An approximation method in numerical
computation of the Leontief's open input-output model

In the Leontief's open input-output model, the numerical value in each industry associated with a given amount of final demand is usually computed by means of the inverse matrix9 It, however, is not a easy job to obtain the inverse of given matrix. Concerning this, it seems to have been considered as necessary to use a large scale computer when the number of industries is large. In this article we shall give simple ways to compute approximate values, in which use is made only of a table-computer.

Let Y_i, denote the given, amount of final demand in the i-th industry, X_i that of out-put to be obtained in the i-th industry associated with these final demands Y_i's and a_ij technical coefficients9 Then the Leontief's system of linear equations is written as follows:

These linear equations can also be written as

where b_ii=1-a_ii

If the amounts of the final demands and those of outputs used in calculating these technical coefficients a_ij are represented by Y_i^' and X_i^' these values must also satisfy the following equations.

Then we put X_i⁽¹⁾

which is taken as an approximation to X_i. These values can be easily calculated, for Y_i^' и X_i^' are known and Y_i are given.

For the purpose of getting a better approximation, we estimate the error of that approximation. From (3) and (4), we obtain

The relative error of the approximation of X_i is then represented as

Since

from (6) and (7), we have

Subtracting (3) from (2) side by side, we get

From (9) and (8) it follows that

As X_i are unknown, we estimate these values by replacing Y_i X_i by Y_i^' X_i^'and by neglecting the second term in the braces of (10), that is, we estimate the values from

Now, a_ijX_j^' is the input of the i-th industry into the j-th industry, and may be positively correlated with X_i^' for fixed i. Therefore, replacing a_ijX_j^' by X_j^', we have

Using these values, we obtain a better approximation of X_i

If has an extra-ordinary value for large X_i^' we should use for the second term in the parchtheses of (13) intead of

From the equations (2) and (3) we have

Subtracting (15) from (14) side by side and neglecting the third term, we have

The second term of the right hand side of (16) can be estimated in the similar way as in obtaining (12), that is,

Then substituting (17) into (16), we get an approximation X_i⁽³⁾ of X_i

The relative error of the approximation X_i⁽¹⁾, is of nearly the same as the value of the second term of the approximation X_i⁽¹⁾. For the short term forecasting, even that approximation may be satisfactory.

For the long range forecasting it is better to use the approximation X_i⁽²⁾ or X_i⁽³⁾. In these two approximations, the second term multiplier of X_i⁽¹⁾ in X_i⁽²⁾ and that of X_i^' in X_i⁽³⁾ are of the same.

For the evaluation of this error, we estimate the difference between weights d_ij by the difference between and for such a j that has an extraordinary value. .

It can be seen that may give the rough estimate of the error in consideration.

Next, we estimate the neglected term by the formula

From two values estimated above, the roughly estimated error can be obtained for X_i⁽²⁾ and X_i⁽³⁾. hen this relative error is negative, we use X_i⁽²⁾. When it is positive, we use X_i⁽³⁾

When some of technical coefficients are changed from those obtained from the survey by taking into account technological change, we should use the values of the final demands computed from the changed technical coefficients and the output data of the survey. These values of the final demands can easily be computed.

If a further better approximation is wanted, we can get it by the successive approximation method, in which X_i⁽²⁾ or X_i⁽³⁾is to be used the first approximation. In this case, stratifying X_i^',we can use in place of each X_i^' the mean of the stratum in which X_i^'lies. In this way the computation is made much simpler.

This table of technical coefficients is concerned with the Japan Economy in 1951 and is presented by Japan Economic Council Board. X_i in 1952 are forecast while Y_i are given.

Industry

1. Agriculture, Forestry and Fisheries
2. Mining
3. Construction
4. Manufacturing
5. Whole sale and Retail Trade
6. Transportation and Communication
7. Public Utilities
8. Service
9. Unknown.

	Xi'	Yi'	1/bii	Yi'/bii	Xi'-Yi'/bii	(Xi'-Yi'/bii)/Xi'	Yi
1	17.386	9.228	1.147667	10.591	6.795	0.39	9.926
2	3.002	200	1.019701	204	2.798	0.93	259
3	5.465	4.704	1.005150	4.728	737	0.16	5.119
4	53.883	24.322	1.704620	41.460	12.423	0.23	24.351
5	8.591	4.997	1.006090	5.027	3.564	0.41	5.349
6	7.395	2.075	1.039061	2.502	4.893	0.66	2.408
7	1.754	432	1.012702	437	1.317	0.75	467
8	10.714	6.205	1.098984	6.819	3.895	0.36	7.955
9	4.185	1.147	1.0	1.147	3.038	0.72	657
Всего	112.375			702.915	39.460

	Yi-Yi'	(Yi-Yi')/bii	∑(Yi-Yi')/bii))/∑Xj'
1	698	801	0.028	0.012
2	59	60	0.031	0.029
3	415	417	0.031	0.005
4	29	49	0.059	0.016
5	352	354	0.031	0.017
6	333	346	0.031	0.020
7	35	35	0.031	0.023
8	1750	1923	0.17	0.006
9	- 490	- 490	0.031	0.022

	Xi1	Error percent	Xi2	Error percent	(Xi'-Yi'/bii)*∑(Yi-Yi')/bii))/∑Xj'
1	18.187	0.8%	18.405	-0.4%	190
2	3.062	2.7	3.151	-0.1	88
3	5.882	0.8	5.911	0.3	23
4	53.932	1.5	54.795	-0.1	733
5	8.945	1.3	9.061	0.03	110
6	7.741	1.7	7.896	-0.3	152
7	1.789	2.1	1.830	-0.1	41
8	12.637	0.2	2.713	-0.4	66
9	3.695	2.9	3.776	0.8	94

	Xi3	Error percent	Xi calculated by means of inverse matrix
1	18.377	-0.3%	18.328
2	3.150	-0.1	3.148
3	5.905	0.4	5.927
4	54.665	0.1	54.734
5	9.055	0.1	9.064
6	7.893	-0.3	7.873
7	1.830	-0.1	1.828
8	12.703	-0.3	12.660
9	3.789	0.4	3.805

B начало