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`: $$ MRS = - \frac{MU_X}{MU_Y} = - \frac{Y}{X} $$

Question

Alice's utility function is `u (X, Y) = X^{709} Y^{434}`.

What is her Marginal Rate of Substitution?

$$ MU_X = \frac{du(X, Y)}{dX} = 709 X^{709 - 1} Y^{434} = 709 X^{708} Y^{434} $$

$$ MU_Y = \frac{du(X,Y)}{dY} = 434 X^{709} Y^{434 - 1} = 434 X^{709} Y^{433} $$

Therefore $$ MRS = - \frac{MU_X}{MU_Y} = - \frac{709 X^{708} Y^{434}}{434 X^{709} Y^{433}} = - \frac{709 Y}{434 X} $$