# Solow-Swan Model and Capital Gains Taxation

[TODO: Make this more legible by someone not intimately familiar with the Solow-Swan model. ]Consider the Solow-Swan model with a fixed labor supply:

$$Y = K^\alpha$$We can model capital accumulation by assuming that $s$ percent of income is saved and $\delta$ percent of capital depreciates each time period. This implies equilibrium capital is given by

$$ K^* = \left( s / \delta \right)^{1/(1-\alpha)} $$So equilibrium output is given by:

$$ Y = \left( s / \delta \right)^{\alpha/(1-\alpha)} $$An interesting tidbit of the Solow-Swan model is that it implies capitalist income is given by $\alpha Y$ and laborer income is given by $(1-\alpha) Y$.

Now, suppose capitalist and laborer income has different savings rates ($s_K$ and $s_L$). Then, we have

$$ s = \alpha s_K + (1-\alpha) s_L $$And suppose the government takes away $t$ percent of output from laborer income via taxes and gives it to capitalists. Then, we have

$$ s = (\alpha + t) s_K + (1 - \alpha - t) s_L $$and so equilibrium output is

$$ Y^* = \left( \frac{(\alpha + t) s_K + (1 - \alpha - t) s_L}{\delta} \right)^{\alpha/(1-\alpha)} $$and equilibrium consumption is

$$ c^* = (1 - (\alpha + t) s_K - (1 - \alpha - t) s_L) \cdot \left( \frac{(\alpha + t) s_K + (1 - \alpha - t) s_L}{\delta} \right)^{\alpha/(1-\alpha)} $$Taking the derivative, setting it equal to zero, and solving yields

$$ t = \frac{\alpha - \alpha s_K - l + \alpha s_L}{s_K - s_L} $$This is just the $t$ that causes $s = \alpha$, so from this it follows that (assuming we value the consumption of capitalists and laborers equally), we should subsidize whichever moves us closer to the optimal savings rate of $\alpha$.