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The Bureau of Labor Statistics routinely checks the prices of 211 different categories of consumption items in 38 geographical areas to compute 8,018 item-area indices.

Many other indices are computed as weighted averages of these base indices.

CPI-W is based on a market basket of goods and services consumed by urban wage earners and clerical workers.

The weights for that index are updated in January of every even-numbered year.

People who say the CPI-W overestimates inflation recommend updating the weights each month; this produces the Chained Consumer Price Index for all urban consumers (C-CPI-U).

People who say the C-CPI-U [or the unchained CPI for All Urban Consumers (CPI-U)] disadvantages the elderly point out that seniors consume more medical care than younger people, and that the costs of medical care have been rising faster than inflation in other parts of the economy.

According to this view, the costs of the things the elderly buy have been rising faster than the market basket averaged to obtain CPI-W, CPI-U or C-CPI-U.

Some have recommended fixing this by using a CPI for the Elderly (CPI-E).

Feldstein, Martin; Jeffrey Liebman (editors) (2002).

The Distributional Aspects of Social Security and Social Security Reform.

A pyramid scheme is a business model that recruits members via a promise of payments or services for enrolling others into the scheme, rather than supplying investments or sale of products.

As recruiting multiplies, recruiting becomes quickly impossible, and most members are unable to profit; as such, pyramid schemes are unsustainable and often illegal.

Pyramid schemes have existed for at least a century in different guises.

Some multi-level marketing plans have been classified as pyramid schemes.

In exchange, the organization promises its new members a share of the money taken from every additional member that they recruit.

The directors of the organization (those at the top of the pyramid) also receive a share of these payments.

For the directors, the scheme is potentially lucrative—whether or not they do any work, the organization's membership has a strong incentive to continue recruiting and funneling money to the top of the pyramid.

Such organizations seldom involve sales of products or services with value.

Without creating any goods or services, the only revenue streams for the scheme are recruiting more members or soliciting more money from current members.

The behavior of pyramid schemes follows the mathematics concerning exponential growth quite closely.

Each level of the pyramid is much larger than the one before it.

For a pyramid scheme to make money for everyone who enrolls in it, it would have to expand indefinitely.

This is not possible because the population of Earth is finite.

When the scheme inevitably runs out of new recruits, lacking other sources of revenue, it collapses.

Since the biggest terms in this geometric sequence are at the end, most people will be in the lower levels of the pyramid; accordingly, the bottom layer of the pyramid contains the most people.

The people working for pyramid schemes try to promote the actual company instead of the product they are selling.

Eventually, all of the people at the lower levels of the pyramid do not make any money; only the people at the top turn a profit.

People in the upper layers of the pyramid typically profit, while those in the lower layers typically lose money.

Since most of the members in the scheme are at the bottom, most participants will not make any money.

In particular, when the scheme collapses, most members will be in the bottom layers and thus will not have any opportunity to profit from the scheme; still, they will have already paid to join.

These recognize that recruiting a large number of others into a scheme can be difficult, so a seemingly simpler model is used.

In this model each person must recruit two others, but the ease of achieving this is offset because the depth required to recoup any money also increases.

The scheme requires a person to recruit two others, who must each recruit two others, and so on.

Prior instances of this scheme have been called the "Airplane Game" and the four tiers labelled as "captain", "co-pilot", "crew", and "passenger" to denote a person's level.

Another instance was called the "Original Dinner Party" which labeled the tiers as "dessert", "main course", "side salad", and "appetizer".

A person on the "dessert" course is the one at the top of the tree.

Another variant, "Treasure Traders", variously used gemology terms such as "polishers", "stone cutters", etc.

One version called the Abundance Fractal uses the four elements as the names of tiers: "Fire", "Air", "Earth", and "Water".

A more recent variation known as the Living Workshop takes the names of plants, calling the tiers "seed", "sapling", "blossom", and "lotus".

Such schemes may try to downplay their pyramid nature by referring to themselves as "gifting circles" with money being "gifted".

Others focus on sharing circles or educational themes around "abundance" to divert attention away from the unsustainable structure.

Popular schemes such as "Women Empowering Women" do exactly this.

Whichever euphemism is used, there are 15 total people in four tiers (1 + 2 + 4 + 8) in the scheme—with the Airplane Game as the example, the person at the top of this tree is the "captain", the two below are "co-pilots", the four below are "crew", and the bottom eight joiners are the "passengers".

The eight passengers must each pay (or "gift") a sum (e.g., $5,000) to join the scheme.

This sum (e.g., $40,000) goes to the captain, who leaves, with everyone remaining moving up one tier.

There are now two new captains so the group splits in two with each group requiring eight new passengers.

A person who joins the scheme as a passenger will not see a return until they advance through the crew and co-pilot tiers and exit the scheme as a captain.

Therefore, the participants in the bottom three tiers of the pyramid lose their money if the scheme collapses.

If a person is using this model as a scam, the confidence trickster would take the majority of the money.

They would do this by filling in the first three tiers (with one, two, and four people) with phony names, ensuring they get the first seven payouts, at eight times the buy-in sum, without paying a single penny themselves.

So if the buy-in were $5,000, they would receive $40,000, paid for by the first eight investors.

They would continue to buy in underneath the real investors, and promote and prolong the scheme for as long as possible to allow them to skim even more from it before it collapses.

This scheme flattens the pyramid into a circle, hence the "Loom" in the name.

Each participant needs to recruit two people without providing them with any service or product.

The scam became viral in April 2020 during the COVID-19 pandemic.

The Blessing Loom scheme has been ruled as illegal in multiple US cities.

The real profit is earned, not by the sale of the product, but by the sale of new distributorships.

Emphasis on selling franchises rather than the product eventually leads to a point where the supply of potential investors is exhausted and the pyramid collapses.

Since matrix schemes follow the same laws of geometric progression as pyramids, they are subsequently as doomed to collapse.

Such schemes operate as a queue, where the person at head of the queue receives an item such as a television, games console, digital camcorder, etc.

For example, ten joiners may be required for the person at the front to receive their item and leave the queue.

Each joiner is required to buy an expensive but potentially worthless item, such as an e-book, for their position in the queue.

The scheme organizer profits because the income from joiners far exceeds the cost of sending out the item to the person at the front.

Organizers can further profit by starting a scheme with a queue with shill names that must be cleared out before genuine people get to the front.

The scheme collapses when no more people are willing to join the queue.

Schemes may not reveal, or may attempt to exaggerate, a prospective joiner's queue position, a condition that essentially means the scheme is a lottery.

Pyramid schemes are based on network marketing, where each person in the pyramid is tasked with bringing in their own subordinates and in turn profiting from their sales or recruitments.

This fails because it essentially requires an infinite number of people to join the company.

If the money you make is based on your sales to the public, it may be a legitimate multilevel marketing plan.

If the money you make is based on the number of people you recruit and your sales to them, it's probably not.

In contrast, pyramid schemes "may purport to sell a product", but often "simply use the product to hide their pyramid structure".

While some people call MLMs in general "pyramid selling", others use the term to denote an illegal pyramid scheme masquerading as an MLM.

Some commentators contend that MLMs in general are nothing more than legalized pyramid schemes.

A pyramid scheme will ask their recruiters to sign up to the business with a big front-up cost.

These are different laws in every state, and they take different actions toward pyramid schemes, but multi-level-marketing is legal.

Multi-level marketing lobbying groups have pressured US government regulators to maintain the legal status of such schemes.

Its complaint states that customers would pay a registration fee to join a program that called itself an "internet mall" and purchase a package of goods and services such as internet mail, and that the company offered "significant commissions" to consumers who purchased and resold the package.

The FTC alleged that the company's program was instead and in reality a pyramid scheme that did not disclose that most consumers' money would be kept, and that it gave affiliates material that allowed them to scam others.

In early 2006, Ireland was hit by a wave of schemes with major activity in Cork and Galway.

Participants were asked to contribute €20,000 each to a "Liberty" scheme which followed the classic eight-ball model.

Payments were made in Munich, Germany, to skirt Irish tax laws concerning gifts.

Spin-off schemes called "Speedball" and "People in Profit" prompted a number of violent incidents and calls were made by politicians to tighten existing legislation.

Ireland has launched a website to better educate consumers to pyramid schemes and other scams.

On 12 November 2008, riots broke out in the municipalities of Pasto, Tumaco, Popayán and Santander de Quilichao, Colombia, after the collapse of several pyramid schemes.

Thousands of victims had invested their money in pyramids that promised them extraordinary interest rates.

The lack of regulation laws allowed those pyramids to grow excessively during several years.

In the United Kingdom in 2008 and 2009, a £21 million pyramid scheme named 'Give and Take' involved at least 10,000 victims in the south-west of England and South Wales.

Leaders of the scheme were prosecuted and served time in jail before being ordered to pay £500,000 in compensation and costs in 2015.

The cost of bringing the prosecution was in excess of £1.4 million.

A number of authorities around the world declared TVI Express to be a pyramid scheme in 2010 and 2011, including the Australian Competition & Consumer Commission, the Bank of Namibia, and the Central Bank of Lesotho.

TVI Express, operated by Tarun Trikha from India, has apparently recruited hundreds of thousands of "investors", very few of whom, it is reported, have recouped any of their investment.

BurnLounge, Inc. was a multi-level marketing online music store founded in 2004 and based in New York City.

By 2006 the company reported 30,000 members using the site to sell music through its network.

In 2007 the company was sued by the Federal Trade Commission for being an illegal pyramid scheme.

The company lost the suit in 2012, and lost appeal in June 2014.

In June 2015, the FTC began returning $1.9 million to people who had lost money in the scheme.

In August 2015, the United States FTC filed a lawsuit against Vemma Nutrition Company, an Arizona-based dietary supplement Multi-Level Marketing firm, accused of operating an illegal pyramid scheme.

In December 2016, Vemma agreed to a $238 million settlement with the FTC, which banned the company from "pyramid scheme practices" including recruitment-focused business ventures, deceptive income claims, and unsubstantiated health claims.

In March 2017, Ufun Store registered as an online business for its members as a direct-sales company was declared operating a pyramid scheme in Thailand.

The Criminal Court handed down prison terms totaling 12,265 to 12,267 years to 22 people convicted over the scheme, which conned about 120,000 people out of more than 20 billion baht.

The Airplane Game scheme resurfaced in 2020 as a savings club by the name of the Blessing Loom, Blessing Circle, Sending Flower, Gifting Flower, and so on.

It is also misrepresented as a Sou-Sou, a legitimate savings club model where the number of people paying into the pot is fixed and each person gets a payout on regular basis.

This is distinct from the pyramid scheme variation making use of the hierarchical flower model, where new members are continually recruited into the group by people at the edges of the flower.

This re-imagining of the old Airplane/8-ball model is making a resurgence on social media platforms such as Instagram and Zoom.

Ragnar Lodbrok was a legendary Viking hero, as well as a legendary Danish and Swedish king.

He is known from Old Norse poetry of the Viking Age, Icelandic sagas, and near-contemporary chronicles.

According to the traditional literature, Ragnar distinguished himself by conducting many raids against the British Isles and the Holy Roman Empire during the 9th century.

He also appears in Norse legends, and according to the legendary sagas Tale of Ragnar's Sons and a Saga about Certain Ancient Kings, Ragnar Lodbrok's father was the legendary king of the Swedes, Sigurd Ring.

Nearly all of the sagas agree that the Danish king Randver was Sigurd's father, with the Hervarar saga citing his wife as Åsa, the daughter of King Harald of the Red Moustache from Norway.

The accounts further tell that Randver was a grandson of the legendary Scandinavian king Ivar Vidfamne by his daughter Aud (whom the Hervarar saga calls Alfhild).

After the death of king Ivar Vidfamne, Aud's eldest son by the Danish king Hrœrekr Ringslinger, Harald, conquered all of his grandfather's territory and became known as Harald Wartooth.

Harald's nephew Sigurd Ring became the chief king of Sweden after Randver's death (Denmark according to Hervarar saga), presumably as the subking of Harald.

Sigurd and Harald fought the Battle of the Brávellir (Bråvalla) on the plains of Östergötland, where Harald and many of his men died.

Sigurd then ruled Sweden and Denmark (being sometimes identified with a Danish king Sigfred who ruled from about 770 until his death prior to 804).

He sired a son with the princess Alfhild of the petty kingdom of Álfheimr, Ragnar Lodbrok, who succeeded him.

Eysteinn Beli, who according to the Hervarar Saga was Harald Wartooth's son, ruled Sweden sometime after Sigurd until he was slain by the sons of Ragnar and Aslaug.

In their accounts of his reign, the Sagas of Scandinavian Prehistory, known as fornaldarsǫgur tell more about Ragnar's marriages than about feats of warfare.

According to the Sögubrot, "he was the biggest and fairest of men that human eyes have seen, and he was like his mother in appearance and took after her kin".

He first killed a giant snake that guarded the abode of the Geatish jarl Herrauð's daughter Thora Borgarhjort, thereby winning her as his wife.

The unusual protective clothes that Ragnar wore, when attacking the serpent, earned him the nickname Lodbrok.

After Thora died, he discovered Kráka, a woman of outstanding beauty and wisdom living with a poor peasant couple in Norway, and married her.

This marriage resulted in the sons Ivar the Boneless, Björn Ironside, Hvitserk, Ragnvald and Sigurd Snake-in-the-Eye.

Kráka was later revealed to actually be Aslaug, a secret daughter of the renowned hero Sigurd Fafnesbane.

As the sons grew up to become renowned warriors, Ragnar, not wishing to be outdone, resolved to conquer England with merely two ships.

He was however defeated by superior English forces and was thrown into a snake pit to die in agony.

The Saga of Ragnar Lodbrok, Tale of Ragnar's Sons, and Heimskringla all tell of the Great Heathen Army that invaded England at around 866, led by the sons of Ragnar Lodbrok to wreak revenge against King Ælla of Northumbria who is told to have captured and executed Ragnar.

They call on the various Danish petty kings to help them ruin the realm of the Franks.

Ywar successfully attacks the kingdoms of Britain, though not as an act of revenge as in the Icelandic sagas.

The chronicle of Sven Aggesen (c. 1190) is the first Danish text that mentions the full name, Regnerus Lothbrogh.

His son Sigurd invades Denmark and kills its king, whose daughter he marries as he takes over the throne.

Their son in turn is Knut, ancestor of the later Danish kings.

Neither of these sources mentions Ragnar Lodbrok as a Danish ruler.

The first to do so is Saxo Grammaticus in his work Gesta Danorum (c. 1200).

This work mixes Norse legend with data about Danish history derived from the chronicle of Adam of Bremen (c. 1075).

Here Ragnar's father Sigurd Ring is a Norwegian prince married to a Danish princess, and different from the victor of Brávellir (who had flourished about thirteen generations earlier).

Sigurd Ring and his cousin and rival Ring (that is, Sigfred and Anulo of recorded history, d. 812) are both killed in battle, whereupon Ragnar is elevated to the Danish kingship (identified by Saxo with Ragnfred, d. 814).

His first deed is the defeat of the Swedish king Frö, who has killed Ragnar's grandfather.

Ragnar is assisted in this by a ferocious shield-maiden named Ladgerda (Lagertha), whom Ragnar forces to marry him.

In this marriage he sires the son Fridleif and two daughters.

Ragnar later repudiates his marriage to Ladgerda and marries Thora Borgarhjort, a daughter of the Swedish king Herrauðr, after killing two venomous giant snakes that guard Thora's residence.

His sons with Thora are Radbard, Dunvat, Sigurd Snake-in-the-Eye, Björn Ironside, Agnar and Ivar the Boneless.

From a non-marital relationship with an unnamed woman (described only as a daughter of a man named Esbjørn), Ragnar fathered Ubbe.

Another, final marriage, to Svanlaug (possibly another name for Aslaug) produces another three sons: Ragnvald, Eric Weatherhat and Hvitserk.

The sons were installed as sub-kings in various conquered territories.

Ragnar led a Viking expedition to England and killed its king, Hama, before killing the earls of Scotland and installing Sigurd Snake-in-the Eye and Radbard as governors.

Norway was also subjugated, and Fridleif was made ruler there and in Orkney.

Later on, Ragnar with three sons invaded Sweden where a new king called Sörle had appeared and withheld the heritage of Thora's sons.

Sörle and his army were massacred and Björn Ironside was installed on the throne.

Some time later Björn was put in charge of Norway, while Ragnar appointed another son, Eric Weatherhat, as ruler in Sweden; he was subsequently killed by a certain Eysteinn.

One of the sons, Ubbe, revolted against his father at the instigation of his maternal grandfather Esbjørn, and could only be defeated and captured with utmost effort.

Finally, the Scythians were forced to accept Hvitserk as their ruler.

In the end Hvitserk was treacherously captured by the Hellespontian prince Daxon and burnt alive with his own admission.

Hearing this, Ragnar led an expedition to Kievan Rus' and captured Daxon who was curiously spared and exiled.

Unlike the Icelandic sources, Saxo's account of Ragnar Lodbrok's reign is largely a catalog of successful Viking invasions over an enormous geographical area.

Among the seaborne expeditions was one against the Bjarmians and Finns (Saami) in the Arctic north.

The Bjarmian use of magic spells caused foul weather and the sudden death of many Danish invaders, and the Finnish archers on skis turned out to be a formidable foe.

Eventually these two tribes were put to flight and the Bjarmian king was slain.

The historical king Harald Klak is by Saxo (based on a passage in Adam's chronicle) made into another persistent enemy of Ragnar, who several times incited the Jutes and Scanians to rebel, but was regularly defeated.

After the last victory over Harald, Ragnar learned that King Ælla had massacred Ragnar's men on Ireland.

Incensed, he attacked the English king with his fleet but was captured and thrown into the snake pit, similar to the Icelandic sagas.

The Ragnarsdrápa, ostensibly composed by Bragi Boddason in the 9th century, praises a Ragnar, son of Sigurd, for a richly decorated shield that the poet has received.

The shield depicts the assault on Jörmunrek, the Hjaðningavíg tale, the ploughing of Gefjon, and Thor's struggle with the Midgard Serpent.

The Knutsdrapa of Sigvat Thordarson (c. 1038) mentions the death of Ælla at the hands of Ivar in York, who "carved the eagle on Ælla's back".

From this the story of the atrocious revenge of Lodbrok's sons already seems to be present.

The reference to a "blood eagle" punishment has however been much debated by modern scholars.

Another lay, Krakumal, put in the mouth of the dying Ragnar in the snake pit, recounts the exploits of Ragnar and mentions battles over a wide geographical area, several relating to the British isles.

The poem's name, "Kráka's lay", alludes to Ragnar's wife's Kráka, though modern philologists commonly date it to the 12th century in its present form.

There is one runic inscription mentioning Lodbrok, carved on the prehistorical tumulus of Maeshowe on Orkney in the early 12th century.

It reads: "This howe was built a long time before Lodbrok's.

Her sons, they were bold; scarcely ever were there such tall men of their hands".

The Viking forces were led by a Norse chieftain named "Reginherus", or Ragnar.

This Ragnar has often been tentatively identified with the legendary saga figure Ragnar Lodbrok, but the accuracy of this is disputed by historians.

Ragnar Lodbrok is also sometimes identified with a Ragnar who was awarded land in Torhout, Flanders, by Charles the Bald in about 841 but eventually lost the land as well as the favour of the King.

Ragnar's Vikings raided Rouen on their way up the Seine in 845 and in response to the invasion, determined not to let the royal Abbey of Saint-Denis (near Paris) be destroyed, Charles assembled an army which he divided into two parts, one for each side of the river.

Ragnar attacked and defeated one of the divisions of the smaller Frankish army, took 111 of their men as prisoners and hanged them on an island on the Seine to honour the Norse god Odin, as well as to incite terror in the remaining Frankish forces.

Ragnar's fleet made it back to his overlord, the Danish King Horik I, but Ragnar soon died from a violent illness that also spread in Denmark.

According to William, the Danish kings of old had the custom to expel the younger sons from the kingdom to have them out of the way.

It was during the time this practice was in fashion that King Lodbrok succeeded his unnamed father on the Danish throne.

After gaining power he honoured the said custom and ordered his junior son Björn Ironside to leave his realm.

Björn thus left Denmark with a considerable fleet and started to ravage in West Francia and later the Mediterranean.

Roughly contemporary with William is Adam of Bremen whose history of the Archbishopric of Hamburg-Bremen contains many traditions about Viking Age Scandinavia.

In a passage referring to the Viking raids of the late 9th century, he mentions the Danish or Norse pirates Horich, Orwig, Gotafrid, Rudolf and Inguar (Ivar).

There the Vikings lost, their king slain and many dead, with few escaping to their ships.

After the battle the Saxons took great plunder, and among other things the banner called "Raven".

This is among the earlier references to the legendary hero Ragnar Lodbrok.

The Irish Cogad Gáedel re Gallaib from the 12th century, with information deriving from earlier annals, mentions king Halfdan (d. 877) under the name "mac Ragnaill".

The form Ragnall may refer to either Ragnvald or Ragnar, and the entry is a strong indication that the name of Ivar's and Halfdan's father was really Ragnar or a similar name.

The early 11th century Three Fragments contains a passage that gives a semi-legendary background to the capture of York by the Vikings in 866.

The two younger sons of Halfdan, King of Lochlann, expelled the eldest son Ragnall who sailed to the Orkney islands with his three sons and settled there.

Two of the sons later raided the English and Franks, proceeding to plunder in the Mediterranean.

One of them learnt from a vision that Ragnall had fought a battle where the third son had been slain and in which he himself had most likely perished.

The two Viking sons then returned home with a lot of dark-skinned captives.

Among the organizers were at least some of the brothers: Ivar the Boneless, Ubba, Halfdan, Björn Ironside, Hvitserk, and Sigurd Snake-in-the-Eye, all of which are known as historical figures, save the slightly more dubious Hvitserk.

Ivar the Boneless was the leader of the Great Heathen Army from 865 to 870, but he disappears from English historical accounts after 870.

The Anglo-Saxon chronicler Æthelweard records Ivar's death as 870.

Halfdan Ragnarsson became the leader of the Great Heathen Army in about 870 and he led it in an invasion of Wessex.

A great number of Viking warriors arrived from Scandinavia, as part of the Great Summer Army, led by King Bagsecg of Denmark, bolstering the ranks of Halfdan's army.

According to the Anglo-Saxon Chronicle, the Danes battled the West Saxons nine times, including the Battle of Ashdown on 8 January 871, where Bagsecg was killed.

Halfdan accepted a truce from the future Alfred the Great, newly crowned king of Wessex.

After Bagsecg's death Halfdan was the only remaining king of the invading host.