1. The Relationship between Inflation and Unemployment
Rate of growth of the price level
Unemployment
Expected inflation

2. The expectations-augmented Phillips relation
The expectation of higher inflation shapes the behavior of firms and individuals in a way that stimulates inflation.
Since the expected rate of inflation is accounted for, the equation is called the expectations-augmented Phillips relation.

3. The adaptive expectations hypothesis
When actual inflation exceeds the expected one this nurtures people’s expectations so
If the projected and the real inflation are the same, people do not expect a change in inflation.

4. The effect of monetary policy
Shows the reverse effect, that of inflation on unemployment
If inflation persists too long, this discourages people’s savings and consequently reduces aggregate investment and employment.

5. Unemployment increases proportionately with the rate of growth of real money
where is the rate of growth of nominal money.

6. The overall model Expectations-augmented Phillips relation
Adaptive expectations
Monetary policy

7. Inflation and Unemployment: and Extended Model
The rate of change of the inflation rate is proportional to the difference between the actual and the natural rate of unemployment.
= const.
Blanchard, Olivier. Macroeconomics. Second Edition, Prentice Hall International, 2000, Chapters 8, 9

8. When the actual rate of unemployment exceeds the natural one, inflation falls and vice versa.
In bad economic times, people lose their jobs and aggregate demand falls.
In a boom people get hired and increase spending, while high aggregate demand pushes prices up.

9. The actual unemployment rate depends on aggregate demand which on its own depends on the real value of money supply.
Thus, actual unemployment depends negatively on real money supply according to the relationship

10. This is identical to the monetary-policy equation:

11. The continuous-time extended model

12. Blanchard’s equation

13. The continuous-time model
Differentiating the first equation further,
and substituting,

14. where

15. The particular integral is
and the characteristic equation is

16. The general solution

17. The time path of a general complementary function
depends on the sine and cosine functions as well as on the term . Since the period of the trigonometric functions is and their amplitude is 1, their graphs will repeat their shape every time the expression increases by .

18. The sine function -1 1

19. The cosine function -1 1

20. Since , inflation rate displays regular oscillations around the rate of growth of money supply, which gives the equilibrium level of inflation.

21. Explosive fluctuation
Equilibrium level

22. Uniform fluctuation

23. Damped fluctuation

24. Convergent oscillation

25. Divergent oscillation

26. Uniform oscillation

27. Nonoscillatory if
Oscillatory if
Divergent if

28. Time path of unemployment
- undefinitized constants

29. Like inflation rate, unemployment rate also displays regular oscillations but its equilibrium is the natural rate of unemployment.
Since again , the time path is neither convergent, nor divergent.

30. A hysteresis system
The rate of change of the inflation rate is a decreasing function not only of the level of the unemployment rate, but also of its rate of change:

32. Solution

33. Since the constants. and
Since the constants and are positive, the roots (or their real part) turn out to be negative and the equilibrium is dynamically stable.

34. The time path of unemployment would also be convergent:

35. An extended model with a reverse effect
With increased aggregate demand prices rise but unemployment falls. Thus,

36. Solution
Since the parameters are positive and the real part of the characteristic roots is also positive, the time path is unstable. The same applies for unemployment.

37. The discrete-time extended model

38. Taking the second difference to express

40. Rearranging and moving one period forward,

41. Again, the intertemporal equilibrium of inflation is the growth rate of nominal money supply:

42. Solution
Distinct real roots

43. The characteristic roots are

44. Since the absolute value of one characteristic root is greater than 1, the time path of inflation is divergent and nonoscillatory.

45. Unemployment depends on inflation in the previous, not in the current period:

47. The roots are complex numbers so the time path of inflation must involve stepped fluctuation.
Since , the fluctuating time path of inflation is explosive.

48. Converting continuous into discrete time

49. Taking a second derivative and then converting it to second difference,

50. where

52. Since the characteristic roots can both be bigger than 1, a convergent time path is possible.
The condition ensures the dynamic stability of inflation.

53. Conclusions:
In both discrete and continuous-time models the equilibrium inflation rate is that of nominal money growth that reflects the monetary policy of the government.
For the same model in discrete and continuous time we do not get only convergence or only divergence.
The dynamic stability of time path of both inflation and actual unemployment would depend on the specific parameters.