Political decision of risk reduction: the role of trust
Meglena Jeleva] & Stéphane Rossignol ⊥∗]GAINS, Université du Maine et CES, Université Paris-I, Panthéon-Sorbonnejeleva@univ-paris1.fr⊥

Université de Versailles et CES, Université Paris-I, Panthéon-Sorbonnerossignol@math.uvsq.fr

July 5, 2007

Abstract
Surveys concerning environmental and health risks point out the lack oftrust of citizens in risk evaluations provided by governments. The aim of
this paper is to take into account this potential distrust in political decisions
concerning risk reduction. We prove that lack of trust reduces the attractivityof risk reduction measures. When heterogeneity in risk exposure and thepossibility of complete risk elimination are introduced, political decisions ofrisk reduction may differ from the preferred decision of any risk and trustgroup. Namely, total risk elimination can be adopted, even if all individuals
prefer null or partial risk reduction measures.
Keywords: risk, political decisions, Choquet expected utility preferences
JEL code: D7, D81.

1

Introduction

Environmental, as well as health and food risks have taken, in the last few decades,a more and more important place in public debate. This is essentially due to uncertainties about their characteristics that render the determination of optimal decisions
concerning these risks’ management particularly difficult. The Precautionary principle has been adopted to guide the authorities’ actions concerning environmental andhealth risks in the presence of scientific uncertainties. However, due to its generaland imprecise formulation, it allows multiple interpretations that make it difficult
to apply as a unique decision criterion (for a discussion on the controversies aroundthe Precautionary Principle, see Godard 2003). Political decisions concerning risk
management are generally guided by two types of consideration: scientific knowledge and individual preferences and beliefs. Beliefs are strongly influenced by public∗

The authors wish to thank J. Eichberger, H. Kempf and the participants of the JMA 2006
annual meeting for helpful comments.

1

information, and trust in this information. Several surveys realized in the Member
States of the European Union on different topics show that the trust of citizens
in their government is far from being complete. For instance, in a Eurobarometer
survey in 2004 on ”The Attitudes of European citizens towards environment” it appears that only 11% of the respondents trust their national government to informthem about environmental problems. Another Eurobarometer survey in 2005 about
radioactive waste confirms this result and even shows a decrease in confidence: itappears that in 2005 19% of the respondents trust their government when it informs
them on the treatment of radioactive waste, whereas it was 29% in 2001.
The aim of this paper is to study the impact of the lack of trust in the available
information on the political decisions concerning risk reduction. More precisely,
we determine the risk reduction level that emerges from a voting process wherevoters differ both in the confidence they have in the risk evaluation given by theauthorities and in their wealth. The obtained choices are discussed with respect todifferent interpretations of the precautionary principle and more precisely with themore radical interpretation that corresponds to a complete risk elimination.
The trust level is taken into account by a model proposed by Jaffray (1988) and
Cohen (1992) under risk and by Eichberger, Kelsey (1999) and Chateauneuf, Eichberger and Grant (2002) under uncertainty. In this model, generalizing expectedutility, beliefs are represented by a weighted sum of a probability distribution, corresponding to the initial information, and a capacity, characterizing complete uncertainty. With this choice criterion, a decision is evaluated by a combination of
the standard expected utility of the decision and its best and worst consequences.
This last element reflects the idea that if one does not believe at all in the available
risk evaluation, one will consider oneself to be in a pure state of uncertainty andwill take into account only the best and worst possible outcomes, according to his
degrees of pessimism-optimism.
The political decision criterion adopted here is that characteristic of a representative democracy modeled by probabilistic voting. The simple direct majority
indeed does not seem to be well adapted for the risk decisions considered here fortwo reasons: (i) it would lead to a complete ignorance of risks affecting a small
minority (which is not realistic), (ii) in the case of several groups differing by morethat one parameter, equilibrium may not exist. We assume that risk reduction isfinanced by a tax on wealth at a uniform rate.
Two types of risk are considered: global risks, that equally affect all the voters(such as global warming, GMO authorization and pandemics) and more specific risks
for which risk groups can be identified, differing both by the estimated probabilitiesof the risk occurrencing and by amounts of loss in the case of risk realization (such as
risks issued from new drugs, new technologies, chemical plant installations etc.). For
global risks, many risk reduction levels are available, going from the absence of any
action to a complete risk elimination (if technologically possible). For specific risks,
we assume that only three decisions are available: no action (status quo), complete
risk elimination (by prohibition of the source of risk) and a given intermediate levelof risk reduction.
The main results are the following.
2

Even if the risk is high, the politically decided level of risk reduction will be
low, if the average trust level is low. The impact of wealth on the risk reductionlevel depends both on the global wealth level in the population and on the wealthdistribution between the different distrust level groups. The difference between
the risk reduction level emerging from a political process and that preferred byan average individual depends on the individuals’ degree of relative risk aversion.
Moreover, it appears that in general, the political decision differs from the socially
optimal one.
When only three decisions are possible: no action (status quo), partial reduction
or complete risk eradication, and when individuals differ not only in their confidence
level but also in their objective risk exposure, the optimal choice may differ fromthe best choices of all the individuals in the population: Condorcet type paradoxes
are possible. These situations may occur when credible prohibition is not too costly
with respect to partial reduction.
The paper starts with the presentation of the preferences representation modeland the determination of the risk reduction level preferred by a given agent. We thenstudy the politically chosen risk reduction level when individuals differ in wealth but
are exposed to the same risk. The fourth section is devoted to the political decision
when individuals differ in risk exposure.

2

The individually preferred risk reduction level

The aim of this section is to determine the level of risk reduction preferred by agiven individual. This would be the implemented risk reduction level if the politicaldecision was taken only with respect to this individual’s preferences (or if this agent
was a dictator). This analysis provides a benchmark for the general study of thepolitical decision of risk reduction.

2.1

Individual preferences representation

We assume in this section that the population is composed of n individuals withpreferences towards wealth characterized by the same increasing and concave utility
function u. Each individual faces a risk of loss, resulting from the occurrence ofa catastrophic event E. The government provides an estimation of the probability
distribution of the individual loss.
We assume that individuals do not completely trust the risk characteristics provided by the government. At least two reasons can explain this distrust:- the risk is new and experts disagree on the estimation of the catastrophe probability that renders the ”official” estimation, often based on an average value, notreliable.
- citizens have more general reasons not to trust the government.
This lack of confidence in probability estimations is well taken into account by amodel in the Choquet Expected utility class proposed by Chateauneuf, Eichberger,
Grant (2002).
3

In this model, beliefs are characterized, not by a probability distribution, but by
a neo-additive capacity, defined in the following way:
Definition 1 Let Ω be a state space and A a σ-algebra of subsets of Ω. µ0 and µ1
are the capacities defined as follows:• µ0 (Ω) = 1 and µ0 (A) = 0 for all A ∈ A, with A 6= Ω;• µ1 (∅) = 0 and µ1 (A) = 1 for all A ∈ A, with A 6= ∅.
For a given finitely additive probability distribution P on (Ω, A), a neo-additive
capacity ν is defined as:
ν(A) = (1 − ε − γ)P (A) + εµ0 (A) + γµ1 (A) with ε ≥ 0, γ ≥ 0 and ε + γ ≤ 1.
A neo-additive capacity is then a convex combination of a probability measure
and two capacities, reflecting complete ignorance.
ε + γ measures to which extent beliefs are far from some estimated probabilities.
In our context, the individual loss is a random variable X, taking its value in{x0 , x1 , ..., xl }, with 0 = x0 < x1 < ... < xl = b, where b is the maximal possible
loss. The government gives the distribution P of X, i.e. the value of P (X = xi ) for
every i. The catastrophic event is E = {X > 0}, its probability estimated by thegovernment is p = P (E), and we set V = E(X/X > 0)
The neo-capacity ν here associated with the official distribution P of X gives:
ν(E) = (1 − ε − γ)p + γ and ν(E) = (1 − ε − γ)(1 − p) + γ.
The preferences representation à la Choquet of an individual with initial wealthy and with beliefs represented by the neo-capacity ν can be written as:Wν (y) = (1 − ε − γ)EP (u(y) − X (ω)) + ε inf (u(y) − X (ω)) + γ sup (u(y) − X (ω))
ω∈Ω

ω∈Ω

= (1 − ε − γ) (u(y) − pE (X/X > 0)) + ε (u(y) − b) + γ(u(y) − 0)
= u(y) − εb − p(1 − ε − γ)V
1 − ε − γ measures the trust (or confidence) level in the government probability
estimation.
Note that, for ε = γ = 0, the beliefs coincide with the official estimation and thestandard expected utility evaluation is obtained. When distrust appears, (ε, γ > 0),ε and γ reflect respectively the individual’s pessimism and optimism. Indeed, themaximal possible loss appears explicitly in the agent’s utility evaluation and its
impact increases with an increase in ε., the minimal possible loss appears too withweight γ but is implicit because its value here is 0.
In the following, we set δ = ε + γ the distrust level and thus the previous
preferences representation becomes:
Wε,δ (y) = u(y) − εb − p(1 − δ)V where 0 ≤ ε ≤ δ ≤ 1.
We see that the utility of the agent depends on the pessimism ε and on the distrustlevel δ. The optimism intervenes only through the distrust δ.
4

2.2

The risk reduction level

A possibility of risk reduction is now introduced for the whole population. This risk
reduction can take two forms, the first one corresponding to self-protection, and theother one to self insurance in the sense of Ehlrich, Becker (1972):- a reduction of the probability of the catastrophic event which decreases from pto (1 − λ)p, with 0 ≤ λ ≤ 1;
- a reduction of the average loss in the case of occurrence of E, which decreases
from V to (1 − λ)V , with 0 ≤ λ ≤ 1.
The difference between the first two types of measure well appears in the following
example concerning the threat of an influenza pandemic. The WHO elaborates
prevention plans with two objectives. The first objective is to avert a pandemic andto control the outbreak in humans. Measures then consist in immediate culling of
infected and exposed birds, quarantine and disinfection of farms, control of animal
movements etc. The second objective is to mitigate the impact of the pandemic (onceit has been declared). The corresponding measures include vaccine development andantiviral drug production and storage.
Note that the two types of risk reduction are equivalent in our model because
the utility function is additively separable.
We assume that the reduction of pV to (1−λ)pV with 0 ≤ λ ≤ 1 costs T (λ).This
cost is financed by a uniform tax rate t(λ, Y ) = T Y(λ) where Y is the total wealth in
the economy. λ 7→ T (λ) is assumed to be an increasing and convex function with
T (0) = 0. Thus if λ = 0, the risk remains at its initial level, and if λ = 1 therisk is completely eliminated according to the government. However, if the agent ispessimistic (i.e. ε > 0), he will believe that the maximal loss is still possible even if
λ = 1 (see the term −εb in Equation (1)).
For an agent with distrust level δ, degree of pessimism ε, and income y, theevaluation of a risk reduction level λ gives:
Wε,δ (y, λ) = u (y(1 − t(λ, Y ))) − εb − (1 − λ)p(1 − δ)V(1)

The optimal risk reduction level λ∗ for this agent is then the solution of thefollowing maximization problem:
max u (y(1 − t(λ, Y ))) − εb − (1 − λ)p(1 − δ)V
λ(2)

The first order condition for an internal solution is:yT 0 (λ) 0u (y(1 − t(λ, Y ))) = p (1 − δ) V
(3)
YThe second order condition is satisfied for all λ ∈ [0, 1] due to the concavity of u
and to the convexity of T.
The optimal risk level equalizes the marginal benefit of risk reduction p (1 − δ) V
0with its marginal cost in terms of utility yTY(λ) u0 (y(1 − t(λ, Y ))) .
Note that the optimal risk reduction level λ∗ = λ∗ (δ, y) does not depend specifically on ε, but only on the global distrust level δ.
5In the special case of logarithmic function: u(x) = ln x, the first order condition
becomes:T 0 (λ)= p (1 − δ) V
(4)(Y − T (λ))
Let us now consider the conditions for corner solutions.
Maintaining risk at its actual level is optimal if and only if the solution of the
optimization problem (2) is λ∗ = 0 which is equivalent to the following condition:yT 0 (0) 0
u (y) ≥ p (1 − δ) V
Y
Thus, preserving the status quo is preferred if the marginal cost of risk reduction
is higher than its marginal benefit ”from the first euro”. This choice will be lesslikely to occur for severe risks, that is when pV is high.
On the other hand, the maximal risk reduction is adopted if and only if thesolution of the maximization program (2) is λ∗ = 1 which is equivalent to thefollowing condition:yT 0 (1) 0u (y(1 − t(1, Y ))) ≤ p (1 − δ) V
Y
The maximal risk reduction level is preferred if the marginal benefit of risk reduction
is higher than its marginal cost until the last risk fraction. This situation is possible
only if T (1) and T 0 (1) are finite, which is not always the case, namely for natural
disasters.

3

The political decision of risk reduction

We consider now a population composed of n individuals differing by their trust in
the estimation of the risk announced by the government, by their pessimism, theirwealth and their political weight. We aim to determine the level of risk reduction
which will be implemented by the government.

3.1

The model of political decision

To simplify, we assume that there are k groups of homogenous individuals of size
ni , with i = 1, .., k; their parameters of distrust and wealth are respectively δiand yi with i = 1, .., k. Pessimism and optimism levels are εi and γi respectively,
with εi + γi = δi . However, within a group, the agents can have preferences on othermatters than the catastrophic risk. We model the political decision with probabilistic
voting (see Persson, Tabellini (2000), Coughlin, Mueller, Murell (1990), Lindbeck,
Weibull (1987, 1993)). Two parties A and B compete in elections. Each party
announces a policy, and it is assumed that the policy announced by the winning
party will be implemented.
More precisely, the agent j of group i votes for party A iff
Wεi ,δi (yi, λA ) + bi,j > Wεi ,δi (yi, λB )6where λA and λB are the policies announced respectively by parties A and B. The
welfare of any agent of group i is Wεi ,δi (yi, λ) if the policy λ is applied. The randomvariable bi,j measures the bias (positive or negative) of elector j in favor of party A,
independently of policy λ. Within group i, the random variables bi,j have the samelaw as a random variable bi , which is assumed to have a continuous law.
The mathematical expectation of the number of votes for A isXX
EW A (λA , λB ) =P (Wεi ,δi (yi, λA ) + bi,j > Wεi ,δi (yi, λB ))ij

As all the bi,j have the same law as bi , we haveX
EW A (λA , λB ) =ni P (Wεi ,δi (yi, λA ) + bi > Wεi ,δi (yi, λB ))iand the mathematical expectation of the number of votes for B is
X
EW B (λA , λB ) =ni P (Wεi ,δi (yi, λA ) + bi < Wεi ,δi (yi, λB ))i

Let Fi denote the cumulative distribution function of bi and fi be its density. We
obtain:X
EW B (λA , λB ) =ni Fi (Wεi ,δi (yi, λB ) − Wεi ,δi (yi, λA ))iandEW A (λA , λB ) = n − EW B (λA , λB )P
where n = ki=1 ni is the total number of agents.
Party B chooses λB to maximize EW B (λA , λB ) (for λA given). Idem for Party A.
Thus the first order conditions are:
X ∂Wε ,δ (yi, λB )
i i
0 =
ni
fi (Wεi ,δi (yi, λB ) − Wεi ,δi (yi, λA ))∂λBi
X ∂Wε ,δ (yi, λA )
i i
0 =
ni
fi (Wεi ,δi (yi, λB ) − Wεi ,δi (yi, λA ))∂λAi
The two parties face the same problem. Thus at the Nash equilibrium, with simultaneous announcement of the policies, we have λA = λB , i.e.:X ∂Wε ,δ (yi, λB )
i i
0=
ni
fi (0)∂λB
iP
which is the FOC corresponding to the maximization ofi ni αi Wεi ,δi (yi, λB ) for
αi = fi (0).
We see that the political equilibrium implements the maximum of a sort ofsocial welfare function, where each elector of group i is considered to have a weight
αi = fi (0).
fi (0) is the density function of bi at the equilibrium. A high fi (0) means thatthe electors of group i will change their vote more easily if the policy proposed ismodified. The political equilibrium then gives a greater weight to the individuals
who are more prompt to change their vote.
73.2

Adopted risk reduction level

With the previous political decision modelization, the adopted risk reduction levelλ∗∗ will be the solution of the following optimization program:max D(λ) =λkXαi ni [u (yi (1 − t(λ, Y ))) − εi b − (1 − λ)p(1 − δi )V ]i=1

where t(λ, Y ) is the uniform tax rate which finances a risk reduction correspondingP
to λ, given by t(λ, Y ) = T Y(λ) with Y = ki=1 ni yi .
The agents choose to vote for A or B. At the equilibrium, parties A and B choose
the same λ to maximize the electoral support function D(λ), taking the parameters
αi , δi , εi , ni , yi , Y , V , p, b as fixed.
The first order condition for an internal solution is:
D0 (λ) = 0

(5)

where
µ µ
¶¶
¸
yi 0
T (λ)
D (λ) =
αi ni −T (λ) u yi 1 −+ p(1 − δi )V
(6)
Y
Y
i=1
" k
µ µ
¶¶#
k
X
T 0 (λ) XT(λ)= −αi ni yi u0 yi 1 −+ p(1 − δ)V
αi ni
Y
Y
i=1
i=1
k
X

0

Pk

·

0

αnδ

i i i
with δ = Pi=1
.
k
αn
i=1

i i

1
Morover, setting u0 (λ) = P
ktion becomes:

i=1

k
P

αi ni i=1¢¡
αi ni yYi u0 yi (1 − Y1 T (λ)) , the first order condi-

T 0 (λ)u0 (λ) = pV (1 − δ)(7)

The second order condition is satisfied for any λ ∈ [0, 1] because of the concavityof u and the convexity of T. As in (3), we obtain in (7) the equality of marginalutility and marginal cost of risk reduction. Note that, as in section 2.2, the adopted
risk reduction level depends here only on δ, and not specifically on the pessimism ε.
It appears that preferences and trust levels of the different groups intervene in
the government decision criterion via an ”average” distrust level δ and an ”average”
marginal cost (in terms of utility) u0 (λ) which both depend not only on the size ofeach group, but also on their respective political weights.
In the particular case of a logarithmic utility function, u(x) = ln x, the first ordercondition becomes:¡¢
T 0 (λ)=p 1−δ V(8)(Y − T (λ))
This case has two specific features:8• only the total wealth in the economy influences the adopted level of risk reduction, the distribution of wealth between the groups of individuals plays norole.
• the political weight of a group has an influence on the adopted risk reductionlevel only via the average trust level in the population.

3.3

Some comparative static results

3.3.1

Impact of risk and trust level

In this section, we study the impact of an increase in the estimated probability ofrisk realization and the impact of an increase in the average distrust level on thepolitically decided investment in risk reduction.
Proposition 1

dλ∗∗
dλ∗∗
> 0 and
< 0.
dp
dδ

Proof
dλ∗∗
∂Dλ0
dλ∗∗
From the first order condition (5), we have
= − D100
and
=
dp
λλ ∂p
dδ
∂Dλ0
00
− D100where Dλλ < 0 from the second order condition of the optimization
λλ ∂δ
program.
P
P
∂Dλ0
∂Dλ0
Moreover,= V ki=1 αi ni (1 − δi ) > 0 and= −pV ki=1 αi ni < 0.¥∂p∂δ• The first result implies that an increase in announced loss probability, ceterisparibus, leads to a higher investment in risk reduction. That corresponds tothe standard results on optimal prevention, obtained by Elrich, Becker (1972).
• From the second result, the optimal level of risk reduction decreases when average trust in government announcements deteriorates, i. e. when δ increases.
This result is specific to the adopted decision model and well emphasizes theparticular role played by the trust level in the individual belief formation andin individual preferences. Indeed, the impacts of p and δ on λ∗∗ are opposite: if the government announces a higher probability of catastrophe, thenthe investment in risk reduction increases, whereas a decrease in the trustlevel decreases the percieved efficiency of risk reduction (only the expected
loss being reduced and not the maximal one), and thus decreases investment
in risk reduction. Note that increases of optimism and of pessimism have thesame impact on λ∗∗ because they both induce a lower trust in risk reduction.
One can note in addition that an increase in δ can come here not only froma decrease in trust due to some government action, but also from an increasein the political weight of the group with the lower trust level (corresponding
here to the higher δi ).

9

3.3.2

Impact of wealth and political weight

Wealth can influence λ∗∗ by two channels: via the global wealth in the economy andvia the distribution of this wealth. We denote by βi = yYi the proportion of the total
wealth belonging to an individual of group i , for i = 1,..., k and determine in thefollowing proposition the impact of two types of wealth modifications: a proportional
increase of all incomes (leaving βi constant for any i), and a simple redistribution
between two groups i and j (leaving Y constant).
¯∂λ∗∗ ¯¯> 0.
Proposition 2 (i)∂Y ¯dβ=0x1−R(ii) for a CRRA utility function (u(x) = ln x for R = 1, and u(x) =for
1−R
R 6= 1),¯∂λ∗∗ ¯¯= 0;- if R = 1 then∂yi ¯¯dY =0µ¶∂λ∗∗ ¯¯αiαj- if R 6= 1, then−> 0.
> 0 ⇔ (R − 1)∂yi ¯dY =0, dyl =0, l6=i, l6=jyiR yjRλ∗∗ increases with the total wealth in the economy, the impact of a modificationin the wealth distribution depends on the relative risk aversion.
Proof

¯
¯
¯
¯
0 ¯
∂λ∗∗ ¯¯∂λ∗∗ ¯¯∂Dλ0 ¯¯
1 ∂Dλ ¯1
From (5) and (6),= − D00
and= − Y D00¯¯∂Y ¯dβ=0∂yi ¯dβ=0λλ ∂Yλλ ∂βi
dβ=0
dY =0
00
where Dλλ < 0 from the second order condition of the optimization program.
kP
αi ni [−T 0 (λ)βi u0 (βi (Y − T (λ))) + p(1 − δi )V ],(i) From (6) and βi = yYi , we have Dλ0 =i=1

so that:0 ¯¯∗∗ ¯P∂Dλ ¯¯∂λk¯= −T 0 (λ) i=1 αi ni βi2 u” (βi [Y − T (λ)]) > 0 and thus>0
∂Y ¯dβ=0∂Y ¯dβ=0¯∂λ∗∗ ¯¯(ii) if R = 1,
= 0 because, from (8), λ∗∗ depends only on Y .
¯
∂y
i
dY =0
¯
µ
¶
αi ni αj nj ni∂Dλ0 ¯¯(1−R)
0
if R 6= 1= −T (λ) (Y −T (λ))R− R
which has the sign of∂βi ¯dY =0
βiR
βj nj
¶
µ
αi
αj
.¥
−
(R − 1)yiR yjR
Consequently, it appears from (i) that risk reduction is as a normal good: investment in it increases with global wealth.
The impact of a redistribution depends on the relative risk aversion R. Whenthe utility function is logarithmic, the risk reduction level is neutral concerning
any redistribution of wealth between individuals in the population: only the total
wealth matters. Thus if a fiscal reform modifies the distribution of wealth, with
global wealth fixed, it will not change the risk reduction level.
On the other hand, when the utility function is CRRA with R 6= 1, the previous
neutrality property no longer holds: a change in wealth distribution influences λ∗∗10even if the total wealth remains constant. This is due to the fact that when R 6= 1
the marginal cost of risk reduction depends not only on the total wealth in theeconomy, but also on wealth distribution. Then, the variation of λ∗∗ in the case of
an increase in yi , Y being constant, will result from two effects: group i becomesricher and thus, for R > 1, prefers more risk reduction (the marginal cost for risk
reduction becomes lower for its members), whereas group j becomes less rich andthus prefers less risk reduction (the marginal cost for risk reduction becomes higher
for its members)1 . Note that, for R = 1, the two effects compensate perfectly.
The following proposition proves an intuitive result: an increase in the politicalweight of the individuals of a given group increases the politically decided level ofrisk reduction if and only if the individuals in this group are in favour of a high risk
reduction level.
Proposition 3dλ∗∗> 0 ⇔ λ∗ (δi , yi ) > λ∗∗ .
dαi

Proof
dλ∗∗
∂D0
∂D0
From (5) and (6),= − D100 ∂αλi which has the sign of ∂αλi .
λλ
dαi
³
´
0
∂Dλ
−yi T 0 (λ∗∗ ) 0
T (λ∗∗ )= ni
u (yi [1 − Y ]) + (1 − δi )pV = ni G0i (λ∗∗ )∂αiYwhere Gi (λ) is defined as Gi (λ) = Wεi ,δi (yi , λ) (see Equation (1)).
G00i (λ) < 0, i.e. G0i is decreasing, and G0i (λ∗ (δi , yi )) = 0 thus G0i (λ∗∗ ) > 0 ⇔∗λ (δi , yi ) > λ∗∗ .
¥3.3.3

Comparison of λ∗∗ with the individually and socially optimal risk
reduction levels

In the following we compare the politically decided risk reduction level λ∗∗ with theindividually prefered risk reduction levels, and particularly
the level preferred by anPi αi ni yiP”average” individual, i.e. of average wealth ȳ =and average trust level δ.
αi n ii

Proposition 4 (i) λ∗∗ ∈ [mini λ∗ (δi , yi ), maxi λ∗ (δi , yi )] ;(ii) for a CRRA utility function u:• if R = 1, then λ∗∗ = λ∗ (δ, ȳ) and λ∗∗ < λ∗ (δi , yi ) ⇔ δi < δi)i• if R 6= 1, then λ∗∗ < λ∗ (δi , yi ) ⇔ ū(yu(y
< 1−δ
1−δ
1 ,...,yk )where ū(y1 , ..., yk ) =Pk

j=1 αj nj u(yj )Pkj=1 αj nj.

Moreover:- if R > 1 then λ∗ (δ, ȳ) > λ∗∗- if 0 < R < 1 then λ∗ (δ, ȳ) < λ∗∗(iii) for any utility function u, if yi = y ∀i then λ∗∗ = λ∗ (δ, y).
Proof
See Appendix
1

Note that the opposite holds for R < 1.

11

Concerning the comparison of λ∗∗ with the individually optimal risk reductionlevels, it appears that λ∗∗ lies in the interval between the minimal and the maximal
individually prefered risk reduction level, which means that it corresponds to acompromise between the individually prefered risk reduction levels. We will provein section 4 that this is not always the case when the risk exposure is differentiated.
For a given individual i, the gap between λ∗∗ and λ∗ (δi , yi ) depends both onindividual wealth and trust level. For instance, for 0 < R < 1, an individual will bein favour of more risk reduction than the politically decided if he is more confident
and less wealthy than the average individual in the population.
Concerning the comparison of λ∗∗ with λ∗ (δ, ȳ), when wealth is equally distributed between individuals or when the utility function is logarithmic (R = 1),
the risk level adopted by the government does not differ from the one preferred byan individual with an average level of trust. It is however important to note that
this average trust level depends not only on the respective sizes of the populationgroups, given by ni , but also on their respective political weights, given by αi . Thus,
the trust level of group i will influence the decision more if this group is big and if
its political weight is important.
For more general preferences, the comparison of λ∗∗ and λ∗ (δ, y) depends on the
degree of concavity of the utility function (measured by R). If R is higher than
1, the marginal utility of wealth is strongly decreasing, and the willingness to pay
for risk reduction is proportionally much lower for poor people than for rich ones.
That is why the political decision corresponds to less risk reduction than in the casewhere an ”average” individual is considered.
In the following, we compare the risk reduction level λ∗∗ resulting from a politicalprocess with the utilitarian risk reduction level λopt which maximizes the social
welfare function D(λ) corresponding to αi = αj ∀i, j, i. e. when the influence ofeach group corresponds to its demographic weight. It means that we compare apositive result λ∗∗ with a normative one λopt .
Remark 1 Let λopt be the utilitarian optimal risk reduction level (obtained withidentical political weight α for every agent). In general, λopt 6= λ∗∗ . However, λopt ∈[mini λ∗ (δi , yi ), maxi λ∗ (δi , yi )] .
More precisely, if αi > α for i such that λ∗ (δi , yi ) > λopt and if αi < α for i suchthat λ∗ (δi , yi ) < λopt then λ∗∗ > λopt .
It appears that in general, the positive and the normative risk reduction levels
are different. In particular, the positive level will be higher than the normative one
if the individuals preferring higher risk reduction are the more politically influential
ones.

4

Political decision with differentiated risk exposure

In this section, we consider risks deriving from a new product or a new technology,
for which different risk groups may be identified: we assume more precisely that12

some individuals, by their location, or by their specific characteristics, are more
exposed than others to risk. Another specificity of these risks is the possibility, forthe public authorities, to eliminate them completely by forbidding the trade of the
products or the use of the technology. In this case, since it is easier to verify thata product is forbidden than to evaluate the risk it generates when it is allowed,
the government will be credible if it announces a total ban. Note however thatforbidding a product or a technology has a cost (of opportunity for example), sincenobody will be able to use the product. This radical solution to the risk exposure
problem can be opposed to two other solutions: risk reduction of level λ (λ ∈]0; 1[),or to totaly acceptance of the risk.
Three scenarios are then possible: authorization, reduction of risk, and prohibition.
We assume here that the agents differ by their risk pi and their degrees of pessimism and optimism εi and γi = δi − εi , where δi is the degree of distrust. Itmeans that, given the individual characteristics (age, profession, localization etc.),
each agent i has an individual probability pi of realization of the risk. The associated utility loss is Vi , and pi is the probability announced by the government for an
agent of type i̇. The trust of agent i in the evaluation of the government is 1 − δi .
In the three cases, we can evaluate the utility of an agent, which is a function ofher income y, of her risk pi , of her degree of pessimism εi and of her degree of trust
1 − δi .
Authorizationi
Wauthor(y) = u(y) − εi b − Vi pi (1 − δi )Reduction of risk by a factor λiWred(y) = u (y(1 − t)) − εi b − (1 − λ)Vi pi (1 − δi )Prohibitioni
Wprohib(y) = u (y(1 − θ))

where θy represents the cost of prohibition.
To simplify, we assume that if the reduction of risk is chosen, this will be ofa factor λ ∈]0; 1[, λ given. To avoid the obvious case of prohibition preferred toreduction for every εi , δi , pi , we assume that θ > t > 0.
In the following, we set Qi = pi Vi (1 − δi ).

4.1

Individual preference

We can examine now which scenario is preferred by an agent of type i: authorization,
reduction of risk or prohibition.
We set: ©
¡ i
¢ª
i
i
i
E1 = (εi , Qi ); Wauthor(y) = max Wauthor(y); Wred(y); Wprohib
(y)
©
¡ i
¢ª
i
i
i
E2 = (εi , Qi ); Wred(y) = max Wauthor(y); Wred(y); Wprohib
(y)
©
¡ i
¢ª
i
i
i
E3 = (εi , Qi ); Wprohib(y) = max Wauthor(y); Wred(y); Wprohib(y)13An agent with individual characteristics (εi , Qi ) prefers the authorization iff(εi , Qi ) ∈ E1 . She prefers reduction of risk by a factor λ if (εi , Qi ) ∈ E2 , andshe prefers the prohibition if (εi , Qi ) ∈ E3 .
Let us denote by z ∗ (εi , Qi ) the individual preference of an individual of characteristics (εi , Qi ), where we set z ∗ (εi , Qi ) = j if (εi , Qi ) ∈ Ej , with j ∈ {1, 2, 3}.
Thus, z ∗ (εi , Qi ) = 1 means that individual i prefers authorization, z ∗ (εi , Qi ) =2 means that individual i prefers risk reduction and z ∗ (εi , Qi ) = 3 means that
individual i prefers prohibition.
The following proposition gives some results about the impact of εi , γi and pi
on the individual preferences z ∗ (εi , Qi ); recall that Qi = pi Vi (1 − εi − γi )Proposition 5 z ∗ (εi , Qi ) is an increasing function with respect to pi and Vi ; a decreasing function of γi , and an increasing function of εi ( Qi being fixed).
Proof
See Appendix
Proposition 5 means that an increase in the estimated risk pi or in the pessimism
level εi leads to a more cautious preferred decision concerning risk. Note that anincrease in εi may modify the decision from authorization to prohibition or fromreduction to prohibition, but never from authorisation to reduction.
Let us compare the results about the influence of the pessimism ε given in Proposition 1 and in Proposition 5. In Proposition 1, an increase of pessimism induces
less risk reduction (since δ = γ + ε). In Proposition 5, an increase of pessimism canlead to prohibition, since here prohibition is a trustable way to eliminate risk.
We can distinguish two cases for the shape of the sets Ej according to the valuesof the costs (θ and t) and benefits (λ) of risk reduction levels.
• Case 1: if u(y)−u(y(1−θ))≤ λ1 , the reduction of risk is never preferred, sinceu(y)−u(y(1−t))prohibition is hardly more expensive than reduction of risk (see Figure 1).
• Case 2: if λ1 ≤ u(y)−u(y(1−θ)), the reduction of risk can be preferred sinceu(y)−u(y(1−t))prohibition is quite more expensive than reduction of risk.(see Figure 2).

4.2

The political choice

In this section, we consider the political choice concerning risk when the three previous possibilities can be chosen by the government. The analysis draws several
scenarios which depend essentially on the relative cost of prohibition with respect
to the reduction of risk, and on the impact of the reduction of risk on the maximum
loss. The main results are summed up at the end of the section.
We keep the model of the political choice by probabilistic voting. The populationis constituted of k groups of homogenous individuals. Group i is composed of agents
14

Figure 1: case 1

Figure 2: case 2

15

having the same characteristics (εi , Qi ). The chosen policy maximizes the politicaldecision function:kXD=ni αi W (εi , Qi )i=1

where αi is the political weight of group i (as in section 3). Three policies are
possible: authorization, reduction of risk by a given factor λ, and prohibition.
We denote by z ∗∗ the political decision. We have z ∗∗ = 1 if authorization isdecided, z ∗∗ = 2 if reduction is decided, and z ∗∗ = 3 if it is prohibition.
Proposition 6 (i) z ∗∗ = z ∗ (ε, Q), i.e. the political decision is that of an average
individual(ii) z ∗ (εi , Qi ) = z ∗ (εj , Qj ), for any i, j = 1, ..., k ⇒ z ∗∗ = z ∗ (εi , Qi ) because Ej
is a convex set for any j = 1, 2, 3. If the individuals have the same preference, thepolitical decision conforms to their wish.
(iii) In case 1, z ∗∗ ∈ conv{z ∗ (εi , Qi ), i = 1, ..., k};(iv) In case 2, z ∗∗ ∈/ conv{z ∗ (εi , Qi ), i = 1, ..., k} is possible because E1 ∪ E2 andE2 ∪ E3 are non-convex sets.
Proof : See Appendix¤∗∗∗
The political decision z = z (ε, Q) is that of an average individual even if
nobody has his preferences. However, in case 1, the preference of the average
agent is a compromise between the preferences of the different agents, i. e. z ∗∗ ∈conv{z ∗ (εi , Qi ), i = 1, ..., k}. For instance, if nobody prefers prohibition, it will not
be adopted.
At the opposite, in case 2, the preference of an average agent can be extreme, i.e.
∗∗z ∈/ conv{z ∗ (εi , Qi ), i = 1, ..., k} is possible. For instance, even if nobody prefers
prohibition, it can be adopted.
Thus, an extreme interpretation of the precautionary principle rejecting any risk
may come from a political process, even when every elector accepts a partial or total
risk. It can be due to a diversity of risk exposures, combined with a difference oftrust levels.
The following remark studies the impact of political weights on the politicaldecision.
Remark 2 When αi increases, with αj fixed for all j 6= i, then (ε, Q) describes a
segment of line and the political decision z ∗∗ = z ∗ (ε, Q) moves consequently. In case1, z ∗∗ variates monotonously. This is no longer true in case 2. For example, we canhave z ∗∗ = 2 for α1 low, z ∗∗ = 3 for α1 medium and z ∗∗ = 1 for α1 high.
Finally, in a last remark, we compare the utilitarian socially optimal decisionz opt with the political decision z ∗∗ . We note that z opt may be an extreme decision,
which was not possible in the framework of section 3 (see Remark 1).
Remark 3 Let z opt be the utilitarian socially optimal decision, i. e. obtained maximizing the social welfare function D(λ) corresponding to α1 = α2 = ... = αk . In general, z ∗∗ 6= z opt . Moreover, in case 2 we can have z opt ∈/ conv{z ∗ (εi , Qi ), i = 1, ..., k}and then the socially optimal decision can be extreme.
16

5Conclusion

This paper is an attempt to introduce non probabilized uncertainty in a political
economy model. More precisely, an estimated probability of a catastrophic event isgiven to the agents who may more or less trust it.
Increases in estimated probability and in distrust do not have the same impact
on risk reduction decisions: if an increase in the estimated probability always leads
to more risk reduction, an increase in distrust leads to less risk reduction.
We show moreover that if all individuals are exposed to the same risk, the political decision and the socially optimal one are different in general but both lie between
the individually preferred risk reduction levels.
This last result is not true for differentiated risk exposure. Political decisions interms of risk reduction may then be extreme and far from optimal if individuals do
not trust in public information. Prohibition can then be politically decided even if
nobody prefers it, which corresponds to a radical interpretation of the precautionary
principle.
To conclude, we think that our model can be considered as a first step in the
introduction of behavioral economics insights in a political economics framework. It
can be developed in several ways: the assumption of purely opportunistic parties
can be relaxed and several risk reduction technologies can be introduced.

17

References[1] CHATEAUNEUF A., EICHBERGER J., GRANT S. (2002) ”Choice under
Uncertainty with the Best and the Worst in Mind: Neo-additive Capacities”,
Discussion Paper n ◦ 393,University of Heidelberg.
[2] COHEN M. (1992) ”Security Level, Potential Level, Expected Utility: A ThreeCriteria Decision Model under Risk”, Theory and Decision 33, 101-134.
[3] COUGHLIN P., MUELLER D.C., MURELL P. (1990). ”Electoral politics, interest groups and the size of government.” Economic Inquiry 28, 682-705.
[4] EICHBERGER J., KELSEY D. (1999) ”E-Capacities and the Ellsberg Paradox”, Theory and Decision 46, 107-140.
[5] ELRICH I., BECKER G. (1972) ”Market insurance, self-insurance and selfprotection”, Journal of Political Economy, 40, 623-648.
[6] GODARD O. (2003) ”Le principe de précaution comme norme de l’action
publique, ou la proportionnalité en question”, Revue économique 54, 1245-1276.
[7] JAFFRAY J. Y. (1988) ”Choice under Risk and the Security Factor: An Axiomatic Model”, Theory and Decision 24.
[8] LINDBECK, A., WEIBULL, J. (1987) ”Balanced-budget redistribution as theoutcome of political competition.”, Public Choice 52, 273-297.
[9] LINDBECK, A., WEIBULL, J. (1993) ”A model of political equilibrium in a
representative democracy” Journal of Public Economics 51, 195-209.
[10] PERSSON, T., TABELLINI, G. (2000) Political Economics. Explaining Economic Policy, MIT Press.

18

Appendix
A

Proof of Proposition 4(i) λ∗∗ is solution of equation (5). Let ej = (0; ..; 0; 1; 0; ..; 0), with 1 uniquely at thejth place. Then λ∗ (δj , yj ) is solution of equation (5) when (α1 , .., αk ) = ej . We set:
h

i
yi 0
H(α1 , .., αk , λ) =αi ni −T (λ) u (yi (1 − t(λ, Y ))) + p(1 − δi )V
Y
i=1
k
X

0

We have H(α1 , .., αk , λ∗∗ ) = 0, and H(ej , λ∗ (δj , yj )) = 0 for any j.
Let j1 , j2 defined by H(ej1 , λ∗∗ ) = minj H(ej , λ∗∗ ) and H(ej2 , λ∗∗ ) = maxj H(ej , λ∗∗ ).
Pk
We have H(α1 , .., αk , λ) =Since H(α1 , .., αk , λ∗∗ ) = 0, thus
j=1 αj H(ej , λ).
H(ej1 , λ∗∗ ) < 0 < H(ej2 , λ∗∗ )We can note that H(ej1 , λ) is decreasing in λ (since the second order condition of program (5) is satisfied) and H(ej1 , λ∗∗ ) < 0 = H(ej1 , λ∗ (δj1 , yj1 )). Thus,
λ∗∗ > λ∗ (δj1 , yj1 ). The same reasoning allows to prove that λ∗∗ < λ∗ (δj2 , yj2 ).
∂D 0(ii) For CRRA utility functions, for any R > 0, λ∗∗ < λ∗ (δi , yi ) ⇔ ∂αλi > 0 according
³
´
0 ∗∗
∗∗
∂D 0
to Proposition 3 and its proof, where ∂αλi = ni −yi TY(λ ) u0 (yi [1 − T (λY ) ]) + (1 − δi )pV
£
¤
0 ∗∗
∂D0
- For R = 1, ∂αλi = ni (− Y T−T(λ(λ∗∗) ) +(1−δi )pV ) = ni −(1 − δ)pV + (1 − δi )pV from∂D 0(8) and thus ∂αλi = ni pV (δ − δi ) i.e. λ∗∗ < λ∗ (δi , yi ) ⇔ δ > δi . Similarly, we haveλ∗∗ = λ∗ (δi , yi ) ⇔ δ = ·δi and thus λ∗∗ = λ∗ (δ, ȳ).
¸
³
´1−R
0
∂Dλ
T (λ∗∗ )T 0 (λ∗∗ )1−R- For R 6= 1, ∂αi = ni − Y
1− Yyi+ (1 − δi )pV
¸
·
Pk
αj nj
= ni −yi1−R pV (1 − δ) Pk j=1
from (5).
1−R + (1 − δi )pV
j=1 αj nj yj
i
h
∂D0
i)
Then, ∂αλi = ni pV − ū(yu(y
(1
−
δ)
+
(1
−
δ
)
and thus λ∗∗ < λ∗ (δi , yi ) ⇔i
1 ,...,yk )u(yi )i< 1−δ.
ū(y1 ,...,yk )1−δConsequently, for R 6= 1, λ∗∗ < λ∗ (δ, ȳ) ⇔ ū(yu(ȳ)< 1.
1 ,...,yk )Notethat, for R > 1 this means that u(ȳ) > ū(y1 , ..., yk ) i. e. that u(ȳ) >Pkj=1 αj nj u(yj )Pkj=1 αj nj. This last inequality is true for any u concave. Then λ∗∗ < λ∗ (δ, ȳ)

for R > 1.
Pk
j=1 αj nj u(yj )PkMoreover, for R < 1, ū(yu(ȳ)
<
1
⇔
u(ȳ)
<
which is never true for u
1 ,...,yk )
α n
j=1

j

j

concave. Then λ∗∗ > λ∗ (δ, ȳ) for R < 1.
(iii) Assume yi = y, ∀i. In this case, the first order condition (7) becomes
µ ·
¸¶
T (λ)
y 0
0
T (λ)u y 1 −= pV (1 − δ)YY
and we obtain the first order condition (3) of the program giving the risk level
preferred by an individual of trust level δ, which implies λ∗∗ = λ∗ (δ, y). ¥19

B

Proof of Proposition 5ii
The reduction is preferred to authorization iff Wauthor(y) ≤ Wred(y), which is equivalent to u(y) − εi b − Qi ≤ u (y(1 − t)) − εi b − (1 − λ)Qi , i.e.
iiWauthor(y) ≤ Wred(y) ⇔ λQi ≥ u(y) − u (y(1 − t))

(9)

i
i
Prohibition is preferred to authorization iff Wauthor(y) ≤ Wprohib(y), which isequivalent to u(y) − εi b − Qi ≤ u (y(1 − θ)), i.e.
ii(y) ⇔ Qi ≥ u(y) − u (y(1 − θ)) − εi b(y) ≤ Wprohib
Wauthor(10)ii(y), which is equivalent(y) ≤ Wprohib
Prohibition is preferred to reduction iff Wredto u (y(1 − t)) − εi b − (1 − λ)Qi ≤ u (y(1 − θ)), i.e.
ii(y) ≤ Wprohib(y) ⇔ (1 − λ)Qi ≥ u (y(1 − t)) − u (y(1 − θ)) − εi b
Wred

(11)

We can represent these 3 inequalities graphically in the plane of coordinates εi , Qi .
Note that:ii
Wauthor(y) ≤ Wred(y) ⇔ Qi ≥ C3

¶
εi
≤
1−
B2
¶
µ
εi
i
i
Wred (y) ≤ Wprohib (y) ⇔ Qi ≥ C1 1 −B1i
Wauthor(y)µi
Wprohib(y) ⇔ Qi ≥ C2

(12)

where we haveu(y) − u (y(1 − t))λu(y) − u (y(1 − θ))=and C2 = u(y) − u (y(1 − θ))bu (y(1 − t)) − u (y(1 − θ))u (y(1 − t)) − u (y(1 − θ))=and C1 =
b
1−λ

C3 =
B2
B1

Those 5 parameters are all positive.
We want now, for each couple (εi , Qi ), to determine the preferred decision ofan agent with type (εi , Qi ). We have Qi = pi Vi (1 − εi ), with εi ∈]0; 1[, thus
(εi , Qi ) ∈ ∆ = {(εi , Qi ); Qi ∈]0; +∞[, εi ∈]0; 1[}
The domain ∆ can be split in 3 zones: ∆ = E1 ∪ E2 ∪ E3 . From the inequalities (9), (10), (11), we can easily see that E1 , E2 and E3 are convex sets. Thus (i)is proven. However, the union of two of these zones is not necessarily a convex set.
We note that for εi and Qi near 0, the inequalities (9), (10), and (11) are notsatisfied. (εi , Qi ) is then in the zone E1 . An agent with loss expectation Qi low isthus always in favor of authorization, if she trusts the government (i.e. εi near 0).
Conversely, if Qi and εi are high, then the agent is in favor of prohibition, because
the risk Qi is high, and the trust is low.
20

Between these two extreme cases, authorization, reduction and prohibition are
possible.
- Case 1: if C1 ≤ C3 , then u(y)−u(y(1−θ))≤ λ1 .
u(y)−u(y(1−t))- Case 2: if C3 ≤ C1 then λ1 ≤ u(y)−u(y(1−θ)).
u(y)−u(y(1−t))With these results, the proof of Proposition 5 is now obvious if we refer to (12)and to the figures corresponding to the different cases.

C

Proof of Proposition 6(i) We set
Dz =

k
X

ni αi Wz (εi , Qi )i=1

where z = 1 refers to authorization, z = 2 to reduction and z = 3 to prohibition.
The policy z ∗∗ is adopted iff Dz∗∗ = max (D1 , D2 , D3 )) . We have
D1 =
=

k
X
i=1
k
X

ni αi [u(y) − εi b − Qi ]ni αi u(y) −i=1= Wauthor (ε, Q)kXni αi εi b −

i=1
k
X

k
X

ni αi Qi

i=1

ni αi

i=1
Pk
ni αi εisetting ε = Pi=1k
i=1 ni αi

Pkni αi Qi
and Q = Pi=1k
i=1 ni αi

We can similarly check that
D2 =

k
X

ni αi Wred (ε, Q)

and

i=1

D3 =

k
X

ni αi Wprohib (ε, Q)

i=1This implies that the political decision corresponds to the preference of an ”average”
agent, i.e. of characteristics (ε, Q), where ε and Q are the averages of the εi and theQi , computed with the political weight αi of each agent.
Authorization is then adopted iff (ε, Q) ∈ E1 , reduction of risk if (ε, Q) ∈ E2 ,and prohibition (ε, Q) ∈ E3 . The point (ε, Q) is in the convex hull of the points(εi , Qi ), for i = 1, ..., k.
(ii) Since E1 , E2 , E3 are clearly convex sets, we find that if all the groups havethe same preferred policy (i.e. belong to the same Es ), then the policy chosen willbe this one.
The union of two Es is not necessarily a convex set, thus the political decision
is less obvious if there is no unanimity. We must distinguish several cases.
21

(iii) In case 1, we note that E2 is empty and so conv{z ∗ (εi , Qi ), i = 1, ..., k} isequal to E1 or E3 or E1 ∪ E3 which are all convex sets.
(iv) In case 2, E2 is not empty and E1 ∪ E2 , E2 ∪ E3 and E1 ∪ E3 are clearly
not convex sets and then even if (ε, Q) ∈ conv{(εi , Qi ), i = 1, ..., k}, we may have/ conv{z ∗ (εi , Qi ), i = 1, ..., k}.
z ∗ (ε, Q) ∈22