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) 6= 0, â® á ०¨¬  II ¯à®¨á室¨â ¯¥à¥å®¤ ­  ०¨¬ IV, ¤ «ì-­¥©è¥¥ à §¢¨â¨¥ ¡ë«® à áᬮâ७® ¢ëè¥. …᫨ ®¡ê¥¬ ¯ áᨢ­ëå à¥áãàᮢ ­¥

¬¥­ï-¥âáï (

T

0

p

(

t

) = 0) ¨ ¨­â¥­á¨¢­®áâì ¢¢®¤   ªâ¨¢­ëå âà㤮¢ëå à¥áãàᮢ (

T

0

a

(

t

)) १ª® ¢®§à áâ ¥â, â® á¨á⥬  ¯¥à¥å®¤¨â ­  ०¨¬ I. …᫨ ®¡ê¥¬ ¯ áᨢ­ëå à¥áãàᮢ

¬¥­ï-¥âáï (

T

0

p

(

t

) 6= 0), â® ¯à®¨á室¨â ¯¥à¥å®¤ ­  ०¨¬ (â ª®© ¢ à¨ ­â 㦥 ¡ë« à áᬮâ-७). …᫨

T

0

p

(

t

) = 0 ¨

T

0

a

(

t

) 6= 0, â® ¤«ï ä㭪樨

'

max

(

t

) á¯à ¢¥¤«¨¢® ­¥à ¢¥­á⢮

'

max

(

t

)

'

(

m

(

t

))

6

T

0 a

(

t

)

'

(

m

(

t

))

. ˆá¯®«ì§ãïãà ¢­¥­¨¥ (3.10), ¯®«ãç ¥¬

'

max

(

t

)

'

(

m

(

t

)) =

1

=

F

(

t

)

F

(

m

(

t

)) 6

T

0

a

(

t

)

'

(

m

(

t

))

:

(3

:

12)

’ ª ª ª ä®­¤ë ­¥ ¢ë¢®¤ïâáï

m

0

(

t

) =0, â®

m

(

t

) =const =

m

¨

F

(

m

(

t

)) =

F

(

m

),

'

(

m

(

t

))=

'

(

m

)¥áâ좥«¨ç¨­ë¯®áâ®ï­­ë¥. ˆ§(3.12)¯®«ãç ¥¬á«¥¤ãî饥­¥à ¢¥­á⢮

F

(

t

)

T

0

a

(

t

) =

F

(

t

)

'

(

t

)

6

1

=

F

(

m

)

'

(

m

)

=const

:

(3

:

13)

’ ª¨¬ ®¡à §®¬, ¤® â¥å ¯®à, ¯®ª  ª ¯¨â «®¢®®à㦥­­®áâì âà㤮¢ëå à¥áãàᮢ ­¥

¯à¥-¢®á室¨â ­¥ª®â®à®©¯®áâ®ï­­®©¢¥«¨ç¨­ë,á¨á⥬  ¡ã¤¥âà §¢¨¢ âìáï ¢à¥¦¨¬¥ I.

Žâ¤¥«ì­®à áᬮâਬ á«ãç ©,ª®£¤ 

'

(

m

(

t

))=0. ’®£¤  ¯¥à¥¬¥­­ë¥¬®¤¥«¨ ®¯à¥-¤¥«ïîâáïãà ¢­¥­¨ï¬¨:

m

0

0 (

t

)=

>

>

>

>

>

<

>

>

>

>

>

: a

T 0

a (t)+T

0

p (t)

;

¥á«¨

(

t

)=

t

¨

T

a

(

t

)+

T

p

(

t

)6=0;

1

;

¥á«¨

(

t

)=

t

¨

T

0

a (

t

)+

T

0

p

(

t

)=0; T

0

a (t)

'((t))

;

¥á«¨

(

t

)

< t

¨

'

(

(

t

))6=0;

1

;

¥á«¨

(

t

)

< t

¨

'

(

(

t

))=0;

'

(

t

)=

T

0

a (

t

)+

T

0

p (

t

)

:

ɇǬ

T

0

a

(

t

) =0¨

T

0

p

(

t

) =0,â®

'

(

t

) =0¨ á¨á⥬  ¯®¯ ¤ ¥â ¢ ०¨¬ ý®¦¨¤ ­¨ïþ, â.¥. ¢â ª®¬á®áâ®ï­¨¨(ä®­¤ë­¥ ¢ë¢®¤ïâá鶴¥â¤¢¨¦¥­¨ï âà㤮¢ëåà¥áãàᮢ)

á¨á-⥬ ¡ã¤¥â­ å®¤¨âá冷â¥å¯®à,¯®ª ­¥ ¨§¬¥­ïâá¥è­¨¥å à ªâ¥à¨á⨪¨á¨á⥬ë

T

0 a

(

t

),

T

0

p

(

t

) (¨­â¥­á¨¢­®áâ좢®¤   ªâ¨¢­ëå¨/¨«¨¯ áᨢ­ëåâà㤮¢ëåà¥áãàᮢ).  áᬮâਬ ®¤¨­ ç áâ­ë© á«ãç ©. ‘¤¥« ¥¬ á«¥¤ãî饥 ¯à¥¤¯®«®¦¥­¨¥: ®¡ê¥¬

âà㤮¢ëå à¥áãàᮢ ª ª  ªâ¨¢­ëå, â ª ¨ ¯ áᨢ­ëå ­¥ ¬¥­ï¥âáï, â. ¥.

T

a

(

t

) = const,

T

p

(

t

) = const, ¨ ¢¢®¤¨¬ë¥ ä®­¤ë ¨¬¥îâ ¬£­®¢¥­­ãî ®â¤ çã, â. ¥.

(

t

) =

t

. ‚ í⮬ á«ãç ¥á¨á⥬  à §¢¨¢ ¥âáï ¯® ®¤­®¬ã ¨§ ०¨¬®¢I, II¨«¨III.

 ¨¡®«¥¥ ¨­â¥à¥á­ë¬ï¢«ï¥âáï०¨¬ II. ‚[10]¯®ª § ­®, ç⮥᫨ äã­ªæ¨ï

F

(

t

) ¨¬¥¥â íªá¯®­¥­æ¨ «ì­ë© ¢¨¤, â. ¥.

F

(

t

) =

f

1=

(

t

)

(

t

) =

Ce

t

, â® äã­ªæ¨ï

m

(

t

) ¯à¨

t

!+1¢ë室¨â­ à¥¦¨¬á¯®áâ®ï­­ë¬§ ¯ §¤ë¢ ­¨¥¬

m

(

t

)

t

,

A

, äã­ªæ¨ï

'

(

t

) á® ¢à¥¬¥­¥¬¢ë室¨â­  ¯¥à¨®¤¨ç¥áª¨© ०¨¬.

 à¨áã­ª å2,3¯à¥¤áâ ¢«¥­ ç¨á«¥­­ ïॠ«¨§ æ¨ï ¤ ­­®£®á«ãç ï.

m t

( )

α

( )

t

L

t

Ðèñ. 2. Ãðàôèêè ôóíêöèé ( ),

α

t m t .

( )

Ðèñ. 3. Ãðàôèê ôóíêöèè

ϕ

( ).

t

ϕ

( )

t

t

t

‚ ⮬ á«ãç ¥, ª®£¤  äã­ªæ¨ï

F

(

t

) ¨¬¥¥â á⥯¥­­®© à®áâ, äã­ªæ¨ï

m

(

t

) ¡ã¤¥â ¯à¨¡«¨¦ âìáïª «¨­¥©­®©

m

(

t

)

at

+

b

,

a <

1 (à¨á.4, 5).

m t

( )

Ðèñ. 4. Ãðàôèêè ôóíêöèé

α

( ),

t m t

( ).

α

( )

t

L

t

t

Ðèñ. 5. Ãðàôèê ôóíêöèè

ϕ

( ).

t

ϕ(

t

)

à¨ í⮬, ¨­â¥à¢ «ë [t

k ;t

k+1

], ­  ª®â®àëå ¯à®¤®«¦ ¥âáï äã­ªæ¨ï '(t) á®

¢à¥-¬¥­¥¬ã¢¥«¨ç¨¢ îâáï,®¤­ ª®®¡ê¥¬ ¢¢®¤¨¬ëå­ íâ¨å ¨­â¥à¢ « å¢à¥¬¥­¨âà㤮¢ëå

à¥áãàᮢ,ª ª¯®ª § «¨à áç¥âë,­¥¬¥­ï¥âáï,â.¥. ¨­â¥£à «®âä㭪樨'(t)­ «î¡®¬

®â१ª¥[t

k ;t

k+1

]¥áâ좥«¨ç¨­  ¯®áâ®ï­­ ï.

‹¨â¥à âãà 

1.

à ã­Œ.

’¥®à¨ï ¨ ¨§¬¥à¥­¨¥ â¥å­¨ç¥áª®£® ¯à®£à¥áá .|Œ.: ‘â â¨á⨪ , 1971.

2.

‹®â®¢€.‚.

‚¢¥¤¥­¨¥ ¢ íª®­®¬¨ª®-¬ â¥¬ â¨ç¥áª®¥ ¬®¤¥«¨à®¢ ­¨¥.|Œ.: 1984.

3. Œ â¥¬ â¨ç¥áª®¥ ¬®¤¥«¨à®¢ ­¨¥ íª®­®¬¨ç¥áª¨å ¯à®æ¥áᮢ / ®¤ ।. ˆ. ‚. Š®â®¢ .|‹.: ˆ§¤-¢®

‹ƒ“, 1980.

4.

¥áªà®¢­ë©ˆ.Œ.

à®¡«¥¬ë ¨á¯®«ì§®¢ ­¨ï ¬ ªà®íª®­®¬¨ç¥áª¨å ¬®¤¥«¥© ¤«ï § ¤ ç

¯« ­¨à®-¢ ­¨ï ¢ ãá«®¢¨ïå ¨­â¥­á¨¢­®£® à §¢¨â¨ï // ‚®¯à. à ¤¨®-í«¥ªâà®­¨ª¨. ‘¥à. €‘“.|1984.|

‚ë¯. 2. ‘. 3{12.

5.

ƒ«ã誮¢‚.Œ.,ˆ¢ ­®¢‚.‚.,Ÿ­¥­ª®‚.Œ.

Œ®¤¥«¨à®¢ ­¨¥ à §¢¨¢ îé¨åáï á¨á⥬.|Œ.:  ãª ,

1983.

6.

Sollow R.

Investment and Technical Progress // Mathematical Methods in the Social Science.|

Stanford Univ. Press, 1960.|P. 89{104.

7.

Š ­â®à®¢¨ç‹.‚.,ƒ®à쪮¢‹.ˆ.

”㭪樮­ «ì­ë¥ ãà ¢­¥­¨ï ®¤­®¯à®¤ãªâ®¢®© ¬®¤¥«¨ // „®ª«.

€ ‘‘‘.|1959.|’. 129, ü 4.|‘. 732{736.

8.

ƒ«ã誮¢‚.Œ.

Ž¡ ®¤­®¬ ª« áᥠ¤¨­ ¬¨ç¥áª¨å ¬ ªà®íª®­®¬¨ç¥áª¨å ¬®¤¥«¥© // “¯à ¢«ïî騥

¬ è¨­ë ¨ á¨á⥬ë.|1977.|ü 2.|‘. 3{6.

9.

Š ­â®à®¢¨ç‹.‚.,†¨ï­®¢‚.ˆ.

Ž¤­®¯à®¤ãªâ®¢ ï ¤¨­ ¬¨ç¥áª ï ¬®¤¥«ì íª®­®¬¨ª¨,

ãç¨â뢠î-é ï áâàãªâãàã ä®­¤®¢ ¯à¨ ­ «¨ç¨¨ â¥å­¨ç¥áª®£® ¯à®£à¥áá  // „®ª«. € ‘‘‘.|1973.|’. 211,

ü 6,|‘. 1280{1283.

10.

Š ­â®à®¢¨ç‹.‚.,†¨ï­®¢‚.ˆ.,•®¢ ­áª¨©€.ƒ.

à¨­æ¨¯ ¤¨ää¥à¥­æ¨ «ì­®© ®¯â¨¬¨§ æ¨¨ ¢

¯à¨¬¥­¥­¨¨ ª ®¤­®¯à®¤ãªâ®¢®© ¤¨­ ¬¨ç¥áª®© ¬®¤¥«¨ íª®­®¬¨ª¨ // ‘¨¡. ¬ â. ¦ãà­.|1978.|

’. 19, ü 5.|‘. 1053{1064.

11.

¥ª« àוֹ.€.,®à¨á®¢ ‘.‚.

Ž¡ ®¤­®© ¤¨­ ¬¨ç¥áª®© ¬®¤¥«¨ § ¬¥é¥­¨ï ¯à®¨§¢®¤á⢥­­ëå

¬®é­®á⥩ // ª®­®¬¨ª  ¨ ¬ â. ¬¥â®¤ë.|2002.|’. 38, ü 3.

12.

¥ª« àוֹ.€,®à¨á®¢ ‘.‚.

Ž¤­®¯à®¤ãªâ®¢ ï ¬®¤¥«ì ¯à®¨§¢®¤á⢠ á ãç¥â®¬ ¨­¥à樮­­ëå

᢮©á⢠¢¢®¤¨¬ëå ¨ ¢ë¢®¤¨¬ëå ä®­¤®¢ // à¥¯à¨­â # WP/2000/093|Œ.: –Œˆ €, 2000.|

56 á.

13.

¥ª« àוֹ.€,®à¨á®¢ ‘.‚.

Ž¤­®¯à®¤ãªâ®¢ ï ¤¨­ ¬¨ç¥áª ï ¬®¤¥«ì ¯à®¨§¢®¤á⢠ á

¨­¥àæ¨-®­­ë¬¨ ᢮©á⢠¬¨ á¨á⥬ë // à¥¯à¨­â # WP/98/045.|Œ.: –Œˆ €, 1998.|22 á.

14.

®à¨á®¢ ‘. ‚.

Œ®¤¥«ì ¯à®¨§¢®¤á⢠ á ãç¥â®¬ ¨­¥à樮­­ëå ᢮©á⢠¢¢®¤¨¬ëå ¨ ¢ë¢®¤¨¬ëå

ä®­¤®¢ // €ã¤¨â ¨ 䨭 ­á®¢ë©  ­ «¨§.|1999.|ü 2.|‘. 5{18.