MICROECONOMICS I.
Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,
Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest
Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest
Authors: Gergely K®hegyi, Dániel Horn, Klára Major Supervised by Gergely K®hegyi
June 2010
ELTE Faculty of Social Sciences, Department of Economics
MICROECONOMICS I.
week 6
Preferences, utility, part 2
Gergely, K®hegyiDániel, HornKlára, Major
The course was prepaerd by Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer and David Hirshleifer (2009) Mikroökonómia. Budapest: Osiris Kiadó, ELTECON-books (henceforth HGH), and Gábor Kertesi (ed.) (2004) Mikroökonómia el®adásvázlatok. http://econ.core.hu/ kertesi/kertesimikro/ (henceforth KG).
Special preferences
Mean returnr on assets is a good, but riskiness of returns is a bad. The preference directions are therefore north and west (up and to the left), so the indierence curves slope upward.
In Zone 1 both commodities, X and Y, are goods, so the indierence curves have negative slope. Zone 2 is the region of satiation for Y; in this region the preference directions are north and west (up and to the left), and the indierence curves have positive slope. In this region an individual would have to be paid to eat another piece of cake.
Y is a good, but X is a neuter commodity. The consumer does not care bout having more or less of X. The only preference direction is up, and so the indierence curves are horizontal.
Example.: Arisztid wants to buy light bulbs. He reckons that the brilliance of the normal and the energy saving bulbs are the same, but the energy saving lasts three longer. Arisztid would like to light up his castle with an unlimited amount of bulbs for the longest possible time. Which function represents his preferences?
Example: The ham sandwich of Arisztid is always a ham and a roll. He would not eat either the roll or the ham alone, but he feels better if he eats more ham. Which function represents his preferences? How would it change if he had two hams with each roll?
Example: Tasziló makes "lettuce-uid". He has unlimited amounts of water and sugar. The only scarce commodity is vinegar. For one deciliter of "lettuce-uid" he needs 2 tablespoons of 10% pure vinegar (x) or 1 tablespoon of 20% pure vinegar (y). The more liquid he makes the better he feels. Which function represents his preferences?
Example: Tasziló throws a garden party, and buys plastic garden furniture for that. At each table (x) 6 guests can sit down on chairs (y). He likes to have as many guests as possible, who can sit, but disprefers those who cannot sit down. He would not invite more quests than chairs he has. Which function represents his preferences in terms of chairs and tables?
Notable utility functions
• Cobb-Douglas utility function:
U(x, y) =xayb
• Perfect substitution:
U(x, y) =ax+by
• Perfect complementarity:
U(x, y) = min{ax;by}
Denition 1. Utility functions U ad U0 are ordinally equivalent if U can be transformed positively and monotonically intoU0, thus the following relation holds: U0 =F(U), whereF:R→Rand dFdU >0
Statement 1. IfU andU0 are ordinally equivalent utility functions thenM RS=M RS0. Proof 1. IfU0=F(U), then
M Ux0 = dF dU
∂U
∂x = dF dUM Ux
and
M Uy0 = dF dU
∂U
∂y = dF dUM Uy
. Thus
M RS0=M Ux0
M Uy0 = M Ux
M Uy
=M RS .
E.g.: LetU(x, y) =x3y5be a utility function. Which of these below is a positive monotonic transforma- tion?
• F(U) = 10U, F(U(x, y)) = 10x3y5
• F(U) =−3U, F(U(x, y)) =−3x3y5
• F(U) =U2, F(U(x, y)) =x6y10
• F(U) = 1/U, F(U(x, y)) = x31y5
• F(U) = lnU, F(U(x, y)) = 3 lnx+ 5 lny
• F(U) =−2/U, F(U(x, y)) =−x32y5
Modeling charity
Charitable giving in 1994 selected income levels
Family income Percentage Average Average as (dollar) contribution contribution percentage (dollar) of income
10 00019 000 64 209 1,36
30 00039 999 80 474 1,37
50 00059 999 84 779 1,44
100 000124 999 92 1846 1,71
150 000199 999 96 3546 2,09
500 000999 999 97 27 491 4,15
more then 1 000 000 100 244 586 4,88
Overall 75 960 2,14
Repeating the math
Denition 2. If Ais a base set then an arbitrary subset of A×A is a binary (two variate) relation:
(a, b)∈R⊆A×A⇔aRb.
E.g.:
1. H: Population of the Earth,R:. . . higher than. . . 2. H: R(real numbers),R:≤
3. H: R(real numbers),R:=
4. H: R(real numbers),R:>
5. H: lines on a plane,R:parallel
6. H: lines on a plane,R:perpendicular
7. H: Rn (vectors ofn-dimensional (Euclidean) space,R:=
8. H: Rn (vectors ofn-dimensional (Euclidean) space,R:≤(pl.: def: x≤y, haxi≤yi, i= 1, . . . , n) 9. H: Hungarian women,R:. . . sister of. . .
10. H: Population of the Earth ,R:. . . relative of. . . 11. H: Hungarian women,R:. . . mother of. . . 12. H: This class,R:. . .friend of. . .
Note 1. The notion of relation can be generalized to many variables easily: R⊆A×A×. . .×A Denition 3 (Characteristics of relations). LetA base set andR a relation within that.
1. Completeness: ∀x, y∈A,xRy oryRxor both 2. Reexivity: ∀x∈A xRx.
3. Transitivity: ∀x, y, z∈A, ifxRy andyRz⇒xRz. 4. Symmetry: ∀x, y∈A, if xRy⇒yRx.
Denition 4. Ordering relation is a relation which is complete, reexive and transitive.
Denition 5. Equivalency relation is a relation which is complete, reexive and transitive and symmetric Example.: Let us decide for the given examples which of the characteristics hold.
relation+set complete reexive transitive symmetric 12
34 56 78 109 1112
Mathematically a little more precise
Denition 6. We call the ⊆ H ×H binary relation dened over H consumption set an ordering of preferences, if
• complete
• reexive
• transitive
Assumption 1. Rationality postulate: we assume that consumer tastes (preferences) can be represented for each consumer with a ordering of preferences. Ifx,y∈H are baskets of a consumption set, thenxy means: the consumer likesy at least as much asx basket.
Denition 7. We call ≺⊆H×H a strict preference relation if x≺y⇔xy,yx holds.
Denition 8. We call ∼⊆H×H relation an indierence preference relation if x∼y⇔xy,yx
holds
Statement 2. A preference relation is an ordering relation, indierence preference relation is an equivalence relation.
Denition 9. • A set weakly preferred to x0:
P(x0)≡ {x|x0x}
• A set indierent from x0:
K(x0)≡ {x|x0∼x}
• We call the border of the weakly preferred set (within the commodity set) the indierence curve.
• A set weakly dispreferred to x0:
D(x0)≡ {x|x0x}
Denition 10. Characteristics of preferences:
• Monotonity: Ifxi≤yi,∀i but for aj xj < yj, thenx≺y
• Convexity: If x∼ythen xtx+ (1−t)y, t∈[0,1]
• Strict convexity: If x∼ythenx≺x+ (1−t)y, t∈[0,1]
• (only for gourmands) Continuity: If for all x0∈H,D(x0)andP(x0) closed connected sets.
Denition 11. The utility function U :H →Rrepresents the⊆H×H ordering of preferences if
• U(x)< U(y)⇔x≺y
• U(x) =U(y)⇔x∼y
Statement 3. representation theorem (G. Debreu) If ⊆H×H ordering of preferences is continuous and monotonic then there exists aU :H →Rutility function which represents it.
Statement 4. LetV(z), V :R→Ran arbitrary strictly monotonic increasing real function and assume that U :H →Rutility function represents ⊆ H×H ordring of preferences. ThenV [U(x)]complex function also represents ⊆H×H ordering of preferences.
Statement 5. Let's assume thatU :H→Rutility function represents⊆H×H ordering of preferences, thenU(x)strictly monotonic and increasing, if monotonic.
Origin of preferences, evolutionary approach e.g. Step children
Food consumption at home, 19721985, as related to family structure (mean=4,305 dollars)
Variable Adjustment of mean (dollars)
Child with adoptive mother −204
Child with stepmother −274
Child with foster mother and father −365 e.g. Heritage
Male Female testator testator to spouse (percentage) 69,8 42,4 to children (percentage) 21,7 47,6
Total 91,5 90,0
Dening preferences methodologically
With statistical-econometrical methods: e.g. linear regression:
If utility function is of e.g. Cobb-Douglas type, then it can be linearized by taking its logarithm assuming ordinal preferences.
U(x1, x2, . . . , xn) =β1x1+β2x2+. . .+βnxn Example. (Varian): Utility of commuting:
• TW: total walking time (to bus or car)
• TT: total travel time, in minutes
• C: cost of travel, in dollars
• A/W: cars/workers within household
• R: race of household (0, if black, 1, if white)
• Z: 1, if white collar, 0, if blue collar worker
U =−0,147T W −0,0411T T−2,24C+ 3,78(A/W)−2,91R−2,36Z
