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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 ∕ ∆Y_{d}.)
marginal propensity to save (MPS): The change in saving
resulting from a small change in disposable income (MPS = ∆S ∕ ∆Y_{d}.)
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.
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The following overview of tax, balancedbudget, and
autonomousspending 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 (T_{a}), investment (I_{a}),
government purchases (G_{a}),
and net exports (X_{a}  M_{a}), and part of
consumption (C_{a}) are
autonomous and that mpc(Y  T_{a}) is induced
consumption. The mpc is based on
disposable income, so
Y = C_{a} + mpc(Y – T_{a}) + I_{a}
+ G_{a} + (X_{a} – M_{a})
We
define total autonomous spending as A =
Ca + Ia + Ga + (Xa – Ma). This leaves Y = A +
mpc(Y – T_{a}), or Y = A + mpcY – mpcT_{a}.
Subtracting mpcY from both sides
leaves Y – mpcY = A – mpcT_{a}.
Factoring Y from the left side
yields Y(1 – mpc) = A – mpcT_{a}
, and then dividing both sides by 1–
mpc:


Y = A

[

1

]

+ T_{a}

[

– mpc

]


1 – mpc

1 – mpc


English translation: Aggregate income (Y) equals autonomous spending [C_{a} = I_{a }+ G_{a} + (X_{a}  M_{a}) = A] times the autonomous spending
multiplier [1/(1  mpc)], plus the
level of taxes (T_{a})
times the autonomous tax multiplier [mpc/(1
 mpc)]. [Because 1mpc = 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 BalancedBudget 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

]

+ ∆T_{a}

[

– mpc

]


1 – mpc

1 – mpc


If the mpc equals
0.8, the spending multiplier equals 5 and the tax multiplier equals 4. Thus,
∆Y = ∆A(5) + ∆T_{a}(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 balancedbudget
multiplier, always equals one because it reflects the numerical sum of
the autonomous spending and tax multipliers:


[

1

] +

[

– mpc

] =

[

1 – mpc

] =

1


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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Wednesday, 11 April 2012
The Mathematics of Simple Keynesian Multipliers
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