As an economy accumulates more capital, the marginal product of each additional unit of capital tends to decline. This phenomenon, known as diminishing returns to capital, suggests that wealthy nations often experience slower growth rates compared to developing nations, which can grow faster by adopting existing technologies and capital.

In the Solow-Swan growth model, the steady state growth path represents a situation where the economy's output, capital, and labor grow at constant rates. When technological progress is present, the economy moves along a balanced growth path where the growth rate of output per effective worker is zero, but total output grows at the rate of technological progress plus population growth.

Convergence occurs because economies with lower capital-to-labor ratios experience higher marginal productivity of capital, leading to faster capital deepening. Additionally, less developed countries can achieve rapid growth by adopting existing technologies from more advanced nations, a process known as the 'catch-up' effect, which facilitates the narrowing of the income gap.

Mankiw, Romer, and Weil (1992) extended the Solow growth model by incorporating human capital. They argued that once you control for the determinants of the steady state—such as savings rates, population growth, and human capital accumulation—poor countries should grow faster than rich countries. This concept is known as conditional convergence, suggesting that countries with lower initial income levels will catch up to their own steady-state income levels more rapidly.

In the Solow-Swan neoclassical growth model, a higher saving rate increases the steady-state level of capital per worker and output per worker, but it does not change the long-run steady-state growth rate of output, which is determined solely by the rate of technological progress.

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Convergence theory suggests that poorer economies, which often have lower capital-to-labor ratios, will experience faster growth rates than wealthier economies. This catch-up effect occurs because of diminishing marginal returns to capital, allowing less developed nations to narrow the income gap with more advanced economies over time.

In the steady state of the Solow model, gross investment is exactly equal to the amount of capital depreciation. This means that net investment is zero, and all investment is effectively used to maintain the existing capital stock at its current level, preventing it from declining due to depreciation.

The golden-rule saving rate is the specific level of saving that results in the highest possible steady-state level of consumption per capita. It represents the optimal balance where the marginal product of capital equals the depreciation rate plus population growth.

In the Solow-Swan neoclassical growth model, the steady state is reached when investment is exactly sufficient to cover both the depreciation of existing capital and the capital requirements for new workers entering the labor force. This keeps the capital-labor ratio (capital per person) constant, allowing the economy to grow at a rate determined by labor force growth and technological progress.