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Suppose there are three kinds of commodities, X Y and Z. We ask an agent about his preferences and receive the following answers:

  • "I prefer Z to Y and Y to X".

  • "For every $n$, I prefer one unit of Z and $n-1$ units of X to $n$ units of Y".

Are these preferences consistent with rational preferences, according to the "rationality" axioms of von-Neumann and Morgenstern?

Initially, I thought that "obviously yes". von-Neumann and Morgenstern only talk about preferences on lotteries, but here, the agent's preferences are only given on sure sequences, so obviously there should be some rational preference relation on lotteries that is consistent with them.

However, one of the rationality axioms is "continuity". It says that there exists a probability $p\in(0,1)$ such that:

$$ p Z + (1-p) X \sim Y $$

The preference relation can be represented by a function $u$ such that: $u(X)=0, u(Y)=p, u(Z)=1$.

But then, if we take $n>1/p$, if the agent receives $n$ units of Y, his utility is more than 1, so the agent should prefer this over receiving one unit of Z and $n-1$ units of X...

What is wrong with these preferences?

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You implicitly assume that the utility of $n$ units of $Y$ equals $n$ times the utility of 1 unit of $Y$, and there is no reason for that. For instance, if $Y$ is a fridge, the gain in utility from having 1 fridge compared to 0 is certainly larger than the gain in utility from having 2 fridges compared to 1. The formula $U("nY")=n*U("Y")$ that you use does not make sense: there are utility functions other than $U$ that represent the same preferences over $\{X,Y,Z\}$, and you will reach different conclusions if you make the same computation with these utilities.

The Von Neumann-Morgenstern theory does not make any assumption on how to value $n$ units of a good. A Von Neumann-Morgenstern representation attributes a utility level to all the consumption goods in the choice set. In your case, the representation would include objects such as $U("Y")$ (the utility of one unit of $Y$), $U("nY")$ (the utility of $n$ units of Y), $U("Z \text{ and }(n-1)X")$ (the utility of $1$ unit of $Z$ and $n-1$ units of $X$), etc. There is nothing that allows you to relate these values with each other. The only behavioral restrictions of the Von Neumann-Morgenstern theory concern preferences over lotteries induced by these goods.

You need other tools to understand whether preferences over a finite set of commodities are rational. These tools are the Weak Axiom of Revealed Preferences and the Generalized Axiom of Revealed Preferences.

On a historical note, the reason why Von Neumann and Morgenstern developed a theory of preferences over lotteries is precisely because they wanted to identify a notion of "how much the good $Y$ is preferred to the good $X$". An obvious possibility was to observe preferences over pairs of goods such as: $n$ units of $Y$ versus $m$ units of $X$. But this is problematic for the reason stated above: nothing relates the utility of one unit of $Y$ to the utility of $n$ units of $Y$. Their trick was to consider preferences over lotteries that deliver $Y$ with a certain probability and $X$ with the complementary probability: the answer to the question "how much are you willing to pay to increase your chance of receiving $Y$ by 1%?" gives a good measure of "how much you prefer $Y$ to $X$". In contrast, the answer to the question "how many units of $X$ are you willing to sacrifice to receive one additional unit of $Y$?" does not make any meaningful sense.

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We need to provide one possible counter example, to show that these are not necessarily efficient.

Consider U(x,y,z)=x+2y+3z

1st condition is obvious, but second condition says 3+n-1>2n, which is incorrect for n>2.

Would these be always inefficient?

Consider U(x,y,z)= x/3 + (2y-y^2/2)+ 4z

Clearly 1 unit of z is preferred to 1 unit of y is preferred to 1 unit of x.

Second condition is also fulfilled.

So it depends on the individual's marginal utiity of each x,y,z.

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