Wednesday, June 3, 2015

The Input Output Table (Closed Economy)

Understanding IO Table for a Closed Economy.

1. Making Sense of Closed Economy

Check this video before you start:


Lets try to understand a closed economy. Say, there are two industry ${{I}_{1}}$ and ${{I}_{2}}$. Say ${{I}_{1}}$ produces commodity ${{S}_{1}}$ and ${{I}_{2}}$ produces commodity ${{S}_{2}}$. To produce, they need the factors of production ($FOP$) say Labor $L$ and capital $K$. These  $FOP$ are supplied by the household ($HH$), let me index household by ${{I}_{0}}$.

Say the total quantity demand of commodity ${{S}_{1}}$ is ${{X}_{1}}$, of that total ${{X}_{1}}$, ${{I}_{1}}$ demands ${{X}_{11}}$, ${{I}_{2}}$ demands  ${{X}_{12}}$ and ${{I}_{0}}$ i.e household demands ${{X}_{10}}$. Or ${{X}_{11}}+{{X}_{12}}+{{X}_{10}}={{X}_{1}}$

Similarly, Say the total quantity demand of commodity ${{S}_{2}}$ is ${{X}_{2}}$, of that total ${{X}_{2}}$, ${{I}_{1}}$ demands ${{X}_{21}}$, ${{I}_{2}}$ demands  ${{X}_{22}}$ and ${{I}_{0}}$ i.e household demands ${{X}_{20}}$. Or ${{X}_{21}}+{{X}_{22}}+{{X}_{20}}={{X}_{2}}$

Now, the total quantity demand of labor $L$ is ${{X}_{3}}$, of that total ${{X}_{3}}$, ${{I}_{1}}$ demands ${{X}_{31}}$, ${{I}_{2}}$ demands  ${{X}_{32}}$ and ${{I}_{0}}$ i.e household demand is zero because the $HH$ is the supplier of $FOP$. Or ${{X}_{31}}+{{X}_{32}}={{X}_{3}}$.
  
Now, the total quantity demand of capital $K$ is ${{X}_{4}}$, of that total ${{X}_{4}}$, ${{I}_{1}}$ demands ${{X}_{41}}$, ${{I}_{2}}$ demands  ${{X}_{42}}$ and ${{I}_{0}}$ i.e household demand is zero because the $HH$ is the supplier of $FOP$. Or ${{X}_{41}}+{{X}_{42}}={{X}_{4}}$.

2. IO Table in Quantities


Simple IO Table
${{I}_{1}}$
${{I}_{2}}$
${{I}_{0}}$
Total Demand
Notation
${{S}_{1}}$
${{ X}_{11}}$
${{ X}_{12}}$
${{ X}_{10}}$
${{ X}_{1}}$
\[\] \[\sum\limits_{j=0}^{2}{{{X}_{i}}_{j}}={{X}_{i}}\] for $i=1$ and $2$
${{S}_{2}}$
${{ X}_{21}}$
${{ X}_{22}}$
${{ X}_{20}}$
${{ X}_{2}}$
$L$
${{ X}_{31}}$
${{ X}_{32}}$

${{ X}_{3}}$
\[\] \[\sum\limits_{j=1}^{2}{{{X}_{i}}_{j}}={{X}_{i}}\] for $i=3$ and $4$

$K$
${{ X}_{41}}$
${{ X}_{42}}$

${{ X}_{4}}$
Total



.\[\sum\limits_{i=1}^{4}{{{X}_{i}}}\]



3. Converting Quantities IO table to Dollar Value IO Table

This IO table is given in format of quantities, therefore say unit of ${{S}_{1}}$ is kilogram and unit of ${{S}_{2}}$ is liters then these quantities are not comparable. Therefore, to make comparable we will express them in dollar values. For that we will multiplies each of the quantities with their respective prices. Say the price of ${{S}_{1}}$, ${{S}_{2}}$, $L$ and $K$ are ${{P}_{1}}$, ${{P}_{2}}$, ${{P}_{3}}$ and ${{P}_{4}}$ respectively. Now, our table would look like:


Simple IO Table
${{I}_{1}}$
${{I}_{2}}$
${{I}_{0}}$
Total Demand
Notation
${{S}_{1}}$
\[{{P}_{1}}{{X}_{11}}\]
\[{{P}_{1}}{{X}_{12}}\]
\[{{P}_{1}}{{X}_{10}}\]
\[{{P}_{1}}{{X}_{1}}\]
$\sum\limits_{j=0}^{2}{{{P}_{i}}{{X}_{ij}}}$ for i=1 and 2
${{S}_{2}}$
\[{{P}_{2}}{{X}_{21}}\]
\[{{P}_{2}}{{X}_{22}}\]
\[{{P}_{2}}{{X}_{20}}\]
\[{{P}_{2}}{{X}_{2}}\]
$L$
\[{{P}_{3}}{{X}_{31}}\]
\[{{P}_{3}}{{X}_{32}}\]

\[{{P}_{3}}{{X}_{3}}\]
$\sum\limits_{j=1}^{2}{{{P}_{i}}{{X}_{ij}}}$ for i=3 and 4
$K$
\[{{P}_{4}}{{X}_{41}}\]
\[{{P}_{4}}{{X}_{42}}\]

\[{{P}_{4}}{{X}_{4}}\]
Total
$\sum\limits_{i=1}^{4}{{{P}_{i}}{{X}_{ij}}}$ for j=1 and 2
 $\sum\limits_{i=1}^{2}{{{P}_{i}}{{X}_{i0}}}$ for i=1 and 2
$\sum\limits_{i=1}^{4}{{{P}_{i}}{{X}_{i}}}$ 
 .$\sum\limits_{j=0}^{2}{{{P}_{i}}{{X}_{ij}}}$ for i=1,2,3 and 4


4. Understanding Each Part of IO Table


Let's extend the concept further.

The sum of dollar value of commodity demand is $\sum\limits_{j=0}^{2}{{{P}_{i}}{{X}_{ij}}}$ for i=1 and 2. Let's name this as DVCOMThe sum of dollar value of factor demanded or say household income is $\sum\limits_{j=1}^{2}{{{P}_{i}}{{X}_{ij}}}$ for i=3 and 4. Lets name this as DVFACThe sum of dollar values of household expenditure is  $\sum\limits_{i=1}^{2}{{{P}_{i}}{{X}_{i0}}}$ for i=1 and 2. Lets name this as HCONS. The total cost of producing ${{S}_{1}}$ and ${{S}_{2}}$ by industries ${{I}_{1}}$ and ${{I}_{2}}$ is $\sum\limits_{i=1}^{4}{{{P}_{i}}{{X}_{ij}}}$ for j=1 and 2. This equation comprise of two major equations

a) Dollar values of intermediate commodities
i.e sum of ${{P}_{1}}{{X}_{11}} + {{P}_{1}}{{X}_{12}} + {{P}_{2}}{{X}_{21}}+ {{P}_{2}}{{X}_{22}}$ , this can be written as $\sum\limits_{i=1}^{2}{{{P}_{i}}{{X}_{ij}}}$ for j=1 and 2. Lets give this name as DVCOMIN.

b) Dollar values of Factors inputs
 i.e sum of ${{P}_{3}}{{X}_{31}} + {{P}_{3}}{{X}_{32}} + {{P}_{4}}{{X}_{41}} + {{P}_{4}}{{X}_{42}}$, this can be written as $\sum\limits_{i=3}^{4}{{{P}_{i}}{{X}_{ij}}}$ for j=3 and 4. Lets give this name as DVFACIN.

Now, once we are given DVCOMIN, DVFACIN and HCONS, all the table can be developed easily.

Watch this video to understand above equations and logic.



In the next blog, based upon the IO table, I will formulate the Consumer's Problem and solve for the optimum solution then I will also provide and derive the linearization of solution.


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