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Break-Even and Pricing

Cost-volume-profit analysis for a single product (contribution margin and its ratio, break-even units and revenue, the volume for a target operating income or an after-tax target net income, margin of safety in units, revenue and percent, degree of operating leverage, the CVP and profit-volume charts and a one-factor-at-a-time sensitivity table); multi-product break-even for a pasted sales mix (weighted-average contribution margin, bundle method, break-even in total units, per product and in revenue); and price elasticity from pasted price, quantity observations (midpoint arc elasticities, a constant-elasticity demand fitted by log-log least squares with the standard error and confidence interval of the elasticity, a linear demand fit, the profit-maximizing price under each model for an entered unit cost by the inverse-elasticity (Lerner) markup rule, and the revenue and profit curves against price). The default case is the Acoustic Concepts example of Garrison, Noreen and Brewer (price 250, variable cost 150, fixed costs 35,000: break-even 350 units and 87,500 of sales, 750 units for a 40,000 profit), with the GMAT and GRE sales mix of Horngren, Datar and Rajan and an illustrative six-point demand table of elasticity about −2.

  • Single product: contribution margin and ratio, break-even units and revenue, target operating income and after-tax target net income, margin of safety, degree of operating leverage

  • CVP chart (revenue, total cost, fixed cost against units), profit-volume chart, sensitivity of profit to ± changes in price, volume, variable cost and fixed cost

  • Sales mix: weighted-average contribution margin, bundle method, break-even and target-profit units per product, break-even revenue

  • Pricing: arc elasticities, constant-elasticity and linear demand fits with standard errors, profit-maximizing price by the inverse-elasticity rule, revenue and profit against price

Break-Even and Pricing

Unit System: /

Analysis (money is a plain number, one currency unit throughout)


Single Product (Garrison, Noreen and Brewer, Acoustic Concepts)









Sales Mix (one product per line, price, variable cost, mix units; Horngren GMAT and GRE example)




Price Elasticity (one price, quantity observation per line; illustrative data of elasticity about −2)



Cost-Volume-Profit Analysis, Sales Mix and Price Elasticity | Method page
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Break-Even and Pricing

Cost-Volume-Profit Analysis, Sales Mix and Price Elasticity

Three questions

A business plan rests on three quantitative questions: how many units must be sold before the venture stops losing money (cost-volume-profit analysis), how that break-even is shared among several products sold together (sales-mix analysis), and what price earns the most when customers buy less as the price rises (demand elasticity and the markup rule). The tool answers each from the managerial-accounting treatment of Garrison, Noreen and Brewer and of Horngren, Datar and Rajan, and from the microeconomics of Mankiw, Pindyck and Rubinfeld and Varian. Money is a plain number in one currency unit; nothing is converted.

Cost-volume-profit analysis

Costs are split into those that vary with volume (materials, direct labor, commissions) and those that do not within the planning range (rent, salaries, depreciation). With a selling price P, a variable cost per unit VC and fixed costs FC, the contribution margin per unit is CM = P − VC, the amount each unit contributes toward the fixed costs and then toward profit; the contribution margin ratio CM/P is the same thing per unit of revenue. The operating income at a volume Q is CM × Q − FC, so:

  • Break-even: CM × Q = FC gives QBE = FC/CM units and, in revenue, FC/(CM ratio). Acoustic Concepts (Garrison): 35,000/100 = 350 speakers, 35,000/0.40 = 87,500 of sales.
  • Target profit: a desired operating income is treated as one more fixed amount to cover, QT = (FC + target)/CM; 750 speakers for a 40,000 profit. An after-tax target net income must first be grossed up to the operating income that leaves it after tax at the rate t, target OI = TNI/(1 − t) (Horngren: 960 after 40 % tax needs 1,600 before tax, 45 units instead of 40).
  • Margin of safety: the excess of the budgeted or actual sales over the break-even sales, in revenue, in units and as a percentage of sales; it is how far sales may fall before a loss appears (12,500 = 12.5 % = 50 speakers at 400 speakers).
  • Degree of operating leverage: DOL = total contribution margin / operating income at the given volume. A percentage change in sales volume produces DOL times that percentage change in operating income, so a business with high fixed costs and a high CM ratio has a high DOL and a volatile profit. At a given volume DOL = 1/(margin-of-safety fraction); it falls toward 1 as sales move away from the break-even point. Acoustic Concepts at 400 speakers: 40,000/5,000 = 8, so 10 % more speakers raise the operating income by 80 %. (Garrison's farms example, Bogside Farm with a DOL of 4 against Sterling Farm with 7 at the same sales and income, shows the effect of the cost structure.)

The CVP graph draws revenue P × Q and total cost FC + VC × Q against units; they cross at the break-even point and the vertical distance between them is the profit or loss. The profit-volume graph draws the profit line itself, CM × Q − FC, which starts at −FC and crosses zero at QBE. The sensitivity table moves the price, the volume, the variable cost per unit and the fixed costs one at a time by the entered percentage and reports the operating income; price is almost always the most powerful lever because a price change goes entirely to the contribution margin, and a volume change is multiplied by the DOL.

Assumptions (Horngren Ch. 3): costs are linear in volume within the relevant range and divide cleanly into fixed and variable; the selling price and the variable cost per unit are constant; the number of units sold is the only cost and revenue driver; for a single product, or for several products at a constant sales mix; inventory levels do not change. Step costs, volume discounts and learning effects break the straight lines and should be handled by re-running the tool for each range.

Multi-product break-even (sales mix)

When several products are sold together, the break-even depends on the sales mix, the proportions in which they sell. At a constant mix the line behaves as a single composite product. Two equivalent ways of working:

  • Weighted-average contribution margin: WACM = Σ (mix sharei × CMi); break-even total units = FC/WACM, split among the products by their mix shares; break-even revenue = FC/(overall CM ratio), where the overall ratio is total contribution margin over total revenue at the mix. Horngren's GMAT Success (CM 80) and GRE Guarantee (CM 30) sold 3 : 2 give WACM = 0.6 × 80 + 0.4 × 30 = 60, so 4,500/60 = 75 units: 45 GMAT and 30 GRE, revenue 12,000.
  • Bundle method: one bundle holds the smallest whole-number mix (3 GMAT + 2 GRE) with CM 3 × 80 + 2 × 30 = 300; 4,500/300 = 15 bundles, the same 45 and 30.

Garrison's Sound Unlimited example works in revenue only: two CD lines with CM ratios of 25 % and 50 % and sales of 20,000 and 80,000 give an overall ratio of 45 % and a break-even of 27,000/0.45 = 60,000. The same numbers are reproduced by entering each line as 1,000 units at prices 20 and 80 with variable costs 15 and 40. A shift of the mix toward the low-margin product raises the break-even even when total sales are unchanged, which is why the tool reports the products' contribution-margin ratios and revenue shares beside the break-even.

Price elasticity of demand

The elasticity ε is the percentage change in quantity demanded per percentage change in price, negative for a normal demand curve. Between two observations it is computed by the midpoint (arc) method, which divides each change by the average of the two values so that the result does not depend on the direction of the change (Mankiw Ch. 5; Allen 1934): ε = [(q2 − q1)/q̄] / [(p2 − p1)/p̄]. Mankiw's ice-cream example, 10 cones at $2.00 and 8 at $2.20, gives a 22 % quantity change over a 9.5 % price change, an elasticity of 2.3 in magnitude (the book prints 2.32 from the rounded percentages; the exact ratio is 2.333). Pindyck and Rubinfeld's example, Q from 6 to 4 as P rises from 8 to 10, gives −1.8 by the arc formula against point elasticities of −1.33 and −2.5 at the two ends. Demand is elastic where |ε| > 1 (revenue falls as the price rises) and inelastic where |ε| < 1 (revenue rises with price).

Over a table of observations the tool fits two demand curves by ordinary least squares:

  • Constant elasticity, q = A pε, fitted as ln q = ln A + ε ln p, so the slope is the elasticity and it is the same at every price. The standard error of the slope, its t statistic and a 95 % confidence interval (Student t on n − 2 degrees of freedom) say how well the data pin ε down; the regression routine reproduces the NIST StRD Norris certified values.
  • Linear demand, q = a − b p: the intercept a is the quantity at a zero price, a/b is the choke price at which demand vanishes, and the point elasticity −b p/q varies from 0 to −∞ along the line, passing −1 at the revenue-maximizing price a/(2b).

Optimal price and the markup rule

With a constant unit (marginal) cost c the profit (p − c) q(p) is largest where marginal revenue equals marginal cost. Since marginal revenue is p (1 + 1/ε), the condition is the inverse-elasticity rule of Lerner (1934), as Pindyck and Rubinfeld write it: (P − MC)/P = −1/ε, that is P = MC/(1 + 1/ε). The left side is the Lerner index of market power, the markup as a fraction of the price. For the constant-elasticity curve the rule gives directly p* = c ε/(ε + 1): an elasticity of −2 means a price of twice the unit cost, −4 means 1.33 times (Pindyck and Rubinfeld Example 10.1: supermarkets near −10 mark up about 11 %, convenience stores near −5 about 25 %). It needs |ε| > 1; where demand is inelastic, raising the price always raises revenue and lowers the quantity (and so the cost), so no finite price maximizes profit, and the tool says so. For the linear curve the optimum is p* = (a + b c)/(2b), half-way between the unit cost and the choke price (Varian Ch. 24: with inverse demand p = α − βy the monopolist produces y* = (α − c)/(2β) at p* = (α + c)/2); the Lerner index at that price again equals −1/ε evaluated there.

The two curves agree on the observed range and extrapolate differently, so their p* differ (11.79 against 14.10 for the default table); a recommended price well outside the observed prices is an extrapolation, and the revenue and profit chart shows how flat or sharp each model's profit peak is. The optimum ignores fixed costs, competitors' reactions, capacity limits and any dependence of the unit cost on volume; it is the single-period, single-product monopoly price of the textbooks.

Reading the results

The single-product section gives the contribution margin and its ratio, the break-even in units and revenue, the target-profit volume (before and, with a tax rate, after tax), the contribution-format income statement at the entered volume, the margin of safety, the degree of operating leverage with its meaning for a percentage change in sales, the sensitivity table and the two charts. The sales-mix section gives a product table with the mix shares, bundle quantities and the break-even and target-profit units per product, the weighted-average margin, the overall ratio and the break-even in units, bundles and revenue, and a bar chart of the entered against the break-even units. The pricing section gives the observations with the arc elasticities, both fits with their statistics, the optimal price under each model with its quantity, revenue, profit, Lerner index and markup, a comparison table, and the demand and revenue-profit charts.

Default case

Acoustic Concepts (Garrison, Noreen and Brewer, Ch. 5): price 250, variable cost 150, fixed costs 35,000, 400 speakers sold, target profit 40,000. The page reproduces the chapter's values: CM 100 and 40 %, break-even 350 speakers and 87,500, 750 speakers for the target, net operating income 5,000, margin of safety 12,500 (12.5 %, 50 speakers), DOL 8. The sales-mix box holds Horngren's GMAT and GRE example (45 and 30 units, 12,000 of revenue at fixed costs of 4,500); the elasticity box holds an illustrative six-point table made with an elasticity near −2 (fitted ε = −2.036 ± 0.058, p* = 11.79 by the constant-elasticity model and 14.10 by the linear model at a unit cost of 6).



References

Garrison, R. H., Noreen, E. W., and Brewer, P. C. (2015). Managerial Accounting, 15th ed. McGraw-Hill Education, New York.

Horngren, C. T., Datar, S. M., and Rajan, M. V. (2015). Cost Accounting: A Managerial Emphasis, 15th ed. Pearson, Boston.

Mankiw, N. G. (2015). Principles of Economics, 7th ed. Cengage Learning, Stamford, CT.

Pindyck, R. S., and Rubinfeld, D. L. (2013). Microeconomics, 8th ed. Pearson, Boston.

Varian, H. R. (2010). Intermediate Microeconomics: A Modern Approach, 8th ed. W. W. Norton, New York.

Lerner, A. P. (1934). "The Concept of Monopoly and the Measurement of Monopoly Power." The Review of Economic Studies, Vol. 1, No. 3, pp. 157–175.

Allen, R. G. D. (1934). "The Concept of Arc Elasticity of Demand." The Review of Economic Studies, Vol. 1, No. 3, pp. 226–229.

NIST/SEMATECH (2012). e-Handbook of Statistical Methods. National Institute of Standards and Technology, Gaithersburg, MD.

Worked Examples

Published examples this tool reproduces; each page lists the inputs to enter and the printed answers.

All worked examples

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