Optimal Public Goods Supply - Samuelson FOC

14.472 – Spring Optimal Public Goods Supply page 1 of 4 A…Samuelson FOC Notation x y z w g u [ x, y , g ] fi p consumption labor supply earnings wage public good utility fraction of population type i price of public good Social welfare maximization: ∑ fiu i  xi , yi , g  E + pg + ∑ f i ( xi − wi yi ) ≤ 0 Maximize x, y , g subject to: (1) FOC: fi u xi [ xi , yi , g ] = λ fi (2) fi u [ xi , yi , g ] = λ fi wi (3) ∑ fiugi  xi , yi , g  = λ p (4) i y Implying: u gi  xi , yi , g  ∑ uxi  xi , yi , g  = p   i u x , y , g ∑ fi wi ugiy  xii , yii , g  = p   fi (5) (6) 14.472 – Spring Optimal Public Goods Supply page 2 of 4 B…Second-Best FOC R. Boadway and M. Keen, “Public Goods, Self-Selection and Optimal Income Taxation,” International Economic Review v34, n3 (August 1993), 463-78. Maximize x, y , g subject to: ∑ fiu i  xi , yi , g  E + pg + ∑ f i ( xi − wi yi ) ≤ 0 (7) u i  xi , yi , g  ≥ u i  x j , y j w j / wi , g  for all i and j Assume two types. Assume the only binding moral hazard constraint is type 1 considering imitating type 2. Maximize x, y , g subject to: f1u1  x1 , y1 , g  + f 2u 2  x2 , y2 , g  E + pg + ∑ f i ( xi − wi yi ) ≤ 0 (8) u1  x1 , y1 , g  ≥ u1  x2 , y2 w2 / w1 , g  FOC: f1u1x  x1 , y1 , g  − λ f1 + µ u1x  x1 , y1 , g  = 0 (9) f1u1y  x1 , y1 , g  − λ f1w1 + µ u1y  x1 , y1 , g  = 0 (10) f 2u x2  x2 , y2 , g  − λ f 2 − µ u1x  x2 , y2 w2 / w1 , g  = 0 (11) f 2u y2  x2 , y2 , g  − λ f 2 w2 − µ u1y  x2 , y2 w2 / w1 , g  = 0 (12) f1u1g  x1 , y1 , g  + f 2u g2  x2 , y2 , g  + µ ( u1g  x1 , y1 , g  − u1g  x2 , y2 w2 / w1 , g  ) = λ p (13) 14.472 – Spring Optimal Public Goods Supply page 3 of 4 Rearranging terms and adding and subtracting  u g2  x2 , y2 , g    to the  u x2  x2 , y2 , g      µ u1x  x2 , y2 w2 / w1 , g   last equation:  ( f1 + µ ) u1g  x1, y1, g  +  f 2 − µ  u1x  x2 , y2 w2 / w1 , g   2 u x , y , g u x2  x2 , y2 , g   g  2 2  (14)   u2  x , y , g    − µ  u1g  x2 , y2 w2 / w1 , g  − u1x  x2 , y2 w2 / w1 , g   g2  2 2    = λ p  u x  x2 , y2 , g          Rearranging terms again: u1  x , y , g  u2  x , y , g  g g 2 2  1 1   + f u 2  x , y , g  − µu1  x , y w2 / w1 , g  f + µ u1  x , y , g  1 2 x 2 2  x  1 1  u1  x , y , g  x 2 2  u2  x , y , g  x 1 1  x 2 2   u1  x , y w / w , g  u 2  x , y , g    g 2 2 2 1  g  2 2  =λp − µ u1  x , y w2 / w1 , g   − x 2 2  u1  x , y w / w , g  u 2  x , y , g    x 2 2 2 1  x  2 2    ( ) ( ) (15) 14.472 – Spring Optimal Public Goods Supply page 4 of 4 Using the other FOC:  u1g  x2 , y2 w2 / w1 , g  u g2  x2 , y2 , g   u1g  x1 , y1 , g  u g2  x2 , y2 , g  1 − µ u x  x2 , y2 w2 / w1 , g   1 λ f1 1 + λ f2 2 − =λp  u x  x2 , y2 w2 / w1 , g  u x2  x2 , y2 , g   u x  x1 , y1 , g  u x  x2 , y2 , g       (16) This reduces to the Samuelson condition if u1g  x2 , y2 w2 / w1 , g  u g2  x2 , y2 , g  = u1x  x2 , y2 w2 / w1 , g  u x2  x2 , y2 , g  For example, this holds with everyone having the same utility function that is additive: a  x  + b  y  + c  g  or everyone has a utility function separable in labor and with the same subadditive form: u i  h  x, g  , y  . Otherwise, the deviation from the Samuelson condition depends on the MRS for the imitated type compared with that for an imitator, that is, it depends on how the MRS varies with labor supply. And the conditions for the Samuelson rule to hold are different for different numeraires. (17)