Question

Assume quantities need not be integers. A monopolist incurs marginal cost equal to $2 per unit. This period, it must pay a $140 unrecoverable fixed cost, and faces demand P(Q)=7 – 0.5 x Q. What are its profits this period?

Answer 1

To determine the monopolist's profits, calculate the quantity where marginal revenue equals marginal cost, use the demand curve to find the corresponding price, and then subtract total costs (fixed plus variable) from total revenue.

Step-by-step explanation:

To calculate the profits of a monopolist, we must first determine the profit-maximizing quantity and price. The monopolist's demand curve is given by P(Q) = 7 - 0.5 x Q, where P is the price and Q is the quantity. The marginal cost (MC) is constant at $2 per unit. To maximize profit, the monopolist will produce where marginal revenue (MR) equals marginal cost (MC). The total cost (TC) is the sum of the variable cost (which is MC multiplied by quantity) and the fixed cost. The total revenue (TR) is the price multiplied by quantity.

To find MR, we need to find the derivative of the TR function. Since TR = P(Q) x Q, and given P(Q) = 7 - 0.5 x Q, the derivative of TR with respect to Q will give us the MR curve. We set MR equal to MC to find the profit-maximizing quantity, then we plug this quantity back into the demand curve to find the profit-maximizing price. Finally, we calculate total profits by subtracting total cost (which includes fixed and variable costs) from total revenue.

The specific steps to identify the profit-maximizing quantity (Q) and price (P) are:

1. Find MR by differentiating the TR function with respect to Q.
2. Set MR equal to the constant MC of $2 to find the profit-maximizing Q.
3. Plug this Q into the demand curve to find the corresponding price P.
4. Calculate total revenue (P x Q).
5. Calculate total cost by adding the fixed cost of $140 to the variable cost (MC x Q).
6. Subtract total cost from total revenue to find the monopolist's profits.