Economics 31401
Fall 2011
Problem Set 4 Suggested Solutions
Questions 1:
1.
The economy has the Phillips curve:
π
=
π
1
– 0.4(u – 0.05).
a.
The natural rate of unemployment is the rate at which the inflation rate does
not deviate from the expected inflation rate.
Here, the expected inflation rate
is just last period’s actual inflation rate. Setting the inflation rate equal to last
period’s inflation rate, that is,
π
=
π
1
, we find that u = 0.05.
Thus, the natural
rate of unemployment is 5 percent.
b.
In the short run (that is, in a single period) the expected inflation rate is fixed
at the level of inflation in the previous period,
π
1
.
Hence, the shortrun
relationship between inflation and unemployment is just the graph of the
Phillips curve:
it has a slope of 0.04, and it passes through the point where
π
=
π
1
and u = 0.05.
In the longrun, expected inflation equals actual inflation,
and so
π
=
π
1
, and output and unemployment equal their natural rates.
The
longrun Phillips curve thus is vertical at an unemployment rate of 5 percent.
The curves are graphed below.
π
LRPC
0.4
π
1
SRPC
0.05
u
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c.
To reduce inflation, the Phillips curve tells us that unemployment must be
above it natural rate of 5 percent for some period of time.
We can write the
Phillips curve in the form
π

π
1
= 0.4(u – 0.05).
Since we want inflation to fall by 2 percentage points, we want
π
–
π
1
= 
0.02.
Plugging this into the lefthand side of the above equations, we find
0.02 = 0.4(u – 0.05).
We now solve the above equation for u:
u = 0.10.
Hence, we need 5 percentage points of cyclical unemployment above the
natural rate of 5 percent.
Okun’s law says that a change of 1 percentage point in unemployment
translates into a change of 2 percentage points in GDP.
Hence, an increase in
unemployment of 5 percentage points corresponds to a fall in output of 10
percentage points.
The sacrifice ratio is the percentage of a year’s GDP that
must be foregone to reduce inflation by 1 percentage point.
Dividing the 10
percentagepoint decrease in GDP by the 2 percentagepoint decrease in
inflation, we find that the sacrifice ration is 10/2 = 5.
d.
One scenario is to have very high unemployment for a short period of time.
For example, we could have 10 have percent unemployment for a single year.
Alternatively, we could have a small amount of cyclical unemployment spread
out over a long period of time.
For example, we could have 6 percent
unemployment for 5 years.
Both of these plans would bring the inflation rate
down from 4 percent to2 percent, although at different speeds.
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