Understanding IO Table for a Closed Economy.
1. Making Sense of Closed Economy
Check this video before you start:
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

$\sum\limits_{i=1}^{4}{{{P}_{i}}{{X}_{i}}}$

4. Understanding Each Part of IO Table
Let's extend the concept further.
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
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.
People Love Colors:
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.
People Love Colors:
Very interesting, thanks for help us.
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