According to demand theorys, make the qualitative analysis of personal service under economic growth and economic recession; A input-output multiplier model is eatablished to find that the emergence of traditional service industries could accelerate economic mobility, and increase the economic multiplier effect.

However,Keynes\'s theory in his masterpiece The General Theory of Employment Interest and Money is untenable and a typical example is multiplier model.

Hopf bifurcaion of a multiplier-accelerator model with progressive income tax and parameter interval of limit cycle existence;