Excise Taxation in a Competitive Market

TheoryGuru applications

Load Economicreasoning package only if it is not already loaded

If[Length@Names["PLTools`*"]<10,
Get["http://economicreasoning.com"]]

Setup

Chain-rule proofs

equilibrium=d[p+t]==s[p];

revenue=t d[p+t];

Global proofs (no calculus)

equilibriumrp=d[(Subscript[p, #])+(Subscript[t, #])]==s[(Subscript[p, #])]&/@{L,H}

supslopesup=(s[(Subscript[p, H])]-s[(Subscript[p, L])])((Subscript[p, H])-(Subscript[p, L]))>0;

dmdslopesdown=(d[(Subscript[p, H])+(Subscript[t, H])]-d[(Subscript[p, L])+(Subscript[t, L])])((Subscript[p, H])+(Subscript[t, H])-((Subscript[p, L])+(Subscript[t, L])))<0;

(Subscript[rev, x_]):=(Subscript[t, x])d[(Subscript[p, x])+(Subscript[t, x])]

Results obtained with chain rule

Quantity depressed

TheoryGuru[{((\[DifferentialD]equilibrium)/(\[DifferentialD]t)),d'[p+t]<0,s'[p]>0},
((\[DifferentialD]s[p])/(\[DifferentialD]t))<0]

Supply price depressed

Demand and supply with no quantitative restrictions

TheoryGuru[{((\[DifferentialD]equilibrium)/(\[DifferentialD]t)),d'[p+t]<0,s'[p]>0},
-1<((\[DifferentialD]p)/(\[DifferentialD]t))<0]

With a specific demand elasticity

TheoryGuru[{((\[DifferentialD]equilibrium)/(\[DifferentialD]t)),d'[p+t]((p+t)/(d[p+t]))==-(1/7),
d[p+t]>0,p+t>0,s'[p]>0},
-1<((\[DifferentialD]p)/(\[DifferentialD]t))<0]

Find sufficient conditions for upward-sloping Laffer curve

TheoryExtra[{((\[DifferentialD]equilibrium)/(\[DifferentialD]t)),d[p+t]>0,d'[p+t]<0,s'[p]>0},
((\[DifferentialD]revenue)/(\[DifferentialD]t))>0,
{t,s'[p],d'[p+t]}]

TheoryGuru[{((\[DifferentialD]equilibrium)/(\[DifferentialD]t)),d[p+t]>0,d'[p+t]<0,s'[p]>0,t==0},
((\[DifferentialD]revenue)/(\[DifferentialD]t))>0]

Variable interpretations

Global results (no calculus)

Supply price depressed

TheoryGuru[{equilibriumrp,supslopesup,dmdslopesdown,(Subscript[t, H])>(Subscript[t, L])},(Subscript[p, L])>(Subscript[p, H])>(Subscript[p, L])-((Subscript[t, H])-(Subscript[t, L]))]

Sufficient conditions for upward-sloping Laffer curve

TheoryOverlapp[{equilibriumrp,dmdslopesdown,supslopesup,d[(Subscript[p, H])+(Subscript[t, H])]>0},
(Subscript[t, L])<(Subscript[t, H])≤0,
(Subscript[rev, H])>(Subscript[rev, L])]

Variable interpretations