# Why Consumer Price Index is linear, when it should be exponential? And what to use instead?

Consumer Price Index looks like a very nice straight line, perfectly approximated with linear function (considering it only after the 1972).

$$cpi(years) = start + rate \cdot years$$

And, we kinda know that there's an inflation. We don't know the exact number, but guessing it's something like 2-6% a year. And it will be approximated by something like exponent

$$inflation(years) = start \cdot rate^{years}$$

We know that CPI and inflation are kinda related. How linear and exponent could be related?

QUESTION 2: Is there any hard measure for inflation? Based on some real prices of something? I was thinking maybe take 1 "unit" of gold, oil, sp500 and real estate, average it and use like a base to compare to USD and calculate the inflation?

UPDATE I figured out why. At first I thought that CPI data is manipulated and can't be trusted, but then I also searched for wages growth history and it's also linear, and it looks reasonable.

So we have dollar loosing its value exponentially, but prices of goods growing linearly, and both are true. That's possible only if we add third hidden variable - prices of goods also changing, decreasing exponentially because of productivity growth. So, exponential inflation is compensated by exponential productivity growth and we have linear growth for prices of goods. The resume - my understanding of inflation was wrong, I assumed its a measure of dollar loosing value, but it's a different thing, it's a measure of prices of goods growth. Inflation can't be used as a measure of dollar loosing its value.

But the main question is still open - what "hard" data could be used to measure dollar loosing its value? (as we discovered the inflation can't be used for that).

CPI and its approximation with $$CPI(year) = 44.46 + 4.58year$$

CPI

Inflation

## 2 Answers

Consumer Price Index looks like a very nice straight line, perfectly approximated with linear function (considering it only after the 1972) ... We know that CPI and inflation are kinda related. How linear and exponent could be related?

The past half century has been characterized as a period of "Great Moderation." Inflation pressure has generally collapsed, due to globalization, technology, demographics, and other secular forces. Cyclical inflation pressure has also been exceptionally muted this cycle. Even in emerging markets (in aggregate), inflation pressure is at secularly low levels. These factors explain why price indices generally have not risen exponentially.

A lot of central banks now have specific inflation targets. For example, the Fed has announced a 2% inflation target (based on core PCE). Assuming the target is consistently reached, then we'd indeed have an exponentially increasing curve that looks like $$(1 + 2\%)^t$$, where $$t$$ is measured in years. But central banks globally have struggled to meet their inflation targets over the past decade, as can be seen below:

QUESTION 2: Is there any hard measure for inflation? Based on some real prices of something? I was thinking maybe take 1 "unit" of gold, oil, sp500 and real estate, average it and use like a base to compare to USD and calculate the inflation?

There really isn't such a thing as the inflation number. It depends on what you're interested in. Are you concerned about goods inflation, services inflation, financial asset inflation, etc.? Who's bearing the cost of the inflation pressure: consumers, producers, exporters, importers, retailers, etc.? You'll get vastly different numbers. In fact, you might get different numbers for the same concept (the UK's RPI vs CPI and US's PCE vs CPI are good examples).

Note that CPI is based on real prices; it uses a very large basket of goods and services.

So we have dollar loosing its value exponentially, but prices of goods growing linearly, and both are true.

You mentioned that "the dollar is losing its value exponentially." The question is "relative to what?" For example, given that goods generally have experienced deflation, dollar relative to goods has risen. The chart below provides another perspective, showing the real value of dollar relative to US trading partners – it's actually near cyclical highs once you account for the relative competition & relative price levels amongst trading partners. So it's important to specify your benchmark.

If your only objective is to show that dollar has lost value, the most natural benchmark is probably gold (which is really a contra-currency) – over the past century, all paper currencies I've looked at have lost nearly 100% of their values against gold.

It's also not clear to me that productivity growth was that exponential. Since at least the financial crisis, productivity growth across developed world has been very weak (although there are lots of measurement issues...). The only reason global potential growth has held up is because emerging markets, where productivity growth is stronger, now represent a much larger share of global output.

P.S. I think we can provide better answers if you let us know what your ultimate goal is. TBH, right now it's a collection of observations. As it stands, it's also not within the scope of quantitative finance; I think you'll get higher quality answers at Economics.

• Thanks. About the goal - I was trying to find a better way to measure price of gold. How high it is today, compared to historical prices. Given that USD is constantly changing I was trying to find more reliable unit of measure. There's a scale-invariant way to measure SP500 - as a relation betweencapitalization / revenue or dividends. I tried to find similar scale-invariant measure for gold. By "normalizing" USD by adjusting inflation, but see now that it won't help. Maybe it's better to ignore USD and find some other base to compare gold to. – Alex Craft Jan 13 '20 at 12:13
• That's much clearer thanks. You're looking for a sort of fair value level for gold. I'm not an expert, but I've seen people using total money divided by gold stock as a proxy. There are also regression models that use several variables; CPI inflation is usually one of the inputs. – Helin Jan 13 '20 at 19:50

This is your chart superimposed on an inflation rate at 2% per annum (most central banks target inflation).

This chart seems to represent (mindful of yearly volatility) the typical behaviour of an exponential chart.

The CPI (and RPI) indexes are based on a measured basket of goods regularly consumed by the consumer. I would suggest this is actually quite a good measure for the value of money to the general consumer.

You can probably create other indexes (the RPI relative to CPI in UK are good examples) but it depends on the purpose for what you want it to measure.

• Take look at this approximation with CPI(year) = 44.46 + 4.58*year it fits CPI perfectly. Also, I think the 1972 is important year as it's when gold standard was removed, so before and after are different behaviours and should be approximated separately. – Alex Craft Jan 12 '20 at 19:26