Online Appendix for "Combating Strategic Counterfeiters in Licit and Illicit Supply Chains"

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1 Online Appendix for "Combating Strategic Counterfeiters in Licit and Illicit Supply Chains" Soo-Haeng Cho Xin Fang Sridhar Tayur Tepper School of usiness, Carnegie Mellon University, Pittsburgh, PA 53 Lee Kong Chian School of usiness, Singapore Management University, Singapore Tepper School of usiness, Carnegie Mellon University, Pittsburgh, PA 53 Appendix A. Proofs of Analytical Results We use A and A to indicate the following assumptions: A q > q N f N + βq and q > q f + βq ; A p > 0 and p p N q q N > 0 so that m > 0. We also use the following definition of expected consumer welfare introduced earlier: ECS N γcs N + γcs and ECS γcs + γcs. Proof of Lemma : We first determine the market shares of the brand-name product and the counterfeit, and then solve the model backwards. The consumer who is indifferent between purchasing the brand-name product and the non-deceptive counterfeit has the taste θ p p N q q N p p N βq f N, which solves θq N p N θq p. Similarly, the consumer who is indifferent between purchasing the non-deceptive counterfeit and not purchasing any product has the taste θ p N qn p N f N +βq. Let m 0 denote the proportion of consumers who do not purchase any product, so that m + m N + m 0. Then: m θ p p N and m N βq f θ θ p p N p N. N βq f N f N + βq In stage 3, the illicit distributor determines p N by solving: p p N p N maxp N w N m N p N w N. 3 p N βq f N f N + βq y noting that the profit of the illicit distributor in 3 is concave in p N, one can easily obtain p N w N, f N βq +f N p +q w N q. In stage, the non-deceptive counterfeiter determines w N to maximize his expected profit given by: p p π N w N, f N γ w N p N N tfn γtf βq f N f N + βq N. 4 Since π N is concave in w N, we can easily obtain wn and π N respectively as follows: wnf N p f N + βq and π q Nf N p γf N + βq 8q βq f N tf N. 5 In stage, the non-deceptive counterfeiter decides f N by solving max π N f N. From 5, f N [f,f] we obtain π N f N γp 4βq f N 3 t, which is positive if t < γp 8βq f N 3. Thus, if t < γp 8βq f 3, π N is convex in f N [f, f], so f N f when π N f π N f. Otherwise, f N

2 can be f or fn f, f that satisfies π N f N fn fn 0. A suffi cient condition for f N > f is γp t <, which can be obtained from π 6fβq f N γp f N fn f tf > 0. 8βq f Remark The initial investment tfn is considered a sunk cost in 4. Our main results continue to hold when the counterfeiter s investment has residual value but gets confiscated if getting caught. Also, whether the confiscation of investment occurs after stage or stage does not affect the counterfeiter s decisions. If confiscation occurs after some units are sold, γ can be interpreted as the fraction of sales the counterfeiter has generated before confiscation. Proof of Lemma : Similar to the proof of Lemma, we can obtain the market shares of the brand-name product and that of the deceptive counterfeit, respectively, as follows: where θ m s θ and m s θ, 6 λp sq +sf +βq λp q and counterfeit products at price p. max s sp w s [0,] represents the aggregate demand for both brand-name In stage 3, the licit distributor solves the following problem to determine s: λp sq + sf + βq λp q sl. 7 From 7, we can show that the profit of the distributor is strictly decreasing in s for s [ ɛ, ], where ɛ is a small and positive constant. Moreover, the profit given in 7 is concave in s for s <. Thus, s is 0 or it satisfies the first order condition in 0, 0.5. In stage, the deceptive counterfeiter decides w to maximize his expected profit given by: π w, f γ [w s λp s q + s f + βq λp ] tf γtf q +h. 8 y noting that π is continuous in w [0, p ], we know that the optimal wholesale price w always exists in [0, p ]. In the case when λ > 0, the closed-form expressions for s and w do not exist. In the case when λ 0, we can obtain from the first-order condition of 7 that s w, f we obtain w lq p w q p. y substituting s into 8 and solving max π w, f, w [0,p ] and π w f p as follows: lp p and π f γ p p q In stage, the counterfeiter decides the functional quality f by solving λ 0, from 9, π f tf < 0, so f l tf γh. 9 max π f. When f [f,f] f. When λ > 0, we next show that f f if t t. For any f f, f], π w f, f π w f, f if t γw f m w f, f m w f, ff f. Suppose t t w f, ff f. Then, for any f f, f], π max γw f m w f, f m f f,f] w f, where the first inequality is due to the optimality of w follows from t t. Therefore, f f. f, f π w f, f π w f given f, and the second inequality f,

3 In the rest of the proof, we show t > 0 in two steps: we first show that s is increasing in f for given w, and then show that the market share of the deceptive counterfeiter, m s θs, is increasing in f for any given w. Then from the definition of t, t > 0. Let π L denote the expected profit of the licit distributor given in 7. Then π L s π L sp w θ s sp w θ l, and λp s s f 3sp w + s sp sq +sf +βq w, sq +sf +βq 3 where the first term is positive because we know from 4. that s < 0.5 and w p, and the s λp sq f βq π second term is also positive according to A. Therefore, L s is increasing in f. Since s satisfies π L s ss 0 due to the concavity of π L with respect to s, s is increasing in f. Next, we show that m increases as f increases from f L to f H for given w. Suppose this does not hold. Then, π L satisfies the following: π L s f H, f H s f H s f H p w θs f H, f H s f H l s f L s f H p w θs f L, f L s f H l < s f L s f H p w θs f L, f H s f H l < s f L s f L p w θs f L, f H s f L l π L s f L, f H, where the first inequality follows from our premise, the second inequality follows from λp s sq +sf +βq θ f < 0 for fixed s, and the last inequality follows from s f > 0. However, this contradicts the condition that s f H maximizes the licit distributor s profit π L given f H. Therefore, m is increasing in f for given w, and t > 0. A suffi cient condition for f ff f. We show this by contradiction. Suppose f > f is t < t max γw fm w f, f m w f, f f,f] f and define f max arg max f f,f] γw fm w f, f m w f, ff f. Then π w f max, f max π w f, f max > π w f, f, where the first inequality is due to the optimality of w f max given f max, and the second inequality follows from t < t. However, this contradicts our premise that f f. Therefore, f > f if t < t. Proof of Proposition : a The proof proceeds as follows: We first obtain m and q, and then derive the condition for > qm. When there is no counterfeiter, the expected profit of the brand-name company is given as follows: π m p c p t q q, 0 where c > 0 is the marginal cost of the brand-name product. From 0, π m p c p 3 t < 0, so we obtain m p c p from the first order condition. When the non-deceptive t m counterfeiter exists, we obtain π after substituting p N and w N into m in as follows: π p c m t p c t. From, when f N f or f, π q p c p γβ this case, from the first order condition of, q p c γp 4 βq f N 3 + γp 4q 3 t < 0 due to A. In. q q N + p 3+γ 3 q 3 γβp t 4β f + 3+γp N q 4

4 We next show by contradiction that > qm when β < qm q N qm m. Suppose β < qm q N qm and qm. For f NH > f NL, from 5, we obtain π N f NH π N f NL q m q γp f NHf NL ββq f NH +f NL / p c γβp t 4β f + 3+γp N p c γβp q 4 t 4β mf + 3+γp N qm 4 m 4βq f NH βq f NL < 0 due to A, so f N is decreasing in q. Then q > p c p q m t m,, and the second inequality where the first inequality follows from qm and f N q f N qm follows from β < qm q N qm m. Thus, there is a contradiction, so > qm when β < qm q N qm m. The case in which β qm q N qm m can be shown similarly. b To establish the result in the proposition, it suffi ces to show that π N is decreasing in q. The proof proceeds in two steps: We first show that π N decreases in q for any given f N, and then show that this result holds even when fn changes with q. First, from 5, we obtain π N q γp β β+q f N β+fn, which is negative by A for any given f 4 βq f N N. Next, we consider the case in which fn changes from f N to f N when q is increased from q L to q H. In this case, π N f N, q L π N f N, q L > π N f N, q H, where the first inequality follows from f N q L f N and the second inequality is due to π N q < 0 f N. c We first prove that ECS N is increasing in q for given f N, and then prove that ECS N decreases when f N is decreased from f NH to f NL for any given q. To prove that ECSN is increasing in q, it suffi ces to show that CS N q > 0 for any given f N because CS q > 0 from the definition of CS. Now suppose that q is increased from q L to q H. Then q N is also increased from q NL to q NH given f N ; θ is decreased from θ L to θ H ; and θ is decreased from θl to θ H. Using p N θ 3p 4q H q NH dθ + θh θq H p dθ > θh θ 3p θl 4q H q NH dθ + θl θh θh θl 3p q N 4q, we can rewrite CS N and find CS N q H > θl θq H p dθ > CS N q L. The first inequality holds because θ L > θ H and θ 3p 4q H q NH dθ + θ 3p 4q H q NH > 0 for θ θ H, θ L. The second inequality holds because θq H p > θq NH p N for θ θ H, θ L. The third inequality follows from the fact that q H > q L and q NH > q NL. Next, suppose fn is decreased from f NH to f NL for fixed q. Then θ remains the same, whereas θ is decreased from θ p 4βq f NH + 3p 4q to θ p 4βq f NL + 3p 4q. Then, ECSN f θ NH γ θ θ 3p 4q f NH + βq dθ + θ θq p dθ + γ p θq p dθ > γ θ θ θ 3p 4q f NL + βq dθ + θ θq p dθ + γ p θq p dθ ECS N f NL. In the above, the first inequality holds because θ 3p 4q f NH + βq > θq p for θ θ, θ, and the second inequality follows from f NH > f NL. Remark When fn f, f, assuming fn condition of, p c t γβ f N / q p 4β f N q becomes γβ f N / q 4βq f N q + 3+γ 4q q m further to the form like β < qm q N qm q m 0, we can still obtain π q < 0. From the first order + 3+γp. The condition for q 4 > qm then > 0. Unfortunately, this condition cannot be simplified because the closed-form expressions for f N and 4

5 fn q are not available. The proof for b does not require fn f or f, so it also holds for fn f, f. For c, suppose q > qm. From, ECS N q [ β6 βq + p β5 + γ 6 + fn p 5 + γ 6q βq fn + p q γ f N / q ] βq fn. Then ECS N is decreasing in q [q m, q ] so that ECS N is lower at q than at qm if f N q < q β6βq +p β5+γ6+f N p 5+γ6q βq f N p q γβq f N. Proof of Proposition : a From 0, π m p q < 0, so we obtain p m q +c from the first order condition. Next, consider the market in which the non-deceptive counterfeiter exists. When fn f or f, from, π γ p q q N 3+γ q < 0 and p q q γ 4q +3q +γq + q +c ; in this case, p < pm due to A. When f N f, f, we show π p p q +c < 0, which then results in p < pm. From, we obtain π p p q +c which is negative because: 4f N γ4f N +4βf N q +4ββ +c q c +q fn / p 6βq f N, 4 βf N q 4 ββq f N p q c q + c < 4f N 4f N βq f N 4f N βq f N p q c q + c f N p q c q + c 0, where the first inequality is due to A and the second inequality is due to f N p 0 and q > p c. b The proof is similar to that of Proposition b, and is hence omitted. c The proof for the case in which f N f or f is similar to that of Proposition c. When f N f, f, from, ECS N p [ βq fn 6βq +p q β5+γ6+p 5+γ 6q fn +p q γ fn / p ] βq fn. efine κ max p [p,pm] βq fn 6 β + p q β5 + γ 6 + p 5 + γ 6q fn p q γ βq fn. Then ECSN is increasing in p [p, pm ] so that ECS N is lower at p than at pm if f N p > κ. Proof of Proposition 3: a When λ 0, we obtain π in 6 as follows: π p c m t p c γ p q + 4q 3p γ l p 8q p 3 after substituting s and w into m l p p + γ p t q q. From, π p c p +γ 3 q p t < 0 due to A, and p c t γ p + lp q 4 p + γp from the first order condition. y the same procedure in the proof of Proposition a, we can prove by contradiction that < qm if and only if l < 4p p q m. Since s l p p > 0, we find that l < p p q m < 4p p q m, so we always have < qm. b To establish the result in the proposition, it suffi ces to show that π is increasing in q. When λ 0, it is easy to see π q π p c When λ 0, from, ECS q γs. Using s p q 3 s + q p q γ > 0 and A, we prove ECS q p q s q for q [q Remark When λ > 0, both q < qm > 0 f from 9. Since f f q, the result follows. p s + 3 p s q q q q + p q, qm ]. and q > qm 5 < 0 when q < q p q γs γ are possible as shown in Table 3. The condition

6 for < qm is t > p c q m γ s m λp m s mqm + s mβqm + f [ γp γ λp m m + λp + s m λp s q m qm +s q m βqm +f qm s q + f β + q + f qm + βqm qm s q qm ], which can be obtained from π q m < 0. In this case, π is increasing in q as in the case when λ 0 shown in the proof of Proposition 3b. The proof follows the same procedure as in that of Lemma, so we provide a sketch of the proof here. For given w and f, we can show that s and m are increasing in q. Then when q is increased from q L to q H, the following inequalities hold in equilibrium: π w q H, f q H, q H π w q L, f q L, q H > π w q L, f q L, q L. Finally, when λ > 0, the condition for ECS q < 0 given in the proof of Proposition 3c is modified to the following: q θ θs q + θ s q < γ γ p q + p θq s θ q +θ f β q s s q Unfortunately, this condition cannot be simplified further since the closed-form expressions of s and f do not exist. The non-monotonicity of ECS Proof of Proposition 4: a From, π and π p p q +c c γ c +q concavity of π, p > pm. b The non-monotonicity of π c When λ 0, from, ECS p ] + γ p + p Remark. Using s p q is shown in Table 3. +γ p q γ l p q p p c + q p c 4q p < 0, l c q c +q > 0 due to q > p c by A. Therefore, by the with respect to p is shown in Table 3. γ[ s p q q q + p < 0 and A, we prove ECS p ]. q p s p for p [p m, p When λ > 0, both p > pm and p < pm + p s p q q > 0 when q < q q p γ p s are possible as shown in Table 3. The condition for p > pm is γ pm + γ s p m Y + p c [ γ γy s p + γ s p m λpm q condition for ECS Y p m λp m s p m q +s p m βq +f pm + λpm s p m f +f p pm +βq q s p s p m q +s p m βq +f q pm ] > 0, where Y π. This can be obtained from p p p m > 0. The p > 0 given in the proof of Proposition 4c is modified to the following: q θs θ p θ s p < γ p γ + p θq s θ p + θ s q s p q p + θ. Unfortunately, this condition cannot be simplified further since the closed-form expressions of s and f do not exist. The non-monotonicity of ECS is established in Table 3. Proof of Proposition 5: When λ 0, we observe from that π does not change with β, and that π is increasing in γ and t. When λ > 0, similar to the proof of Lemma, we can show that the aggregate demand, θ, for the brand-name product and the deceptive counterfeit is increasing in β and decreasing in γ and t, and that the fraction of the brand-name product, s, is decreasing in β and increasing in γ and t. The non-monotonicity of π numerical experiments presented in online appendix. The proofs for π to those of Proposition 3b-c, and hence are omitted. and ECS is shown in our are similar Proof of Corollary : When the non-deceptive counterfeiter exists in the market, it is easy to see. 6

7 that the price decisions of the illicit distributor and the counterfeiter in stages 3 and, respectively, are unchanged. In stage, the counterfeiter chooses his optimal functional quality fn to maximize his expected profit, which is modified from 5 as follows: π N f N p γ+δ f N f N +βq 8q βq f N tfn. Similar to Lemma, we can show that fn f, f or f N f, f that satisfies π N f N fn fn 0, depending on the value of t and whether π N f π N f. When the deceptive counterfeiter exists in the market, in stage 3, the licit distributor chooses its optimal fraction s of counterfeits by solving the following problem which is modified from 7: s δ f l. 3 max s s + δ λp f p w λp s [0,] sq + sq q In stages and, the counterfeiter decides w and f, respectively, to maximize his expected profit given by: π w, f w s λp γ + δ f s q +s f +βq λp q tf γh. When λ 0, by following the procedure similar to that in the base model, we obtain the lq p w q p closed-form expressions of s and w as follows: s +δ f and w lp p p. When λ > 0, similar to the base model, we can show the existence of +δ f s, w and f, but their closed-form expressions are not available. a Suppose fn f or f. We can obtain π q by replacing γ in the base model with γ δ fn. To show that Proposition a continues to hold, we need to prove that fn is decreasing in q. For f NH > f NL, π N f NH q π N f NL q can be expressed as follows: p γ+δ f NH q βq +f NH β+f NH q βq f NH + p 8 βq f NH γ+δ f NL q βq +f NL β+f NL q βq f NL 8 βq f NL < γ+δ f NH [ ] p q βq +f NH β+f NH q βq f NH + p 8 βq f NH q βq +f NL β+f NL q βq f NL 8 βq f NL γ+δ f NH p f NHf NL ββq f NH +f NL /, 4βq f NH βq f NL which is negative due to A. Then, following the same procedure as in the proof of Proposition a, we can show > qm if and only if β < q m qn qm q. For Proposition b, when m fn is given, π N q γ+δ f N p β β+q f N β+fn < 0 due to A. When f 4 βq f N N changes from f N to f N as q is increased from q L to q H, π N f N, q L π N f N, q L > π N f N, q H, where the first inequality follows from fn q L f N and the second inequality is due to π N q < 0 f N. For Proposition c, to prove that ECSN is increasing in q, it suffi ces to show that CS N q > 0. Since CS N q does not depend on γ when f N is given, Proposition c continues to hold. b We can show that the proof for Proposition also applies to the case with γ δ f N similarly to Proposition, except part c when fn f, f. With the extension, ECS N p βq fn [p fn p δ fn βq q +fn q β βδ q +γ β q +fn f N p βδ q γ 5 + δ fn + 6q p + q 6β q p βγ p ]. efine κ max p [p,pm ]p β q + fn f N p βδ q γ 5 + δ fn + 6q p + q 6β q p βγ p δ fn βq q + fn q β βδ q + γ. Then ECSN is increasing in p [p, pm ] so that ECS N is lower at p than at pm if f N p > κ. To show that Proposition 3a continues to hold, when λ 0, we can prove by contradiction that 7

8 q < qm if and only if +δ f l < 4p p q m by the same procedure as in the proof of Proposition a. Since s +δ f l+δ f p p > 0, we obtain l < p p q m +δ f < 4p +δ f p q m, so q < qm so p m c γ+δ f c +q l c +δ f q c +q > 0, π always holds. For Proposition 4a, p p q +c > pm. For Propositions 3b and 4b, we can show that, with the extension, s and are increasing in q and decreasing in p for given w and f. Then, when q is increased from q L to q H, the following inequalities hold in equilibrium: π w q H, f q H, q H π w q L, f q L, q H > π w q L, f q L, q L. Similarly, when p is decreased from p H to p L, the following inequalities hold in equilibrium: π w p L, f p L, p L π w p H, f p H, p L > π w p H, f p H, p H. Propositions 3c and 4c can be shown similarly to Proposition c. c The proofs of π and ECS are similar to those of Proposition 3b-c. When λ 0, the non-monotonicity of π or ECS with respect to γ can be shown numerically as follows. Set q, p 0.5, t 0.0, c 0.0, β 0., l 0.0, h 0 and δ δ 0.. As γ increases from 0. to 0.3, π increases from 0.3 to 0.4, π The non-monotonicity of π increases from 0.74 to 0.8 and ECS increases from to As γ decreases from 0.8 to 0.73 and ECS or ECS decreases from to with respect to t can be shown similarly. The following corollary examines when the quality strategy is more profitable than the pricing strategy for the brand-name company. Corollary Suppose that the brand-name company needs to invest t for developing and setting up facilities to produce a product with quality q. Then: a Consider the market in which a non-deceptive counterfeit with fn f or f exists. There q m exists t N such that if t > t N and β qn qm, the brand-name company is more profitable when setting quality < qm than setting price p < pm ; and if t > t N and q m β < qn qm, the brand-name company is less profitable when setting quality q > m q m than setting price p < pm. b Consider the market in which a deceptive counterfeit exists and there are no proactive consumers with λ 0. There exists t such that if t > t, the brand-name company is more profitable when setting quality < qm than setting price p > pm. Proof of Corollary : a When non-deceptive counterfeits are in the market and β q m qn qm, we know from Propositions and that q m > and pm < p. To determine q m which strategy is more profitable, we compare π q and π π q 3+γp 4q γp c 4q 3+γf 4q βq fn p N +βq 4q q βq fn p p c 4 q m 3+γ p. For given q and p, from, π q p + βγ q βq f N q t, which is decreasing in t. Thus, if t is suffi ciently large, π p < π q so that changing quality from m to q is more profitable than changing price from pm to p. The proofs for q m β < qn qm and for b are similar to that for a, and are hence omitted. q m 8

9 Appendix. Numerical Experiments This section contains our numerical study that examines the effectiveness of the marketing campaign, the enforcement strategy and the technology strategy against the deceptive counterfeiter. Similar to the numerical study presented in 5., we have constructed 04 scenarios for λ 0, 0.5 or 0.5, using the following parameter values: t 0.005, 0.0, 0.05, 0.0, β 0., 0., 0.3, 0.4, γ 0., 0., 0.3, 0.4, l 0.005, 0.0, 0.05, 0.0, c 0.005, 0.0, 0.05, 0.0, h 0, f 0. and f βq 0.0. q, p is fixed at m, pm. We omit the result for increasing t because its effect is essentially the same as increasing γ. We computed the difference in firms expected profits and expected consumer welfare associated with the adjacent values of β or γ for a fixed set of other parameter values. There are 3 increments of β or γ for a set of 56 possible values of other parameters, so there are 768 scenarios for which we can examine the direction of changes with a decrease of β or an increase of γ. The results are summarized in Table 5, which reads as follows: for example, when λ 0.5, reducing β increased π scenarios, and increased ECS in 3.9% of 768 scenarios. in 33.% of 768 scenarios, decreased π Table 5. Effects of Marketing and Enforcement Strategies against eceptive Counterfeits Effects of Reducing β Effects of Increasing γ π π ECS π π ECS λ 0 no change no change 0 λ λ in all Table 5 confirms the results stated in Proposition 5. In addition, similar to the quality and pricing strategies discussed in 5., these strategies are not necessarily more effective as more consumers are proactive with higher λ. Appendix C. Extension: Price ecision of Licit istributor When the deceptive counterfeiter is in the market, suppose that the licit distributor decides on the retail price p, while the brand-name company instead decides the wholesale price w to the licit distributor. The rest of the decisions remain the same as in the base model as follows. After observing the quality q and wholesale price w of the brand-name product, the deceptive counterfeiter decides his functional quality f and wholesale price w in stages and, respectively. In stage 3, the licit distributor decides a fraction of deceptive counterfeits s, and then decides the retail price p. We consider two cases that differ in how the licit distributor determines p : the distributor chooses the retail price that maximizes his expected profit from selling both brand-name and counterfeit goods; and the distributor chooses the optimal retail price as if he does not sell the deceptive counterfeit, for fear that consumers or third parties may identify the deceptive counterfeit from a lower retail price than the price of authentic goods. The second case reflects the fact that a deceptive counterfeit is usually sold at the same price or close to that of its branded 9

10 product so as to deceive consumers see and 3. In the first case, the licit distributor solves the following problem in stage 3 to determine p : λp max s p sw sw p sq +sf +βq λp q sl. We can verify that the distributor profit is concave in p, and obtain p λ sq +sf +βq + λ lower retail price p sw +sw + q. Since w < w and f + βq < q, the distributor charges a when the fraction of counterfeits s is larger or the fraction of proactive consumers λ is larger. To find the optimal fraction s, the licit distributor solves the following problem which is obtained by substituting p max s 4 s λ sq +sf +βq + λ q into the distributor s profit: [ sw + sw λ sq +sf +βq + λ q ] sl. ue to complexity, however, it is not possible to find the closed-form expression of s. Although the part of our analysis in the base model does not rely on its closed-form expression, the impact of any anti-counterfeiting strategy on s becomes prohibitively complex to obtain any analytical result. Thus we conduct extensive numerical experiments to examine the effects of anti-counterfeiting strategies. We use the same parameters in Table 6 as in Table 5. For each case with λ 0 and λ 0.5, there are 04 scenarios in which we can investigate the anti-counterfeiting strategies that change quality or price from the case with no counterfeiter to the optimal levels. On the other hand, similar to Table 5, there are 768 scenarios for which we can examine the anti-counterfeiting strategies that reduce β or increase γ. Table 6 can be read similarly to Table 5. Table 6. Effects of Anti-counterfeiting Strategies in the Extended Model λ 0 λ 0.5 π π ECS π π ECS m q w m w β no change no change γ Table 6 shows that the effects of anti-counterfeiting strategies remain directionally true in this extended model. For example, as the price changes from w m to w, π and ECS can increase or decrease; this is consistent with Proposition 4. Also, as stated in Proposition 5, when λ 0, reducing β has no impact on π and increases π as well as ECS can increase or decrease π and π, but decreases ECS, whereas increasing γ reduces π, but it ; when λ > 0, reducing β or increasing γ decreases π and ECS. One notable exception is that when λ 0, changing qm in the base model. This happens because to can increase π although it always decreases π of the additional lever the licit distributor has i.e., determining p as well as s: in response to the change of q, the distributor can increase the aggregate demand for both brand-name and counterfeit goods by reducing p. As a result, we find that the distributor may increase or decrease s in response to this strategy, which thus creates a non-monotonic effect on π. In the second case, the licit distributor chooses p to maximize p w p the optimal retail price: p w +q. ifferent from the first case, p 0. We obtain does not depend on the

11 fraction of counterfeits s or the fraction of proactive consumers λ. To find the optimal fraction s, the licit distributor solves the following problem: s max w +q λw s sw sw +q sq +sf +βq λw +q q sl. In the case when λ 0, we can obtain from the first-order condition that s y substituting s into 8 and solving max w expected profit of the deceptive counterfeiter π w w [q w +4lq ] q w and π π w, f, we obtain w as follows: q w +4lq 4w w q w. and the corresponding γw w +4lq 8w q w w q tf γh., we can verify that several insights from the base model continue to hold. w From w and π For example, as the risk of the licit distributor selling counterfeits increases with l, the deceptive counterfeiter has to reduce his price w to compensate for the increased risk, resulting in a decrease in his expected profit π. Also, the deceptive counterfeiter always chooses the lower bound f for his functional quality in the market with no proactive consumers. The following corollary shows that the main results from the base model continue to hold in this case. Corollary 3 Suppose the licit distributor decides the retail price p as if he does not sell the deceptive counterfeit. Then there exists l such that if l > l, Propositions 3 and 4 continue to hold except that the conditions in part c are different. Proof: When the deceptive counterfeiter exists in the market with λ 0, we obtain the following π after substituting s and w into m in 6: π w c w q [ γ + w +4lq From the first order condition, q w w c 4t q 8w q w ] + γ t. 4 γ + [ w +4lq ] w q w + γ + γw c [q w 4lq ], which is decreasing in l if l is suffi ciently large. On the other 6t q w q w [q w +4lq ] hand, m w w c, which does not depend on l. Thus, there exists l such that if l > l, we have 4t q < qm. The proof of p 3b, it suffi ces to show that π is similar and hence omitted. To establish the result in Proposition is increasing in q. In the expression of π above, w q is increasing in q and w +4lq 8w q w is increasing in q if l is suffi ciently large. Therefore, there exists l such that if l > l, π is increasing in q. The non-monotonicity of π with respect to p and ECS are shown in Table 7. Corollary 3 shows that our main results in the base model continue to hold as long as the licit distributor faces a significant penalty if getting caught by the authorities, which is true in most countries see 3. The intuition is that in such a case, the counterfeiter offers a low wholesale price to the distributor to compensate for the high risk so that the distributor s margin from selling the counterfeit is high. As a result, the quality strategy discussed in Proposition 3, which reduces the aggregate demand, still discourages the licit distributor from taking the risk of selling the counterfeit and results in a lower s in this case. Similarly, the pricing strategy discussed in Proposition 4 leads to the same behavior of the distributor because it still reduces the aggregate demand and increases the distributor s margin from selling the counterfeit.

12 When λ > 0, the closed-form expressions of s, w and f do not exist. We conduct a numerical study similar to the first case using the same set of parameters. A summary of the results is presented in Table 7. Comparing Table 7 with Table 6, we can verify that the dominant effects are the same in both cases. Therefore, the effects of anti-counterfeiting strategies remain directionally true in all cases. Table 7. Effects of Anti-counterfeiting Strategies in the Extended Model λ 0 λ 0.5 π π ECS π π ECS m q w m w β no change no change γ Appendix. etails of Consumer Survey Respondents of our survey are college students and faculty with ages from 8 to 50. The number of respondents is 86 in the U.S., and it is 80 in China. Two questions were asked in the questionnaire: Are you aware of the sale of counterfeits in each of the above product categories; and For each product category in which you are aware of the sale of counterfeits, do you take into account the risk of getting a counterfeit and therefore discount the value of the product when you purchase a brandname product at a full price in a legal store? Those customers who answered yes to are considered Aware, and those customers who answered yes to both and are considered Proactive. The absolute numbers may be escalated because respondents may be reminded of counterfeits by the questionnaire. Our survey indicates that being aware of the existence of counterfeits differs from being proactive. One may explain such difference from cognitive psychology e.g., endoly et al. 00, Goldsmith and Amir 00, and references therein; for example, it may be due to a positive-outcome bias or wishful thinking caused by overestimating the probability of good things happening. Additional References endoly, E., R. Croson, P. Goncalves, K. Schultz. 00. odies of knowledge for research in behavioral operations. Production and Operations Management Goldsmith, K., O. Amir. 00. Can uncertainty improve promotions? Journal of Marketing Research