Indifference Curve

Consume at the same curve has the same utility.

Slop of indifference curve

Method 1

Intuition

Marginal Rate of Substitution (MRS)
$MRS=\frac{dy/du}{dx/du}$

Maths

$slop=-\frac{dy/du}{dx/du}$

Method 2

$du=\frac{\alpha u}{\alpha x}dx+\frac{\alpha u}{\alpha y}dy=0 \;\;(u\;constant)$
$\therefore \frac{dy}{dx}=-\frac{{MU}_{x}}{{MU}_{y}}$

Properties

1. slop negative, which means x and y are both good goods, not bad goods.
2. ICs can't cross.
3. IC convex.
4. For constant $\bar{y}$, as x increases, MRS decreases. (As x increases, y becomes more and more scarce, so need more x to exchange, thus MRS decreases.)

Functions

The usual form of function to describe the IC is the Cobb-Douglas function
$u={x}^{\alpha}{y}^{1-\alpha}$