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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.

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$u = \max(x, y)$ represents the preferences over two substitute good 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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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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