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Example: Cobb-Douglas production function. f(L,K) = Y = 3 K⅓L⅓. Cost: Labor is $9 and Captal is $27. Y = 3K 3 L 3 ⇔ 1 Y Isoquants: K 3 = 1 ⇔ 3L 3 Y3 K= 27 L To produce 3 units, we have K = 1/L. To produce 1 unit we have K=1/27L, to produce 6 units we have Y = 8/L. Isoquants 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 L 6 7 8 9 10 1 1 Y=6 Y=3 Y=1 C 1 − L . So for $200, we get 3 27 K = 7.4 - ⅓L, and for $100 we get K = 3.7 - ⅓L. Drawing, we get: Isocost curves: C(K,L) = 9L + 27K, so K = K Isoquants and Isocost curves 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 L 6 7 8 9 10 Y=6 Y=3 Y=1 C = 200 C = 100 K Recall that to minimize cost for a given quantity, we had to set MRTS=wL/wK. Suppose we’re producing 6 units. What is the minimum cost? ∂f ( L, K ) 1 −2 3 3 K ∂L = K L so we need MRTS = = 1 −2 ∂f ( L, K ) L K 3L 3 ∂K K 9 1 = ⇔K= L 3 L 27 What now? We wanted to produce 6 units, so 6 = 3 K⅓L⅓ = 3 (⅓L) ⅓L⅓ = 3 (⅓) ⅓ L⅔. Solving for L, we get L = 2 3 2 3 1 3 ⇔L= 2 3 2 1 = 4.9 and K = ⅓L = 1.6 and the total cost is 12 3 12 = 88.18 3 11 2 1 3 3 In general, suppose we want to find the cost as a function of the quantity produced. K and the price ratio is ⅓, so K=⅓L and Q = 3 K⅓L⅓ = 3 (⅓L) ⅓L⅓ = 3 (⅓) ⅓ MRTS = L C =9 2 1 2 3 2 + 27 3 2 ⎛ ⎜ ⎜ Q L⅔. So L = ⎜ 1 ⎜ 1 3 ⎜ 3⎛ ⎞ ⎜ ⎜ 3⎟ ⎝ ⎝ ⎠ ⎞ ⎟ ⎛ ⎟ ⎜Q ⎟ =⎜ 2 ⎜ 3 ⎟ ⎝3 ⎟ ⎟ ⎠ 3 2 3 3 2 ⎞2 Q2 Q2 ⎟ and K = So = ⎟ 9 3 ⎟ ⎠ 3 3 3 3 3 Q Q + 27 = 3Q 2 + 3Q 2 = 6Q 2 3 9 Long-run cost function. C (Q) = 9 Now, what happens if K is fixed at 1.6 and now we wish to produce 3 units? (SHORT RUN) Then we determine how much L we need to produce 3 when K is fixed at the previous level: 3 = 3 ⎛ ⎜ 3 1 1 ⎜ 22 3 = 3K 3 L3 = 3⎜ 1 ⎜ 12 ⎜ 3⋅ ⎝ 3 ⎞3 ⎟ 1 ⎟ 1 1 3 3 ⎟ L ⇔L = ⎟ ⎛ ⎟ ⎜ 3 ⎠ ⎜ 22 ⎜ 1 ⎜ 12 ⎜ 3⋅ ⎝ 3 1 1 ⎛ ⎜ 3⋅ 1 2 ⎜ =⎜ 3 3 1 ⎞3 ⎜ 22 ⎜ ⎟ ⎝ ⎟ ⎟ ⎟ ⎟ ⎠ 3 2 ⎞3 1 ⎟ ⎟ 32 ⎟ ⇔L= 3 ⎟ 22 ⎟ ⎠ 1 Then C = 9 3 2 1 2 3 2 + 27 2 3 2 1 = 48.71 12 3⋅ 3 3 2 In the long run we could have produced this at a cost of C (3) = 6 ⋅ 3 = 31.18 In general we can find the short run cost function like this: We fix K at ĸ. Then to 1 1 1 Q3 Q Q3 3 3 3 produce Q units we have: Q = 3 k L ⇔ L = 1 ⇔ L = Then C = 9 + 27 k . 27 k 27 k 3 3k

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production function, production functions, cobb-douglas production function, cost function, returns to scale, the firm, marginal product of labor, constant returns to scale, marginal product, short run, capital stock, long run, aggregate production function, technical change, cobb douglas

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views: | 12214 |

posted: | 12/27/2009 |

language: | English |

pages: | 3 |

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