Problem 97 - Algebra and Trigonometry

Question

Suppose that, throughout the U.S. economy, individuals spend \(90 \%\) of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is \(0.90 .\) For example, if Jane earns an additional dollar, she will spend \(0.9(1)=\$ 0.90\) of it. The individual who earns \(\$ 0.90\) (from Jane) will spend \(90 \%\) of it, or \(\$ 0.81 .\) This process of spending continues and results in an infinite geometric series as follows: $$1,0.90,0.90^{2}, 0.90^{3}, 0.90^{4}, \ldots$$ The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend \(90 \%\) of every additional dollar that they earn?

Step-by-Step Solution

Answer

The multiplier is 10.

Step 1: Identify the First Term and Common Ratio

The first term of the geometric series is 1 (since the initial dollar earned is taken as the first term). The common ratio is given as 0.90 because each subsequent term is 90% of the previous term.

Step 2: Use the Sum Formula for an Infinite Geometric Series

The formula for the sum of an infinite geometric series is \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio.

Step 3: Substitute the Values

For this series, \( a = 1 \) and \( r = 0.90 \). Substituting these values into the formula gives: \[ S = \frac{1}{1-0.90} \]

Step 4: Calculate the Multiplier

Perform the calculation: \[ S = \frac{1}{0.10} = 10 \]

Key Concepts

Infinite Geometric Series

An infinite geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This series goes on indefinitely without ending. For example, if you start with 1 (the first term) and keep multiplying by 0.90 (the common ratio), you get 1, 0.90, 0.90^2, 0.90^3, and so on.
The remarkable thing about infinite geometric series is that even though they have infinitely many terms, their sum can be finite! This property makes them very useful in economics and finance, among other fields.

Marginal Propensity to Consume

The marginal propensity to consume (MPC) is a measure used in economics to quantify the change in personal consumer spending (consumption) due to a change in disposable income. Essentially, it represents the portion of additional income that an individual is likely to spend rather than save. In this example, the MPC is 0.90, which means individuals spend 90% of every additional dollar they earn.
Let's break it down:

• If Jane earns an extra \(1, she spends \)0.90 (or 90%) of it.
• The person who receives Jane's \(0.90 will spend 90% of this amount, which is \)0.81.
• This spending cycle continues, following the pattern of the geometric series.

Sum Formula for Infinite Geometric Series

To find the sum of an infinite geometric series, you use the following formula: where \(S = \frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio.
In the given exercise, the first term \(a\) is 1, and the common ratio \(r\) is 0.90. By substituting these values into the formula, we get: \( S = \frac{1}{1 - 0.90} = \frac{1}{0.10} = 10 \)
This means the multiplier is 10, indicating the overall impact of the initial dollar spent reverberates through the economy, leading to a total spending of ten dollars throughout the process.

Common Ratio

The common ratio in a geometric series is the factor by which each term is multiplied to obtain the subsequent term. In mathematical notation, if you have a geometric series with first term \(a\) and common ratio \(r\), the series is: \(a, ar, ar^2, ar^3, ...\)
For the given problem, the common ratio is 0.90. This ratio indicates that each term is 90% of the previous term. Understanding the common ratio is crucial as it helps us determine the behavior of the series. In this exercise, the common ratio being less than 1 (0.90 in this case) ensures that the terms get smaller and smaller, allowing the sum of the infinite series to converge to a specific value. This convergence is why we can find a finite sum even for an infinite series!