Author:
(1) Laurence Francis Lacey, Lacey Solutions Ltd, Skerries, County Dublin, Ireland. Editor's Note: This is Part 7 of 7 of a study on how changes in the money supply, economic growth, and savings levels affect inflation. Read the rest below. Table of Links Abstract and 1 Introduction


Methods


2.1 Statistical Methodology
2.2 US Time Series Data
2.3 Hyperinflation Model


Results


3.1 Characterisation of Price Inflation
3.2 Characterisation of Hyperinflation in the Weimar Republic (1922 to 1923)
4. Discussion
5. Conclusion, Supplementary materials, Acknowledgements, and References

Appendix 5. Conclusion The hypothesis that the annual rate of growth in the US consumer price index is a function of the annual growth in the US broad money supply minus the annual growth in US real GDP minus the annual growth in US savings, over the time period 2001 to 2019, has been shown to be the case. However, an exact relationship required the use of a non-zero residual term. A mathematical statistical formulation of a hyperinflationary process has been provided and used to quantify the period of hyperinflation in the Weimar Republic from July 1922 until the end of November 1923. The main mathematical difference between inflation and hyperinflation is that the former is a process with a constant rate of increase, at a constant acceleration, whereas the latter is a process with a constant rate of acceleration. Supplementary materials There is an Appendix providing the mathematical statistical characterization of a hyperinflationary process. Acknowledgements No financial support was received for any aspect of this research. References [1] “Monetary inflation”. Accessed on 11 August 2021 from Wikipedia.com. url: https://en.wikipedia.org/wiki/Monetary_inflation [2] “Consumer Price Index (CPI)”. Accessed on 11 August 2021 from Investopedia.com. url: https://www.investopedia.com/terms/c/consumerpriceindex.asp [3] “Hyperinflation”. Accessed on 11 August 2021 from Investopedia.com. url: https://www.investopedia.com/terms/h/hyperinflation.asp [4] “Broad Money”. Accessed on 11 August 2021 from Investopedia. url: https://www.investopedia.com/terms/b/broad-money.asp [5] Lacey, LF. Characterization of the probability and information entropy of a statistical process with an exponentially increasing sample space. arXiv:2105.14193, 2021. Accessed on 10 August 2021 from arxiv.org. url: https://arxiv.org/abs/2105.14193 [6] Lacey LF. Characterization of the probability and information entropy of a process with an increasing sample space by different functional forms of expansion, with an application to hyperinflation. arXiv:2107.05483, 2021. Accessed on 10 August 2021 from arxiv.org. url: https://arxiv.org/abs/2107.05483 [7] Lacey LF. The relationship between the US broad money supply and US GDP for the time period 2001 to 2019 with that of the corresponding time series for US national property and stock market indices, using an information entropy methodology. arXiv:2106.07354, 2021. Accessed on 10 August 2021 from arxiv.org. url: https://arxiv.org/abs/2106.07354 [8] “The broad money supply of US$ from 2001 to 2019”. Monetary, Broad Money, US Dollars, International Financial Statistics (IFS). Accessed on 13 April 2021 from International Monetary Fund (IMF). url: https://data.imf.org/?sk=B83F71E8-61E3-4CF1-8CF3- 6D7FE04D0930&sId=1393552803658 [9] “Gross Domestic Product for United States, Percent Change, Annual, Not Seasonally Adjusted”. Accessed on 15 April 2021 from Federal Reserve Bank of St. Louis. url: https://fred.stlouisfed.org/series/USANGDPRPCH [10] “US Consumer Price Index (CPI) From 1903”. Federal Reserve Bank of Minneapolis. Accessed on 15 April 2021 from url: https://www.minneapolisfed.org/about-us/monetarypolicy/inflation-calculator/consumer-price-index-1913- [11] Perez, L. “Average U.S. Savings Account Balance 2021: A Demographic Breakdown”. Accessed on 07 August 2021 from valuepenguin.com. url: https://www.valuepenguin.com/banking/average-savings-account-balance [12] Seabold, Skipper, and Josef Perktold. “statsmodels: Econometric and statistical modeling with python.” Proceedings of the 9th Python in Science Conference. 2010. [13] Mizunoa, T; Takayasub, M; Takayasu H. The mechanism of double exponential growth in hyper-inflation. Physica A: Statistical Mechanics and its Applications. 2002, 308, Issues 1–4, 411-419. Accessed on 18 May 2021. url: https://doi.org/10.1016/S0378- 4371(02)00598-8 [14] “Hyperinflation in the Weimar Republic”. Accessed on 18 May 2021 from Wikipedia.com. url: https://en.wikipedia.org/wiki/Hyperinflation_in_the_Weimar_Republic [15] “Velocity of Money”. Accessed on 13 April 2021 from Investopedia. url: https://www.investopedia.com/terms/v/velocity.asp Appendix: Characterization of a hyperinflationary process – one with an initial exponential expansion of the sample space followed a subsequent second exponential expansion of the sample space A process with an exponentially increasing sample space (s) has been characterised [5]. When the exponential expansion occurs with time (t), it has been shown [5] that: s(t) = exp(λ 𝑥 t) H(t) = λ 𝑥 t where, s(t) is the size of the sample space with time, H(t) is the information entropy of the expansionary process with time, and λ is exponential rate constant. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Author: (1) Laurence Francis Lacey, Lacey Solutions Ltd, Skerries, County Dublin, Ireland. Author: Author: (1) Laurence Francis Lacey, Lacey Solutions Ltd, Skerries, County Dublin, Ireland. Editor's Note: This is Part 7 of 7 of a study on how changes in the money supply, economic growth, and savings levels affect inflation. Read the rest below. Editor's Note: This is Part 7 of 7 of a study on how changes in the money supply, economic growth, and savings levels affect inflation. Read the rest below. Editor's Note: This is Part 7 of 7 of a study on how changes in the money supply, economic growth, and savings levels affect inflation. Read the rest below. Editor's Note: This is Part 7 of 7 of a study on how changes in the money supply, economic growth, and savings levels affect inflation. Read the rest below. Table of Links Abstract and 1 Introduction Methods 2.1 Statistical Methodology 2.2 US Time Series Data 2.3 Hyperinflation Model Results 3.1 Characterisation of Price Inflation 3.2 Characterisation of Hyperinflation in the Weimar Republic (1922 to 1923) 4. Discussion 5. Conclusion, Supplementary materials, Acknowledgements, and References

Appendix Abstract and 1 Introduction Abstract and 1 Introduction Methods Methods Methods 2.1 Statistical Methodology 2.1 Statistical Methodology 2.2 US Time Series Data 2.2 US Time Series Data 2.3 Hyperinflation Model 2.3 Hyperinflation Model Results Results Results 3.1 Characterisation of Price Inflation 3.1 Characterisation of Price Inflation 3.2 Characterisation of Hyperinflation in the Weimar Republic (1922 to 1923) 3.2 Characterisation of Hyperinflation in the Weimar Republic (1922 to 1923) 4. Discussion 4. Discussion 5. Conclusion, Supplementary materials, Acknowledgements, and References Appendix 5. Conclusion, Supplementary materials, Acknowledgements, and References Appendix Appendix 5. Conclusion The hypothesis that the annual rate of growth in the US consumer price index is a function of the annual growth in the US broad money supply minus the annual growth in US real GDP minus the annual growth in US savings, over the time period 2001 to 2019, has been shown to be the case. However, an exact relationship required the use of a non-zero residual term. A mathematical statistical formulation of a hyperinflationary process has been provided and used to quantify the period of hyperinflation in the Weimar Republic from July 1922 until the end of November 1923. The main mathematical difference between inflation and hyperinflation is that the former is a process with a constant rate of increase, at a constant acceleration, whereas the latter is a process with a constant rate of acceleration. Supplementary materials There is an Appendix providing the mathematical statistical characterization of a hyperinflationary process. Acknowledgements No financial support was received for any aspect of this research. References [1] “Monetary inflation”. Accessed on 11 August 2021 from Wikipedia.com. url: https://en.wikipedia.org/wiki/Monetary_inflation [2] “Consumer Price Index (CPI)”. Accessed on 11 August 2021 from Investopedia.com. url: https://www.investopedia.com/terms/c/consumerpriceindex.asp [3] “Hyperinflation”. Accessed on 11 August 2021 from Investopedia.com. url: https://www.investopedia.com/terms/h/hyperinflation.asp [4] “Broad Money”. Accessed on 11 August 2021 from Investopedia. url: https://www.investopedia.com/terms/b/broad-money.asp [5] Lacey, LF. Characterization of the probability and information entropy of a statistical process with an exponentially increasing sample space. arXiv:2105.14193, 2021. Accessed on 10 August 2021 from arxiv.org. url: https://arxiv.org/abs/2105.14193 [6] Lacey LF. Characterization of the probability and information entropy of a process with an increasing sample space by different functional forms of expansion, with an application to hyperinflation. arXiv:2107.05483, 2021. Accessed on 10 August 2021 from arxiv.org. url: https://arxiv.org/abs/2107.05483 [7] Lacey LF. The relationship between the US broad money supply and US GDP for the time period 2001 to 2019 with that of the corresponding time series for US national property and stock market indices, using an information entropy methodology. arXiv:2106.07354, 2021. Accessed on 10 August 2021 from arxiv.org. url: https://arxiv.org/abs/2106.07354 [8] “The broad money supply of US$ from 2001 to 2019”. Monetary, Broad Money, US Dollars, International Financial Statistics (IFS). Accessed on 13 April 2021 from International Monetary Fund (IMF). url: https://data.imf.org/?sk=B83F71E8-61E3-4CF1-8CF3- 6D7FE04D0930&sId=1393552803658 [9] “Gross Domestic Product for United States, Percent Change, Annual, Not Seasonally Adjusted”. Accessed on 15 April 2021 from Federal Reserve Bank of St. Louis. url: https://fred.stlouisfed.org/series/USANGDPRPCH [10] “US Consumer Price Index (CPI) From 1903”. Federal Reserve Bank of Minneapolis. Accessed on 15 April 2021 from url: https://www.minneapolisfed.org/about-us/monetarypolicy/inflation-calculator/consumer-price-index-1913- [11] Perez, L. “Average U.S. Savings Account Balance 2021: A Demographic Breakdown”. Accessed on 07 August 2021 from valuepenguin.com. url: https://www.valuepenguin.com/banking/average-savings-account-balance [12] Seabold, Skipper, and Josef Perktold. “statsmodels: Econometric and statistical modeling with python.” Proceedings of the 9th Python in Science Conference. 2010. [13] Mizunoa, T; Takayasub, M; Takayasu H. The mechanism of double exponential growth in hyper-inflation. Physica A: Statistical Mechanics and its Applications. 2002, 308, Issues 1–4, 411-419. Accessed on 18 May 2021. url: https://doi.org/10.1016/S0378- 4371(02)00598-8 [14] “Hyperinflation in the Weimar Republic”. Accessed on 18 May 2021 from Wikipedia.com. url: https://en.wikipedia.org/wiki/Hyperinflation_in_the_Weimar_Republic [15] “Velocity of Money”. Accessed on 13 April 2021 from Investopedia. url: https://www.investopedia.com/terms/v/velocity.asp Appendix: Characterization of a hyperinflationary process – one with an initial exponential expansion of the sample space followed a subsequent second exponential expansion of the sample space A process with an exponentially increasing sample space (s) has been characterised [5]. When the exponential expansion occurs with time (t), it has been shown [5] that: s(t) = exp(λ 𝑥 t) H(t) = λ 𝑥 t where, s(t) is the size of the sample space with time, H(t) is the information entropy of the expansionary process with time, and λ is exponential rate constant. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. available on arxiv

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