In an essay on nonlinear thinking, I wonder if economists spend too much time studying calculus.

economists know that many of the processes that are important in our field, such as compound interest and economic growth, are nonlinear. But the economics curriculum in graduate school is focused on training students to make linear approximations. Studying economics at a leading graduate school involves loading techniques for linear approximations into your analytical tool kit. Naturally, these are the techniques that you use when you do professional research.

…Alan Blinder’s lament about the difficulty of picking up long term trends is a result of over-reliance on linear thinking. The trend that he was talking about was a pickup in the rate of economic growth. This pickup was best anticipated by a nonlinear thinker coming from outside the economics profession — Ray Kurzweil.

*For Discussion*. What are the risks and rewards of trying to forecast exponential trends?

## READER COMMENTS

## Mcwop

## Oct 23 2003 at 9:40am

Linear thinking can have disasterous results. Look at the following example:

Imagine a retiree with $250,000 in an IRA. In a linear draw down example, $250,000 over 30 years assuming a constant annual return of 11.71% and an initial withdrawal rate of 8% increasing annually with inflation will last beyond retirement, leaving a surplus at the end of the 30 years.

Now let’s assume and example of a person retiring in 1969 with the same amount of money and made the same withdrawals, but using actual historic returns. Even though the period 1969-1998 also offered 11.71% average returns, this person would run out of money at the end of 1981 – just 13 years into retirement. The severe market downturn during 1973-1974 (shown by the sharp downturns in the graph) inflicted losses from which the portfolio was never able to recover.

Assumptions: Initial withdrawal amount increases by 3% each year for inflation. Withdrawals are made at the start of each year. Taxes and minimum required distributions from tax-deferred accounts are not considered in this illustration. Performance is based on historical returns for these periods of the S&P 500 Stock Index (60% of the portfolio), U.S. Intermediate Government Bond Index (30%), and 30-day Treasury bills (10%).

Social Security uses many linear assumptions. If there is a 5 year period where incomes decline each year, problems will appear more quickly than projected because of lower payroll tax revenues. Monte carlo analysis anyone?

## David Thomson

## Oct 23 2003 at 10:20am

As an I outsider, it most certainly seems to me that so-called professionally trained economists place far too much emphasis on mathematics. This is probably because so many of them possess an inferiority complex. They are Liberal Arts practitioners, but prefer being perceived as “hard” scientists.

Understanding human nature is far more important. A quintessential example is health care. Those who believe that human beings are innately good argue that the hoi polloi will use free health care in a prudent and thoughtful manner. People like me laugh at such naiveté. We are virtually certain that the masses will soon foolishly bankrupt the system. A mathematician may indeed help to devise an empirical study to verify my cynicism—but the study of human motivation is outside their direct field of expertise.

## Eric Krieg

## Oct 23 2003 at 10:21am

It is a hell of a lot easier to think linearly than it is to think exponentially.

BUT, with that said, economists should be sophisticated enough and have the mathematical tools (even if it is nothing more than MS Excel!) to think non-linearly.

Could it be that trendy ideas like game theory have crowded out more useful subjects like mathematical modeling in economics graduate schools?

Arnold, here is a question that should be asked of every incoming economics graduate student:

If I gave you a differential equation and asked you to solve it numerically, what technique would you use?

Answer: 4th order Runge-Kutta

You don’t even really need to know exactly how a 4ORK works (Mathcad will do it for you), just know that it is the best numerical method for diff equs.

Knowing about 4ORK is an indicator of mathematical sophistication.

BTW, I got a job once at a defense contractor just by knowing that answer!

## Eric Krieg

## Oct 23 2003 at 10:27am

Actually, just re-reading the original post, what is the connection between calculus and linear thinking?

Arnold, you can’t integrate an exponential function? Take its derivative?

Modeling non-linear functions is not much harder than modeling linear ones. So I don’t see the connection between modeling and linear thinking.

If I were to answer Arnold’s bacteria question, I would go right to MS Excel and make a spreadsheet. I would have one column with the time in minutes, and one column with the number of bacteria. Each bacteria count cell would be twice the number of the one above. Start at 0 minutes with one bacteria, and go to 60 minutes.

That’s a quick and dirty model. That’s how engineers do things.

## Boonton

## Oct 23 2003 at 11:43am

The danger of thinking in a non-linear fashion is making an erronous assumption that an expodential function will last forever.

Take a function like Moore’s ‘Law’. Why does such a relationship exist? People with engineering and scientific backgrounds can probably explore this better than I but what if the reason such a relationship exists is because there was a lot of untapped potential in silicon chips? As the potential gets tapped the amount of remaining potential decreases….

Imagine a miner digging down into the earth. When he hits the tip of a stream of ore, he first notices small increases for every inch he digs. That’s a linear relationship. Then when he hits the core of the reserve he starts seeing more and more ore for each inch that he digs…. Now he thinks he has an expodential relationship….but as he approaches the bottom of the reserve the function ceases to operate in his favor and he starts to see declining returns for each inch.

We cannot assume that when engineers tap out the limits of todays chip technology the next source will continue to behave according to Moore’s law. I’d like to believe the optimistic scenero painted but how can we know the relationship isn’t going to turn linear on us?

## Brad Hutchings

## Oct 23 2003 at 3:53pm

Eric wrote:

“If I were to answer Arnold’s bacteria question, I would go right to MS Excel and make a spreadsheet.”

Why not just read the question and realize that answer is… I hate to give it away. Maybe read the question slowly and that will help you. Not helping? Take a minute to read the question, and the answer will come to you while the bacteria fills the petri dish.

Eric, please tell us you’re kidding. If so, it was funny. You know, I read your stuff about using a 4ORK and had no clue what you were talking about even though I have a minor in Math (emphasis on “pure” math). Sure, I’ve probably seen it at some point, and could dig into my Diff Eq textbook and be back up to speed in a couple hours, but really, that is total overkill for this (and just about everything else that doesn’t involve airflow over a curved surface).

Confronting these kinds of problems is about mindset, not your toolchest.

-Brad

## Bruce Cleaver

## Oct 23 2003 at 4:25pm

Boonton – you have a point about the exponential growth continuing forever. The answer you are seeking is called the “logistics curve” in which the first portion of the curve is exponential, then inflects at some point as a critical resource is used up, and becomes asymptotic. It looks like a sort of large “S”.

The trick is to know when the resource is approaching the critical point, and that is where all the guesswork (or insight) comes in. And *that* is the purview of economists and/or sociologists (for certain types of questions).

As an professional engineer, yes, Rung-Kutta is the old workhorse of Diff Eq solving.

## Eric Krieg

## Oct 23 2003 at 4:28pm

>>Why not just read the question and realize that answer is…

Reading comprehension is not my strong point, right Boonton?

Ha!

Hey, after I did the spreadsheet, I said “D’OH! I should have known that”.

Cool thing about Excel is that it took me literally 47 seconds to make the spreadsheet. So what if it was 46 seconds more than it took you to do the problem?

## Eric Krieg

## Oct 23 2003 at 4:44pm

>>Eric, please tell us you’re kidding. If so, it was funny. You know, I read your stuff about using a 4ORK and had no clue what you were talking about even though I have a minor in Math (emphasis on “pure” math). Sure, I’ve probably seen it at some point, and could dig into my Diff Eq textbook and be back up to speed in a couple hours, but really, that is total overkill for this (and just about everything else that doesn’t involve airflow over a curved surface).

Semi-kidding.

You don’t remember our good friend Runge-Kutta because you were a math major. You never took a class in numerical methods. But if you DID want to solve a differential equation numerically, that’s what you would use.

Arnold’s question was simple. But what if you wanted to know the underlying equation, which you were interested in manipulating in a LITTLE more sophisticated way? How would you find it?

## Eric Krieg

## Oct 23 2003 at 4:46pm

>>Confronting these kinds of problems is about mindset, not your toolchest.

“The secret to life, Beavis, is always having the right tool, or something.”

Everything I know I learned from “Beavis and Butthead”. Or maybe it just seems that way sometimes.

## Eric Krieg

## Oct 23 2003 at 4:56pm

I should answer my own question.

IF you know that you function is exponential (I would say thay is a big if), you know that is has the form

Population = A * exp^(B*time)

Where A and B are constants used to tune your equation to the boundary conditions. For example, at time = 0, maybe you have one bacteria. Time = 1 minute, you have 2 bacteria. Then A = 1 and (using the goal seek function in Excel, I found that) B = .693175.

And THAT Mr. Hutchings, took me a minute and 37 seconds to figure out with Excel.

## Lawrance George Lux

## Oct 23 2003 at 8:48pm

A return to the original question Arnold asked must be: The fluxuation of Variable magnitudes, plus the rapidity of flux in those magnitudes; inspire a exponential equation of daily values expressed by the original exponential equations. This means inaccuracy of input data will destroy all output data, by the time a registry can be marked. I hope Someone besides myself can understand that statement. lgl

## Eric Krieg

## Oct 23 2003 at 9:48pm

>>But the economics curriculum in graduate school is focused on training students to make linear approximations

## Arnold Kling

## Oct 24 2003 at 1:02am

**>>But the economics curriculum in graduate school is focused on training students to make linear approximations

## Eric Krieg

## Oct 24 2003 at 9:21am

Arnold, what value is provided by linearizing a non-linear function around the equilibrium? To make a forecast?

In the example you gave, the function IS linear up to and including the two minute point. At three minutes the function is still nearly linear. After that the deviation from linearity is increasingly large.

So a short term forcast using a linear model wouldn’t be too bad. But you wouldn’t want to make long term predictions based on it.

## M

## Oct 24 2003 at 9:46am

Arnold,

Interesting discussion. My feeling is that the real problem in econ training is in the reductionist nature of the math modeling.

A model is a simplified representation of reality (like a roadmap of a city). The value is when we can use this simplified representation to understand how elements of the system work. The problem is that in economics anything of interest is the result of a massive set of interelated factors occuring simultaneously. Therefore most results from analytical models are unlikely to have much “external” validity.

Same sort of issues on the empirical side. The economy is so complex and with so many 2nd, 3rd, or greater order interactions that it is very difficult to specify and then find sufficient data to estimate really good empirical models (this is why I suspect most macro models do relatively poorly).

M

## gerald garvey

## Oct 30 2003 at 11:57am

Glad to see posters who caution against assuming exponential growth rates go on forever. One related point we often overlook is that the magic of compound interest is nothing of the sort. You only get compounding if you reinvest your interest. That is, you cannot eat any of your returns or else you lose your compounding. No magic, just the rewards to old-fashioned abstinence.

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