Homothetic preferences

In consumer theory, a consumer's preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1. For example, in an economy with two goods x , y , homothetic preferences can be represented by an utility function u that has the following property: for every a > 0 :

In mathematics, a homothetic function is a monotonic transformation of a function which is homogeneous; however, since ordinal utility functions are only defined up to a monotonic transformation, there is little distinction between the two concepts in consumer theory.

In a model where competitive consumers optimize homothetic utility functions subject to a budget constraint, the ratios of goods demanded by consumers will depend only on relative prices, not on income or scale. This translates to a linear expansion path in wealth: the slope of indifference curves is constant along rays beginning at the origin.

Furthermore, the indirect utility function can be written as a linear function of the wealth w :

which is a special case of the Gorman polar form. Hence, if all consumers have homothetic preferences (with the same coefficient on the wealth term), the aggregate demand can be calculated by considering a single "representative consumer" who has the same preferences and the same aggregate income.