 # The algebra of income and expenditure

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The algebra of income and expenditure
Variable net exports Algebraic determination of equilibrium GDP Problem

Variable net exports Now we make the more realistic assumption that imports (M) depend on the domestic level of income or GDP(Y). The Marginal Propensity to Import (MPM) is the fraction of the change in income that is spent on imports. That is: Note that:

Net exports and the aggregate expenditure line
(a) Variable net export function (trillions of dollars) Net exports Net exports = X-M X-M 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 Real GDP (trillions of dollars) Net exports are added to C, I, and G to yield AE. The addition of net exports: rotates the spending line about the point where net exports are zero (real GDP is \$10.0 trillion) (b) Aggregate expenditure lines Aggregate expenditure (trillions of dollars) C+I+G C+I+G+(X-M) 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 Real GDP (trillions of dollars)

Let AE denote aggregate expenditure . Thus we can say:
AE = C + I + G + (X – M)  Let NT denote net taxes (taxes minus transfers). Thus we can say: DI = Y – NT The consumption function can be written as: C = a + b(Y – NT) The above and be rewritten as C = a –bNT + bY  Where a-bT is autonomous consumption and b is the marginal propensity to consume.

AE = a – bT + bY + I + G + X – m(Y – NT)
Net exports are described by: X – m(Y – NT)  Where m is the marginal propensity to import Now substitute  and  into  to obtain: AE = a – bT + bY + I + G + X – m(Y – NT) In equilibrium AE = Y. Thus Y = a – bT + bY + I + G + X – m(Y – NT)  We can rearrange  to obtain:

Autonomous expenditure
Multiplier Autonomous expenditure AE AE Slope = b - m a- bNT + I + G + X +mNT Y Y*

Problem Let : C= 100 + 0.75(Y – NT) I = 50 G = 30 X = 40
M = .15(Y – NT) NT = 100

Let Y denote real GDP. Thus we can say: Y = C + I + G + (X – M) 
Let NT denote net taxes (taxes minus transfers). Thus we can say: DI = Y – NT. Let NT = \$1.0 trillion (net taxes are autonomous). Our consumption function is given by: C = (Y – NT) = Y  Induced C Autonomous C

AE AE Slope = = .6 160 400 Y

Effects of a change in (autonomous) I
Let ΔI = 5 . What is the resulting change in Y?

AE AE’ AE 165 160 400 412.5 Y

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