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The savings function shows how much people save at different income levels, expressed as S = Yd − C, where S is savings, Yd is disposable income, and C is consumption. By substituting the consumption function C = a + bY into this formula, we derive S = −a + (1−b)Y. This equation reveals autonomous dissaving (−a) and the marginal propensity to save (1−b), which indicates how much of each additional income unit is saved.

The marginal propensity to save, represented as (1−b) in the savings function, is the slope that shows the change in savings for each additional unit of disposable income. If b = 0.75, then the marginal propensity to save equals 0.25, meaning households save 25 cents of every additional dollar earned. This metric helps economists predict saving behavior across different income groups.

Negative savings, or dissaving, occurs when consumption exceeds disposable income. For example, at a disposable income of 100 with S = −175, households spend more than they earn by borrowing or drawing on existing savings. This typically happens at low-income levels where autonomous consumption (a) is high relative to income, forcing people to finance spending through debt or asset depletion.

The break-even point occurs where savings equal zero, marking the transition from dissaving to positive savings. At this income level, consumption exactly equals disposable income. Using the example S = −200 + 0.25Y, the break-even point is at Y = 800, where households neither save nor borrow. Beyond this point, positive savings increase linearly with income.

The savings function helps economists predict aggregate savings levels across income groups, which directly influences investment, economic growth, and stability. Understanding the relationship between income consumption and saving enables policymakers to forecast how income changes affect total savings in the economy. This relationship is essential for analyzing macroeconomic equilibrium and planning fiscal policy interventions.

Autonomous dissaving, represented as −a in the savings function, is the amount by which consumption exceeds income when disposable income is zero. In the example where a = 200, autonomous dissaving equals −200, meaning households would spend 200 units even with zero income. This negative intercept reflects borrowing or asset depletion necessary to maintain baseline consumption levels.

Changes in disposable income directly affect savings through the marginal propensity to save. With S = −200 + 0.25Y, a 100-unit increase in income raises savings by 25 units. At low incomes (Y = 100), households dissave; at higher incomes (Y = 1000), they save 50 units. This linear relationship demonstrates how income growth enables households to transition from borrowing to accumulating savings.