A Dynamic Approach of the Forest Owner's Self-Insurance Behavior in Risky Forest Management: Saving versus Sylvicultural Practices

A Dynamic Approach of the Forest Owner’sSelf-Insurance Behavior in Risky Forest Management:

Saving versus Sylvicultural Practices

Marielle Brunette 1, Stéphane Couture 2 and Eric Langlais3

the 30th of January 2007

Abstract

The objective of this article is to study the self-insurance behavior of the forestowner in the presence of natural hazards. We develop a theoretical model which con-siders two periods. In this model, the private forest owner can either save a part ofhis income in order to take precautions against natural hazards or adopt sylviculturalpractices. Thus, we want to compare a financial prevention tool with a physical one.At this end, we provide several comparative static analysis especially on initial wealth,on initial forest stock and on timber price. We show that despite the fact that the har-vesting rule is the same whatever the prevention measure adopted by the forest owner,he is confronted to two different problems. If he chooses the saving, he is confronted toa consumption substitution problem between the two periods of the model while, if heopts for the sylvicultural practices, he is faced to a substitution problem between theconsumption in the first period and the amenities in the second one.

JEL Classification: C91, D81, Q23.

Keywords: Experimental Economics, Insurance, Public Policy, Forest, Ambiguity and

Risk.Categories: Renewable Resources: Forestry; Risk and Uncertainty; Experimental Eco-

nomics

1BETA-REGLES, Université Nancy II. [email protected] - ENGREF/INRA. [email protected], Université Nancy II. [email protected]

1

1 Introduction

For several years, the frequency of the extreme climatic events seemed to increase and caused

more and more damage. For example, hurricane Andrew in 1992 costs around 25 billion dol-

lars, earthquake of Kobe in 1995 costs 100 billion dollars and hurricane Katrina in 2005

costs more than 50 billion dollars. The substantial damage developed by these natural dis-

turbances are observable in numerous areas especially in the forest sector. Indeed, in 1999,

storms Anatol, Lothar and Martin hit Denmark with 3,500,000 cubic meters of damage,

France with roughly 140,000,000 cubic meters and Germany with 34,000,000 cubic meters.

In 2003, the forest fire are exceptional in France with 95,000 hectares damaged but particu-

larly in the South of Europe with 150,000 hectars burned in Spain, roughly 500,000 hectares

in Portugal and 120,000 hectares in Italy. More recently, in 2005, Northern Europe was

hit by storm Gudrun which caused 75,000,000 cubic meters of damage in Swedish forest.

Schelhaas, Nabuurs and Schuck (2003) assert that, in European forests, during the period

1950-2000, an annual average of 35 millions of cubic meters of damage of wood are the con-

sequences of natural hazards 1. These damage have principally two origins : windstorm and

fire, liable respectively for 53% and 16% of the total damage. Indeed, the forest sector is very

sensitive to natural hazard because the forest management is a long-term process involving a

long period between the plantation and the harvesting. Moreover, the forest sector provides

services at the forest owner from his harvesting, in financial terms, but also from the ameni-

ties. These amenities refer to recreational aspects, like walk and non-profit services provided

by forest like crop of mushroom but the forest have also an ecological function which is take

into account in the amenities. The forest owner derives more and more utility from these

amenities even if there is no financial incentives to these functions. Thus, the emphasis is

put on this another forest use and on the necessity to consider it in the objective function of

the forest owner 2. Consequently, when a catastrophe occurs, the loss are many and not only

monetary. Therefore, in order to protect himself against these natural hazards, the forest

owner can take out an insurance policy. Nevertheless, we can observe that insurance is a1The damage are very variable depending on the year. For example, in 1999, the damage are very

important due to storms Anatol, Lothar and Martin.2The first paper dealing with the amenities aspects is the Hartman’s article (1976) in a certainty context

and Englin et al. (1990) in a risky framework. They extend the Faustmann’s rule by adding the amenitiesand the risk. They show that the forest owner is face to a trade-off between the gain of not harvest and theloss of not harvest.

2

common practice only in very few regions 3. Generally, the forest owner uses sylvicultural

practices or financial ones rather than insurance to protect his forest assets against natural

disasters. There is no data about the prevention activities of the forest owner but according

to them, to the experts and to the forest owner’s representatives, these prevention measures

are often used. The sylvicultural practices concern the sylvicultural intervention whom aim

is to improve the management of the forest such as reducing density or thinning. The fi-

nancial practices are essentially the saving. Indeed, the forest is a risky productive asset

and the forest owner may hedge his asset against the risk of loss by accumulating a saving.

Nevertheless, the return of the saving is weaker than the forest activities one. These last

practices are usually called self-insurance because they reduce the size of the loss after the

occurence of a catastrophe.

Consequently, the aim of our article is to improve the decision making of the private forest

owner in term of coverage and prevention against natural hazards. The absence of data

about these behavior involve that we adopt a normative approach with modelling. There is

no dynamic theoretical model sufficiently general which allow to analyse the decision making

process of the forest owner related to these coverage decisions.

Our article takes one’s inspiration from the forest management literature. Brunette and

Couture (2006) deal with the prevention against natural hazards in the forest sector. They

developed a static theoretical model favorable to study the natural hazards in the forest area

and study self-insurance activities and market insurance. Nevertheless, this model is static

and it does not consider the saving and the amenities. Amacher, Malik and Haight (2005)

study the forest landowner’s behavior in the presence of fire risk. In a Faustmann-type

framework, they consider that the forest owner can undertake self-insurance actions such

as intermediate fuel treatment and varying initial planting density. They show that based

on simulations, the standard result that fire risk reduces the optimal rotation age does not

hold when landowners use fuel management. Indead, the optimal rotation age rises as fire

risk increases. These authors include the sylvicultural practices but they do not integrate

the amenities and the financial prevention tool. Moreover, they do not consider a dynamic3For example, in Sweden, more than 90% of the private forest area is insured either against fire or against

fire and windstorm and more than 65% of the private forest area in Danemark is insured against windstorm.Inversely, less than 5% of the private forest owner are insured against windstorm in France and approximately2% in Germany.

3

framework. A dynamic approach of the natural hazards management in the forest sector

is introduced by Koskela and Ollikainen (1999). In a Kreps-Porteus-Selden non-expected

utility model, they distinguish uncertainty on initial forest stock from uncertainty on forest

growth and they consider the difference between multiplicative risk and additive risk. They

introduce in their model the amenities. Their aim is to study the effects of biological risk

on harvesting behavior in a two-period model. They show that under biological uncertainty,

a rise in the multiplicative (additive) forest growth risk increases (has no effect on) current

harvesting, while a rise in the forest stock risk always decreases it. However, there is no

dynamic for wealth. Indeed, for consumption, the authors bring together the two periods

in one unique period while they consider utility from amenities in the first period and util-

ity from amenities in the second period. In our article, we generalise to consumption the

adopted approach for amenities. Thus, the article of Koskela and Ollikainen (1999) includes

the amenities and a dynamic apporach but not prevention activities. Moreover, the authors

lay down strong constraints which associate to the Kreps-Porteus-Selden non-expected util-

ity model limit the generalisation of the results.

We propose a dynamic model with two periods which considers the amenities and includes

the self-insurance activities. In this model we compare two types of self-insurance, the saving

and the sylvicultural practices. At this end, we extend the work of Koskela and Ollikainen.

Firstly, we adopt a standard dynamic approach of decision making. Secondly, we integrate

the prevention against natural hazards through the analysis of two self-insurance activities,

the saving and the sylvicultural practices. Finally, we adopt an expected utility approach.

On that account, we propose a self-insurance theoretical model which consider two periods.

In an expected utility framework, we analyse the impact of natural hazards on the private

forest owner’s prevention behavior. We show that the harvesting rule is the same for the two

prevention tools while the forest owner is confronted to different problems. If he opts for

the saving, the owner is confronted to a consumption substitution problem between the two

periods. In return, if he chooses to protect his forest with sylvicultural practices, the owner

is confronted with a substitution problem between the consumption in the first period and

the amenities in the second one. Moreover, we show that the amenities are the main element

of the forest owner’s decision making process when we study the impact of an increase in

4

the initial wealth and the initial forest stock on the owner behavior while, the consumption

is the main one when we analyse the effect of economic variables like price on the forest

owner’s attitude.

The paper is structured as follows. In section 2, we develop two self-insurance models.

The first is an optimal saving decision model and the second one is an optimal sylvicultural

practices decision model both in a risky framework. In section 3, we present the results of the

comparative static analysis by interpreting the difference between the financial prevention

tool and the physical one. Section 4 contains some concluding comments.

2 Optimal prevention activities in a risky context

We study the private forest owner’s prevention behavior, through the saving and the sylvi-

cultural practices, in an expected utility framework. The aim of this two self-insurance

instruments is to reduce the damage born by the owner if the stand is injured by a natural

disturbance. We assume a model with two periods. We think that at first, it is sufficient in

order to better understand the dynamic effects on the forest owner’s decision making and so,

it is not necessary to directly consider a multi-periods modelling. In our model, there is two

categories of decision variables. The first category is connected with the forest management,

it is the harvesting and the second one is connected with the prevention, it is the saving

noted s or the sylvicultural practices noted q. We consider a private forest owner who is

endowed with an exogenous initial wealth y and an even-aged forest with an initial stock

Q. The utilities are separable in the time and so, the forest owner derives utility from the

consumptions (c1, c2) and from the amenities (k1, k2).

The objective of the forest owner is the maximisation of:

V = u(c1) + v(k1) + δE[u(c2) + v(k2)

](1)

We consider that u′(.) > 0, v′(.) > 0, u′′(.) < 0 and v′′(.) < 0. Moreover, δ represents the

discount rate. The existence of the forest stocks k1 and k2 procures the amenities at the

forest owner. This forest stock in the second period is uncertain because we assume that

between the two periods of the model a natural disturbance ε can occur 4. Thus, the forest4For example, this natural disturbance can be a forest fire or a windstorm.

5

stock in the second period depends on the forest stock in the first period by a growth rate γ.

This growth rate belongs to [0, γ̄] with γ̄ > 1 and is exogenously determined (γ̄ is depending

on the considered specie of the stand), and so this growth rate become γ(ε). Consequently,

γ(ε) is the fraction of increased timber volume salvaged in the event of natural risk. Thus,

higher levels of risk decrease salvageable timber: dγ(ε)dε≤ 0. For example, in the case of for-

est fire, if the risk is important, ε will be high and so, the forest growth rate γ(ε) will be weak.

The dynamic of the consumptions and the dynamic of the stocks depend on the prevention

activity adopted by the forest owner : saving or sylvicultural practices.

2.1 Saving

If the forest owner opts for the financial tool in order to protect himself against the conse-

quences of a natural disaster, his decision variables are the harvesting in the first period and

in the second period denoted respectively x1 and x2, and the saving s.

The objective of the forest owner is the maximisation of (1):

V = u(c1) + v(k1) + δE[u(c2) + v(k2)]

with respect to x1, x2 and s and under the following constraints:

c1 = y + p1x1 − s

k1 = Q− x1

c2 = p2x2 + Rs

k2 = γ(ε)(Q− x1)− x2

Thus, the consumption in the first period (c1) represents the forest owner’s initial wealth

(y) plus the revenue obtained by the first period harvesting (p1x1) minus the amount of

saving (s). The forest stock in the first period (k1) corresponds to the initial forest stock (Q)

minus the first period harvesting (x1). His consumption in the second period corresponds

to the outcome obtained by the second period harvesting (p2x2) plus the amount of saving

invests at interest rate (Rs). Indeed, if the forest owner opts for the financial tool, he saves

a part of his income and so his consumption in the first period is reduced. He invests this

6

money at an interest rate R and consequently, he increases his income and his consumption

in the second period. Finally, the forest stock in the second period (k2) corresponds to the

increasing initial forest stock (γ(ε)(Q− x1)) minus the harvesting of the second period (x2).

In order to obtain the optimal first and second period harvesting (x∗1, x

∗2), and the optimal

saving activity (s∗), we calculate the first order conditions by deriving the objective function

(1) by the variables of decision: x1, x2 and s:

p1u′(c1)− v′(k1)− δE[γ(ε)v′(k2)] = 0 (2)

p2u′(c2)− Ev′(k2) = 0 (3)

δRu′(c2)− u′(c1) = 0 (4)

The assumptions made on u and v ensured that the second order conditions are satisfied.

The equation (2) means that the optimal harvesting is reached in the first period when

the marginal benefit of this harvesting (p1u′(c1)) is equal to its marginal cost (v′(k1) +

δE[γ(ε)v′(k2)]). Thus, when x1 increases, the utility of the consumption in the first period

increases through p1, it is the marginal benefit of the first period harvesting. The marginal

cost of this harvesting can be divided in two terms, the first is direct and the second one

is indirect. These effects can be described as follows : an increase in x1 reduces the forest

stock at the first period k1, so v(k1) decreases (direct effect) and through the discount rate,

k2 is reduced and thus v(k2) decreases (indirect effect).

The expression (3) means that the optimal harvesting is reached in the second period when

the marginal benefit of this harvesting (p2u′(c2)) is equal to its marginal cost (Ev′(k2)). Thus,

when x2 increases, the utility of the consumption of the second period increases through p2,

it is the marginal benefit of the second period harvesting. The marginal cost of this har-

vesting represents a direct effect : when x2 increases, the stock in the second period k2 is

reduced, so v(k2) decreases.

Finally, the equation (4) means that the optimal saving is obtained when the marginal benefit

of saving (δRu′(c2)) is equal to its marginal cost (u′(c1)). Nevertheless, (5) can be rewritten

as follow : δR = u′(c1)u′(c2)

. This means that the optimal saving is reached when the marginal

rate of substitution between c1 and c2 is equal to the interest rate R. Consequently, the forest

owner is confronted with a consumption substitution problem between the two periods.

7

The first order conditions given by (2), (3) and (4) allow us to determine the following

harvesting rule:p1

p2

δR =v′(k1)

Ev′(k2)+ δγ(ε) (5)

Thus, the private forest owner will harvest until the real return of the saving (LHS) is equal

to the production growth rate plus the marginal rate of substitution between the amenities

in the first period and in the second period (RHS). This mean that as long as the return of

saving is superior to the production growth rate, the forest owner prioritizes the financial

aspect to the recreational one and so, he decides to cut trees in his forest and to save the

money at rate R. Conversely, as long as the return of the saving is inferior to the production

growth rate, the forest owner privileges the recreational aspect by cutting no trees.

2.2 Sylvicultural Practices

The sylvicultural practices consist for the forest owner to regenerate, in the first period, a

part of his forest. The regenerated part of the forest only procures an outcome in the second

period because the young plantations and the sowing have no financial and ecological value

as they have not reached a certain age and so, a certain size. This regeneration process has

a constitution cost at the first period, cq with c the exogenous marginal cost, which gathers

elements like time, area and seed. So, the parameter of sylvicultural practices q represents

a stock of regeneration that is a physical reserve of trees. Thus, if the owner opts for the

sylvicultural practices rather than for the saving, his decision variables are the harvesting in

the first and in the second period denoted respectively x1 and x2, and the stock of regener-

ation q.

The objective of the forest owner is the maximisation of (1) with respect to x1, x2 and q

8

and under the following constraints:

c1 = y + p1x1 − cq

k1 = Q− x1

c2 = p2x2

k2 = γ(ε)(Q− x1) + q − x2

Thus, the cost of the sylvicultural practices is deducted to the initial wealth y plus the

outcome obtained by the first period harvesting p1x1 while, in the previous case, it is the

amount of the saving which is deducted. The forest stock in the first period k1 is the same

whatever the prevention measure adopted by the forest owner. We can see that the sylvicul-

tural practices act on the forest stock of the second period by increasing it by the amount q

while, we have seen previously, that the saving acts on the consumption of the second period

by increasing it by Rs. Thus, the forest owner removes cq to his consumption in the first

period to set up a stock of regeneration which increase the forest stock in the second period.

In order to obtain the optimal first and second period harvesting (x∗1∗, x∗

2∗), and the op-

timal stock of regeneration (q∗∗), we calculate the first order conditions by deriving the

objective function (1) by the variables of decision: x1, x2 and q:

p1u′(c1)− v′(k1)− δE[γ(ε)v′(k2)] = 0 (6)

p2u′(c2)− Ev′(k2) = 0 (7)

δEv′(k2)− cu′(c1) = 0 (8)

The assumptions made on u and v ensured that the second order conditions are satisfied.

The interpretation of (6) and (7) is similar to the one realised for (2) and (3).

The equation (8) means that the optimal stock of regeneration is reached when the marginal

benefit of this stock (Ev′(k2)) is equal to its marginal cost (cu′(c1)). However, (8) can be

rewritten as follow : c = δEv′(k2)u′(c1)

. This means that the optimal stock of regeneration is

obtained when the marginal rate of substitution between c1 and k2 is equal to the exogenous

marginal cost of constitution c. Consequently, the forest owner is confronted with a substi-

tution problem between the consumption in the first period and the amenities in the second

9

period.

The first order conditions given by (6), (7) and (8) allow us to determine the following

harvesting rule:p1

p2

u′(c1)

u′(c2)=

v′(k1)

Ev′(k2)+ δγ(ε) (9)

We observe that this rule is the same that for the saving context (equation (5)). Indeed, the

expression δR in the harvesting rule (5) is equal to u′(c1)u′(c2)

by (4) and so, the two prevention

measures lead to a similar harvesting rule. Consequently, the interpretation is identical, as

long as the return of the saving is superior to the production growth rate, the owner harvests

and so, he prioritizes the financial aspect to the recreationnal one. Inversely, when the return

of saving is inferior, the owner does not harvest and he preferes the amenities aspect.

These two models allow us to see that despite the fact that the two prevention measures

lead to an identical harvesting rule, the forest owner is faced to two different problems.

In the model with saving, the forest owner is confronted with a consumption substitution

problem between the two periods while, in the model with sylvicultural practices, the forest

owner faces up to a substitution problem between the consumption in the first period and

the amenities in the second period. Consequently, the two prevention measures do not act in

the same way on the parameters of the model and so, it is interesting to see how the forest

owner’s behavior is affected by the variation of these parameters.

3 Analytics of timber supply under risk and multiple-use

of forests

This section is devoted to a comparative static analysis. The objective of this analysis is to

observe if the variables y, Q, p1 and p2 have an impact on the forest owner’s decisions vari-

ables which are the first and the second period harvesting and the self-insurance activities.

The methodology for this analysis is presented in Appendix 1.

During this comparative static analysis, we observe that the decision making is guided by

10

three effects. The first is a risk effect which involved that the more the owner’s wealth is,

the more he tends to exposed him to risk. The second effect is a wealth effect which implied

that the more the owner’s wealth is, the less interest to harvest late he has. The last effect

is a continuation effect which show that delay the harvesting procure to the owner a more

important outcome in the future and so, it increases the consumption in the future and

finally, it procures to the owner a more important utility on the amenities in the future.

These three effects can be opposed and in this case, we obtain ambiguity results.

We present the results that we have obtained in the following table.

Insert Table 1 here.

The ambiguity results discussed above do not appeared directly in the table because we

want it to be readable and not overcharged. For example, when we analyse the impact of an

increase in y on the prevention variables s or q, we have an assumption in order to obtaindsdy≤ 0 for the saving and dq

dy≤ 0 for the sylvicultural practices but we do not say what

is happened when the asssumption is not respected. In fact, without this assumption, we

obtain ambiguity results.

In the table, we can see that the results can be divided in two groups. The first group of

results concerns the static comparative analysis on the variables y and Q while the second

one concerns the analysis made on p1 and p2. Indeed, the analysis on y and Q lets appeared

that the conditions are very similar and the analysis on p1 requires assumptions very close

to the one adopted for p2. Consequently, in a first part we will focus on the interpretation

of the results dealing with the forest owner initial framework and in a second part, we will

take an interest in the economic variables.

3.1 Comparative static analysis on the variables of initial endow-

ment

At first glance, we can observe that the saving presents a more complex mechanism in terms

of analysis because the results are always conditioned to assumptions while it is not the case

for the analysis of the sylvicultural practices. We can also observe that when the assump-

tions are useful in order to obtain the results, generally they bears on the amenities aspects

11

and they can be interpreted in terms of risk aversion coefficient. Consequently, the elements

which are decisive for the forest owner and be taken into consideration by him when he takes

a decision related on the forest management are the amenities in the two periods.

Thus, concerning the forest owner intial wealth y and the decision variable x1, we find a

result without assumption whatever the self-insurance activities undertaken. In fact, when

the initial wealth increases, the owner increases his first period harvesting. For saving, the

explanation is simple. Indeed, the first period harvesting allows the forest owner to save a

part of his outcome and this money being remunerated at rate R, the forest owner becomes

wealthier. The results and the methodology for parameter x1 are detailed in Appendix 1.

Concerning the decision variables x2, no assumption is required to obtain the results for

the sylvicultural practices context. We show that when the initial wealth increases, the

owner decreases his second period harvesting. This result is also true for the saving only

if the following assumption is respected: −v′′(k1)v′(k1)

≤ − δE[γ(ε)v′′(k2)]Ev′(k2)

, with the LHS represent-

ing the absolute risk aversion coefficient of Arrow-Pratt and the RHS being a partial risk

aversion coefficient. This result means that when the forest owner intial wealth increases,

he reduces his second period harvesting. Indeed, as his initial wealth increases, in order to

keep his standard of living constant, he needs to harvest less.

Finally, concerning the prevention variables s or q, we find a similar result both saving and

sylvicultural practices. Thus, when the forest owner’s initial wealth increases, he reduces

his self-insurance activities. This result can be interpreted in terms of substitution effect.

Consequently, the forest owner’s risk aversion is reduced when his initial wealth increases

and then, he decreases his prevention. This result is true for the two prevention activities if

the asbolute risk aversion coefficient is inferior to the partial one.

Now, concerning the forest owner’s initial forest stock Q and the decision variables x1, we

find the same result for the saving and the sylvicultural practices. After an increase in his

initial forest stock, the owner decreases his first period harvesting. Indeed, the higher the

forest stock is, the higher the recreational aspects are. Thus, after an increase in this forest

12

stock, in order to fully takes advantage of his forest, the owner reduces his first period har-

vesting again. This reaction generates an important increase in the forest stock. However,

the result is the same for the two self-insurance activities but the assumption needed in order

to obtain it is different. For the saving, the assumption depends always on the absolute risk

aversion coefficient and it must be inferior to the partial one. In return, for the sylvicultural

practices, the assumption bears on the exogenous marginal cost which must be superior to p22.

Concerning the decision variables x2, no assumption is compulsory for the sylvicutural prac-

tices. We show that after an increase in his initial forest stock, the owner reduces his second

period harvesting. The same result is obtained in the saving context if the absolute risk

aversion coefficient is superior to −p2

p1

Ev′′(k2)Ev′(k2)

. This result means that after an increase in

the forest stock, the forest owner, in order to keep his amenities constant, raises his second

period harvesting. Indeed, as the forest stock increases, the amenities increase and so, if the

owner wants to keep them constant, he increases x2.

Finally, concerning the prevention variables, if the owner’s initial forest stock increases,

he reduces his self-insurance activities both saving and sylvicultural practices. This means

that the larger the forest area is, the less the forest owner’s prevention activities are. In-

deed, as the forest stock increases, the forest owner is less risk averse and so, he reduces his

self-insurance activities. This reasoning is possible if the forest owner’s initial forest stock is

assimilated to his initial wealth but expressed in cubic meters rather than in monetary terms.

This result is true for the two prevention measures if the absolute risk aversion coefficient is

superior to the partial one.

3.2 Comparative static analysis on the economic variables

We can observe two main differences between the first group of variables analysed and this

one. Firstly, in this group, the results are more often conditioned to assumptions and sec-

ondly, the assumptions bear on the parameters of consumption while they deal with amenities

in the group of variables previously analysed. Thus, when we study the economic variables,

we observe that the decisive elements of the forest owner decision making process have also

an economic aspect. We can also see that the assumptions can be interpreted in terms of

13

risk aversion coefficient.

Thus, concerning the timber price in the first period, we obtain the same results for the

three decisions variables both saving and sylvicultural practices. Consequently, when the

timber price increases, the first period harvesting decreases, the second period harvesting

increases and the owner reduces his self-insurance activities. One condition is necessary in

order to obtain these results, either the absolute risk aversion coefficient must be superior

to 1x1

or the partial risk aversion coefficient must be superior to 1. For the saving, these

assumptions can be replaced by another one, δEγ(ε)p2 ≥ R. This condition means that the

return in the second period will be higher than the interest rate and then, the owner reduces

x1 and increases x2. Nevertheless, another assumptions concerning the exogenous marginal

cost must be added to the previous one for the analysis of x2 in the sylvicultural practices

context: c ≤ 1 and c ≤ p1.

Now, concerning the timber price in the second period and the decision variables x1, we

find, without conditions, that in the case of the forest owner saves a part of his wealth, after

an increase in p2, he raises his first period harvesting. This result is true for the sylvicultural

practices either if the absolute risk aversion coefficient is superior to 1x2

or if the partial one

is superior to 1.

Concerning the decision variables x2, we show that if the absolute risk aversion coefficient is

superior to 1x2

or the partial one is superior to 1, then after an increase in the second period

timber price, the owner reduces his second period harvesting.

Finally, for the prevention variables, after an increase in p2, the owner reduces his self-

insurance activities if the partial risk aversion coefficient is inferior to 1 for the saving and

superior to 1 for the sylvicultural practices. It is generally considered that people have a

partial aversion coefficient superior to one but under this condition, we obtain ambiguity

result for the saving case while we obtain that the owner increases prevention after an in-

crease in p2 for the sylvicultural practices context. Thus, in this model, the unique solution

in order to raise the prevention is to increase the timber price in the second period if the

14

forest owner opts for the physical tool of prevention. Nevertheless, it will be necessary to

test if the forest owner are endowed with a partial aversion coefficient superior to one as it

is generally accepted.

The analysis of this comparative statics results allows us to see that for the variables

of initial endowment like the initial wealth and the initial forest stock, the main elements for

the forest owner’s decision are the amenities while these elements concern the consumption

when we analysed the economic variables like the timber price in the two periods. This

analysis also allows us to observe that the forest owner’s behavior towards the natural haz-

ards management depends on variables like risk aversion. Nevertheless, we have no data

about the forest owner’s characteristics which permit us to conclude about the more likely

situation.

4 Concluding Comments

In this article, we develop a dynamic theoretical model in order to analyse the forest owner’s

behavior in a risky forest management. We compare two self-insurance activities that the

forest owner can undertake to protect his forest against natural hazards, the saving and

the sylvicultural practices. In this work, there is several contributions. Firstly, we extend

the model of Koskela and Ollikainen (1999) by considering a standard dynamic approach of

decision making and an expected utility context and by integrating the prevention against

natural hazards. Secondly, our approach is original in the sense of it compares two preven-

tion instruments, the saving which is a financial one and the sylvicultural practices which is

a physical one. Finally, we analyse the saving in a context where the financial tools are more

and more considered in order to cover the important risk. In this framework, we show that

the harvesting rule for the two prevention measures is the same and it implies that the forest

owner harvests until the real return of the saving is equal to the production growth rate. We

also show that even if the aim of the two instruments of prevention analysed in the article

is to reduce the magnitude of the loss after a catastrophe, they do not act in the same way

on the forest owner’s behavior. Indeed, if the forest owner opts for the financial tool, he his

15

confronted with a consumption substitution problem between the two periods of the model

while, if he chooses to protect himself with the sylvicultural practices, he is confronted with

a substitution problem between the consumption in the first period and the amenities in the

second one. Finally, we show that the forest owner’s behavior towards natural disturbances

management depends on numerous assumptions dealing with parameters like risk aversion.

Up to now, we do not have data about the forest owner’s preferences and characteristics

and consequently, it could be a necessary stage in order to have a better understanding of

the natural hazards management in the forest sector. Another future research could deal

with the case where the forest owner opts simultaneously for the saving and the sylvicultural

practices. Nevertheless, in a theoretical point of view, this extension will be very complicated

in our model especially because of the time dimension.

16

Appendix 1

The procedure being similar for all the parameters, rather than listing all the static compar-

ative results, we explain the methodology by analyzing the sign of dx1

dyin the saving context.

Thus, differentiating the first order conditions given by (3), (4) and (5) with respect to y

respectively allows us to obtain the following system of equations:

−p1u′′(c1) = dx1

dy[p2

1u′′(c1) + v′′(k1) + δE[γ(ε)2v′′(k2)]] + dx2

dy[δE[γ(ε)v′′(k2)]]− ds

dy[p1u

′′(c1)]

0 = dx1

dy[δE[γ(ε)v′′(k2)]] + dx2

dy[δp2

2u′′(c2) + δEv′′(k2)] + ds

dy[δp2u

′′(c2)R]

u′′(c1) = dx1

dy[−p1u

′′(c1)] + dx2

dy[δp2u

′′(c2)R] + dsdy

[u′′(c1) + δR2u′′(c2)]

that we can rewrite in a matrix form as follows:K

O

L

=

A B C

E F G

H I J

dx1

dy

dx2

dy

dsdy

with

A = p21u

′′(c1) + v′′(k1) + δE[γ(ε)v′′(k2)] < 0

B = δE[γ(ε)v′′(k2)] < 0

C = −p1u′′(c1) > 0

E = δE[γ(ε)v′′(k2)] < 0

F = δp22u

′′(c2) + δEv′′(k2) < 0

G = δp2u′′(c2)R < 0

H = −p1u′′(c1) > 0

I = p2u′′(c2)R < 0

J = u′′(c1) + δR2u′′(c2) < 0

K = −p1u′′(c1) > 0

O = 0

L = u′′(c1) < 0

17

The sign of dx1

dydepends on the sign of the following expression:

dx1

dy= K(FJ −GI) + L(BG− CF )

Thus, this expression can be rewritten as follow:

−2(u′′(c1)2)p1δ︸ ︷︷ ︸

[p22u

′′(c2)− Ev′′(k2)]︸ ︷︷ ︸?

+ Rδ2u′′(c1)Ev′′(k2)u′′(c2)︸ ︷︷ ︸

[γ(ε)p2 −Rp1]︸ ︷︷ ︸

This expression is positive by the first order conditions (4). Then we have dx1

dy> 0.

This result means that when the forest owner’s initial wealth increases, he increases his first

period harvesting.

We reply this same procedure in order to obtain the static comparative results indicated

in the other propositions.

18

References

Amacher G.S, Malik A.S and Haight R.G (2005), "‘Not Getting Burned: The Importance

of Fire Prevention in Forest Management"’, Land Economics, 81(2), 284-302.

Birot Y. et Gollier C. (2001), " Risk Assessment, Management and Sharing in Forestry,

with Special Emphasis on Wind Storms ", papier présenté lors de la 14ème convocation

de " Academies of Engineering and Technological Sciences " (CAETS, Espoo, Finlande,

juin 2001).

Brunette M. and Couture S. (2006), "‘Auto-assurance, assurance et risques naturels: une

application à la gestion forestière"’, Document de travail LEF n◦ 2006-02.

Ehrlich I. and Becker G. (1972), "‘Market Insurance, Self-Insurance and Self-Protection"’,

Journal of Political Economy, 6, 623-648.

Englin J., Boxall P. and Hauer G.(2000), "An empirical examination of optimal rotations

in a multiple-use forest in the presence of fire risk", Journal of Agricultural and Resource

Economics , volume 25, number 1, pages 14-27.

Gollier C. (2002), "‘Optimal Prevention of Unknown Risks: A Dynamic Approach with

Learning"’, This Paper was written while the author was visiting the Cetelem-Findomestic

Chair of Finance and Consumption at the European University Institute of Florence.

Gollier C. (2003), "‘To Insure or not to Insure: An Insurance Puzzle"’, The Geneva Paper

on Risk and Insurance Theory, 28:5-24.

Hartman R. (1976), "‘The Harvesting Decision when a Standing Forest Has a Value"’,

Economic Inquiry, 14, pp52-58.

Koskela E. and Ollikainen M. (1999), "‘Timber Supply, Amenity Values and Biological

Risk"’, Journal of Forest Economics, 5:2, 285-304.

Schelhaas M.J, Nabuurs G.J and Schuck A. (2003),"’Natural Disturbances in the

European forests in the 19th and 20th centuries"’, Global Change Biology, 9, 1620-1633.

19

Stenger A. (2004), "‘Risk and insurance in forests: an experimental study on non

industrial private forest owners in France in risky and ambiguous contexts"’, preliminary

version.

20

Table 1: Comparative statics results for saving and sylvicultural practices

SAVING SYLVICULTURAL PRACTICES

Variables Decision variables

Conditions Results Conditions Results

x1 None dx1/dy > 0 None dx1/dy > 0 x2

)('

)('')(

)('

)(''

2

2

1

1

kEv

kvE

kv

kv εγδ−≤−

)('

)('')(

)('

)(''

2

2

1

1

kEv

kvE

kv

kv εγδ−≥−

dx2/dy ≤ 0

dx2/dy ≥ 0

None dx2/dy < 0

y s or q

)('

)('')(

)('

)(''

2

2

1

1

kEv

kvE

kv

kv εγδ−≤−

ds/dy ≤ 0

)('

)('')(

)('

)(''

2

2

1

1

kEv

kvE

kv

kv εγδ−≤−

dq/dy ≤ 0

x1

)('

)('')(

)('

)(''

2

2

1

1

kEv

kvE

kv

kv εγδ−≤−

dx1/dQ ≤ 0

²2pc ≥ dx1/dQ < 0

x2 )('

)(''

)('

)(''

2

2

1

2

1

1

kEv

kEv

p

p

kv

kv−≤−

)('

)(''

)('

)(''

2

2

1

2

1

1

kEv

kEv

p

p

kv

kv−≥−

dx2/dQ ≥ 0

dx2/dQ ≤ 0

None dx2/dQ < 0

Q

s or q

)('

)('')(

)('

)(''

2

2

1

1

kEv

kvE

kv

kv εγδ−≥−

ds/dQ ≤ 0

)('

)('')(

)('

)(''

2

2

1

1

kEv

kvE

kv

kv εγδ−≥−

dq/dQ ≤ 0

x1 dx1/dp1≤ 0

11

1 1

)('

)(''

xcu

cu>−

or if )(/1 εγδEc ≥

dx1/dp1≤ 0

x2 dx2/dp1 ≥ 0 11

1 1

)('

)(''

xcu

cu>− , c ≤ 1,

c ≤ p1

dx2/dp1 ≥ 0

p1

s or q

11

1 1

)('

)(''

xcu

cu>−

or if RpE ≥2)(εγδ

ds/dp1≤ 0 11

1 1

)('

)(''

xcu

cu>−

dq/dp1< 0

x1 None dx1/dp2 > 0

22

2 1

)('

)(''

xcu

cu>−

22

2 1

)('

)(''

xcu

cu<−

dx1/dp2 > 0

dx1/dp2 < 0

x2

22

2 1

)('

)(''

xcu

cu>−

dx2/dp2 < 0

22

2 1

)('

)(''

xcu

cu>−

22

2 1

)('

)(''

xcu

cu<−

dx2/dp2 < 0

dx2/dp2 > 0

p2

s or q

22

2 1

)('

)(''

xcu

cu<−

ds/dp2 > 0

22

2 1

)('

)(''

xcu

cu>−

22

2 1

)('

)(''

xcu

cu<−

dq/dp2 > 0

dq/dp2 < 0

21