Almost Complete Macroeconomic Model

Almost Complete Macroeconomic Model

Advanced Macroeconomic 320 Note #5 1

Almost Complete Macroeconomic Model:

So far we have assumed P to be fixed in the short-run.

Now let’s endogenize in the ling-run (and endogenize in the short-run). This means that now P is not longer assumed to be fixed, and will be determined in the model.

The Phillips curve describes the dynamics of inflation in such a way that

. And HN<0

The actual rate of inflation(read as ‘Phi’) is inversely related to the excess unemployment over the natural rate of unemployment and has an inflationary expectation as a shift parameter. Note that u-uNmeasures how much the actual unemployment rate exceeds the natural rate of unemployment. It is the deflation gap.

Can you draw the Phillips curve?

We can express the above in terms of Y and Ybar(=full employment national income) instead of u and uN. Then of course, HY >0

. And H’ or HY>0

It says that the actual rate of inflation is positively related to a) the inflation gap (in percentage terms), and b)the expected inflation rate: the term in the bracket of H measures how much(by what %) the actual national income Y exceeds Y bar or the equilibrium national income, and is expressed in percentage terms.

Basically, the Phillips curve indicates the relationship between the actual Y and the long-run equilibrium Y, and its impact on the inflation. We can also think that Y is the Aggregate Demand and Y bar or the long-run equilibrium national income is the Aggregate Supply.

I will rewrite the rate of inflation, , into P dot over P.

*What is P dot? It is P’s time derivative, or a change in P over one unit of time period.

And thus P dot over P is the percentage rate of inflation, that is, . The Phillips curve can be rewritten as:

Now we have 3 equations for the economic system:

: IS or goods market

: LM or money market

: Phillips Curve ()

We arenot ready yet to endogenize the expected rate of inflation or e.

Thus, we may have a simplifying assumptions for now:.

This is called Static Expectations (not realistic) Later, we will relax this assumption (in the next chapter) by specifying the functional form of the expectations, which should be rational, and thus by endogenizing .

Still the key issue is the impact of the fiscal policy on the real national income:

Now solve for dY/dG (or/and dY/dM) by applying Cramer’s rule:

Step 1: Totally differentiating the above equations, we get 3 X3 matrix:



3) is simply reduced to;


because if is not that large; and

by assumption

Step 2: Rearranging terms into matrix multiplication form:

In order to apply Cramer’s rule. We should have the above total differentials in the form of matrix multiplication such as:

[coefficient matrix ] [ vector of d endogenous variables]

= [coefficient matrix] [ vector of d exogenous variables]

Here we have to determine,”What are the endogenous and exogenous variables?”

As we have discussed before, it depends on the time perspectives:

In the short-run Y is endogenous and P is fixed and thus exogenous.

In the long-run Y = Yf at long-run equilibrium, and P is flexible and thus endogenous.

How about the new variable P dot?

At anyone point of Long-Run equilibrium, because p does not changes.

Thus is equal to zero and thus exogenous.

From one LR equilibrium to another, P changes and thus P dot is endogenous. However the change in P is so small, and thus the P itself has not changed yet: P is fixed and exogenous. In the short run, is fixed, and thus is exogenous.However,(only in approximation), and thus is endogenous.


One more thing: Choice of Monetary Policy Instrument

still we assume that the monetary policy sets the Money Supply, not the Interest Rate. This is an assumption, which makes M or MS exogenous.

In this case, the monetary authority control M or the nominal money supply, and allows the interest rate to be determined endogenously at any rate from the model, M is exogenous and r or i is endogenous.

Of course, in an alternative monetary policy where the monetary authority is engaged in the “interest rate pegging” policy, then r is exogenous and M endogenous.

In general, the monetary authority can choose M or i as their target. Whichever target it chooses, the target variable become exogenous. We will relax the current assumption later, and show that the two targets do not lead to the same qualitative result of the long-run equilibrium.

For now, we will start with the first type of monetary policy. Later we will change to the other monetary policy and examine its implication.

Thus, for now,

endogenous variable ()

exogenous variable ()

In this case, the result of Step 2 is as follows:

Step 3 & 4: Use Cramer’s rule, and LaPlace expansion to get the solution:

> or < 0

Once again, we can rewrite into a slightly familiar form:

The result is the same as before; the sign of dY/dG is ambiguous mainly due to IY term in the denominator.

*Comment: Be reminded of the Crowding-Out Effect.

The last term of the denominator Ir Ly/Lr measures a chain reaction between goods and money markets:

Ir Ly/Lr is the same as dI/dr x d L/dY x dr/ dL = dI/dr x dr /dY, that is,

y md = L(r, y) r, therefore, md > ms I  (y*).

This is the Crowding-out effect. By now we are very familiar with this part.

2) Long-Run (Equilibrium)

Recall that in the long-run equilibrium, there would not be any changes: in other words,

Any variable’s dot should be equal to zero ( P dot is equal to zero).

Because =0 and dY =0, the above system of equations is simplified as

In this case of Long-run, the result of Step 2 is as follows:

Step 3 & 4: Use Cramer’s rule and LaPlace expansion, we get:

There is no point of talking about dY/dG in the long-run. Why?


An expansionary fiscal policy is merely inflationary in the long-run.


In a word, an expansionary fiscal policy raises the interest rate in the long run.

We can see that when G rises, r goes up with the same Y: the IS curve shifts up.

What would be the short-run comparative statics if we assume that the government sets the monetary policy in terms of interest rate?

3)Alternative Assumption about Monetary Policies:“Interest Pegging Policy”

Monetary authority may set monetary policy in terms of ior r (interest rate), rather than in terms of M or MS (nominal quantity of money supply). In the previous version of macroeconomics, we have learned that they are equivalent. Now we will see that they differ in terms of stability implications. At least, in terms of stability, the interest pegging policy seems to be inferior to the monetary policy geared to control the nominal quantity of money supply.

Of course, there are different views. One of them is by W. Poole’s illustration, which we will cover later.


endogenous variables

exogenous variables

Step 2: Rearranging the already-obtained total differentials, we get:




Note: the value of P dot /P square would be close to zero.

In the matrix multiplication, we have:

Again here, the sign of is undetermined: Dependency on the magnitude of, could be of negative sign or positive sign.

So we have got a similar result as the case where M is exogenous.

Comparison of in the above two alternative monetary policy setting:

  • When is exogenous,
  • When is exogenous,

In both cases, the sign of is unclear.

However, apparently there is a very big difference between the two multipliers. What is it?

In the case of “interest pegging policy” there is no crowding out to the government fiscal policy multiplier. Some may think,”This is better as there is no offsetting effect from the money market”. Not really as we will show below.

Although it is not so apparent right now, there is another difference between the two. And it concerns the qualitative nature of equilibrium or dynamic stability. In a word, when r is exogenously controlled by the monetary authority, the long-run equilibrium is a unstable one. There is no such instability characteristics in the case where the monetary authority controls the money supply.

Infact, there are two different perspectives or theories to this point:

(1)Stability (Dynamic); and

(2)William Poole’s analysis

Please, keep the above question- “which monetary policy is superior?” or “do different monetary policies matter?”– in your mind as we are making a deep exploration of the issue of stability.

(1) Dynamic Stability Analysis

*Some Background Information for Dynamic Stability: Condition for Stability

The Correspondence Theorem’ says that if and only if there is a dynamic stability (= tendency to go back to equilibrium), then there is a good comparative static, or comparative static puts on ‘right’ signs.

This means that dynamic stability is a general guarantee of goodness of comparative static. We can check dynamic stability instead of the sign of comparative static.

How do we check ‘stability’ in the given model? Recall that we have AS-AD model where one focus variables are P and Y.

In terms of P, is a requirement for the dynamic stability. It can be illustrated as follows:




The same graph can be translated into a different dimension of P dot and P.


So the Stability requires .You can show that in a system with a unstable equilibrium, or
Now then, what would be the sign of d / d P in the abovetwo alternative monetary policy settings?

Let’s get it by Cramer’s rule respectively:

(1) When :(2) when:

= =

= at all times.

This means

if is exogenous, there is a possibility of stability with , and

if , exogenous, there is no possibility of stability as at any time.

In general, the correspondence theorem would tell us that comparative statics would be of wrong signs.

We can deduct that in generalthe interest pegging monetary policy may be more prone to instability.

(2) However, there is a different view: William Poole’s Analysis: (Reference at Section 2.7 of W.Scarth’s book, Chap. 2. pp34-36.);

The optimal monetary policy depends on the specificnature of uncertainty or Shocks in the economy.

i.)When the economy is subject to good market shocks: (IS shifts around), then is better.


( if )

Relatively stable Relatively unstable

iii.)When an economy is subject to stochastic monetary shocks (LM shifts around), thenis better.

( if ) ( if )

In the short-run, financial asset markets shows significant fluctuations in the national income while the aggregate expenditure (AE = C + I + G + X-M: the basis of IS curve) are quite predictable.

It would be best when “monetary authority should try to peg interest rates in the short-run and the money supply in the long-run.”