Calculus for Economists

Gabriel Sterne complains about economists' loose use of mathematical terminology: 

Of course, it's not just economists who use "increase" and "accelerate" interchangeably. But economics is a mathematical discipline, and in mathematics, "increase" and "accelerate" mean different things. So is Gabriel's observation true, and if it is, is it a problem?

To test Gabriel's hypothesis, I ran a little Twitter test. I asked this question: 

This was of course far from rigorous: the sample was self-selecting, there was no way of restricting it to economists (though I did ban finance tweeps from answering), and it all depended who was on Twitter this morning. And the terminology I used was itself confusing - deliberately so, since this is how economists often write. 

But the results were nevertheless interesting. Most non-economists got the answer right. Physicists, in particular, understood it straight away. But most economists who answered got it wrong. 

As the title of this post indicates, we are back in the realms of calculus again. So I'm going to do something I couldn't really do on Twitter, namely define my terms, starting with what I mean by "inflation." At this point, any goldbugs reading this post can leave the room. Inflation on this occasion has nothing to do with the money supply. 

We define inflation as the rate of increase of the price level. It is therefore the first derivative of the price level P with respect to time, t. We express this mathematically as dP/dt. 

The fact that inflation is the first derivative of price with respect to time is, I think, the reason why economists get this wrong but others get it right. When answering the question, lots of economists took "accelerating" to refer to the price level rather than inflation. I shall return to this later on.

Now, let's define what we mean by "accelerate". To do this, we can take inflation itself as an example. Suppose that last year, the general price level increased by 2%. The annual inflation rate  is thus 2%. Now suppose that this year, the general price level increases by 3%. Prices have increased by 3% and inflation has increased by one percentage point. The rate at which prices are increasing has therefore itself increased. We call this increase "acceleration." It's the second derivative of price with respect to time and is expressed mathematically as d2P/dt2.

When the rate at which prices are increasing itself increases, inflation is increasing. But it is not accelerating. To make this clear, let's extend the example above by another year. Suppose, in the third year, inflation rises to 4%. The rate at which prices are increasing is clearly accelerating - in fact it has doubled in three years, which might be a matter for some concern. But the rate at which inflation is increasing has, as someone has kindly pointed out in the comments, actually fallen.  

And this is where the economists went wrong. Lots of them thought "accelerating inflation" meant accelerating increase of the price level, and therefore said the answer was the second derivative of price with respect to time. But since inflation is the first derivative of price, the correct answer is actually the third derivative of price with respect to time - the change in the rate of acceleration. We express this mathematically as d3P/dt3. Physicists call it "jerk". 

How would the numbers look if inflation really was accelerating? Let's suppose that in the fourth year, price rises really take off. Instead of prices increasing by 4%, they increase by 8%. Inflation has risen by 4%, whereas the previous year it only rose by 1%. So the rate at which inflation is increasing is accelerating. Let's imagine acceleration continues for another three years: inflation rises to 16% in year 1, 32% in year 2 and 64% in year 3 (yes, the numbers get very big very quickly when third-order acceleration takes off). If we chart the change in inflation over the course of all our examples, it looks like this: 

This is what the start of hyperinflation looks like. Hyperinflation is accelerating inflation. The rate at which the price level rises also accelerates, of course, but it is the acceleration of inflation itself that creates the exponential function we can see in this chart. 

This chart is a fine illustration of the difference between accelerating and increasing inflation: 

Venezuela suffered years of increasing inflation. But in 2012-13, inflation started to accelerate. By 2015 it was in full hyperinflation and its economy had tanked. 

So for economists, understanding which derivative of price is in the driving seat really, really matters. Accelerating inflation rapidly destroys an economy. Increasing inflation does not. The policy responses to accelerating and increasing inflation are likely to be quite different. 

And therefore Gabriel's criticism is justified. Economists should not use "accelerate" to mean "increase". Get it right, please, economists. Your countries depend on it. 

Related reading:

Calculus for Journalists 

Arithmetic for Austrians

Covid confusion: going viral - The Mint

Venezuela inflation chart from Wikimedia Commons.

Racing cars picture courtesy of the U.S. Navy via Wikipedia.


  1. I suspect the economists who got it wrong gave up on your article less than halfway through. (Despite the quality of your writing). We live in a society where people like you are labeled, "Too clever by half", and such like.
    Where have all the economists gone? Not you. We need to know more about the Money Tree.

    1. Economist here. Arrived at this comment directly from about four paragraphs in.

  2. This comment has been removed by a blog administrator.

    1. Joe, I removed this comment because it was a duplicate of your first.

  3. The real problem with inflation is that is represented as reflecting the change in prices, rather than the change in the value of money. For the working person, prices relate directly to his/her income. So when I started work in 1970 at an annual salary of £1,050 prices of commodities in terms of minutes worked were:

    Petrol 9 m/litre
    Beer 14 m/pint
    Colour tv 33,400 m
    My current house was built in 1970 and would have cost 6.7 times my salary.
    RPI 73

    If I had started work in 2020 at 21 years old, in a comparable job, my costs would have been
    Petrol 7 m/litre
    Beer 22 m/pint
    Colour tv 724 m
    My current house estimated price would have been15.5 times my salary.
    RPI 1156

    So in effect petrol and beer have not changed much in minutes worked over that time period. Colour TV (like all high-tech stuff) is dramatically cheaper, and house prices are significantly higher. Whether this is caused by lack of supply or the willingness of lenders to lend more money is a moot point. Apart from the fact that housing is such a large proportion of most peoples' living costs one could conclude that costs as felt by the working person have not changed dramatically. Obviously what we buy has changed - in 1970 we didn't buy home computers, for example.

    The underlying issue is that money is being used to measure value without itself having any standard against which to calibrate itself. Imagine measuring the length of your foot in 1970 and finding it was 73 units; then measuring in 2020 at 1156 units.

    Clearly a new measure is required; something that remains constant with time and against which the cost of things can be measured. (KiloWatt-Hours have been suggested ...) That way we would see clearly how the costs of different commodities vary (houses, colour tvs) and how the price of a specific commodity varies with time (so why is beer more expensive now and petrol - surprisingly - a little cheaper?)


    1. No, we don't need a standard against which to measure the value of money. Money is the unit of account and is therefore its own standard.

  4. "Loose use aboot this hoose"

    To be pedantic.....
    Where you say "Suppose that last year, the general price level increased by 2%. The annual inflation rate is thus 2%. Now suppose that this year, the general price level increases by 3%. Prices have increased by 3% and inflation has increased by 1%."

    Wouldn't it be more correct to say "...and inflation has increased by 1 percentage point."? Because if last year inflation was 2% and now it is 3%, the rate of inflation has actually increased by 50%.

    And then at the end of the next paragraph "...But the rate at which inflation is increasing has not changed. It is still 1%." But if inflation in the third year is 4%, then yes it has again risen by 1 percentage point, but the rate at which inflation is increasing hasn't stayed the same; it has actually fallen from 50% to 33%.

    1. Good point, and I plead guilty. I will correct my terminology.

  5. And hyperinflation usually accompanied by crashing production. Why it is usually happening in wartime or aftermath.
    In 1996, the town I worked in at the time experienced a "local hyperinflation" when a storm destroyed the only road and freah milk price in a instant black market shot up in a few hours and crashed back when a supply ship. The road was repaired in a few days and everything went back to normal.


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