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(x_{1}, x_{2}) = U(x_{1}, x_{2}) + λ (M-p_{1}x_{1}-p_{2}x_{2}) Our objective is to maximise this Lagrange function by choosing x_{1}, x_{2} and λ. For that we differentiate the Lagrange function by x_{1}, x_{2} and λ, and set all equal to zero.
is the slope of the indifference curve and p_{1}/p_{2} 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 = p_{1}x_{1}+p_{2}x_{2}, so budget equation holds with equality sign.