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Consider the Solow model (without technology):

$Y = F(K, L) = K^\alpha L^{(1-\alpha)}$

What's the economic interpretation of $\alpha$? Prove and argue the result.

I see it as a share that goes into capital and $1-\alpha$ is the share that goes into labour, but how do I prove this mathematically?

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  • $\begingroup$ "share"? What do you mean? Share of what? $\endgroup$ – Giskard Feb 7 '17 at 11:45
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I don't quite understand what you mean by "share that goes into capital", but the common interpretation is that $\alpha$ is the share of income/output spent on capital.

You can show that the following way: Since the factors will be compensated according to their marginal products, under the assumption of competitive markets, we have (for capital):

$$ \frac{\partial Y}{\partial K}= \alpha K^{\alpha-1}L^{1-\alpha}=\frac{r}{p}, $$ where $\frac{r}{p}$ is the real rent.

So the amount of income spent on capital is equal to price times quantity, which in this case is, $$ \frac{\partial Y}{\partial K}\frac{K}{Y}= \alpha K^{\alpha-1}L^{1-\alpha}\frac{K}{K^{\alpha}L^{1-\alpha}} \\ \frac{\partial Y}{\partial K}\frac{K}{Y}= \alpha $$

Click here for more detail.

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    $\begingroup$ "$\alpha$ is the share of income/output spent on capital." I don't think this is true. You seem to be confusing the production function with a utility function. The Solow model doesn't even have a utility function, only a behavioral one, which tells us that $s$ fraction of the output is saved/spent on capital. $\endgroup$ – Giskard Feb 7 '17 at 13:22
  • $\begingroup$ How am I confusing it with a utility function? What I describe is simply a result stemming from the optimal solution for the producer. $\endgroup$ – option_select Feb 7 '17 at 13:37
  • $\begingroup$ You are right. I finally understood what you show. I still it is ify as I don't think the Solow model assumes anything about competitive capital markets. If you would please make an edit to your question I will retract my downvote. (It is impossible to withdraw a vote without the question being edited.) $\endgroup$ – Giskard Feb 7 '17 at 14:01
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    $\begingroup$ The result was taught to me as put here. But Solow assumes it for only some of the results in his original paper. piketty.pse.ens.fr/files/Solow1956.pdf See page 79. $\endgroup$ – option_select Feb 7 '17 at 14:19
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    $\begingroup$ alternative, $(\partial Y/\partial K)(K/Y)$ is the elasticity and hence, $\alpha$ shows the percentage change of $Y$ at $1\%$ change of capital $\endgroup$ – Yorgos Feb 7 '17 at 15:28
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Following Yorgos's alternative interpretation, (about $\alpha$ which shows the percentage change of $Y$ at $1\%$ change in $K$), one intuition may also be to log-linearize your production function. As follows

$\ln Y = \alpha \ln K + (1-\alpha) \ln L$.

Then recalling that log may be used to compute continuous variations, you could substitute, e.g $K$ for $\exp(\Delta^K_{t+dt}) = \frac{K_{t+dt}}{K_t}$, idem for $Y$ and $L$. This makes obvious, in an econometric framework for example, the "elasticity nature" of $\alpha$.

$\Delta^Y_{t+dt} = \alpha \Delta^K_{t+dt} + (1-\alpha) \Delta^L_{t+dt}$

Evaluated at $\Delta^L_{t+dt} = 0$, we get $\alpha = \frac{\Delta^Y_{t+dt}}{\Delta^K_{t+dt}}$

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It represents two things: (1) the elasticity of output with respect to capital, and (2) capital's share of output.

To show (1), just take the natural log of the production equation, and then take the derivative of the logged equation with respect to time. $$\ln{Y} = \alpha \ln{K} + (1 - \alpha)(\ln{A} + \ln{L})$$ $$\frac{\dot{Y}}{Y} = \alpha \frac{\dot{K}}{K} + (1 - \alpha)\Big(\frac{\dot{A}}{A} + \frac{\dot{L}}{L}\Big).$$

Divide both sides by $\dot{K}/K$. You'll then have the right hand side as $$\frac{\dot{Y}/Y}{\dot{K}/K} = \frac{\dot{Y}}{\dot{K}}\Big(\frac{K}{Y}\Big) = \alpha + (1 - \alpha)\Big(\frac{\dot{A}}{A} + \frac{\dot{L}}{L}\Big)\frac{K}{\dot{K}}.$$

This translates as the elasticity of output with respect to capital is equal to $\alpha$ plus $(1-\alpha)$ times the sum of the elasticities productivity and labor, each with respect to capital. In the basic Solow model, the growth rates of productivity and labor are exogenous constants, thus these latter elasticities are necessarily equal to zero You are left with $$\frac{\dot{Y}}{\dot{K}}\Big(\frac{K}{Y}\Big) = \alpha.$$

To prove (2), take the derivative of output with respect to capital to get the marginal product of capital. This is the interest rate $r$. $$\frac{\partial Y}{\partial K} = \alpha K^{\alpha-1}(AL)^{1-\alpha} = r$$

Then, multiply both sides by $K/Y$. The left hand side is now $rK/Y$, which is capital's share of output. The right hand side will simplify to $\alpha.$

$$\frac{rK}{Y}= \frac{(\alpha K^{\alpha-1}(AL)^{1-\alpha})K}{K^{\alpha}(AL)^{1-\alpha}} = \alpha.$$

The left hand side of the last equation above might look familiar, as it relates to the economy-wide budget constraint: $Y=rK+wL.$

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