L12Optimisation - Introduction to Microeconomics Lecture 12...

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1 Introduction to Microeconomics Lecture 12 More Differentiation and Optimization Simon Cowan Outline z more economic applications z cost functions z profit maximization th d t l th i f ti l d th z the product rule, the inverse function rule and the chain rule z economic applications z elasticity and revenue z the monopolist’s inverse elasticity rule Cost Functions N N 2 Fixed Cost Variable Cost () 9 CQ Q =+ z How do we find marginal cost? z How do we find average cost? Marginal and Average Cost 2 ( ) 9 for 0 Q Q = +≥ () 2 9 MC Q C Q Q C AC Q Q QQ = = == + Relationship of Marginal cost to Minimum Average Cost z Let’s show algebraically that the marginal cost function passes through the average cost function at the minimum AC MC Finding the Minimum Average Cost 1 9 9 C AC Q Q Q Q ==+=+ 2 2 2 9 19 1 0 9 3 ( 3 is ruled out) dAC Q dQ Q Q Q =− = =
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2 How do we know this is a Minimum? 2 2 3 2 19 18 0 dAC Q dQ dAC Q =− = > Because the second derivative is positive everywhere the average cost function is convex. dQ z Does the marginal cost function pass through the average cost function at its minimum? z At Q = 3, MC = 2 Q =6 z At Q = 3 At 3, 99 () 3 6 3 AC Q Q Q = +=+= Sketching the MC and AC functions z Sketching MC = 2 Q is easy: z Slope = 2, intercept = 0 (at Q = 0, MC = 0) z Sketching AC = Q + 9/ Q z We know the minimum is at 3 for positive values of Q . At Q = 1, AC = 10. At Q = 9, AC = 10. The Graph 70 80 90 100 0 10 20 30 40 50 60 0 1 02 03 04 0 Is this general? z If a minimum of average cost exists then marginal cost goes through this point Suppose that ( ) is a differentiable cost function defined for 0. Average cost is /. The stationary point(s) of average cost are defined by () () 0 CQ Q AC C Q Q dAC QC Q C Q > = = z 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 This implies that ( ) ( )/ o dQ Q CQ CQ Q = 23 24 r that marginal cost equals average cost. The second-order condition is
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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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