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QUANTITATIVE METHODS 5
1. Some Basics About Numbers and Quantities
Readings: Marinacci’s “An Axiomatic Approach to Complete Patience and Time Invari-
ance,” Journal of Economic Theory 83, 105-144 (1998). Mas-Colell, Whinston, and Green
on support functions and the supporting hyperplane theorem. §1.6 below is for you to read
and work on, either by yourself or in a study group.
1.1. Lengths and measurements. N and Q from elementary school. As models of mea-
surements of quantities, we’re done.
1.2. Why we want more. Irrationality of easily described lengths, clt and integration.
Sequences in Q, convergence implies settling down, but not the reverse. Subsequences.
Cauchy sequences and R as the completion of Q.
Implications of completeness: decreasing and increasing bounded sequences have limits,
equivalently, every bounded set has a sup and an inf. The idea of completion also shows up
in the major limit theorem in statistics (i.e. the CLT).
1.3. Valuing sequences of rewards. This section is based on classic analyses as well as
the more recent Marinacci’s “An Axiomatic Approach to Complete Patience and Time In-
variance,” Journal of Economic Theory 83, 105-144 (1998). Patience about finite sequences,
(r1, r2,... ,rt), of rewards seems to be about being indifferent between all time permutations
of the sequence. In the dynamic programming models used in game theory and macro, one
often achieves infinite sequences of rewards. These may not be entirely believable,