Mathematical Presentation of Utility maximisation:
Consumer's objective is to maximise her utility by solving UMP. To solve UMP, we set the Lagrange function of the corresponding problem, which is, L(x1, x2) = U(x1, x2) + λ (M-p1x1-p2x2) Our objective is to maximise this Lagrange function by choosing x1, x2 and λ. For that we differentiate the Lagrange function by x1, x2 and λ, and set all equal to zero.
is the slope of the indifference curve and p1/p2 is the slope of the budget line. So, at equilibrium we have a slope of the indifference curve that is equal to the slope of the budget line. Again, from equation (f3) we get M = p1x1+p2x2, so budget equation holds with equality sign.