Mathematics of Keynesian Multipliers

autonomous spending multiplier: The number in simple linear Keynesian models which, when multiplied by the sum of all autonomous spending, yields equilibrium income; in these models, this multiplier equals the reciprocal of the marginal propensity to save.

balanced budget multiplier: Early Keynesian theorists developed the balanced budget multiplier, which suggests that an equal increase in government spending and tax revenue will boost aggregate demand by precisely the increase in the amount spent.

marginal propensity to consume (MPC): The change in consumption generated by a small change in disposable income (MPC = C ∕ Yd.)

marginal propensity to save (MPS): The change in saving resulting from a small change in disposable income (MPS = ∆S ∕ Yd.)

multiplier effect: The total change in spending that results in a Keynesian cross model when new autonomous spending boosts income which, in turn, is spent, creating more income, and so on.

tax multiplier: The coefficient by which aggregate spending is reduced when taxes are increased.

 The following overview of tax, balanced-budget, and autonomous-spending multipliers uses early Keynesian theory to interpret what elected officials may be trying to do the next time they weigh a change in fiscal policy to remedy inflationary pressure or excessive unemployment.   The Autonomous Spending and Tax Multipliers   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: Y=A [ 1/(1-MPC) ] + TA [ -MPC/ (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)]. [Because 1-mpc = mps, 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.   The Balanced-Budget Multiplier   In our simple Keynesian model, all else equal, any change in National Income can be traced to changes in autonomous spending or taxes: ∆Y = ∆A [1/(1-MPC)] + ∆TA [ -MPC/ (1-MPC)] If the mpc equals 0.8, the spending multiplier equals 5 and the tax multiplier equals -4. Thus, ∆Y = ∆A(5) + ∆Ta(-4). If government spending and taxes each grow by \$20 billion, income also rises by \$20 billion because \$20 billion × (5) plus \$20 billion × (– -4) equals \$20 billion. We can generalize: Equal changes in autonomous government spending and taxes cause income to change in the same direction and by the same amount. The applicable multiplier, termed the balanced-budget multiplier, always equals one because it reflects the numerical sum of the autonomous spending and tax multipliers: [1/ (1-MPC)] + [ -MPC/ (1-MPC)] = [(1-MPC)/ (1-MPC)] Much more realistic assumptions than those we have used underpin the sophisticated econometric models used to forecast national economic activity. For example, that income affects taxes, investment, and government outlays is acknowledged. Although serious forecasting models are mathematically far more complex than those considered here, the approaches are basically similar: Assumptions about the behavior of various economic agents are used to predict National Income and Output.

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