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Unformatted text preview: Eco11, Fall 2008 Simon Board Economics 11: Solutions to Practice Problems 7 (Week 8) November 19, 2008 1. Properties of the Profit Function A firm has cost function c ( q ) = q 2 . (a) Calculate the optimal supply function, q * ( p ). (b) Calculate the optimal profit function, π * ( p ). (c) Show that d dp π * ( p ) = q * ( p ). (d) Show that π * ( p ) is convex. (e) Fix a level of output, q , and define the profit the firm makes when the price is p by π ( p ; q ) = pq c ( q ) On a single picture, draw π ( p ; q ) for each q ∈ { , 1 , 2 , 3 , 4 } . Also draw π * ( p ). Discuss your findings and the relationship to (c) and (d). [You may want to use a computer program such as Excel or Mathematica to draw the picture.] Solution (a) The FOC of the profit maximisation problem is p = 2 q Rearranging, q * ( p ) = p/ 2. (b) The profit function is given by π * ( p ) = pq * c ( q * ) = p 2 4 1 Eco11, Fall 2008 Simon Board p 2 4 6 8 10 K 10 10 20 Figure 1: Envelope Property. (c) Differentiating, d dp π * ( p ) = p 2 = y * ( p ) (d) Differentiating again, d 2 dp 2 π * ( p ) = 1 2 ≥ Hence the profit function is convex. (c) See figure 1. The profit function is the maximum of the π ( p ; q ) functions. This implies that the profit function is convex. It also implies that the slope of the profit function at any point equals the slope of π ( p ; q * ( p )), which equals q * ( p ). 2. Profit Maximisation (a) A firm has cost function c ( q ) = 20 q 10 q 2 + q 3 . The output price is p = 8. Solve for the optimal output....
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 Fall '08
 cunningham
 Economics, Supply And Demand, Simon Board

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