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1. Bundles of goods are represented as vectors, where each position in thevector represents a specific good, and the scalar at that positiondenotes the number of units of that good. For convenience, economistsoften operate in two-goods worlds, where one good—thenumeraire—stands for all the other goods (money is themost common numeraire). The countability of such bundles allowsformulating two important preference properties.

Preferences aremonotone if larger bundles are always preferred to smallerbundles of the same goods, i.e. if \(A=\langle a_1,\ldots,a_n\rangle\)is preference ranked and there is a bundle \(B=\langle b_1,\ldots,b_n\rangle\) such that for at least one \(i\): \(a_i\gt b_i\) and for allother \(j\): \(a_j\ge b_j\), then \(A\succ B\).

Preferences arehomothetic if indifference is retained underproportional expansions, i.e. if\(A\sim B,\) then \(aA\sim aB\) for any\(a\ge 0 (aA\) is the vector obtained bymultiplying each element of \(A\) by \(a)\). Theseproperties allow inferring a preference relation over manyalternatives on the basis of only a few observations.

2.A lexicographic preference relation gives absolute priority to onegood over another. In the case of two-goods bundles, \(A\succ B\) if\(a_1\gt b_1\), or \(a_1 =b_1\) and \(a_2 \gt b_2\). Good 1 thencannot be traded off by any amount of good 2. Debreu shows that such apreference relation cannot be represented by a standard utilityfunction.

3. This and thecounterexample below have been dismissed on the ground that properties\(\alpha\) and \(\beta\) were intended only for changes in the set ofalternatives which do not change the agent’s information of thealternatives’ desirability. However, it is not clear how much of adistinction can be made between cases for which the properties hold andthose for which they do not, without recurring to criteria of thesame type as \(\alpha\) or \(\beta\). Such a criticism thus runs the dangerof turning into an immunising defence of the properties.

4.The expansion property \(\gamma\) requires that an element \(X\) that ischosen from every set in a particular class must also be chosen fromtheir union.

As an example of property \(\gamma\), if one of the university’s bestteachers in non-classical logic is also one of its best teachers inclassical logic, then she is one of its best logic teachers.