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I understand that there are two common applications of Lagrangian multipliers in consumer theory: the utility maximization problem (UMP) ans the expenditure minimization problem (EMP).

These seem like the same problem to me but with different parameters being the focus. Why does my professor emphasize the importance of duality between these theories? I don't understand why both are needed.

my question:

To what extent are the UMP and EMP different economic ideas/problems as opposed to just simply being a different way of mathematically stating the same thing?

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The solutions of UMP and EMP only coincide if certain conditions such as monotonicity of preferences is met. Hence they are different problems.

Duality is important because the Lagrangian multipliers frequently have an economic meaning. In the UMP the Lagrangian multiplier of the budget constraint is equal to the marginal utility of money. In the EMP the Lagrangian multiplier of the budget constraint is equal to the marginal cost of utility.

In the EMP you can easily replace the consumer with a corporation deciding what inputs to use to create an output. In this case the Lagrangian multiplier would show the marginal cost of output.

In a similar way you could also alter the UMP to a problem where a corporation has to decide between some kinds of outputs and is trying to allocate resources in a way that will maximize profits. (E.g.: There are $m$ amount of man-hours and several kind of projects that can be taken on.)

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The optimal solutions of UMP and EMP have different uses. EMP leads to welfare analysis for example: compensating variation and equivalent variation use the expenditure function. UMP gives the optimal consumption bundle and allows wealth effects to be studied.

Duality is useful in advanced optimization. Your current UMP and EMP study is a simple introduction to it.

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