How to derive LM Curve from Money Demand and Interest Rate Equilibrium

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Why the LM Curve Is Upward Sloping
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The LM curve is a core concept in macroeconomics tutoring, linking the money market to interest rates and output. Students often struggle because the LM curve is not a demand curve or supply curve — it is an equilibrium locus. This page explains why the LM curve slopes upward and how it fits into the IS–LM and AS–AD frameworks.

The LM curve slopes upward because higher income increases money demand, which raises interest rates to restore equilibrium in the money market.

Formally, the LM curve represents combinations of income \(Y\) and interest rates \(i\) such that:

Money Demand = Money Supply

Why does the LM curve slope upward? Because when income rises, people make more transactions and demand more money. With a fixed money supply, the only way to restore equilibrium is for interest rates to rise, reducing speculative money demand. This positive relationship between income and interest rates generates the upward slope.

  1. Start with the money market equilibrium condition. \(M/P = L(Y, i)\).
  2. Hold real money supply constant. The central bank sets \(M\); price level \(P\) is fixed in the short run.
  3. Increase income \(Y\). Higher income → more transactions → higher money demand.
  4. Money demand now exceeds money supply. The money market is out of equilibrium.
  5. Interest rates must rise. Higher interest rates reduce speculative money demand, restoring equilibrium.
  6. Plot the relationship. Higher \(Y\) → higher \(i\) → upward‑sloping LM curve.

Suppose real money supply is fixed at 100. Money demand is given by:

\(L(Y, i) = 0.5Y – 10i\)

If income increases from 200 to 240:

  • At \(Y = 200\): \(100 = 0.5(200) – 10i \Rightarrow i = 5\%\)
  • At \(Y = 240\): \(100 = 0.5(240) – 10i \Rightarrow i = 7\%\)

Income rises → interest rates rise → LM slopes upward.

  • Thinking the LM curve is a money demand curve (it is not).
  • Ignoring the role of speculative money demand.
  • Assuming the LM curve is vertical (only true in liquidity trap limits).
  • Confusing shifts of the LM curve with movements along it.

The LM curve is essential for understanding monetary policy, interest rate dynamics, and macroeconomic equilibrium. It interacts with the IS curve to determine short‑run output and appears in nearly every graduate macroeconomics exam.

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