1. The constant a comes from the linearization of the Wage Setting relation, F(u,z), into:

F(u,z) = 1 - a u_t + z

[Note: see the Appendix to Chapter 17]

This means that a measures the amount the expected real wage (W/P^e) goes down for every one unit increase in the unemployment rate, i.e. it represents the sensitivity of the expected real wage to changes in the unemployment rate.

The natural rate of unemployment is found when p (the inflation rate) is equal to the p ^e (the expected inflation rate). Setting them both equal to x yields:

x = x + (m +z) - a u_t.

Solving for u_t yields;

u_n = (m +z)/a ,

so increases in a lead to a lower natural rate of unemployment. Also, the natural rate of unemployment occurs where the Wage Setting Relation is equal to the Price Setting Relation (in terms of the model of Chapter 15). In this diagram an increase in a would lead to a wage setting relation with a steeper slope, again yielding a lower natural rate of unemployment.

2. If there were never any errors so that p ^e_t always equaled p _t the unemployment rate would always equal the natural rate of unemployment, and the following graph would be a sensible scatter plot.

3. a. In this model (m +z) = .18 and a = 3, so if u_n = (m +z)/a , then u_n = .18/3 = .06.

b. If q = 0, then the Phillips curve is:

p _t = .18 - 3 (.05), and this yields an inflation rate of .03

p _t+2 = .06 + .18 - 3 (.05), that p _t+2 = .09.

4. Assuming that p ^e_t = q p _t-1, and q = 1, then the natural rate of unemployment is the trigger between increasing inflation and decreasing inflation. If one were slowly decreasing the unemployment rate, when there was a shift from decreasing inflation to increasing inflation one would have just passed the natural rate of unemployment. All this assumes that the process which determines the natural rate of unemployment was constant during the experiment.

5. The answer to this question depends upon how lasting an impression the inflation of the 1970s and early 1980s has on the minds of the general public. If this experience fades rapidly, we may return to a Phillips curve like the one in the 1950s and 1960s, but if this experience has a more lasting impression, then people will not completely eliminate inflationary expectations and the Phillips curve will be to the northeast of the Phillips curve of the 1950s and 1960s.

6. The decrease in the price of oil decreased m , the markup over labor costs. This would decrease the natural rate of unemployment. (Of course, other changes were also at work during this time period, so the effect of oil prices may not be easy to see in the data.)

7. a. From the Phillips curve, with p _t = p _t-1 then u_t would have to equal .06, six percent.

b. Using Okun's law, to keep unemployment constant,

(i) g_yt has to be .03.

And to get an inflation rate of 10%, the aggregate demand relationship shows us that

(ii) g_mt has to be .13.

8. The following table gives the responses to a policy to reduce the inflation rate to 5%.

Year:

t-1 t t+1 t+2

p 10% 5% 5% 5% (given)

u 6% 11% 6% 6% (from Phillips curve)

g_y 3% -9.5% 15.5% 3% (from Okun's law)

g_m 13% -4.5% 20.5% 8% (from Aggregate Demand)

9. a. To solve this problem we need to reduce the number of equations by one. To do this substitute the aggregate demand equation into Okun's law, yielding

(1) u_t - u_t-1 = -.4(g_mt - p _t - .03), and

(2) p _t - p _t-1 = -(u_t - .06)

When g_mt is reduced to 0%, and using u_t-1 =.06 and p _t-1 = .07, we find two equations in two unknowns, u_t and p _t:

u_t -.06 = -.4(0 - p _t - .03)

p _t - .07 = -(u_t - .06)

The solution to these two equations is u_t = .089 and p _t = .041.

b. The solution procedure is identical to part a, but now g_mt = 0, u_t-1 = .089, and

p _t-1 = .041. Making these substitutions into equations (1) and (2) and solving yields:

u_t = 10.1 and p _t = 0.00.

c. In the long run unemployment has to return to it's natural rate of .06, so from equation (1), p has to be -.03. With output growth of .03 and money growth of 0.00, there has to be deflation.

10. Credibility allows the monetary authority to influence p ^e_t by its pronouncements. If the monetary authority is credible, it will get reductions in expected inflation without having to have the people experience the reduction in inflation. This means that they will be able to reduce the inflation rate without having to increase unemployment as much as the case without a credible monetary authority. The length of the recession which accompanies a reduction in inflation will tell you about the credibility of the monetary authority.