In perfectly competitive market,given cost function: C=1/3q3-5q2+30q+10 and market clearing price be 6,obtain profit maximising level of output
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To obtain the profit maximizing level of output in a perfectly competitive market, we need to determine the quantity at which marginal cost equals the market price.
In this case, the cost function is given as C = (1/3)q^3 - 5q^2 + 30q + 10, and the market clearing price is given as 6.
To find the marginal cost (MC), we need to take the derivative of the cost function with respect to quantity (q).
MC = dC/dq = d/dq[(1/3)q^3 - 5q^2 + 30q + 10]
= q^2 - 10q + 30
Now we can set the marginal cost equal to the market price to find the profit maximizing level of output:
6 = q^2 - 10q + 30
Rearranging the equation, we have:
q^2 - 10q + 24 = 0
To solve this quadratic equation, we can factor it:
(q - 4)(q - 6) = 0
Setting each factor equal to zero, we get:
q - 4 = 0 or q - 6 = 0
Solving for q, we find two potential output levels:
q = 4 or q = 6
Since the profit maximizing level of output is the one at which marginal cost equals the market price, we can choose q = 6 as the profit maximizing level of output in this case.
Therefore, the profit maximizing level of output is 6 units.