Deriving the LM Curve Algebraically

The Money Market and the LM Curve

The foundation of the LM curve is the equilibrium in the market for money. Equilibrium occurs when the supply of real money balances is equal to the demand for them. We can express this condition as:

Let's quickly review the two sides of this equation:

• Real Money Supply ($M^s/P$): This is the nominal money supply ($M^s$), which we assume is controlled by the central bank, divided by the overall price level (P). For the purpose of deriving a single LM curve, we treat both $M^s$ and P as fixed, or exogenous. We denote them as $\bar{M}$ and $\bar{P}$.

• Real Money Demand ($M^d/P$): This is the quantity of real balances that people wish to hold. It is often represented by the function $L(Y,r)$, where L stands for liquidity preference. The demand for money is driven by two main factors:

  1. Income (Y): A higher level of real income leads to more transactions, which requires holding more money. Thus, money demand is positively related to income.
  2. Interest Rate (r): The interest rate is the opportunity cost of holding money (which typically earns no interest) instead of other assets like bonds (which do). A higher interest rate increases the opportunity cost of holding money, causing people to demand less of it. Thus, money demand is negatively related to the interest rate.

The LM curve is the set of all points (Y, r) that satisfy the money market equilibrium condition.

Graphical Derivation

Before we proceed to the algebra, let's visualize how the LM curve is constructed. The following diagram illustrates the process.

Let's trace the logic:

1. The Left Panel (Money Market): The real money supply, $\bar{M}/\bar{P}$, is fixed and represented by the vertical line. The money demand curves, $M^d$, are downward sloping. Each money demand curve corresponds to a specific level of income (Y).
2. Initial Equilibrium: Start at an income level $Y_0$. This defines the money demand curve $M^d(Y_0)$. The market clears where this curve intersects the money supply, establishing an equilibrium interest rate of $r_0$. This gives us our first point on the LM curve: $(Y_0,r_0)$.
3. An Increase in Income: Now, suppose income rises to a higher level, $Y_1$. This increases the demand for money for transactions, shifting the entire money demand curve to the right, to $M^d(Y_1)$.
4. New Equilibrium: With a fixed money supply, the increased demand for money creates a temporary shortage. To restore equilibrium, the interest rate must rise to $r_1$. This higher interest rate reduces the quantity of money demanded (by increasing its opportunity cost), bringing it back in line with the fixed supply. This gives us a second point: $(Y_1,r_1)$.
5. The Right Panel (LM Curve): By plotting the equilibrium combinations of income and interest rates—$(Y_0,r_0)$, $(Y_1,r_1)$, and so on—we trace out the upward-sloping LM curve.

The Algebraic Derivation

Now we will formalize this logic. Your preference for mathematical rigor will be well-served here. We begin with the money market equilibrium condition and a specific functional form for money demand.

A common linear form for the real money demand function is:

Here, $k>0$ is a parameter measuring the sensitivity of money demand to income, and $h>0$ measures the sensitivity of money demand to the interest rate.

The equilibrium condition is:

To derive the LM curve, we simply need to rearrange this equation to express the relationship between $r$ and $Y$. It's conventional to solve for $r$ as a function of $Y$, since the interest rate is plotted on the vertical axis.

1. Isolate the interest rate term: $$hr=kY-\frac{\bar{M}}{\bar{P}}$$
2. Solve for r: $$r=\frac{k}{h}Y-\frac{1}{h}\frac{\bar{M}}{\bar{P}}$$

This is the equation for the LM curve. We can see two key properties from this equation:

• Slope: The coefficient on Y, $\frac{k}{h}$, is the slope of the LM curve. Since both $k$ and $h$ are positive, the slope is positive, confirming our graphical derivation.
• Intercept: The intercept depends on the real money supply, $-\frac{1}{h}(\bar{M}/\bar{P})$. This shows mathematically that changes in the nominal money supply ($\bar{M}$) or the price level ($\bar{P}$) will shift the entire curve.

The following reading provides a formal treatment of this derivation and explores the factors influencing the slope.

A Numerical Example

Let's solidify this with a numerical problem. This process mirrors the derivation of the IS curve you completed in the last lesson.

Suppose a money market is described by the following:

• Real money demand: $L(Y,r)=0.4Y-50r$
• Nominal money supply: $\bar{M}=800$
• Price level: $\bar{P}=2$

Our goal is to find the equation of the LM curve.

1. Calculate the real money supply: $$\frac{\bar{M}}{\bar{P}}=\frac{800}{2}=400$$
2. Set real money supply equal to real money demand: $$400=0.4Y-50r$$
3. Solve for r to get the LM equation: $$50r=0.4Y-400$$ $$r=\frac{0.4}{50}Y-\frac{400}{50}$$ $$r=0.008Y-8$$

This is our LM curve. For any given level of income $Y$, it tells us the interest rate $r$ that clears the money market.

The following video works through a very similar problem. It's a great way to reinforce the algebraic steps.

For additional practice, you can use the following handout. Try to solve the numerical problem presented and then check your work against the provided solution.

Conclusion

In this lesson, we have methodically derived the LM curve, which encapsulates equilibrium in the money market. This completes the second half of our core macroeconomic model.

Key Takeaways:

• The LM curve represents all combinations of income (Y) and the interest rate (r) where the money market is in equilibrium.
• It is derived algebraically from the condition that real money supply equals real money demand: $\bar{M}/\bar{P}=L(Y,r)$.
• The LM curve is upward-sloping. This is because a higher income level increases transaction demand for money, which requires a higher interest rate to bring total money demand back into balance with the fixed supply.
• The position of the LM curve is determined by the exogenous real money supply ($\bar{M}/\bar{P}$). An increase in the nominal money supply or a decrease in the price level will shift the LM curve to the right.

We now have two distinct relationships between Y and r: the downward-sloping IS curve (goods market equilibrium) and the upward-sloping LM curve (money market equilibrium). In our next lesson, we will put these two curves together. By solving the IS and LM equations as a system, we can find the single, unique point $(Y^{\ast },r^{\ast })$ that represents the simultaneous equilibrium of the entire economy.