How do we define consumer surplus in the case of perfectly inelastic demand?
This question was inspired by the comments following this answer. For a graph of inelastic demand please also see the linked question.
A motivation for a definition would be that while consumer surplus would be strange in this special case, change in consumer surplus may still be measurable easily with quite reasonable definitions.
From a purely theoretical perspective, if an individual's demand curve is perfectly inelastic, then her willingness to pay for the good is infinite. NB this also implies that she has an infinite budget. Thus, consumer surplus is well defined: it is the willingness to pay minus the price she pays, so as long as the price is finite her consumer surplus is finite.
In practice, no one has an infinite budget. So if the individual's demand curve is truly perfectly inelastic (i.e. the inverse demand is vertical), there exists a price such that beyond that price she can no longer afford to buy the good. This price is her willingness to pay, so consumer surplus is again well defined: the willingness to pay minus the price.
Let $Q^d = D(p)$ be the market demand function, depending on price $p$. Let $p^*$ be equilibrium price (that depends also on supply of course). Then Consumer Surplus is usually defined as
$$\text{CS}=\int_{p^*}^\infty\!D(p)\,dp$$
i.e. the "area under the demand curve", starting from equilibrium price. So it appears, that if $D(p) =\bar q>0$ (perfectly inelastic demand), then we would have
$$\text{CS}=\int_{p^*}^\infty\!\bar q\,dp =\bar q \cdot p\Big|^{\infty}_{p^*}\rightarrow \infty$$
Hmm, this is mathematically sound, but is it useful/meaningful from an economics perspective?
The critical point is the upper limit of the integral of course: by putting it equal to "infinity" we assume that for any price, however high, there will be some demand, however minuscule. In reality, after a price demand will drop to zero (simple linear downward sloping demand curves have this realistic property). Even for totally inelastic demand, after a price all consumers will be "budgeted out" of the market -that they "desperately need" to have the product does not mean that they will indeed get it if the price exceeds, say, their total wealth. In a graph, this means that the vertical line representing demand has an upper end, it does not extend to the sky. So if we realistically take into account this fact, and denote this supremum price by $P_s<\infty$, we can redefine Consumer Surplus as
$$\text{CS}=\int_{p^*}^{P_s}\!D(p)\,dp$$
and for totally inelastic demand we obtain
$$\text{CS}=\int_{p^*}^{P_s}\!\bar q\,dp =\bar q \cdot (P_s-p^*)$$
Graphically,
