Goldin & Homonoff - DataSpace at Princeton University

solution of the standard utility maximization problem:
(xA,yA) = argmax
x,y U(x,y) s.t. BCA. (2)
Because the tax rates do not enter the utility function directly and because they appear symmetrically in
the budget constraint, A’s demand will depend only on the combined tax rate – whether it takes the form
of a register or posted tax does not matter. Hence we can write xA (tp,tr) = xA (tp +tr,0), or xA (tp +tr) for
short. And similarly for y: yA (tp,tr) = yA (tp +tr) = MA −(p +tp +tr)xA (tp +tr), where the last equality
follows from A’s budget constraint. Note that in accordance with the neoclassical model’s prediction, we
have ∂xA
∂tr
= ∂xA
∂tp
= ∂xA
∂ p .
In contrast, B’s intended consumption bundle will not match B’s final consumption bundle whenever
the register tax rate is positive. Because all of the income overspent on x comes out of expenditures
intended for y, we have xB (tp,tr) = xB (tp,tr) for all values of tp and tr. Since B’s intended consumption
of x is insensitive to register taxes, xB (tp,tr) = xB (tp,t ′ r) for all tr and t ′ r, it must also be the case that B’s
final consumption of x is insensitive to register taxes, xB (tp,tr) = xB (tp,t ′ r) for all tr and t ′ r. In particular,
this result holds for t ′ r = 0. Hence, we can write xB (tp,tr) = xB (tp,0) ≡ xB (tp). In words, B’s demand for
x under any non-zero register tax corresponds to B’s optimal demand for x in the special case where the
register tax is zero.
Using B’s true budget constraint, we can derive B’s final consumption of y:
yB (tp,tr) = MB − (p +tp +tr)xB (tp). (3)
Note that in contrast to the neoclassical model, B responds differently to the two types of taxes: ∂xB
∂tp =
∂xB
∂ p
= ∂xB
∂tr
= 0.
To incorporate tax policy into the model, consider a government that must raise a fixed amount of
revenue, R, from register and posted taxes on x. How does the government’s choice between register and
posted taxes affect the well-being of the agents? In particular, we focus on the effects of a revenue-neutral
increase in the register tax – that is, an increase in the register tax coupled with a reduction in the posted
tax by an amount that keeps total revenue constant (at R). Let R denote total revenue collected by the two
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