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^{500} Y^{415}`.
What is the Marginal Rate of Substitution of strawberries for a chocolate in function of X and Y?
$$
MU_X = \frac{du(X, Y)}{dX} = 500 X^{500 - 1} Y^{415} = 500 X^{499} Y^{415}
$$
$$
MU_Y = \frac{du(X,Y)}{dY} = 415 X^{500} Y^{415 - 1} = 415 X^{500} Y^{414}
$$
Therefore
$$ MRS = - \frac{MU_X \left( X, Y \right)}{MU_Y \left( X, Y \right)} = - \frac{500 X^{499} Y^{415}}{415 X^{500} Y^{414}} = - \frac{500 Y}{415 X} $$