Aggregation. Need to aggregate impacts across individuals Welfare Function: Aggregation of individual welfare functions. Individual welfare function for i. U i = U i (X i1 , X i2 , … X ij )
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Ui = Ui(Xi1, Xi2, … Xij)
Max L=Ui(Xi1, Xi2, … Xij) – λi(p1*Xi1+p2*Xi2 …+pj*Xij - Yi)
∂Uj/∂Xij = λi * Pj
∂Uj/∂Xij = ∂Ui/∂Yi * Pj
dUi = Σj∂Ui/∂Xj dXij = Σj∂Ui/∂Yi*Pj dXij
W = W(U1,U2,U3,…Un)
dW = Σi∂W/∂Ui dUi
= ΣjΣi ∂W/∂Ui *∂Ui/∂Y * Pj dXij
= MSUi * MU Incomei * Pj dXij
MSUi = Maginal Social Utility of individual i
MU Incomei = Marginal utility of income for indivual i.