The Marginal Rate of Substitution (MRS)
By Steve Bain
The marginal rate of substitution (MRS) is a concept in economics that relates to the amount of one good that a consumer is willing to sacrifice in order to obtain an extra unit of another good. It is usually used in conjunction with indifference curve analysis, as a way of modelling consumer behavior.
When illustrated via a graph, we express the MRS in terms of how much of the good depicted on the vertical y axis is sacrificed in order to get an additional unit of the good depicted on the horizontal x axis. This is shown in the graph below.
Since much of the analysis on this page assumes an understanding of indifference curves, a quick refresher on that topic may be useful. If so, have a look at my main article at:
In the graph below, we start with a consumer's indifference curve in the two-good model. As usual this is a downward sloping curve, but it slopes downward at a diminishing marginal rate. In other words the curve gets flatter as the consumption of good x increases. Intuitively we can understand why this might be the case, because the more of good x that a consumer enjoys relative to his consumption of good y, the more desirable good y will be compared to good x.
Economists would express this as the consumer having diminishing marginal utility from increasing quantities of a given good. Another way to put it is that, for a fixed amount of utility (utility is fixed along any specific indifference curve), when a consumer has a large amount of one good, he/she will be willing to give up a larger amount of it in order to obtain an extra unit of the other good. That being the case the curve gets flatter as we move along it from left to right.
In the graph, we can calculate the marginal rate of substitution by drawing a straight line that tangentially touches the indifference curve at the consumer's chosen bundle of goods. That bundle occurs at a consumption rate of y for good Y, and x for good X (as shown via the black dashed lines). The straight red tangent line that touches the indifference curve at this consumption bundle has a slope equal to the MRS. We then use the simple geometry of a triangle to deduce that the slope is equal to the length of side a divided by the length of side b as illustrated in the graph.
Diminishing Marginal Rate
At any specific point along the curve, the MRS gets smaller as we move along it from left to right, because the MRS is equal to the slope of the indifference curve at any given point. With a consumption bundle of x,y in the graph below, the MRS line has a steep slope. When the consumer moves to a different bundle, with a change from x to x' and a change from y to y', the x'y' bundle yields a less steep MRS' line.
This illustrates the diminishing marginal rate of utility that the consumer gets from increasing amounts of x over y. As the curve gets flatter, the consumer will only wish to sacrifice a smaller and smaller amount of good y to get more of good x.
Marginal Rate of Substitution Example
To work through a simple marginal rate of substitution example, we need to use some mathematics. Economics is infamous for over-complicating its concepts by using advanced mathematics that are better suited to the physical sciences rather than economic science, but this one is very straight forward if you have a very basic grasp of calculus (if you don't have any knowledge of calculus, don't worry, just skip this section).
We start with a function that estimates the consumer's indifference curve. Let's say that, for quantities of good x between 1 and 16 units, consumption of good y can be approximated by the function: y = (x-20)^2
This quadratic equation can also be written in the form y = x^2 - 40x + 400. Now, using a first order derivative (dy/dx) we can calculate that the slope of the curve will be equal to 2x - 40. From the first equation i.e. y = (x-20)^2, we can calculate that when, for example, 2 units of good x are chosen, the consumer requires 324 units of good y to maintain his/her level of utility. At this point we use the first order derivative (2x - 40) to calculate that the MRS at this consumption bundle is -36. In other words, with 2 units of good x and an MRS of -36, the consumer is happy to give up 36 units of good y in order to get one more unit of good x.