Problem 458 - Calculus Volume 1

Question

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 ?

Step-by-Step Solution

Answer

The model exhibits unsustainable growth as both the rate and acceleration of gains are increasing. By 2014, the prediction is excessively high, indicating unrealistic projections.

Step 1: Understand the Model

The given model is \( y = 40.71 + 1.224^x \). This represents an exponential growth, where \( x \) is the number of years after 1920.

Step 2: Differentiate the Function

Find the first derivative \( y' \) to understand how the stock index changes over time. The first derivative is \( y' = \ln(1.224) \times 1.224^x \). This describes the rate of change of the stock index.

Step 3: Analyze the First Derivative

The expression \( \ln(1.224) \times 1.224^x \) shows that the rate of increase is continuously growing, indicating that the gains in the stock market are becoming faster as time progresses.

Step 4: Calculate the Second Derivative

Find the second derivative \( y'' \) to examine the acceleration of the stock index growth. It is given by \( y'' = (\ln(1.224))^2 \times 1.224^x \). This indicates that the rate of increase itself is increasing, signaling exponential growth in gains.

Step 5: Analyze the Second Derivative

Since \( y'' > 0 \) for all \( x \), it shows that the rate of change is increasing, confirming an unsustainable growth pattern because growth is compounding on itself at an increasing rate.

Step 6: Project the 2014 Value

Substitute \( x = 94 \) (since 2014 is 94 years after 1920) into the model. Using \( y = 40.71 + 1.224^{94} \), the predicted Dow Jones industrial average is exceedingly large, another indication of unsustainable growth.

Key Concepts

Understanding the First Derivative

The first derivative of a function helps us understand the rate of change at any point. In the context of the stock index given by the equation \( y = 40.71 + 1.224^x \), the first derivative \( y' \) illustrates how fast the stock market values are changing over time. In simple terms, it's like looking at the speedometer of a car to see how fast it is going at any moment.
The calculation for the first derivative involves using the natural logarithm. For our function, it becomes \( y' = \ln(1.224) \times 1.224^x \). Here, \( \ln(1.224) \) acts as a constant that modifies the exponential part of the function \( 1.224^x \).
This derivative tells us that the rate of change in the Dow Jones index *itself* is increasing exponentially. The more time passes, the faster the stock market grows, reflecting an ever-accelerating rate of growth.

Exploring the Second Derivative

While the first derivative shows how quickly values grow, the second derivative \( y'' \) helps us understand acceleration, or how the rate of growth is changing. You can think of it like hitting the gas pedal a bit more every second in a car that's already speeding up.
For our stock market model, the second derivative is \( y'' = (\ln(1.224))^2 \times 1.224^x \). This essentially means that the rate at which the first derivative (growth rate) increases is itself increasing. It confirms the notion of exponential growth, where not only are values growing fast, but that growth is also speeding up.
This notion of accelerating growth further highlights the idea that the gains in the stock market during this period were not just growing steadily, but in a manner that compounded on itself, leading to potentially unsustainable growth.

What Makes Growth Unsustainable?

Unsustainable growth happens when the rate of increase becomes too rapid for a system to manage or continue indefinitely. In our exponential model \( y = 40.71 + 1.224^x \), both the first and second derivatives signal such growth.

• The first derivative shows continuously increasing rates of stock gains.
• The second derivative indicates that this increase is accelerating.
  

These characteristics imply that the stock market is on a trajectory that likely cannot be maintained in the long term. Just as a car that speeds up too fast can spiral out of control, the same can happen with financial markets when values rise too steeply.
When the model makes a prediction for 2014, it suggests extremely high values for the Dow Jones index. This hyperbolic prediction aligns with the idea that the rapid and compounded rate of growth is unsustainable, ultimately leading to potential instability, much like what historically occurred during the stock market crash.