Problem 83 - Applied Calculus

Question

BUSINESS: Money Stock Measure From 1964 to 2014 the money stock measure "M1" (currency, traveler's checks, demand deposits, and other checkable deposits) was growing at the rate of approximately \(94 e^{0.56 x}\) billion dollars per decade, where \(x\) is the number of decades since 1964 . Find the total increase in M1 from $$ 1964 \text { to } 2014 $$

Step-by-Step Solution

Answer

The total increase in M1 from 1964 to 2014 is approximately 2591.78 billion dollars.

Step 1: Understand the Problem

We need to find the total increase in money stock, measured by M1, from 1964 to 2014. The problem provides a growth rate formula, which we'll integrate to find the total change over this period.

Step 2: Convert Time Period to Decades

The time from 1964 to 2014 is 50 years. To use our growth formula correctly, convert this time into decades by dividing by 10. Thus, we have \[ x = \frac{50}{10} = 5 \text{ decades} \] since 1964.

Step 3: Identify the Growth Rate Function

The provided growth rate of the money stock M1 is given by the formula \[ 94 e^{0.56x} \] billion dollars per decade. This represents the rate of increase of M1.

Step 4: Setup the Integral for Total Increase

To find the total increase from 1964 to 2014, integrate the growth rate function over the interval from 0 to 5 decades (since \( x = 0 \) represents 1964 and \( x = 5 \) represents 2014):\[ \int_{0}^{5} 94 e^{0.56x} \; dx \]

Step 5: Perform the Integration

Calculate the integral: \[ \int 94 e^{0.56x} \; dx = \frac{94}{0.56} e^{0.56x} = 167.8571 e^{0.56x} + C \]Evaluate this from 0 to 5:\[ \left[ 167.8571 e^{0.56 \times 5} \right] - \left[ 167.8571 e^{0} \right] \]

Step 6: Compute the Definite Integral

Substitute the limits into the integrated result:\[ \left[ 167.8571 e^{2.8} \right] - 167.8571 \]First, compute \( e^{2.8} \approx 16.444 \) and substitute it back:\[ 167.8571 \times 16.444 \approx 2759.64 \]Subtract the lower limit:\[ 2759.64 - 167.8571 = 2591.7829 \]Thus, the total increase in M1 is approximately \( 2591.78 \) billion dollars.

Key Concepts

Exponential Growth

Exponential growth is a powerful concept that appears very often in real-world scenarios, especially when dealing with finance and economics. It's essentially growth where the rate of change is proportional to the current amount. This means, as something grows, it grows faster because its size is increasing. In simple terms, the bigger it gets, the quicker it gets even bigger.

In our problem, we dealt with the money stock measure, M1, which was increasing at a rate given by the formula:

• \[ 94 e^{0.56x} \]

Here, \( e \) represents the exponential function base, approximately equal to 2.718. This formula tells us that as time (in decades) progresses, the M1 money stock grows faster. The example here illustrates an exponential function where the growth rate is continuous and consistently increasing over each decade.

Definite Integral

A definite integral provides a way to compute the total accumulation of change over a specific interval. It is widely used in calculus for finding areas, volumes, and other quantities that accumulate over intervals.

In our given problem, we needed to calculate the total increase in the money stock M1 over a period from 1964 to 2014. This is where the definite integral comes into play. We were tasked with integrating the function:

• \[ \int_{0}^{5} 94 e^{0.56x} \, dx \]

This step involves calculating the area under the curve of the growth rate function from 0 to 5 decades. This area gives us the total increase in the money stock M1 over this time period. The power of calculus allows us to easily determine the accumulation described by our problem, by substituting the limits into our integrated function and effectively measuring the total change.

Money Stock Measure

The money stock measure, particularly M1, is a critical economic indicator used to evaluate the amount of money circulating within an economy. M1 includes cash and other assets that are easily accessible and available for spending, such as:

• Currency in circulation
• Traveler's checks
• Demand deposits
• Other checkable deposits

These components make up the liquid money supply, known as M1. It provides insights into the immediate spending power within an economy. In the given problem, the importance of M1 is highlighted by its exponential growth, giving us a picture of how the economy might be behaving in terms of available liquid finances. Understanding M1 and its growth patterns can aid in economic planning and forecasting.