Equilibrium with Taxes: A Numerical Example

To illustrate equilibrium with taxes, we will again use the example consumption function,

C = 500 + 0.7*(Y-T)

where C is consumption, Y is income before tax, and T is taxes. We are making another simplifying assumption here: the assumption that taxes overall are a lump sum, unrelated to income. Taxes like income taxes and sales taxes are related to income, of course, but on the other hand, the tax rates change from time to time. We are assuming, in effect, that the legislature knows about how much it wants to raise and adjusts the rates to get that amount. We are probably no further off than we would be with any other fairly simple assumption.

We will also assume that

I=1000

G=500

T=500

Since we want to compare the equilibria with and without taxes, first recall the equilibrium with taxes = 0. Using the multiplier of 3.333 and adding autonomous consumption of 500 to investment of 1000 and government purchases of 500 we have

Y = 3.333*(500+1000+500) = 3.333*2000 = 6667

Now let's see what impact the taxes have. Remember, taxes work indirectly by reducing consumption. Our consumption function is

C = 500 + 0.7*(Y-T) = 500 + 0.7*Y - 0.7*T

And since T=500 that is

C = 500 + 0.7*Y - 0.7*500 = 500 + 0.7*Y - 350 = 150 + 0.7*Y

So the new equilibrium income, using the multiplier method, will be

Y = 3.333*(150+1000+500) = 3.333*1650 = 5500

So the introduction of a tax of 500 (billion dollars) has reduced equilibrium income by 1167 (billion dollars), from 6667 to 5500.

An alternative way to get the equilibrium with taxes is to use the multiplier formula

In this example, that is (again)

Y = 3.333*2000 - 2.333*500 = 5500.

As usual, we can visualize that with an equilibrium diagram.

Visualization