Furman ratio without Cobb-Douglas

For discrete changes in the capital-income tax rate.

TheoryGuru applications

Background

Obama administration economists Furman and Summers claimed that only a fraction of the revenue loss from a corporate tax cut benefits labor.  But the standard supply and demand model, which for these purposes is a generalization of long run behavior in the neoclassical growth model, says the opposite.

Here we prove that by machine, without assuming any functional form for the aggregate production function.  k denotes the aggregate capital stock, f[k] aggregate output gross of depreciation (the aggregate quantity of labor is fixed), and τ the capital-income tax rate.

Setup

Get@"http://economicreasoning.com"

Definitions

laborincome[k_]=f[k]-f'[k]k

lrcapitalequilibrium[τ_,k_]:=(* willingness to pay for capital *)(1-τ)f'[k]==ρ+δ (* LR willingness to supply it *)

signconditions={δ>0,ρ>0,(Subscript[k, 1])>0,(Subscript[k, 2])>0,0≤(Subscript[τ, 1])<(Subscript[τ, 2])<1,SameSign[f'[(Subscript[k, 2])]-f'[(Subscript[k, 1])],(Subscript[k, 1])-(Subscript[k, 2])],
(* concave production *)f'[(Subscript[k, 1])]((Subscript[k, 1])-(Subscript[k, 2]))<f[(Subscript[k, 1])]-f[(Subscript[k, 2])]<f'[(Subscript[k, 2])]((Subscript[k, 1])-(Subscript[k, 2]))∨(Subscript[k, 1])==(Subscript[k, 2]),
SameSign[laborincome[(Subscript[k, 2])]-laborincome[(Subscript[k, 1])],(Subscript[k, 2])-(Subscript[k, 1])],
SameSign[f[(Subscript[k, 2])]-f[(Subscript[k, 1])],(Subscript[k, 2])-(Subscript[k, 1])]};

revenue[τ_,k_]:=τ (f'[k]-δ)k

furmanratio:=((laborincome[(Subscript[k, 2])]-laborincome[(Subscript[k, 1])])/(revenue[(Subscript[τ, 1]),(Subscript[k, 1])]-revenue[(Subscript[τ, 2]),(Subscript[k, 2])]))

Interesting but not necessary assumptions

elasticcapitaldemand=((Subscript[k, 2])f'[(Subscript[k, 2])]-(Subscript[k, 1])f'[(Subscript[k, 1])])((Subscript[k, 2])-(Subscript[k, 1]))≥0;

wrongsideoflaffercurve=(revenue[(Subscript[τ, 2]),(Subscript[k, 2])]-revenue[(Subscript[τ, 1]),(Subscript[k, 1])])((Subscript[τ, 2])-(Subscript[τ, 1]))≤0;

Results

Taxation reduces the stock capital and the amount of labor income

TheoryGuru[{lrcapitalequilibrium[(Subscript[τ, 1]),(Subscript[k, 1])],lrcapitalequilibrium[(Subscript[τ, 2]),(Subscript[k, 2])],
Most@signconditions},
(Subscript[k, 2])<(Subscript[k, 1])∧laborincome[(Subscript[k, 1])]>laborincome[(Subscript[k, 2])]]

Taxation reduces labor income more than it increases revenue

TheoryGuru[{lrcapitalequilibrium[(Subscript[τ, 1]),(Subscript[k, 1])],lrcapitalequilibrium[(Subscript[τ, 2]),(Subscript[k, 2])],
Most@signconditions},
revenue[(Subscript[τ, 1]),(Subscript[k, 1])]+laborincome[(Subscript[k, 1])]>revenue[(Subscript[τ, 2]),(Subscript[k, 2])]+laborincome[(Subscript[k, 2])]]

Either the Furman ratio exceeds one or the tax is reducing revenue

TheoryGuru[{lrcapitalequilibrium[(Subscript[τ, 1]),(Subscript[k, 1])],lrcapitalequilibrium[(Subscript[τ, 2]),(Subscript[k, 2])],
Most@Most@signconditions},

furmanratio>1
∨
wrongsideoflaffercurve]

TheoryGuru[{lrcapitalequilibrium[(Subscript[τ, 1]),(Subscript[k, 1])],lrcapitalequilibrium[(Subscript[τ, 2]),(Subscript[k, 2])],
Most@Most@signconditions},

furmanratio>1
∨
furmanratio<0]