Herfindahl-Hirschman Index: Which Market Are You Measuring?

What changes when the list is a market rather than a book?

Squaring each share, adding the squares, the run from near nothing to ten thousand, why the squares rather than the shares, and the curve produced by splitting a list into equal parts: all of that is covered separately under Concentration Risk: How Exposure Clusters and How It Is Measured. Everything left over shows up only when the list being measured stops being a business's own book and starts being a market.

The change from a book to a market sounds small and it is not. Run the index over a business's own customers and the list argues with nobody. My customers is a fact about invoices: every name on the list sent the business money, and the money can be counted. Run it over a market and somebody has to decide which sellers are in that market before a single share can be worked out, and then somebody has to find out what each of them sold. Three questions arrive with that decision and none of them existed on the first list: who drew the boundary, what number would make any difference to anybody, and whether the shares supplied were measured one by one or handed over as an average.

An index is only as good as the boundary somebody drew. Picture ten shops in one mall. Measure the mall and one of them looks large. Measure the district and the same shop is one of two hundred. Measure everything that a phone can order to the same street and the shop is a rounding error. Three numbers, three correct calculations, three completely different pictures, and nobody has lied at any point in the process. The shop did not change. The line around it did.

Why does the scale have to be named every single time?

Because the same index is written two ways and the two differ by a factor of ten thousand. Shares entered as percentages, so thirty goes in as 30, take the total up to 10,000. The same shares entered as fractions, so thirty per cent goes in as 0.30, take the total up to 1. Nothing about the list changed. Only the units of what was fed in.

Anjani Stationers Private Limited, an invented stationer, has a customer book already published in full: thirty six accounts, one of them the Sunrise Public School group at Rs 81,00,000/- of a Rs 2,70,00,000/- year, or 30.00 per cent, and thirty five further accounts at Rs 5,40,000/- each on an average, or 2.00 per cent apiece. Run the index over that book and it scores 1,040 on the 0 to 10,000 scale. The same book scores 0.1040 on the nought-to-one scale. The two figures are not two findings and they are not two lists. One index has been written down in two units, and a reader carrying a remembered number in their head will apply it to whichever of the two figures happens to be in front of them.

So the rule is an instruction rather than a preference. Print the scale or print nothing. A figure of 1,040 with no scale attached is not slightly incomplete. The figure could be four orders of magnitude away from what the reader thinks they are reading.

Field note. The thirty six shares behind both figures are set out under How to Analyse Customer Concentration and Dependence, in the sorted customer list at the foot of its second section. The index is arithmetic on those shares, and the working below lets it be rebuilt rather than taken on trust.

How many equal names would score the same?

One division is all it takes. 10,000 divided by the index gives the number of equal shares that would have produced it. The division converts a shape into a count. Most people want to travel in that direction, and almost nobody is given a tool for it.

Anjani Stationers Private Limited has thirty six accounts and its book scores 1,040 on the 0 to 10,000 scale. Ten thousand divided by 1,040 is 9.62. A head count says thirty six and the index says nine and a half, and only one of those two numbers is about the shape of the list. Put the two ends of the comparison plainly beside each other. Thirty six perfectly equal accounts would score 10,000 divided by 36, or 277.78 on the 0 to 10,000 scale. No thirty six name list can go lower. The real book sits at 1,040, at 3.744 times its own floor.

Think of a street with thirty six shops on it where one shop takes a third of everything anybody spends on that street. Ask how many shops are on the street and the answer is thirty six, correctly. Ask how the street's money is spread and thirty six is a misleading answer. The honest one is closer to nine or ten. The equivalent count is the second answer, said as a number instead of as an impression.

Field note. The head count of thirty six and the 30.00 per cent share both sit in the published customer list; the 9.62 and the 277.78 are two divisions on those figures. The equivalent count carries no scale of its own and cannot be checked without the figure it came from, so it is always written down with the index beside it.

How much of a score can sit on one name?

Take the book apart into the two parts it is actually made of. The Sunrise Public School group is 30.00 per cent of the revenue, and thirty squared is 900. The other thirty five accounts contribute two squared each, four apiece, and thirty five fours are 140. Nine hundred and one hundred and forty make the 1,040. One name of thirty six supplies 900 of the 1,040 points, and 900 is 86.54 per cent of the whole index on the 0 to 10,000 scale.

Now the reading that is worth sitting with, and it is the one the arithmetic gives up without being asked. Strip the largest account out entirely and look at what is left. Thirty five accounts sharing 70.00 per cent, rebasedRecomputing a set of shares against a new total after something has been taken out, so the shares that remain add back up to one hundred. on themselves so they add to a hundred again, come to 2.857142 per cent each. Squared and multiplied by thirty five, that gives exactly 285.71 on the 0 to 10,000 scale. 10,000 divided by 35 is also 285.71, the floor for a thirty five name list. The thirty five accounts left after the largest one are as unconcentrated as a thirty five name list can possibly be, so the entire reading is one account, and the index says so without anybody telling it to look.

Beside that sits a related figure: the same account holds 40.00 per cent of the money still owed against 30.00 per cent of the money earned, a gap of 10.00 points, covered separately under How to Analyse Customer Concentration and Dependence.

What does the index see that a largest four figure does not?

A concentration ratioThe combined share held by the largest few names on a list, usually the top four or the top eight, quoted as one percentage. The ratio stops counting once it reaches the number of names it was defined on. is the older and simpler measure, and on this book it is easy to work out. The largest account is 30.00 per cent. The three behind the Sunrise Public School group are 2.00 per cent each, and thirty plus two plus two plus two is thirty six, so the largest four together are 36.00 per cent.

Now construct a second list purely to test the measure, an arithmetic list and not a business. Four names at 9.00 per cent and thirty two names at 2.00 per cent. Four nines are thirty six and thirty two twos are sixty four, so it sums to 100 and it counts thirty six names, exactly like the real book. Its largest four come to 36.00 per cent, identical to the real book's. And 4 lots of 81 plus 32 lots of 4 is 324 plus 128, or 452 on the 0 to 10,000 scale. Same count of names, same largest four figure to two decimal places, and the index separates the two lists by more than two to one.

Everyday version. Two households both spend thirty six thousand rupees a month across thirty six things. In the first, one of those things is the rent and it takes a third. In the second, four things take a modest slice each and everything else is small. Ask what the four biggest items add up to and both households answer identically. Ask which one has a problem if a single bill changes and only one of them is in trouble.

Field note. The 30.00 per cent and the 2.00 per cent both come off the published customer list, and the largest four figure of 36.00 per cent is one addition on those published shares. The constructed list describes no business.

If the list given is an average, what has actually been computed?

One word in the published customer book changes what the 1,040 on the 0 to 10,000 scale actually is. Go back and read the book one more time, slowly. The thirty five accounts behind the Sunrise Public School group are given as an average of Rs 5,40,000/- each. Not measured one by one and listed. Averaged, and printed as one line.

Averaging matters because of a property of a sum of squaresThe result of squaring every item in a set and adding the results. Squaring makes larger items count for disproportionately more than smaller ones. that has nothing to do with concentration and everything to do with arithmetic: for a fixed group total, the sum of squares is at its smallest when the members of the group are equal, and any movement away from equality raises it. The thirty five accounts total 70.00 per cent whatever their individual sizes are. Fed in as equal, they give the smallest sum of squares that group can produce. So 1,040 on the 0 to 10,000 scale is not the book's index. 1,040 is the lowest index consistent with what was published, and the true figure can only be higher.

Prove it with a second tail that fits the published line exactly as well as the first. Five accounts at 6.00 per cent and thirty accounts at 1.3333 per cent. Five sixes are thirty and thirty times one and a third is forty, so the tail still totals 70.00 per cent across thirty five names, and the average of that tail is still Rs 5,40,000/-. Its index is 900 from the largest account, plus 5 lots of 36 for 180, plus 30 lots of 1.7778 for 53.33. The total comes to 1,133.33 on the 0 to 10,000 scale: 93.33 points above the floor, a rise of 8.97 per cent. The direction is knowable even though the figure is not.

Then answer the one question a reader wants answered on being told a figure is uncertain. How uncertain? Under nine per cent, and there is a specific reason for that rather than a lucky one. One name already carries 86.54 per cent of the index. Everything unknown about the other thirty five is arguing over the remaining 13.46 per cent of the score, so even a badly lopsided tail cannot move the total far. Banded dataFigures published as groups or averages instead of one by one, so what a set of things adds up to is stated without any statement of how it splits inside the set. is a weaker input than a full list, and how much weaker depends entirely on how much of the answer was already settled before the band began.

So the rule, stated as an instruction. Say floor. Say which direction the doubt runs. Say roughly how large it is. A figure written flat is a claim that somebody measured thirty five accounts, and nobody did. So never write 1,040 flat, anywhere.

Who drew the boundary of the market?

Every index needs a list, and every list of sellers needs somebody to have decided which sellers are on it. On a customer book that decision makes itself and nobody notices it happening. My customers is a fact recorded in an invoice ledger, and it settles its own membership. On a market it makes itself nowhere at all.

A market boundary is a decision about what a buyer could turn to instead. Move that decision and every share on the list moves, and so does the whole index. Work the shape on one seller without inventing a single share. The shape is the point, and shares would only distract from it. Take a maker of hard bound registers on one lane. Measure it against the other register makers on that lane and it sits inside a very short list. Measure it against every register maker in the city and the list gets long and its own share drops. Measure it against everybody selling anything a school buys to write in, so notebooks, loose sheets, files and whatever else takes a pen, and the list changes shape again. Measure it against everybody selling anything a school buys at all, and the register maker becomes a small line on a very long list.

Four boundaries, four lists, four indices, and nobody lied at any point. The boundary problem is not a difficulty to be managed carefully. The boundary is the measure's largest failure mode, and what makes it a failure mode rather than an inconvenience is that the arithmetic gives no warning at all: it returns a clean, correctly computed number for every one of the four. There is no flag on the output, no wider range, no footnote the formula adds by itself. The boundary decision was taken before the shares were collected, and nothing downstream of it knows it happened.

The everyday version is the chemist on a residential lane. One chemist, and it looks like the only one. Four chemists in the neighbourhood, and it looks ordinary. Add the delivery application that reaches all four addresses from a warehouse somewhere else and the picture changes again, and none of the three pictures is a lie about the chemist.

What settles a boundary, and what gets printed when nothing does?

One test would settle it in principle, the second of the three set out under Market Structures: Perfect Competition to Monopoly Compared. Send a refused customer down the road and find out whether the thing they wanted is waiting for them there. Notice which way the test faces. The question goes to whoever is doing the buying, never to whoever is doing the selling. No quantity of reading a seller's own books will ever settle it.

Now the honest part. The test has been run on none of the fields Anjani Stationers Private Limited or Setu Bazaar trades in, so no boundary drawn around either is settled by evidence. The honest output is the index with its boundary printed beside it, and an index with no boundary written next to it is an incomplete figure rather than a number in need of context. The difference between those two descriptions is the difference between a reader who knows something is missing and a reader who thinks they have been handed a finding.

The form is four things and it fits on one line each: the index, the scale it is on, the list it was computed over written as a sentence rather than as a word, and the date the shares were read. Anything missing from those four is a hole, and a visible hole is worth more than a number that cannot be checked.

What number would make a difference, and who decides that?

The question of what number makes a difference never arises on a business's own book, and it is worth seeing why not before the answer arrives. A measure of a business's own risk never has to say when somebody should act, and a competition authority's measure always does. When a business computes its own book and dislikes the result, the consequence is its own to choose. When an authority computes a market, the number sits inside a process with steps, and somebody had to write down where the steps begin.

Put the index on two real spend lists. Does it rank them?

Both lists are published in full under The Value Network: Who a Business Depends On, and both total 100.0 per cent by construction, so nothing has to be estimated and nothing has to be averaged. One convention matters before any of it is used. The spend lists give each share of spendThe slice of everything a business pays out to outside parties that goes to one named party, written as a percentage of all outside payments rather than of revenue. to one decimal place while the customer book runs to two, and both conventions are kept rather than harmonised. A quoted share then stays distinguishable from an index computed on it.

Anjani Stationers Private Limited pays six counterpartiesThe parties on the other side of a business's arrangements, each of which supplies something and takes a payment for it. outside: 46.0, 17.0, 14.0, 12.0, 7.0 and 4.0 per cent. Square each and add them, so 2,116 plus 289 plus 196 plus 144 plus 49 plus 16, and the list scores 2,810 on the 0 to 10,000 scale. Ten thousand over 2,810 is 3.56, so six suppliers are behaving like 3.56 equal ones. Setu Bazaar pays six outside as well: 22.0, 21.0, 20.0, 16.0, 12.0 and 9.0 per cent. Its squares, 484, 441, 400, 256, 144 and 81, come to 1,806 on the 0 to 10,000 scale, and ten thousand over 1,806 is 5.54 equal suppliers.

So the index calls Anjani Stationers Private Limited's buying 1.56 times the more concentrated of the two, and every step of that is correct. Both lists were published individually rather than banded, so neither of these two figures is a floor. Both sum to a hundred. Both were squared properly. Hold that finding lightly.

Did the index just rank the wrong column?

Read the same two tables again for what they actually say, using the columns the index was never given. The party that would stop Anjani Stationers Private Limited is the Machine servicing contractor. The contractor is the smallest payee on the list at 4.0 per cent, it has exactly one alternative, and its replacement timeHow long it would take to get the same thing from somebody else, counted from the day the current arrangement ends to the day supply is running again. is 26 weeks against the Paper mill's 2 weeks. Twenty six weeks is thirteen times two weeks. The mill is 11.5 times the contractor's size. The two ratios point in opposite directions.

The party that would stop Setu Bazaar is the Payment provider. The provider is the smallest payee at 9.0 per cent, and 100.0 per cent of the gross merchandise valueWhat all the goods moving over a marketplace in a year are worth, the sellers' share included, as against the smaller sum the marketplace itself keeps. crossing that marketplace in a year passes through it. Both answers are the smallest row in their own list, and the index weights those two rows at 16 of the 2,810 and 81 of the 1,806, both on the 0 to 10,000 scale, or 0.57 per cent and 4.49 per cent of their own scores.

Now exactly what that does and does not prove. The index did not make a mistake and a better index would not fix this. The index was handed two names and two sets of shares, and it returned a fact about the shape of each list. A fact about shape is the only thing it was ever able to return. A replacement time was never among its inputs. A share of flow was never among its inputs. Neither was in the room, so no rearrangement of the inputs, no reweighting and no smarter formula could have produced either of them. The fault is upstream of the arithmetic: the question actually being asked was which business would be stopped, and the shape of a spend list does not answer that question and has never claimed to. Which counterparty would actually stop a business is covered separately under The Value Network: Who a Business Depends On.

Field note. The shares, the replacement times and the counts of alternatives are all printed under The Value Network: Who a Business Depends On, in its two counterparty tables. The 100.0 per cent of flow through the payment provider is stated in the same place, in the paragraph after Setu Bazaar's table. The four figures 2,810, 1,806, 3.56 and 5.54 are arithmetic on those printed shares.

How wide is the range across five lists?

Five lists have now been measured. The span between the ends is larger than most readers expect, and three of the five carry a floor rather than a figure, so all five belong on one scale.

Setu Bazaar's buyer side first. Fifty thousand buyers, of whom the heaviest 5,000 hold 40.00 per cent, so 0.008 per cent each, and the other 45,000 hold 60.00 per cent, so 0.0013333 per cent each. Five thousand lots of 0.008 squared is 0.32, and 45,000 lots of 0.0013333 squared is 0.08, so the list scores 0.40 on the 0 to 10,000 scale, or 0.00004 on the nought-to-one scale, and is equivalent to 25,000 equal buyers. The split inside each of those two bands is not published, and any inequality inside a band can only push the figure up, so 0.40 is a floor.

Then Setu Bazaar's merchant side, where the data is thinner still. Two thousand merchants are published as a count with no split at all. The most that can be said is the floor: 10,000 divided by 2,000, or 5.00 on the 0 to 10,000 scale, with each merchant averaging Rs 24,00,000/- of the Rs 4,80,00,00,000/- that reaches sellers. Any inequality among the two thousand pushes it up and nothing in what was published caps how far. Note what that list is, though. Two thousand merchants are the only population of sellers published for either business, so they are the only seller side list there is.

So the span runs from 0.40 at one end to 2,810 at the other, both on the 0 to 10,000 scale, four orders of magnitude apart, and both ends are real lists belonging to the two businesses. On that same 0 to 10,000 scale the three floors are the customer book at 1,040, the buyer side at 0.40 and the merchant side at 5.00. The two spend lists, at 2,810 and 1,806, are the two exact figures. Their six shares each were published individually rather than banded.

What can be done when the count is all that is known?

The most useful reading of all is a refusal. The paper market that Anjani Stationers Private Limited buys from has nine interchangeable mills within reach beside the one it currently uses, so ten sellers of an identical good are known to exist. Not one of their shares is published anywhere. Which question is being asked has to be said. The same market is refused under The Industry Life Cycle: Emergence to Decline for a completely different reason. A life cycle cannot be placed with no rate of entry, no direction of capacity and no direction of price. The index cannot measure it because it has no share distribution. Two different gaps, two different refusals, and neither substitutes for the other.

A count is not a share distribution. Show it with the two extremes, and label them as exactly that in the same breath. Both are constructions and neither describes the paper market. One seller at 91 per cent with nine at 1 per cent each scores 8,281 plus 9, or 8,290 on the 0 to 10,000 scale. Ten equal tenths score ten lots of a hundred, or 1,000. Both readings fit a count of ten exactly. Two such fields would behave nothing alike in any respect anybody cares about, and no amount of counting heads would separate them.

So no index can be computed for the only market where a full set of interchangeable sellers is known to exist, in either direction, and saying so is worth more than a number would be. Connect it to what is already settled: a count of sellers answers exactly one of the three tests that place a market, covered separately under Market Structures: Perfect Competition to Monopoly Compared. A better count is still a count, and the two extremes above are that ruling with the arithmetic attached.

Field note. The nine mills, the identical weight and finish, the quotes back in a day and the two weeks to first delivery are all printed under The Value Network: Who a Business Depends On, beside the counterparty tables. The 8,290 and the 1,000, both on the 0 to 10,000 scale, describe no market: they are two constructions made to show what one count leaves open.