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Page 1: Simulations for the Paper Tax Compliance and Firms ...darp.lse.ac.uk/resources/data/Bayer-Cowell/final...I af 2abf2 2q1+ 2cq1 3bfq1+ 3bcfq1+ 2q12 + 3bfq12 bfq2+ bcfq2+ 2q1q2+ 4bfq1q2+

Simulations for the Paper"Tax Compliance and Firms' Strategic Interdependance"The simulation in the paper

ü Setup

Invese demand, gross profit, evasion cost, detection rule

P@q1_, q2_D := 1 − q1 − q2gpi@qi_D := HP@q1, q2D − cL qiec@qi_, di_D := Hgpi@qiD − diL^2β@di_, dj_D := a + b Hdj − diL

Expected profit

Π@qi_, qj_, di_, dj_D :=

FullSimplify@H1 − β@di, djDL Hgpi@qiD − t diL + β@di, djDHgpi@qiD − t gpi@qiD − f Hgpi@qiD − diLL − ec@qi, diDD

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Reference b=0Solve@8D@Π@q1, q2, d1, d2D ê. b → 0, d1D 0, D@Π@q2, q1, d2, d1D ê. b → 0, d2D 0,

D@Π@q1, q2, d1, d2D ê. b → 0, q1D 0, D@Π@q2, q1, d2, d1D ê. b → 0, q2D 0<, 8d1, d2, q1, q2<D::d1 →

118

I2 − 4 c + 2 c2 + 9 a f − 9 t + 9 a tM,d2 →

118

I2 − 4 c + 2 c2 + 9 a f − 9 t + 9 a tM, q1 →1 − c3

, q2 →1 − c3

>>FullSimplify@%D::d1 →

118

H2 + 2 H−2 + cL c − 9 t + 9 a Hf + tLL,d2 →

118

H2 + 2 H−2 + cL c − 9 t + 9 a Hf + tLL, q1 →1 − c3

, q2 →1 − c3

>>We get

d1f =118

I2 − 4 c + 2 c2 + 9 a f − 9 t + 9 a tMdf2 =

118

I2 − 4 c + 2 c2 + 9 a f − 9 t + 9 a tMqf1 =

1 − c3

qf2 =1 − c3

118

I2 − 4 c + 2 c2 + 9 a f − 9 t + 9 a tM118

I2 − 4 c + 2 c2 + 9 a f − 9 t + 9 a tM1 − c3

1 − c3

ü Now the case of b>0

Solve@8D@Π@q1, q2, d1, d2D, d1D 0, D@Π@q2, q1, d2, d1D, d2D 0<, 8d1, d2<D;FullSimplify@%D99d1 → −I4 q1 Hq1 + q2L + b2 H−1 + c + q1 + q2L H2 q1 + q2L Hf + tL2 +

2 H2 H−1 + cL q1 + tL + b Hf + tL H2 H−1 + c + q1 + q2L H3 q1 + q2L + 3 tL −

a Hf + tL H2 + 3 b Hf + tLLM ë H4 + b Hf + tL H8 + 3 b Hf + tLLL,d2 → −I4 q2 H−1 + c + q1 + q2L + 2 t + b2 H−1 + c + q1 + q2L Hq1 + 2 q2L Hf + tL2 +

b Hf + tL H2 H−1 + c + q1 + q2L Hq1 + 3 q2L + 3 tL −

a Hf + tL H2 + 3 b Hf + tLLM ë H4 + b Hf + tL H8 + 3 b Hf + tLLL==Declarations as functions of quantities and parameteres

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d1b = d1 ê. Flatten@%Dd2b = −I4 q2 H−1 + c + q1 + q2L + 2 t +

b2 H−1 + c + q1 + q2L Hq1 + 2 q2L Hf + tL2 + b Hf + tL H2 H−1 + c + q1 + q2L Hq1 + 3 q2L + 3 tL −

a Hf + tL H2 + 3 b Hf + tLLM ë H4 + b Hf + tL H8 + 3 b Hf + tLLL−I4 q1 Hq1 + q2L + b2 H−1 + c + q1 + q2L H2 q1 + q2L Hf + tL2 +

2 H2 H−1 + cL q1 + tL + b Hf + tL H2 H−1 + c + q1 + q2L H3 q1 + q2L + 3 tL −

a Hf + tL H2 + 3 b Hf + tLLM ë H4 + b Hf + tL H8 + 3 b Hf + tLLL−I4 q2 H−1 + c + q1 + q2L + 2 t +

b2 H−1 + c + q1 + q2L Hq1 + 2 q2L Hf + tL2 + b Hf + tL H2 H−1 + c + q1 + q2L Hq1 + 3 q2L + 3 tL −

a Hf + tL H2 + 3 b Hf + tLLM ë H4 + b Hf + tL H8 + 3 b Hf + tLLLCheck for the tax rate that induces truthful declarations (in equilibirum for given q1=q2)

d1b − gpi@q1D ê. q2 → q1

−H1 − c − 2 q1L q1 −I8 q12 + 3 b2 q1 H−1 + c + 2 q1L Hf + tL2 + 2 H2 H−1 + cL q1 + tL + b Hf + tL H8 q1 H−1 + c + 2 q1L + 3 tL −

a Hf + tL H2 + 3 b Hf + tLLM ë H4 + b Hf + tL H8 + 3 b Hf + tLLLSolve@% 0, tD::t → −

a f−1 + a

>>Finding the optimal quantities

D@FullSimplify@Π@q1, q2, d1b, d2bDD, q1D 0;% ê. q2 → q1;FullSimplify@%D;Solve@%, q1D;FullSimplify@%D::q1 → −

H−1 + cL H4 H−1 + tL + b Hf + tL H−8 + 6 t + Hf + tL Hb H−3 + tL + 2 a H1 + b Hf + tLLLLL12 H−1 + tL + b Hf + tL H4 H−6 + 5 tL + Hf + tL Hb H−9 + 5 tL + 4 a H1 + b Hf + tLLLL >>

These are the otimal quantities

q1b := −H−1 + cL H4 H−1 + tL + b Hf + tL H−8 + 6 t + Hf + tL Hb H−3 + tL + 2 a H1 + b Hf + tLLLLL12 H−1 + tL + b Hf + tL H4 H−6 + 5 tL + Hf + tL Hb H−9 + 5 tL + 4 a H1 + b Hf + tLLLL

q2b := −H−1 + cL H4 H−1 + tL + b Hf + tL H−8 + 6 t + Hf + tL Hb H−3 + tL + 2 a H1 + b Hf + tLLLLL12 H−1 + tL + b Hf + tL H4 H−6 + 5 tL + Hf + tL Hb H−9 + 5 tL + 4 a H1 + b Hf + tLLLL

ü Parameters

ClearAll@c, a, fD

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c = 0.1a = 0.2f = 0.50.1

0.2

0.5

ü Contourplots

The Aggregate Quantity first

quantity = FullSimplify@q1b + q2bD;Plot3D@%, 8b, 0, 2.5<, 8t, 0.125, 0.4<, ColorFunction → GrayLevel, PlotPoints → 50Dgr1 = ContourPlot@quantity, 8b, 0, 2.5<, 8t, 0.125, 0.4<, Contours → 8,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@Aggregate Quantity, 24, BoldD ,Frame → True, FrameLabel → 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,ImageSize → Large, LabelStyle → LargeD

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Tax

Rat

etAggregateQuantity

Now the revenue

revenue = FullSimplify@2 d1b t ê. 8q2 → q1<D;revenue = revenue ê. q1 → q1b;

gr2 = ContourPlot@revenue, 8b, 0, 2.5<, 8t, 0.125, 0.4<, Contours → 20,ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 100, PlotLabel → Style@Revenue, 24, BoldD , Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

Plot3D@revenue, 8b, 0, 2.5<, 8t, 0.125, 0.4<, ColorFunction → GrayLevel, PlotPoints → 50 D

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Tax

Rat

etRevenue

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Waste

waste = 2 ec@q1b, d1bD ê. q2 → q1;waste = waste ê. q1 → q1b;

gr3 = ContourPlot@waste, 8b, 0, 2.5<, 8t, 0.125, 0.4<, Contours → 10,ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@Waste, 24, BoldD , Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,ImageSize → Large, LabelStyle → LargeD

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Reactivity b

Tax

Rat

et

Waste

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Plot3D@−waste, 8b, 0, 2.5<, 8t, 0.125, 0.4<, ColorFunction → GrayLevel, PlotPoints → 50D

Surplus

surplus = 2 H1 − cL q1b − 2 q1b^2 − waste;

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gr4 = ContourPlot@surplus, 8b, 0, 2.5<, 8t, 0.125, 0.4<, Contours → 35,ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@Surplus, 24, BoldD , Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

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Tax

Rat

et

Surplus

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Plot3D@surplus, 8b, 0, 4<, 8t, 0.125, 0.4<, ColorFunction → GrayLevel, PlotPoints → 50D

Here we check the regions where increasing t or b increases surplus

dsdt = D@surplus, tD;First for t

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gr5 = RegionPlot@dsdt > 0, 8b, 0, 2.5<, 8t, 0.125, 0.35<, ColorFunction → Gray,PlotPoints → 200, PlotLabel → Style@"∂Surplusê∂t", 24, BoldD ,Frame → True, FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

RegionPlot::color : [email protected] is not a valid color or gray-level specification. à

In the shaded region increasing t increases surplus. Increasing t has three effects 1) Increasing the primary evasion incentiveand therefore the waste 2) For positive b increasing the externality and therefore the quantities, which 3) Decreases profitsand therefore the secondary evasion incentives and waste. In the grey region effects 2+3 dominate effect 1

dsdb = D@surplus, bD;Now for b

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RegionPlot@dsdb > 0, 8b, 0, 2.5<, 8t, 0.125, 0.4<,ColorFunction → GrayLevel, PlotPoints → 200, PlotLabel → Surplus , Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<, PlotRange → AllD

Increasing b always increases welfare

Evasion as a fraction of the total profit

evasion = 1 − d1b ê gpi@q1D;evasion = evasion ê. 8q1 → q1b, q2 → q1b<;Plot3D@−evasion, 8b, 0, 2.5<, 8t, .125, .4<, ColorFunction → GrayLevelDgr6 = ContourPlot@−evasion, 8b, 0, 2.5<, 8t, 0.125, 0.4<, Contours → 15,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@Evasion Pecentage, 24, BoldD ,Frame → True, FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

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Tax

Rat

etEvasion Pecentage

CollusionIn what follows, we use the same parameters for a simulation of collusive behaviour

ü Collusion at the Declaration Stage

ClearAll@a, c, fDJointΠ := Π@q1, q2, d1, d2D + Π@q2, q1, d2, d1D

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Solve@8D@JointΠ, d1D 0, D@JointΠ, d2D 0<, 8d1, d2<D::d1 →

−1

2 H1 + 2 b f + 2 b tL I−a f − 2 a b f2 − 2 q1 + 2 c q1 − 3 b f q1 + 3 b c f q1 + 2 q12 + 3 b f q12 − b f q2 +

b c f q2 + 2 q1 q2 + 4 b f q1 q2 + b f q22 + t − a t + 2 b f t − 4 a b f t − 3 b q1 t +

3 b c q1 t + 3 b q12 t − b q2 t + b c q2 t + 4 b q1 q2 t + b q22 t + 2 b t2 − 2 a b t2M,d2 → −

12 H1 + 2 b f + 2 b tL I−a f − 2 a b f2 − b f q1 + b c f q1 + b f q12 − 2 q2 + 2 c q2 − 3 b f q2 +

3 b c f q2 + 2 q1 q2 + 4 b f q1 q2 + 2 q22 + 3 b f q22 + t − a t + 2 b f t − 4 a b f t − b q1 t +

b c q1 t + b q12 t − 3 b q2 t + 3 b c q2 t + 4 b q1 q2 t + 3 b q22 t + 2 b t2 − 2 a b t2M>>FullSimplify@%D::d1 → −

12 + 4 b Hf + tL H2 H−1 + cL q1 + 2 q1 Hq1 + q2L + t +

b Hf + tL HH−1 + c + q1 + q2L H3 q1 + q2L + 2 tL − a Hf + tL H1 + 2 b Hf + tLLL,d2 → −

12 + 4 b Hf + tL H2 q2 H−1 + c + q1 + q2L + t + b Hf + tL HH−1 + c + q1 + q2L Hq1 + 3 q2L + 2 tL −

a Hf + tL H1 + 2 b Hf + tLLL>>d1c = −

12 + 4 b Hf + tL H2 H−1 + cL q1 + 2 q1 Hq1 + q2L + t +

b Hf + tL HH−1 + c + q1 + q2L H3 q1 + q2L + 2 tL − a Hf + tL H1 + 2 b Hf + tLLL;d2c =

−2 q2 H−1 + c + q1 + q2L + t + b Hf + tL HH−1 + c + q1 + q2L Hq1 + 3 q2L + 2 tL − a Hf + tL H1 + 2 b Hf + tLL

2 + 4 b Hf + tL;

These are the jointly optimal declarations

D@FullSimplify@Π@q1, q2, d1c, d2cDD, q1D 0;FullSimplify@Solve@% ê. q2 → q1, q1DD::q1 → −

H−1 + cL H2 H−1 + tL + b Hf + tL H−4 + 3 t + Hf + tL H−b t + a H1 + b Hf + tLLLLL2 H3 H−1 + tL + b Hf + tL H−6 + 5 t + Hf + tL H−b t + a H1 + b Hf + tLLLLL >>

q1c = −H−1 + cL H2 H−1 + tL + b Hf + tL H−4 + 3 t + Hf + tL H−b t + a H1 + b Hf + tLLLLL

2 H3 H−1 + tL + b Hf + tL H−6 + 5 t + Hf + tL H−b t + a H1 + b Hf + tLLLLL ;

q2c = −H−1 + cL H2 H−1 + tL + b Hf + tL H−4 + 3 t + Hf + tL H−b t + a H1 + b Hf + tLLLLL

2 H3 H−1 + tL + b Hf + tL H−6 + 5 t + Hf + tL H−b t + a H1 + b Hf + tLLLLL ;

ü Parameters

c = 0.1a = 0.2f = 0.50.1

0.2

0.5

Quantities

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quantityc = FullSimplify@q1c + q2cD;Plot3D@%, 8b, 0, 2.5<, 8t, 0.125, 0.4<, ColorFunction → GrayLevel, PlotPoints → 50Dgr1c = ContourPlot@quantityc, 8b, 0, 2.5<, 8t, 0.125, 0.4<, Contours → 8,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@Aggregate Quantity, 24, BoldD ,Frame → True, FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,LabelStyle → Large, ImageSize → LargeD

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etAggregateQuantity

Revenue

revenuec = FullSimplify@2 d1c t ê. 8q2 → q1<D;revenuec = revenuec ê. q1 → q1c;gr2c = ContourPlot@revenuec, 8b, 0, 2.5<, 8t, 0.125, 0.35<, Contours → 17,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 100, PlotLabel → Style@Revenue, 24, BoldD , Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<, PlotRange → 80, 0.0272<,ClippingStyle → Black, LabelStyle → Large, ImageSize → LargeD

Plot3D@revenuec, 8b, 0, 2.5<, 8t, 0.125, 0.35<, ColorFunction → GrayLevel,PlotPoints → 50, LabelStyle → Large, ImageSize → LargeD

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Waste

wastec = 2 ec@q1c, d1cD ê. q2 → q1;wastec = wastec ê. q1 → q1c;gr3c = ContourPlot@wastec, 8b, 0, 2.5<, 8t, 0.125, 0.35<, Contours → 15,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@Waste , 24, BoldD, Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

Plot3D@wastec, 8b, 0, 2.5<, 8t, 0.125, 0.4<, ColorFunction → GrayLevel, PlotPoints → 50D

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etWaste

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surplusc = 2 H1 − cL q1c − 2 q1c^2 − wastec;gr4c = ContourPlot@surplusc, 8b, 0, 2.5<, 8t, 0.125, 0.35<, Contours → 40,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@"Surplus Collusion HDeclarationL", 24, BoldD ,Frame → True, FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

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Surplus Collusion HDeclarationL

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dsdtc = D@surplusc, tD;gr5c = RegionPlot@dsdtc > 0, 8b, 0, 2.5<, 8t, 0.125, 0.35<, ColorFunction → GrayLevel,

PlotPoints → 200, PlotLabel → Style@"∂Surplusê∂t", 24, BoldD ,Frame → True, FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

ü Collusion on the production stage (symmetric)

ClearAll@a, c, fDSolve@D@gpi@q1D + gpi@q2D, q1D 0 ê. q2 → q1, q1D::q1 →

1 − c4

>>

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q1p =1 − c4

;

q2p =1 − c4

;

Solve@D@Π@q1p, q2p, d1, d2D, d1D 0 ê. d2 → d1, d1D;FullSimplify@%D::d1 →

14 H2 + b Hf + tLL IH4 a − b H−1 + q1 + q2LL Hf + tL −

2 H−1 + q1 + q2 + 2 tL + c2 H2 + b Hf + tLL + c H−2 + q1 + q2L H2 + b Hf + tLLM>>c = 0.1a = 0.2f = 0.5

d1p =1

4 H2 + b Hf + tLL IH4 a − b H−1 + q1 + q2LL Hf + tL −

2 H−1 + q1 + q2 + 2 tL + c2 H2 + b Hf + tLL + c H−2 + q1 + q2L H2 + b Hf + tLLM;d2p =

dp1;0.1

0.2

0.5

Revenue

revenuep = FullSimplify@2 d1p t ê. 8q2 → q1<D;revenuep = revenuep ê. q1 → q1p;gr1p = ContourPlot@revenuep, 8b, 0, 2.5<, 8t, 0.125, 0.35<, Contours → 17,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 100, PlotLabel → Style@Revenue , 24, BoldD, Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,LabelStyle → Large, ImageSize → LargeD

Plot3D@revenuep, 8b, 0, 2.5<, 8t, 0.125, 0.35<, ColorFunction → GrayLevel, PlotPoints → 50D

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0.0 0.5 1.0 1.5 2.0 2.5

0.15

0.20

0.25

0.30

0.35

Reactivity b

Tax

Rat

etRevenue

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quantityp = FullSimplify@q1p + q2pD;Plot3D@%, 8b, 0, 2.5<, 8t, 0.125, 0.4<, ColorFunction → GrayLevel, PlotPoints → 50DContourPlot@quantityp, 8b, 0, 2.5<, 8t, 0.125, 0.4<, Contours → 8,ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Aggregate Quantity , Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<D

0.0 0.5 1.0 1.5 2.0 2.5

0.15

0.20

0.25

0.30

0.35

0.40

Reactivityb

Tax

Rat

et

Aggregate Quantity

waste

wastep = 2 ec@q1p, d1pD ê. q2 → q1;wastep = wastep ê. q1 → q1p;gr2p = ContourPlot@−wastep, 8b, 0, 2.5<, 8t, 0.125, 0.35<, Contours → 15,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@Waste, 24, BoldD , Frame → True,FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

Plot3D@wastep, 8b, 0, 2.5<, 8t, 0.125, 0.4<, ColorFunction → GrayLevel,PlotPoints → 50, LabelStyle → Large, ImageSize → LargeD

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0.0 0.5 1.0 1.5 2.0 2.5

0.15

0.20

0.25

0.30

0.35

Reactivity b

Tax

Rat

etWaste

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Surplus

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surplusp = 2 H1 − cL q1p − 2 q1p^2 − wastep;gr3p = ContourPlot@surplusp, 8b, 0, 2.5<, 8t, 0.125, 0.35<, Contours → 50,

ContourLabels → None, ContourStyle → Black, ColorFunction → GrayLevel,PlotPoints → 40, PlotLabel → Style@"Surplus Collusion HProductionL", 24, BoldD ,Frame → True, FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → LargeD

0.0 0.5 1.0 1.5 2.0 2.5

0.15

0.20

0.25

0.30

0.35

Reactivity b

Tax

Rat

et

Surplus Collusion HProductionL

In this case for a given b increasing the tax rate always has a negative influence.

Optimal tax rates in the three cases<< PlotLegends`

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graph6 = ContourPlot@8dsdt 0, dsdtc 0, t 0.125<, 8b, 0, 2.5<,8t, 0.08, 0.3<, PlotLabel → Style@"Optimal tax rates", 24, BoldD ,Frame → True, FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → Large,ContourStyle → 88Black<, 8Black, Dashed<, 8Black, DotDashed<<D

ShowLegend@ContourPlot@8dsdt 0, dsdtc 0, t 0.125<, 8b, 0, 2.5<,8t, 0.08, 0.3<, PlotLabel → Style@"Optimal tax rates", 24, BoldD ,Frame → True, FrameLabel −> 88"Tax Rate" t, None<, 8"Reactivity" b, None<<,PlotRange → All, LabelStyle → Large, ImageSize → Large,ContourStyle → 88Black<, 8Black, Dashed<, 8Black, DotDashed<<D,888Graphics@8Black, Line@880, 0<, 81, 0<<D<D, "Competition"<,8Graphics@8Black, Dashed, Line@880, 0<, 81, 0<<D<D, "Collusion on d"<,8Graphics@8Black, DotDashed, Line@880, 0<, 81, 0<<D<D,

"Collusion on q"<<<, LegendPosition → 81, 1<D

0.0 0.5 1.0 1.5 2.0 2.5

0.15

0.20

0.25

0.30

Reactivity b

Tax

Rat

et

Optimal tax rates

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0.0 0.5 1.0 1.5 2.0 2.5

0.15

0.20

0.25

0.30

Reactivity b

Tax

Rat

et

Optimal tax rates

Collusion on q

Collusion on d

Competition

Combine the GraphsExport@"graph.jpg", GraphicsRow@8gr1, gr2<DDgraph.jpg

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Export@"graph2.jpg", GraphicsRow@8gr3, gr4<DDgraph2.jpg

Export@"graph3.jpg", GraphicsRow@8gr4c, gr3p<DDgraph3.jpg

30 finalsimulation.nb