L12Optimisation

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
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 = + = = = = +
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 - - = = + = + = - = - = = = -
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 = + = + =
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
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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