Indifference Curve

An indifference curve depicts all the bundles that provides the same utility.

Example

Alice's utility given `X` chocolates and `Y` strawberries is

$$ u \left( X, Y \right) = X \times Y $$

Her indifference curve at a level of utility equal to 1 satisfies

$$ u \left( X, Y \right) = X \times Y = 1 $$

Solving for Y

$$ Y = 1 / X $$

Similarly, she gets utility 2 if `Y = 2 / X`.

Question

Alice's utility function is `u(X, Y) = X^2Y^3`.

What is the equation of her indifference curve for a level of utility equal to 8?

Draw that indifference curve.

The indifference curve satisfies `u(X, Y) = X^2Y^3 = 8`.

Solving for `Y`.

$$ \begin{align*} Y^3&= \frac{8}{X^2}\\Y &= \frac{8^{\frac{1}{3}}}{X^{\frac{2}{3}}}\\Y &= \frac{2}{X^{\frac{2}{3}}}\end{align*} $$