Derivation of the LM Curve

The LM curve represents all combinations of income and interest rates at which the money market is in equilibrium, meaning money demand equals money supply.

In the money market graph, the interest rate is on the vertical axis, and real money balances are on the horizontal axis. The central bank fixes the money supply, so the supply curve is a vertical line.

The money demand curve slopes downward because when interest rates are high, people prefer to hold less money for purchases and more interest-bearing assets.

But when interest rates are low, holding money becomes less costly, so people are willing to hold more money.

Now consider the role of income on the money demanded. When income is low, people make fewer transactions and require less money for daily purchases. Money demand is so low that the equilibrium interest rate is lower.

When income increases, spending and transactions rise. As a result, money demand increases at every interest rate. Since the money supply is fixed, equilibrium is restored through a higher interest rate.

So, for each income level, there is an interest rate at which money demand equals money supply. Connecting these equilibrium points gives the upward-sloping