# Topic: Modeling Bitcoins Exponential Growth (follow the guidelines and example provided) about the topic of bitcoins exponential growth. Develop a research

Topic: Modeling Bitcoins Exponential Growth (follow the guidelines and example provided) about the topic of bitcoins exponential growth. Develop a research question and explore this question throughout the assessment using graphs, references, etc… Should range between 12-16 pages which includes table of contents, title page, and work cited. Font Arial, Size 12pt, Justified

Refer to the example of the ia provided. International Baccalaureate:

Internal Assessment

Is the Economy Digging Its Own Grave?

Mathematics Standard Level

Table of Contents

_______________________________________________________________

Table of Contents……………………………………………………………….……….…… 1

1. Introduction………………………………………………………………..….……. 2-3

2. Analysis………..…………………………………………..……………….………. 3-10

2.1. Unemployment in the United States..…………………….………..….…….. 3-7

2.2. Private Debt of Consumers in the United States……………..……………. 7-10

2.3. Gross Domestic Product of the United States……….……..……….……. 10-13

3. Conclusion and Evaluation……………….……….………………..……….….…. 13-14

4. Bibliography…………………………………………………..…….……………. 14-15

_______________________________________________________________

1

Introduction

_______________________________________________________________

Do you ever fear another inexplicable economic disaster? According to the historical

background of economic disasters, it is not a thing that people would like to experience. My

fear is to experience a catastrophe like this, which inspired me into researching, and

understanding the mathematical reasons behind the economic crisis in order to get ready for

another crisis just like the one that happened in 2008.

For a brief explanation on the causes of the recession, housing prices fell 31.8 percent,

unemployment was above 9 percent, the interbank borrowing cost or libor increased, and 1

there was an exponential rise in private debt that led to a collapse in asset prices.

There were a few economists that predicted the recession or financial crisis back in

2008, and one economist explained how he predicted it. Steve Keen is an australian

economist who was part of the minority that predicted this crisis, and he voluntarily created

reports each month right before the economic collapse, warning people about what was about

to happen in the future. He claims that his conclusions mainly came from the gatherement of

data, and analysis from looking at public debt numbers of countries including the US and

Australia. This made me wonder about different ways of predicting a future recession, and

the idea of using more than one component to predict the future economic status of a country

sounded accurate, and workable.

I chose this topic to explore for my internal assessment because I am really concerned

about the economic status of my country and the rest of the world. This is a crucial matter

since this concern will impact my life, more specifically the consumption aspect of it. If an

economic disaster occurs in my country, the aftermath of it will have a huge influence on the

firms operating in Morocco. Mass unemployment will happen due the firms going bankrupt,

and less goods and services will be produced which will drive people’s consumption level to

decrease including my own. After doing some research on this matter, I made a list of factors

that may cause a financial crisis and made sure they were quantitative and measurable. I

1 Banks did not want other banks giving them worthless mortgages as collateral, so banks stopped

lending to each other money.

2

discovered that the underlying reasons causing an economic collapse were due to an increase

in unemployment, increase in private debt, and slow gross domestic product rate of a 2

country. I took my time gathering enough data and graphs on those components so I could

further continue my investigation.

My aim is to construct a total of 6 graphs by gathering monthly data points for each of

the years selected, and include mathematical calculations finding the regression function and

derivative for each of them. The reason why I want to make a regression function, is because

the graphs constructed will not grant me a function for me to differentiate. A solution would

be to find an equation that would accurately fit the data, and call them f(x) and g(x)

respectively. Each two graphs with different time periods belonged to different sections of

my economic model; unemployment, private debt, and GDP.

My main objective in this internal assessment is my approach in predicting the future

economic status of a country’s economy based on comparing different time periods of

unemployment, private debt, and GDP. I will compare the data of these factors during the

years of recession, and the data of the recent past 3 to 4 years. I will use my final results to

predict whether a recession is once again on the horizon. I chose to construct my data on the

United States of America, because they are considered to have the strongest economy in the

world, and they recently had a financial crisis back in 2008 leading to a domino effect on the

rest of the countries making it a perfect match for my research and investigation.

Analysis

_______________________________________________________________

My analysis will consist of three important components; unemployment rate, private

debt of consumers, and gross domestic product. My exploration and investigation of these

factors will involve calculus, functions and equations.

2 The total value of goods produced and services provided in a nation commonly in a year.

3

Unemployment in the United States:

Unemployment has been increasing and decreasing continuously, but there have been

times where it remained at a certain level, and sometimes that level was considered to be the

optimal one. A “normal” and recommended unemployment rate should be ranging between

3-5%, but the lower that number, the better.

The unemployment rate after the 2008 financial crisis was above 8%, and since the

beginning of the crisis, the rate of change was higher than the rate of change of the

unemployment rate for when it was at the optimal level.

Chart 1: United States Unemployment Rate

I will use calculus (differentiation) to determine the rate of change at specific points

during the period of the past 4 years, and compare this rate with the rate of change of specific

data points during the period of 2007 and 2012. As seen in Chart 1 above, the red portion of

the chart illustrates the increasing rate, while the arrows on the side represent the positive rate

of change. On the other hand, the green portion to the right is the current rate of change of the

unemployment rate, showing no sign of a recession, and decreasing significantly. However,

the naked eye is not a useful tool to determine the difference between these two rates of

changes, so mathematical tools are required.

4

Mathematical Calculations:

Derivation of f(x):

Below is a calculation in finding the derivative of the function f(x) which represents

the red portion above in Chart 1, and will be referred to as the Crisis Regression Function

from 2007 to 2010. This function has been obtained through the use of a mathematical

software which graphs and plots a regression function that would best fit, which can be seen

in Graph 1 below.

Graph 1: f(x) United States Unemployment Rate (2007-10)

The curve of Graph 1 is , which(x) − .18 x 10 )x .0221x 0.247x 5.08f = ( 3 −4 3 + 0 2 − +

represents the Crisis Regression Function (2007-10).

Mathematical Calculation of the Derivative of f(x):

= dx

d f (x) − .18 0 )x .0221x − .247x) .08( 3 × 1 −4 3 + 0 2 + ( 0 + 5

= + +dx

d f (x) d

dx

(− .18 0 )x[ 3 × 1 −4 3] d dx 0.0221x[

2] d dx − .247x[ 0 ]

∴ = (x)f ′ 500000

−477x −22100x+1235002

Mathematical Calculation of the Second Derivative of f(x):

= dx

d f (x)′′

500000

−477x −22100x+1235002

= + +dx

d f (x)′′ d

dx

[ −477×2500000 ] d dx [ 500000−22100x ] d dx [ 500000123500 ]

∴ = (x)f ′′ 500000

−954x−22100

5

Derivation of g(x):

Below is a calculation to find the derivative of the function g(x) which represents the

green portion above in Chart 1, and will be referred to as the Today’s Regression Function

from 2015 to 2019. This function had been obtained through the use of a mathematical

software which graphs and plots a regression function that would best fit, which can be seen

in Graph 2 below.

Graph 2: g(x) United States Unemployment Rate (2015-19)

The regression function of Graph 2 is which represents the(x) .23 .579ln(x)g = 6 − 0

Today’s Regression Function (2015-19).

Mathematical Calculation of the Derivative of g(x):

=dx

d g(x) .23 − .579ln(x))6 + ( 0

= dx

d g(x) d

dx

[ 100623 − 1000579ln(x) ]

= dx

d g(x) d

dx 100

623 − 5791000 ln(x)

d

dx

∴ = (x)g′ − 5791000x

Mathematical Calculation of the Second Derivative of g(x):

= dx

d g (x)′′ − 5791000x

= dx

d g (x)′′ d

dx

−[ 5791000x ]

∴ = (x)g′′ 5791000×2

6

Contrasting g(x) and f(x):

Index:

➢ f(x) = − .18 0 )x .0221x − .247x) .08( 3 × 1 −4 3 + 0 2 + ( 0 + 5

➢ f’(x) = 500000

−477x −22100x+1235002

➢ f’’(x) = 500000

−954x−22100

➢ g(x) = 6.23 .579ln(x) − 0

➢ g’(x) = − 5791000x

➢ g’’(x) = 5791000×2

Comparing both regression functions we notice that during the crisis the

unemployment rate was continuously rising, and that now for the past four years, the

unemployment rate was slowly decreasing attaining a level lower than 4 percent. I also

compared the derivatives of both graphs telling me that the rate of change of g(x) is greater

than the derivative of f(x). In other words, the rate at which the unemployment rate between

2015 to 2019 changes is decreasing at a rate where it won’t contribute to a recession. In

addition to this analysis, I did the second derivative test to measure the concavity of the

functions to determine whether it was concave up or concave down. This is crucial to my

analysis, because it helps me distinguish the difference in concavity between the two

functions. The positive second derivative: ; indicate that the shape of the function’’(x)g > 0

is concave up. On the other hand, the negative second derivative: ; will have its’’(x)f < 0 concavity downwards. The difference in each of the function’s concavity means that one is more reliant to increase or decrease. Since the is concave up, and is concave’’(x)g ’’(x)f down, this means that there is a low risk of attaining a recession. Private Debt of Consumers in the United States: Increasing private debt is common in most developed countries with fantastic booming economies since everyone wants money to consume. The ratio of Private Debt to GDP in the United States has decreased enormously to 196.70 percent in 2018 from an all time high 212.90 percent in 2009, considering it being 156.20 percent in 1995. We can conclude that the private debt has increased through the years, and we may suggest that it might increase in the upcoming years. 7 Chart 2: Private Debt to GDP in the United States Chart 2 above illustrates the ups’ and downs’ of private debt to GDP, and we can notice that the highest point was right in the year 2008 where coincidentally the financial crisis happened. This indicates that a high percentage of private debt to GDP, and the rate of change is at an increased rate, therefore a financial crisis may occur. I will use the same method as before where I will compare the rate of change of specific points between 2004 and 2008, and the rate of change of specific points in the past 4 years. Mathematical Calculations: Derivation of f(x): Below is a calculation to find the derivative of the function f(x) which represents the red portion above in Chart 2, and will be referred to as the Crisis Line of Best Fit for the percentage ratio of private debt to GDP from 2004 to 2008. This function had been obtained through the use of a mathematical software which graphs and plots a regression function that would best fit, which can be seen in Graph 3 below. Graph 3: f(x) Private Debt to GDP in the United States (2004-08) 8 The slope of Graph 3 is which represents the Crisis Line of Best(x) .05x 926,f = 5 − 9 Fit (2004-08). Mathematical Calculation of the Derivative of f(x): = dx d f (x) .05x 99265 − = 5.05x - 9926dx d f (x) d dx d dx ∴ = 5.05(x)f ′ Derivation of g(x): Below is a calculation to find the derivative of the function g(x) which represents the green portion above in Chart 2, and will be referred to as the Today’s Line of Best Fit for the percentage ratio of private debt to GDP from 2014 to 2018. This function had been obtained through the use of a mathematical software which graphs and plots a line of best fit, which can be seen in Graph 4 below. Graph 4: g(x) Private Debt to GDP in the United States (2014-18) The slope of Graph 4 is , which represents Today’s Line of Best(x) .58x 971g = 0 − Fit (2014-18). Mathematical Calculation of the Derivative of g(x): = 0.58x - 971dx d g(x) = 0.58x - 971dx d g(x) d dx d dx ∴ = 0.58(x)g′ 9 Mathematical Calculation of the Average Percentage Difference Between f(x) and g(x): Δ% Difference = 00% [ 2 f (x)+g (x)′ ′ ] f (x)−g (x)| ′ ′ | × 1 Δ% Difference = 00%[ 25.05+0.58 ] 5.05−0.58| | × 1 ∴ Δ% Difference = 159% Contrasting g(x) and f(x): Index: ➢ f(x) = .05x 9265 − 9 ➢ f’(x) = 5.05 ➢ g(x) = 0.58x 71 − 9 ➢ g’(x) = 0.58 The second part was about the private debt of consumers, and the amount of loans acquired throughout the years plotting it against GDP. During the years before the crisis started, the rate of change of private debt to GDP was very high and had an increasing slope which using a line of best fit gave it as . When I compared the second(x) .05x 926f = 5 − 9 line of best fit for the past 4 years, the slope was slightly different, and(x) .58x 71g = 0 − 9 increasing with a rate of change much lower than the derivative of . To see the difference(x)f using numbers, I calculated the percentage difference (Δ%), which gave me a value of 159%. The calculation can be seen above. This clearly tells me that we have a small risk of a financial crisis since the rate of change of is not increasing as much as the rate of(x)g change back during the years of recession, which is represented through .(x)f Gross United States: The gross domestic product of the United States is one of the largest economies in the world with a GDP exceeding 19 trillion dollars. Although GDP is not the perfect indicator of a nation's well being, it is recommended and a great index for measuring a country’s economic status by many well reputed economists. 10 Chart 3: GDP Growth Rate in the United States Mathematical Calculations: Derivation of f(x): Below is a calculation to find the derivative of the function f(x) which represents the red portion above in Chart 3, and will be referred to as the Crisis Line of Best Fit for the GDP percentage growth rate from 2007 to 2009. This function had been obtained through the use of a mathematical software which graphs and plots a line of best fit, which can be seen in Graph 5 below. Graph 5: f(x) GDP Growth in the United States (2007-09) The slope of Graph 5 is , which represents the Crisis Line of(x) .21x 430f = − 2 + 4 Best Fit (2007-09). 11 Mathematical Calculation of the Derivative of f(x): = dx d f (x) .21x 430− 2 + 4 = -2.21x + 4430dx d f (x) d dx d dx ∴ = (x)f ′ .21− 2 Derivation of g(x): Below is a calculation to find the derivative of the function g(x) which represents the green portion above in Chart 3, and will be referred to as the Today’s Line of Best Fit for the GDP percentage growth rate from 2015 to 2018. This function had been obtained through the use of a mathematical software which graphs and plots a line of best fit, which can be seen in Graph 6 below. Graph 6: g(x) GDP Growth in the United States (2015-18) The slope of Graph 6 is , which represents the Today’s Line of(x) .0578x 14g = 0 − 1 Best Fit (2015-18). Mathematical Calculation of the Derivative of g(x): = dx d g(x) .0578x 140 − 1 = 0.0578x - 114dx d g(x) d dx d dx ∴ = (x)g′ .05780 12 Mathematical Calculation of the Average Percentage Difference Between f’(x) and g’(x): Δ% Difference = 00% [ 2 f (x)+g (x)′ ′ ] f (x)−g (x)| ′ ′ | × 1 Δ% Difference = 00% [ 2 (−2.21)+0.0578 ] (−2.21)−0.0578| | × 1 ∴ Δ% Difference = 211% Contrasting g(x) and f(x): Index: ➢ f(x) = .21x 430− 2 + 4 ➢ f’(x) = .21− 2 ➢ g(x) = .0578x 140 − 1 ➢ g’(x) = .05780 The final part of my analysis is used to illustrate the risk of a financial crisis, with the comparison of the percentage Gross Domestic Product Growth rate. I only took the GDP growth rate for three years of different time periods. One for 2007 to 2009, and the other one for 2015 to 2018. Although I have short amounts of data points, it will still indicate along the other factors whether a recession is approaching. The GDP growth rate during the recession had a constant linear decreasing slope with a constant rate of change of -2.21 showing that during a recession no economic growth is present in the economy. However, when I compare it with Today’s Line of Best Fit, the function was positive and increasing with a derivative of +0.058, which meant that the growth rate was increasing indicating a good sign for the US economy. The percentage difference between the two derivatives was around 211%, showing that the rate at which the growth rate is increasing is far from the recession rate of change of GDP growth rate. All these factors combine with each other to give me a good indication of the economic situation in the United States of America. Conclusion & Evaluation _______________________________________________________________ In conclusion, the economic model that I had set up here using my investigation and exploration skills gave me enough convincing data to suggest that the United States has a less risk of being exposed to another catastrophic financial crisis like the one back in 2008. To sum up my ideas, I chose the main factors that mostly led to a recession, and procured their 13 data to finally compare them with each other using calculus, and functions to decide whether the risk of a recession happening again is very probable. The limitations in my investigation is the strength of my final conclusion which was based on the amount of factors chosen that helped me identify whether the United States might have another recession. A solution to this limitation might be including other factors, like interest rates, inflation rate, consumer confidence, and combining them to strengthen my final approach with more accuracy. Another flaw in my approach to the research question is the methodology employed throughout this assessment. My conclusions in each section depended on the regression function of best fit which gave a rough estimate of what each of the graphs were like. This in itself has a limitation, so applying my mathematical method in it will result in a lower accuracy for my results. This investigation made me learn that there will always be limitations into what we try to answer, no matter the amount of knowledge we know in a field of study such as mathematics. I realized that humans will always approach an answer, but would never reach it completely which is the reason why recognizing limitations is important. This recognition will help counter the flaws through developing a stronger way of approaching a question. Thus, this internal assessment opened my mind, and eyes in many aspects of life which changed the way I understand, and perceive mathematics. Bibliography _______________________________________________________________ Amadeo, Kimberly. “Causes of the 2008 Global Financial Crisis.” The Balance, The Balance, 14 Dec. 2019, “6 Economists Who Predicted the Global Financial Crisis and Why We Should Listen to Them from Now On.” INTHEBLACK, 29 July 2018, 14 “Notice: Data Not Available.” U.S. Bureau of Labor Statistics, U.S. Bureau of Labor Statistics, data.bls.gov/pdq/SurveyOutputServlet, Meserve, Jack. “The Private Debt Crisis.” Democracy Journal, 19 Sept. 2016, democracyjournal.org/magazine/42/the-private-debt-crisis/, “GDP Growth (Annual %) - United States.” Data, data.worldbank.org/indicator/NY.GDP.MKTP.KD.ZG?end=2018&locations=US&start=200 6&view=chart, “United States GDP Growth Rate.” United States GDP Growth Rate | 2019 | Data | Chart | Calendar | Forecast, tradingeconomics.com/united-states/gdp-growth, “Derivative Calculator.” Derivative Calculator • With Steps!, www.derivative-calculator.net/. 15