Linking Decision and Time Utilities

Munich Personal RePEc Archive

Linking Decision and Time Utilities

Kontek, Krzysztof

Artal Investments

18 December 2010

Online at

https://mpra.ub.uni-muenchen.de/27541/

! "# #$

%

& %

%

%

' (#) (* ('# +,) - "

( . / ( ( . 0

1

( . 2 & ')3

4 5

4

(

)

( ) (

)

#

T t t

t T t k

k

U c c d k u c

−

+ =

=

∑

$

"

( )

k d k

ρ

  =  ₊ 

  "$

(

t k

)

u c₊ 4

d k$ 4 : "##"$

ρ 4 + ; (. : % : % ; : "##"$ & 0 (

"# #$ 0 "# #$ < ;

( )

d t t

α

= + )$

2 = '*3$ > ''"$ 2

- ''7$ =

( )

(

)

d t t β

α

= + ,$ 0

( )

d t tβ

α

=

+ 7$

= ? "##@$ 0 ; > ''3$

( )

# # t if t d t if t

β δ

= 

=  _>

 @$

β δ #

( )

( )t

d t e

β

α −

= 3$

5 > '' $

% :

5

/ "##*$

; & "##* "##'$

% 0 % ;

% 8 "##*$ &

+

"##'$ 4

5

A ;

5

! ''"$

5

A B C

%

!

"

#

0 %

= = "

( )

S t % p

$T t

( )

[

]

S t = p T >t " $

5

% ;

,

% : t D #

0 t

; #

! '3' ''"$ 5 ( .

! "# #$

(

%

( )

p=D r " "$

p D r %

= $

( )

(

)

(

max

)

(

minmin

)

v ce v P r

v P v P

− =

− " )$

v ce ; PmaxD 2 % x$ %

Pmin DMin x$ ( .

& =

2 = '7"$ % decision +% .

+; " "$ %

" $

( )

( )

D r =S t " ,$

+; " ,$ ;

( )

( )

+; " 7$ % E# F

rt$ %

D S

$

"

%

%

" 7$

" $ " "$

% 7 >

)

( )

h t

( )

_{( ) ( )}

d S t

S t h t

dt = − ) $

>

( )

h t

τ

= ) "$

& ; ) $ S t

(

=#

)

=

% t≥#

( )

t

S t =

e

−τ ) )$

τ $

6

( ''* "###$

( )

(

r

)

D r =

e

− − α ) ,$

( αD : ) $ : α G

& r= 6e

,

: αH &

;

(

) _{B = C}

,_{. ) ,$ r= 6e ( .}

@

; ; " 7$

( )

(

p

)

D p

e

α − ₌ − −

) 7$

5 : ) $7

% $& ' ' ( ' ) * + ' ) ) * ' + , - ) α_&

& ) )$ ) 7$ " 7$

( )

t e

t

r

e

α τ            

 

−

− −

=

) @$

( )

t t

r

e

α

τ

     

−

=

) 3$

) 3$ % @

3$ 5 : ) " $ : αD % : α

G % tG τ tH τ : αH %

7 _/

! "# #$

@ _{*7, %}

α tDτ 0 6

α G : G τ

6

% : H τ

6

% : αH

% $&! ' ' ( ' ' .( ) α _{* +& ' ' ( '}

' .( ( ' ' ' ( * &/+& τ_{0 $ &}

α

α %

B C

% αD # 7

,$ : ) " $

B C

% %

B C %

*

/

1 (

%

&

% $ > >

)$

( )

t t

r

τ

= + , $

, $ 0

" 7$ t D e t τ

τ

− −   = _{ }   + , "$

0 D−

( )

D p

p

− ₌

− , )$

5 , )$ ) )$ " 7$

( )

t

r t

t e−τ

_τ

( )

(

)

D p

p α

− ₌

− , 3$

( )

r

D r e α

−

−

= , *$

, *$ α : , $

, 7$ : , $

% /& */&2+ ' ( ' ( */&3+ ) )

α. τ_{0 $ ' &}

0 ) α

/

: , *$

A α

& & %

, 7$ α τ

tDτ α

τ ; :

3$

>

7$

( )

r t

t α

τ

=

  +    

# (

t

D e t

τ α

τ

− −  

= _{ }

   

+    

, #$

( )

(

)

D p

p α

− ₌

+ − , $

( )

r

D r e

α −

  −_ −_  

= , "$

, "$ α

, '$ : , "

% /&! */& !+ ' ( ' ( */&4+ ) ) 5

α. τ_{0 $ ' &}

5 )

: p= 6e≈# )3

r D # 7

greater ; $ &

, "$ , '$ %

: B C α G

3

#

)

%

% ) 3$

5

: "###$

( )

r

D r

e

α β   −       −

= 7 $

( )

(

p

)

D p

e

α

β

− ₌ − −

7 "$

& 7 "$ % ) )$ " 7$

( )

t t

r

e

α

β  _τ  

−

=

7 )$

0 β τ 7 )$

( )

I

t t

r

e

α τ       −

=

7 ,$

I α

τ

=

τ β

− ( %

%

2 % >

8 β

8 %

; 8

%

( )

t

S t

e

β τ       −

= 7 7$

: β D 8 %

) )$$ & 7 7$ 4 ) 7$ " 7$

( )

I t t

r

e

α τ       −

=

7 @$

" %

< 4

8

>

( )

(

)

D− p =I p

α α

7 3$

I = $ <

α r

D # 7 rD 6 4 ) ,$ & 7 3$ ) )$

" 7$

( )

t

r t = I e −τ

α α

_

  7 *$

6

8

r " )$ #

" )$

(

)

( )

v PV r

v P

= @ $

PV P v

(

)

( )

v PV =v P r @ "$ 0 r

(

)

( ) ( )

v PV =v P r t @ )$

0 & 4

% @ )$

t

PVγ Pγ

e

α

τ

     

−

γ

I

t

PV P

e

α τ       −

= @ 7$

I 6

α

=

α γ

5 %

2 % = 3

7

" ' )

,

8

%

9 ' %

(

. $ >

> C

%

#

t

PV C e dt C

α τ

_τ

α

  ∞ _{− }    

= = Γ_ + _

 

∫

3 $

Γ : αD % 3 $ Cτ.

.

#

PV C dt t

τ

∞

= = ∞

+

∫

3 "$

J $ .

, 7$

#

PV C dt C t α

τ

α

τ

∞ = = −  ₊     

∫

3 )$

αH 5

.

, '$

#

PV C dt C t α

π τ

π

α

α

τ

∞ = =   +    

,

α H / , '$

B C : , "$

B C , '$

B C

K

α K %

%

< K

increasing

3

decreasing

% 0

B C $ L

, $ , 7$

$

2

9

( . ! "# #$ (

;

%

/

.

%

3 _{B C : ;}

:

0 & / - 8 > 2 ? M + "# #$ Discounting Behavior: A Reconsideration

( N ? "# #$ Survey of time preference, delay discounting models 8 J

& . 0 66 6 D @*7*@

+ N ( "##3$ The fragility of time: Time insensitivity and valuation of the near and far fu,

ture 2 & 7) ,") ,)*

+ : ( / J 0 "##'$ Uncertainty Breeds Decreasing Impatience: The Role of Risk

Preferences in Time Discounting.8 L , " 5 + ? +

. O 66 6 D , @##3

: & > - <4( "##"$ Time Discounting and Time Preference: A criti,

cal Review N + > ,# L " )7 ,#

/ P "##*$ Strotz Meets Allais: Diminishing Impatience and the Certainty Effect 0 +

? '* )$ ,7 @"

! ( 0 '3'$ Prospect theory: An analysis of decisions under risk.+

,3 ) ) )"3

! ? "##'$ An additive,utility model of delay discounting @ @#"

@ '

! J ! O - "##'$ Perception of Anticipatory Time in Temporal Discounting N

L + 1 " ' #

! ! "# #$ Decision Utility Theory: Back to von Neumann, Morgenstern, and Markowitz 8

66 6 D 3 *,",

> - ( ''"$ Anomalies in intertemporal choice: Evidence and interpretation

Q N + #3 "$ 73) 7'3

2 = N + '*3$ An adjusting procedure for studying delayed reinforcement 5 2 > N +

2 = N 0 L / ? + $ Quantitative analysis of behavior 1 7 2 LN +

2 N - > ''7$ Discounting of delayed rewards: Models of individual choice N

+% 0 J @, "@) "3@

( ''#$ On the Shape of the Decision Weight Function / J & 8

''#

( ''*$ The probability weighting function + @# ,'3 7"*

( "###$ Compound Invariant Weighting Functions in Prospect Theory 5 ( ! 0

+ $ Choices, Values, and Frames ? & : .

( > - '' $ Decision Making Over Time and Under Uncertainty: A Common Ap,

proach 2 & )3 3$ 33# 3*@

? / "##@$ Notes on discounting N +% 0 J *7 ,"7

,)7

? ( "## $ Is time discounting hyperbolic or subadditiveR N ? . ") $

7 )"

& ! "##*$ Hyperbolic Discounting: It’s Just Allais Paradox . <+ (

@

& ! "##'$ 0 ? J ? ( ,33 L

.

& ')3$ A note on Measurement of Utility ? + & , 77 @

& = ( ''*$ On hyperbolic discounting and uncertain hazard rates ? & >

J "@7 "# 7 "#"#

0 ! ( ''"$ Advances in Prospect Theory: Cumulative Representation of Uncer,

tainty N ? . 7 ,$ < "'3 )")

8 / "##*$ S,shaped probability weighting and hyperbolic discounting – an intimate relationship

8 1 . + J 0

O - ! ! J 2 & J N ? "##'$ Discounting Time and Time Discounting:

Subjective Time Perception and Intertemporal Preferences N 2 ? 1 S>15