Progressive Policy and Inequality

TheoryGuru applications

Load Economicreasoning package only if it is not already loaded

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

Notes

Interpret PreTaxIncome as the income that people would have without policy intervention.

Tax lists mean-zero additions and subtractions to PreTaxIncome.  Income with policy is therefore PreTaxIncome - Tax

PreTaxIncome and Tax are vectors: one element for each resident of the country (or other policy area).

In the Wolfram Language, x.y refers to the tensor DOT PRODUCT, NOT scalar multiplication.  For TheoryGuru purposes, tensor means vector, so that the result of x.y is a scalar.

Analysis

Setup

inequality[x_]:=(x-average).(x-average)

ProgressivePolicy=Tax.(PreTaxIncome-average)>0 ;
(* on average, low-income people pay a negative tax (i.e., received money)
and high-income people pay tax *)

AfterTaxIncome=PreTaxIncome-Tax;

Result 1: Progressive policy could exacerbate the inequality

TheoryGuru[{ProgressivePolicy,(average^2)≥0 (* clarify that average income is a scalar *)},
inequality[AfterTaxIncome]≥inequality[PreTaxIncome](* policy exacerbates inequality *)]

Text interpretations of key inequalities

rtext={First@MostRecentAssumption->"Progressive policy",
Not@First@MostRecentHypothesis->"policy to reduce inequality."};

Result 2: Progressive policy is necessary but not sufficient for policy to reduce inequality

with formulas

TheoryOverlap[(average^2)≥0,
ProgressivePolicy,
inequality[AfterTaxIncome]<inequality[PreTaxIncome]]

formulas replaced with text interpretations

TheoryOverlap[(average^2)≥0,
ProgressivePolicy,
inequality[AfterTaxIncome]<inequality[PreTaxIncome]]/.rtext

Variable interpretations

Result 3: Results 1 and 2 do not need (a) a nonzero average tax, (b) taxes to be so strong as to swap rich and poor

AverageTaxisZero=Dot[1,Tax]==0;

AverageIncomeDefinition=Dot[1,PreTaxIncome]==average Dot[1,1];

TradingPlaces=(PreTaxIncome-average).(AfterTaxIncome-average)<0;
(* When True: on average, the poor have become the rich and vice versa *)

TheoryOverlap[{AverageTaxisZero,Not@TradingPlaces,AverageIncomeDefinition},
ProgressivePolicy,
inequality[AfterTaxIncome]<inequality[PreTaxIncome]]/.rtext