Dynamic Programming in Economic ModelsNeoclassical Growth Model

Bellman Equation

Dr. Keshab R Bhattarai

Business School, University of Hull

Keshab Bhattarai 2

Neo-classical Growth Model: Current Value Hamiltonian

Preference: dtC

e tt

0

1

1

Technology: 1

tttt NKAY assume 1tA 1tN

Capital accumulation: ttttt KCNYK

Current value Hamiltonian of this problem

1

1

1,,

tttt KCKC

KcH

(1)

C is control, K is state variable, is co-state variable.

Keshab Bhattarai 3

Optimality and Boundary Conditions

First order conditions

0

tC

H ttC

(2)

t

ttt K

H

1tttt K

(3)

ttttt KCNKK (4)

Transversality condition

0

tt

t Ken

Lim

(5)

Keshab Bhattarai 4

Characterisation of the Balanced Growth Path Capital stock, consumption and the shadow price of capital remain constant in the

balanced growth path cgC

C

; KgK

K

and

g

t

t

. From (3)

1

tt

t K

t

ttK

1

(6)

Since the RHS is constant , therefore LHS also should be constant 0K

K . If capital stock

is not growing output is not growing 0Y

Y and consumption is not growing 0C

C .

From (2) t

t

t

t

C

C

0t

t

(7)

Keshab Bhattarai 5

Transitional Dynamics-1Transitional Dynamics-1In tt K, space the transition dynamics of the shadowprice t relative to the steady

state capital stock is that 0t 0t 0t

t K*

1

1

* K

Keshab Bhattarai 6

Transitional Dynamics-2

'* KKK . 0K

0K

0K

*K 'K K

Keshab Bhattarai 7

Transitional Dynamics-2

'* KKK . 0K

0K

0K

*K 'K K

Keshab Bhattarai 8

Saddle Point SolutionPutting all these things together the convergence to the steady state can be summarised inthe following diagram.

0K

0

I II

IV

III

*K 'K KConvergence to the steady state lies in region I and III as shown by the double arrow redline.

Keshab Bhattarai 9

ttt AKCK 1

t

tt CUMax ln 10

Subject to

10

KVCKV 01 ln

Brock-Mirman(1972)dynamic programming problemBellman’s Equations

Value function

Keshab Bhattarai 10

Solution by IterationFirst and Second Iteration of the Value function

01 tK

KAAKCKV lnlnlnln1

KAKAKKVkmacc

lnln'ln,

2

0

'

12

KKAKK

KV ''

1

KKAK

'' KAKK AKK 1'

'ln11

lnln1

1ln2 KAAAKV

Keshab Bhattarai 11

Third Iteration of the Value function

'ln1'ln,max

3 KKAKKVkc

0

'

1

'

13

KKAKK

KV

'

1

'

1

KKAK

'1' KAKK

AKK22

22

1'

'KAKC

AKAKC22

22

1

AKC

221

1

'ln1

1ln1

1ln

1lnln

1ln'

22

22

22

22

223

K

AAAA

AKV

Keshab Bhattarai 12

Fourth Iteration of the Value function

'ln' 34 KVCKV

'ln1'ln 22

,max4 KKAKKV

kc

'

1

'

1 22

KKAK

AKK3322

3322

1'

AKC

33221

1

K

AAA

AAA

AKV

ln11

1ln

1ln1

1ln1

ln1

ln

1ln

1

1ln'

3322

2

22

22

3322

332222

32

2233224

Keshab Bhattarai 13

Limits of the Value Function in Infinite Iterations

kvvkv ln41

404

1..11lim 113322

1jjj

jv

Ax jt 33221

1ln

1

0

j

t

jt

tj xa

2

0

j

t

jt

tj yb

223322

2233222222

..1

..ln..1

jj

jjjjj

t

Ay

1lnlim 1 Ax jj

Ay jtj

ln1

lim

Keshab Bhattarai 14

Limits of the Value Function in Infinite Iterations

1ln11lnlimlimlim 1

1

0

1

0

AAxaj

t

t

j

j

t

jt

t

j

j

j

AAybj

t

t

j

j

t

jt

t

j

j

jln

11ln

1limlimlim 1

2

0

1

0

AAv j

jln

11ln1lim 1

0

kAAkvj

ln1

ln1

1ln1lim 1

kvvkv ln10

Keshab Bhattarai 15

References• Bellman, R (1957) Dynamic Programming, Princeton University Press.• Brock W and L Mirman (1972) Optimal Economic Growth and

Uncertainty: the Discounted Case, Journal of Economic Theory 4(3):479-513.

• Cass, D. (1965): Optimum Growth in Aggregative Model of Capital Accumulation, Review of Economic Studies, 32:233-240.

• Ljungqvist L and T.J. Sargent (2000), Recursive Macroeconomic theory, MIT Press

• Parente S.L.(1994) Technology Adoption, Learning-by-Doing, and Economic Growth, Journal of Economic Theory, 63, pp. 346-369.

• Sargent TJ (1987) Dynamic Macroeconomic Theory, Chapter 1, Harvard University Press.

• Solow, R.M. (1956) “A Contribution to the Theory of Economic Growth.” Quarterly Journal of Economics 70, 65-94.

• Stokey, N. L. and R.E. Lucas (1989) Recursive Methods in Economic Dynamics, Harvard UP, Cambridge, MA.

• Uzawa, H. (1962) “On a Two-Sector Model of Economic Growth,” Review of Economic Studies 29, 40-47.