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I am reading Robert Solow's 1956 paper, entitled 'A Contribution to the Theory of Economic Growth'. I am trying to understand the economic interpretation of the differential equation $$\dot{K}=sY(t),$$ which is located on page 67 (Solow denotes this equation by $(3)$). I have understood each term as: $s$ is the proportion of saved output, $Y(t)$ is the production function at time $t$, and $\dot{K}$ is the rate of increase of capital stock with respect to time $t$. Although, I am having trouble understanding what this equation actually means (I am a novice at economics).

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One interpretation: Think about the equation above in GDP accounting terms. We know that in an economy, Total Savings ($S(t)$) = Total Investment ($I(t)$) (standard GDP accounting exercise).
How is this relevant here? The right hand side is the Total savings in the economy: $\underbrace{s}_{\text{savings rate}}\times\underbrace{Y(t)}_{\text{total output}}$.

The left hand side is related to the total investment in the economy: The increase in capital stock ($\dot{K}$) in the economy is the difference between investment ($I$) and depreciation ($\delta K$). That is, $\dot{K} = I(t) - \delta K(t)$.

Replacing the identity $S=I$ then yields: \begin{align} \dot{K} &= S(t) - \delta K\\ &=sY(t) - \delta K \end{align}

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  • $\begingroup$ Yes! Exactly. At the extreme, $s=1$ represents a situation where all production is going into investment and nothing is consumed. $\endgroup$ – Tomcat Jun 16 at 7:02
  • $\begingroup$ +1 You might just add that $S=I$ holds specifically for a closed economy. $\endgroup$ – Adam Bailey Jun 16 at 11:38

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