Final answer:
The marginal cost function is calculated by differentiating the short run cost function, resulting in MC(q) = 10 + 4q. This reflects the additional cost for each unit of output produced, which is vital for businesses to understand for profitable decision-making.
Step-by-step explanation:
The question is centered on calculating the marginal cost function from a given short-run cost function. The short-run cost function provided is c(q) = 10q + 2q² + 200. Marginal cost (MC) is defined as the additional cost incurred by producing one more unit of output. To find the marginal cost function, we take the derivative of the total cost function concerning quantity, q.
In this case, differentiating the cost function c(q) = 10q + 2q² + 200 gives us the marginal cost function MC(q) = 10 + 4q. This signifies that for each additional unit of output, the cost increases by 10 plus an additional amount multiplied by 4 times the number of units produced. As production increases, the marginal cost is typically expected to rise, reflecting the concept of diminishing marginal productivity.
Businesses often compare the marginal cost to the additional revenue gained from selling an extra unit to determine whether the additional unit is contributing to profit. The calculation of marginal cost is essential for firms to understand their cost structure and make profitable production decisions.