Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

# You are given the following transaction matrix for a two-sector economy. (i) Write the technology matrix - Business Mathematics and Statistics

Sum

You are given the following transaction matrix for a two-sector economy.

 Sector Sales Final demand Gross output 1 2 1 4 3 13 20 2 5 4 3 12
1. Write the technology matrix
2. Determine the output when the final demand for the output sector 1 alone increases to 23 units.

#### Solution

a11 = 4, a12 = 3, x1 = 20

a21 = 5, a22 = 4, x2 = 12

"b"_11 = "a"_11/x_1 = 4/20 = 1/5, "b"_12 = "a"_12/x_2 = 3/12 = 1/4

"b"_21 = "a"_21/x_1 = 5/20 = 1/4, "b"_22 = "a"_22/x_2 = 4/12 = 1/3

The technology matrix is B = [(1/5,1/4),(1/4,1/3)]

I - B = [(1,0),(0,1)] - [(1/5,1/4),(1/4,1/3)]

= [(4/5,-1/4),(-1/4,2/3)], elements of main diagonal are positive

|I - B| = 4/5xx2/3-(-1/4)xx(-1/4)

= 8/15 - 1/16 = (8 xx 16 - 1 xx 15)/(15 xx 16)

= (128 - 15)/(15 xx 16) = 113/240

The main diagonal elements are positive and |I – B| is positive. Therefore the system is viable.

adj(I - B) = [(2/3,1/4),(1/4,4/5)]

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

= 1/(113/240) [(2/3,1/4),(1/4,4/5)] = 240/113[(2/3,1/4),(1/4,4/5)]

X = (I – B)-1D, where

D = (16 xx 15)/113 [(2/3,1/4),(1/4,4/5)] [(23),(3)]

=> 1/113 [(16xx15xx2/3,16xx15xx1/4),(16xx15xx1/4,16xx15xx4/5)] [(23),(3)]

= 1/113 [(16xx5xx2,4xx15xx1),(4xx15xx1,16xx3xx4)] [(23),(3)]

=> 1/113 [(160,60),(60,192)] [(23),(3)]

= 1/113 [(160xx23+60xx3),(60xx23+192xx3)]

= 1/113 [(3680+180),(1380+576)]

= 1/113 [(3860),(1956)] = [(34.159),(17.3097)]

X = [(34.16),(17.31)]

The output of sector 1 should be 34.16 and sector 2 should be 17.31.

Concept: Input–Output Analysis
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