|The concept of bare preference, as developed so far, does not appear to lend itself to a useful formulation of cardinal utility either for individuals or for groups. It certainly does not at this time provide the key to interpersonal comparisons of utility. We gain little toward the construction of an acceptable social welfare function by employing it.|
The Application of Bare Preference to Ordinal Preference Theory
Modifying traditional preference theory with Armstrong's considerations is not easy. Two general courses suggest themselves. We can translate bare preference and the various degrees of intra-threshold preference into a band of fuzziness, or penumbra, 1) around each indifferent curve, or 2) around each alternative of choice. With either course, the implication is that an individual's levels of welfare discrimination are decreased sharply to some finite number (in the realistic case of a saturation area. Cf. J. M. Clark, op. cit., 349-350). But these reductions lead to difficulties.
Under the first course, it might be expected that a possible interpretation would be one in which the indifference curves are viewed as step functions, the step interval being equal to an assumed constant bare preference. This interpretation, however, leads to a contradiction, since it will not necessarily be true that a combination at one welfare level is at least barely preferred to any combination in the next lowest welfare level. A preference intensity less than bare preference can give rise to a welfare change of bare preference. To see this, consider the following diagram.
For graphical purposes we may simplify our alternatives of choice into two-dimensional vectors, i.e., combinations of commodities X and Y. Our curves do not here specify particular preference levels but rather each pair of curves expresses the boundaries of all combinations of A and B which are perceived as giving the same satisfaction. All combinations within the area I are equally satisfactory. Similarly for area II, each combination is deemed barely preferable to any combination in I.
If we could assume that the dimensions of our commodity space were homogeneous, we could take distances in the preference direction, i.e., as measured along gradient vectors, as representing preference intensities.