<p>With the production function you specified you won’t get ‘clean’ per capita production function.</p>
<p>However, to get it in per capita terms you just literally have to divide both sides of equation by <span class="math-container">$L$</span> and that’s it.</p>
<p>In your case per capita output is:</p>
<p><span class="math-container">$$\frac{Y}{L} =\frac{ K^α  L^{\beta}  E^γ }{L} $$</span></p>
<p>Which simplifies to:</p>
<p><span class="math-container">$$\frac{Y}{L} = K^αL^{β-1}  E^γ $$</span></p>
<p>If you want some nicer equation use Cobb-Douglas production function. That is set <span class="math-container">$\beta = 1-\alpha - γ  $</span> and assume <span class="math-container">$\alpha, γ &gt;0$</span>. Then you will get nice per capita output equation:</p>
<p><span class="math-container">$$\frac{Y}{L} = \frac{ K^α  L^{1-\alpha - γ}  E^γ }{L}$$</span></p>
<p>This can be rearranged as:</p>
<p><span class="math-container">$$\frac{Y}{L} = \frac{ K^α   E^γ }{L^\alpha L^ γ }$$</span></p>
<p>Which simplifies to:</p>
<p><span class="math-container">$$\frac{Y}{L} = \left(\frac{ K}{L}\right)^α  \left(\frac{ E}{L}\right)^γ $$</span></p>
<p>Then you can use lower case to denote per capita variables to get ‘nice’ per capita production function:</p>
<p><span class="math-container">$$y= \left(k\right)^α  \left(e\right)^γ $$</span></p>

CC BY-SA 4.0
2023-12-11T14:11:09.330Z
@EliasChrostian no it’s beta- 1 because L^beta / L = L^beta -1
1muflon1
CC BY-SA 4.0
2023-12-11T13:47:39.983Z
2023-12-11T16:40:22.877Z
2023-12-11T16:40:22.877Z
1muflon1
<p>My problem is that I want to construct a Solow model with three input factors; labour, capital and energy. But when trying to divid the equation by labour to get the per capita variables, it doesn’t work link in the model with two inputs.</p>
<p>My equation is:</p>
<p>Y = K^α * L^β * E^γ</p>
<p>I don’t know how I can get the per capita variables by dividing by L.</p>
<p>I also haven’t found anything on the web, so I hope the someone could help me with this little problem.</p>

<production-function><solow><cobb-douglas>
Solow model with three input factors
Elias Chrostian
# Solow model with three input factors

(Asked by: username_0 on 2023-12-11)

My problem is that I want to construct a Solow model with three input factors; labour, capital and energy. But when trying to divid the equation by labour to get the per capita variables, it doesn’t work link in the model with two inputs.

My equation is:

Y = K^α \* L^β \* E^γ

I don’t know how I can get the per capita variables by dividing by L.

I also haven’t found anything on the web, so I hope the someone could help me with this little problem.

**Tags:** production-function, solow, cobb-douglas

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## 1 answers

### Answer by username_1 on 2023-12-11. Score: 1 | Accepted answer

With the production function you specified you won’t get ‘clean’ per capita production function.

However, to get it in per capita terms you just literally have to divide both sides of equation by $L$ and that’s it.

In your case per capita output is:

$$\frac{Y}{L} =\frac{ K^α L^{\beta} E^γ }{L} $$

Which simplifies to:

$$\frac{Y}{L} = K^αL^{β-1} E^γ $$

If you want some nicer equation use Cobb-Douglas production function. That is set $\beta = 1-\alpha - γ $ and assume $\alpha, γ >0$. Then you will get nice per capita output equation:

$$\frac{Y}{L} = \frac{ K^α L^{1-\alpha - γ} E^γ }{L}$$

This can be rearranged as:

$$\frac{Y}{L} = \frac{ K^α E^γ }{L^\alpha L^ γ }$$

Which simplifies to:

$$\frac{Y}{L} = \left(\frac{ K}{L}\right)^α \left(\frac{ E}{L}\right)^γ $$

Then you can use lower case to denote per capita variables to get ‘nice’ per capita production function:

$$y= \left(k\right)^α \left(e\right)^γ $$

> **Comments:**
> **username_1**: @EliasChrostian no it’s beta- 1 because L^beta / L = L^beta -1  

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