Counterfeit Money∗

Elena Quercioli†

Economics Department

Tulane University

Lones Smith‡

Economics Department

University of Michigan

August 16, 2007

Abstract

This paper develops a new tractable positive theory of counterfeit moneybased on a variable intensity costly money verification. Counterfeiters competeagainst police and innocent verifiers, by choosing a quantity and quality of coun-terfeit notes to produce and pass. This induces astrategic complements“hotpotato” game among “good guys” — who exchange currency pairwise, and wishto avoid counterfeit currency passed around. We deduce an equilibrium in thisgame, showing that as the stakes rise in the denomination, counterfeiters producerbetter quality, and verifiers respond with more effort. Resolving the struggle, wededuce from the data that money verification improves in the denomination.

Our entwined counterfeiting and verifying games explain key time series andcross-sectional patterns of counterfeiting:(a) the ratio of theseizedto passedcounterfeit money rises in the denomination, but less than proportionately;(b)the vast majority of counterfeit money used to be seized before circulation, butnow most passes into circulation;(c) the past money prevalence generally risesin denomination, with the least counterfeited notes the lowest; and(d) the shareof passed money found by Federal Reserve Banks generally falls in the note, asdoes the ratio of their internal passed prevalence to the economy average.

We also compute the social cost of counterfeiting, predict both the street priceof counterfeit notes and the costs expended in verifying counterfeit notes. Finally,we describe the determinants of the counterfeiting rate.

∗This paper is wholesale reworking of our 2005 manuscript “Counterfeit $$$” that made restrictivefunctional form assumptions and assumed a fixed quality of money. We have profited from the insights,data, and broad institutional knowledge about counterfeiting of Ruth Judson (Federal Reserve), JohnMackenzie (counterfeit specialist at the Bank of Canada), and Lorelei Pagano (former Special Agentwith the Secret Service). We have also benefited from feedback at the presentations at I.G.I.E.R. atBocconi, the 2006 Bonn Matching Conference, the 2006 SED in Vancouver, the Workshop on Moneyat the Federal Reserve Bank of Cleveland, Tulane, Michigan,and the Bank of Canada, and especiallythe modeling insights of Pierre Duguay (Deputy Governor of the Bank of Canada) and Neil Wallace.

†elenaq@tulane.edu and www.tulane.edu/∼elenaq‡lones@umich.edu and www.umich.edu/∼lones. Lonesthanks the NSF for funding (grant 0550014).

1 Introduction

Fiat currency is almost useless paper or coin that acquires value by legal imperative.

The longstanding problem of counterfeit money strikes at its very foundation, debasing

its value, and undermining its use in transactions. Combatting counterfeiting currency

is increasingly a major concern to governments around the world. The counterfeit rate

of the American dollar is about one per 10,000 notes, and the direct cost to the do-

mestic public is substantial, amounting to $62 million in fiscal year 2006, which is up

69% from 2003. Further, the indirect costs of counterfeiting may be much greater. For

instance, counterfeiting occasioned the first major redesign of the U.S. currency in 60

years in March 1996 for the $100 note; to stay ahead of advancing digital counterfeit-

ing technologies, new designs will be introduced every 7–10years. In addition, there

are tremendous costs borne by the public at large in checkingcurrency.

When we refer to counterfeit money, we have in mind two manifestations of it.

Seizednotes are confiscated before they enter circulation.Passednotes are found at

a later stage, and so cause losses to the public. We have gathered data on seized and

passed money over time and across denominations mostly fromthe Secret Service and

its old statistical abstracts. In the United States, all passed counterfeit currency must

be handed over to the Secret Service, and so very good data is available (in principle).

We develop a tractable economic theory of counterfeit money, and then show that

it successfully explains the facts of both forms of counterfeit money. In so doing, we

make separate contributions to the economics of crime and monetary theory.

Our monetary theory revolves around a simple new decision margin: how much

costly effort individuals expend examining any currency offered to them. They expend

efforts trying to screen out passed counterfeit money unknowingly handed them from

other good guys. This “hot-potato” verification game is novel and interesting in its own

right: The more others protect themselves, the more one should guard oneself. In other

words, this is a game of strategic complements (sometimes called supermodular).

Law enforcement clearly plays a key role in limiting counterfeit money, since any

counterfeiter courts both imprisonment and financial ruin.We thus begin with a simple

model of criminal production, where the main choice variable is how “good” to make

the fake money. Specifically, better quality counterfeits are more costly to verify at

any level. Our theory analyzes both the bad money seized fromcounterfeiters, and the

bad money that successfully passes into circulation, onto unwitting “good guys” in an

anonymous random matching exchange economy. The hot potatoverification game

arises as a collateral battle mostly pitting “good guys” against one another.

1

We formalize two key notions — thepassing fractionof counterfeit goods into

circulation, and the inversely-relatedseized-passed ratio. We show that the conflict

pitting counterfeiters against police and innocent currency verifiers explains the facts

about these ratios. First, no counterfeiting occurs at low enough denominations, where

producer surplus can never pay for legal costs. Above this point, the seized-passed

ratio rises, so that higher denomination notes yield greater profits but must pass less

often. This illustrates the result that criminal behavior falls in the chance of capture and

rises in the criminal payoff (the denomination).1 Second, while the measured passing

fraction falls in the denomination, it does not fall fast enough to compensate for the

greater revenue. Instead, the cost of improved quality is needed to hold profits at bay.

What ultimately transpires in the model is a conflict betweengreater vigilance and

better quality. Both factors help explain the facts about the passed and seized rates. But

to understand the incentives to counterfeit, we must also admit endogenous counterfeit

production levels — since we must keep track of the producer surplus which pays for

the expected legal consequences of any criminal behavior. We show that equilibrium

in the counterfeiting entry game pins down the triple of output, quality, and verification

effort. Next, equilibrium in the verification game fixes the counterfeiting rate.

We prove that if the verification costs and the passing fraction are log-concave,

then counterfeit quality rises in the denomination. We think that this application of

log-concavity in economics beyond the world of probabilitydensities is novel. Also

new is the result that at higher stakes thefts, criminal efforts ramp up, while innocents

grow more vigilant in theft evasion. Only from the data can wefind that the victor in

this struggle between greater quality and vigilance effortis a greater verification rate.

Next, the seized-passed ratio has greatly fallen over time.The vast portion of

counterfeit money used to be seized, while now the reverse holds. This owes to a

technological transformation in counterfeiting, first with office copiers in the 1980s

and then digital means (computers with ink jet printers) in the 1990s. This has pushed

down fixed costs, and lowered production levels enough that marginal production costs

have risen; on balance, scale economies are smaller now, andaverage costs greater.

The equilibrium passing fraction has risen in order to maintain zero profits.

The second conflict amongst innocent verifiers is equilibrated by the counterfeiting

rate. This rate balances the costs and benefits of verification, and falls in the denom-

ination, other things equal. Its relation to the verification rate is subtle. Initially, it

skyrockets, since marginal verification costs in the hot potato game are initially zero.

1See Becker (1968). For simplicity, our theory assumes that the criminal code is implemented aswritten: Namely, the punishment for counterfeiting is the same, irrespective of the denomination.

2

We define thediscovery rateas the chance that any circulating counterfeit money is

found to be passed with a bank or verifier. This translates thecounterfeiting rate into

the observedpassed-circulation ratio. Just like the counterfeiting rate, this ratio dra-

matically rises initially, with the $100 denomination by far the most counterfeited. We

show that with a fixed quality level, the passed-circulationratio would eventually fall.

That this does not occur further underscores why quality must rise in the note.

We conclude by explaining counterfeit money found by Federal Reserve Banks

(FRB). They catch a majority of $1 passed notes, and their share of passed money falls

in the denomination, except for the $100 note. Also, the internal FRB counterfeiting

rate is likewise a decreasing ratio of the overall passed money rate. We argue that both

facts owe to the rising verification rate, and behavior in thehot potato passing game.

For the least valuable notes are most poorly verified and caught.

RELATIONSHIP TO THEL ITERATURES. Despite how common and longstanding a

problem it is, counterfeit money is still very much a blackbox to economists. We have

found no papers sharing either our main novel assumptions orconclusions. Unlike our

matching model, the very few existing papers are theoretical, and are broadly inspired

by Kiyotaki and Wright (1989) and Williamson and Wright (1994). Green and Weber

(1996) explores a random matching model, where only government agents can descry

the counterfeit notes, whose stock is assumed exogenous. Williamson (2002) admits

counterfeits of private bank notes that are discovered withfixed chance; in most of

his equilibria, counterfeiting does not occur. Recognition of counterfeit quality is also

stochastic and exogenous in Nosal and Wallace (2007), who find no counterfeiting in

equilibrium when the cost of counterfeits is high enough. This relates to our discovery

that the counterfeit rate is like the ratio of average verification costs to average produc-

tion costs, and thus counterfeiting spins out of control as production costs vanish.

Our variable intensity verification effort is novel in the counterfeiting literature.

Equilibrium here is secured as individuals adjust this choice variable. By contrast,

in the existing literature, the price of money equilibratesthe model — hence, it is

“general equilibrium”. We feel that a fixed value of notes is agood approximation for

the world we examine where counterfeit notes are extremely rare. It agrees with the

common observation that higher denominations may be declined if verification is too

hard (“No $100 bills accepted”), but are almost never discounted.2 Endogenizing the

price of money cannot explain the variation in counterfeit levels across denominations.

Not surprisingly, there has been no attempt by the existing literature to match the data.

2We have heard of no systematic episodes of notes being discount domestically; however, outsidethe USA, it is true that older $50 and $100 notes may be declined.

3

For a key point of comparison, the papers cited above assume that one observes a

free signal of the money qualityafter acquiring it. We instead posit that individuals

verify when it can affect choice, namely when handed it. Thisis important, producing

the strategic complements hot potato game. It also agrees with how most individuals

behave: When it matters, we check our money; otherwise, it lives in our wallet.

For the economics of crime literature, we focus on the novel variable intensity

struggle between criminals trying to steal — adjusting the quality of their efforts — and

innocents actively seeking to avoid theft by verification. The crime rate (counterfeiting

rate) emerges as an equilibrium quantity balancing these competing interests.3

Our use of supermodular games in monetary economics and the economics of

crime is novel.4 While our main goal is to understand the economics of counterfeit

money, we also identify a new set of stylized facts about counterfeiting across denom-

inations. For those who do not consider these facts important, our explanation of them

should be viewed merely as evidence in favor of our model. We later identify a host of

other economic questions that can in principle be answered by this framework.

The model is laid out in§2, and its equilibrium is explored in§3. We then show how

it explains the behavior of seized money in§4, and of passed money in§5. Technical

proofs are deferred to the appendix, including a novel existence proof.

2 The Model and Preliminary Analysis

We construct a dynamic discrete time model in which notes periodically transact.

Counterfeiting for each denomination∆ plays out as a separate game, and so∆ is fixed

for now. Our data will come from the U.S. dollar denominations $1, $5, . . . , $100.

There are two types of maximizing risk neutral agents: a continuum of bad guys

(counterfeiters) and good guys (transactors). Everyone therefore acts competitively,

believing he is unable to affect the actions of anyone else. Counterfeiters choose

whether to enter, and if so, then select the quantity and quality of money to produce

and distribute, before vanishing or getting jailed. There is free entry of counterfeiters,

and so each earns zero profits. Good guys engage in chance pairwise transactions that

have money changing hands in one or both directions. Counterfeiters must transact

3The literature follows Becker (1968), who derives the number of criminal offenses — analogous toour counterfeiting rate — solely from criminal maximization. His model of crime largely ignores thecrime-fighting role of individuals in defense of their own property, analogous to our verification.

4Diamond (1982) developed a search-matching macroeconomics model that is supermodular in theproduction costs. Our monetary model is supermodular in a pairwise effort choice. Diamond studiesmultiple equilibria, while ours is nested with an entry gamethat forces a unique equilibrium.

4

too, but they are not distinguishable from good guys. A counterfeiter can exchange

multiple times each period, if he wishes. Each good guy chooses an effort level to

examine notes that they are handed. We ignore discounting for the time between ac-

quiring and spending a note is small. Some unknowingly acquire counterfeit currency

and some do not. Below, we flesh out the details of these two enmeshed games.

2.1 The Counterfeiter’s Endogenous Cost Function

While counterfeiting is a dynamic process, we wish for modeling purposes to project

it to a static optimization of a well-behaved increasing andconvex cost function. We

now consider in sequence the three types of costs: production, legal, and distribution.

We simply assume a common technology for producing counterfeit notes of any

given quality. Better quality notes will look and feel more like authentic notes. One

incurs a fixed cost for the human and physical capital, plus a possibly small marginal

cost of production. This increasing returns suggests that the cost function for produc-

ing counterfeit money in quantity and quality might possibly be concave in quantity.

These are not the only costs. A counterfeiter is producing anillegal good, which

may beseizedprior to passing it onto the public: Namely, police may either uncover

the counterfeit note “factory” or catch the criminal in the act of transporting the money.

We assume that the counterfeiter is eventually caught and punished.5 Projecting this

dynamic to a static story, his expected present loss from punishment isL > 0.

Faced with the legal obstacles, a counterfeiter must carefully distribute the money.

Two forces push up distribution costs: how much money one is trying to pass, and

how carefully individuals screen money. First, if he attempts to pass more, then his

unit distribution costs should rise — for law enforcement more quickly catches the

counterfeiter the greater the amount of money he produces.6 In response, counter-

feiters employ different strategies for passing differentamounts of money. At low

production, they may launder their money around a city, buying inexpensive goods

with larger denominations. At greater levels, they may sellthe notes to distributors.

Next, distribution costs should also rise in the difficulty of passing the currency.7

We summarize the hurdles of passing notes by theequilibrium passing fraction0 <

5The Secret Service estimates that the conviction rate for counterfeiting arrests close to 99%.6“If a counterfeiter goes out there and, you know, prints a million dollars, he’s going to get caught

right away because when you flood the market with that much fake currency, the Secret Service is goingto be all over you very quickly. They will find out where it’s coming from.” — interview with JasonKersten, author of Kersten (2005) [All Things Considered, July 23, 2005].

7Questions by Neil Wallace led us to tackle this problem. Withthis formulation, the results willclosely mirror those without any hassle costs. In this sense, this is a robustness test on our theory.

5

f ≤ 1 — namely, the share of production that the counterfeiter passes, or equivalently

the chance that any note passes. We find that higher notes passless readily, so that

the distribution costs are not exogenously known. Working with a cost function that is

only endogenously known is a complex exercise that we have not seen solved before.

To avoid a possibly unsolvable fixed point exercise with a general endogenous cost

function, we assume a tractable separable form. We venture that unit distribution costs

linearly fall in the passing fractionf . Intuitively, to pass a quantityx and qualityq

of counterfeit money, one creates the same distribution network irrespective of the

passing fraction, but incurs a linearly rising “average hassle cost”. In summary, we

posit that thecost functioninclusive of production, distribution, and legal costs, equals

c(x, q) − hfx+ L (1)

We assume monotone, strictly convex, twice-differentiable, and non-negative costs —

so thatc(x, q) ≥ hx + L andcx(0, q) > h for all x > 0. Average production costs

(c(x, q) − hfx+ L)/x have a unique minimum, and explode in the quantity.

Counterfeiters must earn sufficientproducer surplusto pay for their expected legal

costs. With our linear formulation, both the endogenous “hassle” costs and the passing

fraction drop out of the expression for producer surplus:

ψ(x, q) ≡ xcx(x, q) − c(x, q)

A key characteristic of the counterfeiting technology is reflected in the quality

derivatives ofψ. Greater quality may lessen producer surplusψq < 0, insofar as it

disproportionately raises fixed rather than marginal costs. To create that better water-

mark, security thread or color-shifting ink might be accomplished by a more expensive

printing press. We call this technologycapital intensive. For instance, if the quality

level q2 printing press costs an additionalq, then c(x, q) = q2 + x2 + h, whence

ψ(x, q) = x2 − q2 − h, and thusψq < 0. On the other hand, greater quality may

raise producer surplusψq(x, q) > 0, as it largely entails increased attention to de-

tail, and thereby disproportionately lifts marginal rather than fixed costs. We call this

technologylabor-intensive.8 For example, if greater quality scales up costs, as with

c(x, q) = q2x2 + h, thenψ(x, q) = q2x2 − h, and easilyψq > 0.

We cannot yet specify the counterfeiter’s profit function, since his quality choice

affects the passing fractionf that he faces. We must first understand this feedback.

8This was common in earlier decades. Still, as long ago as 1776, printing presses (aboard theH.M.S. Phoenix) were used by the British for counterfeitingContinental currency.

6

2.2 Currency Verification and Counterfeit Quality

To explain passed counterfeit money in circulation, we introduce the next element of

the story. If an innocent individual attempts to spend “hot”money, and it is caught,

then it becomes worthless — since knowingly passing on counterfeit currency is ille-

gal.9 We simply assume that this extra crime of “uttering” is not done.

Faced with this prospect, individuals will choose to verifythe authenticity of any

moneybeforethey accept it.Verificationis a stochastic endeavor that transpires note by

note — as more valuable notes will command closer scrutiny. We write the verification

rate (or intensity) as the chancev ∈ [0, 1] that one correctly identifies a given note as

counterfeit. We assume genuine notes are never mistaken forcounterfeit.

Counterfeiters produce better quality notes since they pass more readily. In our key

modeling insight here, we normalize the meaning of quality so that a note with twice

the quality requires twice as much effort to produce the sameverification intensity.

A verification intensityv ∈ [0, 1] for a qualityq > 0 note therefore costs efforte =

qχ(v). Observe that if quality vanished for a fixede > 0, the induced verification

intensity would explode. This involves only a slight loss ofgenerality insofar as quality

scales verification costs uniformly inv. It also gives a meaning to the units of quality,

and mandates our assumption of a strictly positive lower bound on quality, sayq ≥ 1.

For givenq, costs are smooth, increasing and convex, so thatχ′(v), χ′′(v) > 0 for

all v > 0.10 To ensure a positive optimal verification level, we assume thatχ′(0) = 0.

We also assume that the elasticitylimv→0 vχ′′(v)/χ′(v) exists is positive and finite.

The verification intensity rises in the effort and falls in the counterfeit note quality.

Since verifiers do not know the counterfeit status of any note, we naturally must assume

that they do not descry its qualityq either. Hence, only efforte is a choice variable,

and so we introduce theverification functionV (e, q) — the verification rate induced by

effort e for a note of qualityq. Thenv = V (e, q) is the inverse function toe = qχ(v).11

Lemma 1 (The Verification Function) Letv = V (e, q) and soe = qχ(v).

(a) Verification rises in effort, with slopeVe(e, q) = 1/qχ′(v) > 0.

(b) Verification falls in quality, with slopeVq(e, q) = −χ(v)/qχ′(v) < 0.

(c) Verification becomes perfect as quality vanishes: Ife > 0, V (e, q) ↑ 1 asq ↓ 0.

(d) The verification second derivatives obeyVqq(e, q) > 0 andVee(e, q) < 0.

9Title 18, Section 472 of the U.S. Criminal Code10Weak convexity in this case is remarkably without loss of generality. For one can always secure

an (ex ante) verification chance ofv at cost(χ(v − ε) + χ(v + ε))/2 instead by flipping fair coin, andverifying at ratesv − ε or v + ε. In other words, we must haveχ(v) ≤ (χ(v − ε) + χ(v + ε))/2.

11Since this may not always be defined, we shall defineV (e, q) = 1 if e > qχ(1).

7

2.3 The Passing Fraction and Verification

The passing fraction reconciles the entwined counterfeiting and verification games.

While police seizures are exogenous in our model, we wish to assume that vigilance

by transactors may facilitate police seizures, by providing clues into ongoing counter-

feit operations. To this end, we assume that police seize a fraction0 < s(v) < 1 of

counterfeit money production. The passing fraction thereby reflects seizure and verifi-

cation viaf(v) = (1− s(v))(1− v). So all passing is eventually choked off at perfect

verification (f(1) = 0), and some passing occurs when no one verifies (f(0) > 0).

While verification may be complementary to police seizures,we simply assume that

the passing fraction continuously falls (f ′(v) < 0). Since1 − v > f(v), a “good guy”

successfully passes a counterfeit note more often than a “bad guy”.

If seizures were a fixed fractions of production, then a unit elasticity off(v) =

(1−s)(1−v) would arise:E1−v(f) = 1. If verifier activity enhances police seizures,12

then this elasticity exceeds one. We shall assume that the Secret Service activity is a

fixed function of the verification rate, so thatthe passing elasticity is a fixed number:

Υ ≡ E1−v(f) = −(1 − v)f ′(v)

f(v)∈ [1, 2) (2)

Lemma 2 (The Passing Fraction)The passing fractionf is strictly log-concave.

Proof: By (2),−Υ/(1 − v)=f ′/f=(log f)′, and so(log f)′′=−Υ/(1 − v)2<0. �

2.4 The Counterfeiter’s Problem

We now formulate the counterfeiter’s optimization. Each cares about how much he

produces, at what quality, and how carefully his notes will be examined. Counterfeiters

do not attempt to pass their money at a bank, and so face a verification intensityv =

V (e, q). Their expected revenues for quantityx and qualityq of note∆ aref(v)x∆,

while their costs arec(x, q) − f(v)xh+ L. Hence, theirprofit functionis:

Π(x, q, e) ≡ f(V (e, q))x(∆ + h) − c(x, q) − L (3)

Better quality simultaneously raises the passing fractionand the counterfeiters’ costs.

Maximizing profits here is somewhat nonstandard because theeffort argumente of the

profit function is not a choice variable for the counterfeiters.

12On its web page, the Secret Service also advises anyone receiving suspected counterfeit money:“Do not return it to the passer. Delay the passer if possible.Observe the passer’s description.”

8

We first observe that provided there is any counterfeiting, the optimal quantity

and qualityx, q are positive and finite. This follows from the earlier “cost proviso”

(in §2.1), and the facts that there are bounded returns to greaterquality — indeed, the

passing fractionf never exceeds 1 — while zero quality implies perfect verification

v = 1 (Lemma 1(c)), and thereby precludes all passing.

Since the profit function is smooth, and the solution interior, first order conditions

hold for the counterfeiter’s quantity and quality optimization:

Πx ≡ f(V (e, q))(∆ + h) − cx(x, q) = 0 (4)

Πq ≡ f ′(V (e, q))Vq(e, q)x(∆ + h) − cq(x, q) = 0 (5)

We finally impose the zero profit conditionΠ(x, q, e) = 0. If we first eliminate the

passing fractionf using (4), we find a much simpler and equivalent statement to zero

profits, that the producer surplus pays for the expected legal costs of counterfeiting:

ψ(x, q) = L (6)

This identity says that average production and legal costs equal marginal costs. We

also posit thatL is not so large that (6) has no solution. To generate profits topay

for the legal costsL > 0, the criminal production level is inefficiently higher than in

standard competitive analysis, given that producer surplus rises in quantity.

We now have three equations (4), (5), (6) in three unknownsx, q, e.

2.5 The Hot Potato Game

Each period, innocent transactors either go to the bank (unlike counterfeiters) or meet a

random verifier for transactions. Neither event is a choice,but occur with fixed chances

β and1 − β, respectively. Banks have verifying machines or capable staff who can

better descry counterfeit money than individuals, but still imperfectly. Write their

verification intensity asα ∈ (0, 1). Indeed, from $5–10 million of passed money hits

the Federal Reserve yearly, missed by banks (see Table 4). Altogether, any counterfeit

money is found in a transaction with thediscovery rateρ(v) = αβ + (1 − β)v > 0.

Assume that a fractionκ of all ∆ notes tendered in transaction is counterfeit, with

an average verification ratev. As notes are spent upon acquisition, transactors choose

their intensityv to minimize losses from counterfeit notes and verification efforts:

κ(1 − V (e, q))ρ(v)∆ + e (7)

9

A verifier incurs a loss in the triple event that(i) he is handed a counterfeit note,(ii)

his verifying efforts miss this fact,and(iii) the next transaction catches it. These inde-

pendent events have respective chancesκ, 1 − V (e, q), andρ(v). Fixing the qualityq,

we may re-write their objective function (7) in terms of the induced verification rates:

κ(1 − v)ρ(v)∆ + qχ(v) (8)

This is a doublysupermodular game: One’s verification intensityv is a strategic

complementto the average verification intensityv, and to the counterfeiting rateκ. The

more intensely others check their notes, or the more prevalent they are, the stronger is

the incentive to verify money that one acquires. Namely, theverification best response

function v is increasing inv andκ. Supermodular games in economics often have

multiple ranked equilibria,13 but here there is a unique equilibrium. While we have one

maximum of (8) with no verificationv = v = 0 (a “don’t ask, don’t tell” equilibrium),

this is incompatible with equilibrium in the adjoined counterfeiting game.

The verification game has no asymmetric equilibria: Since (7) is a strictly convex

function of e by Lemma 1(d), it admits a unique solution: All verifiers will choose

the same effort level, inducing the same verification rateV (e, q) = v, for qualityq.

The second order condition for minimizing (8) is met as costsχ are strictly convex.

The first order condition is then justified if a corner solution is not optimal. As we have

argued, agents must choose a common verification intensity,sayv = v. Making this

substitution into the first order condition yields theequilibrium optimality equation:

qχ′(v) = κρ(v)∆ (9)

In other words, the marginal cost of verification equals its marginal benefit. Assuming

κ > 0, this solution is positivev > 0, and is the unique optimum of (8).

3 Equilibrium Analysis

We now analyze the equilibrium in the model, and prove its existence. In so doing, we

allow the denomination∆ to vary, treating it as a parameter of the model, and deduce

some comparative statics. This device not only allows us to prove existence, but also

paves the road for the empirical analysis in sections 4, 5, and 6.

13See Milgrom and Roberts (1990). Diamond (1982) found such a structure in a search economy.

10

3.1 Quantity, Quality, and Effort in the Counterfeit Entry G ame

Without specifying the cost functionsχ(v) andc(x, q), we cannot produce closed form

expressions for the endogenous variables. We derive all comparative statics indirectly.

Before thinking about the feedback between the counterfeiting and verification

games, we deduce a useful property that all notes above a threshold are counterfeited:

For the counterfeiter must pay a fixed legal costL > 0 irrespective of the note that

he counterfeits, because the Secret Service is active even if no one verifies. So if one

is to counterfeit at all, one must choose a high enough note. For greater notes, veri-

fication effort is needed to balance the counterfeiters’ incentives, but these efforts are

vanishingly unimportant as we near the threshold note. Thus, the appendix proves:14

Lemma 3 (Counterfeited Denominations)

(a) There is a unique noteθ > 0, with no ∆ < θ counterfeited, and any∆ > θ

counterfeited and verified (sox[∆]>0 ande[∆]>0). We also havex[θ] > 0, q[θ] = 1.

(b) The verification ratev[∆] and efforte[∆] are positive but vanishing as∆ falls toθ,

while verificationv[∆] rises to 1 as∆ ↑ ∞, but is always imperfect.

A least counterfeited note is consistent with trivial passed rates for the $1 in Figure 3.

From now on, we assume∆ > θ. Then our earlier interior solutions assumption

for (9) was justified: If we hadv = 0, then counterfeiting would be strictly profitable

by Lemma 3, and thus counterfeit money would circulate. But then not verifying at all

would be strictly suboptimal, given positive marginal benefits and zero marginal costs

χ′(0) = 0. Next, if verifying were perfect, then no counterfeiter could pass notes.

As the stakes ramp up in the battle between counterfeiter andverifier — namely,

∆ rises — it is instructive to see whether counterfeit qualityand verification effort

rise too. Let’s first consider how the verification efforte[∆] evolves in the note∆.

Intuitively, individuals should pay greater heed to higherdenomination notes, since

their potential losses from acquiring counterfeit money are greater. In fact, while this

conclusion is correct, the logical road to it is quite different, as the verifiers’ effort

solely choices reflect the entry game. We instead must consider the counterfeiters’

optimization. For a more valuable note must be met with greater scrutiny or it becomes

profitable to counterfeit. This is a robust result, valid forany counterfeiting technology.

Theorem 1 (Effort) Verification effort rises in the denomination, with the elasticity:

E∆(e) =∆e′[∆]

e[∆]=

∆

∆ + h

xcxqcq

(10)

14Reflecting the dependence on∆, let y[∆] be the equilibrium level of any variabley.

11

Proof: The zero-profit identityΠ(x, q, e) ≡ 0 holds for all∆. As ∆ marginally rises,

quantity and quality are already optimized, and so to maintain zero profits, effort must

rise: Namely,Π∆ + Πee′[∆] = 0, sinceΠq = Πx = 0 at the optimum. Intuitively,

profits marginally rise in denomination and fall in verifier effort, or Π∆ > 0 > Πe, so

thate′[∆] > 0. The appendix rewritese′[∆] = −Π∆/Πe as (10) using (3). �

As the denomination rises, the stakes in the counterfeitinggame intensify, and the

marginal benefit of quality is pushed up. On the other hand, we’ve just seen that the

effort rises too, and this has an ambiguous effect on the marginal benefit of quality. To

resolve this ambiguity, we need a new assumption:

VERIFICATION COST PROVISO. Verification costsχ(v) are strictly log-concave.

This assumption precludes verification costs more convex than exponential — for

instance, geometric costsχ(v)=λvr with r>1 work.15

Theorem 2 (Quality) The counterfeit qualityq rises in the denomination.

The proof is in the appendix. This major result of the paper merits some intuition. Let’s

see the role played by log-concavity in the argument: Loosely, it precludes local “near

jumps” of the increasing functionχ, and local “near flats” of the decreasing function

f , wheref(v) moves “much more” thanv.16 Let the note∆ rise a “little”. Then

verification efforte = qχ(v) rises a “little”, by Theorem 1. To sustain zero profits (3),

f(v) must fall “a little”. First, if f is not log-concave, thenv could rise “a lot”, and so

χ(v) could rise “a lot” too. Alternatively, ifχ is not log-concave, then even ifv only

rises “a little”,χ(v) could rise “a lot”. In either case, qualityq=e/χ(v) could fall.

Corollary 1 (Quantity) The quantityx (per counterfeiter) rises in the denomination if

the production technology is physical capital intensive, and falls if it is labor-intensive.

Proof: Assume∆ rises, so that qualityq rises. Recall that producer surplus rises in

quantity: ψx > 0. In a world with a fixed quality level, this would force the same

quantity level for all denominations; here, quantity and quality co-adjust to hold pro-

ducer surplus constant. Ifψq < 0, then quantity must rise in the quality since producer

surplusψ(x, q) = L is constant in∆. Likewise, ifψq > 0, then quantity falls. �

15Log-concavity is a standard assumption for probability densities (see Burdett (1996) and Bagnoliand Bergstrom (2005)). Our application of it to cost functions likeχ or the passing fractionf , is novel.

16Since log-concavity saysχ(v + ε)χ(v − ε) ≤ χ(v)2 for all ε > 0, this precludes “steep rises” inχ, where the ratioχ(v + ε)/χ(v) exceeds the previous ratioχ(v)/χ(v − ε) > 1. It also rules out “nearflats” in the decreasing functionf , sincef(v)/f(v−ε) < 1 provides an upper bound onf(v+ε)/f(v).

12

This result offers some intuition into the nature of counterfeiting. Years long ago,

when counterfeiting was the product of careful handicraft,higher quality entailed

greater care — namely, a greater marginal cost, and so lower quantity. Recently,

greater counterfeit quality has been achieved primarily via a better printing press.

Thus, quality increases fixed costs, which must be amortizedacross larger print runs.

3.2 The Counterfeiting Rate from the Hot Potato Game

From the supermodular structure, the marginal benefit on theleft side of (9) linearly

rises both inκ and inv. This yields an economic expression for the counterfeitingrate:

κ(v) =qχ′(v)

ρ(v)∆=

marginal verification costdiscovery rate× denomination

(11)

The right side vanishes at 0 and explodes atv = ∞, it is a quotient of two increasing

functions ofv, and might therefore rise or fall in the verification rate: Marginal verifi-

cation gains rise linearly inv, while marginal verification costs rise inv by convexity.

Observe that the counterfeit entry game is three dimensional, and therefore is much

harder to solve than the hot potato game. As it turns out, we can exploit a degree

of freedom in the model that allows us to shift the analysis ofchanges in the entry

game to those that occur in the hot potato game. Change verification and production

cost functions toχγ(v) ≡ γχ(v) and cγ(x, q) ≡ c(x, γq). Scaling quality units to

q = q/γ leaves costs unchanged:cγ(x, q) = c(x, γ(q/γ)) = c(x, q) and qχγ(v) =

(q/γ)γχ(v) = qχ(v). To wit, making a currency harder to counterfeitcanbe the same

as making it easier to verify. A new security feature thatuniformlyhalves verification

costs is tantamount to one that raises production costs of quality q to that of2q.17

Theorem 3 (Changing Counterfeiting or Verification Costs)

(a) The counterfeiting rateκ falls if the marginal verification cost functionχ′ falls.

(b) The counterfeiting rateκ falls if production costs uniformly rise toc(x, γq), γ>1.

Proof: The verification costχ does not appear in the profit expression (3), and so

does not affect the solution(x, q, e) to (4), (5), and (6). Any rise in the marginal

verification cost functionχ′ is then entirely met by an increased counterfeiting rate

κ = qχ′(V (e, q))/[ρ(V (e, q))∆] in (11) — the only equation thatκ satisfies.

Next, a rise in the counterfeiting cost functionc(x, q) to c(x, γq) is equivalent to

no change inc(x, q) and a smaller marginal verification costχ′/γ. Thus,κ falls. �

17The Bureau of Printing and Engraving’s motto for the new currency is “Safer. Smarter. MoreSecure.” It asserts on moneyfactory.com that the new money is “harder to fake and easier to check”.

13

This result sheds light on the so-called “cat and mouse” nature of the real world

competition between counterfeiters and governments. Whenmoney becomes more

secure, counterfeiters find it more costly to achieve the same quality. By Theorem 3

and its proof,if the cost increase is uniform across quantities,then the verification

effort holds constant as counterfeiters decrease their quality level (measured in the old

units). In other words, verifiers must relax their efforts, and are only willing to do so

if the counterfeiting rate drops. Data in§4 suggests that insofar as this has occurred, it

has been stymied by technological progress in counterfeiting (see Theorem 8).

In summary, the counterfeiting problem is unambiguously aggravated by a less

readily verifiable currency. Second, it may rise in the counterfeiting costs — but this

channel is more nuanced, as it operates via a changing quality and verification rate.18

3.3 The Problem of Counterfeiting

A. The Social Cost of Counterfeiting.The next result offers a consistency test on the

model, showing that the struggle for the∆ note consumes at most∆ in social costs.

Theorem 4 (Social Costs)The average costs of counterfeiting a∆ note are at most

(1−v)∆, and the average total costs of verifying a circulating∆ note are at mostκv∆.

This is a manifestation of Tullock’s 1967 insight that parties to a transfer (or theft)

of D dollars should be collectively willing to spend up toD to influence the transfer

(or theft). Observe how∆ is a pretty coarse upper bound for the total counterfeiting

expenses(1 − v + κv)∆,19 given the stochastic nature of the verification technology

(namely,κ < 1). Ceteris paribus,the social costs of crime are held down by itsrandom

nature— a key factor absent from Tullock’s analysis, and the subsequent rent-seeking

literature. Social costs arelower when individual prevention effortsv aregreater.20

Proof of Theorem 4:Since counterfeiters earn zero profits (3) in equilibrium, and

f(v) ≤ 1 − v, the average costs of counterfeiting a∆ note are at most(1 − v)∆:

Π = 0 ⇒ [c(x, q) + L− hf(v)x]/x = f(v)∆ ≤ (1 − v)∆ (12)

Next, since verifiers weakly prefer to choosev to no verification, the loss-reduction

benefits of verifying exceed the verification costs in (8). Soκ(v)vρ(v)∆ ≥ qχ(v). Let

18The effect of changing legal costsL on the model is ambiguous, and depends on the level of theverification rate. In the interest of brevity, we omit pursuing this analysis (which parallels§B.4).

19This expression ignores the costs of running law enforcement, but these are fixed in our model.20Laband and Sophocleus (1992) estimate non-exchange transfer activity, like theft, in 1985 at $455

billion. They cannot confirm that Tullock’s bound holds due to unmeasurable attempted thefts.

14

T (v) be the expected number of verifications of a circulating counterfeit note. Then

the expected total verifying costs until a circulating counterfeit∆ note is found are:

qχ(v)T (v) = qχ(v)/ρ(v) ≤ κ(v)v∆ (13)

whereT (v) = 1/ρ(v), since it is the mean of a geometric random variable.21 �

One can see that the counterfeiting cost is farther from its upper bound(1−v)∆ in

(12) the greater is the police seizure rates(v). Curiously, more effective counterfeiting

interdiction lessens the total criminal production costs of counterfeiting.

Second, by the equilibrium first order condition (9), we can simplify the gap in (13):

κ(v)vρ(v)∆ − qχ(v) = q[vχ′(v) − χ(v)]

which reduces to the verification producer surplus. Fix the verification ratev, for

definiteness. If the cost functionχ is more convex — i.e. raising the verification rate

is harder — then producer surplus is larger, and the total cost of verification is farther

from its upper boundκv∆. This upper bound clearly rises in the counterfeiting rate.

B. The Counterfeiting Rate. That κ < 1 is mathematically immaterial in the

verifiers’ optimization (8). To deduceκ < 1, we must remove from theκ expression

in (11) its dependence on the endogenous verification ratev. We next derive a formula

that affords some insights into the fundamental determinants of the counterfeiting rate.

Theorem 5 The counterfeiting rate is approximately given by:

κ ≈ (1 − police seizure rate) ·average verification costsaverage production costs

(14)

Hence,κ < 1 if the counterfeiting costs strictly exceed verification costs — intuitively,

verification costs for a note are incurred many times, but production costs just once.

Theorem 5 sheds light on the development of fiat currency — i.e. non-commodity

money whose face value exceeds its intrinsic cost. This required the technology to

produce large numbers of documents for which counterfeits could be discerned at a

verification cost well below their unit production cost.22

We now find a condition on primitives forκ < 1, and so for our equilibrium to exist

(below). Since average production costs and average verification costs rise in quality,

21If we asked this question for an ex ante counterfeit note, then the expected number of verificationswould be slightly greater, since we assume that counterfeiters do not try to pass their note in a bank.

22An excellent example occurred in Canada. As color was introduced on each denomination in the1970s, the counterfeiting rate massively dropped off.

15

by convexity, and sinceχ(0) = 0, we have:

Corollary 2 (The Counterfeiting) The counterfeiting rate is less than one provided

production and verification costs obey the joint restriction

cq(x, q)/x ≥ χ(1) ∀x, q (15)

3.4 Equilibrium Existence and Uniqueness

Fix ∆. A symmetriccounterfeiting equilibriumis a 5-tuple(x, q, e, v, κ), where:23

(a) Counterfeiters choose quantityx > 0 and qualityq > 0 to maximize profits (3).

(b) Given the verifiers’ efforte > 0, counterfeiters earn no profitsΠ(x, q, e) = 0.

(c) The verification intensityv ∈ (0, 1) satisfiesv = V (e, q).

(d) The verifier’s efforte = qχ(v) solves the optimization (7) for the qualityq, the

verification intensityv, and the counterfeiting rateκ ∈ (0, 1).

This is a dynamic Bayesian game,24 as the counterfeit qualityq and verification

ratev are unobserved, and beliefs about these quantities matter.But deviations are

unobserved, and there is a uniqueq andv for each denomination in equilibrium.

Having completely formulated the equilibrium conditions for the counterfeiting

and verification games, we are ready to attack existence and uniqueness.

Theorem 6 (Existence)Assume (15). For any denomination∆ > θ, a counterfeiting

equilibrium(x, q, e, v, κ) exists, is unique, and is symmetric across agents.

We think that the proof in the appendix is novel in its two-pronged approach: First,

we derive quantity, quality, and effort from the counterfeit entry game. By Lemma 3,

the least counterfeited noteθ is unique, andx[θ], q[θ] > 0 = e[θ]. This solution is the

initial conditions for the dynamical system forx′[∆], q′[∆], e′[∆]. If it is suitably well-

behaved on(θ,∞), then we have a solutionx[∆], q[∆], e[∆] to (4)–(5), andΠ = 0.

The verification rate is thenv[∆] = V (e[∆], q[∆]), for the known functionV .

Note that the hot-potato game solely yields the counterfeitrateκ[∆] from (11) in

terms of the other equilibrium quantities — all of which comefrom the counterfeiting

game. Even the verification effort is determined within the counterfeiting game.

23The specification of equilibrium could also include the number of note∆ counterfeiters. But sincethis supply is infinitely elastic, and is easily computed to be κ[∆]M [∆]/x[∆], we have omitted it. Ourmodel can make predictions about these numbers, but the datais poor (especially by denomination).

24Games with a continuum of players have long been analyzed. See Schmeidler (1973).

16

4 The Economics of Crime: Seized Counterfeit Money

Our model is fortunately testable, and admits expressions for the levels of seized and

passed counterfeit money. We now explore implications of the counterfeit entry game.

4.1 The Passing Fraction

Imagine for a moment a world with a fixed quality level. Corollary 1 then implies

a fixed quantity too, and thus average costs that are invariant to the denomination.

But then higher notes would have to pass less often to ensure zero profits in (3). For

instance, absent hassle costs (h = 0), as the denomination doubles from $5 to $10

or $10 to $20, the equilibrium passing fraction scales by one-half to balance greater

counterfeiting revenues. The elasticity of the passing fraction in∆ is then−1. Finally,

with positive hassle costsh > 0, the elasticity exceeds−1, approaching it as∆ grows.

But when quality is flexible, it optimally rises in the denomination by Theorem 2,

and this pushes up average production costs. However, average costs are pushed down

if the optimal quantity falls, as happens for a labor-intensive technology (Corollary 1).

We show next that on balance, average costs unambiguously rise in the denomination

— even when quantity and quality move in the opposite direction.

Lemma 4 The average counterfeiting costs rise in the note if and onlyif quality does.

Proof: Differentiate average costs in∆, and usexcx(x, q) = c(x, q) + L from (6):

d

d∆

(

c(x, q) + L

x

)

=x(cxx

′ + cqq′) − (c(x, q) + L)x′

x2=cq(x, q)

xq′[∆] �

Intuitively, the cost of rising quality eats into profits at greater denominations. So

the passing fraction falls less than proportionately in∆, and its elasticity exceeds−1.

If quality rises fast enough in the note (because the verification rate falls), then the

passing fraction might even rise to sustain zero profits. In fact, we deducev′[∆] > 0

from the data in§4.2. The premise of the following result is then justified.

Lemma 5 (Passing Fraction) If the verification rate rises in the note (v′[∆] > 0),

then the passing fraction falls, and its elasticityE∆(f) lies in (−1, 0), and equals:25

E∆(f) = −∆

∆ + h·

E∆(χ)

E∆(χ) + E∆(q)(16)

25 If Ex(g) is the elasticity ofg in x, then the product and chain rules of calculus yieldEx(g · h) =Ex(g)+Ex(h), Ex(g/h) = Ex(g)−Ex(h), Ex(f◦g) = Eg(f)Ex(g), andEy(g−1) = 1/Ex(g) if y = g(x).

17

4.2 The Seized-Passed Ratio Across Denominations

We have just produced an expression for the passing fractionelasticity stemming from

its micro foundation in the counterfeit entry game. We now formulate another ex-

pression, based on observables. Counterfeit money is eventually eitherseizedfrom

the criminals by law enforcement or the first verifiers, or successfullypassedonto the

public, and later lost by an unwitting individual. Call these levelsS[∆] andP [∆] —

recalling that we have assumed for simplicity that all aggregates are in steady-state.

The valuesS[∆] and P [∆] obey two steady-state conditions. First, the value

S[∆]+P [∆] of counterfeit production of∆ notes equals the value of counterfeit money

leaving circulation. Second, passed money circulating is constant: To wit, the outflow

of passed money from circulation equals the inflow of new counterfeit money pass-

ing into circulation. We assume that counterfeiters attempt to pass all production, so

that seized money represents failed passed money.26 The inflow of passed money then

equals the passing fraction times the counterfeit production. Altogether, we have:

P [∆] = f [∆] · (production value) = f [∆] · (S[∆] + P [∆])

The importance of theseized-passed ratioS[∆]/P [∆] is apparent, since

1

f [∆]= 1 +

S[∆]

P [∆]≡ R[∆] (17)

The seized-passed ratioR[∆] clearly inherits properties from the passing fraction.

Theorem 7 (Seized-Passed Ratio)The passing fraction has elasticity

E∆(f) = −E∆(R) > −1 (18)

Thus, the verification rate rises in∆ if and only if the seized-passed ratio rises in∆.

Theorems 1 and 2 predict a struggle between better verification efforts and better

counterfeit quality as the denomination rises. Which effect prevails? Observe that

the verification rate increases when efforte ≡ qχ(v) rises proportionately more than

quality q. Otherwise, improved quality overwhelms the greater effort, and depresses

the verification ratev[∆] ≡ V (e[∆], q[∆]). While a verifier may study a $100 note with

greater care than a $5 note, the $100 passes more readily if its quality is sufficiently

26This is an overestimate, because some money might be seized before any passing attempt, perhapsfound in the counterfeiter’s possession or after he is followed back to his plant. Hence, to make senseof our data application below, we assume that this overestimate does not vary in the denomination.

18

($100, 2.43)

($50, 2.29)

($20, 1.93)

($10, 1.55)

($5, 1.38)

1

10

1 10 100

Log DenominationLo

g (

1+

Sei

zed

/Pa

ssed

)

Figure 1: USA Seized Over Passed, Across Denominations.These are the seized-passed ratios, averaged over 1995–2005, for non-Colombiancounterfeits in the USA.Noticeably, they rise in∆. The sample includes almost ten million passed notes, andabout half as many seized notes. Data points are labeled by pairs (∆, 1+S(∆)/P (∆)).So for every passed $5 note, 1.38 have been seized on average.For this log-log graph,slopes are elasticities — positive and well below one.

greater. While our model does not allow us to compare proportionate changes in effort

and quality, the data imply that this has not occurred: Looking at Figure 1,27 we can

conclude from the data that:The seized-passed ratio has risen in the denomination in

the USA 1995–2005(as well as separately for 1995–99 and 2000–04). The verification

rate thus rises in the denomination too, by Theorem 7. Both also hold in Canada over

the span 1980–2005 for all six paper denominations, including the $1000 note.28

In the log-log diagram of Figure 1, the slopes (which are elasticities ofR[∆]) are

not only positive but also less than 1. Thus, one plus the seized-passed ratio less than

doubles when the denomination doubles. This offers a different insight for us. From

Lemma 5 and Theorem 7, we can conclude that higher notes are ofbetter quality.

Intuitively, production costs are greater, and so the rise inR[∆] is less than proportional

27This figure is based on data from Lorelei Pagano. We have excised the Columbian counterfeit data— which are the largest portion of foreign counterfeits (especially for the $100 note), and commanda separate category in the Secret Service accounting. But the seizures for Colombian counterfeits aremostly in Colombia, while our data on passed notes is domestic. The Secret Service has only given usdata on Columbian counterfeits (family C-8094) passed and seized domestically by denomination, aswell as an aggregate across all notes, including both foreign and domestic passed and seized. Since thevast majority of seizures are foreign, either in Columbia oren route to the USA, we have used theseaggregate numbers year by year to scale each denomination’spassed and seized ratio.

28For Canada, from 1980-2005, the seized-passed ratios are respectively 0.095, 0.145, 0.161, 0.184,0.202, and 3.054 over the notes $5, $10, $20, $50, $100, and $1000. Production of the $1000 note wasdiscontinued in 2000 to counter money laundering and organized crime.

19

Table 1:Fraction of Notes Digitally Produced, 1995–2004.This Secret Service dataencompasses all 8,541,972 passed and 5,594,062 seized counterfeit notes in the USA,1995–2004. Observe(a) the growth of inexpensive digital methods of production, and(b) lower denomination notes are more often digitally produced.

Note 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 avg.

$5 .250 .306 .807 .851 .962 .972 .986 .980 .974 .981 .901$10 .041 .095 .506 .851 .908 .911 .961 .963 .971 .978 .756$20 .139 .295 .619 .882 .902 .926 .929 .961 .974 .983 .823$50 .276 .335 .546 .768 .777 .854 .911 .828 .822 .857 .755$100 .059 .066 .147 .263 .239 .314 .267 .251 .307 .399 .250

to the denomination. Table 1 contains (emailed) Secret Service data on counterfeit

notes for the span 1995–2005, and documents how quality rises in the denomination.

In this time span, almost all counterfeit production is by capital intensive means. Also,

digitally-produced notes (eg. using scanners and ink jet printers) are lower quality,

while those made from printing presses are higher quality. One can see that the fraction

of notes produced digitally generally falls in the denomination.

The $100 note stands out in particular. Judson and Porter (2003) find that 73.6% of

passed $100 notes were “circulars” — made by the same source and high quality. The

percentage for the $50 note is 19.2%; other notes are below 3%. Also, the “Supernote”

(circular 14342) is the highest quality counterfeit ever recorded. First found in 1990,

this deceptive North Korean counterfeit $100 note was made from bleached $1 notes,

with the intaglio printing process used by the Bureau of Engraving and Printing.

We can now offer a different implication of the seized-passed analysis concerning

the criminal marketplace. Combining equations (6), (4), and (17), we see that:

(c(x[∆], q[∆]) + L)/x[∆]

∆ + h=cx(x[∆], q[∆])

∆ + h= f [∆] =

1

R[∆](19)

Step back from the single criminal model, and imagine that the producer instead sells to

middlemen. Then the legal costs are partially incurred by both parties, so that average

costs should overstate the “street price” of counterfeit notes (at which they are traded).

Then

street price< average cost≈denomination

1 + seized-passed ratio

The implied street price ceilings can be computed for the denominations from Figure 1,

to get $3.37, $5.95, $9.30, $19.20, $35.70, respectively. Testing this awaits data.29

29 We thank Pierre Duguay for this nice observation. We do have evidence from one recent case: AMexican counterfeiting ring discovered this year sold counterfeit $100 notes at 18% of face value to

20

0

1

2

3

4

5

6

7

8

'64 '67 '70 '73 '76 '79 '82 '85 '88 '91 '94 '97 '00 '03

Figure 2:USA Passed and Seized, 1964–2004.The units here are per thousand dollarsof circulation across all denominations. The dashed line represents seizures, and thesolid line passed money. From 1970–85, the vast majority of counterfeit money (about90%) was seized. The reverse holds (about 20%) for 2000–2004. Two down-spikes in1986 and 1996 roughly correspond to the years of technological shifts.

4.3 The Falling Seized-Passed Ratio Over Time

There has been a sea change in the seized and passed time series since 1980. For the

longest time, seized greatly exceeded passed counterfeit money, as seen in Figure 2.

Starting in 1986, and accelerating in 1995, the seized-passed ratio began to plummet.

Tables have turned: By far, most counterfeit money now is passed.30 We argue that

this is consistent with the technological changes that havetranspired in the industry.

First, in the 1980’s, photocopiers became a tool of choice bycounterfeiters. This is

clearly evidenced by the plant suppressions.31 The numbers of such “plants” (possibly

homes) suppressed was: 11 from 1981–5, 30 in 1986, 345 from 1987–94, and finally

distributors, often gang members, who then resold the counterfeit notes for 25–40% of face value. Themoney was transported across the border by women couriers, carrying the money on their person.

30The Annual Reports of the USSS supplied earlier data, and Lorelei Pagano gave us more recent data.Seized is a more volatile series, as seen in Figure 2, as it owes to random, sometimes large, counterfeitingdiscoveries, and is also perforce contemporaneous counterfeit money. By contrast, passed money istwice averaged: It has been found by thousands of individuals, and may have long been circulating.

31This paragraph is based on the Annual Reports of the USSS until 1996, and thereafter, Table 6.8 inUSTD (2003). This claim is consistent with Chant’s (2004) finding of a digital revolution in the 1990’s.

21

62 in each of 1995 and 1996 (the most recent year for which we have data). Next, in

the 1990s, came a digital counterfeiting technological revolution, using ink jet printers:

No such plants were found through 1994. From 1995–2002, theygrew from 19% to

95% of all plant seizures. This gives us two technological revolutions in counterfeit-

ing: photocopying around 1985, and then digital productionaround 1996. As seen in

Table 1, the digital trend continued past 1996, as digitallyproduced counterfeits have

risen from a very small minority in 1995 to 98% of the $5, $10, and $20 notes.

We will say that there has beenquantity-neutral technological progressof level

t > 1 if for any quantityx, the cost of any quality levelq falls fromc(x, q) to c(x, q/t).

Let us denote the slopes of the quantity and quality int by x and q. Appendix B.8

proves thatx<0<q−q if ψq>0 andx>0>q−q if ψq<0. Summarizing these cases:

Theorem 8 (Technological Change)Assume quantity-neutral technological progress.

Then verification effort and quality both rise, and the resulting verification rate falls.

Production levels fall for a labor-intensive technology (ψq > 0), and rise for a capital-

intensive technology (ψq < 0).

Theorem 8 captures the falling seized-passed ratio described in Figure 2.32 At

first, there was a labor-intensive technology. As quality improved (photocopiers, then

digital production), plants shrunk, and the fixed legal costsL > 0 were amortized over

a smaller production level.33 The seized-passed ratio accordingly fell.34

With the passing elasticity1 ≤ Υ < 2, the unobserved verification rate fell too.

The passing fraction rose roughly from 10% to 80%, and thus1 − v scaled down, and

thus the implied verification rate fell an indeterminant amount.

5 Monetary Economics: Passed Counterfeit Money

5.1 Empirical Analysis of Passed Counterfeit Rates

We now focus on circulating counterfeit money, by fleshing out implications of the hot

potato game. Figure 3 plots the average fractionp[∆] of passed $1 notes in circulation

for 1990–1996, and of the $5, $10, $20, $50, $100 notes for 1990–2004. These ratios

per million have averaged1.49, 19.11, 70.93, 70.27, 42.38, 81.67, respectively.

32This theorem performs a comparative steady-state analysis, and does not do the much harder exer-cise of out of steady-state dynamics. The two approaches will give similar ordinal implications.

33Counterfeiting arrests grew from 1856 in 1995 to 3717 in 2005, plant suppressions from 153 to 611.See USTD (2006).

34The legal seizure effectiveness might have fallen, notationally ruled out by our constant functionf .

22

We used two different measures of stability of the roller coaster shape in Figure 3.

First, with a pooledt-test, we can conclude that the increments from $5 to $10, $5 to

$50, and $50 to $100 note are extremely significant (t = 5.73, t = 6.66, andt = 9.15,

respectively), as well as the drop from $20 to $50 (t = 5.99). Second, checking the

stability of this relation, we looked separately at each of the five year spans 1990–94,

1995–99, 2000–04. The $1 note aside, the $5 and $50 notes are consistently the lowest

and second lowest ranks of the passed counterfeit notes.

There are two problems with the annual passed over circulation ratio as a proxy

for p[∆]. First, we measure domestic passed notes, but circulation is worldwide. Also,

the fraction of notes abroad likely rises in the denomination, possibly substantially.35

Second, we assume that notes trade hands once per “period”. Our results on the seized-

passed ratio do not depend on this assumption, as both measures are flows, but the anal-

ysis of the passed-circulation ratio does (as a flow over a stock). The results should

still obtain if notes trade hands a given randomexpectednumber of times per pe-

riod. Yet the velocity is intuitively rising in the note:36 The higher the note, the less it

transacts on average in a year, and a calendar year represents fewer transaction oppor-

tunities for higher notes. If we interpret the annualized passed data in light of this, the

passed-circulation ratio is eventually rising (from $50 to$100 note), and might well be

monotonically rising in the denomination.37 We seek to explain this using our model.

The common claim that the most counterfeited note domestically on an annualized

basis is the $20 is false over our time span. Accounting for the higher velocity of the

$20,on a per-transaction basis, the$100 note is unambiguously the most counterfeited

denomination.This is the relevant measure for decision-making by the public.

5.2 Equilibrium Passed Money Levels

The total supply of counterfeit and genuine∆ notes has valueM [∆] > 0. Passed

money is circulating counterfeit moneyκ[∆]M [∆] times the discovery rateρ[∆]. Let’s

define thepassed-circulation ratiop[∆] ≡ P [∆]/M [∆]. This is the fraction of all

35Judson and Porter (2003), eg., estimate that3/4 of $100 notes, and2/3 of $50 notes are abroad.36 Lower denomination notes wear out faster, surely due to a higher velocity. Longevity estimates by

the Federal Reserve Bank of NY [www.newyorkfed.org/aboutthefed/fedpoint/fed01.html] are 1.8, 1.3,1.5, 2, 4.6, and 7.4 months, respectively, for $1,. . . ,$100.FRB (2003) has close longevity estimates.

37 Since notes age almost exclusively due to the wear and tear oftransactions, one might argue that anote’s longevity should be inversely proportional to its velocity. Assume so. If we fix the passed rate ofthe $20 note, and scale the passed rates of all other denominations by the relative longevity estimates infootnote 36, then the implied passed rates are strictly rising in the note: 1.4, 12.4, 53.3, 70, 97, and 303.

23

circulating∆-notes per period that will be passed. Then we have from equation (11):

p[∆] =q[∆]χ′(v[∆])

∆=

marginal verification costdenomination

(20)

We will partially explain Figure 3 below by arguing that as∆ rises, initially marginal

verification costs proportionately rise much faster than the denomination does, and

then slower. Once we add endogenous quality, we will see thatit rises faster eventually.

In other words, if quality were constant in the denomination, then the passed rate would

eventually fall. That this does not happen is further evidence of the rising quality.

The implied verification costs are miniscule. The passed-circulation ratio is at

most 1 per 10,000annually. Suppose the $100 note transacts at least four times per

year. Thenp[∆] is at most 1 in 40,000, and marginal verification costs are at most

$100/40,000, or one quarter penny per note. These tiny marginal costs drive the theory.

Let’s explore the implications of relation (20), and its insights for Figure 2. An in-

teresting regularity is the stability of the passed money rates through time, even while

the seized levels have dramatically fallen. From 1970 to 2000 (midpoint in our time

span in Figure 3), the rising cost of living has deflated the real values of each de-

nomination by approximately fourfold. This is attenuated by two other accompanying

changes. First, by Theorem 2, quality should have fallen with the real devaluation. In-

deed, inflation intuitively scales up the cost functionsc andχ, as well as the legal cost

L (time spent in jail); thus, inflation is tantamount to an isolated fall in∆. Second, the

composition of denominations in circulation has shifted upwards through time too.38

Next, consider the technological change identified in§4.3, during which the veri-

fication rate has greatly fallen. But by Theorem 8, quality has risen with this change,

also attenuating the fall inχ′(v[∆]). Altogether, endogenous quality buffers the passed

rates against changes in the real value of the currency, or the counterfeiting technology.

The counterfeiting rateκ is unobserved, and the passed-circulation ratiop is its

observable manifestation. In fact, the Secret Service and the Federal Reserve often

draw inferences aboutκ from p. But p = κ · ρ is at best an imperfect proxy forκ.

If the discovery rateρ falls over time, then the passed rate may hold constant when

the true counterfeiting rate rises. For instance, if digital counterfeits are more easily

recognized, then the verification rate may be higher — pushing upp, but notκ.

For a different insight linking the counterfeit rateκ and passed-circulation ratiop,

note howκ explicitly depends in (11) on the banking verification rateα and banking

chanceβ, whilep in (20) does not. So if banks more effectively verify than thepublic,

38The Secret Service does not have passed or seized levels by denomination before 1995.

24

($1,1.5)

($5,19)

($10,71)

($20,70)

($50,42)

($100,82)

1

10

100

Log DenominationL

og

(P

/M)

Figure 3:USA Passed Over Circulation, Across Denominations.This graph givesthe average ratios of passed domestic counterfeit notes to the (June) circulation of the$1 note for 1990-96, and the $5, $10, $20, $50, $100 notes for 1990–2004, all scaledby 106. The data points are labeled by the pairs(∆, P (∆)/M(∆)). As a log-log graph,slopes are elasticities — positive and around 1.6 until the $10, then zero, about -0.6(−0.6 > −1, and so consistent with (33)), and about 1 until $100. Since the velocityfalls in the note, the passed-circulation ratio is increasingly understated at higher notes.

thenκ falls inα andβ, whilep does not.Ceteris paribus, while better verified markets

have less counterfeit money, it is found at a faster rate withgreaterα orβ. On balance,

these effects exactly cancel, andneutralityobtains: the verification rate only indirectly

affects the passed counterfeit money through the marginal verification cost.

We test this model by looking at its comparative statics for∆. Passed money in (20)

is the ratio of an increasing function of the denomination∆, and∆ itself. Sincev[θ] =

0 by Lemma 3, the marginal verification costχ′(v) begins at zero when∆ = θ, while

the denominator starts atθ > 0. Increments in the numerator initially proportionately

swamp the denominator, while the denominator∆ dominates at large∆.

Lemma 6 The verification elasticityE∆(v) explodes as the verification rate vanishes,

and in fact:

E∆(v) = −1 − v

ΥvE∆(f) (21)

Proof: This owes to the elasticity chain rule,E∆(f) = E1−v(f)Ev(1 − v)E∆(v). �

A simple implication of Lemma 6 is that if the seized-passed ratio varies across

denominations, then so must verification. This precludes models in which verification

is not a choice variable! It cannot be stochastic but exogenous, as in any paper that

presumes a fixed authenticity signal — like Williamson (2002).

Consider the economic factors behind the passed-circulation ratio seen in Figure 3.

Low value notes are neither very profitable to counterfeit, nor very worthy of attention.

25

It is not obvious which force wins this battle — and conversely so for high value notes.

And yet we have seen that the passed-circulation ratio risesmonotonically.

Intuitively, near the lowest counterfeited denominationθ > 0, verification vanishes

to ensure zero profits in the counterfeiting game. The marginal verification cost like-

wise vanishes, and so too the passed-circulation ratio (20). On the other hand, as∆

explodes, verification must become perfect. But if the marginal verification costs are

bounded, then the passed-circulation ratio must vanish. Wesee here the necessity of

endogenous quality in explaining the continuing rise in thepassed rate.

The next formal derivation of the passed rate proceeds instead by looking at slopes.

We take the elasticity form of the passed-circulation ratio(20), and find that it admits

expression in terms of the seized-passed ratio. This interplay between the counter-

feiting and hot potato games allows us to identify the passing fraction. The formulas

below reflect how the verifiers’ cost function depends on the verification ratev, while

counterfeiters’ profits depend on the passing fraction, andthus1 − v.

Theorem 9 (Passed Money)AssumeE∆(R) ≫ 0. As∆ tends down toθ, the passed-

circulation ratio elasticity explodes and so the passed-circulation ratiop[∆] vanishes.

Proof: By the product and quotient rules, the passed-circulationratio (20) has elasticity

E∆(p) = E∆(q) + E∆(χ′) − 1 = E∆(q) + Ev(χ′)E∆(v) − 1

First, E∆(q) > 0 by Corollary 1. Next,Ev(χ′) > 0, while E∆(v) ↑ ∞ by Lemma 6,

sincev[∆] ↓ 0 as∆ ↓ θ by Lemma 3, andE∆(f) = −E∆(R) by Theorem 7. �

To check the premise of Theorem 9 thatE∆(R) ≫ 0, we can see in Figure 1 that

the elasticityE∆(R) is about1/5 over the range of notes $5 to $100.

We now express theunobservedquality elasticity in terms of the elasticities of the

seized-passed ratio and passed money, bothobserved. This result is the analogue of

our deduction in Theorem 7 of the seized-passed ratio elasticity.

Theorem 10 The quality elasticity obeys

E∆(q)

(

1 +Ev(χ

′)

Ev(χ)

E∆(R)

1 − E∆(R)

)

= E∆(p) + 1

This theorem reveals that if quality were fixed, so thatE∆(q) = 0, thenE∆(p) =

−1. In other words,p[∆] is proportional to1/∆. In fact, the data suggest that the

quality elasticity is positive — since the least value ofE∆(p) + 1 is 0.44 in Figure 3.

26

0.00

0.10

0.20

0.30

0.40

0.50

0.60

$1 $5 $10 $20 $50 $100

FR

B P

ass

ed R

ati

o

0%

5%

10%

15%

20%

25%

FR

B s

hare

of

Pass

ed N

ote

s

Figure 4: Counterfeits Detected at Federal Reserve Banks.The “FRB share” (bargraph) is the percentage of all passed notes of that denomination found by FederalReserve Banks (FRB). (The gray bars are 1998 and the slash bars are 2002.) The“FRB passed ratio” (lines) is the ratio of the passed money rate at the FRB divided bythe velocity-adjusted passed-circulation ratiop[∆] for that denomination. (The solidline is 1998, and the dashed line is 2002.) The FRB numbers arefound in Table 6.1 inUSTD (2000), Table 6.3 in USTD (2003), and Table 5 in Judson and Porter (2003).

6 Passed Money Found in the Banking System

We now turn to one final piece of evidence in favor of our probabilistic verification

story. The banks have so far been silent in our story. But theoretically they perform

a nice identifying role, since counterfeit money hitting them has previouslynot been

previously found by verifiers. We use this idea to explain howmuch counterfeit money

banks should find, and then match these implications to some data that we have found.

This yields a joint test on the predictions of the counterfeiting and hot potato games.

Figure 4 gives data about a vast number of counterfeit notes hitting the Federal

Reserve Banks (FRB) — to which commercial banks pass on moneythey do not need,

or which is damaged. The FRB found 21% of all passed counterfeits in 2002, but a

much larger fraction of the low denomination notes. A priori, this reverse monotonicity

might seem surprising since the lowest quality counterfeitnotes presumably should

have been easy for the public to catch. Our model can explain this anomaly.

27

First, let us consider commercial banks. On this front, the data is nonexistent,

but the theoretical prediction is clear. A bank finds a passednote exactly when(i)

it is counterfeit and(ii) the last verifier prior to the commercial bank missed it. A

randomly chosen note∆ — even one about to hit a bank — is counterfeit with chance

κ[∆]. The fraction of notes found by a bank to be passed is therefore:

ξ[∆] =passed notes hitting banktotal notes hitting bank

=κ(1 − v)βα

(1 − κ)β + κ(1 − v)βα≈κ(1 − v)α

The approximation is accurate withinκ≪ 0.0001, or 0.01%. The counterfeiting rateκ

is unobserved, and in principle thisbank passed ratecould be rather ill-behaved. Yet

more passed counterfeit notes should hit a bank when there are more passed counterfeit

notes being found by everyone. So motivated, we normalizeξ by the observed passed-

circulation ratiop[∆] = ρ[∆]κ[∆], eliminatingκ. This leads us to what we call the

commercial bank ratio:ξ[∆]

p[∆]≈

(1 − v[∆])α

ρ[∆]

The measures how much the bank passed rate exceeds the overall passed rate. It should

be falling,39 for the verification and discovery ratesρ[∆] = βα+(1−β)v[∆] are rising.

Now, consider the Federal Reserve Banks, for which we have data. Counterfeits

hitting an FRB have twice escaped earlier detection. An FRB finds a passed note

exactly when(i) it is counterfeit,(ii) the last verifier prior to the commercial bank

missed it, and then(iii) that bank repeated this mistake. We assume that commercial

banks hand over some fractionφ[∆] ∈ (0, 1) of notes each period to an FRB, often

for destruction. We know thatφ[∆] falls in ∆, since the note longevity estimates in

footnote 36 rise in∆. Intuitively, higher denomination notes wear out less often.

The FRB passed rateζ [∆] is the fraction of passed notes hitting an FRB that

are counterfeit. Unlike the commercial banks, the counterfeit buck stops here, and

is found with certainty. Now, a note hits an FRB exactly when it is deposited in a bank

(chanceβ) which then transfers it to an FRB (chanceφ). If that note is counterfeit

(chanceκ), then both verifier and bank must miss this fact (chance(1 − v)(1 − α)).

ζ [∆]=passed notes hitting FRBtotal notes hitting FRB

=κ(1 − v)β(1 − α)φ

(1 − κ)βφ+ κ(1 − v)β(1 − α)φ≈κ(1−v)(1−α)

The approximations are likewise accurate withinκ ≈ 0.0001. We know from available

data that the FRB passed rate is highly non-monotone.40 Proceeding as we did for39Our FOIA to the Secret Service asking for data on passed moneyin the banking sector was ignored.40The non-monotone structure is consistent across 1998, 2002, and 2005: it rises from $1, $5, $10,

28

the commercial banks, we eliminate the unobserved counterfeit rate using the passed-

circulation ratio, and arrive at theFRB ratio:

ζ [∆]

p[∆]=

(1 − v[∆])(1 − α)

ρ[∆](22)

Theoretically, this ratio should be monotonically fallingin the denomination. True

enough, for the only years with available data, 1998, 2002, and 2005, it is falling

monotonically only from the $1 through the $20. But in each case, it turns up at the $50

and further at $100. The rise at the $100 seems especially high, jumping up by a factor

of three. While the overall passed rate is distorted by varying velocities of different

notes, the FRB passed rate is not (as a rate per note processed, rather than per year). If

we scale the passed rates for each of the three years as suggested in footnote 37, then

we can correct for the understatement of passed rates of higher notes.41

For a different perspective on counterfeits in the banking system, we can explore

the proportions of all counterfeit notes that are ultimately found in banks. This exercise

focuses solely on the counterfeit notes. Let thecommercial bank shareµ[∆] denote

the fraction of all passed counterfeit notes of denomination ∆ found by banks. Using

our expressions for passed notes found by verifiers, banks, or an FRB, the reciprocal

bank share is a sum of one increasing term, and one possibly increasing:

µ[∆] =passed notes found by commercial banks

passed notes found by verifiers, commercial banks, or an FRB

=κ(1 − v)βα

κv + κ(1 − v)βα+ κ(1 − v)β(1 − α)φ

The reciprocal of this fraction is the sum of an increasing term, a constant terms, and

then a term falling only due toφ. The shareµ[∆] should then be falling in∆.

Likewise, the analogousFRB shareσ[∆] should be monotonically decreasing too:

µ[∆] =κ(1 − v)β(1 − α)φ

κv + κ(1 − v)βα+ κ(1 − v)β(1 − α)φ

In Figure 4,µ[∆] is falling, and then slightly rising from $50 to $100. We suspect

that the verification system employed by banks misses the highest quality counterfeits

— local bank tellers told us they simply go by the feel of the note, and skip its other

drops at the $20 and $50, and then shoots up at the $50 by a factor of six or more.41Once a counterfeit hits an FRB, it is almost impossible to trace it back to the original depositor. As

such, counterfeit money that is so high quality as to escape earlier detection ought not affect incentivesof individuals in our model. Thus, our model might understate the quality rise at the highest notes.

29

security features. If this is true, then the bank verification rateα drops at the $100

note, thereby explaining the anomalies observed.

7 Conclusion

Summary. Counterfeiting is an interesting crime insofar as it induces two closely

linked conflicts: counterfeiters against verifiers and law enforcement, and verifiers

against verifiers. The typical focus on the first conflict in the small literature bipasses

the key role of the second conflict in explaining passed counterfeit money. Indeed,

since the late 1990s, passed money has greatly exceeded seized money.

We develop a theory of counterfeit money based on costly currency verification

that captures both forms of counterfeit money. This is a new decision margin — as

unwitting innocents strive to avoid acquiring fake money. It pushes up verification

effort for the dearest notes, and so explains the rising seized-passed ratio — especially

at low denominations. But this model ingredient alone wouldforce the seized-passed

rates to rise linearly with the denomination, and would leadthe passed-circulation ratio

to eventually fall. This mandates our second innovation — variable quality counterfeit

production. When quality modeled means higher verificationcosts, we can rationalize

the cross-sectional and time series properties of passed and seized money.

Economics of Crime.We provide a model the battle between criminals and those

they seek to steal from, with variable intensity crime-fighting (verification) and crimi-

nal efforts (quality). Judging from estimates in Laband andSophocleus (1992), efforts

by “good guys” are a significant portion of the social costs ofcrime in the USA.

We show how more valuable counterfeit goods simultaneouslyelicit a greater con-

sumer scrutiny, and a better counterfeit quality. Both effects arise from criminal incen-

tives in the counterfeiting game. The first result is not obvious, and turns on a novel

application of log-concavity to producer theory and our passing fraction. We introduce

the related notions of the passing fraction and the seized-passed ratio, new to the litera-

ture. We find empirically that the latter rises in the value ofthe note. We also show that

positive legal costs force an inefficiently high criminal production level compared to

producer theory. While our paper shows how counterfeiting is a well-calibrated plat-

form for exploring theories about crime, our insights should extend beyond money, to

the widespread counterfeit production of documents, clothing, watches, drugs, art, etc.

Namely, the seized-passed ratio rises in the note, and has massively fallen over time,

owing to a technological shift. We also estimate the “streetprice” of counterfeit notes.

Monetary Theory. As a contribution to monetary economics, this paper explores

30

a supermodular game that arises in the currency verificationefforts. We show how the

interplay of increasing quality and verification effort explains the shape of the passed-

circulation ratio: rising, maybe falling, and then rising.We also show that this theory

also makes sense of the passed money appearing in the federalreserve banks.

Our paper also sheds light on the development of non-commodity fiat currency —

i.e. whose face value greatly exceeds its intrinsic cost. Weshow that the counterfeiting

rate is the ratio of verification to production costs. Fiat currency requires easily verified

characteristics that could not be so easily produced.

Applications and Extensions. Our model should prove a good framework for

thinking about counterfeiting. What is the stock of counterfeit notes? How long do

counterfeit notes circulate? Both questions turn on the passing fraction.42

Finally, one could imagine a complicated general equilibrium setting — combining

the insights of this paper and the earlier literature — having our new decision margin,

where notes would be both verified and discounted.

A Appendix: Heterogeneous Counterfeiters

Observe how the causation flows in the two games. Verifiers choose their verification

effort so that counterfeiters earn zero profits (see Appendix B.3). Equilibrium in the

hot potato game then fixes the counterfeiting rateκ, since there is a unique optimal

verification effort for each counterfeiting rate. This rateis a free variable, given the

counterfeiters’ free entry condition. This is analogous tothe way in which one’s mixed

strategy in a game is chosen to obey the indifference condition of the other players.

This curious causation owes exclusively to the assumption that counterfeiters are

initially homogeneous. Otherwise, the verification rate would also reflect behavior in

the hot potato game. For instance, if counterfeiters’ fixed costs were heterogeneous,

then all firms producing would make the same production and quality choices. Further,

only those with fixed costs below a threshold would enter. Greater verification effort

would then push down this threshold, and thereby diminish the supply of counterfeit

money. Altogether, equilibrium in the entry game would require the counterfeiting

rate to fall in the verification effort (and not remain constant), while the counterfeiting

rate would rise or fall in the verification effort as in (11) tomaintain equilibrium in

42If we knew the annual “velocity” (transaction uses)n[∆] of a denomination∆, then we couldprovide lower bounds on the stocks of circulating counterfeit ∆ notes. For(1−f)(stock) = P [∆]/n[∆],namely the per period amount of passed money. To estimatef properly, one needs to know how muchmoney is seized in the passing attempt. We can say that the stock is at leastP [∆](1+P [∆]/S[∆])/n[∆].

31

the hot potato game. This richer model would thus demand thatboth games be solved

simultaneously. The gains from this exercise do not justifythe substantial costs.

B Appendix: Omitted Analysis and Proofs

B.1 Deriving the Verification Function: Proof of Lemma 1

The first claim follows fromχ(v) > 0 for v > 0. Theq-derivative ofqχ(V (e, q))) ≡ e

assertsqχ′(v)Vq + χ(v) = 0, and thusVq = −χ(v)/qχ′(v), as needed. Next, the

v-derivative ofV (qχ(v), q) ≡ v producesVe(qχ(v), q)qχ′(v) = 1. Sinceχ(0) =

χ′(0) = 0 andχ′(1) = ∞, if we take limits asv vanishes and explodes, we get

Ve(0, q) = ∞ andVe(∞, q) = 0, for anyq > 0. �

B.2 The Least Counterfeited Note: Proof of Lemma 3

PROOF OF PART(a): Let θ+h be the minimum of(c(x, q)+L)/(xf(0)) overx, q ≥ 0.

By assumption, this is realized at a finite and positivex0, q0. Thenθ > 0 because

average costs equal marginal costs, which exceedcx > h at positivex, givenL > 0.

Next, no note∆ < θ can be counterfeited, since profits (3) would be negative, even

absent verification. Conversely, any available note∆ > θ must be counterfeited. For

if not, then producing it withx0, q0 is strictly profitable, which is not possible. �

PROOF OF PART(b): By similar logic, if the passing fraction did not vanish as∆ ↑ ∞,

then high enough denomination notes would become very profitable to counterfeit

by (3). Thus,f(v[∆]) = (1 − v[∆])(1 − s(v[∆])) ↓ 0, and thereforev[∆] ↑ 1.

By the same token, if the verification tended down to somev[θ] > 0 as we neared

∆ = θ, then mimicking this production quantity and quality for a slightly smallerθ−ε

note would yield positive profits since it would not be verified, by assumption. �

B.3 Equilibrium Effort Elasticity: Proof of Theorem 1 Finis hed

We’ve showne′[∆]

e[∆]= −

Π∆

eΠe

> 0 (23)

Substitute from (3), changeVe to Vq with qVq + eVe ≡ 0, and then use (4) and (5):

(∆ + h)e′[∆]

e[∆]= −

f(V )x

ef ′(V )Vex=

f(V )x

qf ′(V )Vqx=xcxqcq

> 0 �

32

B.4 Quality Rises in the Denomination: Proof of Theorem 2

We have used the quantity FOC (4) to derive the law of motion for effort in Theorem 1.

We are left to exploit the information in the quality FOC (5) and the constant producer

surplus condition (6). Totally differentiating them in∆ yields:

0 = ψxx′[∆] + ψqq

′[∆] (24)

−Πqee′[∆] − Πq∆ = Πqxx

′[∆] + Πqqq′[∆]

Simplifying using (23), and then solving these two equations, yields

q′[∆] = −ψx(ΠqeΠ∆/Πe − Πq∆)

Πqxψq − ψxΠqq

(25)

RecallingΠ ≡ f(V )x(∆ + h) − c− L, we getΠq = f ′(V )Vqx(∆ + h) − cq. Hence,

the denominator of (25) is non-negative since the second order Hessian condition is

locally necessary for the optimization:

0 ≤ ΠxxΠqq − Π2qx = −cxxΠqq − Πqx(cq/x− cqx) = (Πqxψq − ψxΠqq)/x (26)

where we have simplifiedΠqx = cq/x− cqx = −ψq/x using the quality FOC (5).

For insight into the numerator of (25), differentiate the identityqχ(V (e, q)) ≡ e in

q and thene, to getqVqχ′ + χ = 0 and thenχ′Ve + qχ′′VeVq + qχ′Veq = 0. These give

Veq

VeVq

= −

(

1

qVq

+χ′′

χ′

)

=χ′

χ−χ′′

χ′

The second numerator factor of (25) is then:

Π∆Πqe

Πe

− Πq∆ = f(V )xf ′′(V )VeVq + f ′(V )Veq

f ′(V )Ve

− xf ′(V )Vq

= fxVq

(

f ′′

f ′−f ′

f+

Veq

VeVq

)

= fxVq

(

f ′′

f ′−f ′

f+χ′

χ−χ′′

χ′

)

(27)

SinceVq < 0, we have from (25) and (26) thatq′[∆] > 0 precisely when

(log f)′′

(log f)′−

(logχ)′′

(logχ)′≡f ′′

f ′−f ′

f+χ′

χ−χ′′

χ′> 0

33

Now, (logχ)′ > 0 > (logχ)′′ asχ is increasing and log-concave. Next, from the proof

of Lemma 2,−Υ/(1−v) = (log f)′ < 0 and(log f)′′ = −Υ/(1−v)2 < 0, whence:43

f ′′

f ′−f ′

f=

(log f)′′

(log f)′=

(−Υ)

(1 − v)2

1 − v

(−Υ)=

1

1 − v> 0 �

B.5 Counterfeiting Rate Formula: Proof of Theorem 5

Substitute from the quality optimality condition (5) into the expression (11):

κ(v) =qχ′(v)

ρ(v)·∆ + h

∆·f ′(v)Vq(e, q)x

cq(x, q)

Next, replaceVq(e, q) from Lemma 1(b), and use the passing elasticity (2), to get

κ = Υ(1 − s(v)) ·qχ(v)/v

qcq(x, q)/x·v

ρ(v)·∆ + h

∆

Since the cost elasticity expressionǫ = qcq(x, q)/c(x, q) > 1 by quality convexity:

κ = (1 − s(v))Υ

ǫ·v

ρ(v)·∆ + h

∆·qχ(v)/v

c(x, q)/x(28)

where the passing elasticity isΥ ∈ (1, 2). Now, (∆ + h)/∆≈1 for small hassle costs

h>0. The ratiov/ρ(v) ≈ 1 whenv is near the bank verification rateα. �

B.6 Existence of Equilibrium: Proof of Theorem 6

We now exploit the initial condition established in Lemma 3 —that the least possible

counterfeited note isθ > 0, and that as∆ falls to θ, the optimal effort level vanishes.

Finally, we can differentiateΠq usingqVqχ′ + χ = 0 and the quality FOC (5) to get

Πqq = (f ′′V 2q + f ′Vqq)x(∆ + h) − cqq = −

(

f ′′χ

f ′χ′+ 1

)

cq/q − cqq

Altogether, the quality elasticity (25) becomes

−(∆ + h)q′[∆]

q[∆]=

(

f ′′

f ′− f ′

f+ χ′

χ− χ′′

χ′

)

f

f ′

(

χ′

χ+ f ′′

f ′

)

χ

χ′+ q

cq

(

cqq − ψ2q/(x

2cxx))

(29)

43Since the passing fraction is explosively log-concave asv ↑ 1, log-concavity ofχ might fail eventhoughq′[∆]>0. The knife-edge verification function is the strictly log-convexχ(v)=ω/(1 − v)δ.

34

Finally, (24) implies thatx′[∆] = −(ψq/ψx)q′[∆], and thus

(∆ + h)x′[∆]

x[∆]=qψq

xψx

(

f ′′

f ′− f ′

f+ χ′

χ− χ′′

χ′

)

f

f ′

(

χ′

χ+ f ′′

f ′

)

χ

χ′+ q

cq

(

cqq − ψ2q/(x

2cxx))

(30)

In other words,x′[∆] ≷ 0 exactly whenψq ≶ 0. Thus, we have a solution(x0, q0, 0) of

(4)–(5) when∆ = θ, with x0, q0 > 0. Next, apply our differential equations (10), (25)

and (30) with this initial condition. Sincex′[∆], q′[∆], e′[∆] are everywhere finite, a

solution exists (by the Fundamental Theorem of Differential Equations). �

B.7 The Elasticity of the Passing Fraction: Proof of Lemma 5

Using (3), write the zero profit identity as:

(∆ + h)f(v[∆]) =c(x[∆], q[∆]) + L

x[∆]

Equate the elasticities in∆ of both sides, using (4). Lemma 4 and (10) then yield

E∆(f) +∆

∆ + h= ∆

(∆ + h)f ′(v)v′ + f

(∆ + h)f=

∆q′cqxcx

=qcqxcx

∆q′[∆]

q=

∆

∆ + h

E∆(q)

E∆(e)

Sincee ≡ qχ(v) impliesE∆(e) ≡ E∆(q) + E∆(χ) > 0, equation (16) follows. �

B.8 The Falling Seized-Passed Ratio: Proof of Theorem 8

We adapt the proof of Theorem 6. For simplicity of differentiation, we instead define

τ = 1/t, and imagine thatτ falls. We differentiate analogues of (4)–(5) inτ at τ = 1:

ψxx+ ψq q = −ψτ = −qψq (31)

Πqxx+ Πqq q = −Πqee− Πqτ = ΠqeΠτ/Πe − Πqτ

where we have used the fact thatΠee + Πτ = 0 — which also impliese < 0, so

verification effort rises with improved technology. Solving for x from these equations:

x = Πτψq

Πqe/Πe + [qΠqq − Πqτ ]/Πτ

Πqxψq − Πqqψx

(32)

35

We haveΠq = f ′(v)Vqx(∆ + h) − τcq, Πτ = −qcq < 0, Πqq = [f ′′(v)(Vq)2 +

f ′(v)(Vqq)]x(∆ + h) − τ 2cqq, andΠqτ = −cq − τqcqq. So atτ = 1:

qΠqq − Πqτ

Πτ

=q[f ′′(v)(Vq)

2 + f ′(v)(Vqq)]x(∆ + h) + cq−qcq

= −Vq

f ′′

f ′

sinceVqq = −Vq/q by Lemma 1 andcq = f ′xVq(∆ + h) by optimal quality (5).

For now, assumeψq > 0. Thenx > 0, since (a) ψq > 0 > Πτ , (b) the denominator

in (32) is non-negative given (26), and (c) the numerator is positive, for by (27):

Πqe/Πe − Vq

f ′′

f ′= Vq

(

f ′′

f ′−f ′

f+χ′

χ−χ′′

χ′

)

+ Vq

f ′

f− Vq

f ′′

f ′= Vq

(

χ′

χ−χ′′

χ′

)

< 0.

where the inequality owes to log-concave costsχ. Likewise,x < 0 whenψq < 0.

Next, q = −q − (ψx/ψq)x by (31), which is negative, sincex/ψq > 0. To wit, as

τ falls (better technology), quality rises, while quantity rises or falls asψq ≶ 0.

Verification moves just likeχ(v) = e/q, which changes according to the sign of

qe− eq = qΠτ

Πe

+ e[q + (ψx/ψq)x] = q

[

e+qcq

∆f ′xVq

]

+ e(ψx/ψq)x = e(ψx/ψq)x

The bracketed term vanishes, using the expressionsVq = −χ/qχ′ ande = qχ. This

shares the sign ofx/ψq > 0. So verification is worse with a better technology. �

B.9 Deducing Quality from the Data: Proof of Theorem 10

First,

E∆(p) = E∆(q) +1 − v

ΥvEv(χ

′)E∆(R) − 1 (33)

SinceE∆(χ) = Ev(χ)E∆(v), Lemma 5 and Lemma 6 imply:

E∆(f) = −∆

∆ + h

E∆(χ)

E∆(q) + E∆(χ)=

1−vv

E∆(f)Υ

Ev(χ)

E∆(q) − 1−vv

E∆(f)Υ

Ev(χ)

Solving this for the quality elasticity using (18) yields

E∆(q) =1 − v

v

Ev(χ)

Υ(1 + E∆(f)) =

1 − v

v

Ev(χ)

Υ(1 − E∆(R))

36

Substitute this into (33), and eliminate the unobserved verification intensityv, to get

E∆(p) =Ev(χ

′)

Ev(χ)

E∆(R)

1 − E∆(R)E∆(q)+ E∆(q)− 1 �

References

Bagnoli, Mark and Ted Bergstrom (2005), “Log-concave Probability and its Applica-tions”, Economic Theory, 26, 445-46.

Becker, Gary S. (1968), “Crime and Punishment: An Economic ApproachThe Journalof Political Economy, 76 (2), 169–217.

Burdett, Kenneth (1996), “Truncated Means and Variances”,Economics Letters, 52(3),263-267.

Chant, John F. (2004), “Canadian Experience with Counterfeiting”, Bank of CanadaReview, Summer, 41–54.

Dennis, Wayne Victor (2005), “The Counterfeit Millionaire: A True Crime AdventureIn The Counterfeiting of $15,000,000.00 U.S.”, internet monograph.

Diamond, Peter (1982), “Aggregate Demand Management in Search Equilibrium”,Journal of Political Economy,90, 881–94.

FRB (2003): Federal Reserve Board, “Lifespan of U.S. Currency and Analysis of anAlternate Substrate”, unpublished report, 2003.

Green, Edward and Warren Weber (1996), “Will the New $100 Bill Decrease Coun-terfeiting?”Federal Reserve Bank of Minneapolis Quarterly Review,20 (3), 3–10.

Judson, Ruth and Richard Porter (2003), “Estimating the Wordwide Volume of Coun-terfeit U.S. Currency: Data and Extrapolation”,Finance and Economics DiscussionSeries,2003-52, September.

Kersten, Jason, “The Art of Making Money: A Profile of Art Williams”,Rolling StoneMagazine, July 28, 2005.

Kiyotaki, Nobuhiro and Randall Wright (1989), “On Money as aMedium of Ex-change”,The Journal of Political Economy, 97 (4), 927–954.

Kultti, Klaus (1996), “A Monetary Economy with Counterfeiting”, Journal of Eco-nomics, 63 (2), 175–186.

Laband, David N., and John P. Sophocleus (1992), “An Estimate of Resource Expen-ditures on Transfer Activity in the United States”,Quarterly Journal of Economics,107 (3) 959–83.

Lengwiler, Yvan (1997), “A Model of Money Counterfeits”,Journal of EconomicsZeitschrift fur Nationalokonomie, 65 (2), 123–132.

37

Milgrom, Paul and John Roberts (1990), “Rationalizability, Learning and Equilibriumin Games with Strategic Complementarities,”Econometrica, 58, 1255–1277.

Nosal, Ed and Neil Wallace (2007), “A Model of (the Threat of)Counterfeiting”,TheJournal of Monetary Economics,54 (4), 994–1001.

Rogerson, Richard, Robert Shimer, and Randall Wright (2005), “Search-TheoreticModels of the Labor Market: A Survey,”Journal of Economic Literature,43 (4), 959–988.

Schmeidler, David (1973), “Equilibrium Points of Non-atomic Games”,Journal ofStatistical Physics, 7, 295–301.

Tullock, Gordon (1967), “The Welfare Costs of Tariffs, Monopolies and Theft”,West-ern Economic Journal, 5, 224–232.

U.S. Secret Service Investigative Activity, ASI: 97, microfiche 8464-1

United States Secret Service,Annual Statistical Report, micro fiche, 1964–1996.

USTD (2000):The Use and Counterfeiting of United States Currency Abroad, UnitedStates Treasury Department. January, 2000.

USTD (2003):The Use and Counterfeiting of United States Currency Abroad, Part 2,United States Treasury Department. March, 2003.

USTD (2006):The Use and Counterfeiting of United States Currency Abroad, Part 3,United States Treasury Department. September, 2006.

Williamson, Stephen (2002), “Private Money and Counterfeiting”, Federal ReserveBank of Richmond Economic Quarterly, 88 (3), 37–57.

Williamson, Stephen and Randall Wright (1994), “Barter andMonetary Exchange Un-der Private Information,”American Economic Review, 84, 104-123.

38