Examples of bounded, positive, inverse demand curves The three most common examples of demand curves I am aware of are
\begin{align}
Q&amp;= b - aP,\\
Q&amp;= bP^a,\\
Q&amp;= b e^{-aP}\\
\end{align}
The first being our classic linear demand curve, the second being the one corresponding to a constant, fixed, elasticity, $a$, and lastly the slightly less common but still useful exponential demand curve. But all of these curves, to me, leave a peculiar feature of their inverses
\begin{align}
P&amp;= b/a - Q/a,\\
P&amp;= (Q/b)^{1/a},\\
P&amp;= -\frac{1}{a}\log{\left(\frac{Q}{b}\right)}\\
\end{align}
For the linear demand curve, price goes negative, as the Quantity produced becomes high. This is conceivable for some products, where it might be costly to store excess, but it is easy to imagine products where negative prices are impossible. The fixed elasticity demand curve solves this, but it has another undesirable property, in my opinion. The price goes to infinity as the quantity produced goes to zero [and, at least in some cases, infinite prices are not plausible in the real-world as we are constrained by finite resources]. The last option, the exponential demand curve, suffers from both issues!
If I want price to be bounded and positive over the domain of all positive quantities, I could start by defining, the inverse demand functions
\begin{align}
P&amp;= \frac{b}{1+aQ},\\
P&amp;= be^{-aQ},\\
\end{align}
In this parameterization, $b$ is the maximum price anyone would be willing to pay for the product no matter how rare it is, and $a$ governs the rate the price declines as the quantity becomes less rare. These price curves would lead to demand curves of the form
\begin{align}
Q&amp;= \left(\frac{b}{P}-1\right)\frac{1}{a},\\
Q&amp;= -\frac{1}{a}\log{\left(\frac{P}{b}\right)},\\
\end{align}
Is anyone aware of demand curves used in the literature like this. I'd assume there must be someone, for some purpose, that might have required a demand curve with such properties. I'm thinking of cases where, due to an externality, quantities can suddenly be taken to zero or very large values. What demand curves do people use in these cases? Does anyone use either of the two above?
You can find demand functions like this in textbook and exam problems. Just a random example from internet:
$$Q = \ln 4 - 0.5 \ln P$$
which is just a special case of $Q=−1/a \ln (P/b)$ where a is 1 and b is also 1 and there is some additional constant $\ln 4$. We just had an exam where I also saw one of the other examples. If I remember right it was $Q = \frac{100}{P}-1$. I also remember seeing more of them in textbooks but it would take forever to find them in thick book full of math problems.