital video recorders, PQR sells flat panel TVs for $3,000. Variable costs of $1,650 per unit leave the company with a contribution margin of $1,350 per unit, or 45 percent. The company is considering a price increase next year on this item. The initial sales level for evaluating the increase (next year’s projected sales) is 4,000 units, which exceeds the company’s current capacity of 3,600 units.

Concern about capacity constraints and the slowing growth of the market for this product have caused management to consider instituting a price increase in order to maximize profits. The break-even sales change for the proposed 5 percent price increase is:

As long as the price increase causes sales to decline by less than 10 percent or 400 units, the price increase will cause the total dollar contribution from this product to increase.

To increase production beyond the current 3,600-unit capacity, additional equipment must be bought at a cost of $150,000. If, however, as a result of the price increase, sales decrease to the point that current capacity is sufficient, purchase of new equipment would not be necessary. The expenditure avoided is a negative change in fixed costs from the level required to achieve the 4,000-unit baseline sales level. A new break-even sales change is calculated as follows:

On the basis of the data in Exhibit 10B-5, we can produce a break-even

EXHIBIT 10B-5 Break-Even Table for PQR Industries’ Proposed 5% Price Increase

graph for this price change (Exhibit 10B-6). The lines representing changes in contribution margin and fixed costs run opposite to the directions we are accustomed to seeing for price cuts, but the change in profit contribution, as indicated by the relative positions of these lines, is still interpreted as in earlier examples.

To verify this, let us refer to the graph and examine the results of a 5 percent, or 200-unit, sales decrease. At this point, there has been no change in fixed costs. Therefore, the change in profitability should equal a positive $300,000, the distance between the line indicating change in contribution margin and the horizontal axis. Reference to the related table will show that this is indeed true.

Summary

Predicting the outcome of a price change is not an exact science, as later chapters will show. A manager should, therefore, consider all possible outcomes of such a change in order to choose the wisest course of action. Tables and graphs of break-even analyses are useful, easily produced tools for this purpose.

Acknowledgment

This chapter was coauthored by Professor Gerald E. Smith of Boston College.