Economic Growth and Dynamic Optimization

- The Comeback -

Rui Mota – rmota@ist.utl.pt Tel. 21 841 9442. Ext. - 3442April 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

( , ) ( , ) ( ) ( ); XF 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

L nL

A gA

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 gK AL

Y n gY

* * * *

* * * *

K L Y LgK 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

0Hu

Hx

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

0cHu

( )( ) ( )( )

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 tc t

0lim ( ) ( )c t

tt e S t

1

01cu c c

Dynamic Optimization: Cake-Eating Economy

S

C

0lim ( ) ( )c t

tt e s t

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

**

* ( )tc c 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

Assignments• Firm supply