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I've seen a derivation of MRS, but it's not immediately obvious to me why letting MRS equal to the price ratio of two goods will give you the solution to an optimization problem. Can anyone link me to a paper that explains this (or explain it themselves)? It would be great to get both the intuition and mathematical derivation! Thanks in advance!

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    $\begingroup$ Read Chapter 5 of "Intermediate microeconomics" by Hal Varian. It provides the intuition and in the appendix you can also find the mathematical derivation. $\endgroup$ – Giskard Apr 10 '16 at 10:13
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The mathematical derivation is straightforward: set up a Lagrangian for the utility maximization the consumer solves subject to a monetary constraint, then divide its partial derivatives with respects to the quantities one with another, and there you have it: MRS equals the price ratio. The intuition behind it is that, given fixed prices (that is, a price taking situation), the ratio of the marginal utilities must also remain constant: suppose by contradiction this isn't the case, and that some good provides relatively high marginal utility. Then the consumer would profit from deviating to consuming a basket in which the quantity of said good is higher, since the increase in utility would be higher than the costs incurred.

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