In this paper, an extension of the theory of consumer behavior to cover situations when prices enter the utility function is considered. In Section 1 we formulate the problem. In Section 2 we derive a generalized Slutsky equation of which the traditional Slutsky equation is a special case. We investigate the class of utility functions that yield homogeneous of degree zero in prices and income demand functions and also see that these can't exist for a certain class of utility functions. We then establish theorems on "substitutable" and "complementary" commodities which are not, in general, valid for the traditional theory. Finally, a set of qualitative restrictions on demand functions is deduced, of which a subset is identical to the traditional restrictions.