3 釜田.pdf, page 16 @ Preflight...- 37 - ¤ G &A æ Ó ùüø ù÷øû åúD 家族における子の数および教育投資の選択と公的教育政策の効果＊釜田公

- 37 -

中京大学経済学論叢25号

2014年 3 月

家族における子の数および教育投資の選択と公的教育政策の効果＊

釜　田　公　良佐　藤　　　隆二　神　律　子

＊本研究は科学研究費補助金（21530319）を受けた。

- 38 -1

( )1990 1980 2.8 1992 3.9

1999 3.7 2011 4.418

1980 1.83 2010 1.70

2006

1

Barham et al., 1995; De Fraja, 2002 Cremer et al., 1992

Blankenau and Camera, 2009

Ihori, Kamada and Sato (2008

Ihori, Kamada and Sato

(2008

Buchanan, 1975

Ihori, Kamada and Sato (2008)

1 Bray and Kwok (2003), Tansel and Bircan (2006)

- 39 -

家族における子の数および教育投資の選択と公的教育政策の効果

2

1980

80

2000 5

2002

2

1

2

3

2

- 40 -3

3

Zhang (2003)

Zhang (1997)2 Borck (2011)

2 3

42

5

n

n pc k�

[(1 ) ]pc I n k T� ��

I � T Y g e

( , )Y y y k g e� � �

y

- 41 -


4

/ ( ) 0ky y k g� � � � � / 0ey y e� � � � 2 2/ [ ( )] 0kky y k g� � � � � 2 2/ ( ) 0eey y e� � � �

2 / ( ) 0key y k g e� � � � � �

kc Y�

( ) ( )p kp pU u c n nU��

( )n� ( ) 0, ( ) ( ) 0,n n n n� � ��

( ) 0, ( ) 0p p p pu c u c� �� kU

( ) ( )kk kU u c v e� �

( )v e� ( ) 0, ( ) 0, ( ) 0, ( ) 0k k k kv e v e u c u c� ��

2

( )T n g k��

n 1 k

0� �

k n e

- 42 -5

k

( ( ) ) ( ) ( ( , )) ( , ) 0p k ku I n k T n u y y k g e y k g e� �� (6)

(6) (7) ( , ; , )k k e n T g�

e n

Max ( , ; , ) Max [ ( ( ( , ; , ) , )) ( )]kke e

V e n T g u y y k e n T g g e v e� � � �

(8) ( )[ ( , ) ( , )] ( ) 0k k e eu Y y k g e k y k g e v e� �� ( , ; , )k k e n T g� (8)

(8) (9) ( ; , )e e n T g�

nMax ( ; , ) Max { ( ( ) ) ( )[ ( ( , )) ( )]}p

p kn nV n T g u I n k T n u y y k g e v e� ��

subject to ( ( ; , ), ; , ), ( ; , )k k e n T g n T g e e n T g� �

( ) ( ( ) ) [ ( ) ( )][ ( ) ( )] 0p kk u I n k T n n n u Y v e� � � �� ( ( ; , ), ; , ), ( ; , )k k e n T g n T g e e n T g� �

(6) (8) (10) n n�( , , ) k e n� � �

2

( , , ) F F Fk e n

( ( ) ) ( ) ( ( , )) ( , ) 0p k ku I n k T n u y y k g e y k g e� ��

(12) ( ) ( , ) ( ) 0k eu Y y k g e v e� �� ( ) ( ( ) ) [ ( ) ( )][ ( ) ( )] 0p kk u I n k T n n n u Y v e� � � ��

n n� ( , , ) k e n� � � ( , , ) F F Fk e n

(6) n n� k

ˆ( , ; )k k e n g�

- 43 -


6

ˆ 0ek � ˆ 0nk � ˆ 1gk � � 2

1

(8) (14) e (15) ˆ( ; )e e n g�

ˆ 0ne � ˆ 0ge � 3

e 1

(16) ˆ[ ( , )][ ( , ) ( , )] ( ) 0k k e eu y y k g e y k g e k y k g e v e��

[0, 1]� � (16) (10) (13) (14)

( ( ), ( ), ( ))e n k� � � 1� �0� �

(1) , (1) , (1)e e n n k k� � �� (0) , (0) , (0)F F Fe e n n k k� � �� 4

0, 0, 0dn de dkd d d� � �

� � �

1 Fn n� � Fe e� � Fk k� �

n

(10) 1 n n�n

n(10) 2

2 3 4

- 44 -7

0g �

k

(1 ) ( [(1 ) ) ] ( ) ( ( , )) ( , ) 0p k ku I n k T n u y y k e y k e� � � ��

(18) ( , ; , )k k e n T ��

e n

Max ( , ; , ) Max [ ( ( ( , ; , ), )) ( )]kke e

V e n T u y y k e n T e v e� ��

(19) ( )[ ( , ) ( , )] ( ) 0k k e eu Y y k e k y k e v e� �� ( , ; , )k k e n T �� (8)

(20) ( ; , )e e n T �� n

Max ( ; , ) Max { ( [(1 ) ] ) ( )[ ( ( , )) ( )]}pp kn n

V n T u I n k T n u y y k e v e� � � ��

subject to ( ( ; , ), ; , ), ( ; , )k k e n T n T e e n T� � ��

[(1 ) ] ( [(1 ) ] ) [ ( ) ( )][ ( ) ( )] 0p kk u I n k T n n n u Y v e� � � � � �� ( ( ; , ), ; , ), ( ; , )k k e n T n T e e n T� � ��

(17) (19) (21) n n� k k�( , , ) k e n� � �

2

( , , ) F F Fk e n

(1 ) ( [(1 ) ) ] ( ) ( ( , )) ( , ) 0p k ku I n k T n u y y k e y k e� � � ��

(23) ( ) ( , ) ( ) 0k eu Y y k e v e� �� [(1 ) ] ( [(1 ) ] ) [ ( ) ( )][ ( ) ( )] 0p kk u I n k T n n n u Y v e� � � � � ��

n n� k k� (17) n n� k k�

- 45 -


8

k ˆ( , ; )k k e n ��

ˆ 0ek � ˆ 0nk � ˆ 0k� � 5 (19) (25)

e (26) ˆ( ; )e e n ��

ˆ 0ne � ˆ 0e� � 6

( , , ) k e n� � � ( , , ) F F Fk e n

1 Fn n� �Fk k� � Fe e� �

4

(10) n n� n g ˆ 1gk � � ˆ 0ge �

7

0ppnn

udndg V

��

( )( ) [2 ( ) ( )][ ( )] 0pnn p kV k k g u n n n u v e� � � ��

(15) n g

ˆ 0nde dnedg dg

� � ˆ 0ne ��

5 6 7

- 46 -9

(14) e n g

ˆ ˆ ˆ 0e n gdk de dnk k kdg dg dg

� � � � ˆ ˆ ˆ0, 0, 1e n gk k k� � � ��

( ) ˆ ˆ 0e nd k g de dnk kdg dg dg�

� � �

2 / 0dn dg � / 0de dg � ( ) / 0d k g dg� �

1 2

3

(10) ,n n k k� � n �

pnpnn

Vdnd V

�

��

[ (1 ) ] [(1 ) ]( )p

n e p e p k kV k k e u n k k e k u n y k u� � � � ��

[(1 ) ]( ) [2 ( ) ( )][ ( )] 0pnn p kV k k u n n n u v e� � � � ��

pnV � 2 3 1

- 47 -


10

k � �

/dn d� (26) ,n �

ˆ ˆnde dne ed d ��

� �

ˆ ˆ0, 0ne e��

0 0

0 > or < 0

dn ded d

dn ded d

� �

� �

� � � ��

� �

�

(25) , , e n �

ˆ ˆ ˆ ˆˆ ˆ( )e e n ndk dnk e k k e kd d� ��

� � � �

ˆ ˆ0, 0ne e�� ˆ ˆ ˆ0, 0, 0e nk k k��

0 0

0 > or < 0

dn dkd d

dn dkd d

� �

� �

� � � ��

�

- 48 -

11

- 49 -


12

(1) gk

(6) n n� , , , k e n g

(A1) 0p p p pkk ke kn kgU dk U de U dn U dg� � � �

, , , p p p pkk ke kn kgU U U U

(A2)2

22 ( )[ ( ) ) 0

( )

ppkk p k k k kk

UU nu n u y u yk

��

(A3)2

( )( )p

pke k e k k ke

UU n u y y u yk e

��

(A4)2

( ) ( ) 0p

pkn p k k

UU k g u n u yk n

� ��

(A5)2

2( )[ ( ) ) 0p

p pkg p k k k kk kk

UU nu n u y u y Uk g

��

(A3) ( , )y k g e�

( )ku � 8 0pkeU �

(A1)-(A5) k , , e n g

ˆ 0pke

e pkk

UkU

� � �

8 1( , ) ( )y k g e k g e� �� ( ) lnk k ku C C�

2 1 1 2 12

11

2

( )[ ]

1 1 ( ) [ (1 )( ) ] [ (1 )( ) ]

( ) (1 )( ) ( )

pke k e k k ke

k k

kk

U n u y y u y

n k g e k g eC C

n k g e C k g eC

� � � �

� ��

�

� � � � �

� � �

� � � �

� ��

��

��

��

0y � 1( )kC k g e� �� 0pkeU �

- 50 -13

ˆ 0pkn

n pkk

UkU

� � �

ˆ 1pkg

g pkk

Uk

U� � � �

e(8) , , e n g

(A6) 0k k kee en egV de V dn V dg� � �

, , k k kee en egV V V

(A7)2

2 22

ˆ ˆ ˆ( ) [ ( ) 2 ] ( ) 0( )

kee k k e e k kk e ke e ee

VV u y k y u y k y k y v ee

� ��

(A8)2 ˆ ˆ

ˆ ˆ ˆ[ ( ) ( )] 0( )p n ek

en n k k k e e k kk e ek

nu k kVV k u y y k y u y k ye n n�

��

(A9)2

ˆ ˆ ˆ ˆ ˆ( 1)( ) [ ( 1) ( 1)] 0keg k k g k e e k kk g e ek g

VV u y k y k y u y k k y ke g

� �� ˆ 1gk � ��

(A8) 2( ) ( ) / ( ), / ( ), p pk k k kk kk p k k e k ek keu y u y U nu n u y y u y U n� ��

ˆ ˆ0, 0e nk k� �

2ˆˆ ˆ[ ( ) ) ( )] [( )( ) ]( )

ˆ ˆ0

( )

pk p pn keen n k k k kk e k k e k ek kk p kep

kk

p n e

k UV k u y u y k u y y u y U nu Un U

nu k kn

�

�

��

��

(A6)-(A9) e ,n g

ˆ 0ken

n kee

VeV�

� �

ˆ 0keg

g kee

Ve

V�

� �

- 51 -


14

(2) �

k

(17) ,n n k k� � , , , k e n �

(A10) 0p p p pkk ke kn kU dk U de U dn U d� ��

, , , p p p pkk ke kn kU U U U �

(A11) 2(1 ) ( )[ ( ) ) 0pkk p k k k kkU n u n u y u y� ��

(A12) ( )( )pke k e k k keU n u y y u y� ��

(A13) (1 )( ) ( ) 0pkn p k kU k u n u y� � ��

(A14) 0pk pU u� ��

(A12) ( , )y k g e� ( )ku �

0pkeU �

(A10)-(A14) k , , e n �

ˆ 0pke

e pkk

UkU

� � �

ˆ 0pkn

n pkk

UkU

� � �

ˆ 0pkpkk

Uk

U�

� � � �

e(19) (25) , , e n �

(A15) 0k k kee en eV de V dn V d� ��

, , k k kee en eV V V �

- 52 -15

(A16) 2 2ˆ ˆ( ) [ ( ) 2 ] ( ) 0kee k k e e k kk e ke e eeV u y k y u y k y k y v e��

(A17)2 ˆ ˆ(1 )ˆ ˆ ˆ[ ( ) ( )] 0

( )p n ek

en n k k k e e k kk e ek

n u k kV k u y y k y u y k y

n��

��

(A18)2 ˆ ˆ(1 )ˆ ˆ ˆ[ ( ) ( )] 0

( )p ek

e k k k e e k kk e ek

n u k kV k u y y k y u y k y

n�

� �

��

��

(A17) 2 2( ) ( (1 ) ) / ( ), / ( ), p pk k k kk kk p k k e k ek keu y u y U n u n u y y u y U n� � ��

ˆ ˆ0, 0e nk k� �

2

22

ˆ ˆ[( ( ) ) ( )]ˆ ˆˆ (1 )

[( (1 ) )( ) ] 0( ) ( )

ken n k k k kk e k k e k ek

pp n ep pn ke

kk p kepkk

V k u y u y k u y y u y

n u k kk UU n u Un U n

��

� �

��

��

(A18)(A15)-(A18) e ,n �

ˆ 0ken

n kee

VeV�

� �

ˆ 0kekee

Ve

V�

�

��

(3) / , / , /dn d de d dk d� � �(16) (10) (14) , , e n �

(A19)ˆ( ) ( )

( ) ( ) 0

k kee en k k ep pne nn

deV V u y k ddnV V ne

� ��

� ��

� � ��

2ˆ ˆ ˆ ˆ( ) ( ) [( ) ] ( )kee k k e e k kk e ke e ek e eeV u y k y u y k y k y k y v e� ��

ˆ ˆ ˆ( ) [ ( ) ( )]ken n k k k e e k kk e ekV k u y y k y u y k y� ��

(A20) ˆ ˆ ˆ( ) ( ) [ ( ) ( )][ ( ) ( )]pne e p e p k k e eV k u n k k u n n n u y k y v e� � � ��

- 53 -


16

ˆ ˆ( ) ( )( ) [2 ( ) ( )][ ( ) ( )]ˆ [ ( ) ( )]

pnn n p n p k

k k n

V k u k k g nk u n n n u Y v e

n n n u y k

� � � � �

� �

� ��

� ��

(16) ˆ( )k e k k eu y v e u y k �� (A20) (6)

ˆ 0ek �

� ��

ˆ( ) ( ) [ ( ) ( )][ (1 )]

ˆ [ ( ) ] ( ) ( ) [ ( ) ( )] 0

pne p p k k e

p k k p k k e

V u n k u n n n u y k

u n u y n k u n n u y n n n k

� � � � �

� � � � � �

� ��

� � ��

( )keeV � ( )p

nnV � 0, =1� ��

( ) 0keeV � � ( ) 0p

nnV � � 0, =1� ��

�

( ) ( ) ( ) ( ) ( ) 0k p k pee nn en neD V V V V� � � � ��

(A19) ( ) 0pnnV � � ( ) 0p

neV � � ( ) 0D � � ˆ 0ek �

(A21)ˆ

( ) 0( )

pk k enn

u y kde Vd D

��

��

(A22)ˆ

( ) 0( )

pk k ene

u y kdn Vd D

��

��

ˆ( ) ( ( ), ( ))k k e n� � ��

(A23) ˆ ˆ 0e ndk de dnk kd d d� � �

� � �

(4) g n

ˆ ˆ1, 0g gk e� � �

(A24)ˆ ˆ ˆ ˆˆ ˆ( ) ( )[ ( ) )]

ˆ ˆ ˆˆ ˆ ˆ [ ( ) ( )] [ ( 1) ] ( )

png e g g p e g g p

k k e g g e g g g p p

V k e k u n k n k e k n u

n n n u y k e k y e v e e k u u

�

� �

� ��

� � � � ��

- 54 -17

0png pp pnn nn

V udndg V V

��

- 55 -


18

Barham, V., Boadway, R., Marchand, M., 1995. Education and the poverty trap. European Economic Review 39, 1257-1275. Becker, G., 1974. A theory of social interactions. Journal of Political Economy 82, 1063-1093. Blankenau, W., Camera, G., 2009. Public spending on education and the incentives for student achievement. Economica 76, 505-527. Borck, R., 2011. Federalism, fertility, and growth. Scandinavian Journal of Economics 113, 30-54. Buchanan, J., 1975. The Samaritan’s dilemma. In: Phelps E. (Ed.), Altruism, Morality and Economic Theory. Russell Sage Foundation, New York, pp. 71-85. Bray, M., Kwok, P., 2003. Demand for private supplementary tutoring: conceptual considerations, and socio-economic patterns in Hong Kong. Economics of Education Review 22, 611-620. Cremer, H., Kessler, D., Pestieau, P., 1992. Intergenerational transfers within the family. European Economic Review 36, 1-16. De Fraja, G., 2002. The design of optimal education polices. Review of Economic Studies 69, 437-466. Ihori, T., Kamada, K., Sato, T., 2008. Altruism, liquidity constraint, and educational investment. Discussion Paper 0803, Institute of Economics, Chukyo University. Tansel, A., Bircan, F., 2006. Demand for education in Turkey: a tobit analysis of private tutoring expenditures. Economics of Education Review 25, 303-313. Zhang, J., 1997. Fertility, growth, and public investments in children. Canadian Journal of Economics 30, 835-843.

- 56 -19

Zhang, J., 2003. Optimal debt, endogenous fertility, and human capital externalities in a model with altruistic bequests. Journal of Public Economics 87, 1825-1835.

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3 釜田.pdf, page 16 @ Preflight...- 37 - ¤ G &A æ Ó ùüø ù÷øû åúD 家族における子の数および教育投資の選択と 公的教育政策の効果＊ 釜 田 公

3 釜田.pdf, page 16 @ Preflight...- 37 - ¤ G &A æ Ó ùüø ù÷øû åúD 家族における子の数および教育投資の選択と公的教育政策の効果＊釜田公