October 21, 2005 12:18masterSheet number 64 Page number 4848Lecture FourProof:In the problem set you will prove that every preference relation thatis monotonic, continuous, and quasi-linear in commodity 1 sat-isfies that for every(x2,. . .,xK)there is some numberv(x2,. . .,xK)so that(v(x2,. . .,xK), 0,. . ., 0)∼(0,x2,. . .,xK).Then, from quasi-linearity in commodity 1, for every bundlex,(x1+v(x2,. . .,xK),0,. . ., 0)∼(x1,x2,. . .,xK), and thus by strong monotonicity in thefirst commodity, the functionx1+v(x2,. . .,xK)represents.Differentiable Preferences (and the Use of Derivatives in EconomicTheory)We often assume in microeconomics that utility functions are dif-ferentiable and thus use standard calculus to analyze the consumer.In this course I (almost) avoid calculus. This is part of a deliberateattempt to steer you away from a “mechanistic” approach to eco-nomic theory.Can we give the differentiability of a utility function an “eco-nomic” interpretation? We introduce a nonconventional definitionof
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