CashInAdvance
Randall Wright
1 Basic Assumptions
We begin with a very simple model: an endowment economy with homoge
neous agents. The representative agent chooses a sequence for consumption
c
t
to solve
max
∞
X
t
=0
β
t
u
(
c
t
)
,
subject to recursive budget and CIA constraints
p
t
c
t
=
p
t
e
t
+
m
t
+
T
t
−
m
t
+1
p
t
c
t
≤
m
t
+
T
t
,
where
p
t
is the nominal price level,
e
t
is the endowment,
m
t
is money hodlings
at the begining of period
t
,and
T
t
is a transfer of money from the government,
that could be negative and that the agent regards as lump sum. The CIA
constraint requires that consumption be
f
nancedoutofcashonhandatthe
start of the period, including the transfer.
1
In particular, one cannot use the
p
t
e
t
dollars one receives from the sale of one’s endowment at
t
to
f
nance
c
t
;
it must be carried forward and used for
c
t
+1
.
There are di
f
erent interpretations of this assumption. One that is similar
to a story Lucas proposed is as follows. First, each household consists of a
pair, say a “worker” and a “shopper.” Second, there are several types of
1
Alternatively, one can assume
T
t
is
not
available that period, so that the CIA con
straint would be
p
t
c
t
≤
m
t
;seebe
low
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View Full Documenthouseholds, and each type lives in a distinct physical location, say an island.
To motivate gains from trade, assume that the consumption good comes in
K
varieties, say di
f
erent colors, and that each type produces good
k
but
wants to consume good
k
+1(mod
K
)
. Soeachper
iod“shoppers”o
feach
type
k
simultaneously take cash to the next island in order to purchase color
k
+1
consumption goods, while their “worker” partners — perhaps better
called “vendors,” really — stay home waiting to sell their endowment of color
k
goods for money. Goods cannot be bartered directly if
K>
2
.
2
Clearly,
cash aquired from today’s sales cannot be used until next period, since the
shopper leaves before the money rolls in. This motivates the CIA constraint.
What is relevant is the timing, not that households come in pairs; e.g., the
“vendor” agent can be replaced by “vending machine” that collects cash
while an individual is out shopping.
We can rewrite the CIA contraint using the budget constraint as
m
t
+1
≥
p
t
e
t
. Hence, the assumption can be reinterpretted as saying that agents are
forced to hold at least the nominal value of their endowment as money from
each period
t
into
t
+1
. Although not usually stated this way, this makes it
prettyobviouswhattheCIAconstraintisdoing—simplyimposingademand
for money. The supply of money at
t
is denoted
M
t
,wherethein
it
ia
lstock
M
0
>
0
is given exogenously. The governm
en
tbudg
e
tcon
s
t
ra
in
th
e
r
ei
s
T
=
M
0
−
M
. Formulating the problem in dynamic programing terms, after
eliminating
c
t
, Bellman’s equation can be written
V
(
m
)= max
m
0
s.t. m
0
≥
pe
½
u
μ
e
+
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 Spring '02
 Krueger
 Steady State, Inflation, CIA constraint

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