Suppose the coal and steel industries form an open economy. Every \$1 produced by the coal industry requires \$0.15 of coal and \$0.20 of steel. Every \$1 produced by steel requires \$0.25 of coal and \$0.10 of steel. Suppose that there is an annual outside demand for \$45 million of coal and $124 million of steel. (a) How much should each industry produce to satisfy the demands?
In my attempt, I chose x1 and x2 represent the amount of output of coal and steel, respectively. Also, external demand + internal demand = what’s being produced, so:
45 + 0.15x1 + 0.20x2 = x1
124 + 0.25x1 + 0.102 = x2
Rearranging gives
0.85x1 - 0.20x2 = 45
-0.25x1 + 0.90x2 = 124
Solving this system gives x1 = 91.3286. . . and x2 = 163.1485. . .
However, I looked at the answer key of the textbook, and it says coal should produce 100 million (x1 = 100) and steel should produce 160 million (x2 = 160). These are suspiciously close to the values I got, but my answers are still wrong.
Is there something wrong with the equations I came up with? I even plugged in my answers to a calculator to check.
We have two equations, one for coal and one for steel.
$\textrm{"Every \$1 produced by steel requires \$0.25 of } \color{red}{\textrm{coal}}"$
This condition has to be in the coal constraint. On the other hand the condition
$\textrm{"Every \$1 produced by the coal industry requires ... \$0.20 of} \color{blue}{\textrm{ steel}}"$
is for the steel constraint:
\begin{align} & x=0.15x+0.25y+45 \quad \color{grey}{\textrm{coal}} \\ & y=0.20x+0.10y+124 \quad \color{grey}{\textrm{steel}}\\ \end{align}
Rearranging the terms gives
\begin{align} & \ \ \ \ 0.85x-0.25y=45 \\ & -0.20x+0.9y=124 \Rightarrow x=4.5y-620 \\ \end{align}
Insterting the value for x into the first constraint:
\begin{align} & \ \ \ \ 0.85\cdot (4.5y-620)-0.25y=45 \\& \\& (0.85\cdot 45-0.25)y=45+0.85\cdot 620 \\& \\& y= \frac{0.85\cdot 620+45}{0.85\cdot 4.5-0.25}=160 \quad \color{grey}{\textrm{steel}} \end{align}