Solving for Equilibrium: The IS-LM Model

The General Equilibrium

The IS curve gives us an infinite set of pairs $(Y,r)$ that clear the goods market. The LM curve gives us another infinite set of pairs $(Y,r)$ that clear the money market. The economy is in a general, short-run equilibrium only when the conditions for both markets are met at the same time.

Graphically, this occurs at the unique intersection of the IS and LM curves.

At any other point, at least one market is out of equilibrium. For example:

• At any point on the IS curve but not on the LM curve, the goods market is in equilibrium, but the money market is not.
• At any point on the LM curve but not on the IS curve, the money market is in equilibrium, but there is either an excess supply of or excess demand for goods.

The equilibrium is the point where there are no forces generating change in either income or the interest rate.

The Algebraic Solution

Finding this equilibrium point is a straightforward exercise in solving a system of two simultaneous equations. As you know from your quantitative background, if we have two independent equations and two unknown variables (in this case, Y and r), we can solve for a unique solution.

The process is as follows:

1. Obtain the IS and LM equations. From our previous lessons, we know how to derive these from the underlying economic functions. They will typically be in a form like:

   • IS curve: $Y=f(r)$, for example, $Y=a-b\cdot r$
   • LM curve: $Y=g(r)$, for example, $Y=c+d\cdot r$

2. Equate the two expressions for Y. Since equilibrium requires that the same Y satisfies both equations, we can set them equal to each other:

   $$f(r)=g(r)$$

   For our linear example:

   $$a-b\cdot r=c+d\cdot r$$

3. Solve for the equilibrium interest rate ($r^{\ast }$). Rearrange the equation from step 2 to isolate $r$.

   $$a-c=(b+d)\cdot r$$ $$r^{\ast }=\frac{a-c}{b+d}$$

4. Solve for the equilibrium income ($Y^{\ast }$). Substitute the value of $r^{\ast }$ back into either the original IS equation or the LM equation. Using the IS equation:

   $$Y^{\ast }=a-b\cdot r^{\ast }$$

   The result would be identical if you used the LM equation.

This algebraic procedure gives us the precise coordinates of the intersection point on the graph.

A Complete Worked Example

Let's walk through this entire process from the fundamental economic equations to the final equilibrium values. The following video provides a clear, step-by-step example. It starts with the familiar components (consumption, investment, etc.), derives the IS and LM equations, and then solves for the equilibrium.

Another Example: A Textual Walkthrough

To reinforce the method, let's examine another numerical example, this time in a written format. This resource uses the same logic and algebraic steps.

Practice Problem

Now it's your turn to apply the technique. The following video presents a slightly different set of equations, where the investment function also depends on income. This is a common and more realistic specification. Try to solve the problem yourself before watching the solution.

The economic components are:

• $C=200+0.25Y_d$ (where $Y_d=Y-T$)
• $I=150+0.25Y-1000i$
• $G=250$
• $T=200$
• $M^d=2Y-8000i$
• $M^s=1600$ (Note: here M is already the real money supply, so P=1)

Your task:

1. Derive the IS equation.
2. Derive the LM equation.
3. Solve the system for the equilibrium interest rate ($i$) and income ($Y$).

Once you have attempted a solution, you can check your work by watching the following video.

Solution spoiler: You should find an equilibrium interest rate of 5% ($i=0.05$) and an equilibrium income of 1000.

Conclusion

Today, we have integrated the goods market and the money market to find the economy's short-run equilibrium. This equilibrium is the unique combination of income and the interest rate that satisfies the equilibrium conditions of both markets simultaneously.

Key Takeaways:

• The overall short-run equilibrium is found at the intersection of the IS and LM curves.
• Algebraically, this corresponds to solving the system of two linear equations that represent the IS and LM curves.
• The standard method is to set the two expressions for $Y$ equal to each other, solve for the equilibrium interest rate $r^{\ast }$, and then substitute $r^{\ast }$ back into either equation to find the equilibrium income $Y^{\ast }$.

This ability to solve for the economy's equilibrium is the foundation for macroeconomic policy analysis. In our next lesson, we will put this model to work. We will analyze how fiscal policy (changes in government spending or taxes) and monetary policy (changes in the money supply) shift the IS and LM curves, respectively, and thereby change the equilibrium levels of income and the interest rate.