Final answer:
The producer surplus is calculated as the area between the market price and the supply curve up to the quantity produced. For a price of $30 and a marginal cost of MC = 10 + 2q, the producer surplus is $100, determined by the base of the triangle (quantity) of 10 and the height (price minus MC at q=0) of $20.
Step-by-step explanation:
To calculate the producer surplus for a profit-maximizing price taker with a marginal cost function MC = 10 + 2q when the price is $30, we must find the quantity where price equals marginal cost. From the profit-maximizing rule P = MC, the firm will produce where the price of $30 equals its marginal cost of 10 + 2q. Solving for q gives us (30 - 10) / 2 = 10 units.
The producer surplus is the area above the supply curve (or MC curve in the case of a price taker) and below the price level up until the produced quantity. Mathematically, this is the integral of the price minus the marginal cost, from zero to the quantity produced. Here, the producer surplus is the area under the price, $30, and above the MC curve from 0 to 10. A graphical approach gives us a triangle with a base of 10 units (0 to q) and a height of $20 ($30 price minus $10, the MC when q=0). Thus, the producer surplus is 0.5 * base * height, which equals 0.5 * 10 * 20 = $100.