Transcript Chapter 04

• Chapter 4 Utility
• Utility: conceptually an indicator of a
person’s overall well-being
• How do we quantify? Can we do
interpersonal comparisons? What does it
mean by “A gives twice as much utility
as B?” Any independent meaning except
that it is something that people maximize?
• Preferences are enough.
• Utility function: a way of assigning
number to every possible consumption
bundle such that more-preferred bundles
get assigned larger number than lesspreferred bundles, i.e. (x1, x2) w (y1, y2) iff
u(x1, x2) ≥ u (y1, y2)
• Utility is a useful way to describe
preferences.
• Ordinal utility (序數): ordering important,
the size of the difference unimportant
• v(x1, x2) = 2u(x1, x2), u and v are equally
good because v(x1, x2) ≥ v(y1, y2) iff u(x1,
x2) ≥ u (y1, y2)
• Monotonic transformation: a way to
transform one set of numbers into another
set such that the order is preserved
• Suppose we plot v vs. u, then the slope is
strictly positive.
• The utility function representing a
preference is not unique as we can always
do a monotonic transformation.
• Cardinal utility (基數): magnitude of
utility matters
• a remote Australian aboriginal tongue, Guugu
Yimithirr, from north Queensland, cardinal directions
(geographic languages) vs egocentric coordinates
• One natural way to construct a utility
function: drawing a diagonal line and
measuring how far each indifference
curve is from the origin
• Some examples of utility functions
• Cobb Douglas, for instance u(x1, x2) =
x1x2 (討論次方) (take log)
• Perfect substitutes: 5-dollar coin (x1) and
10-dollar coin (x2)
u(x1, x2) = 5x1 + 10x2, a units of x1 can
substitute perfectly for b units of x2, u(x1,
x2) = x1/a + x2/b (算有幾組) MRS1, 2 =
∆x2/ ∆x1 = -b/a (intuitively correct)
• Perfect complements: one cup of coffee
(x1) goes with two cubes of sugar (x2),
u(x1, x2) = min{x1 , x2/2}, a units of x1 go
with b units of x2, u(x1, x2) = min{x1/a ,
x2/b} (算成幾套)
• Quasilinear preferences (準線性): u(x1,
x2) = v(x1) + x2, v(x1) = √x1, v(x1) = ln x1
• Marginal utility (evaluated where): the
rate of the utility change with respect to
the change of the consumption of one
good
• MU1 = ∆u/ ∆x1 = (u(x1+ ∆x1, x2) - u(x1,
x2))/ ∆x1
• u(x1, x2) = k
MU1 ∆x1 + MU2 ∆x2 = 0
MRS1, 2= ∆x2/ ∆x1= -MU1 / MU2
• Marginal utility is cardinal, but MRS is
not: v(x1, x2) = f(u(x1, x2)), MRS1, 2 (v) = MV1 / MV2 = -(∆ v/∆x1)/ (∆v/∆x2) = -[(∆
f/∆u)(∆u/ ∆x1)]/ [(∆f/∆u)(∆u/ ∆x2)] =
MRS1, 2 (u)
• Additional materials
• Lexicographic preferences: (x1, x2) w (y1,
y2) if and only if (x1 > y1) or (x1 = y1 and
x2 ≥ y2)
• This is similar to the way the dictionary is
ordered.
• Complete? Reflexive? Transitive?
Monotonic? Indifference curves? Convex?
Strictly convex?
• Cannot be represented by any utility
function