1. Chapter 4 Utility
Key Concept: Utility, an indicator of a person’s overall well-being
We only use its ordinal property in most cases.
Marginal utility depends on the utility function but MRS is not.
MRS1, 2= ∆x2/ ∆x1= -MU1 / MU2

2. 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?

3. 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 less-preferred bundles, i.e. (x1, x2) w (y1, y2) iff u(x1, x2) ≥ u (y1, y2)
Binary preference vs. u(x1, x2)
Utility is a useful way to describe preferences.

4. Ordinal utility (序數) vs Cardinal utility (基數)
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)

5. 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.

6. Table 4.1

7. Fig. 4.1

8. Cardinal utility (基數): magnitude of utility matters
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

9. One natural way to construct a utility function: drawing a diagonal line and measuring how far each indifference curve is from the origin

10. Fig. 4.2

11. Some examples of utility functions
Cobb Douglas, for instance u(x1, x2) = x1x2 (討論次方) (take log)
ln(x1x2)=ln(x1)+ln(x2)
ln(x1ax2b)=aln(x1)+bln(x2)

12. Fig. 4.3

13. Fig. 4.5

14. 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)

15. 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)

16. Quasilinear preferences (準線性): u(x1, x2) = v(x1) + x2 where v(∙) is concave or v’(∙)>0 and v’’(∙)<0
v(x1) = √x1, v(x1) = ln(x1)
√x1 + x2 =4
0 4
1 3
4 2
9 1
16 0

17. Fig. 4.4

18. 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

19. On an indifference curve:
u(x1, x2) = k (a constant)
MU1 ∆x1 + MU2 ∆x2 = 0
MRS1, 2= ∆x2/ ∆x1= -MU1 / MU2
Marginal utility depends on the utility function, 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)

20. 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? Transitive? Monotonic? Indifference curves? Convex? Strictly convex?
Cannot be represented by any utility function

21. Chapter 4 Utility
Key Concept: Utility, an indicator of a person’s overall well-being
We only use its ordinal property in most cases.
Marginal utility depends on the utility function but MRS is not.
MRS1, 2= ∆x2/ ∆x1= -MU1 / MU2