Aggregate spending function shows the relationship between aggregate expenditure and level of national income; it relates the volume of spending to the level of income of an economy. Aggregate spending has two components, viz., C and I. Consumption (C) depends on the level of income as only a certain ratio (or fraction) of income is spent on consumption. This ratio is called marginal propensity to consume and is expressed by the symbol b. But there is also some autonomous consumption expenditure that is unrelated to income level. This is shown by symbol a. Thus, C = a + bY where one part of total consumption expenditure, viz., a is unrelated to income, while the other part, viz., b is related to income and varies with changes in level of income.
Investment expenditure too is partly autonomous and unrelated to changes in income level and therefore called autonomous investment. The other part of investment expenditure called induced investment is related to income level and changes along with income changes. But for the sake of keeping the analysis simple, we assume here all investments to be autonomous and hence independent of income changes.
Aggregate expenditure at each level of income thus comprises of autonomous expenditure (viz., autonomous consumption plus autonomous investment) and induced expenditure (viz., induced consumption expenditure since we assumed all investment to be autonomous and thus ruled out any induced or income related investment).
Now, that's the sum of autonomous and induced expenditure at each level of income. The aggregate spending function is written as
AE = (a+bY) + I
AE = C + I