# IS-LM Tutorial

This page presents a geometrical overview of, and introduction to, the IS-LM model.  For the algebra see any standard textbook, like Branson's Macroeconomics, or The Hicks-Hansen IS-LM Model at the excellent HISTORY OF ECONOMIC THOUGHT site.

I'm not currently teaching a course that uses this tutorial, but I've left it up because a few people out there have found it useful.  Please send any criticisms or suggestions to me at danby@u.washington.edu.  Best, Colin Danby

## 1. Nature of the Model

Any model makes some things endogenous (determined within the model) and some things exogenous (determined outside the model).  Let's go back to the income expenditure model, which you learned in intro macro. In that model Y was endogenous. G and Ip were exogenous. Solving for equilibrium Y required finding the solution to only one equation: Y = C + Ip + G.

Go here to review the macro accounting framework: Macro Flows Tutorial Section 1.2.

Only Y was endogenous in the income-expenditure model. The IS-LM model makes both Y and r endogenous. The key advantage of this is that we can have r determine Ip. Since the level of planned investment is important in the real world and varies a lot, it's nice to have a model in which that is not just set exogenously.

So what determines r? This model attempts to capture Keynes' insights about the money market, which you also studied in intro macro. We regard r as the outcome of the interaction between money demand and money supply. This is why IS-LM is essentially two models stuck together: a model of the goods market, and a model of the money market.

## 2. Equilibrium

This can be a difficult model to learn. There is a danger that you'll concentrate so hard on the mechanics of it that you'll lose sight of how the model relates to the real world.

Please review the concept of equilibrium: Macro Notes Section 1.3. And recall what equilibrium means in the income-expenditure model: Macro Notes Section 1.8.

Macro models do not claim that the economy is always at equilibrium. What they do claim is that if the economy is not at equilibrium, it will move toward equilibrium. (Near the end of this tutorial are some animations that try to show this movement.)  Thus in the income-expenditure model, if G rises a series of events will raise Y, until a new equilibrium is reached at which there is enough extra savings. Review: Macro Notes Section 1.9.

The notion of equilibrium, and how the macroeconomy is supposed to move toward it, is the key to understanding the IS-LM model. This model is composed of a goods market and a money market. You can think of it as embodying two ideas:

a. Depending on the interest rate (which determines Ip), there will be an equilibrium level of Y. If Y is below this equilibrium level, it will tend to rise. If Y is above this equilibrium level, it will tend to fall.

b. Depending on the level of transactions demand for money (set by Y) there will be one interest rate that equilibrates the money market. If r is above this level, it will tend to fall. If r is below this level, it will tend to rise.

What makes things interesting, but also difficult, is that both r and Y can change. Below we will develop our separate models of the goods and money markets, and then put them together. If the terminology of goods and money markets is not familiar, review it here: Macro Notes Section 4.1.

## 3. Goods Market (IS)

Take the income-expenditure model, which you reviewed above (go back if you didn't). If Ip rises, equilibrium Y rises, right?  This was because a higher level of demand for capital goods caused more to be made, more workers got hired, they bought more stuff, and so on.  Add to this the idea that Ip rises when r falls -- the cheaper it gets to borrow money, the more new capital investment projects firms undertake.

Review this important link between r and Ip here: Macro Notes Section 4.2.

Therefore:

• If r falls, Ip rises. When Ip rises, equilibrium Y rises, as shown in the income expenditure model.
• If r rises, Ip falls and equilibrium Y falls.

The different values of r, and the resultant equilibrium values of Y, give us a set of Y,r points that represent "goods market" equilibrium -- a situation in which AD=AS.

Here are some graphs that show how we get this set of points. Click the graphs to enlarge them.

This is theory you already know from intro macro. All we've done is add a new graph with Y on the horizontal axis and r on the vertical axis. The set of (Y,r) combinations that represent goods market equilibrium fall along a line in our graph.

The best way to think about that IS line is as a border -- a boundary in (Y,r) space between points at which AD < AS and Y will tend to fall, and the points at which AD > AS and Y will tend to rise.

Here is another graphical way to represent the IS curve. This picture is essentially the same as the derivation of the IS curve that you looked at above, except that we glued some of the graphs together at their common axes (turning some upside down), to get a "four quadrant" derivation. The depiction of IS itself emphasizes that the line is a frontier between the Y,r points at which Y is below equilibrium, and the (Y,r) points at which it is above its equilibrium level.

It should be apparent from the graphical derivations that if G changes, then the (Y,r) points that equilibrate the goods market change too. In graphical terms, a rise in G will shift IS right, while a fall in G will shift it left.

Additionally, note that the sensitivity of Ip to r will affect the slope of IS. If planned investment is highly sensitive to r, then a small change in r will mean a large change in Y, and IS will be almost horizontal. If planned investment is hardly affected at all by r, then it will take a large change in r to get much change in Y, and IS will be almost vertical.

To recap before moving on: the above is really just the income-expenditure model plus the idea that r affects Ip.  Note that we are reasoning from r to Y, via Ip.

## 4. Money Market (LM)

This is also built out of theory you learned in intro macro, but it's a little harder. Try to keep the reasoning about the money market strictly separate in your mind from what you just learned about the goods market.

The notion of equilibrium here is a situation in which money demand equals money supply.  Money demand comes from two sources: transactions and speculative demand. Money supply we will regard as exogenously set by the central bank, acting on the commercial banking system.

Review Money Demand: Macro Section 3.1
Review Transactions Demand for Money: Macro Notes Section 3.2
Review
Speculative Money Demand: Macro Notes Section 3.3

An extensive review of what money is and how its supply is set can be found here: Macro Notes Part 2.

You should remember that if bond prices rise, that's the same thing as saying interest rates fall. If that's not clear, review it here: Macro Notes Section 3.5.

If the money market is out of equilibrium, the interest rate changes. You may remember that this happens because individuals hold wealth in a portfolio consisting of money and bonds:

• When people finding themselves holding more money than the want, they try to turn some of it into bonds by buying bonds, which pushes up bond prices (which is the same thing as r falling).
• When people find themselves holding less money than they want, they they try to sell bonds to raise money, which pushes down bond prices (which is the same as r rising).

The amount of money held by everyone actually never changes during this story -- rather, the change in r  makes them willing to hold the money they actually hold.  In the first case above, as r falls the advantage of holding wealth as bonds falls, and people stop wanting them as much, which means they're content to hold the money they actually hold.  In the second case above, as r rises the advantage of holding bonds rises, which means people stop wanting to hold more money.  Take some time to think about this -- the key to the argument is this very stark, very simple money/bonds portfolio choice faced by everyone in the economy.  (This is a good time to remember the stock/flow distinction: In the money market we are talking about stock quantities (money supply, money demand, bonds), and our equilibrium is a stock equilibrium.  In the goods market above we are talking about flow quantities (Ip, Y, S, G, T) and our equilibrium is a flow equilibrium.  So keep the two stories straight and strictly separate.)

Got all that?  Then here's the key: the reasoning in this model goes from Y, to transactions demand for money, to an attempt at portfolio adjustment, to a change in the interest rate.

• A rise in output (Y) is also a rise income (remember Y means both).  A rise in income raises people's transactions demand for money.  Higher demand for money means people want to hold portfolios consisting of more money and less bonds at any point in time.  People attempt this readjustment by selling bonds.  Selling bonds lowers the bond price, which is the same thing as raising the interest rate.
• A fall in Y lowers transactions demand for money.  People try to adjust by buying bonds with money -- changing their portfolios so they will hold more bonds, less money.  As they try to do this the bond price rises, which is the same thing as saying r falls.

You can put the pieces together and examine interest rate determination here: Macro Notes Section 3.6.

With those ideas, we can determine the r that will equilibrate the money market for any Y. Here is a series of graphs that derives the LM curve.

The best way to think about that LM line is as a border -- a boundary in Y,r space between points at which Md < Ms and r will tend to fall, and the points at which Md > Ms and r will tend to rise. Here is a four-quadrant derivation that emphasizes this.

It should be apparent from the graphical derivations that a change in the money supply will change the (Y,r) points at which the money market is in equilibrium. In graphical terms an increase in Ms shifts LM down, while a decrease shifts it up.

Additionally, note that the degree to which speculative demand for money responds to changes in r will affect the slope of the LM curve. If only a small change in r brings large changes in speculative demand, then LM will be almost horizontal. If a very large change in r is required to change speculative demand for money, LM will be close to vertical.

Note, before moving on, that in the money side of this model we are reasoning from Y to r.  Depending on Y, r changes because of the effects of a given level of Y on the money market.

## 5. Assembling the Model (IS and LM)

So far, we've drawn two different boundaries in Y,r space:

Note again, as you look at these pictures, that:

• on the goods market, or IS, side of the model we go from r to Y.  Pick any value of r, and draw a horizontal line across the graph at that value of r. The line will cross the IS boundary at some point.  If the interest rate is at that level and output (Y) is to the left of that boundary, it will tend to rise -- output and employment will go up.  If Y is to the right of that boundary, it will tend to fall -- output and employment will go down.
• on the money market, or LM, side of the model we go from Y to r.  Pick any value of Y, and draw a vertical line up the graph at that value of Y. The line will cross the LM boundary at some point.  If output is at that level and r is above that boundary, it will tend to fall.  If r is below that boundary, it will tend to rise.

Putting the two pictures together gives us this geography:

Try to think of this not just as a couple of lines crossing on a graph, but as a 2-dimensional space in which the national economy sort of skates around, with its total output and its interest rate changing.  Once you have that idea, add the notion that the IS and LM boundaries tell you about the forces acting on the national economy at any particular point in this space.  (For another way to visualize this, look at this picture at Prof. Andreas Thiemer's "IS-LM Model: A Dynamic Approach".)

Now we can take the model out for a spin.  Click here for animations of dynamic adjustment.

In intro macro, we noticed that the goods and money markets interacted with each other (Review: Macro Notes Section 4.5). The IS-LM model simply gives us a more formal way to examine these interactions.

These animations show the ways in which fiscal and monetary policy may have effects on output and the interest rate. They also emphasize the fact that adjustment toward a new equilibrium is not instantaneous.

In general, in the stories we start by assuming equilibrium in both markets -- in other words, our macroeconomy has settled down to the one (Y,r) combination that equilibrates both markets at once.  Then we change conditions in one or the other market.  What that means is that the (Y,r) geography shifts and our old (Y,r) point is no longer an equilibrium.  So we start moving, spiralling gently counterclockwise toward the new equilibrium point.

The stories told in the animations boil down to this:

Fiscal Policy Stories:

1. G changes, raising or lowering AD (throwing goods market out of equilibrium)
2. As firms respond, Y changes (and a multiplier process is set off, as AS (Y) adjusts to AD)
3. BUT: as Y changes, Md(t) changes, throwing the money market out of equilibrium.
4. The change in Md(t) changes r
(Note: the degree to which speculative demand for money changes as r changes determines how much r changes as a result of this process.)
5. the change in r affects Ip, which affects Y through the multiplier, modifying initial changes in Y.
(Note: the sensitivity of Ip to r determines how great this change is.)

Monetary Policy Stories:

1. Ms changes, throwing money market out of equilibrium
2. As people either buy or sell bonds to reflect their portfolio preferences, the bond price rises or falls, with r moving the opposite way.
(Note: the degree to which speculative demand for money changes as r changes determines how much r changes as a result of this process.)
3. BUT: as r changes, Ip changes, throwing the goods market out of equilibrium.
(Note: the sensitivity of Ip to r determines how great this change is.)
4. Working through the multiplier, the changed Ip changes Y.
5. The change in Y will affect Md(t), modfiying initial effects on r.

If you can learn these stories as sequences of events, you have a basic grasp of the IS-LM model. Although the pictures are nice, it's more important to be able to follow a concrete story. The geometry is just a supplement, a learning aid. If you memorize the geometry and don't learn what it means, you're not learning economics.

Graphically, the notes in parentheses would affect the steepness or flatness of the IS and LM curves (noted in earlier sections).

## 6. Supplemental Notes for the Curious: History and Critiques

The origin of this model is a paper by John Hicks titled "Mr. Keynes and the Classics," which appeared in the journal Econometrica in 1937.  It was an effort to turn some of the insights in John Maynard Keynes' pathbreaking General Theory of Employment, Interest, and Money (1936) into a mathematical model. Keynes liked Hicks' article, but it’s hardly a full embodiment of Keynes' thinking.  It became popular in textbooks.  (See Kerry A Pearce. and Kevin D. Hoover, “After the Revolution: Paul Samuelson and the Textbook Keynesian Model” pages 183-216 in Allin Cottrell and Michael Lawlor, eds., New Perspectives on Keynes, Duke University Press, 1995.)

The model has some internal problems, particularly because of the way it tries to link a flow model (the IS side, where flows adjust) and a stock model (the LM side, where stocks adjust).  These two models really live in different kinds of time. Equilibration on the LM side is very rapid, while equilibration on the IS side may take months. The "r" on the LM side is a short-term rate; the key "r" for the IS side is going to be a longer-term rate. Hicks addressed these concerns 43 years later in his 1980 article "IS-LM: An explanation" (Journal of Post Keynesian Economics, 3:2, 139-54, reprinted in Hicks, ed., Money, Interest and Wages: Collected Essays on Economic Theory, vol. II, Oxford: Basil Blackwell, pp. 318-331).  He concluded that the problems are important enough that the model is of little use for forward-looking policy analysis.

A more formal presentation of the model, plus incisive discussion of subsequent debates over it, can be found here.    If you’re interested in Keynes’ thought, one of the best introductions is Hyman Minsky’s 1975 book John Maynard Keynes (New York: Columbia University Press), which clarifies the differences between Keynes and the versions of his work that you will encounter in most textbooks.  Robert Skidelsky has written an excellent three-volume biography of Keynes; if you want something shorter try Skidelsky’s 136-page Keynes (Oxford University Press (Past Masters), 1996).  If you’re interested in current heterodox economics, one place to start is the Post-Autistic Economics Network.

©1997,2000 Colin Danby.  Last updated 11/05; original version prepared in 1997 with programming and web design by J. Ringley.