# Numerical Problems Numerical Problems 1

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```					Numerical Problems
1.   Since the natural rate of unemployment is 0.06,    e – 2(u – 0.06), so u – 0.06  0.5( e – ), or
u  0.06  0.5( e – ).
(a) Year 1: u  0.06  0.5(0.08 – 0.04)  0.06  0.02  0.08. The unemployment rate is 0.02 higher
than the natural rate. The percentage that output falls short of full-employment output is
2  0.02  0.04, or 4%.
Year 2: u  0.06  0.5(0.04 – 0.04)  0.06. The unemployment rate equals the natural rate, since
inflation equals expected inflation. Since unemployment is at its natural rate, output is at its
full-employment level.
Since the output loss was 4 percentage points and inflation declined by 8 percentage points, the
sacrifice ratio is 4/8  0.5.
(b) Use equations: u  0.06  0.5( e – ), output shortfall  2 (u – 0.06).

Year               e       u      u – 0.06    Output Shortfall
1        0.08    0.10     0.07      0.01             0.02
2        0.04    0.08     0.08      0.02             0.04
3        0.04    0.06     0.07      0.01             0.02
4        0.04    0.04     0.06      0.0              0.0

The total output shortfall is 0.02  0.04  0.02  0.0  0.08  8-percentage points of output lost.
Inflation fell by 8 percentage points. So the sacrifice ratio is 8/8  1. Notice that, compared with
part a, the sacrifice ratio is higher, for this slower disinflation.

2.   (a) Equating aggregate demand to short-run aggregate supply gives: 300  10(M/P)  500  P – Pe,
or 300  (10  1000/P)  500  P – 50, or 10,000/P  150  P. Multiplying both sides of the
equation by P and rearranging gives P2  150P – 10,000  0, which can be factored as (P – 50)
(P  200)  0. This has the nonnegative solution P  50. Since Pe is also 50, the expected price
level equals the actual price level, so output is at its full-employment level of 500 and the
unemployment rate is at the natural rate of 6%. These are the long-run equilibrium values of the
three variables as well.
(b) When the nominal money supply increases unexpectedly to 1260, we again equate aggregate
demand to short-run aggregate supply, which gives: 300  10(M/P)  500  P – Pe, or 300 
(10  1260/P)  500  P – 50, or 12,600/P  150  P. Multiplying both sides of the equation by P
and rearranging gives P2  150P – 12,600  0, which can be factored as (P – 60)(P  210)  0.
This has the nonnegative solution P  60. When P  60, the short-run aggregate supply curve
gives Y  500  P – Pe  500  60 – 50  510. Output of 510 is 2% above full-employment output
of 500, because (510 – 500)/500  0.02. With a natural unemployment rate of 0.06, Okun’s Law
gives 0.02  – 2(u – 0.06). This can be solved to get u  0.05.
In the long run, Pe adjusts to equal P, output adjusts to its full-employment level of 500, and
unemployment adjusts to the natural rate of 0.06. To find P, use the aggregate demand curve
to get 500  300  10(1260/P), or 200  12,600/P, which can be solved to get P  63.
The results of this exercise are consistent with the existence of an expectations-augmented
Phillips curve. Unexpected inflation reduces unemployment in the short run. In the long run,
however, inflation is higher and unemployment returns to its natural rate.
Chapter 12   Unemployment and Inflation   258

3.   (a)   0.10 – 2(u – 0.06)  0.22 – 2u. This is shown as the Phillips curve labeled PCa in Figure
12.6. If the Fed keeps inflation at 0.10, then u  0.06, the natural rate of unemployment.

Figure 12.6

(b) With expected inflation rising to 12%, the Phillips curve is   0.12 – 2(u – 0.06)  0.24 – 2u.
This is the Phillips curve labeled PC b in the figure. The higher rate of expected inflation has
caused the curve to shift up relative to where it was in part (a). With the actual inflation rate at
10%, the Phillips curve equation is 0.10  0.12 – 2(u – 0.06), which has the solution u  0.07. So
if the Fed tries to maintain the existing rate of inflation after a shock has raised inflation
expectations, the unemployment rate increases. However, if the Fed could convince people that
the inflation rate really would not rise, so that  e remains at 0.10, then the short-run Phillips
curve would remain at PC a, and the unemployment rate would not increase.
(c) With the natural rate of unemployment rising to 0.08 at the same time that expected inflation
rises to 0.12, the Phillips curve equation is   0.12 – 2(u – 0.08)  0.28 – 2u. This is the Phillips
curve labeled PC c in the figure. The new short-run Phillips curve is even higher than those for
parts (a) and (b). With the actual inflation rate held to 10%, the equation becomes 0.10  0.28 –
2u, which can be solved to get u  0.09. The unemployment rate rises both because the Fed holds
inflation below expected inflation and because the natural rate has increased.
This time, even if the Fed convinced people that inflation would remain just 10%, the
unemployment rate would still rise to 8%, since the natural rate has increased to that level.

4.   (a) Beginning in long-run equilibrium, with M  4000, output must be at its full-employment level of
6000 and the unemployment rate must be equal to the natural rate of .05. Using the values for Y
and M in the AD curve, 6000  4000  2(4000/P), which gives P  4. This is also the expected
price level. Because M has been constant for a long time and is expected to remain constant,
 e  0.
Chapter 12   Unemployment and Inflation   259

(b) With P e  4, the SRAS curve is Y  6000  100(P – 4). The AD curve is Y  4000  2(4488/P).
The intersection of the two curves occurs when 6000  100(P – 4)  4000  2(4488/P).
Simplifying terms gives 100P2  1600P – 8976  0, which has the solution P  4.4. Plugging this
into the SRAS curve gives Y  6040. From the Okun’s Law equation we get (6040 – 6000)/6000 
–2 (u – .05), so – 0.00333  u – .05, so u  .0467. Cyclical unemployment is u – u  –.0033.
Unanticipated inflation is (P – Pe)/Pe  .10  10%.
(c) The Phillips curve equation is    e – h(u – u ), which gives .10  0 – h(.0467 – .05). This is
solved to get h  30. So the slope of the Phillips curve is –30.

Analytical Problems
1.   (a) The reduction in structural unemployment would reduce the natural rate of unemployment and
thus would shift both the expectations-augmented Phillips curve and the long-run Phillips curve
to the left.
(b) Despite the expense of the government program to reduce structural unemployment, it would
have a permanent effect. Monetary expansion can work only temporarily—in the long run it
has no effect.

2.   The slope of the short-run aggregate supply curve will be much steeper in economy B, because
producers increase their output only a small amount in response to an increase in price. But economy
A’s short-run aggregate supply curve will be flatter, as people are likely to perceive price changes as
changes in relative price rather than the aggregate price level, and thus will respond more strongly to
changes in prices.
The short-run Phillips curve will also be steeper in economy B, since unemployment, like output,
won’t respond much to a change in inflation. But in economy A, unemployment and output will
respond more strongly to price changes, and the short-run Phillips curve will be flatter.

3.   (a) In Figure 12.7, the SRAS curve shifts up 10% each year, as does the AD curve. Unanticipated
inflation is zero, as both actual and expected inflation are 10%. The economy is at full
employment, since firms set their prices to exactly match the increase in the general price level.

Figure 12.7
Chapter 12   Unemployment and Inflation   260

(b) The surprise increase in the money supply at mid-year leads to a rise in output, as shown in
SRAS curve remains fixed at SRAS1. When the money supply rises by its regular 10% at the end
of the year, the AD curve shifts up to AD3 and firms raise their prices by 15%, shifting the SRAS
curve up to SRAS3. Actual inflation is 15%, but expected inflation was only 10%. As a result,
there was a temporary increase in output above its full-employment level, and a temporary
decline in unemployment below the natural rate. Thus for the year as a whole, cyclical
unemployment was negative and unanticipated inflation was positive, just as in the expectations-
augmented Phillips curve.

Figure 12.8

4.   In the cashless society, there would be no shoe-leather costs, as there would be no cash balances
on which to economize. But menu costs would remain for anticipated inflation. The costs of
unanticipated inflation would remain as well: both the risk of wealth transfers plus confusion
in price signals.

5.   (a) Figure 12.9 shows the effects of increasing the money supply while holding the price level
constant. Beginning at point A, the intersection of aggregate demand curve AD1 and short-run
aggregate supply curve SRAS1, the increase in the money supply shifts the aggregate demand
curve to AD2. Since prices cannot rise, the short-run equilibrium is at point B, with output above
its full-employment level.
Chapter 12   Unemployment and Inflation   261

Figure 12.9

(b) When the price controls are removed, the price level will jump up, with the short-run aggregate
supply curve shifting to SRAS2. The new equilibrium is at point C, where there is full
employment.

6.   (a) A new law that prohibits people from seeking employment before age eighteen is likely to reduce
the natural rate of unemployment because teenagers have a higher-than-average unemployment
rate. With no teenagers allowed in the labor force, the average unemployment rate would be
lower.
(b) A service that makes looking for a job easier is able to match people and jobs more rapidly,
which should reduce the natural rate of unemployment.
(c) If unemployed workers can receive benefits longer, they’ll be in less of a rush to take a job, so the
job-matching process will take longer. As a result, the natural rate of unemployment will rise.
(d) A structural shift in the types of products people buy is likely to raise the natural rate of
unemployment, because it will take time for the economy to shift workers from some types
of occupations to others.
(e) A recession leads to a rise in cyclical unemployment, but doesn’t affect the natural rate of
unemployment. However, if there’s hysteresis, the natural rate of unemployment might rise
in the future.

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