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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 ...


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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 ...


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There are no simple methods for estimating cardinal utility (ordinal utility would be a different matter - you could just observe few of your choices). This is not because cardinal utility would necessary be unmeasurable. Although this is not completely settled question (see Moscati (2018) Measuring Utility: From the Marginal Revolution to Behavioral ...


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