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 ormax TR - TC => Q* = Q3.

Profit Maximization for per Unit Measures

T is maximized:• At Q where MR = MC.

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*

0MCMRdQ

dTCdQ

dTRdQdTM

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 AnalysisApproximation in absence of detailed data on revenue & cost.

Assume both TR & TC are linear.

At the Break-even: TR = TC = TVC + TFCP*QBE = AVC*QBE + TFC(P – AVC)*QBE = TFCQBE = TFC / (P – AVC)

P = $6, AVC = $3.6, TFC = $60KQBE = 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