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Marginal Analysis

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 .

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Marginal Analysis

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  1. Marginal Analysis

  2. 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

  3. Example(1) • The total cost of making x units of a product is TC=2X2+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?

  4. Solution • Total Cost, TC=2X2+4X+500. • Fixed Cost = 500 , not effected by the quantity. • Variable Cost = 2X2 + 4X, changed by quantity. • Marginal Cost MC = dy/dx = 4X + 4 • Average Cost = TC/X = 2X + 4 + 500/X • When x = 500: TC = 502,500 FC = 500 VC = 502,000 MC = 2,004 AC = 1,005

  5. Example(2) • The total revenue and total cost for a product are related to production x by: • TR = 14X – X2 + 2000 • TC = X3 -15X2 + 1000 • How many units should the company make to: • Maximise total revenue • Minimise total cost • Maximise profit

  6. Solution • MR = dy/dx for TR = 14 – 2X, the turning point occurs when MR = 0, so, 14 – 2X = 0 gives X = 7. d2y/dx2 = -2 < 0 , turning point is maximum, when X = 7, TR = 2,049. • MC = dy/dx for TC = 3X2 – 30X, the turning point occurs when MC = 0, so, 3X2 – 30X = 0; when X = 0 or X = 10. d2y/dx2 = 6X – 30, when X=0, d2y/dx2 = -30 < 0 (maximum), when X = 10, d2y/dx2 > 0, (minimum) Then, when X=10, TC = 500.

  7. Solution (Continued) • TP = TR – TC = -X3 + 14X2 + 14x + 1000, dy/dx = -3X2 + 28X + 14 • Use the quadratic equation, the positive root is 9.8 . • d2y/dx2 = -6X + 28, When X = 9.8, d2y/dx2 < 0, confirming a maximum. At this point, the maximum profit = 1540.6

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