Constructing an indifference curve

When I was writing my previous post that applied the consumer choice model (otherwise known as the constrained optimisation model for the consumer), I noticed that in this earlier post applying the same model, I had promised a more detailed discussion of indifference curve. Some sixteen months on, it must be time for me to make good on that promise. So, here's the explanation that I have developed over a number of years, that I use to explain indifference curves in my ECONS101 class.

We'll start by limiting ourselves to one application of the constrained optimisation model - the model for consumer choices. Next, we need some assumptions. We'll assume that the goal of the consumer is to maximise their utility (their satisfaction, or happiness). We'll also assume that the consumer is only buying two goods, Good X and Good Y. And finally, we'll assume that more of each good is always better than less (so, having more of a good will always increase the utility of the consumer). [*]

We can represent the possible bundles of goods that the consumer might choose to consume in a diagram, as shown below. Consider one bundle of goods roughly in the centre of the diagram, Bundle B, which includes XB of Good X, and YB of Good Y.

Now, let's compare Bundle B with other bundles of goods that the consumer might choose. That is shown in the diagram below, by separating the diagram into four quadrants using dotted lines. Now, think about the comparison of Bundle B with bundles in those other quadrants. All of the bundles of goods that lie in the grey shaded quadrant up and to the right of Bundle B must be better than Bundle B (in the sense that they provide the consumer with more utility). That's because those bundles of goods either contain more of Good X, more of Good Y, or more of both goods. And more is always better (higher utility) than less. Next, all of the bundles of goods that lie in the grey shaded quadrant down and to the left of Bundle B must be worse than Bundle B (in the sense that they provide the consumer with less utility). That's because those bundles of goods either contain less of Good X, more of Good Y, or less of both goods. And because more is always better, less is always worse.

What about the other two quadrants? The bundles of goods in those quadrants are not obviously always better, or worse, than Bundle B. To get our head around those, let's draw a big circle around Bundle B, as shown in the next diagram below. Now, think about what happens in the comparison of bundles of goods, as we move anticlockwise around the circle, starting with Bundle D. Bundle D must be better than Bundle B (because Bundle D has more of Good X than Bundle B). That is also true of every bundle of goods as we move around the circle to Bundle E, which is also better than Bundle B (because Bundle E has more of Good Y than Bundle B). Then, we continue around the circle to Bundle F, which is worse than Bundle B (because Bundle F has less of Good X than Bundle B). Somewhere along the way between Bundle E and Bundle F, we moved from bundles of goods that are better than Bundle B to bundles of goods that are worse than Bundle B. So, somewhere along that part of the circle is a bundle of goods that is just as good as Bundle B (because it provides exactly the same amount of utility to the consumer as Bundle B does). Let's say that bundle is Bundle A.

Now, let's go back to our circle. Bundle F was worse than Bundle B. That is also true of every bundle of goods as we move around the circle to Bundle G, which is also worse than Bundle B (because Bundle G has less of Good Y than Bundle B). Then, we continue around the circle to and back to the start at Bundle D, which we recall is better than Bundle B. Somewhere along the way between Bundle G and Bundle D, we moved from bundles of goods that are worse than Bundle B to bundles of goods that are better than Bundle B. So, somewhere along that part of the circle is another bundle of goods that is just as good as Bundle B (because it provides exactly the same amount of utility to the consumer as Bundle B does). Let's say that bundle is Bundle C.

Now, we could repeat this exercise for other circles that are larger, or smaller, than the circle we drew above. And in every case, we would find that there are two bundles of goods on each of our new circles that are just as good as Bundle B - one bundle in the top left quadrant, and one bundle in the bottom right quadrant. If we then draw a curve that joins up all of those bundles of goods that are just as good as Bundle B (because they provide exactly the same amount of utility to the consumer as Bundle B does), we would have a curve that we call the indifference curve. That is shown in the diagram below, and is labelled I0.

It's called an indifference curve because the consumer is indifferent between any of the bundles of goods on that curve, because all of the bundles of goods on the curve provide the consumer with exactly the same amount of utility). If we gave the consumer the choice between Bundle A and Bundle B, they wouldn't care which one they were given - they are indifferent between those two options. Similarly, if we gave the consumer the choice between Bundle B and Bundle C, they would be indifferent. And if we gave the consumer the choice between Bundle A and Bundle C, they would be indifferent. And the same for any other bundle of goods on that indifference curve I0.

So, now you can see where we get indifference curves from. They are a necessary feature of all constrained optimisation models, not just the constrained optimisation model for the consumer. For example, in my ECONS101 class, we also briefly look at the constrained optimisation model of the worker, and the constrained optimisation model of the saver, and both of those feature indifference curves as well. Now, you may be wondering why we draw indifference curves as curves, rather than straight lines. I'll address that point in my next post.

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[*] If this assumption didn't hold, and having more of a good made a consumer worse off, it wouldn't be a good, it would be a bad. We can draw indifference curves for bads. The same principles apply, it's just that higher utility is not up and to the right anymore.