Friday, June 5, 2015

What's that Alpha?

To understand ALPHA ( ${{\alpha}_{ij}}$ ), you have to understand the producer’s problem and it optimum solution. You can find information (here). I stress in that page that we will use equation ${{X}_{ij}}=\frac{{{\alpha }_{ij}}\lambda {{X}_{j}}}{{{P}_{i}}}$ frequently. \[\begin{align} & {{X}_{ij}}=\frac{{{\alpha }_{ij}}\lambda {{X}_{j}}}{{{P}_{i}}} \\ & or,{{P}_{i}}{{X}_{ij}}={{\alpha }_{ij}}\lambda {{X}_{j}} \\ \end{align}\] Taking shares, we get \[\begin{align} & \frac{{{P}_{i}}{{X}_{ij}}}{\sum{{{P}_{i}}{{X}_{ij}}}}=\frac{{{\alpha }_{ij}}\lambda {{X}_{j}}}{\sum{{{\alpha }_{ij}}\lambda {{X}_{j}}}} \\ & or,\frac{{{P}_{i}}{{X}_{ij}}}{\sum{{{P}_{i}}{{X}_{ij}}}}=\frac{{{\alpha }_{ij}}}{\sum{{{\alpha }_{ij}}}} \\ \end{align}\] Remember that \[\sum{{{\alpha }_{ij}}}=1\] , so \[\begin{align} & \frac{{{P}_{i}}{{X}_{ij}}}{\sum{{{P}_{i}}{{X}_{ij}}}}=\frac{{{\alpha }_{ij}}}{1} \\ & \therefore \frac{{{P}_{i}}{{X}_{ij}}}{\sum{{{P}_{i}}{{X}_{ij}}}}={{\alpha }_{ij}} \\ \end{align}\]
Now, I leave you with the table and a question, so what is that Alpha?

Finally, let me correct you, the value $\frac{{{P}_{i}}{{X}_{ij}}}{\sum{{{P}_{i}}{{X}_{ij}}}}={{\alpha }_{ij}}$ holds true only for the Cobb Douglas production function.

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