# Capital income growth equals labor income growth along the balanced growth path

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.

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

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.

• 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. Commented Apr 14 at 10:31