Derivation and Equilibrium in Hicks-Hansen IS-LM Framework

Introduction

The Hicks-Hansen IS-LM model synthesises Classical and Keynesian macroeconomic theory by establishing the simultaneous equilibrium of the goods market and the money market. Proposed originally by John Hicks in 1937 and expanded by Alvin Hansen, the framework determines the unique pair of real output (Y) and the nominal interest rate (r) at which both real expenditure and portfolio choices are jointly balanced.

1. Derivation of the IS Curve (Goods Market Equilibrium)

The IS schedule illustrates combinations of interest rates and real output at which aggregate expenditure equals aggregate output, clearing the goods market.

• Keynesian Cross Foundations: Aggregate demand is specified as AD = C + I + G, where consumption is C = c_0 + c(1 - t)Y, investment is interest-sensitive I = I_0 - br, and government purchases are autonomous G = G_0.
• Algebraic Derivation: In equilibrium, Y = AD. Defining autonomous spending as A = c_0 + I_0 + G_0 and the multiplier as α = 1 / [1 - c(1 - t)], the equilibrium condition is:
  Y = α(A - br)
  Rearranging in inverse form: r = (A / b) - (Y / (αb)).
• Slope and Intuition: An increase in the interest rate raises the cost of capital, dampening planned investment and triggering a multiplier contraction in aggregate demand and equilibrium output. Thus, the IS schedule is downward-sloping.

2. Derivation of the LM Curve (Money Market Equilibrium)

The LM schedule traces combinations of interest rates and real output that clear the money market, equating real money demand to the exogenous real money supply.

• Liquidity Preference: Demand for real balances comprises transaction/precautionary motives (positively dependent on income) and speculative motives (inversely dependent on interest rate): (M/P)^d = kY - hr, where k and h denote sensitivities.
• Algebraic Derivation: Equating real money demand to fixed real money supply M/P yields:
  M/P = kY - hr
  Solving for the interest rate: r = (1 / h)[kY - (M / P)].
• Slope and Intuition: An expansion in real income increases the transaction demand for liquidity. Given a fixed money supply, the interest rate must rise to induce wealth-holders to hold fewer speculative cash balances and restore portfolio balance. Hence, the LM curve slopes upward.

3. Simultaneous General Equilibrium and Policy Multipliers

Simultaneous equilibrium occurs at the intersection of the IS and LM schedules (IS ∩ LM), fixing the unique equilibrium vector (Y*, r*).

• Equilibrium Output: Substituting the LM equation into the IS schedule yields equilibrium output:
  Y* = [αA / (1 + αbk/h)] + [(αb/h) / (1 + αbk/h)] · (M/P)
• Monetary Policy Multiplier: The responsiveness of output to real money supply changes is:
  ∂Y/∂(M/P) = (αb/h) / [1 + (αbk/h)]
• Fiscal Policy Multiplier: The responsiveness of output to autonomous spending changes is:
  ∂Y/∂A = α / [1 + (αbk/h)]

4. Disequilibrium Dynamics

Disequilibrium states adjust back toward (Y*, r*) through differential adjustment speeds: asset markets clear instantaneously via interest rate adjustments, while goods markets clear gradually through inventory and production adjustments.

• Quadrant I (Above IS, Left of LM): Excess supply of goods (ESG → Y↓) and excess supply of money (ESM → r↓).
• Quadrant II (Below IS, Left of LM): Excess demand for goods (EDG → Y↑) and excess supply of money (ESM → r↓).
• Quadrant III (Below IS, Right of LM): Excess demand for goods (EDG → Y↑) and excess demand for money (EDM → r↑).
• Quadrant IV (Above IS, Right of LM): Excess supply of goods (ESG → Y↓) and excess demand for money (EDM → r↑).

Conclusion

The Hicks-Hansen IS-LM framework remains foundational for analyzing short-run macroeconomic fluctuations and policy mixes. By capturing the interaction between fiscal interventions shifting the IS curve and monetary operations shifting the LM curve, it offers vital theoretical grounding for macroeconomic coordination in both advanced and emerging market economies.