Incomplete Risk Sharing Arrangements and the Value ofInformation

by

Bernhard Eckwert and Itzhak Zilcharevised September 2001

Bernhard Eckwert

Department of Economics

University of Chemnitz

D–09107 Chemnitz

Germany

Itzhak Zilcha

The Berglas School of Economics

Tel-Aviv University

Ramat Aviv

Israel

1Financial support from the Stifterverband fur die Deutsche Wissenschaft and theCommerzbank is gratefully acknowledged. We are also grateful to Andreas Szczutkowskiand to an anonymous referee for assistance.

Abstract

The paper constructs a theoretical framework in which the value of information in

general equilibrium is determined by the interaction of two opposing mechanisms:

first, more information about future random events leads to better individual deci-

sions and, therefore, higher welfare. This is the ‘Blackwell effect’ where information

has positive value. Second, more information in advance of trading limits the risk

sharing opportunities in the economy and, therefore, reduces welfare. This is the

‘Hirshleifer effect’ where information has negative value. We demonstrate that in an

economy with production information has positive value if the information refers to

non-tradable risks; hence, such information does not destroy the Blackwell theorem.

Information which refers to tradable risks may invalidate the Blackwell theorem if

the consumers are highly risk averse. The critical level of relative risk aversion

beyond which the value of information becomes negative is less than 0.5.

Keywords: Value of information, general equilibrium, risk sharing markets.

JEL classification numbers: D8, D51, D52.

1 Introduction

In an uncertain economic environment rational agents choose actions based on the

currently available information. Many economic studies use a signalling approach

to model information structures: decision makers observe random signals which

are correlated to an unknown state of nature, and update their beliefs, before the

choice of action. Blackwell has shown, in his seminal (1953) contribution, that more

precise information results in higher welfare. Later, it was demonstrated by many

economists that Blackwell’s result does not hold in various economic circumstances

where information is public and, hence, signals may affect the feasible set of actions

that each agent faces (e.g., by affecting prices).

Dreze (1960) was the first to point out that in some equilibrium situations more

information can be harmful. Later, Hirshleifer (1971, 1975) has demonstrated ex-

plitly that in an exchange economy the value of information may be negative. This

observation was followed by similar results attained by Marshall (1974), Green

(1981), Hakansson et. al. (1982), and others. The main reason for the failure of

Blackwell’s theorem, i.e., the negative value of public information, is that in an

economy with risk sharing mechanisms the release of more information may elimi-

nate opportunities to reallocate risk through trade. As a consequence, agents may

be worse off with better information from an ex ante point of view. For exchange

economies further results were derived by Orosel (1996), Schlee (1998), and Citanna

and Villanacci (2000). Orosel restricts the preferences of consumers to the class of

CARA and shows that under certain conditions more information can be harmful.

Schlee, by contrast, concentrates on risk allocations and demonstrates that under

efficient risk sharing arrangements more information makes all agents worse off.

Citanna and Villanacci consider an asymmetric information setting and show that

informational efficiency may or may not augment allocative efficiency.

The various possible circumstances where the Blackwell theorem fails to hold

were studied by many researchers in partial equilibrium as well. Let us mention

here two examples only: Sulganik and Zilcha (1997) consider signal-dependent fea-

sible sets of actions, and show that more information can be harmful even when

the feasible sets expand with better information. Another example which yields a

similar result was pointed out by Wakker (1988) who considers non-expected util-

1

ity decision makers. Taking a broader view Campbell (2001) analyzes the value

of public information under the solution concept of implementable allocations (in-

stead of competitive equilibrium). In this theoretical setting he derives a fully

general negative relationship between more information and welfare. The possibil-

ity that information may not always be desirable has also been demonstrated in

private information economies. Mailath and Sandroni (1998) show that under cer-

tain conditions poorly informed agents will survive in the asset market while better

informed agents will not (see also Marin and Rahi (1996)).

Almost all of the above mentioned papers consider pure exchange economies in

which agents trade in complete markets after observing the public signal. In such

economies the allocation is ex post (i.e., conditional on the signal) efficient and,

hence, better information cannot be a Pareto improvement (Schlee (1998)). In this

work we reconsider the value of information in a production economy with markets

that are ex post incomplete. We demonstrate that the results attained for pure

exchange economies fail to hold if production is taken into account.1

We take the random state to be composed of two additive shocks: one random

component we call the ‘insurable risk’ (since this type of risk can be hedged using

some futures markets), and the other one is the ‘uninsurable risk’. In equilibrium,

the insurable risk is efficiently allocated while the allocation of the uninsurable

risk is inefficient. Evaluating the welfare implications of better information in such

equilibrium framework we face two possibly opposing effects: on the one hand, by

anticipating the random state in a more reliable way agents are able to improve

the quality of their decisions. This is called the ‘Blackwell-effect’ where more in-

formation has positive impact on economic welfare. On the other hand, the more

accurate is the information revealed by the signal the more limited are the risk

sharing opportunities in this economy; we call this effect of information on risk

sharing the ‘Hirshleifer-effect’. We claim that the robustness of Blackwell’s theo-

rem in general equilibrium is closely related to the efficiency of the risk allocation.

Intuitively, the more informative the signals are the less risks can be shared and the

lower is the welfare attained by risk averse agents. Thus the interaction between

the Blackwell-effect and the Hirshleifer-effect determines the value of information

1The idea that production may be important in getting a positive value of information isalready present in the early work by Marshall (1974).

2

in general equilibrium.

Our main findings can be summarized as follows:

(a) More information regarding risks that cannot be insured within the existing

market structure has always positive value; in this case the Hirshleifer-effect

is absent and, hence, the Blackwell-effect is dominant.

(b) More information regarding the insurable risks has negative value (for all

agents) if the relative measure of risk aversion is not too low (more precisely,

the relative measure of risk aversion is greater or equal to 0.5). In this case,

the Hirshleifer-effect is negative and dominates the Blackwell-effect.

The paper is organized as follows. We present the model in Section 2. In Section

3 we consider first the case where more information is related to risks that cannot

be insured, and later where it is related to the insurable risks. Section 4 concludes

the paper.

2 The Model

We consider an economy which extends over two periods. The economy has two

types of homogeneous agents and a single commodity (capital) that can be either

consumed or used as an input in a production process. We will refer to the first type

of agents as consumers and to the second type of agents as firm owners. Consumers

live for two periods, consume in both periods and supply (inelastically) one unit of

labor in each date t, t = 0, 1. Firm owners also live for two periods but consume

only in the last period. Each firm owner possesses a firm which produces under

constant returns to scale. The production process, which takes place in period 1, is

affected by random shocks. Aggregate output is given by θF (K,L), where K is the

aggregate capital input and L is the aggregate labor input. The random shock θ to

the production technology realizes in period 1. F (K,L) is a neoclassical production

function with constant returns to scale and satisfies: FK(0, L) = ∞, FL(K, 0) =

∞, FK > 0, FL > 0, FKK < 0 and FLL < 0.

We assume that the production shock is composed of two additive components,

θ = λ+ φ. The random variables λ and φ are stochastically independent; they take

3

values in Λ := [λ1, λ2] and Φ := [φ1, φ2], where 0 < λ1 < λ2 < ∞, 0 < φ1 < φ2 <

∞. All agents have access to a futures market in which the λ-risk can be insured.

The φ-risk, by contrast, is uninsurable. On the futures market, which is open at

date 0, contracts for contingent delivery of the commodity are traded. A futures

contract pays off λ units of the commodity at the end of period 1, if the realization

λ occurs.

Let k = K/L and denote by θf(k) per-capita production function. Our previous

specifications imply: f ′ > 0, f ′′ < 0 and f ′(0) =∞. At the outset of period 1, after

the production shock θ has realized, profit maximizing firms rent capital and hire

labor in competitive markets. Assuming that capital depreciates completely in the

production process, the interest factor 1 + r is given by the marginal product of

capital and the wage rate w is equal to the marginal product of labor.

The uncertainty in period 1 is represented by the lack of knowledge of the state

variable (λ, φ) ∈ Λ×Φ. In period 0 all agents observe a signal y which is correlated

to the unknown state variable. Thus the relevant expectation for (λ, φ) is the

updated posterior belief.

The firm owners are risk neutral. Each firm owner possesses some (per-capita)

endowment e at date 1 which is large enough to ensure non-negative total income,

i.e.,2

e ≥ f(I0)(λ2 − λ1) for all (λ, φ) ∈ Λ× Φ. (1)

I0 denotes the total income of each consumer in period 0, which is derived from labor

and capital incomes. Risk neutrality on the part of the firm owners in combination

with (1) implies that the futures market is unbiased, i.e., this market clears at a

price p(y) (which is due at the end of period 1) equal to the conditional mean of

the contract’s payoff,

p(y) = E[λ|y].

Since on average profits from speculation on the futures market are zero, average

consumption of the firm owners at date 1 equals e.

2For simplicity we assume that e is a constant. All of our results remain valid if e is contingenton the state (λ, φ).

4

At the beginning of period 0 the level of the aggregate capital stock is fixed

(which implies fixed factor prices at date 0 in equilibrium). The capital stock at the

outset of date 1 will be determined by the aggregate savings of the consumers in

period 0. Each consumer maximizes expected utility, defined over lifetime consump-

tion. We assume that the von-Neumann Morgenstern utility index is time-separable

and has the following form:

U(c0, c1) = v(c0) + u(c1)

where v(·) and u(·) are strictly concave and strictly increasing utility functions. If s

denotes individual savings and h denotes the sale (or purchase, if negative) on the

futures market, the typical consumer chooses s, h such that:

maxE[v(c0) + u(c1)] s.t. (2)

c0 = I0 − s

c1 = (1 + r)s+ w + h(p(y)− λ),

where E stands for the expectation operator conditional on the information revealed

at date 0.

The timing of events in the economy is as follows:

0︷ ︸︸ ︷ 1︷ ︸︸ ︷

choice of

information

system

signal y realizes;

futures market

opens;

consumers

choose s and h

state variables λ

and φ realize;

firms choose k

production of output;

payment of wage and

capital income;

settlement of futures

contract;

firm owners receive e;

consumption of c1

Let us define a competitive equilibrium in this framework.

5

Definition 1 Given the first period per-capita capital stock k0 and a realization

of the signal y, [c∗0, c∗1(λ, φ), s∗, r∗(λ, φ), w∗(λ, φ)] is a competitive equilibrium, if

c∗0, s∗ ∈ R+; h∗ ∈ R; c∗1(λ, φ), r∗(λ, φ), w∗(λ, φ) are functions defined on Λ× Φ into

R+ and

(i) Given the random interest rate r∗(λ, φ), and wages w∗(λ, φ) for period 1,

consumers’ optimum in problem (2) is attained at [c∗0, c∗1(λ, φ), s∗, h∗], where

I0 = f(k0).

(ii) The aggregate capital stock at date 1 equals aggregate savings at date 0, hence

k∗ = s∗, where k∗ is the per-capita capital at date 1.

(iii) Factors markets are competitive:

1 + r∗(λ, φ) = (λ+ φ)f ′(k∗) for all (λ, φ) ∈ Λ× Φ (3)

w∗(λ, φ) = (λ+ φ)[f(k∗)− k∗f ′(k∗)] for all (λ, φ) ∈ Λ× Φ (4)

The firm owners observe the state of nature prior to their decisions about capital

borrowing and hiring labor. Therefore, according to (3) und (4), the wage rate

equals the marginal product of labor and the rate of return to capital equals the

marginal product of capital for all (λ, φ).

2.1 Information Systems

The uncertainty in period 1 on the part of the consumers and the firm owners is

represented by the lack of knowledge of the state variable (λ, φ) ∈ Λ × Φ. Before

trading on the goods and futures market at date 0, the agents observe a signal

y = (αyλ, βyφ) ∈ Y ⊂ R2 which is correlated to the unknown state variable. Thus

the relevant expectation for (λ, φ) is the updated posterior belief. The constants

α and β take values in {0, 1}. If, e.g., α = 0 and β = 1, then only the second

component of the signal contains information on the unknown state variable (λ, φ).

This specification captures the case where agents can observe only the signal (com-

ponent) yφ but not yλ. α = 1, β = 1 implies that both signals can be observed. We

assume that yλ and yφ are stochastically independent, i.e., yλ contains information

related only to the risk factor λ. Similarly, yφ contains information related only to

the risk factor φ.

6

An information system, denoted (Y, g1), specifies for each state of nature a

conditional probability function over the set of signals: g1(y|λ, φ). The positive real

number g1(y|λ, φ) defines the conditional probability (density) that if the state is

(λ, φ), then the signal y will be sent. The agents do not observe the true state.

Instead they observe a signal and they know perfectly the conditional probability

function g1(y|λ, φ) according to which the signals are generated by the states of

nature. Using Bayes’s rule the agents revise their expectations and maximize utility

on the basis of their updated beliefs.

Let π : Λ× Φ→ R+ be the (Lebesgue-) density function for the prior distribu-

tions over Λ×Φ. Since the random variables λ and φ are stochastically independent,

we can write π(λ, φ) = π(λ)π(φ) in obvious notation.3 The density for the prior

distribution over Y is given by

ν1(y) =

∫Λ×Φ

g1(y|λ, φ)π(λ, φ) d (λ, φ)

=

∫Λ×Φ

g1(y|λ, φ)π(λ)π(φ) dλ dφ for all y. (5)

The density function for the updated posterior distribution over Λ× Φ is

ν1((λ, φ)|y) = ν1(λ|y)ν1(φ|y) = g1(y|λ, φ)π(λ, φ)/ν1(y). (6)

Following Blackwell (1953) we now define a criterion which allows a comparison of

different information systems with respect to their informational contents.

Definition 2 (Informativeness) Let (Y, g1) and (X, g2) be two information sys-

tems. (Y, g1) is said to be more informative than (X, g2), (Y, g1) �inf (X, g2), if

there exists an integrable function g : X × Y → R+ such that∫X

g(x, y) dx = 1 for all y. (7)

g2(x|λ, φ) =

∫Y

g1(y|λ, φ)g(x, y) dy (8)

3To avoid notational clutter we distinguish between the functions π(λ, φ), π(λ) and π(φ) onlyby their arguments. Below we will also apply this notational simplification to the density functionsof all other random variables.

7

holds for all (λ, φ) ∈ Λ× Φ.4

This concept of informativeness is based on a simple and intuitive idea: suppose

that a signal y ∈ Y has realized. Now consider some stochastic mechanism, com-

patible with equation (7), which transforms y into a signal in X according to the

probability density g(·, y). If the signals in X are generated this way then the infor-

mation system (X, g2) can be interpreted as being obtained from the information

system (Y, g1) by adding some noise through a process of randomization.

In economic applications the criterion stated in Definition 2 often creates techni-

cal difficulties. This is also true for the analysis of our model. The following lemma

contains a property of information systems that turns out to be a convenient tool.

To ease the technical derivations we shall state it for the case where the parameter

constellation is given by α = 1, β = 0; i.e., when agents receive information only on

the risk factor λ via the signal component yλ.

Lemma 1 Consider the parameter constellation α = 1, β = 0. Information system

(Y, g1) is more informative than information system (X, g2) if and only if for every

convex function F (·) on the set of density functions over Λ,∫Y

F (ν1(·|y))ν1(y) dy ≥∫X

F (ν2(·|x))ν2(x) dx (9)

holds.

A proof can be found in Kihlstrom (1984). Note that ν1(·|y) and ν2(·|x) are

the posterior beliefs about the λ-risk under the two information systems. Thus,

paraphrasing the technical characterization in Lemma 1, a more informative struc-

ture (weakly) raises the expectation of any convex function of posterior beliefs.

Conversely, for a concave function F this inequality reverses.

2.2 Information and Economic Welfare

The firm owners make their production decision in the absence of uncertainty after

the state of nature (λ, φ) has become known. Since the technology exhibits con-

4By postulating that the function g must be integrable we rule out the possibility that the twoinformation systems are equally informative, i.e., that (Y, g1) �inf (X, g2) and (X, g2) �inf (Y, g1)holds.

8

stant returns to scale, no profits accrue from the production process itself. Also,

speculation on the futures market results in zero average profits since in equilibrium

this market is unbiased. Being risk neutral, the expected utility of the firm owners

therefore does not depend on information and, hence, the firm owners can be ne-

glected in our welfare analysis. By contrast, economic welfare of the consumers is

affected by the underlying information system.

Consider the information system (Y, g1). Due to the strict concavity of the

utility function, optimal savings s and futures market hedging h are determined

uniquely by the first order conditions to problem (2):

v′(I0 − s) = E[(λ+ φ)f ′(s)u′((λ+ φ)f(s) + h(p(y)− λ)) | y

], (y ∈ Y ) (10)

E[(p(y)− λ)u′((λ+ φ)f(s) + h(p(y)− λ)) | y

]= 0, (y ∈ Y ). (11)

In (10) and (11) we have used the market clearing factor prices from (3) and (4);

namely, c1 = (λ + φ)f(s) + h[p(y)− λ]. Since the futures market is unbiased, (11)

implies Cov(λ, u′(c1)) = 0 and hence, due to the strict monotonicity of u′, we must

have h = f(s). Using this equality in (10) we arrive at5

v′(I0 − s) = E[(p(y) + φ)f ′(s)u′((p(y) + φ)f(s)) | y

], (y ∈ Y ). (12)

Denote by s(y) the unique solution to (12) and define for any realization of the signal

y the value function V (y) as the level of a typical consumer’s expected utility,

V (y) := v(I0 − s(y)) + Eφ

[u((p(y) + φ)f(s(y))) | y

]. (13)

Economic welfare, W (Y, g1), is defined as the ex ante expected utility of the con-

sumers at the beginning of period 0,

W (Y, g1) := Ey[V (y)] = Ey

[v(I0 − s(y)) + Eφ[u((p(y) + φ)f(s)) | y]

]. (14)

The value of information is positive if the consumers are better off under a more

informative information system, i.e., if W (Y, g1) ≥ W (X, g2) whenever (Y, g1) �inf

(X, g2).

5Note that equilibrium savings are strictly positive because f satisfies the Inada conditions.

9

3 The Value of Information

In this section we analyze the value of information in two different settings. The

first setting is described by the parameter constellation α = 0, β = 1. Thus the

signal (component) yλ is unobservable while yφ is publicly observed. Recall that yφ

contains information related only to the non-insurable risk factor φ. The second

setting corresponds to the parameter constellation α = 1, β = 0. Under this specifi-

cation agents cannot observe the signal yφ. Instead, they observe yλ which contains

information related only to the insurable risk factor λ.

3.1 The Value of Information on the Non-Insurable Risk

We analyze the value if information under the parameter constellation α = 0, β = 1,

i.e., the agents observe only the signal component yφ. For later reference ob-

serve that in equilibrium the insurable λ-risk is efficiently allocated while the non-

insurable φ-risk is not: future consumption of the consumers c1 = (p(y) + φ)f(s)

does not depend on λ. Hence the insurable risk is allocated efficiently because it

is fully shifted to the risk neutral firm owners. On the other hand, the allocation

of the non-insurable φ-risk is fully inefficient: period 1 consumption of the firm

owners, e+ f(s)(λ− p(y)), does not depend on φ, hence the risk averse consumers

bear all of the non-insurable risk.

Theorem 1 Let (Y, g1) and (X, g2) be two information systems and consider the

parameter constellation α = 0, β = 1; i.e., the agents observe only the signal

yφ which contains information on the non-insurable risk factor φ. If (Y, g1) �inf

(X, g2), economic welfare is higher under (Y, g1) than under (X, g2), i.e., W (Y, g1) ≥W (X, g2).

Proof: See appendix.

Remark: The weak inequalilty in Theorem 1 holds as a strict inequality whenever

the consumers’ utility in the second period is non-logarithmic. With logarithmic

utility optimal savings are independent of the signal and, hence, information has

no value.

10

Theorem 1 contains a surprising result (see the opposing result attained by

Schlee (1998) for the case of pure exchange under efficient risk sharing). Although

in our equilibrium model both the information system and the signals affect the

agents’ opportunity sets via endogenously determined factor prices, information

on the non-insurable risk always has value. In interpreting this finding we will

argue below that the robustness of the Blackwell Theorem in general equilibrium

is intimately connected to the efficiency of the risk allocation: information tends

to have less value if risks, which are affected by the information, are shared more

efficiently.

In economies where risk sharing mechanisms are active more precise information

tends to have two possibly opposing effects: First, by anticipating the uncertain fu-

ture economic environment in a more reliable way, the agents are able to improve

the quality of their decisions. This is the Blackwell-effect where more information

has a positive impact on economic welfare. Second, the more precise is the infor-

mation revealed by the signal, the more limited are the risk sharing opportunities

in this economy. This mechanism, which may have negative welfare implications,

is called the Hirshleifer-effect.

Providing some intuition for the Hirshleifer-effect consider the extreme case

where the signal is fully informative and hence reveals the future state. In such an

informational environment all risks have effectively materialized via the signal when

the current markets open. Thus at the time when the risk sharing mechanism could

be operative there are no risks left to be shared across agents. More generally, if

the signal is less than fully informative, a risk sharing mechanism allows the agents

to share only those risks which have not yet been resolved through the signal. In

other words, the more informative is the signal, the less risks can be shared. This

effect on risk sharing is the Hirshleifer-effect which may have a negative impact on

economic welfare.

The interaction of the Blackwell-effect and the Hirshleifer-effect determines the

value of information in general equilibrium. Under the specification α = 0, β = 1

in Theorem 1 the signal y contains only information on the φ-risk. Accordingly,

while it is true that the Hirshleifer-effect limits the risk sharing opportunities in the

economy, this does not result in welfare losses because the allocation of the φ-risk is

11

perfectly inefficient and, hence, cannot deteriorate any further. As the Blackwell-

effect is always benign, better information increases economic welfare under the

assumption of Theorem 1.

3.2 The Value of Information on the Insurable Risk

Now we consider the other setting α = 1, β = 0, under which the agents observe

only the signal component yλ.6 Define γ(c1) = − c1u′′(c1)

u′(c1)the measure of relative risk

aversion of the consumers for the second period utility function.

Theorem 2 Let (Y, g1) and (X, g2) be two information systems with (Y, g1) �inf

(X, g2), and consider the parameters α = 1, β = 0; i.e., the agents observe only

the signal yλ which contains information on the insurable risk factor λ. Economic

welfare is higher under (X, g2) than under (Y, g1), i.e., W (X, g2) ≥ W (Y, g1), if

relative risk aversion of the consumers, γ(c1), is greater than or equal to 1/2 for all

c1 ≥ 0 .

Proof: See Appendix.7

Comparing theorems 1 and 2, the impact of more precise information on eco-

nomic welfare appears to be less favorable (or even harmful) the better agents can

hedge the risks to which the information is related. This observation is surprising

at first sight because well developed insurance markets are commonly expected to

6After suitable modification of the timing of the markets and of the informational structure athird possibility might be of interest: all trades can be made contingent on both the signal and thestate and take place before news is revealed. Under this specification the information structurebecomes irrelevant because individual decisions in both periods will be conditioned only on thestate variable. As a consequence, the Hirshleifer-effect and the Blackwell-effect both disappear.

7The condition in Theorem 2 is merely sufficient and might be stronger than necessary if theproduction function, utility in period 0, and the priors are given. It is therefore natural to askwhether the condition is necessary if the stated result is to hold for arbitrary parametrizationsof the model. While we do not have a complete answer to this question it is possible to showthat the following is true: Whenever γ(c1) is less than 1/2 but larger than 1/4 then, under somesmoothness conditions, it is possible to construct an economy (satisfying the basic assumptionsin Section 2) such that the assertion in Theorem 2 does not hold for a suitably chosen pair ofinformation systems. The details are tedious and can be obtained from the authors on request.

12

benefit agents by enabling them to capitalize on their information more effectively.

Yet in view of the above theorems, just the opposite is true: information tends to

be less valuable if the market system offers better hedging opportunities against the

risk on which information is revealed. Looked at more closely, however, our results

for this equilibrium economy can be reconciled with traditional views on the value

of information, attained usually for partial equilibrium cases.

According to Theorem 2 better information can (but need not) be harmful if the

risks which are affected by the information are efficiently allocated. The Hirshleifer-

effect now plays an important role. Intuitively, the signal is more informative the

more closely the stochastic fluctuations of the signal are linked to the fluctuations of

the state variable. In particular, a signal that does not fluctuate at all contains no

information. In view of (12) a volatile (i.e., informative) signal induces instabilities

in the savings behavior and thus tends to reduce ex ante expected utility. The

market allocates risks efficiently only conditional on the signal. The risks emanating

from a volatile signal, however, cannot be insured and result in welfare losses. This

is the mechanism through which the Hirshleifer-effect translates more informative

signals into lower ex ante expected utility.

The Hirshleifer-effect may also be interpreted in terms of the ex ante efficiency

of the λ-risk allocation. In our model, the futures market allows agents to share

that part of the λ-risk which has not yet been resolved through the signal. Thus,

from an ex-ante point of view, in equilibrium only part of the insurable λ-risk is

allocated efficiently: the more informative is the signal, the less insurable risks are

efficiently allocated and the lower is economic welfare.

In Theorem 2 the Hirshleifer-effect is complemented by the Blackwell-effect. The

figure below gives the intuition of the interaction between these two effects. In order

to simplify the illustration the figure is drawn for the case of constant relative risk

aversion γ.

13

-

6

γγ∗

impact on eco-nomic welfare

12

Hirshleifer-effect (absolute value)

Blackwell-effect

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Since ex post (i.e., conditional on the signal) the λ-risk is allocated efficiently, the

Hirshleifer-effect causes a deterioration of the ex ante λ-risk allocation and, there-

fore, reduces economic welfare. The strength of the Hirshleifer-effect depends on the

risk aversion of the consumers. If the consumers are only slightly risk averse, then

the welfare loss caused by inefficiently allocated risks is small; hence the Hirshleifer-

effect will be dominated by the Blackwell-effect and overall welfare increases with

better information. Yet, in case the workers are strongly risk averse, their welfare

will be severely affected by inefficiently allocated risks. In the above figure the

critical value of relative risk aversion, beyond which the Hirshleifer-effect outweighs

the Blackwell-effect, is γ∗. In view of Theorem 2, we have γ∗ ≤ 12.8

Schlee (1998) showed that in pure exchange economies better information nec-

essarily makes all agents worse off, if in equilibrium risks are shared efficiently

and if some agents are risk neutral. Theorem 2 demonstrates that this result can-

not be generalized to models with production. In an exchange economy the cake

to be distributed across agents does not grow if, due to better information, the

agents are able to improve the quality of their decisions. Therefore, as long as

better information does not enhance the efficiency of the equilibrium allocation,

8The Hirshleifer-effect and the Blackwell-effect are not independent of one another. Since ourmodel is based on the maximization of the expected value of an intertemporally additive utilityfunction, high relative risk aversion goes hand in hand with low intertemporal substitution in con-sumption (see Hall (1988) and Kocherlakota 1990)). The measure of intertemporal substitution,in turn, affects the Blackwell-effect. E.g., with γ = 1 (logarithmic utility) it is optimal for theworkers not to react to information signals. Since the information is ignored by the agents theBlackwell-effect is absent and, hence, only the harmful Hirshleifer-effect is active.

14

the Blackwell-effect is nil. This is true, in particular, if the allocation is efficient

in the Pareto sense. In models with production, by contrast, the size of the cake

may grow when the economy becomes better informed. This may explain why

the welfare consequences of improved information tend to be less detrimental in a

production economy than in the context of pure exchange.

The informational specifications α = 0, β = 1 in Theorem 1 and α = 1, β = 0

in Theorem 2 are extreme in the sense that the signal y = (αyλ, βyφ) is either un-

informative with respect to the insurable λ-risk or uninformative with respect to

the non-insurable φ-risk. Under the specification α = 1, β = 1 the agents observe

both components of the signal and hence receive information on both risk factors.

In this case the equilibrium allocation of the risks which are affected by the in-

formation signal is neither efficient nor perfectly inefficient. Clearly, in our model

the specification α = 1, β = 1 implies that the effects described in Theorem 1 and

in Theorem 2 show up in combination: better information can but need not lead

to higher economic welfare. Signal components which reduce the exposure of the

agents towards inefficiently allocated risks tend to increase the value of the informa-

tion; signal components which reduce efficiently allocated risks have an ambiguous

impact. Such components tend to increase the value of the information if risk aver-

sion (of the workers) is low; and they tend to decrease the value of the information,

if risk aversion is high.

4 Concluding Remarks

The fact that information may have negative value in equilibrium is well known and

has extensively been discussed in the literature. Technically speaking, in general

equilibrium the individual opportunity sets depend on the information signals via

the price system. The Blackwell theorem therefore lacks applicability and, hence,

the value of information appears to have ambiguous sign.

This paper attempts to explain the value of information by the interaction of

familiar economic mechanisms. In full equilibrium, we have identified market and

information structures where the Blackwell theorem survives as well as market and

information structures where the Blackwell theorem breaks down. Our main re-

sult can be summarized as follows: the value of information is intimatly related to

15

the risk sharing opportunities in an economy. Information about risks for which

hedging markets exist has negative value if consumers are highly risk averse and

positive value if consumers are slightly risk averse.9 By contrast, information about

non-tradable risks always has positive value. Thus, as long as the signal contains in-

formation about non-insurable risks only, the Blackwell theorem is robust in general

equilibrium.

Our findings are not necessarily limited to economies where explicit risk sharing

markets exist. Even in the absence of any hedging instrument some risk sharing may

occur through the operation of goods and labor markets; e.g. part of the production

risk may be shifted to the consumption sector via stochastically fluctuating factor

prices. In principle, our analysis is applicable whenever some risk sharing takes

place in equilibrium. The mechanism at work in our model, which may lead to

lower welfare in better informed economies, has been studied in a different context

by Hart (1975), Milne and Shefrin (1987), Newbery and Stiglitz (1984), and others.

These authors show that the introduction of new securities (or opening new markets)

in an incomplete markets economy may make everybody worse off.

The analysis in this paper is subject to a number of limitations. Firstly, we

have assumed that all consumers and all firm owners are identical. Of course,

by continuity, under suitable smoothness conditions the conclusions of Theorem 1

and Theorem 2 will remain valid as long as the diversity among consumers and

firm owners is sufficiently small. The theorems may also hold in a weak form for

economies with much diversity among agents. Berk and Uhlig (1993) develop a

model with endogenous timing of information. They show that almost surely some

agents exist who prefer more information if there is enough heterogeneity in the

population of agents; these are those agents whose net trades under the initial

information system are sufficiently small. The basic idea underlying the argument

in the paper by Berk and Uhlig can also be applied to the informational structure

we have analyzed in Section 3.1. This suggests that our result in Theorem 1 is

robust with respect to the introduction of heterogeneity among agents in the weak

sense that at least some agents (out of a population with enough diversity) will

9In our model relative risk aversion is the critical measure. This is due to the assumption thatthe production shock is multiplicative. In the presence of an additive shock Theorem 2 can bereformulated in terms of the measure of absolute risk aversion.

16

benefit from better information. Similar considerations apply to Theorem 2.

Secondly, the insurable and the non-insurable risks interact additively. This

specification can be generalized without affecting the results as long as the risks

are stochastically independent. Thirdly, consumer preferences are additively sep-

arable. This is not an innocuous assumption; it shapes the interaction between

the Hirshleifer-effect and the Blackwell-effect as was pointed out in footnote 8 (see

also Hakansson et. al. (1982)). We leave a further clarification of this issue as an

interesting exercise for future research.

So far models of pure exchange have dominated the literature on the value of

information. Yet, in a framework of pure exchange the welfare implications of in-

formation cannot be studied adequately: in such a setting the Blackwell-effect is

absent and hence better information cannot make anybody better off in a com-

petitive equilibrium allocation of risk (Schlee (1998)). In a production economy,

however, more precise information may lead to improved allocation of inputs. As

we have shown this positive Blackwell-effect outweighs the possible negative effects

on risk sharing if risk aversion is sufficiently low.

Appendix

In this appendix we prove theorems 1 and 2.

Proof of Theorem 1: Since the signal y is uninformative with respect to the insurable

risk factor λ, the price of a futures contract is constant, p = E[λ]. Let s(y) be the

equilibrium optimal savings function of the workers if the economy operates under

the information system (Y, g1). s(y) satisfies the optimality condition (12). Each

worker attains a level of expected utility equal to

V (y) = v(I0 − s(y)) + Eφ[u((p+ φ)f(s(y))) | y].

Let us consider the following auxiliary decision problem for a hypothetical decision

maker:

Vh(y) = max0≤s≤I0

{v(I0 − s) + Eφ[u((p+ φ)f(s)) | y]

}(15)

(15) has a unique solution, s(y), which satisfies the necessary and sufficient first

17

order condition

v′(I0 − s(y)) = E[(p+ φ)f ′(s(y))u′((p+ φ)f(s(y))) | y

]. (16)

Comparing (12) and (16) we find that s(y) = s(y) which implies V (y) = Vh(y), y ∈Y. Thus, in equilibrium, a typical worker attains the same welfare level as the

hypothetical decision maker, i.e., W (Y, g1) = Wh(Y, g1) in obvious notation.

Observe that the utility functions v and u in (15) do not depend on the revealed

signal. The problem of the hypothetical decision maker therefore satisfies the con-

ditions of the Blackwell Theorem (1953). Thus the hypothetical decision maker

is weakly better off under the information system (Y, g1) than under (X, g2), i.e.,

Ey[Vh(y)] ≥ Ex[Vh(x)]. Since V (y) = Vh(y) holds for all y and V (x) = Vh(x) holds

for all x, we conclude W (Y, g1) ≥ W (X, g2), which proves the theorem.

Proof of Theorem 2: The informational specification in Theorem 2 describes an eco-

nomic environment where the information system acts on a single risk factor λ. In

view of Lemma 1 we have to show that under the conditions of the theorem the

value function (13) is concave in the updated posterior belief ν1(λ|y).

The first order condition for optimal savings is

v′(I0 − s) = Eφ

[[p(y) + φ]f ′(s)u′

([p(y) + φ]f(s)

) ](y ∈ Y ). (17)

Let s(p(y)) be the solution. The value function in (13) can now be written as

V(p(y)

)= v

(I0 − s(p(y))

)+ Eφ

[u(

[p(y) + φ]f(s(p(y))

) )]. (18)

Since p(y) := E[λ|y] is linear in the posterior belief ν1(λ|y), the value function will

be concave in ν1(λ|y) if it is concave in p(y). Differentiating (18) with respect to

p(y) and using the envelope theorem we get

V ′(·) = Eφ

[f(s(·))u′(

[p(y) + φ]f(s(·)) )]

and hence (omitting arguments of functions)

V ′′ = Eφ[f ′s′u′] + Eφ[fu′′

(f + (p+ φ)f ′s′

)](19)

18

Now, multiply equation (17) by s′ and differentiate with respect to p to obtain

0 = s′2v′′ + Eφ

[ (f ′s′ + (p+ φ)f ′′s′2

)u′]

+ Eφ

[ (f + (p+ φ)f ′s′

)(p+ φ)u′′f ′s′

]. (20)

Adding (20) to the right hand side of (19) and rearranging yields

V ′′ = s′2v′′ + Eφ[(p+ φ)s′2f ′′u′

]+ Eφ

[ (f + (p+ φ)f ′s′

)2u′′ + 2f ′s′u′

]. (21)

The first and second term on the right hand side of (21) are both negative. The

third term can be rewritten as

Eφ

[u′f(1 + A)2

p+ φ

(2A

(1 + A)2− γ(c1)

)], (22)

where A := (p + φ)s′f ′/f , and γ(c1) := −c1u′′(c1)/u′(c1). The above expression is

non-positive, as long as γ(c1) ≥ 2A/(1 +A)2, ∀c1. The last inequality is satisfied if

γ(c1) ≥ 1/2, ∀c1, since 2A/(1 + A)2 is bounded from above by 1/2.

We have shown that under the conditions of Theorem 2 the value function is

strictly concave in the updated posterior beliefs. The proof is complete.

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