Optimal Allocation of Resources in an Open Economic System with Cobb–Douglas Production and Trade Balances

Abstract

This paper develops a nonlinear optimization model for the optimal allocation of labor and investment resources in a three-sector open economy. The model is based on the Cobb–Douglas production function and incorporates sectoral interdependencies, capital depreciation, trade balances, and import quotas. The resource allocation problem is formalized as a constrained optimization task, solved analytically using the Lagrange multipliers method and numerically via the golden section search. The model is calibrated using real statistical data from Kazakhstan (2010–2022), an open resource-exporting economy. The results identify structural thresholds that define balanced growth conditions and resource-efficient configurations. Compared to existing studies, the proposed model uniquely integrates external trade constraints with analytical solvability, filling a methodological gap in the literature. The developed framework is suitable for medium-term planning under stable external conditions and enables sensitivity analysis under alternative scenarios such as sanctions or price shocks. Limitations include the assumption of stationarity and the absence of dynamic or stochastic features. Future research will focus on dynamic extensions and applications in other open economies.

1. Introduction

In the context of the growing complexity of the global economy, accelerated globalization and increasing foreign trade restrictions, the task of efficient allocation of limited production resources is becoming increasingly relevant for open economic systems. This problem is especially acute for developing countries with a raw material focus and structural dependence on the import of capital-intensive components and technologies.

Despite the widespread use of the Cobb-Douglas production function in economic modeling, most of the work is focused either on simplified one- or two-factor models (Tarasyev et al., 2023) or on solving problems without taking into account the constraints typical of open economies (Vasyl’yeva, 2021). A significant part of applied models use linear or simulation approaches (Saraev & Saraeva, 2020), which limits the possibilities of decision-making based on optimality principles.

The issues of simultaneous consideration of intersectoral interactions, restrictions on investment and labor, depreciation of fixed capital, as well as trade restrictions and quotas remain poorly studied. Classical models, as a rule, lack a flexible structure that would allow for external shocks or systemic dependence on imports, which limits their applicability for economic policy.

An example of a three-sector model in an open economy is the work of Kolemayev (2008), who constructed a theoretical structure of balanced growth taking into account intersectoral flows. However, this model has a number of limitations: it is static, focused on equilibrium distribution, and does not contain a formal optimization statement. In addition, there is no numerical solution scheme, which complicates its practical application.

In works (Z. Murzabekov et al., 2020; Z. Murzabekov & Tussupova, 2024; Z. N. Murzabekov et al., 2022) the authors proposed methods and algorithms for solving optimal control problems within the framework of a three-sector model of an economic cluster. Using a special form of the Lagrange multiplier method, the authors simplified the computational procedures, allowing for numerical calculations of a steady state without the use of heavy iterative procedures. These approaches have proven effective in modeling closed economic systems that are not affected by foreign trade. However, modern realities require a transition from closed models to more universal systems that take into account both internal constraints and external economic conditions.

This study aims to develop existing models by formulating a new optimal control problem in a three-sector open economy that simultaneously takes into account:

• limited labor and investment resources;
• depreciation of fixed production capital;
• balance of materials between sectors;
• restrictions on imports of capital-intensive components;
• and the need to maximize the output of the consumer industry as the final result.

The proposed model is formalized as a nonlinear constrained optimization problem solved using the Lagrange multiplier method and a numerical one-dimensional search using the golden section method. This ensures both analytical interpretability and numerical feasibility, allowing parametric analysis to be performed and economically significant scenario results to be obtained.

The Republic of Kazakhstan, an export-oriented country with a high share of the raw materials sector and significant dependence on imports of investment and consumer goods (Hasanli et al., 2024), was chosen as the empirical basis for the model. Kazakhstan is characterized by a high degree of openness: according to the World Bank and the IMF, the share of foreign trade in GDP consistently exceeds 50%, and the level of tariff barriers remains low. In addition, the structure of the country’s economy is fully consistent with the model taking into account the export of raw materials, the import of investment goods and the centralized distribution of resources.

For numerical calibration, statistical data for 2010–2022 were used (Tussupova & Mirzakhmedova, 2024): industrial output, number of employees and investment in fixed assets by sector.

The purpose of this work is to develop and solve a limited nonlinear model for the optimal allocation of resources that reflects the interaction between internal production mechanisms and external trade balances.

In this regard, the following questions are considered within the framework of the study:

• How do industry trade restrictions affect the optimal allocation of capital and labor?
• What are the equilibrium relationships between the distribution of labor and investment?
• How sensitive is the equilibrium to changes in the elasticity of production and trade parameters?

Thus, the proposed model is both a development of economic and mathematical theory and a tool that can be used in strategic analysis and development of economic policy in the context of resource and trade restrictions.

2. Literature Review

The Cobb–Douglas production function, proposed by Cobb and Douglas (1928), remains one of the most widely used tools in applied economics. Due to its analytical simplicity and interpretability, it is widely used in modeling the relationship between the main factors of production (capital and labor) and output. In its classical form, this function is applied to assessing the technological parameters of industries, analyzing the efficiency of resource use, forecasting output, and constructing growth models.

Over time, the application of this function was expanded to multisector, dynamic, and optimization models, which made it possible to describe more complex economic systems. Modern research covers both theoretical aspects of the production structure and applied problems of resource allocation, sustainable growth, and external economic impacts. The review of existing work can be conditionally divided into five areas: (1) computational methods for solving problems with the Cobb–Douglas function; (2) multi-sector and dynamic models; (3) work on open economies and external economic restrictions; (4) applied and empirical research; (5) approaches to designing software systems for management and decision support.

2.1. Computational Approaches to Solving Cobb–Douglas Models

Dinc et al. (2025) proposed using a genetic algorithm to optimize the parameters of the Cobb–Douglas function. The method showed high flexibility in searching for a global optimum, but the approach itself is technical — it does not take into account resource constraints, inter-industry interactions, or the external economic structure. Similarly, in the work of (Betancur-Hinestroza et al., 2025), the Cobb–Douglas function was solved in a quantum setting via the Clairaut differential equation. Although this method expands computational horizons, it is not based on a realistic economic structure and does not include production or trade balances.

2.2. Multisector and Dynamic Models

In work (Matsumoto & Szidarovszky, 2021), the authors considered a two-sector model of economic growth with the Cobb-Douglas function, which takes into account time lags in the dynamics of capital and labor. The authors showed that under certain parameters, the system loses stability through a Hopf bifurcation. However, the model is limited by internal interactions and does not contain constraints related to foreign trade or inter-sector reallocation. Muro (2013) developed a three-sector model of GDP using capital, labor, and land as factors of production. He demonstrated the effect of relative prices on the interest rate and land rent, which is important for structural analysis, but the work does not formulate the problem of optimal resource allocation and does not take into account constraints.

2.3. Optimization and Open Economy Models

The work (Enrique & Garcia-Salazar, 2023) is devoted to a two-sector model taking into account capital flows and subsidies. It emphasizes the complexities of economic policy in an open system, but does not consider production technologies, material balances, and the structure of the production function. The study (Meidute-Kavaliauskiene et al., 2021) demonstrates the application of objective programming to the allocation of natural gas between sectors. However, the approach is linear and specialized—it is not universal for the analysis of total economic output. The work (P. Li & Zhong, 2020) proposes an interregional resource allocation model that takes into account competition. The model is interesting from the point of view of geoeconomics, but is not based on the factor structure of production and does not use Cobb-Douglas-type functions.

2.4. Applied Empirical Research and Estimation Models

Chi et al. (2021) applied an improved Cobb–Douglas function to analyze the relationship between energy consumption and economic growth. The use of cointegration analysis allowed us to identify long-run relationships, but the work is focused on statistical estimation rather than optimization. K. Li et al. (2019) used the Cobb–Douglas function to analyze urban water consumption, but the model does not have a multi-sector structure and does not consider the optimal resource allocation. In (Jin & Zhang, 2011), they proposed an analytical solution to the growth problem in a multi-sector model, but without including trade restrictions and resource balances. And in paper (Kim & Jeon, 2025) a model of consumption and investment with partial borrowing restrictions in a two-sector economy was considered, ignoring the terms of foreign trade and balance.

In their studies (Zmeškal et al., 2023), the authors constructed a model for predicting the distribution of the EVA value based on Levy processes for a small open market. The model has a narrow applied focus and is focused on post-processing of indicators, without forming a controllable system. The work (Zhang, 2020) emphasizes the influence of transportation costs on regional imbalances in China, but it considers exogenous structural constraints without formalizing the problems of optimal resource allocation.

2.5. Architectural and Software Approaches to Management

Recent developments in economic planning and industrial policy emphasize the growing need for integrated decision-support systems and sustainability-oriented frameworks. The work of Gornov et al. (2021) and Muhammad et al. (2023) explores architectures for optimal control and enterprise-level resource management, focusing on data integration, software implementation, and process automation. While these approaches are important from an engineering standpoint, they do not involve formal economic models with multi-sector structures and trade constraints.

Similarly, recent reviews on sustainable manufacturing and circular economy strategies (e.g., Karuppiah et al. 2024a, 2024b) stress the role of digital technologies, resource efficiency, and supply chain resilience. However, these works typically lack quantitative optimization models that connect policy scenarios with sectoral production dynamics. In contrast, the model proposed in this study contributes a mathematically rigorous tool for exploring resource allocation problems under external constraints, bridging the gap between high-level sustainability goals and operational economic modeling.

3. Mathematical Model of an Open Economic System

A macroeconomic model of an open three-sector economy with centralized resource allocation is considered and denote the sectors of the model as follows:

• $i=0$—the material sector, produces resources used in the form of raw materials and semi-finished products by other sectors;
• $i=1$—the capital-forming sector, produces means of production (machinery, equipment, etc.);
• $i=2$—the consumer sector, produces final products for domestic consumption and export.

Each sector has its own production function and interacts with others through intersectoral resource flows and foreign trade.

The output volume in each i-th sector is determined by the Cobb-Douglas production function:

$$X_i=F_i(K_i,L_i)=A_i{K}_i^{{\alpha }_i}{L}_i^{1-{\alpha }_i},(i=0,1,2).$$

(1)

where:

• $X_i$—output volume;
• $K_i$—volume of fixed productive assets;
• $L_i$—volume of labor resources;
• $A_i$—coefficient of neutral technological progress;
• ${\alpha }_i$—capital elasticity coefficient;
• ($1-{\alpha }_i$)—labor elasticity coefficient.

This form allows to reflect both differences in capital intensity of industries and scalability of technology. Parameters $A_i,{\alpha }_i$ are subject to calibration according to industry statistics.

Each sector has its own fixed productive assets (FPA), while labor resources and investments can be freely redistributed between sectors. The change in the FPA of the i-th sector over a period consists of depreciation—${\mu }_i{K}_i$ and an increase due to investments $I_i$:

$${\displaystyle \frac{d{K}_i}{dt}}=I_i-{\mu }_i{K}_i,K_i\left(0\right)=K_i^0,(i=0,1,2).$$

(2)

Let us introduce the following notations:

• ${\theta }_i={\displaystyle \frac{L_i}{L}}$—sectoral shares in labor resource distribution;
• $s_i={\displaystyle \frac{I_i}{X_1}}$—sectoral shares in investment resource distribution;
• $f_i\left(k_i\right)={\displaystyle \frac{X_i}{L_i}}$—labor productivity in the i-th sector;
• $k_i={\displaystyle \frac{K_i}{L_i}}$—capital-labor ratio of the sectors;
• $y_1={\displaystyle \frac{Y_1}{L}}$—share of imported goods for investment;
• $y_2={\displaystyle \frac{Y_2}{L}}$—share of imported goods for consumption;
• $x_i={\theta }_i{f}_i\left(k_i\right)$—specific output of the sectors.

Then, the Equation (2) can be rewritten in the following form for the capital-labor ratio of the sectors:

$${\dot{k}}_i=-{\lambda }_i{k}_i+{\displaystyle \frac{s_i}{{\theta }_i}}(x_1+y_1),k_i\left(0\right)=k_i^0,{\lambda }_i>0,(i=0,1,2),$$

(3)

$$x_i={\theta }_i{A}_i{k}_i^{{\alpha }_i},A_i>0,0<{\alpha }_i<1(i=0,1,2),$$

(4)

the equilibrium of resources is ensured by the following constraints:

• Investment balance:

$$s_0+s_1+s_2=1,s_i>0,(i=0,1,2),$$

(5)

• Labor balance:

$${\theta }_0+{\theta }_1+{\theta }_2=1,{\theta }_i>0,(i=0,1,2),$$

(6)

• Material balance:

$$(1-{\beta }_0)x_0={\beta }_1{x}_1+{\beta }_2{x}_2+y_0,0<{\beta }_i<1,(i=0,1,2).$$

(7)

• Foreign trade balance:

$$q_0{y}_0=q_1{y}_1+q_2{y}_2.$$

(8)

• Industrial security:

$$y_1\le {\gamma }_1{x}_1,y_2\le {\gamma }_2{x}_2.$$

(9)

These conditions model the limits of permissible external pressure.

Here:

• ${\gamma }_1,{\gamma }_2$—maximum permissible shares of imports of investment and consumer goods;
• $y_0$—export of raw materials; $y_1,y_2$—volumes of imports of investment and consumer goods, respectively;
• $q_0$—world price of exported materials;
• $q_1,q_2$—world prices of imported investment and consumer goods;
• ${\lambda }_i$—coefficient of capital-labor ratio reduction due to capital depreciation and employment growth;
• ${\beta }_i$—direct material costs per unit of output in the i-th sector.

Here it is assumed that investment resources come centrally from the capital-forming sector and are supplemented by imports. Such an assumption is typical for economies with high import dependence of capital goods.

4. Formulation of the Problem for Optimal Resource Allocation

The problem consists of finding the optimal allocation of resources between three sectors in an open economy, considering quotas on the import of investment goods. The objective is to maximize the specific output of the consumer sector $(x_2={\theta }_2{A}_2{k}_2^{{\alpha }_2})$, which serves as the primary source of profit.

To achieve this, it is necessary to take into account the share of material exports, global prices of exported materials, as well as the prices of imported investment and consumer goods. This leads to a nonlinear programming problem aimed at determining the stable state of the system.

In this study, industrial security is interpreted as ensuring conditions that maintain equilibrium and stability within the system:

$$y_1={\gamma }_1{x}_1,y_2={\gamma }_2{x}_2.$$

(10)

Then, the equation for the foreign trade balance (8) can be rewritten as follows:

$$y_0={\displaystyle \frac{q_1}{q_0}}{\gamma }_1{x}_1+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2{x}_2.$$

(11)

Thus, the formulation of the problem for optimal resource allocation in the economic system reduces to a nonlinear programming problem:

$$x_2⟶max$$

(12)

subject to the following constraints:

$$s_0+s_1+s_2=1,s_i>0,(i=0,1,2),$$

(13)

$${\theta }_0+{\theta }_1+{\theta }_2=1,{\theta }_i>0,(i=0,1,2),$$

(14)

$$\begin{array}{c}\hfill (1-{\beta }_0)x_0=\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right)x_1+\left({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2\right)x_2,\\ \hfill 0<{\beta }_i<1,(i=0,1,2).\end{array}$$

(15)

$$x_i={\theta }_i{A}_i{k}_i^{{\alpha }_i},A_i>0,0<{\alpha }_i<1(i=0,1,2),$$

(16)

$$-{\lambda }_i{k}_i+{\displaystyle \frac{s_i}{{\theta }_i}}(1+{\gamma }_1)x_1,{\lambda }_i>0,(i=0,1,2).$$

(17)

The developed simplified model enables a more efficient study of the economic system’s stability, emphasizing the conditions and constraints that directly affect the maintenance of equilibrium and stability in the production environment.

5. Solution to the Problem for Optimal Resource Allocation

Since the search is for a stationary solution, the capital-labor ratio of the sectors remains constant over time. In this case, the following system of nonlinear equations can be derived from Equation (17):

$$-{\lambda }_0{k}_0+{\displaystyle \frac{s_0}{{\theta }_0}}(1+{\gamma }_1){\theta }_1{A}_1{k}_1^{{\alpha }_1}=0,$$

(18)

$$-{\lambda }_1{k}_1+s_1(1+{\gamma }_1)A_1{k}_1^{{\alpha }_1}=0,$$

(19)

$$-{\lambda }_2{k}_2+{\displaystyle \frac{s_2}{{\theta }_2}}(1+{\gamma }_1){\theta }_1{A}_1{k}_1^{{\alpha }_1}=0.$$

(20)

Then, find the formulas for $k_i$:

$$k_0={\displaystyle \frac{A_1}{{\lambda }_0}}(1+{\gamma }_1){\left({\displaystyle \frac{A_1}{{\lambda }_1}}(1+{\gamma }_1)\right)}^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}{\displaystyle \frac{s_0{\theta }_1}{{\theta }_0{s}_1}}{s}_1^{{\displaystyle \frac{1}{1-{\alpha }_1}}},$$

(21)

$$k_1={\left({\displaystyle \frac{A_1}{{\lambda }_1}}(1+{\gamma }_1)\right)}^{{\displaystyle \frac{1}{1-{\alpha }_1}}}{s}_1^{{\displaystyle \frac{1}{1-{\alpha }_1}}},$$

(22)

$$k_2={\displaystyle \frac{A_1}{{\lambda }_2}}(1+{\gamma }_1){\left({\displaystyle \frac{A_1}{{\lambda }_1}}(1+{\gamma }_1)\right)}^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}{\displaystyle \frac{s_2{\theta }_1}{{\theta }_2{s}_1}}{s}_1^{{\displaystyle \frac{1}{1-{\alpha }_1}}}.$$

(23)

By substituting the values of $k_0,k_1$ and $k_2$ obtained from Formulas (21)–(23) into the initial formula for the specific output of each sector (16), obtain a transformed system of equations in the form:

$$x_0={\omega }_0{\left({\displaystyle \frac{s_0{\theta }_1}{s_1{\theta }_0}}\right)}^{{\alpha }_0}{\theta }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}},$$

(24)

$$x_1={\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}},$$

(25)

$$x_2={\omega }_2{\left({\displaystyle \frac{s_2{\theta }_1}{s_1{\theta }_2}}\right)}^{{\alpha }_2}{\theta }_2{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}.$$

(26)

where ${\omega }_j,j=0,1,2$ denote the constant coefficients of the system:

$${\omega }_0=A_0{\left({\displaystyle \frac{A_1}{{\lambda }_0}}(1+{\gamma }_1)\right)}^{{\alpha }_0}{\left({\displaystyle \frac{A_1}{{\lambda }_1}}(1+{\gamma }_1)\right)}^{{\displaystyle \frac{{\alpha }_0{\alpha }_1}{1-{\alpha }_1}}},$$

$${\omega }_1=A_1{\left({\displaystyle \frac{A_1}{{\lambda }_1}}(1+{\gamma }_1)\right)}^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}},$$

$${\omega }_2=A_2{\left({\displaystyle \frac{A_1}{{\lambda }_2}}(1+{\gamma }_1)\right)}^{{\alpha }_2}{\left({\displaystyle \frac{A_1}{{\lambda }_1}}(1+{\gamma }_1)\right)}^{{\displaystyle \frac{{\alpha }_2{\alpha }_1}{1-{\alpha }_1}}}.$$

As can be seen, the system of Equations (24)–(26) consists of six exogenous parameters, which are described by three balance Equations (13)–(15).

To maximize the specific output of the consumer sector, the golden ratio principle is applied Jin and Zhang (2011). This principle helps determine the proportions among the exogenous parameters ${\theta }_i,s_i(i=0,1,2)$ that ensure the maximum specific output while considering all constraints and interdependencies between them.

Since $s_1$ and ${\theta }_1$ are free variables, the remaining share of resources for the material and consumer sectors will be $1-s_1$ and $1-{\theta }_1$, respectively.

Denoting m as the share of the consumer sector in the remaining investment resources and h as the share of the consumer sector in the remaining labor resources, obtain the following expressions for distribution:

• for labor resources:

$${\theta }_0=(1-h)(1-{\theta }_1),{\theta }_2=h(1-{\theta }_1),$$

(27)

• for investment resources:

$$s_0=m(1-s_1),s_2=(1-m)(1-s_1).$$

(28)

And thus, by transforming the original problem (12)–(17) using (27) and (28), obtain the following nonlinear programming problem:

$$x_2⟶max$$

(29)

subject to the conditions:

$$\begin{array}{c}\hfill (1-{\beta }_0)x_0=\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right)x_1+\left({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2\right)x_2,\\ \hfill 0<{\beta }_i<1,(i=0,1,2),\end{array}$$

(30)

$$x_0={\omega }_0{\left({\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}\right)}^{{\alpha }_0}(1-h)(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}},$$

(31)

$$x_1={\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}},$$

(32)

$$x_2={\omega }_2{\left({\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}\right)}^{{\alpha }_2}h(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}.$$

(33)

To solve this nonlinear programming problem (29)–(33), the Lagrange multipliers method is applied. The Lagrange function is written as follows:

$$\begin{array}{c}\hfill L({\theta }_1,s_1,x_0,x_1,x_2,c_1,c_2,c_3,c_4)=-x_2+\\ \hfill +c_1\left((1-{\beta }_0)x_0-\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right)x_1-\left({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2\right)x_2\right)+\\ \hfill +c_2\left(x_0-{\omega }_0{\left({\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}\right)}^{{\alpha }_0}(1-h)(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}\right)+\\ \hfill +c_3\left(x_1-{\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}\right)+\\ \hfill +c_4\left(x_2-{\omega }_2{\left({\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}\right)}^{{\alpha }_2}h(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}\right)\end{array}$$

(34)

where $c_j,j=1,2,3,4$—are the Lagrange multipliers. To analyze function (34) for extremum, write the first-order necessary conditions for extremum:

$${\displaystyle \frac{\partial L}{\partial {\theta }_1}}=0,{\displaystyle \frac{\partial L}{\partial s_1}}=0,{\displaystyle \frac{\partial L}{\partial m}}=0,{\displaystyle \frac{\partial L}{\partial h}}=0,{\displaystyle \frac{\partial L}{\partial x_i}}=0,(i=0,1,2).$$

Formulating the stationarity condition:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial L}{\partial {\theta }_1}}=c_2{\omega }_0\left(1-{\displaystyle \frac{{\alpha }_0}{{\theta }_1}}\right){\left({\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}\right)}^{{\alpha }_0}(1-h)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}-\\ \hfill -c_3{\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}+c_4{\omega }_2\left(1-{\displaystyle \frac{{\alpha }_2}{{\theta }_1}}\right){\left({\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}\right)}^{{\alpha }_2}h{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}=0,\end{array}$$

(35)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial L}{\partial s_1}}=-{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}{c}_3{\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}+\\ \hfill +{\alpha }_0{c}_2{\omega }_0{\left({\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}\right)}^{{\alpha }_0}(1-h)(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}\left({\displaystyle \frac{1}{1-s_1}}-{\displaystyle \frac{1}{1-{\alpha }_1}}\right)+\\ \hfill +{\alpha }_2{c}_4{\omega }_2{\left({\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}\right)}^{{\alpha }_2}h(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}\left({\displaystyle \frac{1}{1-s_1}}-{\displaystyle \frac{1}{1-{\alpha }_1}}\right)=0,\end{array}$$

(36)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial L}{\partial m}}=-{\alpha }_0{c}_2{\omega }_0{\left({\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}\right)}^{{\alpha }_0}{\displaystyle \frac{(1-h)(1-{\theta }_1)}{m}}{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+\\ \hfill +{\alpha }_2{c}_4{\omega }_2{\left({\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}\right)}^{{\alpha }_2}{\displaystyle \frac{h(1-{\theta }_1)}{1-m}}{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}=0,\end{array}$$

(37)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial L}{\partial h}}=(1-{\alpha }_0)c_2{\omega }_0{\left({\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}\right)}^{{\alpha }_0}(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}-\\ \hfill -(1-{\alpha }_2)c_4{\omega }_2{\left({\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}\right)}^{{\alpha }_2}(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}=0,\end{array}$$

(38)

$${\displaystyle \frac{\partial L}{\partial x_0}}=(1-{\beta }_0)c_1+c_2=0,$$

(39)

$${\displaystyle \frac{\partial L}{\partial x_1}}=-({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1)c_1+c_3=0,$$

(40)

$${\displaystyle \frac{\partial L}{\partial x_2}}=-1-({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2)c_1+c_4=0,$$

(41)

$$\begin{array}{c}\hfill (1-{\beta }_0){\omega }_0{\left({\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}\right)}^{{\alpha }_0}(1-h)(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}=\\ \hfill =\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}+\\ \hfill +\left({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2\right){\omega }_2{\left({\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}\right)}^{{\alpha }_2}h(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}.\end{array}$$

(42)

Thus, obtain a nonlinear system of eleven Equations (35)–(42) and (31)–(33) with eleven unknown parameters.

To ensure optimal growth between sectors and balanced allocation of labor and investment resources while considering their elasticity, the relationships between the exogenous parameters must satisfy the following equalities:

$${\displaystyle \frac{s_0{\theta }_1}{s_1{\theta }_0}}={\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}={\displaystyle \frac{{\alpha }_0(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_0)}},$$

(43)

$${\displaystyle \frac{s_2{\theta }_1}{s_1{\theta }_2}}={\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}={\displaystyle \frac{{\alpha }_2(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_2)}},$$

(44)

$${\displaystyle \frac{s_0{\theta }_2}{s_2{\theta }_0}}={\displaystyle \frac{m(1-s_1)h(1-{\theta }_1)}{(1-m)(1-s_1)(1-h)(1-{\theta }_1)}}={\displaystyle \frac{{\alpha }_0(1-{\alpha }_2)}{{\alpha }_2(1-{\alpha }_0)}}.$$

(45)

This allows each sector to develop harmoniously, minimizing disproportions in the distribution of production factors.

To ensure that the exogenous parameter relationships (43)–(45) hold, consider the system of Equations (31)–(33) in the form:

$${\omega }_0{\left({\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}\right)}^{{\alpha }_0}{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}={\displaystyle \frac{x_0}{(1-h)(1-{\theta }_1)}}={\displaystyle \frac{x_0}{{\theta }_0}},$$

(46)

$${\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}={\displaystyle \frac{x_1}{{\theta }_1}},$$

(47)

$${\omega }_2{\left({\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}\right)}^{{\alpha }_2}{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}={\displaystyle \frac{x_2}{h(1-{\theta }_1)}}={\displaystyle \frac{x_2}{{\theta }_2}}.$$

(48)

Then, the system (35)–(38) can be rewritten as follows:

$$c_2{\displaystyle \frac{x_0}{1-{\theta }_1}}\left(1-{\displaystyle \frac{{\alpha }_0}{{\theta }_1}}\right)-c_3{\displaystyle \frac{x_1}{{\theta }_1}}+c_4{\displaystyle \frac{x_2}{1-{\theta }_1}}\left(1-{\displaystyle \frac{{\alpha }_2}{{\theta }_1}}\right)=0,$$

(49)

$$\begin{array}{c}\hfill {\alpha }_0{c}_2{x}_0\left({\displaystyle \frac{1}{1-s_1}}-{\displaystyle \frac{1}{1-{\alpha }_1}}\right)-{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}{c}_3{x}_1+\\ \hfill +{\alpha }_2{c}_4{x}_2\left({\displaystyle \frac{1}{1-s_1}}-{\displaystyle \frac{1}{1-{\alpha }_1}}\right)=0,\end{array}$$

(50)

$$-{\alpha }_0{c}_2{x}_0{\displaystyle \frac{1}{m}}+{\alpha }_2{c}_4{x}_2{\displaystyle \frac{1}{1-m}}=0,$$

(51)

$$(1-{\alpha }_0)c_2{x}_0{\displaystyle \frac{1}{1-h}}-(1-{\alpha }_2)c_4{x}_2{\displaystyle \frac{1}{h}}=0.$$

(52)

If express Equations (51) and (52) through $c_4{x}_2$ and equate them, derive:

$${\displaystyle \frac{mh}{(1-m)(1-h)}}={\displaystyle \frac{{\alpha }_0(1-{\alpha }_2)}{{\alpha }_2(1-{\alpha }_0)}}.$$

(53)

If multiply and then divide the left-hand side of this equation by $(1-s_1)(1-{\theta }_1)$ obtain relation (45):

$${\displaystyle \frac{m(1-s_1)h(1-{\theta }_1)}{(1-m)(1-s_1)(1-h)(1-{\theta }_1)}}={\displaystyle \frac{{\alpha }_0(1-{\alpha }_2)}{{\alpha }_2(1-{\alpha }_0)}}.$$

(54)

Next, expanding the brackets in Equation (52), get:

$$c_2{x}_0+c_4{x}_2={\alpha }_0{c}_2{x}_0+{\alpha }_2{c}_4{x}_2+(1-{\alpha }_2)c_4{x}_2{\displaystyle \frac{1}{h}}$$

(55)

Now expanding the brackets in Equation (49) and grouping them as follows:

$${\displaystyle \frac{1}{{\theta }_1}}{c}_3{x}_1={\displaystyle \frac{1}{1-{\theta }_1}}(c_2{x}_0+c_4{x}_2)-{\displaystyle \frac{1}{{\theta }_1(1-{\theta }_1)}}({\alpha }_0{c}_2{x}_0+{\alpha }_2{c}_4{x}_2).$$

Then, substituting the expression obtained from Equation (55) for $c_2{x}_0+c_4{x}_2$ and using Equations (51) and (52), express $c_4{x}_2$ in terms of $c_2{x}_0$, and as a result obtain:

$${\displaystyle \frac{c_3{x}_1}{c_2{x}_0}}m+{\alpha }_0=(1-{\alpha }_0){\displaystyle \frac{m{\theta }_1}{(1-h)(1-{\theta }_1)}}.$$

(56)

Now, if group Equation (50), obtain:

$${\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}{c}_3{x}_1=({\alpha }_0{c}_2{x}_0+{\alpha }_2{c}_4{x}_2)\left({\displaystyle \frac{1}{1-s_1}}-{\displaystyle \frac{1}{1-{\alpha }_1}}\right)$$

As before, using Equation (52), express $c_4{x}_2$ in terms of $c_2{x}_0$, then divide both sides of the equation by $c_2{x}_0$ and then add ${\alpha }_0{\alpha }_1$ to both sides, resulting in the following expression:

$${\displaystyle \frac{c_3{x}_1}{c_2{x}_0}}m+{\alpha }_0={\displaystyle \frac{{\alpha }_0(1-{\alpha }_1)}{{\alpha }_1}}{\displaystyle \frac{s_1}{1-s_1}}.$$

(57)

Since the left-hand sides of Equations (56) and (57) are identical, equating them gives relation (43):

$${\displaystyle \frac{m(1-s_1){\theta }_1}{s_1(1-h)(1-{\theta }_1)}}={\displaystyle \frac{{\alpha }_0(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_0)}}.$$

(58)

To obtain the third relation, find ${\displaystyle \frac{m}{1-h}}$ from Equations (53) and (58) and equate them:

$${\displaystyle \frac{{\alpha }_0(1-{\alpha }_2)}{{\alpha }_2(1-{\alpha }_0)}}{\displaystyle \frac{1-m}{h}}={\displaystyle \frac{{\alpha }_0(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_0)}}{\displaystyle \frac{(1-{\theta }_1)s_1}{{\theta }_1(1-s_1)}}.$$

From this, obtain relation (44):

$${\displaystyle \frac{(1-m)(1-s_1){\theta }_1}{s_{1}h(1-{\theta }_1)}}={\displaystyle \frac{{\alpha }_2(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_2)}}.$$

(59)

Since the validity of the obtained relationships has been confirmed, at the next stage of the study we proceed to the search for key exogenous parameters characterizing the functioning of the economic system under study.

Since these parameters are expressed through the Lagrange multipliers, at the first stage we will determine these multipliers.

First, find the multipliers $c_2,c_3,c_4$ from the system of Equations (39)–(41) as follows:

$$c_2=-(1-{\beta }_0)c_1,$$

(60)

$$c_3=({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1)c_1,$$

(61)

$$c_4=1+({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2)c_1.$$

(62)

As seen, the found multipliers $c_2,c_3,c_4$ are expressed in terms of $c_1$. This multiplier is found from Equation (38). To do this, first substitute the previously found relations (58) and (59) instead of the expressions in parentheses containing powers ${\alpha }_0$ and ${\alpha }_2$. Then, substituting the values of $c_2$ and $c_4$ from (60) and (62), obtain:

$$\begin{array}{c}\hfill -(1-{\alpha }_0){\left({\displaystyle \frac{{\alpha }_0(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_0)}}\right)}^{{\alpha }_0}(1-{\beta }_0){\omega }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}{c}_1=\\ \hfill =(1-{\alpha }_2){\left({\displaystyle \frac{{\alpha }_2(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_2)}}\right)}^{{\alpha }_2}\left(1+\left({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2\right)c_1\right){\omega }_2{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}.\end{array}$$

Now solving this equation for $c_1$, get:

$$c_1=-{\displaystyle \frac{(1-{\alpha }_2){\sigma }_2{\omega }_2{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}}{(1-{\alpha }_0){\sigma }_0(1-{\beta }_0){\omega }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}-(1-{\alpha }_2){\sigma }_2\left({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2\right){\omega }_2{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}}},$$

(63)

here ${\sigma }_0,{\sigma }_2$ means:

$${\sigma }_0={\left({\displaystyle \frac{{\alpha }_0(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_0)}}\right)}^{{\alpha }_0},{\sigma }_2={\left({\displaystyle \frac{{\alpha }_2(1-{\alpha }_1)}{{\alpha }_1(1-{\alpha }_2)}}\right)}^{{\alpha }_2}.$$

Now we proceed directly to the essence of the process of determining exogenous parameters. Based on the previously obtained relationships and expressions for Lagrange multipliers, we will carry out a step-by-step determination of the exogenous parameters of the model.

From (53), find the parameter m as follows:

$$m={\displaystyle \frac{{\alpha }_0(1-{\alpha }_2)(1-h)}{{\alpha }_0(1-{\alpha }_2)+h({\alpha }_2-{\alpha }_0)}}.$$

(64)

Next, determine the parameter ${\theta }_1$ from Equation (35). To do this, first find the expression $c_4{\omega }_2{\sigma }_{2}h{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}$ from Equation (37):

$$c_4{\omega }_2{\sigma }_{2}h{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}={\displaystyle \frac{{\alpha }_0}{{\alpha }_2}}{c}_2{\omega }_0{\sigma }_0{\displaystyle \frac{(1-m)(1-h)}{m}}{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}.$$

(65)

Substituting (65) into Equation (35):

$$\begin{array}{c}\hfill c_2{\omega }_0{\sigma }_0(1-h)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}\left(1-{\displaystyle \frac{{\alpha }_0}{{\theta }_1}}\right)-c_3{\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}+\\ \hfill +{\displaystyle \frac{{\alpha }_0}{{\alpha }_2}}{c}_2{\omega }_0{\sigma }_0{\displaystyle \frac{(1-m)(1-h)}{m}}{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}\left(1-{\displaystyle \frac{{\alpha }_2}{{\theta }_1}}\right)=0,\end{array}$$

$$c_3{\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}=c_2{\omega }_0{\sigma }_0(1-h)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}\left({\theta }_1-{\alpha }_0+{\displaystyle \frac{{\alpha }_0}{{\alpha }_2}}{\displaystyle \frac{(1-m)}{m}}({\theta }_1-{\alpha }_2)\right).$$

(66)

If ${\displaystyle \frac{{\alpha }_0}{{\alpha }_2}}{\displaystyle \frac{(1-m)}{m}}$ is replaced using Equation (53), then (66) can be rewritten as follows:

$$c_3{\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}=c_2{\omega }_0{\sigma }_0(1-h)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}\left({\theta }_1-{\alpha }_0+{\displaystyle \frac{1-{\alpha }_0}{1-{\alpha }_2}}{\displaystyle \frac{h}{1-h}}({\theta }_1-{\alpha }_2)\right).$$

(67)

From this, the exogenous parameter ${\theta }_1$ is found as follows:

$${\theta }_1={\displaystyle \frac{c_2{\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}({\alpha }_0(1-{\alpha }_2)-h({\alpha }_2-{\alpha }_0))}{(1-{\alpha }_2)\left(c_2{\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}-c_3{\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}\right)+c_2{\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}h({\alpha }_2-{\alpha }_0)}}.$$

(68)

To determine the value of the parameter h, the material balance Equation (42) is used.

$$\begin{array}{c}\hfill (1-{\beta }_0){\omega }_0{\sigma }_0(1-h)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}=\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{\displaystyle \frac{{\theta }_1}{1-{\theta }_1}}{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}+\\ \hfill +\left({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2\right){\omega }_2{\sigma }_{2}h{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}.\end{array}$$

(69)

If Equations (60) and (61) are taken into account, the ratio ${\displaystyle \frac{{\theta }_1}{1-{\theta }_1}}$ can be written as follows:

$${\displaystyle \frac{{\theta }_1}{1-{\theta }_1}}={\displaystyle \frac{(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}({\alpha }_0(1-{\alpha }_2)-h({\alpha }_2-{\alpha }_0))}{(1-{\alpha }_2)\left((1-{\alpha }_0)(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}\right)}}.$$

(70)

Then, Equation (69) takes the following form:

$$\begin{array}{c}\hfill (1-{\beta }_0){\omega }_0{\sigma }_0(1-h)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}=\left({\beta }_2+{\displaystyle \frac{q_2}{q_0}}{\gamma }_2\right){\omega }_2{\sigma }_{2}h{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}+\\ \hfill +{\displaystyle \frac{\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}({\alpha }_0(1-{\alpha }_2)-h({\alpha }_2-{\alpha }_0))}{(1-{\alpha }_2)\left((1-{\alpha }_0)(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}\right)}}.\end{array}$$

From this, the exogenous parameter h is determined:

$$h={\displaystyle \frac{(1-{\alpha }_2)\left((1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}\right)}{(1-{\alpha }_2)\left((1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+\left({\beta }_2+{\displaystyle \frac{q_2{\gamma }_2}{q_0}}\right){\omega }_2{\sigma }_2{s}_1^{{\displaystyle \frac{{\alpha }_2}{1-{\alpha }_1}}}\right)+\left({\beta }_1+{\displaystyle \frac{q_1{\gamma }_1}{q_0}}\right){\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}}}.$$

(71)

As seen, the obtained values of the exogenous parameters ${\theta }_1,m$ and h are expressed in terms of $s_1$. This parameter is found from Equation (36). To do this, using Equation (65) and grouping terms, Equation (36) is rewritten as follows:

$$\begin{array}{c}\hfill {\alpha }_0{c}_2{\omega }_0{\sigma }_0(1-h)(1-{\theta }_1)s_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}\left(1+{\displaystyle \frac{1-m}{m}}\right)\left({\displaystyle \frac{1}{1-s_1}}-{\displaystyle \frac{1}{1-{\alpha }_1}}\right)-\\ \hfill -{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}{c}_3{\omega }_1{\theta }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}=0.\end{array}$$

(72)

By bringing the expressions in parentheses to a common denominator and considering expressions (60) and (61), simplifying Equation (72) gives the following:

$$\begin{array}{c}\hfill -(1-{\beta }_0){\alpha }_0{c}_1{\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}\left({\displaystyle \frac{1-h}{m}}\right)\left({\displaystyle \frac{s_1-{\alpha }_1}{1-s_1}}\right)=\\ \hfill =\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\alpha }_1{c}_1{\omega }_1{\displaystyle \frac{{\theta }_1}{1-{\theta }_1}}{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}.\end{array}$$

(73)

Now, substituting Equations (64) and (70) for m and ${\displaystyle \frac{{\theta }_1}{1-{\theta }_1}}$, respectively, obtain:

$${\displaystyle \frac{s_1-{\alpha }_1}{1-s_1}}={\displaystyle \frac{\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\alpha }_1{\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}}{(1-{\alpha }_0)(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}}}.$$

(74)

From Equation (74), the exogenous parameter $s_1$ is determined as follows:

$$s_1={\displaystyle \frac{{\alpha }_1(1-{\alpha }_0)(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}}{(1-{\alpha }_0)(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+(1-{\alpha }_1)\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}}}.$$

(75)

Thus, all exogenous parameters and Lagrange multipliers have been found.

Next, the problem of optimal resource allocation between sectors, considering import quotas on investment goods, is solved by constructing a nonlinear function $f\left(s_1\right)$, which represents the solution for all dependent parameters and variables:

$$\begin{array}{c}\hfill f\left(s_1\right)=-{\alpha }_1(1-{\alpha }_0)(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+\\ \hfill +s_1\left((1-{\alpha }_0)(1-{\beta }_0){\omega }_0{\sigma }_0{s}_1^{{\displaystyle \frac{{\alpha }_0}{1-{\alpha }_1}}}+(1-{\alpha }_1)\left({\beta }_1+{\displaystyle \frac{q_1}{q_0}}{\gamma }_1\right){\omega }_1{s}_1^{{\displaystyle \frac{{\alpha }_1}{1-{\alpha }_1}}}\right).\end{array}$$

(76)

6. Algorithm for Solving the Problem of Optimal Resource Allocation

A step-by-step algorithm for solving the problem of optimal resource allocation within the constraints of an open economic system with an emphasis on the sequential determination of the exogenous parameters of the model:

• Form the Lagrange function $L({\theta }_1,s_1,h,m,x_i,c_j),(i=0,1,2;j=1,2,3,4)$ based on the initial problem statement (Equation (34));
• Write out the necessary first-order conditions and solve the corresponding system of nonlinear algebraic Equations (35)–(42) and (31)–(33);
• Calculate the relationships between the parameters ${\theta }_i$ and $s_i$ that satisfy the conditions of the equilibrium distribution of resources specified by Equations (43)–(45);
• Plot the function $f\left(s_1\right)$ (according to Equation (76)) and determine the optimal value of the parameter $s_1$—the share of investment resources allocated to the capital-forming sector;
• Find the values of the Lagrange multipliers: $c_1$—using Formula (63) and $c_2,c_3,c_4$—using Formulas (60)–(62);
• Determine the labor force h distributed between the material and consumer sectors using Equation (71);
• Determine the share of investments m of the same nodes using Formula (64);
• Find the value of the parameter ${\theta }_1$ using Equation (68);
• Calculate the remaining shares of resources ${\theta }_0,{\theta }_2$ and $s_0,s_2$—using notations (27)–(28);
• Check whether restrictions (13)–(15) are met;
• Determine the stationary state of sectoral capital intensity based on Equations (21)–(23);
• Determine the stationary state of production functions in relative terms using Formulas (31)–(33);
• Calculate the specific material export according to Equation (11).

For the numerical solution of the problem, a procedure was implemented in the Maple software (6.04 Build 756) environment, using built-in optimization and symbolic differentiation tools. The golden section method was used for one-dimensional search by parameter $s_1$, with the first-order conditions (from the Lagrange equations) solved analytically and numerically at each step. The calculations were performed on a standard computer (Intel Core i7, 16 GB RAM), the average calculation time for one optimal distribution was less than 1 s. The resulting scheme is accurate and stable with respect to changes in initial approximations.

7. Results

7.1. Numerical Calculations

To verify the proposed model of optimal resource allocation in an open economic system, a set of numerical experiments was conducted. It should be noted that in numerical calculations the model considers a stable resource allocation with stationary parameters of the external economic environment. This is in line with the practice of strategic and budget planning, where average forecast indicators are used. In particular, in Kazakhstan state programs and budgets are developed for a three-year horizon. The main goal is to determine a stable structure of resource allocation that achieves maximum output in the capital-forming sector while observing all specified constraints.

For numerical calculations, the values of the coefficients ${\alpha }_i$, ${\beta }_i$, ${\lambda }_i$ and $A_i$ were used (see

Table 1. Model coefficients.

i	0	1	2
${\alpha }_i$	0.65	0.7	0.68
${\beta }_i$	0.4	0.11	0.22
${\lambda }_i$	0.07	0.08	0.05
$A_i$	1.92	1.3	2.17

), calculated on the basis of statistical data of the Republic of Kazakhstan for 2010–2022, published on the official website of the Bureau of National Statistics https://stat.gov.kz (accessed between August and November 2023), and reflect information on the volume of industrial production, the number of employees and investments in fixed capital by industry (Tussupova & Mirzakhmedova, 2024).

At the first stage of the numerical analysis, a graph of the function $f\left(s_1\right)$ s constructed, corresponding to Equation (76), which determines the optimal share of investment resources directed to the capital-forming sector. Calculations are carried out in the range from 0 to 1, and the resulting optimal value is used to calculate all other exogenous parameters of the model (see Figure 1).

Table 2. Calculation results of the stationary state of a closed economic system.

i	0	1	2
${\theta }_i$	0.336	0.468	0.194
$s_i$	0.294	0.51	0.165
$k_i$	1058.337	1151.892	1695.374
$x_i$	59.812	84.626	66.350

m = 0.602, h = 0.366.

shows the results of calculating the exogenous parameters of a closed economic system.

Table 3. Calculation results of the stationary state of a open economic system.

i	0	1	2
${\theta }_i$	0.521	0.325	0.153
$s_i$	0.471	0.37	0.158
$k_i$	1390.574	1526.937	2227.593
$x_i$	110.485	71.716	62.781
$y_i$	33.624	35.858	31.390

m = 0.751, h = 0.224.

shows the results of calculating the exogenous parameters of an open economic system.

Figure 2 illustrates the solutions of the differential equations for capital intensity $k_i,(i=0,1,2)$ under given initial conditions.

$${\dot{k}}_0=-{\lambda }_0{k}_0+{\displaystyle \frac{s_0{\theta }_1}{{\theta }_0}}(1+{\gamma }_1)A_1{k}_1^{{\alpha }_1},k_0\left(0\right)=1100,$$

(77)

$${\dot{k}}_1=-{\lambda }_1{k}_1+s_1(1+{\gamma }_1)A_1{k}_1^{{\alpha }_1},k_1\left(0\right)=1250,$$

(78)

$${\dot{k}}_2=-{\lambda }_2{k}_2+{\displaystyle \frac{s_2{\theta }_1}{{\theta }_2}}(1+{\gamma }_1)A_1{k}_1^{{\alpha }_1},k_2\left(0\right)=1600.$$

(79)

Figure 3 illustrates the changes in sectoral output per unit, $x_i,(i=0,1,2)$, over time in an open economy. The graph is constructed based on Equation (4).

7.2. Parametric Analysis

To assess the sensitivity of the model to changes in parameters, a parametric analysis was conducted using two key coefficients-${\alpha }_0$ (see

Table 4. Influence of ${\alpha }_0$ on resource allocation and output in the capital-forming sector.

${\mathit{\alpha }}_0$	${\mathit{x}}_2$	${\mathit{s}}_0$	${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{\theta }}_0$	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$
0.1	2.72	0.6	0.17	0.22	0.933	0.025	0.041
0.2	4	0.64	0.18	0.17	0.902	0.048	0.049
0.3	6.88	0.64	0.21	0.14	0.858	0.082	0.059
0.4	14.32	0.61	0.26	0.13	0.778	0.144	0.076
0.5	37.43	0.53	0.34	0.13	0.626	0.264	0.109
0.6	62.781	0.471	0.37	0.158	0.521	0.325	0.153
0.7	111.64	0.38	0.46	0.15	0.379	0.455	0.164
0.8	266.03	0.22	0.58	0.2	0.139	0.627	0.233

and Figure 4) and ${\beta }_2$ (see

Table 5. Influence of ${\beta }_2$ on resource allocation and output in the consumer sector.

${\mathit{\beta }}_2$	${\mathit{x}}_2$	${\mathit{s}}_0$	${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{\theta }}_0$	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$
0.1	97.1	0.445	0.37	0.184	0.496	0.32	0.18
0.2	62.78	0.471	0.37	0.158	0.521	0.32	0.15
0.3	70.89	0.494	0.37	0.135	0.547	0.32	0.131
0.4	62.46	0.5	0.37	0.12	0.563	0.32	0.115
0.5	55.82	0.522	0.37	0.107	0.576	0.32	0.103
0.6	50.46	0.532	0.37	0.097	0.587	0.32	0.093
0.7	46.03	0.541	0.37	0.088	0.595	0.32	0.085
0.8	42.32	0.548	0.37	0.081	0.602	0.32	0.078

and Figure 5). These coefficients reflect the critical stages of the functioning of the economic system: the initial stage of attracting resources to the material sector and the final phase of product release in the consumer sector. Analysis of the influence of these parameters allows for a comprehensive assessment of the stability and efficiency of the entire model when varying the structural characteristics of the initial and final stages of the production process (see Figure 6).

The model reveals a structural norm. As can be seen from Table 4, with a moderate increase in the capital intensity of the raw materials sector (${\alpha }_0\approx$ 0.4–0.6), the economy reaches a balanced state—output grows ($x_2$ $14.32$ to $62.781$), but not abruptly, and labor and investment remain evenly distributed.

Similarly, from Table 5. we can see that with a moderate material intensity of the consumer sector (${\beta }_2\approx$ 0.2–0.3), high production is maintained without an excessive burden on the raw materials sector. This indicates an optimum zone in which sustainable development is possible with minimal costs and a balanced economic structure.

8. Discussion

This study proposes a formalized nonlinear model of a three-sector open economy, integrating production, labor, capital, and trade balances into a unified optimal resource allocation problem. The obtained results allow for interpreting not only the internal sectoral dynamics but also the system’s response to changes in technological and material constraints.

In contrast to existing works such as the balanced growth model of Kolemayev (2008), which is static and lacks a computational solution, or the study of Tarasyev et al. (2023), limited to CES functions and two-factor interactions, this paper presents a structurally complete and computationally implementable model. In particular, it extends the structural framework to include sectoral trade restrictions and depreciation of fixed capital, while maintaining analytic tractability. Unlike the linear investment allocation in Saraev’s model (Saraev & Saraeva, 2020), the current work solves a fully nonlinear constrained optimization problem with interpretable equilibrium conditions. The use of a Lagrangian framework, coupled with the golden section method, makes the solution both rigorous and efficient.

Further, numerical methods such as those proposed in (Enrique & Garcia-Salazar, 2023), using genetic algorithms to solve the Cobb–Douglas function, lack structural realism and policy relevance. By contrast, our approach preserves the economic meaning of the constraints and balances, making the model suitable for real-world planning scenarios.

The parameter analysis demonstrates a structurally meaningful pattern: an increase in capital productivity in the material sector leads to a reallocation of labor and investment into the capital-forming sector, thereby enabling a significant expansion of consumer output. Conversely, an increase in material consumption per unit of final product leads to a measurable contraction in output and a reorientation of resources toward the material sector. These patterns are consistent with economic theory and confirm the model’s ability to reflect real structural trade-offs.

Although the model assumes stationary external economic conditions, its structure enables sensitivity and scenario analysis. By varying trade parameters such as $q_i,{\gamma }_i$ and $y_i$, or imposing exogenous constraints, one can simulate unfavorable trade conditions, including sanctions, tariff shocks, or fluctuations in world commodity prices. Thus, while not explicitly stochastic or dynamic, the model serves as a flexible framework for applied economic analysis under multiple planning scenarios.

Compared to recent literature on sustainable production and optimization in open systems (e.g., K. Li et al., 2019; Kim & Jeon, 2025), this approach is distinct in its combination of economic interpretability, algorithmic solvability, and policy-relevant structural detail. In this sense, the model fills a methodological gap and offers a scalable tool for examining structural policy alternatives in resource-constrained, trade-dependent economies.

While our model focuses on production efficiency and optimal allocation under trade constraints, it may contribute to the broader sustainability agenda outlined in recent reviews on circular economy and advanced supply chains. Its capability to simulate structural adjustments and trade-related shocks makes it suitable for evaluating industrial resilience and long-term resource sustainability.

9. Conclusions

This paper proposes a novel nonlinear optimization model for resource allocation in a three-sector open economy, including investment, labor, material, and trade constraints. Unlike existing models that are either static, purely empirical, or structurally simplified, this approach integrates intersectoral dynamics with production technologies based on the Cobb–Douglas function, depreciation of capital, and import quotas. The model is analytically tractable and computationally efficient, relying on the Lagrange multiplier method and a golden section search.

The results demonstrate how changes in capital productivity or material intensity can reshape sectoral structure and overall output, providing insight into how production systems respond to technological or trade-induced shifts. This enables practical applications in policy planning, particularly in economies with trade dependencies and resource constraints.

Prospects for further research include the development of a dynamic economic management model, with a transition from the initial state to the vicinity of the stationary regime. This requires the implementation of a differential system with state-dependent control parameters and can be solved using Lyapunov stability theory and optimal control methods. Additionally, it is proposed to expand the model to assess investment efficiency under conditions of stochastic disturbances and unstable global market conditions.