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)
$’s
TC
TR
Profit
Q1
Q
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
$’s
TC
MC
1 unit Q
FC
P
1 unit Q
Break-even
point
Q
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
Solution
Q(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
300
Yield in cords @
rotation
21.3
Total Revenue
@$24/cord
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