a possibilistic and probabilistic approach to precautionary saving final

7/23/2019 A Possibilistic and Probabilistic Approach to Precautionary Saving Final

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A possibilistic and probabilistic approach to precautionary saving

Summary

This paper proposes two mixed models to study a consumers optimal saving in the presence of two

types of risk: income risk and background risk. In the first model the income risk is represented by a fuzzy

number and the background risk by a random variable. In the second model the income risk is represented by a

random variable and the background risk by a fuzzy number. For each of these models three notions of

precautionary saving are defined as indicators of the extra saving induced by the income and the background

risk on the consumers optimal choice. !s a conclusion we can characterize conditions allowing for extra

saving relative to the optimal saving under certainty even when a certain component of risk is modeled with

fuzzy numbers.

Key words: "ptimal saving background risk income risk possibility theory.

JEL: #"$ %&' %('

)owadays it is widely expressed a concern about the huge saving rate of #hinese households which

has far*reaching international implications in terms of #hinas current account surpluses and the +estern

external deficits. ,any analysts and policy makers are trying to trace back the reasons for such behavior and

predict how long it will last. In this regard many of the potential explanations for this phenomenon point to

economic uncertainty as a determinant of precautionary saving -,arcos #hamon ai /iu and 0swar 1rasad

-$2'&33. In contrast to the #hinese example 45 households are well known for their low saving rate whichcalls for some precise explanations as well.

Therefore it is very important to understand the characteristics of such uncertainty in order to model

the households optimal behavior in terms of consumption and saving. 6oth the kind of uncertainty and the

features of the consumers preferences must be analyzed in order to predict micro or macroeconomic saving

patterns. In this context the notion of precautionary saving has long appeared in models of economic decision

under uncertainty. It measures the way adding a source of risk modifies the optimal saving. +hen the

consumers utility function is unidimensional and the risk situation is described by a random variable extra

saving will appear in response to uncertainty when the third order derivative of the utility function is positive.

"n the other hand several authors -)eil !. %oherty and 7arris 5chlesinger -'8(&3 #hristian 9ollier

and ohn +. 1ratt -'88;3 ohn +. 1ratt -'88(33 studied economic decision processes governed by two types of

risk: primary risk -income risk3 and background risk -loss of employment divorce illness etc.3. In our paper

as in ,ario ,enegatti -$2283 the presence of background risk will be associated to a nonfinancial variable and

will be uninsurable nevertheless having an influence on the optimal solution for economic decisions -see e.g.

/ouis 0eckhoudt #hristian 9ollier and 7arris 5chlesinger -$223 and ,ario ,enegatti -$2283. These models assume that the consumers


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activity takes place during two periods and both types of risk act during the second period. The presence or the

absence of one of the two types of risk leads to several possible uncertainty situations which enables the

definition of several notions of precautionary saving -e.g. in ,enegatti -$2283 two of such notions are studied3.

!ll the optimal saving models in the literature are based on probability theory. That is both the

primary and the background risk are modeled as random variables. 7owever there are risk situations for which

probabilistic models are not appropriate -e.g. for small databases3. /otfi !. ?adeh -'8>(3s possibility theory

offers another way to model some risk situations. 7ere risk is modeled with possibility distributions

-particularly with fuzzy numbers3 and the well known probabilistic indicators -e.g. expected value variance

covariance3 are replaced by the corresponding possibilistic indicators.

%ue to the complexity of economic and financial phenomena one can have mixed situations in which

some risk parameters should be probabilistically modeled with random variables and other risk parameters

should be possibilistically modeled with fuzzy numbers. Then we can consider the following four possible

situations: -'3 a random variable captures the primary risk and also a random variable captures the background

risk@ -$3 a fuzzy number captures the primary risk and a fuzzy number captures the background risk@ -&3 a

fuzzy number captures the primary risk and a random variable captures the background risk@ -A3 a random

variable captures the primary risk and a fuzzy number captures the background risk.

The situation -'3 was treated in the abovementioned probabilistic models. The purpose of this paper is

studying the precautionary saving motive in situations -&3 and -A3. In the case of situation -&3 the risk situation

is described by a mixed vector -! B3@ and in the case of situation -A3 the risk situation is described by a mixed

vector -C 63@ where ! 6 are fuzzy numbers and B C are random variables. /et us denote the mixed models

described by situation -&3 as type I models@ we will also denote the mixed models described by situation -A3 as

type II models. For each of those two types of models -type I and type II3 we will define three notions of

precautionary saving and investigate the necessary and sufficient conditions for extra saving to arise after

adding primary risk background risk or both of them. The main results of the paper establish those necessary

and sufficient conditions expressed in terms of third*order partial derivatives of the bidimensional utility

function and in terms of the probabilistic and possibilistic variances associated with the mixed vector.

"ur three notions of precautionary saving will be defined below in detail. 7owever we can anticipate

that the first notion refers to the extra saving arising when a small income risk is introduced relative to the

optimal saving under certainty. The second notion refers to the extra saving arising when a small background

risk is introduced relative to the optimal saving under certainty. !nd finally the third notion refers to the extra

saving arising when both a small income risk and a small background risk are introduced relative to the

optimal saving under certainty. 0ach of the three notions reDuires its own necessary and sufficient conditions

for each kind of precautionary saving to be positive.

+e can offer illustrative examples of type I and type II models in real life. For instance suppose a

man owns a house near the beach surrounded by a wheat field he needs to harvest every year. If the rainfall

during the year is very scarce the annual income from the wheat harvest will be lower. 6ut simultaneously

the renting value of a room in his house could be higher.

!ssume now that only the risk related to the harvest is insurable. 7owever the background risk is

related to the potential lodgers pleasure from sunbathing which could be described by a fuzzy number:

Eweather near the beach is pleasant enough to sunbathe. 6oth risks would be negatively correlated since

under a dry and sunny weather the owner would collect little revenue from the harvest but probably he could


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charge a higher rent to his lodger. Therefore there may or may not be precautionary saving in response to

those two risks. That would be an example we could characterize as a mixed model of type II.

Imagine now that instead the main income source for the owner is the apartment rent. The size of the

rent could be now insurable for him unlike the harvest revenue. In that case the income risk would be

characterized by a fuzzy number with a value dependent on the lodgerGs imprecise perception of the weather.

The situation could be then assimilated to a mixed model of type I. The obHective of this paper is exploring the

conditions for precautionary saving in situations like these two described cases.

+e will describe now briefly the structure of the paper. 5ection ' contains a brief historical

perspective on the contributions dealing with the unidimensional precautionary saving problem and also with

precautionary saving in probabilistic models with background risk. 5ection $ recalls the indicators of fuzzy

numbers and the mixed expected utility of Irina 9eorgescu and ani innunen -$2''3. 5ection & presents the

mathematical framework in which the optimal saving models with background risk are embedded. 5ection A

proposes mixed models of optimal saving of type I with an income risk modeled with a fuzzy number and a

background risk modeled with a random variable. 5ection < deals with mixed models of optimal saving of type

II with an income risk modeled with a random variable and a background risk modeled with a fuzzy number.

Three notions of precautionary saving will be defined for both mixed models of type I and II.

The main results of the paper establish the necessary and sufficient conditions for the positivity of

each notion of precautionary saving in terms of the consumers preferences and the features of risks. ,ore

specifically the existence of each notion of precautionary saving depends on: the signs of third*order partial

derivatives of the utility function and the possibilistic and probabilistic variances associated with the mixed

vectors.

1. Literature Review

5aving under uncertainty is a topic introduced in economic research by 7ayne 0. /eland -'8;(3 and

!gmar 5andmo -'8>23. These papers investigate the way in which the presence of risk modifies the optimal

amount of saving. The variation experienced by the optimal saving after adding risk elements is measured in

/eland -'8;(3 and 5andmo -'8>23 by the concept of precautionary saving. ! central result of those papers

asserts that a positive third*order derivative of a consumers utility function is a necessary and sufficient

condition for an increase in saving as the effect of risk. ,iles 5. imball -'8823 defines the notion of

Eprudence as the sensitivity of optimal saving to the magnitude of risk and introduces the index of absolute

prudence as a measure of this sensitivity. 7e proves that the theory of the absolute prudence index is

isomorphic to the !rrow*1ratt index theory on risk aversion -see e.g. 0eckhoudt 9ollier and 5chlesinger

-$22


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5chlesinger -$22>3 ,ario ,enegatti -$2283 %iego )ocetti and +illiam T. 5mith -$2''3 investigate the effect

of the primary risk and the background risk on the optimal saving decision considering the case of a consumer

with a bidimensional utility function. 5pecifically in the papers by #ourbage and =ey -$22>3 and ,enegatti

-$2283 they established necessary and sufficient conditions for a positive precautionary saving to appear in a

bivariate context. In the particular case of ,enegatti -$2283 he defined two notions of precautionary saving:

the first one explores the impact of a small income risk on saving relative to the optimal saving undercertainty@ the second one explores the impact of both sources of risk -primary and background3 on saving

relative also to the optimal saving under certainty. ,oreover the paper by )ocetti and 5mith -$2''3 develops a

precautionary saving model with an infinite horizon. Finally the paper by 0lyes ouini #lotilde )app and

%iego )ocetti -$2'&3 develops a matrix*measure concept of prudence.

2. Preliminaries

2.1. Preliminaries on Fuzzy Numbers

/et us now introduce some preliminary concepts on fuzzy theory. They will be useful to set up the

consumers optimization problem in which one of the sources of risk will be represented by a fuzzy number.

/et be a nonempty set of states. ! fuzzy subset of is a function

. ! fuzzy set is normal if for some . The support of is defined by

. )ext assume that . For

the level set of is defined by

- is the topological closure of 3.

The fuzzy set is fuzzy convex if is a convex subset of for all .

! fuzzy subset of is called a fuzzy number if it is normal fuzzy convex continuous and with

bounded support. If are fuzzy numbers and then the fuzzy numbers

and are defined by

.

! nonnegative and monotone increasing function is a weighting

function if

+e fix a weighting function and a fuzzy number such that

for . /et be a continuous function

-interpreted as a utility function3.


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The possibilistic expected utility is defined by:

-'3

If is the identity function then from -'3 one obtains the weighted possibilistic expected value

-=obert FullJr and 1eter ,aHlender -$22&33:

-$3

If then one obtains the weighted

possibilistic variance -+ei*9uo ?hang and Cing*/uo +ang -$22>33:

-&3

+hen for and

are the possibilistic mean value and the possibilistic variance of #hrister #arlsson and =obert

FullJr -$22'3.

2.2. i!ed E!"e#ted $tilities

The concept of mixed expected utility was introduced by 9eorgescu and innunen -$2''3 in order to

build a model of risk aversion with mixed parameters: some of them were described by fuzzy numbers and

others by random variables. This same notion has been used by Irina 9eorgescu -$2'$b3 to study mixedinvestment models in the presence of background risk.

In this section we will review this definition of mixed expected utility and some of its properties. For

clarity purposes we will deal only with the bidimensional case. Then a mixed vector will have the form

(A , X) where A is a fuzzy number and X a random variable. +ithout loss of generality we will

only consider the case (A , X) .

/et be a random variable w.r.t. a probability space . +e will denote by

its expected value and by its variance. If is a continuous

function then is a random variable and is the

-probabilistic3 expected utility of w.r.t. .

+e are going to choose a fixed weighting functionf

and a bidimensional continuous utility

function . /et be our particular mixed vector. !ssume that the level sets


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of have the form . For any

will be the random variable defined by

for any .

/et us now define our concept of mixed expected utility which will be subHect to maximization in our

approach to optimal saving.

%e&inition 2.1 (Irina Georgescu, 2012a, Georgescu and Kinnunen 2011)

The mied epected utility associated !ith and the mied

vector is de"ined by

-'3

Remar' 2.2

(i) I" the "u##y number is the constant then

$

-ii3 If the random variable is the constant then

The following two propositions are essential to prove the main theorems to be discussed in the

following sections:

Pro"osition 2.( (Georgescu, 2012a, Georgescu and Kinnunen 2011)

%et be t!o bidimensional utility "unctions and $ I"

then

$

Pro"osition 2.) (Georgescu, 2012a, Georgescu and Kinnunen 2011)


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I" the utility "unction has the "orm

then $

(. * Probabilisti# *""roa#+ to ,"timal Savin-

The optimal saving models presented in #ourbage and =ey -$22>3 and ,enegatti -$2283 consider theexistence of two types of risk background risk and income risk both of them being mathematically

represented by random variables. In this section we will present the general features of these models as a

reference to start building our main models in the following sections.

These two*period models proposed by #ourbage and =ey -$22>3 and ,enegatti -$2283 are

characterized by the following data:

and are consumers utility functions for period resp. .

the variable represents the income and is a nonfinancial variable.

for period the variables and have the certain values and .

for period there is un uncertain income -described by the random variable 3 and a

background risk -described by the random variable 3.

+e denote by and . In ,enegatti -$2283 the author

mentions the following four possible situations for the variables and :

-a3 -simultaneous presence of income risk and background risk3

-b3 -income risk and no background risk3

-c3 -background risk and no income risk3

-d3 -no uncertainty3

#onsider now the following expected lifetime utilities corresponding to the situations -a3 -c3 and -d3

respectively:

-'3

-$3

-&3

wheres

is the level of saving. !ccording to ,enegatti -$2283 the optimization problem can be

formulated as follows:


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-A3

-3.

+e intend to offer in this paper an alternative setting to ,enegatti -$2283s by allowing for the

possibility that either the income risk or the background risk are represented formally by a fuzzy number.

Therefore the structure of the following sections follows a parallel line to ,enegatti -$2283s although instead

of his random vector (Y , X) we will consider mixed vectors of the type - A , X or (Y , B) .

). i!ed odels o& y"e

The mixed models of this section are based on the hypothesis that the income risk is described by a

fuzzy number and the background risk is described by a random variable . +e will preserve the

notation introduced in the previous section. ! fuzzy number corresponds to the variable and the

random variable corresponds to the variable . Thus instead of ,enegatti -$2283s random vector

we have a mixed vector .

+e will fix a weighting function and denote and

. In this case the situations -a3*-d3 of 5ection & become

In this section we will study how the optimal saving changes as we follow the routes

and . The first two are analogous to

the cases studied in ,enegatti -$2283 for the probabilistic models. +e will define three notions of


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Kprecautionary savingK and will establish necessary and sufficient conditions for the positivity of these

indicators.

!ssume that the bidimensional utility functions and are strictly increasing with respect to each

component strictly concave and three times continuously differentiable. +e denote by / /

0resp. 3 the first the second and the third partial derivatives of -resp. 3.

)ext we will use as in ,enegatti -$2283 the following Taylor approximation:

-'3

+++++++ 33--33--3-3-'$'''' savaysavsavsyv

4sing the notion of mixed expected utility from 5ubsection $.$ we introduce the following expected

lifetime utilities:

-$3

-&3

-A3

-


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->3

If we differentiate from -&3-


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and

.

Taking into account ->3-'23 the optimal conditions are written:

-'3

-'(3

Following the line of ,enegatti -$2283 we will introduce three notions of Kmixed precautionary

savingK: .

corresponds to precautionary saving from ,enegatti -$2283 and measures the

modification of the optimal saving when moving from to

i.e. by adding the income risk in the presence of the background risk . The

difference expresses the modification of the optimal saving by moving from the certainty

situation to the situation i. e. by adding

the income risk and the background risk . Finally measures the modification of the

optimal saving by moving from to i.e.

by adding the background risk in the presence of the income risk .

)ext we intend to give necessary and sufficient conditions for the positivity of the three indicators.

Pro"osition ).2%et be a mied vector !ith and

$ The "ollo!ing ine'ualities are e'uivalent

-i3 @

-ii3 .

Proo&.4sing the approximation formula -'3 and 1roposition $.& by applying the mixed expected

utility operator one obtains:


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"ne notices that

and

. !lso thus the

previous relation becomes:

-'83

! similar computation shows that

-$23

Taking into account that is strictly concave it follows that is strictly decreasing

therefore iff . 6y -(3 and -'


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The property -i3 of the previous proposition says that the effect of adding the income risk in the

presence of background risk is the increase in the optimal saving. In particular from 1roposition A.$ it

follows that if then for any

income risk and for any background risk .

)ext we study the change of the optimal saving on the route .

Pro"osition ).(%et be a mied vector !ith and

$ The "ollo!ing are e'uivalent

-i3

-ii3

.

Proo&. Taking into account that is strictly concave iff

6y the formulas -83 and -'&3 one has the eDualities:

Formula -''3 gives the following approximation:

thus

The following eDuivalences follow

iff


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iff

This ends the proof.

#ondition -i3 of 1roposition A.& says that the effect of adding the income risk and the background

risk is the increase in the optimal saving. In particular from 1roposition A.& it follows that if

and then for any mixed vector we have

.

orollary ).)Assume that is a mied vector and

$ I" then

, !here and $

Proo&. The proof follows from 1ropositions A.$ and A.& taking into account that

and . This ends the proof.

Finally consider now the change of the optimal saving on the route .

Pro"osition ).%et be a mied vector !ith and


-i3 @

-ii3 .

Proo&.4sing the approximation formula -'3 and applying 1roposition $.& it follows:

-$$3

6y -'23 and -'


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=eplacing and

with their approximate values obtained by applying -$$3 and -'83 one obtains:

5ince is strictly concave iff

iff .

This ends the proof.

#ondition -i3 of the previous proposition says that adding the background risk in the presence of

the income risk leads to an increase in the optimal saving.

orollary ).3I" and then

$

Proo&.1ropositions A.$ A.& and A.< are applied.

In the next example we will show that there exist mixed vectors with

and and the utility functions for which condition

does not imply

23-3- 'L

' as)As

E!am"le ).4%et be t!o real numbers such that $ %et be a "u##y

number !ith , "or any and the

uni"orm repartition on $ It is *no!n that and


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$ A simple calculation sho!s that and

$ Then

-$&3

!ssume that the utility function has the form:

with

.

"ne notices that

@

.

Then from -$&3 it follows:

-$A3

"ne notices

iff

iff

iff

From -$A3 and these eDuivalences it follows:

-$


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-$;3 iff

For from -$;3 it follows

iff iff

.

Then if we will have

. "n the other hand

thus by 1roposition A.$ .

In particular the above example shows that the converse of #orollary A.; is not true.

. i!ed odels o& y"e

The mixed models of this section assume that the income risk is represented by a random variable

and the background risk by a fuzzy number . +e keep the hypotheses of 5ection A on the bidimensional

utility functions and .

+e fix a weighting function . +e denote and .

5imilarly as in previous sections we consider the following cases:

+e analyze the way the optimal saving changes on the following three routes:

and . For each of these three cases we


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will introduce a notion of Kprecautionary savingK and we will prove necessary and sufficient conditions for the

positivity of these three indicators.

#orresponding to the cases

3-3- $$ da

we introduce four expected lifetime utilities:

-'3

-$3

-&3

-A3

%eriving -'3-A3 it follows

-3

-(3

5imilarly to the previous section it is proved that and are strictly

concave functions. +e form the four maximization problems:

-83

-'23

-'''3

-'$3

in which

and are the optimal solutions.

6y -


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-'&3

-'A3

-'


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=eplacing in -'83 and

with their approximate values from -'(3 and -'83 we find the solution:

-$23

+ith the same type of argument as in the proof of 1roposition A.$ iff

.

Pro"osition .2%et be a mied vector !ith and


-i3

-ii3

.

Pro"osition .(%et be a mied vector !ith and


-i3

-ii3 .

orollary .)%et be a mied vector !ith and

$ I" and

then $

The proofs of 1ropositions


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The positivity conditions of the three notions of precautionary saving

and express the fact that on the routes

and the optimal saving increases. The above results characterize

these conditions in terms of the third partial derivatives of .

3. on#lusions

The optimal saving models of this paper combine methods of probability and possibility theory. For

mixed twoperiod models the way the optimal saving changes is studied in two cases:

-I3 the income risk is a fuzzy number and the background risk is a random variable.

-II3 the income risk is a random variable and the background risk is a fuzzy number.

For each of the two types of models three notions of mixed precautionary saving have been

introduced. These indicators measure the variation of the optimal saving as a result of adding income risk in

the presence of background risk adding background risk in the presence of income risk or simultaneously

adding income risk and background risk to a certain situation. The main results of the paper characterize the

positivity of the three indicators which indicates that by the mentioned modifications the level of optimal

saving increases.

"ur results indicate that also when the consumerGs environment is fuzzy there are potentially reasons

for extra saving relative to certainty. !nd we have characterized the conditions for such precautionary saving.

That characterization could be useful to predict the behavior of aggregate saving in response to well defined

risks affecting a countryGs population. ! possible extension of our paper could be a numerical comparison of

the differences in saving predicted by our model and ,enegatti -$2283Gs.

The mixed models of the paper follow a parallel line with the probabilistic model of ,enegatti -$2283

where the background risk and the income risk are random variables. It remains to study a purely possibilistic

optimal saving model in which both the income risk and the background risk are fuzzy numbers.

Re&eren#es

arlsson/ +rister/ and Robert Full5r. $22'. E"n 1ossibilistic ,ean Malue and Mariance of Fuzzy

)umbers.+u##y ets and ystems '$$: &'


22/24

arlsson/ +rister/ and Robert Full5r. $2''$ -ossibility "or .ecision. 5pringer.

arroll/ +risto"+er %./ and iles S. Kimball. $22(. E1recautionary 5aving and 1recautionary

+ealth. In The /e! -algrave .ictionary o" conomics 0d. 5teven ). %urlauf and /awrence 0. 6lume.

1algrave ,acmillan. 5econd 0dition.

+am6n ar#os/ Kai Liu/ and Eswar Prasad. 0$2'&3. KIncome 4ncertainty and 7ousehold 5avings

in #hina.ournal o" .evelopment conomics'2>.

ourba-e/ +risto"+e/ and 75atri#e Rey. -$22>3. E1recautionary 5aving in the 1resence of "ther

=isks. conomic Theory &$: A'>A$A.

%o+erty/ Neil *./ and 8arris S#+lesin-er. -'8(&3. E"ptimal Insurance in Incomplete ,arkets.ournal o" -olitical conomy 8': '2A


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9eor-es#u/ ;rina. -$2'$b3. E#ombining 1robabilistic and 1ossibilistic !spects of 6ackground =isk.

The '&th I000 International 5ymposium on #omputational Intelligence and Informatics #I)TI $2'$

6udapest 7ungary $2$$ )ovember pp. $$


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>ade+/ Lot&i *. -'8>(3. EFuzzy 5ets as a 6asis for a Theory of 1ossibility. +u##y ets and ystems

': &$(.

>+an-/

a possibilistic and probabilistic approach to precautionary saving final

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