1 Tax Evasion with Ambiguity

Suppose an individual firm with abatement costs function C(e_{i}), strictly decreasing and convex in emissions e_{i}. The firm faces an emissions tax t_{i}. The regulator cannot costlessly observe e_{i}, therefore asks the firm to repot its emissions. The firm can under-report emissions by reporting a level r_{i} < e_{i}. Doing this, the firm can reduce its tax payments but it also faces a chance of being audited and found under-reporting, which is fined with a unit penalty ϕ_{i} > t_{i}.

The objective probability of being inspected is p ∈ (0, 1). But the firm does not know p. So he decides how much to emit and report according to a belief about p. Let’s call π this subjective probability of being inspected. Nevertheless, the firm is uncertain about π. It faces ambiguity with respect to the probability of being inspected. This uncertainty is described by the cumulative distribution function F (π; a, p). The parameter a is an index of ambiguity.

In addition, the firms´ s perception about π, is distorted according to the probability weighting function φ(π, p), which may have a value greater or less than π. However, when π = p, φ(π, p) = p.

The firm chooses a level of emissions and emissions report to minimize its total expected costs

T EC = C(e_{i}) + t_{i} × r_{i} + ∫_{0}^{1}φ(π, p)dF (π; a, p) × [ϕ_{i} (e_{i} − r_{i})]

Subject to e_{i} ≥ r_{i} ≥ 0.

Where

∫_{0}^{1}φ(π, p)dF (π; a, p) ∈ (0, 1)

is defined as the perceived probability of an inspection. This is an expected probability that is determined by the distorted values that the firm believes the probability may take, φ(π, p),and the uncertainty about these values (F (π; a, p)).

In the absence of ambiguity (a = 0), F is the improper distribution function equal to 0 for all π < p, and equal to 1 otherwise. Consequently

∫_{0}^{1}φ(π, p)dF (π; 0, p) = φ(p, p) = p

and the firm´ s objective function reduces to the classic expected cost with known inspection probability p. Snow and Warren assume that an increase in the index of ambiguity results in a mean preserving spread of F. In the presence of ambiguity (a > 0), F (π; a, p) is a mean preserving spread of the improper distribution with mass at π = p.

Assuming (as Snow and Warren) that an increase in p causes a first-order stochastic dominance shift in F, that is F_{p} < 0, the effect of such an increase in the expected costs of the firm is given by

\frac{∂T EC}{∂p} = [∫_{0}^{1}φ_{p}dF + ∫_{0}^{1}φdF_{p}] × [ϕ_{i} (e_{i} − r_{i})]

Integrating by parts (the second integral):

\frac{∂T EC}{∂p} = [∫_{0}^{1}φ_{p}dF − ∫_{0}^{1}φ_{π}F_{p}dπ] × [ϕ_{i} (e_{i} − r_{i})]

To assure that the firm´ s costs increase when p increases, one has to assume (as did Snow and Warren) that φ_{π} and φ_{p} are positive.

Because F (π; a, p) is a mean preserving spread of the improper distribution with mass at π = p, an increase in ambiguity results in an elementary increase in risk.^{1}

∫_{0}^{τ} F_{a}(π; a, p)dπ ≥ 0

for all τ ∈ [0, 1] , with strict inequality at some τ ∈ [0, 1] and with strict equality at τ = 1.

The effect of an increase in ambiguity on the firm’s total expected costs is given by

\frac{∂T EC}{∂a} = [∫_{0}^{1}φdF_{a}] × [ϕ_{i} (e_{i} − r_{i})]

Integrating by parts (two times),

\frac{∂T EC}{∂a} = [∫_{0}^{1}φ_{ππ}∫_{0}^{τ} F_{a}dπdτ] × [ϕ_{i} (e_{i} − r_{i})]

It follows that the firm is ambiguity neutral (∂T EC/∂a = 0) if φ_{ππ} = 0 (φ is linear in π). As in the case of the absence of ambiguity, the objective function of an ambiguity neutral firm reduces to the expected costs function with the probability of an inspection equal to p. This is because the introduction of ambiguity has no effect on the expected costs of the firm if this is ambiguity neutral. Therefore, it must be the case that the expected probability of an audit remains equal to p, implying that φ(π, p) = p for all π ∈ [0, 1] .

1By definition, an elementary increase in risk is a mean preserving spread. We know that if G(.) is a mean-preserving spread of F (.), then ∫_{0}^{x}G(t)dt ≥ ∫_{0}^{x}F (t)dt for all x. (See Mas-Colell, Whinston and Green (1995), pg. 198). The result follows from this definition.

In contrast, the introduction of ambiguity increases the expected costs of the ambiguity averse firm (∂T EC/∂a > 0). (This is a definition, as presented by Snow and Warren). For this to happen, for a firm that under reports, we need φ_{ππ} > 0. Note that if the firm reports truthfully an increase in ambiguity has no effect on the firm´ s expected costs because the firm face no expected penalty. Hence, in the presence of ambiguity the perceived probability of an audit is greater than the true probability, ∫_{0}^{1} φdF > p for taxpayers that are ambiguity averse.^{2}

2 The Effect of Ambiguity on Taxpayer Compliance

Calling L the Lagrange equation, and λ_{i} the multiplier corresponding to the constraint e_{i} − r_{i} ≥ 0, the FOCs for the choice of emissions and emissions report are:

L_{e} = C_{e}(e_{i}) + [∫_{0}^{1}φ(π, p)dF (π; a, p)] × ϕ_{i} − λ_{i} = 0

L_{r} = t_{i} − [∫_{0}^{1}φ(π, p)dF (π; a, p)] × ϕ_{i} + λ_{i} ≥ 0, r_{i} ≥ 0, L_{r} × r_{i} = 0
L_{λ} = r_{i} − e_{i} ≥ 0, λ_{i} ≥ 0, λ_{i} × (r_{i} − e_{i}) = 0)

(SOC to be seen)

With a constant marginal penalty, the firm will under-report (report r_{i} = 0)if the tax is larger than the expected penalty. In this case it will report zero emissions. Assuming we are in this situation, namely

r_{i}^{∗} = 0
t_{i} > [∫_{0}^{1}φ(π, p)dF (π; a, p)] × ϕ_{i}

an increase in ambiguity decreases under-reporting if it makes this inequality to change. This can only happen, obviously, if an increase in ambiguity increases the expected or perceived probability of being inspected. But we have just seen that this is the case if the firm is ambiguity averse. (See above). It follows that, under a constant penalty scheme, a sufficient increase in ambiguity could make an ambiguity averse firm to report truthfully.

2T EC = C(e_{i}) + t_{i}r_{i} + ∫_{0}^{1}φ(π, p)dF (π; a, p) × [ϕ_{i} (e_{i} − r_{i})] . Ambiguity aversity is ∂T EC/∂a > 0. The only way this can happen is that the integral increase with a. The derivative of the integral with respect to a is ∫_{0}^{1}φ_{ππ}∫_{0}^{τ}F_{a}dπdτ, which is positive in the case of ambiguity aversity, as just seen. Q.E.D.

Because experimental tests of cumulative prospect theory suggests that individuals are ambiguity loving with respect to uncertainty about a small probability of loss, this would suggest that, given actual probabilities of being inspected (small), an increase in ambiguity would reduce compliance, contrary to what the IRS would want.

Nevertheless, another brand of the literature treats experimental subjects as individuals instead using the individuals responses to estimate a unique probability weighting function of a representative individual. This literature finds that a considerable proportion of individuals (70% - 80%) are ambiguity averse for low levels of the probability of an audit, a non trivial proportion is ambiguity neutral and less than 10% are ambiguity loving. This raises the question the final effect of an increase in ambiguity on compliance since it could happen that the increase in compliance of ambiguity averse is outweighted by the decrease in compliance by ambiguity lovers.