# Using the GB2 Income Distribution

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## Abstract

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## 1. Introduction

## 2. Inequality and Poverty Measures from the GB2 Distribution

#### 2.1. Inequality Measures

#### 2.1.1. Gini Coefficient

#### 2.1.2. Generalized Entropy Measures

#### 2.1.3. Atkinson Index

#### 2.1.4. Pietra Index

#### 2.1.5. Quintile Share Ratio

#### 2.2. Poverty Measures

#### 2.3. Measures of Pro-Poor Growth

## 3. Estimation

#### 3.1. Estimation with Single Observations

#### 3.2. Estimation with Grouped Data

## 4. Applications

- All inequality measures indicate that inequality increased from 1999 to 2010, and then declined from 2010 to 2013. The recent decline is attributable to a decline in rural inequality; there was an increase in urban inequality in the same period. Also, there is no clear conclusion about how rural inequality changed from 1999 to 2005; the Gini and $I(1)$ suggest a slight decrease, whereas QSR, $I(0)$ and Pietra suggest a slight increase.
- Inequality is much greater in the combined distribution than in its components, reflecting the large discrepancy in mean incomes between the rural and urban areas. Within inequality remains greater than between inequality, however.
- The changes in inequality have been accompanied by large increases in mean income and large decreases in poverty. The decline in poverty was particularly dramatic for rural China where the headcount ratio declined from 57% in 1999 to 3.7% in 2013. Poverty in rural China is uniformly greater than that in urban China.
- The GIC curves show that, from 1999 to 2010, growth has favored the rich more than the poor, but from 2010 to 2013, growth has strongly favored the poor relative to the rich, a result consistent with the decline in inequality over this period. The scalar measures of pro-poor growth are also consistent with this observation. Growth has favured the poor in an absolute sense from 1999 to 2010 $(0<RC<g$, $0<KP<1$, $PEGR<g)$, and in a relative sense after 2010 $(RC>g$, $KP>1$, $PEGR>g)$.

- Urban inequality changed very little from 1999 to 2005, increased dramatically from 2005 to 2010, and then increased more moderately from 2010 to 2016. Rural inequality increased from 1999 to 2010, but declined thereafter. The combined results reflect these changes, with increasing inequality overall, but with Gini coefficients approximately the same in 2010 and 2016.
- Poverty declined from 1999 to 2005, remained roughly constant from 2005 to 2010, when there were large increases in inequality, and then declined again from 2010 to 2016. From 2005 to 2010 a decline in urban poverty was offset by an increase in rural poverty.
- The GIC curves show that growth has favored the rich relative to the poor in all time intervals. From 2005 to 2010 the poor faired very badly; the growth rate for the bottom 15% of the population was negative. This period was also one where the growth in mean incomes was low relative to that in the other two periods. The scalar pro-poor growth measures are in line with the conclusions from the GIC curves. Growth was absolutely but not relatively pro-poor in the first and third time intervals; in the second interval it was not absolutely pro-poor according to the RC measure, and only slightly absolutely pro-poor using the KP measure.

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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1 | McDonald and Xu (1995) and McDonald and Ransom (2008) also consider a five-parameter generalized beta distribution which nests the GB2 and a GB1 distribution. |

2 | The Singh-Maddala distribution is also commonly known as the Burr distribution, and has been described using a variety of other names. See (Kleiber and Kotz 2003, p. 198). |

3 | See, for example, (Butler and McDonald 1989). |

4 | See McDonald and Ransom (2008) or Jenkins (2009) for derivations. Equation (4) in Jenkins (2009) should read $I(1){v}_{1}/\mathsf{\mu}-\mathrm{log}\mathsf{\mu}$. Sarabia et al. (2017) give details of the Theil indices for a wide range of distributions including the GB2. |

5 | It may be better to describe the estimators that minimize $GM{M}_{2}\left(\mathbf{x},\mathsf{\varphi}\right)$ and $GM{M}_{3}\left(\mathbf{x},\mathsf{\varphi}\right)$ as minimum distance estimators rather than GMM estimators because the “moment condition” for ${\overline{y}}_{i}$ is plim ${\overline{y}}_{i}={\mathsf{\mu}}_{i}/{k}_{i}$ not $E\left({\overline{y}}_{i}\right)={\mathsf{\mu}}_{i}/{k}_{i}$. The asymptotic distribution is the same, however. See, for example, (Greene 2012, chp. 13). |

6 | The version of the data that was used was downloaded on 9 March 2018 at http://iresearch.worldbank.org/PovcalNet/povOnDemand.aspx. |

7 | See (McDonald and Xu 1995, p. 139). |

Country/Year | a | b | p | q | $\mathsf{\mu}$ | Population (Millions) | |
---|---|---|---|---|---|---|---|

China rural | |||||||

2013 | 1.5806 | 101.3579 | 3.8613 | 2.1609 | 190.23 | 635.69 | |

2010 | 1.2063 | 21.4069 | 11.6780 | 2.2025 | 131.52 | 697.21 | |

2005 | 1.3443 | 32.0352 | 7.0416 | 2.3558 | 100.07 | 749.35 | |

1999 | 2.0243 | 30.1693 | 3.3733 | 1.3113 | 67.78 | 815.97 | |

China urban | |||||||

2013 | 1.6455 | 261.4467 | 2.3392 | 1.9792 | 373.92 | 721.69 | |

2010 | 1.8842 | 187.8696 | 2.3745 | 1.5884 | 306.81 | 658.50 | |

2005 | 1.8294 | 144.7708 | 2.4059 | 1.7919 | 217.11 | 554.37 | |

1999 | 1.6302 | 95.0994 | 3.2433 | 2.5261 | 134.70 | 436.77 | |

Indonesia rural | |||||||

2016 | 2.0275 | 55.8739 | 3.8536 | 1.3660 | 129.11 | 118.90 | |

2010 | 2.1389 | 36.6977 | 4.4602 | 1.2132 | 96.63 | 121.45 | |

2005 | 2.7720 | 52.1883 | 2.5501 | 1.1926 | 85.84 | 122.57 | |

1999 | 3.0994 | 49.6466 | 2.0371 | 1.2727 | 67.62 | 123.52 | |

Indonesia urban | |||||||

2016 | 0.7417 | 0.0010 | 25,914.0 | 4.0699 | 208.00 | 142.22 | |

2010 | 0.9107 | 0.0094 | 15,488.0 | 3.2802 | 156.25 | 121.08 | |

2005 | 2.0275 | 55.8746 | 3.8535 | 1.3660 | 129.11 | 104.15 | |

1999 | 2.0737 | 35.6598 | 4.7873 | 1.2719 | 96.37 | 85.10 |

Country/Year | Gini | QSR | I(0) | I(1) | Pietra | |
---|---|---|---|---|---|---|

China rural | ||||||

2013 | 0.3349 | 5.4526 | 0.1903 | 0.2086 | 0.2424 | |

2010 | 0.3959 | 7.1456 | 0.2664 | 0.3189 | 0.2901 | |

2005 | 0.3519 | 5.8464 | 0.2097 | 0.2375 | 0.2563 | |

1999 | 0.3638 | 5.6579 | 0.2083 | 0.2495 | 0.2545 | |

China urban | ||||||

2013 | 0.3735 | 6.5286 | 0.2291 | 0.2454 | 0.2628 | |

2010 | 0.3540 | 5.9757 | 0.2126 | 0.2370 | 0.2545 | |

2005 | 0.3436 | 5.7017 | 0.1992 | 0.2163 | 0.2460 | |

1999 | 0.3185 | 4.9247 | 0.1649 | 0.1731 | 0.2246 | |

China combined | ||||||

2013 | 0.4010 | 8.1998 | 0.2659 | 0.2864 | 0.2874 | |

2010 | 0.4323 | 9.5593 | 0.3274 | 0.3451 | 0.3155 | |

2005 | 0.4052 | 6.4547 | 0.2796 | 0.2979 | 0.2959 | |

1999 | 0.3941 | 4.5101 | 0.2495 | 0.2683 | 0.2825 | |

Indonesia rural | ||||||

2016 | 0.3343 | 5.2640 | 0.1912 | 0.2270 | 0.2442 | |

2010 | 0.3502 | 5.2808 | 0.1962 | 0.2412 | 0.2480 | |

2005 | 0.2756 | 3.9165 | 0.1275 | 0.1448 | 0.1980 | |

1999 | 0.2352 | 3.3989 | 0.1002 | 0.1087 | 0.1746 | |

Indonesia urban | ||||||

2016 | 0.4154 | 7.9453 | 0.2920 | 0.3409 | 0.3044 | |

2010 | 0.4070 | 6.7226 | 0.2493 | 0.2930 | 0.2818 | |

2005 | 0.3444 | 5.2640 | 0.1912 | 0.2270 | 0.2442 | |

1999 | 0.3368 | 5.2471 | 0.1939 | 0.2370 | 0.2467 | |

Indonesia combined | ||||||

2016 | 0.4027 | 7.6873 | 0.2737 | 0.3286 | 0.2963 | |

2010 | 0.4042 | 6.5792 | 0.2513 | 0.3013 | 0.2842 | |

2005 | 0.3297 | 4.6841 | 0.1776 | 0.2117 | 0.2357 | |

1999 | 0.2959 | 4.0104 | 0.1539 | 0.1879 | 0.2169 |

Country/Year | ${\mathit{I}}_{\mathit{C}}(0)$ | ${\mathit{I}}_{\mathit{C}}^{\mathit{w}\mathit{i}\mathit{t}\mathit{h}}(0)$ | ${\mathit{I}}_{\mathit{C}}^{\mathit{b}\mathit{e}\mathit{t}\mathit{w}}(0)$ | ${\mathit{I}}_{\mathit{C}}(1)$ | ${\mathit{I}}_{\mathit{C}}^{\mathit{w}\mathit{i}\mathit{t}\mathit{h}}(1)$ | ${\mathit{I}}_{\mathit{C}}^{\mathit{b}\mathit{e}\mathit{t}\mathit{w}}(1)$ | |
---|---|---|---|---|---|---|---|

China combined | |||||||

2013 | 0.2659 | 0.2109 | 0.0550 | 0.2864 | 0.2340 | 0.0523 | |

2010 | 0.3274 | 0.2399 | 0.0875 | 0.3451 | 0.2622 | 0.0829 | |

2005 | 0.2796 | 0.2053 | 0.0743 | 0.2979 | 0.2244 | 0.0735 | |

1999 | 0.2495 | 0.1932 | 0.0563 | 0.2683 | 0.2101 | 0.0582 | |

Indonesia combined | |||||||

2016 | 0.2737 | 0.2461 | 0.0276 | 0.3286 | 0.3019 | 0.0267 | |

2010 | 0.2513 | 0.2227 | 0.0286 | 0.3013 | 0.2732 | 0.0281 | |

2005 | 0.1776 | 0.1568 | 0.0208 | 0.2117 | 0.1910 | 0.0207 | |

1999 | 0.1539 | 0.1385 | 0.0154 | 0.1879 | 0.1723 | 0.0156 |

Country/Year | HC | FGT(1) | FGT(2) | SEN | |
---|---|---|---|---|---|

China rural | |||||

2013 | 0.0374 | 0.0070 | 0.0021 | 0.0099 | |

2010 | 0.2042 | 0.0489 | 0.0171 | 0.0713 | |

2005 | 0.2998 | 0.0786 | 0.0296 | 0.1057 | |

1999 | 0.5702 | 0.1907 | 0.0844 | 0.2568 | |

China urban | |||||

2013 | 0.0077 | 0.0017 | 0.0006 | 0.0020 | |

2010 | 0.0085 | 0.0017 | 0.0005 | 0.0023 | |

2005 | 0.0294 | 0.0062 | 0.0021 | 0.0088 | |

1999 | 0.1064 | 0.0233 | 0.0080 | 0.0324 | |

China combined | |||||

2013 | 0.0216 | 0.0042 | 0.0013 | 0.0083 | |

2010 | 0.1079 | 0.0256 | 0.0089 | 0.0496 | |

2005 | 0.1848 | 0.0478 | 0.0179 | 0.0901 | |

1999 | 0.4084 | 0.1324 | 0.0577 | 0.2289 | |

Indonesia rural | |||||

2016 | 0.1267 | 0.0243 | 0.0073 | 0.0348 | |

2010 | 0.3033 | 0.0700 | 0.0234 | 0.0995 | |

2005 | 0.2917 | 0.0613 | 0.0193 | 0.0883 | |

1999 | 0.4647 | 0.1117 | 0.0385 | 0.1526 | |

Indonesia urban | |||||

2016 | 0.0649 | 0.0122 | 0.0035 | 0.0174 | |

2010 | 0.1142 | 0.0221 | 0.0065 | 0.0313 | |

2005 | 0.1267 | 0.0243 | 0.0073 | 0.0353 | |

1999 | 0.3031 | 0.0700 | 0.0234 | 0.0941 | |

Indonesia combined | |||||

2016 | 0.0931 | 0.0177 | 0.0052 | 0.0345 | |

2010 | 0.2089 | 0.0461 | 0.0150 | 0.0863 | |

2005 | 0.2159 | 0.0443 | 0.0138 | 0.0828 | |

1999 | 0.3988 | 0.0947 | 0.0324 | 0.1659 |

Country/Year | Growth Rate | Growth Rate for the Poor (RC) | KP | PEGR |
---|---|---|---|---|

China | ||||

2010–2013 | 0.3218 | 0.6245 | 1.4251 | 0.3245 |

2005–2010 | 0.4536 | 0.2331 | 0.6503 | 0.2839 |

1999–2005 | 0.6446 | 0.5281 | 0.8702 | 0.4504 |

Indonesia | ||||

2010–2016 | 0.3614 | 0.2836 | 0.8622 | 0.2414 |

2005–2010 | 0.1956 | –0.0107 | 0.0709 | 0.0079 |

1999–2005 | 0.3323 | 0.2449 | 0.8049 | 0.2575 |

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## Share and Cite

**MDPI and ACS Style**

Chotikapanich, D.; Griffiths, W.E.; Hajargasht, G.; Karunarathne, W.; Rao, D.S.P.
Using the GB2 Income Distribution. *Econometrics* **2018**, *6*, 21.
https://doi.org/10.3390/econometrics6020021

**AMA Style**

Chotikapanich D, Griffiths WE, Hajargasht G, Karunarathne W, Rao DSP.
Using the GB2 Income Distribution. *Econometrics*. 2018; 6(2):21.
https://doi.org/10.3390/econometrics6020021

**Chicago/Turabian Style**

Chotikapanich, Duangkamon, William E. Griffiths, Gholamreza Hajargasht, Wasana Karunarathne, and D. S. Prasada Rao.
2018. "Using the GB2 Income Distribution" *Econometrics* 6, no. 2: 21.
https://doi.org/10.3390/econometrics6020021