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Argue that, given your assumptions on the utility function, $x^*$ is the essentially unique (and hence global) maximum. (You need this because there may be local maxima when the assumptions on the utility function is relaxed - this will violate the proposition you're trying to prove). Now simply use the definition of global optima: for any $x\leq x^*$, \$u(x)...

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The second problem is probably for comparison - it is the problem of a benevolent social planner who can freely allocate resources between the two agents.

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It would be really tough to find papers (other than structural models) that empirically determine a particular utility function. Because of their subjective nature, it is incredibly tough to obtain observable data to estimate them. So researchers study the primitive preference relations that utility functions represent. Arguably, the preference relation is ...

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Good question! I would like to address a more general point about utility functions, which is often overlooked. Suppose that such an experiment were conducted on a group of people. Each of them is offered a "basket" containing some combination of goods and it is somehow measured how the utility function for each person depends on each of the goods. ...

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