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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.


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 $g_W = g_R$.


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.

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    $\begingroup$ Dear, I suggest you the Michio Morishima paper Economic expansion and the interest rate in generalized v. Neumann model $\endgroup$ Commented Apr 13 at 8:42

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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.

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    $\begingroup$ 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. $\endgroup$ Commented Apr 14 at 10:31

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