A.Automatic Stabilizers
Economic instability can arise from unforeseen changes in autonomous consumption, gross investment, or net exports. Not only do these changes directly affect desired spending levels (hence production) each of them is subject to the multiplier principle as well: any initial change in spending shifts the aggregate demand curve by a multiple amount. An "automatic stabilizer" can be thought of as any factor that reduces the size of this multiplier effect.
Consider the following simplified aggregate expenditure model. (We will hold the price level fixed, so that changes in equilibrium GDP can be interpreted as horizontal shifts in the aggregate demand curve.)
Y = C + Ig + G + Xn
= a + b(Y -T) + Ig + G + Xn
= a + b(Y -tY) + Ig + G + Xn.
From the model, it is apparent that total tax receipts are assumed to be proportional to GDP at a tax rate of t x 100%.
Solving this for Ye,
the equilibrium level of GDP, we begin by collecting all the terms containing
Y on the left side to obtain Y -bY + btY = a
+ Ig + G + Xn. Next, we factor out
the Y terms while collecting the b terms: Y(1 -b(1
-t)) = A, where A = a + Ig + G
+ Xn is equal to all autonomous spending. Dividing through
by (1 - b(1 -t)), we get the result we seek: Ye
= A.
The multiplier in this model
is 
= 
.
In our previous model with fixed taxes, the multiplier was simply 
.
Comparing the two, it is clear that 
< 
for any
t > 0. That is, at any positive rate of taxation, any change in autonomous
consumption, investment, or net exports will have a smaller impact on GDP than
if taxes are fixed.
What if taxes are not proportional to GDP, as in the previous model? Consider the more general tax system T = f(Y) where f ’(Y) > 0 and f ’’(Y) > 0. The first condition, f ’(Y) > 0, simply states that total taxes rise with income, as before-the tax rate is positive. The second condition, f ’’(Y) > 0, suggests that the rate of taxation also rises with GDP. This suggests that the tax system is progressive. (A regressive tax system would be modeled by the condition f ’’(Y) < 0-the tax rate falls with income.)
Substituting f(Y) for T in the previous model, we obtain Y -bY - bf(Y) = A, an equation that implicitly defines the equilibrium value of Y.
As before, we are interested
in the size of 
,
the multiplier. To find this, we begin by taking the total differential of our
equation for Y: dY -bdY + bf ’(Y)dY
= dA. Now collect the dY and b terms to obtain dY(1
-b(1 -f ’(Y)) = dA. Next, divide both sides by dA
and (1 -b(1 -f ’(Y)) to get our final result:
= 
< for f
’(Y) > 0. As with the proportional tax model, this more general model
shows that for any positive tax rate, the multiplier is reduced and the economy
is automatically stabilized.
To consider how the progressiveness
of the tax system affects the outcome, we need only consider how the size of
the multiplier changes as GDP changes because an increase in GDP will change
the marginal tax rate, f ’(Y). What we require is 
.
To simplify the task, let
Z = 1 -b(1 -f ’(Y)). Then 
= Z -1 and, using the chain rule, 
= -Z -2 
.
From the equation for Z, 
= bf ’’(Y), so substituting back for Z and ,
we get the result:
= 
.
Squaring the bracketed term renders the denominator positive. Consequently, the sign of the derivative depends only on the sign of f ’’(Y). If the tax system is progressive, f ’’(Y) > 0 and the size of the multiplier diminishes as income increases: automatic stabilization is enhanced. Alternatively, a regressive tax system increases the size of the multiplier as income increases, thereby diminishing the automatic stabilization.