<p>Your results are not  at variance with the result of your teacher, there  are just  different definitions of wage, your <span class="math-container">$w$</span> is not the same as the <span class="math-container">$w$</span> of your teacher.</p>
<p>The labor income your teacher considers is of <span class="math-container">$\tilde {w}AL$</span>, instead of <span class="math-container">$wL$</span> as in your case, where <span class="math-container">$\tilde {w}$</span> is the wage for <strong>unit of effective labor</strong> <span class="math-container">$AL$</span>, not <strong>per worker</strong> <span class="math-container">$L$</span> as <span class="math-container">$w$</span>.</p>
<p>In his model the wage per worker is <span class="math-container">$w= \frac {\tilde {w}AL}{L}= \tilde {w}A$</span>.</p>
<p>As a consequence, if the rate of growth of  <span class="math-container">$\tilde {w}$</span> is zero,  the rate of growth of <span class="math-container">$w$</span> is equal to the rate of growth of <span class="math-container">$A$</span>, <span class="math-container">$g$</span>, as in your model: in both cases the result is that the <strong>per worker wage grows at the same rate of productivity</strong>.</p>
<p>Why is the rate of growth of  <span class="math-container">$\tilde {w}$</span> zero?</p>
<p>I suppose that your professor  equates the wage <span class="math-container">$\tilde {w}$</span> to the marginal productivity of <em>effective labor</em> <span class="math-container">$AL$</span>, and the latter depends only on <span class="math-container">$\tilde {k}= K/{AL}$</span>, and <span class="math-container">$\tilde {k}$</span> in steady state is constant:  that is <span class="math-container">$\dot {\tilde {w}}=0$</span>.</p>
<p>Indeed, recalling that that the production function is homogeneous of degree one,   we can write it as <span class="math-container">$$Y= AL f(K/AL)= AL f(\tilde {k}).$$</span>
Differentiating with respect to <span class="math-container">$AL$</span> we have:</p>
<p><span class="math-container">$$\tilde {w}= \frac{\partial (Y)}{\partial (AL)}= AL f’\left(\frac {K}{AL}\right) \left(-\frac {K}{(AL)^2}\right)+f\left(\frac {K}{AL}\right)=f(\tilde {k})-\tilde{k} f’(\tilde {k}).$$</span>
This last  expression  depends uniquely on <span class="math-container">$\tilde {k}$</span>, which in steady state is constant, and therefore <span class="math-container">$\tilde {w}$</span> is constant.</p>

CC BY-SA 4.0
2024-04-14T10:31:59.477Z
I don't doubt that my instructor or I myself did the calculations incorrectly. I'm only trying to figure out which set of definitions is more standard, his or mine.
uninterestedacademic
CC BY-SA 4.0
2024-04-13T12:12:26.070Z
2024-04-13T12:12:26.070Z
BakerStreet
<p>I am trying to show that when technological and population growths are (non-negative) constants, the rate of growth of capital income and labor income in the steady state will be equal. Assume that the production function is given by <span class="math-container">$F(K(t), A(t)L(t))$</span>, and that the growth rates of population and technological progress are <span class="math-container">$n$</span> and <span class="math-container">$g$</span> respectively. All other variables used are standard and as used by authors like Acemoglu.</p>
<hr />
<p>Define <span class="math-container">$W = wL$</span> and <span class="math-container">$R = rK$</span> For brevity, we will avoid writing the <span class="math-container">$t$</span>-argument of every function, that is, we will write the variable <span class="math-container">$L(t)$</span> as <span class="math-container">$L$</span>.</p>
<p><strong>Claim:</strong> <span class="math-container">$g_W = \frac{\dot{W}}{W} = \frac{\dot{R}}{R} = g_R$</span> along the balanced growth path (or in the steady state).</p>
<p><strong>Proof.</strong> The claim <span class="math-container">$g_W = g_R$</span> is equivalent to <span class="math-container">$\frac{\dot w}{w} + \frac{\dot L}{L} = \frac{\dot r}{r} + \frac{\dot K}{K} \iff \frac{\dot w}{w} + n = \frac{\dot r}{r} + g+n \iff \frac{\dot w}{w} = \frac{\dot{r}}{r} + g$</span>. Note that we obtained <span class="math-container">$\frac{\dot{K}}{K} = g+n$</span> from <span class="math-container">$\frac{d}{dt}\left[\ln\left(\frac{K}{AL}\right)\right] = \frac{d}{dt}\left[\left(k^{*}\right)\right] = 0$</span> where <span class="math-container">$k^{*}$</span> is the steady state value of <span class="math-container">$k$</span>.</p>
<p>Now we get, after some tedious calculations, that</p>
<ul>
<li><span class="math-container">$\displaystyle \frac{\dot w}{w} = \frac{\dot{A} [f(\tilde k) - \tilde k f'(\tilde k)]}{A [f(\tilde k) - \tilde k f'(\tilde k)]} = \frac{\dot A}{A} = g$</span> where <span class="math-container">$\tilde k:= \frac{K}{AL}$</span> and <span class="math-container">$w$</span> is given by <span class="math-container">$\frac{\partial F(K,AL)}{\partial L}$</span>.</li>
<li><span class="math-container">$\displaystyle \frac{\dot r}{r} = \frac{f''(\tilde k) \dot{\tilde{k}}}{f'(\tilde k)} = 0$</span> where <span class="math-container">$\tilde k$</span> is defined as above.</li>
</ul>
<p>Combining the two results, it is immediate that <span class="math-container">$g_W = g_R$</span>.</p>
<hr />
<p><strong>Question to the forum:</strong> My instructor considered labor income <span class="math-container">$(W)$</span> as <span class="math-container">$wAL$</span> as opposed to <span class="math-container">$wL$</span>. Further, he deduced, without showing the deductions, that <span class="math-container">$\frac{\dot w}{w} = 0$</span> where <span class="math-container">$w$</span> is the wage rate. Eventually, the two growth rates came out to be equal. However, his growth rate of <span class="math-container">$w$</span> is completely different from mine and so are the definitions. I wonder who did it correctly, me or my instructor.</p>

CC BY-SA 4.0
2024-04-13T08:42:22.843Z
Dear, I suggest you the Michio Morishima paper Economic expansion and the interest rate in generalized v. Neumann model
marco tognoli
<macroeconomics><solow>
Capital income growth equals labor income growth along the balanced growth path
uninterestedacademic
# Capital income growth equals labor income growth along the balanced growth path

(Asked by: username_0 on 2024-04-12)

I am trying to show that when technological and population growths are (non-negative) constants, the rate of growth of capital income and labor income in the steady state will be equal. Assume that the production function is given by $F(K(t), A(t)L(t))$, and that the growth rates of population and technological progress are $n$ and $g$ respectively. All other variables used are standard and as used by authors like Acemoglu.

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Define $W = wL$ and $R = rK$ For brevity, we will avoid writing the $t$-argument of every function, that is, we will write the variable $L(t)$ as $L$.

**Claim:** $g\_W = \frac{\dot{W}}{W} = \frac{\dot{R}}{R} = g\_R$ along the balanced growth path (or in the steady state).

**Proof.** The claim $g\_W = g\_R$ is equivalent to $\frac{\dot w}{w} + \frac{\dot L}{L} = \frac{\dot r}{r} + \frac{\dot K}{K} \iff \frac{\dot w}{w} + n = \frac{\dot r}{r} + g+n \iff \frac{\dot w}{w} = \frac{\dot{r}}{r} + g$. Note that we obtained $\frac{\dot{K}}{K} = g+n$ from $\frac{d}{dt}\left[\ln\left(\frac{K}{AL}\right)\right] = \frac{d}{dt}\left[\left(k^{\*}\right)\right] = 0$ where $k^{\*}$ is the steady state value of $k$.

Now we get, after some tedious calculations, that

* $\displaystyle \frac{\dot w}{w} = \frac{\dot{A} [f(\tilde k) - \tilde k f'(\tilde k)]}{A [f(\tilde k) - \tilde k f'(\tilde k)]} = \frac{\dot A}{A} = g$ where $\tilde k:= \frac{K}{AL}$ and $w$ is given by $\frac{\partial F(K,AL)}{\partial L}$.
* $\displaystyle \frac{\dot r}{r} = \frac{f''(\tilde k) \dot{\tilde{k}}}{f'(\tilde k)} = 0$ where $\tilde k$ is defined as above.

Combining the two results, it is immediate that $g\_W = g\_R$.

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**Question to the forum:** My instructor considered labor income $(W)$ as $wAL$ as opposed to $wL$. Further, he deduced, without showing the deductions, that $\frac{\dot w}{w} = 0$ where $w$ is the wage rate. Eventually, the two growth rates came out to be equal. However, his growth rate of $w$ is completely different from mine and so are the definitions. I wonder who did it correctly, me or my instructor.

**Tags:** macroeconomics, solow
> **Comments:**
> **username_1**: Dear, I suggest you the Michio Morishima paper Economic expansion and the interest rate in generalized v. Neumann model  

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## 1 answers

### Answer by username_2 on 2024-04-13. Score: 1 | Accepted answer

Your results are not at variance with the result of your teacher, there are just different definitions of wage, your $w$ is not the same as the $w$ of your teacher.

The labor income your teacher considers is of $\tilde {w}AL$, instead of $wL$ as in your case, where $\tilde {w}$ is the wage for **unit of effective labor** $AL$, not **per worker** $L$ as $w$.

In his model the wage per worker is $w= \frac {\tilde {w}AL}{L}= \tilde {w}A$.

As a consequence, if the rate of growth of $\tilde {w}$ is zero, the rate of growth of $w$ is equal to the rate of growth of $A$, $g$, as in your model: in both cases the result is that the **per worker wage grows at the same rate of productivity**.

Why is the rate of growth of $\tilde {w}$ zero?

I suppose that your professor equates the wage $\tilde {w}$ to the marginal productivity of *effective labor* $AL$, and the latter depends only on $\tilde {k}= K/{AL}$, and $\tilde {k}$ in steady state is constant: that is $\dot {\tilde {w}}=0$.

Indeed, recalling that that the production function is homogeneous of degree one, we can write it as $$Y= AL f(K/AL)= AL f(\tilde {k}).$$
Differentiating with respect to $AL$ we have:

$$\tilde {w}= \frac{\partial (Y)}{\partial (AL)}= AL f’\left(\frac {K}{AL}\right) \left(-\frac {K}{(AL)^2}\right)+f\left(\frac {K}{AL}\right)=f(\tilde {k})-\tilde{k} f’(\tilde {k}).$$
This last expression depends uniquely on $\tilde {k}$, which in steady state is constant, and therefore $\tilde {w}$ is constant.

> **Comments:**
> **username_0**: I don't doubt that my instructor or I myself did the calculations incorrectly. I'm only trying to figure out which set of definitions is more standard, his or mine.  

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