I am having trouble with the following Cobb-Douglas cost minimization problem. Given the following Cobb-Douglas production function $f(K,L)=y=K^{0.5}L^{0.25}$. The price of labor and capital are given by w and r respectively so I want to minimize $rK+wL$.
After I set up my first order conditions:
$L=rK+wL+\lambda[y-K^{0.5}L^{0.25}]$
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$r=0.5\lambda K^{0.25}L^{-0.5}$
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$w=0.25 \lambda K^{0.5}L^{-0.75}$
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$Y=K^{0.5}L^{0.25}$
In this case I am having problems solving for lambda. I was able to optimize $L^{*}=(\frac{r}{2w})^{\frac{2}{3}}y^{\frac{4}{3}}$ and $K^{*}=(\frac{2w}{r})^{\frac{1}{3}}y^{\frac{4}{3}}$. Any help or insights into solving for lambda the Lagrange multiplier will be greatly appreciated cause I have no idea how to start.
Best Answer
Forming the lagrangian as
$$ L=r K+w L + \lambda(y-K^a L^b) $$
we have the stationarity condition
$$ \nabla L = 0= \left\{ \begin{array}{l} r-a \lambda K^{a-1} L^b \\ w-b \lambda K^a L^{b-1} \\ y-K^a L^b \\ \end{array} \right. $$
and solving for $(K,L,\lambda)$ we get
$$ \left( \begin{array}{rcl} K^* & = & \left(\frac{aw}{br}\right)^{\frac{b}{a+b}} y^{\frac{1}{a+b}}\\ L^* & = & \left(\frac{b r}{a w}\right)^{\frac{a}{a+b}}y^{\frac{1}{a+b}}\\ \lambda^* & = & \left(\frac{r}{ay}\right)^{\frac{a}{a+b}}\left(\frac{w}{by}\right)^{\frac{b}{a+b}}y^{\frac{1}{a+b}} \end{array} \right. $$