Sometimes simple macroeconomic models can help us understand the world better and even though I am not uncritical about the IS/LM model it nonetheless has some interesting features which from time to time makes it useful for policy analysis (if you are careful).

However, a key problem with the IS/LM model is that the model does not take into account – in its basic textbook form – the central bank’s policy rule. However, it is easy to expand the model to include a monetary policy rule.

I will do exactly that in this post and I will use the Federal Reserve’s new policy rule – the Bernanke-Evans rule – to analysis the impact of the so-called fiscal cliff on a (very!) stylised version of the US economy.

We start out with the two standard equations in the IS/LM model.

The money demand function:

(1) m=p+y-α×r

Where m is the money supply/demand, p is prices and y is real GDP. r is the interest rate and α is a coefficient.

Aggregate demand is defined as follows:

(2) y=g-β×r

Aggregate demand y equals public spending and private sector demand (β×r), which is a function of the interest rate r. β is a coefficient. It is assumed that private demand drops when the interest rate increases.

This is basically all you need in the textbook IS/LM model. However, we also need to define a monetary policy rule to be able to say something about the real world.

I will use a stylised version of the Bernanke-Evans rule based on the latest policy announcement from the Fed’s FOMC. The FOMC at it latest meeting argued that it basically would continue to expand the money base (in the IS/LM the money base and the money supply is the same thing) to hit a certain target for the unemployment rate. That means that we can define a simple Bernanke-Evans rule as follows:

(3) m=λ×U

One can think of U as either the unemployment rate or the deviation of the unemployment rate from the Fed’s unemployment target. λ is a coefficient that tells you how aggressive the fed will increase the money supply (m) if U increases.

We now need to model how the labour market works. We simply assume Okun’s law holds (we could also have used a simple production function):

(4) U=-δ×y

This obviously is very simplified as we totally disregard supply side issues on the labour market. However, we are not interested in using this model for analysis of such factors.

It is easy to solve the model. We get the LM curve from (1), (3) and (4):

LM: r= y×(1+δ×λ)⁄α+(1/α)×p

And we get the IS curve by rearranging (2):

IS: r =(1/β)×g-(1/β)×y

Under normal assumptions about the coefficients in the model the LM curve is upward sloping and the IS curve is downward sloping. This is as in the textbook version.

Note, however, that the slope of the LM does not only depend on the money demand’s interest rate elasticity (α), but also on how aggressive (λ) the fed will react to an increase in unemployment.

**The Sumner Critique applies if λ=∞
**

The fact that the slope of the LM curve depends on λ is critical. Hence, if the fed is fully committed to its unemployment target and will do everything to fulfill (as the FOMC signaled when it said it would step up QE until it hit its target) then λ equals infinity (∞) .

Obviously, if λ=∞ then the LM curve is vertical – as in the “monetarist” case in the textbook version of the IS/LM model. However, contrary to the “normal” the LM curve we don’t need α to be zero to ensure a vertical LM curve.

Hence, under a strict Bernanke-Evans rule where the fed will not accept *any* diviation from its unemployment target (λ=∞) the (government) budget multiplier is zero and the so-called Sumner Critique therefore applies: Fiscal policy cannot increase or decrease output (y) or the unemployment (U) as any fiscal “shock” (higher or lower g) will be fully offset by the fed’s actions.

**The Bernanke-Evans rule reduces risks from the fiscal cliff**

It follows that if the fed actually follows through on it commitment to hit its (still fuzzy) unemployment target then in the simple model outlined above the risk from a negative shock to demand from the so-called fiscal cliff is reduced greatly.

This is good news, but it is also a natural experiment of the Sumner Critique. Imagine that we indeed get a 4% of GDP tightening of fiscal policy next year, but at the same time the fed is 100% committed to hitting it unemployment target (that unemployment should drop) then if unemployment then increases anyway then Scott Sumner (and myself) is wrong – or the fed didn’t do it job well enough. Both are obviously very likely…

I am arguing that I believe the model presented above is the correct model of the US economy. The purpose has rather been to demonstrate the critical importance of a the monetary policy rule even in a standard textbook keynesian model and to demonstrate that fiscal policy is much less important than normally assumed by keynesians if we take the monetary policy rule into account.