L12Optimisation - Introduction to Microeconomics Lecture 12...

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Introduction to Microeconomics Lecture 12 More Differentiation and Optimization Simon Cowan
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Outline more economic applications cost functions profit maximization the product rule, the inverse function rule and the chain rule economic applications elasticity and revenue the monopolist’s inverse elasticity rule
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Cost Functions How do we find marginal cost? How do we find average cost? { { 2 Fixed Cost Variable Cost ( ) 9 C Q Q = +
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Marginal and Average Cost 2 ( ) 9 for 0 ( ) ( ) 2 9 ( ) C Q Q Q MC Q C Q Q C AC Q Q Q Q = + = = = = +
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Relationship of Marginal cost to Minimum Average Cost Let’s show algebraically that the marginal cost function passes through the average cost function at the minimum AC MC
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Finding the Minimum Average Cost 1 2 2 2 9 ( ) 9 9 1 9 1 0 9 3 ( 3 is ruled out) C AC Q Q Q Q Q Q dAC Q dQ Q Q Q - - = = + = + = - = - = = = -
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How do we know this is a Minimum? 2 2 3 2 1 9 18 0 Because the second derivative is positive everywhere the average cost function is convex. dAC Q dQ d AC Q dQ - - = - =
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Does the marginal cost function pass through the average cost function at its minimum? At Q = 3, MC = 2 Q =6 At Q = 3, 9 9 ( ) 3 6 3 AC Q Q Q = + = + =
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Sketching the MC and AC functions Sketching MC = 2 Q is easy: Slope = 2, intercept = 0 (at Q = 0, MC = 0) Sketching AC = Q + 9/ Q We know the minimum is at 3 for positive values of Q . At Q = 1, AC = 10. At Q = 9, AC = 10.
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The Graph
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Is this general? If a minimum of average cost exists then marginal cost goes through this point Intuition: if the marginal is above the average this pulls the average up, if the marginal is below the average this drags the average down 2 Suppose that ( ) is a differentiable cost function defined for 0. Average cost is ( )/ . The stationary point(s) of average cost are defined by
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This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.

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L12Optimisation - Introduction to Microeconomics Lecture 12...

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