Question

The Russian-born U.S. economist and Nobel laureate Wassily Leontief (1906-1999) was interested in the following question: What output should each of the industries in an economy produce to satisfy the total demand for all products? Here, we consider a very special example of input-output analysis, an economy with only two industries, A and B. Assume that the consumer demand for their products is, repectively, 1,000 and 780 in millions of dollars per year.

Industry A Industry B

Consumer

1,000 780

What outputs a and b (in millions of dollars per year ) should the two industries generate to satisfy the demand? You may be tempted to say 1,000 and 780, respectively, but things are not quite as simple as that. We have to take into account the interindustry demand as well. Let us say that the industry A produces electricity. Of course, producing almost any product will require electric power.

Suppose that industry B needs 10c/worth of electricity for each \(1 of output B produces and that industry A needs 20c/worth of B's products for each \)1 of output A produces. Find teh outputs a and b needed to satisy both consumer and interindustry demands.

Industry A Industry B

(output a) (output b)

Consumer

1,000 780

Short answer

Consumer and inter-industry demand are $\mathrm{a}=1100\mathrm{and}\mathrm{b}=1000$respectively.

Step-by-step solution

Formation the equation using the given information.

As per the second figure of the question industry, A needs to produce 0.1b

More products in a year and industry B need to produce 0.2a more products in a year.

Since basic outputs were $\mathrm{a}=1000\mathrm{and}\mathrm{b}=780$, the equation for increased productions will be:

$$\begin{gathered} \mathrm{a}=1000+0.1\mathrm{b} \\ \mathrm{b}=780+0.2\mathrm{a} \end{gathered}$$

Now, to solve this linear system, substituting$\mathrm{b}=780+0.2\mathrm{a}$ in first equation

$\begin{array}{rcl}\mathrm{a}& =& 1000+0.1\mathrm{b}\\ & =& 1000+0.1\left(780+0.2\mathrm{a}\right)\\ & =& 1000+78+0.02\mathrm{a}\\ \mathrm{a}-0.02\mathrm{a}& =& 1078\\ 0.98\mathrm{a}& =& 1078\\ \mathrm{a}& =& 1100\end{array}$

Substituting$\mathrm{a}=1100$ in the second equation to get the valu of b,

$\begin{array}{rcl}\mathrm{b}& =& 780+0.2\mathrm{a}\\ & =& 780+0.2\left(1100\right)\\ & =& 780+220\\ & =& 1000\end{array}$

Hence, we get$\mathrm{a}=1100\mathrm{and}\mathrm{b}=780$.

Step 2:-  Final answer

Thus, the ouputs needs to satisy both consumer and inter-industry demand are$\mathrm{a}=1100\mathrm{and}\mathrm{b}=780$.