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1 Cómo debería ser mi modelo de acuerdo a Harford (87)

Firm’s problem when reports emissions:

min(e, e_{r})T EC = C(e, θ_{0}) + π ∗ [G(e − e_{r}) + t ∗ e] + (1 − π) ∗ t ∗ e_{r}

or

min(e, e_{r})T EC = C(e, θ_{0}) + t ∗ e_{r} + π ∗ [G(e − e_{r}) + t ∗ (e − e_{r})]

donde G(e − e_{r}) is the under-reporting fine and t ∗ (e − e_{r}) is the corresponding tax for

the emissions not reported.

The FOC for an interior minimum are:

(1) \frac{∂T EC}{∂e} = C_{e} + π × [G′ + t] = 0
(2) \frac{∂T EC}{∂e_{r}} = t − π × [G′ + t] = 0

>From (1) and (2)

−C_{e} = t

This is Harford’s result; when a firm reports emissions to a regulator who is trying to enforce an evadable tax, the firm’s optimal level of emissions is going to be independent of the enforcement parameters.

Lets call e(t; θ_{0}) the solution to the above equation −C_{e} (e; θ_{0}) = t. The other solution to this problem is e_{r} = e_{r}[e (t; θ_{0}) , G, π] If I change [G(e − e_{r}) + t ∗ (e − e_{r})] for F(e-e_{r}) sucha that F>t, the results derived

above do not change.

2 What happens with this model when the firms reports abatement costs (AC)

T EC = C(e; θ_{0}) + t (θ) × e^{∗} [t (θ)] + π × F [e − e^{∗} (t (θ))]
(1) \frac{∂T EC}{∂e} = C′ + π × F′ = 0

(2) \frac{∂T EC}{∂θ} = t′ (θ) × e^{∗} + t (θ) × e_{t}^{∗} × t′ (θ) − π × F′ × e_{t}^{∗} × t′ (θ) = 0

= t′ (θ) × {e^{∗} + t (θ) × e_{t}^{∗} − π × F′ × e_{t}^{∗}} = 0
= t′ (θ) × {e^{∗} + [e_{t}^{∗} × (t (θ) − π × F′)]} = 0

If t′ (θ) ≠ 0,

e^{∗} = −e_{t}^{∗} × [t (θ) − π × F′]

Using (1)

e^{∗} = −e_{t}^{∗} × [t (θ) + C_{e}]

−C_{e} = [t (θ) + \frac{e^{∗}}{e_{t}^{∗}} ]

This result contradicts Harford’s conlusion about the independence of the emissions choice and the enforcement effort. In this AC reports affects the tax, and therefore the actual level of emissions.

3 Defining the firm’s objective funtion

min(e)C(e; θ_{0}) + t (θ) × e^{∗} [t (θ)] + π × f [e − e^{∗} × t (θ)]

s.t. e ≥ e^{∗}

Lagrange:

C(e; θ_{0}) + t (θ) × e^{∗} [t (θ)] + π × f [e − e^{∗} × t (θ)] + λ × [e^{∗} − e]

Kuhn Tucker conditions:

\frac{∂L}{∂e} = C_{e} + π × f′ − λ ≥ 0; e ≥ 0; \frac{∂L}{∂e} × e = 0
\frac{∂L}{∂λ} = e^{∗} − e ≤ 0; λ ≥ 0; λ × [e^{∗} − e] = 0

The firm is in compliance [e^{∗} = e] ⇔ λ ≥ 0 ⟺ From\frac{∂L}{∂e} :

C_{e} (e^{∗}) + π × f′ (0) − λ = 0 −C_{e} (e^{∗}) = π × f′ (0) − λ

−C_{e} (e^{∗}; θ_{0}) ≤ π × f′ (0)

Figure 1:

For the same reason, when e ≥ e^{∗}, λ = 0, and

−C_{e} (e; θ_{0}) = π × f′ (e − e^{∗}) (1)

Looking at the graph one can see that if the enforcement were weak for some structural reason, the Regulator would know that the firm should be emiting at ê (according to (1), a firm minimizing total expected costs should be emiting such that −C_{e} (e; θ_{0}) = π × f′ (e − e^{∗}). We call ê(θ) the implicit solution to this equation.

Therefore, one could conclude that self-reporting of emissionsis not necessary because it would not be a best strategy for the firm to report a given AC function, and then not being consistent with it by reporting more (or less) than the consistent level of noncompliance.

3.1 Under what conditions this is true?

First, one should notice that it is the firm and not the Regulator who is actually setting the “standard” by reporting AC. Furthermore, the firm also knows that the Regulator knows that in order to be consistent with its report, condition (1) must be satisfied. But if this is the case, the cost of f [ê(θ) − e^{∗} (θ)] becomes certain for the firm.

The problem for the firm becomes:

min(e, θ) C (e, θ_{0})+t (θ)×e^{∗} [t (θ)]+f {ê(θ) − e^{∗} [t (θ)]}+π×{f [e − e^{∗} (θ)] − f [ê(θ) − e^{∗} (θ)]} s.t. e ≥ ê ≥ e^{∗}

The certain cost f {ê(θ) − e^{∗} [t (θ)]} is capturing the issue that the firm should be consistent with its report. ê(θ) act in this case as the firm’s reported emissions.

Under this information structure there is no incentive for the firm to report emissions greater than ê(θ) because it will be charged f [·] for that. And although it surely has incentives to report emissions below ê(θ), this possibility is ruled out by the Regulator by making ê(θ) the firms “indirect” report of emissions.

Now, it is also true that it does not make any sense to charge f {ê(θ) − e^{∗} [t (θ)]} < t(θ). Therefore, we can write f {ê(θ) − e^{∗} [t (θ)]} as t (θ) + F {ê(θ) − e^{∗} [t (θ)]} .

Furthermore, if the firm is audited and found in violation of ê(θ) ,it will have to pay t (θ) × [e − ê(θ)] + F {e − e^{∗} [t (θ)]} − F {ê(θ) − e^{∗} [t (θ)]} + G [e − ê(θ)]; the last term being the fine for under-reporting.

4 The firm’s problem

The firm’s problem would the be:

min(e, θ) C (e, θ) + t (θ) × ê(θ) + F {ê(θ) − e^{∗} [t (θ)]}
+π {[e − ê(θ)] , [ê(θ) − e^{∗} (θ)]}

× {t (θ) × [e − ê(θ)] + F {e − e^{∗} [t (θ)]} − F {ê(θ) − e^{∗} [t (θ)]} + G [e − ê(θ)]}

s.t. e ≥ ê(θ) ≥ e^{∗} (θ)
and θ ≤ θ_{0}

Leaving aside the endogeneity of π, the Lagrange function is:

£(e, θ, λ_{1}, λ_{2}, λ_{3}) = C (e, θ_{0}) + t (θ) × ê(θ) + F {ê(θ) − e^{∗} [t (θ)]}

+π × {t (θ) × [e − ê(θ)] + F {e − e^{∗} [t (θ)]} − F {ê(θ) − e^{∗} [t (θ)]} + G [e − ê(θ)]} +λ_{1} × {e^{∗} [t (θ)] − ê(θ)} + λ_{2} × [ê(θ) − e] + λ_{3} × [θ_{0} − θ]

Kuhn-Tucker conditions:

(1) \frac{∂£}{∂e} = C_{e} (e, θ_{0}) + π × \{t (θ) + F′ \{e − e^{∗} [t (θ)]\} + G′ [e − ê(θ)]\} − λ_{2} ≥ 0; e ≥ 0; \frac{∂£}{∂e} × e = 0

(2) \frac{∂£}{∂θ} = t′ (θ) × ê(θ) + t (θ) × ê′ (θ) + F′ (·) × \{ê′ (θ) − e_{t}^{∗} × t′ (θ)\}

+π×\{ \begin{matrix}t′ (θ) × [e − ê(θ)] − t (θ) × ê′ (θ) − F′ [·] × e_{t}^{∗} × t′ (θ) − F′ [·] × \{ê′ (θ) − e_{t}^{∗} × t′ (θ)\} \\ −G′ [·] × ê′ (θ) + λ_{1} × \{e_{t}^{∗} × t′ (θ) − ê′ (θ)\} + λ_{2} × ê′ (θ) − λ_{3} \end{matrix} \} ≥

0 θ ≥ 0; \frac{∂£}{∂θ} × θ = 0

(3) \frac{∂£}{∂λ_{1}} = e^{∗} [t (θ)] − ê(θ) ≤ 0; λ_{1} ≥ 0; \frac{∂£}{∂λ_{1}} × λ_{1} = 0

(4) \frac{∂£}{∂λ_{2}} = ê(θ) − e ≤ 0; λ_{2} ≥ 0; \frac{∂£}{∂λ_{2}} × λ_{2} = 0

(5) \frac{∂£}{∂λ_{3}} = θ_{0} − θ ≤ 0; λ_{3} ≥ 0; \frac{∂£}{∂λ_{3}} × λ_{3} = 0

4.1 Firm’s choice of real emissions

>From Kuhn-Tucker conditions (1) and (4) we know that:

(a) the firm is giong to emit consistently with its AC iff:

−C_{e} (ê(θ) , θ_{0}) ≤ π × {t (θ) + F′ {ê(θ) − e^{∗} [t (θ)]} + G′ [0]}

Proof. when e = ê(θ) ,by (4) λ_{2} ≥ 0. Susbtituting e by ê(θ) in Kuhn-Tcuker condition (1) and using λ_{2} ≥ 0, we get the above condition.

(b) when the firm does not emit consistently with its AC report:

−C_{e} (e, θ_{0}) = π × {t (θ) + F′ {e − e^{∗} [t (θ)]} + G′ [e − ê(θ)]}

We will call e^{0} (θ) the implicit solution to this equation.

4.2 Firm’s choice of θ

Asuming an interior solution [e^{0} > ê > e^{∗}] and [θ < θ_{0}], Kuhn-Tucker condition (2) becomes:

t′ (θ) × ê(θ) + t (θ) × ê′ (θ) + F′ (·) × {ê′ (θ) − e_{t}^{∗} × t′ (θ)} =

π×\{ \begin{matrix}t′ (θ) × [e^{0} − ê(θ)] − t (θ) × ê′ (θ) − F′ [·] × e_{t}^{∗} × t′ (θ) − F′ [·] × \{ê′ (θ) − e_{t}^{∗} × t′ (θ)\} \\ −G′ [·] × ê′ (θ) \end{matrix} \} =

0 or t′ (θ) × ê(θ) + t (θ) × ê′ (θ) + F′ (·) × ê′ (θ) − F′ (·) × e_{t}^{∗} × t′ (θ) =

+π × \{ \begin{matrix}t′ (θ) × [e^{0} − ê(θ)] − \{t (θ) + F′ [·] + G′ [·]\} × ê′ (θ) \\ \{−F′ [ê(θ) − e^{∗} [t (θ)]] − F′ [e^{0} (θ) − e^{∗} [t (θ)]]\} × e_{t}^{∗} × t′ (θ) \end{matrix} \}

That is, at an optimum in which the firm under-reports its AC, it equates the marginal decrease in tax payments {t′ (θ) × ê(θ) + t (θ) × ê′ (θ)} plus the marginal decrease in the certain penalties {F′ (·) × {ê′ (θ) − e_{t}^{∗} × t′ (θ)}} (I need to prove that {ê′ (θ) − e_{t}^{∗} × t′ (θ)} > 0) to the marginal net increase in the expected penalties.

4.3 Under-report: bounded or unbounded?

In order to compare our results to those of Bulckaen’s, we proceed (asuming an interior solution exists) to rewrite Kuhn-Tucker condition (2) as:

t′ (θ) × ê(θ) + π × [e − ê(θ)] + t (θ) × ê′ (θ) + F′ (·) × {ê′ (θ) − e_{t}^{∗} × t′ (θ)}

+π × \{ \begin{matrix}−t (θ) × ê′ (θ) − F′ [·] × e_{t}^{∗} × t′ (θ) − F′ [·] × \{ê′ (θ) − e_{t}^{∗} × t′ (θ)\} \\ −G′ [·] × ê′ (θ) + λ_{1} × \{e_{t}^{∗} × t′ (θ) − ê′ (θ)\} + λ_{2} × ê′ (θ) − λ_{3} \end{matrix} \} ≥ 0