Question

Prove that the sum of the average propensity to save and the average propensity to consume is always equal to one,

Short answer

The sum of the average propensity to save and the average propensity to consume is always equal to one. We can prove this by defining APS as \(\frac{S}{Y}\) and APC as \(\frac{C}{Y}\), then adding these fractions together: APS + APC = \(\frac{S}{Y}\) + \(\frac{C}{Y}\). We can rewrite this sum as \(\frac{S + C}{Y}\), and recall that total income (Y) is equal to the sum of total savings (S) and total consumption (C). Thus, we can replace the numerator with Y: \(\frac{Y}{Y}\), which simplifies to APS + APC = 1.

Step-by-step solution

Define APS and APC

We'll first define the formulas for APS and APC. The average propensity to save (APS) is the ratio of total savings (S) to total income (Y), and the average propensity to consume (APC) is the ratio of total consumption (C) to total income (Y). In mathematical terms: APS = \(\frac{S}{Y}\) APC = \(\frac{C}{Y}\)

Add APS and APC

Now, let's add APS and APC: APS + APC = \(\frac{S}{Y}\) + \(\frac{C}{Y}\)

Find a common denominator

To add these two fractions, we need to find a common denominator, which in this case is Y. So, we rewrite the sum as: APS + APC = \(\frac{S + C}{Y}\)

Express the sum in terms of total income

Recall that total income (Y) is the sum of total savings (S) and total consumption (C): Y = S + C

Replace the numerator in the fraction

Now, we replace the numerator (S + C) with Y, since they are equal: APS + APC = \(\frac{Y}{Y}\)

Simplify the fraction

Lastly, let's simplify this fraction: APS + APC = 1 Therefore, we have proved that the sum of the average propensity to save and the average propensity to consume is always equal to one.

Key concepts

Average Propensity to Save (APS)

The Average Propensity to Save (APS) is a fascinating concept in economics that helps us understand the portion of income that households save. It is represented by the formula \( APS = \frac{S}{Y} \), where \( S \) is the total savings, and \( Y \) is the total income. Essentially, APS tells you how much of your income you are setting aside as savings.

To put it simply, if you earn \(100 and save \)20, your APS would be 0.2, or 20%. This concept helps economists and policymakers understand saving behaviors and tendencies at different income levels.

Key points about APS:

• It measures savings as a fraction of total income.
• A high APS indicates a strong tendency to save rather than spend.
• Factors such as interest rates, economic stability, and inflation can influence APS.

By examining APS, economists can gauge consumer confidence and future spending patterns, as it generally indicates how much of their additional income people are likely to save.

Average Propensity to Consume (APC)

The Average Propensity to Consume (APC) provides insight into how much of a person's or household's income is spent on goods and services. The formula for this measure is \( APC = \frac{C}{Y} \), where \( C \) represents total consumption and \( Y \) is total income.

This ratio shows us the proportion of income that goes towards spending. For example, if a household earns \(200 and spends \)150 on consumption, the APC would be 0.75, or 75%.

Notable points about APC:

• Shows the consumption habits as a percentage of total income.
• Higher APC suggests less income is being saved.
• It can be affected by income levels, cultural norms, and consumer confidence.

APC is an essential metric for understanding economic activity because it reflects consumer spending trends, which are a crucial driver of economic growth.

Income and Consumption Relationship

The relationship between income and consumption is a cornerstone of economic theory. This relationship is essential because it helps explain how households allocate income between saving and spending.

In general, as income increases, both consumption and savings tend to rise, but at different rates for different people. People with a higher income might save more in terms of absolute values, but their APC might decrease because their consumption doesn't increase proportionally with income.

Important aspects of the income-consumption relationship:

• Income shifts affect savings and spending decisions.
• Changes in income can influence the APC and APS dynamically.
• The marginal propensity to consume (MPC) and save (MPS) can provide further insights.

Understanding this relationship allows economists to make predictions about consumer behavior and the impact of economic policies like tax changes or interest rate adjustments.