5.1 Autonomous Taxation and the Multipliers

An autonomous term in an equation is a term that is independent of any (right-hand side) variable. Normally it is the intercept in the equation. An exogenous variable means the value of the variable is given and could change when the economy is hit with a shock. Obviously, the autonomous term is an example of an exogenous variable, so they are the slope terms in the equations.

Suppose for simplicity that government spending on goods and services, G, the amount of taxes the government collects, T, and the transfers the government provides, Tr, are all exogenous. Then:

Furthermore, since disposable income, YD, is GDP after subtracting taxes and adding transfers, then:

The income–expenditure equilibrium level of GDP, Y*, is equal to planned aggregate expenditure. So:

With a little algebraic manipulation, we get:

This equation gives us the following three multipliers:

So, for example, if the marginal propensity to consume is 0.5, then the autonomous expenditure multiplier is equal to 2. This implies, all else the same, that a $100 billion increase in autonomous consumption (AC), planned investment (IPlanned), or government spending on goods and services (G) will result in a $200 billion increase in the income–expenditure equilibrium level of GDP. Similarly, in this case, the autonomous taxation multiplier is equal to −1, so a $100 billion increase in taxes will result in a $100 billion reduction in the income–expenditure equilibrium level of GDP. Also, in this case, the autonomous transfers multiplier is equal to 1, so a $100 billion increase in transfers results in a $100 billion increase in the income–expenditure equilibrium level of GDP.