# Learning Economics in Three Dimensions

I am trying to teach myself microeconomics via video series. I have a fairly good mathematics foundation, currently studying Partial Differential Equations and having gone through all the prereqs you would expect. One thing that is massively annoying me, especially with regard to the geometry of microeconomics, is the feeling that I'm studying parts of an elephant, but not being shown the whole elephant. As an example, this series I'm watching frequently draws two graphs in $$\mathbb{R}^2$$ on a page, which seem to correspond to two perpendicular slices of a graph in $$\mathbb{R}^3$$, but the graph in $$\mathbb{R}^3$$ is never shown. Presumably, this is because on paper, drawing $$\mathbb{R}^2$$ is easier, but technology has evolved to the point that we no longer need to be constrained by what works on paper. I am seeking a presentation of economics that starts with the full three-dimensional picture and slices out planes as needed for finer analysis, a presentation that shows entire surfaces instead of flat contour plots, a presentation that uses computer animation to demonstrate graph movement over time instead of trying to cram dynamic phenomenon into static drawings.

Does this sort of introduction to economics (especially microeconomics, but perhaps also macroeconomics) exist, or does no one teach it in the above style? Is the three-dimensional picture usually focused on later in economics studies as a more advanced subject, or is it available in some educators' introductory presentations of the subject?

• For most economic problems, three dimensions are not that interesting either. A three-dimensional consumer problem is one in which a consumer consumes three goods. Surely, we consume more than that. Economists even study models with infinitely many commodities and, correspondingly, infinite-dimensional consumer problems. – Michael Greinecker Jun 1 at 6:20
• @MichaelGreinecker Well sure, three dimensions aren't ALWAYS interesting. A simple comparison of relations between $n$ goods may be clear using two. But it seems to me that the goal is often to understand, for example, how two goods relate to each other, and also how one of them relates to a consumer's income. And there are also contour plots used pretty frequently which could convey more information visually by being a surface instead. – user34915 Jun 1 at 20:45

Does this exist?

Of course the 3D representation exist. Below you can see 3D representation of Cobb-Douglas maximization problem with two factors and budget constrain, which is very similar to the problem solved in the video you linked (indeed you would probably not be able to find any qualitative difference in graph for that problem in the video even though this is problem from firm not consumer perspective), from Vuong (2013):

Is the three-dimensional picture usually focused on later in economics as a more advanced presentation, or is it available for introductory presentations of the subject?

Some introductory textbooks will show few 3D graphs, for example, Sydsaeter et al Essential Mathematics for Economics Analysis has few 3D examples.

Actually, paradoxically more advanced textbooks will often have less 3D examples. However, generally in graduate level textbooks you are dealing with cases where you have $$n$$ goods and making graphs in $$n$$ dimensional space would be impossible.

This being said, even in case where graduate textbooks show some simple examples they usually do it using 2D graphs. The 3D examples are nice eye candy but they do not have that much didactic value. While there might be some interesting topologies in some real-life cases, most textbook problems won't have any very interesting topology that would necessitate 3D graph. Once you understand the intuition that when doing constrained optimization with 2 goods there will be some utility sheet intersected and constrained by budget constraint plane (equivalent to picture above) there is not that much more didactic value in these plots and you usually gain more from having 2D representations of part of the problem.

• To clarify, I am not seeking resources which simply add an extra good to a two good problem merely to make things three-dimensional, but resources which combine related concepts often shown on two separate graphs each sharing an axis into a single graph, or pop out contour plots into surfaces, in order to display a more holistic picture of the concept. – user34915 Jun 1 at 20:55
• @user34915 I am not sure if I understand. 1. 2 good problem is already 3 dimensional, since you have to reserve 1 axis for utility. 2. the thing is that in 90% didactic problems you are working with well behaved functions. Often even one with simple one global maxima. The main point of contour plots is that they give you info about topology. Most entry level problems have no interesting topology, so the contour plots do not really offer much didactic value. You will see many examples of level curves but they are typically not colored because typically you are working with – 1muflon1 Jun 1 at 22:25
• well behaved continuously increasing utility. In fact those indifference curves you can see in the video with good 1 on x axis and good 2 on y are level curves of the Cobb-Douglass utility, but there is no point in coloring it to make it into contour plot. These level curves will never intersect and they will monotonically increase as you go from left to right so in that case coloring them would not give you additional info (except perhaps the shade of color informs you about how steep the increase is, but that is superfluous in theoretical problem) – 1muflon1 Jun 1 at 22:28

It's not done in two dimensions because it's easier to draw on paper. It's done in two dimensions because if you can grasp the concepts at work in that minimal $$n=2$$ variable setting, you can generalize the concepts from there to all $$n>2$$ variable settings. So, for pedagogical reasons of simplicity of teaching, instructors keep it to $$n=2$$. I don't have my copy of Varian handy, so I don't remember it's graphs (if any), but if you want the full math treatment of micro (that doesn't involve fundamental proofs of things like choice theory), go with him (https://wwnorton.com/books/9780393957358).