Goldin & Homonoff - DataSpace at Princeton University

∂yB
= −x.
∂tr
Substituting those conditions into the above expression gives the effect of the shift on B’s welfare:
dVB
d(tp
+tr)
= −Uy (xB,yB)xB +
dtr
dtr
∂tp
∂tr
R
R
R
∂xB
∂ p
ε (7)
where ε ≡ Ux (xB,yB) − (p +tr +tp)Uy (xB,yB) represents optimization error from B’s inattention to the
register tax. Because B consumes too much x and too little y relative to her optimal quantity, declining
marginal utility in x and y implies that ε < 0.
From (7), we can see that the net welfare effect on inattentive consumers is ambiguous. Like the
attentive consumer, B benefits because the shift lowers the combined tax rate. On the other hand, by
raising the register tax, the shift pushes B further from her optimal consumption bundle. In general,
either of these effects may dominate.
That even inattentive consumers can be made better off by a shift towards register taxes is somewhat
surprising. The explanation is that for low register tax rates, the welfare loss caused by optimization
error is small (the marginal utilities of expenditures on x and y are similar), but the income effect from
the lower combined tax rate may still be substantial. To develop this intuition, let (x∗ B ,y∗B ) represent the
(interior) optimal bundle in B’s true budget set, that is, the consumption bundle that B would choose were
she to take the register tax into account. Assume that utility is additively separable in x and y so that
ε = Ux (xB) − (p +tp +tr)Uy (yB). Taking first-order Taylor approximations around (x∗ B ,y∗B ) and using
the fact that the pair (x∗ B ,y∗ B ) satisfies the first order condition Ux (x∗ B ) = (p +tr +tp)Uy (y∗ B ), we can write
ε ≈ γ (xB − x ∗ B)
where γ ≡ Uxx (x ∗ B ) + (p +tr +tp) 2 Uyy (y ∗ B ). Additionally, taking the Taylor approximation of x∗ B (tp,tr)
around tr = 0, and using the fact that x ∗ B (tp,0) = xB (tp,0) implies
(xB − x ∗ ∂xB
B) ≈ −tr
∂ p .
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