In the chapter, we develop the connection between the marginal rate of substitution and marginal utility by thinking about how utility from each good changes as the consumer moves a little bit along one of her indifference curves. Here, we work in the opposite direction by starting with utility and seeing how we can use calculus to find the MRS. In this way, we examine what happens when we make small changes to the consumption bundle.

Take the consumer who has utility for goods X and Y, U(X,Y). Any of the consumer’s indifference curves show how the consumer trades off good X for good Y while keeping utility constant. We can choose any one of these indifference curves and call its level of utility :

We are interested in how the utility from each good changes as we change the quantities of X and Y. So, we will totally differentiate the utility function, setting the total change in utility, dU, equal to zero because we are holding the level of utility constant:

The right-hand side of this equation, –dY/dX, is the negative of the slope of the indifference curve or the marginal rate of substitution. Therefore,