Positive Liberty

More paradoxical cakeblogging. A statement of the problem, and a possible solution, below the fold.

In part VIII of the ongoing series, I quoted an example given by Ludwig von Mises, and I suggested that it created a paradox in the ordinal theory of value. Mises wrote,

Let us suppose that the scale of values of the possessor of an apple, a pear, and a glass of lemonade, is as follows:
1. An apple
2. A piece of cake
3. A glass of lemonade
4. A pear

If now this man is given the opportunity of exchanging his pear for a piece of cake, this opportunity will increase the significance that he attaches to the pear. He will now value the pear more highly than the lemonade. If he is given the choice between relinquishing either the pear or the lemonade, he will regard the loss of the lemonade as the lesser evil. But this is balanced by his reduced valuation of the cake. Let us assume that our man possesses a piece of cake, as well as the pear, the apple, and the lemonade. Now if he is asked whether he could better put up with the loss of the cake or of the lemonade, he will in any case prefer to lose the cake, because he can make good this loss by surrendering the pear, which ranks below the lemonade in his scale of values. The possibility of exchange introduces considerations of the objective exchange value of goods into the economic decisions of every individual; the original primary scale of use-values is replaced by the derived secondary scale of exchange values and use-values, in which economic goods are ranked not only with regard to their use-values, but also according to the value of the goods that can be obtained for them in exchange. There has been a transposition of the goods; the order of their significance has been altered. But if one good is placed higher, then—there can be no question of it—some other must be placed lower. This arises simply from the very nature of the scale of values, which constitutes nothing but an arrangement of the subjective valuations in order of the significance of the objects valued.

To which I replied,

But the analysis in the foregoing paragraph seems to turn the linear hierarchy of values into a mobius strip: If we accept that the free exchange of a pear for a piece of cake means that the two are equal in value, and that the one has risen just as the other has fallen, then where do the two terms of this equality stand in relation to lemonade?

Every time I look at (lemonade versus [cake or pear]), I am convinced that I see an inequality. It’s just a different inequality every time I look. It is clear, I think, that outside the market I still prefer the cake to the lemonade, in that I fear the loss of the former more keenly. Within the market, this reverses. Likewise, I introspectively prefer the lemonade to the pear, and again, within the market, this reverses.

This conflict, between the exchange values set in the market and the values I hold within myself, is arguably what impels the individual into the market in the first place. Yet from where I stand, it also creates confusion in my values, for once I enter the market, have I not reached a self-contradictory position: How, after all, can we construct a hierarchy of values in which I desire a piece of cake and a pear both equally, but in which I desire a glass of lemonade some intermediate amount, with the cake above and the pear below?

Let me explain, with thanks to Scott if I’m onto something.

The seeming paradox rests on a comparison between two unequal states: We are mistakenly comparing the desires of the man who has a piece of cake and those of a man who does not. The man who does not have the cake values the pear more highly, as it becomes in effect his first piece of cake. But the man who already has a piece of cake, and who is asked to surrender either it or the lemonade, evidently values the cake less highly, since the cake he is being asked to surrender is his second piece (the pear, with its free convertability into a piece of cake, constitutes his first).

In other words, Mises seems to have missed a question of ordinal marginal utility, and one that compromises his example fairly thoroughly. Mises seems to have implicitly assumed the following extended value hierarchy, even while writing the shorter one above:

1. apple
2. first piece of cake
3. lemonade
4. second piece of cake
5. pear

In the quoted passage above, Mises has only pointed out that the man with this value hierarchy, who possesses apple, pear, cake, and lemonade, will hold the relative trade value of the cake to be less when it becomes in effect his second marginal piece of cake. (Note that for purposes of this example, #4 and #5 could easily be reversed without changing any outcomes.)

This extended hierarchy also determines his decision regarding whether to trade the pear (which, following any loss of a cake, is now more valuable) or the lemonade (in the cakeless state, but with a pear, the lemonade is relatively less valuable). And he behaves accordingly, surrendering that which is of least value to him. He will not trade his first piece of cake — that is, the pear he can exchange in Mises’ initial example of apple, pear, and lemonade — since after all, that piece of cake (the pear, before the trade) has the higher rank than the lemonade.

This is an assumption Mises should not have missed. Why not? Because it is easy to imagine a man who behaves quite differently depending on his placement of the second piece of cake in the value hierarchy: If he possesses the goods (apple, cake, lemonade, pear), this man will part with the lemonade, rather than the cake, and then trade up his pear just afterward. For him, the second marginal piece of cake is also worth more than either the lemonade or the pear. We have:

1. apple
2. first piece of cake
3. second piece of cake
4. lemonade
5. pear

For this man, endowed with one of each of the goods (ie, only one piece of cake), if he is asked to surrender cake or lemonade, he will part with the lemonade, not the cake as in Mises’ example. He will then trade up the pear for his second piece of cake.

And it is surprising that Mises, who used marginalism to attack the naive quantity theory of money, did not see its operation in this much more straightforward example. There is no such thing as valuing “a piece of cake,” and even less so when we’re talking about two of them.

[Apologies if I've bobbled anything here. I've run through it so often in my mind that it's all starting to look the same. And yes, before you ask, I do take forever to decide what I want in a restaurant.]

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