Chapter 4 Utility Key Concept: Utility , an indicator of a person’s overall well-being

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

2. Chapter 4 Utility • Utility is an numeric measure of a person’s happiness.

3. To the top? Contour map good enough.

5. How do we quantify? • What does it mean by “A gives twice as much utility as B?” • Is one person’s utility the same as another’s?

6. Any independent meaning except that it is something that people maximize?

7. We now use utility as a way to describe preferences. • Preferences are enough.

8. Utility function: a way of assigning number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers than less-preferred bundles, • i.e. (x1, x2) w (y1, y2) iff u(x1, x2) ≥ u (y1, y2)

9. Ordinal utility (序數) vs Cardinal utility (基數) • Ordinal number (序數): tells the position of something in the list, 1st, 2nd, 3 • Cardinal number (基數) : how many of something there are, 1, 2, 3

10. 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)

11. Multiplication by 2 is an example of monotonic transformation. • It is a way to transform one set of numbers into another set such that the order is preserved.

12. Table 4.1

13. Suppose v is a monotonic transformation of u. • Because the order is preserved, if we plot v vs. u, then the slope is strictly positive.

14. Fig. 4.1

15. The utility function representing a preference is not unique as we can always do a monotonic transformation.

16. Cardinal utility (基數): the magnitude of utility matters. • Where are we sensitive? At regions that we often have to make decisions.

17. a remote Australian aboriginal tongue, Guugu Yimithirr, from north Queensland • cardinal directions (geographic languages) vs egocentric coordinates

18. A utility function is a way to label indifference curve. • One natural way to construct a utility function is to drawing a diagonal line and measure how far each indifference curve is from the origin

19. Fig. 4.2

20. Some examples of utility functions • Cobb Douglas, for instance • u(x1, x2) = x1x2

21. Fig. 4.3

22. Generalizing, we might have • u(x1, x2) = x1cx2d • Interpret the exponents.

23. Fig. 4.5

24. Perfect substitutes such as • the 5-dollar coin (x1) vs. the 10-dollar coin (x2) • u(x1, x2) = 5x1 + 10x2

25. u(x1, x2) = 5x1 + 10x2 • a units of x1 can substitute perfectly for b units of x2 • u(x1, x2) = x1/a + x2/b • u(x1, x2) = x1/2 + x2/1 • MRS1, 2 = ∆x2/ ∆x1 = -b/a (intuitively correct)

26. Perfect complements such as • 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}

27. 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)

28. √x1 + x2 =4 0 4 1 3 4 2 9 1 16 0

29. Fig. 4.4

30. The indifference curves are parrelle.

31. Marginal utility • If we give the consumer a little more of good 1, how does his utility change?

32. Marginal utility • 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 • Specify where is it evaluated

33. On an indifference curve: u(x1, x2) = k (a constant) MU1∆x1 + MU2∆x2 = 0 MRS1, 2= ∆x2/ ∆x1= -MU1 / MU2

34. Marginal utility depends on the utility function, but MRS does not. • MRS1, 2= ∆x2/ ∆x1= -MU1 / MU2

35. Marginal utility depends on the utility function, but MRS does 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)] = -MU1 / MU2 = MRS1, 2 (u)

36. 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.

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

38. Chapter 4 Utility • Key Concept: Utility, an indicator of a person’s overall well-being • We only use its ordinal property. • MRS does not depends on the utility function used. • MRS1, 2= ∆x2/ ∆x1= -MU1 / MU2