<p>That is the formula of the elasticity of substitution of <span class="math-container">$K_{t+1}$</span> with respect to <span class="math-container">$K_t$</span> expressed differently. Notice that the differential of the function <span class="math-container">$\ln K_{t+1}$</span> is <span class="math-container">$d\ln K_{t+1}=\frac{1}{K_{t+1}}\frac{d K_{t+1}}{d K_{t+1}}d K_{t+1}$</span> (see <a href="https://rads.stackoverflow.com/amzn/click/com/1292074612" rel="nofollow noreferrer" rel="nofollow noreferrer">section 12.9</a> or <a href="https://en.wikipedia.org/wiki/Elasticity_of_substitution#Definition" rel="nofollow noreferrer">this</a>), then <span class="math-container">$d\ln K_{t+1}=\frac{d K_{t+1}}{K_{t+1}}$</span>. Hence, using the same operation with <span class="math-container">$\ln K_{t}$</span> and combining we get the elasticity of substitution of <span class="math-container">$K_{t+1}$</span> with respect to <span class="math-container">$K_t$</span>:</p>
<p><span class="math-container">$$
\frac{d\ln K_{t+1}}{d\ln K_{t}}=\frac{d K_{t+1}}{d K_{t}}\frac{K_{t}}{K_{t+1}}.
$$</span></p>

CC BY-SA 4.0
2023-04-28T16:20:10.297Z
Thank you, that was very helpful for understanding the quick logic behind it :)
Delia
CC BY-SA 4.0
2023-04-26T13:16:32.487Z
2023-04-26T13:35:56.653Z
2023-04-26T13:35:56.653Z
manifold
<p>In the calculation of the equation of motion for capital in the RBC model, I came across this equation:</p>
<p><a href="https://i.sstatic.net/4xXko.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4xXko.png" alt="enter image description here" /></a></p>
<p>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.</p>
<p>Would be thankful for any leads. :)</p>

<elasticity-of-substitution>
Derivative to ln(K(t)) in the RBC model
Delia
# Derivative to ln(K(t)) in the RBC model

(Asked by: username_0 on 2023-04-26)

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

[![enter image description here](https://i.sstatic.net/4xXko.png)](https://i.sstatic.net/4xXko.png)

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. :)

**Tags:** elasticity-of-substitution

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## 1 answers

### Answer by username_1 on 2023-04-26. Score: 0 | Accepted answer

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](https://rads.stackoverflow.com/amzn/click/com/1292074612) or [this](https://en.wikipedia.org/wiki/Elasticity_of_substitution#Definition)), 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 $K\_t$:

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

> **Comments:**
> **username_0**: Thank you, that was very helpful for understanding the quick logic behind it :)  

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