Mathematically, the marginal propensity to consume (MPC) function is expressed as the derivative of the consumption (C) function with respect to disposable income (Y).
, where a is the change in consumption, and b is the change in disposable income that produced the consumption.
Let's give an example. Suppose you receive a bonus with your paycheck, and it's $500 on top of your normal annual earnings. You suddenly have $500 more in income than you did before. If you decide to spend $400 of this marginal increase in income on a new business suit, your marginal propensity to consume will be 0.8 ().
The marginal propensity to consume is measured as the ratio of the change in consumption to the change in income, thus giving us a figure between 0 and 1. The MPC can be more than one if the subject borrowed money to finance expenditures higher than their income. One minus the MPC equals the marginal propensity to save (in a two sector closed economy), both of which are crucial to Keynesian economics and are key variables in determining the value of the multiplier.
The MPC relies heavily upon the real (inflation-adjusted) rate of interest. A high rate of interest causes spending in the future to become increasingly attractive due to the intertemporal substitution effect on consumption. Because a rate increase primarily decreases the present value of lifetime wealth, the consumer relies on becoming a lender to offset this effect. In a two period model, as S(1+r) increases with the interest rate, so does future income[C= -(1+r)c +we(1+r)]. Therefore, every dollar of current income spent by the consumer is 1(1+r) dollars the consumer will not be able to spend in the second period.
Economists often distinguish between the marginal propensity to consume out of permanent income, and the marginal propensity to consume out of temporary income, because if a consumer expects a change in income to be permanent, then they have a greater incentive to increase their consumption (Barro and Grilli, p. 417-8). This implies that the Keynesian multiplier should be smaller in response to permanent changes in income than it is in response to temporary changes in income (though the earliest Keynesian analyses ignored these subtleties). However, the distinction between permanent and temporary changes in income is often subtle in practice, and there is in some cases hardly any way to assign to income data a so-called permanent or temporary character. What is more, the marginal propensity to consume should also be affected by factors such as the prevailing interest rate and the general level of consumer surplus that can be derived from purchasing.