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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