# Utility function types - request for review

I am working on a Wikipedia page that compares several common utility functions. Although I found some information about these topics on the web, I didn't find it all in one place, and often got confused by the different terms. I will be happy for any comments or reviews. Most importantly: are the facts in this page correct?

EDIT: to make the question self-contained, I copy its current contents below. Because StackExchange markdown does not support tables, I used Senseful Solutions to create the text table.

This page compares the properties of several typical utility functions of divisible goods. These functions are commonly used as examples in consumer theory.

The utility functions are exemplified for two commodity types, $x$ and $y$. $p_x$ and $p_y$ are their prices. $w_x$ and $w_y$ are constant parameters.

+---------------+-------------------------------------+-----------------------+-----------------------------------------------------------------------------------+----------------+-------------+-----------------------------------------+----------------------------------+
|     Name      |              Function               |  Indifference curves  |                                   Demand curve                                    |  Monotonicity  |  Convexity  |                Good type                |              Example             |
+---------------+-------------------------------------+-----------------------+-----------------------------------------------------------------------------------+----------------+-------------+-----------------------------------------+----------------------------------+
| Linear        |  ${{x\over w_x}+{y\over w_y}}$      |  Straight lines       |  Step function: only goods with minimum ${w_i p_i}$ are demanded                  |  Strong        |  Weak       |  Substitute good|perfect substitutes    |  Potatoes of two different farms |
| Leontief      |  $\min({x\over w_x},{y\over w_y})$  |  L-shapes             |  hyperbolic: ${\text{Income} \over w_x p_x+w_y p_y}$                              |  Weak          |  Weak       |  Complementary good|perfect complements |  Left and right shoes            |
| Cobb–Douglas  |  $x^{w_x} y^{w_y}$                  |  hyperbolic           |  hyperbolic: $\frac{w_x}{w_x+w_y} {\text{Income} \over p_x}$                      |  Strong        |  Strong     |  Independent good|independent           |  Apples and socks                |
| Maximum       |  $\max({x\over w_x},{y\over w_y})$  |  ר-shapes             |  Discontinuous step function: only one good with minimum ${w_i p_i}$ is demanded  |  Weak          |  Concave    |  Substitute good|substitutes            |  ?                               |
+---------------+-------------------------------------+-----------------------+-----------------------------------------------------------------------------------+----------------+-------------+-----------------------------------------+----------------------------------+


Are the details in the table correct?

EDIT: Many thanks to all commenters and answerers. I revised the page accordingly.

• I feel like voting to close and upvote this question. Have you seen any such questions on a SE website? – VicAche Dec 1 '15 at 9:29
• Perhaps you ought to include that the weights $w_x$ and $w_y$ are positive. It would also be useful to say something about monotonic transformations, e.g. the equivalence of $\ln x + \ln y$ and $x \cdot y$. And how come you include your $\max$ function but no quasilinear functions? (or a CES function) – Giskard Dec 1 '15 at 10:16
• For examples of the Maximum function, you can pick two goods whose consumption interferes with each other. Examples might include music and television shows; trying to consume both (at the same time) reduces the utility that each provides. I agree with Densep that CES and quasilinear should be included. It also might be useful to add a "notes" column for information such as the fact that CES nests various other kinds of utility function. – Ubiquitous Dec 1 '15 at 10:46
• Perhaps this question could be made more "SE firendly" if it were reworded to include an actual question that people can answer. You could, for example, invite people to supply in each answer the details necessary to fill in a missing row of the table. – Ubiquitous Dec 1 '15 at 10:47
• Please post a specific question that does not depend on outside resources (self-contained). For now, I've voted to close that questoin. – FooBar Dec 1 '15 at 11:17

Currently (Dec-1-2015) I see listed Linear, Leontief, Cobb-Douglas, and Maximum.

To incorporate here a comment by @denesp the C.E.S utility function (that nests the Leontief and Cobb-Douglas) should certainly be included

$$u(x,y) = B\big(ax^{-\rho}+(1-a)y^{-\rho}\big)^{-1/\rho}$$,

(which also is a functional form that has non-zero cross partial derivatives),

as should the quasi-linear form

$$u(x,m) = h(x) + m$$

which is very often used, with $m$ standing for residual money income (what's left after purchasing $x$).

I would also include the Translog utility function.

Finally, I think you should increase the scope of the page and also include in a separate section univariate utility functions that are used in environments with uncertainty and in macroeconomics (say, the HARA class of functions). We may keep in our minds these subfields of economics clearly distinct and also from basic consumer theory, but what you build is a wiki page that is likely to be visited by outsiders (e.g. mathematicians or engineers interested in the subject) or just-arrived students, so a broader scope appears useful.

• Thanks! I incorporated most changes. Hopefully I did it correctly. – Erel Segal-Halevi Dec 1 '15 at 15:29

Let me just add some observations which are too long for a comment.

It seems that the description of demand with linear utility as a step function seems not quite correct. In fact, what we get instead is a demand correspondence. The step function "jumps" at the step $w_x=w_y$, while the correct demand correspondence includes all possible combinations of $x$ and $y$ at the step.

Moreover, the selection of demand (possibly better termed Marshallian demand) as an additional feature is good, but I think the table would be more useful if there are (possibly hidden until manually opened) the indirect utility function, the Hicksian demand and the expenditure function in the graph.

Aside from the mentioned CES, quasilinear and translog utility, I would add the isoelastic utility $u(x,y)=x^a+y^b$.

Note that if we start including utility functions over lotteries, we may also expand the tables by preferences over time, i.e. rational/hyperbolic discounting, etc.

Finally, I think the generalization of the current functions to $n$ goods rather than two should not complicate the table but adds generality. (Also possible as additional entries).

Edit: I also just realized that the current demand for the Leontief utility is wrong. It should be $I w_x/(p_yw_y+p_xw_x)$ for good $x$. What is currently given as the demand is instead the indirect utility $I/(p_yw_y+p_xw_x)$.

The demand functions for the linear (and similarly the max function) can be written as $\begin{cases} I/p_x, & p_x w_x \leq p_yw_y \\ 0, & p_x w_x>p_yw_y \end{cases}$

The assumptions of the consumption problem should also be clearly stated (budget constraint, maximization problem)

• Thanks! Regarding isoelastic utility, its Wikipedia page: en.wikipedia.org/wiki/Isoelastic_utility has a different expression - it is a cardinal function of a single variable (rather than an ordinal function of two variables). Is there a different definition of isoelastic utility? – Erel Segal-Halevi Dec 1 '15 at 15:22
• Strange, I always used it for the mentioned function. Could be that I am the only one, of course. :-D – HRSE Mar 24 '16 at 1:33