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## Monday, September 14, 2015

### Economic Illiteracy Grows at Twice the Rate of Inflation

A common trope in economics reporting is to claim that this or that figure had risen at some multiple, usually double, of the rate of inflation. For example, the New York Times reported:

According to statistics collected annually by the Federal Communications Commission, the price paid by consumers for expanded basic cable service has grown at more than twice the rate of inflation annually over the last 17 years. Cable companies say that is because they are offering more services in each package, which causes the overall price to rise. The per-channel price, they argue, has declined.

A quick search will find many similar statements in publications, great and small. Nonetheless, it is a not a meaningful statement, but rather an inadvertent confession of the reporter's economic illiteracy.

A statement that some nominal amount $$x$$ has grown by $$n$$ times the inflation rate says absolutely nothing about the question it purportedly addresses: whether $$x$$ has been growing quickly or slowly in real terms. Saying that $$x$$ has grown at twice the rate of inflation may mean, for example:

• That $$x$$ has grown quickly in real terms. If the rate of inflation was $$10\%$$, $$x$$ would have grown at $$20\%$$ nominally and $$\approx 10\%$$ in real terms.
• That $$x$$ has grown slowly in real terms. If the rate of inflation was $$0.1\%$$, $$x$$ would have grown at $$0.2\%$$ nominally and $$\approx 0.1\%$$ in real terms.
• That $$x$$ has not changed at all in real terms. If the rate of inflation was $$0\%$$, $$x$$ would have grown at $$0\%$$ in both real and nominal terms.
• That $$x$$ has shrunk in real terms. If the rate of inflation was $$-1.0\%$$, $$x$$ would have shrunk at $$-2.0\%$$ nominally and $$\approx -1.0\%$$ in real terms.

More generally, if $$x$$ grew at a constant $$p$$ percent annually and inflation was a constant $$i$$ percent over the period,1 then in real terms $$x$$ annually grew by:

$$\frac{1+p}{1+i}-1$$

That is the only strictly accurate way to calculate the real rate of growth. But if, as is usually the case, the period is short and $$p$$ and $$i$$ single digit percentages, $$p-i$$ is a good first-order approximation.

So if given the a nominal rate of growth and desiring to make a statement about what it means in real terms, either use the formula above or—if the rates involved are small—just subtract out the inflation rate.

Never, ever, divide by the inflation rate. And do not trust any writer who does.

1 If the inflation rate or nominal growth rate vary, as generally they do, one must look up an appropriate inflation index, divide the nominal amounts by it, and then take the $$n$$-th root (where $$n$$ is the length of the period in years).