All spending injections—regardless of whether from more government spending, more investment, more autonomous consumption, or more exports—are perfect substitutes in their impact on equilibrium income. A recessionary gap can be erased (or an inflationary gap created) through new autonomous spending subject to the autonomous spending multiplier.
Alternatively, tax cuts can trigger economic growth. The autonomous tax multiplier equals one minus the autonomous spending multiplier. Algebraically, income is the sum of consumption, investment, government spending, and net exports: Y = C + I + G + (X - M). Suppose all taxes (Ta), investment (Ia), government purchases (Ga), and net exports (Xa - Ma), and part of consumption (Ca) are autonomous and that mpc(Y - Ta) is induced consumption. The mpc is based on disposable income, so
Y = Ca + mpc(Y - Ta) + Ia + Ga + (Xa - Ma)
We define total autonomous spending as A = Ca + Ia + Ga + (Xa - Ma). This leaves Y = A + mpc(Y - Ta), or Y = A + mpcY - mpcTa. Subtracting mpcY from both sides leaves Y - mpcY = A - mpcTa. Factoring Y from the left side yields Y(1 - mpc) = A - mpcTa , and then dividing both sides by 1 - mpc:
1 - mpc
Y = A( 1 - mpc ) + Ta ( 1 - mpc )
English translation: Aggregate income (Y) equals autonomous spending [Ca = Ia+ Ga + (Xa - Ma) = A] times the autonomous spending multiplier [1/(1 - mpc)], plus the level of taxes (Ta) times the autonomous tax multiplier [-mpc/(1 - mpc)]. [The autonomous tax multiplier can be rewritten (-mpc/mps).] Thus, if the spending multiplier is 5, the tax multiplier is -4; if the spending multiplier is 4, the tax multiplier is -3; and so on.