For decades, scholars have attempted to propose models of the linkage between the economy and the environment through Leontief (1970)’s input-output model. It was he who first came up with the idea of combining economics and environment in a scientific activity (seminal paper). Leontief and a number of later scholars such as McNicoll & Blackmore (1993) and McGregor et al. (2001) used a direct production waste coefficient matrix to do empirical studies applying such an approach to modeling air pollution in Scotland. This type of work quantifies the impact of economy to the environment, in terms of emissions. However, these studies do not track further environmental responses to the economy, regarding the activity generated in environmental cleanup. The input-output model can be extended to study this problem with pollution reduction activities.

This research tries an attempt, a more detailed introduction to the literature related to the linkage between the economy and the environment through Leontief’s input-output model.

There are some applications of Leontief’s environmental input-output model to study the impact of economic activities on the environment through air pollutions in Vietnam such as studies of Trinh & Phong (2013), Trinh & Hoa (2017), Hung & Trinh (2019).

Studies around Leontief’s input-output model include Flick (1974); Steenge (1978); Lowe (1979); Qayum (1991); Arrous (1994) and Luptacik & Böhm (1999). Allan, Hanley, McGregor, Swales, & Turner (2004). These studies extend and applying the Leontief system pollution model in both sides: Emission economy and waste treatment.

The using of input-output system to combine economic and environment is also given in “System of Environmental-Economic Accounting (SEEA)” by United Nation. This issue is also mentioned in “Handbook of input-output table compilation and analysis” by United Nation.

However, these extensions do not yet have the perfect combination of environmental and economics by “inter-environmental-economics” type. This study tries an attempt to extend Leontief’s input-output model according to type of Miyazawa’s (1976) inter-regional and demographic-economic models in order to linkage between environmental and economics in a more uniform way

Miyazawa’s concept of the income multipliers was designed to analyze the structure of income distribution by opening columns by consumption type and corresponding income streams in the standard Leontief model. These ideas are also incorporated in the familiar social matrix accounting systems developed by Stone (1961), Pyatt and Roe (1977), and in the parallel development of the demographic-economic model. Relating to Batey and Madden (1981). In a sense, Miyazawa’s system is perhaps the most complex in terms of how it extends the familiar input-output formula. In this study substitute the variables of consumptions and incomes are the amount of cost that the sectors in economy have to buy products (at rows) of waste reduction activity (at columns)

Leontief’s input-output system is expanded with A₁₁ is an endogenous direct input coefficient matrix; A₂₁ is the coefficient matrix for the industries to pay for environmental cleaning, A₁₂ is the cost coefficient matrix. waste treatment activities use products as input costs, A₂₂ is the cost coefficient matrix when waste treatment activities also pay for environmental cleaning (because in the process of disposing of one type of waste it is possible to release another type of waste); X₁ and X₂ are production value vectors for the manufacturing and waste treatment industries. Calling E₁ the direct waste from production, E₂ being the waste generated in the waste treatment process, E₃ being the waste from final consumption and from other sources, Y₁ and Y₂ are the final demand vector of the products of economy and environmental protection costs, V₁ and V₂ are the value added of the manufacturing sectors and waste disposal operations sectors. So, the input-output expansion has structure below (Table 1):

Approach model 1:

From the structure of the expanded input-output table above, we have:

A 11 X 1 + A 12 X 2 + Y 1 = X 1 (1)

A 21 X 1 + A 22 X 2 + Y 2 = X 2 (2)

Suppose to treat a unit of waste requires δ dollar, in the total waste E need to treat the amount of waste is αE (α < 1), so production value of X 2 = δ ⋅ α ⋅ E for abating required amount of waste disposal. Relation (1) and (2) we have:

( ∑ Y 1 + ∑ Y 2 ) = ( ∑ X 1 + ∑ X 2 ) − [ ( ∑ A 11 X 1 + ∑ A 12 X 2 )     + ( ∑ A 21 X 1 + ∑ A 22 X 2 ) ] (3)

Notice that:

( ∑ X 1 + ∑ X 2 ) − ( ∑ A 11 X 1 + ∑ A 21 X 1 ) = GDP

So:

∑ Y 1 = GDP − ( ∑ A 12 X 2 + ∑ A 22 X 2 ) − ∑ Y 2 (4)

where: ∑Y₁ can call “Green GDP” after subtracting the cost of cleaning the environment of both production and final consumption or can be considered as an “environmental tax”.

Rewrite relations (1) and (2) as the matrix form we have:

[ X 1 X 2 ] = [ A 11 A 12 A 21 A 22 ] ⋅ [ X 1 X 2 ] + [ Y 1 Y 2 ] (5)

Put:

B = [ A 11 A 12 A 21 A 22 ]

X = [ X 1 X 2 ]

Y = [ Y 1 Y 2 ]

Relation (5) returns to the standard Leontief relation

X = ( I − B ) − 1 ⋅ Y (6)

Put:

U = ( I − B ) − 1

The U matrix can include sub-matricies as follow:

U = [ U 11 U 12 U 21 U 22 ]

where: U₁₁ and U₂₂ are enlarge Leontief inverse matricies, So U₁₁ and U₂₂ include:

· Direct effects: A₁₁, A₂₂

· Indirect effects: [ ( I − A 11 ) − 1 − A 11 ] and [ ( I − A 22 ) − 1 − A 22 ]

· Induced effects by waste treatment activities: [ U 11 − ( I − A 11 ) − 1 ] and [ U 22 − ( I − A 22 ) − 1 ]

U₂₁ is seen as a matrix of charges (taxes) to be paid per unit increase of the final products in order to reduce the required amount of waste. U₁₂ represents the cost induced by the increase in waste resulting from a changing on the end product.

Approach model 2:

Call: E 1 = ( e 1 i j ) ( k × n ) With: e 1 i j = E 1 i j ⋅ X 1 j ,

where:k is number of residuals and n is number of sectors in economy

And:

E 2 = ( e 2 i j ) ( k × m ) With e 2 i j = X 2 j ,

Where: m is number residual treamnet activities

On the other hand notice that:

A 12 ⋅ X 2 = X 12 presents intermediate input matrix of residual treatment activities

Put: h i j = X 12 i j / E

From the structure of the model we have:

A 11 ⋅ X 1 + h ⋅ E + Y 1 = X 1 (7)

e 1 ⋅ X 2 + e 2 ⋅ X 2 + E 3 = E (8)

Write the system of Equations (7) and (8) as matrix form we have:

[ X 1 E ] = [ A 11 h e 1 e 2 ] ⋅ [ X 1 E ] + [ Y 1 E 3 ] (9)

To implement the above relationship according to the Schur-Miyazawa formula ( Miyazawa, 1976) and Sonis & Hewings (1993), we have:

[ 1 − [ A 11 h e 1 e 2 ] ] − 1 = [ Δ 1 Δ 1 ⋅ h ⋅ ( I − e 2 ) − 1 Δ 2 ⋅ e 1 ⋅ ( I − A 11 ) − 1 Δ 2 ] (10)

In relation (10) Δ₁ and Δ₂ include direct, indirect and diffuse effects. According to the Miyazawa definition Δ₁ and Δ₂ are the combination of an inner factor and an external factor.

Inrternal multipliers for economics: ( I − A 11 ) − 1 .

Inrternal multipliers for environmentalchất thải: ( I − e 2 ) − 1 .

External multipliers: [ I − ( I − A 11 ) − 1 ⋅ h ⋅ ( I − e 2 ) − 1 ⋅ e 1 ] − 1 are considered as the spillover by waste treatment activities to the output of other sectors of the economy.

[ I − ( I − e 2 ) − 1 ⋅ e 1 ⋅ ( I − A 11 ) − 1 ⋅ h ] − 1 are the waste spillover effects of the economy stimulated by the input to waste disposal activities.

Δ 2 ⋅ e 1 ⋅ ( I − A 11 ) − 1 are residuals that induced by a unit increase of final demand

Δ 1 ⋅ h ⋅ ( I − e 2 ) − 1 are increased waste disposal to accommodate an increased unit of waste.

Due to no data for making empirical study, so, the paper provides a theoretical framework that offers a new way for studying the relationship between the economy and the environment.

This study expands the basic relationship of the traditional input-output model with the notions of internal and external factors by Miyazawa to create the relationship between emissions generated by the economy and waste disposal activities. In this study, we attempt to determine the role of the waste multiplier in relation to the sensitivity analysis and the aggregate interactions between waste and waste disposal.

The theoretical framework developments provided here may set the stage for some important empirical analysis. Furthermore, the structure of an economy as revealed by many standard macroeconomic models is often conceal important differences in the nature of internal and external interdependence. For example, Waste reduction activities not only induce to the output of other sectors of the economy but also generate other wastes. The U₂₁ matrix can also allow to determine the costs that industries have to pay for each industry and each type of waste. Moreover, in the approarch model 2 allows to know how amounts residuals increase when a unit final demand increases.

However, limit of this research to be there is no data for making empirical study, so, this research is hoping to get some attention to get the data needed for a later experimental work.

The authors declare no conflicts of interest regarding the publication of this paper.

Bui, T., & Quang, T. N. (2021). An Approach to Economics-Environmental Relations from Expanding the Input-Output System. Theoretical Economics Letters, 11, 56-62. https://doi.org/10.4236/tel.2021.111004