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Mark J. Gibson
Stochastic Dynamic General Equilibrium Models
The state of the economy in period t is a history of events
0 1
t
t
s s s s. The probability of
a particular history is ( )
t
s. The initial event
0
s is given. All equilibrium objects are functions
of the state of the economy.
Pure exchange economy
Consider an economy with heterogeneous consumers in which there is uncertainty about
endowments, ( )
t
i
s. Consumer i has the expected utility function
0
( ) log ( )
t
t t t
i
t s
s c s
An Arrow-Debreu equilibrium is ˆ
t t
i
c s p s such that
t
i
c s solve
0
max ( ) log ( ) t
t t t
i
t s
s c s
0 0
s.t. ( ) ( ) ( ) ( ) t t
t t t t
i i
t s t s
p s c s p s s
t
i
c s
t t
i i
i i
c s s
With a sequential markets structure, ( )
t
i
b s is an Arrow security that delivers one unit of the
consumption good to consumer i in state
t
s. The price of this security is ( )
t
q s.
A sequential markets equilibrium is
t t t
i i
c s b s q s such that
t t
i i
c s b s solve
0
max ( ) log ( )
t
t t t
i
t s
s c s
1
1 1
s.t. ( ) ( , ) ( , ) ( ) ( )
t
t t t t t
i t i t i i
s
c s q s s b s s s b s
t
i
c s , ( )
t
i
b s B ,
0
i
b s
t t
i i
i i
c s s
t
i
i
b s
Production economy
Consider a growth model in which there is uncertainty about total factor productivity. The
representative consumer is endowed with one unit of labor in each period and
0
k units of initial
capital. The consumer’s expected utility is
0
( ) log ( ) (1 ) log 1 ( )
t
t t t t
t s
s c s s
The resource constraint for this economy is
1 1
t t t t t t
c s k s k s z s k s s
An Arrow-Debreu equilibrium is
t t t t t t
c s s k s r s w s p s such that
t t t
c s s k s solve
0
max ( ) log ( ) (1 ) log 1 ( )
t
t t t t
t s
s c s s
1
0 0
s.t. ( ) ( ) ( ) (1 ) ( ) ( ) ( ) ( ) ( )
t t
t t t t t t t t
t s t s
p s c s k s k s w s s r s k s
1 0
0
t t t t
c s s k s k s k s k
1 1
ˆ ˆ
t t t t t
r s p s z s k s s
t t t t t
w s p s z s k s s
1 1
ˆ ˆ ˆ ˆ
t t t t t t
c s k s k s z s k s s
A sequential markets equilibrium is
t t t t t
c s s k s r s w s such that
t t t
c s s k s solve
0
max ( ) log ( ) (1 ) log 1 ( )
t
t t t t
t s
s c s s
1
s.t. ( ) ( ) (1 ) ( ) ( ) ( ) ( ) ( )
t t t t t t t
c s k s k s w s s r s k s
1 0
0
t t t t
c s s k s k s k s k
1 1
ˆ ˆ
t t t t
r s z s k s s
t t t t
w s z s k s s
1 1
ˆ ˆ ˆ ˆ
t t t t t t
c s k s k s z s k s s
