10.5 Homothetic functions

When representing preferences, we usually assume a number of properties, but not every preference order has to have these properties. Typically, we use only complete and transitive preferences, i.e. we assume that the following applies:

1. Completeness: For the two alternatives A and B, either the statement "A is at least as good as B" or "B is at least as good as A" can be made. If even both are valid, then you are indifferent between A and B. But it can never happen that A and B cannot be compared.
2. Transitivity: If "A is at least as good as B" and "B is at least as good as C", then follows "A is at least as good as C". If, as an example, you take numbers and the relation "greater than", this property is trivially fulfilled. But if you take soccer teams and "wins against", you can imagine combinations, where "C wins against A" is the result.

Another property, which is usually intuitively taken for granted, is homotheticism. This means, that if "A is at least as good as B", then this is also valid for any positive multiple, so "$\lambda$A is at least as good as $\lambda$B" for any $\lambda >0$.
Example: If "A = 4 apples and 2 pears" is preferred over "B = 2 apples and 4 pears", then this also applies to "3A = 12 apples and 6 pears" over "3B = 6 apples and 12 pears" or "1/2 A = 2 apples and 1 pear" over "1/2 B = 1 apple and 2 pears". Please note, this relation does not necessarily imply that "1 apple" is preferred over "1 pear".

A function is called homothetic if it can be written as a monotonic transformation of a homogeneous function, i.e. a function $f\left(x,y\right)$ is homothetic if there is a monotonic function $h:ℝ\to ℝ$ and a homogeneous function $g\left(x,y\right)$, so that $f\left(x,y\right)=h\left(g\left(x,y\right)\right)$.
The class of homothetic functions is very extensive. It obviously includes all homogeneous functions, but also many more. For example, the function $f\left(x,y\right)=\mathit{\alpha ln}\left(x\right)+\mathit{\beta ln}\left(y\right)$ is homothetic, but not homogeneous, because

with $g\left(x,y\right)=x^{\alpha }{y}^{\beta }$ and $h\left(x\right)=\mathit{ln}\left(x\right)$.
For example, all linear functions, the Cobb-Douglas-, and the CES- functions are homothetic.
The property of homotheticism for utility- or production- functions has a number of very important implications. Probably the most important one is that the slope of the isoquants along a line of origin is constant, i.e. for a constant ratio of the two goods or factors (= line of origin) the marginal rate of substitution (= MRS = slope of the isoquants) is the same. Since the MRS is fixed by the price ratio (or wage-interest ratio) for all companies in the market, the factor input ratio must also be the same. All companies in an industrial sector, i.e. with the same production technology, therefore have the same factor input ratio, i.e. they use the same amount of labor in relation to the capital input, regardless of the size of the company. It makes no difference for the presentation of the economy whether one large or several small enterprises are considered. This invariance of scale no longer prevails when through technical progress, new processes, or technologies protected by patents, the production function in this industry is no longer homogeneous.
This is illustrated in the graph below. It shows three isoquants of the production- or utility- function $f\left(x,y\right)=\mathit{\alpha ln}\left(x\right)+\mathit{\beta ln}\left(y\right)$. Both, the parameters (at the sliders) as well as the factor input ratio (red cross) can be varied. The slope of the isoquant is always the same for all three points along the line of origin, i.e. with the same factor input ratio.