Question: My students are currently studying demand, supply, and price. When we were discussing the elasticity/inelasticity of demand and supply, a question arose regarding if any good/service exists which possesses both perfectly inelastic demand and supply. I've given it considerable thought and the only goods/services I could think of which would fit the criteria are utilities (ie. electricity), water, waste water treatment, etc. I believe demand for these goods/services is inelastic as would be supply (no substitutes, few suppliers, oligopolistically competitive market structures, etc.). This line of reasoning would apply to their stock as well. Are there any other examples? Am I correct or totally off-base?
You are not totally off-base but your question mentions a common misunderstanding when it comes to the determinants of the elasticity of supply.
In your question you state "perfectly inelastic supply and demand". To that question the answer is no, both functions would be vertical and I know of no good that is perfectly inelastic in demand and supply. One often cited example of perfectly inelastic demand is insulin. Yet insulin is relatively elastic in supply.
Are there goods that are highly inelastic in demand and supply? Yes, but also keep in mind that the usage of elasticity you are referring to is a general, relative measure where one good is compared to another. Examples would be goods that have very few substitutes on the demand side and are very difficult to produce on the supply side (technologically intensive, raw materials are difficult to get quickly, production techniques cannot be modified easily). A good example is gasoline relative to, say, cigarettes or insulin. All are highly inelastic on the demand side but gasoline producers are not able to respond as quickly to a price increase as cigarette or insulin producers. Therefore gasoline has a more inelastic supply than cigarettes or insulin.
The difficulty for students when it comes to the elasticity of supply is to explain the concept without using an elasticity of demand determinant (like available substitutes, time frame for purchase, or proportion of budget). Elasticity of supply is the responsiveness of the producer to a change in the good's price and is therefore determined by production flexibility -ability to quickly and easily/cheaply increase or decrease quantity supplied. The fact that a product is elastic or inelastic to consumers does not necessarily mean anything when it comes to the elasticity of supply. Hence substitutes, few suppliers, or oligopolistic market structures are not determinants, by themselves, of inelastic supply. A monopolist could produce a good that has an elastic supply... think of Google's near monopoly over internet advertising. If the price for internet ads were to increase, it would be relatively easy for Google to respond with an increase in the quantity supplied.
Utilities are inelastic in demand but depending on time of year or day, may be fairly elastic in supply. Again, what's important is that elasticity is a relative concept.