Nonparametric Adverse Selection Problems

Nonparametric adverse selection problems

G. Carlier

December 19, 2000

Abstract

This article is devoted to adverse selection problems in which individual private information is a whole utility function and cannot be reduced to some nite-dimensional parameter. In this case, incentive compatibility conditions can be conveniently expressed using some abstract convexity notions arising in Mass Transfer Theory [8]. After this characterization is provided, an existence result of optimal incentive-compatible contracts is proved. Finally, several economic examples are considered including applications to regulation and labor contracting.

1 Introduction

Most papers on adverse selection problems with a continuum of agents focus on cases where agents' private information reduces to some nite dimensional vector usually labeled . The case of a scalar parameter was rst solved by

Mussa and Rosen [11] in the framework of monopoly nonlinear pricing. In this scalar setting nding almost explicitly optimal incentive-compatible con-tracts is in general possible. However, when the dimension of is larger than

2, important technical diculties arise even in the case of linear preferences (Chone and Rochet [12]) that prevent to solve the principal's program (min-imization of its total cost over the set of incentive-compatible contracts). In [4], a rst attempt was made to study a class of nonlinear preferences and characterize incentive-compatible contracts. However, even in this nite dimensional setting, determination of optimal contracts is out of reach in general.

In many applications, leaving apart technical issues, we believe that there is little justication in writing a particular nite dimensional parametrization of agents' preferences. Therefore it seems much more robust to assume that

the agents' information consists of their whole utility function and not of some nite dimensional vector of types . The very aim of this article is to

study that problem.

Our approach is based on the use of some abstract convexity notions arising in Mass Transfer Theory, see Y. Brenier [2], Mc Cann and Gangbo [3] and V. Levin [8]. We also intend to show that these tools are of very natural use in adverse selection theory. Let us mention that some similarities between Mass Transportation and Adverse-Selection Theory were already observed through abstract cyclical monotonicity arguments [8], [14]. Cyclical monotonicity is not used in this article since we have chosen a more direct approach.

The article is organized as follows. In section 2, we rst recall some basic denitions and then characterize incentive-compatible contracts in a nonparametric setting : incentive-compatible contracts can be derived from some X-convex potential, nally some properties of X-convex functions are

given. In section 3, the principal's program is written as some abstract vari-ational problem subject to anX-convexity constraint. In Section 4, we prove

an existence result. Finally, in section 5, several classic economic examples are revisited including regulation of rms and labor contracting.

2 Incentive compatibility and

X

-convexity

LetX be some convex compact subset of

R

n.

Xis called the space of actions.

In what followsC 0(

X) denotes the space of continuous real-valued

func-tions dened on X, equipped with the topology of uniform convergence, that

is with the norm :

kuk 0 := sup x2X ju(x)j for all u2C 0( X).

Let K be some convex compact subset of C 0(

X), one can think for

in-stance of : K :=fu2C 0( X); u concave ; kuk 0 M 0, u is M 1-Lipschitz g where M 0 and M

1 are positive constants.

In the following,M(X) denotes the space of Radon measures, that is the

topological dual space ofC 0(

X), and notation R

X

udwill be used extensively

to note the duality product between those two spaces. 2

Utility is assumed to be quasilinear, that is of the form:

W(u;x;t) :=u(x)?t for all (u;x;t)2KX

R

: (1)

In other words, if an agent with utility function u is provided an action x

and pays t then his payo is given by (1).

Remark.

The utility function u of an agent is indeed viewed here as his

type. Hence the representation of preferences captured here by the set K is

an ordinal one. Note also that ifucan be interpreted as the agent's parameter

or type, then it is an innite-dimensional one. Let us start with some denitions:

Denition 1

1) A contract is a pair of functions (x(:);t(:)) from K toX

R

. The function x(:) is called the action or physical part of the contract and t(:) is called the monetary of transfer part of the contract.

2) A contract (x(:);t(:)) is incentive-compatible if and only if: 8(u;u 0) 2K 2 ;u(x(u))?t(u)u(x(u 0)) ?t(u 0) : (2)

3) If (x(:);t(:)) is a contract, the potential associated with (x(:);t(:)) is the

function denoted V (x;t)(

:) from K to

R

dened by: V

(x;t)(

u) := u(x(u))?t(u) for all u2K:

Remark.

Condition (2) is called incentive-compatibility condition and ex-presses that it is optimal for every agent to announce his true utility function.

Let us dene now X-convexity and X-subdierentiability:

Denition 2

1) A function V from K to

R

[f+1gisX-convex if and only

if there exists a nonempty subset A of X

R

such that: V(u) = sup

(x;t)2A

u(x)?t, 8u2K:

2) Let V be a function from K to

R

[f+1g and u2 K, x2X is called a X-subgradient of V at u if and only if :

V(u 0) ?V(u)u 0( x)?u(x), 8u 0 2K:

The set of all X-subgradients of V at u is called the X-subdierential of V

at u and is denoted @ X

V(u). Finally, V is said to be X-subdierentiable at u2K if and only if @

X

V(u)6=;:

Remark.

Let V : K !

R

[f+1g be X-convex. With our compactness

assumptions onX and K, either V(u) = +1for allu2K, orV is bounded

in K. In other words, if we dene Dom(V) := fu2K : V(u)<+1g, then

either Dom(V) = ; or Dom(V) = K. In all the following, we shall only

consider the second case.

It can also be convenient in some cases to extendV outsideK by setting V(u) = +1 if u2= K.

The following proposition enables to characterize incentive-compatibility in terms of X-subdierential of someX-convex potential.

Proposition 1

Let (x(:);t(:)) be some contract. It is incentive-compatible,

if and only if the following conditions are satised : 1) V (x;t) is X-convex, 2) x(u)2@ X V (x;t)( u), 8u2K:

Proof.

Assume that (x(:);t(:)) is some incentive-compatible contract, then V

(x;t) is clearly

X-convex since, for all u2K : V (x;t)( u) = sup u 0 2K u(x(u 0)) ?t(u 0)

on the other hand, 8(u;u 0) 2K 2 : V (x;t)( u 0) = u 0( x(u 0)) ?t(u 0) u 0( x(u))?t(u) = V (x;t)( u) + (u 0 ?u)(x(u)) that is exactly x(u)2@ X V (x;t)( u).

Conversely, assume 1) and 2), for all (u;u 0) 2K 2 we have : V (x;t)( u 0) ?V (x;t)( u)(u 0 ?u)(x(u))

the previous can also be written u 0( x(u 0)) ?t(u 0)  u 0( x(u))?t(u) so that (x(:);t(:)) is incentive-compatible.

At this point, it is convenient to prove some properties ofX-convex

func-tions to make the principal's program tractable.

First recall the denition of subdierentiability (see [15]) : 4

Denition 3

Let V be a convex function from K to

R

and u 2 K  2 M(X) is called a subgradient of V at u if and only if :

V(u 0) ?V(u) Z X (u 0 ?u)d, 8u 0 2K:

The set of all subgradients of V at u is called the subdierential of V at u

and is denoted @V(u). Finally, V is said to be subdierentiable at u2 K if

and only if @V(u)6=;:

Proposition 2

Let V : K !

R

be X-convex and u be in K. Then the

following properties hold: 1) @

X

V(u) is a nonempty compact subset of X,

2) V is 1-Lipschitz-continuous and convex in K,

3) The set-valued map u7!@ X

V(u) dened on K has a closed graph,

4) x2@ X V(u) if and only if  x 2@V(u), where  x 2M(X) denotes the Dirac measure at x.

Proof.

By the denition of X-convexity, V can be written as V(u) = sup (x;t)2A u(x)?t, for allu2K (3) with A6=;, AX

R

. 1) Let (x n ;t n)

2Asuch thatV(u) = lim n u(x n) ?t n. Since Xis compact,

up to a subsequence, it may be assumed thatx

nconverges to

x2X, it follows

then that t

n converges to

u(x)?V(u). Let us show that x2@ X V(u). For allu 0 2K,V(u 0) u 0( x n) ?t n= ( u 0 ?u)(x n)+ u(x n) ?t n. Passing to

the limit in the previous yieldsV(u 0) V(u)+(u 0 ?u)(x) so thatx2@ X V(u), hence @ X V(u)6=;. Finally, @ X

V(u) is clearly compact since it is dened as an intersection

of closed subsets of the compact set X.

2) V is clearly convex since it is a supremum of ane functions. Let

(x;t)2A and (u;u 0) 2K 2, we have: u(x)?tku?u 0 k 0+ V(u 0)

taking the supremum in (x;t) 2 A of the leftmost member of the previous

inequality yieldsV(u)?V(u 0) ku?u 0 k 0. Reversing order of uandu 0 shows that V is 1-Lipschitz. 5

3) Let (u n

;x

n) be a sequence of

K  X, such that, for all n, x n 2 @ X V n( u

n). Assume this sequence converges to some (

u;x). Let us check that x2@ X V(u). Fix some u 0

2K, for alln, we have: V(u 0) V(u n) + ( u 0 ?u n)( x n)

which exactly yields the desired result passing to the limit since V is

contin-uous.

4) Proof follows directly from denitions 5.2 and 5.3.

We then have a characterization relating X-convexity to convexity, and X-subdierentiability to subdierentiability:

Proposition 3

Let V be a function from K to

R

then the following

proper-ties are equivalent: 1) V isX-convex,

2)V is convex and, for allu2K there existsx2X such that x

2@V(u),

where  x

2M(X) denotes the Dirac measure at x.

Proof.

By proposition 5.2, we already know that 1))2).

Assume 2), and dene, for allu2K: e V(u) := sup u 0 2K ; x 0 2G(u 0 ) (u?u 0 )(x 0 ) +V(u 0 ) where G(u 0) := fx 0 2X :  x 0 2@V(u 0) g. e V is X-convex and e

V  V. On the other hand, if (u;u 0) 2 K 2 and x 0 2 G(u 0) then V(u) V(u 0) + ( u?u 0)( x 0) so that V  e V. Hence V = e V is X-convex.

We end this section by dening X-subdierentiability with respect to

some nonempty subset of K, !, and constructing canonical extension ofX

-convex potentials. This extension will be implicitely used in the next section, in the case !:= supp() where  2M(K) captures the principal's belief on

the distribution of utility functions among agents.

LetV be some function: !!

R

, V isX-convex in ! if and only if there

exists a nonempty subset A of X

R

such that : V(u) = sup

(x;t)2A

u(x)?t, 8u2!:

IfV isX-convex in!andu2!, we dene theX-subdierential with respect to ! at u as the set : @ X ;! V(u) :=fx2X : V(u 0 )?V(u)u 0 (x)?u(x),8u 0 2!g:

One may check as in Proposition 5.2 that@ X ;!

V(u) is nonempty and compact.

Let us dene now the X-convex canonical extension of V, e V, by: e V(u) := sup u 0 2!; x 0 2@ X ;! V(u 0 ) (u?u 0 )(x 0 ) +V(u 0 ) (4)

for all u2K, so that e

V is X-convex in K and one may easily check: e V =V in! (5) and: @ X ;! V(u) =@ X e V(u), for all u2!: (6)

Remark.

In terms of incentive-compatibility, the previous construction shows that if a contract is incentive-compatible on some subset of agents then it can be extended to some incentive-compatible contract on the whole of K.

3 The principal's program

The principal is not able to observe agents' individual preferences prole, he has yet some a priori concerning the distribution of such proles. More precisely, we assume that the Principal's prior is given by some Radon prob-ability measure  2M(K), 0,

R K

d = 1.

The principal's cost functionC: X !

R

is assumed to be continuous (in

fact lower semi-continuous would be sucient to prove existence). Classically, the principal's program consists in nding contracts that minimize his total cost in the set of incentive-compatible contracts that satisfy an additional condition called the participation constraint. This program then takes the form: (P 0) 8 > > > > < > > > > : inf J(x;t) := Z K [C(x(u))?t(u)]d(u) s.t. : (x;t) is incentive-compatible, V (x;t)( u)0,-a.e. 7

ConstraintV (x;t)

0 is usually called participation constraint since it ensures

that the contract will provide at least 0 level of utility to each agent. Using Proposition 5.1, the previous program is indeed equivalent to:

(P) 8 > > > > > > < > > > > > > : inf F(x;V) := Z K [C(x(u))?u(x(u)) +V(u)]d(u) s.t. : V is X-convex, V 0,-a.e., x(u)2@ X V(u),8u2K:

Of course, F only depends on the restriction of V to ! = supp(). From

an economic point, this means that the admissible contracts only need to be incentive-compatible over ! and we do not assume that ! = K.

How-ever (4), (5) and (6) show that there is no need to explicit that dependence of the incentive-compatibility constraints with respect to !. For notational

purpose, we will therefore write the constraints as in (P) even though the

behavior of V outside ! is irrelevant.

4 Existence of optimal incentive-compatible

contracts

The aim of this section is to prove that (P

0) admits at least one solution

which, in economic terms, means that in our nonparametric setting, there exists in general optimal incentive-compatible contracts.

Theorem 1

There exists at least one solution to the principal's program (P

0).

Before we prove this Theorem, we need rst to study measurability issues and give some technical preliminaries concerning problem (P

0). In particular,

we have to investigate rst the -measurability of u 7! C(x(u))?u(x(u)),

where x(u)2@ X

V(u) and V isX-convex.

LetV be X-convex, since we are interested in minimizingF(:;V), there

is no loss of generality in replacing the inclusion x(:)2@ X V(:) by: x(u)2 V( u),8u2K (7) where V( u) is given by : V( u) := argmin @ X V(u) C(:)?u(:) (8) 8

The set-valued map, V is well dened, nonempty compact-valued and has

a closed graph.

Since we have replaced x(:) 2 @ X

V(:) by (7), C(x(:))?u(x(:)) only

depends on V, namely we have then:

C(x(:))?u(x(:)) =' V(

:) (9)

where ' V(

u) is dened for all u2K by : ' V( u) := min x2@ X V(u) C(x)?u(x) (10)

If V is X-convex, then it is easy to prove that '

V is lower semi-continuous

and bounded (uniformly inu andV) and, since K is compact and separable, '

V is

-measurable.

Now it is convenient to replace (P) by the equivalent program:

(P 1) 8 > > > > < > > > > : inf J(V) := Z K [' V( u) +V(u)]d(u) s.t. : V is X-convex, V 0, -a.e.

Now we are ready to prove the existence theorem:

Proof.

First, it is obvious by compactness of X and K that the value of

(P 1) is nite. Let V n be a minimizing sequence of ( P 1). By Proposition 5.2, V n is

1-Lipschitz, for all n. By Ascoli Theorem, we can extract a subsequence again

denoted V

n converging to some 1-Lipschitz

V in C 0(

K), equiped with the

uniform norm.

Let us show now that V isX-convex.

Dene for allu2K:

F(u) := \ N1 [ nN @ X V n( u)

since X is compact, and @ X

V n(

u)6=;, for alln,F(u)6=;.

Dene also for allu2K : e V(u) := sup u 0 2K, x 0 2F(u 0 ) V(u 0) + u(x 0) ?u 0( x 0) : 9

e

V is X-convex and one has clearly e

V  V. Let us show now the converse

inequality : let (u;u 0) 2 K 2 and x 0 2 F(u 0), by denition of F(u 0), there

exists an increasing sequence of integers N

n, and a sequence x 0 Nn 2 X such that x 0 Nn 2@ X V Nn( u 0) and x 0 Nn !x

0. Then, for all n: V N n( u)V N n( u 0) + ( u?u 0)( x 0 N n)

passing to the limit in the previous inequality yields then

V(u)V(u 0) + ( u?u 0)( x 0)

taking the supremum of the rightmost member in u 0 and

x

0 exactly yields V 

e

V. Finally, this shows that V is X-convex.

The last step of the proof consists in showing thatV is a solution of (P 1). First, we have: Z K V n d ! Z K Vd : (11)

Before going further, we need a technical result that we don't prove since it follows directly from convergence of V

n and of the very denition of X

-subdierential:

Lemma 1

Let u be in K. Assume that x n 2 @ X V n( u) and x n converges to x, then x2@ X V(u).

From Fatou's Lemma, we have: liminf n Z K ' Vn d  Z K liminf n ' Vn d : (12)

Fixu2K, for all n, ' Vn( u) =C(x n) ?u(x n) with x n 2@ X V n( u). Up to

a subsequence, we may assume that x

n converges to some x. The previous lemma implies x2@ X V(u). We get then: lim n ' Vn( u) = C(x)?u(x)' V( u): (13)

Using (11), (12), (13) and the fact that V

n is a minimizing sequence, we

exactly get that V is a solution to (P 1).

5 Examples

This section aims to sketch several applications of our method in various economic areas. First, we brie y show that the class of problems described previously encompasses the class of parametrized-utility models studied in [4]. Then, some specic problems are explicited arising in regulation theory, and labor contracting. Since the model problem of Section 3 actually is the nonparametric extension of the nonlinear pricing model of Chone and Rochet [12] we do not develop any longer this part of the adverse selection theory.

5.1 Parametrized utilities

In this section, we unify the nonparametric framework with the more classical case of parametrized utilities. Let X and K satisfy the same assumptions

as previously. Assume furthermore that each agent is characterized by some nite-dimensional parameter  2 which means that his utility function is v(;:) 2 K, where is some nonempty compact subset of

R

p and

v(:;:) is

continuous. In this parametrized model the principal's belief is usually given by some Radon probability measure on , denoted .

If we dene the continuous mappingT from toK byT() := v(;:) for

all  and the push-forward of through T,: Z K h(u)d(u) := Z h(T())d() (14)

for all continuous test-functionshthen the principal's program in this parametrized

setting is exactly of the form (P) with  given by (14). In other words, the

case of parametrized utilities is essentially the case when  is supported by

some nite-dimensional manifold of K.

5.2 Regulation of a monopolist

We brie y describe how the type of problems studied above can be used in analyzing the regulation of a monopolist with unknown cost function. This problem has received a lot attention in a parametric setting (e.g. the cost of the monopolist is ane with unobservable xed cost and marginal cost) Baron and Myerson [1], Rochet [13] and Laont and Tirole [7]. The case of rms with both unknown demand and costs functions has also been studied by Lewis and Sappington in [10] where the uncertainty is captured by some two-dimensional parameter.

A monopolist's technology is characterized by some cost function cfrom X := [0;x] to [0;+1) and the market demand function is some known

decreasing continuous functionD. A regulator basically aims to set a contract

that species both a selling price p and a monetary compensatory transfer t

with respect to the rm's characteristicc. The regulator's goal is to maximize

the consumer's expected surplus in the set of incentive-compatible regulating schemes.

If a rm with cost cimplements the contract (p;t), its benet is: pD(p)?c(D(p)) +t:

SinceDis invertible there is no loss of generality in considering contracts that

x quantities rather than selling prices by dening x=Dp. For notational

purpose we wil also change cin?c =:uso that the uncertainty on the rm's

technology is now denotedu. The regulator cannot observeubut knows that u belongs to some compact set of continuous functions K and his belief on

the distribution of technologies is given by some Radon probability measure

. Finally we dene for allx2X:

(x) :=D ?1

(x):x:

With those minor changes in mind, we can redene contracts as pair of functions (x;t) fromK to X

R

+ and dene the associated potential : V

(x;t)(

u) := (x(u)) +u(x(u)) +t(u) (15)

for all u2K.

A contract (x;t) is then incentive-compatible if and only if for all (u;u 0) 2 K 2 : V (x;t)( u) (x(u 0)) + u(x(u 0)) + t(u 0) : (16)

One can easily show that (x;t) is an incentive-compatible contract if and

only if V (x;t) is

X-convex and for allu2K: x(u)2@

X

V(u) (17)

V(u)? (x(u))?u(x(u))0: (18)

Condition (18) is the nonnegativity constraint on t.

Finally, the regulator's program takes the form: 12

> > > > > > > > < > > > > > > > > : inf K [V(u)?u(x(u))?S(x(u))]d(u) s.t. : V is X-convex, x(u)2@ X V(u); V 0, -a.e., V(u)? (x(u))?u(x(u))0,-a.e.

where S is the surplus function dened for allx2X by: S(x) := Z x 0 D ?1 (s)ds: (19)

5.3 Labor contracts

We consider now the case of a rm that aims to hire a population of workers with unobservable preferences. Here a contract both species a wage x 2 X = [0;x] and an eective working time t2[0;T].

Each agent is characterized by some utility function u 2 K, where K is

some compact set of continuous functions in X and as usual preferences are

given by:

u(x)?t:

If we assume that the technology of the rm is given by some production function t7! f(t), that the price of output is given equal to 1 and that the

demand for that output is not constrained then the problem of the rm is:

8 > > > > > > < > > > > > > : inf Z K [x(u)?f(t(u))]d(u) s.t. : (x;t) is incentive-compatible, V (x;t) 0, -a.e., t2[0;T],-a.e.

which can be rewritten as:

8 > > > > > > > > < > > > > > > > > : inf Z K [x(u)?f(u(x(u))?V(u)]d(u) s.t. : V is X-convex, x(u)2@ X V(u); V 0,-a.e., u(x(u))?V(u)2[0;T],-a.e. 13

Remark.

In most models of adverse selection in the labor market, the main unobservable parameter is productivity . If productivity is

heteroge-neous among agents then individual production cannot be reduced to eec-tive working time as previously but is equal to t= so that the agent's utility

takes the form :u(x)?t. One could easily extend the previous model by

setting v = (u;),  is only an additional scaling parameter and there is no

diculty in writing the principal's program in a similar way when  is also

unobservable.

References

[1] D. Baron, R. Myerson Regulating a monopolist with unknown costs , Econometrica, vol.50 (1982).

[2] Y. Brenier, Polar Factorization and monotone rearrangements of vector valued functions, Communications in Pure and Applied Mathematics, vol. 44, (1991).

[3] R.J. Mc Cann, W. Gangbo, The Geometry of Optimal Transportation, Acta Math., vol. 177 (1996).

[4] G. Carlier, On an optimal control problem with h-convexity constraint on the state variable and its economic motivation, cahier du CERE-MADE (1999).

[5] P. Chone, Etude de quelques problemes variationnels intervenant en geometrie riemannienne et en economie mathematique, PhD Thesis, Univ. Toulouse I (1999).

[6] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, North-Holland (1972).

[7] J.J. Laont, J. Tirole, Using cost observation to regulate rms, Journal of Political Economy, vol. 94 (1986).

[8] V. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich Problem, Set-Valued Analysis, vol. 7 (1999).

[9] V. Levin, Reduced cost functions and their applications, Journal of Mathematical Economics, vol. 18 (1997). Set-Valued Analysis, vol. 7 (1999).

[10] T.R. Lewis, E.M. Sappington, Regulating a monopolist with unknown demand and cost functions, RAND Journal of Economics, vol. 19 (1988).

[11] M. Mussa, S. Rosen, Monopoly and Product Quality, Journal of Eco-nomic Theory, vol. 18 (1978).

[12] J.-C. Rochet, P. Chone, Ironing, Sweeping and Multidimensional screening, Econometrica (1998).

[13] J.-C. Rochet, Monopoly regulation with two-dimensional uncertainty, Discussion paper, CEREMADE (1984).

[14] J.-C. Rochet, A necessary and Sucient Condition for Rationalizability in a Quasi-linear context, Journal of Mathematical Economics, vol. 16 (1987).

[15] R. T. Rockafellar, Convex Analysis, Princeton University Press (1970).