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									     Public Sector Economics

Measuring Substitution Effects with Multiple
          Margins of Distortion
The Income Tax Distorts on Multiple Margins
• Labor supply, housing demand, charitable
  giving, occupational choice, fringe benefits
• Are there summary indicators of the effects
  of taxes?
• “The” marginal tax rate: summarizes the
  effect on rates of substitution (Barro and
  Sahasakul)
• The compensated taxable income elasticity:
  summarizes the welfare effect (Feldstein,
  Slemrod)
    Tax Favored vs. “Ordinary” Consumption

                                u ( c, f , l )
                   w(1  l )  c  f  t[ w(1  l )  f ]
•   c = ordinary consumption
•   f = tax-favored consumption
•   l = leisure, w(1-l) = pre-tax labor income
•   t = constant tax rate, applied to Taxable Income w(1-l)-f
                      c  (1  t )[w(1  l )  f ]
• compensated demand functions satisfy:
         dc  (1  t )wdl  (1  t )df  dc  (1  t )dTI  0
   Income Tax is Equivalent to an Excise Tax on
             Ordinary Consumption
                   (1   )c  w(1  l )  f
• dwc of the income tax is equivalent to the dwc of
  an excise tax on ordinary consumption
                    1    dc
             dwc    2
                    2 d (1   ) u
               1   1       dc
              2
               2 1   d ln(1   ) u
                1 t2        dc
             
                2 1  t d ln(1  t ) u
                1    dc
               t2
                2 d (1  t ) u
                1               dTI        1       d ln TI       1
               t 2 (1  t )              t 2TI                t 2TI TI
                2             d (1  t ) u 2      d ln(1  t ) u 2

  • can treat TI like it were a composite good  no need to
    separately study each tax-favored activity
Tax Favored Behavior in the Budget Constraint

                                u (c, l )
               w(1  l )  c   ( f )  t[ w(1  l )  f ]
•   c = ordinary consumption
•   f = tax-favored behavior, with cost (f)
•   l = leisure, w(1-l) = pre-tax labor income
•   t = constant tax rate, applied to Taxable Income w(1-l)-f
              c   ( f )  f  c  (1  t )[ w(1  l )  f ]
                   u (c , f , l )  u (c   ( f )  f , l )
• with transformed utility function, this environment is a
  special case of f in the utility function  dwc is
  proportional to the taxable income elasticity
      Statutory vs Empirical Tax Rates
                           u (c,1  n)
                   wn  c   ( f )  T nw  f 
• f = tax-favored behavior, with cost (f)
• T = convex tax function, applied to Taxable Income nw-f
                    MRS  [1  T (nw  f )] w
                      ( f )  T (nw  f )
• T is the statutory marginal tax rate
• the empirical marginal tax rate is dT/d(nw)
           df                df       df        T 
        d (nw) T     d (nw)  d (nw)  T      (0,1)
       1      
               
                     dT        df 
                            d (nw) T   (0, T )
                          1 
                   d (nw)          
                                    

								
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