For the following exercise, consider the stock market crash...

For the following exercise, consider the stock market crash in 1929 in the United States. The table lists the Dow Jones industrial average per year leading up to the crash. $\begin{array}{|c|c|}\hline \text { Years after } 1920 & {\text { Value (S) }} \\ \hline 1 & {63.90} \\ \hline 3 & {100} \\ \hline 5 & {110} \\ \hline 7 & {160} \\ \hline 9 & {381.17} \\ \hline\end{array}$ The best-fit exponential curve to these data is given by $y=40.71+1.224^{x} .$ Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in 2014 ?

For the following exercise, consider the stock market crash in 1929 in the United States. The table lists the Dow Jones industrial average per year leading up to the crash.
$\begin{array}{|c|c|}\hline \text { Years after } 1920 & {\text { Value (S) }} \\ \hline 1 & {63.90} \\ \hline 3 & {100} \\ \hline 5 & {110} \\ \hline 7 & {160} \\ \hline 9 & {381.17} \\ \hline\end{array}$

The best-fit exponential curve to these data is given by $y=40.71+1.224^{x} .$ Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in 2014 ?

Key Concepts

Extrapolation and Model Limitations

Extrapolation involves using a model constructed from historical data to predict future values beyond the observed range. While this can provide insights, particularly for systems that have followed a consistent trend, it also carries the risk that the underlying assumptions of the model may not hold true over longer time frames. In financial markets, unforeseen events or regulatory changes can significantly alter trends, making long-term predictions less reliable.

Second Derivative and Concavity

The second derivative of a function measures the change in the first derivative, effectively indicating whether the rate of growth is accelerating or decelerating. In an exponential model, a positive second derivative implies that the growth rate is itself increasing over time. This accelerating behavior often indicates that the growth is unsustainable in the long run, as it ignores potential real-world constraints.

Unsustainable Growth

Unsustainable growth refers to a situation where the rate of increase in a system becomes so high that it is likely to lead to instability or a collapse. In the context of exponential growth models, continuously accelerating gains, as evidenced by increasingly large first and second derivatives, signal that the growth may not be maintained indefinitely due to external constraints or inherent system limitations.

Exponential Growth Models

Exponential growth models are used to describe situations in which a quantity increases at a rate proportional to its current value. These models are useful in various fields such as finance, biology, and physics, but they can lead to unrealistic long-term predictions when the growth rate is maintained indefinitely without any limiting factors.

First Derivative and Rate of Change

The first derivative of a function indicates the instant rate of change with respect to its independent variable. In the context of exponential growth, the first derivative shows how rapidly the quantity is increasing at a given point in time. An ever-increasing first derivative suggests that the rate of growth is not only high but also accelerating, which can be a signal of unsustainable trends.

Transcript

00:01 In the following question, we want to determine y prime and y double prime of the function y is equal to 40 .71 plus 1 .2 .24 raised to the x power, which models the price of stocks after the year 1920.

00:19 And with this information, we want to justify why the market was unsustainable.

00:23 So let's start off by taking y, f prime of x.

00:28 So f prime of x is going to be the derivative with respect to x of 40.

00:32 1 .71 plus 1 .2 to 4 raised to the x power.

00:38 So in this case this derivative is going to be 0 since this is just a constant.

00:48 And now remember that the derivative of an exponential function, it's itself, times the ellen of the base.

00:59 This is greater than 0, so positive, for all x greater than 0.

01:07 Now f double prime is going to be the derivative with respect to x.

01:11 Of f prime of x so once again we're going to have the derivative itself so 1 .2 to 4 raise to the x power times l on the square of 1 .2 to 4 which is again greater than 0 for all x greater than 0 so now why does why was the market unsustainable well we know that each year oh sorry each year the stock price increase but we also know by the second derivative that this rate rate of price over year also increased.

02:05 So in this case, we know that the price went up.

02:14 And on this case, we know that the rate of the price after each year was also increasing, which is what the second derivative is telling us...

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