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of the analyzed indicators and did not allow them to evaluate the spread of values. The paper also examines the problem of production competitiveness evaluation by the aggregation of the indices characterized by significant heterogeneity of indices relatively to a product chosen as a sample. The degree of such heterogeneity is posed to evaluate with criterion of balanced indices, where reckoning is based on the metrical analysis. Competitiveness of production is a basic indicator for assessment of competitiveness of the enterprise, for making decisions on production, in foreign economic activity.
Keywords: production competitiveness, balanced indices, metrical analysis
References
1. Mingaleva, Zh., A. & Gayfutdinova, O. S. (2007). Formirovanie innovatsionnoq konkurentosposobnosti khozyaystvuyushchikh subyektov [Development of innovative competitiveness of economic entities]. Yekaterinburg, Institut ekonomiki UrO RAN [Institue of Economics, UB RAS], 317.
2. Mingaleva, Zh. A. & Gershanok, G. A. (2012). Ustoychivoye razvitie regiona: innovatsii, ekonomicheskaya bezopasnost, konkurentosposonost [Sustainable development of a region: innovations, economic security, competitiveness]. Ekonomika regiona [Economy of Region], 3, 68-77.
3. Mingaleva, Zh. A. (2008). Osnovnyye metodologicheskiepodkhody k otsenke urovnya konkurentosposobnosti ekonomicheskikh sistem [The main methodological approaches to an assessment of level of innovative competitiveness of economic systems]. St. Petersburg, Politechnical Universitet Pibl., 648-681.
4. Babkina, T. N. (2010). Konukerntosposobnost kak factor ustoychivogo razvitiya promyshlennogo predpriyatiya: avtoref. dicc. ... kand. ekon. nauk [Competitiveness as factor of a sustainable development of the industrial enterprise: PhD thesis]. Izhevsk, Udmurstky State University, 18.
5. Kazaryan, N. V. (2009). Konkurentosposobnost kak faktor ustoychivogo razvitiya mezhotraslevykh kompleksov: na primiere lesopromyshlennogo kompleksa: avtoref. dicc.... kand. ekon. nauk [Competitiveness as factor of a sustainable development of interindustry complexes: on the example of timber processing complex: PhD thesis]. Moscow, Plekhanov Russian University of Economics, 16.
6. Rodionova, L. N., Kantor, O. G. & Khakimova, Yu. R. (2000). Otsenka konkurentosposobnosti produktsii [Assessment of competitiveness of production]. Marketing v Rossii i za rubezhom [Marketing in Russia and abroad], 1, 63-77.
7. Svetunkov, S. G. (1999). Informatsionnoye obespechenie upravleniya konkurentosposobnostyu [Information support of management of competitiveness]. Saint Petersburg State University of Economics and Finance Publ., 227.
8. Gaynanov, D. A., Kantor, O. G. & Kazakov, V. V. (2009). Otsenka urovnya sotsialno-ekonomicheskogo razvitiya territorial-nykh sistem na osnove metricheskogo analiza [Assessment of level of socio-economic development of territorial systems on the basis of the metric analysis]. Vestnik Tomskogo gosudarstvennogo universiteta [Bulleten of the Tomsk State UNiversity], 322, 138-144.
9. Larichev, O. I. (2003). Teoriya i metody prinyatiya resheniy, a takzhe khronika sobytiy v volshebnykh stranakh: uchebnik: izd. 2-e, pererab. i dop. [The theory and methods of decision-making, and also chronicle of events in the magic countries: textbook: prod. the 2nd ed. revised and enlarged edition]. Moscow, Logos, 392.
10. Rodionova, L. N. & Khakimova, Yu. R. (2009). Metodicheskie ukazaniya dlya vypolneniya kursovoy rabity po distsipline «Mrketing» [Methodical instructions for performance of a term paper on discipline "Marketing"]. Ufa State Aviation Technical University, Ufa, 39.
Information about the authors
Rodionova Lyudmila Nikolaevna (Ufa, Russia) — Doctor of Economics, Professor, Head of the Department «Finance, Monetary Circulation and Economic Security», Ufa State Aviation Technical University (12, Karla Marksa str., Ufa, Bashkortostan, 450000, Russia, e-mail: rodion@ufanet.ru).
Kantor Olga Gennadievna (Ufa, Russia) — PhD in Physics and Mathematics, Senior Researcher, Institute of Social and Economic Research of the Ufa Scientific Centre of the Russian Academy of Sciences (71, October av., Ufa, Bashkortostan, 450054, Russia, e-mail: o_kantor@mail.ru).
JEL codes: C61, C62, C53, C02
A. M. Tarasyev, A. A. Usova, Yu. V. Shmotina
PROJECTION OF THE RUSSIAN ECONOMIC DEVELOPMENT IN THE FRAMEWORK OF THE OPTIMAL CONTROL MODEL BY INVESTMENTS IN FIXED ASSETS1
In this paper, we develop an economic growth model taking into account two factors of production: fixed capital and labor force, to study the dynamics of GDP growth. The dependence of the output of these factors is described by a production function of the exponential type. Within the framework of the optimal control theory, the optimization problem for investment levels is being solved to maximize the integral index of con-
1 © Tarasyev A. M., Usova A. A., Shmotina Yu. V. Text. 2014.
sumption. We study the qualitative properties of optimal trajectories as solutions of the Hamiltonian systems arising in Pontryagin's maximum principle. The sensitivity analysis of the equilibrium solutions of the economic system is implemented with respect to the elasticity coefficients of the production function, the depreciation rate of the capital, and the discount factor, and growth trends are indicated. The econometric analysis of the model parameters is provided basing on real data for the Russian economy. In accordance with the results of the regression analysis, the projection of economic development is constructed in conditions of the applicability of the economic growth model.
Keywords: dynamic models of economic growth, optimal control, Pontryagin's maximum principle, economic equilibrium.
Introduction
This paper is devoted to analysis of the economic growth model based on methods of the optimal control theory with a view to construct the optimal investment strategies. The considered model in its design combines classical constructions of the economic growth theory [1], and advanced methods of analysis of the optimal control theory [2], in particular, generalizations of Pontryagin's maximum principle for problems with infinite planning horizon [3-11]. is Investments in fixed assets serve as control parameter and provide the output growth through the production function. In the balance equation the output is redistributed between consumption and investment, and this provides the model clos-edness in terms of control investment parameters. In this statement, the maximized characteristics is consumption, as in all classical models of financial investments [12], for example, in the Fisher stationary model and the Sharpe capital asset pricing model (CAPM) [13-14].
Application of the Pontryagin maximum principle in the model of economic growth generates the Hamiltonian system of differential equations, whose stationary points can be used to describe the equilibrium trajectories of an economy, and whose trajectories — to build prognostic estimates of economic development. It is important to emphasize that the main element of the solution is the optimal programming control which can be interpreted as an optimal investment plan. We develop high accuracy algorithms for constructing optimal investment plans as trajectories of the Hamiltonian systems.
The paper is organized as follows. The pattern of the economic growth model is given the optimal control problem is posed for investments in fixed assets. Qualitative analysis of solutions is provided within Pontryagin's maximum principle, stationary points are constructed for the Hamiltonian system generating economy equilibria and stationary optimal investment plans. In conclusion, the model parameters are set to the real data in the framework of the econometric
analysis of the production function, which complements the study of the properties of the model Hamiltonian systems. A comparative analysis of the prognostic model trajectories and real time series is fulfilled on the basis of the Russian mac-roeconomic data. Basic trends of model trajectories are identified and the key differences between these trends are indicated against the growth trajectories trends of developed economies.
Model of economic growth and the problem of optimal investment control
The basis of the model in the production block is presented by the production function, which is considered in the dynamic process of economic development. The main variables of the model are levels of capital K(t) and labor L(t) at time t, and the production output Y(t), which is given by the exponential production function of the Cobb-Douglas type [12]: Y(t) = F(K(t), L(t)) = aKa (t)L1-a(t). Here a positive parameter a corresponds to the output which is provided by production factors unaccounted in the model — Total Factor Productivity (TFP), and a nonnegative parameter a (0 < a < 1) is the elasticity coefficient.
Using the property of the positive homogeneity of the first order1 for the production function, one can make transition to relative variables: capital k (t) = K (t) / L (t) and GDP y(t) = Y(t) / L (t) per one worker (per capita)
У (t )=
Y (t)_ F (K (t).L (t))
L (t)
= F
Kjt) L (t)
Д
L (t)
= f (к (t )) = ak a(t),
(1)
where the function y (t) = f (k (t)) defines the labor productivity as a function of the capital-labor ratio.
Let us denote by symbols C(t) and I(t) consumption and investments in fixed assets at time t, respectively. Under assumption of the closedness of
1 The property of the positive homogeneity of the first order is defined as follows V v> 0 F (vK ,vL) =vF (K, L).
an economic system, one should require the balance relation in each time period
Y (t ) = C (t) +1 (t ) = C (t) + u (t )Y (t
^ C (t ) = if) = y (t)(! - U (t)),
where the function u(t) corresponds to the share of GDP invested in fixed assets, i.e. I (t) = u(t) Y (t). The relative value of c(t) is the level of consumption per one worker. One can derive restrictions on the investment levels from the balance equation,
0 < c (t ) = y (t )(1 - u (t ))< y (t ), 0 < u (t)< u < 1.
(2)
This version of the model assumes that the growth of the capital stock K(t) is subject to the dynamics introduced by Solow [1]
K (t) = u (t)Y (t)-|K (t), K (0) = K0,
L (t)
L (t )
= n, L (0) = L0,
tional (4) on trajectories of the dynamical system (3).
Analysis of the optimal control problem within the framework of Pontryagin's maximum principle
The posed problem of optimal control is subject to a modification of Pontryagin's maximum principle for the infinite time period [3-11].
Let us construct the stationary Hamiltonian function of the control problem which due to the dynamics (3) and the objective functional of the control process (4) has the following form with the conjugate variable y
H (k, y,u) = ln f (k)+ ln (1 -u) + y(uf (k )-8k). (5)
Let us calculate the control u0, delivering maximum to the Hamiltonian H (5)
dH (k, y,u)
du
1
1 - u
where a positive coefficient | is the depreciation rate of capital, and a constant value of n stands for the growth rate of the labor force L(t).
Passing to the relative values in the last relations, we obtain the following dynamics for k(t)
K
k (t ) = u (t) f (k (t))-8k (t), k (0) = ko = K- (3)
Lo
where the parameter 8 = | + n is the degree of capital dilution due to its depreciation and increase in the labor force.
Let us consider the problem of optimal control of investment in which the objective functional is given by the integral of the logarithmic index of consumption, discounted on the infinite time interval:
J (•)= j e-pt ln c (t) dt =
0
= j e-pt (ln f (k (t))+ ln (1 - u (t))) dt, (4)
0
where p is the discount factor.
Let us note that in the utility theory, the logarithmic function describes the relative increase (in our case — the relative consumption) per unit of time. Under conditions of uncertainty the logarithmic function specifies the constant relative risk aversion.
Problem. It is necessary to construct such an investment strategy (k(t), u(t)) that satisfies the constraints (2) and maximizes the objective func-
u0 (k,y) =
0, 1 -
Vf ( k
0 < f ( k )v< 1,
—, 1 < f ( k , w 1 - u
f ( k )v
1
f(%>—. (6)
v/ 1 - u
The maximum control u0 (k, y) (6) divides the domain of variables (k,y) into three domains with the different structure of the Hamiltonian dynamics defined by relations
dk(t) JH(k(t),y(t),u0 (k(t),y(t))) = dt dy
=u° (k(t),y(t)) f (k(t))-Sk(t), (7)
dH(k(t),y(t),u0 (k(t),y(t)))
d y(t )
dt
= py(t )--f( k )
dk
= pv(t -V(t )(u0 ( k (t ) ,y(t )) f( k (t ))-s).
Let us consider the Hamiltonian system (7) for three different control modes.
Zero control mode
The domain of the zero control mode u0 (k, y) = 0 is determined by the relations 0 < f (k)y < 1. The Hamiltonian system with this control can be written as dk (t )
dt
d y(t )
= -8k (t ), f( k )
dt
= (p + 8)y(t )-
f ( k )■
Passing to the normalized dynamics, we obtain the system of differential equations in variables (k, z) where z = ky
dk (t)
dt
= -8к (t ),
dz (t )=pz (t )- LjQ к =pz (t )-a. (8)
dt
f ( к )
dk(t)_( f (к(t))
dt
1
к(t) z(t)
-8
к (t ),
dz (t ) dt
f(к (t))
p-f(к (t ))^"i(t) z (t )-1
p + (l-a)
f(к (t)) к (t )
z (t )-1.
(9)
Saturated control mode
dh (t )
dz (t ) dt
dt
f
uf (к (t )) U к (t)
-8
к (t ),
p-uf ■ (к (t z (t )-
f ' ( к (t )) к (t ) f(к(t))
p + и (1 -a)
f(к(t)) к (t )
z (t )-a. (10)
In the last formulas, we use the following properties of the Cobb-Douglas production function of the exponential type:
f (k) ^ f (k )k
пк a
f ' ( к ) = пaka-1 = a — = к
a
f ( к )
- = a.
As one can see from the structure of the system (8), its solution declines exponentially to zero at the rate of depreciation of fixed assets 8 with respect to the phase component of capital k(t). Thus, the lack of the investment component leads to reduction of the capital stock up to the zero level.
Transient control mode
The transient control mode is characterized by a non-zero share of output invested in fixed assets that is calculated by the formula
u0 (k,y) =1 -yr^ =1 -77^ z = ky f (k)y f (k) z
f (k)
and meets restrictions: 1 < v '
- < —L_
k < 1 -u'
The Hamiltonian system in the variable control mode has the following structure
irtuUW
Equilibrium of economy in the maximum principle
Stationary levels of the Hamiltonian system
(7) determine the equilibrium positions of the economic system. In order to find these levels we equate the right sides of the Hamiltonian systems
(8)-(10) to zero and solve the system of non-linear algebraic equations in variables (k, z) under conditions k > 0, z > 0, y > 0.
Evidently, the Hamiltonian system (8) for the zero control regimes does not have stationary levels satisfying the constraints k > 0, z > 0. However, in domains of the transient and saturated control regimes steady states exist and their coordinates are determined analytically through the model parameters.
Proposition. The equilibrium points in the domain of the transient control and the domain of the saturated control are given by the formulas
к * =
na 8 + p
1-a * a8 -, и =-< и,
8 + p
V 8 ,
IV
и = и,
z =
a
г, y = п
p + (l -a) 8
c * = y * (l - и" ).
( к • Г •
The saturated control mode is presented by the highest possible level of investments in fixed assets u0 (k, y) = u, and is constrained by the limitations f (k) y > 1/(1 -u). Under these conditions, the Hamiltonian system (7) in the variable (k, z) has the following form
Proof. 1) Let us calculate the coordinates of the equilibrium point in the transient control regime. It is necessary to equate the right sides of the Hamiltonian system (9) to zero
f (k) 1 f (k) 1
p + (l -a)
f ( к )
z-1 = 0 ^
p + (l -a)( 8 + -Il z = 1,
z =
a
f ( к • )
p + 8
p + (l-a)8 к'
a
^ к' =
na
8 + p
Let us calculate the control value corresponding to the equilibrium level
u' = 1 -
f ( * ' У
1 -a(p + (1 -a)ô) = aS < 1 (p + 8)a 5 + p
8 + p
f ( * )
-8 = 0:
^ = 8^ * ' U
\
1
., , au H-a
■* ='т
r -/ f (*) p + u (1 -a)
г-a = 0 ^
(p + (1 -a) 8) г" =a
^ г =-
a
a du 8 du ' p
eu, a = = 1 eu, 8 = ^c; = T""
u da u d8 8 + p
< 1,
eu,p =
p du *
< 0.
The equilibrium point is located in the domain of the transient control if u' < u. Therefore, if the model parameters meet the constraints
a8 _ < u,
the equilibrium point (k, г , u') is attained in the domain of the transient control.
2) Let us calculate the coordinates of the equilibrium point in the domain of the saturated control when u (t) = u. In order to do this, one should resolve the system of nonlinear algebraic equations determined by the right hand sides of the
Hamiltonian dynamics (10) f (k).
u ' dp 8 + p
Coefficients of elasticity show that the equilibrium investments in capital u increases together with the growing elasticity a of the production function and the growing capital depreciation rate 8, and it decreases when the discount factor pincreases.
Elasticity coefficients for the equilibrium output of production y with the respect to the model parameters have the following expressions
e y ,a =
e „ =
y,8
e y,p =
a dy *
a
y* da 1 -a^ 1 -a
1 +
ln a
a
8
Adyl
y d8 p dy"
y * dp 1 -a 8 + p
1-a 8+p p
a
< 0,
< 0.
p + (l -a) 8
The equilibrium control coincides with the maximum possible investment level into capital, i.e. u' = u.
The equilibrium level of the output is calculated by the formula (1)
y = f (k )=a (k )a.
The equilibrium level of consumption, expressed through the balance equation, is given by the following relation
c =(l-u )y\
Proposition is proved.
Analytical expressions for the values of investment u'', capital k*, output volume y* and consumption c allow to analyze the sensitivity of the stationary equilibrium state in terms of the elasticity coefficients for the model parameters a, 8, p. Signs of elasticity coefficient indicate of the growth and decrease trends. Their absolute values indicate the change rates in percentage terms.
Calculation of the logarithmic values of derivatives of investments k* at the equilibrium point provides the following values for the elasticity coefficients with respect to the model parameters
Negative signs of the elasticity of the equilibrium production level y* with respect to changes in the capital depreciation rate 8 and the discount factor p indicate the decline trend. As for the sign of the elasticity of the equilibrium production level y* with respect to changes in the elasticity a of the production function, it is worth to note that small values of the elasticity a make it negative and large values of the elasticity of a make it positive. It means that here both decreasing and increasing trends can be obtained.
If the equilibrium level of capital satisfies the constraint u < u, then one can observe the decline trend over time of the optimal investments to the level of u*. Otherwise, the maximum level of investments is the best possible value on the entire time interval until the time of stabilization of the dynamical system. In all development scenarios the pattern of optimal solutions is configured in the expressed S-shape, which is ensured by the presence of the parameter constraints u on investments u(t). In addition, one can note that the optimal trajectories have a tendency to the saturation growth.
An algorithm for constructing optimal model trajectories
Let us give a brief description of the algorithm for constructing the optimal trajectories as solutions of the Hamiltonian systems (8)-(10). We should note the integration peculiarity of the Hamiltonian systems, which involves boundary conditions for the phase variable of capital k(t), given on the left side, and the boundary condi-
tions for the adjoint variable z(t) are given on the right side of the time interval. In this connection, a two-step integration procedure is developed.
1. The first step. The system is integrated by the Runge-Kutta method in the reverse time from the equilibrium point in the direction of the eigenvector of the Jacobian matrix corresponding to the negative eigenvalue. Integration is implemented until the integrated trajectory reaches the initial point of the phase variable.
2. The second step. Integration is performed in the direct timeline from the initial point. During this process, the components of the optimal model trajectory are reconstructed including the optimal investment plan u0(t).
It is important to emphasize that the two-step integration process provides a highly accurate algorithm for constructing optimal model trajectory with the order higher than the time step of integration.
Econometric analysis of the parameters of the production function
It is worth to emphasize that construction of prognostic model trajectories and econometric analysis of the production function and other parameters of the model are complementary tasks, but they are carried out in separate blocks with different functions. The econometric analysis is based on the classical methods of econometrics [15-16] and is used to identify the model parameters, in particular, the elasticity coefficients of the production function and setting the total factor productivity. The numerical algorithms, after substituting parameters identified in the econometric analysis, work offline producing optimal trajectory of endogenous growth. Another important issue is that the simulation procedure does not attempt to construct the trajectories of the best approximation of the real time series. It builds namely the optimal investment plan and the optimal trajectory of economic growth. If the optimal model trajectory is close to the real data of economic development, then this fact indicates the optimal economic growth of the analyzed economic system in the sense of the integral consumption index. In the case of deviations of the optimal model trajectory from the data in the "upper" or "lower" direction, it is reasonable to discuss a phenomenon of underinvestment or overinvestment in the economic system. In this context, the model constructs the optimal investment plan that indicates imbalances with the real data. Therefore, the model trajectories could be called optimal prognostic trajectories to distinguish them from forecasts of econometric models.
3,5 3 2,5 2 1,5 1
0,5
V
Fig. 1. Comparison of real data and predictive values of GDP (black) on the simple regression model (grey)
1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-Г"
Fig. 2. Share of GDP invested in fixed capital, u, %
The calibration procedure for the production function (1) is fulfilled on the basis of the statistical data [17-18] of the Russian economy in the period from 1990 to 2011 using the paired-regression model
Wi =aXi +p,
where w{ = lnyi, x{ = lnki, p = ln a, i = 1,22.
Estimates of the paired-regression model parameters are calculated by the least squares method:
cov
a = -
( w, X )
a2 ( X)
= 0,836, в = w-a X = 0,249 ^
^ y = 1,283k °'836.
Curves on Figure 1 show the real data (in black) and the trajectory constructed using the regression model (in grey).
To assess the quality of the regression model the coefficient of determination R2 = 0,909 is calculated whose value indicates a good "matching" with the real data. The results of testing the significance hypothesis of the regression model suggest that the fixed capital has a significant impact on
the level of country's GDP, and this is confirmed by the following calculation:
1. Evaluation of the error variance: 1
s = -
N - 2
Yßi = 0'°3.
2. Evaluation of the variance for the elasticity coefficient, ct2 (a) :
CT
(a ) =
x - X )
— = 0,0032.
3. The observed value of the Student's statistics for N - 2 = 20 degrees of freedom:
t =
a
r(â )
= 14,86.
4. The critical value at the level of the statistical reliability, y = 95%:
tk = tN -2 ) = 2,086 ^t > tk.
Prognostic model trajectories and comparison with econometric time series
Returning to the described model of economic growth, let us construct its solutions numerically [4-11], and compare these solutions with real data. To construct the prognostic trajectories we use results of the regression analysis for the production function, as well as data on the Russian economy for the period from 1990 till 2011 [17-18]. The estimated indices of economic development are calculated at the levels:
k * = 5.97,
y
= 5.74, u = 0.443, c * = 3.195,
a8 8 + p
= 0,703 < u.
And selection of a smaller value u for the maximum investment level leads to the economy with practically zero growth.
The model shows that the optimal level of investments in the Russian economy in the current period is implemented on the upper level of constraints. Let us note that a series of experiments, with different restrictions on the maximum level of investments have been conducted and the outcome remained the same: the optimal level of investments is realized at the upper limit. This means that investments in fixed assets in the economy are insufficient and require increase even for the upper threshold. For comparison, in modeling developed economies such as economies of the US [8-11], Japan [4-7], and the UK [4-5], one can observe a qualitatively different picture. Namely, the model investment plans have the tendency to decrease to the stationary equilibrium level u < u, which indicates the situation of over-investment in fixed assets in these countries.
taking into account the economic indicators [1718] given in the list.
1. The level of depreciation of fixed assets according to Rosstat [18] is estimated as 42% in average, thus, the parameter | is equal to 0.42.
2. The growth rate of the labor force n is estimated at the level of 0.004 in the period from 1990 to 2011.
3. The discount factor p according to the data [18] of the Central Bank of the Russian Federation on the refinancing rate is fixed at the level of 0.08.
4. The level of investments in fixed assets, according to Rosstat [18] has a clearly expressed increasing trend (see Figure 2). Therefore, to construct the prognostic trajectory the maximum level of investments u is chosen at a sufficiently high level of 44.3%. However, even the investment limit of 44.3% does not meet the inequality
Fig. 3. Prognostic growth rates (black) of fixed capital per capita of working population in comparison with real data (grey)
This fact indicates that the investment trajectory does not decrease during the forecast period.
Fig. 4. Prognostic growth rates (black) of GDP per capita of working population in comparison with real data (gray)
s
On Figures 3 and 4 the prognostic model tra- decline trend in investments (in percentage) with
jectories of the Russian economic development higher initial levels to lower stationary equilib-
with projections to 2050 are presented in compar- rium levels. For economies of developing countries
ison with the real data. there is a bifurcation and qualitatively different
The presented plots clearly indicate that in trends are observed when the value of the equilib-
the 90th years there is a rapid reduction in pro- rium investments is very high in comparison with
duction and investments in fixed assets, which is the actual current levels. This leads to the situa-
expressed in sharp drops of real statistical trends. tion when the model optimal investments should
At the same time, the prognostic model trajecto- reach the highest possible limit values. ries clearly demonstrate a growing trend, since the C0nclusi0n model is inherently focused on the search for such
an investment strategy that leads the economy The paper deals with analysis of the model of
to the growth domain. The investments growth economic growth within the framework of the op-
is positively reflected in the statistics of the dy- timal control theory. The problem of optimizing
namics, as one can see on the presented graphs. the investment levels is considered, the solutions
Namely, starting from 2000 clear growth trends are constructed within the Pontryagin maximum
are shaped for the main considered factors (ex- principle, and analysis of trends is fulfilled for the
cepting the recession of 2008), and by the end of optimal model trajectories. Identification of pa-
the analyzed period the actual data almost reach rameters and analysis of significance of the re-
the model trends. gression model are implemented for the produc-
It should be noted that the comparative anal- tion function. Basing on algorithms for numerical
ysis of modeling results for the macroeconomic construction of optimal prognostic trajectories,
data on countries' economies in different groups the model growth trends are identified; equilib-
shows that the values of the identified model pa- rium points of major economic indicators are cal-
rameters in these groups are qualitatively differ- culated and compared with real statistical data of
ent by clusters. These differences affect, first of the Russian economy. Major differences are in-
all, the dynamics of investment plans. In econo- dicated between the identified trends and the
mies of developed countries, one can observe the growth trajectories in developed economies.
The study is supported by the grant of the Russian Scientific Fund (project No. 14-18-00574 "Information and analytical "Anticrisis" system: diagnostics of regions, threat assessments, and scenario forecasting for the purpose of maintaining and strengthening of economic security and improve welfare of Russia").
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Information about the authors
Tarasyev Alexander Mikhajlovich (Yekaterinburg, Russia) — Dr. Phys.-Math. Sci., Head of Sector, the Institute of Mathematics and Mechanics named after N. N. Krasovskii of the Ural Branch of the Russian Academy of Sciences; Research Scholar, International Institute for Applied Systems Analysis (IIASA) (16, S. Kovalevskaya str., Yekaterinburg, 620990, Russia, e-mail: tam@imm.uran.ru).
Usova Anastasy Aleksandrovna (Yekaterinburg, Russia) — PhD in Phys.-Math. Sci., Research Scholar, the Institute of Mathematics and Mechanics named after N.N. Krasovskii of the Ural Branch of the Russian Academy of Sciences (16, S. Kovalevskaya str., Yekaterinburg, 620990, Russia, e-mail: anastasy.ousova@gmail.com).
Shmotina Yulia Valer'evna (Yekaterinburg, Russia) — student, the Institute of Mathematics and Computer Science, the Ural Federal University named after the first President of Russia B.N. Yeltsin (19, Mira str., Yekaterinburg, 620002, Russia, e-mail: ju-liashmotina@yandex.ru).
JEL: C610, D210, G130, Q520
A. A. Krasovskii, N. V. Khabarov, M. Obersteiner
IMPACTS OF THE FAIRLY PRICED REDD-BASED CO2 OFFSET OPTIONS ON THE ELECTRICITY PRODUCERS AND CONSUMERS1
This paper deals with the modeling of two sectors of a regional economy: electricity and forestry. We show that CO2 price will impact not only the profits of the CO2 emitting electricity producer (decrease), but also the electricity prices for the consumer (increase), and, hence, some financial instruments might be implemented today in order to be prepared for the uncertain CO2 prices in the future. We elaborate financial instrument based on the Reduced Emissions from Deforestation and Degradation (REDD+) mechanism. We model optimal behavior of forest owner and electricity producer under uncertainty and determine equilibrium fair prices of REDD-based-options.
Keywords: CO2, REDD+, Option Pricing, Optimization, Firm Behavior, Cost Minimizing, Uncertaint
Introduction
This research is focused on developing financial instruments supporting activities within the framework of the Reducing Emissions from Deforestation and Forest Degradation Plus (REDD+) program. The basic idea of the program is that REDD+ would provide payments to jurisdictions (countries, states, or provinces) that reduce forest emissions below agreed-upon benchmark levels2. In a recent review [1] authors discuss the potential of REDD+ and show that there are many research needs and opportunities for ana-
1 © Krasovskii A. A., Khabarov N. V., Obersteiner M. Text. 2014.
2 United Nations Framework Convention on Climate Change [UNFCCC] 2010, https://unfccc.int/.
lyzing REDD+ policy designs at the global, national, and subnational levels including examining land use planning and other applications for ongoing REDD+ policy processes. The economic modeling tools provided in the literature reflect various REDD+ applications, and model the impacts of REDD+ at various scales and dimensions for scenarios of future CO2 prices. The fixed market models are site-specific and mostly estimate the benefits of REDD+ for forests [2]. The partial equilibrium models are sector specific (forest, agriculture) and focus on particular regions, e.g. [3], in long-term perspective 50-100 years. The general equilibrium models are economy-wide and nearer-term (e. g., 20 years) [4]. There are also integrated assessment models which link global economy and biophysical systems at a very long-run
