Oʻzbekcha
IQTISODIYOT MODELLARNI QURISHDA DIFFIRENSIAL TENGLAMALARDAN FOYDALANISH
Аннотация
Ushbu maqolada iqtisodiy jarayonlarni matematik modellashtirishda differensial tenglamalarning tutgan o‘rni va nazariy-amaliy ahamiyati keng tahlil qilinadi. Iqtisodiyot murakkab, ko‘p omilli va vaqt davomida o‘zgarib boruvchi dinamik tizim sifatida qaralib, uning asosiy ko‘rsatkichlari — kapital, investitsiya, milliy daromad, talab va taklif, narxlar darajasi hamda iqtisodiy o‘sish sur’atlarining vaqtga bog‘liq o‘zgarishi differensial tenglamalar yordamida ifodalanishi asoslab beriladi. Maqolada birinchi tartibli oddiy differensial tenglamalar asosida iqtisodiy o‘sish modellari, xususan, Solou modeli misolida kapitalning dinamik harakati va muvozanat holati matematik jihatdan tahlil qilinadi. Shuningdek, talab va taklifning dinamik muvozanati, investitsiya jarayonlari hamda makroiqtisodiy ko‘rsatkichlarning barqarorlik shartlari differensial tenglamalar orqali izohlanadi. Tadqiqot davomida iqtisodiy tizimlarning barqarorlik nuqtalari, o‘sish trayektoriyalari va uzoq muddatli muvozanat holatini aniqlashda differensial tenglamalarning samaradorligi ko‘rsatib beriladi. Nazariy tahlil bilan bir qatorda, differensial tenglamalarning prognozlash, iqtisodiy siyosat samaradorligini baholash va iqtisodiy jarayonlarni optimallashtirishdagi ahamiyati ham yoritiladi.
Ключевые слова
Solou modeli
barqarorlik tahlili
differensial tenglama
dinamik tizim
investitsiya jarayoni
iqtisodiy model
iqtisodiy o‘sish
kapital dinamikasi
matematik modellashtirish
prognozlash
talab va taklif muvozanati
Русский
В данной статье подробно рассматривается роль и теоретико-практическое значение дифференциальных уравнений в математическом моделировании экономических процессов. Экономика исследуется как сложная, многофакторная и динамическая система, изменяющаяся во времени, а основные экономические показатели — капитал, инвестиции, национальный доход, спрос и предложение, уровень цен и темпы экономического роста — описываются с использованием дифференциальных уравнений как функций времени. В статье на основе обыкновенных дифференциальных уравнений первого порядка анализируются модели экономического роста, в частности, динамика накопления капитала и состояние равновесия в модели Солоу. Также рассматриваются динамическое равновесие спроса и предложения, инвестиционные процессы и условия устойчивости макроэкономических показателей, выраженные через дифференциальные уравнения. В ходе исследования показана эффективность применения дифференциальных уравнений для определения точек устойчивости экономических систем, траекторий роста и долгосрочного равновесного состояния. Наряду с теоретическим анализом освещается значение дифференциальных уравнений в прогнозировании, оценке эффективности экономической политики и оптимизации экономических процессов.
анализ устойчивости
динамика капитала
динамическая система
дифференциальное уравнение
инвестиционный процесс
макроэкономические показатели
математическое моделирование
модель Солоу
прогнозирование
равновесие спроса и предложения
экономическая модель
экономический рост
English
This article examines the role and theoretical–practical significance of differential equations in the mathematical modeling of economic processes. The economy is considered as a complex, multi-factor, and dynamically evolving system over time, and its main indicators — capital, investment, national income, supply and demand, price levels, and economic growth rates — are expressed as time-dependent variables described by differential equations. The paper analyzes economic growth models based on first-order ordinary differential equations, particularly focusing on capital accumulation dynamics and equilibrium conditions within the Solow growth model. It also explores the dynamic equilibrium of supply and demand, investment processes, and the stability conditions of macroeconomic indicators through the application of differential equations. The study demonstrates the effectiveness of differential equations in determining stability points of economic systems, growth trajectories, and long-term equilibrium states. Alongside theoretical analysis, the importance of differential equations in forecasting, evaluating the effectiveness of economic policy, and optimizing economic processes is highlighted.
Solow model
capital dynamics
differential equation
dynamic system
economic growth
economic model
forecasting
investment process
macroeconomic indicators
mathematical modeling
stability analysis
supply and demand equilibrium
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