<p>This is just a logaritmization of a Cobb-Douglas type production function assuming small and constant growth rates. For small growth rates <span class="math-container">$x$</span>
<span class="math-container">$$
x \approx \ln(1+x).
$$</span>
The Cobb-Douglas type production function is
<span class="math-container">$$
Y = AK^{\alpha}L^{1-\alpha}.
$$</span>
Assuming constant growth of the variables you have
<span class="math-container">$$
Y(1+y) = A(1+a)\left(K(1+k)\right)^{\alpha}\left(L(1+l)\right)^{1-\alpha},
$$</span>
and so
<span class="math-container">$$
1+y = (1+a)(1+k)^{\alpha}(1+l)^{1-\alpha}.
$$</span>
Taking the logarithm of this you get
<span class="math-container">$$
\ln(1+y) = \ln(1+a) + \alpha\ln(1+k) + (1-\alpha)\ln(1+l).
$$</span>
Using the logarithmic approximation
<span class="math-container">$$
y \approx a + \alpha k + (1-\alpha)l,
$$</span>
and after some algebraic magic
<span class="math-container">$$
y - l \approx a + \alpha (k - l).
$$</span>
The notation for <span class="math-container">$a$</span> was <span class="math-container">$t$</span> in your equation, but otherwise we got it!</p>
<hr />
<p>P.s.: I have not seen a Solow model's production function include <span class="math-container">$R$</span> (resources) before, so I don't know how that would change things, but apparently it didn't?</p>

CC BY-SA 4.0
2024-03-23T11:40:08.777Z
2024-03-26T22:05:28.460Z
2024-03-26T22:05:28.460Z
Giskard
<p><a href="https://i.sstatic.net/9PNBM.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9PNBM.jpg" alt="enter image description here" /></a>In Chapter 6 of the 12th edition of &quot;Economic Development&quot; by Michael P. Todaro &amp; Stephen C. Smith, an equation is introduced to illustrate the consequences of rapid population growth, derived from 'the standard Solow-type neoclassical growth equation'. I understand the components individually, but I'm struggling with the derivation process from the standard production function
Y=f(K,L,R,T).</p>
<p>Could someone kindly provide insight into how this equation is derived? Thanks</p>

<economic-growth><production-function><solow><population-economics>
Derivation from Solow-type neoclassical growth equation?
David Roberts
# Derivation from Solow-type neoclassical growth equation?

(Asked by: username_0 on 2024-03-23)

[![enter image description here](https://i.sstatic.net/9PNBM.jpg)](https://i.sstatic.net/9PNBM.jpg)In Chapter 6 of the 12th edition of "Economic Development" by Michael P. Todaro & Stephen C. Smith, an equation is introduced to illustrate the consequences of rapid population growth, derived from 'the standard Solow-type neoclassical growth equation'. I understand the components individually, but I'm struggling with the derivation process from the standard production function
Y=f(K,L,R,T).

Could someone kindly provide insight into how this equation is derived? Thanks

**Tags:** economic-growth, production-function, solow, population-economics

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## 1 answers

### Answer by username_1 on 2024-03-23. Score: 0

This is just a logaritmization of a Cobb-Douglas type production function assuming small and constant growth rates. For small growth rates $x$
$$
x \approx \ln(1+x).
$$
The Cobb-Douglas type production function is
$$
Y = AK^{\alpha}L^{1-\alpha}.
$$
Assuming constant growth of the variables you have
$$
Y(1+y) = A(1+a)\left(K(1+k)\right)^{\alpha}\left(L(1+l)\right)^{1-\alpha},
$$
and so
$$
1+y = (1+a)(1+k)^{\alpha}(1+l)^{1-\alpha}.
$$
Taking the logarithm of this you get
$$
\ln(1+y) = \ln(1+a) + \alpha\ln(1+k) + (1-\alpha)\ln(1+l).
$$
Using the logarithmic approximation
$$
y \approx a + \alpha k + (1-\alpha)l,
$$
and after some algebraic magic
$$
y - l \approx a + \alpha (k - l).
$$
The notation for $a$ was $t$ in your equation, but otherwise we got it!

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P.s.: I have not seen a Solow model's production function include $R$ (resources) before, so I don't know how that would change things, but apparently it didn't?


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