# Maximized tax revenue & foreign labor

I'm dealing with a tricky assignment, and I have no idea of where to begin.

Person 1 lives in Denmark and has a utility function given by,

$$(1) \ \ u(c,l)=c-\frac{\eta}{\eta+1}(24-l)^{\frac{\eta+1}{\eta}}$$

Where $$\eta>0$$, $$c$$ is consumption and $$l$$ is leisure.

The danish government imposes a tax on the person, $$t, 0 such that his 'post-tax' wage is given by $$w=\bar{w}(1-t)$$. Tax revenue is given by

$$(2) \ \ T=t\cdot \bar{w}\cdot S(w)$$

where $$S(w)$$ is the supply of labor and $$\bar{w}$$ is wage.

First I derived the tax that yields the most tax revenue, $$t^*=\frac{1}{(1+\eta)}$$.

Here comes the problem.

Suppose that a second person (Person 2) with the same utility function is living in Wales. She is only willing to move to Denmark (for example when a big firm want's to import foreign labor because of qualifications) only when she can get a utility (after tax) that is higher than $$\bar{u}$$. Suppose that:

$$(3) \ \ \bar{u}>\frac{1}{\eta+1}\left(\frac{\eta}{1+\eta}\frac{\bar{w}}{p}\right)^{\eta+1}$$

I have to show that the tax that is maximizing tax-revenue ($$t^*$$) is 'too high' for Person 2 to move to Denmark.

I get that I have to show that Person 2's utility is higher when she lives in Wales, than if she lived in Denmark - but how?

• $\bar u$ is function of $w$. I think we can assume that $p$ is the price of consumption $c$. Supply curve $S(w)$ can be derived using the utility function and budget constraint. So we can get utility as function of $w$ and that can be compared with $\bar u$. – Dayne Dec 10 '20 at 18:33
• In the inequality for $\bar u$ is the RHS the term that person from Wales' utility should be higher than? – Jesper Hybel Dec 10 '20 at 20:50
• My guess is that $\bar{u}$ is the utility in Wales, and that the utility in Denmark, given the tax $t^*=\frac{1}{1+\eta}$, is on the right side of the inequality. Now, because Person 2 has the utility $\bar{u}$ she won't be moving to Denmark, given that her utility is higher - as given in the inequality. Therefore it is my job to show that the tax, $t^*$, yields that utility (right side of inequality). – Matt Dec 10 '20 at 21:06
• I don't know if that is the right interpretation of the problem. – Matt Dec 10 '20 at 21:08
• Ok, well solve for $c^\star$ and $l^\star$ and plug into utility function to get value function. Hopefully you will then arrive at the RHS. – Jesper Hybel Dec 10 '20 at 21:09

I give here the procedure but with price $$p=1$$ you can do it yourself without that simplification.

Set up Lagrange

$$\mathcal L(c,l,\lambda) = u(c,l) - \lambda (c-\bar w(1-t)(24-l))$$

clearly

$$\frac{\partial \mathcal L }{\partial c} = \frac{\partial u}{\partial c} - \lambda = 1- \lambda$$

so $$\lambda=1$$ and constraint is binding. Therefore

$$c^\star = \bar w(1-t)(24-l^\star)$$

Furthermore

$$\frac{\partial L}{\partial l} = (24-l)^{1/\eta} - \lambda \bar w (1-t) = 0$$ since $$\lambda = 1$$ it follows that

$$l^\star = 24 - (\bar w(1-t))^\eta$$

plugging $$l^\star$$ and $$c^\star$$ into utility function, starting with $$c^\star$$ you get

$$\bar w(1-t)(24 - l^\star) - \frac{\eta}{\eta + 1}(24 - l^\star)^{\frac{\eta + 1}{\eta}}$$

then $$l^\star$$ to get

$$\bar w(1-t)[\bar w(1-t)]^{\eta} - \frac{\eta}{\eta + 1}(\bar w(1-t))^{\frac{\eta(\eta + 1)}{\eta}},$$

reduce this to get

$$(\bar w (1-t))^{\eta +1}[1/(\eta + 1)]$$

insert optimal tax and get

$$\left(\bar w \frac{\eta}{\eta + 1}\right)^{\eta +1}[1/(\eta + 1)]$$

• Okay! I see what you're doing. I'm more familiar with Lagrange, so this helped a lot. Thank you, Jesper. – Matt Dec 10 '20 at 22:03
• ok, pls accept and upvote then. Happy to help and keep learning. – Jesper Hybel Dec 10 '20 at 22:07