## 1 Introduction

^{1}Moreover, by considering the net tax liability distribution, the tax rate distribution and, finally, the post-tax income distribution, Kakwani and Lambert (1998) suggest that the extent of each overall Axiom violations can be measured by the Atkinson-Kakwani-Plotnick re-ranking index of each distribution. By applying these re-ranking indexes as well as considering the Kakwani (1977) progressivity index and the Kakwani (1984) decomposition of the redistributive effect, Kakwani and Lambert (1998) evaluate the implicit or potential equity in the tax system reachable in the absence of inequities.

## 2 Starting definitions

^{2}

^{3}and \(r(b_i)=\frac{f(b_i)}{b_i}\). We can express the gross tax liability as \(s_i= r(b_i)b_i\), where \(r(b_i)\) is the average tax rate corresponding to the taxable income \(b_i\). Were the marginal tax rates applied to \(x_i\) instead to \(b_i\), the resulting tax liability would be \(v_i= r(x_i)x_i\), being \(r(x_i)\) the average tax rate corresponding to the taxable income \(x_i\). When the rate schedule is not linear and the taxpayer can benefit from deductions, \(r(x_i)\) is greater than \(r(b_i)\). The difference between \(v_i\) and \(s_i\) depends on both \(d_i\) and \(r(x_i) - r(b_i)\), so that \(v_i-s_i= (r(x_i)- r(b_i))x_i+ r(b_i)d_i\). Finally, the net tax liability \(t_i\) is equal to the gross tax liability \(s_i\) minus all the tax credits \(c_i\) from which taxpayer i can benefit. It follows that the net tax liability is computed as \(t_i = r(b_i)(x_i-d_i)-c_i\).

^{4}Similarly, the overall degree of tax progressivity is measured using the Kakwani index \(K=C_{T|X}-G_X\). As it is well known, RS and K are linked by the overall average tax rate \(\theta =\frac{\sum _{i=1}^N t_i}{\sum _{i=1}^N x_i}\), since \(RS=\frac{\mu _T}{\mu _Z}K\) and \(\frac{\mu _T}{\mu _Z}=\frac{\theta }{1-\theta }\).

## 3 Considering axiom violations for the whole tax structure

### 3.1 Non-technical explanation

### 3.2 Technical explanation

^{5}the negative influences of each Axiom violations. More precisely, Kakwani and Lambert (1998) quantify the overall Axiom 1 violations as:

^{6}

### 3.3 A stylized example

i | x | v | d | b | s | c | t | z |
---|---|---|---|---|---|---|---|---|

1 | 500 | 100 | 100 | 400 | 80 | 25 | 55 | 445 |

2 | 600 | 120 | 100 | 500 | 100 | 15 | 85 | 515 |

3 | 700 | 150 | 100 | 600 | 120 | 7 | 113 | 587 |

4 | 1000 | 275 | 100 | 900 | 225 | 150 | 75 | 925 |

Scale | i | j | x | v | d | b | s | c | t | z | \(\tilde{t}\) |
---|---|---|---|---|---|---|---|---|---|---|---|

3.3333 | 4 | 1 | 300 | 82.50 | 30.00 | 270.00 | 67.50 | 45.00 | 22.50 | 277.50 | 0.075 |

1.7142 | 2 | 2 | 350 | 70.00 | 58.33 | 291.67 | 58.33 | 8.75 | 49.58 | 300.42 | 0.142 |

1.2500 | 1 | 3 | 400 | 80.00 | 80.00 | 320.00 | 64.00 | 20.00 | 44.00 | 356.00 | 0.110 |

1.1666 | 3 | 4 | 600 | 128.57 | 85.71 | 514.29 | 102.86 | 6.00 | 96.86 | 503.14 | 0.161 |

## 4 Decomposing axiom violations

### 4.1 The ‘step by step’ decomposition

#### 4.1.1 Non-technical explanation

#### 4.1.2 Technical details

^{7}Axiom 3,

#### 4.1.3 A stylized example

### 4.2 The ‘overall and symultaneous‘ or ‘ex post’ approach

#### 4.2.1 Non-technical explanation

#### 4.2.2 Technical details

#### 4.2.3 A stylized example

## 5 Empirical analysis

### 5.1 Data

^{8}

### 5.2 Results

#### 5.2.1 Basic indexes

^{9}

Index | Value |
---|---|

\(G_{X}\) | 0.42089 |

\(G_{V}\) | 0.48343 |

\(C_{V|X}\) | 0.48245 |

\(C_{V|T}\) | 0.47992 |

\(G_{S}\) | 0.47901 |

\(C_{S|X}\) | 0.47732 |

\(G_{V-S}\) | 0.74211 |

\(C_{(V-S)|X}\) | 0.56811 |

\(C_{(V-S)|T}\) | 0.54614 |

\(C_{(V-S)|Z}\) | 0.57401 |

\(G_{X-V}\) | 0.39786 |

\(C_{(X-V)|X}\) | 0.39768 |

\(C_{(X-V)|Z}\) | 0.39746 |

\(G_{X-S}\) | 0.40147 |

\(C_{(X-S)|X}\) | 0.40123 |

\(G_{C}\) | 0.22783 |

\(C_{C|X}\) | 0.04587 |

\(C_{C|T}\) | 0.02299 |

\(C_{C|Z}\) | 0.05388 |

\(G_{T}\) | 0.64626 |

\(C_{T|X}\) | 0.63954 |

\(G_{Z}\) | 0.37097 |

\(C_{Z|X}\) | 0.37035 |

\(G_{\tilde{V}}\) | 0.07271 |

\(G_{\tilde{T}}\) | 0.42087 |

\(C_{{\tilde{T}}|X}\) | 0.39600 |

\(R_{T|X}\) | 0.00673 |

\(R_{Z|X}=AV_3\) | 0.00062 |

\(R_{Z|X}\) | 0.00062 |

\(R_{V|X}\) | 0.00098 |

\(R_{S|X}\) | 0.00169 |

\(R_{\tilde{V}|X}\) | 0.00586 |

\(R_{\tilde{S}|X}\) | 0.01642 |

\(R_{\tilde{T}|X}\) | 0.02486 |

\(AV_1\) | 0.00155 |

\(AV_{2N}\) | 0.00419 |

\(AV_2\) | 0.00575 |

\(AV_3=R_{Z|X}\) | 0.00062 |

\(RE^P\) | 0.05784 |

RS | 0.05054 |

RE | 0.04992 |

K | 0.21865 |

\(\theta\) | 0.18774 |

\(\frac{\mu _T}{\mu _Z}\) | 0.23113 |

\(\mu _X\) | 21,615.47 |

\(\mu _V\) | 5,918.43 |

\(\mu _S\) | 5,583.88 |

\(\mu _{X-V}\) | 15,697.04 |

\(\mu _{X-S}\) | 16,031.60 |

\(\mu _{V-S}\) | 334.56 |

\(\mu _C\) | 1,525.82 |

\(\mu _T\) | 4,058.06 |

\(\mu _Z\) | 17,557.41 |

\(\mu _{\tilde{V}}\) | 0.24437 |

\(\mu _{\tilde{V-S}}\) | 0.01333 |

\(\mu _{\tilde{C}}\) | 0.11098 |

\(\mu _{\tilde{T}}\) | 0.12006 |

\(r_{V|X}\) | 0.99797 |

\(r_{V|T}\) | 0.99274 |

\(r_{(V-S)|X}\) | 0.76554 |

\(r_{(V-S)|T}\) | 0.73593 |

\(r_{(V-S)|Z}\) | 0.77348 |

\(r_{(X-V)|X}\) | 0.99953 |

\(r_{(X-V)|Z}\) | 0.99899 |

\(r_{C|X}\) | 0.20135 |

\(r_{C|T}\) | 0.10093 |

\(r_{C|Z}\) | 0.23649 |

\(r_{\tilde{V}|X}\) | 0.91947 |

\(r_{\tilde{V}|\tilde{T}}\) | 0.95312 |

\(r_{\tilde{(V-S)}|X}\) | 0.04053 |

\(r_{\tilde{(V-S)}|\tilde{T}}\) | 0.00148 |

\(r_{\tilde{C}|X}\) | \(-\) 0.79120 |

\(r_{\tilde{C}|\tilde{T}}\) | \(-\) 0.84258 |

#### 5.2.2 The ‘step by step’ analysis

\(AV_1^{V}\) | \(AV_1^{D}\) | \(AV_1^{S}\) | \(AV_1^{C}\) | \(AV_1\) | |
---|---|---|---|---|---|

Value | 0.00037 | 0.00022 | 0.00059 | 0.00097 | 0.00155 |

\(AV_2^{V}\) | \(AV_2^{D}\) | \(AV_2^{S}\) | \(AV_2^{C}\) | \(AV_2\) | |
---|---|---|---|---|---|

Value | 0.00221 | 0.00351 | 0.00572 | 0.00003 | 0.00575 |

\(AV_3^{V}\) | \(AV_3^{D}\) | \(AV_3^{S}\) | \(AV_3^{C}\) | \(AV_3\) | |
---|---|---|---|---|---|

Value | 0.00019 | 0.00005 | 0.00024 | 0.00038 | 0.00062 |

#### 5.2.3 The ‘overall and simultaneous’ or ‘ex post’ analysis

\(AAV_1^{V}\) | \(AAV_1^{D}\) | \(AAV_1^{S}\) | \(AAV_1^{C}\) | \(AV_1\) | |
---|---|---|---|---|---|

Value | \(-0.00085\) | 0.00042 | \(-0.00043\) | 0.00199 | 0.00155 |

% of RS | \(-1.69\) | 0.83 | \(-0.86\) | 3.93 | 3.08 |

% of \(AV_1\) | \(-54.83\) | 26.93 | \(-27.90\) | 127.90 | 100.00 |

\(AAV_2^{V}\) | \(AAV_2^{D}\) | \(AAV_2^{S}\) | \(AAV_2^{C}\) | \(AV_2\) | |
---|---|---|---|---|---|

Value | 0.00115 | 0.00065 | 0.00180 | 0.00394 | 0.00575 |

% of RS | 2.28 | 1.29 | 3.57 | 7.81 | 11.37 |

% \(AV_2\) | 20.03 | 11.32 | 31.35 | 68.64 | 100.00 |

\(AAV_3^{V}\) | \(AAV_3^{D}\) | \(AAV_3^{S}\) | \(AAV_3^{C}\) | \(AV_3\) | |
---|---|---|---|---|---|

Value | \(-\) 0.00019 | 0.00011 | \(-\) 0.00008 | 0.00070 | 0.00062 |

% of RS | \(-\) 0.38 | 0.22 | \(-\) 0.16 | 1.38 | 1.22 |

% \(AV_3\) | \(-\) 31.10 | 18.22 | \(-\) 12.88 | 112.88 | 100.00 |

#### 5.2.4 Comparing the two methodologies

^{10}

#### 5.2.5 An overview of the Italian personal income tax through the new methodologies

^{11}whilst the rate schedule and the tax credits for earned incomes and dependent individuals within the household represent 41% and 65% of it, respectively (Barbetta et al. 2018).