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 homogeneous of degree 1. So the particular form is often used, because this representation is homogeneous of degree 1.

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 increasing transformation, so that will not represent the same preferences.

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 ( left to the reader ) to show that is a concave function if . This expression for the Hessian also shows why the requirement was imposed that : if exceeded 1, then preferences would not be convex. In fact, when , CES preferences become example 2, perfect substitutes, which are just on the borderline of representing convex preferences.

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 left to the reader. ]

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. [ Warning : These preferences are also discussed in exercise of the text. I am using somewhat different notation than the text : their 's are my 's, and their 's are my 's. ]

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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