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Unformatted text preview: Eco11, Fall 2008 Simon Board Economics 11: Solutions to Practice Problems 6 (Week 7) November 11, 2008 1. Cost Functions A firms has cost function c ( q ) = 100 10 q + 5 q 2 . a) Find the fixed cost. 1 b) Find the variable cost. 2 c) Find the average cost. d) Find the marginal cost. e) Draw the relationship between MC and AC. Prove that they always intersect at the minimum AC. Solution (a) The fixed cost is 100. (b) The variable cost is 10 q + 5 q 2 . (c) The average cost is AC ( q ) = 100 q 10 + 5 q (d) The marginal cost is MC ( q ) = 10 + 10 q (e) AC is a convex function. Differentiating, it is minimised when d dq AC ( q ) = 100 q 2 + 5 q = 0 1 Definition: The fixed cost is the cost that is independent of output. 2 Definition: The variable cost is the cost that varies with the level of output. 1 Eco11, Fall 2008 Simon Board Rearranging, we find that q = 20. We have AC = MC when 100 q 10 + 5 q = 10 + 10 q Rearranging, we find that q = 20. 2. Cost Functions A firm has production function f ( z 1 ,z 2 ) = z 1 / 2 1 ( z 2 1) 1 / 2 The prices of the inputs are r 1 and r 2 . (a) Find MP 1 , MP 2 , and MRTS . (b) If z 2 is fixed at 5, what is the shortrun cost function? Find the shortrun marginal cost and average cost. (c) What are the longrun input demand functions? What is the longrun cost function? Find the longrun marginal cost and average cost. (d) Does the production function exhibit increasing, constant or decreasing returns to scale. Solution (a) The marginal products are MP 1 = 1 2 z 1 / 2 1 ( z 2 1) 1 / 2 MP 2 = 1 2 z 1 / 2 1 ( z 2 1) 1 / 2 Hence we have MRTS = MP 1 MP 2 = z 2 1 z 1 2 Eco11, Fall 2008 Simon Board (b) The cost minimisation problem is to choose z 1 to minimise r 1 z 1 + 5 r 2 subject to 2...
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 Fall '08
 cunningham
 Economics

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