Mathematical Derivation of ordinary demand function:
Here we present the mathematical and more general proof of the above result. Consider, again, the initial price income situation (p_{0}, M_{0}), where x_{0} is the chosen bundle. Prices change from p_{0} to p_{1}, where p_{0}>p_{1}, and consumer's real income changes (in this case increases). Suppose we adjust consumer's money income in such a way that her purchasing power remains the same. Consumer's money income is changed to M_{1} = p_{1}x_{0}. Suppose x" is the bundle consumer buys with her adjusted money income viz.,
Since with (p_{1}, M_{1}) consumer chooses x" while x0 was available, so x" is revealed preferred to x0 and therefore was not available when x0 was chosen while (p_{0}, M_{0}) prevailed. Thus,
Suppose only price of ith good changes, other prices remaining constant. Therefore,
(p_{i}^{0} - p_{i}^{1}) (x_{i}" - x_{i}^{0}) < 0
So, quantity change is in opposite direction of the price change. Hence, substitution effect is negative. Now, dp = (dp_{1}, dp_{2},...,dp_{i},...,d_{p}^{n}) and dx = (dx_{1}, dx_{2},...,dx_{i},...,dx_{n})^{T}, so that