<p><strong>This question is NOT about how to approximate a CES function to a leontief function.</strong></p>
<p>Knowing that:</p>
<p><span class="math-container">$i= good (\begin{array}{*{20}{c}}
{1}&amp;{or}&amp;{2}
\end{array})$</span></p>
<p><span class="math-container">$j= firm (\begin{array}{*{20}{c}}
{1}&amp;{or}&amp;{2}
\end{array})$</span></p>
<p>I have this optimization problem:</p>
<p><span class="math-container">$\begin{array}{*{20}{c}}
{\begin{array}{*{20}{c}}
{\mathop {Maximize}\limits_{{Z_j},{Y_j},{X_{i,j}}} }&amp;{\pi _j^z}
\end{array} = p_j^z{Z_j} - \left( {p_j^y{Y_j} + \sum\limits_i {p_i^q{X_{i,j}}} } \right)}&amp;{(0.1)}
\end{array}$</span></p>
<p>subject to</p>
<p><span class="math-container">$\begin{array}{*{20}{c}}
{{Z_j} = \min \left( {\frac{{{X_{1,j}}}}{{a{x_{1,j}}}},\frac{{{X_{2,j}}}}{{a{x_{2,j}}}},\frac{{{Y_j}}}{{a{y_j}}}} \right)}&amp;{(0.2)}
\end{array}$</span></p>
<p>In the book that I am following, they arrive at the following first-order conditions:</p>
<p><span class="math-container">$\begin{array}{l}
\begin{array}{*{20}{c}}
{{X_{i,j}} = a{x_{i,j}}{Z_j}}&amp;{\forall i,j}&amp;{(1)}
\end{array}\\
\begin{array}{*{20}{c}}
{{Y_{j}} = a{y_i}Z}&amp;{\forall j}&amp;{(2)}
\end{array}\\
\begin{array}{*{20}{c}}
{{Z_j} = \min \left( {\frac{{{X_{1,j}}}}{{a{x_{1,j}}}},\frac{{{X_{2,j}}}}{{a{x_{2,j}}}},\frac{{{Y_j}}}{{a{y_j}}}} \right)}&amp;{\forall j}&amp;{(3)}
\end{array}
\end{array}$</span></p>
<p>Then in a footnote they say the following: &quot;we cannot solve this profit-maximization problem by simply applying the Lagrange multiplier method because the Leontief-type production function <span class="math-container">$(0)$</span> is not differentiable.To derive the intermediate input demand function <span class="math-container">$(1)$</span> and the composite factor demand function <span class="math-container">$(2)$</span>, we have to use a constant elasticity of substitution (CES) function, which is a generalized functional form of the Leontief-type function, and derive these first-order conditions. <strong>When the elasticity of substitution in the CES function tends to zero, the derived first-order conditions approach the expressions of <span class="math-container">$(0)$</span>, <span class="math-container">$(1)$</span> and <span class="math-container">$(3)$</span>.</strong>&quot;</p>
<p>What I understand from the bolded part above is that you first solve the optimization problem represented by the equations <span class="math-container">$(0.1)$</span> and <span class="math-container">$(0.2)$</span> but using a CES function instead of <span class="math-container">$(0.2)$</span>.  I approach the problem as follows:</p>
<p>Knowing that:</p>
<p><span class="math-container">$\begin{array}{*{20}{c}}
{\sigma  = \frac{1}{{1 - \rho }}}&amp; \wedge &amp;{\rho  = \frac{{\sigma  - 1}}{\sigma }}
\end{array}$</span></p>
<p>The problem is:</p>
<p><span class="math-container">$L({Z_j},{Y_j},{X_{i,j}}) = p_j^z{Z_j} - \left( {p_j^y{Y_j} + \sum\limits_i {p_i^q{X_{i,j}}} } \right) + {\psi _i}\left[ {{{\left( {\frac{{X_{1,j}^\rho }}{{a{x_{1,j}}}}+\frac{{X_{2,j}^\rho }}{{a{x_{2,j}}}}+\frac{{Y_j^\rho }}{{a{y_j}}}} \right)}^{\frac{1}{\rho }}} - {Z_j}} \right]$</span></p>
<p>Later, I understand that once you get the optimal demands with the first order conditions of this problem, then you evaluate what happens to these when sigma tends to 0 and these functions should approximate <span class="math-container">$(1)$</span> and <span class="math-container">$(2)$</span>. I already solved this problem, <strong>but  I don't know how from the first-order conditions of the above problem we arrive at equations (1) and (2). Can someone please tell me how to do this?</strong></p>
<p>This is the first order condition:</p>
<p><span class="math-container">${X_{1,j}} = {Z_j}{\left[ {\frac{1}{{a{x_{1,j}}}} + \frac{1}{{a{x_{2,j}}}}{{\left( {\frac{{p_2^q}}{{p_1^q}}} \right)}^{\frac{\rho }{{\rho  - 1}}}}{{\left( {\frac{{a{x_{2,j}}}}{{a{x_{1,j}}}}} \right)}^{\frac{\rho }{{\rho  - 1}}}} + \frac{1}{{a{y_j}}}{{\left( {\frac{{p_2^y}}{{p_1^q}}} \right)}^{\frac{\rho }{{\rho  - 1}}}}{{\left( {\frac{{a{y_j}}}{{a{x_{1,j}}}}} \right)}^{\frac{\rho }{{\rho  - 1}}}}} \right]^{( - )\frac{1}{\rho }}}$</span></p>

CC BY-SA 4.0
2020-07-03T16:24:09.590Z
What book is this taken from?
Michael Greinecker
CC BY-SA 4.0
2020-07-03T18:17:48.983Z
The book is "Textbook of Computable General  Equilibrium Modelling_ Programming and Simulations" on  the page 90 of the book. This is the reference:                                                       
Hosoe, N., Gasawa, K., & Hashimoto, H. (2010). Textbook of computable general equilibrium modeling: programming and simulations. Springer.
Jose Tapias
CC BY-SA 4.0
2020-07-03T18:35:34.503Z
The way you stated your question left a few things out that are in the book, such as the intermediate goods and the definition of $Y_j$. It should certainly be easier to obtain the conditions directly; see [here](https://economics.stackexchange.com/questions/2969/finding-demand-function-given-a-utility-minx-y-function/2972#2972) how one can work with maximizing $\min$-functions in the context of a consumer problem.
Michael Greinecker
CC BY-SA 4.0
2020-07-03T20:27:11.013Z
I know where to get the functions intuitively, the first stage is not important for the solution of this problem, $Y_j$ is simply another production cost (compound production factor). But in the footnote number 7 of that chapter (chapter 6), they say that they first solve the problem restricted to a CES and then take out the limit of the first order conditions. Can you read it? What I understand from that page request is what I am trying to do because it seems to me a cleaner way to get to the demands.
Jose Tapias
CC BY-SA 4.0
2020-07-03T20:31:17.627Z
In the link, I explain how to formally prove what the optimum is. This is actually more rigorous than proving that some condition is a limit of first-order conditions for CES functions.
Michael Greinecker
CC BY-SA 4.0
2020-07-04T14:22:22.147Z
Thanks Michael, but that is not what I  I'm looking for. Thanks for taking the time to comment.
Jose Tapias
<macroeconomics><microeconomics><mathematical-economics><production-function><leontief>
how to calculate Leontief demand functions from first order conditions of a CES function when sigma tends to 0?
Jose Tapias
# how to calculate Leontief demand functions from first order conditions of a CES function when sigma tends to 0?

(Asked by: username_0 on 2020-07-02)

**This question is NOT about how to approximate a CES function to a leontief function.**

Knowing that:

$i= good (\begin{array}{\*{20}{c}}
{1}&{or}&{2}
\end{array})$

$j= firm (\begin{array}{\*{20}{c}}
{1}&{or}&{2}
\end{array})$

I have this optimization problem:

$\begin{array}{\*{20}{c}}
{\begin{array}{\*{20}{c}}
{\mathop {Maximize}\limits\_{{Z\_j},{Y\_j},{X\_{i,j}}} }&{\pi \_j^z}
\end{array} = p\_j^z{Z\_j} - \left( {p\_j^y{Y\_j} + \sum\limits\_i {p\_i^q{X\_{i,j}}} } \right)}&{(0.1)}
\end{array}$

subject to

$\begin{array}{\*{20}{c}}
{{Z\_j} = \min \left( {\frac{{{X\_{1,j}}}}{{a{x\_{1,j}}}},\frac{{{X\_{2,j}}}}{{a{x\_{2,j}}}},\frac{{{Y\_j}}}{{a{y\_j}}}} \right)}&{(0.2)}
\end{array}$

In the book that I am following, they arrive at the following first-order conditions:

$\begin{array}{l}
\begin{array}{\*{20}{c}}
{{X\_{i,j}} = a{x\_{i,j}}{Z\_j}}&{\forall i,j}&{(1)}
\end{array}\\
\begin{array}{\*{20}{c}}
{{Y\_{j}} = a{y\_i}Z}&{\forall j}&{(2)}
\end{array}\\
\begin{array}{\*{20}{c}}
{{Z\_j} = \min \left( {\frac{{{X\_{1,j}}}}{{a{x\_{1,j}}}},\frac{{{X\_{2,j}}}}{{a{x\_{2,j}}}},\frac{{{Y\_j}}}{{a{y\_j}}}} \right)}&{\forall j}&{(3)}
\end{array}
\end{array}$

Then in a footnote they say the following: "we cannot solve this profit-maximization problem by simply applying the Lagrange multiplier method because the Leontief-type production function $(0)$ is not differentiable.To derive the intermediate input demand function $(1)$ and the composite factor demand function $(2)$, we have to use a constant elasticity of substitution (CES) function, which is a generalized functional form of the Leontief-type function, and derive these first-order conditions. **When the elasticity of substitution in the CES function tends to zero, the derived first-order conditions approach the expressions of $(0)$, $(1)$ and $(3)$.**"

What I understand from the bolded part above is that you first solve the optimization problem represented by the equations $(0.1)$ and $(0.2)$ but using a CES function instead of $(0.2)$. I approach the problem as follows:

Knowing that:

$\begin{array}{\*{20}{c}}
{\sigma = \frac{1}{{1 - \rho }}}& \wedge &{\rho = \frac{{\sigma - 1}}{\sigma }}
\end{array}$

The problem is:

$L({Z\_j},{Y\_j},{X\_{i,j}}) = p\_j^z{Z\_j} - \left( {p\_j^y{Y\_j} + \sum\limits\_i {p\_i^q{X\_{i,j}}} } \right) + {\psi \_i}\left[ {{{\left( {\frac{{X\_{1,j}^\rho }}{{a{x\_{1,j}}}}+\frac{{X\_{2,j}^\rho }}{{a{x\_{2,j}}}}+\frac{{Y\_j^\rho }}{{a{y\_j}}}} \right)}^{\frac{1}{\rho }}} - {Z\_j}} \right]$

Later, I understand that once you get the optimal demands with the first order conditions of this problem, then you evaluate what happens to these when sigma tends to 0 and these functions should approximate $(1)$ and $(2)$. I already solved this problem, **but I don't know how from the first-order conditions of the above problem we arrive at equations (1) and (2). Can someone please tell me how to do this?**

This is the first order condition:

${X\_{1,j}} = {Z\_j}{\left[ {\frac{1}{{a{x\_{1,j}}}} + \frac{1}{{a{x\_{2,j}}}}{{\left( {\frac{{p\_2^q}}{{p\_1^q}}} \right)}^{\frac{\rho }{{\rho - 1}}}}{{\left( {\frac{{a{x\_{2,j}}}}{{a{x\_{1,j}}}}} \right)}^{\frac{\rho }{{\rho - 1}}}} + \frac{1}{{a{y\_j}}}{{\left( {\frac{{p\_2^y}}{{p\_1^q}}} \right)}^{\frac{\rho }{{\rho - 1}}}}{{\left( {\frac{{a{y\_j}}}{{a{x\_{1,j}}}}} \right)}^{\frac{\rho }{{\rho - 1}}}}} \right]^{( - )\frac{1}{\rho }}}$

**Tags:** macroeconomics, microeconomics, mathematical-economics, production-function, leontief
> **Comments:**
> **username_1**: What book is this taken from?  
> **username_0**: The book is "Textbook of Computable General  Equilibrium Modelling_ Programming and Simulations" on  the page 90 of the book. This is the reference:                                                        Hosoe, N., Gasawa, K., & Hashimoto, H. (2010). Textbook of computable general equilibrium modeling: programming and simulations. Springer.  
> **username_1**: The way you stated your question left a few things out that are in the book, such as the intermediate goods and the definition of $Y_j$. It should certainly be easier to obtain the conditions directly; see [here](https://economics.stackexchange.com/questions/2969/finding-demand-function-given-a-utility-minx-y-function/2972#2972) how one can work with maximizing $\min$-functions in the context of a consumer problem.  
> **username_0**: I know where to get the functions intuitively, the first stage is not important for the solution of this problem, $Y_j$ is simply another production cost (compound production factor). But in the footnote number 7 of that chapter (chapter 6), they say that they first solve the problem restricted to a CES and then take out the limit of the first order conditions. Can you read it? What I understand from that page request is what I am trying to do because it seems to me a cleaner way to get to the demands.  
> **username_1**: In the link, I explain how to formally prove what the optimum is. This is actually more rigorous than proving that some condition is a limit of first-order conditions for CES functions.  
> **username_0**: Thanks Michael, but that is not what I  I'm looking for. Thanks for taking the time to comment.  

---

## 0 answers

