budget line and indifference curve on the same plot

Utility maximisation

MRS: marginal rate of substitution

Let an individual have,

Income: I

Goods: x and y

Indifference map is a set of IC

Budget set is $P_xx + P_yy \leq I$

Slope $-Px/Py$

Slope of IC is $MU_x/MU_y$

The utility curve tangent to the budget line gives the maximum utility. Anything giving more utility than that point of intersection, will cost more than the income.

This point is called the equilibrium point. A consumer will prefer not to move away from that point.

The condition, expressed mathematically is,

When the goods are perfect substituents, only one good is purchased, depending on the slope of the indifference curve (which is in this case a set of straight lines) and the budget line. That is, the equilibrium point will be either on the x axis or the y axis.

When the goods are perfect compliments, the equilibrium point is the intersection of budget line and the angled base of the indifference case (which in the case is not a tangent).