econ11_08_ps6_sol

econ11_08_ps6_sol - Eco11, Fall 2008 Simon Board Economics...

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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 short-run cost function? Find the short-run marginal cost and average cost. (c) What are the long-run input demand functions? What is the long-run cost function? Find the long-run 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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econ11_08_ps6_sol - Eco11, Fall 2008 Simon Board Economics...

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