Marginal Analysis

Rules Marginal cost is the rate at which the total cost is changing, so it is the gradient, or the differentiation. Total Cost, TC = y, then Marginal Cost, MC = dy/dx. Marginal Revenue is the rate at which the total revenue is changing, or the gradient or differentiation. Total Revenue, TR = y, then Marginal Revenue, MR = dy/dx. Total Profit, TP = TR - TC

Example(1) The total cost of making x units of a product is TC=2X 2 +4X+500. What are: The fixed cost? The variable cost? The marginal cost? The average cost? What are the costs of making 500 units of the product?

Solution Total Cost, TC=2X 2 +4X+500. Fixed Cost = 500, not effected by the quantity. Variable Cost = 2X 2 + 4X, changed by quantity. Marginal Cost MC = dy/dx = 4X + 4 Average Cost = TC/X = 2X /X When x = 500: TC = 502,500 FC = 500 VC = 502,000 MC = 2,004 AC = 1,005

Example(2) The total revenue and total cost for a product are related to production x by: TR = 14X – X TC = X 3 -15X How many units should the company make to: Maximise total revenue Minimise total cost Maximise profit

Solution MR = dy/dx for TR = 14 – 2X, the turning point occurs when MR = 0, so, 14 – 2X = 0 gives X = 7. d 2 y/dx 2 = -2 < 0, turning point is maximum, when X = 7, TR = 2,049. MC = dy/dx for TC = 3X 2 – 30X, the turning point occurs when MC = 0, so, 3X 2 – 30X = 0; when X = 0 or X = 10. d 2 y/dx 2 = 6X – 30, when X=0, d 2 y/dx 2 = -30 0, (minimum) Then, when X=10, TC = 500.

Solution (Continued) TP = TR – TC = -X X x , dy/dx = -3X X + 14 Use the quadratic equation, the positive root is 9.8. d 2 y/dx 2 = -6X + 28, When X = 9.8, d 2 y/dx 2 < 0, confirming a maximum. At this point, the maximum profit =