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1Kelley School of Business, Indiana University, Bloomington, IN 47405. Ph: (812) 855-3413. Email:

[email protected]

* This research owes an intellectual debt to a few good citizens of Stavropol, Russia, who showed me the

many marvels of unbridled capitalism. I am grateful for a USAID grant through the Eurasia Foundation,

which made possible the trip to Russia in 1993, and to Duke University for financing the trip in 1994.

Klarita Sadiraj helped me understand the Albanian Ponzi schemes, while Oleg Mikhalev provided

perspective on the Russian Ponzi schemes. Frank Acito, Sugato Bhattacharyya, Timothy Crack, Craig

Holden, Rich Rosen, Richard Shockley, Gregg Udell and seminar participants at Indiana, Maryland and

Wisconsin (Madison) provided many thoughtful comments. I am particularly grateful to Mukarram Attari

and Gary Gorton for their insights. All errors are my own.

On the Possibility of Ponzi Schemesin Transition Economies*

by

Utpal Bhattacharya1

First version: May 1998

This version: November 1998


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Abstract

This paper shows that transition economies are breeding grounds for Ponzi schemes. It details how

an unscrupulous profit-maximizing promoter can design classical Ponzi schemes if the following conditions

are met: a large public sector (the proportion of national wealth owned by the state is above a lower bound),

ambiguous laws governing the transfer of property rights from the state to the citizen (victims of a failed

Ponzi scheme may organize to use the states assets for a bailout, the probability of which occurring is above

a lower bound), political connections (the probability of early termination of the Ponzi scheme by a regulator

is below an upper bound) and an inexpensive access to citizens through mass-media (advertising

effectiveness is above a lower bound).

It may not be mere coincidence that some of the spectacular Ponzi schemes in history occurred

during periods of transition -- France (1719), Britain (1720), Russia (1994) and Albania (1997).


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1Summarized from an article appearing in the San Diego Daily Transcript on July 16, 1974.

2For an excellent introduction to Ponzi schemes as well as to many other types of bubbles, see Garber (1990). For detailed expositions,the classics by Mackay (1841) and Kindleberger (1978) are recommended.

3This may be because word-of-mouth referrals have been substituted by the more efficient click-of-mouse referrals. In 1997, the UnitedStates Federal Trade Commission launched "Operation Missed Fortune," a federal and state crackdown on fraudulent Ponzi schemes. Even law

enforcement officers were not spared; 67 employees of the Sacramento Police Department were being investigated. The FTC issued an alert: U.S.

citizens are being asked to report Ponzi and other get-rich-quick schemes to the National Fraud Information Center at 1-800-876-7060 or via the

Internet at http://www.fraud.org. In 1998, the Chinese government banned all businesses that employed some elements of Ponzi schemes,

completely disrupting the selling operations of even legitimate businesses (See Wall Street Journal, May 1, 1998, editorial titled "Avon Ladies Under

Siege.")

-1-

I. Introduction

At the height of his success in Boston in 1920, Charles A. Ponzi was hailed by those he was

cheating as the greatest Italian who ever lived. You're wrong, he said modestly, there's

Columbus, who discovered America, and Marconi, who discovered radio. But, Charlie,

you discovered money, they told him.

The money-making machine that Charles A. Ponzi invented in Boston in June 1919 was elegant in

its simplicity. It had three critical components. First, he convinced a group of people about an investment

idea (coupons issued by the International Postal Union seemingly violated the law of one price and, therefore,

offered an arbitrage opportunity); two, he promised them a high return on their investment (a 50 per cent

interest every ninety days); and, three, he built credibility by initially delivering on his promises ( interest

plus principal of the earlier investments was paid by money invested by those who were recruited into

the scheme later). As his reputation spread by word-of-mouth, people flocked from all over New England

to invest. Ponzi took in about $200,000 a day. The scheme finally crashed when the Boston Globe exposed

him in August 1920.1

Such types of schemes have existed before Ponzi and continue to exist after him. 2 The first

extensively recorded scheme, covered by Mackay (1841), was conceived by a Scotsman, John Law, in France

in 1719. It was immediately followed by the South Sea Bubble in Britain in 1720. Today, thanks to the

Internet, Ponzi schemes are making a dramatic comeback. 3


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4

A partial list of papers in the bubbles literature would be Brock (1979, 1982), Bewley (1980), Tirole (1982, 1985), Scheinkman (1988),Gilles and LeRoy (1992a, 1992b), Kocherlakota (1992), and Huang and Werner (1997). A recent paper by Santos and Woodford (1997)

comprehensively covers this area, and shows that the conditions under which rational bubbles are possible are fragile.

5 Shiller (1981), Shleifer and Summers (1990), and De Long et al. (1990) have explained bubbles using this assumption.

6See Allen and Gorton (1993), Allen, Morris and Postlewaite (1993), and Bhattacharya and Lipman (1995) for examples of rationalbubbles in economies where the number of rounds played and/or time is finite.

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Financial economists have long been puzzled by Ponzi schemes because they seemingly violate the

laws of rationality. An extensive literature has developed to analyze the conditions under which Ponzi

schemes and other types of bubbles can arise in economies that go on forever.4 The existence of these

conditions ensure that it is rational for agents to participate in any round because they expect to close their

position in a later round at a gain. So the Ponzi scheme goes on forever.

To explain Ponzi schemes in economies that do not go on forever, additional assumptions have been

introduced. This literature can be broadly classified into two strands. The first strand is behavioral, and it

assumes that some agents are irrational.5 The second strand, maintaining neo-classical assumptions, has

assumed something specific about the economy (agency problem, asymmetric information, etc.) that drives

the results.6 Our paper belongs to this second strand. Its purpose is to demonstrate how an unscrupulous

promoter can devise a Ponzi scheme in a specific type of finite economy -- a transition economy.

In this paper we argue that a Ponzi scheme is an ingenious method to expropriate state assets by a

politically well-connected promoter in a transition economy. How? The promoter lures citizens with

promises of incredible returns. He exploits their rational belief that, if enough of them take part, the assets

of the state may be used for a bailout if the scheme fails. So the Ponzi scheme in a transition economy is

really a cynical exploitation of the "too big to fail doctrine" by a private citizen. The contribution of this

paper is to detail how this can happen, and then link our hypothesis and its implications to some spectacular

Ponzi schemes that have occurred in transition economies.

A classic Ponzi scheme is an inverted truncated pyramid (see Figure 2). The bottom is the mass of


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citizens participating in the first round, the height is the total number of rounds, and the top is the mass of

citizens participating in the last round. A promoter sells certificates to the citizens in each round, promising

them an attractive return per round on their investment. Since the money raised in a round is used to pay off

the obligation of investors from a previous round, the revenue of the promoter comes mostly from the sales

in the initial round and sales in the last round, when he runs away with the money collected. Also, as a

record of successful payment develops and information about the fantastic scheme spreads by word-of-mouth

(or click-of-mouse), most of the costs of the promoter are the marketing costs of reaching the initial group

of citizens. The risk-neutral promoter designs the scheme to maximize expected profits. He has one

constraint; citizens should participate in each round.

The economic forces at work in the initial rounds is as follows. As the Ponzi scheme is against the

public interest, it is in the interest of the state to intervene immediately. However, there is no representative

government in a transition economy that will do this. Neither are the citizens pivotal enough to coordinate

a stoppage. The only entity that can intervene is a regulator, who trades off the public interest against the

private interest of a political class to which this promoter is linked to. So the probability of intervention is

not unity. More the political connectedness of the promoter, less is this probability of intervention. In the

initial rounds, then, the promoter offers a return that is high enough to ensure that the risk-neutral citizens

expected loss if the regulator intervenes is not greater than the expected gain he achieves if the regulator does

not intervene. So the citizen takes part in these initial rounds.

The economic forces at work in the last two rounds are different. The promoter plans to terminate

the scheme here, if it has not already been terminated before by the regulator. When the scheme is

terminated, the affected citizens get very upset. Anger is the focal point for coordination. They organize to

use the states assets for a bailout, and as their size is large and the size of the state assets is large, the

probability of this bailout is not zero. As our subsequent discussion will reveal, bailouts from state assets

have happened for some failed Ponzi schemes in history. As every citizen has an equal claim on the public


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assets, the bailout amounts to a redistribution of wealth from non-participants to participants. The parameters

of the Ponzi scheme are set such that the expected loss incurred by participating (the price of the certificate

minus the expected net redistribution gain from the bailout) is not greater than the loss incurred by not

participating (the expected redistribution loss from the bailout). So the citizen takes part in these last two

rounds.

The final participatory constraint is the one faced by the promoter himself. We need to address why

the promoter has to wait till the last round to run away with the revenues he collects. The answer is that at

every round, the promoter has to tradeoff the sure revenue he will get if he terminates and runs away now

against the expected revenue he will get if he terminates and runs away a round later. The parameters of the

Ponzi scheme are set to ensure that the latter expected revenue is greater than or equal to the sure former

revenue in all rounds except the last round, and it is lower in the last round. In other words, the Ponzi

scheme is subgame perfect.

An important result of this paper is that under symmetric information, where each citizen knows

which round he is playing, a Ponzi scheme may exist whether the bailout is certain or has a finite probability,

as long as the citizen believes that the bailout will compensate himfor more than what he lost. This is an

unreasonable belief, because we have never seen this occur in history. So we can conclude that, since the

conditions under which Ponzi schemes will germinate in finite economies with symmetric information are

unlikely to exist, Ponzi schemes would be rare in such economies.

To obtain a Ponzi scheme under a more realistic regime of a partial and uncertain bailout, we need

asymmetric information. Under asymmetric information, where the citizen does not know for certain which

round he is playing but has a belief that holds under rational expectations, we give an example of a Ponzi

scheme where the state may give onlypartial compensation. The reason this is possible is because, unlike

in the case of symmetric information where there are two types of citizen participatory constraints (the

constraint for the initial rounds and the constraint for the last round), there is now just one participation


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7It should be noted here that some other episodes % the Tulipmania of Netherlands in 1636-1637 and Ponzis exploits in Boston in 1920% did not have partial bailouts or promises of partial bailouts. So the hypothesis forwarded in this paper can explain Ponzi schemes only in transition

economies.

-5-

constraint (a probability weighted sum of the two types of constraints, where the probability is the rational

expectations belief of the citizen that he is not playing the last round). This allows the promoter an additional

degree of freedom in designing his Ponzi scheme. In particular, he can compensate the citizens increased

expected loss in the last round (caused by a partial bailout) with an increased expected gain in previous

rounds.

We characterize the sufficient conditions under which this constrained maximization problem of the

promoter has a solution. We find that the conditions that breed Ponzi schemes are: a large public sector (the

proportion of national wealth owned by the state is above a lower bound), ambiguous laws governing the

transfer of property rights from the state to the citizen (victims of a failed Ponzi scheme may organize to use

the states assets for a bailout, the probability of which occurring is above a lower bound), political

connections (the probability of early termination of the Ponzi scheme by a regulator is below an upper bound)

and an inexpensive access to citizens through mass-media (advertising effectiveness is above a lower bound).

The above conditions may exist in transition economies. So it may not be mere coincidence that

some of the biggest Ponzi schemes in history have occurred in these economies. A careful examination of

four Ponzi schemes two past and two recent suggests that the four factors described in the last paragraph

may have existed in these countries at that point in time in their history.7

A reading of Mackays (1841) account of the Mississippi Scheme indicates that John Laws scam

had the blessings of France, a state whose finances were in a mess after the death of Louis XIV. According

to Mackay, He proposed to the regent (who could refuse him nothing) to establish a company that should

have the exclusive privilege of trading to the great river Mississippi and the province of Louisiana. In 1719,

Laws company, the Compagnie des Indes,, was further granted the exclusive privilege of trading with the

East Indies, China and the South Seas. John Law started his scheme that year. His scam had the three


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8The observations in this paragraph come from Mackays (1841) account.

9The observations in this paragraph come from Garbers(1990) account.

10Pallada Asset Management in Moscow was entrusted with the responsibility of managing a federal fund set up to compensate victimsof Ponzi schemes (Reuters, February 26, 1997)

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critical ingredients of a classic Ponzi scheme: an investment idea (a share in the profits that were to be made

by trade with exotic lands), a promised attractive return (a 40% annual return on the shares of the

Mississippi Company) and an initial meeting of obligations (he delivered an annual return of 120% initially).

However, it had one feature that we did not observe in Charles Ponzis scheme intimate involvement by

the ruling class. It should be also be noted that when the scheme came crashing down, the holders of useless

Mississippi stock were given 2.5% interest-bearing notes that were secured by the municipal revenues of

the city of Paris.8

The South Sea Bubble in Britain in 1720 was, as Garber (1990) aptly describes, a shadow of the

Mississippi Scheme. The Whig ministry had been dismissed, and public debt was at an astounding ten

millions sterling. In 1720, Parliament granted the South Sea Company monopoly rights over trade with the

South Seas and, in exchange, obtained attractive refinancing terms for the state debt. The South Sea

Company then acted like Laws company: issue successive rounds of stock that promised a share of trading

profits, deliver initial attractive returns (100% return from February to April 1720), and then disintegrate.

Parliament partially bailed out investors by writing off7.1 million sterling of the companys debt.9

History repeated itself in Russia in 1994 as tragedy. Sergei Mavrodi and his MMM scheme

collapsed. He had promised annual returns of 2000%, recruited 5 million Russians, and become the sixth

richest man in Russia. A notable feature of this scheme was the initial non-discouragement by the regulators

and apartial bailoutafter the collapse.10 Bailouts, however, were not promised to upset citizens of the

myriad smaller Ponzi schemes (like Tibet, Ruski Dom Selenga and Khopor) that had also sprouted.

These themes were replayed in a smaller scale in other transition economies in the 1990s. Ponzi

schemes were reported in Romania (600 schemes, the biggest of which was Caritas, which involved twenty


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11The information about the Ponzi schemes in the transition economies of the 1990s is culled from news reports in CNN News, RadioFree Europe, and various issues of Time Magazine. The facts were independently verified by Sadiraj in Albania (see Sadiraj, van Ewijk and van

Wijnbergen (1998)) and by Mikhalev in Moscow. Appendix A shows a copy of a certificate used by MMM in Russia; Appendix B shows a copy

of a certificate used by the foundation Gjallica in Albania. Though we will discuss some salient features of these certificates later in the paper,

interested readers may contact the author for precise translations. In Albania, though the "foundations" were pure Ponzi schemes, "investment

companies" like Vega did have some legitimate economic investments.

-7-

per cent of the population, and which promised 800% return in 100 days), Bulgaria, Slovakia, Serbia and the

Czech Republic.

Then, in 1997, history repeated itself in Albania as farce. Maksude Kademi, Bakshim Driza and

Rapush Xhaferi had promised returns as high as 100% in six months and had sold their certificates to about

half the population. They had attracted a sum which was about four times Albanias state budget, twice its

bank deposits, and roughly equal to its GDP. When their foundations collapsed, about a sixth of the

population lost all their savings, and violent civil unrest erupted. A salient feature of the three Albanian

schemes was the role of the ruling class. State TV actively promoted these funds, giving the impression of

official approval. Political parties endorsedthem. Election posters often included the logos of the funds.

Finally, when the schemes collapsed, the government accepted moral responsibility topay backat least

some of the $370 million lost (which is enormous, considering that the annual state budget is just $500

million ).11

The paper is organized as follows. Section II sets up the optimization problem of an unscrupulous

promoter of a Ponzi scheme, who wants to exploit a citizens belief that a bailout is likely if a critical mass

of citizens are adversely affected. Section III characterizes the sufficient conditions for a solution under the

assumption of symmetric information. We show that only beliefs ofmore than a full bailoutcan lead to

Ponzi schemes. If a Ponzi scheme can exist, we detail its properties. Section IV does the same under the

more realistic assumption of asymmetric information. We give an example to show that beliefs of uncertain

and partial bailouts can also lead to Ponzi schemes. Section V concludes with some sobering policy

implications.


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12There is an implicit "common value" assumption here of the state assets. A "private value" assumption would make the analysis

needlessly complicated without adding much insight.

13Boycko and Shleifer (1993) discuss how special privileges had to be given to managers, workers, and local governments in the earlierstages of Russian privatization.

14As discussed in the introduction of this paper, this did occur in France (1719), Britain (1720) and Russia (1994), and may occur inAlbania (1997). It is interesting to note that Ponzis victims received no compensation from the state in 1920. The victims of "Tulipmania" received

no state compensation in 1637 .

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II. The Model

A. The Political Economy

Let W be the total wealth of a nation, of which a significant fraction is owned by the state.

Normalize W to unity. The state may transfer some of its property rights to its own citizens. Let the

proportion of national wealth that may be so transferred be .

Citizens of this nation are modeled as a continuum of risk-neutral individuals whose mass is N. This

assumption ensures that no citizen is pivotal enough to act strategically. Each citizen has an equal claim on

the state assets. However, if due to any circumstance, a privileged mass M of citizens (M < N) usurps these

claims, the loss per unit mass of citizens excluded from this privileged group is /N , and the gain per unit

mass of citizens included in this privileged group is (1/M - 1/N ). 12 Normalize N to unity.

As laws governing the transfer of property rights from the state to its citizens are unclear, the

possibility of the above exists. Such an event might happen if a powerful clique gets more than its fair share

of the state assets.13 Or it might happen if a significant fraction of the populace unfortunately discover that

they are not being given what has been promised to them in an investment scheme, become very upset, and

demand some compensation from the state.14

We model this situation as follows. When a Ponzi scheme is operating, it is in the public interest

to terminate it. However, as the citizens are atomistic, and there is no representative government to look after

the public interest, there is a coordination failure. There exists, however, a regulator who can terminate the

scheme. This regulator trades off the public interest with the private interest of a ruling class to which the


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promoter of the Ponzi scheme belongs. The probability of intervention is , where is a measure of political

connections of the promoter. Higher is , less is the political connection of the promoter.

If the Ponzi scheme explodes, anger erupts amongst the citizens who have not been paid what had

been promised to them. Anger is the focal point for coordination. They organize to use the states assets for

a bailout, and as their size is large and the size of the state assets is large, there exists a possibility that this

might happen. Let na be the mass of citizens adversely affected, and p(na ) be the resultant probability of

bailout. We study two models: (1) there is no bailout if na is less than a critical mass, n* < N, and certain

bailout if na is greater than or equal to n*, and (2), there is no bailout if na is less than a critical mass, n*, and

uncertain bailout if na is greater than or equal to n*. Under the second regime, we will examine a scenario

where the probability of bailout in the critical region is a constant, . Also, n* $ 0.5 in both the cases.

The above formalization covers a wide range of political regimes. (1) covers situations where

decisions about a bailout are made by majority vote and minorities have no rights. For simple majorities,

we need n* = 0.5N, and for super majorities, we need n* > 0.5N. (2) covers regimes where minority rights

are somewhat respected. Even if a majority of individuals are affected, bailout is not guaranteed.

B. The Promoter of the Ponzi Scheme

A risk-neutral promoter, who is outside the system, devises a classical Ponzi scheme. He has

political connections, and this means that he is endowed with monopoly rights over the Ponzi scheme as well

as the guarantee that the probability of early intervention by the regulator is not unity. We do not model the

rents he pays to obtain these privileges.

In round 0, he sells certificates to a mass n0 of citizens, promising them a return, R. So, if he prices

these certificates at a price P per unit mass of citizens, he promises to redeem them for P(1+R) in round 1.

In round 1, he redeems these original certificates as promised by selling a fresh batch of certificates, again


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15 Appendix A shows a photocopy of a certificate issued by MMM in the Russian Ponzi scheme of 1994. The price of this certificateis a 1000 roubles (shown on the front). Though it is true that this price, P, remained constant for every round, the promised R varied a little(the

back shows the blank column on which the promised dividends were scribbled every round).

16According to Mackay (1841), thousands of working-class people crowded the streets of Paris in 1719 to buy shares as word aboutits success spread. To avoid the crowds, the bourgeoisie rented apartments near the temple of wealth. In 1994, huge lines were reported in front

of MMMs office in Moscow.

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priced at P per unit mass and a promise of a return R, to a mass greater than or equal to n0 (1+R) of citizens.15

And so on. Round L is the last round planned. The size of the population taking part in this last round is

greater than or equal to n0 (1+R)L .

We now make a simplifying assumption. Though the promoter absconds with the last rounds

revenue, we assume that the revenues raised in the other rounds are used to pay off the obligations of the

previous round. So the promoter does not get anything from these rounds. This means that the mass of

recruits is increasing at a constant growth rate every round, which implies that the mass of citizens

participating in round i, i =0, 1, 2, .......L, is n0 (1+R)i . It also means that the only revenue accruing to the

promoter is the sales achieved in the initial round and what he runs away in the last round. So his expected

revenue, if nL is the mass of people in the last round, is n0P + [(1- )L ] nL P, where the term in square

brackets is the probability of survival till round L. Note that since he deliberately plans to run away after

collecting revenues from round L, the citizens in the previous round are not paid as well. This means that the

mass of citizens adversely affected is na = nL + nL-1 = nL (2+R)/(1+R).

The only cost to the promoter is his direct marketing cost. A distinguishing feature of Ponzi schemes

is that the major marketing cost is the initial cost incurred to contact and convince citizens to buy into the

scheme. Once a record of successful payment develops, and information about the fantastic scheme spreads,

marketing costs to reach later recruits are negligible.16 As a matter of fact, if marketing costs did not decrease

as the number of rounds progress, there would be no reason to design these schemes as pyramids. For

simplicity, we will assume that later marketing costs are zero. We adopt the following simple

parameterization of the marketing cost function: the cost is c + f( n0 ) n0 P. The first term c is the fixed cost.


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17Another interpretation of c is that it is the pecuniary penalty imposed on the promoter after the scheme collapses.

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The higher the c, the lower is the effectiveness of the mass-media channel that is being used to reach the

citizens.17 The second term is the variable cost, where f(n0) is the fraction of the revenue being paid out as

a marketing commission. Assume that f(n0) is increasing and convex in the initial mass of citizens contacted,

n0 . The marketing literature on the effectiveness of advertising see, for example, Rao and Miller (1975)

provides strong evidence in favor of this assumption.

The spreading of the news in this economy is modeled as follows. If ni is the mass of citizens aware

of the scheme in round i, then n i+1 = n i d is the mass of citizens aware of the scheme in round i+1. The

parameter d is greater than unity, and it is a measure of how connected the citizens of this economy are.

The higher the d, the faster is the spread of news in the economy. This implies that more wired economies --

wired in terms of communication linkages like the telephone or the internet - have a higher d.

The time line of the Ponzi scheme is given below in Figure 1.

0 1 2 3 L-1 L

n0 n1 contacted n2 contacted n3 contacted nL-1 contacted nL contactedcontacted Revenue = Pn1 Revenue = Pn2 Revenue = Pn3 Revenue=PnL-1 Revenue=PnLby spending Use this to pay Use this to pay Use this to pay .................. Use this to pay Run awayc+f(n0)n0P. off obligations off obligations off obligations off obligations with revenue

Revenue = of n0 of n1 of n2 of nL-2Pn0

Figure 1

The regulator may intervene between any two dates with a probability .


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Figure 2

D. The Participation Constraint of the Citizens

The participation constraint of the citizens depends on whether the bailout is assumed to be certain

or uncertain, and on our assumption of the information distribution in the economy. We will examine two

simple information environments in this paper. In the first scenario, we will assume that everyone has the

same information, that is, everyone knows n0 , P, R and L, and everyone knows which round is being played.

We will call it the symmetric information case. In the second scenario, we will assume that the citizen has

less information than the promoter and the regulator, that is, though the citizen knows n 0 , P, R and L, he

does not know which round is being played. He, of course, has a conjecture that holds under rational

expectations.

Though the second scenario is more realistic and leads to the occurrence of Ponzi schemes under


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n n n nR

Rna L L L= + =

++

12

1

*(2)

Pn

Ia

( )1

(3)

The proof is as follows. Given the political regimes under consideration, there is no bailout if the

mass of citizens involved is less than the critical mass n*. If there is no bailout, participation guarantees a

sure loss of P for the unit mass citizen, whereas non-participation ensures no loss/no gain. So no one will

participate in the last two rounds. If no one participates in the last two rounds, by backward induction, no

one will participate in any round. This leads to an obvious corollary.

COROLLARY 1: Under symmetric information, Ponzi schemes may not exist if the probability of

a bailout is zero.

This does not mean that Ponzi schemes necessarily exist when the probability of a bailout is finite;

we need more conditions for that.

Given that the mass of citizens in the last two rounds is at least n*, the expected payoff of a unit mass

citizen if he participates in the second last round is (-P) + (1- )(-P+ (1/na - 1) p(na)). The first term is

his loss of P if the regulator terminates before the next round. If the regulator does not terminate before the

next round, the second term is his loss of P ameliorated somewhat by the expected gain from the

redistribution inherent in the bailout. The expected payoff of a unit mass citizen if he does not participate

in the second last round is (0)+(1- )(- p(na)). If the regulator terminates before the next round, the first

term is his no loss/no gain. If the regulator does not terminate before the next round, the second term is his

expected loss from the redistribution inherent in the bailout. So the citizen will participate in the second last

round if the former term is greater than or equal to the latter term which gives us the following condition on

the price of the certificate, P:

where I = 1 for the deterministic bailout case and I= for the probabilistic bailout case.


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Pn

Ia

An important point needs to be made here. The citizen participating in the second last round has a

difficult choice: if he participates, he loses, but he loses less than what he would if he did not participate.

This dramatizes the main insight of this paper: it is rational for citizens to take part in spectacular Ponzi

schemes that are certain to explode because they know that, since so many people are taking part, bailouts

are probable, and bailouts are inherently a redistribution of wealth from non-participants to participants. It

is now also clear why it is in the public interest to stop this scheme, but because citizens are atomistic and

there exists no representative government, there is a coordination failure.

The analysis for citizens in the last round proceeds as follows. If a unit mass citizen participates in

the last round, his payoff is -P + (1/na - 1) p(na). It is his loss of P ameliorated somewhat by the expected

gain from the redistribution inherent in the bailout. The expected payoff of a unit mass citizen if he does not

participate in the last round is - p(na). This is his expected loss from the redistribution inherent in the

bailout. So the citizen will participate in the last round if the former term is greater than or equal to the latter

term which gives us the following condition on the price of the certificate, P

Note that inequality (3) subsumes the above inequality.

We now need to address why the promoter intends to terminate at round L, and not terminate later.

The reason is the following. If the promoter terminates at round L, he gets a sure revenue of P nL , and if the

promoter terminates at round L+i, where i=1,2,...,.he gets an expected revenue of (1- )i P nL+i . Substituting

from (2) and (3), it easy to check that if the promoter terminates at L, his sure revenue is less than or equal

to (1- ) I(1+R)/(2+R), and if the promoter terminates at L+i, where i=1,2,...,.his expected revenue is less

than or equal to (1- )i (1- ) I(1+R)/(2+R). The former sure revenue is greater than the latter expected

revenue. So it is optimal to terminate at L.

From (1), it is apparent that the expected profits of the promoter monotonically increases with P.


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-17-

nn

R RL0 1

2 1=

+ + *

( )( )(5)

Max nI

n

I R

Rf n n

I

nc

n Ro

Lo o

{ , ,} * *{ ( ) ( )

( )

( )} { ( ) ( ) }

0

1 11

21

1 + ++

++ (6)

Pn

I= ( )*

1

(4)

Hence, for maximum profit, set P to be as high as possible. So, from (3),

So the maximum revenue in round L is (1- ) I (1+R)/(2+R) , and this is obtained when na = n*

. Further, the

mass of citizens adversely affected is na = nL + nL-1 = nL (2+R)/(1+R) = n0 (1+R)L (2+R)/(1+R) = n*. This

gives us

We now come to the initial round. The optimization problem of the promoter, given in (1), after substituting

P from (4), and nL = n0 (1+R)L = n* (1+R/(2+R) from (5), reduces to:

Notice that there are now only two control parameters, n0 and R. This is because L is determined from the

geometry of the Ponzi scheme once we know n0

and R. (See equation (5)).

What are the participation constraints of the citizens in the initial rounds? Given that there is a

probability that the state will intervene and the unit mass citizen will lose his investment, P, and a

probability 1- that the state will not intervene and the unit mass citizen will make a net profit of PR, R

should be set high enough to ensure citizen participation. So the citizen participation constraint is (1- ) PR

+ (-P) $0. This gives us our second lemma.

LEMMA 2: Under symmetric information, the return offered by the promoter of a Ponzi scheme is

greater than or equal to the ratio of the probability of state intervention to the probability of no state

intervention in the last round, i.e.


20/38

-18-

R

( )1(7)

R d ( )1 (8)

R d= ( )1 (9)

Interestingly, the inequality (7), which is the citizen participation constraint, also ensures that the

Ponzi scheme is subgame perfect. To understand why, realize that the promoter of a Ponzi scheme always

faces a temptation of running away with the money before the planned termination. If the promoter

terminates in round i, he gets a sure revenue of P n i , and if the promoter terminates at i+1, he gets an

expected revenue of (1- ) P ni+1 = (1- )(1+R) P ni . Because of inequality (7), the former sure revenue is

less than or equal to the latter expected revenue. Given our tie breaking rule, it is optimal to wait till round

L.

The inequality (7) tells us that R has a lower bound. R also has an upper bound. According to our

assumption on how fast news can spread in this economy, it is clear that the maximum growth rate of the

mass of citizens participating per round in the Ponzi scheme is d-1. So we get our next lemma.

LEMMA 3: Under symmetric information, the return offered by the promoter of a Ponzi scheme

cannot be greater than the rate at which news travels in this economy i.e.

We are now in a position to solve the optimal control problem of the promoter. Notice, from (5),

that given a n0 , L decreases as R increases. A geometrical way to demonstrate this fact is to notice that in

Figure 2, once we have fixed the top, for any given bottom, the height of the pyramid decreases as the growth

rate of successive layers increases. This implies, from (6), that, given a n0 , the profits of the promoter

increases as R increases. So set R to at its maximum. This gives us:

This has a nice intuitive explanation. Since the promoter knows that there is a probability of

intervention by a regulator every round, it is optimal for the promoter to reach the last two rounds (where

he attracts the critical mass n* of citizens) as soon as possible. So he sets the interest rates (i.e. growth rates


21/38


22/38

-20-

only of the deep parameters n*, d and . As decreases or decreases, from (4), P decreases. This means,

from (10), that the expected profits of the promoter decreases. The expected profit also decreases as c

increases. This implies that if or is below a lower bound or c is above an upper bound, the expected

profits of the promoter are negative, and so he would not initiate a Ponzi scheme. The upper bound for

comes because of the following reason. From, (7), we know that R = d-1$ /(1- ) for existence, which gives

us # (d-1)/d for existence.

The above intuition leads to an obvious corollary of Proposition 1.

COROLLARY 2: The chances of Ponzi schemes occurring decreases in all political regimes if (1)

the proportion of national wealth owned by the state, , decreases, or (2) the probability of a bailout, ,

decreases, or (3) the political connection of the promoter decreases, that is increases or (4) advertising

effectiveness decreases, that is c increases.

Notice that the upper bound of the parameter, , which is (d-1)/d, increases as d increases. This

allows us an interesting insight, which we state in the next corollary.

COROLLARY 3: As an economy becomes more wired (that is, as more and more connections

develop between its citizens, i.e. d increases), regulation has to become more stringent to prevent Ponzi

schemes (that is, the probability of intervention by the regulator has to increase, i.e. has to increase).

Corollary 3 has profound implications for the regulation of investment proposals over the internet.

Though it is unambiguous about its policy prescription -- regulators need to become more vigilant in their

examination of investment schemes as society gets more wired -- it is very important to point out the reason

which drives this policy prescription. The reason is that, as a society gets more wired and it is easier to

contact more people, the too big to fail doctrine becomes easier to exploit. Hence, we either need more

regulatory vigilance or we need more statements of caveat emptor (that is, less belief in the fact that there

will be a bailout).

We now come to the last result in this section, which is really a restatement of (4):


23/38

-21-

Money lost P P

p n nBailout

a a

,( ) ( )

1

10 1

0 1

(19)

PR

R na

+

11

2

( ) (20)

COROLLARY 5: Under asymmetric information, Ponzi schemes may not exist if the probability of

a bailout is zero.

How do we get g(.)? Unfortunately, g(.) is not unique. Many conjectures satisfying inequality (17)

and (18) simultaneously are upheld in a rational expectations equilibrium. We need to add more structure

to the problem to pin g(.) down.

The conjecture we use in this paper is:

There are many reasons we use this conjecture. First, this gives us a well-defined probability measure. We

have 0 # g (.) #1 and

Second, g(.) automatically satisfies restriction (18). Third, g(.) is a decreasing function of R. This is

intuitive, because given the top and bottom of the inverted truncated pyramid of a Ponzi scheme, as the

return, R, increases, the number of rounds, L, decreases. So the probability of being in any of the last two

rounds, 1- g(.), should increase, implying that g(.) should decrease. Fourth, it leads to a simple, closed-form

solution. Last, but not the least, this conjecture is upheld in a rational expectations equilibrium.

Given this conjectured belief of the citizens, the participation constraint of the citizens, (17), reduces

to:

The optimal control problem is analogous to the one analyzed in the previous section. From (1), we


27/38

-25-

Max nn

R

R

n R

R nf n n

n

R

Rc

n R

Lo

{ , ,} *

*

* *{ ( ) } ( )

( ){ ( )} ( ) { ( ) }

0

0 012 1

12

12

12 1

+

+ +

+

(22)

Pn

R

R=

+( )

*1

2

1

1

(21)

R Min d e= { , }1 1 (23)

note that P should be set to its maximum possible value. This gives us,

To ensure that the promoter has the incentive to wait till the planned last round L before running away with

the money, R should have the same lower bound as before. This lower bound of R is given in (7). Further,

as before, the upper bound of R is constrained by the connectedness of the economy. This upper bound

of R is given in (8).

Substituting the expression for P from (21) in (1) and the expression n*(1+R)/(2+R) for nL in (1),

the optimization problem (1), reduces to:

It is apparent that from (22), that, given a n0 , the profits of the promoter increases as R increases. So set R

to at its maximum. This gives us

P is given in (21), where R is substituted from (23). The optimization problem of the promoter, (1),

therefore, becomes the same as (10). The solution procedure is identical. The only difference is that P is

different. For the sake of completion, we restate Proposition 1 and its corollaries.

PROPOSITION 2: Under asymmetric information, a Ponzi scheme will exist if the following

conditions hold: a large public sector ( is bounded below), ambiguous laws governing the transfer of

property rights from the state to the citizen (the assets of the state could be used for a bailout, the probability

of which, , is bounded below), political connections (the probability of early termination of the Ponzi

scheme by a regulator, , is bounded above) and an inexpensive access to citizens through mass-media

(advertising ineffectiveness, as measured by c, is bounded above).


28/38

21Corollary 7 holds only if d < e. This because the upper bound of2

is actually R/(1+R) = (Min(d,e)-1)/(Min(d,e)). If2

is greater than

(e-1)/e, we will not have a Ponzi scheme.

-26-

n solves f n n f nR

RR

n n R R

R0 0 0 0

1 2

11 11 1

11 1

0

( ) ( )ln( ) ln( )

ln( )(( )( ))

/

ln ln ln( ) ln( )

ln( )

*

+ = + + +

+

+

+ + ++ (24)

P RR n

Bailout

( )*

12

1

+ = =

(25)

( )1 2

11

+

bhattacharya 98

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