# homothetic functions in economics

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?

• 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 Nov 30 '20 at 20:27
• @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 Nov 30 '20 at 21:02
• 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 Nov 30 '20 at 21:06
• 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 Nov 30 '20 at 21:35
• @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 Nov 30 '20 at 22:29

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.