Optimal EPA Audit Mechanism With Endogenous Tax

Jui Sen

ECONS 582

1

Motivation Global carbon-dioxide emission

increased by 11.97 percent from 2010 to 2018

In USA greenhouse gas emission increased by 3.1 percent from 2017 to 2018 33500

34000

34500

35000

35500

36000

36500

37000

37500

38000

38500

2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

CO

2 G

lob

al e

mis

sion

in M

ton

Year

CO2 Global Emission Trend

Data source: EDGAR

5120

5140

5160

5180

5200

5220

5240

5260

5280

5300

2014 2015 2016 2017 2018 2019 2020 2021

CO

2 Em

issi

on

in M

ton

Year

US CO2 Emission Trend

2

Motivation Considering both direct and

indirect emission, industrial sector was the largest contributor of greenhouse gas in 2018, which was 28.9 percent

Agriculture10%

Commercial and residential

12%

Transportation28%

Industry 29%

Electricity21%

Us Greenhouse Gas emission sector-wise percentage in 2018 (Indirect Emission)

Data Source: EPA’s Greenhouse Gas Emission, 2018 3

Agriculture10%

Commercial and residential

12%

Transportation29%Industry

22%

Electricity27%

Us Greenhouse Gas emission sector-wise percentage in 2018 (Direct Emission)

Motivation US EPA is at the front line of protecting the environment from emission.

Under GHGRP, approximately 8000 facilities are required to report their emission annually to EPA.

Due to positive relationship between the level of emission and firms’ benefit, how authentic the firms’ reporting is a big concern.

A budget slash has been observed on EPA’s operating budget from FY 2010 to 2018 by 14.3 percent.

Limitation in observing firm’s actual emission and bounded budget of EPA intrigue the need to devise an efficient audit mechanism.

4

Research Question

What is the optimal environmental audit mechanism whenemission tax is endogenous ?

5

Existing Literature and Contribution Macho-Stadler & Perez-Castrillo (2005)

Due to budget and resource constraint, it is optimal to primarily audit easiest-to-monitor firms and to those firms that value pollutions the less.

Evans, Gilpatric & Liu (2009)

Regulator faces a trade-off between inducing truthful self-reporting and deterring emissions.

Oestreich (2015)

CAM creates a reporting contest between the firms, which in turn leads to more truthful reporting compare to RAM.

Oestreich (2017)

Due to fine and budget limitation RAM fails to implement socially efficient emission and author derives an optimal audit mechanism that capable of implementing socially efficient emission level.

6

Existing Literature and ContributionMacho-Stadler & Perez-Castrillo(2005)

Evans, Gilpatric & Liu (2009)

Oestreich (2015) Oestreich (2017) This Paper

• Tax is exogenous.

• Do not ensure socially efficient emission.

• Tax is endogenous

• Do not ensure socially efficient emission.

• Tax is exogenous

• Do not ensure socially efficient emission

• Tax is exogenous.

• Ensures socially efficient emission.

• Does not take into account social cost of untruthful reporting

• Tax is endogenous.

• Ensures socially efficient emission.

• Takes into account social cost of untruthful reporting.

7

Assumptions and Model Assumptions

An industry consisting of two homogenous firms.

Firm-𝑖 chooses emission level, 𝑒𝑖 ∈ 0, 𝐸 , where 𝐸 is the emission level of firms when the pollution is free

of cost.

One unit of final good produces exactly one unit of emission.

Firm-𝑖’s inverse demand function is given by 𝑎 − 𝑒𝑖

EPA cannot perfectly monitor the actual level of emission of a firm without a costly audit.

The number of firms to be audited, 𝑘 ≤ 𝑛.

probability of audit, 𝑝𝑖 of a firm depends on the level of emission self-reported by the firm 𝑖, 𝑟𝑖 for 𝑖 = 1,2

and reference value, 𝑅 (which is exogenous).

The audit mechanism is budget-balancing and symmetric.

Firms have complete information about each other’s emission. 8

Assumptions and Model Assumptions

Unit tax, 𝑡 is endogenous and can be obtained by maximizing the social welfare.

0 < 𝜃 < 𝑡 is levied on unreported emission in addition to tax.

As the unit tax, 𝑡 be endogenous and linear penalty, 𝜃 also endogenous.

Total cost of audit is assumed to be 𝐶 and cost of one audit is normalized to 1.

Model

Expected Profit:

𝐸𝜋 = 𝑎 − 𝑒 𝑒 − 𝑡𝑟 − 𝑝 (𝑟 , 𝑟 )(𝜃 + 𝑡)(𝑒 − 𝑟 ) for 𝑟 ≤ 𝑒 . (1)

Expected Social Welfare

𝐸𝑠𝑤 = 𝐸𝜋1 + 𝐸𝜋2 + 𝑡 𝑟1 + 𝑟2 + 𝑝1 𝜃 + 𝑡 𝑒1 − 𝑟1 + 𝑝2 (𝜃 + 𝑡) 𝑒2 − 𝑟2

𝑅𝑒𝑣𝑒𝑛𝑢𝑒 𝑓𝑟𝑜𝑚 𝑡𝑎𝑥 𝑎𝑛𝑑 𝑝𝑒𝑛𝑎𝑙𝑡𝑦

−𝑑

2𝑒1

2 + 𝑒22

𝐸𝑛𝑣𝑖𝑟𝑜𝑛𝑚𝑒𝑛𝑡𝑎𝑙𝐷𝑎𝑚𝑎𝑔𝑒

+1

2𝑒1

2 +1

2𝑒2

2

𝐶𝑆

−C (2)

9

Assumptions and Model Four-stage Game

Stage:1 EPA assigns an audit probability to each of the firms with an aim to lower to firm’s emission

to the socially efficient level.

Stage:2 EPA chooses the unit tax that maximizes social welfare and then set penalty for untruthful

reporting.

Stage:3 Firms choose the level of emission.

Stage:4 Firms choose emission reports.

10

Model analysis and some results

Stage Four: Reporting equilibrium

max𝑟𝑖≤𝑒𝑖

𝜋1(𝑝1 𝑟1, 𝑟2 , 𝑡, 𝑒1, 𝑟1, 𝑟2) = 𝑎 − 𝑒1 𝑒1 − 𝑡𝑟1 − 𝑝1(𝑟1, 𝑟2)(𝜃 + 𝑡)(𝑒1 − 𝑟1) (3)

Firm-1 to be truthful in reporting if and only if 𝑡(𝑒1 − 𝑟1) ≤ 𝑝1(𝜃 + 𝑡)(𝑒1 − 𝑟1)

𝑝 =𝑡

(𝜃+𝑡) & 𝑝 = 1 −

𝑡

𝜃+𝑡=

𝜃

(𝜃+𝑡).

11

Model analysis and some results

Stage Four: Reporting equilibrium

Lemma 1. As long as 𝑝𝑖 ≥𝑡

(𝜃+𝑡), firms

become truthful in reporting the level of

emission, whereas, if 𝑝𝑖 ≤𝜃

(𝜃+𝑡), firms

report no emissions. It gives audit

probability range, (𝑝, 𝑝) =𝜃

(𝜃+𝑡),

𝑡

(𝜃+𝑡).

E

𝑒 , 𝑟

𝑒∗

𝑒 = 𝑒∗

𝑟 = 0

𝑒

𝑟

𝑒 = 𝑒∗ = 𝑟

𝜃

(𝜃 + 𝑡)

𝑡

(𝜃 + 𝑡)

𝑝

(a) (b) (c)

Fig:1 Firm’s decision about 𝑒 (blue curve) and 𝑟 (orange curve) with probability of auditing.

12

Model analysis and some results

Stage Four: Reporting equilibrium

𝑝1 𝑟1∗, 𝑟2 𝑡 + 𝜃𝐷𝑖𝑟𝑒𝑐𝑡 𝑀𝐵

−𝜕𝑝1 𝑟1

∗,𝑟2

𝜕𝑟1𝑡 + 𝜃 𝑒1 − 𝑟1

∗

𝐼𝑛𝑑𝑖𝑟𝑒𝑐𝑡 𝑀𝐵

= 𝑡⏟𝑀𝐶

(F.O.C) (4)

2𝜕𝑝1 𝑟1,𝑟2

𝜕𝑟1

−

𝑡 + 𝜃

−

−𝜕2𝑝1 𝑟1,𝑟2

𝜕𝑟12

+

𝑡 + 𝜃 𝑒1 − 𝑟1

+

< 0 (S.O.C) (5)

𝑟1∗(𝑒1, 𝑒2, 𝑡, 𝜃) ∈ [0, 𝑒1] if and only if 𝜕𝑝1 .

𝜕𝑟1< 0 at 𝑟1 = 𝑟1

∗

13

Model analysis and some result

Stage Three: Emission equilibrium max

𝑒𝑖≥0𝜋1(𝑒1, 𝑒2, 𝑟1

∗ 𝑒1, 𝑒2, 𝑡, 𝜃 , 𝑟2∗(𝑒1, 𝑒2, 𝑡, 𝜃)) = 𝑎 − 𝑒1 𝑒1 − 𝑡𝑟1

∗ − 𝑝1(𝑟1∗, 𝑟2)(𝜃 + 𝑡)(𝑒1 − 𝑟1

∗) (6)

𝜕𝜋1

𝜕𝑒1= 𝑎 − 2𝑒1 − 𝑝1 𝑡 + 𝜃

𝐷𝑖𝑟𝑒𝑐𝑡 𝑒𝑓𝑓𝑒𝑐𝑡

−𝜕𝑝1

𝜕𝑟2

𝜕𝑟2

𝜕𝑒1𝑡 + 𝜃 𝑒1 − 𝑟1

𝑆𝑡𝑟𝑎𝑡𝑒𝑔𝑖𝑐 𝑒𝑓𝑓𝑒𝑐𝑡

= 0 (7)

Considering budget-balanced audit mechanism from (4) and (7) we obtain:

𝑎 − 2𝑒1

𝑀𝐵

= 𝑝1 𝑡 + 𝜃 +

𝜕𝑝2𝜕𝑟2𝜕𝑝1𝜕𝑟1

𝜕𝑟2

𝜕𝑒1(𝑡 − 𝑝1 𝑡 + 𝜃 )

𝑀𝐶

(8)

= 1 (9)

If equation (9) holds true, firm-1’s equilibrium level of emission: 𝑒1∗ =

(𝑎−𝑡)

2(10)

14

Model analysis and some result

Stage Two: Tax equilibrium

max𝑡≥0

𝐸𝑠𝑤 = 𝑎 − 𝑡 −1

4𝑎 − 𝑡 2 −

𝑑

4𝑎 − 𝑡 2 − 𝐶 (11)

𝑡∗ = 𝑎(𝑑−1)

(𝑑+1)(12)

The penalty, 𝜃 is being set as, 0 < 𝜃 < 𝑎(𝑑−1)

(𝑑+1)

15

Model analysis and some results Stage One: Designing optimal audit mechanism

As our audit mechanism is symmetric, we get,

From equation (9), we have

To check, totally differentiate first order condition of firm-1 (equation (4)) and firm-2, which yields,

2𝜕𝑝1

𝜕𝑟1𝑡 + 𝜃 −

𝜕2𝑝1

𝜕𝑟12

𝑡 + 𝜃 𝑒1 − 𝑟1

𝜕𝑝1

𝜕𝑟2𝑡 + 𝜃 −

𝜕2𝑝1

𝜕𝑟1𝜕𝑟2𝑡 + 𝜃 𝑒1 − 𝑟1

𝜕𝑝2

𝜕𝑟1𝑡 + 𝜃 −

𝜕2𝑝2

𝜕𝑟2𝜕𝑟1𝑡 + 𝜃 𝑒2 − 𝑟2 2

𝜕𝑝2

𝜕𝑟2𝑡 + 𝜃 −

𝜕2𝑝2

𝜕𝑟22

𝑡 + 𝜃 𝑒2 − 𝑟2

𝜕𝑟1

𝜕𝑟2=

𝜕𝑝1

𝜕𝑟1𝑡 + 𝜃 0

0𝜕𝑝2

𝜕𝑟2𝑡 + 𝜃

𝜕𝑒1

𝜕𝑒2

𝜕𝑟2

𝜕𝑒1=

−𝜕𝑝1𝜕𝑟1

𝑡+𝜃𝜕𝑝2𝜕𝑟1

𝑡+𝜃 −𝜕2𝑝2

𝜕𝑟2𝜕𝑟1𝑡+𝜃 𝑒2−𝑟2

𝑡+𝜃 2 2𝜕𝑝1𝜕𝑟1

−𝜕2𝑝1

𝜕𝑟12 𝑒1−𝑟1 2

𝜕𝑝2𝜕𝑟2

−𝜕2𝑝2

𝜕𝑟22 𝑒2−𝑟2 −

𝜕𝑝1𝜕𝑟2

𝑡+𝜃 −𝜕2𝑝1

𝜕𝑟1𝜕𝑟2𝑡+𝜃 𝑒1−𝑟1

𝜕𝑝2𝜕𝑟1

𝑡+𝜃 −𝜕2𝑝2

𝜕𝑟2𝜕𝑟1𝑡+𝜃 𝑒2−𝑟2

16

Model analysis and some results Stage One: Designing optimal audit mechanism

⟹𝜕𝑟2

𝜕𝑒1 𝑟1=𝑟2

=−

𝜕𝑝1𝜕𝑟1

𝑡+𝜃𝜕𝑝2𝜕𝑟1

𝑡+𝜃

𝑡+𝜃 2 2𝜕𝑝1𝜕𝑟1

−𝜕2𝑝1

𝜕𝑟12 𝑒1−𝑟1 2

𝜕𝑝2𝜕𝑟2

−𝜕2𝑝2

𝜕𝑟22 𝑒2−𝑟2 −

𝜕𝑝1𝜕𝑟2

𝜕𝑝2𝜕𝑟1

[Considering both budget-balance and symmetry,

𝜕2𝑝𝑖

𝜕𝑟𝑖𝜕𝑟𝑗 𝑟𝑖=𝑟𝑗

= 0]

⟹ 1 =−

𝜕𝑝1𝜕𝑟1

𝜕𝑝2𝜕𝑟1

2𝜕𝑝1𝜕𝑟1

−𝜕2𝑝1

𝜕𝑟12 𝑒1−𝑟1 2

𝜕𝑝2𝜕𝑟2

−𝜕2𝑝2

𝜕𝑟22 𝑒2−𝑟2 −

𝜕𝑝1𝜕𝑟2

𝜕𝑝2𝜕𝑟1

⟹ 1 =

𝜕𝑝1𝜕𝑟1

𝜕𝑝1𝜕𝑟1

2𝜕𝑝1𝜕𝑟1

−𝜕2𝑝1

𝜕𝑟12 𝑒1−𝑟1 2

𝜕𝑝2𝜕𝑟2

−𝜕2𝑝2

𝜕𝑟22 𝑒2−𝑟2 −

𝜕𝑝1𝜕𝑟1

𝜕𝑝2𝜕𝑟2

[as from budget balancing 𝜕𝑝1

𝜕𝑟2= −

𝜕𝑝2

𝜕𝑟2& 𝜕𝑝2

𝜕𝑟1= −

𝜕𝑝1

𝜕𝑟1]

⟹ 1 =1

2−

𝜕2𝑝1

𝜕𝑟12

𝜕𝑝1𝜕𝑟1

𝑒1−𝑟1

2

−1

[from symmetry 𝜕𝑝1

𝜕𝑟1=

𝜕𝑝2

𝜕𝑟2]

⟹ 1 =1

2−

𝜕2𝑝1

𝜕𝑟12

(𝜕𝑝1𝜕𝑟1

)2

𝑝1 𝑡+𝜃 −𝑡

𝑡+𝜃

2

−1

[from (4) we get, 𝑒1 − 𝑟1 =1

𝜕𝑝1𝜕𝑟1

𝑝1 𝑡+𝜃 −𝑡

𝑡+𝜃]

17

Model analysis and some results Stage One: Designing optimal audit mechanism

⟹ 2 −

𝜕2𝑝1

𝜕𝑟12

𝜕𝑝1𝜕𝑟1

2

𝑝1 𝑡+𝜃 −𝑡

𝑡+𝜃= 2

⟹

𝜕2𝑝1

𝜕𝑟12

𝜕𝑝1𝜕𝑟1

2 =2−2

−𝑝1 𝑡+𝜃 −𝑡

𝑡+𝜃

=2−2

− 𝑝1−𝑡

𝑡+𝜃

=2−2

− 𝑝1−𝑡

𝑡+𝜃

=2−2𝑡−𝜃

𝑡+𝜃

=2−2 𝑡+𝜃

𝑡−𝜃=

2−2 𝑡+𝜃

𝑡−𝜃=

2−2𝑎 𝑑−1

𝑑+1+𝜃 𝑡

𝑎 𝑑−1

𝑑+1−𝜃 𝑡

⟹

𝜕2𝑝1

𝜕𝑟12

𝜕𝑝1𝜕𝑟1

2 =2−2 𝑎 𝑑−1 +𝜃 𝑡 (𝑑+1

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1

⟹ −1

𝜕𝑝1𝜕𝑟1

+𝟐−𝟐 𝒂 𝒅−𝟏 +𝜽 𝒕 (𝒅+𝟏

𝒂 𝒅−𝟏 −𝜽 𝒕 (𝒅+𝟏𝑹

𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝒅𝒆𝒓𝒊𝒗𝒆𝒅 𝒇𝒓𝒐𝒎 𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒆 𝒊𝒏𝒕𝒆𝒈𝒓𝒂𝒕𝒊𝒐𝒏

=2−2 𝑎 𝑑−1 +𝜃 𝑡 (𝑑+1

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1𝑟1

⟹𝜕𝑝1

𝜕𝑟1= −

1

2− 2 𝑎 𝑑−1 +𝜃 𝑡 (𝑑+1

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1𝑅−𝑟1

⟹ 𝑝1 𝑟1, 𝑟2 =𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1

2− 2 𝑎 𝑑−1 +𝜃 𝑡 (𝑑+1ln(𝑅 − 𝑟1) + 𝑣

18

Model analysis and some results Stage One: Designing optimal audit mechanism

As the audit mechanism is both budget-balancing and symmetric,

𝑝1 𝑟1, 𝑟2 =𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1

2− 2 𝑎 𝑑−1 +𝜃 𝑡 (𝑑+1ln 𝑅 − 𝑟1 +

1

2−

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1

2− 2 𝑎 𝑑−1 +𝜃 𝑡 (𝑑+1ln 𝑅 − 𝑟2

𝑝1 𝑟1, 𝑟2 =1

2+

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1

2− 2 𝑎 𝑑−1 +𝜃 𝑡 (𝑑+1ln

𝑅−𝑟1

𝑅−𝑟2at 𝑟1 = 𝑟2 = 𝑟∗ (13)

From lemma 1, 𝑝1 =𝑡

(𝜃(𝑡)+𝑡)iff 𝑝1 ≥

𝑡

𝜃(𝑡)+𝑡⟹ 𝑟 ≤ 𝑅 − 𝑅 − 𝑟 exp

And, 𝑝1 =𝜃(𝑡)

(𝜃(𝑡)+𝑡)iff 𝑝1 ≤

𝜃(𝑡)

𝜃(𝑡)+𝑡⟹ 𝑟1 ≥ 𝑅 − (𝑅 − 𝑟2)exp

2− 2 𝜃(𝑡) 𝑑+1

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1

Proposition 1. A speculation for the optimal audit mechanism, which assigns an audit probability to firm-𝑖 is given by:

𝑝𝑖 𝑟𝑖, 𝑟𝑗 =

𝜃(𝑡) 𝑑+1

𝜃(𝑡)(𝑑+1)+𝑎 𝑑−1 𝑖𝑓 𝑟𝑖 > 𝑅 − (𝑅 − 𝑟𝑗)exp

2− 2 𝜃(𝑡) 𝑑+1

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1

𝑎 𝑑−1

𝜃(𝑡)(𝑑+1)+𝑎 𝑑−1 𝑖𝑓 𝑟𝑖 < 𝑅 − 𝑅 − 𝑟𝑗 exp

𝑎 𝑑−1 2− 2

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1)

1

2+

𝑎 𝑑−1 −𝜃 𝑡 (𝑑+1

2− 2 𝑎 𝑑−1 +𝜃 𝑡 (𝑑+1ln

𝑅−𝑟𝑖

𝑅−𝑟𝑗𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

(14)

19

Model analysis and some results

Stage One: Designing optimal audit mechanism Analyzing upper limit and lower limit of audit probability depending on change in

penalty, 𝜃 with tax, 𝑡 = 𝑡∗

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8Up

pe

r lim

it a

nd

low

er l

imit

of p

rob

ab

ility

𝜃 and 𝑡=𝑡∗

Change in range of probability with 𝜃Upper_p Lower_p

𝑡=𝑡*

Proposition 2. If imposed penalty on unreported emission is exactly equal to tax, then RAM ensures socially efficient emission.

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Further Research

Reporting behavior of firms in response to other firms’ reporting, own emission and audit probability. (working…)

Designing audit mechanism using signaling game.

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Question?

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