<p>I give here the procedure but with price <span class="math-container">$p=1$</span> you can do it yourself without that simplification.</p>
<p>Set up Lagrange</p>
<p><span class="math-container">$$\mathcal L(c,l,\lambda) = u(c,l) - \lambda (c-\bar w(1-t)(24-l)) $$</span></p>
<p>clearly</p>
<p><span class="math-container">$$\frac{\partial \mathcal L }{\partial c} = \frac{\partial u}{\partial c} - \lambda = 1- \lambda$$</span></p>
<p>so <span class="math-container">$\lambda=1$</span> and constraint is binding. Therefore</p>
<p><span class="math-container">$$c^\star = \bar w(1-t)(24-l^\star)$$</span></p>
<p>Furthermore</p>
<p><span class="math-container">$$\frac{\partial L}{\partial l} = (24-l)^{1/\eta} - \lambda \bar w (1-t) = 0$$</span>
since <span class="math-container">$\lambda = 1$</span> it follows that</p>
<p><span class="math-container">$$l^\star = 24 - (\bar w(1-t))^\eta$$</span></p>
<p>plugging <span class="math-container">$l^\star$</span> and <span class="math-container">$c^\star$</span> into utility function, starting with <span class="math-container">$c^\star$</span> you get</p>
<p><span class="math-container">$$\bar w(1-t)(24 - l^\star) - \frac{\eta}{\eta + 1}(24 - l^\star)^{\frac{\eta + 1}{\eta}}$$</span></p>
<p>then <span class="math-container">$l^\star$</span> to get</p>
<p><span class="math-container">$$\bar w(1-t)[\bar w(1-t)]^{\eta} -  \frac{\eta}{\eta + 1}(\bar w(1-t))^{\frac{\eta(\eta + 1)}{\eta}},$$</span></p>
<p>reduce this to get</p>
<p><span class="math-container">$$(\bar w (1-t))^{\eta +1}[1/(\eta + 1)]$$</span></p>
<p>insert optimal tax and get</p>
<p><span class="math-container">$$\left(\bar w \frac{\eta}{\eta + 1}\right)^{\eta +1}[1/(\eta + 1)]$$</span></p>

CC BY-SA 4.0
2020-12-10T22:03:16.527Z
Okay! I see what you're doing. I'm more familiar with Lagrange, so this helped a lot. 
Thank you, Jesper.
Matt
CC BY-SA 4.0
2020-12-10T22:07:16.607Z
ok, pls accept and upvote then. Happy to help and keep learning.
Jesper Hybel
CC BY-SA 4.0
2020-12-10T21:39:45.313Z
2020-12-10T21:39:45.313Z
Jesper Hybel
<p>I'm dealing with a tricky assignment, and I have no idea of where to begin.</p>
<p>Person 1 lives in Denmark and has a utility function given by,</p>
<p><span class="math-container">$$(1) \ \ u(c,l)=c-\frac{\eta}{\eta+1}(24-l)^{\frac{\eta+1}{\eta}}$$</span></p>
<p>Where <span class="math-container">$\eta&gt;0$</span>, <span class="math-container">$c$</span> is consumption and <span class="math-container">$l$</span> is leisure.</p>
<p>The danish government imposes a tax on the person, <span class="math-container">$t, 0&lt;t&lt;1$</span> such that his 'post-tax' wage is given by <span class="math-container">$w=\bar{w}(1-t)$</span>. Tax revenue is given by</p>
<p><span class="math-container">$$(2) \ \ T=t\cdot \bar{w}\cdot S(w)$$</span></p>
<p>where <span class="math-container">$S(w)$</span> is the supply of labor and <span class="math-container">$\bar{w}$</span> is wage.</p>
<p><strong>First</strong> I derived the tax that yields the most tax revenue, <span class="math-container">$t^*=\frac{1}{(1+\eta)}$</span>.</p>
<p>Here comes the problem.</p>
<p>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 <span class="math-container">$\bar{u}$</span>. Suppose that:</p>
<p><span class="math-container">$$(3) \ \ \bar{u}&gt;\frac{1}{\eta+1}\left(\frac{\eta}{1+\eta}\frac{\bar{w}}{p}\right)^{\eta+1}$$</span></p>
<p><strong>I have to show</strong> that the tax that is maximizing tax-revenue (<span class="math-container">$t^*$</span>) is 'too high' for Person 2 to move to Denmark.</p>
<p>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?</p>

CC BY-SA 4.0
2020-12-10T18:33:09.733Z
$\bar u$ 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
CC BY-SA 4.0
2020-12-10T20:50:56.967Z
In the inequality for $\bar u$ is the RHS the term that person from Wales' utility should be higher than?
Jesper Hybel
CC BY-SA 4.0
2020-12-10T21:06:05.113Z
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
CC BY-SA 4.0
2020-12-10T21:08:58.547Z
I don't know if that is the right interpretation of the problem.
Matt
CC BY-SA 4.0
2020-12-10T21:09:15.790Z
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
CC BY-SA 4.0
2020-12-10T21:16:26.257Z
From where would you suggest that I solve for $c^*$ and $l^*$?
Matt
<microeconomics><labor-economics><taxation><optimal-taxation>
Maximized tax revenue & foreign labor
Matt
# Maximized tax revenue & foreign labor

(Asked by: username_0 on 2020-12-10)

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?

**Tags:** microeconomics, labor-economics, taxation, optimal-taxation

> **Comments:**
> **username_1**: $\bar u$ 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 $\bar u$.  
> **username_2**: In the inequality for $\bar u$ is the RHS the term that person from Wales' utility should be higher than?  
> **username_0**: 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).  
> **username_0**: I don't know if that is the right interpretation of the problem.  
> **username_2**: 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.  
> **username_0**: From where would you suggest that I solve for $c^*$ and $l^*$?  

---

## 1 answers

### Answer by username_2 on 2020-12-10. Score: 3 | Accepted answer

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

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

> **Comments:**
> **username_0**: Okay! I see what you're doing. I'm more familiar with Lagrange, so this helped a lot.  Thank you, Jesper.  
> **username_2**: ok, pls accept and upvote then. Happy to help and keep learning.  

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