# Relation between linear utility function and U=max{x,y}

I'm studying general equilibrium theory, and in the study guide I came across a utility function of the type $U=\max\{x,y\}$, which I'm not that familiar with. I study mainly from two books: Intermediate microeconomics by Varian and Nicholson's Microeconomic Theory and couldn't find any details about the nature of this kind of utility function in either.

My teachers over the semesters have mentioned some details about it as a curious fact, so I know the choice of the consumer is driven by the relative price of the goods, in this case x and y, so he'll choose to consume the cheaper good, and if the relative prices are equal he'll be indifferent towards both. Also I know the indifference curves have the shape of an inverted L.

What is not clear to me is how is it related to the Perfect Substitutes utility function since both seem extremely similar to me, but the indifference curves are very different.

• Is the utility function $U=\max\{x,y\}$ a more general case?

• More importantly, when the relative prices are the same, how can we illustrate consumer's choice in that inverted L indifference curve, doesn't it turn linear?

The optimal choice set for a max function and a perfect substitutes function with equal relative prices share some solutions [i.e, boundary solutions], but in general, the indifference curves, and hence non-boundary solutions, are different.

# Main Idea

For both a max(x1 x2) and perfect_sub(x1 x2) utility function, the point, say, m/p1 (or m/p2) would maximize utility. So both utility functions share boundary solutions. But think about the IC of m/p1 for a perf_subs consumer and think about the IC of m/p1 for a max consumer. You see the points that the two consumers think are 'just as good' as that m/p1 point are very different.

That is, the same boundary bundle might be a solution to both utility functions, but the other solutions will differ.

# Logic

The reason is that U = perfect_subs is a (not-strictly) convex utility function, whereas U = max isn't. That is: the consumer is either indifferent between, or actually prefers, less extreme combinations to more extreme combinations for the former. As for the latter? Well, they just care about the max; they like extreme combinations (e.g, [C = (m/p1, 0)])

That's why for U = perfect_subs with relative prices the same, both A = (100,0) and B = (ß(x1), (1-ß)x2) have the same utility. That is,B lies on a straight line IC connecting (100, 0) to (0, 100). Whereas for U = max(x1 x2), B would not be on the same IC as A (unless ß is 0 or 1, in which case, we're back to talking about boundary solutions!)

The word, max is confusing here. Essentially, the perfect_subs (x1 x2) consumer wants to maximize total goods. The max(x1 x2) consumer just cares about maximizing the larger quantity in his {x1, x2} bundle.

# Tl;DR

To be more concrete: If A = (100,0) and B = (75, 25), a perf_subs consumer is indifferent between A and B; a max consumer is not.

• Thank you for your answer. I find problematic that your answer and the last user's answer are uncompatible, and thats the root of my confusion with this function. When the relative prices are the same, wouldn't max consumer and perf subs consumers share the same IC? after all, he (max consumer) feels indifferent towards both A and B because the prices no longer tilt his choice towards any of the corner solutions. I get that when prices are uneven, their respective IC's are different. Do you have any literature to back up your answer? I'm sorry, im very grateful, but also quite confused. – José Julián Parra Mar 15 '17 at 17:04
• I tried to explain the intuition a little in the logic section. By virtue of one being convex and the other being nonconvex, the two consumers feel different about combinations. The max consumer will view (100,0) as better than (75,25); perf_subs consumer won't. While both get utility of 100 from A, note that max consumer gets only U = 75 from B. OR: (1) perf_subs is convex -- > its ICs are convex; (2) max is nonconvex -- > its ICs are ¬convex; (3) A convex (OR nonconvex) utility function would never have nonconvex (OR convex) ICs; (4) A perf_subs IC would never be the same as a max IC. – thewhitetie Mar 15 '17 at 17:25
• (I would suggest re-reading Varian's discussion of convexity. And make sure you can describe, in English, what the difference in behavior is between a $max$ and $perf_sub$ consumer.) – thewhitetie Mar 15 '17 at 17:37
• I appreciate your patience. I understood your convexity argument, the reason I was confused was because I thought that the fact that the consumer was indifferent between two corner bundles when Px/Py=1 meant that he was indifferent between those two bundles and a linear combination of them, so his IC will somehow turn linear. Thinking back, that is pretty arbitraty. So, let me se if i get this: Let it be U={x,y}, if Px/Py=1 then the consumer will be indifferent between the bundles A={x,0} and B={0,y} but NEVER to a linear combination of them. Correct? – José Julián Parra Mar 19 '17 at 2:46
• I messed up the utility function, i meant U=max{x,y}. – José Julián Parra Mar 19 '17 at 3:06

HerrK. was correct in the comments. Sorry for the lapse.

What is happening is that $U(x,y)=$max{$x,y$} with $P_x=P_y=P$ causes a consumer to be indifferent between consumption bundles: $(\frac{w}{P},0) \equiv(0,\frac{w}{p})$

Whenever you have unequal prices, say $P_x<P_y$ then the consumer will strictly prefer the cheaper good. Here, this means the consumer prefers the consumption bundle $(x,y) \equiv (\frac{w}{P},0)$

Perfect substitute functions are of the form $U(x,y)=aX+bY$,$x,y \in \mathbb{R^2_+}$ . Suppose $P_x=P_y=P$ then the consumer is indifferent between any mixing of the goods $x,y$ that exhausts his wealth $w$ (assuming the divisibility of goods etc.). If instead prices are unequal, the consumer will equate $MRS = \frac{MU_x}{MU_y}$.

This can yield corner solutions that are similar to the optimal bundles chosen by the agent facing $U(x,y)=$max{$x,y$}. Specifically, whenever the budget line and indifference curves have different slopes. For example: