Question 16.9.1: Once upon a time, in an ancient land perhaps not too far fro......
Once upon a time, in an ancient land perhaps not too far from Norway, an economy consisted of three industries—fishing, forestry, and boat building.
(i) To produce 1 ton of fish requires the services of α fishing boats.
(ii) To produce 1 ton of timber requires β tons of fish, as extra food for the energetic foresters.
(iii) To produce 1 fishing boat requires γ tons of timber.
These are the only inputs needed for each of these three industries. Suppose there is no final (external) demand for fishing boats. Find what gross outputs each of the three industries must produce in order to meet the final demands of d_{1} tons of fish to feed the general population, plus d_{2}tons of timber to build houses.
Let x_{1} denote the total number of tons of fish to be produced, x_{2} the total number of tons of timber, and x_{3} the total number of fishing boats.
Consider first the demand for fish. Because βx_{2} tons of fish are needed to produce x_{2} units of timber, and because the final demand for fish is d_{1}, we must have x_{1} = βx_{2} + d_{1}. Note that producing fishing boats does not require fish as an input, so there is no term with x_{3}.
In the case of timber, a similar argument shows that the equation x_{2} = γ x_{3} + d_{2} must be satisfied. Finally, for boat building, only the fishing industry needs boats; there is no final demand in this case, and so x_{3}= αx_{1}. Thus, the following three equations must be satisfied:
(\mathrm{i})\ x_{1}=\beta x_{2}+d_{1}\qquad\mathrm{~(ii)\ x_{2}=}\gamma x_{3}+d_{2}\qquad\quad(\mathrm{iii})\ x_{3}=\alpha x_{1}. (∗)
One way to solve these equations begins by using (iii) to insert x_{3} = αx_{1} into (ii). This gives x_{2} = γαx_{1} + d_{2}, which inserted into (i) yields x_{1} = αβγ x_{1} + βd_{2} + d_{1}. Solving this last equation for x_{1} gives x_{1} = (d_{1} + βd_{2})/(1 − αβγ ). The corresponding expressions for the two other variables are easily found, and the results are:
x_{1}={\frac{d_{1}+\beta d_{2}}{1-\alpha\beta\gamma}}, x_{2}={\frac{\alpha\gamma d_{1}+d_{2}}{1-\alpha\beta\gamma}}, \mathrm{~and~}\quad x_{3}=\frac{\alpha d_{1}+\alpha\beta d_{2}}{1-\alpha\beta\gamma}\qquad\qquad(**)Clearly, this solution for (x_{1}, x_{2}, x_{3}) only makes sense when αβγ < 1. In fact, if αβγ ≥ 1, it is impossible for this economy to meet any positive final demands for fish and timber—production in the economy is too inefficient.