Profit maximisation under perfect competition and monopoly: formula sheet

Key formulas

Total revenue

TR = P × Q

Price times quantity sold.

Average revenue

AR = TR ÷ Q = P

Equals price, so AR is the demand curve for the firm.

Marginal revenue

MR = ΔTR ÷ ΔQ

For a price taker, MR = P. For a firm facing a downward-sloping demand curve, MR < P.

Total cost

TC = FC + VC

FC is constant in the short run. No fixed cost exists in the long run.

Average costs

AC = TC ÷ Q = AFC + AVC

AFC = FC ÷ Q and AVC = VC ÷ Q.

Marginal cost

MC = ΔTC ÷ ΔQ = ΔVC ÷ ΔQ

Fixed cost drops out because it does not change.

Economic profit

Economic profit = TR − TC, where TC includes implicit costs (normal profit)

Zero economic profit means the firm earns normal profit.

MC and average relationship

MC < AC ⇒ AC falls; MC > AC ⇒ AC rises; MC = AC at minimum AC

The same holds for AVC.

Profit

π = TR − TC

TR = P × Q. TC includes all costs, including normal profit in the usual economic treatment.

Marginal revenue and marginal cost

MR = dTR/dQ; MC = dTC/dQ

With discrete data, use the change in total divided by the change in quantity.

First order condition

dπ/dQ = 0, so MR = MC

Necessary, not sufficient. It also holds at a profit minimum.

Second order condition

d²π/dQ² < 0, so slope of MC > slope of MR

MC cuts MR from below. For a price taker, this means MC is rising.

Profit per unit

Average profit = P − AC; total profit = (P − AC) × Q

Use AC at the profit-maximising output, not at any other output.

Price taker

MR = P

So the rule becomes P = MC.

Price-taker revenue

AR = MR = P

Holds for a firm facing a horizontal demand curve. Market demand still slopes down.

Profit-maximising rule

MR = MC, so P = MC (MC rising)

The second-order condition is that MC cuts MR from below.

Profit

π = (P − ATC) × Q

Positive means supernormal profit, zero means normal profit, negative means a loss.

Shutdown rule (short run)

Produce if P ≥ min AVC; shut down if P < min AVC

At P = AVC the firm is indifferent, because it loses its fixed costs either way.

Short-run supply curve

Firm supply = MC curve above min AVC

Market supply is the horizontal sum of firms' supply curves.

Long-run equilibrium

P = MR = MC = min ATC

Zero economic profit, so no entry or exit.

Total revenue

TR = P × Q

For a monopolist, P depends on Q through the demand curve.

Marginal revenue

MR = d(TR)/dQ

Always below price for a downward-sloping demand curve.

MR for linear demand

If P = a − bQ, then TR = aQ − bQ² and MR = a − 2bQ

MR has the same intercept as demand and twice the slope.

Profit-maximising condition

MR = MC, with MC cutting MR from below

Use the second condition to reject any other crossing point.

MR and elasticity

MR = P × (1 − 1/|e|), where e is price elasticity of demand

MR > 0 only if |e| > 1. So a monopolist operates where demand is elastic.

Mark-up rule

(P − MC) ÷ P = 1 ÷ |e|

Follows from MR = MC. Lower elasticity means a bigger mark-up.

Profit

Profit = TR − TC = (P − AC) × Q

Supernormal if P > AC at the chosen output.

Profit-maximising rule

MR = MC (with MC cutting MR from below)

Applies to every firm. Only MR differs between the two structures.

Perfect competition condition

P = MR = MC

Price taker. Long-run equilibrium also has P = minimum AC and zero economic profit.

Monopoly condition

MR = MC, then P from demand, so P > MC

Output is set by MR = MC, price is read off the demand curve above that output.

Linear demand and MR

If P = a − bQ, then TR = aQ − bQ² and MR = a − 2bQ

MR has the same intercept and twice the slope. Valid for straight-line demand.

Deadweight loss (linear case)

DWL = ½ × (Q_c − Q_m) × (P_m − MC)

Valid for linear demand and constant MC. Q_c is the competitive output where P = MC. Q_m is the monopoly output.

Monopoly profit

π = (P − AC) × Q

Use AC at the monopoly output.

Perfect price discrimination

Each unit sold at the buyer's maximum willingness to pay; output where P = MC

Output equals the competitive level. The firm takes all surplus and DWL is zero.

Third-degree discrimination

MR₁ = MR₂ = MC

Set across separate markets. The market with less elastic demand gets the higher price.

Regulation benchmarks

Marginal cost price: P = MC. Average cost price: P = AC

For a natural monopoly, P = MC may give a loss, since MC < AC when AC is falling.

Quick revision

• Profit-maximising output is where MR = MC and MC cuts MR from below.
• Profit = TR − TC, or (P − AC) × Q.
• MR = change in TR ÷ change in Q; MC = change in TC ÷ change in Q.
• Perfect competition: many sellers, identical product, free entry and exit, price taker, so P = MR = AR.
• Perfectly competitive firm's short-run supply curve is its MC curve above minimum AVC.
• Shut down in the short run if P is below minimum AVC.
• Long-run competitive equilibrium: P = MC = minimum AC, with zero economic profit.
• Monopoly: one seller, barriers to entry, faces the downward-sloping market demand curve.
• For a straight-line demand curve, MR has the same intercept and twice the slope.
• Monopolist sets output at MR = MC, then charges the price on the demand curve at that output, so P > MC.
• A monopolist can earn economic profit in the long run because entry is blocked.
• Compared with perfect competition, monopoly usually gives higher price, lower output and a deadweight loss, unless large cost savings from scale exist.

Common mistakes

• Including fixed cost when calculating MC. Fix: Use MC = ΔTC ÷ ΔQ. Fixed cost does not change, so it cancels out.
• Saying MR = P for every firm. Fix: MR = P only for a price taker. If the firm must cut price to sell more, MR is below P.
• Stopping at MR = MC without checking that MC cuts MR from below. Fix: Always test the slope of MC against the slope of MR, or check that d²π/dQ² < 0. Write the check as a line in your answer.
• Using price as MR for a monopoly or any firm facing a downward-sloping demand curve. Fix: Derive TR = P × Q first, then differentiate. For a linear demand P = a − bQ, MR = a − 2bQ.
• Drawing a downward-sloping demand curve for the individual firm. Fix: Draw two side-by-side diagrams: the market with sloping curves, and the firm with a horizontal line at the market price.
• Saying a loss-making firm should always shut down immediately. Fix: Compare P with AVC, not ATC. If P ≥ AVC, the firm covers variable costs and part of fixed costs, so it continues in the short run.
• Reading the price off the MR curve at MR = MC. Fix: The MR = MC point only gives output. Go vertically up to the demand curve for the price.
• Using MR = P, as in perfect competition. Fix: For a monopolist, MR is below AR. Differentiate TR = P × Q with P depending on Q.
• Setting a monopolist's price equal to MR or reading price from the MR curve. Fix: After finding Q from MR = MC, always substitute Q into the demand curve to get P.
• Writing MR = P for a monopoly. Fix: For a monopoly, MR < P because a price cut applies to all units. For linear demand, MR has twice the slope.

Exam tips

• Show the formula before substituting. Examiners give marks for method even if the arithmetic slips.
• In MCQs, check the wording: MC uses change in total cost, not average cost.
• Label axes and mark where MC crosses AC and AVC at their minima in any diagram.
• State clearly whether you are answering for the short run or the long run.
• When asked about normal profit, say it is included in costs and means zero economic profit.
• Write the rule in full: MR = MC with MC cutting MR from below. Examiners give marks for the second condition.
• In diagram questions, label Q at the MR = MC intersection, then draw a vertical line to the demand curve for price and to AC for cost per unit. Shade the profit rectangle.
• For MCQs, check whether the firm is a price taker. If so, MR = P and the rule is P = MC. If not, derive MR from the demand curve.