Challenging the law of supply and demand

According to conventional wisdom an increase in demand for a commodity increases the price it commands in the market, and this in turn leads to an increase in supply, to match the increase in demand. A simple analysis of the optimal supply given a demand curve leads me to believe that it is not impossible, or even implausible, to have the paradoxical situation that an increase in demand at every price can result in a decrease of supply.

Describing demand
Let d(p) be the average amount of the commodity that will be sold (per unit time) at the price p. Note that, even if the commodity is sold in indivisible units, this average is a continuous variable. One can safely assume that this demand function is monotonically decreasing and, generically although not necessarily, strictly decreasing. The reason is that, at a lower price, one is at least as likely to purchase the commodity (and at least as often) as at a higher price. The demand function can therefore be inverted, to give the supply
function p(s), which is the price that the commodity commands in the market if it is supplied (and consumed) at the rate s.

Optimizing supply
For the supplier, the problem is to determine the rate at which to supply the commodity so as to maximize his profit. The revenue that can be expected from selling the commodity at the rate s is sp(s). Let c(s) be the cost of bringing the commodity to market at the rate s. The form of the cost function is, interestingly, not important for our current analysis.
The profit derived from bringing the commodity to market at the rate s is, therefore, P(s)=sp(s)-c(s). The commodity will
be supplied at the rate s₀ maximizing the profit. This satisfies P'(s₀)=0 and, generically, P"(s₀)<0.

The response of the supply to changes in the demand
It is well-known from the mean-value theory of phase transitions that s₀ does not necessarily depend continuously on small changes in the profit function P(s). For instance, it is possible that a small change in the unit cost leads to a large change in the optimal supply. This behaviour is not generic, though.

In any case, suppose that the demand changes by δp(s), with a concommitant change δs₀ in the optimal supply. Then, P'(s₀)=0 implies
P"(s₀)δs₀+[s₀δp(s₀)]'=0.
In other words, since P"(s₀)<0, the optimal supply s₀ increases if, and only if, sδp(s) is an increasing function of s at s=s₀. This makes sense, since this function represents
the increase in expected revenue for a given supply level s. It is also worth noting that this result does not depend on the specific form of the profit function or any of its components, only on the variation of the revenue function.
Moreover, p(s) is itself everywhere decreasing. It is therefore likely that δp(s) will also be decreasing, and not impossible that sδp(s) will be decreasing at least for some range of values of s.

Conclusion
The conclusion of this analysis is that, if the demand changes in such a way that the expected revenue grows much faster at larger prices than the current market price than at smaller prices, the price of the commodity will increase in response to the increase in demand (as expected) while at the same time supply will contract. This is true even if the demand increases across the board!