What is homothetic function


Motivation of the homothetic function
What should a production function look like that preserves as much of the homogeneity as possible, but allows more options for output change with total factor change? To do this, imagine the isoquantas as contour lines in a production mountain range. This mountain range is distorted by shifting individual isoquants up or down while maintaining their shape. This distortion is achieved through a monotonous transformation. In this way, expansion paths can be generated in the production mountains, which correspond to increasing and then decreasing economies of scale. That would enable an S-shaped cost function. Another reason for considering homothetic instead of homogeneous functions is provided by utility theory: it is theoretically possible to define utility functions homogeneously, but in the case of (ordinal) utility functions, it makes no sense (mathematically: not well-defined). The five utility functions \ begin {eqnarray *} U_1 (x_1, x_2) & = & \ root 4 \ of {x_1 x_2} \ U_2 (x_1, x_2) & = & \ sqrt {x_1x_2} \ U_3 (x_1, x_2 ) & = & x_1x_2 \ U_4 (x_1, x_2) & = & \ log x_1 + \ log x_2 \ U_5 (x_1, x_2) & = & K \ cdot \ ln \ left (x_1 ^ {r / 2} \ cdot x_2 ^ {r / 2} +1 \ right) \ quad \ hbox {with} \ quad r> 0 \ end {eqnarray *} are equivalent ordinal utility functions. Their degrees of homogeneity are different with the values ​​$ {1 \ over 2} $ for $ U_1 $, $ 1 $ for $ U_2 $ and $ 2 $ for $ U_3 $. The fourth and fifth functions are not homogeneous. Therefore, if one wants to consider utility functions that have a constant rate of substitution on a path through the origin, then one will define them as homothetic.
Homothetic function
A function $ f (\ vec {x}) $ is called homothetic if there is a monotonic function $ g $ such that \ begin {equation *} g (f (\ vec {x})) \ end {equation *} is linearly homogeneous, i.e. $ g (f (\ lambda \ vec {x})) = \ lambda g (f (\ vec {x})) $. In economics it is usually assumed that $ g $ is monotonically increasing.