Chapter 3.4, Problem 78E

This problem refers to Leontief’s input—output model, first discussed in the Exercises 1.1.24 and 1.2.39. Consider three industries $I_1,I_2,I_3$ , each of which produces only one good, with unit prices $p_1=2,p_2=5,p_3=10$ (in U.S. dollars), respectively. Let the three products be labeled good 1, good 2, and good 3. Let
$A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]=\left[\begin{array}{ccc}0.3& 0.2& 0.1\\ 0.1& 0.3& 0.3\\ 0.2& 0.2& 0.1\end{array}\right]$

be the matrix that lists the interindustry demand in terms of dollar amounts. The entry $a_{ij}$ tells us how many dollars’ worth of good i are required to produce one dollar’s worth of good j. Alternatively, the interindustry demand can be measured in units of goods by means of the matrix
$B=\left[\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right]$ ,
where $b_{ij}$ tells us how many units of good i are required to produce one unit of good j. Find the matrix B for the economy discussed here. Also, write an equation relating the three matrices A, B, and S. where
$S=\left[\begin{array}{ccc}2& 0& 0\\ 0& 5& 0\\ 0& 0& 10\end{array}\right]$

is the diagonal matrix listing the unit prices on the diagonal. Justify your answer carefully.