Assuming a profit function as p=90-2q and the cost function as c=10+0.5q2.find the profile maximization output and price
To find the output level and price that maximize profit, we need to determine the quantity (output) at which the profit function reaches its maximum value.
The profit function is given by p(q) = 90 - 2q, and the cost function is given by c(q) = 10 + 0.5q^2.
The profit (π) is calculated as follows: π(q) = p(q) - c(q).
Substituting the profit and cost functions into the profit equation, we have:
π(q) = (90 - 2q) - (10 + 0.5q^2)
= 90 - 2q - 10 - 0.5q^2
= 80 - 2q - 0.5q^2
To find the maximum profit, we need to take the derivative of the profit function with respect to q and set it equal to zero.
dπ(q)/dq = -2 - q = 0
Solving for q:
-2 - q = 0
q = -2
As we obtain a negative value for q, it means there is no maximum profit level. This indicates that the profit function does not have a maximum point.
However, we can still find the corresponding price at a given quantity using the demand function p(q) = 90 - 2q.
Substituting q = -2 into the demand function:
p(-2) = 90 - 2 * (-2)
= 90 + 4
= 94
Therefore, at a quantity of -2 (which does not maximize profit), the corresponding price is 94.