Matrix multiplication
Resource allocation in economics

As an example, a fictitious factory uses 4 kinds of basic commodities, b 1, b 2, b 3, b 4 {\displaystyle b_{1},b_{2},b_{3},b_{4}} to produce 3 kinds of intermediate goods, m 1, m 2, m 3 {\displaystyle m_{1},m_{2},m_{3}}, which in turn are used to produce 3 kinds of final products, f 1, f 2, f 3 {\displaystyle f_{1},f_{2},f_{3}}. The matrices

provide the amount of basic commodities needed for a given amount of intermediate goods, and the amount of intermediate goods needed for a given amount of final products, respectively.For example, to produce one unit of intermediate good m 1 {\displaystyle m_{1}}, one unit of basic commodity b 1 {\displaystyle b_{1}}, two units of b 2 {\displaystyle b_{2}}, no units of b 3 {\displaystyle b_{3}}, and one unit of b 4 {\displaystyle b_{4}} are needed, corresponding to the first column of A {\displaystyle \mathbf {A} }.

Using matrix multiplication, compute

this matrix directly provides the amounts of basic commodities needed for given amounts of final goods. For example, the bottom left entry of A B {\displaystyle \mathbf {AB} } is computed as 1 ⋅ 1 + 1 ⋅ 2 + 2 ⋅ 4 = 11 {\displaystyle 1\cdot 1+1\cdot 2+2\cdot 4=11}, reflecting that 11 {\displaystyle 11} units of b 4 {\displaystyle b_{4}} are needed to produce one unit of f 1 {\displaystyle f_{1}}. Indeed, one b 4 {\displaystyle b_{4}} unit is needed for m 1 {\displaystyle m_{1}}, one for each of two m 2 {\displaystyle m_{2}}, and 2 {\displaystyle 2} for each of the four m 3 {\displaystyle m_{3}} units that go into the f 1 {\displaystyle f_{1}} unit, see picture.

In order to produce e.g. 100 units of the final product f 1 {\displaystyle f_{1}}, 80 units of f 2 {\displaystyle f_{2}}, and 60 units of f 3 {\displaystyle f_{3}}, the necessary amounts of basic goods can be computed as

that is, 1000 {\displaystyle 1000} units of b 1 {\displaystyle b_{1}}, 1820 {\displaystyle 1820} units of b 2 {\displaystyle b_{2}}, 1180 {\displaystyle 1180} units of b 3 {\displaystyle b_{3}}, 2180 {\displaystyle 2180} units of b 4 {\displaystyle b_{4}} are needed.Similarly, the product matrix A B {\displaystyle \mathbf {AB} } can be used to compute the needed amounts of basic goods for other final-good amount data.
combination:repeat_prompt, detectable_format:constrained_response