A firm in a perfectly competitive market has the following cost function: c=1/3q3-5q2+30q+10 in the market- clearing price is 6, obtain the profit maximising level of output?
To find the profit-maximizing level of output in a perfectly competitive market, the firm should produce the quantity at which marginal cost equals the market price. In this case, the market-clearing price is given as 6.
The marginal cost function can be obtained by taking the derivative of the total cost function with respect to quantity (q). Let's start by finding the derivative of the cost function:
c = (1/3)q^3 - 5q^2 + 30q + 10
To find the derivative, differentiate each term with respect to q:
c' = (1/3 * 3)q^(3-1) - 5 * 2q^(2-1) + 30 * 1q^(1-1) + 0
Simplifying the derivative results in:
c' = q^2 - 10q + 30
Now that we have the marginal cost function, we need to set it equal to the market price and solve for the quantity (q):
q^2 - 10q + 30 = 6
Rearranging the equation:
q^2 - 10q + 24 = 0
This is a quadratic equation that can be factored as:
(q - 4)(q - 6) = 0
Setting each factor equal to zero:
q - 4 = 0 or q - 6 = 0
Solving for q:
q = 4 or q = 6
Therefore, the profit-maximizing level of output is either 4 or 6 units, depending on economic conditions and other factors.