ECO370Y5 Lecture Notes - Lecture 1: Marshallian Demand Function, Indirect Utility Function, If And Only If

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Published on 30 Apr 2020
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Economics 9010. Name___________________________
Fall 2015
Exam I. There are 4 questions worth 25 points each.
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1. Answer the following question in three parts.
a) Let f:ℝ⟶ℝ be a strictly increasing function and u: X⟶ℝ be a utility function representing
preference relation ≿. Show that the function v: X⟶ℝ defined by v=f(u(x)) is also a utility
function representing .
b) Consider a rational preference relation . Show that if u(x)=u(y) implies x~y and if u(x)>u(y)
implies xy then u( ) is a utility function representing .
c) Evaluate the following statement as true, false, or uncertain and explain your answer. "If u is
a differentiable ordinal utility function, then goods x and y are complements if 2u/xy > 0."
a) We must show x y iff v(x) v(y). Suppose x y. Since u represents ≿, u(x) u(y). Since
f( ) is increasing, f(u(x)) f(u(y)). So v(x) v(y). We have shown v(x) v(y) if x y. We must
now go the other way. Suppose v(x) v(y). Since f( ) is increasing, u(x) u(y). Since u
represents ≿, x y. Thus, we have shown x y if v(x) v(y). This completes the proof.
b) Note u(x) u(y) implies either u(x) = u(y) or u(x) > u(y). Thus, u(x) u(y) implies either x ~
y or x y. So, u(x) u(y) implies x y. We must now go the other way. Since is complete,
either x ~ y, x y, or y x. Note not[y x] implies not[u(y) > u(x)]. Thus, x y implies u(x)
u(y).
c) False. Ordinal utility functions are subject to monotonic transformations, which can change
the sign of second derivatives. For example, consider the case of Cobb-Douglas utility with two
goods: u = x0.5y0.5. The cross derivative is 2u/xy = 0.25x-0.5y-0.5 > 0. Yet we cannot have
complements with two goods, by the fact that the Slutsky matrix is NSD. Note we could also
have written Cobb-Douglas utility as u = 0.5lnx + 0.5lny, in which case 2u/xy = 0. Cross
derivatives are meaningless. For two goods x, y, complements are defined as dxh/dpy < 0.
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