Finally, we have undertaken a disaggregated analysis of urban taxation of various goods. We obtained some criteria that urban taxes or subsidies need to satisfy if they are to be Pareto-efficient. This result takes into account the induced effects of taxes on migration, the urban wage, and urban employment. It appears similar to the multi-good Ramsey tax result, but is based on considerations altogether different from those of Ramsey-type analyses.
The derivative of (13.7) with respect to pr is(13.A1)
Recall that ∂Vr/∂pr = λrQ and βr = λr∂W(Vr)/∂Vr. Thus, substituting (13.8) and (13.9) into (13.A1), we obtain (13.12).
The derivative of (13.11) with respect to pr, keeping Ne unchanged, is(13.A2)
We have substituted ∂A/∂Nr = − A/Nr in deriving the last step of (13.A2). Now, a substitution of (13.14) and of the definitions of tr and the elasticities into (13.A2) yields (13.13).
Multiply both sides of (13.5) by (N − Nr). The derivative of the resulting expression with respect to pr is(13.A3)
From n. 4 ,. Substituting this into (13.A3), we obtain (13.A4)