Market Structure and Competitive Equilibrium

Cournot oligopoly equilibrium

A Cournot oligopoly is a market in which a small number of firms sell a homogeneous good and each firm chooses its output simultaneously, taking the outputs of its rivals as given. Each firm maximises its profit \pi_i = P(Q)q_i - C_i(q_i) with respect to its own q_i, which gives a best-response, or reaction, function. The Cournot-Nash equilibrium is the point where all reaction functions intersect. For linear inverse demand P = a - Q and n symmetric firms with constant marginal cost c < a, the first-order condition is a - Q - q - c = 0. Imposing symmetry gives q = \frac{a-c}{n+1}, Q = \frac{n(a-c)}{n+1} and P = \frac{a+nc}{n+1}. The price lies between marginal cost and the monopoly price: as n rises the price falls toward c, and when n = 1 it becomes the monopoly price \frac{a+c}{2}. When firms have different constant marginal costs, their reaction functions have different intercepts, so the lower-cost firm produces more and the higher-cost firm produces less. For a duopoly with P = a - Q, firm i produces q_i = \frac{a - 2c_i + c_j}{3} when this is positive. Fixed costs do not appear in the first-order conditions, so they change profit but not equilibrium output. With n firms, constant marginal cost c and P = a - bQ, total output is Q = \frac{n(a-c)}{b(n+1)} regardless of the fixed cost of each firm. Under incomplete information, a firm that does not know its rival's cost maximises expected profit. Its best response depends on the expected output of the rival, E[q_j], which is the probability-weighted average of the rival's type-specific best responses. In a Bayesian Nash equilibrium, each type of the informed rival then chooses its own best response to the uninformed firm's output.

Monopoly profit maximization

A monopolist maximises profit by choosing the output at which marginal revenue equals marginal cost, MR = MC, and then reads the price from the demand curve at that output. For linear inverse demand P = a - bQ, marginal revenue is MR = a - 2bQ, which lies below the demand curve. Marginal cost depends only on variable cost, so a fixed cost lowers profit but does not change the profit-maximising quantity. If the average variable cost is given as a function of output, total variable cost is AVC \times Q and marginal cost is its derivative with respect to Q. When the optimal quantity is stated, the condition MR = MC can be solved for an unknown demand parameter, and profit is then computed as PQ - C(Q). A monopolist's optimal price depends only on the objective being maximised. If a party earns a fixed share of revenue, its objective is that share, so the other party's costs are irrelevant to the decision, and the optimum is found by maximising the share of revenue. A multi-plant monopolist minimises the cost of producing a given total output by equalising marginal costs across plants, MC_X = MC_Y, subject to the output constraint. If this equality would require a negative output at one plant, the solution is a corner: all output is produced at the plant whose marginal cost is lower at the relevant output level, and the other plant produces nothing.

Long-run competitive equilibrium

In long-run perfect competition, free entry and exit drive economic profit to zero. Price therefore equals the minimum of long-run average cost, P = \min LRAC, and each firm produces at its efficient scale, where average cost is minimised. The minimum can be found by setting \frac{dAC}{dq} = 0 or by solving AC = MC. The number of firms is found by dividing market demand at the long-run price by the output of one firm, n = \frac{Q_d(P^*)}{q^*}. For example, if minimum average cost is 10 at q^* = 4, the price is 10, and market demand at that price is 400 units, then n = 100. If a demand-shifting variable such as average wealth enters the demand function, its value must be substituted into demand before the number of firms is computed. The long-run supply curve depends on how input costs respond to industry output. With external economies, each firm's cost curves shift down as total industry output expands, so the long-run supply curve slopes downward and the industry is a decreasing-cost industry. After an increase in demand, the new long-run equilibrium has a larger quantity and a price below the original price.

Bertrand duopoly equilibrium

In a Bertrand model, firms compete by choosing prices rather than quantities. With homogeneous products and identical constant marginal costs, the firm that sets the lower price captures the whole market, so each firm has an incentive to undercut its rival whenever price exceeds marginal cost. The only Nash equilibrium is therefore P = MC, at which each firm earns zero economic profit. This result is known as the Bertrand paradox because two firms are enough to drive price to the competitive level. With differentiated products, demand for each firm depends on both prices, for example q_i = a - bp_i + dp_j with b > 0 and d > 0. Firm i maximises (p_i - c)(a - bp_i + dp_j), and the first-order condition gives the best response p_i = \frac{a + bc + dp_j}{2b}. The symmetric Nash equilibrium is found by solving the two best-response equations together. Unlike the homogeneous case, the equilibrium price exceeds marginal cost when products are differentiated.

Linear demand and supply equilibrium

Market equilibrium occurs where quantity demanded equals quantity supplied, D(p) = S(p). For linear functions, this is a single linear equation in price, and the equilibrium quantity is found by substituting the equilibrium price into either function. Consumer surplus is the area between the demand curve and the price actually paid, measured over the quantity traded. For linear inverse demand P = a - bQ, consumer surplus at price P and quantity Q is the triangle \frac{1}{2}(a - P)Q. A price ceiling set below the equilibrium price creates a shortage, and the quantity traded is limited to the smaller of quantity demanded and quantity supplied at the ceiling price, Q = \min(D(\bar p), S(\bar p)). Consumer surplus must be computed with the price actually paid, not the equilibrium price. A specific tax t on buyers shifts the effective demand price: the buyer pays p_b = p_s + t, where p_s is the price received by sellers. The quantity traded falls to the level where the buyer price and seller price both satisfy their curves. The decrease in consumer surplus is the difference between the consumer surplus before and after the tax, computed with the buyer price.

Monopoly total revenue maximization

Total revenue is TR = PQ. Maximising total revenue requires setting marginal revenue equal to zero, \frac{dTR}{dQ} = MR = 0, which is the point where demand has unit price elasticity. For linear inverse demand P = a - bQ, total revenue is TR = aQ - bQ^2, which is maximised at Q = \frac{a}{2b} and P = \frac{a}{2}, giving TR_{max} = \frac{a^2}{4b}. The maximum total revenue is therefore a fixed number that depends on the parameters of demand, and any claim that total revenue can exceed it can be checked by comparing with this value. A monopolist with zero cost would choose this point, but a monopolist with positive cost chooses output where MR = MC, which lies at a lower output than \frac{a}{2b}.