2. 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 corresponding quantity of output that achieves this optimal level.
a) To maximize total revenue, we need to find the quantity of output at which the product of price and quantity is maximized. Total revenue can be calculated as the product of price (P) and quantity (Q). So, total revenue (TR) = P * Q.
Given the demand function P = 24 - 0.5Q, we can substitute this into the total revenue equation to get TR = (24 - 0.5Q) * Q.
To find the quantity that maximizes total revenue, we differentiate TR with respect to Q and set it equal to zero.
d(TR)/dQ = 24 - Q - 0.5Q = 24 - 1.5Q
Setting this derivative equal to zero and solving for Q:
24 - 1.5Q = 0
1.5Q = 24
Q = 24 / 1.5
Q = 16
Therefore, the level of output that maximizes total revenue is Q = 16.
b) To maximize profits, we need to find the level of output at which the difference between total revenue (TR) and total cost (TC) is maximized. Profit (π) is given by the equation: π = TR - TC.
Given the demand function P = 24 - 0.5Q and the average cost function AC = Q^2 - 8Q + 36 + 3/Q, we can substitute these into the profit equation to get:
π = (24 - 0.5Q) * Q - (Q^2 - 8Q + 36 + 3/Q) * Q
Simplifying the equation, we get:
π = 24Q - 0.5Q^2 - Q^3 + 8Q^2 - 36Q - 3Q + 36
To maximize profits, we differentiate π with respect to Q and set it equal to zero.
d(π)/dQ = 24 - Q - 3Q^2 + 16Q - 36 - 3 = 0
-3Q^2 + 15Q - 15 = 0
Next, we solve this quadratic equation to find the value(s) of Q that maximize profits. Using the quadratic formula:
Q = (-b ± √(b^2 - 4ac)) / 2a
After substituting the values, we get:
Q = (-15 ± √(15^2 - 4(-3)(-15))) / (2(-3))
Q = (-15 ± √(225 + 180)) / -6
Q = (-15 ± √405) / -6
Simplifying further, we get:
Q = (-15 ± 20.12) / -6
There are two possible solutions:
1. Q = (-15 + 20.12) / -6 = -5.12 / -6 ≈ 0.85
2. Q = (-15 - 20.12) / -6 = -35.12 / -6 ≈ 5.85
Since the quantity of output cannot be negative, the level of output that maximizes profits is Q ≈ 5.85.
In summary:
a) The level of output that maximizes total revenue is Q = 16.
b) The level of output that maximizes profits is Q ≈ 5.85.