Apuntes de Macroeconomía Matemática

��

�� . �� 2010

A�� M�� M�� J�� A�� D.

�� C�� ............................................. ............................................. ..................... 2 1. A�� C�� M��.......................................................................................3 1.1. F�� H��..............................................................................................3 1.2. H�� M�� ....................................................................4 1.3. L�� E�� D��..................................................................8 1.4. F�� H�� A�� ................................................18 2. M�� C�� .................................................. .................................. 20 2.1. D�� M�� ................................................ ............................................. . 20 2.2. E�� E�� ............................................................................24 2.3. C�� B�� K��...........................................................................25 3. M�� C�� .................................................................................27 3.1. D�� M�� ................................................ ............................................. . 27 3.2. �� E�� E�� ...................................................... ................ 30 3.3. E�� C�� D��................................................................33 3.4. I�� E��.......................................................................37 4. O�� C��............................................................................................38 4.1. E�� ..................................... .......................................... ............. 38 4.2. E�� A��......................................................................................................40 4.3. I�� ...................................................................................................................43 5. M�� G�� ............................................................................45 5.1. C�� A�� ............................................. .......................................... ............. 45 5.2. C�� .................................................................................................49 6. A�� I��.................................................................................................53 6.1. �� A�� ..................................... ............................................. ................... 53 6.2. M�� A��..............................................................................56 6.3. M�� .............................................................................................59 6.4. �� C�� ...................................................... .................................. 65

2

A�� M�� M�� J�� A�� D.

1. �� 1.1. �� r �� :



n

 

x  f x ,

�� x



 x 1 , . . . , x n T .

� �� t  0 : t x



t r f x .

E� �� , �� r  1 �� . E� �� , �� r  1 �� 1. ��1. A�� , �� r �� . �� 1: ��  �� 1 �� n

x



  f  xx x i i1

i

.

L�� . E� ��, �� 1. ��: �� maxK , L U K , L



 wL  pF K , L   rK 

s. a. K  0 L  0 .

3

A�� M�� M�� J�� A�� D.

 �� , �� , �� F  w , r , � �� , �� (��) ��)

F K

F  L

r p ,





w p .

E�� K � L �� .

 �� HG1, �� 1 �� A�� , �� F  F K , L 

F K  F L K  L



F K , L



r w p K  p L .

 �� , �� E�� F  �� K � �� L .

1.2. �� L�� . E� ��, �� . �� , �� , �� T

maxy t   F x t , y t dt , 0

x t  y t 

m



x i,t , i



n



y j, t , j

 1,2,... 1,2,... , n,





x



1,2,... m , 0  t  T

0

 t  T

s. a. g i x t , y t  .

A��, �� x i �� , � �� y j ��  �� y t  . �� 2: �� 

�� y t  ��:  (�) y t  ��

4

A�� M�� M�� J�� A�� D.

m

H t = F (x t , y t ) + ∑ λ i , t g i x t , y t  . i =1

(�) ��

 i,t



  H t t  x i,t .

L� �� . L� �� , �� . A�� , �� (�) �� , x 0   . (�) C�� . ��: C�� , �� : 

maxc t   e t U c t dt 0

s. a. k t   c t  k  t .

A�� (�) � (�) (�) H t t



e t U c t    t  f k t   c t  ,

 t



  H t t k t

(�)



 t f  k t 

.

N��  t �� k t � U c t  , �� ,  t �� .

��  , �� y t  � �� 2, �� J  ��. A�� J x 0 



 0T F x t , y t  dt .

5

A�� M�� M�� J�� A�� D. L� �� , �� J x 0 



 0T

1

F x t , y t  dt  J x T 1 , 0

 T 1  T .

E�� , �� . �� , �� . �� , �� T

max y t  F  x t , y t h, t 0

h : tamaño del intervalo x t h



s. a. g x t , y t h  x t .

L� �� J  x t 



max y t F  x t , y t h  J  x t h  

F  x t , y t h  J g x t , y t h  x t   J  x t  h



0



max y t



0



max y t F  x t , y t  

J g x t , y t h  x t   J  x t  h

.

D�� J g x t , y t h  x t   J  x t  h 

limg  x t , y y t h  0 



J g x t , y t h  x t   J  x t  g x t , y t  g x t , y t h

J g x t , y t h  x t   J  x t  g x t , y t  g x t , y t h

0





J   x t g x t , y t 

max y t F  x t , y t   J   x t g x t , y t   .

�� (�� (�)) ��  t C�� , �� y t 



y   x t  , 6



J   x t  .

A�� M�� M�� J�� A�� D.

�� 0  F  x t , y   x t    J   x t g x t , y   x t   



F x x

F  x t



 F y y y 

 J  g  g J   x t   x t  x t



0

  J  g   g  g y   J   x y

F x x



 J

g  g x J 



0.

�� J  g

    J  x t   J  x t t t



F x x

   g x J    J

   J

   J

  J

0

F x x  g x J  

  F   J  g   x t .

�� (�) �� . � ��. ��: �� T

max  1  tx t  U t 2 dt 0

s. a. x t  U t t x 0   x t libre

 T



0.

E� �� H t t



1  tx t x t  U t 2

� ��

7

  t U t t

0

A�� M�� M�� J�� A�� D.

 H t t U t t



2U t t   t  0

U t t



 t 2 .

  H t t  x t



t 



maxU t t H t t



A�� , �� (�)  t





 t

t 2 2





t

 C

.

� ��  T

T 2 2



 C 

 t



0 2

C 

2

 T

2

2

 T

t





2

.

�� U t t �� U t 

t 2



4

2

t

  x t  

x t 



 T 2 4

3

t 3

1 4

 T 2

 tT 2

 C

.

� �� x 0 

x t



C  



t 3  tT 2 12 4

 

.

1.3. �� x



f t , x 

y



gt , y

x t 0 



x0

yt 0 



y0

8

,

A�� M�� M�� J�� A�� D. ��  � g �� . �� f t , x i 



gt , x i 



x 0



y 0

x i

x 0



y i

df

df

dx

dy

x  x i

dg

dg

dx

dy

y  y i

x  x i

 A

y 0

y  y i

.

I�� A . �� , �� A  P  P1 . �� , �� detA Tr A





det  

Tr   

  1  2

  1   2 ,

��  1 �  2 �� A , � ��  . A�� : � �� A ��

 

1

1

0

2

1

0

0

2

.

� �� A ��

 

�� . � �� J��

9

A�� M�� M�� J�� A�� D.

 

J

 1



0 

.

A�� , �� A �� , ��  1 �  2 . �� , ��

det

1

0

0

2

  I



0



det

 1    2   

 1  

0

0

 2  



 1  2   1    2    2



0



2



0





   1   2    1  2

2

  Tr A  detA

Tr A 



0











0

Tr A  2  4detA

.

2

E�� , �� 1



2



Tr A 2  4 det detA



0



1 Tr A  2  4

detA .

�� 1. �� .

10

0

A�� M�� M�� J�� A�� D. �� , �� x i

x 0



y i



x i



y i

P

y 0

1

0

0

2

ż 



w

P 1

P1

x  x i

 A

y  y i

x  x i



y  y i

x i



P

1

0

z

0

2

w

1

0

z

0

2

w

y i

.

N�� ż   1 z  zt   z 0 e  1 t

w



2 w



wt 



w 0 e  2 t ,

�� z � w �� x � y ��. L��, �� z �� w dz dw



 t ż   1 z 0 e 1   1 z 0 exp expt  1  t

w

2 w0 e

2 w 0

2

  2  

.

�� : � E� ��  1 �  2 . � L� ��  1 �  2 . E� ��, ��  1 ,  2 

11



A�� M�� M�� J�� A�� D.

�� 2. �� .

� ��  1 ,  2 

 ��

��

�� 3. �� .

��: ��

A



3 4 0 2

��

12

A�� M�� M�� J�� A�� D.

x



y

x

A

y

.

C�� A det A   I 



0

3   2   









0

3



2

.

E�� 3 4



0 2

3 0

P

0 2

P1

.

� ��

P

a c





v1 v2

b d

,

�� v 1 � v 2 �� A 3 4 0 2



3a  4 b 2b





v1

3a



3v 1



v1

3b

3 4 0 2

v2

13



2v 2



1 0

.

A�� M�� M�� J�� A�� D. 3c  4d  2c





v2

2d  2d

4



1

.

��

P



1

4

0

1

P 1



1 4 0 1

�

.

E� �� x



y



1

4

3 0

1 4

x

0

1

0 2

0 1

y

3 0

1 4

x

0 2

0 1

y

3 0

z

0 2

w

1 4

x

0 1

y



ż 



w

,

� 1

4

z

0

1

w



x y

.

�� z

z 0 e 3 t



w 0 e 2t

w



x y



z 0 e 3t

1 0

 w 0 e 2 t

4 1

.

N�� , �� x � y ��

14

A�� M�� M�� J�� A�� D.

y

x

�� 4. �� .

E� �� x � y �� z � w . E� �� , �� , �� x �� .

��: ��

A

8



3

3 2

.

N�� A ��  �� J��

J



5 1

8



3

3 2

0 5



P

5 1 0 5



P1

D�� , �� P



v1 v2

�� . A��

15

5 . ��

.

A�� M�� M�� J�� A�� D. 8

3

3 2 3

a c

3 2

b d

8



8a  3 b 





5 1

v1 v2



a  b

5a

8c  3 d  a  5 c

3a  2b



v1 v2



0 5

a c

5 1

b d

0 5

0

v1

3c  3 d  a



a  b

5b

3c  2d  b  5d





v2

1 3

1 0

�

P1



0

1

3

3

.

E� �� x



y



1 3

5 1

0

1

x

1 0

0 5

3

3

y

5 1

0

1

x

0 5

3

3

y

5 1

z

0 5

w

1

0

1

x

3

3

y



ż 



w

�

16

1

0

��

P

1



3c  3d  b

1



,



1 3

0

.

A�� M�� M�� J�� A�� D.

1 3

z

1 0

w

1

x



y

.

�� : ��: ż  5 z  w

w



ż  5 z  w 0 e 5t



5w



w

w 0 e 5 t

.

�� z



 t  t

   t   t   ż   ;

��   t   t 



5 z  w 0 e 5 t

  t   5 z   t    

w 0 e 5 t

  t   5   5  t  t   t  

w 0 e 5 t

t   5 t    t 

w 0 e 5 t .





� �� (�)   5 t   t  e 5 t 5 t 5 t (��) e   w 0 e    w 0 t 

z



w 0 te 5 t .

E� �� z



w 0 e 5 t

w



x y

w 0 te 5 t



1 3

w 0 te 5 t

1 0

w 0 e 5t

1

17

A�� M�� M�� J�� A�� D.



x



y

w 0 te

1

5 t

1

 w 0 e

5 t

1 3

0

E� �� , �� y �� :

. 

0 � x



w 0 ��

y

x

�� 5. �� .

1.4. �� E� �� . �� , � �� , �� . E� �� , ��

 X i P,  i   X i

P,   i i

,

�� X i �� i , X �� , P ��  i �� .

18

A�� M�� M�� J�� A�� D. �� 3: �� , �� Ū i P,  i   a i P  b P i .

E� �� G�� , �� , �� ; � �� . E� �� , ��  �� : (�) x 1 , x 2  Dom f , x 1



x2

f  x 1   f  x 2  .



(�) �� g : HG1. ��: �� w i , �� C�� D�� N �� N

U i  X 1 , . . . , X N N 



  X n,i  

n

n 1



X n,i



wi Pn

n N n 1



n

; n



0,

n .

C�� N �� . A�� N

Ū i P, w i  

 n 1

N

A��  n 1  n



n

n

wi Pn

 N n 1  n 

n

N



n

 n 1

 N n 1  n

n

N

wi Pn



n

.

1 ��  nn

Ū i P, w i   w i

N

  n 

n

n 1

1 Pn



wi  n 1

n Pn

n

.

N�� G��  i  w i , a i P  0 , � bP   n  n / P n  n . E� ��, �� .

19

A�� M�� M�� J�� A�� D.

2. �� 2.1. �� E� �� N�� E�� 1987. �� HG1 �� Y  F K , AL , �� A �� L . �� A � L �� A A



 g, L L



n

.

�� , �� s �� , ��  �� K   I sF K , AL  K  K .

�� k  K / AL �� , ��  sF K / AL, 1   k  K AL



 sf k    k  K AL .

�� k ��   k



k t



 AL  KA L  KAL K  AL 2

 K t AL 

 K AL

 KL  KA  2 2

A L

   k

 K  k n  g  AL

 K AL

  k n  g   k



� �� 20

.

AL

A�� M�� M�� J�� A�� D.

sf k    k  k  k n  g  

k  sf k   k n  g    .

E�� , �� k  ��   0 k



sf k  



k  n  g    .

�� 6. �� .

N�� . �� ; �� s �� k  �� . L��, �� n , ��  � �� g ; �� . A�� , �� Y  F K , AL



ALf k  ,

� �� Y 



ALf k  

21

A�� M�� M�� J�� A�� D.



 Y Y 



A Lf k    AL f k   ALf k  



n  g

.

E�� . �� , ��  Y Y



A Lf k   AL f k   ALk  f  k  ALf k 



n  g 

f  k   k  n  g   f k  .

�� k  k  , �� , �� , ��   0 . �� k  k  , �� . E� �� , �� : � �� s �� . � �� . � L� �� HG1 �� . � E� �� , �� , �� n � g . �� ?

22

A�� M�� M�� J�� A�� D.

�� 7. �� .

N�� c �� c

1  s f k  ,



� �� c



1  s f k  

f k    n  g   k  ,



�� k 



gs .

A��, �� dc  ds





f  k   dk ds 

f  k  

 n  g   



dk  ds



0

n  g   .

E�� .

23

A�� M�� M�� J�� A�� D.

2.2. �� E� �� k   0 �� , �� . �� , �� :

�� 8. �� .

A��, K��I�� I�� Y  F K , L , �� : K K , L   . (�) K  0  F K K K , L  0 . (�) K    F K (�) L  0  F L L K , L   . (�) L    F L L K , L  0 . �� , �� sf k  � n  g   k , �� , �� . E� �� , �� : (�) F   0, F    0 . 

�� , �� . �� :

24

A�� M�� M�� J�� A�� D.

�� 9. �� . 





�� , k 1 � k 3 �� , �� k 2   �� . E�� k 1 � k 3 ��  �� k 2 (�.�. �� ). 2.3. �� E�� N�� K��, �� , �� : (�) L� �� . (�) L�� 70% � 30% �� . (�) L� �� C�� . � ��. (�) L�� . C�� , �� . �� : (�) �� F K , AL �� HG1, ��

25

A�� M�� M�� J�� A�� D. �� k  . C�� , ��. A�� , F K , AL 





k   ��

ALf k 

F K



ALf  k  1 AL

F K



r p



f  k 

.

� ��

E� �� r 





r  pf  k 

.

pf  k   �� .

(�) C�� (�) � �� 1 F ( K , AL) =

∂F ∂K

K +

∂F ∂ L

L ,

�� 100%. L��, �� : r  K Y 



pf  k  k  AL ALf k  



pf  k    k f k   ,

�� . � �� w  L Y 



 f k    pf  k  k   AL ALf k  



1 

pf  k    k f k   ,

�� . (�) L� �� C�� K Y 



k  AL ALf k  



k  f k   ,

�� . (�) E� �� y ��. E� �� 26



Af k   , � �� y / y



g ��

A�� M�� M�� J�� A�� D.

w p



 f k    pf  k  k   A

,

� �� g . A�� . E� �� , �� A t



A t 1

  AY t t  .

3. �� 3.1. �� E� �� F�� C��. �� 

maxc t   e t U c t dt , U  c t  0



0, U  c t 



0.

A��, �� HG1. D�� , �� . A��, Y t t



F K t t , L t 



 L t c t   K t t  K t ,

�� L t c t �� ,  ��  t �� . E� �� K t t , � K �� , �� . O�� t �� L t . A��, �� . D�� , �� k t 



c t   k t 

�

27

 K L t , t L t L t



n

A�� M�� M�� J�� A�� D.

 k t



 K t  nk t L t



f k t 



c t   k t  k  t  nk t



f k t 



c t  k  t  n   k t

   k t

 n   k t . f k t   c t 

E�� : 

maxc t   e t U c t dt  maxc t  J 0

 k t



s. a. f k t   c t   n   k t k 0  0 limt   t k k t  0 .

� �� H t t



  n  k t  , e t U c t    t  f k t   c t 

��  t



 J k t ,

�� . �� H t t �� H t t



 n   k t , e t U c t    t  f k t   c t   t



e t  t ,

��  t �� . A�� , �� 2 ��

28

A�� M�� M�� J�� A�� D. (�)

∂ H t ∂ct

 t  . E�� ,  t �� = e U c t    t   0  U c t    t .

��

�� .

  H t t k t (�)



e t  t  f  k t   n    

,

�  t



 e t  t  e t  t  e t  t  e t  t   t  . t

�� ,  t    t     t





 t  f  k t   n    

 t n      f  k t  

  c  U t  U c t 



n      f  k t 

.

�� , �� c t � k t , �� . ��   c t  U U  c t   k t

�� ċ t (�) k  t



0





 0 � k t   f k t

 A��, �� k t









n      f  k t 

f k t   c t  n  k t

,

0 . E� ��

 c t  n   k t .

0 �� ĉ t  0 . D�   maxk  t t �� , �� , �� k   0 ĉ t  0 . E�� c t / k . L�� t 



  k t



c t  n    c t  k  t � k t

29



0

A�� M�� M�� J�� A�� D. 

  f  k t



n  

�� . (��) ċ t  0 



n    

U  ĉ t  



  f  k t

cte

  ĉ t    U

   k t

0

cte . (�� ).

  ĉ t   U  c t ċ t   ċ  0 N�� U �� k t  k , �� t t �� . A�� , ��    n    n      f  k     t t �� k t  k t . ��  k

�� 10. �� .

3.2. �� C�� limt   t k t



0



limt  e t  t k t



0,

� ��  t 

 t





 t n      f  k t  

 0 exp

 t 0 n      f k z z  dz 

30

.

A�� M�� M�� J�� A�� D. C�� , �� , �� , �� 2? L� �� . �� : (1) L�� c t



0 � k t



maxk t  .

�� 11. �� .

E� �� limt  e t  t k t 



limt  e t k t  0 exp exp

maxk t   0 exp exp

L� �� .



k t 



 t 0 n      f k z z  dz 

maxk t   0 li limt   exp exp 

 0t n      f k z z  dz

 0t  n    dz 





.

0 ��

(2) L�� k t  0 � �� c t

31



0.

A�� M�� M�� J�� A�� D.

�� 12. �� .

L� �� c t



0

  c t     t  U

0 ;

�� k t



  c    0  f  k t      t  U t .

�� . L� �� :

32

A�� M�� M�� J�� A�� D.

�� 12. �� .   t . N�� n      f  k �� n ,  , �  , �� , �� k  .

F��, �� c t � k t , ��  k t

ċ t





1

k t   k t

0

c t   c t

f k t  R c t 

,

�� R �� . E�� . 3.3. �� E� �� (�� ). A �� k t  �� . A�� k t 



w t  r t t k t .

33

A�� M�� M�� J�� A�� D.

L� �� 

maxc t   e t U c t dt 0

s. a. w t  r t t a t  c t  a t  na t a t  k t  b t ,

�� a t �� b t �� . N�� . L� �� . E�� , �� . �� limt  a t exp exp

t

  0 r z z  n dz  0 .

A�� , �� limt  a t exp exp

t

  0 r z z  n dz



0.

A�� , �� . I�� 0 � T T  0T w t dt    r t t  n a t dt  a 0 0



t  0T w t dt  a 0 exp exp  r z z  ndz 0



 0T c t dt  a T



 0T c t dt  a T .

E� �� 0 � , �� . E� �� , � �� . A�� , �� T   , �� t  0 w t exp exp   r z z  ndz 0



h0



dt  a 0



t  0 c t exp exp   r z z  ndz 0


dt

dt

.

E��

34

A�� M�� M�� J�� A�� D. (�� h 0 ). �� , ��



H t t



 n a t   c t  e t U c t    t w t  r t t 

H t t



 na t   c t  ; e t U c t    t w t  r t t 

� �� 2

 H t t (�) c t



  H t t (�) a t  t



e t U  c t    t 



e t  t r t t  n 



0

U  c t 





 t

.

 e t  t  e t  t  e t  t  e t  t   t  . t

    t

 t n    r t t  .

N��  a t   r t t , � �� , �� . E�� . ��:  c c �� U c t   e t ,   0 �� U c t   e t ��

  c t  U

U  c t ċ t







0 , � U  c t    2 e c t  0 . A��

 2 e c t ċ t ,

� �� (�) �� 2   c  U t U  c t  



c t



 2 e c t ċ t e c t



ċ t

ċ t  n    r t t 

1  t r  n  dz  0 z z .

�� . E� ��  �� , ��

35

A�� M�� M�� J�� A�� D. �� 0 � t . c t 1  1 U c t    lnc t  ��   1 . �� 1   �� . �� , U  c t 



1 c t



0, U  c t 



 12



c t

0

,

�   c U

t 



U c t 

 12 ċ t c t 1 c t





 ċc t t .

E��

 ċc t t



n    r t t



c t



c 0 exp

 t 0 r z z  n  dz

.

�� . �� c 0 ? �� 

h0 

h0





h0

t  0 ct exp exp   r z z  ndz 0



 0t r z z  n  dz

c 0  exp exp 0





c 0  exp exp 0





h0

h0 



exp exp

dt t

  0 r z z  n dz dt

 0t r z z  n  dz   0t r z z  ndz 

exp c 0  exp 0

 0t dz

 c c 0  e t dt  0 0



c0

dt

dt 

h 0

.

E� �� , �� .

36

A�� M�� M�� J�� A�� D. 3.4. ��  t �� G t , ��

  t w t  r t a t 



c t  a t  na t . .

N�� (�� ). �� , �� k t  �� , �� .



w t  r t t k t ��  t , �

�� , �� t  0 w t   t  exp exp   r z z  n dz 0



h 0



dt  a 0



t  0 ct exp exp   r z z  n dz 0


dt

dt .

E� �� . D� �� : �� :

�� 13. �� .

A�� , �� ; �� k t �� 1  z k r t t k t  Ingreso por Capital . ��

37

A�� M�� M�� J�� A�� D. �� w t  1  z k r t t k t



c t  k  t  nk t ; ;

� �� f ′( k z∗ ) =

n + ρ 1 − z k

∗

∗

> n + ρ ⇒ k > k z

.

�� 14. �� .

L� �� , �� .

4. �� 4.1. �� E� �� D�� , �� , �� , �� . L� �� . A��, ��

38

A�� M�� M�� J�� A�� D. g

b t

g

 nb t 

g

G t    t  r t t b t ,

g

�� b t �� t . �� , �� a t



g

 b t  b t , k t 

� �� c t  a t  na t



w t  r t t a t   t . .

D� �� h 0



t  0 w t   t  exp exp   r z z  ndz 0

dt  a 0



t  0 ct exp exp   r z z  ndz 0

dt

,

� �� g

b0



t  0 G t exp exp   r z z  n dz 0



dt    t exp exp 0

t

  0 r z z  ndz dt .

��

 0 c t Rt dt  a 0   0 w t Rt dt   

g

b0

 0 G t Rt dt



  0 c t Rt dt  k 0  b 0   0 w t Rt dt   0 G t Rt dt , dt 

�� Rt



exp exp

t

  0 r z z  ndz .

E�� . A�� (�� ), �� G t . A�� , �� , �� . �� , �� . L�� : � L�� . 39

A�� M�� M�� J�� A�� D. � L�� . � E� �� . L��, �� . 4.2. �� H�� . �� : �� , �� . A �� t �� b t � �� . �� , �� : 

maxc t , I t   e t U c t dt 0

b t



s. a. k t  I t t b t  c t  I t t 1  T  I t / k t    f k t  t k

limt  e t  t b t  0 limt   e t  t q t k t  0 , k t  �� t k �� I t t �� , T  I t / �  �� . N�� ; �� , � �� .

L��, �� H t t



e t U c t    t b t  c t  I t t 1  T  I t / k t    f k t     t q t I t t  , t k

�� q t �� q



Valor de Mercado Mercado Valor Contable .

A��, �� q t  1 �� , �� q t �� . D�� :

40



1 ��

A�� M�� M�� J�� A�� D. (�)

 H t t c t  H t t  I t t 

e t U  c t    t 



1 

0



U  c t 



 t

.

 t   t T   I t t / k k t    t T  I t t / k t    t q t k

e t





k t

T   I t t / k t  k k t

k t    T  I t t / k

q t



0

.

(�)

  H t t b t



e t  t 

 e t  t  t 

 t 



 t

  H t t k t



     t   t   t 



 t    

 0 e  .

 t  e 

 e t  t q t  t 

e t  t  e t  t  e t  t   t 





T  I t t / k t  k

 t I t 2 k t 2

 f  k t 

e t  t q t  e t  t q t   t q t 

 t q t   t q t   t q t  f  k t   T   I t t / k k t 

 t I t 2 k t 2 .

E� �� , �� b t ��    ��  t



0



U  c t 



0



c t

C�� , ��

41



c 0 .

A�� M�� M�� J�� A�� D.

�� 15. �� .  E�� 0, t  �� , ��  �� t ,  �� .

A�� , �� c 0 , �� ? �� 

b 0   0 e t c 0  I t t 1  T  I t t / k k t    f k t  dt 

 b 0  0 et  f k t   I t t 1  T  I t t / k k t  dt 



c0









c 0  e t dt 0

 0 et  f k t   I t t 1  T  I t t / k k t  dt   b 0

.

E� �� . �� , �� , �� 1 

T   I t t / k t  k k t

k t    T  I t t / k

0



1

T 0



0

42

q t



 I t t / k t  k

,

A�� M�� M�� J�� A�� D. T   I t / k t  t k



0

T   I t / k t  t k



0

   I t / k t  t k



0.

1 D��   q t  ��, ��

 1 q t 



q t 



I t t k t



I t t



k t q t 

.

4.3. �� L� �� . �� , ��  �� K t t , �� K t t   K t t ,   K t t   0 . L��, �� K t t k t , �� HG1. �� . �� , �� k t  I t t , � ��

 0 e rt K t t k t  I t t dt . C�� C  I t t  �� C 0 C   I t t 





C  0



0,

0, C   I t t 



0,

��

 0 e rt K t t k t   I t t   C  I t t  dt . �� H t t



e rt K t t k t   I t t  C  I t t     t  I t t ,  t



H t t





e rt  t

e rt K t t k t  1   t  I t t   C  I t t   ,

43

A�� M�� M�� J�� A�� D. � ��

 H t t (�)  I t t

 t  1



 C   I t t   e rt 1   t 





C   I t t 



 t





0

C   I t t   1 .

��  t �� k t , ��  t �� .

  H t t (�) k t



 t





K t t 

e rt K t t 

,

rert  t  e rt  t  e rt r  t   t  

  t . r  t 

E�� . � ��,  t



 C   I   1  t t t



K t t 





C   I t t 

r  t  C   I t t 

 I t t t

 I t t t .

I�� . A��, ��  t



K t t 

   1 C  k t 

r  t   t

,

�� K t t







k t f d 



k t  f d  �  �� 

�� k t . C�� k t  0   t  1 , �� , �� 1 . C��  t  0  K t t   r  t � �� K t t �  ,  k t  f d  



r  t ,   



0 . C�� , �� :

44

A�� M�� M�� J�� A�� D.

�� 16. �� .

��  t �� , �� k t . ��

 t  K z z e r  zt  dz   t  r  z   z e r  zt  dz 



 t  K z z e r  zt  dz   z e r  zt  t 

 t  K z z e r  zt  dz  limT   T e r T t    t 



 t  K z z e r  zt  dz   t .

5. �� 5.1. �� E� ��, �� , �� , � �� . L�� M�� A�� , �� , �� .

45

A�� M�� M�� J�� A�� D.

�� 17. �� .

O�� , �� t  1 �� , �� 1 �� , �� 2 �� . A�� . D�� , �� : maxc 1,t ,s t  U c 1,t 1, t  

U c 2,t 2, t 1  , U   1  



0, U  



0

s. a. w t  c 1, t  s t c 2, t 1



s t 1  r t t 1  ,

�� c i, j j �� i �� s t �� . �� c 2, t 1



w t  c 1, t 1  r t t 1  ,

� �� . E�� maxc 1,t  U c 1,t 1, t  

U w t  c 1,t 1, t 1  r t t 1   1  

46



maxc 1,t  J

A�� M�� M�� J�� A�� D.

 J c 1,t 1, t



U  c 1,t 1, t  



1  r t t 1  U w t  c 1,t 1, t 1  r t t 1   1  

U  c 1,t 1, t 





0

1  r t t 1  U c 2,t 2, t 1  1   .

L� �� , �� , �� . A�� , �� w t �� :

 U  c 1, t   w t 



c 1,t 1, t U c 1,t 1, t  w t 





 1  r t t 1 U  c 2, t 1  w t 1  

c 1,t 1  r t t 1 2  1, t U c 2,t 2, t 1  1  1   w t

c 1, t 1  r t t 1 2   U c 1, t   U c 2, t 1  1   w t 

c 1,t 1, t w t



1  r t 1 2  U c 2, t 1  1  

1



1 

1  

1  r t t 1 2

U  c 1,t 1, t  U  c 2,t 2, t 1 



0

.

� �� :

 U  w t  s t  

w t 





s t U w t  s t  1  w t 



w t 

1  r t t 1  U s t 1  r t t 1  1  

1  r t t 1 2   s U s t 1  r t t 1  t 1   w t

s t 1  r t t 1 2 U  s 1  r   U  w  s  t t t t t 1 1   w t 

s t w t

1



1 

1  r t t 1  1  

2

U  c 2,t 2, t 1  U  c 1,t 1, t 





U  w t  s t 

0

.

L�� , ��

47

A�� M�� M�� J�� A�� D. �� . �� , �� r t t 1 ?

 U  c 1, t  



r t t 1



1 1  

1  

r t t 1

U  c 1, t 

c 1, t r t t 1

U  c 2, t 1    1  r t t 1 U  c 2,t 2, t 1 



1

1  r t t 1  U c 2, t 1  1  



U  c 1,t 1, t 





 w t  c 1,t 1, t 1  r t 1 

r t t 1

c 1,t 1, t r t t 1

U  c 2, t 1   1  r t t 1 U  c 2, t 1 





c 1, t 1  r t t 1   w t  c 1, t  r t t 1

c 1, t 1  r t 1 2   U c 1, t   U c 2, t 1  1   r t t 1 U  c 2,t 2, t 1  1  





1  r t t 1  U c 2,t 2, t 1 w t  c 1,t 1, t  1   .

� ��

s 1,t 1  r t t 1 2  1, t  U c 1,t U c 2,t 1, t   2, t 1  1   r t t 1



U  c 2, t 1  1  r t t 1  U c 2, t 1 w t     c 1, t  1   1   .

N� �� . L�� . M�� (� �� ), �� . L� �� , �� .

48

A�� M�� M�� J�� A�� D.

�� 18. �� .

5.2. �� HG1 F K t t , L t  , �� n . �� F K t t , L t 

�� k t 



w t  r k k k t �



k t 





L t f k t  ,

r t t .

�� , �� . A��, s  sw t , r t t 1  . E� �� k t 1



k t 1





sw t , r t t 1  L t L t 1



sw t , r t t 1  1  n

s f k t   f  k t k t , f  k t   1  n



H k t 

.

�� , �� , � ��, �� :

49

A�� M�� M�� J�� A�� D.

�� 19. �� .

E� �� . ��: �� U  x   ln x � �� F K t t , L t  �� : maxc 1,t ,s t  U c 1, t  

U c 2, t 1  1  



maxc 1,t ,s t  lnc 1, t  

s. a. w t  c 1, t  s t c 2, t 1



s t 1  r t t 1 

E��, �� 5.1.: 1  1  r t t 1 c 1, t 1   



1 c 1,t 1, t c 1, t 

1 c 2, t 1



1 1  



 c 1, t  1  w t 

c 1, t



1 w t  c 1,t 1, t

1   w t 2   .

� �� s t



w t 

1   w t 2  

50



1 w t 2   .

lnc 2, t 1  1  



K t  L t 1  . E�

A�� M�� M�� J�� A�� D.

�� F K t t , L t 



K t  L t 1 



f k t 



k t  ;

�� k t 



w t  f  k t k t



w t



w t



k t    k t 

k t   f  k t k t



1   k t  .



A��, �� s t

1 w t 2  



1   k  2   t ,



� 

k t 1

1   k t  2   1  n  .

E� �� , �� : �� , �� . E� �� , �� F K t t , L t 



L t c t  I t t





F K t t , L t   K t



f k t   k t

E� �� , �� c t





c  � k t

k    k  



L t c t  K t 1  K t



L t c t  K t 1

c t  1  n k t 1 . 

k t 1



k  ��

c   1  n k 

f k    nk 



c .

�� , ��



k  



n.

L�� .

51

A�� M�� M�� J�� A�� D.

�� 20. �� .

��: C�� , �� k  

1   k  2   1  n 

1 1

1   2   1  n 

  k 

.

� ��

k   

f  k  



k 1 





k 



1  

2   1  n 

 

n

.

L� �� . E�� . ��. A�� , ��  �� , �� 

1   

f  k  

2   1  n  

f  k  



k 





n k  .

E� ��, �� , �� .

E� �� . 52

A�� M�� M�� J�� A�� D. �� , �� . E�� E�� .

6. �� 6.1. �� L�� N�� G�� A��, M�� J�� . �� , �� . �� . E� ��, �� . M�� , ��  �� : �      min ,  max  . �   F  . �� P  1 � �� . A��, �� . �� , ��  ��  �� i ��. E�� w i    i . �� , �� L  dF  



L   f  d  , 

�� L �� . A�� , �� v  , �� w   v   . E� �� , �� L  l  1  lv   dF   , 

�� l �� .

53

A�� M�� M�� J�� A�� D.

�� , �� w �� , �� . A��, ��  

  v   w  .

E��, �� ; 



w

0,  ; 



w



w

0 Z w



 ��   E     ��









E 

w



; 

,

 v   w . L��, ��





E 

 v   w   .

�� 21. �� .

D�� , ��

54

A�� M�� M�� J�� A�� D. �� . �� , �� . E� �� , �� . �� :

�� 22. �� .

� �� :

�� 23. �� .

55

A�� M�� M�� J�� A�� D.

E� �� w 3 . �� , �� , �� . ��: ��   Uniforme1, 4  � v    , �� 0   E   v   w   �� E   v   w   E  

w 

 w



E     w  





w    3



   1 . E� �� w 

w 

 1



3 w









w    3

 3  1 .

A�� , �� , 1 



 3  1

4









1 2

 12 12  4







4 11

3  1

,

� �� 



.

�� 4 11



1 2

.

6.2. �� C�� , �� . C�� (�� ), �� (��). A�� . �� , � ��, �� ? ��  �� e �� , ��      min ,  max  �   F   e , �� e  e bajo , e alto  . �� F   e alto  �� 56

A�� M�� M�� J�� A�� D. �� F   e bajo  , �� , ��  �� . L�� F 

 e alto   F   e bajo  .

�� 24. �� .

A��, �� , ��





 f 

 e alto d 







 f 

 e bajo d  .

�� U w, e



v w  g e  Ū ,

�� Ū �� . L��, �� maxe,w     w   f   ed  

s. a.





v w  ge  f   ed   Ū .

57

A�� M�� M�� J�� A�� D.

C�� e ��  �� , �� maxe,w     w   f   ed  

s. a.





 g e v w f   ed   Ū  .

E� �� , �� minw   w f   ed  

s. a.





v w f   ed   Ū   g e .

A��  









w   v w  Ū   g e  f   ed 

 w



1  v  w







1   v  w   f   ed  0



v  w



1 





0

w  : constante

.

A�� , �� w  �� v w  e

 Ū  g e ,

� ��





 f 

 e alto d   w  e alto 







 f 

 e bajo d   w  e bajo  ,

�� e alto . E�� , �� . A��, �� minw   w f   ed  

58

A�� M�� M�� J�� A�� D. s. a.

 e



 g e v w f   ed   Ū 

arg maxe  arg



v w f   e d   g e  .



�� , �� minw   w f  

 e alto d 

s. a.







v w f 



v w f   e alto d 



Ū   g e alto 

 e alto d   g e alto     v w f   e bajo d   g e bajo  .

�� ,



 

w   v w 

 Ū   ge  

 f   e alto   f   e bajo   v w



1  v w



   1 

f   e alto d 

 e bajo  f   e alto  . f 

��  �� . �� ,   e bajo 



f 

 e alto 

 v



w



0

 w 

0.

N�� J�� v  �� , �� , �� . A�� v w   



w

v 

 w f   ed 

 w f  

ed  .

6.3. �� E� �� , �� 59

A�� M�� M�� J�� A�� D. �� , �� . A �� J�� C�� , �� . �� , �� L max � N , �� . A��, �� Ū  

 t e U t dt ,

0





0,

�� U t  ��

U t 



wt   e t  ; empleado

; desempleado

0

,

� wt  � et  �� t ��. E� �� et   0, ē  , �� U L L t   wt   ē E�� ( L ): U S E�� ( S ): S t   wt  U U U t   0 . D�� ( U ): A�� , ��  �� Pr; b



0



e b  ,

�� b �� . A��, �� Pr; q



0



e q  ,

�� q �� . A�� , �� , ��  �� e b q  .

60

A�� M�� M�� J�� A�� D. E� �� , �� a �� , �� , �� . E� �� , �� t 

Rē Lt    w t  Lt   S t  ,



R 



N   1. 0, R   0, R ē L /

E� �� , �� wt  � �� Lt  �� ē . �� V i �� i   L, S , U  . F�� V i  E Ū i  . �� w V L L T  







V L L T 

V L L T 

V L L T 

bT V T    0T e bt e t w  ē dt   e T e bT V L L T   1  e U U 





 0T e b e

bT V T   w  ē dt  e T e bT V L L T    1  e U U

t

b t

b  

T

w  ē 

  bT V T    e T e bT V L L T    1  e U U

0

1  e b T w  ē   e b T V T   e T 1  e bT V T  L L U U b   

V L L T 



w  ē b  



e T 1  e bT 

1  e b T

V U U T 

.

�� T  0 � �� V L L

limT  0 V L L T 



V L L



w  ē  bV U b   .

��

61

A�� M�� M�� J�� A�� D. V S S



w  b  qV S S  V U 



V S S



w  b  q V U U b  q   ;

aV L L  V U 



V U U



aV L L a   .

� V U U

V L L





L� �� , �� w �� V S S . . A�� w  ē  bV U b  



V U U





w   b  qV U U b  q  

 q1 bē   q ē   ē   qw  V L L  V U





ē  0

q

.

E�� , �� . A��, �� . A�� , �� , �� 

V L L

w  ē  bV U b  

�� b  V L L 

w  ē  bV U

 b V L w  ē  L  V U 

V L L





V L L



w

w











 b ē w  ē  q

ē V L L  ē  b q

ē V U U  ē  b   q

62

A�� M�� M�� J�� A�� D.



w





ē aV L L  V U   ē   b   q  ē    a  b  

w

ē

q .

E�� w �� , � �� , � �� . � �� . �� , �� L . A�� , �� . F�� a L max  NL 



NLb



a



NLb L max  NL



a  b



L max b L max  NL ,

� �� w

 ē  

L max b L max  NL

 

ē

q.

E�� (�� (�� ). L� �� : A �� , �� , �� . �� , �� . C�� , �� w �� , ��

  L



 R ē   w ē L



0



w

   /  ē R ē L  ē R ē L N   ē

.

E�� R , �� . �� , �� w  ē � �� NL  L max . �� 63

A�� M�� M�� J�� A�� D. �� , ��  ��, �� R ē L / N   1 . N�� (� ��). E� �� , �� N�C � �� .

�� 25. �� .

N�� , �� . A��, �� w �� , �� , �� . E� �� , �� q . �� q ��, �� , �� , �� N�C �� . A��, � �� q  1  NSC  S L L � �� . �� , �� b  0 , �� , �� , ��

64

A�� M�� M�� J�� A�� D. w

 ē

 1  q

.

E� �� , � �� . ��, �� . �� . �� , �� ē R ē L / N   ē , �� , �� . �� , �� , �� , �� . E�� , �� . 6.4. �� E� �� . A��, �� . �� 1 �� . �� , �� . E� �� R  VP L c , �� L c �� , �� Re �� 1  p .



R ��

 � R f



0 ��

A��, �� , ��  Re   1  i 2  L c  ,

� �� R   1  i 1  L c .

E� �� , �� i1



R  1, i 2 L c

65



Re 1 L c

A�� M�� M�� J�� A�� D.

 i 1  i 2 .

D�� , �� , �� L , ��

Di j 



2 L c ; j



1

L c



2

; j

,

� �� i �� i 2  i 1 





p1  i 2  L c  L c

1 p1  i 1  L c  12 1  i 1  L c 2

 L c ,

N�� i 1 , �� L c � �� , �� . A�� , �� i 1  1 p1  i 1  L c  12 1  i 1  L c 2









p

p

i 2  ? D��



 L c

1 pR  12 R  2 1 pR 2 1 R 2

 pRe



 R e

1 R 2   Re 12 R



p1  i 2  L c  L c



pRe

 12 R



 12 R R 

2 Re  R

.

E� ��, �� , �� . �� . �� 2 E�� J�� A�� . �� B �� i � �� w �� , �� 66

A�� M�� M�� J�� A�� D. L c



B  w



0 . E� �� L c



0 ��

��. �� R �� R



 1  p i  R f i ; p i

p i Rie

i

�� , ��

g p i  : pdf de proyectos ,



�� Rie



1  r  L c

f



Ri .

A�� , �� r 







 p0 p i 1  r  L c g p i dp i   p0 1  p i  R f i g p i dp i ,

� �� U r 

U r    p i



p i  Rie  1  r  L c  f



U r 



R  1  p i  Ri  p i 1  r  L c



U r 



R  R i

f

 p i 1  r  L c  R f i ;

 0 � �� r  0 . E�� , �� , ��  �� U r 

�� 26. �� .

� ��  �� U r ; p i

  p

 1  w

67

A�� M�� M�� J�� A�� D. 

f

R  R i

 p 1  r  L c  R f i  1  w ,

��  �� (�� ). � ��

 p  r

0.

�� , �� , �� : i  p maxr 0  r  

r  r



p

L c  p i g p i dp i  L c 1  r g p  0

 p  p f   Ri g p    1  p r r



0

.

E� �� , �� . A��, �� : (�) C�� (M�� ).

�� 27. �� .

E�� r ��  , �� , �� , �� .

68

A�� M�� M�� J�� A�� D.

�� 28. �� .

(�) C�� .

�� 29. �� . ��.

E� ��, �� S c � ��  . E� �� .

69

A�� M�� M�� J�� A�� D.

�� 30. �� .

L� �� : �� r 0 ��  , �� . E�� . E� �� r 0 .

70

Apuntes de Macroeconomía Matemática

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