Example
Alice's utility function is `u \left( X, Y \right) = XY`.
Her marginal rate of substitution is the ratio of the marginal utilities `MU_X` and `MU_Y`:
$$ MRS = - \frac{MU_X \left( X, Y \right)}{MU_Y \left( X, Y \right)} = - \frac{Y}{X} $$Question
Alice's utility function is `u (X, Y) = X^{668} Y^{810}`.
What is the Marginal Rate of Substitution of strawberries for a chocolate in function of X and Y?
$$
MU_X = \frac{du(X, Y)}{dX} = 668 X^{668 - 1} Y^{810} = 668 X^{667} Y^{810}
$$
$$
MU_Y = \frac{du(X,Y)}{dY} = 810 X^{668} Y^{810 - 1} = 810 X^{668} Y^{809}
$$
Therefore
$$ MRS = - \frac{MU_X \left( X, Y \right)}{MU_Y \left( X, Y \right)} = - \frac{668 X^{667} Y^{810}}{810 X^{668} Y^{809}} = - \frac{668 Y}{810 X} $$