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, $t, 0<t<1$ 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?

  • $\begingroup$ $\bar u$ 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 $\bar u$. $\endgroup$ – Dayne Dec 10 '20 at 18:33
  • $\begingroup$ In the inequality for $\bar u$ is the RHS the term that person from Wales' utility should be higher than? $\endgroup$ – Jesper Hybel Dec 10 '20 at 20:50
  • $\begingroup$ 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). $\endgroup$ – Matt Dec 10 '20 at 21:06
  • $\begingroup$ I don't know if that is the right interpretation of the problem. $\endgroup$ – Matt Dec 10 '20 at 21:08
  • $\begingroup$ 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. $\endgroup$ – 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

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


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

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

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


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

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

  • $\begingroup$ Okay! I see what you're doing. I'm more familiar with Lagrange, so this helped a lot. Thank you, Jesper. $\endgroup$ – Matt Dec 10 '20 at 22:03
  • 1
    $\begingroup$ ok, pls accept and upvote then. Happy to help and keep learning. $\endgroup$ – Jesper Hybel Dec 10 '20 at 22:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.