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

enter image description here

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. :)


1 Answer 1


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 $K_t$:

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.