Indifference curve for complements

The indifference curve of perfect complements are L shaped.

Most goods are between these two extremes. The curve will look like an intermediate between the two. i.e: looking like $1/x$, and are convex. That is, linear combinations (or a convex combination) of A and B will be preferable to A or B, where A, B are two states in $U(x,y)$.

For example, 5% A and 95% B will be preferable to either.

Utility function

One usually considered utility function is initially positively sloped and then becomes negatively sloped after a saturation point. In the first region, the marginal utility is positive and then negative. We will be restricting ourselves to the positive marginal utility region.

The positive marginal utility can take different forms such as linear, exponential, logarithmic etc.

Slope of indifference curve

The slope is called the marginal rate of substitution, and is usually non constant.

When we have a lot of y and little x, we are willing to spend several y to get some x. This is because as we have higher and higher y, we approach its saturation point. This is called diminishing marginal utility.

When the utility is held constant (that is, on an indifference curve), an implicit relationship between x and y is created. Let

We can write,

Then,

Income and prices

Let the income of a consumer be I, and the market price of x and y are $P_x$ and $P_y$. Then,

This is called the budget equation.

If we spend the whole income on x, then the amount of x we have will be $I/P_x$, and similarly for y, it will be $I/P_y$. We can plot this on a graph. The line joining the two points $(I/P_x, 0)$ and $(0, I/P_y)$ is called the budget line. The division to the right of the budget line is not affordable, and that to left is affordable and is called the feasible set.