10.2 A formal analysis of monopoly menu pricing
10.2.1 Quality-dependent prices
We assume that all consumers prefer higher quality and lower prices but that they di¤er in their relative valuations of these two attributes. We de…ne a consumer’s indirect utility as
v = (
U ( ; s) p if consumer buys one unit of quality s at price p, 0 if he does not buy,
where s 0 is the quality of the product, p 0 is its price, and is the consumer’s taste parameter. As all consumers value higher quality, we assume that U ( ; s) is an increasing function of s. As for the distribution of tastes , we assume for simplicity that it is concentrated at two points: there are two types of consumers, indexed by i = 1; 2, in respective proportions 1 and .a We take 2> 1and accordingly, we refer to type 1 as the ‘low type’and to type 2 as the ‘high type’. We assume that high-type consumers care more about quality than low-type consumers: for any quality level s, U ( 2; s) > U ( 1; s). Furthermore, we also assume that high-type consumers value more any increase in quality than low-type consumers: for any s2 > s1,
U ( 2; s2) U ( 2; s1) > U ( 1; s2) U ( 1; s1) . (SC)
This condition is the standard single-crossing property according to which higher types exhibit a greater willingness to pay for every increment in quality.b As we will see, this condition is necessary for menu pricing to be possible.
The monopolist is able to produce two exogenously given qualities s1 and s2, with s2 > s1, at respective constant unit costs c1 and c2 (with ci < U ( 1; si) to make the problem non-trivial). The question is whether the monopolist will choose to price-discriminate by o¤ering the two qualities priced appropriately, or whether he will prefer to o¤er a single quality. In
aThe results presented here continue to hold in settings with continuous distributions
of consumers along such that the probability density function is positive everywhere.
bIn the previous example, the single-crossing property translated in the fact that uni-
versities were willing to pay more than businesses for upgrading from the basic to the pro version.
the latter case, let us assume that the monopolist always prefers to o¤er the high quality s2. A su¢ cient condition for the monopolist to select the high quality is
U ( 1; s2) U ( 1; s1) > c2 c1; (HQ)
according to which the value low-type consumers attribute to an increase in quality is larger than the cost di¤erence between the two qualities. Then, the monopolist has two options: either he charges the high price equal to U ( 2; s2) and sells to high-type consumers only, or he lowers the price to U ( 1; s2) and sells to all consumers. The former option is more pro…table if the proportion of high types, , is large enough, namely if
> U ( 1; s2) c2 U ( 2; s2) c2 0
.
The pro…t from selling only the high-quality can thus be written as
s=
(
(U ( 2; s2) c2) if 0; U ( 1; s2) c2 if < 0:
Under menu pricing, the monopolist must …nd the pro…t maximising price pair (p1; p2) that induces type i consumers to select quality si? There are two concerns: participation (each consumer must do at least as well consuming the good as not consuming it) and self-selection (or incentive compatibility, each type of consumers must prefer their consumption to the consumption of the other type of consumers). For the low-type group, the participation and incentive compatibility constraints respectively write as
U ( 1; s1) p1 0 , p1 U ( 1; s1) (PC1)
U ( 1; s1) p1 U ( 1; s2) p2 , p1 p2 [U ( 1; s2) U ( 1; s1)] (IC1) Similarly for the high-type group:
U ( 2; s2) p2 0 , p2 U ( 2; s2) (PC2)
U ( 2; s2) p2 U ( 2; s1) p1 , p2 p1+ [U ( 2; s2) U ( 2; s1)] (IC1) Of course, the monopolist wants to choose p1 and p2 to be as large as possible. It follows that, in general, one of the …rst two inequalities and one of the second two inequalities will be binding. Our previous simple example taught us that what matters is participation of the low type and self-selection
of the high type; we expect thus (PC1) and (IC2) to bind. Let us demon- strate this. Suppose …rst, by contradiction, that (PC2) is binding. Then (IC2) implies that p2 p1+ p2 U ( 2; s1) or U ( 2; s1) p1. Using the as- sumption that high types care more about quality, we can write U ( 1; s1) < U ( 2; s1) p1, which contradicts (PC1). It follows that (PC2) is not bind- ing and that (IC2) is binding; that is, p2 = p1+ [U ( 2; s2) U ( 2; s1)] :
Now consider (PC1) and (IC1). If (IC1) were binding, we would have p1= p2 [U ( 1; s2) U ( 1; s1)]. Using the binding (IC2), the latter equality rewrites as p1 = p1+ [U ( 2; s2) U ( 2; s1)] [U ( 1; s2) U ( 1; s1)], which implies U ( 1; s2) U ( 1; s1) = U ( 2; s2) U ( 2; s1). As menu pricing supposes s2 > s1, this contradicts our initial assumption (SC). It follows that (IC1) is not binding and that (PC1) is binding, so
p1 = U ( 1; s1) , and
p2 = U ( 2; s2) [U ( 2; s1) U ( 1; s1)] :
Because U ( 2; s1) > U ( 1; s1), we observe that p2 < U ( 2; s2): the monopolist is not able to extract full surplus from high-type consumers.
Lesson 10.2 Consider a monopolist who o¤ ers two pairs of price and qual- ity to two types of consumers. Prices are chosen so as to fully appropriate the low-type’s consumer surplus. On the other hand, high-type consumers obtain some surplus (a so-called “information rent”) because they can always choose the low-quality o¤ ering instead.
When is menu pricing optimal?
We need now to compare pro…ts when the monopolist only sells the high quality and when it price discriminates by selling both qualities. In the latter case, pro…ts are given by
m= (1 ) [U ( 1; s1) c1] + [U ( 2; s2) (U ( 2; s1) U ( 1; s1)) c2] :
Consider …rst the case where the proportion of high-type consumers is large enough, so that the monopolist sells to them only when it produces a single quality ( 0). Then, menu pricing modi…es pro…ts as follows:
Menu pricing involves two opposite e¤ects. First, it increases pro…ts through market expansion: low-type consumers now buy the low quality, which yields a margin of U ( 1; s1) c1per consumer. Second, it decreases pro…ts because of cannibalization: high-type consumers still buy the high quality but now, at a price reduced by U ( 2; s1) U ( 1; s1). The net e¤ect is positive pro- vided that high-type consumers are not too numerous:
> 0 , < U ( 1; s1) c1 U ( 2; s1) c1
: (10.1)
The latter condition is compatible with our starting point if and only if > 0, which is equivalent to U ( 2; s2) c2 U ( 2; s1) c1 > U ( 1; s2) c2 U ( 1; s1) c1 :
Let us now examine the other case (i.e., < 0). Here, the monopolist sells the high quality at a low price to everyone if he decides to sell only one quality. The change in pro…ts induced by menu pricing is then given by
= m s= (1 ) [(U ( 1; s1) c1) (U ( 1; s2) c2)] + [(U ( 2; s2) U ( 2; s1)) (U ( 1; s2) U ( 1; s1))] :
There are again two opposite e¤ects: (i) pro…t from low-type consumers decreases (because they buy the low quality instead of the high quality, which is detrimental for the monopolist according to assumption (HQ)), but (ii) pro…t from high-type consumers increases (they continue to buy the high quality but pay now a higher price according to assumption (SC)). Here, the net e¤ect is positive as long as high-type agents are numerous enough:
> 0 , > U ( 1; s2) U ( 1; s1) (c2 c1) U ( 2; s2) U ( 2; s1) (c2 c1)
: (10.2)
For this condition to be compatible with our starting point, we need 0> , or U ( 2; s2) c2 U ( 2; s1) c1 > U ( 1; s2) c2 U ( 1; s1) c1 ; (10.3)
which is the exact same condition as in the previous case. Condition (10.3) says that going from low to high quality increases surplus proportionally
more for high-type consumers than for low-type consumers. We can there- fore conclude:
Lesson 10.3 Menu pricing is optimal if the proportion of high-type con- sumers, , is comprised between and , which supposes that going from low to high quality increases surplus proportionally more for high-type con- sumers than for low-type consumers.
Welfare e¤ects of menu pricing
Social welfare is computed as the sum of consumer surplus and the monop- olist’s pro…t. From the above analysis, we easily compute the following:
Ws = (
(U ( 2; s2) c2) if 0;
U ( 1; s2) c2+ [U ( 2; s2) U ( 1; s2)] if < 0: Wm = (1 ) (U ( 1; s1) c1) + (U ( 2; s2) c2) :
The change in welfare induced by menu pricing can then be computed as W = Wm Ws, with
W = (
(1 ) (U ( 1; s1) c1) > 0 if 0;
(1 ) [U ( 1; s2) U ( 1; s1) (c2 c1)] < 0 if < 0:
We observe that welfare increases when 0. In that case, menu pricing expands the market as low-type consumers are sold the low quality, whereas they are left out of the market when only the high quality is sold. In contrast, welfare decreases when < 0. Here, the monopolist chooses to cover the whole market when he sells only the high quality; so, under menu pricing, low type consumers are sold the low quality instead of the high one, although, according to condition (HQ), what they are willing to pay for higher quality is larger than the extra cost of producing higher quality; gains of trade are thus left unexploited and welfare is lower.
Lesson 10.4 Menu pricing improves welfare if selling the low quality leads to an expansion of the market; otherwise, menu pricing deteriorates welfare.