A Primer on Equilibrium in Spatial Models of Money
1
Competitive Equilibrium with Fiat Money on
the Townsend Turnpike
•
HH
i
0
s
problem
max
{
c
i
t
≥
0
,m
i
t
+1
≥
0
}
∞
t
=0
∞
X
t
=0
β
t
u
(
c
i
t
)(
1
)
s.t.p
t
c
i
t
+
m
i
t
+1
=
p
t
y
i
t
+
m
i
t
−
z
i
t
where
z
i
t
are lump sum taxes or transfers of money to agent type
i.
•
Given
u
0
(0) =
∞
,
agent will never choose
c
i
t
=0
,
but may choose
m
i
t
+1
,
so attach multiplier
θ
t
to nonnegativity constraint on
m
i
t
+1
.
•
The Lagrangian for this problem is:
£
=
∞
X
t
=0
β
t
·
u
μ
y
i
t
+
m
i
t
−
m
i
t
+1
−
z
i
t
p
t
¶
+
θ
i
t
m
i
t
+1
¸
•
The foc wrt
m
i
t
+1
is
u
0
(
c
i
t
)
p
t
=
β
u
0
(
c
i
t
+1
)
p
t
+1
+
θ
i
t
(2)
⇐⇒
u
0
(
c
i
t
)
βu
0
(
c
i
t
+1
)
=
p
t
p
t
+1
+
p
t
θ
i
t
0
(
c
i
t
+1
)
u
0
(
c
i
t
)
0
(
c
i
t
+1
)
≥
p
t
p
t
+1
where the latter inequality holds since
θ
i
t
≥
0
.
Thus the MRS
6
=MRT (the
rate of return on money between periods
t
and
t
+ 1) in periods in which
θ
i
t
is binding.
De
f
nition 1
A
competitive monetary equilibrium
is a sequence of
f
nite
positive prices
{
p
∗
t
}
∞
t
=0
and sequences of consumptions
{
c
i
∗
t
}
∞
t
=0
,
money balances
{
m
i
∗
t
+1
}
∞
t
=0
,
and lump sum taxes/transfers
{
z
i
∗
t
}
∞
t
=0
for each agent type
i
such
that: (i) Given
{
p
∗
t
}
∞
t
=0
and
{
z
i
∗
t
}
∞
t
=0
,
{
c
i
∗
t
}
∞
t
=0
and
{
m
i
∗
t
+1
}
∞
t
=0
solve the house
hold problem (1); (ii) markets clear:
c
A
∗
t
+
c
B
∗
t
=1
(goods) and
m
A
∗
t
+
m
B
∗
t
=
M
t
; and (iii) the government budget constraint is satis
f
ed
z
A
∗
t
+
z
B
∗
t
=
M
t
−
M
t
−
1
.
•
Can we support an optimal allocation without intervention?
Proposition 2
An (interior) optimum cannot be supported in a competitive
monetary equilibrium without intervention (i.e. when
z
i
∗
t
,
∀
i, t
).
1
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By contradiction. Suppose we try to support (
c
A
∗
t
=
λ, c
B
∗
t
=1
−
λ
)
∀
t
w/o intervention. Since
y
A
even
=0and
u
0
(0) =
∞
,
type
A
hh will enter an
even period with money accumulated in the odd period when
y
A
odd
= 1 (i.e.
m
A
even
>
0). But then the nonnegativity constraint is not binding in the odd
period (i.e.
θ
A
odd
= 0). In that case, from (2) we have
u
0
(
λ
)
βu
0
(
λ
)
=
p
∗
t
p
∗
t
+1
,t
odd.
(3)
A similar argument which applies for type
B
agents implies
u
0
(1
−
λ
)
0
(1
−
λ
)
=
p
∗
t
p
∗
t
+1
even.
(4)
But (3) and (4) imply
1
β
=
p
∗
t
p
∗
t
+1
,
∀
t
(5)
⇐⇒
p
∗
t
+1
=
βp
∗
t
which implies that the price level should be decreasing (i.e. de
f
ation) at rate
(1
−
β
) in order to satisfy the optimal allocation. Given this price sequence,
what is happening to household money balances. Again take agent type
A.
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 Spring '07
 CORBAE
 Equilibrium, Steady State, Order theory, Economic equilibrium, Agent Type

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