Given a firm’s demand function, P = 24 - 0.5Q and the average cost function, AC = Q2 – 8Q + 36 + 3/Q, calculate the level of output Q which a) maximizes total revenue
b) maximizes profits
To calculate the level of output that maximizes total revenue and profits, we need to find the level of output where the derivative of the total revenue (TR) and the derivative of the total profit (π) with respect to Q are equal to zero.
a) To maximize total revenue, we need to find the level of output where the derivative of total revenue with respect to Q is equal to zero.
Total Revenue (TR) = Price (P) * Quantity (Q)
TR = P * Q
Given the demand function P = 24 - 0.5Q, we can substitute the value of P into the TR equation.
TR = (24 - 0.5Q) * Q
TR = 24Q - 0.5Q^2
To find the derivative of TR with respect to Q, we differentiate the equation with respect to Q:
dTR/dQ = 24 - Q
To maximize total revenue, we set dTR/dQ = 0 and solve for Q:
24 - Q = 0
Q = 24
Therefore, the level of output Q that maximizes total revenue is 24 units.
b) To maximize profits, we need to find the level of output where the derivative of total profit (π) with respect to Q is equal to zero.
Total Profit (π) = Total Revenue (TR) - Total Cost (TC)
π = TR - TC
Given that average cost (AC) is Q^2 – 8Q + 36 + 3/Q, we can find the total cost (TC) as follows:
Total Cost (TC) = Average Cost (AC) * Quantity (Q)
TC = AC * Q
Substituting the value of AC into the TC equation:
TC = (Q^2 – 8Q + 36 + 3/Q) * Q
TC = Q^3 – 8Q^2 + 36Q + 3
Now, let's find the derivative of π with respect to Q:
dπ/dQ = d(TR - TC)/dQ
dπ/dQ = (24Q - 0.5Q^2) - (Q^3 - 8Q^2 + 36Q + 3)
Simplifying the equation:
dπ/dQ = 24Q - 0.5Q^2 - Q^3 + 8Q^2 - 36Q - 3
To maximize profits, we set dπ/dQ = 0 and solve for Q:
24Q - 0.5Q^2 - Q^3 + 8Q^2 - 36Q - 3 = 0
Solving this equation may require the use of numerical techniques or computational software to find the value of Q that maximizes profits.