economic growth and dynamic optimization - the comeback - rui mota – [email protected] tel. 21 841...
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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
– consume a constant proportion of income0 1( ) ( ),S t sY t s
1( ) ( ) ( )C t s Y t
Solow Model – Assumptions
• Labour augmenting production function:• Constant returns to scale
• Positive and diminishing returns to inputs:
• Inada (1964) conditions:
– Ensures the existence of equilibrium.
• Example of a neoclassical production function:– Cobb-Douglas:
– Intensive form:
( ) ( ( ), ( ) ( ))Y t F K t A t L t
1( , ) aF K AL K AL
0 0 0 0( ) , '( ) , ''( )f f k f k
( , ) ( , ) ( ) ( );X
F K AL F K AL y t f k xAL
00' 'lim ( ) , lim ( )
k kf k f k
( ) af k k
Solow Model – Dynamics
• Labour and knowledge (exogenous):
• Dynamics of man-made Capital– Fraction of output devoted to investment
• Dynamics per unit of effective labor
• - actual investment per unit of effective labour
• - break-even investment.
( ) ( )dK
K sY t K tdt
( ) ( ) ( )k t sf k t n g k t
( )sf k t
( )n g k t
Ln
L
Ag
A
Solow Model – Balanced Growth Path
t
0k 0k
0 0
*lim ( )k t
k t k
How do the variables of the
model behave in the steady
state? *
*
*
*
K ALn g
K AL
Yn g
Y
* * * *
* * * *
K L Y Lg
K L Y L
k
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 continuous r-vector such as to :
• 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.
( ) ru t
00( )max ( ( ), ( ), ) . .u t
f x t u t t dt s t
00( ( ), ( ), ), ( )x f x t u t t x x
Pontryagin’s Maximum Principle – Usual Procedure
• Step 1 – Construct the present value Hamiltonian
• Step 2 – Maximize the Hamiltonian in w.r.t the controls
• Step 3 – Write the Euler equations
• Step 4 – Transversality condition
0( , , , ) ( )H x u p t f x
0H
u
H
x
0lim ( ) ( )t
t x t
Pontryagin’s Maximum Principle – With discount
• Step 1 – Construct the current value Hamiltonian
• Step 2 – Maximize the Hamiltonian in w.r.t the controls
• Step 3 – Write the Euler equations
• Step 4 – Transversality condition
1( , , , ) ( )c t cH x u p t f e x
0cH
u
( )
( ) ( )( )
cc c Ht t
x t
0lim ( ) ( )c t
tt e x t
10( )max ( ( ), ( ), ) . .t
u tf x t u t t e dt s t 00( ( ), ( ), ), ( )x f x t u t t x x
Dynamic Optimization: Cake-Eating Economy
• What is the optimal path for an economy “eating” a cake?
• Optimal System:
• Transversality condition:
0max ( ) t
Cu c e dt 00( ) ( ), ( )S t c t S S
subject to
* *
*
*
S t c t
c t
c t
0lim ( ) ( )c t
tt e S t
1
01
cu c c
Dynamic Optimization: Cake-Eating Economy
• Explicit Solution: – From the dynamics of consumption
– Resource stock constraint: • The remaining stock of cake is the sum of all future consumption of
cake, i.e.,
• In the planning horizon, all the cake is to be consumed, i.e,
0
**
*( )
tcc t c e
c
0 0 0* * * * *( ) ( )
t
t tt
S t c d c e d c e c e
0 0 0 00 0
* *( )t
S c t dt c e dt S c
0
0
*
*
( )
( )
t
t
c t S e
S t S e
* *( ) ( )c t S t
The optimal strategy is to consume a fixed portion of the cake