Thursday, June 4, 2015

Producer's Problem and Solution

In our economy, there are two industries ${{I}_{1}}$ and ${{I}_{2}}$. Each of these industries tries to minimize their cost subjected to their production function.
Here is the video to have fun with these derivations.


Let’s formulate the problem:
$\min {{C}_{j}}=\sum\limits_{i=1}^{4}{{{P}_{i}}{{X}_{ij}}}$
$s.t.$
${{X}_{j}}={{A}_{j}}\prod\limits_{i=1}^{4}{X_{ij}^{{{\alpha }_{ij}}}}$
where, j=1 and 2 and $\sum\limits_{i=1}^{4}{{{\alpha }_{i1}}=1;}\sum\limits_{i=1}^{4}{{{\alpha }_{i2}}=1;}{{\alpha }_{ij}}>0$

Can you figure out what are the cell of these equation represents in following Colorful IO Table:



Let’s implement the Lagrange function:

$\begin{align}
  & L=\sum\limits_{i=1}^{4}{{{P}_{i}}{{X}_{ij}}}+\lambda \left( {{X}_{j}}-{{A}_{j}}\prod\limits_{i=1}^{4}{X_{ij}^{{{\alpha }_{ij}}}} \right) \\
 & or,L={{P}_{1}}{{X}_{1j}}+{{P}_{2}}{{X}_{2j}}+{{P}_{3}}{{X}_{3j}}+{{P}_{4}}{{X}_{4j}}+\lambda \left( {{X}_{j}}-{{A}_{j}}X_{1j}^{{{\alpha }_{1j}}}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}} \right) \\
 & or,L={{P}_{1}}{{X}_{1j}}+{{P}_{2}}{{X}_{2j}}+{{P}_{3}}{{X}_{3j}}+{{P}_{4}}{{X}_{4j}}+\lambda {{X}_{j}}-\lambda {{A}_{j}}X_{1j}^{{{\alpha }_{1j}}}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}} \\
\end{align}$

Taking first order derivative w.r.t ${{X}_{1}}$ :
$\begin{align}
  & \frac{dL}{d{{X}_{1}}}={{P}_{1}}+0+0+0+0-\lambda {{\alpha }_{1j}}{{A}_{j}}X_{1j}^{{{\alpha }_{1j}}-1}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}} \\
 & or,\frac{dL}{d{{X}_{1}}}={{P}_{1}}-\frac{{{\alpha }_{1j}}\lambda {{A}_{j}}X_{1j}^{{{\alpha }_{1j}}}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}}}{X_{1j}^{{{\alpha }_{1j}}}} \\
 & or,\frac{dL}{d{{X}_{1}}}={{P}_{1}}-\frac{{{\alpha }_{1j}}\lambda }{X_{1j}^{{{\alpha }_{1j}}}}{{A}_{j}}X_{1j}^{{{\alpha }_{1j}}}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}} \\
 & or,\frac{dL}{d{{X}_{1}}}={{P}_{1}}-\frac{{{\alpha }_{1j}}\lambda }{X_{1j}^{{{\alpha }_{1j}}}}{{X}_{j}} \\
\end{align}$
$\because {{X}_{j}}={{A}_{j}}\prod\limits_{i=1}^{4}{X_{ij}^{{{\alpha }_{ij}}}}={{A}_{j}}X_{1j}^{{{\alpha }_{1j}}}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}}$

First order condition,
$\begin{align}
  & or,0={{P}_{1}}-\frac{{{\alpha }_{1j}}\lambda }{{{X}_{1j}}}{{X}_{j}} \\
 & or,\frac{{{\alpha }_{1j}}\lambda }{{{X}_{1j}}}{{X}_{j}}={{P}_{1}} \\
 & \therefore {{X}_{1j}}=\frac{{{\alpha }_{1j}}\lambda {{X}_{j}}}{{{P}_{1}}} \\
\end{align}$

Now, let’s put this in general form,

${{X}_{ij}}=\frac{{{\alpha }_{ij}}\lambda {{X}_{j}}}{{{P}_{i}}}$   (i)

This equation (i) is most important equation and we will use this again and again.

Taking first order derivative w.r.t $\lambda $

$\frac{dL}{d\lambda }={{X}_{j}}-{{A}_{j}}X_{1j}^{{{\alpha }_{1j}}}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}}$

First order condition,
$\begin{align}
  & 0={{X}_{j}}-{{A}_{j}}X_{1j}^{{{\alpha }_{1j}}}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}} \\
 & or,{{X}_{j}}={{A}_{j}}X_{1j}^{{{\alpha }_{1j}}}X_{2j}^{{{\alpha }_{2j}}}X_{3j}^{{{\alpha }_{3j}}}X_{4j}^{{{\alpha }_{4j}}} \\
 & or,{{X}_{j}}={{A}_{j}}\prod\limits_{i=1}^{4}{X_{ij}^{{{\alpha }_{ij}}}} \\
\end{align}$

We know, ${{X}_{ij}}=\frac{{{\alpha }_{ij}}\lambda {{X}_{j}}}{{{P}_{i}}}$, let’s put value of ${{X}_{ij}}=\frac{{{\alpha }_{ij}}\lambda {{X}_{j}}}{{{P}_{i}}}$ in ${{X}_{j}}={{A}_{j}}\prod\limits_{i=1}^{4}{X_{ij}^{{{\alpha }_{ij}}}}$, we get
 \[or,{{X}_{j}}={{A}_{j}}\prod\limits_{i=1}^{4}{{{\left( \frac{{{\alpha }_{ij}}\lambda {{X}_{j}}}{{{P}_{i}}} \right)}^{{{\alpha }_{ij}}}}}\]

Since, ${{X}_{j}}$and $\lambda $  doesnot have ${{i}^{th}}$ index, we can take them out from the product.
\[\begin{align}
  & or,{{X}_{j}}={{A}_{j}}{{X}_{j}}\lambda \prod\limits_{i=1}^{4}{{{\left( \frac{{{\alpha }_{ij}}}{{{P}_{i}}} \right)}^{{{\alpha }_{ij}}}}} \\
 & or,1={{A}_{j}}\lambda \prod\limits_{i=1}^{4}{{{\left( \frac{{{\alpha }_{ij}}}{{{P}_{i}}} \right)}^{{{\alpha }_{ij}}}}} \\
 & or,\frac{1}{\lambda }={{A}_{j}}\prod\limits_{i=1}^{4}{\left( \frac{\alpha _{ij}^{{{\alpha }_{ij}}}}{P_{i}^{{{\alpha }_{ij}}}} \right)} \\
  & or,\lambda =\frac{1}{{{A}_{j}}}\prod\limits_{i=1}^{4}{\left( \frac{P_{i}^{{{\alpha }_{ij}}}}{\alpha _{ij}^{{{\alpha }_{ij}}}} \right)} \\
 & or,\lambda =\frac{1}{{{A}_{j}}}\prod\limits_{i=1}^{4}{\alpha _{ij}^{-{{\alpha }_{ij}}}}\prod\limits_{i=1}^{4}{P_{i}^{{{\alpha }_{ij}}}} \\
 & or,\lambda =\left( \frac{1}{{{A}_{j}}}\prod\limits_{i=1}^{4}{\alpha _{ij}^{-{{\alpha }_{ij}}}} \right)\prod\limits_{i=1}^{4}{P_{i}^{{{\alpha }_{ij}}}} \\
\end{align}\]

Let’s assume, \[Q=\left( \frac{1}{{{A}_{j}}}\prod\limits_{i=1}^{4}{\alpha _{ij}^{-{{\alpha }_{ij}}}} \right)\] then,
\[\therefore \lambda =Q\prod\limits_{i=1}^{4}{P_{i}^{{{\alpha }_{ij}}}}\]

Since we have value for $\lambda $ , now let’s put the value in equation (i), we get
\[\begin{align}
  & {{X}_{ij}}=\frac{{{\alpha }_{ij}}\lambda {{X}_{j}}}{{{P}_{i}}} \\
 & or,{{X}_{ij}}=\frac{{{\alpha }_{ij}}Q\prod\limits_{i=1}^{4}{P_{i}^{{{\alpha }_{ij}}}}}{{{P}_{i}}} \\
 & \therefore {{X}_{ij}}={{\alpha }_{ij}}{{X}_{j}}Q\prod\limits_{i=1}^{4}{P_{i}^{{{\alpha }_{ij}}}}/{{P}_{i}} \\
\end{align}\]

Now, in the research papers, you will directly see the values of \[{{X}_{ij}}={{\alpha }_{ij}}{{X}_{j}}Q\prod\limits_{i=1}^{4}{P_{i}^{{{\alpha }_{ij}}}}/{{P}_{i}}\] Where,  \[Q=\left( \frac{1}{{{A}_{j}}}\prod\limits_{i=1}^{4}{\alpha _{ij}^{-{{\alpha }_{ij}}}} \right)\]This represents the optimum level of input required for the production to minimize the cost subjected to given technology.

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