The geometry of community indifference curves
While all of our proofs assumed only one or two residents, the following summary applies to the case of an N person community for any N> 0 and for many commodities. Community indifference curves will be non-intersecting and convex if we assume that the underlying individual indifference curves are convex and (i) there is only one resident of the economy or only one "dictator" who makes all consumption decisions or if (ii) the government continually redistributes income in order to reach the highest possible Bergsonian welfare contour (and the BWCs are convex with the individual utility functions concave) or if (iii) all residents have indifference curve maps which are identical and hamothetic or if (iv) all residents have homothetic preferences and incomes are always distributed in the same proportion between them, or if (v) they have identical indifference curve maps and incomes are always distributed evenly between them, or if (vi) all individuals have the same marginal propensity to consume any given good and in equilibrium all individuals consume all goods, or if (vii) value shares are characterized by PIGL and incomes are always distributed proportionately. If these curves are rationalized by (i), the community preference map will be identical to that of the individual or dictator and normative significance can be attached to it in that it represents his welfare. If rationalized by route (ii), these curves are the BWCs drawn with goods on the axes instead of utilities, i.e., the social indifference curves sO that again normative significance is automatically attached to them. In this case, the community indifference curve corresponding to a Bergsonian welfare level W*, when drawn with consumption of the two goods on the axes, is the envelope of the Scitovsky curves which permit the government to reach the Bergsonian welfare level W*. Also, when the Bergsonian welfare contours are right-angled, the community indifference curves are a consistent, non-intersecting family of Scitovsky indifference curves. If rationalized by (iii) or (v) the community indifference curve map will be identical to that of any individual with all dimensions multiplied by the community's population and this will also be true of (vii) except that a different scale factor must be used. If rationalized by route (iv), the map is homothetic and correctly indicates Bergsonian welfare, provided that the Bergsonian welfare function is of the Cobb-Douglas type and that the distribution shares are the respective exponents of the welfare function. If rationalized by route (v), normative significance is automatically attached to it, for movement to a higher curve implies that every resident is better off. Finally, if rationalized by (vi), the community's Engel curves are parallel to those of each individual and if rationalized by either (iii), (v) or (vi) each community indifference curve is a Scitovsky indifference curve and represents a contour of potential welfare. Finally, there is no normative significance attached to the community indifference curves in general if they are rationalized according to (iv) or (vii), for in these cases they simply serve to indicate the relative prices which will prevail as a function of the community's stocks of goods. © 1979 Institut fur Weltwirtschaft an der Universitat Kiel.
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