IS-LM Model Notes

IS-LM Model

Overview

The IS-LM model combines insights from the goods and money markets to provide a short-run view of the economy, assuming fixed prices. It serves as a general equilibrium model, illustrating the interaction between these markets and focusing on aggregate demand to determine output.

Equilibrium in the IS-LM Model

Core Concepts

  • IS-LM Model Definition: A short-run macroeconomic model where prices are fixed.
  • General Equilibrium: Achieved when all markets (goods and money) are in equilibrium.
  • Aggregate Demand Focus: The model primarily describes aggregate demand, assuming supply adjusts to meet demand at fixed prices.

The IS Curve

The IS curve represents the equilibrium in the goods market.

  • Equation: r = C0 + I0 \frac{1}{b} - \frac{c}{b}T + \frac{G}{b} - \frac{1-c}{b}Y
  • Shape: Typically linear.
  • Slope: Downward sloping.
Factors Influencing the IS Curve Slope
  • Sensitivity of Investment to Interest Rate (b):
    • High b (b \uparrow): Investment is highly sensitive to changes in the interest rate (\Delta r), making the IS curve more elastic (flatter).
    • Low b (b \downarrow): Investment is less sensitive to changes in the interest rate, making the IS curve more inelastic (steeper).

The LM Curve

The LM curve represents the equilibrium in the money market.

  • Equation: r = \frac{k}{h}Y - \frac{1}{h} \frac{M}{P}
  • Shape: Typically linear.
  • Slope: Upward sloping.
Factors Influencing the LM Curve Slope
  • Sensitivity of Money Demand to Interest Rate (h):
    • High h (h \uparrow): Money demand is highly sensitive to changes in the interest rate (\Delta r), making the LM curve more elastic (flatter).
    • Low h (h \downarrow): Money demand is less sensitive to changes in the interest rate, making the LM curve more inelastic (steeper).

IS-LM Equilibrium

  • Definition: The point where both goods and money markets are in equilibrium.
  • Uniqueness: Usually unique due to the opposing slopes of the IS and LM curves.

Equilibrium Algebra

The equilibrium is found by solving the system of equations for the IS and LM curves simultaneously.

  • IS Curve: r = C0 + I0 \frac{1}{b} - \frac{c}{b}T + \frac{G}{b} - \frac{1-c}{b}Y
  • LM Curve: r = \frac{k}{h}Y - \frac{1}{h} \frac{M}{P}

Solving these equations yields equilibrium output (Y^) and interest rate (r^):

  • Y^* = \frac{h}{bk + (1-c)h}(C0 + I0 + G - cT) + \frac{b}{bk + (1-c)h} \frac{M}{P}
  • r^* = \frac{k}{bk + (1-c)h}(C0 + I0 + G - cT) + \frac{1-c}{bk + (1-c)h} \frac{M}{P}

Stability of IS-LM Equilibrium

The IS-LM model tends to return to equilibrium naturally due to market forces.

Scenario 1: Goods Market in Equilibrium, Money Market Not
  • If the interest rate is too high, the demand for money is too low, leading to an excess supply of money (\frac{M}{P} > L(Y, r)).
  • This excess supply increases demand for bonds, pushing bond prices up (P_I \uparrow) and interest rates down (r \downarrow).
  • Lower interest rates increase investment, boosting output (Y \uparrow) until equilibrium is restored.
Scenario 2: Money Market in Equilibrium, Goods Market Not
  • If the interest rate is too low, production is too low relative to demand.
  • Output increases (Y \uparrow), raising the demand for money (L(Y, r) \uparrow).
  • This creates an excess demand for money (\frac{M}{P} < L(Y, r)), decreasing demand for bonds, reducing bond prices (P_I \downarrow), and increasing interest rates (r \uparrow).
  • Higher interest rates decrease investment (I \downarrow), eventually stabilizing the economy.

Desirable Equilibrium

  • Full Employment Equilibrium: Occurs when the number of unemployed workers equals the number of job vacancies, indicating all unemployment is voluntary.
  • Natural Level of Output: The output level associated with full employment.

Achieving Desirable Equilibrium

  • The IS-LM equilibrium may not always result in full employment due to market rigidities and failures.
  • This justifies government intervention to steer the economy toward a more desirable outcome.

Fiscal Policy in the IS-LM Model

Impact on Output

  • Government Spending (G): An increase in G increases Y: \frac{\partial Y}{\partial G} = \frac{h}{bk + (1-c)h} > 0
  • Taxes (T): An increase in T decreases Y: \frac{\partial Y}{\partial T} = -\frac{ch}{bk + (1-c)h} < 0

Impact on Interest Rate

  • Government Spending (G): An increase in G increases r: \frac{\partial r}{\partial G} = \frac{k}{bk + (1-c)h} > 0
  • Taxes (T): An increase in T decreases r: \frac{\partial r}{\partial T} = -\frac{ck}{bk + (1-c)h} < 0

Multiplier Effect

The multiplier effect in the IS-LM model is smaller than in the Keynesian Cross model because it accounts for the interaction between the goods and money markets: \frac{h}{bk + (1-c)h} < \frac{1}{1-c}

Crowding Out

  • Definition: An increase in G increases the interest rate, which reduces private investment.
  • Elasticity Impact:
    • More inelastic (steeper) LM curve: Higher crowding out, as a greater change in r is required to restore equilibrium in the money market.
    • More elastic (flatter) IS curve: Higher crowding out, as investment is more responsive to changes in r.

Effectiveness of Fiscal Policy

The effectiveness of fiscal policy depends on the degree of crowding out.

  • Less crowding out means more effective government spending.
    • More elastic LM curve: More effective government spending.
    • More inelastic IS curve: More effective government spending.

Monetary Policy in the IS-LM Model

Impact on Output and Interest Rate

  • Money Supply (M): An increase in M increases Y and decreases r:
    • \frac{\partial Y}{\partial M} = \frac{b}{bk + (1-c)h} \frac{1}{P} > 0
    • \frac{\partial r}{\partial M} = -\frac{1-c}{bk + (1-c)h} \frac{1}{P} < 0

Effectiveness of Monetary Policy

The effectiveness of monetary policy also depends on the degree of crowding out.

  • For a given IS curve slope, monetary policy is more effective the more inelastic (steeper) the LM curve.
  • For a given LM curve slope, monetary policy is more effective the more elastic (flatter) the IS curve.

Policy Mix

Fiscal and monetary policies can be used in tandem to achieve specific economic goals.

Liquidity Trap

Key Elements

  • Interest Rates Near Zero: Interest rates are close to or equal to 0.
  • Horizontal LM Curve: LM curve is perfectly elastic.
  • Investment and Animal Spirits: Investment depends more on expectations (animal spirits) than on interest rates, making the IS curve steep and inelastic.

Consequences

  • Ineffective Monetary Policy: Conventional monetary policy is unable to stimulate the economy.
  • Non-Zero Interest Rate Floor: Monetary authority can't reduce interest rates below zero.
  • Idle Cash: Increases in the money supply result in idle cash.
  • Full Employment Challenges: Full-employment equilibrium may not be achievable.

Escape from Liquidity Trap

  • Expansionary fiscal policy is key.
  • Government spending increases demand and income without crowding out, leading to a multiplier effect.
  • Improved consumer optimism can boost investment.

Quantitative Easing

  • Quantitative easing (QE) which refers to the central bank purchasing longer-term securities to increase the money supply and encourage lending and investment, can be effective under certain assumptions.
  • If an increase in M leads to an increase in expected inflation (\pi^e), the real interest rate (r) can become negative, stimulating investment and output.
  • This assumes that investment depends on the real interest rate, I(r), and r = i - \pi^e (Fisher equation).

Classical-Monetarist Case

  • Money demand depends mostly on transaction motives (high k, low h).
  • LM curve is inelastic, while the IS curve is elastic/flat.
  • Monetary policy is effective, while fiscal policy is ineffective due to high crowding out.