A towns economy consists of three industries, agriculture, manufacturing and gold mining. To grow one unit of food the agriculture industry needs 0.5 units of its own output and 0.4 units of manufacturing goods, but nothing from the mine. To manufacture one unit of industrial equipment the manufacturing industry requires 0.2 units of agricultural produce and 0.4 units of manufactured goods, but nothing from the mine. To mine one unit of gold the mine requires 0.2 units of agriculture and 0.6 units of manufacturing from the agriculture and manufacturing industries, respectively, but none of its own output.

(a) Tabulate the above information for this economy, with columns corresponding to each industrys required inputs for one unit of production.

(b) Write the input-output matrix A for this economy.

(c) The mine owners want to determine the maximum possible output of the mine. To do this they assume that there is no external demand for agriculture or manufacturing (that is, assume these industries only exist to support the mine). The maximum output from agriculture is known to be 300 units. Using the formula (I A)x = d, show that

2y + 2z = 1500

6y 6z = 1200

z d = 0

where the demand for gold is d, the output from manufacturing is y and the output from the mine is z. Hint: set the demand vector to d = (0, 0, d) and the output vector to x = (x, y, z) with x = 300

(d) Solve for y, z and d using a matrix inverse