The last problem set 
The dynamic aggregate demand/aggregate supply model
Problem 1
Assume that the aggregate demand ( IS ) equation is given by
Y*=100, r*=2% and a=1. The Central Bank follows the Taylor rule:
and
The inflation target is π*=2%
a) Using the simple Fisher equation:
and assuming adaptive expectations
, derive the dynamic aggregate demand (DAD). Sketch it on the (Y, π) diagram
b) The dynamic aggregate supply (DAS) is given by the Phillips curve of the form
. Sketch it on the (Y, π) diagram.
c) Calculate the long run equilibrium values of output, inflation and the nominal interest rate
(assuming shock e and
are zero).
d) Using the DAD/SAS lines, show the consequences of a demand shock equal to 1 that lasts 2
periods.
Problem 2 (from Mankiw)
Explain the statement:” If the central bank wants to achieve lower interest rates, it should raise
the interest rate.”
Problem 3 (from Mankiw)
Suppose that the central bank has the wrong estimates of the equilibrium real interest rate r*. That is; it
follows the Taylor rule of the form
Where r**>r*, where r* is the true equilibrium real interest rate.
Solve for long run equilibrium under this policy rule. Explain in words the intuition behind this
solution.
Answer:
First, write down the simple aggregate demand equation:
For ease of exposition, assume that a=1
Recall the Fisher equation:
Combine these equations to get:
Use the Taylor rule to get:
This gives us:
Rearrange:
And finally:
Note that the emergence of a gap between the true r* and the r** (that is the value of equilibrium r
that the central bank thinks is true) works like a shock. If r**>r*, then it is like a negative shock. The
reason is the following: the nominal interest rate set by the central bank is too high, because r** is
“too high”, what depresses the demand (just like a negative shock). In equilibrium Y=Y*, et=0 but
inflation does not hit the π* - it will be lower.
Problem 4.
In year 2007 the US economy was hit by two major, negative shocks: an increase in the price of oil
and a significant decrease in housing prices. Assume that both shocks lasted for 1 period. Show the
consequences.
In period 1, the shocks shift DAD and DAS left. In period 2, the DAD comes back to old position, and
the DAS slowly adjusts. See the graph.
Problem 5.
The dynamic aggregate demand curve is given by:
The dynamic aggregate supply is:
The central bank target is π*=4% and the economy is in LR equilibrium. In period t+1 the Bank
reduces the target to π*=2,8%; in period t+2 the Bank reduces the target further to π*=2,6%, in t+3 to
π*=2,2%. Expectations are backward looping (adaptive).
a) Using the DAD-SAS, show the dynamics of inflation and output after the changes in inflation
target.
The calculations are not difficult, but tedious.
Firstly, the DAD and SAS shocks are zero, so you can forget about them.
Secondly, combine the DAD and DAS; for example, take the DAS and plug it into the DAD
(substitute current inflation in the DAD with the DAS equation).
Do the calculations for period t+1: calculate the output gap
and then πt+1. Remember that
your previous period inflation πt (in DAS) is equal to 4% and your π*= 2,8.
Repeat for t+2, changing the target inflation to 2,6% and changing the previous period inflation
from 4% to the value that you calculated above (i.e. inflation in t+1). Continue until you get the
true inflation to be equal to the target.
b) Calculate the cumulative output loss (the sum of output gaps) required to reduce inflation to
2%
Problem 6.
Suppose that people’s expectations are subject to random shocks. That is, expected inflation in period
t-1 is:
The shock h is normally zero, but it deviates from zero when some event beyond past inflation causes
expected inflation to change. Similarly:
a) Derive the DAD and SAS curves.
In the DAD equation, the only change stems from the Fisher equation, that now looks like:
Which implies:
Assume a=1.
Plugging in the Taylor rule, we get:
The very basic DAS is unchanged:
The new element (h)appears here:
Hence, the DAS becomes:
b) Suppose that the economy experiences an inflation scarce. That is, in period t, for some reason
people come to believe that inflation in period t+1 will be higher, that is ht>0 (for this period
only).Show this shock using the DAD/SAS diagram. Explain the results
Let’s start from period t=1;in this period h1=1 (and ho=0). Note that this changes the position of
DAD in period t=1. A positive h1 is like a positive demand shock in period t=1. Hence, in t=1 the
DAD shifts to the right. In t=1, DAS does nothing, as ht-1= ho=0
Now, we move to t=2. In t=2, h2=0, so DAD shifts back to the old position (shifts left, back to DAD0).
In t=2, the DAS shifts left. Firstly, ht-1 = h1=1, which shifts DAS left. Additionally, πt-1 =/= πt-2 or
equivalently π1 =/=π0, so this will also shift the DAS left. So you end up with a big shift left of the DAS
(see the graph)
c) In what sense are inflation scares self-fulfilling?.
There was an inflation scare and yet, the “true” inflation went up as a result. The reason is that
people changed their behaviour – they started to buy more, as they thought that the real interest rate
is lower and the firms started to produce less, as the true inflation turned out to be lower than
expected. So it turns out that if we think something will happen – it happens.
Problem 7
Assume that due to higher savings, the amount of capital went up. Show the consequences, using the
DAD/SAS model.
ANSWER:
If saving goes up, the amount of capital goes up and the amount of long-run output Y*=F(K,N) goes
up (Solow model explains why! As a matter of fact the neoclassical theory of investment also explains
why) .
This implies that Y*, DAD and SAS all shift to the right – see Prof. Brzozowski’s lecture notes – he
has nice graphs that show this change .