# Consumption function

Not to be confused with demand function.

In economics, the consumption function describes a relationship between consumption and disposable income.[1] Algebraically, this means ${\displaystyle C=f(Y_{d})}$ where ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ is a function that maps levels of disposable income ${\displaystyle Y_{d}}$—income after government intervention, such as taxes or transfer payments—into levels of consumption ${\displaystyle C}$. The concept is believed to have been introduced into macroeconomics by John Maynard Keynes in 1936, who used it to develop the notion of a government spending multiplier.[2] Its simplest form is the linear consumption function used frequently in simple Keynesian models:[3]

${\displaystyle C=a+b\times Y_{d}}$

where ${\displaystyle a}$ is the autonomous consumption that is independent of disposable income; in other words, consumption when income is zero. The term ${\displaystyle b\times Y_{d}}$ is the induced consumption that is influenced by the economy's income level. The parameter ${\displaystyle b}$ is known as the marginal propensity to consume, i.e. the increase in consumption due to an incremental increase in disposable income, since ${\displaystyle \partial C/\partial Y_{d}=b}$. Geometrically, ${\displaystyle b}$ is the slope of the consumption function. One of the key assumptions of Keynesian economics is that this parameter is positive but smaller than one, i.e. ${\displaystyle b\in (0,1)}$.[4]

Criticism of the simplicity and irreality of this assumption lead to the development of Milton Friedman's permanent income hypothesis, and Richard Brumberg and Franco Modigliani's life-cycle hypothesis. But none of them developed a definitive consumption function. Friedman, although he got the Nobel prize for his book A Theory of the Consumption Function (1957), presented several different definitions of the permanent income in his approach, making it impossible to develop a more sophisticated function. Modigliani and Brumberg tried to develop a better consumption function using the income got in the whole life of consumers, but them and their followers ended in a formulation lacking economic theory and therefore full of proxies that do not account for the complex changes of today's economic systems.[citation needed]

Until recently, the three main existing theories, based on the income dependent Consumption Expenditure Function pointed by Keynes in 1936, were Duesenberry's (1949) relative consumption expenditure,[5] Modigliani and Brumberg's (1954) life-cycle income, and Friedman's (1957) permanent income.[6]

Some new theoretical works are based, following Duesenberry's one, on behavioral economics and suggest that a number of behavioural principles can be taken as microeconomic foundations for a behaviorally-based aggregate consumption function.[7]