Question

Assume that a competitive firm has the total cost function:

TC= 1q³ − 40q² + 820q + 1900

Suppose the price of the firm's output (sold in integer units) is $600 per unit.

Use calculus and formulas to find a solution

a. How many integer units should the firm produce to maximize profit?

b. What is the total profit at the optimal integer output level?

Answer 1

To find the optimal quantity that maximizes profit, we need to take the derivative of the profit function with respect to quantity and set it equal to zero. Once we have the optimal quantity, we can find the total profit by plugging the optimal quantity into the profit function.

Step-by-step explanation:

To find the optimal quantity that maximizes profit, we need to find the quantity that corresponds to the maximum point on the profit curve. The profit function can be derived by subtracting the total cost function from the total revenue function. In this case, the profit function is given by:

Profit = Revenue - Cost

Profit = (Price * Quantity) - (1q³ - 40q² + 820q + 1900)

To find the maximum point on the profit curve, we need to take the derivative of the profit function with respect to quantity and set it equal to zero:

d(Profit)/dq = 0

Solving this equation will give us the optimal quantity that maximizes profit.

Once we have the optimal quantity, we can find the total profit by plugging the optimal quantity into the profit function.