ADONGO'S- LEONTIEF'S MODEL

MULTI- COMBINATIONAL SYSTEM ALGEBRAL AS A FITTED MODEL IN ECONOMY

 The economy system of one society consists of several industries in which each industry depends on its own and others to produce its own products. Empirically, the interdependent relationship between one industry and others varies from time to time and the time period each industry depends on others industries to produce its own products varies from industry to industry. Because of this, it is difficult for an economist to determine the gross production of each industry: unless the gross production of one industry is given that the gross production of others industries can be determined at a specified future time. This can be done by applying the concept of the multi- combinational system algebra to construct an input- output economical models based on past period production and interdependent relationship to forecast future gross production of each of the industry.

MULTI-COMBINATIONAL INPUT- OUTPUT MATRIX

Consider a simple open economy as being based on agricultural products, manufactured goods and fuels is given as:

	OUTPUTS
INPUTS	Agricultural products	Manufactured goods	Fuels
Agric	a11	a12	a13
Manufactured	a21	a22	a23
Fuels	a31	a32	a33

If we know the surpluses or final demands of agricultural products to beD₁, manufactured goods to be D₂ and fuels to be D₃: then the multi-combinational input- output matrix of above open economy is given as:

	A	M	F	Surpluses
A M F	a11 a21 a31	a12 a22 a32	a13 a23 a33	D1 D2 D3
AM MF	a11a12a21a22 a21a22a31a32		a12a13a22a23 a22a23a32a33	D1D2 D2D3
	AM		MF

Where the gross production matrix is

X =	(X1X2,   X2X3)

All is represented as; X- AX= D or (I – A)X =D, where I is identity matrix.

OPEN ECONOMY

The economy of Ghana is based on agricultural industry, mining industry and oil industry. An output of 1ton of agricultural products requires an input of 0.1ton of agricultural products, 0.02 ton of minerals, and 0.05 ton of oil. An output of 1 ton minerals requires an input of 0.01 ton of agricultural products, 0.13 ton of minerals and 0.18 ton oil. An output of 1 ton of oil requires an input of 0.01 ton of agricultural products, 0.2 ton of minerals and 0.05 ton of oil. The multi-combinational input-output matrix of the above open economy 

 which have surpluses of 40 units of agric, 35 units of mineral and 25 units of oil.

	Agricultural(A)	Manufactured(M)	Oil(O)	Surpluses(D)
A               M O	0.1 0.02 0.05	0.01 0.13 0.18	0.01 0.20 0.05	40 35 25
AM MO	0.1*0.01*0.02*0.13 0.02*0.13*0.05*0.18		0.01*0.01*0.13*0.2 0.13*0.2*0.18*0.05	40*35 35*25
	AM		MO

Find the gross production of oil industry if the gross production of agricultural product is 80

                                 

 SOLUTION

A =

0.0000026             0.00000234

0.00000026 0.000234

X = (I-A) ^-1 D,

                                                                

X₁x₂= 1399.97

X₂x₃= 875.03

X₃= 80[875.03/1399.97]= 50.

Hence, the gross production of x₃ is 50, if the gross production of x₁ is 80.

NOTE: The value of x₂ is non-dependent variable and has no effect in the computation.

CLOSED ECONOMY

In this model no external demands are needed:  inputs and outputs are used within the system, then such a model is called closed economical models. In such a model labor is needed so, there is no surplus and D=0.





 REFERENCE
Dietzenbacher, Eric and Michael L. Labr, eds. Wassilly Leontief and Input-Output Economics. Cambridge University Press, 2004.