# Examples of bounded, positive, inverse demand curves

The three most common examples of demand curves I am aware of are

\begin{align} Q&= b - aP,\\ Q&= bP^a,\\ Q&= 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&= b/a - Q/a,\\ P&= (Q/b)^{1/a},\\ P&= -\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&= \frac{b}{1+aQ},\\ P&= 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&= \left(\frac{b}{P}-1\right)\frac{1}{a},\\ Q&= -\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?

• But why are they less parsimonious, they have the same number of parameters as the other models. Is there something specific about these functional forms that are bad, or is it that these functions aren't common in the literature? – WetlabStudent Nov 17 '20 at 0:09
• You are actually right my bad - the change in notation made me think you made some addition there - I reviewed it too hastily (my bad I am sorry for that - I will delete my previous comment as I am retracting that statement). But now looking at this for a second time there is actually no additional difference between them - I dont understand what is point of rewriting b as $p_0$ - if you are only asking if inverse demand curves are common in economics then yes they are - in fact they are often even more common then standard ones if this is your question I can give you pointers to sources – 1muflon1 Nov 17 '20 at 0:26
• My question is whether there are examples of inverse demand curves that are strictly positive with respect to quantity, smooth [I do get that you can define things piecewise to avoid infinite or negative prices, but these aren't differentiable functions then], and bounded (no infinity allowed). Some background - I'm a mathematician, and I have been toying with some applied problems for research and teaching. From a mathematical perspective, my new equations are nicer (yes you can replace p0 w b, the variable name is not important). – WetlabStudent Nov 17 '20 at 5:37
• sorry I was not tagged so I did not noticed your comment I guess you already got an answer – 1muflon1 Nov 17 '20 at 19:10

$$Q = \ln 4 - 0.5 \ln P$$
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.
• in our classes we used the essential mathematics for economic analysis and mathematics for economists. I think I seen problems like this in both - in our courses professor gave extra points for anyone who would bring solution to all exercises at the end of the chapters so I ended up doing all of them and seeing all sorts of functions. I recall seeing some with logarithms and demand functions in form of Q=a/p - b were also common - but you don't see these functions often in assigned readings (papers for econ classes) there people usually just say demand is some $q(p)$ and leave it at that – csilvia Nov 19 '20 at 23:01