# Show that this income tax is effectively a lump sum tax

This is a standard income, leisure tradeoff model. $$\max_{c,l} \min\{c; l\}$$

$$s.t. \space c = w(1-t)(1-l)$$ $$l$$ is leisure (where total time is 1), $$w$$ is wage, $$c$$ is consumption, and $$t$$ is the labor income tax.

I was asked to show that this income tax is effectively a lump sum tax.

The definition for lump sum tax is as follows: A tax whose value is independent of the individual's behavior.

Individuals can do nothing to avoid the lump sum tax, and there are no relative price changes. I guess that the utility function had something to do with it, but I do not see how the tax can be viewed as a lump sum tax.

• 1. What is $c$? 2. Did you try to solve the maximization problem? – Giskard Jun 8 '19 at 10:17
• Fixed. I'll give it a try. – Yejin Jun 8 '19 at 10:19

A lump sum tax is a tax which only has an income effect, it has the equivalent effect of reducing the agent's wealth. In particular, it has no substitution effect, it does not incentivise the agent to switch between $$c$$ and $$l$$.
In your case the agent's optimal choice of $$c$$ and $$l$$ is given by: $$(c^*,l^*)=\left(\frac{w(1-t)}{1+w(1-t)},\frac{w(1-t)}{1+w(1-t)}\right)$$ (show this)
It is easy to see that increasing $$t$$ is equivalent to decreasing $$w$$.