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A model of (the threat of) counterfeiting
ARTICLE IN PRESS
Journal of Monetary Economics 54 (2007) 994–1001 www.elsevier.com/locate/jme
A model of (the threat of) counterfeiting$ Ed Nosala,, Neil Wallaceb a
Federal Reserve Bank of Cleveland, Research Department, P.O. Box 6387, Cleveland, OH 44101-1387, USA b Department of Economics, The Pennsylvania State University, 612 Kern Graduate Building, University Park, PA 16802-3306, USA Received 9 December 2005; received in revised form 20 February 2006; accepted 27 February 2006
Abstract A simple matching-model of money with the potential for counterfeiting is constructed. In contrast to the existing literature, lotteries are included. These provide scope for the operation of the intuitive criterion of Cho and Kreps. The application of that refinement is shown to imply that there is no equilibrium with counterfeiting. If the cost of producing counterfeits is low enough, then there is no monetary equilibrium. Otherwise, there is a monetary equilibrium without counterfeiting. In other words, the threat of counterfeiting can eliminate the monetary equilibrium. r 2006 Elsevier B.V. All rights reserved. JEL classification: E-40 Keywords: Counterfeiting; Matching model; Cho–Kreps refinement
1. Introduction Almost by definition, a model of counterfeiting has to include imperfect recognizability of the object being counterfeited. Most such models assume asymmetric information of the following sort: the owner of an object knows more about it than potential owners—and, in particular, more than the one potential owner in a pairwise meeting. Some models assume exogenous stocks of high- and low-quality objects (see Velde et al., 1999; Kulti, 1996). Others, which deal more directly with counterfeiting, allow agents in the model to choose $
We thank Guillaume Rocheteau for his comments and discussion.
Corresponding author. Tel.: +1 216 579 3050.
E-mail address: [email protected] (E. Nosal). 0304-3932/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmoneco.2006.02.006
ARTICLE IN PRESS E. Nosal, N. Wallace / Journal of Monetary Economics 54 (2007) 994–1001
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whether or not to produce the low-quality objects (see Williamson and Wright, 1994; Green and Weber, 1996; Williamson, 2002). The previous work on recognizability done in the context of matching models includes two additional assumptions: each agent is restricted to hold either zero or one unit of some object and trades are deterministic; that is, lotteries are not permitted.1 Those two assumptions have an important and previously unnoticed consequence. Although trade in those models is formulated as a signaling game in which holders of the high-quality object, the high types, would like to signal their type, the above models allow essentially no scope for such signaling. A high type who holds one unit of the object and cannot offer lotteries can deviate from a candidate pooling equilibrium, an equilibrium in which both types make the same offer to an uninformed potential owner, only by not trading. Although our model shares many features with those cited above, we permit lotteries and, therefore, small deviations from a candidate pooling equilibrium. (The small deviations by a high type involve offering the object with a lower probability than that in the candidate pooling equilibrium.) An implication is that there is no pooling equilibrium that satisfies a commonly applied refinement: the Intuitive Criterion of Cho and Kreps (1987). Consequently, in our model of counterfeiting there is no such equilibrium in which counterfeiting occurs. After presenting the model and the results, we discuss how we reconcile our model and its nonexistence implication with two related observations: instances of counterfeiting and public efforts to thwart counterfeiting.
2. The model We borrow most of the components of the model from existing work. The background matching environment is that in Shi (1995) and Trejos and Wright (1995), a setting with divisible and costly to-produce goods and money holdings restricted to be in the set f0; 1g. The informational features are taken from Williamson and Wright (1994), in which there is a recognizability problem regarding the quality of goods and none regarding money. Here, we apply their formulation of the recognizability problem to fiat money—to the problem of distinguishing genuine fiat money from counterfeits. And we assume that people, at a cost, can decide to produce counterfeits. Finally, in order to provide a motivation for people to try to avoid receiving counterfeits and to limit the state space, we assume that counterfeits last just one period.2 As in Shi (1995) and Trejos and Wright (1995), at each date there are N42 types of perishable goods and N specialization types of people. There is a unit measure of each specialization type. A type-n person, n 2 f1; 2; . . . ; Ng, consumes only good-n and produces only good-n þ 1 (modulo N). Each person maximizes expected discounted utility. Period utility for a type-n person who does not produce a counterfeit is uðxÞ y and is uðxÞ y c for a type-n person who does produce a counterfeit, where x is the amount of good-n consumed, y is the amount of good-n þ 1 produced, and c40 is the utility cost of 1 The first use of lotteries in matching models of money is in Berentsen et al. (2002). However, they do not deal with asymmetric information. 2 In many economies, currency of all but the lowest denominations is traded very few times before being deposited in the banking system (see Cramer, 1981). And in the banking system, counterfeits are almost always detected.
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producing a counterfeit. The function u is strictly increasing and unbounded, twice differentiable, and strictly concave, and uð0Þ ¼ 0, u0 ð0Þ ¼ 1 and u0 ð1Þ ¼ 0. The discount factor is b 2 ð0; 1Þ. We also assume that a person who counterfeits gives up the ability to produce the specialized good.3 People cannot commit to future actions and trading histories are private information. Those two assumptions and the absence of a double-coincidence in any pairwise meeting imply that trade must involve money. We assume that money is indivisible and that each person’s holding at any time is in the set f0; 1g. There are two kinds of money: genuine money and counterfeits. The economy is endowed with a fixed stock of genuine money, the amount m 2 ð0; 1Þ per specialization type. The sequence of actions within a period is as follows. A fraction m of each specialization type starts with genuine money. Those without genuine money decide whether or not to produce a counterfeit. Let y denote the fraction of the population who produce counterfeits so that people are split as follows: the fraction m have a unit of genuine money, the fraction y have a counterfeit, and the remainder, 1 m y, have neither. People are then randomly paired off. The only meetings that matter are trade meetings, single-coincidence meetings in which the potential consumer, the buyer, has a unit of money, genuine or counterfeit, and the potential producer, the seller, does not. As in Williamson and Wright (1994), in any trade meeting, the pair receives a signal regarding the ‘‘quality’’ of the money of the buyer: with probability f the signal reveals the type of money held by the buyer—either genuine or counterfeit—and with probability 1 f the signal is uninformative and leaves the seller with the posterior probability m=ðm þ yÞ that the money is genuine. After the signal is realized, the buyer makes a take-it-or-leave-it lottery offer to the seller. After meetings dissolve, any counterfeit money disintegrates. We study only allocations that are symmetric across specialization types. For such allocations, our economy has no potentially varying ‘‘state:’’ each date starts with a fraction m of each specialization type holding genuine money. Therefore, we also restrict our analysis to constant allocations. 3. Equilibrium allocations In order to simplify the notation, we begin by showing that there is no separating equilibrium. Lemma 1. There is no equilibrium in which y40 and holders of genuine money distinguish themselves in trade meetings from holders of counterfeits when the uninformative signal is realized. Proof. Suppose, by way of contradiction, that there is such an equilibrium. Then sellers do not produce in exchange for a counterfeit because, by assumption, counterfeits do not last beyond the current trade. However, separation implies that sellers produce for genuine money. But, then, a holder of a counterfeit gains by duplicating the strategy of the holder of genuine money, a contradiction. & Hence, the only candidates for equilibria are pooling allocations in which holders of counterfeits emulate the actions of holders of genuine money when the uninformative 3
As noted below, nothing important depends on that assumption.
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signal is realized. Therefore, it is sufficient to consider five quantities: the trade in meetings when the informative signal is realized, ðqI ; pI Þ (output and the probability that money is transferred, respectively), the trade in meetings when the uninformative signal is realized, ðqU ; pU Þ, and the fraction of people who produce counterfeits, y. Let v0 denote the discounted utility of beginning a date without money, let v1 denote that of beginning with genuine money, and let D bðv1 v0 Þ. Without loss of generality, we describe the buyer take-it-or-leave-it problems for holders of genuine money under the assumption that DX0 and under the assumption that there is no lottery over output, only over whether the buyer surrenders money. Problem 1. Choose ðqj ; pj Þ 2 Rþ ½0; 1 to maximize uðqj Þ pj D subject to qj þ pj lj DX0,
(1)
for j ¼ I; U with lI ¼ 1 and lU ¼ m=ðm þ yÞ: The objective in this problem is the buyer’s gain from trade. Because ðqj ; pj Þ ¼ ð0; 0Þ is feasible in this problem and implies a zero value for the objective, any solution to this problem implies a nonnegative gain from trade for the buyer. The constraint, (1), is that the seller receive a nonnegative gain from trade taking into account that when the uninformative signal is realized, j ¼ U, the seller’s posterior distribution is that the buyer’s money is genuine with probability m=ðm þ yÞ. Next we express v0 and v1 in terms of the trades and the choice of whether to counterfeit. We have, ~ ð1 bÞv0 ¼ maxfC; Cg,
(2)
where 1my ð1 fÞuðqU Þ, N is the flow pay-off to counterfeiting, and C ¼ c þ
m y C~ ¼ ½fðqI þ pI DÞ þ ð1 fÞðqU þ pU DÞ ð1 fÞqU , N N is the flow pay-off to not counterfeiting. And we have 1my ff½uðqI Þ pI DÞ þ ð1 fÞ½uðqU Þ pU Dg. N We can now define a monetary equilibrium. ð1 bÞv1 ¼
(3)
(4)
(5)
Definition 1. An allocation ðqI ; qU ; pI ; pU ; yÞ is a monetary equilibrium if it satisfies the following conditions: (i) maxfqI ; qU g 4 0; (ii) there exists ðv0 ; v1 Þ that satisfies (2)–(5) and is ~ ¼ CXC ~ such that ðqI ; pI Þ and ðqU ; pU Þ solve Problem 1 and maxfC; Cg and with equality if y40; and (iii) ðqU ; pU Þ is immune to individual deviations when out-of-equilibrium beliefs are consistent with the Cho–Kreps refinement. ~ ¼ C~ in (ii) is necessary in order that not everyone counterfeit. The condition maxfC; Cg If they do, then there are no sellers.4 Condition (iii) requires that there is no deviation that 4
If counterfeiters could produce output, then the above formulation would change as follows. The flow payoff ~ to a counterfeiter, denoted C 0 , would satisfy C 0 ¼ C þ C~ and the condition for counterfeting would be C 0 XC.
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is profitable for a holder of genuine money and that signals that the money is genuine because it is a deviation that is unprofitable for a holder of a counterfeit. Proposition 1. There is no equilibrium with counterfeiting. Proof. Assume, by contradiction that there is such an equilibrium. Then ðy; D; qU ; pU Þ440. Consider a deviation ðq; pÞ with qoqU . Any such deviation comes only from a holder of genuine money because a counterfeiter values only output. Profitability for a holder of genuine money implies uðqÞ þ ð1 pÞD4uðqU Þ þ ð1 pU ÞD, while a seller will surely accept ðq; pÞ if m D. q þ pD4 qU þ pU mþy
(6)
(7)
(Note that the seller places a probability equal to one that the deviant offer comes from a buyer with genuine money.) Let p ¼ pU m=ðm þ yÞ and q ¼ qU e, where e40 and is arbitrarily small. Then inequalities (6) and (7) are satisfied. & Notice that this proof only requires the single-crossing property—that a holder of counterfeit money value retaining money less than does a holder of genuine money. Given that there is no equilibrium with counterfeiting, we expect to find that there is a monetary equilibrium only for parameters that make counterfeiting unprofitable. To describe such parameters, we begin by setting out the solution to Problem 1. Lemma 2. Let q^ j ¼ lj D and let qj be the unique solution to u0 ðqj Þ ¼ 1=lj : The solution to Problem 1 is qj ¼ minfq^ j ; qj g and (1) at equality. Proof. Any solution to Problem 1 satisfies (1) at equality. It follows that if the constraint pj p1 is not binding, then the solution is qj ¼ qj : If it is binding, then q^ j oqj and the solution is qj ¼ q^ j and pj ¼ 1. & Next, let q^ be the unique positive solution to ^ uðqÞ Nð1 bÞ . ¼1þ bð1 mÞ q^
(8)
^ (This expression for q^ is the positive solution to (5) after setting y ¼ 0, qI ¼ qU ¼ q, ^ ðpI ; pU Þ ¼ ð1; 1Þ, and ðv0 ; v1 Þ ¼ ð0; q=bÞ.) Also, let q be the unique solution to u0 ðq Þ ¼ 1, ^ q g, and q¯ ¼ minfq; c 1m X uð¯qÞ . (9) A ¼ ðc; fÞ : 1f N Then we have: Proposition 2. The condition ðc; fÞ 2 A is necessary and sufficient for existence of a monetary equilibrium. (footnote continued) ~ (The counterfeiter who becomes a Moreover, nothing would preclude y ¼ 1 m, which would happen if C 0 4C. producer disposes of the counterfeit held.)
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^ qn g ¼ q, ^ then we propose p ¼ 1 and Proof. Sufficiency. We propose pI ¼ pU ¼ p: If minfq; ^ ^ q g ¼ q , then we propose v0 ¼ 0, ð1 bÞv1 ¼ ð1 mÞ= ðv0 ; v1 Þ ¼ ð0; q=bÞ: If minfq; N½uðq Þ q , and p given by (1) at equality. These imply C~ ¼ 0 and, by the definition of A, Cp0. It follows that conditions (i) and (ii) of Definition 1 are satisfied. Condition (iii) is vacuous because y ¼ 0. Necessity. Assume that ðc; fÞeA and that a monetary equilibrium exists. Either y ¼ 0 or y40: The latter is ruled out by Proposition 1. The former implies that the equilibrium is ~ which contradicts y ¼ 0. & that in the sufficiency part. But ðc; fÞeA implies C4C, Finally, to emphasize the role of the Cho–Kreps refinement, condition (iii) in Definition 1, we show that without that condition there can be equilibria with counterfeiting. We show this in two steps. First, we show the following. Lemma 3. For some parameters, there is an allocation with counterfeiting that satisfies conditions (i) and (ii) of Definition 1. ^ q —where q^ Proof. We show that the claim is satisfied if ðc; fÞeA and is near A, and if qo 0 ^ is defined in (8) and q , as above, is the unique solution to u ðq Þ ¼ 1. (The inequality qo q restricts the parameters other than ðc; fÞ and is obviously not vacuous.) In particular, we show that the claim is satisfied with pI ¼ pU ¼ 1. It follows from the bindingness of constraint (1) in Problem 1 that C~ ¼ 0. This implies that any equilibrium with counterfeiting satisfies C ¼ 0, v0 ¼ 0 and, hence, v1 ¼ D=b. With pI ¼ pU ¼ 1, C ¼ 0 is equivalent to c 1my m ¼ u D (10) 1f N mþy (see (3)), and (5) can be written as ð1 bÞ 1my m D¼ f½uðDÞ D þ ð1 fÞ u D D , b N mþy
(11)
or as Nð1 bÞ=b m D . 1þ D ¼ fuðDÞ þ ð1 fÞu 1my mþy
(12)
First, consider the pairs ðy; DÞ that satisfy (10). Because u is unbounded, strictly increasing, and uð0Þ ¼ 0, for each y 2 ½0; 1 mÞ, there is a unique D, denoted f ðyÞ , that ^ and f ðyÞ ! 1 satisfies (10). Moreover, f is differentiable, strictly increasing, f ð0Þ 2 ð0; qÞ, as y ! 1 m. Next, consider the pairs ðy; DÞ that satisfy (12) and D40. It follows from u0 ð0Þ ¼ 1 , u0 ð1Þ ¼ 0, and strict concavity of u that for each y 2 ½0; 1 mÞ, there is a unique positive D, denoted gðyÞ, that satisfies (12). Moreover, gð0Þ ¼ q^ (see the proof of Proposition 2), gðyÞ ! 0 as y ! 1 m, and g is decreasing. (The last conclusion uses the fact that at the point ðy; gðyÞÞ, the derivative w.r.t D of the left-hand side of (12) exceeds the derivative w.r.t D of the right-hand side of (12), a kind of second-order condition.) ^ DÞ, ^ that satisfies (10) and (11). Therefore, there exists a unique ðy; DÞ, denoted ðy; ^ ^ ^ and ^ It remains to confirm that ðpI ; qI Þ ¼ ð1; DÞ Moreover, y 2 ð0; 1 mÞ and D 2 ð0; qÞ. ^ ^ ^ ðpU ; qU Þ ¼ ð1; ðm=ðm þ yÞÞDÞ solve Problem 1. The former is true because Doqoq ¼ qI . ^ ^ The latter requires ðm=ðm þ yÞÞDpqU , where qU is the unique solution to ^ u0 ðqU Þ ¼ ðm þ yÞ=m. To guarantee this, we use ðc; fÞ near A. Let x c=ð1 fÞ. Because
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^ But, as x ! ðð1 mÞ=NÞuðqÞ, ^ f ð0Þ ! q. ^ That and the ðc; fÞeA, xoðð1 mÞ=NÞuðqÞ. ^ Therefore, for x sufficiently fact that f 0 ð0Þ40 imply that y^ ! 0 as x ! ðð1 mÞ=NÞuðqÞ. ^ qU is sufficiently close to q , which implies that ðm=ðmþ close to ðð1 mÞ=NÞuðqÞ, 5 ^ Dpq ^ yÞÞ & U. The second step is to confirm that the pU ; qU shown to exist in Lemma 3 is immune to individual deviations for some beliefs about the response to deviations. One such belief is that the seller believes with probability one that any deviator holds counterfeit money. And if pU ¼ 1, as is the case for the allocation shown to exist in the proof of the lemma, another such belief is that the seller believes that any deviator has genuine money with probability m=ðm þ yÞ. (Notice that in both cases, the seller’s belief does not depend on the nature of the deviation.) Thus, nonexistence of a counterfeiting equilibrium depends on imposing some restriction on out-of-equilibrium beliefs.6 Two features of our model permit the Cho–Kreps refinement to have bite: first, holders of counterfeits value output as least as much as do holders of genuine money and value retaining what they hold less than do holders of genuine money; second, holders of genuine money have a rich possible set of deviations. The refinement would not have bite in the absence of lotteries because, then, there would be no profitable deviation for a holder of genuine money that would be accepted by the seller. 4. Conclusion We have set out a simple model in which people can at a cost produce counterfeits. However, such production never happens in equilibrium. Our model has three extreme assumptions: the perishable nature of counterfeits, asset holdings in the set f0; 1g, and the assumption that buyers make take-it-or-leave-it offers. The assumption that counterfeits last only one period is used primarily to conclude that any equilibrium with counterfeiting has pooling. However, pooling could, instead, be taken to be a definition of trade involving counterfeits. (In the absence of pooling, sellers would be accepting counterfeits knowingly—as if counterfeits were simply another acceptable money—which does not conform to what we think of as trade involving counterfeits.) If it is, then, nonexistence of an equilibrium with counterfieitng depends only counterfeits being worth less than genuine money. The f0; 1g assumption plays a significant role because it gives rise to two types, holders of genuine money and holders of counterfeits. Those types allow us to easily apply the Cho–Kreps refinement. How do we reconcile counterfeiting in the actual economy with our nonexistence result? The model is about steady states in which the actual and perceived stocks of counterfeits are equal. Much of the counterfeiting we see in actual economies consists of sporadic episodes, each one of which is short-lived and region-specific. Hence, those do not have to 5 We suspect that the conclusion of Lemma 3 holds for all ðc; fÞeA, but the argument would be lengthy and the result is not important for the message of this paper. 6 This is very different from the no-countefeiting result in Williamson (2002, pp. 52 and 53). Williamson does not have lotteries and, therefore, does not consider refinements. He has as a condition for a steady state that people with investment opportunities have to be indifferent among three options: produce no asset, produce the lowquality asset, produce the high-quality asset. In our model, we require only indifference between the first two options.
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be interpreted as steady states. More to the point, counterfeiting is almost always specific to one or several denominations and some counterfeiting episodes are accompanied by the disappearance in trade of the denominations being counterfeited. We are tempted, therefore, to interpret such disappearance as the analogue of the nonexistence implied by the model.7 Finally, our nonexistence result is derived from a model that embodies a very specific mechanism for exchange; namely, buyers make take-it-or-leave-it offers. A model with a different mechanism may have an equilibrium with positive trade in the presence of counterfeit money. Finally, the model implies that counterfeiting can be a threat even if we do not observe it. According to the model, actions taken that increase the cost of producing counterfeits, c, and/or that increase the probability that counterfeits can be recognized, f, can be worthwhile. In particular, because the magnitude of ð1 mÞ=Nuð¯qÞ in (9) does not depend on c or f, increases in c or f enlarge the set of magnitudes of the other parameters for which a monetary equilibrium exists. Thus, the model is consistent with efforts to thwart counterfeiting. References Berentsen, A., Molico, M., Wright, R., 2002. Indivisibilities, lotteries, and monetary exchange. Journal of Economic Theory 107, 70–94. Cho, I.-K., Kreps, D., 1987. Signaling games and stable equilibria. Quarterly Journal of Economics 102, 179–221. Cramer, J.S., 1981. The work that money does: the transaction velocity of circulation in the Netherlands, 1950–1978. European Economic Review 15, 307–326. Green, E., Weber, W., 1996. Will the new $100 decrease counterfeiting? Federal Reserve Bank of Minneapolis Quarterly Review 20 (3), 3–10. Kulti, K., 1996. A monetary economy with counterfeiting. Journal of Economics 63, 175–186. Lee, M., Wallace, N., Zhu, T., 2005. Modeling denomination structures. Econometrica 73, 949–960. Shi, S., 1995. Money and prices: a model of search and bargaining. Journal of Economic Theory 67, 467–498. Trejos, A., Wright, R., 1995. Search, bargaining, money and prices. Journal of Political Economy 103, 118–141. Velde, F., Weber, W., Wright, R., 1999. A model of commodity money, with applications to Gresham’s law and the debasement puzzle. Review of Economic Dynamics 2, 291–323. Williamson, S., 2002. Private money and counterfeiting. Federal Reserve Bank of Richmond Economic Quarterly 88/3, 37–57. Williamson, S., Wright, R., 1994. Barter and monetary exchange under private information. American Economic Review 84, 104–123.
7 Of course, whether such disappearance is implied by a model with several denominations and with denomination-specific counterfeiting technologies remains to be determined. For a model of multiple denominations (without a recognizability problem), see Lee et al. (2005).
Journal of Monetary Economics 54 (2007) 994–1001 www.elsevier.com/locate/jme
A model of (the threat of) counterfeiting$ Ed Nosala,, Neil Wallaceb a
Federal Reserve Bank of Cleveland, Research Department, P.O. Box 6387, Cleveland, OH 44101-1387, USA b Department of Economics, The Pennsylvania State University, 612 Kern Graduate Building, University Park, PA 16802-3306, USA Received 9 December 2005; received in revised form 20 February 2006; accepted 27 February 2006
Abstract A simple matching-model of money with the potential for counterfeiting is constructed. In contrast to the existing literature, lotteries are included. These provide scope for the operation of the intuitive criterion of Cho and Kreps. The application of that refinement is shown to imply that there is no equilibrium with counterfeiting. If the cost of producing counterfeits is low enough, then there is no monetary equilibrium. Otherwise, there is a monetary equilibrium without counterfeiting. In other words, the threat of counterfeiting can eliminate the monetary equilibrium. r 2006 Elsevier B.V. All rights reserved. JEL classification: E-40 Keywords: Counterfeiting; Matching model; Cho–Kreps refinement
1. Introduction Almost by definition, a model of counterfeiting has to include imperfect recognizability of the object being counterfeited. Most such models assume asymmetric information of the following sort: the owner of an object knows more about it than potential owners—and, in particular, more than the one potential owner in a pairwise meeting. Some models assume exogenous stocks of high- and low-quality objects (see Velde et al., 1999; Kulti, 1996). Others, which deal more directly with counterfeiting, allow agents in the model to choose $
We thank Guillaume Rocheteau for his comments and discussion.
Corresponding author. Tel.: +1 216 579 3050.
E-mail address: [email protected] (E. Nosal). 0304-3932/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmoneco.2006.02.006
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whether or not to produce the low-quality objects (see Williamson and Wright, 1994; Green and Weber, 1996; Williamson, 2002). The previous work on recognizability done in the context of matching models includes two additional assumptions: each agent is restricted to hold either zero or one unit of some object and trades are deterministic; that is, lotteries are not permitted.1 Those two assumptions have an important and previously unnoticed consequence. Although trade in those models is formulated as a signaling game in which holders of the high-quality object, the high types, would like to signal their type, the above models allow essentially no scope for such signaling. A high type who holds one unit of the object and cannot offer lotteries can deviate from a candidate pooling equilibrium, an equilibrium in which both types make the same offer to an uninformed potential owner, only by not trading. Although our model shares many features with those cited above, we permit lotteries and, therefore, small deviations from a candidate pooling equilibrium. (The small deviations by a high type involve offering the object with a lower probability than that in the candidate pooling equilibrium.) An implication is that there is no pooling equilibrium that satisfies a commonly applied refinement: the Intuitive Criterion of Cho and Kreps (1987). Consequently, in our model of counterfeiting there is no such equilibrium in which counterfeiting occurs. After presenting the model and the results, we discuss how we reconcile our model and its nonexistence implication with two related observations: instances of counterfeiting and public efforts to thwart counterfeiting.
2. The model We borrow most of the components of the model from existing work. The background matching environment is that in Shi (1995) and Trejos and Wright (1995), a setting with divisible and costly to-produce goods and money holdings restricted to be in the set f0; 1g. The informational features are taken from Williamson and Wright (1994), in which there is a recognizability problem regarding the quality of goods and none regarding money. Here, we apply their formulation of the recognizability problem to fiat money—to the problem of distinguishing genuine fiat money from counterfeits. And we assume that people, at a cost, can decide to produce counterfeits. Finally, in order to provide a motivation for people to try to avoid receiving counterfeits and to limit the state space, we assume that counterfeits last just one period.2 As in Shi (1995) and Trejos and Wright (1995), at each date there are N42 types of perishable goods and N specialization types of people. There is a unit measure of each specialization type. A type-n person, n 2 f1; 2; . . . ; Ng, consumes only good-n and produces only good-n þ 1 (modulo N). Each person maximizes expected discounted utility. Period utility for a type-n person who does not produce a counterfeit is uðxÞ y and is uðxÞ y c for a type-n person who does produce a counterfeit, where x is the amount of good-n consumed, y is the amount of good-n þ 1 produced, and c40 is the utility cost of 1 The first use of lotteries in matching models of money is in Berentsen et al. (2002). However, they do not deal with asymmetric information. 2 In many economies, currency of all but the lowest denominations is traded very few times before being deposited in the banking system (see Cramer, 1981). And in the banking system, counterfeits are almost always detected.
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producing a counterfeit. The function u is strictly increasing and unbounded, twice differentiable, and strictly concave, and uð0Þ ¼ 0, u0 ð0Þ ¼ 1 and u0 ð1Þ ¼ 0. The discount factor is b 2 ð0; 1Þ. We also assume that a person who counterfeits gives up the ability to produce the specialized good.3 People cannot commit to future actions and trading histories are private information. Those two assumptions and the absence of a double-coincidence in any pairwise meeting imply that trade must involve money. We assume that money is indivisible and that each person’s holding at any time is in the set f0; 1g. There are two kinds of money: genuine money and counterfeits. The economy is endowed with a fixed stock of genuine money, the amount m 2 ð0; 1Þ per specialization type. The sequence of actions within a period is as follows. A fraction m of each specialization type starts with genuine money. Those without genuine money decide whether or not to produce a counterfeit. Let y denote the fraction of the population who produce counterfeits so that people are split as follows: the fraction m have a unit of genuine money, the fraction y have a counterfeit, and the remainder, 1 m y, have neither. People are then randomly paired off. The only meetings that matter are trade meetings, single-coincidence meetings in which the potential consumer, the buyer, has a unit of money, genuine or counterfeit, and the potential producer, the seller, does not. As in Williamson and Wright (1994), in any trade meeting, the pair receives a signal regarding the ‘‘quality’’ of the money of the buyer: with probability f the signal reveals the type of money held by the buyer—either genuine or counterfeit—and with probability 1 f the signal is uninformative and leaves the seller with the posterior probability m=ðm þ yÞ that the money is genuine. After the signal is realized, the buyer makes a take-it-or-leave-it lottery offer to the seller. After meetings dissolve, any counterfeit money disintegrates. We study only allocations that are symmetric across specialization types. For such allocations, our economy has no potentially varying ‘‘state:’’ each date starts with a fraction m of each specialization type holding genuine money. Therefore, we also restrict our analysis to constant allocations. 3. Equilibrium allocations In order to simplify the notation, we begin by showing that there is no separating equilibrium. Lemma 1. There is no equilibrium in which y40 and holders of genuine money distinguish themselves in trade meetings from holders of counterfeits when the uninformative signal is realized. Proof. Suppose, by way of contradiction, that there is such an equilibrium. Then sellers do not produce in exchange for a counterfeit because, by assumption, counterfeits do not last beyond the current trade. However, separation implies that sellers produce for genuine money. But, then, a holder of a counterfeit gains by duplicating the strategy of the holder of genuine money, a contradiction. & Hence, the only candidates for equilibria are pooling allocations in which holders of counterfeits emulate the actions of holders of genuine money when the uninformative 3
As noted below, nothing important depends on that assumption.
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signal is realized. Therefore, it is sufficient to consider five quantities: the trade in meetings when the informative signal is realized, ðqI ; pI Þ (output and the probability that money is transferred, respectively), the trade in meetings when the uninformative signal is realized, ðqU ; pU Þ, and the fraction of people who produce counterfeits, y. Let v0 denote the discounted utility of beginning a date without money, let v1 denote that of beginning with genuine money, and let D bðv1 v0 Þ. Without loss of generality, we describe the buyer take-it-or-leave-it problems for holders of genuine money under the assumption that DX0 and under the assumption that there is no lottery over output, only over whether the buyer surrenders money. Problem 1. Choose ðqj ; pj Þ 2 Rþ ½0; 1 to maximize uðqj Þ pj D subject to qj þ pj lj DX0,
(1)
for j ¼ I; U with lI ¼ 1 and lU ¼ m=ðm þ yÞ: The objective in this problem is the buyer’s gain from trade. Because ðqj ; pj Þ ¼ ð0; 0Þ is feasible in this problem and implies a zero value for the objective, any solution to this problem implies a nonnegative gain from trade for the buyer. The constraint, (1), is that the seller receive a nonnegative gain from trade taking into account that when the uninformative signal is realized, j ¼ U, the seller’s posterior distribution is that the buyer’s money is genuine with probability m=ðm þ yÞ. Next we express v0 and v1 in terms of the trades and the choice of whether to counterfeit. We have, ~ ð1 bÞv0 ¼ maxfC; Cg,
(2)
where 1my ð1 fÞuðqU Þ, N is the flow pay-off to counterfeiting, and C ¼ c þ
m y C~ ¼ ½fðqI þ pI DÞ þ ð1 fÞðqU þ pU DÞ ð1 fÞqU , N N is the flow pay-off to not counterfeiting. And we have 1my ff½uðqI Þ pI DÞ þ ð1 fÞ½uðqU Þ pU Dg. N We can now define a monetary equilibrium. ð1 bÞv1 ¼
(3)
(4)
(5)
Definition 1. An allocation ðqI ; qU ; pI ; pU ; yÞ is a monetary equilibrium if it satisfies the following conditions: (i) maxfqI ; qU g 4 0; (ii) there exists ðv0 ; v1 Þ that satisfies (2)–(5) and is ~ ¼ CXC ~ such that ðqI ; pI Þ and ðqU ; pU Þ solve Problem 1 and maxfC; Cg and with equality if y40; and (iii) ðqU ; pU Þ is immune to individual deviations when out-of-equilibrium beliefs are consistent with the Cho–Kreps refinement. ~ ¼ C~ in (ii) is necessary in order that not everyone counterfeit. The condition maxfC; Cg If they do, then there are no sellers.4 Condition (iii) requires that there is no deviation that 4
If counterfeiters could produce output, then the above formulation would change as follows. The flow payoff ~ to a counterfeiter, denoted C 0 , would satisfy C 0 ¼ C þ C~ and the condition for counterfeting would be C 0 XC.
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is profitable for a holder of genuine money and that signals that the money is genuine because it is a deviation that is unprofitable for a holder of a counterfeit. Proposition 1. There is no equilibrium with counterfeiting. Proof. Assume, by contradiction that there is such an equilibrium. Then ðy; D; qU ; pU Þ440. Consider a deviation ðq; pÞ with qoqU . Any such deviation comes only from a holder of genuine money because a counterfeiter values only output. Profitability for a holder of genuine money implies uðqÞ þ ð1 pÞD4uðqU Þ þ ð1 pU ÞD, while a seller will surely accept ðq; pÞ if m D. q þ pD4 qU þ pU mþy
(6)
(7)
(Note that the seller places a probability equal to one that the deviant offer comes from a buyer with genuine money.) Let p ¼ pU m=ðm þ yÞ and q ¼ qU e, where e40 and is arbitrarily small. Then inequalities (6) and (7) are satisfied. & Notice that this proof only requires the single-crossing property—that a holder of counterfeit money value retaining money less than does a holder of genuine money. Given that there is no equilibrium with counterfeiting, we expect to find that there is a monetary equilibrium only for parameters that make counterfeiting unprofitable. To describe such parameters, we begin by setting out the solution to Problem 1. Lemma 2. Let q^ j ¼ lj D and let qj be the unique solution to u0 ðqj Þ ¼ 1=lj : The solution to Problem 1 is qj ¼ minfq^ j ; qj g and (1) at equality. Proof. Any solution to Problem 1 satisfies (1) at equality. It follows that if the constraint pj p1 is not binding, then the solution is qj ¼ qj : If it is binding, then q^ j oqj and the solution is qj ¼ q^ j and pj ¼ 1. & Next, let q^ be the unique positive solution to ^ uðqÞ Nð1 bÞ . ¼1þ bð1 mÞ q^
(8)
^ (This expression for q^ is the positive solution to (5) after setting y ¼ 0, qI ¼ qU ¼ q, ^ ðpI ; pU Þ ¼ ð1; 1Þ, and ðv0 ; v1 Þ ¼ ð0; q=bÞ.) Also, let q be the unique solution to u0 ðq Þ ¼ 1, ^ q g, and q¯ ¼ minfq; c 1m X uð¯qÞ . (9) A ¼ ðc; fÞ : 1f N Then we have: Proposition 2. The condition ðc; fÞ 2 A is necessary and sufficient for existence of a monetary equilibrium. (footnote continued) ~ (The counterfeiter who becomes a Moreover, nothing would preclude y ¼ 1 m, which would happen if C 0 4C. producer disposes of the counterfeit held.)
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^ qn g ¼ q, ^ then we propose p ¼ 1 and Proof. Sufficiency. We propose pI ¼ pU ¼ p: If minfq; ^ ^ q g ¼ q , then we propose v0 ¼ 0, ð1 bÞv1 ¼ ð1 mÞ= ðv0 ; v1 Þ ¼ ð0; q=bÞ: If minfq; N½uðq Þ q , and p given by (1) at equality. These imply C~ ¼ 0 and, by the definition of A, Cp0. It follows that conditions (i) and (ii) of Definition 1 are satisfied. Condition (iii) is vacuous because y ¼ 0. Necessity. Assume that ðc; fÞeA and that a monetary equilibrium exists. Either y ¼ 0 or y40: The latter is ruled out by Proposition 1. The former implies that the equilibrium is ~ which contradicts y ¼ 0. & that in the sufficiency part. But ðc; fÞeA implies C4C, Finally, to emphasize the role of the Cho–Kreps refinement, condition (iii) in Definition 1, we show that without that condition there can be equilibria with counterfeiting. We show this in two steps. First, we show the following. Lemma 3. For some parameters, there is an allocation with counterfeiting that satisfies conditions (i) and (ii) of Definition 1. ^ q —where q^ Proof. We show that the claim is satisfied if ðc; fÞeA and is near A, and if qo 0 ^ is defined in (8) and q , as above, is the unique solution to u ðq Þ ¼ 1. (The inequality qo q restricts the parameters other than ðc; fÞ and is obviously not vacuous.) In particular, we show that the claim is satisfied with pI ¼ pU ¼ 1. It follows from the bindingness of constraint (1) in Problem 1 that C~ ¼ 0. This implies that any equilibrium with counterfeiting satisfies C ¼ 0, v0 ¼ 0 and, hence, v1 ¼ D=b. With pI ¼ pU ¼ 1, C ¼ 0 is equivalent to c 1my m ¼ u D (10) 1f N mþy (see (3)), and (5) can be written as ð1 bÞ 1my m D¼ f½uðDÞ D þ ð1 fÞ u D D , b N mþy
(11)
or as Nð1 bÞ=b m D . 1þ D ¼ fuðDÞ þ ð1 fÞu 1my mþy
(12)
First, consider the pairs ðy; DÞ that satisfy (10). Because u is unbounded, strictly increasing, and uð0Þ ¼ 0, for each y 2 ½0; 1 mÞ, there is a unique D, denoted f ðyÞ , that ^ and f ðyÞ ! 1 satisfies (10). Moreover, f is differentiable, strictly increasing, f ð0Þ 2 ð0; qÞ, as y ! 1 m. Next, consider the pairs ðy; DÞ that satisfy (12) and D40. It follows from u0 ð0Þ ¼ 1 , u0 ð1Þ ¼ 0, and strict concavity of u that for each y 2 ½0; 1 mÞ, there is a unique positive D, denoted gðyÞ, that satisfies (12). Moreover, gð0Þ ¼ q^ (see the proof of Proposition 2), gðyÞ ! 0 as y ! 1 m, and g is decreasing. (The last conclusion uses the fact that at the point ðy; gðyÞÞ, the derivative w.r.t D of the left-hand side of (12) exceeds the derivative w.r.t D of the right-hand side of (12), a kind of second-order condition.) ^ DÞ, ^ that satisfies (10) and (11). Therefore, there exists a unique ðy; DÞ, denoted ðy; ^ ^ ^ and ^ It remains to confirm that ðpI ; qI Þ ¼ ð1; DÞ Moreover, y 2 ð0; 1 mÞ and D 2 ð0; qÞ. ^ ^ ^ ðpU ; qU Þ ¼ ð1; ðm=ðm þ yÞÞDÞ solve Problem 1. The former is true because Doqoq ¼ qI . ^ ^ The latter requires ðm=ðm þ yÞÞDpqU , where qU is the unique solution to ^ u0 ðqU Þ ¼ ðm þ yÞ=m. To guarantee this, we use ðc; fÞ near A. Let x c=ð1 fÞ. Because
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^ But, as x ! ðð1 mÞ=NÞuðqÞ, ^ f ð0Þ ! q. ^ That and the ðc; fÞeA, xoðð1 mÞ=NÞuðqÞ. ^ Therefore, for x sufficiently fact that f 0 ð0Þ40 imply that y^ ! 0 as x ! ðð1 mÞ=NÞuðqÞ. ^ qU is sufficiently close to q , which implies that ðm=ðmþ close to ðð1 mÞ=NÞuðqÞ, 5 ^ Dpq ^ yÞÞ & U. The second step is to confirm that the pU ; qU shown to exist in Lemma 3 is immune to individual deviations for some beliefs about the response to deviations. One such belief is that the seller believes with probability one that any deviator holds counterfeit money. And if pU ¼ 1, as is the case for the allocation shown to exist in the proof of the lemma, another such belief is that the seller believes that any deviator has genuine money with probability m=ðm þ yÞ. (Notice that in both cases, the seller’s belief does not depend on the nature of the deviation.) Thus, nonexistence of a counterfeiting equilibrium depends on imposing some restriction on out-of-equilibrium beliefs.6 Two features of our model permit the Cho–Kreps refinement to have bite: first, holders of counterfeits value output as least as much as do holders of genuine money and value retaining what they hold less than do holders of genuine money; second, holders of genuine money have a rich possible set of deviations. The refinement would not have bite in the absence of lotteries because, then, there would be no profitable deviation for a holder of genuine money that would be accepted by the seller. 4. Conclusion We have set out a simple model in which people can at a cost produce counterfeits. However, such production never happens in equilibrium. Our model has three extreme assumptions: the perishable nature of counterfeits, asset holdings in the set f0; 1g, and the assumption that buyers make take-it-or-leave-it offers. The assumption that counterfeits last only one period is used primarily to conclude that any equilibrium with counterfeiting has pooling. However, pooling could, instead, be taken to be a definition of trade involving counterfeits. (In the absence of pooling, sellers would be accepting counterfeits knowingly—as if counterfeits were simply another acceptable money—which does not conform to what we think of as trade involving counterfeits.) If it is, then, nonexistence of an equilibrium with counterfieitng depends only counterfeits being worth less than genuine money. The f0; 1g assumption plays a significant role because it gives rise to two types, holders of genuine money and holders of counterfeits. Those types allow us to easily apply the Cho–Kreps refinement. How do we reconcile counterfeiting in the actual economy with our nonexistence result? The model is about steady states in which the actual and perceived stocks of counterfeits are equal. Much of the counterfeiting we see in actual economies consists of sporadic episodes, each one of which is short-lived and region-specific. Hence, those do not have to 5 We suspect that the conclusion of Lemma 3 holds for all ðc; fÞeA, but the argument would be lengthy and the result is not important for the message of this paper. 6 This is very different from the no-countefeiting result in Williamson (2002, pp. 52 and 53). Williamson does not have lotteries and, therefore, does not consider refinements. He has as a condition for a steady state that people with investment opportunities have to be indifferent among three options: produce no asset, produce the lowquality asset, produce the high-quality asset. In our model, we require only indifference between the first two options.
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be interpreted as steady states. More to the point, counterfeiting is almost always specific to one or several denominations and some counterfeiting episodes are accompanied by the disappearance in trade of the denominations being counterfeited. We are tempted, therefore, to interpret such disappearance as the analogue of the nonexistence implied by the model.7 Finally, our nonexistence result is derived from a model that embodies a very specific mechanism for exchange; namely, buyers make take-it-or-leave-it offers. A model with a different mechanism may have an equilibrium with positive trade in the presence of counterfeit money. Finally, the model implies that counterfeiting can be a threat even if we do not observe it. According to the model, actions taken that increase the cost of producing counterfeits, c, and/or that increase the probability that counterfeits can be recognized, f, can be worthwhile. In particular, because the magnitude of ð1 mÞ=Nuð¯qÞ in (9) does not depend on c or f, increases in c or f enlarge the set of magnitudes of the other parameters for which a monetary equilibrium exists. Thus, the model is consistent with efforts to thwart counterfeiting. References Berentsen, A., Molico, M., Wright, R., 2002. Indivisibilities, lotteries, and monetary exchange. Journal of Economic Theory 107, 70–94. Cho, I.-K., Kreps, D., 1987. Signaling games and stable equilibria. Quarterly Journal of Economics 102, 179–221. Cramer, J.S., 1981. The work that money does: the transaction velocity of circulation in the Netherlands, 1950–1978. European Economic Review 15, 307–326. Green, E., Weber, W., 1996. Will the new $100 decrease counterfeiting? Federal Reserve Bank of Minneapolis Quarterly Review 20 (3), 3–10. Kulti, K., 1996. A monetary economy with counterfeiting. Journal of Economics 63, 175–186. Lee, M., Wallace, N., Zhu, T., 2005. Modeling denomination structures. Econometrica 73, 949–960. Shi, S., 1995. Money and prices: a model of search and bargaining. Journal of Economic Theory 67, 467–498. Trejos, A., Wright, R., 1995. Search, bargaining, money and prices. Journal of Political Economy 103, 118–141. Velde, F., Weber, W., Wright, R., 1999. A model of commodity money, with applications to Gresham’s law and the debasement puzzle. Review of Economic Dynamics 2, 291–323. Williamson, S., 2002. Private money and counterfeiting. Federal Reserve Bank of Richmond Economic Quarterly 88/3, 37–57. Williamson, S., Wright, R., 1994. Barter and monetary exchange under private information. American Economic Review 84, 104–123.
7 Of course, whether such disappearance is implied by a model with several denominations and with denomination-specific counterfeiting technologies remains to be determined. For a model of multiple denominations (without a recognizability problem), see Lee et al. (2005).