# Linear Algebra/Topic: Input-Output Analysis/Solutions

## Solutions

Hint: these systems are easiest to solve on a computer.

Problem 1

With the steel-auto system given above, estimate next year's total productions in these cases.

1. Next year's external demands are: up ${\displaystyle 200}$ from this year for steel, and unchanged for autos.
2. Next year's external demands are: up ${\displaystyle 100}$ for steel, and up ${\displaystyle 200}$ for autos.
3. Next year's external demands are: up ${\displaystyle 200}$ for steel, and up ${\displaystyle 200}$ for autos.

These answers were given by Octave.

1. With the external use of steel as ${\displaystyle 17\,789}$ and the external use of autos as ${\displaystyle 21\,243}$, we get ${\displaystyle s=25\,952}$, ${\displaystyle a=30\,312}$.
2. ${\displaystyle s=25\,857}$, ${\displaystyle a=30\,596}$
3. ${\displaystyle s=25\,984}$, ${\displaystyle a=30\,597}$
Problem 2

In the steel-auto system, the ratio for the use of steel by the auto industry is ${\displaystyle 2\,664/30\,346}$, about ${\displaystyle 0.0878}$. Imagine that a new process for making autos reduces this ratio to ${\displaystyle .0500}$.

1. How will the predictions for next year's total productions change compared to the first example discussed above (i.e., taking next year's external demands to be ${\displaystyle 17,589}$ for steel and ${\displaystyle 21,243}$ for autos)?
2. Predict next year's totals if, in addition, the external demand for autos rises to be ${\displaystyle 21,500}$ because the new cars are cheaper.

1. ${\displaystyle s=24\,244}$, ${\displaystyle a=30\,307}$
2. ${\displaystyle s=24\,267}$, ${\displaystyle a=30\,673}$
Problem 3

This table gives the numbers for the auto-steel system from a different year, 1947 (see Leontief 1951). The units here are billions of 1947 dollars.

 used by    steel used by     auto used by    others total value of    steel 6.90 1.28 18.69 value of    auto 0 4.40 14.27
1. Solve for total output if next year's external demands are: steel's demand up 10% and auto's demand up 15%.
2. How do the ratios compare to those given above in the discussion for the 1958 economy?
3. 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?
1. These are the equations.
${\displaystyle {\begin{array}{*{2}{rc}r}(11.79/18.69)s&-&(1.28/14.27)a&=&11.56\\-(0/18.69)s&+&(9.87/14.27)a&=&11.35\end{array}}}$
Octave gives ${\displaystyle s=20.66}$ and ${\displaystyle a=16.41}$.
2. These are the ratios.
 1947 by steel by autos use of steel 0.63 0.09 use of autos 0.00 0.69
 1958 by steel by autos use of steel 0.79 0.09 use of autos 0.00 0.70
3. Octave gives (in billions of 1947 dollars) ${\displaystyle s=24.82}$ and ${\displaystyle a=23.63}$. In billions of 1958 dollars that is ${\displaystyle s=32.26}$ and ${\displaystyle a=30.71}$.
Problem 4

Predict next year's total productions of each of the three sectors of the hypothetical economy shown below

 used by    farm used by     rail used by     shipping used by    others total value of    farm 25 50 100 500 value of    rail 25 50 50 300 value of    shipping 15 10 0 500

if next year's external demands are as stated.

1. ${\displaystyle 625}$ for farm, ${\displaystyle 200}$ for rail, ${\displaystyle 475}$ for shipping
2. ${\displaystyle 650}$ for farm, ${\displaystyle 150}$ for rail, ${\displaystyle 450}$ for shipping
Problem 5

This table gives the interrelationships among three segments of an economy (see Clark & Coupe 1967).

 used by    food used by     wholesale used by     retail used by    others total value of    food 0 2 318 4 679 11 869 value of    wholesale 393 1 089 22 459 122 242 value of    retail 3 53 75 116 041

We will do an Input-Output analysis on this system.

1. Fill in the numbers for this year's external demands.
2. Set up the linear system, leaving next year's external demands blank.
3. 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?
4. 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.)

## References

• Leontief, Wassily W. (Oct. 1951), "Input-Output Economics", Scientific American 185 (4): 15 .
• Clark, David H.; Coupe, John D. (Mar. 1967), "The Bangor Area Economy Its Present and Future", Reprot to the City of Bangor, ME .