<p><span class="math-container">$u = \max(x, y)$</span> represents the preferences over two substitute goods that cannot be consumed together. For example - tea and coffee. In the event that the consumer gets x quantity of tea and y quantity of coffee, consumer choose to consume only one of the them depending on the quantity. He always choose the one that is offered in larger quantity and throws the one that is offered in smaller quantity.</p>

CC BY-SA 4.0
2020-11-22T09:25:53.710Z
2020-12-11T11:28:03.433Z
2020-12-11T11:28:03.433Z
Amit
<p>Your thinking is correct that, in some ways, <span class="math-container">$x_1, x_2$</span> are substitute goods. We define substitute goods which have the following property:</p>
<p><span class="math-container">$$\left.\frac{\partial x_i}{\partial p_j}\right|_{u=\bar u}&gt;0$$</span></p>
<p>The case of <span class="math-container">$U(x_1,x_2)=\max\{x_1,x_2\}$</span> is that of a boundary solution as the indifference curves are now <em>concave</em> to the origin.</p>
<p>So equilibrium solution is:</p>
<p><span class="math-container">\begin{align}
x_i^*(p_i,p_j)=
\begin{cases}
0 &amp; p_i\geq p_j \\
B/p_i &amp; p_i \leq p_j
\end{cases}
\end{align}</span></p>
<p>where, <span class="math-container">$B$</span> is the total expenditure. Note that I have taken equality at both because when prices are same, the consumer will (randomly) choose one the two products and consume only that.</p>
<p>It can be seen that, for a given <span class="math-container">$p_i$</span>, <span class="math-container">$x_i^*(p_i,p_j)$</span> is a step function w.r.t <span class="math-container">$p_j$</span> which increases from <span class="math-container">$0$</span> to <span class="math-container">$B/p_i$</span> as <span class="math-container">$p_j$</span> increases beyond <span class="math-container">$p_i$</span>. Therefore, the function <span class="math-container">$x_i^*(p_i,p_j)$</span> is increasing in <span class="math-container">$p_j$</span> (though not strictly).</p>

CC BY-SA 4.0
2020-11-22T08:17:00.387Z
2020-11-22T08:36:42.643Z
2020-11-22T08:36:42.643Z
Dayne
<p>I am not sure, which type of goods does the maximum utility function represent i.e., <span class="math-container">$U(X_1, X_2) =\max(X_1, X_2)$</span>.</p>
<p>As the <span class="math-container">$U(X_1, X_2) =\min(X_1, X_2)$</span> represent the complementary goods, and <span class="math-container">$U(X_1, X_2) =X_1+ X_2$</span> represent the substitute goods, I think it represents the substitute goods as the maximum of both matter. So am I correct?</p>
<p>Please clarify this doubt, thank you.</p>

CC BY-SA 4.0
2020-11-22T13:59:44.467Z
There are a couple answers to this already on Econ SE. For example, see [this, for an answer to what this function (anti-Rawls or maximax) represents](https://economics.stackexchange.com/questions/9495/a-term-for-utility-functions-based-on-the-max-operator).  And see [this for a good explanation on how this differs from perfect substitutes](https://economics.stackexchange.com/questions/15766/relation-between-linear-utility-function-and-u-maxx-y).
anguyen1210
<microeconomics><utility><supply-and-demand><preferences><measuring-utility>
which type of goods maximum utility function represent?
Anishka Mishra
# which type of goods maximum utility function represent?

(Asked by: username_0 on 2020-11-22)

I am not sure, which type of goods does the maximum utility function represent i.e., $U(X\_1, X\_2) =\max(X\_1, X\_2)$.

As the $U(X\_1, X\_2) =\min(X\_1, X\_2)$ represent the complementary goods, and $U(X\_1, X\_2) =X\_1+ X\_2$ represent the substitute goods, I think it represents the substitute goods as the maximum of both matter. So am I correct?

Please clarify this doubt, thank you.

**Tags:** microeconomics, utility, supply-and-demand, preferences, measuring-utility

> **Comments:**
> **username_1**: There are a couple answers to this already on Econ SE. For example, see [this, for an answer to what this function (anti-Rawls or maximax) represents](https://economics.stackexchange.com/questions/9495/a-term-for-utility-functions-based-on-the-max-operator).  And see [this for a good explanation on how this differs from perfect substitutes](https://economics.stackexchange.com/questions/15766/relation-between-linear-utility-function-and-u-maxx-y).  

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## 2 answers

### Answer by username_3 on 2020-11-22. Score: 3 | Accepted answer

Your thinking is correct that, in some ways, $x\_1, x\_2$ are substitute goods. We define substitute goods which have the following property:

$$\left.\frac{\partial x\_i}{\partial p\_j}\right|\_{u=\bar u}>0$$

The case of $U(x\_1,x\_2)=\max\{x\_1,x\_2\}$ is that of a boundary solution as the indifference curves are now *concave* to the origin.

So equilibrium solution is:

\begin{align}
x\_i^\*(p\_i,p\_j)=
\begin{cases}
0 & p\_i\geq p\_j \\
B/p\_i & p\_i \leq p\_j
\end{cases}
\end{align}

where, $B$ is the total expenditure. Note that I have taken equality at both because when prices are same, the consumer will (randomly) choose one the two products and consume only that.

It can be seen that, for a given $p\_i$, $x\_i^\*(p\_i,p\_j)$ is a step function w.r.t $p\_j$ which increases from $0$ to $B/p\_i$ as $p\_j$ increases beyond $p\_i$. Therefore, the function $x\_i^\*(p\_i,p\_j)$ is increasing in $p\_j$ (though not strictly).


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### Answer by username_2 on 2020-11-22. Score: 5

$u = \max(x, y)$ represents the preferences over two substitute goods that cannot be consumed together. For example - tea and coffee. In the event that the consumer gets x quantity of tea and y quantity of coffee, consumer choose to consume only one of the them depending on the quantity. He always choose the one that is offered in larger quantity and throws the one that is offered in smaller quantity.


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