Fit a price adjustment to the shortage between two curves

The given diagram has quantity $Q$ tonnes on the horizontal axis and price $P$ dollars per kg on the vertical axis. Consumer curve $C_1$ and producer line $C_2$ meet at $A=(1,4)$. Consumer quantity is inversely proportional to price, while producer quantity is directly proportional to price.

(i) Interpret $A$ in context.

(ii) Find both equations, expressing $Q$ in terms of $P$.

A shortage occurs when consumer quantity exceeds producer quantity. The rate of increase of price after $t$ months is proportional to this shortage.

(iii) Given an initial price of $3 and a price of $3.65 after one month, find $P(t)$ and sketch it, showing its long-term behaviour. Suggest why producers might use $P=aQ+b$, $a,b>0$, instead of $C_2$.