<h2>Part 1 of the solution</h2>
<p>The fundamental equation of Solow model is (neglecting the <span class="math-container">$t$</span> subscripts):</p>
<p><span class="math-container">$$ \Delta k= sf(k) -(n+d),$$</span></p>
<p>where  <span class="math-container">$k= K/N$</span>, <span class="math-container">$\Delta k= k_{t+1}-k_t$</span> and <span class="math-container">$f(k)$</span> is the intensive production function.   And this equation must be set equal to <span class="math-container">$0$</span> in order to find the steady state value of <span class="math-container">$k$</span>.</p>
<p>But the first step is to calculate the intensive production function <span class="math-container">$f(k)$</span>.</p>
<p>As in your result you didn't obtain the part <span class="math-container">$(\frac {1} {1-\alpha}) ^{-1/\rho}$</span>, I suppose that you lost somewhere in your calculations <span class="math-container">$(1-\alpha)$</span>.
As <span class="math-container">$(1-\alpha)$</span> is in the production function, I suppose that your mistake could be in the calculation of the intensive production function.</p>
<p>Let's calculate the <strong>intensive production function</strong> from the production function in your question, which is (again neglecting the <span class="math-container">$t$</span> subscripts):</p>
<p><span class="math-container">$$Y=z[\alpha K^\rho + (1-\alpha)N^\rho) ^{1/\rho}.$$</span></p>
<p>Dividing both sides by <span class="math-container">$N$</span> we have:</p>
<p><span class="math-container">$$\frac{Y}{N}= f(k)= z[\alpha \frac {K^\rho}{N^\rho}+(1-\alpha)\frac {N^\rho}{N^\rho} ]^{-1/\rho}=$$</span>
<span class="math-container">$$\;\;\;=z[\alpha k^\rho+ (1-\alpha)]^{1/\rho}.$$</span></p>
<p>Using this last equation, making the calculations and rearranging, I found the result in your question.</p>
<p>You can try, verifying if you manage to obtain it.<span class="math-container">$^1$</span></p>
<hr />
<p><span class="math-container">$^1$</span> <sup> I didn’t post all the passages because they are enough long, but enough trivial algebraic passages, and I don’t know if it is worthwhile to post them,  if you reach the result yourself.
But I can post them subsequently, if you prefer.
</sup></p>

CC BY-SA 4.0
2022-12-04T16:44:04.700Z
Thanks for the great answer, your time is much appreciated. I am working on it, however, I started by plugging f(k) into the sf(k)-(n+d) equation (i.e change in capital), and then setting it equal to 0. I am note sure whether this step is incorrect, or is it my algebra that is failing, but I obtain a result that is almost similar, but not similar to the one requested in the setup. For example, my results in regards of the alpha, as well as the (1-alpha) are different than on the proof.
René González
CC BY-SA 4.0
2022-12-05T12:49:12.650Z
It's easy to make mistakes in this kind of calculations. I have posted in a separate answer a step by step solution.
BakerStreet
CC BY-SA 4.0
2022-12-04T14:40:09.627Z
2022-12-06T17:23:09.217Z
2022-12-06T17:23:09.217Z
BakerStreet
<h2>Part 2 of the solution</h2>
<p>I post a step by step solution, slow passages are the best way to avoid mistakes (I hope).</p>
<p>In the previuos answer we calculated the intensive production function:</p>
<p><span class="math-container">$$f(k)=z[\alpha k^\rho+ (1-\alpha)]^{1/\rho}\;\;\;\; (1)$$</span></p>
<p>The fundamental equation, set equal to <span class="math-container">$0$</span> in order to find the steady state <span class="math-container">$k$</span>, is:</p>
<p><span class="math-container">$$ \Delta k= sf(k) -(n+d)=0 \;\;\;\; \;\;(2)$$</span></p>
<p>Substituting <span class="math-container">$(1)$</span> into <span class="math-container">$(2)$</span> we have:</p>
<p><span class="math-container">$$sz [\alpha k^\rho+ (1-\alpha)]^{1/\rho} - (n+d) k=0 \Leftrightarrow$$</span></p>
<p><span class="math-container">$$sz [\alpha k^\rho+ (1-\alpha)]^{1/\rho} = (n+d) k \Leftrightarrow$$</span>(raising to <span class="math-container">$\rho$</span>-th power) <span class="math-container">$$s^\rho z^\rho  [\alpha k^\rho+ (1-\alpha)] = (n+d)^\rho  k^\rho  \Leftrightarrow$$</span>
<span class="math-container">$$s^\rho z^\rho  \alpha k^\rho+ (1-\alpha)s^\rho z^\rho  = (n+d)^\rho  k^\rho  \Leftrightarrow$$</span>
<span class="math-container">$$k^\rho (s ^\rho z^\rho \alpha -(n+d)^\rho)= -(1-\alpha) s ^\rho z^\rho \Leftrightarrow  $$</span>
<span class="math-container">$$k^\rho=      \frac {-(1-\alpha) s^\rho z^\rho}{s^\rho z^\rho \alpha -(n+d)^\rho} \Leftrightarrow$$</span>
<span class="math-container">$$k=     \left( \frac {-(1-\alpha) s^\rho z^\rho}{s^\rho z^\rho \alpha -(n+d)^\rho}\right)^{1/\rho}=$$</span>
<span class="math-container">$$=  (1-\alpha)^{1/\rho}   \left( \frac {- s^\rho z^\rho}{s^\rho z^\rho \alpha -(n+d)^\rho}\right)^{1/\rho}=$$</span></p>
<p><span class="math-container">$$=\left(\frac{1}{1-\alpha}\right) ^{-1/\rho} \left(\frac{s^\rho z^\rho \alpha- (n+d)^\rho}{- s^\rho z^\rho}\right)^{-1/\rho}=$$</span></p>
<p><span class="math-container">$$=\left(\frac{1}{1-\alpha}\right) ^{-1/\rho}   \left( \left( \frac {n+d} {sz}\right)^\rho - \alpha \right)^{-1/\rho}$$</span></p>
<p><span class="math-container">$\Box$</span></p>

CC BY-SA 4.0
2022-12-06T16:52:38.757Z
Thanks for such a great and well explained answer. The more I do these exercises, the more I get better with my first time immersion into algebra.
René González
CC BY-SA 4.0
2022-12-06T17:01:28.060Z
It is just question of practice. Thanks to you!
BakerStreet
CC BY-SA 4.0
2022-12-05T12:47:49.630Z
2022-12-05T14:26:59.380Z
2022-12-05T14:26:59.380Z
BakerStreet
<p><a href="https://i.sstatic.net/IShZ3.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/IShZ3.png" alt="SOLOW GROWTH MODEL WITH POPULATION GROWTH - PROOF OF STEADY STATE OF KAPITAL PER WORKER" /></a></p>
<p>Hello everyone, I am trying to obtain the requested solution as shown in the image (last equation of the image attached), however, after I calculate the law of motion, I only obtain only part of the equation that I am requested to show (i.e I only get n+d/sz part of the equation). What am I missing to do in this case?</p>

<macroeconomics><economic-growth><solow><steady-state>
Solow model with population growth - proof of steady state level of capital per worker
René González
# Solow model with population growth - proof of steady state level of capital per worker

(Asked by: username_0 on 2022-12-04)

[![SOLOW GROWTH MODEL WITH POPULATION GROWTH - PROOF OF STEADY STATE OF KAPITAL PER WORKER](https://i.sstatic.net/IShZ3.png)](https://i.sstatic.net/IShZ3.png)

Hello everyone, I am trying to obtain the requested solution as shown in the image (last equation of the image attached), however, after I calculate the law of motion, I only obtain only part of the equation that I am requested to show (i.e I only get n+d/sz part of the equation). What am I missing to do in this case?

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


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## 2 answers

### Answer by username_1 on 2022-12-05. Score: 2 | Accepted answer

## Part 2 of the solution

I post a step by step solution, slow passages are the best way to avoid mistakes (I hope).

In the previuos answer we calculated the intensive production function:

$$f(k)=z[\alpha k^\rho+ (1-\alpha)]^{1/\rho}\;\;\;\; (1)$$

The fundamental equation, set equal to $0$ in order to find the steady state $k$, is:

$$ \Delta k= sf(k) -(n+d)=0 \;\;\;\; \;\;(2)$$

Substituting $(1)$ into $(2)$ we have:

$$sz [\alpha k^\rho+ (1-\alpha)]^{1/\rho} - (n+d) k=0 \Leftrightarrow$$

$$sz [\alpha k^\rho+ (1-\alpha)]^{1/\rho} = (n+d) k \Leftrightarrow$$(raising to $\rho$-th power) $$s^\rho z^\rho [\alpha k^\rho+ (1-\alpha)] = (n+d)^\rho k^\rho \Leftrightarrow$$
$$s^\rho z^\rho \alpha k^\rho+ (1-\alpha)s^\rho z^\rho = (n+d)^\rho k^\rho \Leftrightarrow$$
$$k^\rho (s ^\rho z^\rho \alpha -(n+d)^\rho)= -(1-\alpha) s ^\rho z^\rho \Leftrightarrow $$
$$k^\rho= \frac {-(1-\alpha) s^\rho z^\rho}{s^\rho z^\rho \alpha -(n+d)^\rho} \Leftrightarrow$$
$$k= \left( \frac {-(1-\alpha) s^\rho z^\rho}{s^\rho z^\rho \alpha -(n+d)^\rho}\right)^{1/\rho}=$$
$$= (1-\alpha)^{1/\rho} \left( \frac {- s^\rho z^\rho}{s^\rho z^\rho \alpha -(n+d)^\rho}\right)^{1/\rho}=$$

$$=\left(\frac{1}{1-\alpha}\right) ^{-1/\rho} \left(\frac{s^\rho z^\rho \alpha- (n+d)^\rho}{- s^\rho z^\rho}\right)^{-1/\rho}=$$

$$=\left(\frac{1}{1-\alpha}\right) ^{-1/\rho} \left( \left( \frac {n+d} {sz}\right)^\rho - \alpha \right)^{-1/\rho}$$

$\Box$

> **Comments:**
> **username_0**: Thanks for such a great and well explained answer. The more I do these exercises, the more I get better with my first time immersion into algebra.  
> **username_1**: It is just question of practice. Thanks to you!  

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### Answer by username_1 on 2022-12-04. Score: 4

## Part 1 of the solution

The fundamental equation of Solow model is (neglecting the $t$ subscripts):

$$ \Delta k= sf(k) -(n+d),$$

where $k= K/N$, $\Delta k= k\_{t+1}-k\_t$ and $f(k)$ is the intensive production function. And this equation must be set equal to $0$ in order to find the steady state value of $k$.

But the first step is to calculate the intensive production function $f(k)$.

As in your result you didn't obtain the part $(\frac {1} {1-\alpha}) ^{-1/\rho}$, I suppose that you lost somewhere in your calculations $(1-\alpha)$.
As $(1-\alpha)$ is in the production function, I suppose that your mistake could be in the calculation of the intensive production function.

Let's calculate the **intensive production function** from the production function in your question, which is (again neglecting the $t$ subscripts):

$$Y=z[\alpha K^\rho + (1-\alpha)N^\rho) ^{1/\rho}.$$

Dividing both sides by $N$ we have:

$$\frac{Y}{N}= f(k)= z[\alpha \frac {K^\rho}{N^\rho}+(1-\alpha)\frac {N^\rho}{N^\rho} ]^{-1/\rho}=$$
$$\;\;\;=z[\alpha k^\rho+ (1-\alpha)]^{1/\rho}.$$

Using this last equation, making the calculations and rearranging, I found the result in your question.

You can try, verifying if you manage to obtain it.$^1$

---

$^1$  I didn’t post all the passages because they are enough long, but enough trivial algebraic passages, and I don’t know if it is worthwhile to post them, if you reach the result yourself.
But I can post them subsequently, if you prefer.

> **Comments:**
> **username_0**: Thanks for the great answer, your time is much appreciated. I am working on it, however, I started by plugging f(k) into the sf(k)-(n+d) equation (i.e change in capital), and then setting it equal to 0. I am note sure whether this step is incorrect, or is it my algebra that is failing, but I obtain a result that is almost similar, but not similar to the one requested in the setup. For example, my results in regards of the alpha, as well as the (1-alpha) are different than on the proof.  
> **username_1**: It's easy to make mistakes in this kind of calculations. I have posted in a separate answer a step by step solution.  

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