# 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.
* Next year's external demands are: up $200$ from this year for steel, and unchanged for autos.
* Next year's external demands are: up $100$ for steel, and up $200$ for autos.
* Next year's external demands are: up $200$ for steel, and up $200$ for autos.

### Answer

These answers were given by Octave.
* With the external use of steel as $17\,789$ and the external use of autos as $21\,243$, we get $s=25\,952$, $a=30\,312$.
* $s=25\,857$, $a=30\,596$
* $s=25\,984$, $a=30\,597$

### Problem 2

In the steel-auto system, the ratio for the use of steel by the auto industry is $2\,664/30\,346$, about $0.0878$. Imagine that a new process for making autos reduces this ratio to $.0500$.
* 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 $17,589$ for steel and $21,243$ for autos)?
* Predict next year's totals if, in addition, the external demand for autos rises to be $21,500$ because the new cars are cheaper.

### Answer

Octave gives these answers.
* $s=24\,244$, $a=30\,307$
* $s=24\,267$, $a=30\,673$

### 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 |
* Solve for total output if next year's external demands are: steel's demand up 10% and auto's demand up 15%.
* How do the ratios compare to those given above in the discussion for the 1958 economy?
* 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?

### Answer

* These are the equations. Octave gives $s=20.66$ and $a=16.41$.
    ${\begin{array}{*{2}{rc}r}(11.79/18.69)s&-&(1.28/14.27)a&=&11.56\\-(0/18.69)s&+&(9.87/14.27)a&=&11.35\end{array}}$
* 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
* Octave gives (in billions of 1947 dollars) $s=24.82$ and $a=23.63$. In billions of 1958 dollars that is $s=32.26$ and $a=30.71$.

### 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 | 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.
* $625$ for farm, $200$ for rail, $475$ for shipping
* $650$ for farm, $150$ for rail, $450$ 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 | 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.
* Fill in the numbers for this year's external demands.
* Set up the linear system, leaving next year's external demands blank.
* 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?
* 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. (1951), "Input-Output Economics", Scientific American, 185 (4): 15 {{citation}}: Unknown parameter |month= ignored (help).
* Clark, David H.; Coupe, John D. (1967), "The Bangor Area Economy Its Present and Future", Reprot to the City of Bangor, ME {{citation}}: Cite has empty unknown parameter: |1= (help); Unknown parameter |month= ignored (help).