I am working with Rubinstein's book. It states there that if preferences are differentiable, then value per dollar at a bundle of a commodity is as large as value per dollar of the bundle of any other commodity.

Then we are given that if preferences are represented by a Utility function that is differentiable at x* (optimal solution) then $D_k U(x*)/ p_k ≥ D_i U(x*)/ p_i$.

I do not understand the mathematical part of the definition. X* is an optimal solution (bundle of $x_k$ and $x_j$ commodities). They are both in the bundle so why would the value per dollar of $x_k$ be higher than value per dollar of $x_j$?



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