# ECO206Y1 Lecture Notes - Indifference Curve, Utility, Substitute Good

Utility Function: CU = u(x1, x2, x3, … , xn)

Math representation of preference ordering:

a.

X>Y U(x)>U(y)

b.

X～Y U(x)=U(y)

* numbers in the utility function are “ordinal” not “cardinal”*

From the utility function, we only know the order of the preference (whether X is before Y),

not the strength of the preference (whether the utility of X is 3 times greater than Y)

* These utility functions are only unique to a “positive monotonic transformation”.

e.g. V[U(x) = [U(X)]2; 3 + 2U(X); 5 + lnU(X)

-- “Positive monotonic transformation” is like “relabeling of the preferences”;

-- It does not change the order of preferences;

Indifference Set/curve

Set of commodity bundles that satisfies U(x1, x2) = Uo(some level of utility)

x1

x2

-- Indifference curve pass to the right indicate higher utility

(assume “more is better”)

-- Slope of the indifference curve:

1.

Total Differentiate “U(x1, x2) = Uo”

MU of x1 MU of x2

*MU of x1: the utility one gets from consuming one more unit of x1, holding x2 constant.

Implication:

Measure the rate at which the consumer would be willing to substitute one good for another

e.g.

Suppose U(x1, x2) = x10.5 * x20.5

MRS (x2for x1) = U1/U2 = x2/x1

Assume U=100.

X1

X2

MRS

100

100

1

225

44.44

0.1975

400

25

0.0625

Topic 2 - Utility and Indifference Curves

September 21, 2011

10:29 AM

## Document Summary

Utility function: cu = u(x1, x2, x3, , xn) Math representation of preference ordering: x>y u(x)>u(y, x y u(x)=u(y) * numbers in the utility function are ordinal not cardinal * From the utility function, we only know the order of the preference (whether x is before y), not the strength of the preference (whether the utility of x is 3 times greater than y) * these utility functions are only unique to a positive monotonic transformation . e. g. v[u(x) = [u(x)]2; 3 + 2u(x); 5 + lnu(x) - positive monotonic transformation is like relabeling of the preferences ; - it does not change the order of preferences; Set of commodity bundles that satisfies u(x 1, x2) = uo (some level of utility) x2. - indifference curve pass to the right indicate higher utility (assume more is better ) *mu of x1: the utility one gets from consuming one more unit of x1, holding x2 constant.