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^{265} Y^{492}`.
What is the Marginal Rate of Substitution of strawberries for a chocolate in function of X and Y?
$$
MU_X = \frac{du(X, Y)}{dX} = 265 X^{265 - 1} Y^{492} = 265 X^{264} Y^{492}
$$
$$
MU_Y = \frac{du(X,Y)}{dY} = 492 X^{265} Y^{492 - 1} = 492 X^{265} Y^{491}
$$
Therefore
$$ MRS = - \frac{MU_X \left( X, Y \right)}{MU_Y \left( X, Y \right)} = - \frac{265 X^{264} Y^{492}}{492 X^{265} Y^{491}} = - \frac{265 Y}{492 X} $$