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 maximizes profits, we need to analyze the given demand function and average cost function.
a) Maximizing total revenue:
Total revenue (TR) is calculated by multiplying the quantity (Q) by the price (P). In this case, the demand function gives us the price as a function of quantity, so we can substitute the demand function into the TR equation to maximize it.
TR = P * Q = (24 - 0.5Q) * Q = 24Q - 0.5Q^2
To maximize total revenue, we need to find the level of output (Q) that results in the highest value of TR. This can be done by differentiating the TR equation with respect to Q and setting it equal to zero to find the critical point:
d(TR)/dQ = 24 - Q = 0
Solving for Q, we get Q = 24.
Therefore, the level of output that maximizes total revenue is Q = 24.
b) Maximizing profits:
Profits (π) are calculated as total revenue (TR) minus total cost (TC). Total cost is the product of average cost (AC) and quantity (Q).
π = TR - TC = TR - (AC * Q)
To maximize profits, we need to find the level of output (Q) that results in the highest value of profits. This can be done by differentiating the profit equation with respect to Q and setting it equal to zero to find the critical point:
d(π)/dQ = d(TR)/dQ - d(AC)/dQ = 24 - Q - (2Q - 8 + 3/Q) = 0
Rearranging the equation, we get:
3/Q^2 - Q + 16 = 0
Unfortunately, this equation does not have a simple analytical solution. So we need to solve it numerically or graphically.
Once we have the value of Q that satisfies the above equation, we can substitute it back into the profit equation to calculate the maximum profit.
Please note that in this particular problem, the average cost function includes a quadratic term and a fractional term. The presence of a fractional term might cause unusual cost behavior at certain levels of output. Thus, we need to ensure that the value of Q that maximizes profits lies within the feasible range of production.