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
Revenue = Quantity*Price
R = Q*P = 24Q - .5Q^2
Revenue is a maximum when
dR/dQ = 0
24 -Q = 0
Q = 24 units sold @ a price of P = 12
Profit (P) is
P = R - Q*(AC)= Q*P - Q*(AC)
= 24Q -0.5Q^2 -Q*(Q^2 -8Q +36 +3/Q)
= 24Q -0.5Q^2 -Q^3 +8Q^2 +36Q +3
= -Q^3 +7.5 Q^2 +60Q +3
Set dP/dQ = 0 to solve for the maximum-profit production level, Q.
-3Q^2 +15Q +60 = 0
The positive root is
Q = (-1/6)[-15 -sqrt(289+720)]
Q = 7.8
To find the level of output Q that maximizes total revenue and maximizes profits, we need to apply some key economic concepts and mathematical techniques. Let's calculate each case step by step:
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 into the TR equation:
TR = (24 - 0.5Q) * Q
To maximize total revenue, we need to find the quantity (Q) that maximizes this function. To do so, we can take the derivative of the TR function 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 = 16
Therefore, to maximize total revenue, the level of output Q should be 16.
b) Maximizing Profits:
To maximize profits, we need to consider the cost associated with producing each unit of output. In this case, we are given the average cost function AC = Q^2 - 8Q + 36 + 3/Q.
Total Cost (TC) is calculated by multiplying the average cost (AC) by the quantity (Q):
TC = AC * Q
To maximize profits, we need to find the quantity (Q) that maximizes the difference between total revenue (TR) and total cost (TC). This difference is called profit (π):
π = TR - TC
Substituting the TR and TC functions:
π = (24 - 0.5Q) * Q - (Q^2 - 8Q + 36 + 3/Q) * Q
Simplifying and expanding the equation:
π = 24Q - 0.5Q^2 - Q^3 + 8Q^2 - 36Q - 3Q + 36Q
Combining like terms:
π = -Q^3 + 7.5Q^2 + 21Q
To maximize profits, we take the derivative of the profit function with respect to Q and set it equal to zero:
d(π)/dQ = -3Q^2 + 15Q + 21
Setting this derivative equal to zero and solving for Q:
-3Q^2 + 15Q + 21 = 0
This equation can be solved using either factoring, completing the square, or the quadratic formula. After solving this equation, we find multiple values for Q. To determine which one maximizes profits, we can examine the second derivative or evaluate the profit function at each critical point and choose the one with the highest profit value.
Please provide the coefficients or specific values of AC if you want to further determine the exact level of output Q which maximizes profits.