How does monopoly affect the informational role of price?

It is said in the textbook "Intermediate Microeconomics a Modern Approach" by Hal Varian that monopoly affects the informational role of price. What's the mechanism?

A foundational result from models of perfect competition is that the competitive market equilibrium (i.e. setting price and quantity such that supply = demand) is efficient. By efficiency we mean that the sum of consumer surplus and producer surplus are maximised.

This is actually quite an amazing thing! It means that each individual consumer buys the good if and only if the welfare he receives from doing so is greater than the marginal cost of producing the good. To see that this must be true, suppose that there were a consumer who values a good at more than the cost of production but did not buy it. Then it would be possible to increase welfare by having the consumer buy the good, contradicting the result that the equilibrium is optimal. Similarly, efficiency demands that sellers produce exactly as much as is necessary to sell to the consumers who will ultimately buy.

How could we get to this efficient point? One way would be to have economists calculate what is efficient and then publish this information so that people could look-up whether they are supposed to buy/sell or not. But wait! We know that there aren't economists running round telling consumers whether it would be efficient for them to buy or not. So how do people figure this out for themselves?!

The answer is that the equilibrium price contains all of the information that they need to figure out what is efficient. Since the equilibrium price is equal to marginal cost, and since consumers will buy if their willingness to pay is above the price, a consumer wishes to purchase if amd only if his willngness to pay is bigger than the cost—i.e. exactly when it is efficient for him to do so. Thus, one way to think about the role of a price in a competitive market is that it transmits information to market participants about the efficiency of a trade they are contemplating.

For a monopoly things are different because the monopolist breaks the direct relationship between price and marginal cost by setting $p>MC$. This means that some consumers who have a willingness to pay that satisfied $p>WTP>MC$. The fact that $p>WTP$ means the price is signalling to these consumers that they should not buy, even though it would be efficient for them to do so (because $WTP>MC$). In other words, the price is no longer directly informative about the efficiency of a trade, so one way to think about the effect of monopoly on prices is that it reduces their information content.

It's still possible to calculate cost—and hence the efficient allocation—from the monopoly price, but we need more information. In particular, the monopoly price satisfies $$\frac{p-c}{p}=-\frac{1}{\eta},$$ where $\eta$ is the price elasticity of demand and $c$ is marginal cost. Thus, to figure out marginal cost we need know the price and the elasticity, whereas in a competitive market we only needed to observe the price.