ECO370Y5 Lecture Notes - Lecture 1: Marshallian Demand Function, Indirect Utility Function, If And Only If
41 views6 pages
Economics 9010. Name___________________________
Exam I. There are 4 questions worth 25 points each.
GSU Honor Code:
As members of the academic community, students are expected to recognize and uphold
standards of intellectual and academic integrity. The University assumes as a basic and minimum
standard of conduct in academic matters that students be honest and that they submit for credit
only the products of their own efforts. Both the ideals of scholarship and the need for fairness
require that all dishonest work be rejected as a basis for academic credit. They also require that
students refrain from any and all forms of dishonorable or unethical conduct related to their
Cheating on Examinations. Cheating on examinations involves giving or receiving unauthorized
help before, during, or after an examination. Examples of unauthorized help include the use of
notes, texts, or “crib sheets” during an examination (unless specifically approved by the faculty
member), or sharing information with another student during an examination (unless specifically
approved by the faculty member). Other examples include intentionally allowing another student
to view one’s own examination and collaboration before or after an examination if such
collaboration is specifically forbidden by the faculty member.
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."
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) ≥
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.