Question

Suppose there are \(n\) identical firms in a Cournot equilibrium. Show that the absolute value of the elasticity of the market demand curve must be greater than \(1 / n\). (Hint: in the case of a monopolist, \(n=1\), and this simply says that a monopolist operates at an elastic part of the demand curve. Apply the logic that we used to establish that fact to this problem.)

Short answer

The market demand elasticity \( |\varepsilon| \) must be greater than \( 1/n \) in a Cournot equilibrium with \( n \) identical firms.

Step-by-step solution

Understand the Cournot Model

In a Cournot equilibrium, each firm decides its output level assuming the output of the other firms remains constant. The market price is determined by the total output of all firms. For this exercise, we consider there are \( n \) identical firms.

Apply Market Demand Elasticity

The elasticity of the market demand curve \( \varepsilon \) is defined as \( \varepsilon = \frac{dQ}{dP} \cdot \frac{P}{Q} \), where \( Q \) is the total quantity and \( P \) the price. The absolute value of \( \varepsilon \) must exceed \( \frac{1}{n} \) for a Cournot equilibrium to sustain.

Relate to a Single Firm's Output Decision

For one of the firms in Cournot equilibrium, the marginal revenue should be equal to the marginal cost. When considering the effect of its output on the market, the firm perceives an effective demand elasticity \( n \varepsilon \) (since there are \( n \) firms), resulting in a perceived marginal revenue \( MR = P(1 - \frac{1}{n} \varepsilon) \).

Establish Condition for Profit Maximization

For profit maximization, the firm sets its marginal cost equal to its perceived marginal revenue. If the elasticity \( \varepsilon \) were \( \leq \frac{1}{n} \), the perceived marginal revenue would become negative or zero, making production unprofitable. Thus, as long as the firm is active in the market, \( |\varepsilon| > \frac{1}{n} \).

Conclusion

Thus, in a Cournot equilibrium with \( n \) identical firms, the condition \( |\varepsilon| > \frac{1}{n} \) ensures that each firm operates where marginal cost equals marginal revenue, maintaining profitability.

Key concepts

Market Demand Elasticity

Market Demand Elasticity is a crucial concept in economics, especially in the context of a Cournot Equilibrium. It describes how sensitive the quantity demanded is to a change in price. In mathematical terms, elasticity \( \varepsilon \) can be expressed as:
\[ \varepsilon = \frac{dQ}{dP} \cdot \frac{P}{Q} \]
Here, \( Q \) represents the total quantity demanded, and \( P \) is the price level.

• Elastic demand means consumers are very responsive to price changes.
• Inelastic demand indicates consumers are not very responsive.

In a Cournot equilibrium, where multiple firms decide on their output based on others' behavior, each firm must consider this elasticity. While a monopolist operates where demand is elastic, in a market with \( n \) firms, the total market demand elasticity must be greater than the reciprocal of the number of firms, or \( |\varepsilon| > \frac{1}{n} \). This ensures firms remain profitable despite competitive pressures.
The rationale here is that if elasticity were less than or equal to \( 1/n \), the perceived revenue from producing additional output would not cover costs, driving firms to adjust their production strategy.

Marginal Cost

Marginal Cost (MC) is the cost of producing one more unit of a good. For a firm, it represents the extra expense incurred when increasing production by a small amount. It's a critical factor in determining pricing and output levels.

• To calculate MC, consider the change in total cost when one additional unit is produced.
• It helps managers decide the optimal level of production to maximize profit.

In a Cournot equilibrium, firms align their production decisions such that the marginal cost equals the marginal revenue (MR). Since firms assume competitors' outputs remain constant, the output decision is heavily influenced by MC.
For instance, if a firm's MC is lower than the price at which they can sell additional units, they are encouraged to produce more. However, the concept becomes complex in a Cournot setting as each firm's perceived demand elasticity, impacted by all firms, becomes \( n\varepsilon \). Therefore, the condition for sustainability and profitability within these competitive settings is that the absolute market demand elasticity \( |\varepsilon| \) must exceed \( 1/n \).
This ensures that increasing production will still yield a profitable price above marginal costs, driving output and revenue expansion.

Profit Maximization

Profit Maximization is the primary goal of firms in any economic model, including Cournot competition. It refers to achieving the highest possible profit with the resources available. This is where a firm's revenue exceeds its costs by the largest possible margin.

• The firm's objective is to find the level of output where marginal revenue (MR) equals marginal cost (MC).
• This balance ensures no additional unit of production will decrease profit.

In a Cournot equilibrium, this condition is met by each firm independently. However, because of competitive assumptions, each business perceives market demand differently. They consider the elasticity of market demand and the output of other firms.
For a Cournot firm, the MR is affected by its own output's effect on total market sales. It perceives a demand elasticity \( n \cdot \varepsilon \), leading them to adjust production such that MC equals their perceived MR which is \( P(1 - \frac{1}{n}\varepsilon ) \). This is sustainable only when \( |\varepsilon| > \frac{1}{n} \), ensuring competition does not drive profits to zero. Maintaining this elasticity allows firms to sustainably increase output until profit conditions are maximized without incurring losses.