Lecture 8 · Slides L8 · Case, Fair and Oster ch. 8, 9, 19 · about 50 min

The Core of Macroeconomic Theory: The Keynesian Cross and Multipliers

Keynes's answer to the Great Depression: in the short run, planned spending determines output. How equilibrium output is found, why a change in spending is multiplied, and the government, tax, balanced-budget and open-economy multipliers.

By the end you can

Write the consumption and saving functions and use MPC + MPS = 1
Find equilibrium output from Y = AE, from a table, and from S = I
Explain the multiplier process and compute 1/MPS
Compute the government, tax and balanced-budget multipliers
Explain why imports make the multiplier smaller

Why a new model was needed

Classical theory says national income depends on factor supplies and technology. Between 1929 and 1933 neither changed much, yet output collapsed during the Great Depression. Classical theory could not explain it. John Maynard Keynes’s General Theory of Employment, Interest, and Money (1936) offered a new answer: in the short run, output is driven by aggregate demand.

Throughout, aggregate output = aggregate income = $Y$ , in real terms: every unit produced is income for someone.

Consumption and saving

For Keynes, current income is the key determinant of consumption. As income rises, consumption rises, but by less than income.

Keynesian consumption functionFormula sheet →

C = a + bY

$a$: autonomous consumption, spending at zero income
$b$: MPC, marginal propensity to consume, 0 < b < 1

Use it when you need consumption at a given income, or the slope of the consumption line.

Saving is whatever is not consumed. The triple bar means an identity: true by definition.

Saving and the propensitiesFormula sheet →

S \equiv Y - C \qquad MPS = \frac{\Delta S}{\Delta Y} \qquad MPC + MPS \equiv 1

$MPS$: marginal propensity to save

Use it when a question gives the MPC and needs the MPS, or asks for saving at a given income.

Why must $MPC + MPS = 1$ ? Since $Y \equiv C + S$ , a change in income is split between the two: $\Delta Y = \Delta C + \Delta S$ . Divide by $\Delta Y$ .

The lecture’s example is $C = 100 + 0.75Y$ :

Income $Y$	Consumption $C$	Saving $S = Y - C$
0	100	−100
80	160	−80
100	175	−75
200	250	−50
400	400	0
600	550	50
800	700	100
1000	850	150

Every 100 of extra income raises consumption by 75 and saving by 25. At $Y = 400$ the household spends exactly its income.

Planned versus actual investment

Firms hold inventories: goods awaiting sale. Planned investment $I$ is the additions to capital and inventories firms intend to make. Actual investment also includes unplanned changes in inventories: if a firm overestimates sales, unsold goods pile up and actual investment exceeds planned.

Planned investment falls when the interest rate rises (borrowing is dearer) and depends on expected future sales and on business optimism, which Keynes called animal spirits. In this lecture $I$ is taken as given.

Equilibrium output

Planned aggregate expenditure is what the economy plans to spend: $AE \equiv C + I$ (adding $G$ and $NX$ later).

Goods market equilibriumConditionFormula sheet →

Y = AE = C + I

$Y$: aggregate output
$AE$: planned aggregate expenditure

Use it when you find equilibrium output. It is a condition, not an identity: it holds only in equilibrium.

What if output is not at equilibrium?

$Y > C + I$ : output exceeds planned spending. Inventories rise unexpectedly, so firms cut output.
$Y \lt C + I$ : planned spending exceeds output. Inventories fall, so firms raise output.

The lecture’s table with $C = 100 + 0.75Y$ and $I = 25$ :

$Y$	$C$	$I$	$AE = C + I$	Unplanned inventory $Y - AE$	Equilibrium?
100	175	25	200	−100	No: output rises
200	250	25	275	−75	No: output rises
400	400	25	425	−25	No: output rises
500	475	25	500	0	Yes
600	550	25	575	+25	No: output falls
800	700	25	725	+75	No: output falls

Drag the orange point away from equilibrium and press Let firms adjust. The bar is the unplanned inventory change that pushes firms back.

The planned-expenditure line crosses the 45° line at one point, $Y = 500$ . That crossing is the Keynesian cross.

The saving and investment approach

Since $Y \equiv C + S$ always, and $Y = C + I$ in equilibrium, substitute:

C + S = C + I \quad\Longrightarrow\quad S = I

Equilibrium occurs only when saving equals planned investment. Check at $Y = 500$ : $S = 500 - 475 = 25 = I$ .

The multiplier

A multiplier is the ratio of the change in equilibrium output to the change in an exogenous variable, one that does not depend on the state of the economy.

Raise planned investment by 25, from 25 to 50. At first, spending exceeds output by 25. Firms produce more, which raises income, which raises consumption, which raises spending again, and so on. The new equilibrium is $Y = 600$ : output rose by 100, four times the rise in investment.

Simple multiplierKey formulaFormula sheet →

\text{Multiplier} = \frac{1}{MPS} = \frac{1}{1 - MPC} \qquad \Delta Y = \Delta I \times \frac{1}{1 - MPC}

$MPS$: 1 - MPC

Use it when a question changes an exogenous spending item (I, G, autonomous C) and asks for the change in output. The steeper the AE line (the higher the MPC), the bigger the multiplier.

Why $1/MPS$ ? In equilibrium $S = I$ , so any change in saving equals the change in investment: $\Delta S = \Delta I$ . And $\Delta S = MPS \times \Delta Y$ . So $\Delta Y = \Delta I / MPS$ .

In the real world the multiplier is smaller than $1/MPS$ because taxes depend on income, the central bank reacts through interest rates, prices adjust, and part of spending goes on imports. We add these one at a time.

Adding the government

Fiscal policy is the government’s spending and taxing. Monetary policy is the central bank’s control of money. Discretionary fiscal policy means deliberate changes in $G$ or $T$ .

Net taxes $T$ are taxes minus transfers. Disposable income is $Y_d \equiv Y - T$ , and consumption now depends on it:

C = a + b(Y - T) \qquad AE \equiv C + I + G \qquad \text{budget deficit} \equiv G - T

The lecture’s example: $C = 100 + 0.75Y_d$ , $I = 100$ , $G = 100$ , $T = 100$ .

Leakages equal injectionsConditionFormula sheet →

S + T = I + G

$S + T$: leakages from the spending stream
$I + G$: injections

Use it when you check equilibrium with government, or solve without writing out C.

Switch the diagram to + G, T and move $G$ up by 50: output rises from 900 to 1,100, an increase of 200, four times the change in $G$ .

Three multipliers

Government spending multiplierFormula sheet →

\frac{\Delta Y}{\Delta G} = \frac{1}{1 - MPC} = \frac{1}{MPS}

Use it when G changes and taxes are fixed.

A tax change works indirectly: $T \uparrow \Rightarrow Y_d \downarrow \Rightarrow C \downarrow \Rightarrow Y \downarrow$ . The first-round change in spending is only $-MPC \times \Delta T$ , because part of the tax is paid out of saving.

Tax multiplierFormula sheet →

\frac{\Delta Y}{\Delta T} = -\frac{MPC}{1 - MPC} = -\frac{MPC}{MPS}

Use it when net taxes change and G is fixed. It is negative and smaller in size than the spending multiplier.

Balanced-budget multiplierFormula sheet →

\frac{\Delta Y}{\Delta G}\Big|_{\Delta G = \Delta T} = \frac{1}{MPS} - \frac{MPC}{MPS} = \frac{1 - MPC}{MPS} = 1

Use it when G and T change by the same amount. Output changes by exactly the change in G.

In the lecture’s third table, $G$ and $T$ both rise from 100 to 300 and output rises from 900 to 1,100: by 200, the same as $\Delta G$ .

Multiplier	Policy	Formula	Effect on $Y$
Government spending	change in $G$	$1/MPS$	$\Delta G \times 1/MPS$
Tax	change in net taxes	$-MPC/MPS$	$\Delta T \times (-MPC/MPS)$
Balanced budget	$\Delta G = \Delta T$	$1$	$\Delta G$

A warning: taxes depend on income

So far $T$ was a lump sum. In reality tax revenue rises with income: $T = T_0 + tY$ . Then:

Equilibrium with a proportional taxFormula sheet →

Y = \frac{1}{1 - b + bt}\,(a + I + G - bT_0)

$b$: MPC
$t$: marginal tax rate
$T_0$: autonomous (lump-sum) taxes

Use it when taxes are T0 + tY. The multiplier becomes 1/(1 - b + bt), smaller than 1/(1 - b): taxes absorb part of every rise in income. Lecture 10 works through this model in detail.

The open economy

With trade, spending on domestic output is $AE \equiv C + I + G + EX - IM$ . Imports rise with income:

IM = mY

where $m$ is the marginal propensity to import (MPM).

Open-economy multiplierFormula sheet →

\frac{1}{1 - (MPC - MPM)}

$MPM$: marginal propensity to import

Use it when the economy trades. Part of each extra lira of spending goes on foreign goods, so the multiplier is smaller than in a closed economy.

What drives imports? Their composition (consumption goods, capital goods, intermediate goods and inputs), the relative price of domestic and foreign goods (the terms of trade $P_X/P_M$ ) and the exchange rate. What drives exports? Economic activity in the rest of the world and the relative price of domestic goods: exports rise when foreign output rises or when domestic prices fall relative to world prices.

Summary and review

$C = a + bY$ ; $S \equiv Y - C$ ; $MPC + MPS \equiv 1$ .
Equilibrium: $Y = AE$ , or $S = I$ (with government: $S + T = I + G$ ).
Unplanned inventory changes push output toward equilibrium.
Multipliers: $1/MPS$ for $G$ or $I$ ; $-MPC/MPS$ for $T$ ; 1 for a balanced budget.
Paradox of thrift: trying to save more lowers income, not saving.
Income taxes and imports shrink the multiplier: $1/(1 - b + bt)$ , $1/(1 - (MPC - MPM))$ .