The Multiplier, Growth and Money and Banking: Question 7

Worked solution

Part (a): Deriving and solving the equilibrium condition

Planned aggregate expenditure is:

$AE = C + I + G + (X - M)$

Substituting the given functions:

$AE = (60 + 0.7Y) + 150 + 200 + \big(130 - (40 + 0.1Y)\big)$

Collecting the constant terms:

$60 + 150 + 200 + 130 - 40 = 500$

Collecting the $Y$ terms (consumption adds $0.7Y$, but imports subtract $0.1Y$):

$0.7Y - 0.1Y = 0.6Y$

So:

$AE = 500 + 0.6Y$

Equilibrium national income occurs where planned expenditure equals actual output, $Y = AE$:

$Y = 500 + 0.6Y \implies Y - 0.6Y = 500 \implies 0.4Y = 500 \implies Y = 1250$

Check: at $Y=1250$, $C=60+0.7(1250)=935$, $I=150$, $G=200$, $M=40+0.1(1250)=165$, so $AE=935+150+200+(130-165)=1250=Y$. ✓

So Kestria’s equilibrium level of national income is $\boxed{1250}$, i.e. $1,250 million.

Part (b): The implied multiplier

The multiplier is the reciprocal of the marginal propensity to withdraw (leak) out of each extra $1 of income. Here, out of every extra $1 of national income, $0.6 is re-spent on domestic output (via consumption net of the import leakage), so $0.4 leaks out (as saving, tax and imports combined):

$k = \frac{1}{1-0.6} = \frac{1}{0.4} = 2.5$

This is exactly consistent with the $0.6Y$ term found in (a): the coefficient on $Y$ in the $AE$ equation, $0.6$, is the fraction of extra income re-spent domestically, so $1$ minus this coefficient, $0.4$, is the marginal leakage whose reciprocal gives the multiplier, $k=2.5$.

Part (c): Effect of a rise in investment

An autonomous rise in investment of $\Delta I = 50$ ($50 million), with $k=2.5$, changes equilibrium national income by:

$\Delta Y = k \times \Delta I = 2.5 \times 50 = 125$

So the new equilibrium level of national income is:

$Y' = 1250 + 125 = 1375$

Check: with $I=200$, the new intercept is $500+50=550$, so $Y'=550+0.6Y' \implies 0.4Y'=550 \implies Y'=1375$. ✓

So the new equilibrium level of national income is $1,375 million.

Part (d): Equilibrium national income and full employment

Reaching $Y=AE$ only means that planned spending equals the current level of output. It says nothing about whether that level of output is large enough to employ everyone in Kestria’s labour force who wants to work at the going wage. The equilibrium level of national income found in (a) could therefore settle below the full-employment (potential) level of output, since there is no automatic mechanism in this model that pulls $Y$ up to full employment; if it does, the shortfall in planned spending relative to full-employment output leaves an output gap and demand-deficient unemployment.

Final answers

• (a) $AE = 500 + 0.6Y$; equilibrium national income $=$ $1,250 million
• (b) Multiplier $k =$ 2.5, consistent with the $0.6Y$ term since $k=\dfrac{1}{1-0.6}$
• (c) New equilibrium national income $=$ $1,375 million
• (d) Equilibrium $Y=AE$ need not coincide with the full-employment level of national income, since there is no guarantee planned spending is high enough to employ the whole labour force