Consider a Cournot oligopoly consisting of four identical firms producing good X.If the firms produce good X at a marginal cost of $7 per unit and the market elasticity of demand is −2,determine the profit-maximizing price.
A) $6 per unit
B) $8 per unit
C) $10 per unit
D) $12 per unit

Question

Quizplus · Accepted Answer

To determine the profit-maximizing price in a Cournot oligopoly, we need to use the following formula:

P = MC / (1 + elasticity of demand * (total industry output / individual firm's output))

In this case, we know that:
- Marginal cost (MC) = $7 per unit
- Elasticity of demand = -2
- There are four identical firms, so each firm's output is 1/4 of the total industry output

Plugging these values into the formula, we get:

P = 7 / (1 - 2 * (1/4))
P = 7 / (1/2)
P = 14

Therefore, the profit-maximizing price would be $14 per unit. However, we need to check if this price is feasible, given the cost structure and demand conditions. Since the marginal cost is $7 per unit, a price of $14 would result in a loss of $7 per unit. This means that we need to lower the price to increase profit.

The next step is to consider how the other firms will react to this price. In a Cournot oligopoly, each firm assumes that its rivals' outputs will remain constant when it sets its own output level. This means that each firm needs to consider the effect of its own output on the market price.

Assuming that each firm produces the same amount (1/4 of the total industry output), the total industry output will be 4 * (1/4) = 1. Therefore, the formula for the profit-maximizing price becomes:

P = 7 / (1 - 2 * (1))
P = 7 / (-1)
P = -7

This price is negative, which doesn't make sense in the context of the problem. It indicates that the demand is not strong enough to support a price that covers the firms' costs. Therefore, each firm will need to lower its output to increase the market price.

Assuming that each firm reduces its output by half (to 1/8 of the total industry output), the total industry output will be 4 * (1/8) = 1/2. The formula for the profit-maximizing price becomes:

P = 7 / (1 - 2 * (1/2))
P = 7 / (0)
P = undefined

This price is undefined, which also doesn't make sense. It indicates that the firms cannot collectively produce enough to satisfy demand, so the market price would be extremely high.

Given these results, we can conclude that the profit-maximizing price must be somewhere between $7 and $14 per unit. We can try different prices within this range to see which one yields the highest profit.

When we plug in a price of $8 per unit, we get:

P = 7 / (1 - 2 * (1))
P = 7 / (-1)
P = -7

This price is still negative, but we can assume that the firms will continue to reduce output until the price reaches a positive value.

Therefore, the most reasonable answer is B) $8 per unit, as it is within the feasible price range and it is a reasonable assumption that the firms will adjust their output to reach a positive market price.

Consider a Cournot Oligopoly Consisting of Four Identical Firms Producing