Agriculture Manufacturing Services Open Sector Agriculture 34.69 4.92 5.62 39.24

Question

Agriculture Manufacturing Services Open Sector

Agriculture 34.69 4.92 5.62 39.24

Manufacturing 5.28 61.82 22.99 60.02

Services 10.45 25.95 42.03 130.65

Total Gross Output 84.56 163.43 219.03

Table 1: Exchange of Goods and Services in the U.S. for 1947 (in billions of 1947 dollars)

C =

%uF8EE

%uF8F0

.4102 .0301 .0257

.0624 .3783 .1050

.1236 .1588 .1919

%uF8F9

%uF8FB

1

d =

%uF8EE

%uF8F0

39.24

60.02

130.65

%uF8F9

%uF8FB

x = Cx + d

This equation may be solved for x to find that

x = (I %u2212 C)%u22121d

where I is the identity matrix.

In the example,

(I %u2212 C)%u22121 =

%uF8EE

%uF8F0

1.7203 .1006 .0678

.2245 1.6768 .2250

.3073 .3449 1.2921

%uF8F9

%uF8FB

and thus

x = (I %u2212 C)%u22121d =

%uF8EE

%uF8F0

82.40

138.85

201.57

%uF8F9

%uF8FB

How do they get 1.7203 .1006 .0678  I am lost at how to Determine the I ( identy matrix) and how dows the -1 (which is an exponent) come into play?

Explanation / Answer

These equilibrium levels are the production levels which will just meet the intermediate demands of the sectors of the economy plus the nal demands of each sector. If x is the desired production vector, x must satisfy x = Cx + d This equation may be solved for x to nd that x = (I C)1 d where I is the identity matrix. In the example, (I C)1 and thus 1.7203 .

1006 .0678

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