Axioms of Rational Choice

Completeness If A and B are any two situations, an individual

can always specify exactly one of these possibilities: A is preferred to B B is preferred to A A and B are equally attractive

Axioms of Rational Choice

Transitivity If A is preferred to B, and B is preferred to C,

then A is preferred to C Assumes that the individual’s choices are

internally consistent

Axioms of Rational Choice

Continuity If A is preferred to B, then situations “close to” A

must also be preferred to B Used to analyze individuals’ responses to

relatively small changes in income and prices

Utility Given these assumptions, it is possible to

show that people are able to rank in order all possible situations from least desirable to most

Economists call this ranking utility If A is preferred to B, then the utility assigned to

A exceeds the utility assigned to B

U(A) > U(B)

Utility Utility rankings are ordinal in nature

They record the relative desirability of commodity bundles

Because utility measures are nonunique, it makes no sense to consider how much more utility is gained from A than from B

It is also impossible to compare utilities between people

Utility Utility is affected by the consumption of

physical commodities, psychological attitudes, peer group pressures, personal experiences, and the general cultural environment

Economists generally devote attention to quantifiable options while holding constant the other things that affect utility ceteris paribus assumption

Utility Assume that an individual must choose

among consumption goods X1, X2,…, Xn

The individual’s rankings can be shown by a utility function of the form:

utility = U(X1, X2,…, Xn) Keep in mind that everything is being held

constant except X1, X2,…, Xn

Economic Goods In the utility function, the X’s are assumed to

be “goods” more is preferred to less

Quantity of X

Quantity of Y

X*

Y*

Preferred to X*, Y*

Worsethan

X*, Y*

?

?

Indifference Curves

An indifference curve shows a set of consumption bundles among which the individual is indifferent

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U1

Combinations (X1, Y1) and (X2, Y2)provide the same level of utility

Marginal Rate of Substitution The negative of the slope of the indifference

curve at any point is called the marginal rate of substitution (MRS)

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U1

1UUdX

dYMRS

Marginal Rate of Substitution MRS changes as X and Y change

reflects the individual’s willingness to trade Y for X

Quantity of X

Quantity of Y

X1

Y1

Y2

X2

U1

At (X1, Y1), the indifference curve is steeper.The person would be willing to give up more

Y to gain additional units of X

At (X2, Y2), the indifference curveis flatter. The person would bewilling to give up less Y to gainadditional units of X

Indifference Curve Map Each point must have an indifference curve through it

Quantity of X

Quantity of Y

U1

U2

U3 U1 < U2 < U3

Increasing utility

Transitivity Can two of an individual’s indifference curves intersect?

Quantity of X

Quantity of Y

U1

U2

A

BC

The individual is indifferent between A and C.The individual is indifferent between B and C.

Transitivity suggests that the individualshould be indifferent between A and B

But B is preferred to Abecause B contains more

X and Y than A

Convexity A set of points is convex if any two points can be joined by a straight

line that is contained completely within the set

Quantity of X

Quantity of Y

U1

The assumption of a diminishing MRS isequivalent to the assumption that allcombinations of X and Y which are preferred to X* and Y* form a convex set

X*

Y*

Convexity If the indifference curve is convex, then the combination (X1 + X2)/2, (Y1 + Y2)/2

will be preferred to either (X1,Y1) or (X2,Y2)

Quantity of X

Quantity of Y

U1

X2

Y1

Y2

X1

This implies that “well-balanced” bundles are preferredto bundles that are heavily weighted toward onecommodity

(X1 + X2)/2

(Y1 + Y2)/2

Utility and the MRS

Suppose an individual’s preferences for hamburgers (Y) and soft drinks (X) can be represented by

YX 10 utility Solving for Y, we get

Y = 100/X

• Solving for MRS = -dY/dX:MRS = -dY/dX = 100/X2

Utility and the MRS

MRS = -dY/dX = 100/X2

Note that as X rises, MRS falls When X = 5, MRS = 4 When X = 20, MRS = 0.25

Marginal Utility

Suppose that an individual has a utility function of the form

utility = U(X1, X2,…, Xn) We can define the marginal utility of good

X1 by

marginal utility of X1 = MUX1 = U/X1 The marginal utility is the extra utility

obtained from slightly more X1 (all else constant)

Marginal Utility

The total differential of U is

n

n

dXX

UdX

X

UdX

X

UdU

...2

2

1

1

nXXX dXMUdXMUdXMUdUn

...21 21

The extra utility obtainable from slightly more X1, X2,…, Xn is the sum of the additional utility provided by each of these increments

Deriving the MRS

Suppose we change X and Y but keep utility constant (dU = 0)

dU = 0 = MUXdX + MUYdY Rearranging, we get:

YU

XU

MU

MU

dX

dY

Y

X

/

/

constantU

MRS is the ratio of the marginal utility of X to the marginal utility of Y

Diminishing Marginal Utility and the MRS

Intuitively, it seems that the assumption of decreasing marginal utility is related to the concept of a diminishing MRS Diminishing MRS requires that the utility function

be quasi-concave This is independent of how utility is measured

Diminishing marginal utility depends on how utility is measured

Thus, these two concepts are different

Marginal Utility and the MRS

Again, we will use the utility function5050 .. utility YXYX

The marginal utility of a soft drink is

marginal utility = MUX = U/X = 0.5X-0.5Y0.5

The marginal utility of a hamburger is

marginal utility = MUY = U/Y = 0.5X0.5Y-0.5

X

Y

YX

YX

MU

MU

dX

dYMRS

Y

X

5050

5050

5

5..

..

constantU .

.

Examples of Utility Functions Cobb-Douglas Utility

utility = U(X,Y) = XY

where and are positive constants The relative sizes of and indicate the relative

importance of the goods

Examples of Utility Functions Perfect Substitutes

utility = U(X,Y) = X + Y

Quantity of X

Quantity of Y

U1

U2

U3

The indifference curves will be linear.The MRS will be constant along the indifference curve.

Examples of Utility Functions Perfect Complements

utility = U(X,Y) = min (X, Y)

Quantity of X

Quantity of YThe indifference curves will be L-shaped. Only by choosing more of the two goods together can utility be increased.

U1

U2

U3