Marginal Rate of Substitution (MRS), Marginal Utility (MU), and How They Relate

Posted by Emily on 2/3/17 5:52 PM

blog-2.jpgIntroduction

In this post, I start off explaining the Marginal Rate of Substitution (Sections II-IV). Then, I cover the concept of Marginal Utility (Sections V-VII). In both cases, I start with a story explanation, then give a formal definition, and finally provide some other useful information about the concept. After that, I connect the two concepts (Marginal Utility and Marginal Rate of Substitution) and show how they relate mathematically, first without calculus (Section VIII) and then with calculus (Section IX). Finally, I demonstrate that the Marginal Rate of Substitution has an advantage over Marginal Utility in terms of describing preferences and behavior (Section X), because it is less sensitive to the exact utility function you choose to use!

Story Explanation of the Marginal Rate of Substitution

Let’s imagine that I have some jelly beans and some M&Ms. I like both types of candy and I like having the choice between fruity and chocolatey, so I’m pretty happy right now. Now imagine someone comes along and wants one of my jelly beans. Maybe this person only wants half a jelly bean. The point is that the person wants a very very small amount of jelly beans.

If I give the person half a jelly bean, I’m a little less happy than I was before. But! The person could give me some amount of M&Ms that would make me exactly as happy as I was before I gave up that tiny bit of jelly beans. The amount of M&Ms that would make me exactly as happy might be one third of an M&M, it might be two M&Ms, or maybe it would be half an M&M. The point is, a very small amount of M&Ms would make me equally as happy as I was before, and this amount of M&Ms is not necessarily equal to the amount of jelly beans I gave up.

The Marginal Rate of Substitution captures the rate at which I would be willing to exchange a tiny bit of jelly beans for M&Ms.

Formal Definition of the Marginal Rate of Substitution

The Marginal Rate of Substitution (MRS) is the rate at which a consumer would be willing to give up a very small amount of good 2 (which we call Screen Shot 2017-02-03 at 2.25.48 PM.png) for some of good 1 (which we call Screen Shot 2017-02-03 at 2.25.58 PM.png) in order to be exactly as happy after the trade as before the trade. Let Screen Shot 2017-02-03 at 2.27.19 PM.png and Screen Shot 2017-02-03 at 2.27.24 PM.png be very small changes (e.g. “marginal” changes) in Screen Shot 2017-02-03 at 2.25.58 PM-1.png and Screen Shot 2017-02-03 at 2.25.48 PM-1.png. Then, the MRS equals Screen Shot 2017-02-03 at 2.30.19 PM.png.

Note that the MRS is negative, because we are giving up some of Screen Shot 2017-02-03 at 2.25.48 PM-2.png(soScreen Shot 2017-02-03 at 2.27.24 PM.png is negative) to get some ofScreen Shot 2017-02-03 at 2.27.19 PM.png (soScreen Shot 2017-02-03 at 2.27.19 PM.png is positive). A negative divided by a positive is a negative, so it follows that the MRS is negative.

Relationship Between the MRS and Indifference Curves

At any given point along an indifference curve, the MRS is the slope of the indifference curve at that point. Note that most indifference curves are actually curves, so their slopes are changing as you move along them. That means that the MRS is also changing!

To find the slope of a curve at a specific point, you use calculus. Take the first derivative of the equation for the indifference curve, then plug in the values of Screen Shot 2017-02-03 at 2.25.58 PM-1.png andScreen Shot 2017-02-03 at 2.25.48 PM-2.png for the point you are interested in. That will give you the MRS at that point.

What do you think happens to the MRS along the indifference curve? When I have a lot ofScreen Shot 2017-02-03 at 2.25.48 PM-2.png, I’m willing to give up quite a bit ofScreen Shot 2017-02-03 at 2.25.48 PM-2.png to get a little bit of Screen Shot 2017-02-03 at 2.25.58 PM-1.png. However, this changes as I move along my indifference curve. When I get to a point where I’m just as happy as before but now I have tons ofScreen Shot 2017-02-03 at 2.25.58 PM-1.png and almost noScreen Shot 2017-02-03 at 2.25.48 PM-2.png, I no longer want to give up muchScreen Shot 2017-02-03 at 2.25.48 PM-2.png to get a littleScreen Shot 2017-02-03 at 2.25.58 PM-1.png. This phenomenon is known as the diminishing rate of marginal substitution.

The Marginal Rate of Substitution (MRS) is the slope of the indifference curve

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Story Explanation of the Marginal Utility

Let’s imagine again that I have some jelly beans and some M&Ms. If someone takes a tiny (“marginal”) amount of jelly beans away from me, I’m slightly less happy. Similarly, if someone gives me a tiny bit more jelly beans, I’m a little happier. My marginal utility of jelly beans is the change in happiness I experience from a tiny (e.g. “marginal”) change in the amount of jelly beans I have.

Similarly, my happiness (which economists call “utility”) would change if someone changed the amount of M&Ms I had. When the change in M&Ms is tiny (“marginal”) then the resulting change in my utility is known as my marginal utility of M&Ms.

Note that in both cases, marginal utility is defined with respect to a specific type of candy that I have. We considered the marginal utility of jelly beans and the marginal utility of M&Ms. We can combine these ideas to figure out what would happen if I experienced simultaneous changes in the amount of jelly beans and M&Ms in my possession, but marginal utility is always defined with respect to a specific good. 

Formal Definition of Marginal Utility

The marginal utility with respect to good 1 is the change in utility a consumer experiences when the amount ofScreen Shot 2017-02-03 at 2.25.58 PM-1.png the consumer has changes by a tiny bit while the amount ofScreen Shot 2017-02-03 at 2.25.48 PM-2.png the consumer has remains constant. We can represent this marginal utility as Screen Shot 2017-02-03 at 2.41.16 PM.png. Therefore, Screen Shot 2017-02-03 at 2.42.02 PM.png is the rate of change in utility Screen Shot 2017-02-03 at 2.42.42 PM.png resulting from a small change in good 1 (Screen Shot 2017-02-03 at 2.25.58 PM-1.png).

Similarly, the marginal utility with respect to good 2 is the rate at which utility changes when the consumer’s amount ofScreen Shot 2017-02-03 at 2.25.48 PM-2.png is changed by a marginal amount while his/her amount ofScreen Shot 2017-02-03 at 2.25.58 PM-1.png remains fixed at a constant amount. The equation for Screen Shot 2017-02-03 at 2.44.15 PM.png is Screen Shot 2017-02-03 at 2.45.06 PM.png.

Marginal utility will always be positive when we are dealing with goods (as opposed to bads or neutrals). This is because getting more will make us happier, so when the denominator (Screen Shot 2017-02-03 at 2.27.19 PM.png) is positive, the numeratorScreen Shot 2017-02-03 at 2.42.42 PM.png is also positive. Similarly, when we lose some of good 1,Screen Shot 2017-02-03 at 2.27.19 PM.png is negative and we are less happy, so Screen Shot 2017-02-03 at 2.47.28 PM.png is also negative. A negative divided by a negative is positive, so the marginal utility of a good will always be a positive value.

Marginal Utility v. Actual Change in Utility

Note that in both cases, we can do a little algebra to find the total change in utility resulting from a marginal change in one good while the amount of the other good is held constant.

Let’s use good 1 as our example. Above, we saw that Screen Shot 2017-02-03 at 3.54.29 PM.png. If we multiply both sides byScreen Shot 2017-02-03 at 2.27.19 PM.png, then we have Screen Shot 2017-02-03 at 3.55.25 PM.png. Therefore, the change in utility resulting from a tiny change in good 1 and no change in good 2 is just the product of that tiny change in good 1 and the marginal utility with respect to good 1.

Connection Between Marginal Utility & Marginal Rate of Substitution

The Marginal Rate of Substitution looks at the balance in changes of good 1 and good 2 required for the consumer to be indifferent between his/her consumption bundles before and after trade. But what does indifference mean? It means that utility for both bundles is exactly equal. Therefore, Screen Shot 2017-02-03 at 3.56.34 PM.png

There is some (negative) change in utility resulting from giving up a little bit of good 2, and as we saw in the previous section, this change equals Screen Shot 2017-02-03 at 3.57.52 PM.png Similarly, there is some (positive) change in utility from getting a little more of good 1, which equals Screen Shot 2017-02-03 at 3.58.40 PM.png Since we want to be indifferent before and after the trade, it must be that the sum of these changes equals zero. That is, Screen Shot 2017-02-03 at 3.59.57 PM.png With a little algebra, we can find the MRS from this equation of marginal utilities!

First, subtract Screen Shot 2017-02-03 at 5.24.19 PM.png from both sides. Then, Screen Shot 2017-02-03 at 5.23.55 PM.png Next, divide both sides byScreen Shot 2017-02-03 at 2.27.19 PM.png and byScreen Shot 2017-02-03 at 2.44.15 PM.png. The result is Screen Shot 2017-02-03 at 5.26.06 PM.png The left hand side is just the MRS, and the right hand side is the negative ratio of marginal utilities. In the MRS section, we learned why the left hand side would automatically be negative. The right hand side needs the negative sign because marginal utility is positive for goods, so the ratio of marginal utilities is always positive.

MRS and Marginal Utility Relationship – Calculus Edition

When using calculus, the marginal utility of good 1 is defined by the partial derivative of the utility function with respect toScreen Shot 2017-02-03 at 2.27.19 PM.png. That is, Screen Shot 2017-02-03 at 5.27.43 PM.pngWe want to consider a tiny change in our consumption bundle, and we represent this change as Screen Shot 2017-02-03 at 5.28.49 PM.pngWe want the change to be such that our utility does not change (e.g. Screen Shot 2017-02-03 at 5.29.32 PM.png). Therefore, we want to solve Screen Shot 2017-02-03 at 5.30.22 PM.pngRearranging terms as before, we find Screen Shot 2017-02-03 at 5.31.24 PM.png which is just the calculus version of Screen Shot 2017-02-03 at 5.32.35 PM.png

Instead of using derivatives, we could use implicit functions. Any given indifference curve can be represented as Screen Shot 2017-02-03 at 5.33.43 PM.png where Screen Shot 2017-02-03 at 5.34.58 PM.png is a constant and the level of utility held constant along the indifference curve. We use the notation Screen Shot 2017-02-03 at 5.36.28 PM.png simply to illustrate that Screen Shot 2017-02-03 at 5.37.15 PM.png is a function ofScreen Shot 2017-02-03 at 2.25.58 PM-1.png.

If we differentiate both sides of Screen Shot 2017-02-03 at 5.38.16 PM.png with respect toScreen Shot 2017-02-03 at 2.25.58 PM-1.png, we get: Screen Shot 2017-02-03 at 5.38.57 PM.png

We can again rearrange terms and the result is the same as what we found before: Screen Shot 2017-02-03 at 5.39.50 PM.png

Monotonic Transformations Affect MU but Not MRS

The downside of marginal utility is that its magnitude depends on the utility function we’re using. This is not ideal, because utility functions are usually ordinal, which means we don’t care exactly what numbers the utility function spits out, we just care that the utility function gives us higher numbers for bundles the consumer likes better.

The great thing about the MRS is that even though it is function of the marginal utilities with respect to goods 1 and 2, it doesn’t change if apply a positive monotonic transformation to our utility function. (Positive monotonic transformations are any functions that preserve the original order when applied, like adding a constant to the original utility function, raising the original utility function to an odd power, taking the natural log, etc.) To see why this is so, let’s pretend Screen Shot 2017-02-03 at 5.43.23 PM.png was our original utility function and Screen Shot 2017-02-03 at 5.44.12 PM.png is our monotonically transformed utility function (so Screen Shot 2017-02-03 at 5.44.40 PM.png is a monotonic function). Then, using our calculus definition of the MRS, we have Screen Shot 2017-02-03 at 5.46.41 PM.png before the transformation and Screen Shot 2017-02-03 at 5.47.23 PM.png

So the MRS is completely unchanged by any monotonic transformation!

Acknowledgements: Much of this post was inspired by chapters 3 and 4 of Hal Varian’s textbook Intermediate Microeconomics: A Modern Approach.

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Tags: economics