4.1 Ordinal versus cardinal utility

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Micro I. Lesson 4. Utility

In the previous lesson we have developed a method to rank consistently all bundles in the (x,y) space and we have introduced a concept –the indifference curve- to help us in this analysis. Now we introduce a related concept to rank bundles –the utility function- that will be useful to solve the equilibrium of the consumer in terms of calculus.

4.1 Ordinal versus cardinal utility

In the past, utility was conceived as a quantitative measure of a person’s welfare out of consuming goods. Now it is recognised that utility cannot be quantified because of the impossibility of interpersonal comparisons. So we do now as we did in Lesson 3, we only rank bundles.

Utility function: A way of assigning a number to every possible consumption bundle such that more- preferred bundles get assigned larger numbers than less-preferred bundles.

( , )x y′ ′ f ( ,x y′′ ′′) if and only if ( ,U x y′ ′) fU x y( ,′′ ′′) The only property about the numbers the utility function generates which is important is how it orders the bundles; how it ranks them. The size of the difference between the numbers assigned to each

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bundle does not matter. Thus we talk about ordinal utility.

Since only the ranking matters, there can be no unique way to assign utility to bundles. If U(x,y) represents one way of ranking goods, 2U(x,y) is equally acceptable: it ranks bundles in the same manner. Multiplying by 2 is an example of a monotonic transformation.

Example:

U=xy

Ranking: B>A>C

U=2xy

Ranking: B>A>C

x y U=2xy

A 1 4 8

B 2 3 12

C 1 2 4

Bundle

x y U=xy

A 1 4 4

B 2 3 6

C 1 2 2

Bundle

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A monotonic transformation is a way of transforming one set of numbers into another set of numbers in a way that the order of the numbers is preserved.

If the original utility function is U(x,y), we represent a monotonic transformation by ^{f U x y}

[

^{( , )}

]

^{. The}

property the function f[.] has to have is that

1 2 1

If (U >U ⇒ f U ) > f U( ) Examples of monotonic transformations:

f(U) = 2U f(U) = 3U f(U) = U+2 f(U) = U+10 f(U) = 5+3U

f(U) = U³ (What about f(U) = U²?)

See that to preserve the order, f(U) must be a strictly increasing function of U.

Utility functions have indifference curves too; they are the level curves in the space (x,y) of the three dimensional function U=f(x,y). The indifference curves of a monotonic transformation of a utility function are the same as the indifference curves of the original utility function, only that the numbers attached to each indifference curve are different.

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4.2 From utility functions to indifference curves If you are given a utility function U(x,y), it is easy to derive a given indifference curve from it: simply plot all points (x,y) such that U(x,y) equals a constant.

Examples: U(x,y)=xy k=xy y=k/x

2 2

( , )

U x y = x y

Notice that since xy cannot be negative (we are in the positive quadrant), x y² ² = (xy)² preserves the same order. So this utility function is a monotonic transformation of the previous function. The formula for the indifference curve is:

k=1 k=2 k=3

x

y Rectangular hyperbola

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2 2 1 2

1 2 / x y

xy

y x

φ φ

φ

=

This is the same indifference curve map as before, only that the levels of the indifference curves are the squared of the previous levels

Perfect substitutes (blue pencils, red pencils) ( , )

U x y x y k x y

y k x

= +

= −

φ=1 φ=4 φ=9

x

y Rectangular hyperbola

Slope -1

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Perfect substitutes but at differents proportions: for example, suppose for the consumer x is twice as valuable as y.

( , ) 2 2

2

U x y x y

k x y

y k x

= +

= −

In general,

( , )

U x y ax by k ax by

k a

y x

b b

= +

= −

This is a utility function in which the consumer values x as much as a/b units of y.

Slope -2

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Perfect complements (left shoe, right shoe)

{ }

( , ) min , U x y = x y

If I have 2x (two right shoes) and 1y (one left shoe) it is like if I had only one pair of shoes: I get the same utility as with 1x and 1y.

The proportion need not be 1 to 1. Say, a consumer uses always 1x (cup of cofee) with 2y (two sugars), then

( , ) min ,1

u x y = x 2 y

 

Slope 1

1 2

Slope 2

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In general

{ }

( , ) min , U x y = ax by

The slope of the axis is then a/b. Check you understand the function ^min

{}

^⋅ ^.

Cobb-Douglas utility function

( , ) ^a ^b; 0; and 0 U x y = x y a > b >

This is a well known function which generates well behaved indifference curves (smooth, negative and convex).

Good x is relatively preferred to good y Good y is relatively preferred to good x

Cobb-Douglas functions are frequently used in production theory, where instead of utility we talk of output, and instead of goods we talk of inputs.

a>b a<b

y

x

y

x

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4.3 Marginal Utility and the MRS

Consider a consumer that consumes the bundle (x,y).

How does this consumer’s utility change when we maintain the amount of y and give him a little more of x? The change in utility per unit of change in x is the marginal utility of x

(

MUx

)

.

( , ) ( , )

x

U U x x y U x y

MU x x

∆ + ∆ −

= =

∆ ∆

MUx measures by how much utility changes when we change x by a small amount holding y constant.

In the limit, if the change in x is infinitesimal,

x

MU U

x δ

= δ

From the definition of marginal utility it follows that the change in utility that results from a small

increase in x, holding y constant is:

U MUx x

∆ = ⋅∆

We have the same sort of definitions for good y;

marginal utility of y

(

^MU^y

)

^.

( , ) ( , )

y

U U x y y U x y

MU y y

MU U

y

U MU y

δ δ

∆ + ∆ −

= =

∆ ∆

=

∆ = ⋅ ∆

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The utility function can be used to measure the MRS defined in the previous lesson.

Suppose that, for a given utility function, both x and y change. In general, when we change the quantities consumed of x and y, the level of utility will change.

The total change in utility will be the sum of the change in utility generated by the change in x plus the change in utility generated by the change in y.

x y

U MU x MU y

∆ = ⋅∆ + ⋅ ∆

Suppose additionally that this change in x and y is a movement along a given indifference curve. This means that after the movement, the level of utility must be the same, and that ∆ =U 0. Therefore,

0 = MU_x ⋅∆ +x MU_y ⋅ ∆y Or

x y

y MU

x MU

∆ = −

∆

But ∆ ∆y x, measured along a given indifference curve, is (minus) the MRS. Therefore,

x y

MU MRS y

x MU

= −∆ =

∆

The MRS can be measured by the ratio of the respective marginal utilities of the two goods.

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Annex (Some calculus)

( , ) (1) ( , )

( , )

x

y

U U x y U U x y

MU x x

U U x y

MU y y

δ δ

=

= =

Totally differentiating (1) we find

x y

U U

dU dx dy

x y

dU MU dx MU dy

δ δ

= +

Along an indifference curve dU = 0. Therefore,

0 _x _y

x y

MU dx MU dy dy MU

dx MU

MU MRS dy

dx MU

= +

= −

= − =