Adalat Muradovd1
Azerbaijan State University of Economics
Yadulla Hasanli2
Institute of Control Systems of ANAS
Nazim Hajiyev3
Azerbaijan State University of Economics
Azerbaijan’s dynamic input-output model with inverse recursion
Abstract
In this article Azerbaijan’s dynamic "input-output" model with the methodological basis of
the inverse models of recursion was developed. The interbranch relations in Azerbaijan economy
have been studied by means of a developed model, including mutually related oil and non-oil
sectors.
Dynamic model with inverse recursion refers to the type of dynamic "input-output" models.
In this model, endogenous (found) parameters are total output and investments sectors. More
precisely, the forecasted output volume for the final year of the sectors’ and the total volume of
capital investments for the entire period in each and every sector are found. The yearly
distribution of capital investment is realized via an exogenously given parameter.
The calculation in this model is implemented in two stages. In the first stage, the total output
volume of the sectors for the final year and the total volume of capital investments in each sector
for the entire forescasted period are determined. In the second stage, the volume indicator of the
gross output for each year of the forecasting period is calculated.
The realization of the model has been implemented based on Azerbaijan’s National
Accounts and statistical input-output tables, as well as other relevant data.
Keywords: reverse recursion, dynamic input-output model, endogenous and exogenous variables,
oil and non-oil sectors, the total output, capital investment.
1. Induction
Reverse recursion and dynamic models are simple types of dynamic “expense-output”
model. In this model, endogenous parameters are presented by the production and capital
investment. In other words, for each sector the production volume for the last year and the capital
investments for the entire period are forecasted. The allocation of capital investments is
implemented by the means of exogenous factor j (t).
∑𝑠𝑡=1 𝜑𝑗 (𝑡) = 1
1
𝑗 = 1,2.
(1)
Azerbaijan State University of Economics, rector, Dr.Prof., adalet_muradov@yahoo.com
“Modelling of Social and Economic Processes” Laboratory head of Institute Control Systems of ANAS,
Dr.Prof., yadulla_hasanli@yahoo.com, yadulla59@mail.ru, yadullahasanli@box.az
2
3
Director of MBA Department in ASUE, Dr. Associate prof., n.hajiyev@aseu.edu.az;
nazimhajiyev2013@gmail.com
Here, 𝜑𝑗 (𝑡) – is the portion of the capital investment in total capital investment in sector j in some
particular year t; s – the index for the last forecasted year; t is indicator of time: 𝑡 = 1,2, … , 𝑒𝑡𝑐.; jsector index, for example in our model 1 is for oil and gas and 2 is for non-oil and gas.
Note that this type of dynamic model was applied at the first time in the institute of
Economy of Sciences of the Republic of Lithuania4.
This model is implemented in two calculation phases. Last year, the total volume of the
projected total volume of capital investments in the sector is defined for each and all sectors as
part of first phase. In the second stage, forecasting, production volume index is calculated for each
year of the period.
2. The economic-mathematical model of the case
The average annual volume of funds is calculated based on the following formula5.
𝐸𝐹𝑗 (𝑡) = (1 − 𝑟𝑗 )(𝐸𝐹𝑗 (𝑡) + 𝛼𝑗 ∆𝐸𝐹𝑗 (𝑡)),
j=1,2.
(2)
Hore, 𝐸𝐹𝑗 (𝑡)- the major production funds in sector j in year t; 𝐸𝐹𝑗 (𝑡)- the volume of the major
production funds in sector j at the beginning of the year t; ∆𝐸𝐹𝑗 (𝑡) – the utilization of the major
production funds in year t in sector j; 𝑟𝑗 – the consumption index of the major production funds,
i.e. their depreciation; 𝛼𝑗 – the index of the average annual additions to the production funds in
sector j.
It is assumed that, indexes 𝛼𝑗 𝑣ə 𝑟𝑗 are constant throughout the entire period. The equal
depreciation of the major production funds is achieved by the following formula:
𝐸𝐹𝑗 (𝑡 + 1) = (1 − 2𝑟𝑗 )𝐸𝐹𝑗 (𝑡) + (1 − 2𝑟𝑗 )∆𝐸𝐹𝑗 (𝑡),
j=1,2.
(3)
For the year, where the average annual depreciation of major production funds is equal
means that the depreciation is two times higher than the average annual depreciation.
In the recurrent formula above (2) the annual volume of the funds at the beginning of the
forecasted year can be expressed as a sum of the funds existing at the beginning the previous year
and the funds came into the service during the previous year.
См. Сатуновский Л. Валансовые динамические модели планирования и распределения продукции в
республике (на примере Литовской ССР). В сб. «Динамические модели территориального планирования». М.,
«Наука», 1972.
5
“Моделирование народного хозяйственных процессов» (коллек авторы), М., «Экономика», 1973. 478с. С.
258.
4
𝐸𝐹𝑗 (2) = (1 − 2𝑟𝑗 )𝐸𝐹𝑗 (1) + (1 − 2𝑟𝑗 )∆𝐸𝐹𝑗 (1);
2
2
3
3
𝐸𝐹𝑗 (3) = (1 − 2𝑟𝑗 ) 𝐸𝐹𝑗 (1) + (1 − 2𝑟𝑗 ) ∆𝐸𝐹𝑗 (1) + (1 − 2𝑟𝑗 )∆𝐸𝐹𝑗 (2);
2
𝐸𝐹𝑗 (4) = (1 − 2𝑟𝑗 ) 𝐸𝐹𝑗 (1) + (1 − 2𝑟𝑗 ) ∆𝐸𝐹𝑗 (1) + (1 − 2𝑟𝑗 ) ∆𝐸𝐹𝑗 (2) + (1 − 2𝑟𝑗 )∆𝐸𝐹𝑗 (3);
.................................................................................................
𝐸𝐹𝑗 (𝑡) = (1 − 2𝑟𝑗 )
𝑡−1
𝐸𝐹𝑗 (1) + ∑𝑡−1
𝑘=1(1 − 2𝑟𝑗 )
𝑡−𝑘
∆𝐸𝐹𝑗 (𝑘) ; 𝑡 > 1.
(4)
(2) –dən 𝐸𝐹𝑗 (𝑡)-i tapıb (4)-də yerinə qoysaq aşağıdakı münasibətləri alarıq,
𝐸𝐹𝑗 (1) = (1 − 𝑟𝑗 )𝐸𝐹𝑗 (1) + (1 − 𝑟𝑗 )𝛼𝑗 ∆𝐸𝐹𝑗 (1);
𝐸𝐹𝑗 (2) = (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 )𝐸𝐹𝑗 (1) + (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 )∆𝐸𝐹𝑗 (1)+(1 − 𝑟𝑗 )𝛼𝑗 ∆𝐸𝐹𝑗 (2);
2
2
𝐸𝐹𝑗 (3) = (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 ) 𝐸𝐹𝑗 (1) + (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 ) ∆𝐸𝐹𝑗 (1) + (1 − 𝑟𝑗 )(1 −
3
2𝑟𝑗 )∆𝐸𝐹𝑗 (2) + (1 − 𝑟𝑗 )𝛼𝑗 ∆𝐸𝐹𝑗 (3);
3
𝐸𝐹𝑗 (4) = (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 ) 𝐸𝐹𝑗 (1) + (1 − 𝑟𝑗 )(1 −
2
2𝑟𝑗 ) ∆𝐸𝐹𝑗 (1) + (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 ) ∆𝐸𝐹𝑗 (2) + (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 )∆𝐸𝐹𝑗 (3) + (1 −
𝑟𝑗 )𝛼𝑗 ∆𝐸𝐹𝑗 (4);
..................................................................................................
𝐸𝐹𝑗 (𝑡) = (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 )
𝑡−1
𝐸𝐹𝑗 (1) + ∑𝑡−1
𝑘=1(𝑡 − 𝑘 + 1) ∆𝐸𝐹𝑗 (𝑘) + (1 − 𝑟𝑗 )𝛼𝑗 ∆𝐸𝐹𝑗 (𝑡);
(5)
The putting into the service the major production funds in practice is aligned with the total capital
investment in statistical and accounting practises.
𝐶 = ∆𝐸𝐹 ∓ 𝐴𝑍,
(6)
Here, C is capital investment, AZ is completed construction additions (+) or disposals (-).
the equation (6) shows the funds put into the service using the following 𝛾𝑗 proportion:
𝛾𝑗 =
∆𝐸𝐹𝑗 (𝑡)
(7)
𝐾𝑗 (𝑡)
In our case, the index 𝛾𝑗 does not change throughout the years. 𝐾𝑗 (𝑡) is capital investment in
sectors in year t, by the means of index 𝜑𝑗 (𝑡) we define the total capital investment 𝐾𝑗 required
for the entire period.
𝐾𝑗 (𝑡) = 𝜑𝑗 (𝑡)𝐾𝑗 ,
𝑗 = 1,2, … , 𝑛.
(8)
Therefore, considering the equations (7) and (8) the exploitation of the funds in year t will be as
following:
∆𝐸𝐹𝑗 (𝑡) = 𝜑𝑗 (𝑡)𝛾𝑗 𝐾𝑗 ,
𝑗 = 1,2, … , 𝑛.
(9)
If we put (9) into the (5) instead of ∆𝐸𝐹𝑗 (𝑡), then the volume of the average annual funds
( 𝐸𝐹𝑗 (𝑡)) will be defined by the means of major production funds at the beginning of the period
(𝐸𝐹𝑗 (1)) and capital investments made during the year (𝐾𝑗 ).
𝐸𝐹𝑗 (𝑡) = (1 − 𝑟𝑗 )(1 − 2𝑟𝑗 )
(1 − 𝑟𝑗 )𝛼𝑗 𝜑𝑗 (𝑡)𝛾𝑗 𝐾𝑗 ;
𝑡−1
𝐸𝐹𝑗 (1) + ∑𝑡−1
𝑘=1(1 − 𝑟𝑗 )(1 − 2𝑟𝑗 )
𝑡−𝑘
𝜑𝑗 (𝑘)𝛾𝑗 𝐾𝑗 +
(10)
As a result of implemented calculations and transformations we can show the clear dynamic
model with the reverse recursion. The production and dynamic allocation of the balance is as
following:
𝑥𝑖 (𝑡) = ∑𝑛𝑗=1 𝑎𝑖𝑗 𝑥𝑗 (𝑡) + ∑𝑛𝑗=1 𝑏𝑖𝑗 (𝑡) 𝐾𝑗 (𝑡) + 𝑦𝑖 (𝑡);
(11)
If we consider (8) and (9) in (11), we will get the following
𝑥𝑖 (𝑡) = ∑𝑛𝑗=1 𝑎𝑖𝑗 𝑥𝑗 (𝑡) + ∑𝑛𝑗=1 𝑏𝑖𝑗 (𝑡) 𝜑𝑗 (𝑡)𝐾𝑗 + 𝑦𝑖 (𝑡);
(12)
The balance of the production and distribution for the last forecast year will be as follows.
𝒙𝒊 (𝒔) = ∑𝒏𝒋=𝟏 𝒂𝒊𝒋 𝒙𝒋 (𝒔) + ∑𝒏𝒋=𝟏 𝒃𝒊𝒋 (𝒔) 𝝋𝒋 (𝒔)𝑲𝒋 + 𝒚𝒊 (𝒔);
(13)
Here, 𝑥𝑖 (𝑠) – the production volume in sector j during the last year;
𝑦𝑖 (𝑠) – the net production volume in sector j during the last year6;
𝑏𝑖𝑗 (𝑠)- the structure index of the capital contributions, in other words the portion of the capital
contribution in total contribution in j sector.
Each sector is identified in accordance with the demand for the fund to the amount of total capital
contribution for the particular year. The demand for the funds ( 𝐸𝐹𝑗 (𝑡)) is calculated as fund
index (𝑓𝑗 (𝑡)) multiplied by output volume (𝑥𝑗 (t)).
𝐸𝐹𝑗 (𝑡) = 𝑓𝑗 (𝑡)𝑥𝑗 (𝑡) ,
𝑗 = 1,2, … , 𝑛.
(14)
Considering (10) the equation can be written as following:
6
We would like to note that the net final product in this model means the small portion of the capital contribution
derived from the ultimate product.
𝒇𝒋 (𝒔)𝒙𝒋 (𝒔) = (𝟏 − 𝒓𝒋 )(𝟏 − 𝟐𝒓𝒋 )
𝒔−𝟏
𝑬𝑭𝒋 (𝟏) + [∑𝒔−𝟏
𝒌=𝟏(𝟏 − 𝒓𝒋 )(𝟏 − 𝟐𝒓𝒋 )
𝒔−𝒌
𝝋𝒋 (𝒌)𝜸𝒋 +
(𝟏 − 𝒓𝒋 )𝜶𝒋 𝝋𝒋 (𝒔)𝜸𝒋 ] 𝑲𝒋 , 𝒋 = 𝟏, 𝟐, … , 𝒏. (15)
The know index of 𝐸𝐹𝑗 (1) (the balance of the major production funds at the beginning of the
forecasted period) is found based on the exogenous data 𝑦𝑖 (𝑠) (13) (the ultimate product volume
for the last year of forecasted period), 𝑥𝑗 (𝑠) (15) (production volume during the last year of the
forecasted period) and 𝐾𝑗 (the capital investment for the entire forecasted period). After the
capital investment required for the entire period is defined, using the formula (8) we can
determine the capital contribution for the each year within the period.
Existence of the non-negative output and capital contributions provide the stable exogenous factor
of the factor 𝜑𝑗 (𝑡). The forecasting of 𝜑𝑗 (𝑡) can be implemented by the extrapolation of prior
year prices or using the trend model. If we accept that the capital contributions will be made by
the equal temps (𝛿𝑗 ), then the factor 𝜑𝑗 (𝑠) for the last year of the forecasted period will be
calculated as the following:
𝜑𝑗 (𝑠) =
𝛿𝑗 (1 + 𝛿𝑗 )𝑠−1
(1 + 𝛿𝑗 )𝑠 − 1