Okun's law

1. Consider an economy in long-run equilibrium with an inflation rate, π, of 12% (0.12) per year and a natural unemployment rate, ū, of 6% (0.06). The expectations-augmented Phillips curve is

π = π^e - 2(u - ū).

Assume that Okun's law holds so that a 1 percentage point increase in the unemployment rate maintained for one year reduces GDP by 2% of full-employment output.

a) Consider two-year disinflation. In the first year π = 0.04 and π^e = 0.08. In the second year
π = 0.04 and π^e = 0.04. In the first year, what is the unemployment rate? By what percentage does output fall short of full-employment output? In the second year, what is the unemployment rate? By what percentage does output fall short of full-employment output? What is the sacrifice ratio for this disinflation?

b) Now consider a four-year disinflation according to the following table:

Year 1 2 3 4
π 0.08 0.04 0.04 0.04
π^e 0.10 0.08 0.06 0.04

What is the unemployment rate in each of the four years? By what percentage does output fall short of full-employment output each year? What is the sacrifice ratio for this disinflation?

2. Consider the following extended classical economy (in which the misperceptions theory holds):

AD Y = 300 + 10(M/P).
SRAS Y = Ỹ+ P - P^e
Okun's law (Y - Ỹ)/Ỹ = -2(u - ū)
Full-employment output Ỹ = 500.
Natural unemployment rate ū = 0.06.

a) Suppose that the money supply M = 1000 and that the expected price level P^e = 50. What are the short-run equilibrium values of output, Y, the price level, P, and the unemployment rate, u? What are the long-run equilibrium values of these three variables?

b) Now suppose that an unanticipated increase raises the nominal money supply to M = 1260. What are the new short-run equilibrium values of output, Y, the price level, P, and the unemployment rate, u? What are the new long-run equilibrium values of these three variables? In general, are your results consistent with an expectations-augmented Phillips curve?

3. In a certain economy the expectations-augmented Phillips curve is

π = π^e - 2(u - ū) and ū = 0.06.
a) Graph the Phillips curve of this economy for an expected inflation rate of 0.10. If the Fed chooses to keep the actual inflation rate at 0.10, what will be the unemployment rate?

b) An aggregate demand shock (resulting from increased military spending) raises expected inflation to 0.12 (the natural unemployment rate is unaffected). Graph the new Phillips curve and compare it to the curve you drew in part (a). What happens to the unemployment rate if the Fed holds actual inflation at 0.10? What happens to the Phillips curve and the unemployment rate if the Fed announces that it will hold inflation at 0.10 after the aggregate demand shock, and this announcement is fully believed by the public?

c) Suppose that a supply shock (a drought) raises expected inflation to 0.12 and raises the natural unemployment rate to 0.08. Repeat part (b).

4. An economy is described by the following equations:

AD Y = 4000 + 2(M/P)
SRAS Y = Ỹ + 100(P - P^e)
Okun's law (Y - Ỹ)/Ỹ = -2(u - ū)

In this economy full-employment output Ỹ equals 6000 and the natural unemployment rate ū equals 0.05.

a) Suppose that the nominal money supply has long been constant at M = 4000 and is expected by the public to remain constant forever. What are the equilibrium values of the price level P, the expected price level P^e, expected inflation π^e, output Y, and the unemployment rate u?

b) A totally unexpected increase in the money supply occurs, raising it form 4000 to 4488. What are the short-run equilibrium values of the price level, expected price level, output, and unemployment rate? What are the values of cyclical unemployment and unanticipated inflation?

c) What is the slope of the expectations-augmented Phillips curve (equal to -h in equation,
π = π^e - h(u - ū)) in this economy?