The strategic role of public authority in the control of - - PowerPoint PPT Presentation

The strategic role of public authority in the control

• f countereinfing. A differential game approach.

Marta Biancardi* Andrea Di Liddo* Giovanni Villani**

*Department of Economics - University of Foggia Largo Papa Giovanni Paolo II, 1 71100 Foggia - Italy **Department of Economics and Finance - University of Bari Largo Abbazia S.Scolastica, 53 70124 Bari - Italy e-mail: giovanni.villani@uniba.it

NED2019 - KIEV, 4-6 September 2019

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Introduction

• The counterfeiting is the unauthorized use of a registered

trademark on a product that is identical or similar to the product for which is registered and used.

• Counterfeiting has consequences on legal producers and

consumers.

• We have two types of counterfeiting: deceptive

counterfeiting and non-deceptive counterfeiting.

• To combat the counterfeting there are two importants

measures: the monitoring control and the sanctions.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Introduction-Literature review

• Grossman and Shapiro (1988) show a model of

counterfeiting in luxury market. They present a dynamic model between two countries and two policies are analyzed: enforcement policy and imposition of a tariff on low-quality products. These measures are exogenous.

• Banerjee (2003) examines the government’s role in

restricting piracy in a software market. The author assumes government chooses these measures to maximize domestic social-welfare subject to a balanced budget constraint. In these two paper the difference in production cost between the legal firm and counterfeiter is not considered.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Introduction-Literature review

• Yao (2015) examines the price and welfare implications of

demand side penalties in the context of deceptive counterfeiting showing that the penalties are not recommended in some countries. The public measures are exogenous.

• Tsai (2012) considers a vertical product differentiation

model to discuss the influences caused by counterfeiting

• n price and outputs of original products, counsumer

surplus and social welfare.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

The Model: assumptions

We consider a planning horizon [0, T]. In the absence of counterfeiting, the demand for the genuine firm is: Dg,nc(t) = R(t)

• α − βpg(t)
• +

where

• pg(t) is the price of a unity of the genuine product;
• R(t) the manufacturers brand reputation;

In the presence of counterfeiting, the demand of the fake is: Dc(t) = R(t)[ρ(pg(t) − pc(t))]+ where pc(t) is the price of a unity of the counterfeited product. The parameter ρ can be assumed as a measure of the competition between the two firms. Obviously 0 ≤ pc < pg

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

The Model: assumptions

In the presence of counterfeiting, the demand of the legal firm is: Dg(t) = R(t)

• α − βpg (t) − ρ
• pg (t) − pc (t)
• +

We assume that: 0 ≤ ρ ≤ β that is, the direct-price effect is larger than the cross-price effect. Moreover:

• The genuine firm and the counterfeiters incurs linear

production costs and that c > 0 and b > 0 for a genuine item and a fake, respectively. Obviously b < c.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

The Model: assumptions

• The evolution of the brand’s reputation is described by the

following linear differential equation: ˙ R = ka(t) − σR(t); R(0) = R0 ≥ 0, where a(t) is the advertising effort, k > 0 is an efficiency parameter and σ > 0 is the decay rate;

• The advertising cost is convex increasing and given by the

quadratic function ω 2 a2(t) where ω is a positive parameter

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Role of public authority: Confiscation

• The (average) counterfeiter profit is

Πc = T {(1 − φ)pc(t) − b} Dc(t) dt, where φ ∈ [0, 1] is the monitoring level.

• Since counterfeiters can enter/exit the market freely, they

sell the fakes at a price pc such that their profit is zero: pc = b 1 − φ

• The (average) genuine firm profit is given by

Πg = T

• (pg(t) − c)Dg(t) − ω

2 a2(t)

• dt + sR(T)

where sR(T) is the salvage value of the brand at T, with s ≥ 0.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Role of public authority: Fixed Fine

If the counterfeiters are caught, they are forced to pay a penalty

• f an amount F. For instance the software piracy and so b = 0.
• The (average) counterfeiter profit is

Πc = T [(1 − φ)pc(t)Dc(t) − φF] dt

• The price pc must be a solution of the following quadratic

equation pc(pg − pc) = φF (1 − φ)ρR(t). (1)

• Eq. (1) has two positive solutions if

pg ≥

• 4φF

(1 − φ)ρR(t),

• therwise it has no solutions.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Let ˜ pc the price of the pirated software. The (average) genuine firm payoff is given by Πg = T

• (pg − c)R(t)[α − βpg − ρ(pg − ˜

pc)] − ω 2 a2(t)

• dt + sR(T).

Let pg <

• 4φF

(1 − φ)ρR(t). Then pirates cannot stay in the market so that the demand of the genuine is given by Dg(t) = R(t)

• α − βpg (t)
• .

The (average) genuine firm payoff is given by Πg = T

• (pg(t) − c)R(t)
• α − βpg (t)
• − ω

2 a2(t)

• dt + sR(T).

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Role of Public Authority: Fines counterfeiters

• The (average) counterfeiter profit is

Πc = T

• [(1 − φ)pc(t) − b]Dc(t) − φfpg(t)Dc(t)
• dt,

where f is the ratio of fines. This implies that pc = b + φfpg 1 − φ . (2)

• The (average) genuine firm profit is given by

Πg = T

• (pg(t) − c)Dg(t) + yφfpg(t)Dc(t) − ω

2 a2(t)

• dt + sR(T

where y ∈ [0, 1].

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Role of Public Authority: Fines counterfeiters

• The social planner chooses the level of enforcement φ that

maximizes the welfare. A type of cost function widely used is λ(φ) = hφ2; h > 0.

• The welfare W is the unweighted sum of the following

components: the profit of the genuine producer Πg, the surplus of consumers S (both original and fake consumers), the amount of sanctions eventually collected by the social planner, minus the enforcement costs λ(φ).

• The game is solved as a leader-follower game. The social

planner (leader) chooses φ; the genuine firm (the follower), having observed the choice of the leader, chooses the price pg and the advertising.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Role of Public Authority: Assumptions

• We assume that counterfeiters set the price of a fake as a

fixed fraction of the price of a genuine item, that is pc = δpg; 0 < δ < 1. (3)

• We assume that y = 0, and so all fines are taken by public
• authority. The (average) genuine firm profit is given by

Πg = T

• (pg(t) − c)Dg(t) − ω

2 a2(t)

• dt + sR(T)
• Social Welfare is:

W = T [S(t) + πg(t) + φfpgDc(t) − λ(φ)] dt + sR(T)

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Surplus of consumers

• Surplus of legal consumers; We determine pgmax, the price

such that the demand is equal to zero: pgmax = α β + ρ(1 − δ) and so: S∗

g =

pgmax

pg

Dg(t) dp = R(t){c(β + ρ(1 − δ)) − α}2 8(β + ρ(1 − δ)) .

• Surplus of fake consumers. We have that pcmax = pg and

so: S∗

c = (1 − φ)

pg

pc

Dc(t) dp and so: S∗

c = (1 − φ)ρ(1 − δ)2R(t)[c(β + ρ(1 − δ)) + α]2

8[β + ρ(1 − δ)]2 .

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Feedback Nash Equilibria

First step: max

pg,a Πg

subject to: ˙ R(t) = ka(t) − σR(t) HJB equation for legal firm: −∂Vg

∂t (t, R(t)) =

max

pg, a (pg − c)Dg − ω

2 a2(t) + ∂Vg ∂R (t, R(t))(ka(t) − σR(t))

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Feedback Nash Equilibria

Second step: max

φ

W subject to ˙ R(t) = ka(t) − σR(t) HJB equation for public authority: −∂VP ∂t = max

φ {S(p∗ g) + πg(p∗ g) + φfp∗ gDc(p∗ g) − kφ2 + ∂VP

∂R (ka∗ − σR)} (4)

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Feedback Nash Equilibria

Proposition The unique Feedback Nash equilibrium exists and it is give by: p∗

g =α + c(β + ρ(1 − δ))

2(β + ρ(1 − δ)) (5) φ∗ = R(t)ρ(1 − δ) [c(β + ρ(1 − δ)) + α]2 (2f + δ − 1) 16h[β + ρ(1 − δ)]2 . (6)

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Feedback Nash Equilibria

The optimal advertising level: a∗(t) = k ω Γ1 σ +

• s − Γ1

σ

• e−σ(T−t)
• (7)

and: R∗(t) = e−σtψ(t) 2ωσ2 where: ψ(t) = k[(sσ − Γ1)eσ(2t−T) − (sσ − Γ1)e−σT + 2Γ1eσt − 2Γ1] +2σ2G0ω. and: Γ1 = [α − c(β + ρ(1 − δ))]2 4(β + ρ(1 − δ))

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Feedback Nash Equilibria

• The advertising effort is a∗(t) > 0 for all t ∈ [0, T] and it is

increasing if s > Γ1 σ , decreasing if s < Γ1 σ , constant if s = Γ1 σ .

• The monitoring control: φ∗ ∈ [0, 1] ⇐

⇒ f ∈ [1−δ

2 , f ∗]

• ∂φ∗

∂h < 0; ∂φ∗ ∂c > 0; ∂φ∗ ∂ρ > 0 ⇐ ⇒ ρ ∈ [α − cβ −

• α2 − 8αcβ

2c(1 − ω) , α − cβ +

• α2 − 8αcβ

2c(1 − ω) ]

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Case with y = 1

• All fines are given to genuine firm. It is not possible to

compute analytically the Feedback Nash equilibria.

• The (average) genuine firm profit is given by

Πg = T

• (pg(t) − c)Dg(t) + φfpgDc(t) − ω

2 a2(t)

• dt + sR(T)
• Social Welfare is:

W = T [S(t) + πg(t) − λ(φ)] dt + sR(T)

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Case with y = 1- Ex-post analysis - No strategic monitoring control

Based on results of our previous paper Biancardi et al. (2019), assuming profit-maximization behaviour and b = 0, the players’

• ptimization problem is given by:

max

pc(t) Πc = max pc(t)

T

• (1 − φ)pc(t)Dc(t) − φfpg(t)Dc(t)
• dt

max

pg(t),a(t)

T

• (pg(t) − c)Dg(t) + φfpg(t)Dc(t) − w

2 a2(t)

• dt + sR(T)

subject to the dynamics in ˙ R.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Proposition The unique FeedBack Nash equilibrium is given by: p∗

c = (1 − φ + fφ) µ;

p∗

g = 2 (1 − φ) µ;

and the optimal trajectory about the advertising effort of legal producer is: a∗(t) = k w

• −kg

σ +

• s + kg

σ

• e−σ(T−t)
• where:

µ = α + c(β + ρ) γ ; γ = f 2φ2ρ + fφρ(3φ − 4) + (1 − φ)(4β + 3ρ) kg = cα + 2(1 − φ)[f 2φ2ρ − fφρ(2 − φ) + (1 − φ)(2β + ρ)]µ2 + ... [fφρc − c(1 − φ)(2β + ρ) − 2α(1 − φ)]µ

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Welfare Analysis

The horizon time T = 1; the advertising cost ω = 1; the initial level of brand R0 = 1; the decay rate σ = 1; the adverting efficiency on brand k = 1; the demand parametres α = 1 and β = 1.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

(a) 0.3c; 0.3ρ (b) 0.3c; 0.6ρ (c) 0.3c; 0.9ρ (d) 0.6c; 0.3ρ (e) 0.6c; 0.6ρ (f) 0.6c; 0.9ρ (g) 0.8c; 0.3ρ (h) 0.8c; 0.6ρ (i) 0.8c; 0.9ρ

Figura: The green surface denotes the combination of (φ, f) such that the social welfare W(φ > 0) is larger than W(φ = 0) assuming h = 0.10

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

(a) 0.3c; 0.3ρ (b) 0.3c; 0.6ρ (c) 0.3c; 0.9ρ (d) 0.6c; 0.3ρ (e) 0.6c; 0.6ρ (f) 0.6c; 0.9ρ (g) 0.8c; 0.3ρ (h) 0.8c; 0.6ρ (i) 0.8c; 0.9ρ

Figura: The green surface denotes the combination of (f, φ) such that the social welfare W(φ > 0) is larger then W(φ = 0) assuming h = 0.30.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019

Conclusions

• Three policies in order to cambact the counterfeintig. The

determination of φ in order to maximize the social welfare when a fraction of fines are given to legal firm is very hard to determine;

• The assumption of production cost c = 0 influences the

welfare;

• The implication of φ in the welfare analysis making an

ex-post analysis.

MARTA BIANCARDI*, ANDREA DI LIDDO*, GIOVANNI VILLANI** NED 2019 - KIEV, 4-6 SEPTEMBER 2019