A monopolist faces the price equation P = 1,000 – 0.5Q, and cost is given as TC = 400 + 100Q +2.5Q^2.Determine the profit at the revenue maximizing level and the profit maximizing level. Compare the answers above and comment on the appropriate goal of the firm.
Always always always, maximize where MC=MR
Total revenue is P*Q = 1000Q- 0.5Q^2. MR is the first derivative of TR. So, MR=1000 - Q
MC is the first derivative of TC. So, MC=100 + 5Q
MC=MR is 100+5Q = 1000-Q. solve for Q. then plug the optimized Q into TC and TR to derive the profit.
To determine the profit at the revenue maximizing level and the profit maximizing level, we need to find the quantity that maximizes revenue and the quantity that maximizes profit.
1. Revenue Maximization:
To maximize revenue, we need to find the quantity that maximizes the total revenue (TR). Total revenue is calculated by multiplying the quantity (Q) by the price (P).
TR = P * Q
Given the price equation P = 1,000 – 0.5Q, we can substitute this into the revenue equation:
TR = (1,000 – 0.5Q) * Q
To find the quantity that maximizes revenue, we differentiate the revenue equation with respect to Q and set the derivative equal to zero:
dTR/dQ = 1,000 – Q = 0
Solving for Q, we get:
Q = 1,000
Now, substitute this value of Q back into the price equation to find the corresponding price:
P = 1,000 – 0.5(1,000) = 1,000 – 500 = 500
So, at the revenue maximizing level, the quantity is 1,000 and the price is 500.
To calculate the profit at the revenue maximizing level, we need to subtract the total cost (TC) from the total revenue (TR):
Profit = TR - TC
Substituting the values we found:
Profit = (500 * 1,000) - (400 + 100(1,000) + 2.5(1,000)^2)
After calculating the expression, we will have the profit at the revenue maximizing level.
2. Profit Maximization:
To maximize profit, we need to find the quantity that maximizes the difference between total revenue (TR) and total cost (TC). Again, total revenue is given by multiplying Q by P.
TR = P * Q = (1,000 – 0.5Q) * Q
Now, we need to calculate the total cost (TC) using the given equation:
TC = 400 + 100Q + 2.5Q^2
To find the quantity that maximizes profit, we differentiate the profit equation (TR - TC) with respect to Q and set the derivative equal to zero:
d(Profit)/dQ = d(TR - TC)/dQ = 0
Then, solve the equation for Q.
Once we have the value of Q, substitute it back into the price equation to find the corresponding price.
Now, calculate the profit using the profit equation:
Profit = TR - TC
Substitute the obtained values of Q and P into the equation to find the profit at the profit maximizing level.
Once we have both profit values, we can compare them and comment on the appropriate goal of the firm. If the profit at the revenue maximizing level is higher than the profit at the profit maximizing level, the firm's goal might be to maximize revenue. Conversely, if the profit at the profit maximizing level is higher, the firm's goal might be to maximize profit.
Please note that to provide an accurate comparison and comment, the actual calculations are necessary and depend on the given cost equation TC.