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

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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 x≻y 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/∂x∂y > 0."

Answer

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/∂x∂y = 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/∂x∂y = 0. Cross

derivatives are meaningless. For two goods x, y, complements are defined as dxh/dpy < 0.