Counterfeit Money∗ by arw15539


									                               Counterfeit Money∗
                  Elena Quercioli†                            Lones Smith‡
             Economics Department                      Economics Department
                 Tulane University                     University of Michigan

                                       August 16, 2007


           This paper develops a new tractable positive theory of counterfeit money
       based on a variable intensity costly money verification. Counterfeiters compete
       against police and innocent verifiers, by choosing a quantity and quality of coun-
       terfeit notes to produce and pass. This induces a strategic complements “hot
       potato” game among “good guys” — who exchange currency pairwise, and wish
       to avoid counterfeit currency passed around. We deduce an equilibrium in this
       game, showing that as the stakes rise in the denomination, counterfeiters producer
       better quality, and verifiers respond with more effort. Resolving the struggle, we
       deduce from the data that money verification improves in the denomination.
           Our entwined counterfeiting and verifying games explain key time series and
       cross-sectional patterns of counterfeiting: (a) the ratio of the seized to passed
       counterfeit money rises in the denomination, but less than proportionately; (b)
       the vast majority of counterfeit money used to be seized before circulation, but
       now most passes into circulation; (c) the past money prevalence generally rises
       in denomination, with the least counterfeited notes the lowest; and (d) the share
       of passed money found by Federal Reserve Banks generally falls in the note, as
       does the ratio of their internal passed prevalence to the economy average.
           We also compute the social cost of counterfeiting, predict both the street price
       of 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 restrictive
functional 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), John
Mackenzie (counterfeit specialist at the Bank of Canada), and Lorelei Pagano (former Special Agent
with the Secret Service). We have also benefited from feedback at the presentations at I.G.I.E.R. at
Bocconi, the 2006 Bonn Matching Conference, the 2006 SED in Vancouver, the Workshop on Money
at the Federal Reserve Bank of Cleveland, Tulane, Michigan, and the Bank of Canada, and especially
the modeling insights of Pierre Duguay (Deputy Governor of the Bank of Canada) and Neil Wallace.
    † and∼elenaq
    ‡ and∼lones. Lones thanks 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 7–10 years. In addition, there
are tremendous costs borne by the public at large in checking currency.
    When we refer to counterfeit money, we have in mind two manifestations of it.
Seized notes are confiscated before they enter circulation. Passed notes 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 from the 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 from counterfeiters, and the
bad money that successfully passes into circulation, onto unwitting “good guys” in an
anonymous random matching exchange economy. The hot potato verification game
arises as a collateral battle mostly pitting “good guys” against one another.

    We formalize two key notions — the passing fraction of counterfeit goods into
circulation, and the inversely-related seized-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 between greater 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 probability densities 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 we find that the victor in
this struggle between greater quality and vigilance effort is 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, and average 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.
     See Becker (1968). For simplicity, our theory assumes that the criminal code is implemented as
written: Namely, the punishment for counterfeiting is the same, irrespective of the denomination.

We define the discovery rate as the chance that any circulating counterfeit money is
found to be passed with a bank or verifier. This translates the counterfeiting rate into
the observed passed-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-circulation ratio 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 the hot potato passing game.
For the least valuable notes are most poorly verified and caught.
   R ELATIONSHIP TO THE L 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 or conclusions. 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 with fixed 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 equilibrates the model — hence, it is
“general equilibrium”. We feel that a fixed value of notes is a good 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.
    We have heard of no systematic episodes of notes being discount domestically; however, outside
the USA, it is true that older $50 and $100 notes may be declined.

    For a key point of comparison, the papers cited above assume that one observes a
free signal of the money quality after acquiring it. We instead posit that individuals
verify when it can affect choice, namely when handed it. This is 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
     The literature follows Becker (1968), who derives the number of criminal offenses — analogous to
our counterfeiting rate — solely from criminal maximization. His model of crime largely ignores the
crime-fighting role of individuals in defense of their own property, analogous to our verification.
     Diamond (1982) developed a search-matching macroeconomics model that is supermodular in the
production costs. Our monetary model is supermodular in a pairwise effort choice. Diamond studies
multiple equilibria, while ours is nested with an entry game that forces a unique equilibrium.

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 and convex 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 an illegal good, which
may be seized prior 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 is L > 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 different amounts of money. At low
production, they may launder their money around a city, buying inexpensive goods
with larger denominations. At greater levels, they may sell the 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 the equilibrium passing fraction 0 <
      The Secret Service estimates that the conviction rate for counterfeiting arrests close to 99%.
      “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 going
to be all over you very quickly. They will find out where it’s coming from.” — interview with Jason
Kersten, author of Kersten (2005) [All Things Considered, July 23, 2005].
      Questions by Neil Wallace led us to tackle this problem. With this formulation, the results will
closely mirror those without any hassle costs. In this sense, this is a robustness test on our theory.

f ≤ 1 — namely, the share of production that the counterfeiter passes, or equivalently
the chance that any note passes. We find that higher notes pass less 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 fraction f . Intuitively, to pass a quantity x and quality q
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 the cost function inclusive of production, distribution, and legal costs, equals

                                     c(x, q) − hf x + L                                    (1)

We assume monotone, strictly convex, twice-differentiable, and non-negative costs —
so that c(x, q) ≥ hx + L and cx (0, q) > h for all x > 0. Average production costs
(c(x, q) − hf x + L)/x have a unique minimum, and explode in the quantity.
    Counterfeiters must earn sufficient producer surplus to 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 of ψ. Greater quality may lessen producer surplus ψq < 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 technology capital intensive. For instance, if the quality
level q 2 printing press costs an additional q, then c(x, q) = q 2 + x2 + h, whence
ψ(x, q) = x2 − q 2 − h, and thus ψq < 0. On the other hand, greater quality may
raise producer surplus ψq (x, q) > 0, as it largely entails increased attention to de-
tail, and thereby disproportionately lifts marginal rather than fixed costs. We call this
technology labor-intensive.8 For example, if greater quality scales up costs, as with
c(x, q) = q 2 x2 + h, then ψ(x, q) = q 2 x2 − h, and easily ψq > 0.
    We cannot yet specify the counterfeiter’s profit function, since his quality choice
affects the passing fraction f that he faces. We must first understand this feedback.
   This was common in earlier decades. Still, as long ago as 1776, printing presses (aboard the
H.M.S. Phoenix) were used by the British for counterfeiting Continental currency.

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 verify the authenticity of any
money before they accept it. Verification is 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 chance v ∈ [0, 1] that one correctly identifies a given note as
counterfeit. We assume genuine notes are never mistaken for counterfeit.
   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 same verification intensity.
A verification intensity v ∈ [0, 1] for a quality q > 0 note therefore costs effort e =
qχ(v). Observe that if quality vanished for a fixed e > 0, the induced verification
intensity would explode. This involves only a slight loss of generality insofar as quality
scales verification costs uniformly in v. It also gives a meaning to the units of quality,
and mandates our assumption of a strictly positive lower bound on quality, say q ≥ 1.
   For given q, costs are smooth, increasing and convex, so that χ′ (v), χ′′ (v) > 0 for
all v > 0.10 To ensure a positive optimal verification level, we assume that χ′ (0) = 0.
We also assume that the elasticity limv→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 quality q either. Hence, only effort e is a choice variable,
and so we introduce the verification function V (e, q) — the verification rate induced by
effort e for a note of quality q. Then v = V (e, q) is the inverse function to e = qχ(v).11

Lemma 1 (The Verification Function) Let v = V (e, q) and so e = qχ(v).
(a) Verification rises in effort, with slope Ve (e, q) = 1/qχ′ (v) > 0.
(b) Verification falls in quality, with slope Vq (e, q) = −χ(v)/qχ′ (v) < 0.
(c) Verification becomes perfect as quality vanishes: If e > 0, V (e, q) ↑ 1 as q ↓ 0.
(d) The verification second derivatives obey Vqq (e, q) > 0 and Vee (e, q) < 0.
     Title 18, Section 472 of the U.S. Criminal Code
     Weak convexity in this case is remarkably without loss of generality. For one can always secure
an (ex ante) verification chance of v at cost (χ(v − ε) + χ(v + ε))/2 instead by flipping fair coin, and
verifying at rates v − ε or v + ε. In other words, we must have χ(v) ≤ (χ(v − ε) + χ(v + ε))/2.
     Since this may not always be defined, we shall define V (e, q) = 1 if e > qχ(1).

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 fraction 0 < s(v) < 1 of
counterfeit money production. The passing fraction thereby reflects seizure and verifi-
cation via f (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). Since 1 − v > f (v), a “good guy”
successfully passes a counterfeit note more often than a “bad guy”.
    If seizures were a fixed fraction s of production, then a unit elasticity of f (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 that the passing elasticity is a fixed number:

                                               (1 − v)f ′ (v)
                          Υ ≡ E1−v (f ) = −                   ∈ [1, 2)                        (2)
                                                   f (v)

Lemma 2 (The Passing Fraction) The passing fraction f 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 intensity v =
V (e, q). Their expected revenues for quantity x and quality q of note ∆ are f (v)x∆,
while their costs are c(x, q) − f (v)xh + L. Hence, their profit function is:

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

Better quality simultaneously raises the passing fraction and the counterfeiters’ costs.
Maximizing profits here is somewhat nonstandard because the effort argument e of the
profit function is not a choice variable for the counterfeiters.
    On 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.”

   We first observe that provided there is any counterfeiting, the optimal quantity
and quality x, q are positive and finite. This follows from the earlier “cost proviso”
(in §2.1), and the facts that there are bounded returns to greater quality — indeed, the
passing fraction f 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 condition Π(x, q, e) = 0. If we first eliminate the
passing fraction f 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 that L is not so large that (6) has no solution. To generate profits to pay
for the legal costs L > 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 unknowns x, 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 and 1 − β, respectively. Banks have verifying machines or capable staff who can
better descry counterfeit money than individuals, but still imperfectly. Write their
verification intensity as α ∈ (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 the discovery rate ρ(v) = αβ + (1 − β)v > 0.
    Assume that a fraction κ of all ∆ notes tendered in transaction is counterfeit, with
an average verification rate v. As notes are spent upon acquisition, transactors choose
their intensity v to minimize losses from counterfeit notes and verification efforts:

                               κ(1 − V (e, q))ρ(v)∆ + e                              (7)

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 chances κ, 1 − V (e, q), and ρ(v). Fixing the quality q,
we may re-write their objective function (7) in terms of the induced verification rates:

                                           ˆ            v
                                     κ(1 − v)ρ(v)∆ + qχ(ˆ)                                      (8)

   This is a doubly supermodular game: One’s verification intensity v is a strategic
complement to the average verification intensity v, and to the counterfeiting rate κ. 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, the verification best response
function v is increasing in v and κ. 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 verification v = 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 rate V (e, q) = v, for quality q.
    The second order condition for minimizing (8) is met as costs χ 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, say v = v. Making this
substitution into the first order condition yields the equilibrium 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 positive v > 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.
       See Milgrom and Roberts (1990). Diamond (1982) found such a structure in a search economy.

3.1 Quantity, Quality, and Effort in the Counterfeit Entry Game
Without specifying the cost functions χ(v) and c(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 cost L > 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 note θ > 0, with no ∆ < θ counterfeited, and any ∆ > θ
counterfeited and verified (so x[∆] > 0 and e[∆] > 0). We also have x[θ] > 0, q[θ] = 1.
(b) The verification rate v[∆] and effort e[∆] are positive but vanishing as ∆ falls to θ,
while verification v[∆] 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 had v = 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 and verifier — namely,
∆ rises — it is instructive to see whether counterfeit quality and verification effort
rise too. Let’s first consider how the verification effort e[∆] evolves in the note ∆.
Intuitively, individuals should pay greater heed to higher denomination 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 for any counterfeiting technology.

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

                                              ∆e′ [∆]    ∆ xcx
                                  E∆ (e) =            =                                      (10)
                                               e[∆]     ∆ + h qcq
       Reflecting the dependence on ∆, let y[∆] be the equilibrium level of any variable y.

Proof: The zero-profit identity Π(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∆ + Πe e′ [∆] = 0, since Πq = Πx = 0 at the optimum. Intuitively,
profits marginally rise in denomination and fall in verifier effort, or Π∆ > 0 > Πe , so
that e′ [∆] > 0. The appendix rewrites e′ [∆] = −Π∆ /Πe as (10) using (3).
    As the denomination rises, the stakes in the counterfeiting game 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:
       V ERIFICATION C OST P ROVISO . Verification costs χ(v) are strictly log-concave.
       This assumption precludes verification costs more convex than exponential — for
instance, geometric costs χ(v) = λv r with r > 1 work.15

Theorem 2 (Quality) The counterfeit quality q 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 function χ, and local “near flats” of the decreasing function
f , where f (v) moves “much more” than v.16 Let the note ∆ rise a “little”. Then
verification effort e = 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, then v could rise “a lot”, and so
χ(v) could rise “a lot” too. Alternatively, if χ is not log-concave, then even if v only
rises “a little”, χ(v) could rise “a lot”. In either case, quality q = e/χ(v) could fall.

Corollary 1 (Quantity) The quantity x (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 quality q 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. If ψq < 0, then quantity must rise in the quality since producer
surplus ψ(x, q) = L is constant in ∆. Likewise, if ψq > 0, then quantity falls.
      Log-concavity is a standard assumption for probability densities (see Burdett (1996) and Bagnoli
and Bergstrom (2005)). Our application of it to cost functions like χ or the passing fraction f , is novel.
      Since log-concavity says χ(v + ε)χ(v − ε) ≤ χ(v)2 for all ε > 0, this precludes “steep rises” in
χ, where the ratio χ(v + ε)/χ(v) exceeds the previous ratio χ(v)/χ(v − ε) > 1. It also rules out “near
flats” in the decreasing function f , since f (v)/f (v − ε) < 1 provides an upper bound on f (v + ε)/f (v).

   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 amortized across larger print runs.

3.2 The Counterfeiting Rate from the Hot Potato Game
From the supermodular structure, the marginal benefit on the left side of (9) linearly
rises both in κ and in v. This yields an economic expression for the counterfeiting rate:

                           qχ′ (v)      marginal verification cost
                    κ(v) =         =                                                          (11)
                           ρ(v)∆     discovery rate × denomination

The right side vanishes at 0 and explodes at v = ∞, it is a quotient of two increasing
functions of v, and might therefore rise or fall in the verification rate: Marginal verifi-
cation gains rise linearly in v, while marginal verification costs rise in v 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 of changes in the entry
game to those that occur in the hot potato game. Change verification and production
cost functions to χγ (v) ≡ γχ(v) and cγ (x, q) ≡ c(x, γq). Scaling quality units to
ˆ                                      ˆ                                  ˆ
q = q/γ leaves costs unchanged: cγ (x, q) = c(x, γ(q/γ)) = c(x, q) and q χγ (v) =
(q/γ)γχ(v) = qχ(v). To wit, making a currency harder to counterfeit can be the same
as making it easier to verify. A new security feature that uniformly halves verification
costs is tantamount to one that raises production costs of quality q to that of 2q.17

Theorem 3 (Changing Counterfeiting or Verification Costs)
(a) The counterfeiting rate κ falls if the marginal verification cost function χ′ falls.
(b) The counterfeiting rate κ falls if production costs uniformly rise to c(x, γq), γ > 1.

Proof: The verification cost χ 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 function χ′ is then entirely met by an increased counterfeiting rate
κ = qχ′ (V (e, q))/[ρ(V (e, q))∆] in (11) — the only equation that κ satisfies.
   Next, a rise in the counterfeiting cost function c(x, q) to c(x, γq) is equivalent to
no change in c(x, q) and a smaller marginal verification cost χ′ /γ. Thus, κ falls.
    The Bureau of Printing and Engraving’s motto for the new currency is “Safer. Smarter. More
Secure.” It asserts on that the new money is “harder to fake and easier to check”.

   This result sheds light on the so-called “cat and mouse” nature of the real world
competition between counterfeiters and governments. When money 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 most κv∆.

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 to D 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 its random
nature — a key factor absent from Tullock’s analysis, and the subsequent rent-seeking
literature. Social costs are lower when individual prevention efforts v are greater.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 choose v to no verification, the loss-reduction
benefits of verifying exceed the verification costs in (8). So κ(v)vρ(v)∆ ≥ qχ(v). Let
      The effect of changing legal costs L on the model is ambiguous, and depends on the level of the
verification rate. In the interest of brevity, we omit pursuing this analysis (which parallels §B.4).
      This expression ignores the costs of running law enforcement, but these are fixed in our model.
      Laband 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.

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)

where T (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 rate s(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 rate v, for
definiteness. If the cost function χ 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 bound κv∆. 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 deduce κ < 1, we must remove from the κ expression
in (11) its dependence on the endogenous verification rate v. 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:

                                                       average verification costs
                  κ ≈ (1 − police seizure rate) ·                                                 (14)
                                                       average production costs

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 for κ < 1, and so for our equilibrium to exist
(below). Since average production costs and average verification costs rise in quality,
     If we asked this question for an ex ante counterfeit note, then the expected number of verifications
would be slightly greater, since we assume that counterfeiters do not try to pass their note in a bank.
     An excellent example occurred in Canada. As color was introduced on each denomination in the
1970s, the counterfeiting rate massively dropped off.

by convexity, and since χ(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 symmetric counterfeiting equilibrium is a 5-tuple (x, q, e, v, κ), where:23

  (a) Counterfeiters choose quantity x > 0 and quality q > 0 to maximize profits (3).
  (b) Given the verifiers’ effort e > 0, counterfeiters earn no profits Π(x, q, e) = 0.
  (c) The verification intensity v ∈ (0, 1) satisfies v = V (e, q).
  (d) The verifier’s effort e = qχ(v) solves the optimization (7) for the quality q, the
       verification intensity v, and the counterfeiting rate κ ∈ (0, 1).

    This is a dynamic Bayesian game,24 as the counterfeit quality q and verification
rate v are unobserved, and beliefs about these quantities matter. But deviations are
unobserved, and there is a unique q and v 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 note θ is unique, and x[θ], q[θ] > 0 = e[θ]. This solution is the
initial conditions for the dynamical system for x′ [∆], q ′ [∆], e′ [∆]. If it is suitably well-
behaved on (θ, ∞), then we have a solution x[∆], q[∆], e[∆] to (4)–(5), and Π = 0.
The verification rate is then v[∆] = V (e[∆], q[∆]), for the known function V .
   Note that the hot-potato game solely yields the counterfeit rate κ[∆] from (11) in
terms of the other equilibrium quantities — all of which come from the counterfeiting
game. Even the verification effort is determined within the counterfeiting game.
      The specification of equilibrium could also include the number of note ∆ counterfeiters. But since
this supply is infinitely elastic, and is easily computed to be κ[∆]M [∆]/x[∆], we have omitted it. Our
model can make predictions about these numbers, but the data is poor (especially by denomination).
      Games with a continuum of players have long been analyzed. See Schmeidler (1973).

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 costs h > 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 only if quality does.

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

        d     c(x, q) + L         x(cx x′ + cq q ′ ) − (c(x, q) + L)x′   cq (x, q) ′
                                =                                      =          q [∆]
       d∆          x                                 x2                      x

    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 deduce v ′ [∆] > 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 elasticity E∆ (f ) lies in (−1, 0), and equals:25

                                              ∆        E∆ (χ)
                              E∆ (f ) = −         ·                                                  (16)
                                             ∆ + h E∆ (χ) + E∆ (q)
     If Ex (g) is the elasticity of g in x, then the product and chain rules of calculus yield Ex (g · h) =
Ex (g)+Ex (h), Ex (g/h) = Ex (g)−Ex (h), Ex (f ◦g) = Eg (f )Ex (g), and Ey (g −1 ) = 1/Ex (g) if y = g(x).

4.2 The Seized-Passed Ratio Across Denominations
We have just produced an expression for the passing fraction elasticity stemming from
its micro foundation in the counterfeit entry game. We now formulate another ex-
pression, based on observables. Counterfeit money is eventually either seized from
the criminals by law enforcement or the first verifiers, or successfully passed onto the
public, and later lost by an unwitting individual. Call these levels S[∆] and P [∆] —
recalling that we have assumed for simplicity that all aggregates are in steady-state.
   The values S[∆] 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 the seized-passed ratio S[∆]/P [∆] is apparent, since

                                      1      S[∆]
                                         =1+       ≡ R[∆]                                        (17)
                                   f [∆]     P [∆]

The seized-passed ratio R[∆] 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 effort e ≡ qχ(v) rises proportionately more than
quality q. Otherwise, improved quality overwhelms the greater effort, and depresses
the verification rate v[∆] ≡ 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
    This is an overestimate, because some money might be seized before any passing attempt, perhaps
found in the counterfeiter’s possession or after he is followed back to his plant. Hence, to make sense
of our data application below, we assume that this overestimate does not vary in the denomination.


               Log (1+ Seized/Passed)
                                                                                ($100, 2.43)
                                                             ($20, 1.93)
                                                                            ($50, 2.29)
                                            ($5, 1.38)
                                                         ($10, 1.55)

                                                                       Log Denomination
                                        1                              10                      100

Figure 1: USA Seized Over Passed, Across Denominations. These are the seized-
passed ratios, averaged over 1995–2005, for non-Colombian counterfeits in the USA.
Noticeably, they rise in ∆. The sample includes almost ten million passed notes, and
about 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 of R[∆]) 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 of better quality.
Intuitively, production costs are greater, and so the rise in R[∆] is less than proportional
      This 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 command
a separate category in the Secret Service accounting. But the seizures for Colombian counterfeits are
mostly in Colombia, while our data on passed notes is domestic. The Secret Service has only given us
data on Columbian counterfeits (family C-8094) passed and seized domestically by denomination, as
well as an aggregate across all notes, including both foreign and domestic passed and seized. Since the
vast majority of seizures are foreign, either in Columbia or en route to the USA, we have used these
aggregate numbers year by year to scale each denomination’s passed and seized ratio.
      For 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 was
discontinued in 2000 to counter money laundering and organized crime.

Table 1: Fraction of Notes Digitally Produced, 1995–2004. This Secret Service data
encompasses 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[∆]   cx (x[∆], q[∆])            1
                                       =                 = f [∆] =                             (19)
                        ∆+h                   ∆+h                  R[∆]

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).
                street price < average cost ≈
                                                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
    We thank Pierre Duguay for this nice observation. We do have evidence from one recent case: A
Mexican counterfeiting ring discovered this year sold counterfeit $100 notes at 18% of face value to









     '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 dollars
of circulation across all denominations. The dashed line represents seizures, and the
solid line passed money. From 1970–85, the vast majority of counterfeit money (about
90%) was seized. The reverse holds (about 20%) for 2000–2004. Two down-spikes in
1986 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 have transpired in the industry.
    First, in the 1980’s, photocopiers became a tool of choice by counterfeiters. 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. The
money was transported across the border by women couriers, carrying the money on their person.
      The 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, counterfeiting
discoveries, and is also perforce contemporaneous counterfeit money. By contrast, passed money is
twice averaged: It has been found by thousands of individuals, and may have long been circulating.
      This paragraph is based on the Annual Reports of the USSS until 1996, and thereafter, Table 6.8 in
USTD (2003). This claim is consistent with Chant’s (2004) finding of a digital revolution in the 1990’s.

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, they grew from 19% to
95% of all plant seizures. This gives us two technological revolutions in counterfeit-
ing: photocopying around 1985, and then digital production around 1996. As seen in
Table 1, the digital trend continued past 1996, as digitally produced 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 been quantity-neutral technological progress of level
t > 1 if for any quantity x, the cost of any quality level q falls from c(x, q) to c(x, q/t).
                                                                    ˙     ˙
Let us denote the slopes of the quantity and quality in t by x and q. Appendix B.8
            ˙       ˙                   ˙       ˙
proves that x < 0 < q − q if ψq > 0 and x > 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 costs L > 0 were amortized over
a smaller production level.33 The seized-passed ratio accordingly fell.34
    With the passing elasticity 1 ≤ Υ < 2, the unobserved verification rate fell too.
The passing fraction rose roughly from 10% to 80%, and thus 1 − 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 fraction p[∆] 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 averaged 1.49, 19.11, 70.93, 70.27, 42.38, 81.67, respectively.
      This 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.
      Counterfeiting arrests grew from 1856 in 1995 to 3717 in 2005, plant suppressions from 153 to 611.
See USTD (2006).
      The legal seizure effectiveness might have fallen, notationally ruled out by our constant function f .

    We used two different measures of stability of the roller coaster shape in Figure 3.
First, with a pooled t-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, and t = 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 random expected number 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 value M[∆] > 0. Passed
money is circulating counterfeit money κ[∆]M[∆] times the discovery rate ρ[∆]. Let’s
define the passed-circulation ratio p[∆] ≡ P [∆]/M[∆]. This is the fraction of all
      Judson and Porter (2003), eg., estimate that 3/4 of $100 notes, and 2/3 of $50 notes are abroad.
       Lower denomination notes wear out faster, surely due to a higher velocity. Longevity estimates by
the Federal Reserve Bank of NY [] are 1.8, 1.3,
1.5, 2, 4.6, and 7.4 months, respectively, for $1,. . . ,$100. FRB (2003) has close longevity estimates.
       Since notes age almost exclusively due to the wear and tear of transactions, one might argue that a
note’s longevity should be inversely proportional to its velocity. Assume so. If we fix the passed rate of
the $20 note, and scale the passed rates of all other denominations by the relative longevity estimates in
footnote 36, then the implied passed rates are strictly rising in the note: 1.4, 12.4, 53.3, 70, 97, and 303.

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

                                q[∆]χ′ (v[∆])   marginal verification cost
                       p[∆] =                 =                                                (20)
                                     ∆               denomination

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 that it 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 10,000 annually. Suppose the $100 note transacts at least four times per
year. Then p[∆] 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 functions c and χ, 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 in χ′ (v[∆]). Altogether, endogenous quality buffers the passed
rates against changes in the real value of the currency, or the counterfeiting technology.
    The counterfeiting rate κ is unobserved, and the passed-circulation ratio p is its
observable manifestation. In fact, the Secret Service and the Federal Reserve often
draw inferences about κ from p. But p = κ · ρ is at best an imperfect proxy for κ.
If the discovery rate ρ 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 up p, but not κ.
    For a different insight linking the counterfeit rate κ and passed-circulation ratio p,
note how κ explicitly depends in (11) on the banking verification rate α and banking
chance β, while p in (20) does not. So if banks more effectively verify than the public,
       The Secret Service does not have passed or seized levels by denomination before 1995.

                         100                                                    ($100,82)

                           Log (P/M)


                                       ($1,1.5)                    Log Denomination
Figure 3: USA Passed Over Circulation, Across Denominations. This graph gives
the 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 scaled
by 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 velocity
falls in the note, the passed-circulation ratio is increasingly understated at higher notes.

then κ falls in α and β, while p does not. Ceteris paribus, while better verified markets
have less counterfeit money, it is found at a faster rate with greater α or β. On balance,
these effects exactly cancel, and neutrality obtains: 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. Since v[θ] =
0 by Lemma 3, the marginal verification cost χ′ (v) begins at zero when ∆ = θ, while
the denominator starts at θ > 0. Increments in the numerator initially proportionately
swamp the denominator, while the denominator ∆ dominates at large ∆.

Lemma 6 The verification elasticity E∆ (v) explodes as the verification rate vanishes,
and in fact:
                           E∆ (v) = −         E∆ (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.

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 rises monotonically.
    Intuitively, near the lowest counterfeited denomination θ > 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. We see here the necessity of
endogenous quality in explaining the continuing rise in the passed 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 rate v, while
counterfeiters’ profits depend on the passing fraction, and thus 1 − v.

Theorem 9 (Passed Money) Assume E∆ (R) ≫ 0. As ∆ tends down to θ, the passed-
circulation ratio elasticity explodes and so the passed-circulation ratio p[∆] vanishes.

Proof : By the product and quotient rules, the passed-circulation ratio (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,
since v[∆] ↓ 0 as ∆ ↓ θ by Lemma 3, and E∆ (f ) = −E∆ (R) by Theorem 7.
    To check the premise of Theorem 9 that E∆ (R) ≫ 0, we can see in Figure 1 that
the elasticity E∆ (R) is about 1/5 over the range of notes $5 to $100.
    We now express the unobserved quality elasticity in terms of the elasticities of the
seized-passed ratio and passed money, both observed. This result is the analogue of
our deduction in Theorem 7 of the seized-passed ratio elasticity.

Theorem 10 The quality elasticity obeys

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

   This theorem reveals that if quality were fixed, so that E∆ (q) = 0, then E∆ (p) =
−1. In other words, p[∆] is proportional to 1/∆. In fact, the data suggest that the
quality elasticity is positive — since the least value of E∆ (p) + 1 is 0.44 in Figure 3.

                              0.60                                    25%

                                                                             FRB share of Passed Notes
           FRB Passed Ratio                                           20%



                              0.00                                    0%
                                     $1   $5   $10   $20   $50 $100
Figure 4: Counterfeits Detected at Federal Reserve Banks. The “FRB share” (bar
graph) is the percentage of all passed notes of that denomination found by Federal
Reserve 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 by
the velocity-adjusted passed-circulation ratio p[∆] for that denomination. (The solid
line is 1998, and the dashed line is 2002.) The FRB numbers are found in Table 6.1 in
USTD (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 previously not been
previously found by verifiers. We use this idea to explain how much 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 money they 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 counterfeit notes presumably should
have been easy for the public to catch. Our model can explain this anomaly.

    First, let us consider commercial banks. On this front, the data is nonexistent,
but the theoretical prediction is clear. A bank finds a passed note 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:

                   passed notes hitting bank         κ(1 − v)βα
         ξ[∆] =                              =                       ≈ κ(1 − v)α
                    total notes hitting bank   (1 − κ)β + κ(1 − v)βα

The approximation is accurate within κ ≪ 0.0001, or 0.01%. The counterfeiting rate κ
is unobserved, and in principle this bank passed rate could 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 normalize ξ by the observed passed-
circulation ratio p[∆] = ρ[∆]κ[∆], eliminating κ. This leads us to what we call the
commercial bank ratio:
                               ξ[∆]    (1 − v[∆])α
                               p[∆]        ρ[∆]
The measures how much the bank passed rate exceeds the overall passed rate. It should
be falling,39 for the verification and discovery rates ρ[∆] = 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 fraction φ[∆] ∈ (0, 1) of notes each period to an FRB, often
for destruction. We know that φ[∆] falls in ∆, since the note longevity estimates in
footnote 36 rise in ∆. Intuitively, higher denomination notes wear out less often.
    The FRB passed rate ζ[∆] 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
(chance β) which then transfers it to an FRB (chance φ). If that note is counterfeit
(chance κ), then both verifier and bank must miss this fact (chance (1 − v)(1 − α)).

          passed notes hitting FRB         κ(1 − v)β(1 − α)φ
ζ[∆] =                             =                               ≈ κ(1−v)(1−α)
           total notes hitting FRB   (1 − κ)βφ + κ(1 − v)β(1 − α)φ

The approximations are likewise accurate within κ ≈ 0.0001. We know from available
data that the FRB passed rate is highly non-monotone.40 Proceeding as we did for
       Our FOIA to the Secret Service asking for data on passed money in the banking sector was ignored.
       The non-monotone structure is consistent across 1998, 2002, and 2005: it rises from $1, $5, $10,

the commercial banks, we eliminate the unobserved counterfeit rate using the passed-
circulation ratio, and arrive at the FRB ratio:

                                    ζ[∆]   (1 − v[∆])(1 − α)
                                         =                                                            (22)
                                    p[∆]          ρ[∆]

Theoretically, this ratio should be monotonically falling in 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 the commercial bank share µ[∆] 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:

                            passed notes found by commercial banks
       µ[∆] =
                  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 to φ. The share µ[∆] should then be falling in ∆.
    Likewise, the analogous FRB share σ[∆] should be monotonically decreasing too:

                                          κ(1 − v)β(1 − α)φ
                       µ[∆] =
                                 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.
      Once 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 incentives
of individuals in our model. Thus, our model might understate the quality rise at the highest notes.

security features. If this is true, then the bank verification rate α 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 would force the seized-passed
rates to rise linearly with the denomination, and would lead the passed-circulation ratio
to eventually fall. This mandates our second innovation — variable quality counterfeit
production. When quality modeled means higher verification costs, 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 and Sophocleus (1992), efforts
by “good guys” are a significant portion of the social costs of crime in the USA.
    We show how more valuable counterfeit goods simultaneously elicit 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 of the 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 “street price” of counterfeit notes.
   Monetary Theory. As a contribution to monetary economics, this paper explores

a supermodular game that arises in the currency verification efforts. 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 federal reserve banks.
    Our paper also sheds light on the development of non-commodity fiat currency —
i.e. whose face value greatly exceeds its intrinsic cost. We show 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 rate κ, since there is a unique optimal
verification effort for each counterfeiting rate. This rate is a free variable, given the
counterfeiters’ free entry condition. This is analogous to the 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) to maintain equilibrium in
    If we knew the annual “velocity” (transaction uses) n[∆] of a denomination ∆, then we could
provide lower bounds on the stocks of circulating counterfeit ∆ notes. For (1−f )(stock) = P [∆]/n[∆],
namely the per period amount of passed money. To estimate f properly, one needs to know how much
money is seized in the passing attempt. We can say that the stock is at least P [∆](1+P [∆]/S[∆])/n[∆].

the hot potato game. This richer model would thus demand that both games be solved
simultaneously. The gains from this exercise do not justify the substantial costs.

B Appendix: Omitted Analysis and Proofs

B.1 Deriving the Verification Function: Proof of Lemma 1
The first claim follows from χ(v) > 0 for v > 0. The q-derivative of qχ(V (e, q))) ≡ e
asserts qχ′ (v)Vq + χ(v) = 0, and thus Vq = −χ(v)/qχ′ (v), as needed. Next, the
v-derivative of V (qχ(v), q) ≡ v produces Ve (qχ(v), q)qχ′ (v) = 1. Since χ(0) =
χ′ (0) = 0 and χ′ (1) = ∞, if we take limits as v vanishes and explodes, we get
Ve (0, q) = ∞ and Ve (∞, q) = 0, for any q > 0.

B.2 The Least Counterfeited Note: Proof of Lemma 3
P ROOF OF PART (a): Let θ+h be the minimum of (c(x, q)+L)/(xf (0)) over x, q ≥ 0.
By assumption, this is realized at a finite and positive x0 , q0 . Then θ > 0 because
average costs equal marginal costs, which exceed cx > h at positive x, given L > 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 with x0 , q0 is strictly profitable, which is not possible.
P ROOF 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 therefore v[∆] ↑ 1.
  By the same token, if the verification tended down to some v[θ] > 0 as we neared
∆ = θ, then mimicking this production quantity and quality for a slightly smaller θ − ε
note would yield positive profits since it would not be verified, by assumption.

B.3 Equilibrium Effort Elasticity: Proof of Theorem 1 Finished
We’ve shown
                                 e′ [∆]    Π∆
                                        =−     >0                                 (23)
                                 e[∆]      eΠe
Substitute from (3), change Ve to Vq with qVq + eVe ≡ 0, and then use (4) and (5):

              (∆ + h)e′ [∆]     f (V )x      f (V )x     xcx
                            =− ′          = ′          =     >0
                 e[∆]         ef (V )Ve x  qf (V )Vq x   qcq

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 = ψx x′ [∆] + ψq q ′ [∆]                    (24)
                    −Πqe e′ [∆] − Πq∆ = Πqx x′ [∆] + Πqq q ′ [∆]

Simplifying using (23), and then solving these two equations, yields

                                         ψx (Πqe Π∆ /Πe − Πq∆ )
                           q ′ [∆] = −                                              (25)
                                             Πqx ψq − ψx Πqq

Recalling Π ≡ f (V )x(∆ + h) − c − L, we get Πq = f ′ (V )Vq x(∆ + h) − cq . Hence,
the denominator of (25) is non-negative since the second order Hessian condition is
locally necessary for the optimization:

   0 ≤ Πxx Πqq − Π2 = −cxx Πqq − Πqx (cq /x − cqx ) = (Πqx ψq − ψx Πqq )/x
                  qx                                                                (26)

where we have simplified Πqx = cq /x − cqx = −ψq /x using the quality FOC (5).
   For insight into the numerator of (25), differentiate the identity qχ(V (e, q)) ≡ e in
q and then e, to get qVq χ′ + χ = 0 and then χ′ Ve + qχ′′ Ve Vq + qχ′ Veq = 0. These give

                           Veq            1   χ′′        χ′ χ′′
                                =−           + ′     =     − ′
                          Ve Vq          qVq  χ          χ  χ

The second numerator factor of (25) is then:

                Πqe                 f ′′ (V )Ve Vq + f ′ (V )Veq
           Π∆       − Πq∆ = f (V )x                              − xf ′ (V )Vq
                Πe                            f ′ (V )Ve
                                     f ′′ f ′         Veq
                          = f xVq        ′
                                           − +
                                     f        f      Ve Vq
                                        ′′     ′
                                     f       f       χ′ χ′′
                          = f xVq          − + − ′                                  (27)
                                     f′       f      χ     χ

Since Vq < 0, we have from (25) and (26) that q ′ [∆] > 0 precisely when

                    (log f )′′ (log χ)′′  f ′′ f ′ χ′ χ′′
                              −          ≡ ′ − + − ′ >0
                    (log f )′   (log χ)′  f    f   χ  χ

Now, (log χ)′ > 0 > (log χ)′′ as χ 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 ′   (log f )′′    (−Υ) 1 − v      1
                      −    =            =               =     >0
                  f′    f    (log f )′    (1 − v)2 (−Υ)   1−v

B.5 Counterfeiting Rate Formula: Proof of Theorem 5
Substitute from the quality optimality condition (5) into the expression (11):

                                      qχ′ (v) ∆ + h f ′ (v)Vq (e, q)x
                         κ(v) =              ·     ·
                                       ρ(v)    ∆         cq (x, q)

Next, replace Vq (e, q) from Lemma 1(b), and use the passing elasticity (2), to get

                                                   qχ(v)/v      v     ∆+h
                       κ = Υ(1 − s(v)) ·                      ·     ·
                                                  qcq (x, q)/x ρ(v)    ∆

Since the cost elasticity expression ǫ = qcq (x, q)/c(x, q) > 1 by quality convexity:

                                               Υ   v    ∆ + h qχ(v)/v
                      κ = (1 − s(v))             ·    ·      ·                                                (28)
                                               ǫ ρ(v)    ∆     c(x, q)/x

where the passing elasticity is Υ ∈ (1, 2). Now, (∆ + h)/∆ ≈ 1 for small hassle costs
h > 0. The ratio v/ρ(v) ≈ 1 when v is near the bank verification rate α.

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 is θ > 0, and that as ∆ falls to θ, the optimal effort level vanishes.
Finally, we can differentiate Πq using qVq χ′ + χ = 0 and the quality FOC (5) to get

                                                                                     f ′′ χ
           Πqq = (f ′′ Vq2 + f ′ Vqq )x(∆ + h) − cqq = −                                    + 1 cq /q − cqq
                                                                                     f ′ χ′

Altogether, the quality elasticity (25) becomes

                                                      f ′′        f′        χ′       χ′′   f
                      q ′ [∆]                         f′
                                                             −    f
                                                                       +    χ
                                                                                 −   χ′    f′
             −(∆ + h)         =                                                                               (29)
                      q[∆]            χ′
                                           +   f ′′          χ
                                                                  +    q
                                                                            cqq − ψq /(x2 cxx )
                                      χ        f′            χ′        cq

    Since the passing fraction is explosively log-concave as v ↑ 1, log-concavity of χ might fail even
though q ′ [∆] > 0. The knife-edge verification function is the strictly log-convex χ(v) = ω/(1 − v)δ .

   Finally, (24) implies that x′ [∆] = −(ψq /ψx )q ′ [∆], and thus

                                                       f ′′        f′        χ′       χ′′   f
              (∆ + h)x′ [∆]   qψq                      f′
                                                              −    f
                                                                        +    χ
                                                                                  −   χ′    f′
                            =                                                                      (30)
                 x[∆]         xψx      χ′
                                            +   f ′′          χ
                                                                   +    q
                                                                             cqq − ψq /(x2 cxx )
                                       χ        f′            χ′        cq

In other words, x′ [∆] ≷ 0 exactly when ψq ≶ 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. Since x′ [∆], 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:

                                                       c(x[∆], q[∆]) + L
                         (∆ + h)f (v[∆]) =

Equate the elasticities in ∆ of both sides, using (4). Lemma 4 and (10) then yield

              ∆     (∆ + h)f ′ (v)v ′ + f   ∆q ′ cq   qcq ∆q ′ [∆]    ∆ E∆ (q)
 E∆ (f ) +       =∆                       =         =              =
             ∆+h        (∆ + h)f             xcx      xcx q          ∆ + h E∆ (e)

Since e ≡ qχ(v) implies E∆ (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 that τ falls. We differentiate analogues of (4)–(5) in τ at τ = 1:

                       ˙      ˙
                    ψx x + ψq q = −ψτ = −qψq                                                       (31)
                      ˙       ˙        ˙
                  Πqx x + Πqq q = −Πqe e − Πqτ = Πqe Πτ /Πe − Πqτ

                                       ˙                                    ˙
where we have used the fact that Πe e + Πτ = 0 — which also implies e < 0, so
verification effort rises with improved technology. Solving for x from these equations:

                                     Πqe /Πe + [qΠqq − Πqτ ]/Πτ
                         x = Πτ ψq                                                                 (32)
                                           Πqx ψq − Πqq ψx

We have Πq = f ′ (v)Vq x(∆ + h) − τ cq , Πτ = −qcq < 0, Πqq = [f ′′ (v)(Vq )2 +
f ′ (v)(Vqq )]x(∆ + h) − τ 2 cqq , and Πqτ = −cq − τ qcqq . So at τ = 1:

         qΠqq − Πqτ             q[f ′′ (v)(Vq )2 + f ′ (v)(Vqq )]x(∆ + h) + cq      f ′′
                           =                                                   = −Vq ′
             Πτ                                        −qcq                         f

since Vqq = −Vq /q by Lemma 1 and cq = f ′ xVq (∆ + h) by optimal quality (5).
    For now, assume ψq > 0. Then x > 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):

               f ′′            f ′′ f ′ χ′ χ′′               f′     f ′′        χ′ χ′′
Πqe /Πe − Vq        = Vq           − + − ′            + Vq      − Vq ′ = Vq       − ′      < 0.
               f′              f′   f   χ  χ                 f      f           χ  χ

where the inequality owes to log-concave costs χ. Likewise, x < 0 when ψq < 0.
          ˙                  ˙                                 ˙
   Next, q = −q − (ψx /ψq )x by (31), which is negative, since x/ψq > 0. To wit, as
τ falls (better technology), quality rises, while quantity rises or falls as ψq ≶ 0.
    Verification moves just like χ(v) = e/q, which changes according to the sign of

                Πτ                              qcq
   ˙    ˙
 q e − eq = q                       ˙
                   + e[q + (ψx /ψq )x] = q e +                       ˙             ˙
                                                      Vq + e(ψx /ψq )x = e(ψx /ψq )x
                Πe                             ∆f ′ x

The bracketed term vanishes, using the expressions Vq = −χ/qχ′ and e = qχ. This
shares the sign of x/ψq > 0. So verification is worse with a better technology.

B.9 Deducing Quality from the Data: Proof of Theorem 10
                       ` ∆ (p) = E∆ (q) +
                       E                   Ev (χ′ )E∆ (R) − 1                              (33)
Since E∆ (χ) = Ev (χ)E∆ (v), Lemma 5 and Lemma 6 imply:

                                                     1−v          E∆ (f )
                         ∆         E∆ (χ)             v             Υ
                                                                          Ev (χ)
            E∆ (f ) = −                       =                    1−v E∆ (f )
                        ∆ + h E∆ (q) + E∆ (χ)   E∆ (q) −                       Ev (χ)
                                                                    v      Υ

Solving this for the quality elasticity using (18) yields

                       1 − v Ev (χ)                 1 − v Ev (χ)
            E∆ (q) =                (1 + E∆ (f )) =              (1 − E∆ (R))
                         v     Υ                      v     Υ

Substitute this into (33), and eliminate the unobserved verification intensity v, to get

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


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