22 The national economy of Seriland manufactures three products: steel, cars, and machines. (1) To produce $1 of steel requires 30¢ of steel, 15¢ of cars, and 40¢ of machines.

(2) To produce $1 of cars requires 45¢ of steel, 20¢ of cars, and 10¢ of machines. (3) To produce $1 of machines requires 40¢ of steel, 10¢ of cars, and 45¢ of machines. During the coming year, Seriland wants to consume ds dollars of steel, dc dollars of cars, and dm dollars of machinery.

For the coming year, let s = dollar value of steel produced c = dollar value of cars produced m = dollar value of machines produced Define A to be the 3 * 3 matrix whose ijth element is the dollar value of product i required to produce $1 of product j (steel= product 1, cars = product 2, machinery =product 3).
a Determine A.
b Show that

(Hint: Observe that the value of next year’s steel production = (next year’s consumer steel demand) + (steel needed to make next year’s steel) (steel needed to make next year’s cars) + (steel needed to make next year’s machines). This should give you the general idea.)
c Show that Equation (24) may be rewritten as

d Given values for ds, dc, and dm, describe how you can use (I - A)^-1 to determine if Seriland can meet next year’s consumer demand.
e Suppose next year’s demand for steel increases by $1. This will increase the value of the steel, cars, and machines that must be produced next year. In terms of (I - A)^-1, determine the change in next year’s production requirements.

Transcribed Image Text:

S C S Ac+ ds dc m m [dm