<p>I'll use <span class="math-container">$(w_1, w_2)$</span> to denote the factor prices instead of <span class="math-container">$(p_1, p_2)$</span> as the latter is traditionally used for output prices.</p>
<p><span class="math-container">$c(w_1, w_2, y)$</span> solves the maximization problem: <span class="math-container">$$\max_{x_1, x_2 \ \geq \ 0} [w_1 x_1 + w_2 x_2] \text{ subject to } \left(\lambda x_1^{\rho} + (1-\lambda) x_2^{\rho}\right)^{1/\rho} \geq y$$</span></p>
<p>Write the Lagrangian as <span class="math-container">$\mathcal{L}(x_1, x_2, \Lambda) = w_1 x_1 + w_2 x_2 - \Lambda \left[\left(\lambda x_1^{\rho} + (1-\lambda) x_2^{\rho}\right) - y^\rho\right]$</span> after modifying the constraint.</p>
<p>The first-order-conditions are:</p>
<ol>
<li><span class="math-container">$\frac{\partial \mathcal{L}}{\partial x_1} = w_1 - \Lambda \lambda \rho x_1^{\rho - 1} = 0$</span></li>
<li><span class="math-container">$\frac{\partial \mathcal{L}}{\partial x_2} = w_2- \Lambda (1-\lambda) \rho x_2^{\rho - 1} = 0$</span></li>
<li><span class="math-container">$\lambda x_1^{\rho} + (1-\lambda) x_2^{\rho} = y^\rho$</span></li>
</ol>
<p><strong>Steps to find the conditional factor demands:</strong></p>
<ul>
<li>Replace <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> in (3) using the results obtained in (1) and (2). This gives us <span class="math-container">$$\lambda \left[ \frac{w_1}{\Lambda \lambda \rho} \right]^{\frac{1}{\rho - 1}} + (1-\lambda) \left[ \frac{w_2}{\Lambda (1-\lambda) \rho} \right]^{\frac{1}{\rho - 1}} = y^\rho.$$</span></li>
<li>From the above equation, calculate <span class="math-container">$\Lambda$</span> in terms of the other (exogenous) variables.</li>
<li>Now that you have the value of <span class="math-container">$\Lambda$</span>, replace it in (1) and (2) to get the factor demands <span class="math-container">$(x_1^{*}, x_2^{*})$</span> in terms of the desired exogenous variables.</li>
</ul>
<p>The cost function is then given by <span class="math-container">$c(w_1, w_2, y) = w_1 x_1^{*} + w_2 x_2^{*}$</span>.</p>

CC BY-SA 4.0
2024-04-02T22:17:54.340Z
2024-04-02T22:17:54.340Z
uninterestedacademic
<p>I want to find the cost function  given the CES production function:
<span class="math-container">$$
Y = F(x_1,x_2) = (\lambda x_1^ \rho+(1-\lambda)x_2^\rho)^\frac{1}{\rho}
$$</span>
with <span class="math-container">$0&lt;\rho&lt;1$</span>.
So far I have set up the Lagrangian and I have relative prices equal to MRS
<span class="math-container">$$
\frac{p_1}{p_2} = \frac{\lambda x_1^{\rho-1}}{(1-\lambda)x_2^{\rho-1}}
$$</span></p>
<p>and tried to rearrange it, but I only get to:</p>
<p><span class="math-container">$$
(p_1^{\frac{\rho}{\rho-1}}+p_2^{\frac{\rho}{\rho-1}})^{\frac{1}{\rho}}(1-\lambda)^{\frac{1}{\rho-1}}x_2^\rho= p_2^{\frac{1}{\rho-1}}(\lambda^{\frac{\rho}{\rho-1}}x_1^\rho+(1-\lambda)^{\frac{\rho}{\rho-1}}x_2^\rho)^{\frac{1}{\rho}}
$$</span></p>
<p>I tried this since without the weights you get <span class="math-container">$Y$</span> on the right hand side and can then continue to solve for conditional demand <span class="math-container">$\hat{x}(p1,p2,Y)$</span> but I don't know what to do with the weights <span class="math-container">$\lambda$</span> and <span class="math-container">$(1-\lambda)$</span></p>
<p>Do you know what I can do here.</p>

<microeconomics><mathematical-economics><optimization><cost-functions><ces-function>
Cost function from a weighted CES production function
fabs
# Cost function from a weighted CES production function

(Asked by: username_0 on 2024-02-11)

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.

**Tags:** microeconomics, mathematical-economics, optimization, cost-functions, ces-function

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## 1 answers

### Answer by username_1 on 2024-04-02. Score: 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 $c(w\_1, w\_2, y) = w\_1 x\_1^{\*} + w\_2 x\_2^{\*}$.


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