Question

If, in a monopoly market, the demand for a product is p = 400 − x and the revenue function is R = px, where x is the number of units sold, what price will maximize revenue?

Answer 1

The price that maximizes revenue in a monopoly market, given the demand equation p = 400 - x, is calculated to be $200. This is found by deriving the revenue function, setting the marginal revenue (MR) to zero, and solving for the quantity (x), and then using this quantity to find the corresponding price.

Step-by-step explanation:

To determine the price that will maximize revenue for a monopolist, we can first derive the revenue function from the given demand equation, p = 400 - x, and then find its maximum value. The revenue function, R, is defined as R = px, where p represents the price of the product and x is the quantity sold.

We can substitute the demand equation into the revenue function to get R = x(400 - x). This simplifies to R = 400x - x^2. To maximize revenue, we need to calculate the derivative of R with respect to x to find the marginal revenue and set it equal to zero, as this is where the maximum occurs. Deriving R with respect to x gives us MR = 400 - 2x. Setting MR equal to 0 and solving for x: 0 = 400 - 2x implies x = 200.

Once we have the quantity that maximizes revenue, we can find the corresponding price by substituting x back into the demand equation: p = 400 - x = 400 - 200 = $200. Therefore, the price that maximizes revenue in this monopoly market is $200.