Marginal Cost using Calculus: Marginal cost is the rate of change of cost as we make an infinitesimally small change in output. Marginal cost is MC = dC/dq Given that C = VC + F, it follows that MC = dVC/dq + dF/dq = dVC/dq, because fixed costs do not change as output changes: dF/dq = 0. For example, suppose that the variable cost is: VC = 4q + 6q2 and the fixed cost is F = 10, TC = VC + F = 4q + 6q2 + 10. Using the variable cost, MC = dVC/dq = d(4q + 6q2)/dq = 4 + 12q. We get the same expression for marginal cost if we use the total cost: dC/dq = d(4q + 6q2 + 10)/dq = 4 + 12q

Production Functions and the Shapes of Cost Curves: Short-Run Costs The production function determines the shape of a firm’s cost curves. In the short run, diminishing marginal returns to labor determine the shape of the production function. The firm increases output by using more labor. However, each extra worker increases output by a smaller amount. The production function determines the shape of the variable cost curve and its related curves. As output increases, variable cost increases more than proportionally because of diminishing marginal returns. The production function determines the shape of the marginal cost, average variable cost, and average cost curves. Cost Curves The average total-cost curve is U-shaped. At very low levels of output average total cost is high because fixed cost is spread over only a few units.

Average total cost declines as output increases. Average total cost starts rising because average variable cost rises substantially. The bottom of the U-shaped ATC curve occurs at the quantity that minimizes average total cost. This quantity is sometimes called the efficient scale of the firm. Average and Marginal Costs Average cost is increasing when marginal cost is greater than average cost, average cost is decreasing when marginal cost is less than average cost, and average cost equals marginal cost when average cost is minimized. To find the quantity that minimizes average total cost, set the first derivative of ATC with respect to Q equal to zero or … find the quantity where MC = ATC Using Calculus Suppose a firm’s total cost curve is TC = 15 Q 2 + 8Q + 45. MC = 30Q + 8 Minimum of ATC??? Set ATC = MC or Set first derivative of ATC wrt Q = 0 6.2 Short-Run Costs (10 of 10)

Short-Run Cost Summary In the short run, the cost associated with fixed inputs is fixed, while the cost from inputs that can be adjusted is variable. Given that input prices are constant, the shapes of the variable cost and the cost-per-unit curves are determined by the production function. Where there are diminishing marginal returns, the variable cost and cost curves become relatively steep as output increases, so the average cost, average variable cost, and marginal cost curves rise with output. The average cost and average variable cost curves fall when marginal cost is below them and rise when marginal cost is above them, so the marginal cost cuts both these average cost curves at their minimum points. Long Run Costs Cost Minimization Isoquants/Isocosts