The Theory of Value

  • R. G. D. Allen


An individual consumer is considered. There are n consumer goods (X1, X2, …Xn) 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 :
$$u = u\left( {{x_1},{x_2}, \ldots {x_n}} \right)$$
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, … xn) 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 :
$$utility = \phi \left( u \right)$$
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.


Utility Function Utility Level Indifference Curve Consumption Activity Substitution Effect 
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© R. G. D. Allen 1959

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