Developing a dynamic mathematical model to analyze the impact of technological progress on productivity using differential equations and the Cobb-Douglas production function

Abstract

The study formulates an advanced mathematical model based on a system of nonlinear differential equations to investigate the dynamic interdependencies among productivity (Y), capital (K), and labor (L). Anchored in the Cobb–Douglas production function, the model extends its classical formulation by incorporating temporal dynamics and technological progress as an endogenous factor. It explicitly accounts for capital accumulation through investment and depreciation mechanisms, as well as variations in labor growth rates over time. Analytical exploration of the system identifies the steady-state equilibrium conditions under which capital variation approaches zero, allowing for the determination of long-term sustainable productivity levels. Furthermore, numerical simulations employing hypothetical and empirical parameters are conducted to assess the model’s sensitivity to changes in technological efficiency, investment ratios, and depreciation rates. The results reveal that technological advancement and innovation act as stabilizing forces that enhance system convergence and improve equilibrium productivity. Hence, the proposed model provides a rigorous quantitative framework for evaluating the dynamic effects of technological evolution on macroeconomic performance and long-run growth stability.