Question

Consider a Bertrand oligopoly consisting of four firms that produce an identical product at a marginal cost of $$260$. The inverse market demand for this product is $P=800‐4Q$

a. Determine the equilibrium level of output in the market.

b. Determine the equilibrium market price.

c. Determine the profits of each firm$1$

Short answer

a. The equilibrium level of the output in the market is 135 units.

b. The equilibrium market price is $260.

c. The profit of each firm is zero.

Step-by-step solution

Given information

Bertrand's oligopoly model is an alternative to Cournot's model, which is characterized as a simultaneous game where the strategic choice is based on the price rather than the quantity.

In this case, there are four firms in the market that produce the same product at a marginal cost of $\cancel{S}260$, and they have the following inverse demand.

Bertrand's inverse demand:

$P=800‐4Q$

Determining the equilibrium level of the output in the market

a.

The equilibrium level of the output in the market occurs when the price is equal to the marginal cost. If it produces below the marginal cost, it will generate losses. If it produces above the marginal cost, it will decrease its sales as the products are homogeneous.

Therefore, the Bertrand condition establishes that the following condition must be fulfilled to obtain the optimal output level.

$P=MC$

Substituting and solving for Q, we get

.$$\begin{gathered} 800‐4Q=2604Q \\ 4Q=800‐260Q \\ Q=540/4 \\ =135 \end{gathered}$$

Hence, the equilibrium level of the output in the market is 135 units.

Determining the equilibrium market price

b.

To obtain the equilibrium market price, the output level obtained in Exercise (a) should be substituted in the inverse demand function of the oligopoly.

$$\begin{gathered} P=800‐4Q \\ P=800‐4\left(135\right) \\ P=800‐540 \\ P=260 \end{gathered}$$

Hence, the equilibrium market price is $260.

Determining the profit of each firm

As the marginal costs are equal to the prices in this model, all four firms will obtain the same level of profit. To calculate the profit, the following way is required.

$$\begin{gathered} Benefit\left(\right)=TotalRevenue‐TotalCost \\ Benefit\left(\right)=Price‵Quantity‐\left(MarginalCost\right)Q \\ =260\left(135\right)‐260\left(135\right) \\ =35,100‐35,100 \\ \left(P\right)=0 \end{gathered}$$

If the result were greater than 0, it would be divided by $4$to obtain the unit value of the profit for each firm.