Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that the government pumps an extra \(\$ 1\) billion into the economy. Assume that each business and individual saves \(25 \%\) of its income and spends the rest, so of the initial \(\$ 1\) billion, \(75 \%\) is respent by individuals and businesses. Of that amount, \(75 \%\) is spent, and so forth. What is the total increase in spending due to the government action? (This is called the multiplier effect in economics.)

Step-by-Step Solution

Answer

The total increase in spending is $4 billion.

Step 1: Understand the Problem

The government introduces a $1 billion stimulus into the economy. With each cycle, individuals and businesses save 25% of their income and spend 75%. We need to calculate the total increased spending, taking into account repeated cycles of spending and saving.

Step 2: Identify the Series

This is a geometric series problem because each spending cycle results in 75% spending of the previous cycle's total. The first term, \( a \), is the initial $1 billion, and each subsequent term is 75% of the previous term.

Step 3: Define the Geometric Series

The general form for a geometric series is \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. In this problem, \( a = 1 \text{ billion} \) and \( r = 0.75 \), since 75% of the money is respent in the economy in each cycle.

Step 4: Calculate the Multiplier Effect

Using the formula for the sum of an infinite geometric series, \( S = \frac{1 \text{ billion}}{1 - 0.75} = \frac{1 \text{ billion}}{0.25} = 4 \text{ billion} \). Thus, the total increase in spending is $4 billion.

Key Concepts

Multiplier Effect in Economics

The multiplier effect is a fascinating concept in economics. Imagine the government injects money into the economy—like an extra $1 billion. This action doesn't just increase spending by $1 billion. Instead, it triggers a chain reaction of spending, saving, and respending.
The key to understanding this lies in how people use the money. When individuals and businesses receive extra money, they save a portion and spend the rest. For instance, in our example, they spend 75% and save 25%.
Each time money is spent, it moves through the economy, generating further spending. This cycle continues, creating a greater total effect from the initial government spending.

• The multiplier effect shows how one action can amplify through repeated consumption cycles.
• It relies heavily on the propensity to consume and save within the economy.
• Ultimately, the effect measures the total change in economic activity due to an increase in initial spending.

Understanding the multiplier effect helps policymakers predict economic changes and plan budgetary actions to stimulate growth.

Understanding Geometric Sequences

A geometric sequence is a series of numbers where each term is a fixed multiple of the previous one. In our economic example, we're dealing with a situation where each term represents the amount spent after each cycle.
In a geometric sequence, you define two main pieces:

• The first term (\(a\)), which is where the sequence starts.
• The common ratio (\(r\)), which tells you how much to multiply each term by to get the next term.

Our sequence begins at 1 billion, represented as \(a = 1 \text{ billion}\), and with every spending cycle, 75% or \(r = 0.75\), is reused. This multiplier or ratio is what keeps the money flowing through each wave of spending.
Understanding this concept is crucial because it helps us determine overall financial growth or decline in processes like economic spending. Each term represents an economic decision made by individuals and businesses.

Calculating Infinite Series

An infinite series is when terms in a sequence go on indefinitely. The total sum can seem endless and hard to calculate without the right formula.
In problems like ours, where a series continues without stopping, the infinite geometric series formula simplifies the process:

• \[ S = \frac{a}{1 - r} \], where \(S\) is the sum of the series, \(a\) is the first term, and \(r\) is the common ratio.

In the context of economic stimulus, this formula is applied to determine the total effect of repeated spending cycles.
Our calculation shows the formula in action. First, the \(1 billion injection, followed by progressive spending of 75% at each stage, using \(r = 0.75\). By substituting our values in, we calculate:\[ S = \frac{1 \text{ billion}}{1-0.75} = \frac{1 \text{ billion}}{0.25} = 4 \text{ billion} \].
This calculation reveals that the original billion dollars leads to a total spending increase of \)4 billion. It illustrates how cumulative effects of small, continuous actions result in significant economic impact.

Economic Stimulus and Its Impact

Economic stimulus is a strategy used by governments to encourage growth and increase consumption. By injecting funds, they aim to jumpstart or boost the economy, counteracting downturns and promoting activity.
The purpose of a stimulus is multifaceted:

• It increases immediate consumption by providing businesses and individuals with more money to spend.
• Encourages economic growth by fostering more production, jobs, and income cycles.
• Targets specific areas within the economy to manage comprehensive effects of downturns effectively.

In our scenario, the government's $1 billion initiative sought to amplify spending, and it worked through the mechanics of the multiplier effect. The end result expands over time, as calculated, to $4 billion.
This strategy's success underscores why understanding concepts like geometric sequences and infinite series is key. They determine how well such initiatives can predictably stimulate an economy.