# Linear Algebra/Topic: Input-Output Analysis

|current=Topic: Input-Output Analysis
|previous=Topic: Computer Algebra Systems
An economy is an immensely complicated network of interdependences.
Changes in one part can ripple out to affect other parts. Economists
have struggled to be able to describe, and to make predictions about,
such a complicated object. Mathematical models using systems of linear
equations have emerged as a key tool. One is Input-Output Analysis,
pioneered by W. Leontief, who won the 1973 Nobel Prize in Economics.

Consider an economy with many parts, two of which are the steel industry
and the auto industry. As they work to meet the demand for their product
from other parts of the economy, that is, from users external to the
steel and auto sectors, these two interact tightly. For instance, should
the external demand for autos go up, that would lead to an increase in
the auto industry\'s usage of steel. Or, should the external demand for
steel fall, then it would lead to a fall in steel\'s purchase of autos.
The type of Input-Output model we will consider takes in the external
demands and then predicts how the two interact to meet those demands.

We start with a listing of production and consumption statistics. (These
numbers, giving dollar values in millions, are excerpted from
, describing the 1958 U.S. economy.
Today\'s statistics would be quite different, both because of inflation
and because of technical changes in the industries.)

```{=html}
<center>
```
+-----------------+----------------+----------------+----------------+-------------+
|                 |   *used by*  \ |   *used by*  \ |   *used by*  \ |   *total*   |
|                 |   *steel*      |   *auto*       |   *others*     |             |
+-----------------+----------------+----------------+----------------+-------------+
|   *value of*  \ |   5 395        |   2 664        |                |   25 448    |
|   *steel*       |                |                |                |             |
+-----------------+----------------+----------------+----------------+-------------+
|   *value of*  \ |   48           |   9 030        |                |   30 346    |
|   *auto*        |                |                |                |             |
+-----------------+----------------+----------------+----------------+-------------+

```{=html}
</center>
```
For instance, the dollar value of steel used by the auto industry in
this year is $2,664$ million. Note that industries may consume some of
their own output.

We can fill in the blanks for the external demand. This year\'s value of
the steel used by others this year is $17,389$ and this year\'s value of
the auto used by others is $21,268$. With that, we have a complete
description of the external demands and of how auto and steel interact,
this year, to meet them.

Now, imagine that the external demand for steel has recently been going
up by $200$ per year and so we estimate that next year it will be
$17,589$. Imagine also that for similar reasons we estimate that next
year\'s external demand for autos will be down $25$ to $21,243$. We wish
to predict next year\'s total outputs.

That prediction isn\'t as simple as adding $200$ to this year\'s steel
total and subtracting $25$ from this year\'s auto total. For one thing,
a rise in steel will cause that industry to have an increased demand for
autos, which will mitigate, to some extent, the loss in external demand
for autos. On the other hand, the drop in external demand for autos will
cause the auto industry to use less steel, and so lessen somewhat the
upswing in steel\'s business. In short, these two industries form a
system, and we need to predict the totals at which the system as a whole
will settle.

For that prediction, let $s$ be next years total production of steel and
let $a$ be next year\'s total output of autos. We form these equations.

$$\begin{array}{rl}
\text{next year}\textrm{'}\text{s production of steel}
&=\text{next year}\textrm{'}\text{s use of steel by steel}   \\
&\quad+\text{next year}\textrm{'}\text{s use of steel by auto}  \\             
&\quad+\text{next year}\textrm{'}\text{s use of steel by others} \\
\text{next year}\textrm{'}\text{s production of autos}
&=\text{next year}\textrm{'}\text{s use of autos by steel}   \\
&\quad+\text{next year}\textrm{'}\text{s use of autos by auto}  \\             
&\quad+\text{next year}\textrm{'}\text{s use of autos by others} 
\end{array}$$

On the left side of those equations go the unknowns $s$ and $a$. At the
ends of the right sides go our external demand estimates for next year
$17,589$ and $21,243$. For the remaining four terms, we look to the
table of this year\'s information about how the industries interact.

For instance, for next year\'s use of steel by steel, we note that this
year the steel industry used $5395$ units of steel input to produce
$25,448$ units of steel output. So next year, when the steel industry
will produce $s$ units out, we expect that doing so will take
$s\cdot (5395)/(25\,448)$ units of steel input--- this is simply the
assumption that input is proportional to output. (We are assuming that
the ratio of input to output remains constant over time; in practice,
models may try to take account of trends of change in the ratios.)

Next year\'s use of steel by the auto industry is similar. This year,
the auto industry uses $2664$ units of steel input to produce $30346$
units of auto output. So next year, when the auto industry\'s total
output is $a$, we expect it to consume $a\cdot (2664)/(30346)$ units of
steel.

Filling in the other equation in the same way, we get this system of
linear equation.

$$\begin{array}{*{3}{rc}r}
{\displaystyle\frac{5\,395}{25\,448}}\cdot s 
&+ &{\displaystyle\frac{2\,664}{30\,346}}\cdot a &+ &17\,589 
&= &s \\[1em]  
{\displaystyle\frac{48}{25\,448}}\cdot s     
&+ &{\displaystyle\frac{9\,030}{30\,346}}\cdot a &+ &21\,243 
&= &a
\end{array}$$

Gauss\' method on this system.

$$\begin{array}{*{2}{rc}r}
(20\,053/25\,448)s &- &(2\,664/30\,346)a &= &17\,589 \\ 
-(48/25\,448)s      &+ &(21\,316/30\,346)a &= &21\,243 
\end{array}$$

gives $s=25\,698$ and $a=30\,311$.

Looking back, recall that above we described why the prediction of next
year\'s totals isn\'t as simple as adding $200$ to last year\'s steel
total and subtracting $25$ from last year\'s auto total. In fact,
comparing these totals for next year to the ones given at the start for
the current year shows that, despite the drop in external demand, the
total production of the auto industry is predicted to rise. The increase
in internal demand for autos caused by steel\'s sharp rise in business
more than makes up for the loss in external demand for autos.

One of the advantages of having a mathematical model is that we can ask
\"What if \...?\" questions. For instance, we can ask \"What if the
estimates for next year\'s external demands are somewhat off?\" To try
to understand how much the model\'s predictions change in reaction to
changes in our estimates, we can try revising our estimate of next
year\'s external steel demand from $17,589$ down to $17,489$, while
keeping the assumption of next year\'s external demand for autos fixed
at $21,243$. The resulting system

$$\begin{array}{*{2}{rc}r}
(20\,053/25\,448)s &- &(2\,664/30\,346)a &= &17\,489 \\ 
-(48/25\,448)s      &+ &(21\,316/30\,346)a &= &21\,243 
\end{array}$$

when solved gives $s=25\,571$ and $a=30\,311$. This kind of exploration
of the model is **sensitivity analysis**. We are seeing how sensitive
the predictions of our model are to the accuracy of the assumptions.

Obviously, we can consider larger models that detail the interactions
among more sectors of an economy. These models are typically solved on a
computer, using the techniques of matrix algebra that we will develop in
Chapter Three. Some examples are given in the exercises. Obviously also,
a single model does not suit every case; expert judgment is needed to
see if the assumptions underlying the model are reasonable for a
particular case. With those caveats, however, this model has proven in
practice to be a useful and accurate tool for economic analysis. For
further reading, try  and
.

## Exercises

*Hint: these systems are easiest to solve on a computer.*   `{{TextBox|1=
;Problem 3: 
This table gives the numbers for the auto-steel system from
a different year, 1947 (see {{harvnb|Leontief|1951}}).
The units here are billions of 1947 dollars.

<center>
<TABLE style="text-align:center;" border=0px cellpadding=0px cellspacing=0px>
<TR>
<TD style="border-right:2px solid black; border-bottom:2px solid black;">
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''used by''&nbsp;&nbsp;<br>&nbsp;&nbsp;''steel''&nbsp;&nbsp; 
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''used by''&nbsp;&nbsp; <br>&nbsp;&nbsp;''auto''&nbsp;&nbsp; 
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''used by''&nbsp;&nbsp;<br>&nbsp;&nbsp;''others''&nbsp;&nbsp;
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''total''&nbsp;&nbsp;
<TR> 
<TD style="border-right:2px solid black;"> &nbsp;&nbsp;''value of''&nbsp;&nbsp;<br>&nbsp;&nbsp;''steel''&nbsp;&nbsp;
<TD>&nbsp;&nbsp;6.90&nbsp;&nbsp;
<TD>&nbsp;&nbsp;1.28&nbsp;&nbsp;
<TD>
<TD>&nbsp;&nbsp;18.69&nbsp;&nbsp;
<TR>
<TD style="border-right:2px solid black;">&nbsp;&nbsp;''value of''&nbsp;&nbsp;<br>&nbsp;&nbsp;''auto''&nbsp;&nbsp;
<TD>&nbsp;&nbsp;0&nbsp;&nbsp;
<TD>&nbsp;&nbsp;4.40&nbsp;&nbsp;
<TD>
<TD>&nbsp;&nbsp;14.27&nbsp;&nbsp;
</TABLE>
</center>
<ol type=1 start=1>
<li> Solve for total output if next year's external
demands are: steel's demand up 10% and auto's demand up
15%.
<li> How do the ratios compare to those given above in the
discussion for the 1958 economy?
<li> Solve the 1947 equations with the 1958 external demands
(note the difference in units; a 1947 dollar buys about
what $1.30 in 1958 dollars buys).
How far off are the predictions for total output?
</ol>
}}`{=mediawiki}  `{{TextBox|1=
;Problem 5: 
This table gives the interrelationships among three segments of an 
economy (see {{harvnb|Clark|Coupe|1967}}).
<center>
<TABLE style="text-align:center;" border=0px cellpadding=0px cellspacing=0px>
<TR>
<TD style="border-right:2px solid black; border-bottom:2px solid black;">
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''used by''&nbsp;&nbsp;<br>&nbsp;&nbsp;''food''&nbsp;&nbsp; 
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''used by''&nbsp;&nbsp; <br>&nbsp;&nbsp;''wholesale''&nbsp;&nbsp; 
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''used by''&nbsp;&nbsp; <br>&nbsp;&nbsp;''retail''&nbsp;&nbsp; 
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''used by''&nbsp;&nbsp;<br>&nbsp;&nbsp;''others''&nbsp;&nbsp;
<TD style="border-bottom:2px solid black;">&nbsp;&nbsp;''total''&nbsp;&nbsp;
<TR> 
<TD style="border-right:2px solid black;"> &nbsp;&nbsp;''value of''&nbsp;&nbsp;<br>&nbsp;&nbsp;''food''&nbsp;&nbsp;
<TD>&nbsp;&nbsp;0&nbsp;&nbsp;
<TD>&nbsp;&nbsp;2&thinsp;318&nbsp;&nbsp;
<TD>&nbsp;&nbsp;4&thinsp;679&nbsp;&nbsp;
<TD>
<TD>&nbsp;&nbsp;11&thinsp;869&nbsp;&nbsp;
<TR>
<TD style="border-right:2px solid black;">&nbsp;&nbsp;''value of''&nbsp;&nbsp;<br>&nbsp;&nbsp;''wholesale''&nbsp;&nbsp;
<TD>&nbsp;&nbsp;393&nbsp;&nbsp;
<TD>&nbsp;&nbsp;1&thinsp;089&nbsp;&nbsp;
<TD>&nbsp;&nbsp;22&thinsp;459&nbsp;&nbsp;
<TD>
<TD>&nbsp;&nbsp;122&thinsp;242&nbsp;&nbsp;
<TR>
<TD style="border-right:2px solid black;">&nbsp;&nbsp;''value of''&nbsp;&nbsp;<br>&nbsp;&nbsp;''retail''&nbsp;&nbsp;
<TD>&nbsp;&nbsp;3&nbsp;&nbsp;
<TD>&nbsp;&nbsp;53&nbsp;&nbsp;
<TD>&nbsp;&nbsp;75&nbsp;&nbsp;
<TD>
<TD>&nbsp;&nbsp;116&thinsp;041&nbsp;&nbsp;
</TABLE>
</center>
We will do an Input-Output analysis on this
system.
<ol type=1 start=1>
<li> Fill in the numbers for this year's external
demands.
<li> Set up the linear system, leaving next year's
external demands blank.
<li> Solve the system where next year's external demands
are calculated by taking this year's external demands and
inflating them 10%.
Do all three sectors increase their total business by
10%?
Do they all even increase at the same rate? 
<li> Solve the system where next year's external demands
are calculated by taking this year's external demands and
reducing them 7%.
(The study from which these numbers are taken concluded that
because of the closing of a local military facility, overall
personal income in the area would fall 7%, so this might 
be a first guess at what would actually happen.)
/Solutions/

## References

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