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Counterfeiting and Negative Consumption Externalities – A Closer Look


We extend the work of Grossman and Shapiro (J Econ 103:79–100, 1988) on consumption externalities in prestige goods markets, and model a general aversion towards large levels of output interacting with an aversion towards copies in particular. These externalities play the role of protecting the market share of the producer of originals. We show that the well-established result under positive network externalities, that piracy is an equilibrium, extends to the case of negative consumption externalities. When externalities are pronounced enforcement should be strict, while in markets subject to moderate externalities there are no strong arguments in favor of a strict policy.

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  1. 1.

    OECD (2008) estimates international trade in counterfeit and pirated products to as much as USD 200 billion per year, perhaps the double if products distributed domestically and via the Internet is included.

  2. 2.

    See Shy (2011) for an overview of network economics and Belleflamme and Peitz (2012) and Peitz and Waelbroeck (2006) for overviews of piracy in particular.

  3. 3.

    We focus on prestige goods but the analysis extends to the case without consumption externalities and thus applies to all non-deceptive, or secondary markets, counterfeiting.

  4. 4.

    There are a few other related references. Yao (2005), studies counterfeiting and investment incentives in absence of so called snob effects while Higgins and Rubin (1986), study private versus public enforcement in a model with snob or prestige externalities. Grilo et al. (2001) analyze the effects of conformity and vanity in a model without piracy. Finally, Tsai and Chiou (2012) has a short section on counterfeiting under negative consumption externalities. Their model resembles ours, although it is based on more restrictive parameter assumptions.

  5. 5.

    See Shy and Thisse (1999), Slive and Bernhardt (1998), and Takeyama (1994). For an analysis of the role of expectations in network markets, see Griva and Vettas (2011).

  6. 6.

    See Tsai and Chiou (2012), Martínez-Sánchez (2010) and Banerjee (2003).

  7. 7.

    Empirical studies of counterfeiting in these markets find that piracy rates vary substantially across countries. In software markets, for example, they can be less than 40 % in some countries and almost 100 % in others. This variation can be explained by country-specific differences in intellectual property rights protection which, in turn, to a large extent depends on cultural and institutional factors. See Marron and Steel (2000). Qian (2008) studies the importance of counterfeiting on the price and quality of authentic products, using data from Chinese shoe manufacturing.

  8. 8.

    The US Government Accountability Office (2010) notes explicitly the need to investigate factors that seem to be crucial to the effects of counterfeits, e.g., consumers’ willingness to substitute between the legitimate and the counterfeit good.

  9. 9.

    The parameter α used below could be thought of as capturing a negative network externality. Since the term network is typically associated with positive externalities, we use the term prestige externality instead.

  10. 10.

    Even when products are identical it is conceivable that consumers put a higher value on owning the original product.

  11. 11.

    See See Griva and Vettas (2011) for a discussion on the formation of expectations in this kind of setting. In Grossman and Shapiro (1988) the assumption of perfectly correlated valuations of quality and prestige implies that network size is determined by the marginal consumer who is indifferent between buying and not buying, and expectations of quantities have no effect on other consumers’ decisions.

  12. 12.

    In other words, the costs associated with building a strong brand name are assumed to be mainly fixed.

  13. 13.

    It can be noted that many models with positive network externalities are characterized by multiplicity of equilibria. See e.g., Griva and Vettas (2011) and Grilo et al. (2001). The fact that we arrive at a unique equilibrium can be attributed to the assumption of negative externalities.

  14. 14.

    In Häckner and Nyberg (1996) negative externalities are firm specific, capturing e.g., waiting time externalities in firms with capacity constraints. By raising its price, a firm can then achieve a competitive advantage over another firm. This in turn increases the average price level. Similar results are found in Ben El Hadj-Ben Brahmin et al. (2012).

  15. 15.

    The direct effect on welfare from prestige- and pirate externalities are unambiguously negative as they reduce quantities (Proposition (3.2)) and lower the consumption value.

  16. 16.

    There is no aggregation problem here since utility is measured in monetary terms.

  17. 17.

    In all subsequent proofs, the letter S will be used to denote “sign”.

  18. 18.

    Note that the result differs from Häckner and Nyberg (1996) who show that free market prices are always too high in a model with firm specific congestion externalities and identical consumers.

  19. 19.

    Introducing an tariff on low-quality goods is conceivable in a scenario where fake trade-mark labels are added once products have been imported. This policy would have a negative impact on welfare also in any existing markets for legitimate low-quality goods, but such effects are not accounted for in the analysis.

  20. 20.

    We are grateful to an anonymous referee for suggesting this extension.

  21. 21.

    Obviously, in order to derive \( \overline{c} \) , as well as \( \tilde{c} \) later on, equilibrium prices and quantities have to be recalculated. This is straightforward so we refrain from presenting the new expressions here. Full proofs are available from the authors on request.

  22. 22.

    Note that Tsai and Chiou (2012) also find a u-shaped relationship between welfare and the strictness of government enforcement in a numerical example.

  23. 23.

    Maximizing firm profits when demand equals 1 − V 1 (assuming Q L  = 0 and substituting P H for P L ) yields the monopoly price, S H /2 .

  24. 24.

    Martínez-Sánchez (2010) reaches a similar result on the strategic value of competition in a model without externalities.

  25. 25.

    Proofs of these robustness checks are available from the authors on request.


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We are grateful to two anonymous referees for constructive comments and suggestions.

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Corresponding author

Correspondence to Astri Muren.



Proof (Proposition 4.2)

It is straightforward to aggregate welfare and to insert equilibrium quantities and prices given c≥0. Let welfare be denoted by W. \( {\scriptscriptstyle \frac{\partial^2W}{\partial {c}^2}} \) is independent of c, so we may conclude that W is either strictly convex or strictly concave in c. The next step is to evaluate the slope of W at \( c=\overline{c} \) and c = 0. It is straightforward to verify that \( S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=\overline{c}\right)=\alpha \left(3z{S}_H-{S}_L\left(2z+1\right)\right)+{S}_L\left({S}_H-{S}_L\right) \) which is positive for z > 1. Moreover,

$$ S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=0\right)=S\left({z}^2{\alpha}^2\left(\alpha +2{S}_H\right)\left(3{S}_H-2{S}_L\right)+z\alpha \left[\begin{array}{l}2{\alpha}^2\left({S}_H-2{S}_L\right)+\\ {}\alpha \left(4{S}_H^2-7{S}_H{S}_L+2{S}_L^2\right)+{S}_L\left(4{S}_H^2+{S}_H{S}_L-2{S}_L^2\right)\end{array}\right]+{S}_L\left[\begin{array}{l}{\alpha}^3+{\alpha}^2\left(5{S}_H-6{S}_L\right)\\ {}+\alpha \left(6{S}_H^2-14{S}_H{S}_L+5{S}_L\right)-2{S}_L\left({S}_H-{S}_L\right)\left(2{S}_H-{S}_L\right)\end{array}\right]\right)\equiv S(A). $$

The function A is convex in α. A < 0 for α = 0 while A > 0 for α = S L . Hence, \( S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=0\right)>0 \) for α sufficiently close to S L .

Proof (Proposition 4.3)

It is straightforward to aggregate welfare and to insert equilibrium quantities and prices given c≥0. Let welfare be denoted by W. \( {\scriptscriptstyle \frac{\partial^2W}{\partial {c}^2}} \) is independent of c, so we may conclude W is either strictly convex or strictly concave in c. The next step is to evaluate the slope of W at \( c=\overline{c} \) and c = 0.

$$ S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=0\right)=S\left({z}^2{\alpha}^2\left(\alpha +2{S}_H\right)\left(3{S}_H-2{S}_L\right)-z\alpha \left[\begin{array}{l}{\alpha}^2\left({S}_H+{S}_L\right)+\\ {}\alpha {S}_L\left(2{S}_L-{S}_H\right)-{S}_L\left(8{S}_H^2-4{S}_H{S}_L-{S}_L^2\right)\end{array}\right]+{S}_L\left[\begin{array}{l}{\alpha}^3+{\alpha}^2\left(2{S}_H-3{S}_L\right)\\ {}+\alpha \left(2{S}_H^2-6{S}_H{S}_L+{S}_L^2\right)+{S}_L^2\left({S}_H-{S}_L\right)\end{array}\right]\right)\equiv S(B). $$

Although B is a complex expression it is easy to establish that 1) B is strictly convex in α , 2) \( {\scriptscriptstyle \frac{\partial B}{\partial \alpha }}>0 \) at α = 0 and 3) B > 0 for α = 0. Hence, \( S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=0\right)>0 \). \( S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=\overline{c}\right)=S\left(\begin{array}{l}z\alpha {S}_H\left(3{S}_H-2{S}_L\right)+{\alpha}^2\left({S}_H-{S}_L\right)+\\ {}\alpha \left(2{S}_H^2-5{S}_H{S}_L+2{S}_L^2\right)-{S}_L\left({S}_H^2-2{S}_H{S}_L+{S}_L^2\right)\equiv S(C).\end{array}\right. \)

The function C is increasing in α. Moreover, C is negative for α = 0 and positive for α = S L . Hence, \( S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=\overline{c}\right)>0 \) for α sufficiently close to S L .

Proof (Proposition 4.4)

It is straightforward to aggregate welfare and to insert equilibrium quantities and prices given c≥0. Let welfare be denoted by W. \( {\scriptscriptstyle \frac{\partial^2W}{\partial {c}^2}} \) is negative so W is strictly concave in c. The next step is to evaluate the slope of W at \( c=\tilde{c} \) and c = 0. \( S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=0\right)=S\left(\alpha z\left(\alpha +2{S}_H\right)-2{\alpha}^2+\alpha \left(3{S}_L-2{S}_H\right)+4{S}_H{S}_L\right) \) which is positive even for z = 1. \( S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=\tilde{c}\right)=S\left({\alpha}^2+\alpha \left(3{S}_L-2{S}_H\right)-4{S}_H\left({S}_H-{S}_L\right)-z\alpha \left(\alpha +2{S}_H\right)\left(\alpha +2{S}_H-{S}_L\right)\right)\equiv S(D). \) The function D is concave in α and negatively sloped both for α = 0 and for α = S L . Hence D is maximal for α = 0. and it is easy to check that D < 0 for α = 0. Hence, \( S\left({\scriptscriptstyle \frac{\partial W}{\partial c}}\kern0.5em \Big|\kern0.5em c=\tilde{c}\right)<0 \).

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Häckner, J., Muren, A. Counterfeiting and Negative Consumption Externalities – A Closer Look. J Ind Compet Trade 15, 337–350 (2015).

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JEL Classification

  • L13
  • D62


  • Consumption externalities
  • Counterfeiting
  • Product differentiation