Proposition 2 (How Does Excess Burden Depend on the Budget Adjustment Rule?). Assume that the government introduces a small misperceived tax tjinto an untaxed market. Behavior is characterized by the function ˆt(tj), ρ, and equations (1) through (4). The excess burden of this tax is approximately
EB(tj)' −1
and all expressions are evaluated at t = 0.
Proof. The result follows directly from lemma 1 and proposition 1. Re-write the equation for excess burden from proposition 1 in the presence of no existing taxes, using the definition of θc, as
EB(tj) ' −1
Then insert the expression for ∂x∂tjc from lemma 1 to obtain the result.
Proposition 3 (How does the desirability of low-salience taxes depend on the budget adjustment rule?). Suppose that the government has recourse to two types of taxes, tI and tE, and that
1. Consumers only notice tI: ˆt = tI. 2. No individuals debias (see section 5.2).
Then the introduction of a small tax individuals ignore, tE, causes less excess burden than the introduction of a rate-equivalent tax that individuals perceive fully, tI, if and only if ρ < −εx,q|p, where εx,q|pis evaluated at tI = tE = 0.
Proof. The excess burden of a price inclusive tax will be
EB(tI)' −1 2t2I∂xc
∂p (43)
The excess burden of a price exclusive tax will be
EB(tE)' −1 Re-arrange this equation and use the Slutsky equation εx,q|p= εcx,q|p− ωxηx,Z to obtain the result.
Figure 8: Budget Adjustment Under Ironing
x y
(x, y)
Budget (no tax) Budget (ATR)
Budget (tax)
(ˆx, ˆy)
Budget (ATR) with virtual income
Budget(no tax) with virtual income
Budget (MTR)
Notes: The budget line with tax is drawn for a two-bracket linear tax on x with a higher marginal tax rate in the top bracket. We assume for comparison with figure 3 that u0(x) > 0, i.e. that x is a good and not income. An ironing individual assuming she faces a linear tax equal to the ATR she ends up paying will consume (x, y). Virtual income, the income not lost to taxation due to a lower marginal tax rate for some levels of x is equal to the length of the blue dotted line. This is calculated by comparing the budget line under the non-linear tax with the budget line for a linear tax with marginal tax rate equal to the top marginal tax rate (note that the price of y is normalized to 1). We add virtual income to the budget and assume the individual faces a linear tax equal to ATR to find the planned consumption bundle (ˆx, ˆy). Movement from (ˆx, ˆy) to (x, y) is determined by the income elasticity and x’s share of the budget, just like in the third panel of figure 3.
Ironing
Figure 8 describes how the budget adjustment rule ρ3 works in the Schmeduling model, where there are non-linear taxes. The figure illustrates why assuming that an individual assumes she faces a linear tax with rate equal to her average tax rate (and no exemption) is the same as assuming that she accounts for the impact of taxes on her virtual income but misperceives relative prices. The same techniques could be used to examine other budget adjustment rules.
Proposition 4. Assume that
1. Individuals engage in ironing, so ˆτ = AT R ≡Rx
0 τ (s)ds/x.
2. The budget adjustment rule is ρ = 0.
3. There are no income effects on demand for x.
4. No individuals debias (see section 5.2).
Then the excess burden of a non-linear tax τ(x) will be given by
EB(τ (x)) ' −1
where εx,1−τ|τ is the elasticity of taxable income with respect to the net-of-tax rate, and εx,1−τ|AT Ris the elasticity of taxable income under full optimization, i.e. the elasticity of demand with respect to the net of tax rate following the introduction of a linear tax equal to ATR (which also equals the elasticity of x with respect to the price). Derivatives and elasticities are evaluated at the no-tax equilibrium τ = AT R = 0.
Proof. Applying proposition 2, adapted to the income tax case, normalizing to p = 1 so t = τ, and assuming ρ = ωxη = 0gives42
42One difference between this and the formulation in proposition 2 is that the denominator of the elasticity is evaluated at ˆτ instead of at τ0= 0. We are free to use the latter, to be consistent with Liebman and Zeckhauser (2004), because to a second order approximation this will not matter. To see why note that
1
1− ˆτ = 1− ˆτ + ˆτ2− ˆτ3+ ...
so multiplying by τ2and ignoring third order terms we will have
τ2' τ2 1− AT R
Debiasing
Proposition 5 (Extension to Non-Linearities in Tax Rates). Assume there is only one type of misperceived tax, tj. Under assumptionsA1 and A2, and the assumption that demand is locally linear in (p + ˆt) and Z but not in tax rates, the excess burden at a tax rate tjis initial tax rate t0is approximately:
EB(∆tj|t0) ' −1
Proof. The proof is virtually identical to the proof in Chetty (2009), but instead of approximating the difference between fully optimizing demand and actual demand by x∗− x ' (1 − θj)∂x∂p(t− ˆt), I approximate it, using that there is only one type of misperceived tax, as
x∗(p, t, Z)− x(p, t, Z) = (x∗(p, t, Z)− x(p + ˆt, 0, Z)) − (x(p, t, Z) − x(p + ˆt, 0, Z))
to allow for nonlinearity in tax rates. The rest of the proof follows the one in Chetty (2009), almost to the letter.
For the marginal excess burden approximation, note that by the fundamental theorem of calculus,
θj,t0 = ∂Θ
Taking a Taylor series expansion of equation (51) about t0, applying this approximation and ignoring third-order or higher terms yields the desired result.
Proposition 6 (The Curse of Debiasing). Suppose that the government has recourse to two taxes, tI and tE, and that 1. Consumers respond perfectly to changes in tI, so θctI = 1.
2. Consumers initially under-respond to changes in tE, so θctE < 1when tE= 0.
3. AssumptionsA1, A2, and A3 are true.
4. Optimal demand x∗is strictly positive at all prices and tax rates.
Then there exists a tax rate t0Eat which an increase in tEcauses higher marginal excess burden than an increase in the noticed tax tI.
Proof. The first step of the proof shows that whenever the marginal degree of error for a change in tE, θt0E > 1, an increase in tE causes higher marginal excess burden than an identical increase in tI. The second step proves the existence of a tax rate t0Eat which θt0E > 1using the mean value theorem.
Step 1. Write the difference between the marginal excess burden of a small increase in tE minus the marginal excess burden of a small increase in tI, given initial tax rates tI0 and tE0 as follows, using proposition 4/ 5 and the fact that ∂x/∂tI = ∂x/∂p43
where derivatives are evaluated at (t0I, t0E). Now imposing that the increase in the per-unit tax rate be the same for both taxes, ∆tE= ∆tI = ∆t, and recalling that θctj =∂x∂tjc
Before the second part of the proof can be completed, we need to derive a relationship between θtc0E and θt0E. Write
θtc0E =θ∂x∂p + x∂Z∂x
∂x
∂p + x∂Z∂x
and use the negativity of the compensated price effect to derive that for non-Giffen goods (which are ruled out by additive separability), we will have Step 2 Write the difference between optimal demand and true demand as
x∗(p + t0I, t0E, Z)− x(p + t0I, t0E, Z)' (1 − Θ)∂x
∂pt0E (59)
When tE0 = 0, x∗− x = 0 by A2. By A3, there is a tax rate t0such that when tE= t0E, x∗− x = 0. Take a derivative
43Applying proposition 4 in this way requires that we let ˆt be any components of the tax for which the individual optimizes fully at all tax rates. That is x(p, ˆt, Z) = x∗(p + ˆt, Z).
of equation (59) to obtain44
∂(x∗− x)
∂tE = (1− θt0E)∂x
∂p (60)
By the second condition of the proposition, we can take a small tax rate tE such that θt0E < 1. At this tax rate, x∗− x < 0. The mean value theorem therefore implies the existence of a tax rate t∗Esuch that
∂(x∗− x) First consider the marginal degree of error:
θct0E ≡
Applying our simplifying assumptions again, we can approximate (x∗− x) by
x∗− x ' (∂x/∂p)tE0= (∂xc/∂p)tE0
Second, consider the average degree of compensated error at some tax rate t0:
Θc ≡
Using the expression we just derived for θtE0we observe that the anti-derivative of the argument of the integral Rt
ˆt0θct0dt0will be F (G)(t − ˆt) = F (G)t0E, and by evaluating this expression at the limits of integration, noting that F (G(ˆt− ˆt)) = F (G(0)) = 0 and dividing by t − ˆt = t0Ewe obtain that Θ = F (G).
Note that Proposition 6 is proven in the text of the paper, prior to the statement of the proposition.