Some Examples of Utility Functions

Each of these functions represents preferences which are
continuous, strictly monotonic, and convex.

Example 1 : Fixed Coefficients

2 goods

where and are positive constants ( and where ``'' means ``the minimum of the two items''. So if , then , and if , then . The indifference curves for these preferences are -shaped, with a kink along the diagonal line through the origin with equation . ( This line has a slope . )

goods

where each is a positive constant.

This utility function is continuous, although it is not continuously differentiable at the kinks. It is increasing, in the sense that if -- although it is not necessarily true that if . The utility function is quasi-concave, since the preferences it represents are convex ( although not strictly convex ). ( This convexity can be seen by drawing a line between any two points on or above some -shaped indifference curve ; the line connecting these points must also be above the indifference curve. )

Example 2 : Perfect Substitutes

where the 's are all positive constants. Since the gradient of this function,

the preferences the function represents are strictly monotonic if each of the 's is positive.The Hessian matrix of second deriatives of the function is just the zero matrix, since all of the 's are constants. That means that the function is not just quasi-concave, it's actually concave. It's actually convex, as well, since it's linear. ( But it's not strictly concave, and the preferences it represents are not strictly convex. )

In 2 dimensions, the indifference curves for these preferences are straight lines, with slope . In higher dimensions, the indifference surfaces are planes, or hyperplanes.

Example 3 : Quasi-Linear Preferences

where is any increasing, concave ( not just quasi-concave ) function mapping -dimensional vectors into real numbers. The partial derivatives of this function are , and for any . So if is non-decreasing in all its arguments, then the preferences that this utility function represents are strictly monotonic. The matrix of second derivatives of this utility function is

so that

and for any direction vector , if the function is concave.

In two dimensions, the slope of an indifference curve through any bundle is . Concavity of the function means that , so that the indifference urve gets steeper as we move up it ( and to the left ). In this case, notice that the slope of an indifference curve is independent of the level of consumption of good 1, since it depends only on . That means that if we move right horizontally, the slopes of the indifference curves stay constant.

Example 4 : Cobb-Douglas Preferences

There are several ways of representing the same preferences here. One
way is

where the 's are all positive constants. I could take the natural logarithm of above, which is a monotonically increasing transformation, to get

For given , the functions and represent exactly the same preferences. A third transformation is to take to the power , where

to get

where

Notice that the new exponents have been constructed so that

which turns out to be a fairly convenient property of the representation .

Why is this property of
so convenient? Suppose that
I multiply each element in the consumption bundle
by the same constant . Then

where, in the above string of equations, I used the fact that , and the fact that the 's sum to 1. A function such that for all is called

Note that the other functions and representing
Cobb-Douglas preferences are *not* homogeneous of degree 1. A
function, such as or , which is a monotonically
increasing transformation of a homogeneous-of-degree-1 function is
called *homothetic*. So ``homothetic'' is a property of the
underlying Cobb-Douglas preferences. ``Homogeneous of degree 1'' is a
property only of one of the functions (
)
representing those preferences.

**Left to the reader** : check whether the preferences in examples
1,2 and 3 are homothetic.

To check that Cobb-Douglas preferences are strictly monotonic and
convex, it is easiest to use the logarithmic representation
. Taking the first partial derivatives,

so that all the partial derivatives are positive ( at least, when the consumption levels of each good are positive ) and the preferences are strictly monotonic. Taking derivatives yet again, the Hessian matrix of second derivatives of the function is

so that is a concave function, which means that all other representations of these preferences are quasi-concave, and that the preferences themselves are convex.

In two dimensions, the slope of an indifference curve is ,
or

since

**Left to the reader** : check that
.

As we move up and to the left along an indifference curve, falls and rises, so that the curve gest steeper.

Also, the , the slope of the indifference curve, depends only on
the *ratio* of consumption of the two goods, . This
propperty, that the slope of an indifference curve does not
vary as all elements of the consumption bundle are increased by the
same prroportion, will hold for *any* homothetic preferences.

In two dimensions, what this means is that the slope of an indifference curve is unchanged as we move along any diagonal through the origin, since the ratio is constant along any such diagonal.

Example 5 : Constant Elasticity of Substitution

where the 's are positive constants, and is a constant, which is less than or equal to 1.

So it's o.k. for to be negative -- but it cannot exceed 1.

Now if , I can take a monotonically increasing transform of
by taking it to the power , to get

but this only works if . If , then taking to the power is not a monotonically

Using the chain rule to take the partial derivatives of ,

which simplifies ( a little ) to

which must be non-negative, so that the preferences represented by are strictly monotonic.

**Left to the reader** : so that is
homogeneous
of degree 1, and the preferences represented by are
homothetic.

Using the other representation ,

( again demonstrating that CES preferences are strictly monotonic ). This means that the matrix of second derivatives of is

which makes it a straightforward exercise (

When , taking to the power is no longer a
monotonically increasing transformation. But letting
is a monotonically increasing
transformation. [ Why? If , then the derivative with respect
to of is
. ] The matrix of
second derivatives of is

which must be negative definite when . [ Again, this demonstration is

So Constant Elasticity of Substitution preferences are strictly monotonic and convex if .

A problem : the only restriction imposed on is that it be less than or equal to 1. But when , the expression for does not make much sense.

A solution : Look at the slope of the indifference curves, in two
dimensions, the , When
, then

**Left to the reader :** This is also the correct expression for the
if and is used as a representation of the
preferences, or if and is used as the
representation of the preferences.

So what happens to this as gets close to zero? This
expression approaches

which is an expression that's been used before here, in the section immediately above. That's the for Cobb-Douglas preferences.

It turns out that Cobb-Douglas preferences are a special case of CES preferences, the limiting case of CES preferences as approaches 0.

This first graph illustrates indifference curves when the elasticity of substitution is greater than 1.

This second graph illustrates indifference curves when the elasticity of substitution is less than 1.

6. Stone-Geary Preferences

These preferences are **not** defined on the whole of . They
are only defined on consumption bundles for which for each
element , where the 's are constants, usually described as
*subsistence* levels of consumption. Stone-Geary preferences can
be represented by the utility function

where the 's are again posiitve constants. [

Notice that Cobb-Douglas preferences are a special case of
Stone-Geary preferences : simply set all the subsistence levels
equal to 0.I can also use the same monotonically increasing
transformations of as I used for Cobb-Douglas, namely,
so that

and , with , so that

where .

From the representation , it should be clear [ but is **left
to the reader** ] that Stone-Geary preferences are strictly monotonic
on the consumption set
.

Since the Hessian of is a diagonal matrix, with elements of the
form

along the diagonal, the function is concave, so that Stone-Geary preferences are quasi-concave.

But, unlike Cobb-Douglas ( or CES ) preferences, Stone-Geary
preferences are not necessarily homothetic.[**Left to the reader** :
what restrictions on the 's would make these preferences
homothetic? ]

The slope of an indifference curve for these preferences is

so that, in general, the slopes of the indifference curves are not constant along a diagonal line through the origin. But these slopes are constant along any diagonal line through the ``subsistence point'' .

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