Profit Analysis of the Firm
Profit Maximization for Total Measures
T is maximized:
• Where the slope of T is 0
(TR and TC are parallel
or their slopes are equal).
dT / dQ = M = 0
2 such points (Q1, Q3) require:
2. d2T / dQ2 is negative or
max TR - TC => Q* = Q3.
Profit Maximization for per Unit Measures
T is maximized:
• At Q where MR = MC.
dT dTR dTC
M 


 MR  MC  0
dQ dQ dQ
2 such points require:
•
MR < MC for any Q > Q* = Q3
(Q* is one of FONC candidates)
or when MC is increasing.
T = [(TR – TC)/Q]Q
= (AR – AC)Q = (P – AC)Q
Max T = area of the rectangle
= (AR|Q* - AC|Q*)Q*
= (P|Q* - AC|Q*)Q*
A Numerical Example
• Given estimates of
• P = 10 - Q
• C(Q) = 6 + 2Q
• Optimal output?
• MR = 10 - 2Q = 2 = MC
• Q = 4 units
• Optimal price?
• P = 10 - (4) = $6
• Maximum profits?
• PQ - C(Q) = 6(4) - (6 + 8) = $10
Shut-Down Point
• In the long run all cost must be recovered.
• In the short run fixed cost incurred before
production begins and do not change regardless
of the level of production (even for Q = 0).
• Shut down only if: –TFC > max T
(total)
P < AVC
(per unit).
• TFC = AFC*Q = (SAC – AVC)*Q
• Operate with loss if: max T > –TFC (total)
SAC > P  AVC (per unit).
– This is the third T maximizing condition.
Break-Even Analysis
Approximation in absence of
detailed data on revenue & cost.
Assume both TR & TC are linear.
At the Break-even:
TR = TC = TVC + TFC
P*QBE = AVC*QBE + TFC
(P – AVC)*QBE = TFC
QBE = TFC / (P – AVC)
P = $6, AVC = $3.6, TFC = $60K
QBE = 60,000 / (6 – 3.6)
QBE = $25,000
(P – AVC) unit contribution margin.
1 – P/AVC contribution margin ratio
(fraction of P to recover TFC)
Types of Business Analysis
• Profit Maximization
– Requires complete knowledge of
Revenue and Cost Functions.
• Break-Even Analysis
– Simplified profit maximization
analysis with limited applications
• Incremental Profit Analysis
– Variation of profit maximization
analysis used to evaluate proposed
projects by comparing incremental
revenues and cost associated with project