I Think I Found an Error On Page 42 of MWG

 

The definition of local nonsatiatedness requires that for every point P in the set, and for every real-valued epsilon >= 0, there exists another point Q in the set such that the distance between P and Q is less than epsilon, and, the utility function at Q is higher than the utility function at P.


The problem is, it should read, "...for every real-valued epsilon > 0", not greater than or equal to.  When the border of the curve enclosing the set of points is included in the set of points, the situation changes to a situation where the border, which must be included if the definition reads "epsilon >= 0" as opposed to "epsilon > 0", is full of points that violate the definition of local nonsatiatedness.  In particular, every point on the border of the filled-in circle shown in the diagram on p. 43 violates local nonsatiatedness.  When epsilon >= 0, the border must be included, but that same border violates the property, meaning that with this definition, there is no such thing as a curve with local nonsatiatedness.


The correct way to write this definition is in a way where the set is always open...that way, the epsilon definition will hold for every point that approaches the border, and the border itself would not have to adhere to the local nonsatiatedness condition, since it wouldn't be included in the set of points.



Update:


Maybe the argument would be that the points on the border to conform to local nonsatiatedness, because points that lie within the circle might have higher utility than the border points.  At the top right of the circle at the border, though, we would expect the points emanating out from the center of the circle to be progressively higher in utility.  Also, we shouldn't have to have it be the case that every curve with local nonsatiatedness be such that lower values of each commodity lead to a higher utility; if we assume that the goods are desirable and not "garbage" or "bad," which is what usually is the case, then this definition of local nonsatiatedness is erroneous.






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