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I want to find the cost function given the CES production function: $$ Y = F(x_1,x_2) = (\lambda x_1^ \rho+(1-\lambda)x_2^\rho)^\frac{1}{\rho} $$ with $0<\rho<1$. So far I have set up the Lagrangian and I have relative prices equal to MRS $$ \frac{p_1}{p_2} = \frac{\lambda x_1^{\rho-1}}{(1-\lambda)x_2^{\rho-1}} $$

and tried to rearrange it, but I only get to:

$$ (p_1^{\frac{\rho}{\rho-1}}+p_2^{\frac{\rho}{\rho-1}})^{\frac{1}{\rho}}(1-\lambda)^{\frac{1}{\rho-1}}x_2^\rho= p_2^{\frac{1}{\rho-1}}(\lambda^{\frac{\rho}{\rho-1}}x_1^\rho+(1-\lambda)^{\frac{\rho}{\rho-1}}x_2^\rho)^{\frac{1}{\rho}} $$

I tried this since without the weights you get $Y$ on the right hand side and can then continue to solve for conditional demand $\hat{x}(p1,p2,Y)$ but I don't know what to do with the weights $\lambda$ and $(1-\lambda)$

Do you know what I can do here.

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