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 less- preferred bundles, i.e. (x 1, x 2 ) w (y 1, y 2 ) iff u(x 1, x 2 ) ≥ u (y 1, y 2 ) Utility is a useful way to describe preferences.

Ordinal utility ( 序數 ): ordering important, the size of the difference unimportant v(x 1, x 2 ) = 2u(x 1, x 2 ), u and v are equally good because v(x 1, x 2 ) ≥ v(y 1, y 2 ) iff u(x 1, x 2 ) ≥ u (y 1, y 2 ) 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.

Table 4.1

Fig. 4.1

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

Fig. 4.2

Some examples of utility functions Cobb Douglas, for instance u(x 1, x 2 ) = x 1 x 2 ( 討論次方 ) (take log) Perfect substitutes: 5-dollar coin (x 1 ) and 10-dollar coin (x 2 ) u(x 1, x 2 ) = 5x x 2, a units of x 1 can substitute perfectly for b units of x 2, u(x 1, x 2 ) = x 1 /a + x 2 /b ( 算有幾組 ) MRS 1, 2 = ∆x 2 / ∆x 1 = -b/a (intuitively correct)

Fig. 4.3

Fig. 4.5

Perfect complements: one cup of coffee (x 1 ) goes with two cubes of sugar (x 2 ), u(x 1, x 2 ) = min{x 1, x 2 /2}, a units of x 1 go with b units of x 2, u(x 1, x 2 ) = min{x 1 /a, x 2 /b} ( 算成幾套 ) Quasilinear preferences ( 準線性 ): u(x 1, x 2 ) = v(x 1 ) + x 2, v(x 1 ) = √x 1, v(x 1 ) = ln x 1

Fig. 4.4

Marginal utility (evaluated where): the rate of the utility change with respect to the change of the consumption of one good MU 1 = ∆u/ ∆x 1 = (u(x 1 + ∆x 1, x 2 ) - u(x 1, x 2 ))/ ∆x 1

u(x 1, x 2 ) = k MU 1 ∆x 1 + MU 2 ∆x 2 = 0 MRS 1, 2 = ∆x 2 / ∆x 1 = -MU 1 / MU 2 Marginal utility is cardinal, but MRS is not: v(x 1, x 2 ) = f(u(x 1, x 2 )), MRS 1, 2 (v) = - MV 1 / MV 2 = -(∆ v/∆x 1 )/ (∆v/∆x 2 ) = -[(∆ f/∆u)(∆u/ ∆x 1 )]/ [(∆f/∆u)(∆u/ ∆x 2 )] = MRS 1, 2 (u)

Additional materials Lexicographic preferences: (x 1, x 2 ) w (y 1, y 2 ) if and only if (x 1 > y 1 ) or (x 1 = y 1 and x 2 ≥ y 2 ) 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