IAI Actuarial Core Principles · Business Economics

Profit maximisation under imperfect competition: formula sheet

Full chapter guide

Key formulas

Profit-maximising rule: MR = MC (with MC cutting MR from below); Applies in every market structure. What changes is how MR relates to price.
Price taker: P = AR = MR; Holds only under perfect competition, where the firm faces a horizontal demand curve.
Firm with market power: MR < P = AR; Holds for a firm facing a downward-sloping demand curve that sells at a single price.
Linear demand and MR: If P = a − bQ, then MR = a − 2bQ; The MR curve has the same intercept and twice the slope. Use it for monopoly-type calculations.
Lerner index of market power: L = (P − MC) ÷ P; Zero for perfect competition. Higher values mean more market power. For a profit-maximiser, L = 1 ÷ |elasticity of demand|.
Supernormal profit: (AR − AC) × Q; Positive when price exceeds average cost. Entry can remove it in the long run if barriers are low.
Total revenue: TR = P × Q; P is read from the demand curve at quantity Q.
Average revenue: AR = TR ÷ Q = P; The demand curve is the AR curve.
Marginal revenue (definition): MR = dTR/dQ (or ΔTR ÷ ΔQ for discrete changes); Use the derivative for a continuous demand function.
Linear demand and MR: If P = a − bQ, then TR = aQ − bQ² and MR = a − 2bQ; MR has the same intercept a and twice the slope. MR = 0 at Q = a ÷ (2b).
MR and elasticity: MR = P × (1 + 1/e) = P × (1 − 1/|e|); e is the price elasticity of demand, which is negative on a downward-sloping curve.
Point elasticity (linear demand): |e| = (P ÷ Q) × (1 ÷ b) = P ÷ (bQ); Equals 1 at the midpoint of the demand line, where Q = a ÷ (2b).
Perfect competition: MR = AR = P; Demand facing the firm is horizontal.
Profit-maximising rule: MR = MC, with MC cutting MR from below; Gives the output. Then find the price from the demand curve.
Total and marginal revenue: TR = P × Q; MR = dTR ÷ dQ; For discrete changes, MR = change in TR ÷ change in Q.
Linear demand and MR: If P = a − bQ, then TR = aQ − bQ² and MR = a − 2bQ; MR has the same intercept as demand and twice the slope.
Supernormal profit: Profit = (P − AC) × Q = TR − TC; AC is read at the profit-maximising output, not at the price.
Deadweight loss (linear curves): DWL = ½ × (Qc − Qm) × (Pm − MCm); Qc is the competitive output where P = MC. Qm and Pm are monopoly output and price. This formula applies when MC is constant or linear and the triangle is a true triangle.
Lerner index (MR = MC link): (P − MC) ÷ P = 1 ÷ |PED|; Follows from MR = P(1 − 1/|PED|). Shows a monopolist operates where demand is elastic.
Profit maximisation with one cost curve: MR₁ = MR₂ = MC (total output Q = Q₁ + Q₂); Third degree with two markets. MC is evaluated at total output, not per market.
Marginal revenue and elasticity: MR = P × (1 − 1/|ε|); Here ε is price elasticity of demand. Valid for a point on a demand curve.
Price ratio across markets: P₁ ÷ P₂ = (1 − 1/|ε₂|) ÷ (1 − 1/|ε₁|); Follows from MR₁ = MR₂. The market with the lower |ε| gets the higher price. Needs |ε| > 1 in both.
Linear demand and MR: If P = a − bQ, then MR = a − 2bQ; MR has the same intercept and twice the slope.
First degree outcome: P = willingness to pay for each unit; output where P = MC; consumer surplus = 0; Total surplus equals the perfectly competitive level.
Profit: π = P₁Q₁ + P₂Q₂ − C(Q₁ + Q₂); Compare with the single-price profit to show the gain.
Profit-maximising rule: MR = MC (with MC cutting MR from below); Set output here, then read price from the demand (AR) curve.
Profit per unit and total profit: Total profit = (P − ATC) × Q; Positive means supernormal profit, zero means normal profit, negative means a loss.
Long-run equilibrium condition: MR = MC and P = AR = ATC; Demand is tangent to ATC, so supernormal profit is zero.
Excess capacity: Excess capacity = Q at minimum ATC − Q at long-run equilibrium; Positive because the tangency lies on the falling part of ATC.
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 of demand.
Profit maximisation: MR = MC; Every firm still uses this rule. Under kinked demand, check whether MC lies in the gap of the MR curve.
Linear inverse demand (two firms): P = a − b(q₁ + q₂); Standard set-up for Cournot questions. Market quantity Q = q₁ + q₂.
Cournot best response (zero or constant MC = c): q₁ = (a − c − b·q₂) ÷ (2b); Comes from MR₁ = a − 2b·q₁ − b·q₂ set equal to c. Firm 2's response is the mirror image.
Cournot equilibrium, n identical firms: qᵢ = (a − c) ÷ (b(n + 1)); P = (a + n·c) ÷ (n + 1); For two firms, each produces one third of (a − c) ÷ b. Valid for linear demand and identical constant MC.
Bertrand equilibrium (identical products, same constant MC): P = MC; Each firm earns zero economic profit. The result needs identical goods and no capacity limits.
Nash equilibrium test: Each player's strategy is a best response to the other's strategy; In a matrix, mark each player's best payoff in response to each rival choice. A cell with both marks is a Nash equilibrium.
Cartel (joint monopoly) output: Set MR of total market = MC, then share output; Total profit is maximised, but each member is tempted to produce more than its quota.

Quick revision

Profit is maximised where MR = MC, provided MC cuts MR from below.
For a firm with market power, MR is less than price because selling more requires lowering the price on all units.
For linear demand P = a − bQ, MR = a − 2bQ: same intercept, twice the slope.
Find the price from the demand curve at the profit-maximising output, not from the MR curve.
A monopolist does not have a supply curve, because output depends on both demand and costs.
Price discrimination needs market power, separable markets or customers, and no resale between them.
In price discrimination across markets, set MR = MC in each market, so the market with less elastic demand gets the higher price.
In monopolistic competition, firms earn supernormal profit in the short run, but entry shifts demand until long-run profit is normal.
Oligopoly firms are interdependent, so each firm's best choice depends on its rivals' choices.
A Nash equilibrium is where no player can gain by changing strategy alone, given the others' strategies.
In the prisoner's dilemma, each player's dominant strategy can leave both worse off than cooperating.

Common mistakes

Saying a monopoly can charge any price it likes. Fix: A monopolist chooses price or output, not both. Higher price means lower quantity demanded. It sets output where MR = MC and reads price off demand.
Treating oligopoly and monopolistic competition as the same because both have differentiated products. Fix: Separate them by number of firms and interdependence. Monopolistic competition has many firms and free entry. Oligopoly has few firms, high barriers and strategic reactions.
Drawing the MR curve above the demand curve, or equal to it, for an imperfect competitor. Fix: MR = AR only when price does not fall with output. With downward-sloping demand, MR is always below AR.
Writing MR = a − bQ by differentiating P instead of TR. Fix: Always multiply by Q first to get TR, then differentiate. Check that the MR slope is 2b.
Reading the price from the MR curve at the MR = MC point. Fix: After finding Q, always go vertically up to the demand (AR) curve and read the price there.
Using MR = P for a monopolist. Fix: For a monopolist, P > MR because the price cut applies to all units. Derive MR from TR.
Setting MR in each market equal to the price of the other market, or setting prices equal. Fix: Equalise MR across markets and to MC. Prices will differ.
Substituting Q into the MR function to get the price. Fix: MR is used only to find Q. Always read price from the demand curve.
Reading the price off the MC curve instead of the demand curve. Fix: Go up from the MR = MC output to the demand (AR) curve. That point gives the price.
Saying long-run profit is zero in the sense of no profit at all. Fix: Say zero supernormal (abnormal) profit. The firm still earns normal profit, which is included in costs.

Exam tips

Always give the three tests: number of firms, product type and entry barriers. Examiners award marks for linking features to the structure.
In written answers, state that MR = MC applies to every structure and that only the shape of the demand curve changes.
For MCQs, watch conditions: P = MR is true only for price takers, and MR = a − 2bQ only for linear demand with a single price.
Name specific sources of market power rather than writing just 'barriers'. Use scale, legal rights, input control, brands and capital cost.
Do not rely on real-world examples alone. Describe the features, because the exam may use a hypothetical market.
For a written question, always show TR first, then differentiate. Marks are usually given for the method, not just the final MR.
Use the elasticity check MR = P(1 − 1/|e|) to verify your answer. It takes seconds and catches algebra errors.
In diagram questions, label demand (AR), MR, the MR = 0 quantity, and where |e| = 1. Mark the elastic and inelastic sections clearly.