A monopolist considers whether to disclose some information about its product. The information will affect how the consumer values the product but its impossible to predict in advance how the consumer will react. With probability q the consumer will view it as good news and he would be willing to pay a high price V for the product. But with probability 1-q it will be viewed as bad news and the consumer would only be willing to pay a low price v where 0 < v < V.
The consumer’s reaction to the information is subjective and cannot be observed by the monopolist. That is, after disclosing the information, the monopolist can’t tell whether the consumer’s willingness to pay has risen to V or fallen to v.
In the absence of disclosure, the consumer is uncertain whether his the value is V or v and so his willingness to pay is equal to the expected value of the product, i.e. qV + (1-q)v. This is therefore the price the monopolist can earn.
Supposing that the monopolist can costlessly disclose the information, what would its profits be then? It won’t continue to charge the same price. Because with probability (1-q) the consumer’s willingness to pay has dropped to v and he would refuse to buy at a price of qV +(1-q)v. At that price he will buy only with probability q and since that would be true at any price up to V, the monopolist would do better setting a price of V and earning expected profit qV.
Alternatively he could set a price of v. For sure the consumer would agree to that price (whether his willingness to pay is V or v) and so profits will be v. And since this is the highest price that would be agreed to for sure, v and V are the only prices the monopoly would consider. The choice will depend on which is larger qV or v.
But note that both qV and v are smaller than qV +(1-q)v. Disclosing information lowers monopoly profits and so the information will be kept hidden.
This little model can play a role in the debate about mandatory calorie labeling.
Cheap Talk 
12 comments
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November 26, 2012 at 12:41 am
JW Mason
So the implicit assumptions here are that both the monopolist and the consumer know the true value of q.
In other words, even though the monopolist knows nothing about the consumer’s preferences over the hidden property, and the consumer knows nothing about the true value of the concealed property, both of them somehow arrive at the same estimate of the distribution of demand given full information. In other words, it just so happens that the consumer’s priors about the chance that the good has the property that the consumer prefers, is *exactly the same* as the monopolist’s prior that the property that the good has, is the one that the consumer prefers. The only reason this works is because you are using labeling both priors “q”, even though there is no reason to think they will be the same.
November 27, 2012 at 12:02 am
jeff
You are right I am assuming a common prior. But it would be even easier to construct an example like this without it.
November 26, 2012 at 2:28 am
Danny Lynch
This part puzzled me:
“The consumer’s reaction to the information is subjective and cannot be observed by the monopolist. That is, after disclosing the information, the monopolist can’t tell whether the consumer’s willingness to pay has risen to V or fallen to v.”
Why wouldn’t the monopolist be able to gather this information? Is that modeling something I’m not thinking of?
November 26, 2012 at 4:06 am
alex
Crucial in the analysis is the marginal cost of the good. If it is high, it may be that the monopolist prefers to disclose the information and charge a high price. The condition is
qV+(1-q)v-c
November 26, 2012 at 4:08 am
alex
oops, didn’t work… the solution is, after a line of computation: if c is greater than v the solution is to disclose the information and charge qV, otherwise don’t say anything and charge qV+(1-q)v
November 26, 2012 at 8:55 am
Hb
cf Lewis and Sappington 1994 in IER and Johnson and Myatt on 2006 on demand rotations and the result that a monopolist would like to release extremal information when such release leads to rotation-ordered demand curves, where as Alex writes which extreme you want to go to does indeed depend on the marginal cost. The bite of course is on intermediate information release (what does it mean, when does it lead to rotation-ordered demand). Johnson and Myatt provide some micro-foundations.
There is a fair bit of literature on this both in econ and marketing – some of it fairly recent.
(This is closely related also to Gentzkow and Kamenica’s work: if you have full flexibility on consistent posteriors then you want to make sure that efficient trades occur and allow consumers to know only that it is an efficient trade – this allows you to extract all the surplus.)
November 26, 2012 at 9:16 am
Steven
There’s a strange assumption in your model, and I’m not sure if it’s deliberate. For comparison, here’s how I would model the situation.
The product either contains phlebotinum (http://tvtropes.org/pmwiki/pmwiki.php/Main/AppliedPhlebotinum) or does not. The monopolist knows which is the case, while the consumer places prior probability of p that the product contains phlebotinum.
The monopolist doesn’t know whether considers phlebotinum to be a good or a bad. With probability q, the consumer values the product with phlebotinum at V and with probability 1-q he values the product with phlebotinum at v (0<v<V). He values the product without phlebotinum at u.
Your version of the model is a restriction of this model. Eliminating p and u from the model certainly has the virtue of simplicity, but I can't figure out any justification for the choice.
November 26, 2012 at 11:47 am
JW Mason
Steven-
Right, that’s exactly my response. Jeff implicitly assumes that p=q, but there’s no justification for this.
November 27, 2012 at 12:07 am
jeff
Steven I agree with what you are saying and no it was not deliberate. There is no good reason for q=p.
November 26, 2012 at 4:32 pm
Enrique
I too question the assumptions of the model, especially the lack of information of the monopolist after the consumer has made his choice. Presumably, the consumer’s choice reveals information to the monopolist, although the information revealed might be noise or a deceptive signal
November 27, 2012 at 7:13 am
twicker
As it happens, with the calorie thing, it’s assuming that the consumer will be disappointed to find out what the actual calorie count is. In fact, consumers are often surprised by how *few* calories certain restaurant foods have (in comparison to what they thought) – thus making purchases *more* likely.
McDonalds is using this to great effect:
McDonald’s: Favorites Under 400
December 2, 2012 at 10:06 pm
Anonymous
Here is a closely related work “The optimal private information in single unit monopoly” by Saak: http://www.sciencedirect.com/science/article/pii/S0165176505004088