1. that indifference curves can never cross.  This should be obvious - they are contour lines, and contour lines cannot cross by definition.  This property is known as "transitivity", and preference maps which show this property are said to be transitive - that is, you can reliably "walk" up and down them, or along a contour without getting higher or lower.  An intransitive preference map would be as meaningless as an intransitive geographical map. No one would know which way was up - a nonsense.
2. that indifference curves should be universal (that is, include of all the options and of all possible and relevant choices open to the population under consideration), and also inclusive of everyones preferences
3. that indifference curves can only show the ranking of preferences, but not the real perceived value of the bundles of goods - that indifference curves are ordinal, and not cardinal:  that is we can judge something better or worse (or, in the limit or at the boundary - on the indifference curve - as practically equivalent) but cannot (or need not be asked to) put a quantitative value on this bundle of goods (and its judged equivalents) in relation to other (better or worse) bundles.
4. that a "social indifference curve" - one which shows society's collective preferences between alternatives, is very likely to be either intransitive or exclusive or non-universal, or all three.  This is because social preferences are obtained through voting procedures (in democratic societies at least), with each individual voting as to whether option A is better or worse than option B.  Unanimity is very unlikely - so that the majority rules.  In which case, the minority preferences are discounted, and minorities are excluded from the social preference. In addition, we are seldom asked to vote on all the options available, but are only given a limited choice about what to vote on.  Even if we were given the chance of voting on all the options, we would be quite likely to run into trouble.

Suppose that there are just three people (me, you and someone else) - three, so that we can get a majority vote - and only three options (A, B and C).  We each get to rank these three options.  My ranking is A > B> C.  Suppose your ranking is C > A > B, and the other person's ranking is B > C > A.  Now, lets count up the votes. Option A beats option B by 2:1; Option B beats option C by 2:1; and option C beats option A by 2:1.  Our attempt at making a collective choice is frustrated by intransitivity - we have voted that A > B > C > A - we go round in circles and cannot make up our minds what to do.  This, known as the voting paradox, or the cycling phenomenon, is a well known conundrum in the theory of social choice (see, for example, Phelps, E.S. Political Economy , an Introductory Text.  Norton, 1985. (p198 - 202)).  Kenneth Arrow has proved (in what is known as the Impossibility Theorem) that we cannot simultaneously respect the diversity of individual preferences while at the same time ensuring that the collective (public) choice will be coherent (i.e. be transitive).

Indifference curves can only be ordinal, not cardinal.  This is certainly true (as in mathematically or logically true) if we take indifference curves in isolation from everything else.  It is impossible to defend a proposition that every individual would (still less should) subscribe to the same value scale as everyone else for all possible choices or options.  We are all different, and will value things differently, even if we want to or are prepared to actually put a price on them at all - which in many cases we are not.  The best we can hope for is that people can rank (judge between) alternatives, and this is all that is necessary for the construction of an indifference curve. So far, so good.

But we do not, at least not in any real sense, express preferences in isolation from what is possible.  We only bother to even think about what our preferences are when there is some point in so doing.  Otherwise, we don't bother.  The point, in our construction of our indifference curve above, is to decide what bundle of goods we prefer amongst all those bundles which are possible - the ones defined by the PPF.  Once we choose such a preferred bundle, we reveal our preferences. The bundle we actually choose is demonstrably worth at least what we have to pay to get it (its supply cost), otherwise we would not have chosen it - so we have necessarily put a value (at least a minimum value) on it. That is what our choice does - it puts at least a minimum value on the choice we choose.  And, if we have put a value on this bundle, then we have also put the same value on all the other bundles which lie on this indifference curve as well.  So this indifference curve is cardinal.  And, if this one is cardinal, then so, too, must all the others be (even if we don't bother to put actual values on them until it is necessary or worthwhile so to do - that is, until we are faced with the real choice, rather than some artificial or hypothetical possibility.)

In other words, the assertion that indifference curves only measure ranks and not values is a partial argument, rather than a general logical consequence of our behaviour patterns and actions.

Collective preference maps are impossible - there is no such thing as a "well-behaved" social preference map:  any such attempt will be partial, incomplete, incoherent, intransitive, exclusive etc.  What did we draw above, then?

Our system - the market system - doesn't just count votes, it weighs them.  Which requires that we actually do put values on our choices -  we have to answer the question of how much more or less do we prefer this option rather than that in order to trade.  The market system goes even further - it also provides the mechanisms, the processes, through which we can and do adjust and adapt our preferences in response to what other people think and do, both in terms of what other people are prepared to pay for things and what it costs to produce them.  The market system is reflexive.

What we drew was an aggregation (a sum) of individual preferences - each and everyone expressing what they were willing to pay for particular bundles, and trading off their willingness' to pay with each other in a reflexive fashion (i.e. taking notice of what others think, in terms of what they are prepared to pay).

The impossibility theorem does not apply to such a sum of individual preferences, because it denies the possibility of trade and reflexivity in making our choices and preferences.  It takes individual preferences as given, and then tries to arrive at some collective choice or preference map on the basis of these fixed and given preferences, whatever they are.  Not surprisingly, it finds this to be generally impossible - it will only happen as an improbable accident of everyone having mutually compatible preferences.  Trade and exchange is necessary to facilitate the adaptation and adjustment of individual preferences so that they become mutually compatible.  Without trade and exchange, we will almost certainly just go round in circles.  Even with it, is hard enough not to.

Nevertheless, much of the political-economic literature treats the impossibility theorem as being a substantial problem.  So it is, if we restrict our attention to a set of fixed preferences (for or against equality, public ownership, private wealth, etc.) and to democratic voting procedures for arriving at a majority rule.  But there is more to life, and even to our conceptions and operation of markets, than this.  This is simplistic, rather than a sensible simplification, which is what we are trying to do here.

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