Optimal allocation of resources in an open economic system with Cobb-Douglas production and trade balances

Tussupova, Kamshat; Murzabekov, Zainelkhriet Article Optimal allocation of resources in an open economic system with Cobb-Douglas production and trade balances Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Tussupova, Kamshat; Murzabekov, Zainelkhriet (2025) : Optimal allocation of resources in an open economic system with Cobb-Douglas production and trade balances, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13, Iss. 7, pp. 1-22, https://doi.org/10.3390/economies13070184 This Version is available at: https://hdl.handle.net/10419/329464 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Academic Editor: Gheorghe H. Popescu Received: 2 May 2025 Revised: 14 June 2025 Accepted: 15 June 2025 Published: 26 June 2025 Citation: Tussupova, K., & Murzabekov, Z. (2025). Optimal Allocation of Resources in an Open Economic System with Cobb–Douglas Production and Trade Balances. Economies,13(7), 184. https://doi.org/ 10.3390/economies13070184 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article Optimal Allocation of Resources in an Open Economic System with Cobb–Douglas Production and Trade Balances Kamshat Tussupova * and Zainelkhriet Murzabekov Department of Information Systems, Al-Farabi Kazakh National University, Al-Farabi Ave. 71/23, Almaty 050040, Kazakhstan; [email protected] *Correspondence: [email protected]u 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 resourceefficient 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. Keywords: open economy; resource allocation; nonlinear programming; stationary state; sustainable growth 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 oneor 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 Economies 2025,13, 184 https://doi.org/10.3390/economies13070184 Economies 2025,13, 184 2 of 22 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? Economies 2025,13, 184 3 of 22 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 Economies 2025,13, 184 4 of 22 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 highlevel 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. Economies 2025,13, 184 5 of 22 The output volume in each i-th sector is determined by the Cobb-Douglas production function: Xi=Fi(Ki,Li) = AiKαi iL1−αi i,(i=0, 1, 2). (1) where: •Xi—output volume; •Ki—volume of fixed productive assets; •Li—volume of labor resources; •Ai—coefficient of neutral technological progress; •αi—capital elasticity coefficient; • (1 −αi)—labor elasticity coefficient. This form allows to reflect both differences in capital intensity of industries and scalability of technology. Parameters Ai , αi 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—µiKiand an increase due to investments Ii: dKi dt =Ii−µiKi,Ki(0) = K0 i,(i=0, 1, 2). (2) Let us introduce the following notations: •θi=Li L—sectoral shares in labor resource distribution; •si=Ii X1—sectoral shares in investment resource distribution; •fi(ki) = Xi Li—labor productivity in the i-th sector; •ki=Ki Li—capital-labor ratio of the sectors; •y1=Y1 L—share of imported goods for investment; •y2=Y2 L—share of imported goods for consumption; •xi=θifi(ki)—specific output of the sectors. Then, the Equation (2) can be rewritten in the following form for the capital-labor ratio of the sectors: ˙ ki=−λiki+si θi (x1+y1),ki(0) = k0 i,λi>0, (i=0, 1, 2), (3) xi=θiAikαi i,Ai>0, 0 <αi<1(i=0, 1, 2), (4) the equilibrium of resources is ensured by the following constraints: • Investment balance: s0+s1+s2=1, si>0, (i=0, 1, 2), (5) • Labor balance: θ0+θ1+θ2=1, θi>0, (i=0, 1, 2), (6) • Material balance: (1−β0)x0=β1x1+β2x2+y0, 0 <βi<1, (i=0,1,2). (7) • Foreign trade balance: q0y0=q1y1+q2y2. (8) • Industrial security: y1≤γ1x1,y2≤γ2x2. (9) Economies 2025,13, 184 6 of 22 These conditions model the limits of permissible external pressure. Here: •γ1,γ2—maximum permissible shares of imports of investment and consumer goods; •y0 —export of raw materials; y1 , y2 —volumes of imports of investment and consumer goods, respectively; •q0—world price of exported materials; •q1,q2—world prices of imported investment and consumer goods; •λi —coefficient of capital-labor ratio reduction due to capital depreciation and employment growth; •βi—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 (x2=θ2A2kα2 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: y1=γ1x1,y2=γ2x2. (10) Then, the equation for the foreign trade balance (8) can be rewritten as follows: y0=q1 q0γ1x1+q2 q0γ2x2. (11) Thus, the formulation of the problem for optimal resource allocation in the economic system reduces to a nonlinear programming problem: x2−→ max (12) subject to the following constraints: s0+s1+s2=1, si>0, (i=0, 1, 2), (13) θ0+θ1+θ2=1, θi>0, (i=0, 1, 2), (14) (1−β0)x0=β1+q1 q0γ1x1+β2+q2 q0γ2x2, 0<βi<1, (i=0, 1, 2). (15) xi=θiAikαi i,Ai>0, 0 <αi<1(i=0, 1, 2), (16) −λiki+si θi (1+γ1)x1,λi>0, (i=0, 1, 2). (17) Economies 2025,13, 184 7 of 22 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): −λ0k0+s0 θ0(1+γ1)θ1A1kα1 1=0, (18) −λ1k1+s1(1+γ1)A1kα1 1=0, (19) −λ2k2+s2 θ2(1+γ1)θ1A1kα1 1=0. (20) Then, find the formulas for ki: k0=A1 λ0(1+γ1)A1 λ1(1+γ1) α1 1−α1s0θ1 θ0s1s 1 1−α1 1, (21) k1=A1 λ1(1+γ1) 1 1−α1s 1 1−α1 1, (22) k2=A1 λ2(1+γ1)A1 λ1(1+γ1) α1 1−α1s2θ1 θ2s1s 1 1−α1 1. (23) By substituting the values of k0 , k1 and k2 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: x0=ω0s0θ1 s1θ0α0 θ0s α0 1−α1 1, (24) x1=ω1θ1s α1 1−α1 1, (25) x2=ω2s2θ1 s1θ2α2 θ2s α2 1−α1 1. (26) where ωj,j=0, 1, 2 denote the constant coefficients of the system: ω0=A0A1 λ0(1+γ1)α0A1 λ1(1+γ1) α0α1 1−α1, ω1=A1A1 λ1(1+γ1) α1 1−α1, ω2=A2A1 λ2(1+γ1)α2A1 λ1(1+γ1) α2α1 1−α1. 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 Economies 2025,13, 184 8 of 22 exogenous parameters θi , si(i= 0, 1, 2 ) that ensure the maximum specific output while considering all constraints and interdependencies between them. Since s1 and θ1 are free variables, the remaining share of resources for the material and consumer sectors will be 1 −s1and 1 −θ1, 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: θ0= (1−h)(1−θ1),θ2=h(1−θ1), (27) • for investment resources: s0=m(1−s1),s2= (1−m)(1−s1). (28) And thus, by transforming the original problem (12) – (17) using (27) and (28) , obtain the following nonlinear programming problem: x2−→ max (29) subject to the conditions: (1−β0)x0=β1+q1 q0γ1x1+β2+q2 q0γ2x2, 0<βi<1, (i=0, 1, 2), (30) x0=ω0m(1−s1)θ1 s1(1−h)(1−θ1)α0 (1−h)(1−θ1)s α0 1−α1 1, (31) x1=ω1θ1s α1 1−α1 1, (32) x2=ω2(1−m)(1−s1)θ1 s1h(1−θ1)α2 h(1−θ1)s α2 1−α1 1. (33) To solve this nonlinear programming problem (29) – (33) , the Lagrange multipliers method is applied. The Lagrange function is written as follows: L(θ1,s1,x0,x1,x2,c1,c2,c3,c4) = −x2+ +c1(1−β0)x0−β1+q1 q0γ1x1−β2+q2 q0γ2x2+ +c2  x0−ω0m(1−s1)θ1 s1(1−h)(1−θ1)α0 (1−h)(1−θ1)s α0 1−α1 1  + +c3  x1−ω1θ1s α1 1−α1 1  + +c4  x2−ω2(1−m)(1−s1)θ1 s1h(1−θ1)α2 h(1−θ1)s α2 1−α1 1  (34) Economies 2025,13, 184 15 of 22 2. Write out the necessary first-order conditions and solve the corresponding system of nonlinear algebraic Equations (35)–(42) and (31)–(33); 3. Calculate the relationships between the parameters θi and si that satisfy the conditions of the equilibrium distribution of resources specified by Equations (43)–(45); 4. Plot the function f(s1) (according to Equation (76) ) and determine the optimal value of the parameter s1 —the share of investment resources allocated to the capitalforming sector; 5. Find the values of the Lagrange multipliers: c1 —using Formula (63) and c2 , c3 , c4 —using Formulas (60)–(62); 6. Determine the labor force h distributed between the material and consumer sectors using Equation (71); 7. Determine the share of investments mof the same nodes using Formula (64); 8. Find the value of the parameter θ1using Equation (68); 9. Calculate the remaining shares of resources θ0 , θ2 and s0 , s2 —using notations (27) – (28) ; 10. Check whether restrictions (13)–(15) are met; 11. Determine the stationary state of sectoral capital intensity based on Equations (21) – (23) ; 12. Determine the stationary state of production functions in relative terms using Formulas (31)–(33); 13. 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 s1 , 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 αi , βi , λi and Ai were used (see Table 1), 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). Economies 2025,13, 184 16 of 22 Table 1. Model coefficients. i 0 1 2 αi0.65 0.7 0.68 βi0.4 0.11 0.22 λi0.07 0.08 0.05 Ai1.92 1.3 2.17 At the first stage of the numerical analysis, a graph of the function f(s1) 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 2shows the results of calculating the exogenous parameters of a closed economic system. Figure 1. Graph of the function f(s1). Table 2. Calculation results of the stationary state of a closed economic system. i 0 1 2 θi0.336 0.468 0.194 si0.294 0.51 0.165 ki1058.337 1151.892 1695.374 xi59.812 84.626 66.350 m= 0.602, h= 0.366. Table 3shows the results of calculating the exogenous parameters of an open economic system. Table 3. Calculation results of the stationary state of a open economic system. i 0 1 2 θi0.521 0.325 0.153 si0.471 0.37 0.158 ki1390.574 1526.937 2227.593 xi110.485 71.716 62.781 yi33.624 35.858 31.390 m= 0.751, h= 0.224. Economies 2025,13, 184 17 of 22 Figure 2illustrates the solutions of the differential equations for capital intensity ki,(i=0, 1, 2)under given initial conditions. ˙ k0=−λ0k0+s0θ1 θ0(1+γ1)A1kα1 1,k0(0) = 1100, (77) ˙ k1=−λ1k1+s1(1+γ1)A1kα1 1,k1(0) = 1250, (78) ˙ k2=−λ2k2+s2θ1 θ2(1+γ1)A1kα1 1,k2(0) = 1600. (79) Figure 2. Capital intensity dynamics in the sectors of an open economy. Figure 3illustrates the changes in sectoral output per unit, xi , (i= 0, 1, 2 ) , over time in an open economy. The graph is constructed based on Equation (4). Figure 3. Dynamics of sectoral output in an open economy. Economies 2025,13, 184 18 of 22 7.2. Parametric Analysis To assess the sensitivity of the model to changes in parameters, a parametric analysis was conducted using two key coefficientsα0 (see Table 4and Figure 4) and β2 (see Table 5 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). Table 4. Influence of α0on resource allocation and output in the capital-forming sector. α0x2s0s1s2θ0θ1θ2 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 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 ( α0≈ 0.4–0.6), the economy reaches a balanced state—output grows ( x2 14.32 to 62.781), but not abruptly, and labor and investment remain evenly distributed. (a) (b) Figure 4. Analysis of the impact of changes in the elasticity coefficient on the distribution of investment (a) and labor (b) resources across industries of the material sector. Economies 2025,13, 184 19 of 22 (a) (b) Figure 5. Analysis of the distribution of investments (a) and labor (b) resources by consumer sector industries depending on the volume of material costs. (a) (b) Figure 6. The influence of α0(a) and β2(b) on the specific output of the consumer sector x2. Table 5. Influence of β2on resource allocation and output in the consumer sector. β2x2s0s1s2θ0θ1θ2 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 Similarly, from Table 5. we can see that with a moderate material intensity of the consumer sector ( β2≈ 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. Economies 2025,13, 184 20 of 22 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 qi , γi and yi , 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. Economies 2025,13, 184 21 of 22 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. Author Contributions: Conceptualization, K.T. and Z.M.; methodology, K.T.; software, K.T.; validation, K.T. and Z.M.; formal analysis, Z.M.; investigation, K.T.; data curation, Z.M.; writing—original draft preparation, K.T.; writing—review and editing, K.T. and Z.M.; visualization, K.T.; supervision, K.T.; project administration, K.T.; funding acquisition, K.T. All authors have read and agreed to the published version of the manuscript. Funding: This research is funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (grant no. AP22684879). Institutional Review Board Statement: Not applicable. 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