Keynesian System with Money, Income, and Interest

A central innovation of Keynes's General Theory was the integration of money and finance into the determination of aggregate output and employment. Classical and neoclassical economics had largely accepted the neutrality of money in the long run: changes in the money supply were thought to alter only the general price level, leaving real magnitudes—output, employment, the real interest rate—unaffected. Keynes rejected this dichotomy. He argued that money plays a vital role in a world of fundamental uncertainty, where the future is unknowable and economic agents form fragile expectations. In such a world, money is not merely a medium of exchange; it is a store of value and a shield against the anxiety of an uncertain tomorrow.

In Keynes's system, the demand for money—or liquidity preference—arises from three motives. The transactions motive: money is needed to bridge the gap between the receipt of income and making expenditures. The volume of such planned spending is closely related to the level of income, so the transactions demand is written as an increasing function of income. The precautionary motive: individuals hold money to meet unforeseen contingencies—sudden illness, repair costs, or unexpected investment opportunities. Keynes treated this motive as also depending positively on income. The speculative motive: this is the distinctively Keynesian addition. Wealth can be held in the form of money (which pays no interest but is perfectly liquid and free of capital risk) or in long-term bonds (which pay a fixed coupon but whose price varies inversely with the interest rate). Each individual forms a subjective normal rate of interest—a long-run average to which the current rate is expected to revert. If the current rate is below this normal rate, the individual expects bond prices to fall. If the expected capital loss exceeds the interest earned, holding money dominates bonds. Hence, as the interest rate falls, more wealth-holders will regard bonds as overpriced and switch into money. At a sufficiently low rate, everyone becomes convinced that the rate can only rise, and the speculative demand for money becomes virtually infinite. This is the famous liquidity trap, where the demand for money is perfectly interest-elastic.

Aggregating these motives yields the total real money demand as a function of income and the interest rate. The money supply is set exogenously by the monetary authority. Equilibrium in the money market requires that the real money supply equals real money demand. This equation is the LM relation, which defines combinations of income and the interest rate that clear the money market.

The IS curve is a fundamental construct in the Keynesian model that integrates money, income, and the interest rate. It depicts all combinations of the real interest rate and real output such that the goods market is in equilibrium—that is, planned aggregate expenditure equals actual output. In a closed economy with a government sector, the equilibrium condition is Y = C + I + G.

To transform this into a curve linking the interest rate and output, we must specify the behavioral relationships that determine each component of aggregate demand. Consumption depends positively on households' disposable income (total income minus net taxes). The marginal propensity to consume lies strictly between zero and one. Investment spending is taken to be a negative function of the real interest rate. Given the expected profitability of capital projects, a higher interest rate reduces the net present value of marginal investment projects, lowering the volume of planned investment. Government purchases and net taxes are treated as exogenous fiscal policy instruments.

Substituting the behavioral equations into the equilibrium condition yields a single equation relating the interest rate and output. For any given interest rate, it determines the level of income that clears the goods market. The traditional method constructs the IS curve from the income-expenditure diagram (the "Keynesian cross"). In the upper panel, aggregate expenditure is plotted as a function of output for a fixed interest rate. The equilibrium output is found at the intersection with the 45-degree line. If the interest rate falls, investment rises, shifting the aggregate expenditure schedule upward and leading, via the multiplier process, to a higher equilibrium income. Plotting all such pairs traces a downward-sloping locus—the IS curve. The negative slope reflects the chain: a lower interest rate stimulates investment, which raises aggregate demand, which through the multiplier raises output.

The IS curve is downward sloping. The magnitude of this slope depends on two key parameters. First, the interest sensitivity of investment: a larger sensitivity means investment responds more to a change in the interest rate, causing a larger shift in aggregate demand and a larger change in equilibrium income. Thus, the IS curve is flatter. Conversely, if investment is interest-inelastic, the IS curve becomes nearly vertical. Second, the marginal propensity to consume: a higher MPC raises the multiplier, amplifying any initial change in investment. The result is a flatter IS curve. Conversely, a low MPC steepens the IS curve.

The label "IS" originates from the alternative equilibrium condition: in a closed economy with government, the goods market equilibrium is equivalent to the equality of national saving and investment. Total saving is the part of income not consumed by the private sector or absorbed by taxes net of government purchases. From the equilibrium condition, we have saving equals investment. Now, consumption (and thus saving) is a function of income, while investment is a function of the interest rate. For a given interest rate, investment is fixed; equilibrium income must adjust so that saving equals that level of investment. If the interest rate rises, investment falls; to restore equilibrium, output must fall to reduce saving. This yields the same downward-sloping locus.

The slope of the IS curve is determined by the interest sensitivity of investment and the multiplier. A larger interest sensitivity of investment flattens the IS curve: a given reduction in the interest rate elicits a larger jump in investment, which then generates a larger multiplier expansion of output. The value of this sensitivity depends on the marginal efficiency of capital schedule, the degree of competition in banking, the prevalence of credit constraints, and the sensitivity of firms' cost of capital to policy rates. In an open economy, the real exchange rate channel adds further interest sensitivity: a lower interest rate depreciates the currency, boosting net exports, thereby making the IS curve even flatter.

The multiplier amplifies the effect of any autonomous change in spending. With a proportional income tax rate and an import propensity, the multiplier becomes 1/[1-b(1-t)+m]. A higher MPC (or lower tax rate and import propensity) raises the multiplier, thereby flattening the IS curve because any initial stimulus—including one originating from an interest-rate cut—is propagated more strongly. Thus tax policy can influence not only the position but also the slope of the IS curve: a reduction in the marginal tax rate increases the multiplier and rotates the IS curve counter-clockwise.

The IS curve is drawn for given values of fiscal policy variables and the exogenous components of consumption and investment. Changes in these factors shift the entire curve. An increase in government spending raises autonomous spending, shifting the IS curve to the right by the multiplier times the change in spending. A lump-sum tax cut shifts the IS curve right by a smaller amount. An increase in autonomous consumption (due to improved consumer confidence) or autonomous investment (due to technological optimism) shifts the IS curve right.

Expectations matter through intertemporal channels. If households expect higher future income, they may increase current consumption even at unchanged current disposable income, raising effective autonomous consumption. If firms expect higher future demand, they may invest now, shifting autonomous investment. In richer models the user cost of capital depends on expected future interest rates; a shift in expectations about the path of monetary policy hence affects investment at the current interest rate, again shifting the IS curve.

Financial frictions can also shift the IS curve. Suppose investment depends on the borrowing rate, which equals the policy rate plus an external finance premium. An increase in this premium—caused by a disruption in credit markets, a rise in risk aversion, or a tightening of lending standards—reduces investment at every policy rate. This is equivalent to a leftward shift of the IS curve. The external finance premium is thus an important factor that can displace the IS curve, even if fiscal policy and autonomous private demands remain unchanged.

The LM curve is the locus of all combinations of income and the interest rate that satisfy the money market equilibrium condition for a given real money supply. To derive its slope, we totally differentiate the equilibrium condition, holding the money supply and price level constant. Because money demand rises with income and falls with the interest rate, the ratio of the income sensitivity to the interest sensitivity (with a negative sign) is strictly positive. The economic intuition is straightforward: starting from an equilibrium position, a rise in income raises the transactions demand for money. For the money market to remain in equilibrium, the opportunity cost of holding money must rise sufficiently to induce households and firms to economize on their cash holdings. That rise in the interest rate reduces speculative balances and restores equality between money supply and demand.

The steeper the LM curve, the larger the required increase in the interest rate for a given change in income, a situation that arises when the income sensitivity of money demand is relatively large or the interest elasticity of money demand is small. Two special cases deserve mention. When the interest elasticity of money demand approaches infinity—the liquidity trap—the slope of the LM curve goes to zero and the LM curve becomes horizontal. In this classical Keynesian case, at very low interest rates the public is willing to hold any amount of money because the yield on bonds has fallen to the point where capital gains are not expected; the opportunity cost of holding money is negligible. Conversely, if money demand is completely interest-inelastic, the LM curve is vertical, corresponding to the classical quantity-theory world where velocity is constant and the interest rate plays no role in money demand.

The position of the LM curve depends on the real money supply and on any exogenous factors that alter money demand for given income and interest rate. An increase in the nominal money supply, given a fixed price level, raises real balances and shifts the LM curve to the right. A decline in the price level has the same effect. A decrease in money demand—say due to the widespread adoption of credit cards—shifts the LM curve rightward. In contrast, a reduction in the money supply or a rise in the price level shifts the LM curve to the left.

The steepness of the LM curve is governed entirely by the income and interest sensitivities of money demand, which reflect the technological and institutional features of the payments system, the degree of financial sophistication, and the asset-market behavior of economic agents. If the interest elasticity of money demand is large, a small rise in the interest rate induces a substantial shift out of money into bonds, so a given increase in income—and therefore in transactions demand—can be accommodated with only a modest interest-rate increase. Consequently, the LM curve is relatively flat. Conversely, when money demand is highly interest-insensitive, the same income-induced increase in money demand forces a large rise in the interest rate to restore equilibrium; the LM curve is steep.

A larger income elasticity of money demand means that money demand is more sensitive to income fluctuations. For any given interest elasticity, a higher income elasticity makes the LM steeper because a given change in income provokes a larger excess demand for money, necessitating a larger compensating movement in the interest rate. Empirically, the income elasticity of money demand is often close to unity in the long run, but it can vary with financial innovation.

The LM curve shifts when, for a given pair of income and interest rate, either the real money supply changes or the money demand function itself alters. An increase in the nominal money supply, implemented by the central bank through open-market purchases of government securities, raises the real money stock at a given price level. At the initial equilibrium, real balances exceed demand; the interest rate must fall to restore equilibrium. Graphically, the LM curve shifts rightward. A contractionary policy shifts LM left.

A rise in the price level reduces real money balances and shifts the LM curve to the left. Deflation has the opposite effect. Exogenous shifts in money demand also matter: for given income and interest rate, any factor that increases the demand for real balances shifts LM to the left, because it requires a higher interest rate to offset the increased desire to hold money. A rise in real wealth—whether due to a stock-market boom, housing appreciation, or a positive shock to permanent income—tends to raise the demand for money as a store of value, shifting LM left. Expectations operate through several channels. A widely anticipated rise in future interest rates increases the speculative demand for money today, shifting LM left. Financial innovation—the introduction of credit cards, electronic payments, and sweep accounts—reduces transactions demand for conventional money aggregates, effectively shifting LM right.

The IS-LM model formalizes the simultaneous determination of real income and the interest rate by the interaction of the goods market and the money market. A general short-run equilibrium is a pair of income and interest rate that lies on both curves, clearing both markets simultaneously.

The goods market is in flow equilibrium when aggregate output matches aggregate demand in each period; the money market is in flow equilibrium when the existing stock of money is willingly held at the prevailing interest rate. There is no requirement that stocks of capital or wealth be at their desired long-run levels. Hence the IS-LM framework is a model of short-run equilibrium; long-run equilibrium would additionally require stock equilibrium in the capital account and full employment.

The intersection of the IS and LM curves determines the unique pair of income and interest rate at which the goods and money markets clear simultaneously for a given price level and policy stance. At any point off the curves, at least one market is in disequilibrium and endogenous forces drive the system toward equilibrium. The money market adjusts quickly via interest-rate changes, while the goods market responds more slowly through changes in output. The sign pattern of the Jacobian, combined with the stability condition that the determinant is negative, guarantees that the equilibrium is a stable node or focus: trajectories spiral or converge monotonically toward the intersection.

The comparative-static multipliers, derived from the Jacobian of the system, quantify how monetary expansion lowers the interest rate, stimulates investment, and raises income through the Keynesian transmission mechanism. The potency of this mechanism is governed by the interest sensitivities of investment and money demand, giving rise to the liquidity trap, classical, and intermediate regimes. Because the IS-LM equilibrium is a flow equilibrium, its validity is intrinsically short-run; long-run analysis must embed the model within a framework that tracks the evolution of capital, debt, and money stocks.