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I don't understand why the Marshallian and Hicksian demand have such different properties. Both are functions from $$\Bbb{R}^n_+ \times \Bbb{R}_+ \rightarrow \Bbb{R}_+^n$$both are solved using the method of Lagrangian multipliers. Yet the properties of the two are very different.

Does anyone know of any resources that discuss these differences in depth and in an organized way? My professor has us using his lecture notes. These are a good introduction but just don't rigorously discuss the differences.

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Technically they are related but are not the same Marshallian (or Walrasian henceforth) demand is the result of $x(p,w)\in argmax_{y\in\mathbb{R}^{L}_{+}}u(y)$ subject to $p'y(p,w)\leq w$. The Hicksian demand is the result of the dual problem $h(p,u)\in argmin_{y\in\mathbb{R}^{L}_{+}}p'y$ subject to $u(y)\geq u$. The formal relationship is established trough the value functions of both problems, for the walrasian demand we have the value function or indirect utility $v(p,w)=u(x(p,w))$ and the value function of the second problem (hicksian demand) is the expenditure function $e(p,u)=p'h(p,u)$. Now, the indirect utility and the expenditure are inverses of each other in the following sense:

  1. $e(p,v(p,w))=w$
  2. $v(p,e(p,u))=u$

And to answer your question $x(p,e(p,u))=h(p,u)$ and $h(p,v(p,w))=x(p,w).$ This means that this demands have different properties but can be made equal by adjusting the wealth in the case of the walrasian demand such that I maintain the consumer in a fixed utility level u, thus obtaining the hickisan demand and vice-verse changing the utility level with changes in wealth using to go from the hicksian to the the walrasian. The properties of each demand comes from the properties of the value functions and are a consequence of the theorem of the maximum. Like continuity of the objective.
A good source is Nolan Miller notes http://business.illinois.edu/nmiller/documents/notes/notes4.pdf. But every micro theory books covers this material.

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Marshallian and Hicksian demands are very different objects. If you want a very rigorous treatment, Jehle and Reny's graduate textbook on microeconomics is very good. For a more intermediate level check the book by Hal Varian

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