# Are there any examples for $u(x_1, x_2) = \max\{x_1, x_2\}$ in real word?

I know how its graph looks like, and it's like when you want to choose between 2 inferior goods you choose the cheaper one so you can have more, but is there another examples?

• How about useful iPhone apps and useful Android apps? You are only going to buy one smartphone plus the useful apps you can afford Nov 2, 2019 at 14:34
• Technically, shouldn't it be $u(x_1,x_2) = \max \{u(x_1), u(x_2) \}$? Nov 8, 2019 at 4:28

One way to think about this utility function is "I can only use one of these two goods, and I will use the one that I have more of"

Some examples:

• Choosing between two sets of equally-valued but non-matching dishware. Once you have one set, the other one is useless to you.

• If your budget constraint is such that you can only have one kind of platform of some kind, then "items to be used with that platform" works. As commented above, Android apps and iPhone apps, or multi-platform video games, or DVDs vs Blu-Rays.

• If you were an opportunistic politician deciding whether to pander to group A or group B, knowing that whichever group you pick the other will hate you, then "number of voters in group A" and "number of voters in group B" would go into your utility function as max(A, B).

• The political example is stimulating, but the way you describe makes it more compicated, since "the others will hate you" indicates the generation of disutility - i.e. of a "bad" due to the consumption of the good - like environmental pollution. Nov 7, 2019 at 22:33
• I basically mean it as "they won't vote for you" so not a bad as long as they're smaller than your favored group Nov 8, 2019 at 23:23

Both are goods, so each provides utility if consumed alone. The utility function

$$u(x_1, x_2) = \max\{x_1, x_2\}$$

or more generally

$$u(x_1, x_2) = \max\{f(x_1), g(x_2)\}$$

for some functions $$f,g$$, may describe either

a) a situation where it is impossible to consume the two goods together. For example, concerts happening at the same time. Note that the fundamental aspect here is not the discreteness of the goods in question. It could be short-term vacations in a 4-day window where you can adjust the duration by one-two days but in any case cannot choose two destinations. By the way, this is an extreme foundation for a binomial choice model, although the final choice may be the good offering the lesser utility if the other is very expensive (in other words, this utility function does not reflect "lexicographic preferences"). But the utility function is about preferences, not actual choice.

b) a situation where consuming one of them "even a little", makes the consumption of the other to offer zero utility (in a static/synchronous, non-intertemporal framework). Note that we are talking about actual consumption, not "buying and leave it unconsumed". It is not easy to come up with examples in this category, because one has to find two goods where the actual consumption of the one "neutralizes" the concurrent actual consumption of the other. The relation need not be symmetric. A possible example is a very spicy food and a food with a very subtle taste (assuming that the consumer does not experience utility from the nutritional value of the food, i.e. this is a recreational eating situation). There is no point eating them together.

• The b) explanation does not seem to work as "actual consumption". If I consume 1 unit of spicy food I will not feel the taste of the more subtle tasting food, regardless of the amount consumed, even if it is larger than that of the spicy food. Nov 8, 2019 at 6:39
• @Giskard Indeed. Look at the utility function I wrote, it is not necessarily one-to-one in terms of units of consumption. Nov 8, 2019 at 7:00
• Yes, but a unit of spicy food seems to invalidate the utility gained from the consumption of any amount of the subtle food. You could construct such $f$,$g$ functions, but they would have to be somewhat contrived. Nov 8, 2019 at 12:46
• @Giskard Then you would rather choose to consume 0 units of spicy food (if you have 1 unit, throw it out) and 100 units of subtle food, than 1 unit of spicy food and 100 units of subtle food. Feb 26, 2020 at 17:41
• @user253751 Utility is not a matter of choice. Feb 26, 2020 at 17:51

For an exam you have to write an essay on each of two topics. The instructor checks the quality of your essays and for grading discards the lower-quality essay. You choose quality $$x_i$$ for essay $$i$$.