A manufacturing firm produces output using a single plant. The relevant cost function is TC=45,000+8Q squared and the demand function is Q=100-0.02P
a)What is the minimum level of AC
b)What is the firms profit when AC is at a minimum?
c)What is the firms maximum profit
To find the minimum level of average cost (AC), we need to first find the expression for AC and then determine the output level that minimizes it.
The average cost (AC) is calculated by dividing the total cost (TC) by the quantity produced (Q). Therefore, we can express the AC function as AC = TC / Q.
Given that the cost function is TC = 45,000 + 8Q^2, we substitute this into the AC expression:
AC = (45,000 + 8Q^2) / Q
To find the minimum level of AC, we need to find the value of Q that minimizes this expression.
To do this, we can take the derivative of AC with respect to Q and set it equal to zero. So, let's find the derivative:
d(AC)/dQ = [d(45,000)/dQ + d(8Q^2)/dQ] / Q
= [0 + 16Q] / Q
= 16
Setting the derivative equal to zero:
16 = 0
Since this equation has no solution, it suggests that the average cost does not have a minimum value.
Therefore, there is no minimum level of average cost (AC) for this manufacturing firm.
Moving on to the next part of the question:
b) To calculate the firm's profit when AC is at a minimum, we need to know the expression for profit.
Profit (π) is determined by subtracting total cost (TC) from the total revenue (TR). The total revenue is calculated by multiplying the quantity produced (Q) by the price (P):
TR = Q * P
Given the demand function Q = 100 - 0.02P, we can substitute it into the TR expression:
TR = Q * P
= (100 - 0.02P) * P
= 100P - 0.02P^2
Now, we can express the profit function (π) as:
π = TR - TC
= (100P - 0.02P^2) - (45,000 + 8Q^2)
Since we don't have the value of P, we can't find the exact profit when AC is at a minimum. We need the price (P) to calculate it.
c) Finally, to find the firm's maximum profit, we need to find the output level (Q) that maximizes the profit function (π).
To do this, we can take the derivative of π with respect to Q and set it equal to zero. However, since we don't have the price (P), we can't calculate the maximum profit either.
In summary:
a) The minimum level of average cost (AC) does not exist for this manufacturing firm.
b) The firm's profit when AC is at a minimum can't be determined without knowing the price (P).
c) The firm's maximum profit can't be determined without knowing the price (P).