Analyzing Some Economic Relations Based on Expansion Input-output Model

This paper is trial attempt for introducing the concept of Leontief inverse matrix and the Leontief extended system for Keynes multipliers, which can analyze the relationship between income groups and consumer groups, respectively. The model is also used to analyze the structure of income in order to describe quantitatively the relationship between income from production and income not from production. The empirical study used the Vietnam input-output table, 2005.


Introduction
In the previous decades, it was a lot of studies in extension of basic I/O model, including Social Accounting Matrix-SAM (Richard Stone, 1961), System of National Account -SNA, Demographic Model-economic modeling (Miyazawa, 1966), and inter -regional model (Miyazawa and other authors, 1976).These extension I/O models were built and applied by most countries in the world for analyzing and forecasting the economy (Pyatt and Roe, 1977;Cohen, 1988;Pyatt and Round, 1985).There are many different uses on this model such as I/O analysis, SAM analysis and CGE model.These analyzes are based mainly on the basic relationships in I/O model and SAM.

Demographic-Economic Model
Demographic-Economic model is created by Miyazawa (1966), it's a similar form to the Social Accounting Matrix, in order to describe the distribution and redistribution of the economy.Essentially, the Demographic-Economic model and the Social Accounting Matrix are similar and it could easily be changed from one model to another depending on other study purposes.In this study, Demographic-Economic model is developed in institutional regions (households, other type of enterprise, State region is divided by type of tax).These institutional regions are considered as endogenous variables: saving and relations with foreign countries are considered as exogenous variables.This model is a combination between the notion of inter-regional I/O model and demographic-economic model, as presented in matrix form below: With: A is the coefficient matrix direct costs; x1 is the vector of the output value of economic activity; x2 is the total income of households; x3 is the total income of the state sector; x4 is the total income of enterprise type; h is the matrix (vector) of income coefficient from the production of the group of households, income from production is understood as workers' income from production divided by 2 types of household; g is the matrix (vector) of coefficient on revenues from production (added value tax, special consumption tax, other taxes and fees); e is the coefficient matrix of income from the production of various types of enterprises (state enterprises, state owned enterprises and enterprises with foreign investments), income from the production here is understood including producer surplus and depreciation of fixed assets; c1 is the coefficient matrix of consumption by household groups corresponding to income groups; g1 is the vector of consumption coefficients of the State corresponding to type of state budget revenues; c2 is the coefficient matrix represented by the redistribution of income between State sector and household sector; c3 is a coefficient matrix represented by the redistribution between the enterprise sector and household sector; g2, g3 represented by state expenditures transfer to the household sectors and enterprise sector; e1, e2, e3 is a coefficient matrix represented by the redistribution from the enterprise sector to the household sector, the State sector and to the other types of enterprises.And f1, f2, f3 f4 are the exogenous variables.

Symbols
In which, vector v, c and B could be refined as follows: From that, relation (1) could be rewrite in form: Based on the theory of the Miyazawa and development of demographic-economic model of Batey and Madden (1983), relation (8) represent as: In which: ∆ 1 is consider as Leontief extended matrix, each element of matrix ∆ 1 includes: direct costs, indirect costs, dispersion effect by final consumption of households and spending for usual activities of the Government.These elements are greater than corresponding elements of popular Leontief's matrix (I-A) -1, because it includes the requirement of more production to meet production affect cause of requirement of final consumption.∆ 2 is known as extended multiplier Keynesian matrix and it can be decayed as follows: Of which: (I-B) -1 is considered as multiplier matrix and internally spread in redistribution processes: if the matrix B is the direct expenditure matrix of regions to create a unit income from redistribution, matrix (I-B) -1 represents total expenditure, direct redistribution to create 1 unit of income from redistribution (influence between regions).Factor (I-(I-B) -1 .v.(I-A) -1 .c) - represents the external spread from the manufacturing process to the redistribution process, which means income from redistribution not only depends on the internal relations in the redistribution process but also depends on income from the production of each region caused by the influence of final consumption.
∆ 1 .c is matrix showing the influence of production by final consumption.

v.(I-A)
-1 is an income matrix received from production.
Note that Equation ( 9) can be rewritten as follows: In which: ∆ 1 = ∆ 11 .(I-A) - and ∆ 2 = ∆ 22 .(I-B) - Equation ( 11) introduced the level of various type of effects, first of all is the influence of production regional and redistribution region, second of all is the effect of final consumption to production and income spread from production to non-production, and finally the external spread effect to areas of production and distribution regions.
In addition this model also allows quantification of the inversed effect from redistribution to production areas.From formula (8), ( 9) and ( 11), this relationship between X 1 and X' is represented as follows: Equation ( 12) and ( 13) describes the inverse relationship inter-regional (between sectors and regions and between production and non-production).
Above is the general model, in addition to depending on purposes of researches, internal and external variable could be changed.For example, to consider the impact of taxes, relations (1) can be rewritten as follows: In which, f' 1 , f' 2 , f' 4 is external variable including f i (i=1,2,4)-is the matrix of tax.
Here: L 'is the matrix transpose of the matrix L; f'' i include taxes and other external variables.

I/O analysis
In an economy, changes in the structure of the sectors are often closely related to each other: some sectors heavily depend on other sectors while a few of them not depend on others too much.Thus the change of some sectors will affect to the economy more than other sectors.I/O analysis is usually based on the backward Linkages and forward linkages.These linkages are tools to measure the relationship of a sector with other sectors, with the role of a sector using the input or supplying the input.
Reverse link is used to measure the relative importance of a sector as the role to use products and services to be the input for the entire production system.Reverse link is defined as the ratio of the sum of the elements (by column) of the Leontief matrix compared to the average of the entire production system.This ratio is called the index of the power of dispersion and is defined as follows: Where:  ij -elements of Leontief's matrix.The higher rates mean larger backward linkages of the industry.
And when the industry developed, it will lead to the growth of the entire system.The policy makers can rely on this index as an important reference in decision making.
Forward linkages implies that the importance of the sector as a source of material products and services for the entire production system is seen as the sensitivity of the economy, which is measured by the sum of the elements in row of the Leontief inverse matrix compared to the average of the entire system.
There are 2 type of I/O table: competitive-import type and non-competitive-import type.In competitive-import I/O table, coefficient matrix of intermediate direct costs includes: import cost as domestic product and imported product.So that, Analyzing the power of dispersion and the sensitivity of the economy will be confused with the import path.A sector with a high power of dispersion doesn't mean that the sector affects well to production, but stimulating import.In the non-competitive-import I/O table, coefficient matrix of intermediate direct cost does not include import cost as import products, so that, the survey of the power of dispersion and sensitivity of an industry will reflect the impact of that industry to domestic product.

Impacts of Non-production Income to Savings by Institutional Regions
The calculation of impacts of non-production income to savings of five institutional regions is based on the multipliers in the Keynes matrix of the model.Calculation results are presented in Table 1: Picture 2. Impacts from production to income redistribution In addition, values of Δ 22 (analysis of Δ 2 ) shows impacts from production to income redistribution.This table shows that the household region is most affected from the production process with a coefficient of 2.13, and then the state with 1.36.

Income Impacts of Non-production to the Production Process
This impact is calculated based on the formula ∆ 1 .c. (I-B) -1 in relation ( 9).This formula shows the impact of our distribution outside the manufacturing process to final consumption and spread to the production process.For example, if the Government region obtained 1 unit from non-production, including direct taxes (personal income tax and corporate income tax), collection of social insurance and health insurance will lead to budget changed and stimulate on production is 1.20 units.The calculated result of income effects from non-production income to the production process in each institutional regions are presented in Table 2.The results showed that this impact of the household region is the largest, next is the Government region, however, the impact of industry structure from the two regions is different.While 1 unit of non -production is used for "food, tourist, and travel…" by the household region, the Government region focuses on "military", "management" and "food".But 3 regions of industry got quite small index of this impact, and it's nearly the same.
From past till now, Vietnam government always has expected the State (E1) and FDI (E3) are 2 main regions for helping and directing other institutional regions.However, The calculated result in Table 02 also showed one special thing: in 3 institutional regions, FDI (E3) is the smallest impact of non -production income to the production process.The second region is E1, and the third is E2.

Impacts of Production to Income Redistribution of Institutional Regions
This section calculates the ability of income redistribution from production to institutional regions (Households, Government and enterprises) by sector groups (combined for 30 sectors).The calculation based on formula ∆ 2 .v.(I-A) -1 and results showed in table 3. Results showed that the household region has the highest ability of income redistribution among most sectors.However, income redistribution between institutional regions and sectors are not even.This may suggest that the Government should tax personal income tax evenly at a certain level, while industrial income tax should be charged depend on type of industry.The result in Table 03 showed that the ability of income redistribution of FDI -E3 is the smallest among 3 regions (E1-E2-E3) and in almost economic activities.However, the ability of FDI -E3 in 16.Mines is high, because of this calculation included income which have to refund to foreign investors in crude-oil exploitation.

Interactive Impacts of Economic Activities in the Production Process (Analysis Leontief Expanded Inverse Matrix).
Using relation ( 11) and formula ∆ 1 = ∆ 11 .(I-A) - , we can calculate internal impacts in production process and external impacts from activities of non -production to the production process.Activities of non -production is understood as spending of institutional regions (Households, Government, enterprises), including final consumption, transfer and ownership.The calculation results are presented in appendix 3. The results showed that the production of products for consumption purposes and the processing industry often have higher level of external impacts than internal impacts.
Especially, industries with very high inter-impacts in production such as food processing industry, textile industry and parts production of motorcycle.This is an interesting point for policy makers, particularly industrial policy.Calculation results of this impact for 30 combined sectors are reported in appendix 3.

I/O Analysis for Sector's Impacts
As In theory, the sectors with large index of power of dispersion should be preferred because it has a strong impact over an entire production.However, due to the index of power of dispersion change over time, prioritized policy have to change to suit: many industry sectors may be the key sector in this period but is not the key sector in the next period.Compare two I/O models, shows that indexes of power of dispersion of the key service industries in the period 2005 I/O model are increased compared to the previous period.This is a good sign: does it the expression of modernization and industrialization to the service sectors?
If you look at the index of power of dispersion in competitive types and non-competitive types, you could see an interesting thing in competitive types (including imports in import cost), the index of power of dispersion is larger than 1 unit in sectors.Sectors such as: medicine, rubber and rubber products, chemicals (all kinds), precision equipment, home tools, machine tools, common machine, specialized machinery, transport equipment, transformer machine, electrical equipment, broadcasting equipments, black metal, thread (all kinds), textiles and leather have index of power of dispersion to domestic products is smaller than 1 unit; This means if these sectors are developed, it will stimulate stronger on import than domestic production.In contrast, sectors with import cost including imported products have index of power of dispersion smaller than 1 unit, but in non-competitive type (input only include domestic products), it's larger than 1 unit.According to Rasmussen-Hrishman, the industry in non-competitive types with the higher index of power of dispersion should be as key sectors.
Appendix 2 shows that these sectors belong to industry group of meat processing and meat products, processed fruits and vegetables, sugar, coffee, tea types, alcohol, tobacco, seafood processing, other foods...

Conclusion and Policy Suggestion
The analysis of I/O models and demographic -economic model showed the changes of the economy cause of different impacts to sectors and institutional regions.So, calculation on this element is necessary to plan the tax policy and other policies.Such as, analyze the index of power of dispersion shows that this index of the sector is very large, then, if you stimulate development of this sector, it highly impacts other sectors in the economy.Calculations show that these sectors are almost processing sectors: meat processing, vegetable, coffee, seafood, etc… For example on coffee, mainly used to export but in raw form.Fruit and vegetable processing industry in Vietnam is still very weak, so there is no production stimulated and it created low "added value".Thus, the potential of processing industry is great, the scale as well as the economic impacts, calculated in terms of the overall economy.On the other hand, the development of these sectors helps to stimulate and enlarge the value of agricultural labor, to minimize the negative impact of the integration process on the lives of more than 10 million rural households in Vietnam.
Appendix 1. Economic structure changed through the index of power of dispersion we presented in Section II, Part 2, here are results of calculations of backward linkage -BL of two I/O model in 2000 and 2005 in Appendix 1. I/O model in 2000 represents the economic structure of the period 1998-2002 and I/O model in 2005 represents the period from 2003 to 2007.Through Appendix 2, we can clearly see the structural change of the economy through changes in the index of power of dispersion of sectors (112 sectors).

Table 1 .
Calculation results of Keynes extended multiplier (∆ 22 ) Picture 1. Backward linkage of Keynes extended matrix Calculation results of Keynes extended matrix shows the impact of non-production income to saving of each institutional region (Household region, Government region, and industrial regions).In which, the most clearly effect in Government region is 2.21, then Household region is 2.207 and the less one in FDI industry is 1.72.This suggests that if the Government region received 2.21 units of income from non -production, they will have 1 unit of saving.While if FDI industry region received 1.72 units of property and transfer, they will have 1 unit of saving and transfer capital broad.

Table 2 .
Impact of non-production income to the production process by institutional regions

Table 3 .
The ability of income redistribution of institutional regions