Suppose we have following regression models of quantity:

\displaystyle  \begin{array}{rcl}  \ln Q &=& \alpha + \beta P,\\ \ln Q &=& \alpha + \beta\ln P, \\ \ln Q &=&\alpha + \beta\ln P + \gamma (\ln P)^2. \end{array}

What happens to value {V=Q\cdot P} when we change price? Since {\ln V= \ln Q+\ln P} we get following value models:

\displaystyle  \begin{array}{rcl}  \ln V &=& \alpha + \beta P + \ln P,\\ \ln V &=& \alpha + (\beta+1)\ln P, \\ \ln V &=&\alpha + (\beta+1)\ln P + \gamma (\ln P)^2. \end{array}

Now we can dance the usual dance of adding {\Delta} quantities in both sides of the equations:

\displaystyle  \begin{array}{rcl}  \ln(X+\Delta X)=\ln X + \ln\left(1+ \frac{\Delta X}{X} \right)=\ln X+ \frac{\Delta X}{X}. \end{array}

Thus we get:

\displaystyle  \begin{array}{rcl}  \frac{\Delta V}{V} &=& \beta \Delta P + \frac{\Delta P}{P},\\ \frac{\Delta V}{V} &=& (\beta+1)\frac{\Delta P}{P} , \\ \frac{\Delta V}{V} &=& (\beta+1)\frac{\Delta P}{P} + 2\gamma \left(\frac{\Delta P}{P} \ln P\right) + \gamma \left(\frac{\Delta P}{P}\right)^2. \end{array}

So the conclusions are:

  1. If we have log-linear quantity-price model and the coefficient {\beta} is negative, then if we increase the price, the value will diminish, unless {\frac{\Delta P}{P}} dominates {\Delta P}. This can happen for small values of {P}.
  2. If we have log-log quantity-price model, then we have a paradox. If {\beta>-1}, no matter how much we increase the price, the value will increase. In other words, loss of unit sales is compensated by the price increase. The dream of every seller.
  3. If we have log-quadratic-log quantity-price model, the relationship is more complicated. But unless {P=1}, negative coefficient of {\gamma} ensures the decrease of value in the event of unreasonable increase of price.