Denote by x* the optimal basic solution of the original instance of the problem. x* corresponds to a particular submatrix B. Now consider a small increment Db of one or more components of the vector b.
If the changes are sufficiently small, the basic solution of the new problem hatx* corresponding to the same matrix B is still feasible. In this case, it is optimal for the new problem since a change in the r.h.s. does not affect the reduced costs. Which is the value of the new optimal solution?
One easily derives that the new optimal value is equal to the old one plus the quantity delataz given by the scalar product of Db the dual optimal vector u*, which has not changed. From the last equality we deduce the economic interpretation of the dual vector u*. Indeed this equality says that each component of u* represents the increment in the primal optimal value corresponding to an unit increase of the same component.
In other words, the optimal dual variables represent how much on looses in the optimal value of the primal problem because of the rigidity of each single constraint. For this reason the dual variables are also called shadow prices or marginal prices.