Marginal Rate of Substitution

The Marginal Rate of Substitution (MRS) tells how much a unit of a good is worth to the consumer in terms of the other good. $$MRS = - \frac{MU_X}{MU_Y}$$

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^{805} Y^{456}`.

What is the Marginal Rate of Substitution of strawberries for a chocolate in function of X and Y?

$$ MU_X = \frac{du(X, Y)}{dX} = 805 X^{805 - 1} Y^{456} = 805 X^{804} Y^{456} $$ $$ MU_Y = \frac{du(X,Y)}{dY} = 456 X^{805} Y^{456 - 1} = 456 X^{805} Y^{455} $$

Therefore

$$ MRS = - \frac{MU_X \left( X, Y \right)}{MU_Y \left( X, Y \right)} = - \frac{805 X^{804} Y^{456}}{456 X^{805} Y^{455}} = - \frac{805 Y}{456 X} $$