Wednesday, June 24, 2015

Alphas and Betas in IO table assuming CD function

We have already derived the Alphas (here) and Beta (here). The main reason for this post is to find out the Alphas and Betas in the Input Output Table under the assumption of Cobb Douglas functions.

In this IO table in algebraic expression is given below.

In the values, Lets define this above table as:

The Alphas are the parameters in Cobb Douglas Function. Remember we assumed both the utility and production as CD function:

  \[\begin{align} & \frac{{{P}_{i}}{{X}_{ij}}}{\sum{{{P}_{i}}{{X}_{ij}}}}={{\alpha }_{ij}} \\ \end{align}\]

Or, in other words for CD function, the alphas are the initial share of ith commodity/factors in total inputs of j-th industry.Therefore, in the table we can derive the alphas as:

Note:  Within this table of Alphas, we have named ALPHACOM (share of ith commodity in total inputs of j-th industry) and ALPHAFAC (share of ith factors in total inputs of j-th industry).

The definition for Betas are shares. See my previous post for Betas (here). In this IO table the Betas are:

Note: Within the above table of Betas, We have named BCOM (share of j-th industry in total demand for i-th commodity), BHOUS(share of household in total demand for i-th commodity) and BFAC(share of j-th industry in total demand for i-th factor).

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