Problem 73 - Calculus: Early Transcendentals

Question

When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending \( D \) dollars. Suppose that each recipient of spent money spends \( 100c% \) and saves \( 100s% \) of the money that he or she receives. The values \( c \) and \( s \) are called the marginal propensity to consume and the marginal propensity to save and, of course, \( c + s = 1. \) (a) Let \( S_n \) be the total spending that has been generated after \( n \) transactions. Find an equation for \( S_n. \) (b) Show that \( \lim_{n \to \infty} S_n = kD, \) where \( k = 1/s. \) The number \( k \) is called the multiplier. What is the multiplier if the marginal propensity to consume is \( 80%? \) Note: The federal government uses this principle to justify lending a large percentage of the money that they receive in deposits.

Step-by-Step Solution

Answer

(a) \( S_n = D \frac{1 - c^n}{1 - c} \) (b) \( \lim_{n \to \infty} S_n = \frac{D}{s}, k = 5 \text{ if } c = 0.8.\)

Step 1: Understanding the Spending Process

Start with the initial spending by the government which is \( D \) dollars. This will be the initial amount in the community's economy.

Step 2: Calculating Spending After Transactions

Each recipient of the initial \( D \) dollars spends a fraction \( c \) and saves \( s \). After the first transaction, the money spent is \( Dc \). The same fraction \( c \) continues to be spent in each subsequent transaction.

Step 3: Formulating the Equation for Total Spending

Total spending after \( n \) transactions, \( S_n \), is a geometric series: \[ S_n = D(1 + c + c^2 + \, ... \, + c^{n-1}). \] Since \( c < 1 \), this series converges as \( n \to \infty \).

Step 4: Summation of Geometric Series

The sum of the first \( n \) terms of a geometric series \( 1 + c + c^2 + \, ... \, + c^{n-1} \) is given by: \[ S_n = D \frac{1 - c^n}{1 - c} \]

Step 5: Calculating the Limit of Total Spending

As \( n \to \infty \), \( c^n \to 0 \). Thus, \( S_n \to D \frac{1}{1-c} \). Since \( s = 1 - c \), then \( S_n \to \frac{D}{s}. \)

Step 6: Determining the Multiplier

The multiplier \( k \) is defined as \( \frac{1}{s} \). To find the multiplier when \( c = 0.8 \), calculate \( s = 1 - 0.8 = 0.2 \). Thus, the multiplier is \( \frac{1}{0.2} = 5 \).

Key Concepts

Marginal Propensity to Consume

When individuals receive money, they have choices on how to use it. A crucial economic concept here is the marginal propensity to consume, commonly abbreviated as MPC. This refers to the percentage of additional income that a consumer spends on goods and services, rather than saving it. The MPC is expressed as a decimal and can vary based on a person's financial situation and economic confidence.

For instance, if someone receives an extra \(100 and spends \)80 of it, their marginal propensity to consume ( \( c \) ) is 0.8. This choice directly affects the flow of money in the economy. A higher MPC means people are spending a larger portion of their income, which can stimulate economic activity.

• Encourages higher levels of consumption
• Amplifies the multiplier effect in an economy
• Varies based on socioeconomic factors

Marginal Propensity to Save

On the flip side of spending is saving. Marginal Propensity to Save, or MPS, is another key economic measure. This represents the portion of additional income that households save instead of spending on consumption. Similar to MPC, it's expressed as a decimal.

If you receive a $100 windfall and decide to save $20, your MPS is 0.2. It's vital to note that MPC and MPS will always add up to 1, implying all funds are either saved or spent.

• A higher MPS results in more money being saved
• Reduces the immediate effect of increased income on the economy
• Significantly influences personal financial wellbeing and economic stability

Geometric Series

Understanding a geometric series is crucial to grasp how spending impacts the economy through the multiplier effect. A geometric series is a sequence of numbers where each term after the first is found by multiplying or dividing the previous one by a fixed, non-zero number known as the common ratio.

When discussing economic spending, each round of spending results in a fraction of the original amount being spent again in subsequent rounds. Hence, when the initial spending is repeated over transactions, it forms a geometric sequence:\[ S_n = D(1 + c + c^2 + \, ... \, + c^{n-1}) \]where each term is influenced by the marginal propensity to consume, \( c \).

• Facilitates the calculation of total spending over multiple transactions
• Useful in economic modeling and predictions
• Provides insight into the cumulative effect of spending in an economy

Economic Theory

Economic theory encompasses a broad range of concepts that describe how economies function and grow. Within this framework, the multiplier effect illustrates how an increase in spending results in an even greater increase in national income and productivity.

The initial spending, such as by a government or local authority, sets off a chain reaction of further spending. This principle is based on the understanding of MPC and MPS. In theoretical terms, the multiplier (\( k \) ) quantifies this chain reaction:\[ k = \frac{1}{s} \]With a given example, a multiplier of 5 implies that the initial injection of money will lead to a total increase in economic activity by five times that original amount.

• Shows the interdependence of spending and saving choices
• Helps governments plan fiscal policies
• Crucial for understanding economic cycles and growth