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Numerical Example of the Fischer Model

The aggregate demand equation for the Fischer model is

(1) Yt = Mt - Pt + vt

This can be rewritten as

(2) Pt = Mt - Yt + vt

The vertical shift in the aggregate demand curve is by definition

the change in the price level holding output constant. This is

equal to the change in M plus the change in V.

The aggregate supply equation is

(3) Yt = Pt - (.5t-1Pt + .5t-2Pt) + ut

This can again be rewritten with the price level on the left-hand

side to derive the vertical shift in the curve:

(4) Pt = Yt + (.5t-1Pt + .5t-2Pt) - ut

Increases in the expected price level shift the aggregate supply

curve upwards while a supply shock shifts the aggregate supply

curve downwards. An increase in the expected price level one

period in advance will shift the aggregate supply curve by only

half this amount because only half the workers are able to

renegotiate their contracts at any given period.

Fischer showed that the Fed can stabilize output by changing

the money supply to offset demand and supply shocks. To see this,

take an example where the demand shock was zero at period 1. At

time period 2 there is a demand shock of .1. Assume first that

the Fed does not try to stabilize output and follows a constant

money supply rule (In the example below, the money supply has been

set to equal 1, but any other constant would give the same

variance of output). Assume that the autoregressive process

governing the demand shock is

(5) vt = .6vt-1 + nt

This would lead to the following sequence of prices and output:

Vertical

Intercept

Period v n t-1v t-2v M t-1M t-2M P Y AD AS

0 0 0 0 0 1 1 1 1 0 1 1

1 .1 .1 0 0 1 1 1 1.05 .05 1.1 1

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2 .06 0 .06 0 1 1 1 1.04 .02 1.06 1.02

3 .036 0 .036 .036 1 1 1 1.036 0 1.036 1.036

Notes: The vertical intercept of the aggregate demand curve is

just M + v. The vertical intercept of the aggregate supply curve

is (.5t-1Pt + .5t-2Pt) - ut, where t-1Pt = .666t-1Mt + .333t-2Mt and t-2Pt= t-2Mt .

Since the shock is a surprise in period one, expectations, an thus

the aggregate supply will be unaffected. Aggregate demand will

shift up by the amount of the shock, and given equal slopes of the

supply and demand curves, half of this shift will go into prices

and half into output.

In subsequent periods the shock will decay at the assumed rate

of .6. In period 2, the shock will be expected at time period 1

but not two periods prior at time period 0. The aggregate supply

curve will shift up by two percent (half of the workers anticipatea four percent increase in the price level) and aggregate demand

will increase by the amount of the shock, six percent. The price

level will increase by an average of the supply and demand shifts,

or 4 percent, and output by 2 percent. Notice that the division

of the aggregate demand shock 2/3 into prices and 1/3 into output

is exactly the same division when there is an increase in the

money supply that is expected one period in advance but not two

periods.

After two periods, the shock will be expected by both groups

of workers and thus will affect only the price level. This again

is the same result as an increase in the money supply anticipated

two periods in advance.Now see what the sequence of events would be if the Fed

followed the optimal money supply rule. The effect in period one

would be exactly the same as in the previous example because the

Fed is unable to react contemporaneously to the shock. The effect

on output is also the same in period three because all demand

changes are neutral in the long run (in this case after two

periods). The difference between the optimal money supply rule

and the constant money supply rule occurs in period two. The rule

says that the Fed should offset demand shocks by the anticipated

amount of the shock. The anticipated value of the shock in period

two is .6 so the Fed should reduce the money supply by thisamount:

Period v n t-1v t-2v M t-1M t-2M P Y AD AS

0 0 0 0 0 1 1 1 1 0 1 1

1 .1 .1 0 0 1 1 1 1.05 .05 1.1 1

2 .06 0 .06 0 .94 .94 1 1 0 1 1

3 .036 0 .036 .036 .964 .964 1 1 0 1 1

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By offsetting the demand shock, the Fed has kept the position of

the aggregate demand curve constant and stabilized output. While

the optimal monetary rule was defined in terms of stabilizing

output and not prices, note that in this case the rule also

stabilizes the price level. Note that the aggregate supply curve

does not shift because workers understand that the Fed's rule will

stabilize the price level so their price expectations do not

change.

Now let's examine the same case with a supply shock. In

period one the shock will increase output but decrease the price

level, since the aggregate supply curve shifts. The optimal

monetary rule calls for the Fed to offset supply shocks by a two-

to-one ratio. Thus in period 2, since the supply shock is

expected to equal .06, the Fed should decrease the money supply by

.12. The reason that supply shocks have to be offset at a higher

ratio is that the price level will be changing in this case (both

aggregate supply and aggregate demand are shifting downwards).The Fed has to offset both the original shock and the worker's

expectation of a lower price level due to the shock. An example

is as follows:

Period u t-1u t-2u M t-1M t-2M P Y AD AS

0 0 0 0 0 1 1 1 1 0 1 1

1 .1 .1 0 0 1 1 1 .95 .05 1 .9

2 .06 0 .06 0 .88 .88 1 .88 0 .88 .88

3 .036 0 .036 .036 .928 .928 .928 .892 .036 .928 .856

Unlike demand shocks, output does not return to natural output in

period 3 in the case of supply shocks. Thus demand shocks are

neutral in the long run but supply shocks have real effects, in

keeping with the long-run classical properties of this model.