0, implying that higher consumption
requires larger real-money holdings.
Since, from equation (8.18),
∂m/∂c
= −
(T
c
+
1
)/(T
m
+
π)
and this must
be positive, we conclude that
T
m
+
π <
0, which confirms our assumption
that
∆
>
0 and shows that d
c/
d
R <
0. Hence, the direction of the responses
of consumption to increases in
x
,
b
, and
R
are the same as those of money
holdings.
8.8
Cash and Credit Purchases
In practice, while some goods and services must be purchased for cash, others
may be bought on credit. Suppose that
c
1
,t
is the consumption of goods that
are only available for cash and
c
2
,t
is the consumption of other goods that may
be purchased, at the buyer’s choice, either by cash or by credit. For the cash-
only goods there is a cash-in-advance constraint,
m
1
,t
=
c
1
,t
. For the goods that
may be purchased using credit, households can either use cash or issue bonds,
which, in effect, is what credit is. If borrowing and lending rates are the same,
then
B
t
in the household budget constraint becomes the net stock of bonds,
i.e., total bonds less credit. To illustrate, we consider both a CIA and an MIU
setup.
8.8.1
CIA
Suppose that for cash goods a cash-in-advance constraint holds and for credit
goods a credit-in-advance constraint holds, so that
M
1
t
=
P
1
t
c
1
t
is cash and
M
2
t
=
P
2
t
c
2
t
is credit extended in period
t
at the interest rate
R
t
. Total
expenditure is
P
t
c
t
=
P
1
t
c
1
t
+
P
2
t
c
2
t
,
(8.20)
where
P
1
t
and
P
2
t
are the prices of cash and credit goods, which are taken as
given,
P
t
is the general price level, and total consumption is assumed to be
c
t
=
(c
1
t
)
α
(c
2
t
)
1
−
α
α
α
(
1
−
α)
1
−
α
,
(8.21)
which implies a constant elasticity of substitution between cash and credit. The
nominal budget constraint is
∆
B
t
+
1
+
∆
M
1
,t
+
1
+
(
1
+
R
t
)M
2
t
+
P
t
c
t
=
P
t
x
t
+
R
t
B
t
+
M
2
t
,
(8.22)
where
x
t
is exogenous income. This can be rewritten as
B
t
+
1
+
P
1
,t
+
1
c
1
,t
+
1
+
(
1
+
R
t
)P
2
t
c
2
t
=
P
t
x
t
+
(
1
+
R
t
)B
t
.
(8.23)
We assume that households maximize
∑
∞
s
=
0
β
s
U(c
t
+
s
)
, where
U(c
t
)
=
ln
c
t
,
with respect to
{
c
1
,t
+
s
, c
2
,t
+
s
, B
t
+
s
+
1
;
s
0
}
, subject to the budget constraint
equation (8.23) and the two constraints equations (8.20) and (8.21).

192
8.
The Monetary Economy
The Lagrangian can be written as
L =
∞
s
=
0
{
β
s
ln
c
t
+
s
+
λ
t
+
s
[P
t
+
s
x
t
+
s
+
(
1
+
R
t
+
s
)B
t
+
s
−
B
t
+
s
+
1
−
P
1
,t
+
s
+
1
c
1
,t
+
s
+
1
−
(
1
+
R
t
+
s
)P
2
,t
+
s
c
2
,t
+
s
]
}
.
The first-order conditions are
∂
L
∂c
1
,t
+
s
=
β
s
α
c
1
,t
+
s
−
λ
t
+
s
−
1
P
1
,t
+
s
=
0
,
s
0
,
∂
L
∂c
2
,t
+
s
=
β
s
1
−
α
c
2
,t
+
s
−
λ
t
+
s
(
1
+
R
t
+
s
)P
2
,t
+
s
=
0
,
s
0
,
∂
L
∂B
t
+
s
=
λ
t
+
s
(
1
+
R
t
+
s
)
−
λ
t
+
s
−
1
=
0
,
s >
0
.
It can be shown that the ratio of cash to credit is
M
1
t
M
2
t
=
P
1
t
c
1
t
P
2
t
c
2
t
=
α
1
−
α
.
(8.24)
From equations (8.20), (8.21), and (8.24) the general price level is
P
t
=
P
α
1
t
P
1
−
α
2
t
.
It follows that the total demand for money is
M
t
=
M
1
t
+
M
2
t
=
P
t
c
t
=
P
α
1
t
P
1
−
α
2
t
.

#### You've reached the end of your free preview.

Want to read all 489 pages?

- Fall '13
- Caetano
- Macroeconomics, The Land, Keynesian economics, general equilibrium, DGE models, Traditional Macroeconomics