Development and growth III:
Facoltà di Scienze Politiche Università di Macerata paolo.sospiro@unimc.it
Macerata 13 March 2013 Development Economics University of Macerata
2
Formalization of the SM
getting 6.3
and 6.4
Thinking that
6.5
( ) ( ) S t = I t
( 1) (1 ) ( ) ( ) K t + = − σ K t + I t
( ) ( ) S t s Y t
=
Formalization of the SM
Then we have 8.1
Dividing everything per Assuming P(t) growth constant
8.2
Then the 8.1 becomes
8.3
( 1) (1 ) ( ) ( ) K t + = − σ K t + sY t
P t ( )
( 1) (1 )
tP t + = + n P
(1 + n k t ) ( + 1) = (1 − σ ) ( ) k t + sy t ( )
4
Formalization of the SM
8.3
where and are the Kapital and Income p.c.
6.6 e 6.7
k
(1 + n k t ) ( + 1) = (1 − σ ) ( ) k t + sy t ( ) y
( ) ( ) K t P t ( ) ( ) Y t P t
5
The intuition of Solow
8.3
where
where saving rate p.c.
The sum is
The stock of Kapital p.c.
(1 + n k t ) ( + 1) = (1 − σ ) ( ) k t + sy t ( ) (1 − σ ) ( ) k t
( ) sy t
( 1) k t +
6
The relationship between Kapital and Population growth and the Steady State
(1+n) /k
(1−δ)k+sy
k
* k
7
Il rapporto capitale/popolazione e lo stato stazionario
(1+n) /k
(1+δ)k+sy
k
* k
8
The Steady State
In the steady State
8.4
Then the H-D could be written and take into account of some endogenuos variable such as the Kapital
8.5
( ) ( 1) * k t = k t + = k
*
*
k s
y = n + δ
The Progress and efficiency of the workers
Now it’s necessary to separate the workers from the population
8.6
where is the worker productivity Therefore assume the population growth and the increasing of the workers efficiency given by
8.7
( ) ( ) ( ) L t = E t P t E t ( )
π
( 1) (1 ) ( )
E t + = + π E t
10
Formalization of the new SM
Taking again
8.3
Instead of and having and that are Kapital and Income per unit of product o per worker
8.8 and 8.9
k
(1 + n k t ) ( + 1) = (1 − σ ) ( ) k t + sy t ( ) y
( ) k E t
^
k
^
y
( ) y E t
11
The formalization of the new SM
Then 8.3
8.3
becomes
8.10
(1 + n k t ) ( + 1) = (1 − σ ) ( ) k t + sy t ( )
^ ^ ^
(1 + n )(1 + π ) ( k t + 1) = (1 − σ ) ( ) k t + s y t ( )
12
The new model
(1+n)(1+g)
(1+δ)k+sy
k
* k
13
Conditional convergence
E
D F
A
B
C
14
Unconditional Convergence
' A
D ' B
E
F
C
A
B
