3. Models for optimizing the active base of contributors per categories

#### 3.1. Initial model

considered in the tributary and management report [1, 2] and fiscal regimes [32–34]. Tables 2

It was important to establish within each category the ratio of participation in monetary units

of contributors)

Fourth quarter of 2015 (millions

First quarter of 2016 (millions

Percentage (%) of increase

of contributors)

with respect to three classes of taxes as shown in the following tables (Tables 4 and 5):

Source: Compiled by authors with data from the tributary and management report up to the first quarter of 2016.

Tax Millions of pesos Relation with respect to total (%)

Federal income law of 2016 (millions of pesos)

Types of contributors ISR (%) IVA (%) IEPS (%) Total (%) Individuals 21 14.12 3.21 39 Employees 32 21.21 4.82 58 Entities 2 1.28 0.29 3 Total 55 37 8 100

Table 4. Ratio of participation in taxes (LIF 2015) per number contributors up to the fourth quarter of 2015.

249299.5 17.95

Individuals 19.4 19.9 19.9 Employees 29.5 29.9 30.2 Entities 1.8 1.8 1.8 Total 50.7 51.6 51.9

Table 1. Contributors according to the class reported in the tributary and management report.

IVA 703848.50 741988.7 5.42 IEPS 159970.60 348945.2 18.13 Source: Compiled by authors with data from the federal income law in the 2015 and 2016 fiscal years.

ISR 1059206.20 55 IVA 703848.50 37 IEPS 159970.60 8 Total tax collection 1923025.30 100 Source: Compiled by authors with data from federal income law for the fiscal year of 2015 [4].

Table 2. Collection of ISR, IVA, and IEPS according to the LIF of 2015.

059206.20 1<sup>0</sup>

Table 3. Variation in the estimated collection for ISR, IVA, and IEPS (2015–2016).

Concept Federal income law of 2015 (millions of pesos)

ISR 1<sup>0</sup>

Source: Compiled by authors.

and 3 include the collection of ISR, IVA, and IEPS.

Third quarter of 2015 (millions

of contributors)

Types of contributors

214 Taxes and Taxation Trends

#### 3.1.1. Structure and assessment

The first approach was to develop a model applicable to the fiscal year of 2016 and the previous ones. The OF will consider the constant returns per million contributors up to the fourth quarter of 2015, and this will multiply the optimized number of contributors with the three tax categories considered in the tributary and management reports. The final results are the incomes by ISR, IVA, and IEPS included in the fiscal year of 2015 (Table 2)

$$\text{Maximize} = R\_{47201597}\pi + R\_{472015AS}\varphi + R\_{472015PM}\omega \tag{1}$$

The initial model should be permitted to display the time evolution of the active base of contributors with respect to the three tax categories. The returns considered the total revenues of 2015 with respect to the census or active base of contributors up to the fourth quarter of 2015, and the restrictions were with respect to the previous quarter. The restrictions are the following<sup>1</sup> :


<sup>1</sup> The first restriction considers the number of contributors at the end of the financial year, i.e., at the fourth quarter of 2015; however, the rest of restrictions consider the number of contributors at the previous exercise (third quarter of 2015). This allows to obtain the optimal combination and evolution of the model and to compare it with the official information at the end of the financial year (fourth quarter). The comparison constitutes an indicator of the diversity or not of the tax burden, if there is a need to increase the number of contributors in a category of whether the active base should not be increased.

• Entities: millions of active contributors according to the third quarter tributary and management report of 2015

$$
\pi + \varphi + \omega \le \theta \tag{2}
$$

$$
\pi \succeq \theta\_1 \tag{3}
$$

<sup>R</sup>2015<sup>τ</sup> <sup>¼</sup> <sup>1923025</sup>:3∗0:<sup>39</sup>

<sup>R</sup>2015<sup>φ</sup> <sup>¼</sup> <sup>1923025</sup>:3∗0:<sup>58</sup>

<sup>R</sup>2015<sup>φ</sup> <sup>¼</sup> <sup>1923025</sup>:3∗0:<sup>03</sup>

8 >>>>><

>>>>>:

contributorsÞ, and ω ¼ 1:80 ð Þ entities; in million contributors .

Based on the above, the OF is given by

Subjected to restrictions

and IEPS within the LIF for 2015.

million contributors [2].

million contributors; otherwise, the result would be exact.

3

19:9 � �

29:9 � �

1:8 � �

τ þ 1φ þ 1ω ≤ 51:60 τ þ 0φ þ 0ω ≥ 19:40 τ þ 1φ þ 0ω ≥ 29:50 τ þ 0φ þ 1ω ≥ 1:80

τ,φ, ω ≥ 0

The results obtained by using the PHP Simplex tool [35] and replicated with Solver in Excel were the following: τ ¼ 20:30 ð Þ individuals; in million contributors , φ ¼ 29:50 ðemployees, in million

Based on the optimized number of contributors, the product of these variables with the returns, i.e., the maximized results of (6), is tested for equality with total revenue by ISR, IVA,

Maximize ¼ 37267:93 20 ð Þþ :30 37267:93 29 ð Þþ :5 37267:93 1ð Þ :8

Maximize ¼ 1923025:20

As can be noted, the optimized results for the total revenue by ISR, IVA, and IEPS are the same with respect to the approved LIF for the fiscal year 20153 (Table 2). The proposed model indicates, however, in this scenario of constant returns per million contributors, that a better choice would be to increase the number of individuals to 20.3 million instead of the one reported in the tributary form of the fourth quarter of 2015 in which this number reaches 19.9 million. The difference, however, was in the number of employees that went from 29.5 to 29.9

A ten-decimal place's difference exists due to the fact that only two decimal points were considered for the returns per

¼ 37726:93

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¼ 37726:93

¼ 37726:93

Maximize ¼ 37267:93τ þ 37267:93φ þ 37267:93ω (6)

$$
\varphi \succeq \theta\_2 \tag{4}
$$

$$
\omega \ge \theta\_3 \tag{5}
$$

where τ ¼ individuals, φ ¼ employees, ω ¼ entities, θ ¼ active contributors roll, in millions; θτ ¼ active roll of individuals, in millions; θφ ¼ active roll of employees, in millions; and θω ¼ active roll of entities, in millions.

Based on the above, the initial model is the following:


The estimated returns, R4T2015, (for 2015) per million contributors, is determined by the total returns considered in the LIF of 2015 multiplied for each class of contributors with respect to the total, and the result of this is divided by the number of contributors for each class according to the tributary and management report of the fourth quarter of 20152 :

<sup>2</sup> The data is contained in Tables 1 and 2 and in pages 30 and 31. It is important to note that for the initial model, the return per million contributors is constant in all categories but not for the following scenarios:

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$$R\_{2015}\tau = \left[\frac{1923025.3 \ast 0.39}{19.9}\right] = 37726.93$$

$$R\_{2015}\varphi = \left[\frac{1923025.3 \ast 0.58}{29.9}\right] = 37726.93$$

$$R\_{2015}\varphi = \left[\frac{1923025.3 \ast 0.03}{1.8}\right] = 37726.93$$

Based on the above, the OF is given by

$$\text{Maximize} = 37267.93\pi + 37267.93\varphi + 37267.93\omega \tag{6}$$

Subjected to restrictions

• Entities: millions of active contributors according to the third quarter tributary and man-

where τ ¼ individuals, φ ¼ employees, ω ¼ entities, θ ¼ active contributors roll, in millions;

The estimated returns, R4T2015, (for 2015) per million contributors, is determined by the total returns considered in the LIF of 2015 multiplied for each class of contributors with respect to the total, and the result of this is divided by the number of contributors for each class

The data is contained in Tables 1 and 2 and in pages 30 and 31. It is important to note that for the initial model, the return

according to the tributary and management report of the fourth quarter of 20152

per million contributors is constant in all categories but not for the following scenarios:

θτ ¼ active roll of individuals, in millions; θφ ¼ active roll of employees, in millions; and

Active base of contributors up to the fourth quarter of 2015

(active contributors up to the third quarter of 2015, in millions)

(active contributors up to the third quarter of 2015, in millions)

(active contributors up to the third quarter of 2015, in millions)

τ þ φ þ ω ≤ θ (2)

τ ≥ θ<sup>1</sup> (3)

φ≥ θ<sup>2</sup> (4)

ω ≥ θ<sup>3</sup> (5)

Maximize

R<sup>4</sup>T2015PFτ þ R<sup>4</sup>T2015ASφ þ R<sup>4</sup>T2015PMω

τφω ≤ θ

τ ≥ θ<sup>1</sup>

φ ≥ θ<sup>2</sup>

ω ≥ θ<sup>3</sup>

:

agement report of 2015

216 Taxes and Taxation Trends

θω ¼ active roll of entities, in millions.

Subjected to

Individuals

Employees

Entities

(millions of contributors)

Returns per million contributors R<sup>4</sup>T2015PF ¼ Fourth quarter return, individuals R<sup>4</sup>T2015AS ¼ Fourth quarter return, employees R<sup>4</sup>T2015PM ¼ Fourth quarter return, entities

Model 1

Restriction variables

1 θ

2 θ<sup>1</sup>

3 θ<sup>2</sup>

4 θ<sup>3</sup>

2

Source: Compiled by authors

Based on the above, the initial model is the following:

$$\begin{cases} 1\tau + 1\varrho + 1\omega \le 51.60 \\ 1\tau + 0\varrho + 0\omega \ge 19.40 \\ 0\tau + 1\varrho + 0\omega \ge 29.50 \\ 0\tau + 0\varrho + 1\omega \ge 1.80 \end{cases}$$
 
$$\tau, \varrho, \omega \ge 0$$

The results obtained by using the PHP Simplex tool [35] and replicated with Solver in Excel were the following: τ ¼ 20:30 ð Þ individuals; in million contributors , φ ¼ 29:50 ðemployees, in million contributorsÞ, and ω ¼ 1:80 ð Þ entities; in million contributors .

Based on the optimized number of contributors, the product of these variables with the returns, i.e., the maximized results of (6), is tested for equality with total revenue by ISR, IVA, and IEPS within the LIF for 2015.

$$\text{Maximize} = \text{37267.93}(20.30) + \text{37267.93}(29.5) + \text{37267.93}(1.8)$$

$$\text{Maximize} = 1923025.20$$

As can be noted, the optimized results for the total revenue by ISR, IVA, and IEPS are the same with respect to the approved LIF for the fiscal year 20153 (Table 2). The proposed model indicates, however, in this scenario of constant returns per million contributors, that a better choice would be to increase the number of individuals to 20.3 million instead of the one reported in the tributary form of the fourth quarter of 2015 in which this number reaches 19.9 million. The difference, however, was in the number of employees that went from 29.5 to 29.9 million contributors [2].

<sup>3</sup> A ten-decimal place's difference exists due to the fact that only two decimal points were considered for the returns per million contributors; otherwise, the result would be exact.

#### 3.2. A model with differentiated returns

Using the proposed model of the above section, the next model considers several types of returns per million contributors with respect to three categories, a condition that can be well estimated and updated by SAT. For this, \$34,000 is considered for individuals, \$35,963.57 for employees, and \$75,000 for entities which results in an OF of the following form:

$$\text{Maximize} = 34000\pi + 35963.57\varphi + 75000\omega \tag{7}$$

ð Þ in million contributors , and ω ¼ 2:70 Entities in million contributors ð Þ. Based on the optimized number of contributors, the product of these variables by the returns, i.e., the maximized result of (7), is compared with the total revenues by ISR, IVA, an IEPS within the LIF of 2015 (Table 2),

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Maximize ¼ 34; 000 19 ð Þþ :4 35963:57 29 ð Þþ :5 75; 000 2ð Þ :70

Maximize ¼ 1923025:31

Unlike the model with constant returns, in this model that considers distinct returns, the increase should have been registered in entities, and if this is not the case, the original way of considering contributors is preferred instead, even though this situation is uneven with respect

In the following, an approach called prototype model (PM), whose objective is to give tax authorities a better idea of the capacity to adequate tax policies to obtain optimized results, is presented. The first approach is a model with constant returns<sup>4</sup> and whose objective function

modified. Also, an additional restriction concerning the total number of active contributors and distributed in two classes (employees and entities) is that this should be at least 34.43

> τ þ 1φ þ 1ω ≤ 61:70 τ þ 0φ þ 0ω ≥ 19:40 τ þ 1φ þ 0ω ≥ 29:50 τ þ 0φ þ 1ω ≥ 2:05 τ þ 1φ þ 1ω ≥ 34:43

> > �τ,φ, ω ≥ 0

With the purpose of verifying the time evolution of the results in a broader range, in this scenario the returns are obtained by dividing the total revenues by ISR, IVA, and IEPS within the LIF of 2015 by the total number of contributors up to the third quarter of 2015 in the tributary and management report. Unlike the model derived above, this model considers to

The number 61.7 in the first restriction represents the total number of active contributors estimated for late 2016. The

8

>>>>>>>><

>>>>>>>>:

quantities 19.40, 29.50, and 1.80 correspond to the active census up to the third quarter of 2015.

Maximize ¼ 37929:49τ þ 37929:49φ þ 37929:49ω (8)

, the restriction for entities to be at least 2.5 million contributors will be

Using the maximized OF of Eq. (7), the following is obtained:

i.e., \$1923025.30.

to tax participation.

to maximize is

million.

4

5

In this new proposal<sup>5</sup>

3.3. Prototype model with constant returns

obtain the returns by ISR, IVA, and IEPS in the LIF of 2016.

The restrictions were 51.60 million contributors as the maximum allowed and that corresponds to the total number of contributors of the tributary and management report of the fourth quarter of 2015 and also the restriction which corresponds to the official number of contributors up to the third quarter of 2015 and that will allow to know the optimal change in each category.


Source: Compiled by the authors

1τ þ 1φ þ 1ω ≤ 51:60 1τ þ 0φ þ 0ω ≥ 19:40 0τ þ 1φ þ 0ω ≥ 29:50 0τ þ 0φ þ 1ω ≥ 1:80 8 >>>>>< >>>>>: τ,φ, ω ≥ 0

The results obtained by using the PHP Simplex tool [35] and replicated with Solver in Excel were the following: τ ¼ 19:40 Individuals in million contributors ð Þ, φ ¼ 29:50 Employees

ð Þ in million contributors , and ω ¼ 2:70 Entities in million contributors ð Þ. Based on the optimized number of contributors, the product of these variables by the returns, i.e., the maximized result of (7), is compared with the total revenues by ISR, IVA, an IEPS within the LIF of 2015 (Table 2), i.e., \$1923025.30.

Using the maximized OF of Eq. (7), the following is obtained:

$$\text{Maximize} = \mathbf{34}, \mathbf{000}(\mathbf{19.4}) + \mathbf{35} \mathbf{963.57}(\mathbf{29.5}) + \mathbf{75}, \mathbf{000}(\mathbf{2.70})$$

$$\text{Maximize} = \mathbf{1923025.31}$$

Unlike the model with constant returns, in this model that considers distinct returns, the increase should have been registered in entities, and if this is not the case, the original way of considering contributors is preferred instead, even though this situation is uneven with respect to tax participation.

#### 3.3. Prototype model with constant returns

3.2. A model with differentiated returns

218 Taxes and Taxation Trends

Model 2

Restriction variables

1 θ

2 θ<sup>1</sup>

3 θ<sup>2</sup>

4 θ<sup>3</sup>

Returns per million contributors Rdif <sup>2015</sup>PF ¼ Differenced returns, individuals Rdif <sup>2015</sup> AS ¼ Differenced returns, employees Rdif <sup>2015</sup>PM ¼ Differenced returns, entities

Subjected to

Individuals

Employees

Entities

Source: Compiled by the authors

(millions of contributors)

Using the proposed model of the above section, the next model considers several types of returns per million contributors with respect to three categories, a condition that can be well estimated and updated by SAT. For this, \$34,000 is considered for individuals, \$35,963.57 for

The restrictions were 51.60 million contributors as the maximum allowed and that corresponds to the total number of contributors of the tributary and management report of the fourth quarter of 2015 and also the restriction which corresponds to the official number of contributors up to the

> τ þ 1φ þ 1ω ≤ 51:60 τ þ 0φ þ 0ω ≥ 19:40 τ þ 1φ þ 0ω ≥ 29:50 τ þ 0φ þ 1ω ≥ 1:80

> > τ,φ, ω ≥ 0

The results obtained by using the PHP Simplex tool [35] and replicated with Solver in Excel were the following: τ ¼ 19:40 Individuals in million contributors ð Þ, φ ¼ 29:50 Employees

8 >>>>><

Active base of contributors up to the fourth quarter of 2015

(active contributors up to the third quarter of 2015, in millions)

(active contributors up to the third quarter of 2015, in millions)

(active contributors up to the third quarter of 2015, in millions)

>>>>>:

Maximize ¼ 34000τ þ 35963:57φ þ 75000ω (7)

Maximize

Rdif <sup>2015</sup>PFτ þ Rdif <sup>2015</sup>ASφ þ Rdif <sup>2015</sup>PMω

τφω ≤ θ

τ ≥ θ<sup>1</sup>

φ ≥ θ<sup>2</sup>

ω ≥ θ<sup>3</sup>

employees, and \$75,000 for entities which results in an OF of the following form:

third quarter of 2015 and that will allow to know the optimal change in each category.

In the following, an approach called prototype model (PM), whose objective is to give tax authorities a better idea of the capacity to adequate tax policies to obtain optimized results, is presented. The first approach is a model with constant returns<sup>4</sup> and whose objective function to maximize is

$$\text{Maximize} = \text{37929.49}\pi + \text{37929.49}\phi + \text{37929.49}\omega \tag{8}$$

In this new proposal<sup>5</sup> , the restriction for entities to be at least 2.5 million contributors will be modified. Also, an additional restriction concerning the total number of active contributors and distributed in two classes (employees and entities) is that this should be at least 34.43 million.

$$\begin{cases} 1\tau + 1\varrho + 1\omega \le 61.70 \\\\ 1\tau + 0\varrho + 0\omega \ge 19.40 \\\\ 0\tau + 1\varrho + 0\omega \ge 29.50 \\\\ 0\tau + 0\varrho + 1\omega \ge 2.05 \\\\ 0\tau + 1\varrho + 1\omega \ge 34.43 \end{cases}$$

$$-\tau, \varrho, \omega \ge 0$$

<sup>4</sup> With the purpose of verifying the time evolution of the results in a broader range, in this scenario the returns are obtained by dividing the total revenues by ISR, IVA, and IEPS within the LIF of 2015 by the total number of contributors up to the third quarter of 2015 in the tributary and management report. Unlike the model derived above, this model considers to obtain the returns by ISR, IVA, and IEPS in the LIF of 2016.

<sup>5</sup> The number 61.7 in the first restriction represents the total number of active contributors estimated for late 2016. The quantities 19.40, 29.50, and 1.80 correspond to the active census up to the third quarter of 2015.


The results obtained with the PHP Simplex tool [35] and replicated with Solver of MS Excel were the following: τ ¼ 27:27 million contributors, individuals; φ ¼ 32:38 million contributors, employees; and ω ¼ 2:05 million contributors, entities.

Based on the optimized number of contributors, the product of these variables by the returns, i.e., the maximized result of (19), is compared with the total revenue by ISR, IVA, and IEPS, in this case, considering the LIF of 2016. Maximizing, again, the OF of (19) results in

3.4. Prototype model with differentiated returns

Model 4

Restriction variables

1 θ

2 θ<sup>1</sup>

3 θ<sup>2</sup>

4 δ<sup>1</sup>

5 δ<sup>2</sup>

6 δ<sup>3</sup>

Source: Compiled by authors

Returns per million contributors

Rdif <sup>2016</sup>PF ¼ 2016 differenced returns, Individuals Rdif <sup>2016</sup> AS ¼ 2016 differenced returns, Employees Rdif <sup>2016</sup>PM ¼ 2016 differenced returns, Entities

Subjected to

Individuals

Employees

8

Estimated active base of contributors for 2016

Minimum number of contributors (entities)

(millions of active contributors up to the fourth quarter of 2015)

(millions of active contributors up to the fourth quarter of 2015)

Minimum number of contributors (employees and entities)

Minimum number of contributors (individuals and entities)

(in millions of contributors)

(in million contributors)

(in million contributors)

(in million contributors)

maximize the following objective function:

contributors) will reach at least 22.50

8

In the following a model which considers increments per million contributors, where each contributor may lie within three categories, is presented (the categories may be adjusted by fiscal authorities when needed). Moreover, a restriction which considers a minimum number of contributors in the employees and entities categories is added (as before these categories may be adjusted by the goals and objectives of the fiscal authorities). The model proposes to

An additional restriction is the condition that the total number of contributors (in million

The variable δ<sup>n</sup> will be used for restrictions that depend upon goals and objectives of the fiscal authorities.

Maximize ¼ 40239:19τ þ 35963:57φ þ 37000ω (9)

Maximize

A Model for Estimating the Number of Taxpayer That Fullfill Mexican Income Law

Rdif <sup>2016</sup>PFτ þ Rdif <sup>2016</sup>ASφ þ Rdif <sup>2016</sup>PMω

221

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τφω ≤ θ

τ ≥ θ<sup>1</sup>

φ ≥ θ<sup>2</sup>

φ ω ≥ δ<sup>2</sup>

φ ω ≥ δ<sup>3</sup>

ω ≥ δ<sup>1</sup>

Maximize ¼ 37929:49 27 ð Þþ :27 37929:49 32 ð Þþ :38 37929:49 2ð Þ :05 Maximize <sup>¼</sup> <sup>2340249</sup>:537

The above result represents the total expected tax collection for 2016 considering ISR, IVA, and IEPS. The results present differences in decimals due to the fact that only two decimal points were considered in the returns; however, by using the complete decimals, the result would be exact.

<sup>6</sup> The variable δ<sup>n</sup> will be used for restrictions that are set in accordance with goals and objectives of tax authorities. 7 Value corresponds to the sum of the revenues for 2016 included in the federal law of incomes (LIF) (Table 3).

#### Model 4

The results obtained with the PHP Simplex tool [35] and replicated with Solver of MS Excel were the following: τ ¼ 27:27 million contributors, individuals; φ ¼ 32:38 million contributors,

Maximize

R<sup>3</sup>T2015PFτ þ R<sup>3</sup>T2015ASφ þ R<sup>3</sup>T2015PMω

τφω ≤ θ

τ ≥ θ<sup>1</sup>

φ ≥ θ<sup>2</sup>

φ ω ≥ δ<sup>2</sup>

ω ≥ δ<sup>1</sup>

Based on the optimized number of contributors, the product of these variables by the returns, i.e., the maximized result of (19), is compared with the total revenue by ISR, IVA, and IEPS, in

Maximize ¼ 37929:49 27 ð Þþ :27 37929:49 32 ð Þþ :38 37929:49 2ð Þ :05

Maximize <sup>¼</sup> <sup>2340249</sup>:537

The above result represents the total expected tax collection for 2016 considering ISR, IVA, and IEPS. The results present differences in decimals due to the fact that only two decimal points were considered in the returns; however, by using the complete decimals, the result would be

The variable δ<sup>n</sup> will be used for restrictions that are set in accordance with goals and objectives of tax authorities.

Value corresponds to the sum of the revenues for 2016 included in the federal law of incomes (LIF) (Table 3).

this case, considering the LIF of 2016. Maximizing, again, the OF of (19) results in

employees; and ω ¼ 2:05 million contributors, entities.

exact.

Model 3

220 Taxes and Taxation Trends

Restriction Variables

1 θ

2 θ<sup>1</sup>

3 θ<sup>2</sup>

4 δ<sup>1</sup>

5 δ<sup>2</sup>

Source: Compiled by the authors

Return per million contributors

R<sup>3</sup>T2015PF ¼ Return for the third quarter of 2015, Individuals R<sup>3</sup>T<sup>2015</sup> AS ¼ Return for the third quarter of 2015, Employees R<sup>3</sup>T2015PM ¼ Return for the third quarter of 2015, Entities

Subjected to

Individuals

Employees

and entities

6

(in million contributors)

(in million contributors)

(in million contributors)

Active base of contributors, estimated for 2016

(millions of active contributors, third quarter of 2015)

(millions of active contributors, third quarter of 2015)

Minimum required number of contributors for employees

Minimum number of contributors for entities

6

7


#### 3.4. Prototype model with differentiated returns

In the following a model which considers increments per million contributors, where each contributor may lie within three categories, is presented (the categories may be adjusted by fiscal authorities when needed). Moreover, a restriction which considers a minimum number of contributors in the employees and entities categories is added (as before these categories may be adjusted by the goals and objectives of the fiscal authorities). The model proposes to maximize the following objective function:

$$\text{Maximize} = 40239.19\pi + 35963.57q + 37000\omega \tag{9}$$

An additional restriction is the condition that the total number of contributors (in million contributors) will reach at least 22.50

<sup>8</sup> The variable δ<sup>n</sup> will be used for restrictions that depend upon goals and objectives of the fiscal authorities.

1τ þ 1φ þ 1ω ≤ 61:70 1τ þ 0φ þ 0ω ≥ 19:90 0τ þ 1φ þ 0ω ≥ 29:90 0τ þ 0φ þ 1ω ≥ 2:05 0τ þ 1φ þ 1ω ≥ 34:43 1τ þ 0φ þ 1ω ≥ 22:50 8 >>>>>>>>>>>< >>>>>>>>>>>: �τ,φ, ω ≥ 0

dividing the total collection of ISR in the fiscal year 2015 which is \$1059206.20 (in million pesos) by the total number of contributors which is 16,752,516 and which results in an approximated return per contributor of 0.0632 million pesos11. The objective function will be represented by the returns of each fiscal regime reported by the authority. Restrictions are composed of the total number of contributors (for this case it is greater than the one reported on September 2015 which is 17,000,00012), and consequently a better tax collection is expected than the one that was considered for the base of the returns. The following restrictions (14 in total) will correspond each to the total number of contributors per regime13; the number of residents abroad without a permanent establishment in Mexico is at least 200. Also, the restriction, wages, salaries, and similar regime together with the fiscal incorporation regime

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R<sup>θ</sup><sup>n</sup> ¼ Return per restriction variable

θ ¼ Total number of contributors for ISR in accordance to official goals and objectives ð Þ

θ<sup>1</sup> ¼ Wages and salaries regime and wages like incomes official data ð Þ

θ<sup>2</sup> ¼ Fiscal incorporation regime official data ð Þ

θ<sup>3</sup> ¼ Individuals with enterprise and professional activities regime official data ð Þ

θ<sup>4</sup> ¼ General regime for the law of entities official data ð Þ

θ<sup>5</sup> ¼ Ŕegimen de Arrendamiento dato oficial ð Þ

<sup>θ</sup><sup>6</sup> <sup>¼</sup> Incomes by dividends regime partners and shareholders , official data

θ<sup>7</sup> ¼ Agriculture, forestry, livestock and PF and PM fishing regime official data ð Þ

θ<sup>8</sup> ¼ Regime for the rest of incomes official data ð Þ

θ<sup>9</sup> ¼ Incomes by interests regime official data ð Þ

θ<sup>10</sup> ¼ Entities with non � profit purposes official data ð Þ

θ<sup>11</sup> ¼ Producers cooperatives that defer their incomes official data ð Þ

θ<sup>12</sup> ¼ Regime of coordinated official data ð Þ

θ<sup>13</sup> ¼ Corporate groups regime official data ð Þ

11For all the proposed models, the returns may be updated with constant quantities for each contributor or with differentiated quantities with respect to each regime and in accordance to the latest information of the fiscal authorities. 12The maximum expected number of contributors could be set according to the goals and objectives of the fiscal author-

14These two restrictions represent examples in which additional restrictions may be derived (in accordance to the goals

ities; the model presented in this work is exemplified.

and objectives of the fiscal authorities).

13In accordance to the official number of contributors reported by the authority [33].

15The number of contributors for each regime is found in the inequalities of the presented notation.

are at least 10,600,000 contributors14. In the following, the notation is presented15:

The results obtained by using the PHP Simplex tool [35] and replicated with MS Excel Solver are the following: τ ¼ 27:27 million contributors, individuals; φ ¼ 29:90 million contributors, employees; and ω ¼ 4:53 million contributors, entities.

Based on the optimized number of contributors, the next steps are to multiply these variables by the returns, i.e., the maximized result of (9), and to compare it with the total revenues by ISR, IVA, and IEPS for equality.

Using the maximized OF, (9) results in

$$\text{Maximize} = 40239.19(27.27) + 35963.57(29.90) + 37000(4.53)$$

$$\text{Maximize} = 2340243.40^9$$

The above result represents the total tax collection expected for 2016 for taxes ISR, IVA, and IEPS. Using exact quantities with all decimals will result in an exact value.
