A duopoly face market demand Q= 100 - P. The marginal cost of each firm is 40 and fixed costs are zero.
a) calculate optimal profits
To calculate the optimal profits of each firm in a duopoly, we need to determine the optimal level of output and price for each firm. This can be achieved using the profit-maximizing condition, which states that firms should produce where marginal cost (MC) equals marginal revenue (MR).
Given:
Market demand: Q = 100 - P
Marginal cost: MC = 40
Fixed costs: FC = 0
Step 1: Calculate market price
To determine the market price, we need to find the quantity demanded at the equilibrium. At equilibrium, the total quantity supplied by both firms will equal the market demand.
Q1 + Q2 = 100 - P (Equation 1)
Step 2: Calculate firm-level quantities
Each firm faces the same market demand, which means they have half of the total market demand. Therefore, we can determine each firm's quantity as:
Q1 = Q2 = (Q1 + Q2)/2 (Equation 2)
Step 3: Calculate marginal revenue
To calculate marginal revenue (MR), we need to find the derivative of the market demand function with respect to quantity and multiply it by the market price.
MR = (dQ/dP) * P
Differentiating the market demand equation: Q = 100 - P
dQ/dP = -1
Substituting dQ/dP = -1 and P = (Q1 + Q2)/2 into the MR equation, we get:
MR = (-1) * (Q1 + Q2)/2
Step 4: Set MC = MR to find the optimal quantity
Since both firms have the same marginal cost (MC = 40) and the same market price (P), we can set MC equal to MR:
40 = (-1) * (Q1 + Q2)/2
Solving this equation, we get:
Q1 + Q2 = -80
Step 5: Substitute the optimal quantity into the market demand equation to find the market price
From Equation 1, we know that:
Q1 + Q2 = 100 - P
Substituting Q1 + Q2 = -80, we get:
-80 = 100 - P
Rearranging the equation, we find:
P = 100 + 80
P = 180
Step 6: Calculate the optimal quantity for each firm
Using Equation 2, we know that:
Q1 = Q2 = (Q1 + Q2)/2 = -80/2
Q1 = Q2 = -40
Step 7: Calculate the optimal profits for each firm
To calculate the profits, we need to subtract the total costs from the total revenue.
Total revenue (TR) = Price * Quantity
Total costs (TC) = MC * Quantity
For each firm:
TR = P * Q
TC = MC * Q
Substituting the values:
TR1 = P * Q1 = 180 * (-40) = -7200
TR2 = P * Q2 = 180 * (-40) = -7200
TC1 = MC * Q1 = 40 * (-40) = -1600
TC2 = MC * Q2 = 40 * (-40) = -1600
Profit (π) = TR - TC
π1 = TR1 - TC1 = (-7200) - (-1600) = -5600
π2 = TR2 - TC2 = (-7200) - (-1600) = -5600
Therefore, the optimal profits for each firm are -5600.