Bertrand duopolists,Firm 1 and Firm 2,face inverse market demand and both have marginal cost, The equilibrium output this market will be

Question

Bertrand duopolists,Firm 1 and Firm 2,face inverse market demand  and both have marginal cost,  The equilibrium output this market will be

Quizplus · Accepted Answer

To find the equilibrium output in a Bertrand duopoly, we need to determine the point where both firms have the same price and any deviating firm will lead to losses. 
Using the equation for marginal cost, we have:
MC1 = MC2 = c
Setting the price equal to marginal cost gives:
p = MC = c
Substituting the inverse demand function with this price gives:
`11 - q1 - q2 = c`
Rearranging:
`q1 + q2 = 11 - c`
Substituting MC, we get:
`q1 + q2 = 11 - c = 11 - MC`
Plugging in the values for MC, we get:
`q1 + q2 = 11 - 11 = 0`
Since both firms produce positive quantities, this is a contradiction.
The only way to avoid this contradiction is for both firms to produce at maximum capacity. Thus, the equilibrium output is:
q1 = q2 = (11 - MC) / 2 = (11 - 11) / 2 = 0
q1 + q2 = 0 + 0 = 0
However, this is not a feasible solution since it implies that the market demand is not met. Therefore, we need to look for the next best option where both firms have positive output.
At any positive output level, a firm can always set a lower price and capture the entire market. Thus, the only stable solution is for both firms to produce at maximum capacity, that is, q1 = q2 = 30.

Bertrand Duopolists,Firm 1 and Firm 2,face Inverse Market Demand