# Solow model and economic interpretation of $\dot{K}=sY(t)$

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).

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