3.4. BICONTINUOUS TYPE 57
in the previous section, where we have a continuous and a discrete parameter for the type,
now we have that both dimensions are continuous, that is, the type space is the rectangle
Θ = (0, α) × (0, β), with α, β > 0.
As the decision q is a function defined from Θ ⊆ R
2
into R
+
, we expect that its level
sets are one-dimensional curves with images contained in Θ. The level sets for q represents a
continuous pooling of types, and the marginal condition states that pooling types must have
identical marginal utility at the same chosen decision. The result is that both q(a, b) and
v
q
(q(a, b), a, b) should have common level sets. This principle may be expressed in a convenient
way by using partial differential equation (PDE), as we will see in what follows.
Analysing the customer’s maximization problem we derive a partial differential equation
(PDE) whose solution can be interpreted as resolving this relation among the level sets. This
is a quasilinear first-order partial differential equation and we use the method of characteristic
curves to solve it. This method is adequate for our problem because the characteristic curves
for this particular PDE is exactly the level sets of both q(a, b) and v
q
(q(a, b), a, b). In the end
it provides us a natural change of variables that reduces the problem from two dimensions to
only one dimension. Then we apply variational calculus techniques to find the solution of this
one-dimensional problem and finally we go back to the original variables.
The parameter representing the customer’s type (a, b) ∈ Θ has a positive and continuous
density p(a, b). For simplicity, we assume that the monopolist’s production costs are zero.
We also assume that the monetary transference t is positive. Consequently the tariff T is
positive for all units consumed q. This means that the monopolist ever has positive profits when
selling to any customer (a, b). Consider an incentive compatible allocation rule (q, t). From the
(IC) condition, each customer has the following maximization problem:
max
(a
,b
)∈Θ
{v(q(a
, b
), a, b) − t(a
, b
)}.
The first-order conditions for the maximization problem above give us an analogous UC condi-
tion of the discrete-continuous problem
6
:
Proposition 8. (The Quasilinear Equation.) Suppose that (q, t) : Θ → R
+
×R is incentive
6
Actually the UC condition is equivalent to the PDE (3.11). The PDE formulation is more convenient to
the bicontinuous model.