Teaching Economics Without Calculus

[bhobba]

Just a quick question. Can you really teach economics without calculus? Reviewing it now just to refresh my memory since we have a budget being delivered tomorrow here in Aus. I have reached the point where it is proven that maximizing overall profit is different from profit per unit of whatever you are considering - you can maximize one or the other - but not necessarily both - obviously you usually want to maximize overall profit. It is easy with calculus but without it how the bejesus would you do it. And for those that studied without calculus I wonder how many even know its true?

BTW for those interested the course I am using for review is Economics With Calculus on Edx exactly as taught at Caltec,

Thanks
Bill

 

[jedishrfu]

Well one way to solve is to ask someone who knows Calculus...

Another way might be by numerical methods where you try to tune your parameters until you get the best result.

A third way might be via genetic algorithms (similar to the preview idea but better organized) where you create solution vectors that you score keeping the better scored solutions after each iteration.

 

[vela]

As my econ prof would tell the class, think marginal, i.e., consider what happens when you increase or decrease production by one unit.

 

[bhobba]

Well one way to solve is to ask someone who knows Calculus...

I think those doing more advanced Economics can't escape calculus. I certainly learned it in Mathematical Methods Of Economics and calculus was a required perquisite. Strangely I was the only student which says a lot. Actuaries of course do it. Very in demand course as the pay is so high and job prospects so good - but without a stimulus like that it seems unloved. Advanced degrees in economics must have it - no choice. Just disappointing its not at the early level nearly everyone does at uni so they understand something every citizen should know. The economic stuff you hear certainly would, hopefully anyway, be elevated.

Thanks
Bill

 

[jedishrfu]

While not directly related to this question, here are some lectures using python and julia to do economics problems.

https://lectures.quantecon.org/

 

[Helios]

I've never had a mathematical economics course. What is the very greatest thing that calculus has ever done for economics? What tops the list as the one greatest feat and that calculus was the hero?

 

[gmax137]

My only Econ course was 101 - the intro. I remember thinking that this whole course could be taught in a few weeks if we used calculus instead of words.

 

[jasonRF]

The short answer is yes, it is possible to learn basic economics without calculus. At least in the US, it is typical for introductory level courses (even those for economics majors) to use no calculus. I certainly took courses that worked that way. Would it be more efficient with calculus? Sure, but you can still learn quite a bit and get an understanding for how things work and why without calculus. A deeper understanding probably does require extra tools like calculus, statistics, etc. which is why upper division courses do require a calculus background and economics students need to learn serious probability and statistics as well.

You can even work as an economist without using much, if any, calculus. My father was an economics professor, published regularly, and had a successful consulting business. Very little of his research and consulting required any calculus. He seemed to think that one of the primary benefits of studying economics is that you learn distinct, useful ways to formulate and analyze problems that was not just mathematics. It is a social science after all. For the types of problems he was interested in, a lot of math just wasn't required. Of course, he advised his students to take as much math as possible if they planned to go to grad school.

jason

 

[Helios]

I asked what the one great thing that calculus did for economics and no one answered, and that was two days of thinking time. I think that calculus isn't the first thing that comes to mind when it comes to economics.
People don't act and chose, buy and sell, according to calculus. The criticism the calculus-economist faces is the factuality of his models. Do they represent reality?
When calculus was introduced to the study of planetary motion, the physics immensely improved. The same thing cannot be said for economics. The calculus really didn't solve any mystery or give us any new understanding of economics.
I don't see economics getting to be called "advanced economics" just because it loads on a bunch of advanced mathematics. I don't see economics getting called "caveman economics" just because it omits calculus. Math can make economics look like physics, but we should consider that there's a false sophistication economics can assume if it misuses or excessively uses mathematics. There are certainly a lot of economic quacks with their mathematical fantasies and there is a lot of institutional quack economics that goes unchallenged.

 

[gmax137]

I think that calculus isn't the first thing that comes to mind when it comes to economics.

Well, it was the first thing that came to my mind as I sat in Econ101 and heard the professor saying things like "ceteris paribus" and "elasticity of demand"

he criticism the calculus-economist faces is the factuality of his models. Do they represent reality?

That's pretty much the question for any model, isn't it?

I can't really speak to any of your other points.

 

[Sorcerer]

At my school we had economics students taking analysis, let alone calculus. Analysis is, IMHO, the grown up version of calculus, where you rigorously prove every little piece of calculus bit by bit in very general ways starting from scratch.

Remember the epsilon delta stuff everyone glossed over in calculus? In analysis you take that to the extreme, based on the text I have, and it makes elementary calculus look like child’s play.

And economics students were taking that course at my school (although to be fair, I think they were first year post grads or senior undergrads).

 

[gibberingmouther]

At my school we had economics students taking analysis, let alone calculus. Analysis is, IMHO, the grown up version of calculus, where you rigorously prove every little piece of calculus bit by bit in very general ways starting from scratch.

Remember the epsilon delta stuff everyone glossed over in calculus? In analysis you take that to the extreme, based on the text I have, and it makes elementary calculus look like child’s play.

And economics students were taking that course at my school (although to be fair, I think they were first year post grads or senior undergrads).

i just today decided to start a review of calculus with the intent of being more theoretical about it this time (at least looking at proofs, and trying to understand the logic behind the definitions and theorems). i got stuck on the epsilon and delta using definition of the limit. i'll probably take a second look at it tomorrow. it sounds like i might have trouble with analysis then if it is as you say ... that and abstract algebra are subjects i know little of but have a strong curiosity about. got to buff up on regular calc. and trig. before i get into that though!

anyway, i have never taken an economics course and i doubt i will because of the cost, unless i can take it as a required elective. can anyone suggest books to read or textbooks that are good for getting an introduction to the subject? if i ever have time i would like to learn a little about it, out of curiosity and so i can become a more educated citizen. i do have "New Ideas From Dead Economists" and "Freakonomics" - these are the only two economics related books i have. is it a really hard subject to learn?

 

[bhobba]

I've never had a mathematical economics course. What is the very greatest thing that calculus has ever done for economics? What tops the list as the one greatest feat and that calculus was the hero?

I gave an example - your view of it is? ie can you explain it easily, in a few lines without calculus? I am not sure how you would explain it at all without calculus - but willing to listen to answers.

BTW the answer is things are easier if you use calculus often only taking a couple lof lines. As an example explain optimizing some utility function without calculus? Trivial with calculus - otherwise - I have no idea.

It's called advanced economics because it often has concepts you can't do without calculus eg how do you explain to concept of force of interest which is used in more advanced work.

Thanks
Bill

 

[bhobba]

My only Econ course was 101 - the intro. I remember thinking that this whole course could be taught in a few weeks if we used calculus instead of words.

When I did Mathematical Economics strictly speaking I would have had to do intro to Macroeconomics and Microeconomics. But my professor said you can read about it over the break - just get a calculus based book. Non calculus based books, for example, so he claimed, spend the first chapter discussing a straight line - with your background that would just be boring and of no value. The course I am currently doing does it all in 9 sections - say a section a day and its a week and a bit - I learned it all in about 2 weeks over the break with just a couple of hours each day before doing the course at uni.

Thanks
Bill

 

[bhobba]

Analysis is, IMHO, the grown up version of calculus, where you rigorously prove every little piece of calculus bit by bit in very general ways starting from scratch.

Discussing why you should do analysis really requires a new thread. But only as a comment it allows you to understand issues that would otherwise confuse a thinking student eg exactly how do you resolve Zeno's paradox. You sometimes see questions on this forum that Zeno's paradox is an unresolved problem. I did a math degree - most students hated it - only a few nuts like me took to it.

Thanks
'Bill

 

[bhobba]

i got stuck on the epsilon and delta using definition of the limit.

That's analysis - most people find it hard so don't worry if you do. You just need a bit of calculus for beginning economics - the following will suffice:
https://www.amazon.com/dp/0471827223/?tag=pfamazon01-20

Then do the course I am doing
https://www.edx.org/course/principles-economics-calculus-caltechx-ec1011x-0

Its free.

Thanks
Bill

 

[bhobba]

Very little of his research and consulting required any calculus.

The use of calculus is an understanding thing - in practice you use computer programs to do the grunt work such as give the results of some complicated model or pound out statistics. What was it, I think it was Marvin Minsky said, when asked what programming language do you use - he said - graduate student.

Being a professor he would have access to packages like SAS, Simula etc that do the grunt work for him - it's uderstanding what they say and what implicit assumptions they make - now that's hard. He would have learned the theory behind that understanding and that would have required calculus and advanced statistics - for which, if you do it properly requires calculus as well. But having gone beyond that he would rarely need it.

And yes Economics is not just applied math - it significantly requires a good intuitive understanding as well. Even the course I am doing emphasized it does not matter what the math says you must subject it to the intuitive understanding you have developed - and a big part of the course is developing that understanding. Somethings however, like what I mentioned about profit per item and overall profit are not intuitive - that's when one must triple check the math to make sure its true.

Thanks
Bill

 

[Auto-Didact]

In the universities in most European countries, only in STEM courses does calculus get taught, with the caveat that sciences stands for the physical sciences. Applied/technical mathematics students get the same calculus course as physics students (Calc I, II and III), while pure mathematics students get taught analysis as well. Computer science students and econometrics students usually only have to take Calc I with Calc II and III remaining as electives.

All other sciences (life, social, cognitive, etc) including economics, either do not get taught calculus or get a heavily simplified version without any serious mathematical theory such as rigorous definitions about continuity, theorems, the epsilon-delta definition of the limit and so on.

They basically get taught how to differentiate and how to integrate simple functions by doing a bunch of problems following the sum, product and chain rules; this is not unlike how high school level calculus is taught except at a higher pace. Usually this course is combined with a heavily simplified and shortened version of linear algebra as well.

The only two semi-rigorous concepts they sometimes do get taught in this course, much earlier than physics/math students, are 1) a truncated version of Taylor expansion, wherein a function needs to be constructed based on the relevant quantities in the function given as part of a word problem, and 2) constrained optimization using Lagrangian multipliers.

 

[bhobba]

All other sciences (life, social, cognitive, etc) including economics, either do not get taught calculus or get a heavily simplified version without any serious mathematical theory such as rigorous definitions about continuity, theorems, the epsilon-delta definition of the limit and so on.

Well here in Aus it depends on the course. One course you do calculus to quite an advanced level and use it in economics and finance, and pass rigorous exams on it, is Actuarial Science.

The typical prepatory math they take is the same as any science or math student in first year - or what godd students do if they take accelerated math at HS:
https://handbook.unimelb.edu.au/subjects/mast10008
https://handbook.unimelb.edu.au/subjects/mast10009

The reason of course is those actuarial exams are both rigorous - hence the math - and tough - they really only want dedicated students.

Thanks
Bill

 

[Sorcerer]

Discussing why you should do analysis really requires a new thread. But only as a comment it allows you to understand issues that would otherwise confuse a thinking student eg exactly how do you resolve Zeno's paradox. You sometimes see questions on this forum that Zeno's paradox is an unresolved problem. I did a math degree - most students hated it - only a few nuts like me took to it.

Thanks
'Bill

Yeah it does, but I DID notice something cool looking at analysis (at least with the textbooks I own): it sometimes carries a resemblance to topology. It’s been a while since I looked at those books, but I distinctly remember an “oh my” moment when I realized that, at least in that particular case (which I regret I do not recall), it appeared the two were more or less doing the same thing.


Edit- I think it involved a particular proof, and I think the topology relevance was related to neighborhoods, but as I said, I can’t recall. But I swear it felt like more or less a different take on the same process.

 

[andrewkirk]

People don't act and chose, buy and sell, according to calculus.

Yes they do.

Derivatives traders use calculus to determine prices at which they will trade a given security.
Investors use calculus to work out how much to hedge their risk.
Large, sophisticated manufacturers use calculus to determine which products to manufacture in order to maximise their profit.

I note that these are all microeconomic applications of calculus. Perhaps you are thinking more of macroeconomics. It is certainly the case that many uses of calculus in macroeconomics are questionable, but that is because the assumptions made are always so debatable. That's really a feature of macroeconomics generally rather than specifically the use of calculus in macroeconomics.

 

[Sorcerer]

In the universities in most European countries, only in STEM courses does calculus get taught, with the caveat that sciences stands for the physical sciences. Applied/technical mathematics students get the same calculus course as physics students (Calc I, II and III), while pure mathematics students get taught analysis as well. Computer science students and econometrics students usually only have to take Calc I with Calc II and III remaining as electives.

All other sciences (life, social, cognitive, etc) including economics, either do not get taught calculus or get a heavily simplified version without any serious mathematical theory such as rigorous definitions about continuity, theorems, the epsilon-delta definition of the limit and so on.

They basically get taught how to differentiate and how to integrate simple functions by doing a bunch of problems following the sum, product and chain rules; this is not unlike how high school level calculus is taught except at a higher pace. Usually this course is combined with a heavily simplified and shortened version of linear algebra as well.

The only two semi-rigorous concepts they sometimes do get taught in this course, much earlier than physics/math students, are 1) a truncated version of Taylor expansion, wherein a function needs to be constructed based on the relevant quantities in the function given as part of a word problem, and 2) constrained optimization using Lagrangian multipliers.

As I said earlier, at my school we had economics grad students taking analysis and other upper division and early grad math courses. We’re they doing it as electives? I don’t know. But I do know that peer reviewed publications in economics sometimes have some fairly sophisticated math in them.

 

[Auto-Didact]

As I said earlier, at my school we had economics grad students taking analysis and other upper division and early grad math courses. We’re they doing it as electives? I don’t know. But I do know that peer reviewed publications in economics sometimes have some fairly sophisticated math in them.

I was describing the standard undergraduate program, given that calculus and analysis are after all undergraduate subjects. Hell, when I was taking calculus there was a senior psychology student taking analysis with the mathematics students, wiping the floor with my freshmen physics colleagues.

Of course, there are also some specialized undergraduate and graduate programs which expand significantly upon mathematics training outside of physics/math but these tend to be fairly rare, specific to only particular universities and/or specialized graduate courses or tracks such as quantitative finance.

As for the advanced math in economics publications, that's no big surprise given that many big name researchers in the late 20th century were/are applied mathematicians and/or physicists or economics grad students under mathematicians/physicists (e.g. Fischer Black from the Black-Scholes equation and Eugene Fama under the influence of Benoit Mandelbrot). Moreover many economists and other social scientists that do end up being quantitative researchers working in academia will eventually learn much of the math required, whether or not they formally learned it at university.

 

[Sorcerer]

I was describing the standard undergraduate program, given that calculus and analysis are after all undergraduate subjects. Hell, when I was taking calculus there was a senior psychology student taking analysis with the mathematics students, wiping the floor with my freshmen physics colleagues.

Of course, there are also some specialized undergraduate and graduate programs which expand significantly upon mathematics training outside of physics/math but these tend to be fairly rare, specific to only particular universities and/or specialized graduate courses or tracks such as quantitative finance.

As for the advanced math in economics publications, that's no big surprise given that many big name researchers in the late 20th century were/are applied mathematicians and/or physicists or economics grad students under mathematicians/physicists (e.g. Fischer Black from the Black-Scholes equation and Eugene Fama under the influence of Benoit Mandelbrot). Moreover many economists and other social scientists that do end up being quantitative researchers working in academia will eventually learn much of the math required, whether or not they formally learned it at university.

Now that is impressive. Usually it works the other way around.

 

[Kurt Heckman]

I believe Economics (Micro and perhaps Macro) are now required for every university degree in the United States. I helped review the Microeconomics textbook for one college and there was no calculus requirement there. A student put all of the math formulas in vCalc's Microeconomics Calculator. The most complicated math was the formula for Midpoint Method of Price Elasticity of Demand :

PED= [(Q2−Q1)÷(Q2+Q1)/2] / [(P2−P1)÷(P2+P1)/2]

where:

• PED is the Price Elasticity of Demand
• P1 this is the first price point
• P2 this is the second price point
• Q1 is the quantity point associated with the first price point (P1)
• Q2 is the quantity point associated with the second price point (P2)

 

[FactChecker]

I asked what the one great thing that calculus did for economics and no one answered, and that was two days of thinking time. I think that calculus isn't the first thing that comes to mind when it comes to economics.

I guess one could avoid calling it calculus, but how can a person do much in economics without discussing rates (derivatives) and cumulative amounts (integrals)?

 

[Sorcerer]

I asked what the one great thing that calculus did for economics and no one answered, and that was two days of thinking time. I think that calculus isn't the first thing that comes to mind when it comes to economics.
People don't act and chose, buy and sell, according to calculus. The criticism the calculus-economist faces is the factuality of his models. Do they represent reality?
When calculus was introduced to the study of planetary motion, the physics immensely improved. The same thing cannot be said for economics. The calculus really didn't solve any mystery or give us any new understanding of economics.
I don't see economics getting to be called "advanced economics" just because it loads on a bunch of advanced mathematics. I don't see economics getting called "caveman economics" just because it omits calculus. Math can make economics look like physics, but we should consider that there's a false sophistication economics can assume if it misuses or excessively uses mathematics. There are certainly a lot of economic quacks with their mathematical fantasies and there is a lot of institutional quack economics that goes unchallenged.

Well, based on what very little I know of economics, it makes things easier to look at.

Price elasticity of demand, for example, is easier to see like this:

$$e_p = \frac{\frac{dQ}{Q}}{\frac{dP}{P}}$$

than other ways I've seen it written. How much does demand change if price changes a little bit? Calculus is the exact branch of math to efficiently explore that.

Or optimization. I mean, calculus is like custom made for optimization, giving handy little functions that can approximate whatever real world application you need.



Or if you study economic growth. As I understand it, calculus of variations is highly useful for that sort of thing.



Furthermore, I think the notion of a continuum of preference makes sense, at least as an approximation. And of course, with continuous functions, calculus is the go-to math. I mean, which is easier? A little calculus, or counting 5000 different types of candy bar and finding a function to represent whatever economics relation you desire? I mean, when you look at how many goods and services there are, why would you want to have a bazillion little terms in your equations when you can just use an integral or derivative to get a function that is very approximate?



Granted, again, I barely have any knowledge of economics. But I do know for a fact that calculus makes things that deal with change MUCH easier. Try calculating the area of a curve using a sum of shapes underneath it. You'll spend several minutes or longer. Or you could look at the function, find the function that looks closest to it, and then integrate it. Done in a minute or less.

 

[Helios]

Thanks for these examples. I'm all for optimization and the calculus of variations. These problems are straightforward and instructive. However, like andrewkirk said, uses of calculus in macroeconomics can be questionable. I see macroeconomist turning economics into a monumental physics problem with words like elasticity, force, and temperature all used in their terminology and explanations. When I said "People don't act and chose, buy and sell, according to calculus.", I mean people don't act and chose, buy and sell, according to the equations of macroeconomics calculus that look like physics equations." I don't know how they test their own models or if that is even possible. Who knows?
I'm still saying there's no one single great paradigm epiphany that calculus brought forth for understanding economics at large, macroeconomics, comparable to say Newton's Principia.

 

[FactChecker]

I'm still saying there's no one single great paradigm epiphany that calculus brought forth for understanding economics at large, macroeconomics, comparable to say Newton's Principia.

It is not right to blame calculous for that. Economics is not physics and does not play by such hard and fast rules. But I don't think you can say much about economics without using calculous some way. Anything with rates, slopes, cumulative totals, probabilities, etc. will almost ceratainly involve calculous. You may not mention calculous, but it is there.

 

[Sorcerer]

Thanks for these examples. I'm all for optimization and the calculus of variations. These problems are straightforward and instructive. However, like andrewkirk said, uses of calculus in macroeconomics can be questionable. I see macroeconomist turning economics into a monumental physics problem with words like elasticity, force, and temperature all used in their terminology and explanations. When I said "People don't act and chose, buy and sell, according to calculus.", I mean people don't act and chose, buy and sell, according to the equations of macroeconomics calculus that look like physics equations." I don't know how they test their own models or if that is even possible. Who knows?
I'm still saying there's no one single great paradigm epiphany that calculus brought forth for understanding economics at large, macroeconomics, comparable to say Newton's Principia.

I get most of this post, but why is elasticity such a problem? It’s a pretty straight forward concept: for example, health care is always going to be relatively expensive because it is by its nature low in elasticity (i.e. changes in price lead to very small changes in demand- primarily because most people would rather not die).

I mean, it’s a pretty vital part of the Law of Demand (in that it sharpens it, or rather, gives more detail than just a proportionality relation), and if there is anything in economic that mimics the importance of Newton to physics, it’s that.

 

[Helios]

But I don't think you can say much about economics without using calculous some way.

A contrary opinion would come from the Austrian school of economics. They say a lot about economics with a heavy attention to epistemology. Start with axioms, make deductive conclusions, develop praxeology and then economics. There's far to go philosophically even without any math.

 

[Sorcerer]

A contrary opinion would come from the Austrian school of economics. They say a lot about economics with a heavy attention to epistemology. Start with axioms, make deductive conclusions, develop praxeology and then economics. There's far to go philosophically even without any math.

Then what becomes of anything that deals with rates of change or continuous curves?

Here is a fairly big issue without calculus that the Austrian school glosses over (from the article below):

“One obvious problem arises here. Without continuous preferences, it is also highly unlikely that e.g. supply and demand can ever be equal. If you draw the supply and demand curves continuously, then they are (almost) bound to intersect. But if you draw them as a discrete set of points, supply and demand in general don't have to intersect. Thus, the argument against calculus based upon the rejection of continuity also argues against even the use of simple algebraic constructs - like intersecting supply and demand lines - that fill Rothbard's works.”


Also as I understand it, there are plenty of other issues with the Austrian school, such as the belief that indifference does not exist. Yet I’m willing to flip a coin quite often when I experience a very real inability to choose one option over another, and submit my choice to the results of random chance.

http://econfaculty.gmu.edu/bcaplan/whyaust.htm

 

[FactChecker]

A contrary opinion would come from the Austrian school of economics. They say a lot about economics with a heavy attention to epistemology. Start with axioms, make deductive conclusions, develop praxeology and then economics. There's far to go philosophically even without any math.

Ok. I'll grant you that point. As a mathematician, I was not aware of a branch of economics that did not believe in mathematical modeling.

 

[Auto-Didact]

Ok. I'll grant you that point. As a mathematician, I was not aware of a branch of economics that did not believe in mathematical modeling.

There are also schools of economic thought on the other end of the spectrum, i.e. believing that the modeling of economics based on the mathematics of Newtonian physics, i.e. modeling as path-independent linear, time invariant continuous systems is a hopelessly inadequate approach giving an inaccurate and incomplete fiction at best, scarcely resembling empirical observations. This school of economic thought does not have a name or moniker per se and is possibly even being researched not by economists, but predominantly by physicists and mathematicians who were not trained in economics, but ended up working in it.

Having no definite name, this new scientific field/school of thought goes under many names in the literature (e.g. econophysics) and is associated with novel fields of science and mathematics, mainly complexity science, nonlinear dynamics and fractal mathematics. The main idea is that economic phenomena are far more complicated than the classical equations imply or can describe and that this description therefore requires far more advanced mathematical and computational tools in order to actually make accurate economic models and so enable a complete rewrite of economic theory based more closely on empirical data instead of highly overreaching and oversimplifying axioms.

 

[Sorcerer]

There are also schools of economic thought on the other end of the spectrum, i.e. believing that the modeling of economics based on the mathematics of Newtonian physics, i.e. modeling as path-independent linear, time invariant continuous systems is a hopelessly inadequate approach giving an inaccurate and incomplete fiction at best, scarcely resembling empirical observations. This school of economic thought does not have a name or moniker per se and is possibly even being researched not by economists, but predominantly by physicists and mathematicians who were not trained in economics, but ended up working in it.

Having no definite name, this new scientific field/school of thought goes under many names in the literature (e.g. econophysics) and is associated with novel fields of science and mathematics, mainly complexity science, nonlinear dynamics and fractal mathematics. The main idea is that economic phenomena are far more complicated than the classical equations imply or can describe and that this description therefore requires far more advanced mathematical and computational tools in order to actually make accurate economic models and so enable a complete rewrite of economic theory based more closely on empirical data instead of highly overreaching and oversimplifying axioms.

Doesn’t the Austrian school reject emperical data, though (I mean, they seem fairly adverse to statistics, which are in fact emperical data)?