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:

- 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.
- 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
$h:\mathbb{R}\to \mathbb{R}$ 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

$$f\left(x,y\right)=\mathit{\alpha ln}\left(x\right)+\mathit{\beta ln}\left(y\right)=\mathit{ln}\left({x}^{\alpha}{y}^{\beta}\right)=h\left(g\left(x,y\right)\right)$$ |

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.

(c) by Christian Bauer

Prof. Dr. Christian Bauer

Chair of monetary economics

Trier University

D-54296 Trier

Tel.: +49 (0)651/201-2743

E-mail: Bauer@uni-trier.de

URL: https://www.cbauer.de