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.


1 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 $c(w_1, w_2, y) = w_1 x_1^{*} + w_2 x_2^{*}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.