Break-Even Analysis

• Study of interrelationships among a firm’s sales, costs, and operating profit at various levels of output

• Break-even point is the Q where TR = TC (Q1 to Q2 on graph)

TR

TC

Q

$’s

Profit

Q1 Q2

Linear Break-Even Analysis

• Over small enough range of output levels TR and TC may be linear, assuming– Constant selling price (MR)– Constant marginal cost (MC)– Firm produces only one product– No time lags between investment and resulting

revenue stream

Graphic Solution Method

• Draw a line through origin with a slope of P (product price) to represent TR function

• Draw a line that intersects vertical axis at level of fixed cost and has a slope of MC

• Intersection of TC and TR is break-even point

TR

TC

FC

Break-even point

MC

P

1 unit Q

1 unit Q Q

$’s

Algebraic Solution

• Equate total revenue and total cost functions and solve for Q

TR = P x Q

TC = FC + (VC x Q)

TR = TC

P x QB = FC + VC x QB

(P x QB) – (VC x QB) = FC

QB (P – VC) = FC

QB = FC/(P – VC), or in terms of total dollar sales,

PQ = (FxP)/(P-VC) = ((FxP)/P)/((P-VC)/P) = F/((P/P) – (VC/P))

= F/(1-VC/P)

Related Concepts

• Profit contribution = P – VC– The amount per unit of sale contributed to

fixed costs and profit

• Target volume = (FC + Profit)/(P – VC)– Output at which a targeted total profit would

be achieved

Example 1 – how many Christmas trees need to be sold

• Wholesale price per tree is $8.00

• Fixed cost is $30,000

• Variable cost per tree is $5.00

• SolutionQ(break-even) = F/(P – VC) = $30,000/($8 - $5)

= $30,000/$3 = 10,000 trees

Example 2 – two production methods to accomplish same task

• Method I : TC1 = FC1 + VC1 x Q

• Method II : TC2 = FC2 + VC2 x Q

• At break-even point:FC1 + (VC1 x Q) = FC2 + (VC2 x Q)

(VC1 x Q) – (VC2 x Q) = FC2 – FC1

Q x (VC1 – VC2) = FC2 – FC1

Q = (FC2 – FC1)/(VC1 – VC2)

Example 2 continued: bowsaw or chainsaw to cut Christmas trees

• Bowsaw

– Fixed cost is $5.00

– Variable cost is $0.40 per

• Chainsaw

• Fixed cost is $305

• Variable cost is $0.10 per tree

• Solution

Q(break-even) = ($305 - $5)/($0.40 - $0.10)

= 300/.30 = 1,000 trees

Revenue and Cost as Function of a Single Input

• Marginal Revenue Product– Increase in revenue per unit increase in one

input

– = MR / (Xt+1 – X t )

• Marginal Factor Cost– Increase in cost per unit increase in one input

– =MC / (Xt+1 – X t )

• Go to Spreadsheet

Example 3: Optimal planting density

No. Trees per A @ 5 years

Yield in cords @ rotation

Total Revenue @$24/cord

300 21.3 511

400 23.4 562

500 25.2 605

600 26.7 641

700 28.0 672

800 29.1 698

900 30.1 722

Example 3: Continued

• Fixed costs per acre:– Land . . . . . . . $300

– Site prep . . . . 100

– Annual . . . . 60

– Set-up . . . . . 5

– Total . . . . 465

• Variable costs per 100 seedlings– Seedlings . . . . $ 5

– Planting . . . . 20

– Total . . . . 25

TC = 465 + 25 x (# trees per A/100)

Go to Spreadsheet