Cost function from a weighted CES production function

Question

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.

Answer 1

I'll use $(w_1, w_2)$ to denote the factor prices instead of $(p_1, p_2)$ as the latter is traditionally used for output prices.

$c(w_1, w_2, y)$ solves the maximization problem: $$\max_{x_1, x_2 \ \geq \ 0} [w_1 x_1 + w_2 x_2] \text{ subject to } \left(\lambda x_1^{\rho} + (1-\lambda) x_2^{\rho}\right)^{1/\rho} \geq y$$

Write the Lagrangian as $\mathcal{L}(x_1, x_2, \Lambda) = w_1 x_1 + w_2 x_2 - \Lambda \left[\left(\lambda x_1^{\rho} + (1-\lambda) x_2^{\rho}\right) - y^\rho\right]$ after modifying the constraint.

The first-order-conditions are:

1. $\frac{\partial \mathcal{L}}{\partial x_1} = w_1 - \Lambda \lambda \rho x_1^{\rho - 1} = 0$
2. $\frac{\partial \mathcal{L}}{\partial x_2} = w_2- \Lambda (1-\lambda) \rho x_2^{\rho - 1} = 0$
3. $\lambda x_1^{\rho} + (1-\lambda) x_2^{\rho} = y^\rho$

Steps to find the conditional factor demands:

• Replace $x_1$ and $x_2$ in (3) using the results obtained in (1) and (2). This gives us $$\lambda \left[ \frac{w_1}{\Lambda \lambda \rho} \right]^{\frac{1}{\rho - 1}} + (1-\lambda) \left[ \frac{w_2}{\Lambda (1-\lambda) \rho} \right]^{\frac{1}{\rho - 1}} = y^\rho.$$
• From the above equation, calculate $\Lambda$ in terms of the other (exogenous) variables.
• Now that you have the value of $\Lambda$, replace it in (1) and (2) to get the factor demands $(x_1^{*}, x_2^{*})$ in terms of the desired exogenous variables.

The cost function is then given by $c(w_1, w_2, y) = w_1 x_1^{*} + w_2 x_2^{*}$.