(a) Explain the Cournot model of duopoly using reaction functions and interpret it as a Nash equilibrium. (20 marks) (b)…

(a) The Cournot model of duopoly as a Nash equilibrium

Two firms produce an identical product, choose quantities simultaneously and each takes the other's output as given. Let market demand be P = a − b(q₁ + q₂) and marginal cost c for both, with a > c.

Firm 1's profit is π₁ = [a − b(q₁ + q₂) − c]q₁. Setting ∂π₁/∂q₁ = a − 2bq₁ − bq₂ − c = 0 gives its reaction function q₁ = (a − c − bq₂)/2b. By symmetry, q₂ = (a − c − bq₁)/2b. Each reaction function shows the profit-maximising output for every possible output of the rival, and is downward sloping, since more output by one firm reduces the other's marginal revenue.

The equilibrium is where the two reaction functions intersect. Solving gives q₁* = q₂* = (a − c)/3b. Industry output is Q* = 2(a − c)/3b and price P* = (a + 2c)/3. Each firm earns (a − c)²/9b.

Nash interpretation: at (q₁*, q₂*) each output is the best response to the other, so neither firm can gain by changing its output alone. This is a pure-strategy Nash equilibrium of the simultaneous quantity game. It is stable, since the reaction functions have slopes of −1/2 (less than 1 in absolute value), so any departure converges back.

Comparison: Cournot output (2/3 of the competitive output (a − c)/b) is larger than monopoly output (a − c)/2b, so price lies between the monopoly and competitive prices. With n firms the output is n(a − c)/(n + 1)b, approaching the competitive result as n grows. The joint profit is below the collusive monopoly profit, which is why each firm is tempted to cheat on a cartel.

(b) "Perfect competition is incompatible with increasing returns to scale"

The statement is correct for increasing returns internal to the firm, but not if the increasing returns are external.

Internal increasing returns. Perfect competition requires many price-taking firms, each earning normal profit at P = LMC. If a firm has increasing returns at all outputs, LAC falls continuously and LMC < LAC, so a firm setting P = LMC makes a loss. A firm that expands lowers its cost and undercuts rivals, and the industry moves to a few large firms, or to natural monopoly. Competition cannot survive, and price must be regulated. So internal increasing returns across the relevant range contradict the many-firm structure.

Qualifications. First, returns to scale are increasing only up to the minimum efficient scale. If that scale is small relative to the market, many firms can operate at the bottom of a U-shaped LAC curve, and competition is consistent. Second, external economies (Marshall) lower costs for all firms as the industry grows, without giving any single firm an advantage, so competition and a falling long-run industry supply curve coexist. Third, with product differentiation, increasing returns are compatible with monopolistic competition; and contestable-market theory suggests that threat of entry can discipline even a natural monopolist.

So the conclusion holds for internal increasing returns in a homogeneous product market, and fails for external economies and limited-range scale economies.

(c) A model of oligopoly explaining price stickiness: the kinked demand curve

Paul Sweezy's kinked demand curve model explains why prices in an oligopoly tend to be rigid. A firm assumes its rivals react asymmetrically. If it raises its price, rivals will not follow, so it loses many customers and demand above the current price is highly elastic. If it cuts price, rivals will match the cut to protect their shares, so it gains few customers and demand below the current price is inelastic. The demand curve therefore has a kink at the prevailing price P₀ and output Q₀.

The kinked demand curve gives a marginal revenue curve with a vertical gap directly below the kink, because the two segments have different slopes. As long as the marginal cost curve passes through this gap, the profit-maximising output stays at Q₀ and price at P₀, even if costs change within the range. Hence the price is sticky, and firms prefer non-price competition such as advertising and service.

Limitations: the model does not explain how the original price is set, and empirical studies such as Stigler's found that prices are sometimes changed when costs change, particularly in a recession or under price leadership. It remains useful as a description of price rigidity under interdependence.