Informational Equilibrium
Author(s): John G. Riley
Source: Econometrica, Vol. 47, No. 2 (Mar., 1979), pp. 331-359
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1914187
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Econometrica, Vol. 47, No. 2 (March, 1979)
INFORMATIONAL EQUILIBRIUM
BY JOHN G. RILEY1
If buyers are less well-informed about product quality than sellers, market prices will
reflect average quality. Sellers of high quality products therefore have an incentive to
engage in some distinguishing activity which operates as a signal to potential buyers.
This paper explores the viability of such signalling or "informational equilibria." It is
established that with a continuum of quality levels there is no Nash equilibrium. An
alternative non-cooperative equilibrium concept is then developed in which potential price
searching agents take account of possible reactions by other agents. It is shown that there is
a unique "reactive" informational equilibrium.
1. INTRODUCTION
AMONG RECENT ADVANCES in the economics of information, one of the more
controversial has been the development of the notion of an informational equilibrium. By this is meant an equilibrium in a world of incomplete markets, where
either the observed actions of better-informed agents or the resulting equilibrium
prices yield valuable information to worse-informed agents.
For example, in his path-breaking dissertation, Spence [12] suggested that
difficulties in observing human traits correlated with labor productivity, and in
mionitoring productivity, would result in an equilibrium where wage offers were
based on the educational credentials of the job seeker. That is, firms would use
education as a screening device to sift out workers of lower productivity. As
Spence emphasized, a crucial precondition for such an equilibrium is that those
with greater productivity are also faster learners in school and hence have lower
opportunity costs. Given this assumption, the higher productivity individuals,
facing wage offers contingent upon educational performance, find it in their
interest to accumulate higher credentials, and thereby provide a signal to potential employers.
Working independently, Rothschild and Stiglitz [11] and Wilson [16] have
proposed a similar model of information transmission in insurance markets.
Instead of offering a single price per unit of coverage, they show that firms have an
incentive to charge higher prices for greater coverage. In this way higher risk
individuals, for whom additional coverage yields greater marginal benefits, are
separated from lower risk individuals.
Several aspects of these models have aroused considerable debate. First, there
are those (e.g., Barzel [2]) who have criticized Spence's conclusion that in general
there will be "overinvestment" in the signal. The core of this criticism is that in the
absence of alternative means of communicating information, there may be no
feasible Pareto improving system of taxes and lump sum transfers. Then unless
one imposes a specific welfare criterion, Spence's normative inference is replaced
l I should particularly like to thank Charles Wilson and Michael Rothschild for their many helpful
comments. Discussions with James Mirrlees, Lloyd Shapley, Joseph Stiglitz, Earl Thompson, and
Michael Walker are also gratefully acknowledged. This research was supported by National Science
Foundation grant SOC-76-13443.
331
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332
JOHN
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RILEY
by the positive (and trivial) proposition that investment in the signal will be greater
than in a world of costless information.
Without disputing the above, it should be pointed out that there is a system of
taxes and transfers that, if adopted, eliminates the divergence between the private
and social marginal benefit of signalling. This provides some justification for
describing informational equilibria as "production inefficient."
Second, especially in reference to Spence's labor market model, it has been
argued (Layard and Psacharopoulos [7]) that firms are able to monitor an
employee's productivity at relatively low cost. If so, signalling in labor markets is
at most a phenomenon of minor importance. Ongoing empirical research may
help to resolve this issue.
Third, and most fundamental, has been the doubt raised about the viability of
informational equilibria. Spence's initial notion of an informational equilibrium
yielded an infinite number of equilibrium states. But Riley [9] showed that even
naive experimentation by buyers might "unravel" all but the Pareto dominating
member of this class. Independently, Rothschild/Stiglitz established that for a
simple model consistent with the Spencian framework, there may be no Nash
equilibrium set of contracts!
To establish such a result is, of course, to establish an inconsistency in the
assumptions made about the decision rules of the individual agents. One possible
way out of this dilemma, proposed by Rothschild/Stiglitz, is to assume that firms
consider only the alternatives that are in some sense "close" to an initial set of
offers. As will be shown below, however, there is no "local" Nash informational
equilibrium for a continuum of classes. This suggests that any definition of
"closeness" which ensures existence in finite class models is sensitive to the
number of these classes, that is, essentially ad hoc.
Given such a conclusion, it is natural to ask whether there exists a strategic
equilibrium in which each agent takes the reactions of other agents at least
partially into account. Rather than attempt to specify a truly dynamic interaction
model, the approach taken below is to search for a plausible "imperfectly
competitive" equilibrium concept.
The paper is organized as follows. In Section 2 the notion of an informationally
consistent price function is developed and a class of imperfect information models
is described. Spence's labor market model and the Rothschild/Stiglitz insurance
model are shown to be members of this class.
Section 3 considers the existence of "informationally consistent" Walrasian
equilibrium price functions. By this is meant a schedule of prices, conditional upon
the level of the signal, with the property that when sellers acting as price takers
choose their maximizing levels of the signal, buyers find that their quality forecasts
are correct. Given the assumptions of the model it is shown that there exists a
differentiable family of such functions.2 In addition, it is shown that all informa-
tionally consistent price functions must belong to this family.
2 In neither the discussion by Spence [13] nor Riley [9] is existence formally established, although in
the latter, strong necessary conditions are obtained.
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INFORMATIONAL EQUILIBRIUM 333
Since the proofs for this section are the most difficult, some readers may prefer
to jump from the description of Theorem 1 to Section 4. There it is established
that no members of the above family are noncooperative Nash equilibria. It is
demonstrated that for any informationally consistent price function, there is
always an alternative offer at the lowest quality level (e.g., lowest skill level)
which, if introduced by a single agent, yields the latter increased profits.3 Conditions are also established under which there are new profitable offers at higher
quality levels.
Finally, in Section 5, two alternative noncooperative equilibrium concepts are
examined. The notion central to both is that if buyers act as Nash competitors,
ignoring possible responses of other buyers, the actual and anticipated outcomes
of their actions will continually diverge. Given such divergence, it seems reason-
able to suppose that buyers eventually learn to predict how other buyers will react.
Actions are then taken only if profitable after the predicted reactions.
Specifically, it is argued that each buyer will learn to expect at least one other
buyer to react and exploit profitable opportunities opened up as a result of any
new offer. Such beliefs are shown to sustain the Pareto dominating informational
equilibrium.
Elsewhere, Wilson [16] has suggested a more defensive strategic equilibrium
concept. Instead of supposing that at least one buyer exploits new opportunities, it
is assumed that all buyers simply drop the contracts that would result in losses
given the presence of a new offer. For the continuum of agents model considered
in this paper, it is shown that Wilson's "anticipatory equilibrium" is never
informationally consistent.
Concluding remarks appear in Section 6.
2. A MODEL OF INFORMATION TRANSFER
Consider a market in which each of the set of potential sellers, I, provides one
unit of a commodity. Each seller i E I, may also engage in a sales-related activity at
an intensity yi E Y Writing the product price as p, the preferences of sellers,
defined over all ordered pairs {(y, p)ly E Y, p E1 R,}, are assumed representable by
the family of utility functions,
Ui = Ui(y,p), iEI.
Furthermore, differences among sellers are assumed to be parameterized by a
single unobservable characteristic 0 E 0. Then the family of utility functions can
be written in the alternative form,
Ui=U(0i;y,p), iEI.
If seller i has 0i units of the characteristic we may therefore re
of "type Oi." A seller of type 0i brings to market a product that
each of the set of notential buyers. J. This valuation. Vi. measured in dollars. is an
3The method of proof follows closely that suggested in correspondence with Stiglitz early in 1974.
This corrects an error in a later and apparently more general proof appearing in Riley [9].
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334
JOHN
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RILEY
increasing function of 0 and (possibly) also of the intensity of sales-related
activity, y:
Ve = VA0; Y).
Buyers are assumed to begin at an informational disadvantage in that prior to
purchase they are unable to distinguish directly variations in product value. Then
in the absence of any additional information and assuming buyers to be risk-
neutral, equilibrium trade must all take place at a single price, p, equal to the
average value of traded products.
The exact nature of the equilibrium, however, critically depends on the
reservation prices of the different sellers. If reservation prices are an increasing
function of product value and the highest reservation prices exceed average value,
the most valuable products are not traded. Because this lowers the average value
of products offered for sale more high quality products are withdrawn. As Akerlof
[1] has noted, such an "adverse selection" process may even continue until only
the lowest quality "lemons" are actually traded.
The question posed by Spence and others is whether there are conditions under
which price differentials, reflecting quality differentials, are viable even in this
world of extreme informational asymmetry. Specifically, if there is some activity y
whose marginal cost is lower for sellers of higher quality products, might not those
sellers "signal" that they have a superior product by selecting a higher level of y?
To answer this question we must first define an informational equilibrium.
The approach (at least implicitly) taken by Spence was to extend the Walrasian
price taker model of modern general equilibrium theory. With prices conditioned
on y, that is, p = p (y), it is assumed that seller i e I, acting as a price taker chooses a
utility maximizing value of y. Formally,
(I) Vi E I, yi is chosen to maximize U(0j; y, p(y)).
On the other side of the market buyer i e J, also acting as a price taker purchases
a commodity from seller i in the belief that the price p(yi) does reflect the value of
the commodity. For such conditional forecasts to be realized we then require
(II) Vi Ei I, P(Yi) = V(Oi; Yi).
Following Rothschild and Stiglitz, we shall describe a price function p(y) as
informationally consistent (INC) if conditions (I) and (II) are satisfied.
Assumptions
ASSUMPTION 1: The unobservable characteristic 0 is distributed on 0= [0, 01
according to the strictly increasing function F(O) E C'.
ASSUMPTION 2: U(6; y, p) E C on 0 x Y x R, and V(O; y) E C on 0 x Y,
where Y = [O, y ].
ASSUMPTION 3: U(O; y, p) is strictly increasing in p.
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INFORMATIONAL EQUILIBRIUM 335
ASSUMPTION 4: V(O; y) > 0, 1(0; y) > 0, V2(0; y)O .
ASSUMPTION 5: d Ug) <?
ASSUMPTION 6: V 0 E 0, U(0; y, V1(0; y)) is either a strictly decreasing function
of y, or it has a unique turning point at y*(0), the utility maximizing level of the
signal. Moreover VO E( 0 U(O; y, V(fi; y)) < U(0; y*(O), V(9; y*(O))).
Assumption 1 specifies that there is a continuum of different types of sellers. In
addition, along with Assumption 2, it imposes regularity restrictions upon
functional forms.4 Assumption 3 requires that all sellers strictly prefer higher
prices. Assumption 4 requires that all products brought to market have positive
value, that for any given y, a higher 0 is associated with a more valuable product,
and that additional inputs of y never reduce product quality. Assumption 5 is
crucial. The opportunity cost, in monetary units, of increasing the level of the
signal y is (- U2/ U3). Without the restriction that this be smaller for those capable
of producing a higher quality product (those with higher 0), it would always pay
those with lower 0 to mimic, and thereby obtain the higher price. Finally,
Assumption 6 is a further regularity condition eliminating the possibility of
multiple local maxima. It also requires, in essence, that every seller would prefer
to mimic the decision of the lowest quality seller rather than increase the level of
the signal to the largest feasible level in order to be paid the actual value of his
product.
Before turning to a discussion of informationally consistent price functions, we
note that the widely discussed labor market and insurance market models are both
special cases. In Spence [12], 0 is "natural ability"; y is education; V(0; y) is
marginal value product; and p(y) is the market wage. The welfare of type 0 is the
difference between the wage and the cost of obtaining an education y, that is,
U(0; y, p(y)) = p(y) - c(0; y).
The alternative welfare function
U(0; y, p(y)) = p(y)[i - t(0; y)]
allows an interpretation more explicitly in terms of the time costs of education;
p(y) becomes the wage per year, i is the life span, and t(0, y) is the time type 0
needs to obtain an education y.5
In the insurance model, insurance companies are assumed unable to observe
the odds, 0, that an individual with income I will not occur a loss L. However,
they obtain indirect information from the type of policy an individual purchases.
With full information and fair insurance each individual would choose full
4 It should be noted that, except for the discussion of pooling in the latter part of Section 4, t
requirement is that F(6), U(6; y, p), and V(6; y) E C2.
S This alternative model was suggested originally by Mirrlees. For another human capital interpretation, see Riley [10].
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336
JOHN
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RILEY
coverage. With imperfect information individuals are offered the opportunity of
reducing their coverage by an amount y to L - y. As an incentive, insurance
companies offer a higher indemnity-premium ratio p(y). The premium, paid in
both states, is then (L - y)/p(y) and expected utility of type 0 is given by
U(O; y, p(y))-= +oU I- u Y)l I-y- )
Assuming insurance is actuarially fair the ratio of the premium to the indemnity
( = l/p) must equal the probability of a loss. Then condition (II) reduces simply to
p(y) = 1 + 0.
A third application suggested by Leland and Pyle [8] involves signalling in
financial markets. The usual capital asset pricing model is modified to give buyers
(outsiders) less information about the mean return of an asset than the initial
owners (insiders). The latter prefer to spread their risk by selling shares. However
buyers use a larger insider position, y, as a signal of a higher mean return. As a
result the insider position is larger for a superior asset and this is reflected in a
higher market price p(y).
It is easy to show that for each of the above interpretations of the basic signalling
model, restrictions that ensure Assumptions 1-6 are natural.6
3. EXISTENCE AND DIFFERENTIABILITY OF INFORMATIONALLY CONSISTENT
PRICE FUNCTIONS
While earlier papers by Spence [13] and Riley [9] described necessary conditions for the existence of informationally consistent price functions, this has been
formally established only for examples. In the first part of this section it is shown
that if Assumptions 1-6 are satisfied, there is a family of INC price functions, each
member of which belongs to C'. It is then established that under the same
assumptions every INC price function must belong to this family.
It will be convenient to express a particular bid p, for a commodity with signal
level y, as the ordered pair (y, p). Then the expression (y, p) -d.(y', p') is a
statement that type 0 is indifferent between the two "offers," (y, p) and (y', p'),
that is, U(0; y, p) = U(0; y', p').
In Figure 1, indifference curves are depicted for types 01 and 02. Since U is
strictly increasing in p, all points lying above an indifference curve are strictly
preferred.
LEMMA 1: At every offer (y, p), the slope of the indifference curve of type 01 is
greater than that of type 02, if and only if 01 < 02.
6The latter part of Assumption 6 is violated for the insurance case, if the difference between the
most and least favorable odds (9 - 9) is sufficiently large. In such cases, the lowest risk types may prefer
to drop out of the insurance market altogether. That is, signalling may not completely offset the
problem of adverse selection.
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INFORMATIONAL EQUILIBRIUM 337
PROOF: The indifference curve of type 0 through the offer (y', p') has slope
-U2(0; y', p')/U3(0; y', p'). From (A4) this is everywhere decreasing in 9.
Q.E.D.
Product price
p
<Y" P>W1L< Y' Pt>.
Pg~~~~e
P'~~~~~~~~~~~e <' 1e2
,,,,,,,,,.1,,~~~~ ,,, , < , , >
Signal
FIGURE 1.-Sellers' preferences.
It is this restriction on preference maps which opens the possibility of signalling.
As an immediate implication of Lemma 1, we have:
LEMMA 2: If types 01 and 92, where09 < 92, face the same set of offers and th
signal-price pair (y', p') is maximalfor type 92, the level of the signal chosen by t
01 never exceeds y'.
PROOF: Consider again Figure 1. If (y', p') is maximal for type 02, all other
offers must lie in the vertically shaded region. Either (y', p') is maximal for type 01,
or there is some alternative offer lying in his preferred set (the horizontally shaded
region). From Lemma 1 the indifference curve {(y, p)R(y, p) 1-(y'p')} has greater
slope at (y', p') than {(y, p)(yY, P) 2-(y', p')}. Furthermore, these indifference
curves cannot again cross. Then the intersection of the two shaded regions must lie
to
the
left
of
y
=
y'.
Q.E.D.
Next consider the ordinary differential equation
(1) -= b-(y, ).
dy
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338
JOHN
G.
RILEY
Suppose
(a) b(y, 9)e C' on Yx0.
(,3) For each 9 e 0, either b(y, 9) > 0, Vy e Y, or b(y, 9) = 0 at a unique point in
Y and b(y, 9) > 0 only for larger y.
From (,1) it follows that for any fixed level of 9 in Y x 0 (a horizontal line in
Figure 2), d9/dy is either positive for all y or is first negative and then positive.
Then the family of solutions to the ordinary differential equation (1) can be
written as
(2) 9 = g(y; k) with k > ?
ak
where g(y; k) e C', has a unique minimum in Y, and no interior maximum.
It is a straightforward matter to check that the function
Af a) -(a/ay){U(9; y, V(; y))}
by 0-U3(9; y, V(9; y))Vl(9; y)
satisfies conditions (a) and (,1). From Assumption 3 the denominator is str
positive. From Assumption 2 both numerator and denominator belong to C',
hence so must b(y, 9). Finally, from Assumption 5, the numerator changes sign at
most once on Y for each 9 e 0, and is positive for sufficiently large y. It follows
that the solutions to the ordinary differential equation
(3) d9 -(a/ay){U(9; y, V(; y))}
dy U3 V1
take the form 9 = g(y; k) as depicted in Figure 2.
For each value of the constant of integration k, we can define on Y, a price
function
(4) p(y;k)= V(g(y; k);y).
Since g(y; k) e C' on Y and is strictly increasing in k, the funct
these same properties. Moreover, differentiating (4) yields:
(5) dp= Vdg+ V2.
dy dy
Substituting for dg/dy (= d9/dy) from (3) an
(6) U2 dy
+ ayU3-= d U(9; y, p(y; k))} = 0
Therefore, for all k, the first order conditions for utility maximization are satisfied
along 9 = g(y; k).
Note that for each 9, Assumption 6 requires that
-U(9; y, V(9; y))}<O
ay
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INFORMATIONAL EQUILIBRIUM 339
Unobservable characteristic
&
0 Y = Y * (9) &g- g ( (y; k)
=~~~~~~~ >; e k'
Pareto dominating
informationally
el. consistent signaling
profi I e
l
y
?. Y (8)
Y Signal
FIGURE 2.-Informational consistency.
if and only if y > y*(0). Then from (3), the locus of minimum points of the f
9 = g(y; k), is y = y*(9). This locus, depicted in Figure 2, is the level of the sig
that would be chosen by each type, if the product value, V(9; y), could be
costlessly determined by potential buyers.
Let k be the highest value of k for which the curves 9 = g(y; k) intersect 9 = 0.
Also define
(7) Ok = O k],
where 9k min {@, g(y; k)}. The non-negatively sloped segmen
for which k - k can be inverted to yield a mapping,
(8) y = f(9; k)
from 0k (the closure of Ek), onto a subinterval of Y. For each k we shall denote
the range of this mapping [f(q; k), f(k; k)] by Yk.
By construction the family of functions, y = f(k; k), is defined over the
unshaded region of Y x 0 in Figure 2. Except at the point (y*(9), 9), each curve
9 = g(y; k) has a strictly positive slope. Hence, the inverse function y = f(9; k) i
strictly increasing on 0k and differentiable on Ek*
With this background, we can now prove our first major result.
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340
JOHN
G.
THEOREM
p
RILEY
1:
For
all
(y{p(y;k)=
k
-
k
and
su
V(g(y;k);y
O, Y -E Yk,
is informationally consistent.
PROOF: See Appendix.
Theorem 1 demonstrates the existence of a family of informationally consistent
price functions. We now ask whether there might be other INC price functions.
The proof that this is not the case is established in the following Lemmas.
LEMMA 3: If p(y) is informationally consistent the level of the signal selected by
each type is unique. Moreover this level y =f(9), is an increasing continuous
function of 9.
PROOF: See Appendix.
As an immediate consequence of Lemmas 2 and 3 we also have:
LEMMA 4: If y = f(9) is the level of the signal chosen by type 9 and p(y) is
informationally consistent the utility of type 9 is strictly lower for all y ? f(9).
From Lemma 3, y = f(9) is an increasing continuous function of 9 with
range Yo = [f(9), f(Q)]. Hence it can be inverted and from condition (II) we must
have
(9) Vy E Yo, p(y)= V(F1 (y); y).
Since f1(y) is increasing and continuous over Yo, p(y) is a non-decreasing
continuous function of y. The following stronger result is proved in the appendix.
LEMMA 5: If p(y) is informationally consistentand Yo is the setof observed levels
of the signal, p(y) is differentiable on Yo.
Given Lemma 5 it is a straightforward matter to show that any INC price
function must belong to the family described in Theorem 1.
The optimal choice of type 6, f(6), must satisfy
a10
dp
0
(1 0) { U(0; Y, P (Y))} = U2 + U3=O.
ay
dy
Also, to satisfy Condition (II),
(11) p(y) = V(f (y); y).
Then, since p(y) is differentiable on Yo, so is the inverse function 6 = f1(y).
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INFORMATIONAL EQUILIBRIUM 341
Differentiating (11), we then have:
dp dO
-= v1-+ V2.
dy dy
Combining this with (10) and (11) yields
(3) dO -(a/ay){U(o; y, V(O; y))}
dy U3V1
But we have already described the solutions to (3) as belonging to the family
O = g(y; k). Hence:
THEOREM 2: If p(y) is informationally consistent, then over the set of chosen
levels of the signal Yo, p(y) is a member of the family of functions p = p(y; k).
4. INSTABILITY OF INFORMATIONALLY CONSISTENT PRICE FUNCTIONS
Spence, in his initial exposition of signalling, interpreted informational consistency as a necessary and sufficient condition for equilibrium. Indeed, the concept
of an informationally consistent price function is a natural extension of the
concept of a Walrasian equilibrium price vector. When only the seller knows
product quality, marketed commodities are characterized (labeled) by the level of
the signal y. Each potential seller facing the same set of parametric prices
{p = p(y)Iy E Y}, chooses to bring to market the product that maximizes his utility.
On the other side of the market, buyers, acting as price takers, find that the
products they purchase have the anticipated characteristics.
However, in contrast with the standard Walrasian equilibrium, informational
consistency is not sufficient to eliminate opportunities for potential gain by price
searching buyers.7
In the first part of this section, it is demonstrated that for all members of the
family of INC price functions, a single buyer can expect to gain from offering a
higher price to sellers who signal at the lowest level y(G). Of course, the buyer
makes losses on those products sold by type 0. However, the new offer is also
attractive to agents selling higher quality products. We shall see that there is a
price -, which attracts agents selling products with an average value exceeding p.
Consider the profile of welfare levels associated with an INC price function
P(Y):
U*(O) = max U(O; y, p(y)).
y
This has a slope at the lower end point of the distribution, 9, given by:
(12) dd1= Ul+ U2+U3dPd
dO 1[2 y dyJdO
= Ui (G; y, p(y))I
7 Rothschild and Stiglitz [11] describe the issues in the con
considers a continuum of agents. For a further discussion of ca
see Spence [14].
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342
JOHN
G.
RILEY
since from (6) the bracketed expression is zero.
Next consider the average value, V(8), of products offered by individuals in
[0, 6] assuming that they choose the same level of the signal, y, as type 0:
V(8) = V(x; y)F'(x) dx
Integrating by parts, this can be rewritten as
V(8) = V(6, y)-_ J Vl(x; y)F(x) dx
Then differentiating totally with respect to 6, we have
V ( ) = F'(){ f" V1(x; y)F(x) dx }
At 0 both the numerator and the denominator of the bracketed expression and
their first derivatives disappear. Applying l'H6pital's Rule twice we therefore
have
(13) V'(@)=F'(G)J Vii(; y)F(9)+ Vi(O; y)F'(8)
-I 2F(P)F"(9) +2F92
since F(6) = 0 and F(6) > 0.
Suppose an agent offers the average value V(8), over some interval [p,
anyone with the minimum level of the signal. The welfare of type 6, if he accepts
this new offer, is
(14) U(9) = U(6; y, V(8)).
Consider now the class of offers of the form (y, V(8)) for diffe
Clearly, U(9) = U*(9). Moreover, differentiating (14) with resp
d- U(9) = U1 + U3 V'(0)
U1 + U3 V1(6; y) from (13),
> Ui(6; y, V(6, y)) since V1, U3>0,
d
=Ud U(9) from (1 2).
dO-
The profile U(6) is, therefore, steeper at 0 and hence lies above U*(O) for some
interval (9, 01]. Then suppose the new offer is of the form
(y, V(W)) where G E (6, 01].
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INFORMATIONAL EQUILIBRIUM 343
For those in (6, 6], the welfare resulting from takin
U(6; y, V(8)) : U(6, y, V(8)) since 6 - 6,
= U() from (14),
> U*(6) since 6 E (, 1].
Therefore, all those individuals with 6 E [6, J] prefer the new offer. Since the
average value of the products offered by those in [6, 6] is V(8), the agent making
the new offer would just break even were no one else to accept. However,
0 E (6, 01] which implies that U(O) > U*(O). It follows that there is some interval
(6, 62) over which the new offer is also preferred. Sellers from this interval raise t
average value of products to V(02), thereby yielding an expected gain to the buyer
making the new offer.
We have, therefore, proved Theorem 3.
THEOREM 3: Given an informationally consistent set of offers (y, p(y)), there is
an alternative offer in the neighborhood of that accepted by those at the lower
endpoint of the distribution, which, if made by a single agent, results in purchases of
products with an expected value strictly greater than the price paid.
For the intuition behind this result, consider Figure 3. With each seller choosing
his utility maximizing offer (y(0), p(y(6))), p(y) is the lower envelope of the set of
indifference curves {(y, p) -@(y(6), p(y(6)))}. Then to a first approximation p(y) is
the indifference curve for any type 6 close to 0. To a first approximation, type 6 is
therefore indifferent between (y(6), p(y(6))) and (y, V(P; y)). Hence, any offer
(y, p) such that - exceeds V(P; y) is strictly preferred by both type 0 and type 6,
for 6 sufficiently close to 0.
However, if both types choose y, the average value of their products is a
weighted average of V(Q; y) and V(6; y) where the weights reflect the relati
frequencies of the two types in the population. Therefore, for any positive weigh
there is an offer (y, p), strictly preferred by both types, and yielding an expecte
gain to the buyer making the offer.
This argument cannot, however, be extended to alternative offers at levels of
the signal other than the minimum, y. The difference is that for any such "interior"
offer the higher price attracts not only sellers of superior quality products, but also
those with inferior products. Intuitively one might anticipate that any conclusions
about whether such an offer yields an expected gain will depend on the distribution of the unobservable 6. Indeed, this is the case. As long as the density
function declines sufficiently rapidly with increasing 6, the only alternative offers
yielding an expected gain to buyers are those in the neighborhood of the lower
endpoint of the distribution. However, if the density function increases with 6
sufficiently rapidly, the losses incurred as a result of attracting inferior quality
products are more than offset by the large number offering superior quality
products.
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344
JOHN
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RILEY
Product
P
price
p
(y)
<y, p>(<y(e), v(e;Y(e))>
v(8;y(8))
v(8,y)
v
p=
(8',Y)
y
p=v(;Y)
-_
V
;
Y)
_
y(e)
Y(8)
Signal
FIGURE 3.-Unraveling at the lower endpoint.
For a proof of these statements we consider an alternative offer at some leve
the signal - greater than the minimum, that is,
(y, where y = y(0), > , and p >p(y).
Clearly this offer is preferred by type 6, and for - sufficiently close to p(y) t
exist types 01( < ) and 62(> 6) who are just indifferent to the new offer. Let y
be the best offer for type Oi prior to the introduction of the alternative. Also
p =P(Ylji)
be the indifference curve for this type through (y(0i), p(y(0i))). Substituting fo
we may write
(15) p = p(y(0)IOi) p( I oi).
Such indifference curves are depicted in Figure 4 for types 01 and 62. Each
indicates the price that type Oi would have to be paid to make it worthwhile
signalling at the level initially chosen by type 6. By construction the curves jus
touch the profile of informationally consistent offers p*(O)=p(y(6)) at Oi and
intersect at (6, p).
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346
JOHN
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RILEY
Therefore,
U(1i; y(6),P) P -2 6k
and the indifference curve (15) becomes
p(0lai) =- 2(ai+
For types 01 and 62 this curve must pass through (6, p). We can, therefore
for -, obtaining:
P = (1 + 02).
The expected gain as a result of selecting at random from those preferring
offer is therefore proportional to
02
r@2
{J:2[
If
F(6)
losses.
is
v(;
everywher
However,
alternative
Of
course,
the
7Y)-p
Appendix
function
is
the neighborhood of p(y) yields an expected gain.
if
offers
w
in
gener
we
d
sufficien
THEOREM 4: If the distribution function F(6) is sufficiently convex at 6 there are
alternative offers (y(6), p) close to (y(6), p(y(6))) which yield expected gains on
each product purchased. If F(6) is sufficiently concave all such offers yield expected
losses.
This last result is important, because it indicates that in order to ensure the
viability of equilibrium, it is not sufficient to impose some special assumption
about the lower endpoint. However, given a density function which declines
sufficiently rapidly to make all new "local" interior offers unprofitable, what
assumptions would eliminate the endpoint problem?
The simplest approach is to assume that there is a mass point at the lower end of
the distribution (in the labor market, the army of the proletariat!). Then the
greater value of some of the products for sale at any higher bid price has a
negligible effect on average value. As a result, any agent making such a bid suffers
losses.
A more subtle approach is to modify the assumptions of the model in such a way
that the marginal cost of signalling increases without bound as 6 approach 0 from
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INFORMATIONAL EQUILIBRIUM 345
Product price
P
p
(ee2)
p(&t&i)/
/ P*(e)= p (y(e
/ I I | s (e& e2)
s(eiei)
I ' \> /// roUnobservable
ei e e 2 characteristic
FIGURE 4.-Interior unraveling.
Assumption (A5) implies that at any intersection of two indifference curves
satisfying (15), the steeper curve belongs to the individual with the lower value of
6. It follows that the new offer is preferred by type 6 if and only if 6 E (61, 02).
Before discussing the possibility of expected gains for the general case, consider
the following simple example first discussed by Spence [13] in his discussion of
labor market signalling:8
U(; y, p) =p-Y, V(6; y) = a,
where p is to be interpreted as the wage, and y/ G the cost of an education y for
individual of type 6. Spence shows that for informational consistency,
p(y) = (2y +k)/
Since p(y) = V(6; y) =, we can substitute for p(y) to obtain:
(16) y (6) = 102 - k.
8I am grateful to Mirrlees for this example.
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INFORMATIONAL
EQUILIBRIUM
347
above.9 Returning to the special labor market example, in which the marginal cost
of signalling is 1/0 for type 0, recall from equation (16) that
y(0) =102-k
The highest possible value of k is 02/2, since y (0) - 0. In this case, the net welfare
associated with the informationally consistent price function is
U*(0) = p(y(0)) - y(0)/0 = 2(0 + 02/0)
Note that for 0 > 0, (d/d@) U*(0) = 0. That is, those types with slightly greater 6
are no better off than type 0. Clearly, all would prefer to be offered a price equal to
the average value of their products.
In contrast, for the limiting case 0 = 0, U*(6) = 0/2. Moreover, if the distribution function is strictly concave, the average value over any interval [0, 6] is
less than the median value, 0/2. Then if 0 = 0, a new pooling offer at the lower
endpoint of the distribution yields expected losses. We have already seen that for
this simple model, concavity implies that alternative interior offers result in
expected losses. Therefore, if =0, the INC price function p(y) = (2y)1/2 is
viable. 10
Whether this result is generalizable remains an open question. However, while
it indicates the possibility of a "competitive" informational equilibrium, the
special assumptions required are hardly satisfactory. We now explore the viability
issue, in the absence of such assumptions.
5. ALTERNATIVE EQUILIBRIUM CONCEPTS
In Section 4 we examined the nature of informationally consistent (INC) price
functions, that is, price schedules for which the signal provided accurate information as to the value of the product offered for sale. Here it is useful to extend the
notion of informational consistency. A set of contracts will be described as weakly
informationally consistent (WINC) if the price paid to agents with a given signal is
equal to the average value of the products traded.
For example, if all agents choose the same level of the signal, -, and the single
price offered,
p= V(O y) dF(O)
(the average value of all traded products), ()7, a) is WINC. Alternatively, a WINC
set of contracts may separate out some subset of the agents, while the others are in
one or more heterogeneous pools.
We now analyze the viability of informationally consistent sets of contracts.
9Spence [15] uses an assumption of this type but does not consider the possibility of interior
unraveling.
t0For 0=O, the "utility" function U(G; y, p) = p-yl/ is undefined, thereby violating
Assumption 1.
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348
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First, consider the usual Nash noncooperative equilibrium concept. In our
context, this can be summarized as follows:
NASH EQUILIBRIUM: A set of weakly informationally consistent contracts is a
Nash equilibrium if and only if there exists no alternative offer which, if made by a
single buyer, would result in his purchasing products with an expected value in excess
of the price paid.
From Theorem 3 it follows immediately that there is no INC Nash equilibrium.
Moreover, it is easy to show that, whenever heterogeneous types are pooled, there
exist alternative profitable offers, which draw away the sellers of higher valued
products.
Consider a pooling contract, (y, 5), as depicted in Figure 5. Given Assumption
5, the types for which this is maximal are those on some closed interval [01, 02].
Moreover, for the contract to be WINC, the price paid, -, must be equal to the
average value,
02
J V(0; y) dF(0),
O1
Product price
p(y)
<Y, p>p>(2<y, p/
I <y,>>2 I < / py
Y (Gi1 ) Y E Ys YW82e) Signal
FIGURE 5.-Unraveling a pooling contract.
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INFORMATIONAL EQUILIBRIUM 349
of the products sold by types 0 E [01, 02]. Then the valuation curve for the highest
quality sellers in this interval, p = V(02; y) lies strictly above (3, p). Also the
valuation curve for the lowest type, p = V(01; y), lies strictly below (3, ).
Therefore, for each E > 0, and sufficiently small, there is some 0 in the interval
[01, 02] such that
V(0; Y) =P +?.
Consider the valuation curve for type 0E, p = V(06; y), and this type's
indifference curve through (, p5). Both curves are depicted in Figure 5. For all
y >3 the latter lies above the indifference curve of type 02 through (3, 5). Also
V(06; Y) < V(02; y). Therefore the two curves must meet to the right of (3, p)
some point (yS, pE). Suppose a firm offers (yS, pE). By construction: (i) type 0, is
just indifferent between (3, -) and (Ye, pE); (ii) if type 09 accepts the new offer, th
price paid, PE, is just equal to the value of the product purchased, V(06,; yE).
From Lemma 1, (i) implies that the new offer will not be chosen by any type
0< 0e. Furthermore, Assumption 5 implies that (yES, p) 4<(Y, -P) for every 0 E
(0e, 02]. Finally, since (3, p) is, by assumption, initially WINC, those types left in
the pool generate a loss. Combining this with the previous result, we have
therefore proved the following theorem.
THEOREM 5: There is no Nash equilibrium set of (weakly) informationally
consistent contracts.
As a possible alternative equilibrium concept, Rothschild and Stiglitz have
suggested that, in a world of imperfect information, agents might only consider
new offers "close to" the initial set of contracts. However, the results summarized
in Theorem 3 apply in any arbitrarily small neighborhood of the lower endpoint of
the distribution. Moreover, for all offers (y, p) on the indifference curve AB in
Figure 5, the argument made in the proof of Theorem 5 continues to hold. In
particular, it is true for offers arbitrarily close to (3, -). Hence:
THEOREM 6: There is no (weakly) informationally consistent local Nash equilibrium.
Given this negative conclusion, how might such a market in fact behave?
Presumably, in the absence of collusion, each agent would eventually learn to
expect some reaction by other agents to changes in his own list of offers. The
question we shall now consider is whether there are relatively simple reaction
functions which, if followed by all agents, support an equilibrium.1"
Consider again the method by which an INC price function is "unraveled." The
agent making the new offer benefits by attracting a pool of heterogeneous types.
But, as we have seen (Figure 5), it is then possible for a second agent to benefit by
attracting the sellers of the highest quality products from this pool. As a result, the
1 "The remainder of this section has been influenced by Wilson's elegant discussion of the viability
of equilibrium in insurance markets.
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350
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RILEY
first agent suffers losses. Then if agents learn to anticipate such reactions, it is
surely plausible that they will choose not to make this type of alternative offer.
These ideas suggest the following strategic equilibrium concept.
REACTIVE EQUILIBRIUM: A set of offers is a reactive equilibrium if, for any
additional offer which generates an expected gain to the agent making the offer, the
is another which yields a gain to a second agent and losses to the first. Moreover,
further addition to the set of offers generates losses to the second agent.
To be more precise, let Si be a feasible strategy of agent j. Also let O?(S, ... ., Sn)
be the outcome for agent j when the strategies adopted by agents 1 . . . n, are
51, S2, * * *, Sn.
Then So = (S, S, ... , SO) is a reactive equilibrium if for all j, and all Si, s
that
O?(Sol... , So1, Si, S+l1 ... , Son) ?i 0i(S0),
there exists an agent k and feasible strategy Sk, such that
(i) Ok(Sl, S2 *... Si ... Sk ... Son) 'k Ok(S0),
(ii) Ok (Sol, S52 .. Si * Sj * Sk . .. Son) >k Ok(Sob S20 . .. S;. . . Sok . . . Son),
(iii) Oi(Sob S( ... Sj... Sk ... Sn) -< Oj(So),
(iv) Vl ? j, k, and all feasible Si,
Ok (Sol, S20 ... Si ... Sk * 51 St .. Son)
'-k 0k ?(S 5?2 * 02 .. Si . S k ..5?1 . Son)
Note first that, in contrast to a game theoretic solution with strategic threats, the
reaction envisaged there is restricted to be directly advantageous. Recognizing the
deterrent effect of the reaction, it seems reasonable that the reacting agent be one
who does not gain from the initial "defection."
Second, the reaction is not a response by all agents in the market, but simply by
another agent who "undercuts" the new offer as long as it is offered.
Finally, the reacting agent, unlike the initial defector, cannot suffer losses as a
result of further defections. There are many additional offers that would benefit a
third agent and reduce the gains to the second reacting agent. However, the worst
that can happen is that the third agent bids away all those who might have been
attracted by the reacting agent. The essential point is that there is no particular
reason why a third agent should choose to make this kind of bid. Therefore, the
reacting agent can anticipate being strictly better off if he adds the reactive bid to
his original set of offers.12
In Riley [9] it is demonstrated that for each INC price function other than the
Pareto dominating price function p = p(y; k) there is an alternative offer which
12 While I have not been able to provide a proof, it is my conjecture that the results described below
also hold if the defector announces a complete new price function rather than a finite number of
additional bids.
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INFORMATIONAL EQUILIBRIUM 351
yields products whose value to a buyer exceeds price on all purchases. While the
proof was derived for the labor market application, it is readily extended.
The essence of the argument is that for all k < k, the lowest type 0 choose a level
of the signal y, such that the marginal benefit of signalling p'(y) exceeds V2(9; y).
But since marginal benefit and marginal opportunity cost are equated at the
optimal level of the signal, the private marginal cost of signalling is greater than
the social marginal benefit. This opens up an opportunity for a buyer to make an
offer "profitable" to himself and preferred by type 0 at a lower level of the signal.
Since type 0 sells the lowest quality product, any others accepting the offer only
enhance profitability.
Then the new offer is profitable for all purchases and in the face of reactive
offers, the worst that can occur is that all buyers are attracted elsewhere.
Next consider the Pareto dominating INC price function, p = p(y; k). In
Section 3 it was shown that for type 0, the utility level achieved is equal to the
utility level attainable in a world of costless information. Therefore, there can be
no alternative preferred offer at levels of the signal lower than y(9) which yields a
buyer profit on type 0. Any addition to the set of offers which yields expected
profits must therefore result in pooling of agents, some of whom generate losses.
But we can again apply the argument used in the proof of Theorem 5.
For each such pool, a second buyer can react by making an offer (yES, p) which
yields profits on all those accepting. Moreover, by choosing e sufficiently small,
the average value of the products sold by those for whom the original pool is still
attractive, can be reduced below the price paid, p. We have therefore proved:
THEOREM 7: There is a unique reactive equilibrium which is the Pareto
dominating member of the family of informationally consistent price functions.
While this theorem is somewhat reassuring, there is an obvious objection which
merits further attention. Any attempt to describe simple decision rules that result
in an equilibrium in which expectations based on signals are fulfilled, no matter
how plausible, is to some degree ad hoc. We must therefore ask whether there are
other decision rules which also imply the viability of an informational equilibrium.
And if so, do these alternative rules lead to different equilibria?
Wilson's discussion of insurance markets, with a finite number of risk classes,
suggests that the answer to both questions is in the affirmative. His suggestion is
that instead of one or more agents aggressively reacting to new opportunities for
profit, all buyers develop a defense mechanism. Each agent is assumed to be able
to anticipate the implications of a new offer, upon the profitability of all his own
offers. Offers which become unprofitable, as a result of the new offer, are simply
withdrawn. Given this behavior, the agent contemplating the introduction of a
new offer must ask whether it will remain profitable in the face of such a
withdrawal. We can summarize this equilibrium concept as follows:
WILSON ANTICIPATORY EQUILIBRIUM: Let E be an enlargement of a set of
WINC offers, such that no element of E is unprofitable and at least one is stric
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352
JOHN
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profitable. Let S(E) be the subset of the original offers which, as a result of the
enlargement, yield losses. Then the original set is an anticipatory equilibrium if there
is no enlargement E, such that the elements of E remain profitable after the
withdrawal of sufficient elements of S to ensure that none of those remaining gen
losses.
Focusing on the insurance market model, with a finite number of risk classes,
Wilson has shown that in general there is a WINC anticipatory equilibrium.
Unfortunately, his method of proof does not extend to the case discussed in this
paper in which an unobservable characteristic is continuously distributed.
However, it is conjectured that the theorem continues to hold.
From Theorem 3 we know that for any INC price function there is an alternative
profitable offer which does not result in losses to any agents. Instead, there are
simply no takers for other offers close to the bottom of the distribution. Then
withdrawing these offers has no effect on the profitability of those remaining. We
therefore have the following theorem:
THEOREM 8: Given Assumptions 1-6, there is no informationally consistent
Wilson anticipatory equilibrium.
Moreover, the above argument establishes that for any WINC Wilson equi-
librium, there must be some interval of types 0 E [0, 01] at the lower tail of the
distribution who will accept the same "pooling" offer.
The result summarized in Theorem 4 is also revealing. Once again, the new
offer does not create losses for any of those transacting at prices on the INC price
function. Then, if the distribution function is sufficiently convex over [0, 6], there
can be no subinterval over which the Wilson equilibrium is informationally
consistent. It follows that for such cases the Wilson equilibrium set of contracts
consists of one or more contracts, accepted by pools of heterogeneous types.
6. FINAL REMARKS
The question posed in the introduction concerned the viability of equilibria in
which agents' actions reveal valuable information to other agents. The first stage
in the analysis involved a characterization of a set of informationally consistent
contracts, that is, contracts for which each agent's expectations are realized. It was
then shown that, for a fairly general class of economies, there is no set of
informationally consistent contracts with the property that every additional
contract at best breaks even.
It should be noted that the lack of a Nash equilibrium is not a phenomenon
specific to this class of models. Recently, Grossman [4] and Grossman and Stiglitz
[5] have explored the implications of agents using commodity futures and asset
price movements, as signals of "insider" information. It is shown that if these
signals are "consistent" in the sense that they provide outsiders with all useful
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INFORMATIONAL EQUILIBRIUM 353
information, there may be no competitive "rational expectations" equilibrium.
Related results can also be found in Green [3] and Hart [6].
Given the unsatisfactory nature of the Nash equilibrium concept, two attempts
to characterize an imperfectly competitive equilibrium were then examined. In
each, the assumption that agents ignore any response by other agents was relaxed.
Since in both cases the modifications were relatively modest in scope, it could be
argued that it was not necessary to stray far from the security of the neo-Walrasian
fold in a search for a market equilibrium. However, the results of Section 5
indicate that the nature of the equilibrium is very sensitive to the type of response
function built into the model. The reactive equilibrium set of contracts is an
informationally consistent differentiable price function. In contrast, Wilson's
anticipatory equilibrium is never informationally consistent across all types, and
under strong assumptions, may involve trading at only a single price.
The constrast between the imperfectly competitive equilibria analyzed above
and the Arrow-Debreu, complete market equilibrium, is further sharpened by a
consideration of welfare implications. From Theorem 3, there is always a set of
contracts which would be preferred by all agents to the reactive equilibrium set,
and strictly preferred by those near the lower tail of the distribution. Moreover,
Wilson has shown that it is relatively easy to construct examples in which an
alternative set of contracts is strictly preferred by all agents, to the anticipatory
equilibrium set.13
It therefore seems fair to conclude that the imperfectly competitive market
structures described above are a major departure from the standard neo-Walrasian model.
This paper has indicated that the transmission of information via markets can b
explained as a noncooperative equilibrium phenomenon. However, the extent to
which information is transferred, even in our highly structured model, remains
unclear. In the reactive equilibrium the information transfer is complete. In the
anticipatory equilibrium it may be very imperfect. Further resolution of this
issue must await an exploration of the robustness of alternative non-myopic
equilibrium concepts.
University of California, Los Angeles
Manuscript received April, 1977; final revision received April, 1978.
APPENDIX
THEOREM 1: For all k -! k and sufficiently close to k the price function,
= p(y; k)= V(g(y; k); y), y E Yk,
0, Y Yk,
is informationally consistent.
13 However, it is the case that Wilson's anticipatory equilibrium is Pareto efficie
sets of weakly informationally consistent contracts.
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354
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PROOF: From equation (6), the first order condition for utility maximization is satisfied at
y(0) = f(6; k). Moreover, if type 6 chooses this level of the signal it follows from the definition of p(y)
that
p(y(6))= p(f(6; k); k)= V(g(f(6; k); k); f(6; k))
= V(6; y(0)) since g' = f.
Hence, for all 6 E Ok and k S k, Condition (II) is satisfied. The proof is completed in three steps. First it
is shown that y = f(Q; k) is the unique global maximum over Yk for all 6 E Ok* It is then established that
for k sufficiently close to k, Ok = 0. Finally it is demonstrated that for k sufficiently close to k,
y = f(6; k) is also the global maximum over Y.
Suppose that for some 6 E Ok, Y = f(6; k) is not the unique global maximum. Let y' be a global
maximum for 6. If y' is in the interior of Yk, that is, f(q; k) < y'< f(k; k), there is some 6' such that
y'= f(O'; k). Since the first order condition is satisfied at y'= f(O'; k), we have
U2(0'; y', p(y')) + U3(6'; y', p d(Y))p= 0.
dy
Furthermore, since y' is the maximum for 6, we also have
U2(0; y', p(y'))+ U3(; y, p(y'))-= 0.
dy
But this violates Assumption 5.
If y'= f(Q; k), then
dy
However,
U2(9; y', p(y')) + U3(6; Y% p(y'))-= 0.
dy
Since 0 > 6, Assumption 5 is again violated. By an almost identical argument y' $ f(Ok; k).
Next, we must show that for all k sufficiently close to k, 0k = 0. Suppose this to be false. Then from
the definition of Ok, given by (7), we must have 0k = g(y; k) < 6. But if type Ok signals at the level y we
have
U(6k; y, V(y; k)) = U(6k; y, p(y; k)) since Condition (II) must be satisfied,
> U(6k; y*(9), p(y*(Q); k)) since y yields the unique global
maximum for type Ok,
= U(6k; y*(9), V(Q; y*(9)) since y*(Q) is maximal for type 0.
But this inequality contradicts Assumption 6. Then 0 k = 0, and since all functions are continuous in
we must have E0k = 0 for all k sufficiently close to k.
Finally we must establish that y(6) = f(6; k) is the maximum not only over Yk but also over Y. F
each type 6, we have
U(6; y(6), p(y(6))) = U(6; y(6), V(6; y(6))) since Condition (II) must be satisfied,
> U(6; y, V(6; y)), Vy > y(6), from Assumption 6 since y(0)
> Y*(O),
> U(6; y, 0) since V(6; y)>0.
Therefore no potential seller will ever prefer a zero price for a level of the signal greater than
y(6) = f(6; k).
It remains to consider levels of the signal lower than f(Q; k). From Lemma 2, the level of the signal
optimal for all other types is at least as large as the level optimal for type 0. Then we need to show that
the latter will prefer y = f(Q; k). By construction f(q; k) = y*(0) (see Figure 2). From Assumption 6,
y*(6) yields the solution of maxy {U(9; y, V(9; y))}. In particular, since V(6; y) > 0, we have
U(Q; y*(g), p(y*(6); k)) = U(Q; y*(6), V(G; y*(6)))> U(Q; y, O), Vy E Y.
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INFORMATIONAL EQUILIBRIUM 355
Then y = f(Q; k) is indeed optimal for type 0. Given the
that for all k sufficiently close to k, y = f(g; k) is opt
LEMMA 3: If p(y) is informationally consistent the level of the signal selected by each type is unique.
Moreover this level is an increasing continuous function of 6.
PROOF: Let y = 1(6) be the lowest and y = h (6) be the highest level of the signal selected by type 6.
From Lemma 2 we have for any 61,
h(6) S l(61), 0 < 61,
and
1(8) _-h (01), 0 >01.
We now establish that 1(61) = h(61). Suppose this to be false, that is,
(17) y' = 1(61) < h(61) = y".
We now show that an implication of (17) is that for 6< 61 and sufficiently close, informational
consistency requires y(6) > y'.
First of all, since both y' and y" are utility maximizing levels of the signal for type 61, we have
U(61; y', p(y')) = U(61; y", p(y')).
Moreover, p(y) is INC therefore the price paid must equal product quality. Therefore,
(18) U(61; y', p(y'))= U(61; y', V(61; y'))= U(61; y", V(61; y")).
From Assumption 6, U(6; y, V(6; y)) is a strictly quasi-concave function of y. Therefore, for all
y e (y', y"),
U(61; y', p(y')) < U(61; y, V(61; y)).
Moreover, since U is continuous in 6, there exists some interval (60, 61) and some y E (y', y") such that
U(O; y', p(y')) < U(O; y, V(O; y)), vo E= (0o, 01)Also,
(19) U(6; y', V(6; y')) < U(6; y', p(y')), VO E (6o, 1), since V(6; y') < V(01; y') = p(y').
These two inequalities combined with the strict quasi-concavity of U(6; y, V(6, y)) imply that
(20) U(6; y, V(6; y)) < U(6; y', p(y')), Vy : y' and 6 E (60, 61).
But p(y) is informationally consistent. Thus type 6 must choose an offer of the form (y
(20) such offers are dominated by (y', p(y')) for all y less than or equal to y'. Therefo
choose y > y'. But this contradicts Lemma 2.
Therefore for each 6 there is a unique chosen level of the signal y (6) which is increasing in 6. Suppose
this is discontinuous from the right at 61. Then y' = Y(61) < inf,0,0 y(0) = y". Since y(0) is maximal f
9 we must have
U(01; y', p(y')) = U(01; y', V(01; y')) >- U(01; y(o), P(Y(O))), VO > 01,
and
U(6; y', p(y')) S U(6; y(O), p(y(6))) = U(6; y(O), V(6; y(O))), V6> 61Letting 6 -e 61 from above, it therefore follows that
U(61; y', p(y')) = U(61; y', V(61; y')) = U(61; y"; V(61; y")).
But this is precisely condition (18). We may therefore reapply the argument used to prove that
l(6l) = h(61), obtaining the same contradiction.
Finally, an almost identical argument establishes that there can be no discontinuity from the left.
Q.E.D.
LEMMA 5: If p(y) is informationally consistentand Yo is the setof observed levels f the sig
differentiable on YO.
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356
JOHN
G.
RILEY
PROOF: Suppose an INC price function, p(y), is not differentiable at yo e Yo. Since p(y) is INC,
there is some type 60E 0 for which p(yo) = V(00; yo). Then there is some ko for which the function
O = g(y; k) defined by (2) passes through (yo, Oo).
Consider the differentiable function p*(y) = p(y; ko). From Equation (4) p*(yo) = V(00; yo) = p(yo).
Since p(y) is not differentiable at yo,p(y)-p*(y) is not differentiable at yo. Then (p(yo+h)p*(yo + h))/h has no limit point as h - 0. In particular, it does not approach zero. Hence, there exists
an ? > 0 and sequence hT of real numbers such that h -0 O and for all r
p(yo + h) -p*(yo + h,)>
hT
That is, either
(a) p(yo + hT) - (y? + h,) E 1h,|
or
(b) p(yo+h7,)-p*(yo+h,.)< -EIhTI.
Suppose (a) is true, then we have (dropping the arguments from the right hand side):
U(0; yo + hk, p(yO + h,)) -U(; yo, p(yo))
hTU2 + [p*(yo + hT) p()]3+[(+ T) p(+ T]3+OhT)
>hjU2? u/i-] +eI hTI~~~-p*yo)U3+Op(yohhp(oh).3 (
>hT[ U2 +-U3 dyJ+E|h|U + 0(h t)
Now recall from (6), that
U2 + U3 = 0.
dy
Therefore,
U(O; yO+ hr, p(yo+ h)) -U(O; yo, p(yo))> E jhTU3+ O(h1)
>0
for T sufficiently large.
Thus for all 0, (yo + h., p(yo + hT)) is strictly preferred to (yo, p(yo)). But this is im
an observed level of the signal.
Suppose (b) is true. Then define OT = f (yo + hT). Since f l(y) is continuous on Y, OT -- Oo as hT * 0.
Then
U(O, YO + hr; p(yo + h,)) -U(O9, yo, p(yo))
= kU2 + [P*(yo + hT) p()]3+ (O+ T) p y T) U3 + OhT)
< [U2+ U3 dy TY)]. TIU3+ (T)
U2 and U3 are evaluated at 0T rather than 00. However, both are differentiable. Therefore, we also
have
U2+ U3dp* (yo U +U+ dp*+(yo) +0(T- O0).
dy 0=0_ dy =60o
Substituting then yields
U(ffr; yO+ hT, p(yo+ lT)) - U(OT; yo, p(yO)) < IhTI{O(9T
<0
for r sufficiently large. In this case, type OT iS better off accepting the offer (yo, p(yo
(f(OT), p(f(OT))). But this is impossible, since f(OT) is, by definition, the maximizing level o
type
OT,
Q.E.D.
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INFORMATIONAL EQUILIBRIUM 357
LEMMA 6: If 0 = '(61 + 02), 6E [61, 62], and ar(6) is
integral
r02
2 r(0) dO
can be approximated as follows:
e2
{ i(6) d6 = ?r(i)(02 - 61) + ir'(j)(j- j)(02 - 61)
?+2i7r"(j)[1#2(02
2 r()[28-1)3 +-(j6~+(-0)2(6_ j)2(828)6i)1?
+ (80(62
1)4. PROOF: Taking a Taylor's series expansion of ir(6) around 6, we have, VO E [01, 62],
(21) m| m inr"Th(6u }(- )3 S r(6) - r(0) - r'(0)( - @) -2ir"(0)(6 - )2
(2) 6 ILE[01,02] r
-| max Vr"'() }(o - )3.
6 toAe(021
Also,
02
J (6 -) d= (0-0)(02- 61)
91
and
02
1 (6 _ J)2 dO = 112(02 - 61)3 + (_- 0)2(02 - 6k)
Finally, since (6- _)3 (6- 1), VO E [61, 621,
02
J (6- J)3 dO z41(02- 8 )4
Integrating over the inequalities in (21) then yields the desired result.
THEOREM 4: If the distribution function F(6) is sufficiently convex at 6 there are alternative offers
(y(6), p) close to (y(6), p(y(6))) which yield expected gains on each product purchased. If F(6) is
sufficiently concave all such offers yield expected losses.
PROOF: From Section 3, all informationally consistent price functions p = p(y) E C'. Moreover,
the utility maximizing signal y = y(6) E Ca on (9, Q). Then p*(6) = p(y(6)) E Ca on (0, 9). Also, given
Assumption 2 the indifference curve p = p(6 I6) E5 CO.
Consider the difference between these two curves
s(6I60) = P(6I60) -p*(6).
For all alternative offers A such that p-p*(0) is sufficiently small, there are types 61 and 62 for which, as
depicted in Figure 4,
(22) s(0l 1) = S(i1l2).
By construction, p(6I60) and p*(6) are tangential at 6 = Oi, and, from Lemma 4, the former lies strictly
above the latter for all 6 ? Oi. Then if the first nOn-zero derivative of s(010i), evaluated at Oi, is the mth,
m must be an even number.
While the analysis is easily generalized, we shall suppose that m = 2, at 6 = 61. Applying Taylor's
expansion,
(23) s(6161) = sii(611)(6 -_ 1)2+ O(6 -1)3.
Similarly,
(24) s(0102) = s11(02102)(0 _ 02)2+ 0(6- 02)3.
Combining (22), (23), and (24) then yields
(25) sli(6101)(- _1)2 + 0(0 - 61)3 = S11(02102)(0- 02)2+ 0(0- 02)3.
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358
JOHN
We
have
chosen
G.
RILEY
already
by
type
noted
6i,
that
s(6i,6)
s(010
E
C'.
obtain:
s11(02102) = s11(61161) + 0(62 - 6)Combining this with (25), we then have
s11(6116)(6-_ 61)2 + 0(j-_ 1)3
= si(01101)(j- 62)2+ (6- 02)20 (02- 1) + 0(6- 02)3.
Since s11(6116D)>0 and 6E4Ol, 62], this implies
1(- _ 61)2 - (6 - 02)21 = 0(02 - 01)
But the left-hand side is 1(2 - 61)(26 - 61 - 02)1. Therefore
(26) 6 = 2(01 + 62) + 0(02 - 61)2.
That is, to a first-order of approximation, 6 lies midway between 01 and 62. Al
Therefore, from (23) and (26)
(27) p - V(6; )=p- p*() = s(0) = 0(02 - 1)2.
Since the new offer (y, p) is preferred by all those in (01, 62), the resulting
is proportional to
r02 r02
(28) f 7r(6) do=J [ V(6; y) -1]F'(0) do.
From Assumption 2, the integrand is an element of CO; therefore we can
third-order approximation. From (28)
,I'(6)= V1(6; ?)F'(j)+[V(6; y)-j3]F"(j)
and
Ir"(6)= V 1(6; ?)F'(6) + 2 V1(6; )F"(6) + [V(6; y) - #]F"'(6).
Noting from (26) and (27) that both (V(6, y) -p#) and (6- 6) are of second order in (62- 61) and
ignoring all terms of fourth and higher order in (62 - 61), we have
(6)(02- 61) = [V(&; 7) - ]F'(j)(02- 61),
X(j) (02- 61)( - 6) = V1(6; Y))F'(6)(02 - 61)(6- 6),
XT"(j)[12(0- 281)3 (6 _)2(02 - 61)] = 11?[ V1 (J; 7)F'(j) + 2 V1(6; 9)F"(j)](62 - o1)3
Combining these expressions (24) can be rewritten as
2
J 7r(6) dO=F'(6) [V(J; 3)-1](02-01)+ V1(6, ?)(02-01)(6-6)
?4Vl (0; Y)(02 - 61) ? Vl (j; 9)(02 - 61) F'(j) + ?0(2Since V1(6, yD > 0 the sign of this expression is determined by the sign of F"(j)/F'(j) if the absolute
value of the latter is sufficiently large. Q.E.D.
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INFORMATIONAL EQUILIBRIUM 359
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