Abstract
The article discusses the issues of shortterm forecasting of gross regional product values, which is a key indicator of the functioning and development of regional socioeconomic systems using the apparatus of production functions. Neoclassical twofactor neoclassical production functions with constant (CES  function) and variable (VES  function) elasticity of substitution of production cost factors of the system under consideration were used as tools. The economic indicators of the average number of people employed in the economy of the regional socioeconomic system and the cost of its fixed assets, taking into account the speed of their retirement at the end of the year, were used, as factors of production costs. To build the above production functions, the index values of these indicators were used. The construction was carried out by the methods of mathematical statistics with the subsequent assessment of accuracy and adequacy by the corresponding generally accepted statistical criteria. After the forecast of changes in the values of indicators characterizing the factors of production costs, the latter were substituted into the constructed production functions to obtain a predictive estimate of the regional gross product. By the example of the economic system of the KhantyMansiysk Autonomous Okrug  Yugra, estimates were made of the changes in the values of this economic indicator with a forecast horizon of one year. The forecasting results were compared with the data of socioeconomic statistics presented by the Federal State Statistics Service (Rosstat) in a regional context.
Keywords: Production function, regional economic system
Introduction
Gross Regional Product (GRP) is a key economic indicator that characterizes the functioning and development (Alferova & Tretyakova, 2018; Drohobytsky, 2004; Kolemaev, 2005; Kleiner, 2012; Sokol et al., 2017) of the regional socioeconomic system (RSES). To assess the value of GRP in modeling the functioning of RSES, the apparatus of production functions (PF) is often used (Alferova & Tretyakova, 2018; Drohobytsky, 2004; Kolemaev, 2005; Kleiner, 2012; Sokol & Kutyshkin, 2016; Sokol et al., 2017). The identified production functions, as a rule, are used to analyze the efficiency and effectiveness of spending aggregated factors of labor and capital costs in RSES (Alferova & Tretyakova, 2018). At the same time, the PF can also be considered as a tool for shortterm forecasting of GRP RSES values. The main types of PFs used to solve the above problems are twofactor neoclassical production functions of the CES type — a function (CES PF), including the PF CobbDouglas (CD PF) (Alferova & Tretyakova, 2018; Drohobytsky, 2004; Kleiner, 2012; Sokol et al., 2017). Production functions of the type VESfunction (VES PF) are used much less frequently (Sokol & Kutyshkin, 2016; Sokol et al., 2017).
Problem Statement
For shortterm forecasting of changes in GRP RSES KhMAO  Yugra, we assume that the functioning of its economy is characterized by the following assumptions:
1. The functioning of the economy of this system can be described by twofactor neoclassical production functions and changes little over time, including for the adopted forecasting horizon.
2. The costs of the labor factor in the considered time interval change with a constant growth rate.
3. The lag between investments in fixed assets and the value of fixed assets of the regional economy is not taken into account.
4. The coefficient characterizing the disposal of fixed assets is considered known.
5. The economy of the region is considered closed, i.e. the direct influence of foreign trade on it is not directly taken into account.
Research Questions
Twofactor linearly homogeneous neoclassical PFs that are used to describe the functioning of the economy of the RSES of the RSES KhMAO  Yugra are in (Alferova & Tretyakova, 2018; Drohobytsky, 2004; Kleiner, 2012; Sokol et al., 2017):
 Constant elasticity substitution PF:
$Y=f(K,L)=A{\left(b{K}^{?}+(1b){L}^{?}\right)}^{5/?},?=\frac{1s}{s},0=s=1,s=\text{const}$(1)
$Y=f(K,L)=A\left(K{)}^{a}\right(L{)}^{\xdf},a>0,\xdf>0,a+\xdf=1,s=1$(2)
In expressions (1.2), K, L are the values of the factors of capital and labor costs in the economic subsystem of the RSES under consideration; α, β  elasticity of the final product Y by cost factors K, L; δ is an indicator of the uniformity of the FS: δ = 1; σ is the elasticity of substitution of the cost factor K by cost factor L; A  a constant, the values of which, like the values of the variables α, β, σ, are determined by the methods of mathematical statistics.
To describe the cost factors in the construction of PF, as a rule, use the following economic indicators (Sokol et al., 2017). For the capital cost, factor K is the cost of fixed assets of the RSES economic subsystem without or taking into account their load, as well as investments in its fixed capital. For the factor of labor costs L, the average number of people employed in the economy of the region, labor costs of workers employed in the region's economy, actual hours worked in the sectors of the region’s economy are used. To solve problems of an analytical nature with an appropriate justification, the construction of a production function allows the use of combinations of all of the above economic indicators. If it is necessary to solve the tasks of forecasting GRP, then it is possible to use only those indicators for which models for assessing changes in their values are developed and tested taking into account the adopted forecasting horizons. In this context, from the above indicators for L, it is advisable to use only the indicator of the average number of people employed in the region’s economy, and for K, the cost of its fixed assets.
Changes in the values of the factor of labor costs L are proposed to evaluate the dependence (Kolemaev, 2005):
${L}_{\text{teall}}={L}_{\text{O.act}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{v}t\right),t\in [\mathrm{0},\dots ,T]$.(3)
Here Lt.calc, L0.act  the calculated and actual values of the average number of people employed in the regional economic subsystem of the RSES at the times t and t = 0 (for the initial year of the considered retrospective time period (t = [0, ..., T] of the functioning of the RSES) respectively; ν is a constant determined by the methods of mathematical statistics in the analysis of retrospective data L0.act for the same period of time.
Changes in the values of the capital cost factor K for the period t = [0,…,Т], in turn, are proposed to be determined by the dependence (Kolemaev, 2005):
${K}_{\text{t.calc}}=\left(\mathrm{1}{\mu}_{t\mathrm{1}.act}\right){K}_{t\mathrm{1}.act}+I{n}_{t\mathrm{1}.act}$,(4)
where Kt.calc, Kt1.act  the estimated and actual cost of fixed assets of the economic subsystem of the RSES in the year t and t1, respectively; Int1.act  the actual value of investments in fixed assets of the RSES economic subsystem in year t1; μt1.act  the actual coefficient characterizing the disposal of fixed assets of this subsystem at the end of the year t1.
PF identification is carried out by presenting Y, K and L in the form of a series of basic indices relative to the selected year T0, the retrospective series of the source data of the corresponding economic indicators under consideration. If for the GRP value Y and the number of employees in the regional economic subsystem the formation of the basic indices IY, IL is not particularly difficult, then it is advisable to use the methodology proposed in (Bessonov & Voskoboinikov, 2006) to determine the values of the basic indices of the physical volume of the value of the basic production assets of the economic system_{}.
Purpose of the Study
This paper considers a number of issues related to the use of production functions for shortterm forecasting of the gross regional product of a regional socioeconomic system using the example of the RSES of the KhantyMansiysk Autonomous Okrug  Yugra (RSES KhMAO  Yugra).
Research Methods
Shortterm forecasting of the GRP is proposed to be carried out as a result of the following sequence of procedures:
1. For each $t\in [\mathrm{0},\dots ,T]$according to the official socioeconomic statistics presented in a regional context, the values of economic indicators of the final product Yt.ф f (GRP), cost factors of production Kt.act, Lt.act and the value of investments in fixed assets Int.act are identified for considered economic subsystem RSES.
2. The sequences of absolute values of Yt.act, Kt.act and Lt.act are converted into series of basic indices IY.act(t), IK act(t) and IL.act (t).
3. For the time interval t = [0,…,Т] constructing the CES PF (1) includes the CD PF (2) (Sokol & Kutyshkin, 2016), and the VES PF (Sokol et al., 2017).
4. According to (3.4), for t = T+1, the calculated values KТ+1.calc and LТ+1.calc are determined, which are being transformed by the basic indices IK.calc(T+1) and IL.calc(t) of the corresponding series formed earlier indices (paragraph 2).
5. Using the constructed PFs (paragraph 3), the GRP index IY.calc(T+1) is calculated based on the values of IK.calc(T+1) and IL.calc(t).
6. The relative forecast error of the gross regional product of the RSES economic subsystem under consideration is determined:
$e(T+1)=\frac{{I}_{Y.calc}(T+1){I}_{Y.act}(T+1)}{{I}_{Y.act}(T+1)}$.(5)
The initial statistics of GRP values (Yt.act), cost factors of production Kt.act, Lt.act, as well as the values of investments in fixed assets Int.act are given in the statistical compilation “Regions of Russia Socialeconomic indicators ”(https://www.gks.ru/folder/210/document/13204). The absolute values of Yt.act, Kt.act, Lt.act according to paragraph 2 of the previous section were converted into series of basic indices of the physical volume IY.act(t), IK.act(t), IL.act(t). The year 2001 was chosen as the base one (t = 0). Table 1 shows the values of these indices, which were subsequently used to identify PF.
The data in Table 1 regarding the baseline index of the average number of people employed in the economy_{}() show the presence of significant “jumps” of this indicator in the interval of 2012 2017 against the background of stagnation in the value of_{}() and a uniform increase in_{}(). Table 2 shows the results of constructing models (3) of changes in the average number of people employed in the RSES KhMAO  Yugra economy at_{} = 868,7 thousand people.
The coefficients of disposal of fixed assets μ_{} RSES KhMAO  Yugra according to the statistical collection “Regions of Russia Socioeconomic indicators” are equal: μ_{} = 0,6; μ_{} = 0,7; μ_{} = 0,7; μ_{} = 0,8; μ_{} = 0,8.
Findings
Identification of the above PFs and prediction of_{}() (item 2) values were carried out in MatLab15 ™. Table 3 presents the identified CobbDouglas production functions (2) for each of the considered time intervals. Here are the values of the determination coefficient^{} for the identified PFs and the corresponding calculated values of the Fisher criterion_{}, as well as tabular values of this criterion_{} for the significance level of 0.05 and two degrees of freedom (Korolyuk et al., 1985).
Table 4, the structure of which is similar to Table 3, in turn, presents the results of constructing CES production functions (1) for the same time intervals (values of the_{} criterion for the significance level of 0.05 and three degrees of freedom (Korolyuk et al., 1985).
Identification of PFs of type VES  a function according to the methodology [4] allows you to create only a tabular form for representing this function for the considered time interval. Due to the limitations of the scope of this work, the authors cite only one such table for the interval of 2001–2016 (table 5). Relative error ε_{} between_{} and_{} (PF VES) calculated using the constructed PF VES is determined by the dependence, the structure of which is similar to (5).
Relative error ${\stackrel{\u203e}{\epsilon}}_{t}$= 0,0449. Coefficient^{} = 0,7871.
The calculated values of the elasticity of substitution of production cost factors σ for the considered RSES show that starting from 2011 the used VES PF is very close to the Leontyev PF, for which σ → 0. Table 6 together shows:
The calculated values of ε() for all used PF show steady growth. This is due to the fact that since 2012 year, in the economic system under consideration, against the background of the growth in the capitallabor ratio ${k}_{\text{t.act}}={I}_{K\text{.act}}/{I}_{\text{L.act}}$, the growth in the average labor productivity ${y}_{\text{t.act}}={I}_{Y\xb7act}/{I}_{\text{L.act}}$has practically stopped. This is also evidenced by the values of_{} and_{} Table 1. The construction of neoclassical linearly homogeneous PFs provides for the fulfillment of the requirement associated with their growth (the first derivative of with respect to cost factors and must be greater than zero), as well as ensuring the convexity of this function upward (the second derivative of with respect to and should be less than zero). As a result of this, the construction of PFs based on_{} data, which are characterized by a fairly stable downward trend (2011 – 2016 years), leads to an increase in the discrepancy between the calculated_{} and source_{} data.
Conclusion
The results obtained (table 6) show that forecasting the GRP values of the considered RSES using a VES type PF  the function gives a better approximation of the calculated values of this economic indicator to its actual values, which are presented in the materials of the Federal State Statistics Service of the Russian Federation (Rosstat). It should also be noted that the use of the apparatus of production functions to describe economic systems in the functioning of which there is stagnation of the values of the final product with a steady increase in the costs of production factors will not give a close approximation of the calculated values of the final product to its actual values. At the same time, the authors believe that shortterm forecasts of the GRP value using PFs of the VEStype function can be considered an upper estimate of the values of this indicator.
Acknowledgments
This work was supported by a grant from the Russian Federal Property Fund and the Government of the KhantyMansi Autonomous OkrugUgra for the implementation of scientific project No. 1847860016 «Computer simulation of the dynamics of the socioeconomic system of the resourceproducing region of the north of Russia using growth theory, agentbased approach and GIS technology».
References
Alferova, T. V., & Tretyakova, E. A. (2018). The production function of the regional economy: an example of the Ural economic region. Bulletin of Ural State Economic University, 5, 7283.
Bessonov, V. A., & Voskoboinikov, I. B. (2006). Dynamics of fixed assets and investments in the Russian transitional economy. Moscow, IET.
Drohobytsky, I. N. (2004). Economic and mathematical modeling: a textbook for university students. Moscow, Publishing house «Examination».
Kleiner, G. B. (2012). Production Functions: Theory, Methods, Application. Moscow, Demand Book.
Kolemaev, V. A. (2005). Economic and mathematical modeling. Modeling of macroeconomic processes and systems. Moscow, UNITYDANA.
Korolyuk, V. S., Portenko, N. I., Skorohod, A. V., & Turbin, A. F. (1985). Handbook of probability theory and mathematical statistics. Moscow, Science. The main edition of the physical and mathematical literature.
Sokol, G. A., & Kutyshkin, A. V. (2016). Using production VESfunctions for modeling the functioning of economic systems. Bulletin of Ural State University of Economics, 2, 123134.
Sokol, G. A., Kutyshkin, A. V., & Petrov, A. A. (2017). On the use of production functions for modeling the functioning of the regional economy. Bulletin of South Ural State University. Series: Computer Technologies, Management, Electronics, 4, 8597.
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01 July 2021
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Cite this article as:
Kutyshkin, A. V., & Shulgin, O. V. (2021). ShortTerm Forecasting Of Gross Regional Product Using Production Functions. In D. S. Nardin, O. V. Stepanova, & V. V. Kuznetsova (Eds.), Land Economy and Rural Studies Essentials, vol 113. European Proceedings of Social and Behavioural Sciences (pp. 486493). European Publisher. https://doi.org/10.15405/epsbs.2021.07.59