<p>We have that:
<span class="math-container">$$
\dot K = s K^\alpha L_0^b e^{nbt}
$$</span>
Rewriting the differential equation gives:
<span class="math-container">$$
K^{-\alpha} \frac{dK}{dt} = s L_0^b e^{nbt} 
$$</span>
Integrate both sides with respect to <span class="math-container">$t$</span> from <span class="math-container">$0$</span> to <span class="math-container">$T$</span> gives:
<span class="math-container">$$
\frac{1}{b} [K^b]^T_0 = s L_0^b \frac{1}{nb}[e^{nbt}]^T_0
$$</span>
So:
<span class="math-container">$$
K^b_T = K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT}
$$</span>
Equivalently:
<span class="math-container">$$
K_T = \left[K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT}\right]^{\frac{1}{b}}
$$</span></p>

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2021-10-12T13:32:43.220Z
2021-10-12T13:32:43.220Z
tdm
<p>In Solow's &quot;A Contribution to the Theory of Economic Growth&quot; paper, how do we go from this equation</p>
<p><a href="https://i.sstatic.net/GPKN4.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GPKN4.png" alt="enter image description here" /></a></p>
<p>to</p>
<p><a href="https://i.sstatic.net/SbObF.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/SbObF.png" alt="enter image description here" /></a></p>
<p>such that</p>
<p><a href="https://i.sstatic.net/xIkUt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/xIkUt.png" alt="enter image description here" /></a></p>
<p>Could you please give me a hint about that derivation?</p>

<macroeconomics><economic-growth><solow>
Derivation of Solow Growth Model (Solow, 1956)
Twister
# Derivation of Solow Growth Model (Solow, 1956)

(Asked by: username_0 on 2021-10-12)

In Solow's "A Contribution to the Theory of Economic Growth" paper, how do we go from this equation

[![enter image description here](https://i.sstatic.net/GPKN4.png)](https://i.sstatic.net/GPKN4.png)

to

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

such that

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

Could you please give me a hint about that derivation?

**Tags:** macroeconomics, economic-growth, solow

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## 1 answers

### Answer by username_1 on 2021-10-12. Score: 6 | Accepted answer

We have that:
$$
\dot K = s K^\alpha L\_0^b e^{nbt}
$$
Rewriting the differential equation gives:
$$
K^{-\alpha} \frac{dK}{dt} = s L\_0^b e^{nbt}
$$
Integrate both sides with respect to $t$ from $0$ to $T$ gives:
$$
\frac{1}{b} [K^b]^T\_0 = s L\_0^b \frac{1}{nb}[e^{nbt}]^T\_0
$$
So:
$$
K^b\_T = K^b\_0 - \frac{s}{n} L\_0^b + \frac{s}{n} L\_0^b e^{nbT}
$$
Equivalently:
$$
K\_T = \left[K^b\_0 - \frac{s}{n} L\_0^b + \frac{s}{n} L\_0^b e^{nbT}\right]^{\frac{1}{b}}
$$


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