Break-Even AnalysisStudy of interrelationships among a firms sales, costs, and operating profit at various levels of outputBreak-even point is the Q where TR = TC (Q1 to Q2 on graph)TRTCQ$sProfitQ1Q2

Linear Break-Even AnalysisOver small enough range of output levels TR and TC may be linear, assumingConstant selling price (MR)Constant marginal cost (MC)Firm produces only one productNo time lags between investment and resulting revenue stream

Graphic Solution MethodDraw a line through origin with a slope of P (product price) to represent TR functionDraw a line that intersects vertical axis at level of fixed cost and has a slope of MCIntersection of TC and TR is break-even pointTRTCFCBreak-even pointMCP1 unit Q1 unit Q

Q$s

Algebraic SolutionEquate total revenue and total cost functions and solve for QTR = P x QTC = FC + (VC x Q)TR = TCP x QB = FC + VC x QB(P x QB) (VC x QB) = FCQB (P VC) = FCQB = 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 ConceptsProfit contribution = P VCThe amount per unit of sale contributed to fixed costs and profitTarget volume = (FC + Profit)/(P VC)Output at which a targeted total profit would be achieved

Example 1 how many Christmas trees need to be soldWholesale price per tree is $8.00Fixed cost is $30,000Variable cost per tree is $5.00SolutionQ(break-even) = F/(P VC) = $30,000/($8 - $5) = $30,000/$3 = 10,000 trees

Example 2 two production methods to accomplish same taskMethod I : TC1 = FC1 + VC1 x QMethod II : TC2 = FC2 + VC2 x QAt 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 treesBowsawFixed cost is $5.00Variable cost is $0.40 per ChainsawFixed cost is $305Variable cost is $0.10 per treeSolutionQ(break-even) = ($305 - $5)/($0.40 - $0.10) = 300/.30 = 1,000 trees

Example 3: Optimal planting density

No. Trees per A @ 5 yearsYield in cords @ rotationTotal Revenue @$24/cord30021.351140023.456250025.260560026.764170028.067280029.169890030.1722

Example 3: ContinuedFixed costs per acre:Land . . . . . . . $300Site prep . . . . 100Annual . . . . 60Set-up . . . . . 5 Total . . . . 465Variable costs per 100 seedlingsSeedlings . . . . $ 5Planting . . . . 20 Total . . . . 25TC = 465 + 25 x (# trees per A/100)

Example 3: ContinuedSee Excel spreadsheet