Corporate-income Tax Incidence with the Furman Ratio

Model includes noncorporate capital.

TheoryGuru applications

Background

Obama administration economists Furman and Summers claimed that only a fraction of the revenue loss from a corporate-income tax cut benefits labor.  But the standard supply and demand model says the opposite.

Summers, as well as Nobel Laureate Paul Krugman, rejected this result, asserting that it depends on “what share of the capital stock is even affected by the corporate tax rate.”

The supply and demand model readily accommodates the fact that the statutory corporate rate does not apply to much of the nation’s capital.  Here a machine proves the supply-demand result, without assuming any functional form for the aggregate production function, and without restricting the share of capital that is subject to the tax (except that the share cannot be zero or negative).  k denotes the aggregate capital stock, which is a composite of capital k1 that is subject to the statutory rate and capital k2 that is not.

Setup

Get@"http://economicreasoning.com"

SetOptions[TheoryGuru,method->"Parallel"];
(* these proofs run quicker in parallel mode, if available *)

Definitions

k=k1 g[(k2/k1)];(* k is a homogeneous aggregate of k1 and k2 *)

taxablesh=(k1/(k1+k2));

laborincome[k1_,k2_]=f[k]-f'[k]k

lrcapitalequilibrium=(1-τ)D[f[k]-δ k1-δ k2,k1]==D[f[k]-δ k1-δ k2,k2]==ρ;

signconditions={δ>0,ρ>0,k1>0,k2≥0,f'[k]>δ,0≤τ<1,f''[k]<0,g[(k2/k1)]>0,g'[(k2/k1)]>0,g''[(k2/k1)]<0};

revenue[τ_,k1_,k2_]:=τ (D[f[k],k1]-δ)k1

furmanratio[τ_,k1_,k2_]:=(("((\[DifferentialD]laborincome[k1, k2])/(\[DifferentialD]τ))")/(-"((\[DifferentialD]revenue[τ, k1, k2])/(\[DifferentialD]τ))"))

Interesting but not necessary assumptions

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

wrongsideoflaffercurve=((\[DifferentialD]revenue[τ,k1,k2])/(\[DifferentialD]τ))≤0;

Results

Taxation reduces:
the stock of taxed capital,
total capital services,
the share of capital that is taxable, and
the amount of labor income.

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),lrcapitalequilibrium,
((\[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},
((\[DifferentialD]k1)/(\[DifferentialD]τ))<0∧((\[DifferentialD]k)/(\[DifferentialD]τ))<0∧((\[DifferentialD]taxablesh)/(\[DifferentialD]τ))<0∧((\[DifferentialD]laborincome[k1,k2])/(\[DifferentialD]τ))<0]

Taxation reduces labor income more than it increases revenue

TheoryDifficulty=5;(* helps run faster by not searching for easy proofs *)

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),lrcapitalequilibrium,
τ>0,((\[DifferentialD]ρ)/(\[DifferentialD]τ))≥0==((\[DifferentialD]δ)/(\[DifferentialD]τ)),signconditions},
((\[DifferentialD](laborincome[k1,k2]+revenue[τ,k1,k2]))/(\[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[k1,k2]+revenue[τ,k1,k2]))/(\[DifferentialD]τ)),-((\[DifferentialD]ρ)/(\[DifferentialD]τ))]]

Either the Furman ratio exceeds one or the tax is not raising revenue

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]τ)),lrcapitalequilibrium,τ>0,((\[DifferentialD]ρ)/(\[DifferentialD]τ))==0==((\[DifferentialD]δ)/(\[DifferentialD]τ)),
signconditions,elasticcapitaldemand,Not[wrongsideoflaffercurve]},

furmanratio[τ,k1,k2]>1]

From the aggregate perspective, the tax increases the required rate of return

The stock-weighted average marginal capital-income tax rate is taxablesh τ, where taxablesh is the share of capital that is taxable.
In order to have comparable average-marginal rate effects, let dx be a policy that raises τ in inverse proportion to taxablesh.

If all capital were uniformly taxed (taxablesh == 1), then a compensated reduction in time preference ((dρ)/(dx))==-(ρ/(1-τ taxablesh)) would keep capital services constant

In other words, an increase in the average marginal capital-income tax rate, without exempting any capital income from it, has the same effect on k as increasing the rate of time preference by (ρ/(1-τ taxablesh)).

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]x)),lrcapitalequilibrium,
((\[DifferentialD]ρ)/(\[DifferentialD]x))==-(ρ/(1-τ taxablesh)),((\[DifferentialD]δ)/(\[DifferentialD]x))==0,taxablesh"((\[DifferentialD]τ)/(\[DifferentialD]x))"==1,
signconditions,taxablesh==1},
((\[DifferentialD]k)/(\[DifferentialD]x))==0]

If some of the capital is exempt from tax (taxablesh < 1), and τ > 0, then the above compensation is not enough to keep capital services constant.

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]x)),lrcapitalequilibrium,
((\[DifferentialD]ρ)/(\[DifferentialD]x))==-(ρ/(1-τ taxablesh)),((\[DifferentialD]δ)/(\[DifferentialD]x))==0,taxablesh"((\[DifferentialD]τ)/(\[DifferentialD]x))"==1,
signconditions,taxablesh<1},

SameSign[((\[DifferentialD]k)/(\[DifferentialD]x)),-τ]]

Achieving the same increase in the average marginal capital-income tax rate with an uneven rate change has a greater effect k.

The same argument applies to labor income

TheoryGuru[{((\[DifferentialD]lrcapitalequilibrium)/(\[DifferentialD]x)),lrcapitalequilibrium,
((\[DifferentialD]ρ)/(\[DifferentialD]x))==-(ρ/(1-τ taxablesh)),((\[DifferentialD]δ)/(\[DifferentialD]x))==0,taxablesh"((\[DifferentialD]τ)/(\[DifferentialD]x))"==1,
signconditions,taxablesh<1},

SameSign[((\[DifferentialD]laborincome[k1,k2])/(\[DifferentialD]x)),-τ]]

Variable interpretations