Simulations for the Paper Tax Compliance and Firms...
Transcript of Simulations for the Paper Tax Compliance and Firms...
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
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
2 finalsimulation.nb
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
finalsimulation.nb 3
c = 0.1a = 0.2f = 0.50.1
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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
4 finalsimulation.nb
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Tax
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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
finalsimulation.nb 5
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etRevenue
6 finalsimulation.nb
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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Waste
finalsimulation.nb 7
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;
8 finalsimulation.nb
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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Surplus
finalsimulation.nb 9
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
10 finalsimulation.nb
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
finalsimulation.nb 11
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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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
finalsimulation.nb 13
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
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Quantities
14 finalsimulation.nb
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
finalsimulation.nb 15
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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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etRevenue
finalsimulation.nb 17
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
finalsimulation.nb 19
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
20 finalsimulation.nb
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
>>
finalsimulation.nb 21
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
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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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etRevenue
finalsimulation.nb 23
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
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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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etWaste
finalsimulation.nb 25
Surplus
26 finalsimulation.nb
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
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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`
finalsimulation.nb 27
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
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Optimal tax rates
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Optimal tax rates
Collusion on q
Collusion on d
Competition
Combine the GraphsExport@"graph.jpg", GraphicsRow@8gr1, gr2<DDgraph.jpg
finalsimulation.nb 29
Export@"graph2.jpg", GraphicsRow@8gr3, gr4<DDgraph2.jpg
Export@"graph3.jpg", GraphicsRow@8gr4c, gr3p<DDgraph3.jpg
30 finalsimulation.nb