# Optimal capital income tax

Imagine we have a representative consumer that maximizes $$\sum_{t=0}^{\infty}\beta^{t}u\left(c_{t},n_{t}\right)$$ s.t. $$c_{t}+k_{t+1}=w_{t}n_{t}\left(1-\tau_{t}^{labour}\right)+r_{t}k_{t}\left(1-\tau_{t}^{capital}\right)+\left(1-\delta\right)k_{t}$$ where we have that $$n_{t}$$ is total labour supplied. We get F.O.Cs for capital accumulation (Euler) and labor-leisure $$\frac{\partial U}{\partial c_{t}}\left(c_{t},l_{t}\right)=\beta\left(r_{t}\left(1-\tau_{t}^{capital}\right)+\left(1-\delta\right)\right)\frac{\partial U}{\partial c_{t+1}}\left(c_{t+1},l_{t+1}\right)$$ $$\frac{\partial U}{\partial c_{t}}\left(c_{t},l_{t}\right)=\frac{\partial U}{\partial n_{t}}w_{t}\left(1-\tau_{t}^{labour}\right)$$ A classic result from Chamley (1986) is that capital taxation is distortionary. In other words, optimal $$\tau_{t}^{capital}=0$$ from the social planner's perspective. What is the intuition behind the result?

• There is a 2020 AER paper that explains why that conclusion is not warranted. (wp-version here) Jan 24 at 12:33

A classic result from Chamley (1986) is that capital taxation is distortionary.

This is not actually result from Chamley. In virtually every model of optimal taxation, capital taxation is distortionary. What is the classic Chamley result is that the optimal capital tax is zero.

What is the intuition behind the result?

The intuition behind the result of the model is that burden of taxation depends on relative elasticity of supply of labor and capital. Since supply of capital is perfectly elastic in the long run all tax burden gets shifted onto labour which is not perfectly elastic in long run. Since all the burden gets shift onto labor it makes no sense to tax capital in the model as you can just tax labor directly.

However, as mentioned in the comments the results got challenged more recently. This being said, empirical studies find very large tax burden shift onto labour but not 100% shift such as predicted by Chamley & Judd result (e.g. see Roy-Cesar & Vaillancourt (2010), Gruber, 1997 etc for some examples).

• Before I accept this, could you please clarify how the inelasticity of labour in the LR makes the burden of taxation shift? Jan 24 at 18:23
• @KwameBrown I mean how detailed explanation you are looking for? If you took 101 economics you probably learned about model supply and demand and tax burden being shifted to the side (supply or demand) depending on which side is more inelastic. Inelastic supply and elastic demand <- all burden goes onto supply side. Elastic supply but inelastic demand all burden is on demand side. If they are both not perfectly inelastic the bigger share of burden goes to less elastic side. Its the same principle just effect is stretched across multiple markets
– 1muflon1
Jan 24 at 19:54