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Working Paper Series
Market-Based Regulation and the
Informational Content of Prices
WP 06-12
Philip Bond
University of Pennsylvania
Itay Goldstein
University of Pennsylvania
Edward Simpson Prescott
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Market-Based Regulation
and the Informational Content of Prices1
Philip Bond, University of Pennsylvania
Itay Goldstein, University of Pennsylvania
Edward Simpson Prescott,2 Federal Reserve Bank of Richmond
November 2006
Working Paper Number 06-12
JEL Codes: D82, D84, G14, G21
Key Words: Rational expectations, regulation, market discipline
1
We thank Beth Allen, Franklin Allen, Mitchell Berlin, Alon Brav, Douglas Diamond, Andrea
Eisfeldt, Gary Gorton, Richard Kihlstrom, Rajdeep Sengupta, Annette Vissing-Jorgensen, and seminar participants at CEMFI, Duke University, the Federal Reserve Bank of Philadelphia, the Federal
Reserve Bank of Richmond, IDEI (Toulouse), Imperial College, INSEAD, Northwestern University,
Rutgers University, SIFR (Stockholm), the University of Pennsylvania, the University of Virginia,
the Washington University Conference on Corporate Governance, and Yale University for their comments.
2
The views expressed in this paper do not necessarily reflect the views of the Federal Reserve
Bank of Richmond or the Federal Reserve System.
Abstract
Various laws and policy proposals call for regulators to make use of the information reflected
in market prices. We focus on a leading example of such a proposal, namely that bank
supervision should make use of the market prices of traded bank securities. We study the
theoretical underpinnings of this proposal in light of a key problem: if the regulator uses
market prices, prices adjust to reflect this use and potentially become less revealing. We
show that the feasibility of this proposal depends critically on the information gap between
the market and the regulator. Thus, there is a strong complementarity between market
information and the regulator’s information, which suggests that regulators should not
abandon other sources of information when learning from market prices. We demonstrate
that the type of security being traded matters for the observed equilibrium outcome and
discuss other policy measures that can increase the ability of regulators to make use of
market information.
1
Introduction
A basic premise in economics is that prices in financial markets aggregate useful information.
This premise underlies several recent laws and policy proposals that call for regulatory
agencies to use the information in market prices when making various decisions.
One
example is the Sarbanes-Oxley Act of 2002, which changed how publicly traded corporations
are governed in the United States.
Section 408 of the act calls for the Securities and
Exchange Commission to consider market data — namely, share price volatility and priceto-earnings ratios — when deciding whether to review the legality of a firm’s disclosures.
Another example is class action securities litigation. Courts in the United States use share
price changes as a guide for determining damages.1
Despite the attraction of such laws and proposals, policymakers must confront the following problem: if regulators use market prices, prices adjust to reflect this use and might
become less revealing.
In this paper, we analyze the extent to which the feedback from
regulatory actions to market prices prevents effective inference in equilibrium. Our analysis
focuses on arguably the most important policy proposal of this type, namely, that supervision of banks should make use of the information conveyed by the market prices of traded
bank securities. We demonstrate that the feasibility of this policy proposal depends critically on the information gap between the market and the bank regulator. Implementing a
successful policy of learning from market prices requires that the regulator has reasonably
precise information from other sources.
An important responsibility of bank supervisors is to assess the probability of a bank
failing and, if needed and possible, to take actions to reduce it. Presently, a supervisor’s
main tool for assessing the risk of failure is periodic direct examination.
Supervisors
evaluate all the important facets of a bank, including financial measures such as the strength
of its loan portfolio and balance sheet, but also non-financial measures such as the quality
of its management. This direct supervision is expensive. In the United States, the federal
and state governments spent nearly 3 billion dollars in 2005 supervising banks and similar
1
See, e.g., Cooper Alexander (1994).
1
institutions such as thrifts and credit unions.2 There are also limits to direct supervision.
Evaluating a bank’s balance sheet is more complex than it used to be, and the technical
skills for such evaluation are in short supply.
In addition, the supervisor may not be
able to obtain relevant information held by other market participants.
If market prices
contain information on a bank’s fundamentals, then the supervisor may be able to learn
this information from market prices, saving substantial expenses and improving the quality
of its information.
For these reasons, many recent proposals call for bank supervisors to use prices of traded
bank securities.
For example, one of the three pillars of the Basel II reform of capital
regulations is the use of market discipline, one form of which is the use of information in
market prices.3 A specific proposal, most recently advocated by Evanoff and Wall (2000),
would require a bank to regularly issue subordinated debt, partly so that supervisors could
use the price of it to monitor the health of the issuing bank.4 Finally, many policymakers
emphasize the benefits of using information conveyed by market prices. Speaking in 2001,
Gary Stern, President of the Federal Reserve Bank of Minneapolis, argued,5
Market data are generated by a very large number of participants.
Market
participants have their funds at risk of loss. A monetary incentive provides a
perspective on risk taking that is difficult to replicate in a supervisory context.
Unlike accounting-based measures, market data are generated on a nearly con2
Number is from authors’ calculations. It does not include costs borne by the examined banks themselves.
Including these indirect costs would increase the estimated cost substantially.
3
More generally, market discipline is the reliance on and use of private counterparty supervision to monitor
and limit bank risk. Information produced by market participants and reflected in market prices provides
one form of market discipline.
4
Closely related, the Gramm-Leach-Bliley Act (1999) mandates that large banks wishing to engage in
certain activities possess at least one outstanding issue of debt receiving a high rating from a credit rating
agency (Section 5136A). This rule allows supervisors to make use of the assessment of one type of market
participant, namely, credit rating agencies.
5
See http://www.minneapolisfed.org/pubs/region/01-09/stern.cfm
2
tinuous basis and to a considerable extent anticipates future performance and
conditions. Raw market prices are nearly free to supervisors. This characteristic seems particularly important given that supervisory resources are limited and
are diminishing in comparison to the complexity of large banking organizations.
Similarly, according to Alan Greenspan (2001),6
The Federal Reserve and other regulatory agencies already monitor subordinated debt yields and issuance patterns in evaluating the condition of large
banking organizations. ... This use of subordinated debt is one example of the
effort supervisors should undertake to employ data from a variety of markets.7
We evaluate the theory behind these proposals by studying a rational expectations model
with a financial market that trades the securities of a bank and with a regulator who can
take costly actions that improve a bank’s health. Such actions affect the value of the bank’s
securities. The price that is set in the financial market reflects the expected value of the
securities given the information available in the market. The bank regulator decides on his
action based on his own information about the bank’s fundamentals, and that which he can
infer from the prices of the bank’s securities.
A key problem in implementing a supervision policy that is based on market prices is
the following. If the regulator plans to act on the information the security prices convey
about the bank’s fundamentals, then the prices will not only reflect information about the
fundamentals, but also the effect of the regulator’s resulting action. Prices will simultaneously affect and reflect the regulator’s action, so both the prices and the price-dependent
6
See: http://www.minneapolisfed.org/pubs/region/01-09/greenspan.cfm
7
Federal Reserve use of market data is described in Feldman and Schmidt (2003). They surveyed Federal
Reserve supervisors to determine whether they used market data. They found that market signals were
frequently used by supervisors to help form their overall opinion of a bank as well as to assess the quality of
a bank’s borrowers. The Federal Deposit Insurance Corporation’s use of market data is described in Burton
and Seale (2005). They report that market signals are used and that these signals are frequently used in
off-site surveillance of banks and can be used to help target more detailed exams. For both bank supervisors,
however, the degree to which market signals were used varied across supervisory personnel.
3
action of the regulator will need to be consistent. This consistency condition can interfere
with the ability to infer information in equilibrium. For example, a low price may indicate
that the fundamentals are bad and thus call for the regulator’s intervention. It may also
indicate that the fundamentals are not bad enough to justify intervention, in which case the
price is low just because no intervention is expected (this assumes that intervention has a
positive effect on the value of the security).
Our analysis highlights that the ability of the regulator to make use of market information depends critically on the quality of his own information. When the regulator has
relatively precise information, he is able to learn from market prices and implement his
preferred intervention rule as a unique equilibrium. When the regulator’s information is
moderately precise, additional undesirable equilibria exist in which the regulator intervenes
either too much, or too little. Interestingly, in this range, the type of equilibrium — i.e.,
whether there is too much or too little intervention — depends on whether the traded security has a convex or a concave payoff. Thus, our model generates different implications for
learning from the price of equity than for learning from the price of debt. Finally, when the
regulator’s information is imprecise, he is unable to implement his preferred intervention
strategy in equilibrium. Overall, our results suggest that there is a strong complementarity
between the regulator’s information and the market’s information. Thus, regulators should
not completely abandon direct examination and should use it together with inference from
market prices.
We discuss additional policy measures that help to implement the desired market-based
intervention policy, even when the information gap between the market and the regulator
is not small.
These measures include observing the prices of multiple traded securities,
improving the transparency on the side of the regulator, introducing a security that pays
off in the event that the regulator intervenes, and taxing security holders to change the
effect of the regulator’s action on the value of their securities.
As mentioned in our opening paragraph, our analysis is based on the idea that market
prices provide useful information to regulators.
The usual justification for this is that
markets gather information from many different participants who trade on their private
4
information on bank fundamentals and who do not communicate with the regulators outside the trading process. This idea goes back to Hayek (1945), who argues that markets
provide an efficient mechanism for information production and aggregation.
The ability
of financial markets to produce information that accurately predicts future events has been
demonstrated empirically.
For example, Roll (1984) shows that private information of
citrus futures traders regarding weather conditions is impounded into citrus futures prices,
so that prices even improve upon public weather forecasts.
In the corporate-finance lit-
erature, papers by Luo (2005), Chen, Goldstein and Jiang (2006), and Bakke and Whited
(2006) provide evidence that information in prices guides real investment decisions. In the
banking literature, many papers document that bank security prices reflect underlying risk
(see Flannery [1998] and Furlong and Williams [2006] for surveys) and that markets have
information that regulators do not have. For example, works by Krainer and Lopez (2004)
and others find that market prices can forecast ratings downgrades by bank supervisors.8
Our analysis is also based on the idea that market prices reflect the expected result of the
subsequent regulatory action. DeYoung, Flannery, Lang, and Sorescu (2001) find that the
market prices of bank securities indeed reflect the likely regulatory actions that result from
information generated by bank supervisors. Papers by Covitz, Hancock, and Kwast (2004)
and Gropp, Vesala, and Vulpes (2004) find that the connection between market prices and
risk depends on the regulatory regime, i.e., if government support of debt holders is more
likely, then a weak connection between the price of debt and risk is observed.
Our model belongs to a class of models recently developed in the finance literature, in
which an economic agent seeks to glean information from a firm’s market price and then
takes an action that affects the firm’s value. Models of this type were analyzed by Khanna,
Slezak, and Bradley (1994), Dow and Gorton (1997), Subrahmanyam and Titman (1999),
Goldstein and Guembel (2005), Bond and Eraslan (2006), and Dow, Goldstein and Guembel
(2006). The difficulty in these models stems from the fact that prices affect and reflect firm
8
This effect is not found, however, in all markets; Gilbert, Vaughan, and Meyer (2003) find that prices
in the jumbo CD market does not contain information not already contained in the supervisory surveillance
model.
5
value at the same time — a feature that is missing from the vast majority of models of
financial markets, where the value of the firm is assumed to be exogenous.9
Our model
deals with a problem that is absent from existing models of this class: inferring information
from the price is complicated by the fact that one price may be consistent with different
fundamentals. This feature arises because, unlike in other models in this class, the agent
in our model (regulator) acts to maximize his own objective function rather than the value
of the traded security.
Three existing papers — namely, Sunder (1989), Bernanke and Woodford (1997), and
Birchler and Facchinetti (2004) — note that market-based regulation is prone to an inference
problem.
They do not, however, analyze and characterize equilibrium outcomes as a
function of the information gap between the market and the regulator or of the type of
security being traded. This equilibrium analysis along with the resulting policy implications
are the main results of our paper. Other papers study different dimensions of market-based
regulation. Rochet (2004) and Lehar, Seppi, and Strobl (2005) study how market prices
can help the regulator commit to an optimal regulatory policy.
Faure-Grimaud (2002)
points out that the availability of free stock price information reduces expropriation by the
regulator.
Morris and Shin (2005) argue that transparency by the central bank may be
detrimental as it reduces the ability of the central bank to learn from the market.
The remainder of the paper is organized as follows. In Section 2, we present the model.
Section 3 defines an equilibrium in our model. In Section 4, we characterize equilibrium
outcomes as a function of the information gap between the market and the regulator and
of the type of security being traded. Section 5 discusses policy measures that may restore
optimal intervention in equilibrium. Section 6 concludes. All proofs are relegated to the
Appendix.
9
For example, in papers such as Grossman and Stigltiz (1980), prices may reveal the information possessed
by some investors to other investors, but the value of the asset being traded remains unaffected.
6
2
The model
The model has one bank, a regulator, and a financial market that trades the bank’s securities.
There are three dates, t = 0, 1, 2.
At date 0, the prices of bank securities are
determined in the market. At date 1, the regulator may intervene in the bank’s operations.
Finally, at date 2, all security holders are paid.
2.1
The bank
In the absence of regulatory intervention, the bank’s assets generate a gross cash flow of
θ+ε−T at date 2. The component θ is stochastic and is realized at date 0. It represents the
information available in the economy at date 0 regarding the bank’s future cash flows. We
will often refer to θ as the fundamental of the bank. The component ε is also stochastic
and is realized at date 2.
It represents the component of the bank’s cash flow about
which no information is available at date 0. The component T is deterministic and can be
interpreted as an amount stolen by the bank’s manager or any other inefficiency involved
in the management of the bank. Throughout, we assume that the fundamental θ is drawn
£ ¤
uniformly from some interval θ, θ̄ . The shock ε is drawn from a single-peaked density
function g (the cumulative distribution function is G) with mean 0.
The bank has two types of securities: insured deposits and uninsured claims. We let D
denote the face value of deposits. This amount is insured by the regulator, i.e., if the bank
does not generate enough cash flow to pay D to depositors, the regulator will provide the
resources to ensure that depositors are paid in full. Other security holders are not insured.
They receive payment from the bank at date 2 if the bank has resources left after paying
D to depositors. The order in which they get paid is determined by a prespecified priority
rule.
2.2
The regulator
In practice, a bank regulator who believes that a bank is performing poorly can take several
actions to attempt to improve the bank’s health. He can directly improve a bank’s cash
7
flow by limiting capital distributions. He can also change its operations by restricting the
bank’s growth.
He can dismiss existing board members and managers and require the
hiring of new employees.
If the regulator believes the bank is suffering from temporary
liquidity problems, he can offer to provide funding at a below-market rate.
Formally, in our model the regulator has an opportunity to intervene in the bank’s business at date 1. For tractability, we assume that whatever the specific form of intervention,
the effect is to increase the bank’s date 2 cash flow by an amount T . Specifically, if the
regulator intervenes, the bank’s date 2 cash flow is θ + ε instead of θ + ε − T .10
When deciding whether to intervene, the regulator weighs the cost against the benefit.
We assume a fixed cost of intervention C. The benefit of intervention is that it reduces the
regulator’s expected payment to insured depositors. Let Γ (θ) denote the expected payment
to insured depositors absent regulatory intervention as a function of the fundamental θ. It
is given by
Γ (θ) = E [max {0, D − (θ + ε − T )}] = G (T − θ) D +
Z
D+T −θ
T −θ
(D − (θ + ε − T )) g (ε) dε.
(1)
Then, using V (θ) to denote the gain to the regulator from intervention, we get
V (θ) ≡ Γ (θ) − Γ (θ + T ) .
(2)
If the regulator were fully informed about θ, it would intervene when V (θ) > C and not
when V (θ) < C. Lemma 1 establishes an important property of the function V (θ):
Lemma 1 The value of regulatory intervention V (θ) is first increasing and then decreasing
in the fundamental θ.
Intuitively, at low fundamentals the regulator pays out on its deposit insurance obligation regardless of whether or not it intervenes, while at high fundamentals the bank has
10
We assume that the type of intervention performed by the regulator cannot be replicated by the banks’
security holders. We make this assumption because bank supervisors have legal powers that enable them
to control bank actions. Governance by security holders, on the other hand, is limited, as emphasized by
the literature on corporate governance.
8
enough resources to pay depositors in full, again independent of whether the regulator intervenes. Thus, the value of intervention is maximized at intermediate values of θ, where
intervention is most likely to reduce the regulator’s deposit insurance obligations.
For
tractability and to make the problem of economic interest, we assume that
¡ ¢
V (θ) − C > 0 > V θ̄ − C.
(3)
£ ¤
It follows that there exists a unique θ̂ ∈ θ, θ̄ at which the regulator is indifferent between
intervening and not intervening, i.e., V (θ̂) = C. For fundamentals below (above) θ̂, a fully
informed regulator would strictly prefer to intervene (not intervene).
It is worth stressing that our main results would be qualitatively unchanged if the
intervention cost C and/or the gross intervention benefit T were allowed to depend on θ.
In particular, our results would still hold if T exceeds C only when bank fundamentals
are poor, but C exceeds T when bank fundamentals are good. The key ingredient for our
analysis is the existence of a threshold fundamental θ̂, above (below) which the net benefit
of intervention to the regulator is negative (positive).
Finally, our focus in this paper is on a regulator that attempts to obtain his preferred outcome (minimizing the costs involved in deposit insurance), as opposed to the maximization
of a social welfare function. It is important to note, however, that the two maximization
problems coincide when the regulator’s funds are raised by distortionary taxes and when
the regulator is obliged to provide deposit insurance, for reasons exogenous to the model
(such as preventing bank runs).
2.3
The value of bank securities
While the regulator may intervene in the bank to reduce the expected cost of a bail-out, the
intervention also affects the value of the uninsured securities. Some uninsured securities
may be traded in the financial market at date 0, in which case their price reflects their
expected equilibrium value. Before turning to the equilibrium analysis, we characterize the
value of securities with and without regulatory intervention.
We let X (θ) denote the expected value of a security absent regulatory intervention.
9
We focus on securities whose value X (θ) is strictly increasing in θ.
A key property for
our equilibrium results will be whether X (·) is concave or convex.11 Two leading examples
of uninsured bank securities are equity and debt.
intervention is
X (θ) ≡
Z
D+T −θ
The expected value of equity absent
(ε − (D + T − θ)) g (ε) dε,
which is easily shown to be strictly increasing and convex in the fundamental θ.12 Likewise,
the expected value of uninsured debt with a face value of B is
Z D+B+T −θ
ε − (D + T − θ) g (ε) dε + (1 − G (D + B + T − θ)) B,
X (θ) ≡
D+T −θ
which can be shown to be strictly increasing, convex for low fundamentals, and concave for
high fundamentals.
Economically, the convex then concave shape arises because debt is
junior to deposits but senior to residual equity claims. In what follows, we will sometimes
refer to a convex security as equity and to a concave security as debt (assuming that we are
in the range where debt is concave). We conduct most of our analysis for the case in which
the regulator observes the price of only one traded security. In Section 5.2.1, we consider
the case in which multiple security prices can be observed.
We denote the expected value of regulatory intervention for investors holding the security
by U (θ). Our paper deals primarily with the case in which intervention affects the value
of the security only through its effect on the bank’s cash flows.
Most of the forms of
intervention discussed above fall within this category. In this case, the effect of regulatory
intervention is equivalent to an exogenous addition of T to the bank’s date 2 cash flow, so
U (θ) = X (θ + T ) − X (θ) .
(4)
In Section 5.2.4, we discuss a case in which intervention has an additional effect on the value
of the security, e.g., by taxing the security holders. As we demonstrate there, such a form
11
Our analysis applies equally to securities whose expected value is strictly decreasing in the fundamental
θ. In our analysis, a decreasing concave (convex) security is equivalent to an increasing convex (concave)
security.
12
If the bank has also issued debt with a face value of B, the expected value of equity absent intervention
is given by an analogous expression, with D simply replaced by D + B.
10
of intervention can resolve the inference problems analyzed in our paper, and this leads to
one of our policy implications.
2.4
Information
A key point in our analysis is that the regulator does not know θ, and may learn it from
market prices.
We assume that the realization of θ is known in the market at date 0,
and that it serves as a basis for the price formation. In addition, at date 0, the regulator
observes a private noisy signal of θ: φ = θ + ξ. We assume that ξ, the noise with which
the regulator observes the fundamental, is uniformly distributed over [−κ, κ].
One limitation of our informational structure is that it assumes that the regulator always
knows less than the information collectively possessed by market participants (i.e., the
information of market participants aggregates to θ, while the regulator only observes a noisy
signal of θ). To assess the robustness of the model, we have also analyzed an extension in
which the regulator sometimes has more information than the market. We model this by
assuming that the regulator sometimes observes θ + ε (recall that ε is not observed by the
market). Our analysis indicates that this extension does not affect the qualitative results
of the model. Details of this analysis are available upon request.
3
Equilibrium
3.1
Market prices
Before making its intervention decision, the regulator possesses two pieces of information:
its own signal φ and the observed prices of market securities P (if there are several traded
securities, P is a vector). An intervention policy is thus a function I (P, φ), where I ∈ [0, 1]
is the probability of intervention.
For a given intervention policy I (·, ·), the price of a traded security incorporates the intervention probability. Specifically, the equilibrium pricing function P satisfies the rational
expectations equilibrium (REE) condition
£ ¤
P (θ) = X (θ) + Eφ [I (P (θ) , φ) |θ] U (θ) for all θ ∈ θ, θ̄ .
11
(5)
The first component in this expression is the expected value of the security absent regulatory
intervention given the fundamental θ.
The second component is the additional value
stemming from the possibility of regulatory intervention, the probability of which depends
on the price P (θ) and the regulator’s own signal φ.
3.2
Time consistency
For most of the analysis in the paper (Section 5.2.5 is the exception), we also require the
regulator’s intervention policy to constitute a “best response” to the market price. That
is, we require the intervention policy to be time consistent. This implies that the regulator
intervenes with probability 1 when the expected benefit from intervention is greater than
the cost and intervenes with probability 0 when the expected benefit is smaller than the
cost.
Thus, under time consistency, I is either 0 or 1, except for the case in which the
expected benefit is exactly equal to the cost. In this case, the regulator may choose to play
a mixed strategy.
For a given pricing function P (·), the observation of a particular price P tells the
¡ ¢
regulator that the market observed a fundamental θ0 such that P θ0 = P . Formally,
the intervention policy I (·, ·) is time consistent given a pricing function P (·), when for all
³
´
equilibrium realizations P̃ , φ̃ of the price-signal pair,
⎧
h
i
³
´ ⎨ 1 if Eθ V (θ) |P (θ) = P̃ and φ̃ > C
h
i
.
I P̃ , φ̃ =
⎩ 0 if E V (θ) |P (θ) = P̃ and φ̃ < C
θ
(6)
(Note that if the expected value of intervention exactly equals the cost C, any probability
of intervention is time consistent.)
3.3
Equilibrium definition
The formal definition of an equilibrium is as follows:
Definition 1 A pricing function P (·) and an intervention policy I (·, ·) together constitute
an equilibrium if they satisfy the REE condition (5) and the time-consistency condition (6).
12
4
Market-based intervention
We now explore the equilibrium outcomes when there is one traded security and the regulator attempts to learn the fundamental θ from the price of this security. Along the way,
we will make a distinction between the equilibrium outcomes for a convex security (e.g.,
equity) and those for a concave security (e.g., debt). We start by defining an important
class of equilibria for which the regulator can perfectly infer the market’s information.
Definition 2 A fully revealing equilibrium is an equilibrium in which each price is associated with one fundamental, and thus the fundamental can be inferred from the price. An
equilibrium is essentially fully revealing if when a price is associated with more than one
fundamental the regulator can still distinguish among the different fundamentals based on
his private information.
In both fully revealing and essentially fully revealing equilibria, the regulator chooses
the optimal action based on θ: he intervenes when θ is less than the cutoff value θ̂ and
does not intervene when θ is above θ̂.
Thus, we will sometimes refer to fully revealing
and essentially fully revealing equilibria as equilibria with optimal intervention.
Clearly,
any other equilibrium does not feature optimal intervention since in such equilibrium, the
combination of the price and the regulator’s private signal does not enable the regulator to
infer θ perfectly.
From (4), the price function for the security under optimal intervention is given by
⎧
⎨ X (θ + T ) if θ < θ̂
P (θ) =
.
(7)
⎩ X (θ)
if θ > θ̂
This function is graphically depicted in Figure 1. (Note that Figure 1 and the other figures
in the paper are only schematic. In particular, the functions X and X + U need not be
linear.)
Inspection of Figure 1 reveals the difficulty in obtaining an equilibrium with optimal
intervention when the security holder’s gain from intervention satisfies (4). The difficulty
stems from the fact that under optimal intervention, the price function is non-monotone and
13
X (θ ) + U (θ )
P (θ )
X (θ )
θˆ
θˆ − T
θˆ + T
θ
Figure 1: Security price under optimal intervention
that the non-monotonicity occurs around θ̂.
This is a result of the optimal intervention
rule employed by the regulator – to intervene only when the fundamentals are below θ̂
– and the increase in the value of the security from intervention.
As a result of this
non-monotonicity, fundamentals on both sides of θ̂ have the same price, so the regulator
can infer neither the level of the fundamental, nor the the optimal action, from the price
alone. Essentially, the fact that the price reflects the expected reaction of the regulator to
the price makes learning from the price more difficult. Following this logic, the possibility
of obtaining the optimal intervention rule in equilibrium depends on the precision of the
regulator’s private signal. A precise private signal will enable the regulator to distinguish
between different fundamentals that have the same price.
We now provide a complete
analysis of equilibrium outcomes based on the precision of the regulator’s signal.13
13
The above discussion should make clear that the key force that makes inference hard in our model is the
non-monotonicity of the value of bank securities under optimal intervention. In our model, non-monotonicity
arises in part from the discreteness of the intervention decision, but this feature is certainly not necessary
for non-monotoncity.
14
4.1
The regulator’s signal is precise: unique equilibrium
We start with the case in which the regulator’s signal φ is relatively precise. Our first result
is as follows:
Proposition 1 For κ < T /2, an equilibrium with optimal intervention exists.
The intuition behind this result is that under the optimal intervention rule, there are at
most two fundamentals associated with each price. Moreover, these fundamentals are at a
distance T from each other (see Figure 1). Since the regulator’s signal is relatively precise,
the regulator can use the signal to perfectly infer the realization of the fundamental when
the price is consistent with two different fundamentals. Thus, he can follow the optimal
intervention rule.
It is worth stressing that in this equilibrium, both the price and the
private signal serve an important role: the price tells the regulator that one of two different
fundamentals may have been realized, while the private signal enables the regulator to
differentiate between these two fundamentals. Thus, the regulator uses both the price and
the private signal to infer the underlying fundamental.
Our next result shows that if the regulator’s signal is sufficiently accurate whenever
two fundamentals have the same equilibrium price, the regulator’s signal is sufficient to
distinguish them.
As such, the optimal intervention equilibrium is the only equilibrium.
Although intuitive, the proof of this result is involved.
The key difficulty is the need to
rule out equilibria in which there are an infinite number of fundamentals associated with
the same price.
Proposition 2 There is a κ̄ > 0, such that regardless of whether the traded security is convex or concave, when κ ≤ κ̄ the optimal intervention equilibrium is the unique equilibrium.
4.2
The regulator’s signal is moderately precise: multiple equilibria
As the regulator’s information precision worsens, in the sense that κ increases beyond the
point κ̄ defined by Proposition 2 and approaches T /2, the optimal intervention equilibrium
remains an equilibrium. However, additional and less desirable equilibria emerge. In such
15
X (θ ) + U (θ )
P (θ )
X (θ )
θˆ
θˆ − T
θˆ + T
θ
Figure 2: Security price in an equilibrium with too much intervention
equilibria, the regulator cannot perfectly infer the fundamental from market prices. As we
will establish, the form of these non-optimal intervention equilibria depends critically on
whether the expected security payoff X is convex or concave. Specifically, if X is convex,
then these alternate equilibria feature excessive intervention, while if X is concave, the
reverse is true.
4.2.1
The case of a convex security: equilibrium with too much intervention
We consider first the case in which the regulator observes the market price of a convex security. We establish that, in this case, there are equilibria in which the regulator intervenes
too much relative to the optimal intervention equilibrium depicted in Figure 1. Figure 2
depicts an example of such an over-intervention equilibrium.
As we can see in Figure 2, in this equilibrium, the regulator always intervenes when
the fundamentals are below θ̂ (the left line in the price function), never intervenes at some
fundamentals above θ̂ (the right line in the price function), and intervenes with positive
probability for other fundamentals above θ̂ (the middle line in the price function).
16
This
last feature reflects that the regulator intervenes too much in this equilibrium. This happens
because, at the fundamentals associated with the middle line, the price does not fully reveal
that the fundamentals are above θ̂, even after the price is combined with the regulator’s
private signal. That is, the distance between the middle line and the left line is too small
for the regulator to be able to differentiate between fundamentals associated with these
lines by using its private signal, since κ > κ̄. Because at fundamentals associated with the
middle line the regulator cannot rule out that the fundamental is below θ̂, he intervenes
with positive probability. This increases the price at these fundamentals. In turn, it is this
price increase that pushes the left and middle lines closer together, preventing the regulator
from being able to distinguish them.
We now formally prove the existence of an equilibrium with too much intervention. The
main result here is Proposition 3, which establishes the existence of such an equilibrium
and provides a full characterization of it.
proposition.
We start with some algebra leading to the
In an equilibrium with too much intervention, over-intervention occurs for
fundamentals lying in some interval to the right of θ̂.
As a first step, consider what is
required to generate too much intervention at a point θ̂+ infinitesimally to the right of θ̂.
Over-intervention occurs if the market price at θ̂+ is the same as the market price at some
´
³
θ ∈ θ̂+ − T, θ̂ . The probability of intervention at θ̂+ in this case is the probability that
the regulator receives a signal φ that is consistent with the fundamental θ:
´ θ + κ − θ̂ − (−κ)
³
θ̂+ − θ
+
=1−
.
Pr φ ≤ θ + κ|θ̂+ =
2κ
2κ
As such, the market prices at θ < θ̂ and θ̂+ coincide if and only if
Ã
!
³ ´
³ ´
θ̂+ − θ
X θ̂+ + 1 −
U θ̂+ = X (θ) + U (θ) .
2κ
(8)
We first show that there is a fundamental θ satisfying (8) whenever the security is convex,
2κ < T , and 2κ is sufficiently close to T :
Lemma 2 For κ < T /2 sufficiently close to T /2, there exists a unique θ01 < θ̂ such that
(8) holds.
17
Lemma 2 establishes that too much intervention is possible at a point immediately to
the right of θ̂. The convexity of the security is important here. To see why, let us inspect
equation (8).
Obviously, the LHS in (8) is equal to the RHS when θ = θ̂+ .
because 2κ < T , the LHS is smaller than the RHS when θ = θ̂+ − 2κ.14
Moreover,
Since the LHS
is linear and increasing in θ, a necessary condition for (8) to have a solution below θ̂ is
that its RHS is convex in θ. This implies that the security has to be convex for too much
intervention to occur.
Using the result in Lemma 2, the following proposition fully establishes and characterizes
the equilibrium. To better understand the proposition, it is useful to look again at Figure
2, which depicts an equilibrium of the type described here. As Figure 2 shows, there are at
most three fundamentals associated with each price. If one takes this property as given, it
is straightforward to evaluate the intervention probabilities and to then show that multiple
fundamentals are associated with the same price.
The key difficulty that we deal with
in the proof is to show that, in equilibrium, there are indeed at most three fundamentals
associated with each price.
Proposition 3 Suppose the regulator observes the price of a convex security. For κ < T /2
sufficiently close to T /2, there exist equilibria in which the regulator intervenes with positive
probability at fundamentals above θ̂. In particular, there exists an interval (θ01 , θ11 ) in which
θ01 is as defined in Lemma 2 and θ11 < θ̂ such that the following holds:
For any set Y1 ⊂ [θ01 , θ11 ], there exists a strictly increasing function θ∗2 : Y1 → < such that
θ∗2 (θ01 ) = θ̂, and such that the following is an equilibrium:
1. [Optimal intervention below θ̂] If θ ≤ θ̂, the regulator intervenes with probability 1,
and the price is X (θ) + U (θ).
2. [Over-intervention for some θ > θ̂] If θ ∈ θ∗2 (Y1 ) the regulator intervenes with proba´
³
θ−θ∗−1
(θ)
θ−θ∗−1
(θ)
2
2
U (θ).
>
0,
and
the
price
is
X
(θ)
+
1
−
bility 1 −
2κ
2κ
14
³
´
³
´
³
´
³ ´
The RHS equals X θ̂+ − 2κ + U θ̂+ − 2κ = X θ̂+ − 2κ + T , while the LHS equals X θ̂+ , which
is smaller.
18
3. [Optimal intervention for some θ > θ̂] If θ > θ̂ and θ ∈
/ θ∗2 (Y1 ), the regulator never
intervenes, and the price is X (θ).
The result in Proposition 3 begs the question of whether, when the regulator observes the
price of a convex security, there also exist equilibria in which too little intervention occurs.
The following proposition provides a negative answer to this question.
Any equilibrium
entails too much intervention in the following sense:
Proposition 4 Suppose the regulator observes the price of a convex security, and κ <
T /2. Then any equilibrium other than the optimal intervention equilibrium entails a strictly
positive probability of intervention at some fundamental θ > θ̂.
4.2.2
The case of a concave security: equilibrium with too little intervention
We now consider the case in which the regulator observes the market price of a concave
security.
Parallel but opposite results hold in this case relative to the case of a convex
security. Specifically, we establish that in this case there are equilibria in which the regulator
intervenes too little relative to the optimal intervention equilibrium depicted in Figure 1.
Figure 3 depicts an example of such an under-intervention equilibrium.
As we can see, the equilibrium depicted in Figure 3 exhibits too little intervention. The
regulator intervenes optimally at fundamentals associated with the left line and the right
line of the pricing function, but intervenes too little at fundamentals associated with the
middle line. This happens because at these fundamentals, the price does not fully reveal
that the fundamental is below θ̂, even after it is combined with the regulator’s private signal.
Proposition 5 establishes the existence of equilibria with too little intervention and
provides a full characterization of them. The proof is parallel to the proof of Proposition
3. To clarify the mathematical intuition and the role of concavity for equilibria with too
little intervention, we now go over the first steps leading to the proof. Consider what is
required to generate too little intervention at a point θ̂− infinitesimally to the left of θ̂.
Under-intervention occurs if the market price at θ̂− is the same as the market price at some
´
³
θ ∈ θ̂, θ̂− + T . The probability of intervention at θ̂− in this case is the probability that
19
X (θ ) + U (θ )
P (θ )
X (θ )
θˆ
θˆ − T
θˆ + T
θ
Figure 3: Security price in an equilibrium with too little intervention
the regulator receives a signal φ that is not consistent with the fundamental θ:
´ θ − κ − θ̂ − (−κ)
³
θ − θ̂−
−
=
.
Pr φ ≤ θ − κ|θ̂− =
2κ
2κ
As such, the market prices at θ > θ̂ and θ̂− coincide if and only if
Ã
!
³ ´
³ ´
θ − θ̂−
X θ̂− +
U θ̂− = X (θ) .
2κ
(9)
For this equation to hold, the value of the security has to be concave. To see why, note
that the LHS in (9) is equal to the RHS when θ = θ̂− .
Moreover, because 2κ < T , the
LHS is larger than the RHS when θ = θ̂− + 2κ. Since the LHS is linear and increasing in
θ, a necessary condition for (9) to have a solution above θ̂ is that its RHS is concave in θ.
This implies that the security has to be concave for too little intervention to occur. We
now turn to the proposition.
Proposition 5 Suppose the regulator observes the price of a concave security.
For κ <
T /2 sufficiently close to T /2, there exist equilibria in which the regulator intervenes with
20
probability less than one at fundamentals below θ̂.
In particular, there exists an interval
(θ02 , θ12 ) where θ02 > θ̂ such that the following holds:
For any set Y2 ⊂ [θ02 , θ12 ], there exists a strictly increasing function θ∗1 : Y2 → < such that
θ∗1 (θ12 ) = θ̂, and such that the following is an equilibrium:
1. [Optimal intervention above θ̂] If θ ≥ θ̂, the regulator intervenes with probability 0,
and the price is X (θ).
2. [Under-intervention for some θ < θ̂] If θ ∈ θ∗1 (Y2 ) the regulator intervenes with probability
θ∗−1
(θ)−θ
1
2κ
> 0, and the price is X (θ) +
θ∗−1
(θ)−θ
1
U
2κ
(θ).
3. [Optimal intervention for some θ < θ̂] If θ < θ̂ and θ ∈
/ θ∗1 (Y2 ), the regulator always
intervenes, and the price is X (θ) + U (θ).
To complete the analysis of a concave security, the next proposition provides a result
that is parallel to the one in Proposition 4.
Essentially, for the case of a security with
a concave payoff, any equilibrium that does not exhibit optimal intervention has too little
intervention.
Proposition 6 Suppose the regulator observes the price of a concave security, and κ < T /2.
Then any equilibrium other than the optimal intervention equilibrium entails an intervention
probability strictly less than 1 at some fundamental θ < θ̂.
4.2.3
Convex vs. concave: some intuition
The results we just described demonstrate that the shape of the security matters for the
type of equilibrium one can expect. We now provide some intuition for these results.
Why does under-intervention never occur when the regulator tracks the price of a convex security?
In general, under-intervention occurs when the regulator intervenes with
probability less than 1 at some fundamental θ1 < θ̂, and probability 0 at some fundamental
θ2 > θ̂ with a matching price. To gain intuition, it is useful to consider what is required
for the intervention probability at θ1 to equal 1/2.
21
First, observe that the intervention
probability at θ1 is Pr (θ1 + ξ < θ2 − κ). Given our uniformity assumption, we can evaluate this expression explicitly as
only if θ2 − θ1 = κ < T /2.
θ2 −θ1
2κ ,
and so the intervention probability at θ1 is 1/2
Although the specifics of this calculation clearly depend on
our distributional assumptions, the general point is that when the regulator’s signal is relatively accurate, the fundamentals θ1 and θ2 must be close to each other for the intervention
probability at θ1 to equal 1/2. But then, since θ2 is relatively close to θ1 , if X were convex,
X (θ2 ) would certainly lie below the average of X (θ1 ) and X (θ1 + T ). This implies that
the prices at θ1 and θ2 do not coincide as the price at θ1 in the proposed equilibrium is
exactly the average of X (θ1 ) and X (θ1 + T ), and the price at θ2 is X (θ2 ). As such, X must
be concave for an equilibrium of this type to exist. More generally, an under-intervention
equilibrium only exists if X is concave.
For the over-intervention case, a parallel argument applies. This time, consider what
is required for an equilibrium with an intervention probability of 1 at θ1 < θ̂ and 1/2 at
θ2 > θ̂, with matching prices at the two fundamentals. Parallel to the previous analysis, θ1
and θ2 must lie at a distance of only κ < T /2 apart. For the prices to match, the average
of X (θ2 ) and X (θ2 + T ) must equal X (θ1 + T ). But this cannot occur if X is concave,
for since θ1 + T is relatively close to θ2 + T , certainly X (θ1 + T ) exceeds the average of
X (θ2 ) and X (θ2 + T ).
4.3
The regulator’s signal is imprecise: no equilibrium
Finally, consider the case when κ > T /2, that is, when the regulator’s signal is imprecise
so that the information gap between the market and the regulator is large. The first thing
to note is that when κ > T /2, optimal intervention cannot occur in equilibrium. To see
this, look again at Figure 1. As we can see in the figure, in an equilibrium with optimal
intervention, there are fundamentals at a distance of T from each other on both sides of θ̂
that have the same price. Since the regulator’s signal is imprecise, i.e., since 2κ > T , the
signal does not enable the regulator to always distinguish between two fundamentals that
have the same price. Thus, given a price that is associated with two fundamentals, it is
impossible for the regulator to always intervene at one fundamental and never intervene at
22
the other and therefore optimal intervention cannot occur.
Our main result in this subsection is in fact much stronger. Proposition 7 shows that
when κ > T /2, not only is there no equilibrium with optimal intervention, but there is also
no other rational-expectations time-consistent equilibrium.
Proposition 7 Suppose that the regulator’s information is relatively poor (κ > T /2) and
that it observes the price of a single security. Regardless of whether the security’s expected
payoff is convex or concave in the fundamental θ, no equilibrium exists.
Although the proof of Proposition 7 is long and involved, in the limiting case in which
the regulator receives no information at all (i.e., κ → ∞) it is possible to give the following
straightforward and intuitive proof.
First, we claim that the only candidate equilibrium
in this case is one with fully revealing prices.
To see this, suppose instead that there is
an equilibrium in which two fundamentals θ1 and θ2 6= θ1 are associated with the same
price. Since the regulator has no information, its intervention policy must be the same at
θ1 and θ2 . But then the prices are not equal, giving a contradiction. (It is important to
note that both Proposition 7 and this simple limit argument cover mixed strategies by the
regulator.) However, there is no fully revealing equilibrium either: given time-consistency,
a fully revealing equilibrium features optimal intervention, a possibility ruled out by the
text preceding Proposition 7.
5
Policy implications: implementing market-based intervention
5.1
Main implication: the importance of the regulator’s information
The results in the previous section demonstrate the difficulty in implementing an intervention policy that is based on the market price of a bank’s security. The problem stems from
the fact that the market price adjusts to reflect the expected regulator’s action, and this
reduces the extent to which the fundamental, which is what the regulator is trying to learn,
can be inferred from the price.
23
A key determinant of whether optimal intervention can be implemented based on market
price is the quality of the private signal that is observed by the regulator in addition to the
price. We show that when the regulator has a very precise signal, optimal intervention is
obtained as a unique equilibrium. When the regulator has a moderately precise signal, optimal intervention is still an equilibrium, but there are also other equilibria with suboptimal
intervention. Finally, when the regulator’s signal is imprecise, there is no equilibrium in
the model. Of course, one has to be cautious when interpreting a no-equilibrium result. At
the very least, however, we believe it points to the problem associated with implementing
a market-based intervention when the quality of the regulator’s own signal is poor.
For the environment studied in the previous section, these results imply that there is a
strong complementarity between the market’s information and the regulator’s information.
To be able to implement a successful market-based intervention policy, the regulator still
needs to produce a reasonably precise signal of his own. Thus, market-based intervention
cannot perfectly substitute for direct supervision but instead is a complement.
In the following subsection, we study whether there are alternatives to the regulator
generating a precise private signal for which market-based intervention will work. The first
alternative we consider is for the regulator to learn from the prices of multiple securities.
The second alternative is to improve transparency by disclosing the regulator’s signal to
the market.
The third alternative is to issue a security that directly predicts whether
the regulator is going to intervene. The fourth alternative is to impose taxes on security
holders that change their payoffs in case of regulatory intervention. Finally, we consider
the possibility that the regulator can commit ex ante to a policy rule based on the realized
price.
5.2
5.2.1
Other implications
Multiple securities
Thus far we have restricted attention to the case in which the regulator observes only one
price, that of a convex security (e.g., equity), or that of a concave security (e.g., debt). A
key question is whether it helps if both these securities trade publicly, and the regulator
24
learns from the prices of both.
It turns out that observing the prices of both securities resolves the problem of multiple
equilibria when the regulator’s signal is moderately precise, but does not solve the problem
of no equilibrium when the regulator’s signal is imprecise. We start by proving the first
result.
Proposition 8 Suppose that κ < T /2 and that the regulator observes the price of both a
strictly concave and a strictly convex security. Then the optimal intervention equilibrium
is the unique equilibrium.
To gain intuition for this result, recall the results of the previous section.
There,
we showed that when the regulator’s information is moderately precise, there may exist
equilibria with too much or too little intervention, in addition to the equilibrium with
optimal intervention.
We also showed that an equilibrium with too much intervention
requires that the security whose information the regulator observes be convex, while an
equilibrium with too little intervention requires that the security be concave. Thus, in this
range, observing both the price of a concave security and the price of a convex security
eliminates the equilibria with suboptimal intervention and leaves the optimal intervention
equilibrium as the unique equilibrium.
Although in theory the observation of multiple security prices helps the regulator infer
the bank’s fundamental, in practice the feasibility of this measure clearly depends on the
existence of liquid and well-functioning markets in distinct securities. Additionally, even
multiple security prices do not help the regulator when his own information is of low quality:
Proposition 9 Suppose that κ > T /2 and that the regulator observes the price of both a
concave and a convex security. Then no equilibrium exists.
Essentially, once equilibrium fails to exist with one traded security, it will not be generated by adding another security.
25
5.2.2
Transparency
We now return to the case of one traded convex or concave security and assume that the
regulator makes public its own signal φ before the market price is formed.
Our analysis
implies that this form of transparency improves the regulator’s ability to make use of market
information. Specifically, transparency resolves the problem of multiple equilibria when the
regulator’s signal is moderately precise, but it does not solve the problem of no equilibrium
when the regulator’s signal is imprecise. The argument is as follows.
Under the “transparency” regime in which the regulator truthfully announces his signal
φ, the equilibrium pricing function depends on both the fundamental θ and the regulator
signal φ. Consider a specific realization φ∗ of the regulator’s signal, along with any pair
of fundamentals θ1 and θ2 such that φ∗ is possible after both. The prices at (θ1 , φ∗ ) and
(θ2 , φ∗ ) must differ.
If, instead, the prices coincided, the intervention decisions would
also coincide, but in this case the prices would not be equal after all. It follows that all
fundamentals θ for which the regulator’s signal φ∗ is possible must have different prices given
realization φ∗ , that is, given φ∗ prices are fully revealing.
This argument together with
time-consistency implies that the only candidate equilibrium features optimal intervention.
As such, transparency eliminates the suboptimal equilibria of Propositions 3 and 5. The
intuition is that these equilibria were based on the market not knowing the regulator’s
action, a problem that is solved once the regulator discloses its signal truthfully.
Now, when κ < T /2, optimal intervention is indeed an equilibrium, with prices P (θ, φ) =
X (θ) + U (θ) for θ ≤ θ̂ and P (θ, φ) = X (θ) for θ > θ̂. On the other hand, when κ > T /2,
optimal intervention is not an equilibrium. To see this, if we suppose to the contrary that
it was an equilibrium, then there exist fundamentals θ1 and θ2 and a regulator’s signal
realization φ ∈ [θ1 − κ, θ1 + κ] ∩ [θ2 − κ, θ2 + κ] such that (θ1 , φ) and (θ2 , φ) have the same
price, in contradiction to above. It follows that for κ > T /2, there is no equilibrium.
Although a policy of transparency improves the regulator’s ability to infer bank fundamentals from market prices, in practice there may be limits to its viability.
In particular,
if a bank knows that the regulator will make its information public, it may be less inclined
to grant easy access to the regulator in the first place.
26
In this sense, it is possible that
transparency would serve to increase κ, potentially making the regulator’s inference problem
worse instead of better.
5.2.3
Regulator security
Neither of the measures discussed so far allows the regulator to infer the bank’s fundamental
when its own information is poor (κ > T /2). The next possibility we discuss is the creation
of an “event market” in which market participants trade a security that pays 1 if the
regulator intervenes, and 0 otherwise.
We will refer to such a security as a regulator
security. Clearly such a market is feasible only if the regulator’s intervention is publicly
observable and verifiable – a condition that is not required in any of our analysis to this
point, and in practice may fail to hold.
However, if such a market could be created, its
existence would render optimal intervention as the unique equilibrium irrespective of the
quality of the regulator’s information.
More formally, suppose that in addition to a standard security market, an event market
of the type described is feasible and exists. Let Q be the price of the regulator security, with
P being the price of the standard security as before. The regulator’s intervention policy
I can now depend on Q in addition to P and his own signal φ. The rational expectations
equilibrium pricing condition for the regulator security is
Q(θ) = Eφ [I (P (θ) , Q (θ) , φ) |θ] .
Under these conditions we obtain:
Proposition 10 If the market trades both a standard bank security and the regulator security, then for all κ the unique equilibrium of the economy features optimal intervention.
The intuition behind this result is the following: a regular bank security may have the
same price for different fundamentals because the probability of intervention is different
across these fundamentals. But, once the regulator security is traded, the probability of
intervention can be inferred from its price, and thus the fundamental can be inferred from
the combination of its price and the prices of regular bank securities. This implies that the
regulator will intervene optimally in equilibrium.
27
Note that the regulator security specified here is not a standard security, i.e., it is not a
security typically traded in financial markets. This is because the price of this security is
not linked to the value that goes to any class of bank claim holders. Instead, the regulator
security aggregates information regarding whether the regulator is going to intervene in the
bank or not.15 The result in Proposition 10 implies that issuing such a unique security is a
very powerful tool: with this security in place, the unique equilibrium of the model entails
optimal intervention for all κ.
5.2.4
Taxation
The next policy tool we consider is taxation.
Suppose that there is one security, whose
value absent intervention is X(θ) and with intervention is X(θ + T ). The regulator would
like to intervene if and only if θ < θ̂, but is unable to do so because the value of the security
under this intervention policy is not monotonic in θ, and so is not fully revealing.
In
principle, the regulator could restore monotonicity by taxing the investors who hold this
security when it intervenes. A simple way to do this is to tax the investors by an amount T
when intervention takes place. If the regulator intervenes if and only if θ < θ̂, the market
price of the security is then X(θ) + U (θ) − T for θ < θ̂, and X (θ) otherwise. Since this
price is monotone in θ, the regulator is indeed able to learn the fundamentals perfectly from
the price and to intervene precisely when fundamentals are below θ̂.16
The above argument establishes that when intervention-contingent taxation is possible,
there is always an equilibrium with intervention if and only if θ < θ̂, regardless of the
precision of the regulator’s information.
In fact, we now show that when the regulator
observes the price of a convex security, this equilibrium is unique:
Proposition 11 Suppose the regulator observes the price of a convex security and imposes
a tax of T whenever intervention occurs.
15
Then the unique equilibrium is intervention if
In this sense, it resembles securities traded in prediction markets, which try to predict the probability
of an event.
16
More generally, the price is monotone under this intervention policy if the regulator imposes a tax of at
least X (θ + T ) − X (θ) whenever intervention occurs.
28
and only if θ < θ̂.
It is important to note that, relative to the other measures discussed in this section,
imposing an intervention-dependent tax on bank security holders is a more drastic policy
response.
Taxation of this form clearly has distributional consequences and is likely to
be costly to implement – both in terms of direct costs, and in terms of other distortions.
Moreover, intervention-contingent taxation is possible only if intervention is publicly observable – under which circumstances a regulator security of the type discussed above
would also solve the regulator’s inference problem.
5.2.5
Commitment
Thus far in the paper, we have assumed that the regulator’s intervention decision is ex
post efficient, i.e., the regulator does what is optimal for him to do given the prices and its
private signal. A natural question is whether the regulator can achieve optimal intervention
by committing ex ante to a policy rule as a function of the realized price. To answer this
question, we reconsider the case of one traded security (convex or concave), and assume
that the regulator can commit ex ante to an intervention policy that is a function of the
price only.
This last assumption is natural given that committing to a policy rule that
is based on the publicly observed price may be feasible, while committing to a policy rule
that is based on a privately observed signal is probably not.
In view of the regulator’s
commitment, for this subsection only we drop the requirement that the time-consistency
condition (6) must be satisfied in equilibrium.
The main thing to note about this case is that an equilibrium under regulatory commitment must entail fully revealing prices, i.e., in such an equilibrium every fundamental must
be associated with a different price. This is because the regulator’s intervention decision
is now based only on the price.
As a result, if two fundamentals had the same price,
they would also have the same probability of intervention, and this would generate different
prices. Thus, finding the optimal commitment policy boils down to finding the price function that maximizes the regulator’s ex ante value function, subject to the constraint that
the price function is fully revealing of the fundamentals.
29
The fact that the price function must be fully revealing implies that the regulator cannot
achieve optimal intervention under commitment. This is because, as we saw in Figure 1,
optimal intervention generates a price function that is not fully revealing — it has different
fundamentals associated with the same price.
The following proposition establishes a
stronger result on the effectiveness of commitment. It says that, under commitment, the
regulator will end up deviating from the benchmark full-information optimal intervention
policy over a set of fundamentals that is at least of size T . Thus, commitment is not very
effective in solving the problems raised in this paper.
Proposition 12 If the regulator commits ex ante to an intervention policy based on the
realization of the price of one security, it will not be able to achieve optimal intervention.
The set of fundamentals at which the regulator deviates from the full-information optimal
intervention policy is at least of size T .
6
Conclusion
We study a rational expectations model of regulatory supervision of banks based on the
market prices of bank securities. Because prices reflect bank fundamentals and expectations
of regulatory actions at the same time, the regulator cannot always extract information from
the price to make an efficient intervention decision. The ability of the regulator to extract
such information depends on the gap between the information available to the market
and the information available to the regulator. When the gap is large, our model has no
rational-expectations equilibrium. When the gap is moderate, there are multiple equilibria,
some of which exhibit sub-optimal intervention. When the gap is small, there is a unique
equilibrium with optimal intervention.
These results imply that a regulator cannot rely
on market-based intervention alone, but instead should use market prices to supplement
his own signal. Moreover, our analysis demonstrates that the equilibrium outcome under
market based regulation depends on the type of security whose price the regulator observes.
Convex securities lead to too much intervention and concave securities to too little.
We also analyze four policy measures that improve the ability of the regulator to use
30
market information for bank supervision. Two of these measures — learning from multiple
securities and disclosing the regulator’s information — resolve the problem of multiple equilibria when the regulator’s information is moderately precise, but do not solve the problem
of no equilibrium when the regulator’s information is imprecise. The other two measures
– issuing a regulator security and taxation – solve both problems. As we have noted, the
implementation of each of these policy measures requires nontrivial conditions to be met.
Finally, we study the case in which the regulator can commit ex ante to a policy rule that
depends on the price. We show that this is not effective in obtaining optimal intervention.
Empirical exploration of the ideas presented in the paper is likely to revolve around
the relation between market prices and the regulators’ intervention decisions — two key
variables in our model that are easily observable.
Our theoretical analysis points out
that such empirical exploration should take into account two key features of the model.
First, if the regulator uses the market price in his intervention decision, there will be dual
causality between the two variables: market prices will reflect the regulator’s action and
affect it at the same time. A similar setting, in which prices affect and reflect an action of
market participants, has been empirically analyzed by Bradley, Brav, Goldstein, and Jiang
(2006). Second, when the information that the regulator has outside the financial market
is not precise enough, our model generates equilibrium indeterminacy, which might make
the relation between the two variables more difficult to detect.
Finally, we noted before that the insights from our analysis can be applied to many other
settings in which regulators intervene on the basis of information they glean from market
prices.
In fact, our analysis is more general than that; it can apply to other settings in
which agents use information from market prices to take actions that affect the value of the
security. Importantly, the analysis requires that, under optimal intervention, the value of
the security is non monotone in the fundamentals. As we argued before, this is the result of
a setting where the agent acts to maximize his own objective function rather than the value
of the security. Examples that satisfy this criterion and to which our analysis can apply
include the decision of bond holders on whether to monitor the firm based on information
gleaned from its stock price; the decision of lenders on whether to lend money to the firm
31
based on the same information; and the design and inclusion of market-price triggers in
debt covenants.
32
Mathematical Appendix
We start with a couple of preliminaries. First, we have favored expositional clarity
over mathematical precision whenever no loss is associated with the former. Specifically,
even though each combination of a fundamental and regulator signal has measure zero, we
routinely evaluate the conditional probability Pr (θ|θ ∈ {θ1 , θ2 }) as 1/2 if θ ∈ {θ1 , θ2 } (recall
that the fundamental θ is distributed uniformly). Likewise, we evaluate Pr (θ|θ ∈ Θ) = 0
if θ ∈
/ Θ.
We use parallel calculations for conditional expectations. The only proof in
which it is important for us to proceed more formally in taking conditional probabilities
and expectations is that of Proposition 2 (see the proof for the relevant details).
Second, it is convenient to state the following straightforward result separately.
Lemma 3 In any equilibrium,
⎧
n
o
⎨ 1 if θ < max θ̂ − 2κ, θ̂ − T
n
o .
Pr (I|θ) =
⎩ 0 if θ > min θ̂ + 2κ, θ̂ + T
Proof of Lemma 3: Consider a fundamental θ < θ̂ − 2κ.
regulator observes only signals below θ̂ − κ.
fundamental θ̃ ≥ θ̂.
At this fundamental, the
Such signals are never observed after any
As such, when the fundamental is θ the regulator knows that the
fundamental lies to the left of θ̂. By time-consistency, it intervenes with probability 1.
Next, consider a fundamental θ < θ̂ − T . In any equilibrium the price at θ is bounded
above by X (θ) + U (θ) = X(θ + T ) < X(θ̂). Moreover, any fundamental θ̃ ≥ θ̂ has a price
that satisfies P (θ̃) ≥ X(θ̃) ≥ X(θ̂). Thus, in any equilibrium, if θ < θ̂ − T then θ cannot
share a price with any fundamental above θ̂.
Again, by time-consistency the regulator
intervenes with probability 1. The argument is similar for θ > θ̂ + 2κ and θ > θ + T , and
the result follows.
Proof of Lemma 1: Observe first that
Γ0 (θ) = − (G (D + T − θ) − G (T − θ)) < 0,
Γ00 (θ) = g (D + T − θ) − g (T − θ) .
33
This implies that −Γ0 (θ) is single-peaked since (given that g is itself single-peaked) −Γ00 (θ)
is negative if and only if D + T − θ is below some cutoff level, i.e., if and only if θ is above
some cutoff level. Second, since
¡
¢
V 0 (θ) = −Γ0 (θ + T ) − −Γ0 (θ)
it follows that V is itself single-peaked, since V 0 (θ) is first positive and then negative as a
function of θ.
Proof of Proposition 1: In an optimal intervention equilibrium there is a cutoff fundamental, θ̂, for which the regulator only intervenes for θ ≤ θ̂.
What we need to check is
that this policy is feasible. Under this intervention policy, P (θ) = X (θ) + U (θ) for θ ≤ θ̂,
and P (θ) = X(θ) for θ > θ̂. As such, there are at most two fundamentals related to each
price. For prices that are related to just one fundamental, the equilibrium price is trivially
fully revealing. For prices that are related to two fundamentals, e.g., θ1 < θ̂ < θ2 ,
X (θ1 ) + U (θ1 ) = X (θ2 ) .
But from (4), this means that X (θ1 + T ) = X (θ2 ), and thus θ2 = θ1 + T . Since T > 2κ,
the regulator can distinguish between θ1 and θ2 with its own signal and follow its optimal
intervention rule.
Proof of Proposition 2: We prove the result for the case in which X is convex.
The
case in which X is concave follows similarly.
The proof requires us to be more mathematically precise in our treatment of probabilities
and expectations than is the case elsewhere in the paper. In particular, unlike elsewhere
in the paper, we must assign conditional expectations and probabilities in cases where the
conditioning set has infinitely many members yet is still null. Formally, let B denote the
¡£ ¤ ¢
£ ¤
Borel algebra of θ, θ̄ , so that θ, θ̄ , B is a measurable space. Let µ : B → [0, 1] be the
£ ¤
probability measure associated with the uniform distribution on θ, θ̄ .
h
i
0
Let κ̄ > 0 be such that U(θ)
2κ̄ − X (θ + T ) > 0 for all θ ∈ θ̂, θ̂ + T , and fix an arbitrary
κ ∈ [0, κ̄]. We will show that in any equilibrium optimal intervention occurs almost surely.
The proof is by contradiction: suppose to the contrary that there exists an equilibrium in
34
which the regulator intervenes suboptimally over a non-null set of fundamentals. Clearly
suboptimal intervention can only occur at non-revealing prices; and by Lemma 3, suboptimal
h
i
intervention and non-revealing prices can only occur in θ̂ − 2κ, θ̂ + 2κ .
Throughout the proof we use the following definitions. Let P be the set of non-revealing
prices. For each non-revealing price P ∈ P let ΘP be the set of fundamentals associated
[
ΘP be the set of all fundamentals with a non-revealing price.
with that price. Let Θ =
P ∈P
By hypothesis, Θ has strictly positive measure.
Claim A: In an equilibrium, in which the regulator intervenes suboptimally over a non-null
h
i
set of fundamentals, Θ ∩ θ̂, θ̂ + 2κ has strictly positive measure.
´
h
i
³
Proof of Claim A: Consider the conditional probability Pr ΘP ∩ θ̂, θ̂ + 2κ |ΘP .
´
³
h
i
Clearly it equals Pr Θ ∩ θ̂, θ̂ + 2κ |ΘP . Moreover,
Z
θ∈Θ
´
³
h
i ´
³
h
i
Pr Θ ∩ θ̂, θ̂ + 2κ |ΘP (θ) µ (dθ) = Pr Θ ∩ θ̂, θ̂ + 2κ |Θ .
h
i
³
h
i
´
Suppose that contrary to the claim Θ∩ θ̂, θ̂ + 2κ is null. In this case, Pr Θ ∩ θ̂, θ̂ + 2κ |ΘP (θ) =
0 for almost all θ in Θ. But then the regulator would intervene optimally for almost all
θ ∈ Θ: it would intervene with probability 1 at almost all θ ∈ Θ, since almost all members
of Θ lie below θ̂. Since suboptimal intervention can potentially happen only at θ ∈ Θ, this
contradicts an equilibrium, in which the regulator intervenes suboptimally over a non-null
set of fundamentals, and completes the proof of Claim A.
For any signal realization φ, the regulator knows the true fundamental lies in the interval
[φ − κ, φ + κ]. As such, for a price P ∈ P and signal φ the regulator’s expected payoff (net
of costs) from intervention is
v (P, φ) ≡ Eθ [V (θ) − C|θ ∈ ΘP ∩ [φ − κ, φ + κ]] .
(10)
The heart of the proof lies in establishing:
Claim B: In an equilibrium of the kind described above, for any P ∈ P: (1) sup ΘP ∩
h
i
h
i
θ̂ − 2κ, θ̂ = θ̂ and (2) v (P, φ = θ + κ) ≥ 0 for any θ ∈ ΘP ∩ θ̂ − 2κ, θ̂ .
Proof of Claim B: Let θ1 and θ2 ∈ (θ1 , θ1 + 2κ] be an arbitrary pair of members of ΘP
such that θ1 ≤ θ̂ and θ2 ≥ θ̂ (clearly all members of ΘP cannot lie to the same side of θ̂,
35
and at least one such pair must lie within 2κ of each other). Since θ1 and θ2 have the same
price
U (θ1 )
X (θ1 ) +
2κ
Z
θ1 +κ
θ1 −κ
U (θ2 )
I (P, φ) dφ = X (θ2 ) +
2κ
Z
θ2 +κ
I (P, φ) dφ.
θ2 −κ
By convexity of X, U (θ2 ) > U (θ1 ). It follows that
¶
µZ θ2 +κ
Z θ1 +κ
U (θ2 )
X (θ1 ) + U (θ1 ) ≤ X (θ2 ) +
I (P, φ) dφ +
(1 − I (P, φ)) dφ .
2κ
θ2 −κ
θ1 −κ
Equivalently,
µ
¶
Z θ2 −κ
Z θ2 +κ
U (θ2 )
X (θ1 + T ) ≤ X (θ2 )+
θ1 − θ2 + 2κ +
(1 − I (P, φ)) dφ +
I (P, φ) dφ .
2κ
θ1 −κ
θ1 +κ
(11)
h
i
h
i
Define θ∗1 = sup ΘP ∩ θ̂ − 2κ, θ̂ and θ∗2 = inf ΘP ∩ θ̂, θ̂ + 2κ .
Suppose that either v (P, φ = θ1 + κ) < 0 or θ∗1 < θ̂. In the former case, v (P, φ) < 0
for any signal φ above θ1 + κ (since v (P, φ) is monotonically decreasing in φ).
In the
latter case, any signal φ above θ∗1 + κ rules out that θ ≤ θ̂. As such, by time-consistency
I (P, φ) = 0 for all φ > θ1 + κ in the former case, and φ > θ∗1 + κ in the latter case. Since
both sides of (11) are continuous in θ1 and θ2 , it follows that
!
Ã
Z θ∗2 −κ
U (θ∗2 )
∗
∗
(1 − I (P, φ)) dφ
X (θ + T ) ≤ X (θ2 ) +
θ − θ2 + 2κ +
2κ
θ−κ
for θ = θ1 in the former case, and θ = θ∗1 in the latter case. Certainly I (P, φ) = 1 for all
φ < θ∗2 − κ, since for these signal values the regulator knows that the fundamental lies to
the left of θ̂. Thus the function Z defined by
Z (θ, θ2 ) ≡ X (θ2 ) +
U (θ2 )
(θ − θ2 + 2κ) − X (θ + T )
2κ
is weakly positive at (θ, θ2 ) = (θ1 , θ∗2 ) in the former case, and at (θ∗1 , θ∗2 ) in the latter case.
However,
Z (θ∗2 , θ∗2 ) = X (θ∗2 ) + U (θ∗2 ) − X (θ∗2 + T ) = 0
U (θ∗2 )
− X 0 (θ∗2 + T ) > 0,
Z1 (θ∗2 , θ∗2 ) =
2κ
where the strict inequality follows since θ∗2 ≤ θ̂ + T (see Lemma 3) and κ ≤ κ̄. Since Z is
concave in its first argument, it follows that Z (θ, θ∗2 ) < 0 for all θ < θ∗2 , which contradicts
36
Z (θ1 , θ∗2 ) ≥ 0 in the former case, and Z (θ∗1 , θ∗2 ) ≥ 0 in the latter case. This completes the
proof of Claim B.
We are now ready to complete the proof. By Claim B, for any ε > 0 and any P ∈ P
h
i
there exists θP,ε ∈ ΘP ∩ θ̂ − ε, θ̂ such that v (P, φ = θP,ε + κ) ≥ 0. As such, the integral
Z
∪P ∈P (ΘP ∩[θP,ε ,θP,ε +2κ])
¢
¡
v P (θ) , φ = θP (θ),ε + κ µ (dθ)
(12)
is weakly positive. Since v is a conditional expectation (see its definition (10)), the integral
is also equal to
Z
∪P ∈P (ΘP ∩[θP,ε ,θP,ε +2κ])
(V (θ) − C) µ (dθ) .
The domain of the integral (12) can be expanded as
h
i´ [ ³
h
i´
³
h
i´ [ ³
ΘP ∩ θP,ε , θ̂ ∪
ΘP ∩ θ̂ + 2κ − ε, θP,ε + 2κ .
Θ ∩ θ̂, θ̂ + 2κ − ε ∪
P ∈P
P ∈P
The term V (θ) − C is strictly negative over the first set above, with the single exception
of at θ̂. For all ε small enough and by Claim A, the first set has strictly positive measure,
while the other two have measures that approach zero. As such, the integral in expression
(12) is strictly negative for ε small enough. The contradiction completes the proof.
Proof of Lemma 2: Define the function
¶
µ
θ2 − θ1
U (θ2 ) − X (θ1 ) − U (θ1 )
Z (θ1 , θ2 ) = X (θ2 ) + 1 −
2κ
¶
µ
θ2 − θ1
= X (θ2 ) + 1 −
U (θ2 ) − X (θ1 + T ) ,
2κ
where θ2 ≥ θ1 . Intuitively, this is the difference between the price at fundamental θ2 given
−θ1
, and the price at fundamental θ1 given an intervention
an intervention probability 1 − θ22κ
probability 1.
Observe that Z has the following properties:
Z11 (θ1 , θ2 ) < 0,
U 0 (θ2 )
> 0,
Z12 (θ1 , θ2 ) =
2κ
Z (θ, θ) = 0.
37
Moreover, for any θ,
Z (θ − 2κ, θ) = X (θ) − X (θ − 2κ + T ) < 0.
³
´
³ ´
We know that Z θ̂ − 2κ, θ̂ < 0 and Z θ̂, θ̂ = 0.
³ ´
provided Z1 θ̂, θ̂ < 0. We know that,
³ ´
=
Z1 θ̂, θ̂
Since Z11 < 0, the result follows
³ ´
U θ̂
³
´
− X 0 θ̂ + T
2κ
³
´
³ ´
X θ̂ + T − X θ̂
³
´
− X 0 θ̂ + T
2κ
Z θ̂+T µ
´¶
1
2κ 0 ³
0
X θ̂ + T
X (θ) −
=
dθ.
2κ θ=θ̂
T
³ ´
Since X is a convex function, Z1 θ̂, θ̂ < 0 for all 2κ close enough to T .
=
Proof of Proposition 3: Define Z as in the proof of Lemma 2. Observe first that
´
³ ´
³
´
³
´
³
since Z θ01 , θ̂ = Z θ̂, θ̂ = 0, and Z11 < 0, then Z θ1 , θ̂ > 0 for any θ1 ∈ θ01 , θ̂ .
³
´
³ ´
Moreover, Z ·, θ̂ is single-peaked. Let θ̂11 ∈ θ01 , θ̂ be its maximum. Since for any
³
´
´
³
θ1 ∈ θ01 , θ̂11 , Z θ1 , θ̂ > 0 and Z (θ1 , θ1 + 2κ) < 0, by continuity there exists some
θ2 > θ̂, for which Z (θ1 , θ2 ) = 0. We define a function, θ∗2 (θ1 ), where θ∗2 is the smallest θ2 ,
above θ̂, for which Z (θ1 , θ2 ) = 0. Economically, θ∗2 (θ1 ) is the fundamental which has the
same market price as θ1 . We know that θ∗2 (θ01 ) = θ̂.
We know θ∗2 (θ1 ) is a strictly increasing function. To see this, note that
³
´ Z θ2
Z2 (θ1 , y) dy.
Z (θ1 , θ2 ) = Z θ1 , θ̂ +
θ̂
³
´
Since Z θ1 , θ̂ is increasing over the range [θ01 , θ̂11 ], and Z12 > 0, it follows that for
any θ2 ≥ θ̂, Z (θ1 , θ2 ) is increasing in θ1 over [θ01 , θ̂11 ].
Thus, the smallest θ2 , at which
Z (θ1 , θ2 ) = 0, is strictly increasing in θ1 , implying that θ∗2 (θ1 ) is a strictly increasing
function.
Since θ∗2 (θ01 ) = θ̂, the function
V (θ1 )+V (θ∗2 (θ1 ))
2
is strictly positive at θ1 = θ01 . Define
θ11 as the minimum of θ̂11 and the infimum value such that
θ∗2 (·) is increasing and
V (·)+V
2
(θ∗2 (·))
V (θ1 )+V (θ∗2 (θ1 ))
2
> C over the interval [θ01 , θ11 ].
38
≤ C. As such,
We have now defined the values θ01 and θ11 of the proposition statement. It remains
to show that there is an equilibrium of the type described. This requires showing that the
prices are rational given the intervention probabilities, and that the intervention probabilities result from the regulator’s optimal behavior given the information in the price and its
own private signal. It is immediate to show that the prices in the proposition statement
are rational given the corresponding intervention probabilities. Thus, we turn to show that
the intervention probabilities result from the regulator’s optimal behavior. We will do this
by analyzing different ranges of the fundamentals separately.
For a fundamental θ ≤ θ̂ and θ ∈
/ Y1 , the price is X (θ) + U (θ) = X (θ + T ).
same price may be observed at the fundamental θ + T .
The
Since 2κ < T , the regulator’s
private signal will indicate for sure that the fundamental is θ and not θ + T . Hence, the
regulator will optimally choose to intervene, generating intervention probability of 1, as is
stated in the proposition. Note that the same price cannot be observed at any fundamental
below θ + T . Observing such a price at a fundamental below θ + T would imply that the
/ Y1 .
fundamental belongs to the set θ∗2 (Y1 ), but this contradicts the fact that θ ∈
For a fundamental θ ≤ θ̂ and θ ∈ Y1 , the price is again X (θ)+U (θ). As before, the same
price may be observed at the fundamental θ + T without having an effect on the decision of
the regulator to intervene at θ, given that 2κ < T . Here, however, the same price will also
be observed at the fundamental θ∗2 (θ). This is because the fundamental θ∗2 (θ) ∈ θ∗2 (Y1 )
´
³
θ∗ (θ)−θ
U (θ∗2 (θ)), which by construction is equal to
generates a price of X (θ∗2 (θ)) + 1 − 2 2κ
X (θ) + U (θ). (Note that the same price will not be observed at any other fundamental
in the set θ∗2 (Y1 ), since X (θ) + U (θ) and θ∗2 (θ) are strictly increasing in θ.) Thus, at the
fundamental θ, the regulator observes a price that is consistent with both θ and θ∗2 (θ), and
may observe a private signal that is also consistent with both of them.
If this happens,
given the uniform distribution of noise in the regulator’s signal, the regulator will intervene
as long as
V (θ)+V (θ∗2 (θ))
2
≥ C. By construction, this is true for all θ ∈ Y1 , and thus, at the
fundamental θ, the regulator will intervene with probability 1, as is stated in the proposition.
For a fundamental θ > θ̂ and θ ∈
/ θ∗2 (Y1 ), the price is X (θ) = X (θ − T )+U (θ − T ). The
0
0
same price may be observed at the fundamental θ − T and also at some θ > θ̂; θ ∈ θ∗2 (Y1 ).
39
Since 2κ < T , the regulator’s private signal at the fundamental θ will indicate for sure that
the fundamental is not θ −T . Hence, the regulator will know that the fundamental is above
θ̂, and will optimally choose not to intervene, generating intervention probability of 0, as is
stated in the proposition.
³
Finally, for a fundamental θ > θ̂ and θ ∈ θ∗2 (Y1 ), the price is X (θ)+ 1 −
θ−θ∗−1
(θ)
2
2κ
´
U (θ).
As follows from the arguments above, the same price will be observed at the fundamental
00
00
(θ), and also may be observed at some fundamental θ > θ̂; θ ∈
/ θ∗2 (Y1 ). (As argued
θ∗−1
2
before, two fundamentals in the set θ∗2 (Y1 ) cannot have the same price.) As also follows
from the arguments above, the regulator will optimally choose to intervene if and only if
(θ) (the signal cannot be consistent with both
its signal is consistent with both θ and θ∗−1
2
00
(θ) and θ ).
θ∗−1
2
Due to the uniform distribution of noise in the regulator’s signal, this
generates an intervention probability of 1 −
θ−θ∗−1
(θ)
2
,
2κ
as is stated in the proposition.
Proof of Proposition 4: Suppose to the contrary that there exists an equilibrium without
optimal intervention and in which the probability of intervention for all θ > θ̂ is 0. In this
equilibrium there must exist some θ1 < θ̂ such that E [I|θ1 ] < 1.
Because θ1 < θ̂, it
follows that there must exist θ2 ∈ (θ̂, θ1 + 2κ] with the same price as θ1 . Moreover, because
E [I|θ] = 0 for all θ > θ̂, the fundamental θ2 is the unique fundamental to the right of θ̂
with the same price as θ1 . So the intervention policy I in this equilibrium must satisfy
⎧
⎨ 0 if φ ∈ (θ2 − κ, θ2 + κ)
I (P (θ1 ) , φ) =
.
⎩ 1 if φ ∈ (θ − κ, θ + κ) and φ ∈
/
(θ
−
κ,
θ
+
κ)
1
1
2
2
As such, the expected intervention probability at θ1 is
E [I|θ1 ] = Pr ((θ1 + ξ) ∈ (θ1 − κ, θ2 − κ)) =
θ2 − θ1
.
2κ
Define a function
Z (θ) = X (θ1 ) +
θ − θ1
U (θ1 ) − X (θ) .
2κ
On the one hand, observe that Z (θ2 ) = X (θ2 ) + E [I|θ1 ] U (θ1 ) − X (θ2 ) = 0, since by
hypothesis θ1 and θ2 have the same price. But on the other hand, Z (θ1 ) = 0, Z (θ1 + 2κ) =
X (θ1 + T ) − X (θ1 + 2κ) > 0, and Z is concave since X is convex. As such, there is no
40
value of θ ∈ (θ1 , θ1 + 2κ] for which Z (θ) = 0. The resultant contradiction completes the
proof.
Proof of Proposition 5: The proof is omitted, and is available from the authors. It is
parallel to the proof of Proposition 3.
Proof of Proposition 6: The proof is omitted, and is available from the authors. It is
parallel to the proof of Proposition 4.
Proof of Proposition 7: We prove the result for the case in which X is concave. The
case in which X is convex follows similarly.
Suppose to the contrary that an equilibrium exists. Let P (·) be the equilibrium price
function. We know that there cannot be a fully-revealing equilibrium (see the main text
immediately prior to the proposition statement).
Define Θ∗ to be the non-empty set of
fundamentals at which the price is not fully-revealing, i.e.,
©
¡ ¢ª
Θ∗ = θ : ∃θ0 6= θ such that P (θ) = P θ0 .
Given Θ∗ , define θ∗ = sup Θ∗ . We prove the following claims.
¡ ¢
¡ ¢
Claim 1: If two fundamentals θ0 and θ00 have the same price, i.e., P θ0 = P θ00 , then
¯ 0
¯
¯θ − θ00 ¯ ≤ T < 2κ.
Proof of Claim 1: Let θ0 and θ00 > θ0 be two fundamentals with the same price.
¡ ¢
¡ ¢
know that P θ00 ≥ X θ00 and
We
¡ ¢
¡ ¢
¡
¢ ¡ ¢
¡ ¢
¢
¡ ¢
¡
X θ0 + T = X θ0 + U θ0 ≥ X θ0 + E I|θ0 U θ0 = P θ0 .
¡ ¢
¢
¡ ¢
¡
So if θ00 > θ0 + T then P θ00 > X θ0 + T ≥ P θ0 , a contradiction. Thus θ00 ≤ θ0 + T .
o
n
o
n
Claim 2: If θ > max θ∗ , θ̂ then P (θ) = X (θ); and if θ ≤ max θ∗ , θ̂ then P (θ) ≤
o´
³
n
X max θ∗ , θ̂ .
Proof of Claim 2: By definition, if θ > θ∗ the price is fully-revealing. So if θ > θ̂ also,
o
´
³
n
the regulator does not intervene, and P (θ) = X (θ). So for any θ ∈ max θ∗ , θ̂ , ∞ ,
the price is X (θ).
³
n
o´
¡ ¢
for some θ0 ≤
Next, suppose that contrary to the claim P θ0 > X max θ∗ , θ̂
o
n
o
n
¡ ¢
max θ∗ , θ̂ . But then there exists θ > max θ∗ , θ̂ ≥ θ∗ such that P (θ) = P θ0 ,
contradicting the fact that θ∗ = sup Θ∗ . This completes the proof of Claim 2.
41
Claim 3: θ∗ > θ̂.
n
o
Proof of Claim 3: Suppose to the contrary that θ∗ ≤ θ̂, so that max θ∗ , θ̂ = θ̂. By
³ ´
Claim 2, P (θ) = X (θ) if θ > θ̂, and P (θ) ≤ X θ̂ for θ ≤ θ̂. As such, whenever the true
fundamental is strictly below θ̂ the regulator knows either that the fundamental is strictly
below θ̂; or that the fundamental is either strictly below θ̂ or equal to θ̂, with a positive
probability of both. So the regulator intervenes with probability one for any θ < θ̂. But
³ ´
then the price is not below X θ̂ for any θ close to θ̂. This contradiction completes the
proof of the Claim 3.
Claim 4: P (θ∗ ) = X (θ∗ ), and so E (I|θ∗ ) = 0.
Proof of Claim 4: From Claims 2 and 3, P (θ) ≤ X (θ∗ ) for θ ≤ θ∗ . The claim follows
since certainly P (θ∗ ) ≥ X (θ∗ ).
Now, consider first the case where θ∗ ∈ Θ∗ . By construction, there exists a fundamental
¡ ¢
¡ ¢
¡
¢ ¡ ¢
θ0 < θ∗ such that: P θ0 = X θ0 + E I|θ0 U θ0 = X (θ∗ ). By Claim 1 and by T < 2κ,
θ∗ < θ0 + 2κ. Since E (I|θ∗ ) = 0, the regulator does not intervene at signals above θ∗ − κ.
¢
¡
¢
¡
¢
¡
∗
−θ0
. Define the function Z θ0 , θ∗ as follows:
Thus, E I|θ0 ≤ Pr θ0 + ξ ≤ θ∗ − κ = θ 2κ
¢
¡ ¢
¡ ¢ θ∗ − θ0
¡
− X (θ∗ ) .
Z θ0 , θ∗ ≡ X θ0 + U θ0
2κ
¢
¡
By the above arguments, in the proposed equilibrium, Z θ0 , θ∗ ≥ 0. We know that
¡
¢
¡
¢
¡
¢
¢
¡
Z θ0 , θ0 = 0, and that Z θ0 , θ0 + 2κ = X θ0 + T − X θ0 + 2κ < 0. Since the security
¡
¢
¡
¢
is concave, Z22 > 0. Thus, there are no θ0 and θ∗ ∈ θ0 , θ0 + 2κ for which Z θ0 , θ∗ ≥ 0.
This is a contradiction to the proposed equilibrium.
∗
/ Θ∗ . There exists some sequence (θi )∞
Suppose now that θ∗ ∈
i=0 ⊂ Θ that converges to
θ∗ . Moreover, by Claims 2, 3, and 4, E (I|θi ) → 0 as i → ∞: for if this is not true, there is a
θi ≤ θ∗ at which the price is above X (θ∗ ). For each θi in this sequence there exists at least
one fundamental, θ0i , at which the price is the same and which lies to the left of θ̂. (If instead
all fundamentals with price P (θi ) were to the right of θ̂, no intervention would occur, and
¡
¢ ¡ ¢
¡ ¢
they could not have the same price.) So X θ0i + E I|θ0i U θ0 = X (θi ) + E (I|θi ) U (θi ).
42
Note that θi − θ0i is bounded away from 0 as i → ∞ since θi → θ∗ > θ̂. We know that
¡
¢
E I|θ0i =
≤
Z
θ0i +κ
θ0i −κ
θi −κ
Z
θ0i −κ
I (P (θi ) , φ)
1
dφ
2κ
1
I (P (θi ) , φ) dφ +
2κ
θi − θ0i
+ E (I|θi ) .
≤
2κ
¡
¢
Define the function Z θ0i , θi as follows:
Z
θi +κ
I (P (θi ) , φ)
θi −κ
1
dφ
2κ
¢
¡ ¢
¡ ¢ θi − θ0i
¡ ¡ ¢
¢
¡
− X (θi ) + E (I|θi ) U θ0i − U (θi ) .
Z θ0i , θi ≡ X θ0i + U θ0i
2κ
¡
¢
By the above arguments, in the proposed equilibrium, Z θ0i , θi ≥ 0. We use εi to de¡ ¡ ¢
¢
note E (I|θi ) U θ0i − U (θi ) . We know that ε approaches 0 (the value of intervention,
¢
¡
¢
¡
U (θ), is bounded above by T ). We know that Z θ0i , θ0i = εi , and that Z θ0i , θ0i + 2κ =
¢
¡
¢
¡
X θ0i + T − X θ0i + 2κ + εi < 0 for all i large enough. Since the security is concave,
¡
¢
(θ −θ0 )(X(θ0i +T )−X(θ0i +2κ))
.
Z22 > 0. Thus, for any θi between θ0i and θ0i + 2κ, Z θ0i , θi ≤ εi + i i
2κ
¡ 0 ¢
2κεi
. Then,
This implies that Z θi , θi ≥ 0 can hold only if θ0i ≤ θi ≤ θ0i + X(θ0 +2κ)−X(θ
0
+T )
i
i
since εi approaches 0, there are no θ0i and θi that are bounded away from each other, for
¢
¡
which Z θ0i , θi ≥ 0. This is a contradiction to the proposed equilibrium.
Proof of Proposition 8:
Suppose the regulator observes the price of securities A and
B, where security A is strictly convex and security B is strictly concave. The heart of the
proof is the following straightforward claim:
Claim: For any pair of fundamentals θ1 and θ2 6= θ1 there is no probability q ∈ (0, 1) such
that
Xs (θ1 ) + qUs (θ1 ) = Xs (θ2 ) for securities s = A, B
(13)
Xs (θ1 ) + qUs (θ1 ) = Xs (θ2 + T ) for securities s = A, B.
(14)
or
Proof of Claim: Observe that
⎧
⎧
⎫
⎫
⎨ > ⎬
⎨ convex ⎬
Xs (θ1 )+qUs (θ1 ) = (1 − q) Xs (θ1 )+qXs (θ1 + T )
X (θ + qT ) if security s is
⎩ < ⎭ s 1
⎩ concave ⎭
43
Since Xs is monotone strictly increasing for both securities, it is immediate that neither
(13) nor (14) can hold.
The proof of the main result applies this Claim.
Consider any equilibrium, and let
Θ be the set of fundamentals that share the same price vector as a fundamental at which
intervention is suboptimal.
non-empty.
Suppose that (contrary to the claimed result) the set Θ is
Let θ∗ be its supremum.
Clearly if θ∗ ≤ θ̂ then for all equilibrium prices
associated with fundamentals Θ the regulator would know the true fundamental lies below
θ̂, and would intervene optimally. So θ∗ > θ̂. Moreover, by Lemma 3, θ∗ ≤ θ̂ + 2κ < θ̂ + T .
By construction, for fundamentals θ > θ∗ the regulator intervenes optimally, so P (θ) =
X (θ). Therefore, for all fundamentals θ ∈ Θ the equilibrium price vector satisfies P (θ) ≤
X (θ∗ ). Consider an arbitrary sequence {θi } ⊂ Θ such that θi → θ∗ .
The intervention
probabilities converge to zero along this sequence, E [I|θi ] → 0 (otherwise, the equilibrium
price would strictly exceed X (θ∗ ) for some θi ). There are two cases to consider:
Case A: On the one hand, suppose there exists some ε > 0 and some infinite subsequence {θj } ⊂ {θi } such that for each θj there is a fundamental θ0j 6= θj with the same
¤
£
price, and E I|θ0j ∈ [ε, 1 − ε]. It follows that there is a subsequence {θk } ⊂ {θj } such
£
¤
that for each θk there is a fundamental θ0k 6= θk with the same price, and E I|θ0k converges
to q ∈ [ε, 1 − ε] as k → ∞. Since for all k
¡ ¢
£
¤ ¡ ¢
Xs (θk ) + E [I|θk ] Us (θk ) = Xs θ0k + E I|θ0k Us θ0k
© ª
for securities s = A, B, and the left-hand side converges to Xs (θ∗ ), it follows that θ0k
¡ ¢
¡ ¢
must converge also, to θ0 say. Thus Xs (θ∗ ) = Xs θ0 + qUs θ0 for securities s = A, B,
directly contradicting the above Claim.
Case B: On the other hand, suppose that Case A does not hold. So there exists an
infinite subsequence {θj } ⊂ {θi } such that for each fundamental θ0j possessing the same
£
¤
price as θj the intervention probability E I|θ0j is either less than 1/j or greater than
1 − 1/j. It follows that for j large, all fundamentals with the same price vector as θj are
close to either θ∗ (if the intervention probability is close to 0) or θ∗ − T (if the intervention
probability is close to 1): formally, there exists some sequence εj such that εj → 0 and such
44
that θ0j ∈ [θ∗ − T − εj , θ∗ − T + εj ] ∪ [θ∗ − εj , θ∗ ].
But for j large enough, θ∗ − εj > θ̂,
θ∗ − T + εj < θ̂, and (θ∗ − εj ) − (θ∗ − T + εj ) = T − 2εj > 2κ. That is, for j large, if the
regulator observes price vector P (θj ) and its own signal, it knows with certainty which side
of θ̂ the fundamental lies. As such, it intervenes optimally, giving a contradiction.
Proof of Proposition 9: Exactly as in Proposition 7 a fully-revealing equilibrium cannot
exist. Suppose a non-fully revealing equilibrium exists. So at some set of fundamentals
Θ∗ the prices of both the concave and convex securities must be the same for at least two
distinct fundamentals. That is, the set
©
¡ ¢
ª
Θ∗ ≡ θ : ∃θ0 6= θ such that Pi (θ) = Pi θ0 for all securities i
is non-empty. The proof of the concave half of Proposition 7 applies, and gives a contradiction.
Proof of Proposition 10: First, in any equilibrium where there exist θ1 < θ2 with
the same price, the expected intervention probabilities E [θ1 |I] and E [θ2 |I] must differ
(otherwise prices would not be identical). Given that the probability of intervention can be
directly inferred from Q(θ), then the regulator can always infer θ based on P (θ) and Q(θ).
Then, the regulator will choose to intervene when θ ≤ θ̂, and not intervene otherwise. The
same is true if the equilibrium is fully revealing. Thus, if there is an equilibrium, it must
feature optimal intervention.
Now, let us show that optimal intervention is indeed an equilibrium. In such an equilibrium, the price of any bank security is X(θ + T ) for θ < θ̂ and X(θ) for θ > θ̂.
regulator security has a price of 1 for θ < θ̂ and 0 for θ > θ̂.
The
Then, independent of the
regulator’s private signal, the regulator will choose to intervene below θ̂ and not intervene
above θ̂. This is indeed consistent with the prices, so optimal intervention is an equilibrium.
Proof of Proposition 11: Suppose that contrary to the claimed result there is another
equilibrium.
It cannot entail fully revealing prices.
Let Θ be a set of (at least two)
h i
fundamentals that are all associated with the same price. Clearly the sets Θ ∩ θ, θ̂ and
Θ ∩ (θ̂, θ̄] are both non-empty, since otherwise all fundamentals in Θ lie to the same side
45
of θ̂ – in which case intervention occurs at all members of Θ with the same probability
(either 0 or 1), and so the fundamentals cannot be associated with the same price.
h i
Let θ∗ be the highest fundamental in Θ that is still below θ̂, i.e., θ∗ = sup Θ ∩ θ, θ̂ .
Take θ0 ∈ Θ ∩ (θ̂, θ̄]. Let (θi ) be a sequence in Θ converging to θ∗ such that θi ≤ θ∗ . (The
degenerate case in which θ∗ ∈ Θ and θi = θ∗ for all i is a special case.)
Observe that for any θi in the sequence,
Z θ0 +κ
£ 0¤
1
I (P (Θ) , φ) dφ
=
E I|θ
0
θ −κ 2κ
Z θ∗ +κ
Z θ0 +κ
1
1
I (P (Θ) , φ) dφ +
I (P (Θ) , φ) dφ
=
θ∗ +κ 2κ
θi +κ 2κ
Z θi +κ
Z θ0 −κ
1
1
+
I (P (Θ) , φ) dφ −
I (P (Θ) , φ) dφ.
θi −κ 2κ
θi −κ 2κ
Since there is no intervention at regulator signals above θ∗ + κ, it follows that
¤
£
E I|θ0 ≤ E [I|θi ] + εi ,
where εi → 0. In the conjectured equilibrium, for all i
¤¡ ¡ ¢
¢
£
¡ ¢
X θ0 − X (θi ) = E [I|θi ] (U (θi ) − T ) − E I|θ0 U θ0 − T .
¡ ¢
The LHS is strictly positive, and is bounded away from 0. Since U θ0 − T ≤ 0, the RHS
is bounded above by
¡ ¡ ¢
¡
¡ ¢¢
¢
¡ ¡ ¢
¢
E [I|θi ] (U (θi ) − T )−(E [I|θi ] + εi ) U θ0 − T = E [I|θi ] U (θi ) − U θ0 −εi U θ0 − T .
¡
¡ ¢¢
As i → ∞ this upper bound converges to E [I|θi ] U (θi ) − U θ0 . This term is weakly
negative since X is convex and so U is increasing. This gives the required contradiction
and completes the proof.
Proof of Proposition 12:
h
i
Denote the size of the set of parameters in θ̂ − T, θ̂ over
which the regulator intervenes optimally as λ− , and the size of the set of parameters in
h
i
θ̂, θ̂ + T over which the regulator intervenes optimally as λ+ .
By the shape of the price function under optimal intervention (see Figure 1), every
h
i
fundamental θ ∈ θ̂ − T, θ̂ that exhibits optimal intervention decision implies that the
46
h
i
intervention decision at θ + T ∈ θ̂, θ̂ + T is suboptimal. This is because optimal interven-
tion at both θ and θ + T implies that the two fundamentals have the same price, but this
is impossible in a commitment equilibrium. Thus, the set of fundamentals with optimal
h
i
intervention in θ̂ − T, θ̂ cannot be greater than the set of fundamentals with suboptimal
h
i
intervention in θ̂, θ̂ + T . That is, λ− ≤ T − λ+ , which implies that λ− + λ+ ≤ T . This
completes the proof.
47
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