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.: 146 For example, in an economy with two goods x , y {\displaystyle x,y} , homothetic preferences can be represented by a utility function u {\displaystyle u} that has the following property: for every a > 0 {\displaystyle 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 an increasing monotonic transformation, there is a small distinction between the two concepts in consumer theory.: 147
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 income: the slope of indifference curves is constant along rays beginning at the origin.: 482 This is to say, the Engel curve for each good is linear.
Furthermore, the indirect utility function can be written as a linear function of wealth w {\displaystyle 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), aggregate demand can be calculated by considering a single "representative consumer" who has the same preferences and the same aggregate income.: 152–154