Explain about the duality between direct and indirect utility.

Duality between Direct and Indirect Utility:

While seen how one can recover an indirect utility function by observed demand functions through solving the integerability equations. Now see how to resolve for the direct utility function.

The answer demonstrates quite nicely the duality between direct and indirect utility functions. This is most convenient to define the computations in terms of the normalized indirect utility function, there prices divided by income therefore expenditure is identically one. Hence the normalized indirect utility function is specified by

v(p) = max_x u(x)

that is px = 1