recipes2 - ADDITIONAL RECIPES AND FORMULAS FOR EXAM 2...

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ADDITIONAL RECIPES AND FORMULAS FOR EXAM 2 Short-cuts for Cobb-Douglas production functions q = AL α K β Average product of labor: AP L = AK β L 1 - α since AP L = q L = AL α K β L = AK β L 1 - α Average product of capital: AP K = AL α K 1 - β since AP K = q K = AL α K β K = AL α K 1 - β Marginal product of labor: MP L = α q L since MP L = ∂q ∂L = αAL α - 1 K β = α AL α K β L = α q L Marginal product of capital: MP K = β q K since MP K = ∂q ∂K = βA L α K β - 1 = β AL α K β K = β q K Slope along isoquants: d K d L = - α β K L since d K d L = - MP L MP K = - ∂q ∂L ∂q ∂K = - α q L β q K = - α β q L q K = - α β q L × K q = - α β K L Returns to scale: if q 1 = AL α K β and q 2 = A (2 L ) α (2 K ) β , then q 2 = 2 α + β q 1 since q 2 = A (2 L ) α (2 K ) β = A 2 α L α 2 β K β = 2 α 2 β ( AL α K β ) = 2 α + β q 1 so returns to scale are decreasing if α + β < 1, constant if α + β = 1, and increasing if α + β > 1 1
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Useful formulas for manipulating equations with exponents: ( x a ) b = x a × b x a x b = x a + b ( xy ) a = x a y a x y a = x a y a x - a = 1 x a x a - 1 = x a x =
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  • Spring '08
  • Spam
  • average variable cost, APL, Cobb-Douglas case

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