Econ 4010 Lecture 14

Econ 4010 Lecture 14 - 7 The Cost of...

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<Lecture 14> 7. The Cost of Production Q1. Suppose that a firm's production function is q = 10L 1/2 K 1/2 . The cost of a unit of labor is $20 and the cost of a unit of capital is $80. a. The firm is currently producing 100 units of output, and has determined that the cost-minimizing quantities of labor and capital are 20 and 5 respectively. Graphically illustrate this situation on a graph using isoquants and isocost lines. The isoquant is convex: 100 = 10L 1/2 K 1/2 . TC = $20*20 + $80*5 = $800, so the isocost line is: 20L + 80K = $800. At the optimal quantities of L and K, the isocost line is tangent to the isoquant. b. The firm now wants to increase output to 140 units. If capital is fixed in the short run, how much labor will the firm require? Illustrate this point on your graph and find the new cost. The new level of labor is L = 39.2 (q = 140 = 10 * √L * √5 ). c. Graphically identify the cost-minimizing level of capital and labor in the long run if the firm wants to produce 140 units. d. If the marginal rate of technical substitution is K/L, find the optimal level of capital and labor required to produce the 140 units of output. Set the MRTS = the ratio of the input costs: K/L = 20/80. Now substitute this into the production function, set q equal to 140, and solve: 140 = 10√L√(L/4), then L=28, K=7. Capital Labor isocost line isoquant line optimal point Capital Labor A B 1
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Expansion Path and Long-Run Total Cost (p.233) Expansion Path The curve passing through the points of tangency between the firm's isocost lines and its isoquants is its expansion path . The expansion path describes the combinations of L and K that the firm will choose to minimize costs at each output level. As long as the use of both L and K increases with output, the expansion path curve will be upward sloping. The slope of the expansion path: As output increases 100 to 200 units, K increases from 25 to 50 units, while labor increases from 50 to 100 units.
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