Economic Growth and Dynamic
Optimization
- The Comeback -
Rui Mota – [email protected] Tel. 21 841 9442. Ext. - 3442
April 2009
Solow Model – Assumptions
• Can capital accumulation explain observed growth?
• How does the capital accumulation behaves along time and
what are the explanatory variables?
• Consumers:
– Receive income Y(t) from labour supply and ownership of firms
S (t )  sY (t ), 0  s  1
– consume a constant proportion of income
C (t )  (1  s )Y (t )
Solow Model – Assumptions
Y (t )  F ( K (t ), A(t ) L(t ))
• Labour augmenting production function:
• Constant returns to scale
F ( K ,  AL )   F ( K , AL )  y(t )  f ( k ); x 
• Positive and diminishing returns to inputs:
X
AL
• Inada (1964)
f (0)conditions:
 0, f '( k )  0, f ''( k )  0
– Ensures the existence of equilibrium.
'
lim f (k )  , lim f ' (k )  0
0
k 
• Example of a neoclassical kproduction
function:
– Cobb-Douglas:
– Intensive form:
F ( K , AL )  K a  AL 
1 
f (k )  k a
Solow Model – Dynamics
• Labour and knowledge (exogenous): L
L
n
A
g
A
• Dynamics of man-made Capital
– Fraction of output devoted to investment
dK
 K  sY (t )   K (t )
dtunit of effective labor
• Dynamics per
•
•
sf  k (t ) 
k (t ) -actual
sf  kinvestment
(t )    n per
g
 ofk (effective
t)
unit
labour
 n  g    k (t )
- break-even investment.
Solow Model – Balanced Growth Path
k
 lim k (t )  k *
k0  0 t 
How do the variables of the
model behave in the steady
state?
K*
AL

n

g

K*
AL
k0
k0
Y*
 n g
*
Y
K * L*
Y * L*
g * *
*
*
K L
Y L
t
Solow Model – Central questions of growth theory
• Only changes in technological progress have growth effects on
per capita variables.
• Convergence occurs because savings allow for net capital
accumulation, but the presence of decreasing marginal returns
imply that the this effect decreases with increases in the level of
capital.
• Two possible sources of variation of Y/L:
– Changes in K/L;
– Changes in g.
• Variations in accumulation of capital do not explain a significant
part of:
– Worldwide economic growth differences;
– Cross-country income differences.
• Identified source of growth is exogenous (assumed growth).
Dynamic Optimization: Infinite Horizon
• Optimal control: Pontryagin’s maximum principle
• Find a control vector
for some class of piece-wise
u(t )  r
continuous r-vector such as to :

max  f 0 ( x(t ), u(t ), t )dt
u(t )
0
x  f ( x(t ), u (t ), t ), x(0)  x0
s .t .
• Control variables are instruments whose value can be choosen by
the decision-maker to steer the evolution of the state-variables.
• Most economic growth models consider a problem of the above
form.
Pontryagin’s Maximum Principle – Usual Procedure
• Step 1 – Construct the present value Hamiltonian
H ( x, u , p , t )  f 0 ( )   x
• Step 2 – Maximize the Hamiltonian in w.r.t the
controls
H
0
u
• Step 3 – Write the Euler equations    H
x
• Step 4 – Transversality condition
lim  (t ) x(t )  0
t 
Pontryagin’s Maximum Principle – With discount

max  f1 ( x(t ), u(t ), t )e   t dt
u(t )
0
s.t .
x  f ( x(t ), u (t ), t ), x(0)  x0
• Step 1 – Construct the current value Hamiltonian
H c ( x, u , p, t )  f1 ( )e   t   c x
• Step 2 – Maximize the Hamiltonian in w.r.t the controls
• Step 3 – Write
H c

0
the Euler
equations
u
H c ( )
 (t )   (t ) 
x(t )
c
• Step 4 – Transversality condition
c
lim  c (t )e t x(t )  0
t 
Dynamic Optimization: Cake-Eating Economy
• What is the optimal path for an economy “eating” a
cake? 
 t
max  u(c)e dt
C
0
subject to
S (t )  c(t ), S (0)  S0
c1
u c 
  c    0
1 
• Optimal System:
S *  t   c*  t 
c*  t 
c*  t 



• Transversality condition:
lim  c (t )e t S (t )  0
t 
Dynamic Optimization: Cake-Eating Economy
C
S
lim  c (t )e t s(t )  0
t 
Dynamic Optimization: Cake-Eating Economy
• Explicit Solution:
– From the dynamics of consumption
– Resource stock constraint:

 t
c*

*




c
(
t
)

c
e
0
c*

• The remaining stock of cake is the sum of all future consumption of
cake, i.e.,



t
t
S * (t )   c* ( )d   c0*e


 
d 


c0*e


 
t
 *   t
 c0 e

• In the planning horizon, all the cake is to be consumed, i.e,


0
0
S0   c(t )dt   c0*e

 t

c* (t )  S 0 e 

S (t )  S 0 e
*


 t


 t
 *
c0

dt  S0 

The optimal strategy is to consume
a fixed portion of the cake
 *
c (t )  S (t )

*
Assignments
• Firm supply