homothetic functions in economics

Question

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?

Bertrand · 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.

homothetic functions in economics

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