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Counterfeit Money∗ Elena Quercioli† Lones Smith‡ Economics Department Economics Department Tulane University University of Michigan August 16, 2007 Abstract 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. † elenaq@tulane.edu and www.tulane.edu/∼elenaq ‡ lones@umich.edu and www.umich.edu/∼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. 1 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. 1 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. 2 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. 2 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. 3 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 3 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. 4 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. 4 too, but they are not distinguishable from good guys. A counterfeiter can exchange multiple times each period, if he wishes. Each good guy chooses an effort level to examine notes that they are handed. We ignore discounting for the time between ac- quiring and spending a note is small. Some unknowingly acquire counterfeit currency and some do not. Below, we ﬂesh out the details of these two enmeshed games. 2.1 The Counterfeiter’s Endogenous Cost Function While counterfeiting is a dynamic process, we wish for modeling purposes to project it to a static optimization of a well-behaved increasing 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 < 5 The Secret Service estimates that the conviction rate for counterfeiting arrests close to 99%. 6 “If a counterfeiter goes out there and, you know, prints a million dollars, he’s going to get caught right away because when you ﬂood the market with that much fake currency, the Secret Service is 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]. 7 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. 5 f ≤ 1 — namely, the share of production that the counterfeiter passes, or equivalently the chance that any note passes. We ﬁnd that higher notes 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. 8 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. 6 2.2 Currency Veriﬁcation and Counterfeit Quality To explain passed counterfeit money in circulation, we introduce the next element of the story. If an innocent individual attempts to spend “hot” money, and it is caught, then it becomes worthless — since knowingly passing on counterfeit currency is ille- gal.9 We simply assume that this extra crime of “uttering” is not done. Faced with this prospect, individuals will choose to 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. 9 Title 18, Section 472 of the U.S. Criminal Code 10 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. 11 Since this may not always be deﬁned, we shall deﬁne V (e, q) = 1 if e > qχ(1). 7 2.3 The Passing Fraction and Veriﬁcation The passing fraction reconciles the entwined counterfeiting and veriﬁcation games. While police seizures are exogenous in our model, we wish to assume that vigilance by transactors may facilitate police seizures, by providing clues into ongoing counter- feit operations. To this end, we assume that police seize a 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. 12 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.” 8 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 ˆ their intensity v to minimize losses from counterfeit notes and veriﬁcation efforts: κ(1 − V (e, q))ρ(v)∆ + e (7) 9 A veriﬁer incurs a loss in the triple event that (i) he is handed a counterfeit note, (ii) his verifying efforts miss this fact, and (iii) the next transaction catches it. These inde- pendent events have respective 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) ˆ 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 ˆ 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 ˆ 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 ˆ 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. 13 See Milgrom and Roberts (1990). Diamond (1982) found such a structure in a search economy. 10 3.1 Quantity, Quality, and Effort in the Counterfeit Entry 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 14 Reﬂecting the dependence on ∆, let y[∆] be the equilibrium level of any variable y. 11 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. 15 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. 16 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). 12 This result offers some intuition into the nature of counterfeiting. Years long ago, when counterfeiting was the product of careful handicraft, higher quality entailed greater care — namely, a greater marginal cost, and so lower quantity. Recently, greater counterfeit quality has been achieved primarily via a better printing press. Thus, quality increases ﬁxed costs, which must be 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. 17 The Bureau of Printing and Engraving’s motto for the new currency is “Safer. Smarter. More Secure.” It asserts on moneyfactory.com that the new money is “harder to fake and easier to check”. 13 This result sheds light on the so-called “cat and mouse” nature of the real world competition between counterfeiters and governments. 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 18 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). 19 This expression ignores the costs of running law enforcement, but these are ﬁxed in our model. 20 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. 14 T (v) be the expected number of veriﬁcations of a circulating counterfeit note. Then the expected total verifying costs until a circulating counterfeit ∆ note is found are: qχ(v)T (v) = qχ(v)/ρ(v) ≤ κ(v)v∆ (13) 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, 21 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. 22 An excellent example occurred in Canada. As color was introduced on each denomination in the 1970s, the counterfeiting rate massively dropped off. 15 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. 23 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). 24 Games with a continuum of players have long been analyzed. See Schmeidler (1973). 16 4 The Economics of Crime: Seized Counterfeit Money Our model is fortunately testable, and admits expressions for the levels of seized and passed counterfeit money. We now explore implications of the counterfeit entry game. 4.1 The Passing Fraction Imagine for a moment a world with a ﬁxed quality level. Corollary 1 then implies a ﬁxed quantity too, and thus average costs that are invariant to the denomination. But then higher notes would have to pass less often to ensure zero proﬁts in (3). For instance, absent hassle costs (h = 0), as the denomination doubles from $5 to $10 or $10 to $20, the equilibrium passing fraction scales by one-half to balance greater counterfeiting revenues. The elasticity of the passing fraction in ∆ is then −1. Finally, with positive hassle 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) 25 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). 17 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 26 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. 18 10 Log (1+ Seized/Passed) ($100, 2.43) ($20, 1.93) ($50, 2.29) ($5, 1.38) ($10, 1.55) Log Denomination 1 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 27 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. 28 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. 19 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). Then denomination 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 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 20 8 7 6 5 4 3 2 1 0 '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. 30 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. 31 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. 21 62 in each of 1995 and 1996 (the most recent year for which we have data). Next, in the 1990s, came a digital counterfeiting technological revolution, using ink jet printers: No such plants were found through 1994. From 1995–2002, 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. 32 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. 33 Counterfeiting arrests grew from 1856 in 1995 to 3717 in 2005, plant suppressions from 153 to 611. See USTD (2006). 34 The legal seizure effectiveness might have fallen, notationally ruled out by our constant function f . 22 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 35 Judson and Porter (2003), eg., estimate that 3/4 of $100 notes, and 2/3 of $50 notes are abroad. 36 Lower denomination notes wear out faster, surely due to a higher velocity. Longevity estimates by the Federal Reserve Bank of NY [www.newyorkfed.org/aboutthefed/fedpoint/fed01.html] are 1.8, 1.3, 1.5, 2, 4.6, and 7.4 months, respectively, for $1,. . . ,$100. FRB (2003) has close longevity estimates. 37 Since notes age almost exclusively due to the wear and tear 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. 23 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, 38 The Secret Service does not have passed or seized levels by denomination before 1995. 24 100 ($100,82) ($10,71) Log (P/M) ($20,70) ($50,42) ($5,19) 10 ($1,1.5) Log Denomination 1 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: 1−v E∆ (v) = − E∆ (f ) (21) Υv Proof: This owes to the elasticity chain rule, E∆ (f ) = E1−v (f )Ev (1 − v)E∆ (v). A simple implication of Lemma 6 is that if the seized-passed ratio varies across denominations, then so must veriﬁcation. This precludes models in which veriﬁcation is not a choice variable! It cannot be stochastic but exogenous, as in any paper that presumes a ﬁxed authenticity signal — like Williamson (2002). Consider the economic factors behind the passed-circulation ratio seen in Figure 3. Low value notes are neither very proﬁtable to counterfeit, nor very worthy of attention. 25 It is not obvious which force wins this battle — and conversely so for high value notes. And yet we have seen that the passed-circulation ratio 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. 26 0.60 25% FRB share of Passed Notes 0.50 FRB Passed Ratio 20% 0.40 15% 0.30 10% 0.20 5% 0.10 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. 27 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 39 Our FOIA to the Secret Service asking for data on passed money in the banking sector was ignored. 40 The non-monotone structure is consistent across 1998, 2002, and 2005: it rises from $1, $5, $10, 28 the commercial banks, we eliminate the unobserved counterfeit rate using the passed- circulation ratio, and arrive at 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. 41 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. 29 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 30 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 42 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[∆]. 31 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 32 B.4 Quality Rises in the Denomination: Proof of Theorem 2 We have used the quantity FOC (4) to derive the law of motion for effort in Theorem 1. We are left to exploit the information in the quality FOC (5) and the constant producer surplus condition (6). Totally differentiating them in ∆ yields: 0 = 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 33 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 ) 2 χ f′ χ′ cq 43 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 . 34 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 ) 2 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[∆]) = x[∆] 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 35 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. 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