• Chapter 3 Preferences
• Key Concept: We want to characterize
preferences of a consumer by a binary
relationship called at least as good as.
• We then derive the indifference curves
for a given preference.
• The marginal rate of substitution (MRS)
measures the marginal willingness to pay
for x in terms of y.
• Chapter 3 Preferences
• Choose the “best” thing one can “afford.”
• Start with consumption bundles (a complete
list of the goods that is involved in consumer’s
choice problem)
• Mount Everest is higher than Jade Mountain.
We don’t care how high each is.
• A binary relation : w
• (x1, x2) w (y1, y2) is read as (x1, x2) is at least as
good as (y1, y2)
• This binary relation w is complete and
transitive.
• Complete: for any (x1, x2), (y1, y2), either (x1, x2)
w (y1, y2), (y1, y2) w (x1, x2) or both (every two
bundles can be compared)
• Transitive: for any (x1, x2), (y1, y2), (z1, z2), if
(x1, x2) w (y1, y2) and (y1, y2) w (z1, z2), then (x1,
x2) w (z1, z2)
• Complete + Transitive = Rational preference
• One can experimentally test whether these two
axioms are satisfied (kids, societal preferences).
• From this binary relation w, one can derive
two other binary relations s and i.
• (x1, x2) s (y1, y2) if and only if
(x1, x2) w (y1, y2) and
it is not the case that (y1, y2) w (x1, x2).
• Read this as the consumer strictly prefers (x1,
x2) to (y1, y2).
• (x1, x2) i (y1, y2) if and only if
(x1, x2) w (y1, y2) and (y1, y2) w (x1, x2).
• Read this as the consumer is indifferent
between (x1, x2) and (y1, y2).
• Given a binary relation w and for (x1, x2),
can list all the bundles that are at least as good
as it -- the weakly preferred set
Similarly, can list all the bundles for which the
consumer is indifferent to it -- the indifference
curve
• We don’t need to use the idea of utility.
Preferences are enough.
• Some examples:
• Two distinct indifference curves cannot
cross.
• Perfect substitutes: ten dollar coins and
five dollar coins
• Perfect complements: left shoe and right
shoe
• Bads, neutrals
• Satiation
• Discrete goods
• Let us constrain preferences a bit.
• Some useful assumptions
• Monotonicity:
• if x1 ≥ y1, x2 ≥ y2 and (x1, x2) ≠ (y1, y2),
then (x1, x2) s (y1, y2) (the more, the better)
• This implies indifference curves have
negative slopes (examine).
• Convexity:
• if (y1, y2) w (x1, x2) and (z1, z2) w (x1, x2),
then for any weight t between 0 and 1,
(ty1+(1-t)z1, ty2+(1-t)z2) w (x1, x2)
• This means averages are preferred to
extremes (examine).
• We often assume it to get an interior
solution instead of a corner solution. For
non convex preferences, one can ref to a
¼ circle.
• Strict convexity: obviously stronger than
convexity.
• if (y1, y2) w (x1, x2), (z1, z2) w (x1, x2), and
(y1, y2) ≠ (z1, z2), then for any weight t
strictly in between 0 and 1, (ty1+(1-t)z1,
ty2+(1-t)z2) s (x1, x2)
• To describe preferences, a useful way is
to calculate the marginal rate of
substitution (MRS).
• The MRS (one thing for another thing,
evaluated where) measures the rate at
which the consumer is “just” willing to
substitute one thing for the other
• MRS1, 2: for a little of good 1, the amount
of good 2 that the consumer is willing to
give up to stay indifferent about this
change, ∆x2/ ∆x1
• The MRS1, 2 at a point is the slope of the
indifference curve at that point (to stay
put) and measures the marginal
willingness to pay for good 1 in terms of
good 2.
• If good 2 is money, then it is often called
the marginal willingness to pay.
• Useful assumption: diminishing MRS
(when you have more of x1, it can
substitute for x2 less)
• Chapter 3 Preferences
• Key Concept: We want to characterize
preferences of a consumer by a binary
relationship called at least as good as.
• We then derive the indifference curves
for a given preference.
• The marginal rate of substitution (MRS)
measures the marginal willingness to pay
for x in terms of y.