Furman ratio without Cobb-Douglas

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 specific 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.

For discrete changes in the capital-income tax rate, see this notebook.

Setup

Get@"http://economicreasoning.com"

Definitions

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

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

signconditions={δ>0,ρ>0,k>0,f'[k]-δ>0,0≤τ<1,f''[k]≤0};

revenue[τ_,k_]:=τ (f'[k]-δ)k

furmanratio[τ_,k_]:=(("((\[DifferentialD]laborincome[k])/(\[DifferentialD]τ))")/(-"((\[DifferentialD]revenue[τ, k])/(\[DifferentialD]τ))"))(* Furman and Summers asserted that it does not make sense for this ratio to exceed one *)

Interesting but not necessary assumptions

elasticcapitaldemand=D[k f'[k],k]>0 (* i.e., more capital means more aggregate capital income *)

wrongsideoflaffercurve=((\[DifferentialD]revenue[τ,k])/(\[DifferentialD]τ))<0;

fcd[k_]=(k^α);

Results

Taxation reduces the capital stock and the amount of labor income

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),
((\[DifferentialD]ρ)/(\[DifferentialD]τ))≥0==((\[DifferentialD]δ)/(\[DifferentialD]τ))(* the tax does not increase the willingness to supply capital or affect the depreciation rate *),
signconditions[[3;;-1]]},
((\[DifferentialD]k)/(\[DifferentialD]τ))<0∧((\[DifferentialD]laborincome[k])/(\[DifferentialD]τ))<0]

Taxation reduces labor income more than it increases revenue

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),lrcapitalequilibrium,τ>0,((\[DifferentialD]ρ)/(\[DifferentialD]τ))≥0==((\[DifferentialD]δ)/(\[DifferentialD]τ)),
signconditions},
((\[DifferentialD](laborincome[k]+revenue[τ,k]))/(\[DifferentialD]τ))<0]

In the neighborhood of a zero tax rate, this comes from the effect (if any) of the tax on ρ

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),lrcapitalequilibrium,τ==0,((\[DifferentialD]ρ)/(\[DifferentialD]τ))≥0==((\[DifferentialD]δ)/(\[DifferentialD]τ)),
signconditions},
SameSign[((\[DifferentialD](laborincome[k]+revenue[τ,k]))/(\[DifferentialD]τ)),-((\[DifferentialD]ρ)/(\[DifferentialD]τ))]]

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

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),lrcapitalequilibrium,τ>0,((\[DifferentialD]ρ)/(\[DifferentialD]τ))≥0==((\[DifferentialD]δ)/(\[DifferentialD]τ)),
signconditions,elasticcapitaldemand},

furmanratio[τ,k]>1
∨
wrongsideoflaffercurve]

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),lrcapitalequilibrium,τ>0,((\[DifferentialD]ρ)/(\[DifferentialD]τ))≥0==((\[DifferentialD]δ)/(\[DifferentialD]τ)),
signconditions},

furmanratio[τ,k]>1
∨
furmanratio[τ,k]<0]

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),lrcapitalequilibrium,τ==0,((\[DifferentialD]ρ)/(\[DifferentialD]τ))≥0==((\[DifferentialD]δ)/(\[DifferentialD]τ)),
signconditions,elasticcapitaldemand},

furmanratio[τ,k]≥1
∨
wrongsideoflaffercurve]

Cobb-Douglas, with 0 < α < 1 ∧ k > 0, satisfies the sign conditions

FullSimplify[f'[k]>0∧
f''[k]≤0∧
laborincome'[k]>0∧
elasticcapitaldemand
/.f->fcd (* replace f with a Cobb-Douglas function *),

0<α<1∧k>0]

Local Cobb-Douglas is a stronger assumption than elasticcapitaldemand

capitalshare[k_]=((k f'[k])/(f[k]));

LocalCobbDouglas={0<capitalshare<1,capitalshare'[k]==0};

TheoryOverlap[{k>0,δ>0,f[k]>0,f'[k]≥δ,f''[k]≤0},elasticcapitaldemand,LocalCobbDouglas]

Variable interpretations

Extensions of this model

Let part of the capital stock escape tax: pdf here and executable Mathematica notebook here.
Let the sellers of final goods have market power: pdf here and executable Mathematica notebook here.
General discussion of extensions here.