Basic Keynesian Model

The simplest Keynesian model strips the macroeconomy down to its two fundamental private sectors: households and business firms. Households consume and save; firms invest and produce. There is no government, foreign trade, or monetary sector. The model's core insight is that, at given prices, output and employment are determined entirely by the level of aggregate demand.

In the two-sector model the aggregate supply schedule is taken to be perfectly elastic up to the full-employment output level. That is, firms are willing to supply any amount of output demanded without a rise in the general price level—the economy operates with substantial spare capacity and constant nominal production costs. Consequently, the price level is treated as exogenously fixed. Under this assumption, changes in nominal income are changes in real income, and the determination of output can be studied using the aggregate demand function alone. This is the hallmark of the simple Keynesian model: the supply side is passive, so attention focuses on the construction and behavior of aggregate demand.

Aggregate demand comprises private consumption and private investment. Keynes posited a "fundamental psychological law" according to which current real consumption expenditure is a stable, increasing function of current real disposable income. In the two-sector economy, taxes and transfer payments are absent, so disposable income equals total real income. The simplest linear representation is C = a + bY, where a is autonomous consumption (the amount consumed when income is zero, financed out of past saving or credit) and b is the marginal propensity to consume (MPC). The complement 1-b is the marginal propensity to save (MPS). The average propensity to consume, C/Y = a/Y + b, declines as income rises, a property that Keynes held to be empirically important.

Investment expenditure is taken as autonomous (exogenous). Firms base their planned investment on long-run profit expectations, animal spirits, and the state of technology—none of which are immediately tied to current income. Hence investment is an exogenously fixed value. The goods market is in equilibrium when actual output equals aggregate demand: Y = C + I. Substituting the behavioral functions and solving yields the equilibrium level of output as the product of the expenditure multiplier and the sum of autonomous spending components. The multiplier exceeds unity: a one-unit increase in autonomous spending raises equilibrium income by more than one unit. The multiplier captures the successive rounds of induced consumption that follow from an initial injection of spending.

Graphically, equilibrium is depicted by the intersection of the aggregate demand schedule—a straight line with intercept equal to autonomous spending and slope equal to the MPC—and a 45-degree line from the origin. At any output below equilibrium, aggregate demand exceeds output, inventories are drawn down unintentionally, and firms expand production. At any output above equilibrium, aggregate demand falls short of output, unintended inventory accumulation occurs, and output contracts. Thus equilibrium is a stable, demand-determined outcome.

The same result can be derived through the equality of planned saving and planned investment. Saving, the part of income not consumed, is S = -a + (1-b)Y. Equilibrium requires that leakages equal injections. In the two-sector model, saving is the only leakage and investment the only injection, so S = I. Graphically, the saving schedule (slope equal to MPS) and the horizontal investment line intersect at equilibrium. In disequilibrium actual investment differs from planned investment because of unintended changes in inventories. When output exceeds equilibrium, households save more than firms plan to invest, so goods pile up as unintended inventory investment is positive; when output falls short of equilibrium, saving is less than investment and unintended inventory investment is negative. Equilibrium is the unique income level at which unintended inventory investment is zero and therefore actual investment matches plans.

One striking implication of the two-sector model is the paradox of thrift. Suppose households attempt to increase saving—for instance, by decreasing autonomous consumption (a downward shift of the consumption function). The saving schedule shifts upward. With autonomous investment unchanged, the initial equilibrium is disturbed: at the former income level, saving exceeds investment, causing unintended inventory accumulation and a contraction of output. The new equilibrium income is lower, and, because saving must equal the still-fixed investment, the level of saving in the new equilibrium is exactly the same as before. The attempt to save more ex ante has been completely frustrated by the fall in income, leaving ex post saving unchanged. This paradox disappears once investment is allowed to respond to changes in income or the interest rate, but in the simple model it underscores the fallacy of composition: what is prudent for an individual household may be contractionary for the economy as a whole.

Government adds two new autonomous spending flows—purchases of goods and services G, and net taxes T—and a new leakage channel. With lump-sum taxation, taxes are exogenous. Disposable income is Y - T, and consumption becomes C = a + b(Y - T). With investment and government spending as autonomous components, aggregate expenditure is the sum of these components plus induced consumption. The equilibrium output is the product of the multiplier and the sum of autonomous spending minus the tax-induced reduction in consumption.

The multiplier for government spending is identical to the simple expenditure multiplier: 1/(1-b). By contrast, a lump-sum tax change gives the tax multiplier, which is -b/(1-b). Its absolute value is smaller than the spending multiplier by a factor of b: a tax cut raises disposable income and hence consumption, but the initial injection is only b times the tax change, while government spending enters directly.

A particularly instructive result is the balanced budget multiplier. If the government increases spending and taxes by the same amount, the net change in income is exactly equal to the change in spending. The multiplier equals exactly one. The intuition is straightforward: the initial government spending adds fully to aggregate demand, while the tax hike reduces consumption by only b times the tax change; the net autonomous boost is (1-b) times the balanced budget change, which, when multiplied by 1/(1-b), yields exactly the balanced budget change. This result implies that a balanced-budget fiscal expansion is not neutral—it raises output by the amount of the spending increase.

A more realistic tax system ties net taxes to income. Let taxes be a proportion t of income, where t is the marginal tax rate. Disposable income is (1-t)Y, so consumption becomes C = a + b(1-t)Y. The multiplier is now 1/[1-b(1-t)]. Because the marginal tax rate is positive, the multiplier is smaller than in the lump-sum case. The leakage now includes both saving and induced tax payments; a fraction t of each additional dollar of income is diverted to the government before it can be spent. This provides the first glimpse of automatic stabilizers: a higher marginal tax rate reduces the multiplier and thereby dampens the output response to demand shocks.

If the tax function includes a lump-sum component, the multiplier for government spending or investment is 1/[1-b(1-t)], while the multiplier for lump-sum taxes is -b/[1-b(1-t)]. Transfer payments that are independent of income enter through disposable income and have a multiplier of +b/[1-b(1-t)], symmetric to the lump-sum tax multiplier.

The open economy adds exports and imports to aggregate expenditure. Exports are determined primarily by foreign income and are treated as autonomous from the perspective of the domestic economy. Imports, however, rise with domestic income. The simplest formulation is M = M0 + mY, where M0 is autonomous imports and m is the marginal propensity to import.

With a proportional tax system, consumption remains C = a + b(1-t)Y. Aggregate expenditure is the sum of consumption, autonomous investment, government spending, autonomous exports, minus autonomous imports and induced imports. Setting output equal to aggregate expenditure and solving yields the open-economy equilibrium output. The open-economy multiplier is 1/[1-b(1-t)+m]. Comparing this with the closed-economy multiplier, we see that m enters the denominator as an additional leakage. A larger marginal propensity to import shrinks the multiplier, because part of each round of induced spending leaks abroad. In the extreme case of a small open economy with very high m, the multiplier can be close to unity, implying that domestic fiscal expansion largely benefits foreign producers rather than raising domestic output.

If we abstract from government, the foreign trade multiplier reduces to 1/(1-b+m)—the classic result of early open-economy Keynesian models. The twin leakages are saving and imports, and the multiplier is smaller the larger either the MPS or the marginal propensity to import.

The multiplier effect is one of the central insights of Keynesian macroeconomics: a change in autonomous expenditure—spending that does not depend on current income—leads to a more than proportional change in equilibrium output. The size of this effect depends critically on the structure of the economy, specifically on the number and nature of leakages from the circular flow of income.

The systematic reduction of the multiplier as we move from the two-sector to the four-sector model underscores a general principle: the multiplier is the reciprocal of the marginal leakage rate. In a closed economy with lump-sum taxes, leakages equal the marginal propensity to save. With proportional taxes, the leakage rate rises to s + bt, where s is the MPS and t is the marginal tax rate. In the open economy, the leakage rate is s + bt + m, where m is the marginal propensity to import. Each additional leakage channel reduces the multiplier and attenuates the potency of autonomous expenditure changes.

The multiplier process can be understood as an infinite geometric series. An initial injection of autonomous spending raises income by that amount. A fraction equal to the MPC times one minus the tax rate of this additional income is spent on consumption (in the closed economy), generating further income. This second-round income increment induces a third round of consumption, and so on. The total change in income is the sum of this infinite series, which converges to the initial injection divided by the leakage rate.

Three comparative-static insights unify the models. First, the multiplier is a decreasing function of the sum of marginal leakages. Economies with high saving rates, high marginal tax rates, or high propensities to import exhibit small multipliers and hence limited scope for demand management through fiscal expansion. Second, the composition of leakages matters for policy design. A cut in the marginal tax rate raises the multiplier and amplifies the response to any subsequent shock, whereas a cut in the lump-sum component shifts the intercept of the aggregate-expenditure schedule without altering its slope. Policymakers thus face a trade-off: automatic stabilizers (a high marginal tax rate) reduce volatility but also reduce the efficacy of discretionary fiscal interventions. Third, in the four-sector model, fiscal expansion generates a twin-deficit dynamic. An increase in government spending raises output, which raises imports, worsening the trade balance.

The multiplier analysis has profound implications for fiscal policy. The magnitude of the fiscal multiplier determines whether government spending can effectively stimulate the economy during recessions. When the economy operates below full employment and the price level is fixed, the multiplier process works through real quantities. However, as the economy approaches full employment, the aggregate supply curve becomes upward-sloping, and increases in aggregate demand translate into price inflation rather than real output growth. The simple multiplier analysis also abstracts from monetary feedback: in the IS-LM framework, expansionary fiscal policy raises the interest rate, which crowds out private investment, reducing the effective multiplier below the simple Keynesian value. The extent of this crowding out depends on the interest sensitivity of money demand and investment, parameters that vary across economies and over time.