Home > Micro concepts, Preferences and Indifference Curves > Marginal utilities and the marginal rate of substitution

## Marginal utilities and the marginal rate of substitution

January 19, 2012

The marginal rate of substitution is the rate at which the consumer is willing to substitute one good for another in order to retain the same level of utility.

Lets say our goods are are X and Y, and the total utility derived from having a bundle that is a combination of some X and some Y is U.

We will have a utility function of the form $U(X,Y)$.

The marginal utility we get by adding a unit more of X will be $\frac{\partial U}{\partial X} = MU_X$.

The marginal utility we get by adding a unit more of Y will be $\frac{\partial U}{\partial Y} = MU_Y$.

The marginal utility of X is also the change in total utility we get divided by the change in X, $MU_X =\frac{\Delta U}{\Delta X} = \frac{U(X+\Delta X, Y) - U(X,Y)}{\Delta X}$.

When the change in X is small then we can simply approximate to $\Delta U = MU_X \Delta X$, and by the same logic when the change in Y is small we can approximate to $\Delta U = MU_Y \Delta Y$.

If we were increasing both X and Y then the total change in utility would be $\Delta U = MU_X \Delta X + MU_Y \Delta Y$.

The concept of marginal rate of substitution is that it tells us how much we are willing to substitute of one good in order to get more of another, whilst keeping our overall utility constant. So the key thing here is that overall utility is being unchanged. This means that if for instance we are adding a unit more of X, then we are having to give up some of Y to make up for it.

So if overall utility is unchanged, $\Delta U = 0$ so $0 = MU_X \Delta X + MU_Y \Delta Y$. Hence $- MU_Y \Delta Y = MU_X \Delta X \Rightarrow \frac{\Delta Y}{\Delta X} = \frac{- MU_X}{MU_Y}$. $\frac{\Delta Y}{\Delta X}$ is $\frac{dY}{dX}$ so this gives us an expression for the marginal rate of substitution: it is simply the ratio of the marginal utilities. In the context of an indifference curve, this is the slope of the indifference curve, which makes sense as it is $\frac{dY}{dX}$.