Using multiple price indexes to measure changes in inequality is not a good idea

Although I often disagree with him, I very much enjoy Will Wilkinson as a writer. But I really have to object when he claims in desperate italics that

the use of multiple price indexes to measure trends in inequality…is the correct thing to do.

It is not at all the correct thing to do.

Wilkinson’s post is called “The Indeterminacy of Income Growth”. Putting aside measurement error in tracking nominal income (not Wilkinson’s point), the current distribution of income and its relative growth in different quantiles is not at all indeterminate. Many consequences of inequality follow from nominal disparities in income, independently of how changes alter consumption. Kevin Drum makes this point very well.

“Consumption inequality” is in general the wrong way to think about the consequences of inequality. As Mark Thoma recently pointed out, wealth is largely not about what one consumes, but about the time and freedom one has to sacrifice to sustain “normal” consumption. The price of freedom is nowhere in ones “consumption basket”. It is related to the gap between income and ordinary consumption, and also to the price of what one does not ordinarily consume, the cost of deviation. Indebtedness also entails a cost in freedom that we miss if we focus on consumption. In my view, freedom, not consumption, is the central distinction between rich and poor. It is odd that I should argue this point with libertarian Wilkinson.

But, let’s put that question aside and focus on consumption inequality. Here’s a thought experiment that I think captures Wilkinson’s view of why we should “use multiple price indexes” when thinking about changes in inequality.

Suppose that two gentleman, Richie Rich and Peter Poor have incomes respectively of $1,000,000 and $10,000 in the year 2000. There are four goods in our economy, caviar, opera, hot dogs, and sit-coms. Each month, Rich purchases 73 servings of caviar, 24 operas, two hot dogs, and a sit-com. Poor purchases 38 hot dogs and 15 sit-coms. Since we are focusing in consumption inequality alone, we’ll say both Rich and Poor spend their entire incomes each month on these goods. They are both budget constrained: they would consume more if they had more cash.

Now suppose that, between 2000 and 2010 Rich’s income doubles while Poor’s income increases by 10%. Further, suppose that the price of caviar and opera both double, while the price of hot dogs and soap operas increase by 10%. What Wilkinson wants us to conclude from this experiment is that inequality has not, in fact increased. Ignoring rounding errors, Rich can still consume basically what he could consume in 2000, as can Poor. They are both in the same situation they were in 2000, no?

No, they are not. Wilkinson is ignoring substitution effects. As the price difference between caviar and hot dogs expands, Rich will shift his consumption basket, foregoing some caviar for hot dogs. Doing so will make Rich strictly better off than he was in 2000: he could have maintained his old consumption basket, but the opportunity presented by cheap hot dogs gave him a better deal. Poor, on the other hand, will not shift any of his consumption towards caviar and opera, and he cannot shift away, since he was already consuming none of the now more expensive luxuries. Poor’s consumption basket will have gone nowhere over the aughts, while Rich’s will have improved. If we use multiple price indices to claim that the two groups’ “real incomes” stayed the same over the period, we will have missed this change. It is an error of elementary microeconomics.

It would be an error to claim that consumption inequality does not change with a changing price level even if incomes are fixed and prices move uniformly. This is easy to see when inflation outpaces income. Suppose there is no income growth in our economy, but the price of all goods double. Using price indices to measure “real income”, there will appear to have been no change in inequality. But Poor would have been forced to cut his consumption of sit-coms entirely and to carefully ration hot dog calories. Rich will have partially substituted from caviar to hot dogs and from opera to sit-coms, but his nutritional needs will remain met. In “utility” terms, the change in the price level will have impacted Poor far more than Rich, both because the marginal utility of consumption diminishes with wealth and because Rich can cushion the loss of real income with substitutions unavailable to Poor.

For a reductio ad absurdam, assume that incomes remain constant while the price of rich goods increases 100 fold, but that of poor goods increases only 50 fold. “Real income inequality” will appear to have fallen by half, but Poor will have starved to death while Rich gorges on hot dogs. I’d say inequality had in fact increased.

In his initial piece, Wilkinson frequently alludes to the conventional inflation rate, suggesting by analogy that if using a consumption-basket-based price index is sensible at a national level, it ought to be sensible for smaller groups too, and perhaps we should even imagine different inflation rates for each individual. But the basic problem with price indices, that they ignore substitution, gets more pronounced the more finely you slice and dice the population. Patterns of national consumption over the short-term are much less sensitive to changes in relative price than an individual’s or a subgroup’s consumption. An individual will switch to hot dogs when caviar prices skyrocket, but the economy as a whole will consume what the economy produces, while prices adjust to ensure equilibrium. Over longer terms, national consumption baskets change, and BLS has to account for shifts in the overall consumption basket with unempirical estimates of changes in value (“hedonics”). These questionable fudges would have to be used at a very high frequency for subgroup CPIs to track lived experience.

Usually the substitution problem leads to overstatements of inflation or understatements of real income growth. Rich people and poor people have different abilities to make substitutions in their consumption basket. Suppose I’m rich and I work in New York, so I commute by taxi. If taxi fares rise, I can substitute subway commutes. But if I’m poor, my “consumption basket” was already all subway for the trip to work. I can’t substitute when the cost of my cheapest feasible option rises. This asymmetry means that a rich-person CPI will be overstated due to neglected substitutions much more than the poor person’s CPI. Measurement error will bias us against perceiving inequality.

In general, trying to capture changes in “real income inequality” via price indices, single or multiple, will fail to track a big source of inequality, the differential ability of the wealthy to respond to price and income changes via substitution. This mismeasurement can go both ways: the approach can understate the relative gains of the poor as their incomes increase relative to the price of “rich goods”, creating new substitution options for the poor without much changing the consumption options of the rich. (Note that the poor are harmed, always in absolute terms and sometimes in relative terms, when the price of “rich goods” increases, unless we take their income as permanently fixed. The multiple price index approach assumes the poor should be indifferent to the price of goods outside their typical basket.)

It is basically a bad idea to try to measure “real income inequality” with price indices, because the consumption-related welfare of the poor is so much more sensitive to changes in income than that of the rich. Variations in consumption or spending that generate small changes in quality of life among the wealthy generate large variations in nutrition, health, education, and shelter among the poor. If you think consumption inequality is all that matters, you should just not pay any attention to what’s going on with the rich and focus on increasing real consumption among the poor. If that’s what you think, then the multiple price index stuff is just a fancy way of persuading us to ignore the burgeoning nominal incomes of the rich. But it’s not “correct”, and not helpful except as smart-sounding obfuscation. If we should concentrate on the absolute consumption of the poor, just make a simple case based on decreasing marginal utility. I won’t be persuaded, because I think the benefits of relative wealth are much larger than what is visible in consumption. But messing around with price indexes won’t help us resolve that argument.

Wilkinson’s bottom line, I think, is this:

[T]here is no fact of the matter about real income growth. And this means that there is no fact of the matter about the trend in income inequality.

I’ve ultimately supported Wilkinson’s case, in a sense. He talks up multiple price indices as the “correct” tool to measure inequality ultimately to persuade us that it is too hard to do properly. I agree that it is hard to do properly, and think claims based on income-level specific price-indices should be taken with boulders of salt. Wilkinson then continues

[I]f you’re committed to a story about American political economy that requires a great deal of confidence about how much median income or income inequality has or hasn’t risen since the 1970s, you should know that no such confidence is warranted.

That’s a step too far. Most social phenomena lack precise measures. We build social theories on differences along dimensions of freedom, individualism, racism, “happiness”, quality of institutions, trust, relationship-strength, formal versus familial affiliations, etc. etc. etc. We’ve made up measures for all of these things, but they are all poor. We do empirical work with our measures anyway. Smart readers aren’t snowed by the false precision of a regression and a t-statistic. We should require a lot of evidence to take seriously results whose fancy math obscures messy data. Nevertheless, I’m pretty sure that “quality of institutions” has had something to do with economic growth and that cross-sectional differences in freedom have important implications for welfare.

We always observe qualitative differences first, and then try to quantify those with our measures. (If you don’t believe me, look at the tendentious manner by which measures of social phenomena are actually constructed). Qualitatively, I’m as certain that income inequality has increased in meaningful ways as I am certain that the United States has better institutions than post-Communist Eastern Europe or that the character of racism in America has changed since the 1970s. I can come up with quantitative measures that would capture some aspects of those changes. (If I couldn’t come up with anything, I might revisit my qualitative certainty.) But I can’t come up with a definitive measure, something objective that I can use as uncontroversially as a chemist uses temperature. Wilkinson tries simultaneously to anoint one true measure and then trash it as too noisy to be reliable. He is right that his measure is bad. We have better measures, and our evidence that inequality has consequentially increased in America is at least as good as our evidence for many other social phenomena that we widely assume are real.

Note: See Karl Smith for an excellent discussion of the ambiguous line between price inflation and quality distinctions, and how that can contributes to mismeasurements of inflation even using conventional, national price indexes. If rich people are paying more and getting more value for their money, then an inflation measure that saw the price change but ignored the increase value would understate rich-person income growth.

Update History:

  • 28-September-2010, 12:15 p.m. EDT: Cleaned up some embarrassing repeated phrases and grammatical mistakes. No substantive changes.
  • 28-September-2010, 2:50 p.m. EDT: Fixed a couple of typos pointed out by gappy in the comments. Thanks gappy!

30 Responses to “Using multiple price indexes to measure changes in inequality is not a good idea”

  1. Ha ha … I had to laugh! I never thought substitution and ‘Hedonics’ could be the objects of humor … excellent!

    Of course the subject requires more than a comment (forthcoming @ Economic Undertow but QE first) but substitution is a pillar of the Ponzi economy and demands debunking.

    $100 crude oil will – by itself – create substitutes such as hydrogen ‘economies’, electric cars, wind and solar power, etc. We can see how all that is turning out … well, can we?

    (Insert joke here!)


  2. john personna writes:

    I kind of worry that Will has people lost in inflation sophistry.

    My argument is that inequality isn’t about income, it’s about net worth.

  3. gappy writes:

    Steve, it’s reductio ad absurdum, not reducto ad absurdam. And, “anoint” needs two n’s, not three. Aside from this, it seems that you are criticizing WIlikinson for not being precise enough in his modelling of the measurement of inequality. Fine, but that is better than doing nothing, and a step in the right direction. I have yet to see a paper that measures inequality taking into account income uncertainty, net worth, consumption (including substitution effects) measurement, and income growth within one’s life cycle. I am not stating there is a unique way to do this, but these are not second-order effects. Most papers are happy to mention increases in Gini indices or percentiles ratios to score a partisan point. That’s fine too, but then let’s just complain that inequality has gone qualitatively up and get done with it.

    And, as you mention alternative objectives in your post, I believe that there is far too little discussion on the validity of inequality as a welfare function. Rawls did it, Sen did it, most everyone else is happy using it.

  4. I think it’s a great post.

  5. john personna writes:

    Oh think the post is excellent, given the income misdirection, of course.

  6. Evan writes:

    Gappy, I think that it’s fine to argue how much welfare benefit might be derived from a particular level of inequality-reducing policy, but with things as bad as they are, it seems to me that getting to the point where the trend is reversed is more important than how we do it. Later, and hopefully before GDP growth lowers and inflation starts, you can start looking at the policies you have in place and cutting the worst of them.

    But in the midst of a crisis with the trend pointing up, I don’t think that it’s the time to let the perfect be the enemy of the good.

    Unless you’re talking about the moral argument that inequality is bad for people, in which case I am not sure there’s much more to say. I am totally willing to let inequality go where it may assuming a basic level of welfare for everyone and an ironclad guarantee of equal access to influence over the government. Unfortunately, I cannot imagine a world in which the latter exists, and I doubt that the former could survive long in a political environment entirely responsive to the rich.

  7. Steve Roth writes:

    > In my view, freedom, not consumption, is the central distinction between rich and poor. It is odd that I should argue this point with libertarian Wilkinson.

    Somewhat tangential to the main thrust of this post, but yes. For the vast majority of people, the primary constraints on their freedom/liberty/choice/ability to satisfy preferences (you choose) emerge not from some ebil gubmint man, but from their bank balances. I’m always amazed that libertarians don’t seem to grasp this.

    Will’s post is a valiant effort at waving away inequality. “We can’t measure it accurately so it might not even exist.” Feh.

    John Personna: I agree 100% about wealth inequality. The magnitude utterly dwarfs both income and consumption inequality, however measured, and the freedom that comes from wealth is different in both quality and quantity from that derived from income.

    There’s a reason all those characters in 19th century novels are so obsessed with capital. The downside of not having it is profound, and for them back then –poorhouses, debtor’s prisons, all that — it was even more so.

  8. Another nice post.

    One thought:

    both because the marginal utility of consumption diminishes with wealth and because Rich can cushion the loss of real income with substitutions unavailable to Poor.

    Some people might get the impression that these are two independent phenomena, but they are actually very closely related. The diminishing marginal utility of individual goods is what enables substitution to be a utility-maximising strategy. And the existence and range of substitution is one of the key determinants of the rate of decline in the marginal utility of wealth.

    An underexplored issue in classical treatments, which is very relevant to this debate, is the granularity of goods. Continuity in consumption, along with classical assumptions about marginal utility, would imply that even the poorest people should probably buy 0.1% of an iPod along with 0.05% of a Playstation and 0.001% of a Ferrari. In reality, Peter Poor must make many more trade-offs among whole units of goods than Richie Rich. Richie gets a whole iPod, Playstation and Ferrari (though probably no more than a single unit of any of them) while Peter gets none at all. A small increase or decrease in Peter’s wealth could therefore result in a substantial discontinuous jump in utility. A similar change in Richie’s wealth is much less likely to cause such a jump. The cognitive cost for Peter of optimising his consumption under this constraint is high, and he’s quite likely to end up with suboptimal utility as a result.

  9. Dan writes:

    Wilkinson fan here. Allow me to make a few quick points.

    You acknowledge that it’s theoretically possible for different inflation rates to cancel out rising nominal income growth. Then you build a hypothetical to explain why you think it’s unlikely. Fair enough. Substitution effects do come into play (it’s why economists think inflation tends to be overstated).

    But here’s the deal. Once you admit that it’s theoretically possible, it becomes an empirical question. It’s probably easier for the rich to employ substitution effects (any empirical studies of this?), but that doesn’t tell us how much it actually occurs in real life.

    Your example of the subway vs. a cab is convenient, because it supports your view that the poor can’t substitute away. Alternate view: the rich person drives a car from a suburb with no train or bus serivce nearby, whereas the poor person can substitute away from the train in favor of the (cheaper) bus.

    How often does this actually happen? We don’t know. That’s the point of having multiple price indices. If we have two indices in time period 1, we can measure the impact of an increase in the price of caviar on the two different groups in time period 2. If the rich substitute away from caviar, we can measure how much is consumed in time period 2 and give caviar less weight in the basket of goods. Thus, the substitution effect could be measured.

    It’s a ‘just so’ story to say that different price indices would account for the entire rise in nominal inequality. I don’t think anyone is making that argument, certainly not Wilkinson. However, I find it very plausible that the price of goods in the consumption of the poor has risen slower than that of the rich. We don’t really know whether, and to what degree, that’s true. Which was Wilkinson’s entire point, contra his co-blogger at The Economist.

  10. […] Using multiple price indexes to measure changes in inequality is not a good idea Steve Waldman […]

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  12. liberal writes:

    In my view, freedom, not consumption, is the central distinction between rich and poor. It is odd that I should argue this point with libertarian Wilkinson.

    Odd? You must be kidding. It’s odd only if you don’t understand an important part of economics, namely the role of rent in a modern economy.

    (Most) so-called libertarians are nothing more than crypto-feudalists who think that the use of force (via the government or private actors) in the collection of economic rents (primarily land rent, but there are others of course like those accruing to “intellectual property rights”) is fine and dandy.

  13. winterspeak writes:

    SRW: This also works in reverse of course. If the price of caviar falls, the rich get very little benefit, but the poor get a lot as their opportunity set is larger.

    I’m not sure if, from the 1950s to day, “caviar” has net fallen or risen. Items like cellphones, computers, the internet, cars, healthcare (I’m serious — you can get 1950s healthcare very cheaply, you just wouldn’t want it) are all luxuries that have become commonplace today. Gordon Gekko is happy to trade in his motorola brick for an iPhone, but not as happy as someone who went from nothing to an iPhone.

    In general, any new invention brings the price from infinity — out of reach to everyone no matter how rich they are — to something within the reach of at least a few, thus increasing the opportunity set of the poor. This is not captured by standard CPI measures at all.

  14. Indy writes:

    Scott Sumner’s got a related good post on this where I made this comment.

    I stand by it – I think that what originally motivated this meme is the ancient historical and ongoing need of the progressive idealist to have a compelling justification for the urgent enactment of their particular and preferred political program. “Inequality” and “The Inexcusable Condition Of Our Poor” anything short of perfect equality can almost always fulfill this function … until it jumps the shark and starts to appear unreasonable, hysterical, and fundamentalist – always raising the bar, shifting the standard, and snatching away the football, so that any actual satisfaction before the achievement of “Total Victory!” is impossible.

  15. […] price indexes should be used for rich and poor to better measure income inequality.  Steve Waldman wrote a long post attempting to refute Wilkinson’s assertion: As the price difference between caviar and hot […]

  16. […] interfluidity » Using multiple price indexes to measure changes in … […]

  17. Free Spirit writes:

    I’ll second Winterspeak’s comment and add another: How does your argument change if Rich is at the cusp of entering retirement with a full asset load, while Poor is a teen about to become independent of his parents?

    The studies of income that use longitudinal data from the IRS point to a “stage-of-life” component in income and wealth inequality. Demographics are very important here, as are distribution of heritable wealth (the common metric of inequality) and jobs (the determinant of income for working types). Any significant gain from the sale of an asset, whether that gain is real or nominal, whether that asset was held for decades of inflationary pressure, results in a short-term peak in the income numbers (without a change in wealth). All of this has to be accounted for and corrected.

  18. Ted K writes:

    Terrific post. Wilkinson certainly isn’t shy in having it out with bloggers. If he doesn’t respond here somewhere in the near-term I will take that as the waving of a white flag. It’s great seeing someone in the higher-income bracket with the intellectual muscle to have it out with this garbage/right-wing propaganda, that way they can’t just cry “class resentment!!” after they’ve lost the argument.

    Sure do appreciate this stuff Mr. Waldman. You’re one of the best in the econ/fin blogging world.

  19. JKH writes:


    Wilkinson draws the analogy of tracking inflation for different countries rather than different income groups – and then treating an income group as a country.

    Imports for a country will change over time as a partial function of inflation differentials. This is akin to substitution. Isn’t this captured in inflation measures?

    Given his use of such an analogy, why would you assume he’s ignoring substitution effects?

  20. […] on issues like this—what, concretely, is stopping people from getting ahead—and less quibbling about the data. There are concrete steps we could take to make people better-off, and we should take […]

  21. […] too boring to write a post about) and confused indignation from those that want to talk about everything except real income inequality. Anyway, this prompted me to take another look at Broda’s (the intellectual ring master of […]

  22. […] already seen these, but for recording keeping sake here’s Matt Steinglass, Will Wilkinson, Steve Waldmen, Will again, and Karl. My take on it is that each income inequality is a multidimensional […]

  23. […] already seen these, but for recording keeping sake here’s Matt Steinglass, Will Wilkinson, Steve Waldmen, Will again, and Karl. My take on it is that each income inequality is a multidimensional […]

  24. […] Using multiple price indexes to measure changes in inequality is not a good idea – But politically it probably make sense. […]

  25. Oliver writes:


    Not strictly to your or SRW’s point, but even if Wilkinson did not ignore substitution when comparing income classes with nations it seems to me that he is committing an error. There is a large qualitative difference in the decision of one consumer to voluntarily forgo the consumption of a lower class of goods in favour of a higher one within the same, transparent market than for the consumer in one country to wait for mechanisms outside of his/her immediate control to create a level playing field at some theoretical future time. For the analogy between classes of consumers and classes of nations (or even regions within a country) to work, one would have to assume equal ‘frictions’ (is that the word?) on all accounts. An example: how can one account for a thing like a language barrier or a differing legal system between countries (both exist within the EU e.g.) with an impersonal, transparent domestic goods market? Are regional and ethnic affiliations comparable to consumption habits? Is the reason for someone born in say Paris, Palermo or Pitcairn to remain there despite glaring distortions in international comparison that often are not aligned with strict personal preferences not far more profound than the reason for someone to purchase the Mac instead of the Dell? Obviously, there are also ‘frictions’ within goods markets such as peer pressure among youths or golf club members that work the other direction. But either way, I think it’s safe to say: substitution ≠ substitution.

  26. Oliver writes:

    …or am I muddling different economic concepts?

  27. JKH writes:


    I agree with your point. It’s a crude analogy, and international trade frictions are stickier than intra-national ones.

    I’m no expert in the construction of price indexes. But the argument following from his international analogy seemed to imply a price index construction process that followed the actual composition of imports and therefore changes in that composition. So I’m assuming he implied a similar dynamic in his domestic argument based on income levels.

  28. Noumenon writes:

    I’m trying to find a post on this blog. The one with a thought experiment about society having a huge financial investment vehicle with no actual value, but that provided the wealthy with feelings of security. Can anyone help me find it?

  29. Noumenon writes:

    Wow, that was a lot more complicated than the way I remembered it.

    That post’s influence might be broadened just by giving the model a name so people could talk about (and Google for) “Steve Waldman’s ‘Inequality Dragdown’ model” or whatever you’d call it. Thanks for helping me find it the hard way.