Motivating linear algebra for economics students?

I'm a tutor for the introductory linear algebra course at my school; this course is required for most upper division economics classes, so a lot of my tutees are economics majors.

This is a typical linear algebra course that focuses on things like linear dependence, subspaces, eigenvalues, etc. and does not spend time on "practical applications". As a result, a lot of the economics students have no idea why they should be taking the course. Since I don't know the first thing about economics, I also have no idea why they should be taking the course.

Is it possible to convince economics students that linear algebra is important for their field? More precisely:

Are there any motivating examples where linear algebra is used in economics?

• Calculating the price of your shopping is the same thing as evaluating an element of the dual base of a free $\mathbb Z$-module on a vector of the module. I do suspect that this is not what you had in mind, however. ;) – Gunnar Þór Magnússon Jul 9 '11 at 6:29
• Linear programming? Really, linear algebra is so ubiquitous it's hard to come up with examples :D – Alexei Averchenko Jul 9 '11 at 6:36
• Linear programming is used as a problem solving tool. The sought level of understanding may leave something to be desired. A colleague took a position at a school of economics, and showed me some of their (former) course material. A linear programming problem was described all right, as were several motivating examples. The 'theory' consisted of an explanation of how to use M$Excel to find an optimal solution. May be you can do a toy example, though? – Jyrki Lahtonen Jul 9 '11 at 6:39 • Why the -1? Anything I can do to improve this question? – Elliott Jul 9 '11 at 7:56 • Think about Control Theory as the light at the (somewhat) end of the tunnel. – r.g. Jul 9 '11 at 11:15 6 Answers In linear regression linear algebra is used to determine the coeffecients of the predictor equation from the data. Linear regression is the backbone of econometrics. In modern Portfolio Theory the optimal portfolio is defined in terms of the covariance matrix of asset returns, and the expected volatility of the portfolio is a quadratic form. Given a matrix showing how each of$n\$ sectors depends on resources from the others, the intermediate consumption and demand of each sector is expressed by solving a system of linear equations (i.e. inverting a matrix).

Apart from these examples, I use linear algebra every day at work - I work in high frequency trading for an investment bank. So it's clearly relevant, at least for some people, some of the time.

A standard example is the Leontief input-output model, see http://en.wikipedia.org/wiki/Input-output_model

They may well be doing it to raise their expected incomes (typically; not in every case, but typically).

So, now the question is: can you find evidence that knowing linear algebra will increase their expected incomes? Have a look at some job ads for econometricians, financial modellers, quant analysts in financial centres; jobs where having a good handle on linear algebra and optimisation are essential. You should be seeing salaries a long way above the norm.

So there's your answer - appeal to the economist in each of them. Do it right, and you might even be able to shoehorn the motivation into the explanation of a basic model: create an equation to predict their future salary, and add some variables, one of which is knowledge of linear algebra. [edit:] I think a linear regression model of their future salaries would be a good way to introduce them to the application of maths to economics, and a good way to motivate them, at the same time [end edit]

So you'll use linear algebra to show them the economic benefits of learning linear algebra.

• I think the OP is interested in (easy examples of) applications of linear algebra to economics. I don't think "knowing linear algebra increases your expected income" qualifies as such. Did you mean to imply that the "equation to predict their future salary" should involve some linear algebra? – Rasmus Jul 9 '11 at 12:56
• Yes - I think a linear regression model of their future salaries would be a good way to introduce them to the application of maths to economics, and a good way to motivate them, at the same time. Although it's a question about maths, it's also a question about pedagogy and motivating students. I regret that my highlighting of the non-maths aspects irritates some people, but I think the OP is better served by an answer that considers both the maths and the motivation aspects together. – EnergyNumbers Jul 9 '11 at 13:21
• -1 As an economist, I think it is somewhat insulting that this is supposed to be the main motivation. Pretty much everyone I studied with was driven by intelectual curiosity or an urge to make the world a better place. – Michael Greinecker Jul 10 '12 at 6:24

As already mentioned, the most important use of linear algebra for economists is for dealing with linear econometric models. For a sound geometric understanding of them, students should ideally learn about orthogonal projections. Excellent introductions to the geometry of least squares estimation can be found in the first chapters of An Introduction to Classical Econometric Theory by Paul A. Ruud and Econometric Theory and Methods by Russell Davidson.

The second main use of linear algebra for economics students is as a foundation for multivariate calculus and optimization.

There are of course many other uses of linear algebra. David Gale has written a beautiful book on The Theory of Linear Economic Models. But if you go dow that road, you will also have to motivate the point of simplifying mathematical models in economics too. Otherwise, you might alienate the students even more. I'm not sure this is a good investment of time for such a course.

Lastly, even economics students will think that PageRank is fun.

Keeping aside the fact that working with vectors and matrices is handy in advanced courses, the most used application is probably ODEs and dynamics. Economists are obsessed with steady states and they want them to be (asymptotically) stable in their models: think of determining stability based on eigenvalues or solving linear ODEs.

Second, students are for sure going to need linear algebra in econometrics. OLS requires linear independence of the regressors and there are even (advanced) tests based on eigenvalues, etc.

• I don't think it's fair to say that economists are "obsessed" with steady states. For sure, neoclassical equilibrium economics studies steady states. But more and more work is focused on non-equilibrium outcomes, for example in evolutionary game theory. – Chris Taylor Jul 10 '12 at 8:37

This may be more of a niche motivation and is meant to complement rather than replace the above, but for those economics students who like economic theory, linear algebra is important because the a great deal of math that is interesting to economic theory eventually becomes linear algebra.

Algebraic topology is clearly relevant (if a topic economics have classically shied away from) due to the connections to fixed point theory.

Differential methods (say, even at the level of the implicit function theorem) rely on linear algebraic notions of the being able to invert the Jacobian matrix.

Any student of economics with the curiosity to 'take apart' the tools they'll learn as economists will inexorably be led to linear algebra.