a monopoly firm is faced with the following demand functio P=13-0.5Q the marginal cost function is given by 3+4Q and a fixed cost is 4 determine the profit maximization formula
To determine the profit maximization formula for a monopoly firm, we need to find the quantity and price at which the firm maximizes its profit. This can be done by equating the firm's marginal cost (MC) to the marginal revenue (MR) and solving for the quantity (Q).
1. Start by finding the marginal revenue (MR) function. In the case of a monopoly, the MR is given by the derivative of the total revenue (TR) with respect to quantity (Q). TR is obtained by multiplying the price (P) by the quantity (Q):
TR = P * Q
So, the expression for TR is: TR = (13 - 0.5Q) * Q
To find the MR, take the derivative of TR with respect to Q:
MR = d(TR) / d(Q)
= d((13 - 0.5Q) * Q) / d(Q)
= 13 - Q
2. Set the MR equal to the MC to find the profit-maximizing quantity:
MR = MC
13 - Q = 3 + 4Q
Simplify the equation:
13 - 3 = 4Q + Q
10 = 5Q
Q = 2
So, the profit-maximizing quantity is Q = 2.
3. Substitute this quantity back into the demand function to find the corresponding price:
P = 13 - 0.5Q
P = 13 - 0.5(2)
P = 13 - 1
P = 12
Therefore, the profit-maximizing price is P = 12.
4. Now, we can calculate the profit. Profit (π) is given by the difference between total revenue (TR) and total cost (TC):
TR = P * Q
= 12 * 2
= 24
TC = FC + VC
= 4 + (3 + 4Q)
= 4 + (3 + 4(2))
= 4 + 3 + 8
= 15
π = TR - TC
= 24 - 15
= 9
Therefore, the profit-maximizing formula for this monopoly firm is: Profit (π) = 9.