# 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^{475} Y^{92}.

What is her Marginal Rate of Substitution?

$$MU_X = \frac{du(X, Y)}{dX} = 475 X^{475 - 1} Y^{92} = 475 X^{474} Y^{92}$$

$$MU_Y = \frac{du(X,Y)}{dY} = 92 X^{475} Y^{92 - 1} = 92 X^{475} Y^{91}$$

Therefore $$MRS = - \frac{MU_X}{MU_Y} = - \frac{475 X^{474} Y^{92}}{92 X^{475} Y^{91}} = - \frac{475 Y}{92 X}$$