1. Chapter 3 Preferences Intermediate Microeconomics:
A Tool-Building Approach
Routledge, UK
© 2016 Samiran Banerjee

2. Preferences as rankings
• How would you rank an apple and a banana?
(1) An apple is better than a banana
(2) A banana is better than an apple
(3) An apple is as good as a banana

3. Rankings as binary relations
• Set of objects: X
• Binary relation: R defined over any two elements of X
• Ex. 1: X = students in class
R = is taller than
R relates any pair from X unless they are the same height
• Ex. 2: X = students in class
R = is a sibling of
R does not relate any pair from X unless they are siblings
“is related to”

4. Preferences as binary relations
For a decision-maker, we have the set of objects to choose from,
X = {A, B, C, D, …}
and preferences (binary relations) over these objects
= is at least as good as
Then,
A B means [A B & not B A]
A B means [A B & B A]
E.g., fruits, students, candidates in an election
Primitive (given)
Strict preference
(derived from primitive)
“is preferred to”
Indifference
(derived from primitive)
“is indifferent to”

5. Properties of binary relations
• Ex. 3: X = students in class
R = is the same age (born on the same day)
or older than (born on an earlier day)
P1 • Pick any single item, x, from X. Is x R x?
Yes: R is reflexive
P2 • Pick any two different items, x and y, from X. Is either x R y, or y R x, or both?
Yes: R is total
P3 • Pick any three items, x, y, and z, satisfying x R y and
y R z. Is x R z?
Yes: R is transitive

6. Properties of preferences
Ex. 4: For consumers, the ranking is of commodity bundles
X = the commodity space,
= is at least as good as
We say preferences are regular if is reflexive, total and transitive, i.e., satisfies P1, P2, and P3.
Two more properties
P4 • Monotonicity
P5 • Convexity (to be covered later)

7. Strict monotonicity • Captures the idea that “goods are good”
• More of at least one good is better
Strictly better
than A = (4, 3)
B A
C A
D A
Strictly worse
than A = (4, 3)

8. (Weak) Monotonicity • Captures the idea that “more cannot be worse”
• More of at least one good is at least as good
Weakly better
than A = (4, 3)
B A
C A
D A
Weakly worse
than A = (4, 3)

9. Strict monotonicity and indifference
• Suppose preferences are strictly monotonic
• Pick a bundle, A
• Suppose bundles B or C are indifferent to A. Where might these bundles lie?
Indifference curve

10. Utility Representation
• Suppose there is a utility function u that associates a real number with every object in X
• Then u(P) is the utility level of object P
Given preferences on X, a utility function u represents the preferences means
if P Q, then u(P) ≥ u(Q), and vice versa
• The “vice versa” part means that given a utility function, it is possible to figure out the underlying preferences
• A utility function preserves the same ranking over the objects as the underlying preferences

11. Deriving indifference curves
• Suppose there is a utility function on :
u(x1, x2) = x1 + x2
• Fix the utility level at, say, = 12, so 12 = x1 + x2
• Plot all combinations of x1 and x2 that yield a utility of 12
Indifference curve for utility level of 16
Indifference curve for utility level of 12

12. Slope of an indifference curve
• Suppose u(x1, x2) is a utility function
• Fix the utility level at = u(x1, x2)
• Take the total differential:
d = dx dx2
• Since d = 0, we obtain
Marginal utility of good 2, MU2 ∂u
∂x1 . ∂u
∂x2 .
Marginal utility of good 1, MU1
Marginal rate of substitution
(MRS)
dx2
dx1
∂u/∂x1
∂u/∂x2
MU1
MU2 – = =
Negative slope of
indifference curve

13. Types of preferences
•There are many preferences satisfying P1–P5 • Four basic ones – Linear: u(x1, x2) = ax1 + bx2 – Leontief: u(x1, x2) = min {ax1, bx2} – Quasilinear: u(x1, x2) = f(x1) + x2, or u(x1, x2) = x1 + f(x2) – Cobb-Douglas: u(x1, x2) = A(x1)a(x2)b
*The function f is generally increasing and strictly concave, i.e., f ’ > 0 and f ” < 0

14. Linear preferences MRS = a/b MRS = a/b a > 0 and b > 0
“Goods”
(perfect substitutes)
“Bads”
(perfect substitutes)
“Neutral”
a > 0 and b = 0
a = 0 and b > 0
MRS = 0
MRS = ∞
“Neutral”

15. Leontief preferences • Suppose u(x1, x2) = min {2x1, 3x2}
• Draw the indifference curve for = 6
• Set 6 = min {2x1, 3x2}
• Solve 6 = 2x1 and plot
• Solve: 6 = 3x2 and plot
• Erase the lower envelope

16. Leontief preferences • Suppose u(x1, x2) = min {ax1, bx2}
• Draw the ray through the origin with slope a/b
• All indifference curves will have kinks along this ray
• The goods are perfect complements
x1 and x2 are consumed in
fixed proportions
• There is no MRS!

17. Quasilinear preferences
• Suppose u(x1, x2) = √x1 + x2
• Draw the indifference curves for utility levels 4, 8, 12, 20
• Indifference curves are ‘vertically parallel’
• The MRS is the same along any vertical line

18. Cobb-Douglas preferences
• Suppose u(x1, x2) = x1x2, i.e., A = 1, a = 1, and b = 1
• Draw the indifference curves for utility levels 0, 25, 64, 121
• The indifference curve for = 0 is L-shaped; there’s no MRS
• The MRS is the decreasing along other indifference curves

19. Utility is ordinal • Suppose u = x1 + x2 for consumer Teri
• Suppose v = 2x1 + 2x2 for consumer Toni, Teri’s twin
• How do their preferences, i.e., ranking of bundles, differ?
• They don’t!
• They both like A over B
• They are both indifferent between
A and C
• Teri and Toni’s MRS = 1
• Their indifference curves just have
different values
• They have the same preferences

20. Positive monotonic transformations
• If v = f(u) where f ’ > 0 and f ” < 0 for positive u, then v is a positive monotonic transformation (PMT) of u
• Ex. 1: v = ur, where r > 0
• Ex. 2: v = ln u
• Ex. 3: v = a + bu, where a > 0 and b > 0
• Ex. 4: v = eu
• If v is a PMT of u, then they represent the same preferences
— the set of indifference curves under u are the same as the set of indifference curves under v
— at any bundle, the MRS is the same

21. Convex sets • Pick any two points in the set S
• Join them with a straight line
• If the line lies within S, then S is a convex set
A strictly convex set
Not a convex set
A weakly convex set

22. Preferences are convex if the set is a convex set
Convex preferences
P5 • Pick any commodity bundle A in
• Ex.1: Leontief preferences
• Draw the indifference curve that passes through A
• Find all bundles that are at least as good as A, the weakly-better-than A set,
Preferences are convex if the set is a convex set

23. Strictly convex preferences
• Pick any commodity bundle A in
• Ex.2: Cobb-Douglas preferences
• Draw the indifference curve that passes through A
• Find all bundles that are at least as good as A, the weakly-better-than A set,
Preferences are strictly convex if the set is a strictly convex set

24. Meaning of convex preferences
1. Consumers prefer averages to extremes
• Pick A on an indifference curve with little x1 and lots of x2
• Pick B on an indifference curve with little x2 and lots of x1
• Join them
• Any point on this line like C is preferred to A or B

25. Meaning of convex preferences
2. Consumers have diminishing MRS
As you go from left to right along an indifference curve…
from A to B to C to D to E…
the consumer gives up more and more units of x2:
a > b > c > d