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In the calculation of the equation of motion for capital in the RBC model, I came across this equation:

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Can someone explain what are the mathematical steps in between? I don't see how exactly the derivative to ln(K(t)) gets us an almost elasticity-like equation.

Would be thankful for any leads. :)

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That is the formula of the elasticity of substitution of $K_{t+1}$ with respect to $K_t$ expressed differently. Notice that the differential of the function $\ln K_{t+1}$ is $d\ln K_{t+1}=\frac{1}{K_{t+1}}\frac{d K_{t+1}}{d K_{t+1}}d K_{t+1}$ (see section 12.9 or this), then $d\ln K_{t+1}=\frac{d K_{t+1}}{K_{t+1}}$. Hence, using the same operation with $\ln K_{t}$ and combining we get the elasticity of substitution of $K_{t+1}$ with respect to $K_t$:

$$ \frac{d\ln K_{t+1}}{d\ln K_{t}}=\frac{d K_{t+1}}{d K_{t}}\frac{K_{t}}{K_{t+1}}. $$

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  • $\begingroup$ Thank you, that was very helpful for understanding the quick logic behind it :) $\endgroup$
    – Delia
    Apr 28, 2023 at 16:20

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