Question

Cournot Oligopoly and Number of Firms In a Cournot oligopoly, each firm assumes that its rivals do not change their output based on the output that it produces. Ilustration: A Cournot oligopoly has two firms, YandZ. Yobservesthe market demand curve and the number of units that Z produces. It assumes that Z does notchange its output regardless of the number of units that it (Y) produces, so chooses a production level that maximizes its profits. The general effects of a Cournot oligopoly do not depend on the size of the firms, the shape of the market demand curve, or the shape of the marginal cost curve. The mathematics is easiest for firms of the same size, linear demand curves, and flat marginal cost curves. Suppose an industry has two firms, a linear demand curve, and marginal costs, and no fixed costs: Demand curve: Q α-β P Marginal cost curve: MCk In a competitive industry, what is the equilibrium quantityfor the industry? (Setting price equal to marginal cost gives Q = α-β x k. Since the industry is competitive, price equals marginal cost, and the supply curve for the industry is P k; this gives the same result.) What is the equilibrium quantity for the firm? (With two identical firms, each produces half the industry quantity.) If the two firms merge into a monopoly, what is the monopoly price? (Show that the marginal revenue curve is MR a-2p P, by setting total revenue P x Qand differentiating with respect to Q. Setting marginal revenue equal to marginal cost gives A total of 2,400 units are produced. If there were three firms in this Cournot oligopoly, how many units would be produced? A. 1,800 units B. 2,400 units C. 2,700 units D. 3,000 units E. 3,600 units (In a two Cournot oligopoly, each firm produces Vs the competitive quantity; in a three fim Cournot oligopoly, each firm produces the competitive quantity.)

Answer 1

Case 1: Competitive Industry

In the scenario, where industry is competitive, firms are assumed to be price takers i.e. price is given as an exogenous quantity.

Total profit in industry = Total revenue - Total cost

$\pi = P(Q)Q - TC = PQ - MC.Q$   

(Since P is exogenous and there are no fixed costs) and Q is the industry output.

Profit maximization is given by -

$\frac{\partial \pi}{\partial Q} = 0 \Rightarrow P = MC$

Since MC = k, substituting this in demand curve, we get:

$Q = \alpha - \beta P = \alpha - \beta k$

This is the equilibrium quantity in the industry,

Since, both firms have equal size (same marginal cost), thus they have equal share in the market.

Hence, equilibrium quantity of the firm = Q/2

= $q = \frac{\alpha - \beta k}{2}$

Case 2: Monopolistic Industry

In the scenario,

Total profit in industry = Total revenue - Total cost

$\pi = P(Q)Q - TC$   

and Q is the industry output = firm output (monopoly)

Since $Q = \alpha - \beta P \Rightarrow \beta P = \alpha - Q \Rightarrow P = \frac{\alpha}{\beta} - \frac{Q}{\beta}$

Thus $\pi = P(Q)Q - TC = (\frac{\alpha}{\beta} - \frac{Q}{\beta})Q - kQ$

Profit maximization is given by -

$\frac{\partial \pi}{\partial Q} = 0 \Rightarrow MR = MC$where $MR = \frac{\partial TR}{\partial Q}$

$\frac{\partial \pi}{\partial Q} = 0 \Rightarrow$$\frac{\alpha}{\beta} - \frac{2Q}{\beta} = k$$\Rightarrow \frac{2Q}{\beta} = \frac{\alpha}{\beta} - k \Rightarrow 2Q = \alpha - \beta k \Rightarrow Q =$$\frac{\alpha - \beta k}{2}$

This is the equilibrium quantity in the industry,

Equilibrium price P is given by demand curve -

$P = \frac{\alpha}{\beta} - \frac{Q}{\beta} = \frac{\alpha}{\beta} - \frac{\alpha - \beta k}{2\beta} = \frac{\alpha + \beta k}{2 \beta}$

(Notice: Wrong answer given in the hints, MR is not given by derivative of TR w.r.t P but w.r.t Q)

Case 2: Cournot Tripoly

Quantity produced in competition = $q = \frac{\alpha - \beta k}{2}$

In Cournot model of 3 firms with equal marginal cost c,

q = $q_{cournot} = \frac{1}{4}q_{competitive} = \frac{1}{4}\frac{\alpha - \beta k}{2} = \frac{\alpha - \beta k}{8}$

Total output produced = 2400 (assumed to be arising from competition)

If it is a cournot tripoly, quantity produced by each firm = 2400 / 4 = 600

Thus total output produced by three firms = 600*3 = 1800

(Since each firm has same marginal costs, they hold equal share and produce equal quantity).