National Income Determination

National-income determination is the process by which the equilibrium level of national income (real GDP) is established in an economy. It is the analytical destination of this whole course: the circular flow gave us the accounting, aggregate demand and supply gave us the framework, the multiplier gave us the amplifier — and here they combine to answer the central question of Keynesian macroeconomics, at what level of output does the economy come to rest, and is it full employment? Two equivalent tools deliver the answer. The Keynesian cross (the 45-degree-line model, popularised by Paul Samuelson, 1948, as a teaching simplification of Keynes's 1936 General Theory) shows equilibrium where planned spending equals output; the injections-withdrawals (J = W) approach shows the same equilibrium from the circular-flow side. The model's revolutionary implication — that equilibrium can sit below full employment, leaving a deflationary gap that the economy will not close on its own — is the intellectual foundation of the entire case for active fiscal policy.

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Spec Mapping

This lesson maps to AQA 7136 section 4.2.2 — How the macroeconomy works: the determination of equilibrium national income, the inflationary and deflationary (output) gaps, drawing together the multiplier (4.2.2), and feeding 4.2.3 (output gaps and inflation) and 4.2.5 (fiscal policy). It is examined in Paper 2 (National and international economy) and is synoptic with Paper 3. All four assessment objectives apply: AO1 for the equilibrium conditions (AE = Y and J = W) and the definitions of the gaps; AO2 for applying the gap analysis and the gap/multiplier calculation to UK episodes; AO3 for the chain from disequilibrium through the adjustment mechanism to a new equilibrium, and through the multiplier to the size of injection needed to close a gap; and AO4 for evaluating whether the economy self-corrects (classical) or stays stuck (Keynesian) and the limits of the model.

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The Keynesian Cross Model

The 45-degree line and the AE function

The 45-degree line plots every point at which planned aggregate expenditure (AE) equals actual output (Y) — it is the equilibrium locus, because the economy is in balance only when what firms produce is exactly what people plan to spend. The aggregate expenditure function plots planned spending at each level of national income:

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

Using the Keynesian consumption function $C = a + b Y_d$C=a+bYd​ (where $a$a is autonomous consumption, $b$b the MPC and $Y_d = Y - T$Yd​=Y−T disposable income), and treating $I$I, $G$G and $X$X as autonomous in the simple model:

$AE = (a + bY_d) + I + G + (X - M)$AE=(a+bYd​)+I+G+(X−M)

Equilibrium income is where the AE line cuts the 45-degree line.

Feature of AE	Explanation
Intercept	Total autonomous spending ($a + I + G + X$a+I+G+X minus autonomous imports) — planned spending even when income is zero
Slope	The marginal propensity to spend domestically out of national income. In a closed economy this is the MPC; in an open economy with tax it is $MPC(1-t) - MPM$MPC(1−t)−MPM, which is flatter than the 45-degree line
Position	A rise in any autonomous component (G, I, X, autonomous C) shifts AE up; a rise in autonomous T or M shifts it down

Two features of the AE line carry the whole analysis and deserve unpacking. The intercept is positive because some spending happens even at zero income: households run down savings or borrow to maintain a minimum standard of living (autonomous consumption $a$a), and government spending, investment and exports do not vanish just because national income is low. This autonomous spending is the floor beneath aggregate expenditure. The slope — flatter than the 45-degree line — reflects induced spending: as income rises, consumption rises, but by less than the rise in income, because some of each extra pound leaks into saving, tax and imports. The slope is therefore the marginal propensity to spend domestically out of national income, which is exactly $1 - MPW$1−MPW (one minus the leakage rate). This is the crucial link to the multiplier: the flatter the AE line (the larger the leakages), the less the equilibrium income moves when AE shifts, which is just the geometric face of a smaller multiplier. A steep AE line (small leakages) means a large multiplier and a big swing in equilibrium income for any given shift; a flat AE line (large leakages) means a small multiplier and a muted response. Seeing the multiplier in the geometry of the diagram — as the ratio of the change in equilibrium income to the vertical shift of AE — is what distinguishes a candidate who understands the model from one who has merely memorised it.

How equilibrium is reached: the inventory mechanism

The economy is pushed to equilibrium by unplanned changes in inventories (stocks).

Condition	What it means	Adjustment
AE = Y	Planned spending equals output	Firms sell exactly what they make — no change
AE > Y	Planned spending exceeds output	Inventories fall unexpectedly → firms raise production → Y rises towards equilibrium
AE < Y	Output exceeds planned spending	Inventories pile up unexpectedly → firms cut production → Y falls towards equilibrium

The inventory mechanism is worth dwelling on because it is how the model actually works and because it has a clear real-world counterpart. Firms cannot observe "aggregate expenditure" directly; what they observe is their order books and stock levels. When planned spending exceeds output, the first symptom is that stocks fall faster than expected — shelves empty, order backlogs build — and firms read this, correctly, as a signal to raise production and hire. When output exceeds planned spending, the symptom is unsold inventory piling up in warehouses, and firms cut production and shed labour. The change in unplanned inventories is thus the economy's thermostat: it is the signal that drives output towards the level at which planned spending is exactly met. This is also why stockbuilding is a closely watched component of the national accounts — a sharp swing in inventories often signals a turning point in the cycle, as firms over- or under-estimate demand and then correct. Crucially, the adjustment is in quantities (output and employment), not prices: the Keynesian cross holds the price level fixed and lets output do all the adjusting, which is precisely its defining assumption and its central limitation.

Exam Tip: Always explain equilibrium dynamically via the inventory mechanism — "if AE > Y, stocks run down unexpectedly, so firms expand output until AE = Y". Stating only the static condition AE = Y misses the AO3 marks for the process.

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The Injections-Withdrawals (J = W) Approach

The same equilibrium can be found from the circular-flow side. National income is in equilibrium when total injections equal total withdrawals:

$J = W \quad\Longleftrightarrow\quad I + G + X = S + T + M$J=W⟺I+G+X=S+T+M

This is equivalent to AE = Y — they are two views of the one equilibrium. Withdrawals rise with income (people save more, pay more tax and import more as income rises), while injections are largely autonomous, so the W line slopes up and the J line is roughly flat; they cross at equilibrium income.

Why are the two conditions equivalent? Start from the expenditure identity. National income (output) is either spent on consumption or withdrawn from the flow: $Y = C + S + T + M$Y=C+S+T+M. Planned expenditure on that output is $AE = C + I + G + X$AE=C+I+G+X (treating imports as a leakage from the spending side). Setting $AE = Y$AE=Y for equilibrium and cancelling the common $C$C on both sides leaves $I + G + X = S + T + M$I+G+X=S+T+M — that is, $J = W$J=W. So the two conditions are not merely consistent; they are algebraically the same statement, viewed once from the expenditure side (planned spending equals output) and once from the circular-flow side (injections into the flow equal leakages out of it). This is reassuring rather than redundant: it means whichever approach a question invites, you can cross-check the answer with the other, and you can explain equilibrium in whichever language the data favour. The slope difference between the lines also mirrors the AE diagram: withdrawals rise with income because saving, tax and imports are all induced by income, so the W line's slope is exactly the MPW — the same leakage rate that determines the slope of the AE line and the size of the multiplier.

Condition	Disequilibrium	Adjustment
J > W	Injections exceed withdrawals	Income rises → saving, tax and imports rise → W rises until J = W
J < W	Withdrawals exceed injections	Income falls → saving, tax and imports fall → W falls until J = W
J = W	Equilibrium	No tendency to change

Exam Tip: AE = Y and J = W give the identical equilibrium — they are not rival models but two angles on the same result. Being able to switch between them, and to say why they coincide (planned spending equals output exactly when the leakages from the flow are matched by the injections into it), is a quick way to show command of the material.

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The Inflationary and Deflationary Gaps

The model's most important payoff is that equilibrium income ($Y_e$Ye​) need not equal the full-employment level ($Y_{FE}$YFE​). The difference is an output gap.

The deflationary (recessionary) gap

A deflationary gap exists when equilibrium income is below full employment — planned spending is too low to employ everyone.

Feature	Explanation
Definition	The amount by which planned expenditure falls short, at $Y_{FE}$YFE​, of the level needed for full employment
Cause	Deficient aggregate demand — households, firms and/or government are not spending enough
Symptoms	Demand-deficient (cyclical) unemployment, spare capacity, low inflation or deflation
Keynesian cure	Raise injections (higher G, lower T, lower interest rates) to shift AE up and close the gap
Multiplier point	The required injection is smaller than the gap, because the multiplier amplifies it

Calculating the injection needed to close a gap

This is the model's signature calculation, and it ties straight back to the multiplier. The required injection is the gap divided by the multiplier:

$\text{Required injection} = \frac{\text{Output gap}}{k}$Required injection=kOutput gap​

Illustratively, if the deflationary gap is £50bn and the multiplier is $k = 2$k=2:

$\text{Required injection} = \frac{£50\text{bn}}{2} = £25\text{bn}$Required injection=2£50bn​=£25bn

A £25bn rise in autonomous spending raises equilibrium income by $£25\text{bn} \times 2 = £50\text{bn}$£25bn×2=£50bn, exactly closing the gap. Note the elegance: you do not need to inject the whole £50bn, because the multiplier does the rest of the work.

This calculation is the single most examined numerical skill in this topic, and candidates lose marks in two predictable ways. The first error is to inject the whole gap (£50bn here) rather than $\text{gap}/k$gap/k — this over-shoots, pushing equilibrium income past full employment and opening an inflationary gap on the other side. The second error is to confuse the output gap (a horizontal distance on the diagram, measured in £ of output) with the vertical distance between the AE line and the 45-degree line at $Y_{FE}$YFE​, which is the required injection ($\text{gap}/k$gap/k). On the Keynesian cross, the deflationary gap is the vertical shortfall of planned spending at full employment — the amount by which AE must rise to reach the 45-degree line at $Y_{FE}$YFE​ — and that vertical amount, multiplied up, closes the horizontal output gap. Keeping the two distances straight (vertical = required injection; horizontal = output gap, equal to the injection times $k$k) is exactly what the precise definitions are protecting against. A neat way to state it in an answer: "the deflationary gap, measured as the vertical shortfall of AE at $Y_{FE}$YFE​, is $\Delta J$ΔJ; the resulting rise in income is $k\,\Delta J$kΔJ, which equals the horizontal output gap."

It also matters which autonomous component does the injecting, because they are not equivalent. Raising government spending (G) injects demand directly — the full £25bn enters the flow at once. Cutting taxes raises disposable income, but households save, tax-back and import part of it, so the first-round injection is smaller than the tax cut, and the tax multiplier is correspondingly weaker than the spending multiplier. This is the logic behind the "balanced-budget multiplier" result and, more practically, behind the Keynesian preference for direct spending over tax cuts when the priority is to close a gap quickly: a pound of G works harder than a pound of tax cut, because none of the G leaks away before it enters the flow.

The inflationary gap

An inflationary gap exists when equilibrium income exceeds the full-employment level — planned spending outstrips what the economy can sustainably produce.