8.Ìàòåìàòè÷åñêèå ìåòîäû â ýêîíîìèêå

Nickolay Zosimovich, Olga Kozhushko

National Aviation University, Institute of International Relations (Kiev, Ukraine)

Modeling interbranch balance in a country with Leontief model

 

Summary. Analysis of the sources on the task of input-output model was made by means of Mathcad with Leontief model. For a visual demonstration of the Leontief model, calculate balance for  the state of  Washington in 2002 (USA). Calculations show that if one of the indicators is large enough then the end result will be even higher.

Key words: economic analysis, balance, economic sector, industry, product, mathematical simulate, data, equilibrium, matrix, dynamic model, production capacity.

I. Introduction. The historical precursors to input-output analysis were in evidence as far back as the first half of the XVII-th-century. Most economic historians cite Francois Quesnay's (1694-1774) Tableau Economique as the earliest recorded examples to depict the importance of mutual interindustry flows or, in more modem parlance, systematic economic interdependence (Addition 1).         The model Quesnay created consisted of three economic movers [6]:                  
1. The “Proprietary” class, which consisted of just the landowners.   
2. The “Productive” class, which contained all agricultural laborers.  
3. The “Sterile” class, which contained artisans (craftsmen) and merchants.
The flow of production and/or cash between the three classes started with the Proprietary class because they own the land and they buy from both of the other classes. The process has these steps (Fig. 1):      
1. The farmer produces 1500 food on land leased from the landlord. Of that 1500, he retains 600 food to feed himself, his livestock, and any laborers he hires. He sells the remaining 900 in the market for $1 per unit of food. He keeps $300 ($150 for himself, $150 for his laborer) to buy non-farm goods (clothes, household goods, etc) from the merchants and artisans. This produces $600 of net profit, to which Quesnay refers as product net.    
2. The artisan produces 750 units of crafts. To produce at that level, he needs 300 units of food and 150 units of foreign goods. He also has subsistence need of 150 units of food and 150 units of crafts to keep himself alive during the year. The total is 450 units of food, 150 units of crafts, and 150 units of foreign goods. He buys $450 of food from the farmer and $150 of goods from the merchant, and he sells 600 units of crafts at the market for $600. Because the artisan must use the cash he made selling his crafts to buy raw materials for the next year’s production, he has not net profit.         

Fig. 1. Production Flow Diagram for Quesnay’s Tableau [6]

 

3. The landlord is only a consumer of food and crafts and produces no product at all. His contribution to the production process is the lease of the land the farmer uses, which costs $600 per year. The landlord uses $300 of the rent to buy food from the farmer in the market and $300 to buy crafts from the artisan. Because he is purely a consumer, Quesnay considers the landlord the prime mover of economic activity. It is his desire to consume which causes him to expend his entire lease income on food and crafts and which provides income to the other classes.
4. The merchant is the mechanism for exporting food in exchange for foreign imports. The merchant uses the $150 he received from the artisan to buy food from the market, and it is assumed that he takes the food out of the country to exchange it for more foreign goods [6].

II. Statement of the problem. Suppose an economy has n industries each producing a single unique product. (There is a generalization of input output analysis, called activity analysis, in which an industry may produce more than one product, some of which could be pollutants.) Let the product input requirements per unit of product output be expressed as an  matrix A. Let X be the n dimensional vector of outputs and F the n dimensional vector of final demands. The amounts of production used up in producing output X is AX. This is called the intermediary demand. The total demand is thus AX+F. The supply of products is just the vector X. For an equilibrium between supply and demand the following equations must be satisfied [2]:

 

The equilibium production is then given by

A viable economy is one in which any vector of nonnegative final demand induces a vector of nonnegative industrial productions. In order for this to be true the elements of must all be positive. For this to be true  has to satisfy certain coditions.

A minor of a matrix is the value of a determinant. The principal leading minors of an  matrix are evaluated on what is left after the last m rows and columes are deleted, where m runs from (n-1) down to 0.

The condition for the  matrix of  to have an inverse of nonnegative elements is that its principal leading minors be positive. This is known as the Hawkins-Simon conditions [2].

Synthesizing knowledge about reproduction theory of Marx and Engels, cybernetics, Norbert Wiener and the economic and mathematical model of input-output balance (IOB) Leontief, as well as a huge personal experience as a mechanical engineer and an organizer of production at different levels of management corporation of the USSR, the creator of the first Automated Control Systems (ACS), founder of the school strategic planning Veduta Nikolai Ivanovich (1913-1998) developed a dynamic model of the IOB which suggests the inclusion of the impact of the market (equilibrium prices) to determine the proportions of the plan.

In the scheme of IOB for the first time systematically coordinated balance of revenues and outlays of producers and consumers - the state (inter-state block), households, exporters and importers (the external economic balance). A dynamic model of IOB his method of economic cybernetics. It is a system of algorithms, effectively linking the tasks of end users with capabilities (material, human and financial), producers of all forms of ownership. Based on the model is determined by the effective allocation of productive government investment. Introducing a dynamic model of IOB, the government is able to adjust the mode «online» for development based on refinement of the production capacity of residents and demand dynamics of end users to meet the requirements of national and global security [3].

III. Simulation methodics of Leontief model. For a visual demonstration of the Leontief model, calculate balance for  the state of  Washington. From the "input-output" table of Washington in 2002 [1] choose the input data for calculations (Table 1).

Table 1

The input data for task

 

Mining

Electric Utilities

Gas Utilities

Air Transportation      

Inter-industry subtotal

Total final demand

Mining

0,8   

124,8   

14,2   

0,0   

139,8

139

Electric Utilities

10,5   

1773,0   

0,4   

3,2   

1787,1

14,1

Gas Utilities

2,8   

134,1   

1,0   

0,3   

138,2

137,2

Air Transportation      

0,2

5,6

0,7

0,2

6,7

6,5

Total intermediate input

 

14,3

2037,5

16,3

3,7

2071,8

 

Fig. 2. Results of Table 1 calculation

 

For the final demand Y=(139;14,1;137,2;6,5) total output for Mining is 278,8, Electric Utilities – 1.801õ103 , Gas Utilities – 275,4, Air Transportation – 132 (Fig. 2). Because of too much the end result of  Electric Utilities, reduce its  input rate by 2 times and then change the vector of final demand (Table 2).

Table 2

A modified input data for task

 

Mining

Electric Utilities

Gas Utilities

Air Transportation      

Inter-industry subtotal

Total final demand

Mining

0,8   

124,8   

14,2   

0,0   

139,8

139

Electric Utilities

10,5   

886,5   

0,4   

3,2   

1787,1

900,6

Gas Utilities

2,8   

134,1   

1,0   

0,3   

138,2

137,2

Air Transportation      

0,2

5,6

0,7

0,2

6,7

6,5

Total intermediate input

 

14,3

2037,5

16,3

3,7

2071,8

 

Fig. 3. Results of Table 2 calculation

 

Repeating calculation with a reduced rate Electric Utilities 2 times showed that this significant reduction caused a small variation of certain  parameters, but the end  result is not affected. It means that for the final demand Y=(139;900,6;137,2;6,5) total output is the same: Mining – 278,8, Electric Utilities – 1.801õ103 , Gas Utilities – 275,4, Air Transportation – 132 (Fig. 3).

IV. Conclusions. Leontief model has a strong historical background which shows that interest in the issue of interbranch relations in the economy concerned scientists for many years.

1.     There are related methods that specify the calculations and give a more accurate result.

2.     Leontief model is very useful because it includes the important factors that significantly affect the economy as a whole.

3.     Calculations show that if one of the indicators is large enough then the end result will be even higher.

4.     If this indicator reduce significantly the end result does not change. In the calculation process   certain parameters change only.

The calculations produced above have a illustrative purpose. In order to trace the significant changes and get more specific and accurate results it is necessary to make a matrix which will include a greater number of industries. Here it is necessary to change indicators separately to follow the trend. In practice use an additional parameters such as value added, imports, total employment etc.

 

References

1.     http://www.ofm.wa.gov/economy/io/2002/default.asp  - Office of Financial Management.

2.     http://www.sjsu.edu/faculty/watkins/inputoutput.htm#INTRO  - San José State University Department of Economics.

3.     http://ru.wikipedia.org/wiki/ - Ìåæîòðàñëåâîé_áàëàíñ.

4.     http://en.wikipedia.org/wiki/Tableau_%C3%A9 – economique.

5.     http://www.referenceforbusiness.com/encyclopedia/Inc-Int/Input-Output-Analysis.html  - Reference for Business. Encyclopedia of Business, 2nd ed.

6.     http://econ-thought.blogspot.com/2010/01/francois-quesnays-tableau-economique.html  - Modern Economic Thought.