3.7: The Saving Function

The savings function explains how individuals allocate a portion of their income to savings after meeting consumption needs. It establishes a mathematical relationship between income (Y), consumption (C), and savings (S).

S = Y − C

Where:

• S = Savings
• Y = Income
• C = Consumption

This identity simply states that savings are part of income and are not used for consumption.

The Consumption Function

Consumption is typically expressed in linear form as:

C = a + bY

Where:

• a = Autonomous consumption (the amount spent even when income is zero)
• b = Marginal propensity to consume (MPC), representing how much of each additional unit of income is spent on consumption

Deriving the Savings Function

Substituting the consumption function into the savings identity:

S = Y− (a + bY)S

S = −a + (1−b)Y

This is the savings function, where:

• –a represents autonomous dissaving—the amount by which consumption exceeds income when income is zero.
• (1 – b) is the slope of the function and is known as the marginal propensity to save (MPS). It indicates how much of each additional unit of income is saved.

Numerical Example

Let us assume: a = 200 and b = 0.75.

Then the savings function becomes:

S = − 200 + 0.25Y

Case 1: When Income (Y) = 100

S=−200 + 0.25 × 100

S = −175

At this income level, savings are negative. This indicates dissaving, meaning the individual is spending more than they earn, likely by borrowing or drawing on existing savings.

Case 2: When Income (Y) = 1000

S = −200 + 0.25 × 1000

S = 50

Here, savings are positive, indicating that a portion of income is now being set aside after covering consumption.

Graphical Interpretation

The savings function is a straight line:

• Intercept: –200, which represents autonomous dissaving
• Slope: 0.25, indicating the increase in savings for every additional unit of income

The graph of the function shows:

• Dissaving at low-income levels
• A break-even point where savings are zero
• Positive savings at higher income levels

Importance of the Savings Function

The savings function plays a critical role in both microeconomic and macroeconomic analysis:

• It helps economists and policymakers predict saving behavior across different income groups.
• It assists in determining the aggregate level of savings in an economy, which influences investment, growth, and economic stability.
• For individuals, it serves as a guide for personal financial planning, indicating when income levels may permit saving after meeting essential consumption.

The savings function illustrates how much people save at different levels of disposable income.

Economists express savings as the difference between disposable income and consumption, which can be represented using the formula S = Yd - C. By substituting the consumption function into this formula, we get a new equation.

In this equation, “–a” is the intercept, which shows savings when disposable income is zero. This value is usually negative, indicating borrowing.

The term “(1 – b),” or the marginal propensity to save, is the slope. It shows the change in savings for each additional unit of disposable income.

Consider a = 200 and b = 0.75. Then, the savings function becomes as shown. At a disposable income of 100, savings S = –175. This negative value indicates that people are still dissaving—spending more than they earn.

However, at a disposable income of 1000, savings S = 50. This positive value means households are now saving—setting aside part of their income.