et's make this rigorous with two good k and l, where we look at labor demand, the price of labor is w, the price of capital is R Just plugging into the formula gives us 斧=?爷斧 is more useful to also use the two first order conditions and note that 作+B and PkK /K=0. /W Manipulating these equations yields PfulfKK? Ph o
Let’s make this rigorous with two good K and L, where we look at labor demand, the price of labor is W, the price of capital is R. Just plugging into the formula gives us: W L /L /W = 1 L ? RK WL W K /K /W It is more useful to also use the two first order conditions and note that fLL /L /W + PfKL /K /W = 1 and PfKL /L /W + PfKK /K /W = 0. Manipulating these equations yields: /L /W = fKK PfLLfKK ? PfKL 2 < 0
and /K PfiLfkx pfx KL
and /K /W = fKL PfLLfKK ? PfKL 2
This still isn,'t all that helpful- let's try a separable production function YK, Lp=ak+ bLk 里=" PALK? L PbKYK? 1pLK?2 LPbKYK? 1PL K?2 K? Well-that isnt all that interesting. It certainly tells us that the unimportance result can be general(Hicks point)
This still isn’t all that helpful– let’s try a separable production function: fÝK,LÞ = aKJ + bLK W L /L /W = W L 1 PbKÝK ? 1ÞL K?2 = PbKL K?1 LPbKÝK ? 1ÞL K?2 = 1 K ? 1 Well– that isn’t all that interesting. It certainly tells us that the unimportance result can be general (Hicks’ point)
How about Cobb-Douglas: N\K, LP=KLK In the Crs case, you cant solve for scale, only for factor proportions Use cost minimization to solve min RK+WL+VKL??0 This gives uS: JRK= Y1? JpWL or, using the g constraint, K=O W1?p11?J RJ And L=Q WY1?Jp )0r"h=?J Thats a little bit better - in this case the elasticity of demand for labor is equal to one minus labors share in the production function
How about Cobb-Douglas: fÝK,LÞ = KJL K In the CRS case, you can’t solve for scale, only for factor proportions. Use cost minimization to solve: min K,L,V RK + WL + V KJL 1?J ? Q This gives us: JRK = Ý1 ? JÞWL or, using the Q constraint, K = Q WÝ1?JÞ RJ 1?J And L = Q RJ WÝ1?JÞ J or W L /L /W = ?J That’s a little bit better– in this case, the elasticity of demand for labor is equal to one minus labor’s share in the production function
Of course, we can't say anything about the degree of substitutability
Of course, we can’t say anything about the degree of substitutability