Question

Suppose the inter-industry flow of the product of two sectors X and Y are given as under.

Production Sector	Consumption sector		Domestic demand	Gross output
	X	Y
X	15	10	10	35
Y	20	30	15	65

Find the gross output when the domestic demand changes to 12 for X and 18 for Y.

Answer 1

a₁₁ = 15, a₁₂ = 10, x₁ = 35

a₂₁ = 20, a₂₂ = 30, x₂ = 65

`"b"_11 = "a"_11/x_1 = 15/35 = 3/7`, `"b"_12 = "a"_12/x_2 = 10/65 = 2/13`

`"b"_21 = "a"_21/x_1 = 20/35 = 4/7`, `"b"_22 = "a"_22/x_2 = 30/65 = 6/13`

The technology matrix is B = `[(3/7,2/13),(4/7,6/13)]`

I - B = `[(1,0),(0,1)] - [(3/7,2/13),(4/7,6/13)]`

`= [(4/7,-2/13),(-4/7,7/13)]`, The main diagonal elements are positive

|I - B| = `= [(4/7,-2/13),(-4/7,7/13)]`

`= 4/7 xx 7/13 - (- 4/7) xx ((- 2)/13) => 28/91 - 8/91 = 20/91 > 0`

Since the main diagonal elements of I – B are positive and |I – B| is positive the problem has a solution.

adj (I - B) = `[(7/13,2/13),(4/7,4/7)]`

`("I - B")^-1 = 1/|"I - B"|` adj (I - B)

`= 1/(20/91) [(7/13,2/13),(4/7,4/7)] = 91/20 [(7/13,2/13),(4/7,4/7)]`

`= 1/20 [(91xx7/13,91xx2/13),(91xx4/7,91xx4/7)]`

`= 1/20 [(49,14),(52,52)]`

X = (I – B)^-1D, where D = `[(12),(18)]`

`= 1/20 [(49,14),(52,52)] [(12),(18)]`

`=> 1/20 [(588 + 252),(624 + 936)]`

`= 1/20 [(840),(1560)] = [(42),(78)]`