To determine the new equilibrium interest rate, equate the IS curve from part (a) with the new LM curve from above: 1,700 – 100r = 250 + 100r 1,450 = 200r 7.25 = r.
Substituting this interest rate into either the IS or the LM equation, we find Y = 975.
Therefore, the new equilibrium interest rate is 7.25, and the new equilibrium level of output is 975, as depicted in Figure 11–14.
g. The aggregate demand curve is a relationship between the price level and the level of income. To derive the aggregate demand curve, we want to solve the IS and the LM equations for Y as a function of
P. That is, we want to substitute out for the interest rate. We can do this by solving the IS and the LM equations for the interest rate:
IS: Y = 1,700 – 100r 100r = 1,700 – Y.
LM: (M/P) = Y – 100r 100r = Y – (M/P).
Combining these two equations, we find 1,700 – Y = Y – (M/P) 2Y = 1,700 + M/P Y = 850 + M/2P.
Since the nominal money supply M equals 1,000, this becomes
Y = 850 + 500/P.
This aggregate demand equation is graphed in Figure 11–15.
How does the increase in fiscal policy of part (d) affect the aggregate demand curve? We can see this by deriving the aggregate demand curve using the IS equation from part (d) and the LM curve from part (b):
IS: Y = 1,900 – 100r 100r = 1,900 – Y.
LM: (1,000/P) = Y – 100r 100r = Y – (1,000/P).
Combining and solving for Y: 1,900 – Y = Y – (1,000/P), or
Y = 950 + 500/P.
By comparing this new aggregate demand equation to the one previously derived, we can see that the increase in government purchases by 50 shifts the aggregate demand curve to the right by 100. How does the increase in the money supply of part (e) affect the aggregate demand curve? Because the
AD curve is Y = 850 + M/2P, the increase in the money supply from 1,000 to 1,200 causes it to become Y = 850 + 600/P.
By comparing this new aggregate demand curve to the one originally derived, we see that the increase in the money supply shifts the aggregate demand curve to the right.
Chapter 13
PROBLEMS AND APPLICATION:3,6
3.According to the rational-expectations approach, if everyone believes that policymakers are committed to reducing inflation, the cost of reducing inflation—the sacrifice ratio—will be lower than if the public is skeptical about the policymakers’ intentions. Why might this be true? How might credibility be achieved?
The cost of reducing inflation comes from the cost of changing people’s expectations
about inflation. If expectations can be changed costlessly, then reducing inflation is also costless. Algebraically, the Phillips curve tells us that
Π= EΠ?–?(u – un) + v..
If the government can lower expected inflation EΠto the desired level of inflation, then there is no need for unemployment to rise above its natural rate.
According to the rational-expectations approach, people form expectations about inflation using all of the information that is available to them. This includes information about current policies in effect. If everyone believes that the government is committed to reducing inflation, then expected inflation will immediately fall. In terms of the Phillips curve, EΠfalls immediately with little or no cost to the economy. That is, the sacrifice ratio will be very small.
On the other hand, if people do not believe that the government will carry out its
intentions, then EΠ?remains high. Expectations will not adjust because people are skeptical that the government will follow through on its plans.
Thus, according to the rational-expectations approach, the cost of reducing inflation depends on how resolute and credible the government is. An important issue is how the government can make its commitment to reducing inflation more credible. One possibility, for example, is to appoint people who have a reputation as inflation fighters. A second possibility is to have Congress pass a law requiring the Federal Reserve to lower inflation. Of course, people might expect the Fed to ignore this law, or expect Congress to change the law later. A third possibility is to pass a constitutional amendment limiting monetary growth. People might rationally believe that a constitutional amendment is relatively difficult to change.
6. Suppose that an economy has the Phillips curve
Π ??Π-1– 0.5(u – u),
n
and that the natural rate of unemployment is given by an average of the past two years’ unemployment:
un ??0.5(u-1??u-2).
a. Why might the natural rate of unemployment depend on recent unemployment (as is assumed in the preceding equation)?
b. Suppose that the Fed follows a policy to reduce permanently the inflation rate by 1 percentage point. What effect will that policy have on the unemployment rate over time? c. What is the sacrifice ratio in this economy? Explain.
d. What do these equations imply about the short-run and long-run tradeoffs between inflation and unemployment?
In this model, the natural rate of unemployment is an average of the unemployment rates in the past two years. Hence, if a recession raises the unemployment rate in some year, then the natural rate of unemployment rises as well. This means that the model exhibits hysteresis: short-term cyclical unemployment affects the long-term natural rate of unemployment.
a. The natural rate of unemployment might depend on recent unemployment for at least two reasons, suggested by the theory of hysteresis. First, recent unemployment rates might affect the level of frictional unemployment. Unemployed workers lose job skills and find it harder to get jobs; also,
unemployed workers might lose some of their desire to work, and hence search less hard for a job. Second, recent unemployment rates might affect the level of structural unemployment. If labor negotiations give a greater voice to ―insiders‖ than ―outsiders,‖ then the insiders might push for high wages at the expense of jobs. This will be especially true in industries in which negotiations take place between firms and unions.
b. If the Fed seeks to reduce inflation permanently by 1 percentage point, then the Phillips curve tells us that in the first period we require ?1 –?0 = –1 = –0.5(u1 – un1), or
(u1 –un1) = 2.
That is, we require an unemployment rate 2 percentage points above the original natural rate u . Next period, however, the natural rate will rise as a result of the cyclical unemployment. The new natural rate u will be
u = 0.5[u1 + u0]
= 0.5[(un1 + 2) + un1] = un1 + 1.
Hence, the natural rate of unemployment rises by 1 percentage point. If the Fed wants to keep inflation at its new level, then unemployment in period 2 must equal the new natural rate u . Hence,
u2 = un1+ 1.
In every subsequent period, it remains true that the unemployment rate must equal the natural rate. This natural rate never returns to its original level: we can show this by deriving the sequence of unemployment rates:
u3 = (1/2)u2 + (1/2)u1 = u + 1.5 u4 = (1/2)u3 + (1/2)u2 = u + 1.25 u5 = (1/2)u4 + (1/2)u3 = u + 1.375.
Unemployment always remains above its original natural rate. In fact, we can show that it is always at least 1 percent above its original natural rate. Thus, to reduce inflation by 1 percentage point,
unemployment rises above its original level by 2 percentage points in the first year, and by 1 or more percentage points in every year after that.
c. Because unemployment is always higher than it started, output is always lower than it would have been. Hence, the sacrifice ratio is infinite.
d. Without hysteresis, we found that there was a short-run tradeoff but no long-run tradeoff between inflation and unemployment. With hysteresis, we find that there is a long-run tradeoff between inflation and unemployment: to reduce inflation, unemployment must rise permanently.