Goldin & Homonoff - DataSpace at Princeton University

Again, the first term represents a positive income effect and the second term represents a negative
welfare effect stemming from optimization error, which grows in size as register taxes increase.
Appendix B: Welfare Analysis under Endogenous Producer Prices
This Appendix expands the model developed in Part I to the setting in which firms adjust their prices
in response to changes in the type of tax imposed. Like before, the policy change we consider is an
increase in the register tax accompanied by a reduction in the posted tax designed to keep government
revenue unchanged.
For the attentive consumer (A), the welfare effect of the shift may be found by differentiating the
indirect utility function:
dVa
dtr
R
∂ px
= −Uy(xA,yA)(xA)
∂tr
R
= −Uy(xA,yA)(xA) 1 + ∂tp
∂tr
R
+ ∂ p
∂tr
This equation differs from (6) in that it reflects changes in the pre-tax price of x in addition to changes in
the combined tax on x. In words, the welfare effect of the shift for A stems entirely from the change in
d px
the after-tax price of x, px = p +tp +tr. When the net effect is negative, < 0, the resulting income
dtr R
effect makes A better off ( dVa
> 0). In turn, when the net effect of the shift is to raise the after-tax price
dtr R
of x, the associated income effect reduces A’s welfare.
To solve for ∂tp
∂ p
and , consider the market clearing condition x(p,tp,tr) = s(p), in which
R R
∂tr
∂tr
aggregate demand is the sum of demand by attentive and inattentive consumers, x(p,tp,tr) ≡ xa(p +tp +
tr) + xb(p +tp), and s(p) represents producer supply. Totally differentiating this condition as well as the
government’s budget constraint yields:
∂ p
∂tr
R
∂tp
∂tr
R
=
=
∂xb
∂ p
∂x ∂s tr+tp
∂ p − ∂ p − x
∂s ∂x tr+tp
∂ p − ∂ p + x
∂x ∂s tr+tp
∂ p − ∂ p − x
33
∂x ∂s
∂ p
∂ p
∂s ∂xA
∂ p ∂ p
∂X
∂ p
∂s
∂ p