Question
How is profit maximization calculated. And what formular is used
Answers
GPT-5 mini
Profit maximization is found by choosing the output (or price) that makes the difference between total revenue and total cost as large as possible.
Key formulas and steps
1) Define profit:
π(q) = TR(q) − C(q)
where TR(q) is total revenue as a function of quantity q, and C(q) is total cost.
2) First-order condition (FOC) — set marginal revenue equal to marginal cost:
dπ/dq = dTR/dq − dC/dq = MR(q) − MC(q) = 0 => MR = MC
- MR = dTR/dq (marginal revenue)
- MC = dC/dq (marginal cost)
For a price-taking firm (perfect competition), MR = P, so the condition simplifies to:
P = MC
For a monopolist with inverse demand P(Q), MR = P(Q) + Q·P'(Q), so the condition is:
P(Q) + Q·P'(Q) = MC
Using demand elasticity ε (ε = (dQ/dP)·P/Q, typically negative), the monopoly markup (Lerner index) is:
(P − MC)/P = −1/ε (or equivalently (P − MC)/P = 1/|ε|)
3) Second-order condition (SOC) — ensure a maximum:
d^2π/dq^2 = dMR/dq − dMC/dq < 0
Practical steps
- Write TR(q) and C(q).
- Compute MR and MC.
- Solve MR = MC for q.
- Check second derivative (or check neighboring values) to confirm a maximum.
- Compute profit π(q) at that q.
Quick numeric example
TR = 20q − q^2, C = 4q + 10
π = TR − C = 20q − q^2 − 4q − 10 = 16q − q^2 − 10
dπ/dq = 16 − 2q → set = 0 → q = 8
d^2π/dq^2 = −2 < 0 so it’s a maximum
Profit at q = 8: π = 16·8 − 8^2 − 10 = 54
If you want, tell me your TR and C (or demand and cost functions) and I’ll find the profit-maximizing output/price and profit.
Key formulas and steps
1) Define profit:
π(q) = TR(q) − C(q)
where TR(q) is total revenue as a function of quantity q, and C(q) is total cost.
2) First-order condition (FOC) — set marginal revenue equal to marginal cost:
dπ/dq = dTR/dq − dC/dq = MR(q) − MC(q) = 0 => MR = MC
- MR = dTR/dq (marginal revenue)
- MC = dC/dq (marginal cost)
For a price-taking firm (perfect competition), MR = P, so the condition simplifies to:
P = MC
For a monopolist with inverse demand P(Q), MR = P(Q) + Q·P'(Q), so the condition is:
P(Q) + Q·P'(Q) = MC
Using demand elasticity ε (ε = (dQ/dP)·P/Q, typically negative), the monopoly markup (Lerner index) is:
(P − MC)/P = −1/ε (or equivalently (P − MC)/P = 1/|ε|)
3) Second-order condition (SOC) — ensure a maximum:
d^2π/dq^2 = dMR/dq − dMC/dq < 0
Practical steps
- Write TR(q) and C(q).
- Compute MR and MC.
- Solve MR = MC for q.
- Check second derivative (or check neighboring values) to confirm a maximum.
- Compute profit π(q) at that q.
Quick numeric example
TR = 20q − q^2, C = 4q + 10
π = TR − C = 20q − q^2 − 4q − 10 = 16q − q^2 − 10
dπ/dq = 16 − 2q → set = 0 → q = 8
d^2π/dq^2 = −2 < 0 so it’s a maximum
Profit at q = 8: π = 16·8 − 8^2 − 10 = 54
If you want, tell me your TR and C (or demand and cost functions) and I’ll find the profit-maximizing output/price and profit.