## Thursday, May 27, 2021

### Cost of Living Indices: Why Use Them?

One technical issue that comes up when looking at consumer price indices is that the concept of a “cost of living index” comes up. This is a technical definition, as opposed to the vaguely defined notion of a “cost of living” that is used in popular discussion.

In most of my writing, I aim to explain technical concepts and debates in plain English. In this case, I am not going to attempt it. My feeling is that the specifics of this technical debate are not going to be meaningful to a general audience. Meanwhile, my attempt at explaining it may be more confusing that just reading the literature for someone with a technical background. Instead, I am just going to give a simplified explanation of why this debate happened in the first place, with just a tease of the underlying issues (as I understand it).

Note: this is a very preliminary draft of a section that follows up from an earlier piece that discusses the more vague common usage of the phrase "cost of living." This technical piece is not aimed at any current debate about inflation. The Bloomberg article referenced in the tweet below discusses the current "is what is happening now 'inflation'?" debate.

## Need for a Mathematical Methodology

The main thing to keep in mind is that statisticians and economists at national statistical agencies are tasked with calculating a consumer price index (which implies an inflation rate). They are expected to provide a number that is relatively precise, not an essay whining about how hard it is to come up with an aggregate price level. So they need to come up with a methodology (algorithm) to take price observations and turn it into a price level.

At the same time, the creation of price indices is a subject of academic enquiry. The main job of an academic is to produce new research. This means that they need to develop new algorithms, which are supposed to be better than the old ones. This means that the academic with the new algorithm starts arguing with the academics who developed the old one, generating controversy that is the fun part of academic research.

Let us imagine that we need to calculate the monthly inflation rate between January and February. We have all the prices for goods and services for the two months, with adjustments made for things like changing product sizes. We also have the exceptional luck that there was nothing new to purchase in February, and every product that was available in January was also available in February.

We see that 98% of price changes are between -0.1% and 0.6% on the month. We then need to come up with a single monthly inflation number. There are decisions as to how to construct that aggregate number.
• We need to decide what weight to assign to each item. The usual logic would be base the weighting on how large a weight the item has in overall consumer spending. Theoretically, we could use either the weights from January or February. This is a concern since a large price increase for a particular item in February could mean that consumers bought less.
• We need to have a mathematical function to create an average inflation rate.
• Under normal circumstances, we need to have a methodology to deal with items entering/leaving the list of prices.
There are several alternate methods to come up with the monthly index inflation number. In general, the different methods will be close to each other. Using the example numbers, it would be entirely possible for one method to result in an average 0.2% inflation, while another gives 0.3%.

An entirely sensible reaction to this would be: 0.2% or 0.3%, what’s the big difference? And in the abstract, that is probably the most sensible response. Unfortunately, life is not that simple – we are not only interested in one month’s inflation numbers. We need to keep doing the calculation every month. If the exact same outcome happened every month, the two estimates for inflation would be (roughly) 2.4% versus 3.6% - which is a disagreement that is half the size of the lower inflation rate.

## Methodologies Have a Bias

Statisticians need to pick a single methodology. And if we look at two competing methodologies, we will see that one method will be consistently higher or lower than the other.

People who complain about statisticians manipulating inflation lower will probably insist on using the higher value. However, this is not particularly helpful.

If we look at an historical period, prices were what they were. Changing methodologies to get a new inflation rate does not change what the true change in the cost of living people experienced. Let us imagine that the measured CPI change over a decade was an annual average of 2%, but there is an alternative measure that sets it at 3%. All this means is that you need to associate the cost of living increase you experienced over the decade with 3%, not 2%. Other than for the people who receive payments indexed to the CPI, this does not affect their financial situation.

Meanwhile, using too high an inflation rate poses problems. If we imagine that such a thing as a “true” inflation rate exists (which is a debatable point), if we choose a methodology that is on average 1% too high versus the “true” inflation rate, we have an error in the price level that will grow at 1% forever. If that inflation methodology is used elsewhere in the national statistics, it could result in real GDP growing on average 1% below its “true” value. This means that if we look at the volumes of produced output – for example, the number of cars produced – the components may end up growing faster than the overall economy (which should be impossible).

A related concern is the measurement of productivity. Since we can estimate the number of hours worked reasonably well over time, the higher inflation rate implies lower output, hence the output per hour worked (worker productivity) is lower. I am unconvinced that we can measure concepts like the price level over the span of decades, so I am not concerned about the effect on long-run productivity statistics. However, mainstream economists study a version of growth theory that turns them into bugs about productivity: the only way to raise living standards over time is to raise worker productivity. Reducing productivity by mangling the inflation measurement is something that they are not happy about, since it implies that society is not progressing (or whatever).

## Cost-of-Living Indices

With all that preamble out of the way, we can finally turn to what a cost-of-living index is. I am working from the discussion in a paper by Jack E. Triplett (“Should the Cost‐of‐living Index Provide the Conceptual Framework for a Consumer Price Index?”) and will paraphrase his definitions as follows. A cost-of-living index is one that measures the cost of maintaining a constant standard of living, as defined by a utility function. That is, if a household wants to keep its “utility” constant, how much does it cost? (For readers who are unfamiliar with the concept of utility, you can think of it as a numerical “satisfaction” index that is based on what is consumed in a period.)

Utility is a disputed idea within economics; mainstream economists build their theory around it, while heterodox critics argue that it is a nonsensical concept. However, I have some sympathy with Triplett’s views in this case. He argues that alternative statistical measures of “standard of living” are fuzzily defined.

Why do I agree with Triplett? Any price index implicitly defines a constant “standard of living” that is associated with the index level. We can label that “standard of living” to be utility, or whatever. I think that any version will end up having problems, but you need to pick one.

The reason why the format matters is the question of substitution. If the price of fish rises due to a dropping fish stocks but the price of beef does not move, an entirely reasonable reaction is to not buy fish but instead buy beef. The cost of living in some sense rose, but how much of a weight should be attached to each item? Should we weigh the price increase of fish by its previous consumption weight (when it was cheaper) or after the consumption fell due to the price hike?

Utility functions give an answer to this question. By assuming a particular form for the utility function, it generates the trade-off needed to keep the same total “satisfaction.” The resulting inflation rate ends up being lower than would be the case if people did not substitute away from goods experiencing rapid price rises. To the extent that official CPI measures diverge from proposed cost-of-living indices, the official CPI inflation rates end up overstating inflation.

Although I enjoy arguments about mathematical methodologies, I do not see any one being the “correct” answer – they all have flaws. Rather than vent off steam about statisticians conspiring to lower the measured CPI, we just need to accept that the measures are what they are, and we should not expect statistical perfection.

## References and Further Reading

• Triplett, Jack E. “Should the Cost‐of‐living Index Provide the Conceptual Framework for a Consumer Price Index?.” The Economic Journal 111.472 (2001): 311-334.
(c) Brian Romanchuk 2021

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