# Power Laws and Inequalities: The Case of British District House Price Dispersion

## Abstract

**:**

## 1. Introduction

## 2. Economic Systems and Power Laws

## 3. Finance and the Distribution of Local House Prices

## 4. Method

_{R}is the population of a city ranked as R and P

_{1}is that of the largest city. For Zipf’s law to hold, the −α exponent, hereafter referred to as the Zipf-Pareto (Z-P) exponent, should be 1. Lavalette’s ranking power law can be expressed as ${P}_{R}={P}_{1}{\left[\frac{N\times R}{N-R+1}\right]}^{-q}$ (Gray 2022) (2), where N is the total number of cities. In both cases, P

_{1}is the calibrating value (CV). The formula describes a semi-logarithmic S-shape (Chlebus and Divgi 2007).

## 5. Data

^{2}0.936; non-PUA values CV £562,624, −q 0.226, R

^{2}0.986) for 2017 reveal the former has a higher CV and a steeper distribution. Fusing these two together generates a third series of 370 values. Also in Figure 3 are stripes highlighting expected prices for primary urban areas, or cities (positive) and non-PUAs, or rural areas (negative). The layout entails how the third series weaves the two sets of data together by rank order so the spacing between stripes varies. As they have greater high values, reflecting the CV, the PUA stripes are denser in the lower ranks. The corresponding all-district values are reported in Table 1. Combining the corresponding Z-P values also produces something that reflects the expected values of the full series.

## 6. Discussion

#### 6.1. Median and Growth

#### 6.2. Comment on Aggregation and Time Profiles

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Zipf-Pareto | Lavalette | ||
---|---|---|---|

Expression | ${P}_{R}={P}_{1}{R}^{-\alpha}$ | ${P}_{R}={P}_{1}{\left(\frac{NR}{N-R+1}\right)}^{-q}$ | |

10th City by size compared with the 15th City | α = 1 q = 1 and N = 146 q = 1 and N = 224 | 1.5 | 1.557 1.536 |

10th City by size compared with the 15th City | α = 0.25 q = 0.25 and N = 146 q = 0.25 and N = 224 | 1.107 | 1.117 1.113 |

maximum ÷ minimum | N^{α} | N^{2q} | |

maximum ÷ median | ([N + 1]÷2)^{α} | N^{q} | |

Median | ${P}_{1}$([N + 1]÷2)^{−}^{α} | ${P}_{1}{N}^{-q}$ | |

Slope ratio | Quartile ratio would be assessed as ω = 75%; QR 80%; and DR at 90% | ${\left(\frac{100-\omega}{\omega}\right)}^{-\alpha}$ | ${\left(\frac{100-\omega}{\omega}\right)}^{-2q}$ |

α = 0.5 q = 0.25 | Quartile ratio = 1.732 QR= 2, DR = 3 | Quartile ratio = 1.732 QR= 2, DR = 3 | |

90:50 ratio | ${\left(\frac{10}{50}\right)}^{-\alpha}$ | ${\left(\frac{10}{90}\right)}^{-q}$ | |

Coefficient of Variation. If the data follows a t-distribution | CoV of 0.398 is associated with a DR of 3.1; a QR of 2.01; and a Quartile ratio of 1.73. |

## Note

1 | https://www.centreforcities.org/wp-content/uploads/2016/01/2016-PUA-Table.pdf (accessed on 3 March 2023). |

## References

- Amaral, Francisco, Martin Dohmen, Sebastian Kohl, and Moritz Schularick. 2023. Interest Rates and the Spatial Polarization of Housing Markets. CEPR Discussion Paper No. 17780. American Economic Review: Insights, Forthcoming. Available online: https://www.aeaweb.org/articles?id=10.1257/aeri.20220367 (accessed on 7 July 2023).
- Aoki, Kosuke, James Proudman, and Gertjan Vlieghe. 2004. House prices, consumption and monetary policy: A financial accelerator approach. Journal of Financial Intermediation 13: 414–35. [Google Scholar] [CrossRef]
- Arshad, Sidra, Shougeng Hu, and Badar Nadeem Ashraf. 2018. Zipf’s law and city size distribution: A survey of the literature and future research agenda. Physica A: Statistical Mechanics and Its Applications 492: 75–92. [Google Scholar] [CrossRef]
- Bogin, Alexander, William Doerner, and William Larson. 2017. Local House Price Paths: Accelerations, Declines, and Recoveries. The Journal of Real Estate Finance and Economics 58: 201–22. [Google Scholar] [CrossRef] [Green Version]
- Brakman, Steven, Harry Garretsen, and Charles van Marrewijk. 2020. An Introduction to Geographical and Urban Economics: A Spiky World. Cambridge: Cambridge University Press. [Google Scholar]
- Brandily, Paul, Mimosa Distefano, Hélène Donnat, Immanuel Feld, Henry Overman, and Krishan Shah. 2022. Bridging the Gap: What Would It Take to Narrow the UK’s Productivity Disparities? London: Resolution Foundation, June 30. [Google Scholar]
- Brown, David, Tony Champion, Mike Coombes, and Colin Wymer. 2015. The Migration-commuting nexus in rural England. A longitudinal analysis. Journal of Rural Studies 41: 118–28. [Google Scholar] [CrossRef] [Green Version]
- Carlino, Gerald, and Albert Saiz. 2019. Beautiful City: Leisure Amenities and Urban Growth. Journal of Regional Science 59: 369–408. [Google Scholar] [CrossRef] [Green Version]
- Cerqueti, Roy, and Marcel Ausloos. 2015a. Evidence of economic regularities and disparities of Italian regions from aggregated tax income size data. Physica A: Statistical Mechanics and Its Applications 421: 187–207. [Google Scholar] [CrossRef] [Green Version]
- Cerqueti, Roy, and Marcel Ausloos. 2015b. Cross ranking of cities and regions: Population versus income. Journal of Statistical Mechanics: Theory and Experiment 2015: P07002. [Google Scholar] [CrossRef] [Green Version]
- Chlebus, Edward, and Gautam Divgi. 2007. A Novel Probability Distribution for Modeling Internet Traffic and its Parameter Estimation. Paper presented at the IEEE GLOBECOM 2007—IEEE Global Telecommunications Conference, Washington, DC, USA, November 26–30; pp. 4670–4675. Available online: https://ieeexplore.ieee.org/document/4411796 (accessed on 30 March 2020).
- Coulson, N. Edward, Crocker H. Liu, and Sriram V. Villupuram. 2013. Urban economic base as a catalyst for movements in real estate prices. Regional Science and Urban Economics 43: 1023–40. [Google Scholar] [CrossRef] [Green Version]
- Cristelli, Matthieu, Michael Batty, and Luciano Pietronero. 2012. There is more than a power law in Zipf. Scientific Reports 2: 812. [Google Scholar] [CrossRef] [Green Version]
- DEFRA. 2020. Statistical Digest of Rural England; London: Department for Environment, Food & Rural Affairs. Available online: https://www.gov.uk/government/statistics/statistical-digest-of-rural-england (accessed on 13 October 2020).
- DiPasquale, Denise, and William Wheaton. 1996. Urban Economics and Real Estate Markets. Englewood Cliffs: Prentice Hall. [Google Scholar]
- Eliazar, Iddo, and Morrel H. Cohen. 2013. On the physical interpretation of statistical data from black-box systems. Physica A: Statistical Mechanics and Its Applications 392: 2924–39. [Google Scholar] [CrossRef]
- Fernandez, Rodrigo, Annelore Hofman, and Manuel B. Aalbers. 2016. London and New York as a Safe Deposit Box for the Transnational Wealth Elite. Environment and Planning A 48: 2443–61. [Google Scholar] [CrossRef]
- Fontanelli, Oscar, Pedro Miramontes, Germinal Cocho, and Wentian Li. 2017. Population Patterns in World’s Administrative Units. Royal Society Open Science 4: 170281. [Google Scholar] [CrossRef] [Green Version]
- Gal, Peter, and Jagoda Egeland. 2018. Reducing Regional Disparities in Productivity in the United Kingdom. Working Paper 1456. Paris, France: OECD Economics Department. [Google Scholar] [CrossRef]
- Glaeser, Edward L., and Joseph Gyourko. 2005. Urban Decline and Durable Housing. Journal of Political Economy 113: 345–75. [Google Scholar] [CrossRef] [Green Version]
- Gray, David. 2018. Convergence and divergence in British housing space. Regional Studies 52: 901–10. [Google Scholar] [CrossRef]
- Gray, David. 2022. Do House Price-Earnings Ratios in England and Wales follow a Power Law? An Application of Lavalette’s Law to District Data. Environment and Planning B: Urban Analytics and City Science 49: 1184–96. [Google Scholar] [CrossRef]
- Hanushek, Eric A., Steven G. Rivkin, and Lori L. Taylor. 1996. Aggregation and the Estimated Effects of School Resources. The Review of Economics and Statistics 78: 611–27. [Google Scholar] [CrossRef]
- Himmelberg, Charles, Christopher Mayer, and Todd Sinai. 2005. Assessing High House Prices: Bubbles, Fundamentals and Misperceptions. Journal of Economic Perspectives 19: 67–92. [Google Scholar] [CrossRef] [Green Version]
- Hsu, Wen-Tai. 2012. Central Place Theory and City Size Distribution. The Economic Journal 122: 903–32. [Google Scholar] [CrossRef]
- Huang, Daisy J., Charles K. Leung, and Baozhi Qu. 2015. Do Bank Loans and Local Amenities Explain Chinese Urban House Prices? China Economic Review 34: 19–38. [Google Scholar] [CrossRef] [Green Version]
- Leung, Charles Ka Yui Leong, Youngman Chun Fai Wong, and Siu Kei. 2006. Housing Price Dispersion: An Empirical Investigation. The Journal of Real Estate Finance and Economics 32: 357–85. [Google Scholar] [CrossRef]
- Martin, Ron, Ben Gardiner, and Peter Tyler. 2014. The Evolving Economic Performance of UK Cities: City Growth Patterns, 1981–2011; Working Paper, Foresight Programme on the Future of Cities; London: UK Government Office for Science, Department of Business, Innovation and Skills.
- Martin, Ron, Peter Sunley, Ben Gardiner, Emil Evenhuis, and Peter Tyler. 2018. The city dimension of the productivity growth puzzle: The relative role of structural change and within-sector slowdown. Journal of Economic Geography 18: 539–70. [Google Scholar] [CrossRef] [Green Version]
- McCann, Philip. 2013. Modern Urban and Regional Economics, 2nd ed. Oxford: Oxford University Press. [Google Scholar]
- McCann, Philip. 2020. Perceptions of regional inequality and the geography of discontent: Insights from the UK. Regional Studies 54: 256–67. [Google Scholar] [CrossRef]
- McCombie, John. 1988. A synoptic view of regional growth and unemployment: II The post-Keynesian theory. Urban Studies 25: 399–417. [Google Scholar] [CrossRef]
- Miles, David, and Victoria Monro. 2019. UK House Prices and Three Decades of Decline in the Risk Free Real Interest Rate. Staff Working Paper No. 837. London: Bank of England. [Google Scholar]
- Montagnoli, Alberto, and Jun Nagayasu. 2015. UK house price convergence clubs and spillovers. Journal of Housing Economics 30: 50–58. [Google Scholar] [CrossRef] [Green Version]
- Pike, Andy Pike, Danny MacKinnon, Mike Coombes, Tony Champion, David Bradley, Andrew Cumbers, Liz Robson, and Colin Wymer. 2016. Uneven Growth: Tackling City Decline. York: Joseph Rowntree Foundation. Available online: https://www.jrf.org.uk/report/uneven-growth-tackling-city-decline (accessed on 30 March 2020).
- Rae, Alasdair. 2015. The illusion of transparency: The geography of mortgage lending in Great Britain. Journal of European Real Estate Research 8: 172–95. [Google Scholar] [CrossRef]
- Richmond, Peter. 2007. A roof over your head; house price peaks in the UK and Ireland. Physica A: Statistical Mechanics and Its Applications 375: 281–87. [Google Scholar] [CrossRef] [Green Version]
- Roback, Jennifer. 1982. Wages, rents, and the quality of life. Journal of Political Economy 90: 1257–78. Available online: https://www.jstor.org/stable/1830947 (accessed on 30 March 2020). [CrossRef]
- Shao, Jia, Plamen Ch. Ivanov, Branko Urošević, H. Eugene Stanley, and Boris Podobnik. 2011. Zipf rank approach and cross-country convergence of incomes. Europhysics Letters 94: 48001. [Google Scholar]
- Sinai, Todd. 2010. Feedback Between Real Estate and Urban Economics. Journal of Regional Science 50: 423–48. [Google Scholar] [CrossRef]
- Szumilo, Nikodem. 2019. The spatial consequences of the housing affordability crisis in England. Environment & Planning A 51: 1264–86. [Google Scholar] [CrossRef]
- Tang, Pan, Ying Zhang, Belal E. Baaquie, and Boris Podobnik. 2016. Classical convergence versus Zipf rank approach: Evidence from China’s local-level data. Physica A: Statistical Mechanics and Its Applications 443: 246–53. [Google Scholar] [CrossRef]
- Van Nieuwerburgh, Stijn, and Pierre-Olivier Weill. 2010. Why Has House Price Dispersion Gone Up? The Review of Economic Studies 77: 1567–606. [Google Scholar] [CrossRef] [Green Version]
- Varian, Hal R. 2014. Intermediate Microeconomics: A Modern Approach, 9th ed. New York: W. W. Norton & Company. [Google Scholar]

EWS ^{q} | R^{2 q} | CV ^{q} | K-S ^{q} p-Val. | EWS ^{α} | R^{2 α} | CV ^{α} | K-S ^{α} p-Val. | PUA ^{q} | PUA ^{α} | Non-PUA ^{q} | Non-PUA ^{α} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

2004 | 0.215 | 0.986 | 487,030 | 0.472 | 0.349 | 0.775 | 759,075 | <0.001 | 0.239 | 0.400 | 0.204 | 0.325 |

2005 | 0.196 | 0.991 | 458,939 | 0.774 | 0.321 | 0.798 | 700,390 | <0.001 | 0.218 | 0.371 | 0.184 | 0.297 |

2006 | 0.183 | 0.991 | 437,908 | 0.774 | 0.309 | 0.849 | 680,811 | <0.001 | 0.204 | 0.360 | 0.172 | 0.284 |

2007 | 0.183 | 0.984 | 460,314 | 0.53 | 0.316 | 0.880 | 739,517 | <0.001 | 0.210 | 0.378 | 0.167 | 0.279 |

2008 | 0.187 | 0.977 | 462,056 | 0.24 | 0.326 | 0.893 | 763,492 | <0.001 | 0.217 | 0.393 | 0.169 | 0.286 |

2009 | 0.185 | 0.979 | 412,394 | 0.53 | 0.325 | 0.909 | 686,198 | <0.001 | 0.217 | 0.398 | 0.164 | 0.280 |

2010 | 0.198 | 0.981 | 472,642 | 0.278 | 0.348 | 0.909 | 813,374 | <0.001 | 0.233 | 0.426 | 0.176 | 0.300 |

2011 | 0.208 | 0.977 | 492,685 | 0.418 | 0.367 | 0.913 | 877,254 | <0.001 | 0.246 | 0.450 | 0.184 | 0.314 |

2012 | 0.211 | 0.975 | 497,050 | 0.32 | 0.373 | 0.915 | 897,356 | <0.001 | 0.251 | 0.459 | 0.186 | 0.318 |

2013 | 0.218 | 0.971 | 520,783 | 0.278 | 0.390 | 0.926 | 974,644 | <0.001 | 0.263 | 0.487 | 0.189 | 0.323 |

2014 | 0.229 | 0.963 | 580,549 | 0.206 | 0.413 | 0.940 | 1,146,962 | <0.001 | 0.279 | 0.519 | 0.195 | 0.337 |

2015 | 0.242 | 0.964 | 664,024 | 0.206 | 0.435 | 0.933 | 1,349,860 | <0.001 | 0.294 | 0.543 | 0.206 | 0.355 |

2016 | 0.255 | 0.964 | 751,260 | 0.088 | 0.456 | 0.922 | 1,568,640 | <0.001 | 0.307 | 0.561 | 0.220 | 0.379 |

2017 | 0.261 | 0.967 | 806,241 | 0.105 | 0.463 | 0.912 | 1,685,219 | <0.001 | 0.312 | 0.569 | 0.226 | 0.386 |

2018 | 0.258 | 0.971 | 802,638 | 0.149 | 0.455 | 0.905 | 1,640,559 | <0.001 | 0.307 | 0.559 | 0.224 | 0.379 |

2019 | 0.252 | 0.973 | 775,617 | 0.24 | 0.443 | 0.900 | 1,547,266 | <0.001 | 0.301 | 0.544 | 0.219 | 0.369 |

^{q}drawn from the Lavalettean function (2),

^{α}drawn from a Z-P function (1). Kolmogorov–Smirnov test p-values.

K-S p-Val. | Coefficient | Quartile | QR | DR | Skew | Kurt | ||
---|---|---|---|---|---|---|---|---|

TTWA 2006 | CoV | 0.257 | 1.423 | 1.527 | 1.792 | 0.38 | −0.16 | |

Lavalette | 0.876 | Min | 0.13 | 1.331 | 1.434 | 1.771 | ||

Max | 0.139 | 1.357 | 1.470 | 1.842 | ||||

Z-P | 0.002 | Min | 0.198 | 1.243 | 1.316 | 1.545 | ||

Max | 0.26 | 1.331 | 1.434 | 1.771 | ||||

TTWA 2017 | CoV | 0.346 | 1.509 | 1.718 | 2.139 | 0.89 | 0.48 | |

Lavalette | 0.881 | Min | 0.166 | 1.440 | 1.584 | 2.074 | ||

Max | 0.181 | 1.488 | 1.652 | 2.215 | ||||

Z-P | 0.038 | Min | 0.285 | 1.368 | 1.485 | 1.871 | ||

Max | 0.338 | 1.450 | 1.598 | 2.102 | ||||

ITL3 2006 | CoV | 0.348 | 1.627 | 1.832 | 2.263 | 1.69 | 4.7 | |

Lavalette | 0.623 | Min | 0.171 | 1.456 | 1.607 | 2.120 | ||

Max | 0.186 | 1.505 | 1.675 | 2.265 | ||||

Z-P | 0.003 | Min | 0.28 | 1.360 | 1.474 | 1.850 | ||

Max | 0.352 | 1.472 | 1.629 | 2.167 | ||||

ITL3 2017 | CoV | 0.598 | 1.957 | 2.373 | 3.332 | 2.37 | 7.93 | |

Lavalette | 0.214 | Min | 0.246 | 1.717 | 1.978 | 2.948 | ||

Max | 0.288 | 1.883 | 2.222 | 3.545 | ||||

Z-P | 0.052 | Min | 0.463 | 1.663 | 1.900 | 2.766 | ||

Max | 0.531 | 1.792 | 2.088 | 3.211 | ||||

LAD 2006 | CoV | 0.328 | 1.534 | 1.758 | 2.129 | 1.35 | 3.97 | |

Lavalette | 0.66 | Min | 0.173 | 1.462 | 1.616 | 2.139 | ||

Max | 0.178 | 1.479 | 1.638 | 2.186 | ||||

Z-P | 0.000 | Min | 0.278 | 1.357 | 1.470 | 1.842 | ||

Max | 0.324 | 1.428 | 1.567 | 2.038 | ||||

LAD 2017 | CoV | 0.546 | 1.969 | 2.357 | 3.308 | 2.52 | 11.42 | |

Lavalette | 0.000 | Min | 0.249 | 1.728 | 1.994 | 2.987 | ||

Max | 0.267 | 1.798 | 2.097 | 3.233 | ||||

Z-P | <0.001 | Min | 0.43 | 1.604 | 1.815 | 2.572 | ||

Max | 0.491 | 1.715 | 1.975 | 2.941 |

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**MDPI and ACS Style**

Gray, D.P.
Power Laws and Inequalities: The Case of British District House Price Dispersion. *Risks* **2023**, *11*, 136.
https://doi.org/10.3390/risks11070136

**AMA Style**

Gray DP.
Power Laws and Inequalities: The Case of British District House Price Dispersion. *Risks*. 2023; 11(7):136.
https://doi.org/10.3390/risks11070136

**Chicago/Turabian Style**

Gray, David Paul.
2023. "Power Laws and Inequalities: The Case of British District House Price Dispersion" *Risks* 11, no. 7: 136.
https://doi.org/10.3390/risks11070136