The Theory of Value

Abstract

An individual consumer is considered. There are n consumer goods (X ₁, X ₂, …X _n) so that the individual’s consumption is shown by the vector :

$$x = \left( {{x_1},{x_2}, \ldots {x_n}} \right)$$

and hence by a point P in commodity space of n dimensions. Assume that the individual’s level of satisfaction or utility is a function of his consumption :

$$u = u\left( {{x_1},{x_2}, \ldots {x_n}} \right)$$

((1))

where u is taken to vary continuously, and with continuous derivatives of the first and second order. However, the relation of the utility level u to consumption x is taken in the ordinal sense, i.e. u(x₁, x₂, … x_n) is only one of many functions which can represent utility, and any other function which orders consumption in the same way will serve. This means that u is determined only up to an increasing (monotonie) transformation :

$$utility = \phi \left( u \right)$$

((2))

where ϕ is any function such that ϕ′(u)>0. For example :

$$au + b;\quad a{u^2};\quad a\,\log \,u\quad \left( {a > 0} \right)$$

are all possible functions to represent ordinal utility.