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Lecture 5Econ 340H

Managerial EconomicsWeb Appendix 5A

Cost Theory Break-Even versus

Contribution Analysis

Christopher Michael

Trent UniversityDepartment of Economics

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• Break-even analysis and operating leverage

• Risk assessment

Topics

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• We can have multiple B/E (break-even) points with non-linear costs & revenues.

• If linear total cost and total revenue:

TR = P•Q

TC = F + V•Q• where V is Average Variable

Cost• F is Fixed Cost• Q is Output

• Cost-Volume-Profit analysis

TotalCost

TotalRevenue

B/E B/EQ

Break-Even Analysis

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The Break-Even Quantity: Q B/E

• At break-even: TR = TCSo, P•Q = F + V•Q

• Q B/E = F / ( P - V) = F/CM

where contribution margin is: CM = ( P - V)

TR

TC

B/E Q

PROBLEM: As a garagecontractor, find Q B/E

if: P = $9,000 per garage V = $7,000 per garage& F = $40,000 per year

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• Amount of sales revenues that breaks even

• P•Q B/E = P•[F/(P-V)]

= F / [ 1 - V/P ]

Break-Even Sales Volume

Variable Cost Ratio

Ex: At Q = 20, B/E Sales Volume is

$9,000•20 = $180,000 Sales Volume

Answer: QB/E = 40,000/(2,000)= 40/2 = 20 garages at the break-even point.

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Quantity needed to attain a target profit

If is the target profit, Q target = [ F + ] / (P-V)

Suppose want to attain $50,000 profit, then,

Q target = ($40,000 + $50,000)/$2,000

= $90,000/$2,000 = 45 garages

Target Profit Output

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Degree of Operating Leverageor Operating Profit Elasticity

• DOL = E sensitivity of operating profit (EBIT) to

changes in output

• Operating = TR-TC = (P-V)•Q - F

• Hence, DOL = (Q)•(Q/) =

(P-V)•(Q/) = (P-V)•Q / [(P-V)•Q - F]

A measure of the importance of Fixed Costor Business Risk to fluctuations in output

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DOL as Operating Profit Elasticity

DOL = (P - V) Q / [ (P - V) Q - F ]• We can use empirical estimation methods to find

operating leverage • Elasticities can be estimated with double log

functional forms• Use a time series of data on operating profit and

outputLn EBIT = a + b• Ln Q, where b is the DOLthen a 1% increase in output increases EBIT by b%b tends to be greater than or equal to 1

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• DOL = (9,000-7,000) • 45 .

[(9,000-7,000)•45 – 40,000]

= 90,000 / 50,000 = 1.8

• A 1% INCREASE in Q 1.8% INCREASE in operating profit.

• At the break-even point, DOL is INFINITE. – A small change in Q increases EBIT by an

astronomically large percentage rate

Suppose the Contractor Builds 45 Garages, What is the D.O.L?

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• Dependent Variable: Ln EBIT uses 20 quarterly observations N = 20

The log-linear regression equation isLn EBIT = - .75 + 1.23 Ln Q

Predictor Coeff Stdev t-ratio pConstant -.7521 0.04805 -15.650 0.001Ln Q 1.2341 0.1345 9.175 0.001s = 0.0876 R-square= 98.2% R-sq(adj) = 98.0%

The DOL for this firm is 1.23. So, a 1% increase in output leads to a 1.23% increase in operating profit

Regression Output

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Output

recession

TIME

EBIT =operating profit

Trough

peak

1. EBIT is more volatilethan output over cycle

2. EBIT tends to collapse late in recessions

Operating Profit and the Business Cycle

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• One approach to risk, is the probability of losing money.

• Let QB/E be the break-even quantity, and Q is the expected quantity produced.

• z is the number of standard deviations away from the mean

• z = (QB/E - Q )/• 68% of the time within 1 standard deviation• 95% of the time within 2 standard deviations• 99% of the time within 3 standard deviations

Break-Even Analysis and Risk Assessment

Continued…

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Problem: If the break-even quantity is 5,000, and the expected number produced is 6,000, what is the chance of losing money if the standard deviation is 500?

Answer: z = (5,000 – 6,000) / 500 = -2.

There is less than 2.5% chance of losing money. Using table 13B.1, the exact answer is 0.0228 or 2.28% chance of losing money.

Break-Even Analysis and Risk Assessment (concluded)