<p>You are correct. The condition is true for homogeneous functions but not for homothetic ones.</p>
<p>If <span class="math-container">$U$</span> is homogeneous of degree <span class="math-container">$\alpha$</span> then <span class="math-container">$U_c$</span> and <span class="math-container">$U_\ell$</span> are homogeneous of degree <span class="math-container">$\alpha - 1$</span>.</p>
<p>The equations (42) and (43) are obtained applying Euler's theorem to these two functions.</p>

CC BY-SA 4.0
2024-11-13T11:28:58.480Z
2024-11-13T11:28:58.480Z
tdm
<p>Proposition 3: If the utility function is homothetic with respect to consumption and hours worked, then the steady-state capital tax is zero.</p>
<p>Proof: This class of preference satisfies:
<span class="math-container">\begin{equation}\tag{42}\label{42}
c^{i}_{t}U_{cc^i_t}+l^i_{t}U_{lc^i_t}=AU_{c^i_t}
\end{equation}</span>
<span class="math-container">\begin{equation}\tag{43}\label{43}
l^i_tU_{ll^i_t}+c^i_tU_{cl^i_t}=BU_{l^i_t}
\end{equation}</span></p>
<p>where <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are different constant...</p>
<p><strong>My Question</strong>
The above is part of the article ''Optimal Fiscal Policy in Overlapping Generations Models'' (Garriga 2019). I would like to understand why he states equations \eqref{42} and \eqref{43}.</p>
<p><strong>My Attempt</strong></p>
<p>Affirmation: If <span class="math-container">$U(c,l)$</span>  is a homothetic function, it doesn't necessarily hold \eqref{42} and \eqref{43}.</p>
<p>Proof: Consider the function <span class="math-container">$f(x,y)=e^{x+y}$</span>. This function is a homothetic.</p>
<p><span class="math-container">$\bullet$</span> If <span class="math-container">$ x+y=g(x,y) $</span>, and <span class="math-container">$ f(x,y)=e^{g(x,y)} $</span>,</p>
<p><span class="math-container">$\bullet$</span> <span class="math-container">$ \frac{\partial f}{\partial g}=e^g &gt; 0 $</span>, so <span class="math-container">$ f $</span> is increasing in <span class="math-container">$ g $</span>.</p>
<p><span class="math-container">$\bullet$</span> On the other hand, <span class="math-container">$g(\lambda x, \lambda y)=\lambda x + \lambda y=\lambda g(x,y) $</span>, so <span class="math-container">$g$</span> is homogeneous of degree 1.</p>
<p>Therefore, it is verified that <span class="math-container">$ f(x,y)=e^{x+y} $</span> is homothetic.</p>
<p>Let's check if the function <span class="math-container">$ f(x,y) = e^{x+y} $</span> satisfies equation \eqref{42}.</p>
<p><span class="math-container">$\bullet$</span> <span class="math-container">$ f_x = f_y = f_{xx} = f_{yx} = e^{x+y} $</span>.</p>
<p><span class="math-container">$\bullet$</span> Substituting into equation \eqref{42}, we get:
<span class="math-container">$ xe^{x+y} + ye^{x+y} = (x+y)e^{x+y} $</span></p>
<p>This result indicates that a homothetic function does not necessarily satisfy equation \eqref{42}.</p>

<macroeconomics><utility><taxation><optimal-taxation>
Does a Homothetic Function with Respect to Consumption and Hours Worked Satisfy Certain Conditions in Fiscal Policy Models?
Franciscolli
# Does a Homothetic Function with Respect to Consumption and Hours Worked Satisfy Certain Conditions in Fiscal Policy Models?

(Asked by: username_0 on 2024-07-25)

Proposition 3: If the utility function is homothetic with respect to consumption and hours worked, then the steady-state capital tax is zero.

Proof: This class of preference satisfies:
\begin{equation}\tag{42}\label{42}
c^{i}\_{t}U\_{cc^i\_t}+l^i\_{t}U\_{lc^i\_t}=AU\_{c^i\_t}
\end{equation}
\begin{equation}\tag{43}\label{43}
l^i\_tU\_{ll^i\_t}+c^i\_tU\_{cl^i\_t}=BU\_{l^i\_t}
\end{equation}

where $A$ and $B$ are different constant...

**My Question**
The above is part of the article ''Optimal Fiscal Policy in Overlapping Generations Models'' (Garriga 2019). I would like to understand why he states equations \eqref{42} and \eqref{43}.

**My Attempt**

Affirmation: If $U(c,l)$ is a homothetic function, it doesn't necessarily hold \eqref{42} and \eqref{43}.

Proof: Consider the function $f(x,y)=e^{x+y}$. This function is a homothetic.

$\bullet$ If $ x+y=g(x,y) $, and $ f(x,y)=e^{g(x,y)} $,

$\bullet$ $ \frac{\partial f}{\partial g}=e^g > 0 $, so $ f $ is increasing in $ g $.

$\bullet$ On the other hand, $g(\lambda x, \lambda y)=\lambda x + \lambda y=\lambda g(x,y) $, so $g$ is homogeneous of degree 1.

Therefore, it is verified that $ f(x,y)=e^{x+y} $ is homothetic.

Let's check if the function $ f(x,y) = e^{x+y} $ satisfies equation \eqref{42}.

$\bullet$ $ f\_x = f\_y = f\_{xx} = f\_{yx} = e^{x+y} $.

$\bullet$ Substituting into equation \eqref{42}, we get:
$ xe^{x+y} + ye^{x+y} = (x+y)e^{x+y} $

This result indicates that a homothetic function does not necessarily satisfy equation \eqref{42}.

**Tags:** macroeconomics, utility, taxation, optimal-taxation

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## 1 answers

### Answer by username_1 on 2024-11-13. Score: 0

You are correct. The condition is true for homogeneous functions but not for homothetic ones.

If $U$ is homogeneous of degree $\alpha$ then $U\_c$ and $U\_\ell$ are homogeneous of degree $\alpha - 1$.

The equations (42) and (43) are obtained applying Euler's theorem to these two functions.


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