MICROECONOMICS I.
week 6
PREFERENCES, UTILITY, PART 2 Authors:
Gergely K®hegyi, Dániel Horn, Klára Major Supervised by
Gergely K®hegyi
June 2010
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K®hegyi-Horn-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).
Mean return r on assets is a good, but riskiness of return s is a bad. The preference directions are therefore north and west (up and to the left), so the indierence curves slope upward.
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K®hegyi-Horn-Major
Special preferences (cont.)
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.
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K®hegyi-Horn-Major
Special preferences (cont.)
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.: 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?
x: normal bulb, y: energy saving bulb U(x,y) =x+3y
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?
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K®hegyi-Horn-Major
Special preferences (cont.)
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. x: roll (piece), y: ham (slice)
U(x,y) =min{x;y}
How would it change if he had two hams with each roll?
U(x,y) =min{2x;y}
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?
y =x+2y
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?
y =min{6x;y}
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K®hegyi-Horn-Major
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
Utility functions U ad U0 are ordinally equivalent if U can be transformed positively and monotonically into U0, thus the following relation holds: U0=F(U), where F :R→Rand
dFdU >0
If U and U are ordinally equivalent utility functions then MRS =MRS0.
Proof
IfU0 =F(U), then
MUx0 = dF dU
∂U
∂x = dF
dUMUx and
MUy0 = dF dU
∂U
∂y = dF
dUMUy . Thus
MRS0 =MUx0
MUy0 = MUx
MUy =MRS .
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K®hegyi-Horn-Major
Notable utility functions (cont.)
E.g.: Let U(x,y) =x3y5be a utility function. Which of these below is a positive monotonic transformation?
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) =ln U,F(U(x,y)) =3 ln x+5 ln y F(U) =−2/U,F(U(x,y)) =−x32y5
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K®hegyi-Horn-Major
Modeling charity (cont.)
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
Denition
If A is a base set then an arbitrary subset of A×A is a binary (two variate) relation:
(a,b)∈R⊆A×A⇔aRb.
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K®hegyi-Horn-Major
Repeating the math (cont.)
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 of n-dimensional (Euclidean) space, R :=
8 H: Rn (vectors of n-dimensional (Euclidean) space, R :≤(pl.:
def: x≤y, ha xi ≤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 . . .
The notion of relation can be generalized to many variables easily:
R⊆A×A×. . .×A
Denition (Characteristics of relations)
Let A base set and R a relation within that.
1 Completeness: ∀x,y∈A, xRy or yRx or both
2 Reexivity: ∀x∈A xRx.
3 Transitivity: ∀x,y,z ∈A, if xRy and yRz⇒xRz.
4 Symmetry: ∀x,y ∈A, if xRy⇒yRx.
Denition
Ordering relation is a relation which is complete, reexive and transitive.
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K®hegyi-Horn-Major
Repeating the math (cont.)
Denition
Equivalency relation is a relation which is complete, reexive and transitive and symmetric
characteristics hold.
relation+set complete reexive transitive symmetric 12
34 56 78 109 1112
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K®hegyi-Horn-Major
Repeating the math (cont.)
relation+set complete reexive transitive symmetric
1 No No X No
2 X X X No
3 No X X X
4 No No X No
5 No X X X
6 No No No X
7 No X X X
8 No X X No
9 No No ? X
10 No ? No X
11 No No No No
12 No ? No ?
Denition
We call the⊆H×H binary relation dened over H consumption set an ordering of preferences, if
complete reexive transitive
Assumption
Rationality postulate: we assume that consumer tastes (preferences) can be represented for each consumer with a ordering of preferences. If x,y∈H are baskets of a consumption set, then xy means: the consumer likes y at least as much as x basket.
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K®hegyi-Horn-Major
Mathematically a little more precise (cont.)
Denition
We call ≺⊆H×H a strict preference relation if x≺y⇔xy,yx holds.
Denition
We call ∼⊆H×H relation an indierence preference relation if x∼y⇔xy,yx
holds
Statement
A preference relation is an ordering relation, indierence preference relation is an equivalence relation.
Denition
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}
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K®hegyi-Horn-Major
Mathematically a little more precise (cont.)
Denition
Characteristics of preferences:
Monotonity: If xi ≤yi,∀i but for a j xj <yj, then x≺y Convexity: If x∼y then xtx+ (1−t)y,t∈[0,1] Strict convexity: If x∼y then x≺x+ (1−t)y,t∈[0,1] (only for gourmands) Continuity: If for all x0∈H, D(x0)and P(x0)closed connected sets.
Denition
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
preferences is continuous and monotonic then there exists a U :H→Rutility function which represents it.
Statement
Let V(z),V :R→Ran arbitrary strictly monotonic increasing real function and assume that U:H→Rutility function represents⊆H×H ordring of preferences. Then V[U(x)]
complex function also represents ⊆H×H ordering of preferences.
Statement
Let's assume that U:H→Rutility function represents
⊆H×H ordering of preferences, then U(x)strictly monotonic and increasing, ifmonotonic.
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K®hegyi-Horn-Major
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
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K®hegyi-Horn-Major
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 =−0,147TW−0,0411TT−2,24C+3,78(A/W)−2,91R−2,36Z
