<p>In the theory of production (and similarly for consumption), a homothetic production function is compatible with the occurrence of fixed costs, while a homogeneous production function is not.
In both cases (in standard notations), the production can be written as <span class="math-container">$$y=F(h(x)),$$</span> with <span class="math-container">$h$</span> linearly homogeneous. It is also common to impose <span class="math-container">$h(0)=0$</span>. Then, the cost function inherits the form <span class="math-container">$$c(w,y)=g(w)F^{-1}(y).$$</span>
When the production function is homogeneous of degree <span class="math-container">$k$</span> it turns out that <span class="math-container">$F^{-1}(y)=y^{1/k}$</span> and so <span class="math-container">$c(w,0)=0$</span>. For homothetic production functions however, <span class="math-container">$c(w,0)=g(w)F^{-1}(0) &gt; 0$</span> in general.<br />
A further advantage of the homothetic case, is that the degree of returns to scale can depend on <span class="math-container">$y$</span>, while it is constant for the (degree <span class="math-container">$k$</span>-) homogeneous technology.</p>

CC BY-SA 4.0
2020-12-01T13:49:54.547Z
2020-12-01T21:47:26.260Z
2020-12-01T21:47:26.260Z
Bertrand
<p>Are there any results in economics that require function to be homothetic? The textbook I am using (Essential Mathematics for Economic Analysis) says that function is homothetic when &quot; <span class="math-container">$f(x)=f(y)$</span> and <span class="math-container">$t&gt;0$</span>, then <span class="math-container">$f(tx)=f(ty)$</span>&quot;. It also mentions that there are homothetic functions which are not homogenous like <span class="math-container">$F=xy+1$</span>.</p>
<p>But then all economic examples in the  book where homothetic function is used turn out to work with homogenous functions too. Then why are they special? Is there some economic example where having homogenous function would not be enough so there must be homothetic function for it to work?</p>

CC BY-SA 4.0
2020-11-30T20:27:52.987Z
How do you define homothetic functions? And of course, if every homogenous function is homothetic, then the assumption that a function is homogenous is more or equally restrictive as the assumption that the function is homothetic. There is no nonhomogenous homothetic function.
Michael Greinecker
CC BY-SA 4.0
2020-11-30T21:02:50.407Z
@MichaelGreinecker but I put the definition from textbook there. Is there a mistake? Also, I get that every homothetic function is homogenous but the opposite doesn't hold right or wrong? I am asking if there is any example that needs function to be homothetic in addition to homogenous to work.
WilliamT
CC BY-SA 4.0
2020-11-30T21:06:04.847Z
For the usual definition, every homogenous function is homothetic, but not every homothetic function is homogenous. And no, you've neither put a definition there nor a reference to the texbook.
Michael Greinecker
CC BY-SA 4.0
2020-11-30T21:18:08.193Z
@MichaelGreinecker I added the book title to my question. But then I don't get it. I thought that citation says what functions are homothetic
WilliamT
CC BY-SA 4.0
2020-11-30T21:35:01.993Z
At least in the 5th edition of the book, the definition of a homothetic function is that if $f(\mathbf{x})=f(\mathbf{y})$ and $t>0$, then $f(t\mathbf{x})=f(t\mathbf{y})$. Example 12.7.5. is of function that is homothetic, but not homogenous. It is given by $F(x,y)=xy+1$
Michael Greinecker
CC BY-SA 4.0
2020-11-30T22:29:32.000Z
@MichaelGreinecker I am so sorry the book written this in blue and they use that often for definitions. But my question is exactly about cases like $F(x,y)=xy+1$ - so there are functions that are homothetic but not homogenous. But all economic examples in the book are when the function is both homogenous and homothetic. I wan't to know if there is any result for which function has to be homothetic not homogenous. I am asking this because I don't get what is the point of homothetic functions. What is their economic significance? Why are they not excluded from the book like trig functions?
WilliamT
<mathematical-economics>
homothetic functions in economics
WilliamT
# homothetic functions in economics

(Asked by: username_0 on 2020-11-30)

Are there any results in economics that require function to be homothetic? The textbook I am using (Essential Mathematics for Economic Analysis) says that function is homothetic when " $f(x)=f(y)$ and $t>0$, then $f(tx)=f(ty)$". It also mentions that there are homothetic functions which are not homogenous like $F=xy+1$.

But then all economic examples in the book where homothetic function is used turn out to work with homogenous functions too. Then why are they special? Is there some economic example where having homogenous function would not be enough so there must be homothetic function for it to work?

**Tags:** mathematical-economics

> **Comments:**
> **username_1**: How do you define homothetic functions? And of course, if every homogenous function is homothetic, then the assumption that a function is homogenous is more or equally restrictive as the assumption that the function is homothetic. There is no nonhomogenous homothetic function.  
> **username_0**: @MichaelGreinecker but I put the definition from textbook there. Is there a mistake? Also, I get that every homothetic function is homogenous but the opposite doesn't hold right or wrong? I am asking if there is any example that needs function to be homothetic in addition to homogenous to work.  
> **username_1**: For the usual definition, every homogenous function is homothetic, but not every homothetic function is homogenous. And no, you've neither put a definition there nor a reference to the texbook.  
> **username_0**: @MichaelGreinecker I added the book title to my question. But then I don't get it. I thought that citation says what functions are homothetic  
> **username_1**: At least in the 5th edition of the book, the definition of a homothetic function is that if $f(\mathbf{x})=f(\mathbf{y})$ and $t>0$, then $f(t\mathbf{x})=f(t\mathbf{y})$. Example 12.7.5. is of function that is homothetic, but not homogenous. It is given by $F(x,y)=xy+1$  
> **username_0**: @MichaelGreinecker I am so sorry the book written this in blue and they use that often for definitions. But my question is exactly about cases like $F(x,y)=xy+1$ - so there are functions that are homothetic but not homogenous. But all economic examples in the book are when the function is both homogenous and homothetic. I wan't to know if there is any result for which function has to be homothetic not homogenous. I am asking this because I don't get what is the point of homothetic functions. What is their economic significance? Why are they not excluded from the book like trig functions?  

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## 1 answers

### Answer by username_2 on 2020-12-01. Score: 4 | Accepted answer

In the theory of production (and similarly for consumption), a homothetic production function is compatible with the occurrence of fixed costs, while a homogeneous production function is not.
In both cases (in standard notations), the production can be written as $$y=F(h(x)),$$ with $h$ linearly homogeneous. It is also common to impose $h(0)=0$. Then, the cost function inherits the form $$c(w,y)=g(w)F^{-1}(y).$$
When the production function is homogeneous of degree $k$ it turns out that $F^{-1}(y)=y^{1/k}$ and so $c(w,0)=0$. For homothetic production functions however, $c(w,0)=g(w)F^{-1}(0) > 0$ in general.  
A further advantage of the homothetic case, is that the degree of returns to scale can depend on $y$, while it is constant for the (degree $k$-) homogeneous technology.


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