iven 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 pr
To maximize total revenue and profit, we need to find the level of output (Q) that corresponds to those objectives.
a) Maximizing Total Revenue:
Total Revenue (TR) is calculated by multiplying the price (P) by the quantity (Q): TR = P * Q.
Given the demand function P = 24 - 0.5Q, we can substitute this equation into the total revenue equation to obtain: TR = (24 - 0.5Q) * Q.
To find the level of output that maximizes total revenue, we need to determine the value of Q that yields the highest total revenue. This occurs when the derivative of total revenue (dTR/dQ) with respect to Q is equal to zero.
Let's calculate the derivative of TR with respect to Q:
dTR/dQ = 24Q - 0.5Q^2.
To find the value of Q that maximizes TR, we need to set dTR/dQ equal to zero and solve for Q:
24Q - 0.5Q^2 = 0
0.5Q^2 - 24Q = 0
Q(0.5Q - 24) = 0
From this equation, we have two possible solutions:
Q = 0 (which is not meaningful in this context)
0.5Q - 24 = 0
0.5Q = 24
Q = 48
Therefore, to maximize total revenue, the firm should produce an output level of Q = 48.
b) Maximizing Profit:
Profit (π) is calculated by subtracting the total cost (TC) from the total revenue (TR): π = TR - TC.
To maximize profit, we need to find the level of output (Q) that maximizes the difference between total revenue and total cost. This occurs when the derivative of profit (dπ/dQ) with respect to Q is equal to zero.
The total cost function is given as AC = Q^2 - 8Q + 36 + 3/Q. Integrating this equation with respect to Q gives us the Total Cost (TC) function.
Let's differentiate the profit equation (π = TR - TC) with respect to Q to find the value of Q that maximizes profit:
dπ/dQ = dTR/dQ - dTC/dQ
We already have the derivative of TR with respect to Q (dTR/dQ = 24Q - 0.5Q^2). To find the derivative of TC with respect to Q (dTC/dQ), we differentiate AC with respect to Q.
dTC/dQ = 2Q - 8 - 3/Q^2
Now, let's set dπ/dQ equal to zero and solve for Q:
24Q - 0.5Q^2 - (2Q - 8 - 3/Q^2) = 0
-0.5Q^2 + 22Q - 3/Q^2 - 8 = 0
To solve this equation, we can use numerical approximation methods, such as the Newton-Raphson method, to find the value of Q that maximizes profit.
Note: The numerical approximation for maximizing profit is beyond the scope of this explanation.