1

2

1 1 1 1

1 1

10 1 11 1

1 10 11

1

Aurelius Augustinus

Isaac Newton

9

2

19 20

Henri-Louis Bergson

18 19

Nicolas Léonard Sadi Carnot

William Thomson

Rudolf Clausius

3

1927

Werner Karl Heisenberg

(John von

Neumann)

( 1999 )

200

4

19 (John

Rae)

Eugen von Böhm-Bawerk William S.

Jevons

20

Irving Fisher 1929

(discount rate)

1000

1000 1000

1000

1000 20%

1200

1000 1200

500 500 100 600

5

1

2 3 3

4 4 2

2

(C0) (C1)

0C0C1 I0

C0 I0

C1

C0C1 1

(C0,C1)

U0(C0,C1)

(C0*,C1

*)

1+r r

r 1

r

< 1>

2

6

100 1

120 100 =120 /(1+r) r

=0.2 20%

100 10

20 100 =20 /(1+r)+20

/(1+r)2+ +20 /(1+r)10 r=0.15 15%

20% 102% 138%

7

(Jerry Hausman)

8.9% 39%

11%

6%

15%

8

1989

4

0.5 1 2 4 4 $40 $200 $1000 $5000

4

4 3 64

1 $1000

15%

15%

1

1 2 4

10% 0.5 20%

2

$1000 $5000

$40 $200 20%

3

$1000 1 40%

10%

9

20%

3

100 1 120

20%

100 =120 /(1+r)

100 1

110 1 110

1 120 1

1

C U(C) dU(C)/dC

d2U(C)/d2C<0

20

Paul A. Samuelson C0 C1

(discounted utility)

10

DU(C0, C1) U(C0)

U(C1)/(1+ )

DU(C0, C1)=U(C0)+U(C1)/(1+ )

U(C0) r

(time preference rate)

100

100 log

log(100)=2 log(120)=2.08 100 1 120

=2.08/2-1=0.04 log

4%

20%

11

100 110 1

100 1 1 110

4

4

1

1+r r

r C1*>C0

*

r

(r>0)

12

( >0)

1 0

(C1>C0)

(C1<C0)

BOX1

<BOX1>

(r= )

(C1*=C0

*) 2

0 45 45

r

45

< 2>

log

r g

10% 5% 5%

13

5% 10% 5%

(Alan R.

Rogers)

(ln )/ 0.5 30

5% 7%

( )

( )

5

14

1970

(Robert H. Strotz) 1955

( )

(Homeros)

(precomittoment)

15

(consistent planning)

6

1 10 5 20 5

6 10 10 20

1 10 5 20 10 20

6 10

BOX2

<BOX2>

10 3

20 10 4 20

1 10 6 20 5 20

1 10

DU( ,10 ) = DU(3 ,20 ) > DU(4 ,20 )

16

DU(1 ,10 ) = DU(6 ,20 ) < DU(5 ,20 )

DU( ,10 )>DU(4 ,20 )

1 DU(1 ,10 )<DU(5 ,20 )

(immediacy effect)

( )

7

17

(hyperbolic discounting)

(George Ainslie)

DU(C0, C1) U(C0)

U(C1)/(1+ t) DU(C0, C1)=U(C0)+U(C1)/(1+ t)

(1+ t) (1+t) (1+ t) /

0 100 1

10 3( =0.2 =0.3)

< 3>

0 100

1 4

5 40 6 10

4

100 3

18

500 10

< 4>

( =1) 3

log

4 8 16 2 3

4 n n

BOX3

<BOX3>

8

19

1 1 2 1

1 1 5

100% 50%

1 1 2

10 1 12

100%

20%

1 6

16 6 11

12 6 16

2 6 2 20

4 6 9

8 6 4

1 2

20

(reference point)

R C-R DU(C-R)

0

1

1 10

9

1 1 1

George Loewenstein

1

86% 14%

2 1

1

21

2

80% 20%

3 1

1 2

1 2

43% 57%

1

2

3

U( )/(1+ /12)+U( )/(1+ /12)2

< U( )/(1+ /12)+U( )/(1+ /12)2

[U( ) U( )]/(1+ /12)

< [U( ) U( )]/(1+ /12)2

<0 3

U( ) U(

)

U( )

22

10

U1(C0,C1)

2U1(C0 ,C1) / C0 C1 >0

d 1 (C0)/dC0<0

23

3

3

1

(Richard Thaler) ( )

24

19

Robert L.B. Stevenson

(f-MRI)

25

1

0

(I0)

1+r

(C1)

(C0)

(U0)

C0*

C1*

C0

C1

26

2

0

(I0)

1+r

(C1)

(C0)

(U0)

C0*

C1*

C0

C1 45

27

3

28

4

29

BOX1

DU(C0, C1)

U(C0) U(C1)/(1+ ) DU(C0,

C1)=U(C0)+U(C1)/(1+ )

(1+ ) [dU(C0)/dC0]/[dU(C1)/dC1]

[dU(C0)/dC0]/[dU(C1)/dC1] r

C0* C1

*

(1) ( >0)

(2) (C1*> C0

*)

30

BOX2

t x (t,x) t<s x<y c>0

U(x)/(1+ )t = U(y)/(1+ )s U(x)/(1+ )t+c = U(y)/(1+ )s+c

U(x)/(1+ t )t = U(y)/(1+ s )s U(x)/(1+ t +c )t+c < U(y)/(1+ s+c )s+c

t s t +c s+c

t - t +c > s - s+c (c)

(t<s)

U(x)/(1+ t )t > U(y)/(1+ s )s U(x)/(1+ t +c )t+c < U(y)/(1+ s+c )s+c

( )

31

BOX3

t T

T=log(1+t)

exp

exp[- T] U(X) = exp[- log(1+t)]U(X)

= exp[log(1+t)- ]U(X) = U(X)/(1+t)