Question

Use the first-order condition (Equation 15.2 ) for a Cournot firm to show that the usual inverse elasticity rule from Chapter 11 holds under Cournot competition (where the elasticity is associated with an individual firm's residual demand, the demand left after all rivals sell their output on the market). Manipulate Equation 15.2 in a different way to obtain an equivalent version of the inverse elasticity rule: \\[ \frac{P-M C}{P}=-\frac{s_{i}}{e_{Q, P}} \\] where \(s_{i}=q_{i} / Q\) is firm i's market share and \(e_{Q, p}\) is the elasticity of market demand. Compare this version of the inverse elasticity rule with that for a monopolist from the previous chapter.

Short answer

Answer: The inverse elasticity rule for a Cournot firm can be written as: \\[ \frac{P-MC(q_i)}{P} = -\frac{s_i}{e_{Q,P}} \\] Here, \(P\) is the price, \(MC(q_i)\) is the marginal cost function for firm i, \(s_i\) is firm i's market share, and \(e_{Q,P}\) is the elasticity of market demand. This formula shows how the markup of a firm is related to its market share and the price elasticity of demand. Comparing this formula to the inverse elasticity rule for a monopolist: \\[ \frac{P-MC(q)}{P} = -\frac{1}{e_Q} \\] we can see that they have the same structure. The main difference lies in the fact that the Cournot firm's rule uses residual demand elasticity (\(e_{Q,P}\)) and the firm's market share (\(s_i\)), while the monopolist's rule uses the total demand elasticity (\(e_Q\)).

Step-by-step solution

Write down Equation 15.2 for the first-order condition of a Cournot firm

Given Equation 15.2 as the first-order condition for a Cournot firm: \\[ MR(q_i) = MC(q_i) \\] where \(MR(q_i)\) is the marginal revenue for firm i, and \(MC(q_i)\) is the marginal cost function for firm i.

Write the marginal revenue function as a function of the price, quantity, and market demand

The marginal revenue function for a Cournot firm can be written in terms of price, output of firm i (\(q_i\)), and elasticity of market demand: \\[ MR(q_i) = P\left(1+\frac{1}{e_{Q,P}}\right) \\]

Substitute Equation 2 into Equation 1 and solve for the inverse elasticity rule

Substituting the marginal revenue function equation into the first-order condition equation and then solving for the inverse elasticity rule give us: \\[ P\left(1+\frac{1}{e_{Q,P}}\right) = MC(q_i) \\] Now, divide both sides by price: \\[ 1+\frac{1}{e_{Q,P}} = \frac{MC(q_i)}{P} \\] Subtract one from both sides: \\[ \frac{1}{e_{Q,P}} = \frac{MC(q_i)-P}{P} \\] Now, we can define firm i's market share (\(s_i\)) as \(s_i=\frac{q_i}{Q}\), and the inverse elasticity rule becomes: \\[ \frac{P-MC(q_i)}{P} = -\frac{s_i}{e_{Q,P}} \\]

Compare this version of the inverse elasticity rule with that for a monopolist

In the case of a monopolist, the inverse elasticity rule developed in the previous chapters would be: \\[ \frac{P-MC(q)}{P} = -\frac{1}{e_Q} \\] where \(MC(q)\) is the monopolist's marginal cost and \(e_Q\) is the elasticity of demand. Comparing both the Cournot and the monopolist inverse elasticity rules, we can see that they have the same structure, with the only difference being that for Cournot competition, we use a firm's residual demand (\(e_{Q,P}\)) and its market share (\(s_i\)), while in the monopoly case we use total demand elasticity (\(e_Q\)).