Fit a price adjustment to the shortage between two curves

At a price of $4 per kg, producers supply $1$ tonne of rice and consumers buy all of it. Supply and demand match: this is the equilibrium point.

$C_1:Q=k_1/P$ and $C_2:Q=k_2P$. Using $Q=1$, $P=4$ gives $k_1=4$, $k_2=1/4$. Thus consumer quantity is $Q_c=4/P$ and producer quantity is $Q_s=P/4$.

For the observed prices below equilibrium, shortage is $4/P-P/4$. Thus $P^{\prime}=k_3(4/P-P/4)=k_3(16-P^2)/(4P)$, $k_3>0$. Separating gives $\int4P/(16-P^2)dP=\int k_3dt$, hence $-2\ln|16-P^2|=k_3t+C$. Therefore $16-P^2=Ae^{Bt}$, where $B=-k_3/2<0$. Take the positive price root $P=\sqrt{16-Ae^{Bt}}$.

$P(0)=3$ gives $A=7$. Then $P(1)=3.65$ gives $e^B=(16-3.65^2)/7=1071/2800$ and $B=\ln(1071/2800)\approx-0.96102663$. The exact fitted expression is $P(t)=\sqrt{16-7(1071/2800)^t}$ for $t\ge0$. The source’s rounded display $P\approx\sqrt{16-7e^{-0.961t}}$ is retained as an approximation; the square root, visible in the source, is lost by native extraction.

The price curve starts at $(0,3)$, passes through $(1,3.65)$, increases with decreasing slope, and approaches $P=4$ dollars per kg from below. For $P=aQ+b$, a positive intercept $b$ can represent a minimum price needed to cover fixed initial investment costs; the source notes that production requires the price to cover those costs. Direct proportionality alone puts the producer line through the origin.