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Step-by-step mathematical derivation with typeset equations and expandable detail.
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The 45-degree line and aggregate expenditure
The Keynesian cross is the simplest closed-economy, fixed-price model of output determination. It operates entirely in the goods market and asks: at what level of output does planned aggregate expenditure (AE) equal actual output? The 45-degree line in space represents the identity , the set of points where planned spending exactly absorbs production. Any point off this line implies unplanned inventory accumulation (if ) or depletion (if ), triggering firms to adjust production toward the line.
Aggregate expenditure is decomposed into autonomous spending and an induced component. Consumption follows the Keynesian consumption function , where is the marginal propensity to consume and is disposable income. Investment , government spending , and taxes are treated as exogenous. Adding these components yields total planned expenditure as a linear function of output with slope less than one.
Aggregate expenditure is the sum of consumption, planned investment, and government purchases.
The Keynesian consumption function: autonomous consumption plus the marginal propensity to consume times disposable income.
Planned expenditure as a linear function of output. The bracketed term is total autonomous expenditure; is the slope.
Goods-market equilibrium
Equilibrium requires that planned aggregate expenditure equal actual output: . Substituting the expenditure function into this condition produces a single equation in that can be solved algebraically. Setting as shorthand for total autonomous expenditure, the equilibrium condition becomes . Collecting terms on and dividing through yields the equilibrium output level .
Graphically, the equilibrium is the intersection of the upward-sloping AE line (slope ) and the 45-degree line (slope 1). Because the AE line is flatter than the 45-degree line, they cross exactly once, guaranteeing a unique equilibrium. If output is below , planned spending exceeds production, inventories fall, and firms expand output. If output is above , production outpaces spending, inventories pile up, and firms cut back. The economy thus converges to through the inventory-adjustment mechanism.
The goods-market equilibrium condition: output equals autonomous spending plus induced consumption.
Equilibrium output: autonomous expenditure scaled up by the Keynesian multiplier.
The expenditure multiplier
The Keynesian multiplier captures the total output response to a one-unit increase in autonomous spending. When the government raises by one dollar, output initially rises by one dollar. That extra income induces dollars of new consumption, which becomes someone else's income, generating additional dollars of spending, and so on. The infinite geometric series converges to because . With a marginal propensity to consume of 0.8, for example, the multiplier is 5: each dollar of new government spending raises equilibrium output by five dollars.
Comparative statics follow directly. An increase in or raises and shifts the AE line upward, moving the equilibrium right along the 45-degree line by the multiplier times the shock. A tax increase lowers by (not the full amount, because only the consumed portion of the tax matters), so the tax multiplier is , smaller in absolute value than the spending multiplier. This asymmetry is the foundation of the balanced-budget multiplier result: a simultaneous equal increase in and still raises output by exactly one dollar.
The spending multiplier: the sum of successive rounds of induced consumption.
The tax multiplier: smaller in absolute value than the spending multiplier because taxes affect spending only through the MPC.
Balanced-budget multiplier: equal increases in spending and taxes raise output by exactly the size of the fiscal change.