Question: The market for a product is given by the following demand and supply equations.

\[\begin{aligned} & {{q}_{d}}\left( t \right)=500-p\left( t \right)-p'\left( t \right) \\ & {{q}_{s}}\left( t \right)=3p\left( t \right)+\frac{1}{2}\text{ }p'\left( t \right)-16 \\ \end{aligned}\]

Assume that the market always clears.

(a) Obtain a first order differential equation for price. [5 marks]

(b) Solve the differential equation assuming initial values \(p\left( 0 \right)=20\) [7 marks]

(c) Calculate the steady state for price. [6 marks]

(d) Plot the phase diagram for this system, marking on it the steady state and arrows indicting the direction of motion. [7 marks]

(e) Calculate how long it takes (in units of time, which have not been specified), for the process to complete half the adjustment from its initial value towards steady state. [5 marks]