1
Problem 1:
Core Inflation and Non-Core Inflation in the Two-Period Economy (30 points).
Two distinct measures of inflation – called core inflation and non-core inflation – generally
attract attention by policy-makers and the media.
The core inflation rate is the rate of growth of
prices of so-called “core goods” (such as food, clothing, and shelter), while the non-core
inflation rate is the rate of growth of prices of so-called “non-core goods” (generally energy
items).
Consider our usual two-period economy (with no government), in which the representative
consumer has no control over his
nominal
income.
Rather than there being only one “type” of
good the consumer purchases each period, however, suppose that each period there are two
“types” of goods:
core good and non-core goods.
The lifetime utility function of the
representative consumer is
!"
! " ! " ! " ! "
1
1
2
2
1
1
2
2
,
,
,
ln
ln
ln
ln
CORE
NONCORE
CORE
NONCORE
CORE
NONCORE
CORE
NONCORE
u c
c
c
c
c
c
c
c
#
$
$
$
,
where
ln
stands for the natural logarithm,
1
CORE
c
stands for consumption of core goods in period
1,
1
NONCORE
c
stands for consumption of non-core goods in period 1, and similarly for
2
CORE
c
and
2
NONCORE
c
.
The representative consumer begins period one with zero assets (i.e.,
A
0
= 0).
The period-by-
period budget constraints of the representative consumer are thus
1
1
1
1
1
1
2
2
2
2
2
2
1
(1
)
CORE
CORE
NONCORE
NONCORE
CORE
CORE
NONCORE
NONCORE
P
c
P
c
A
Y
P
c
P
c
A
Y
i A
$
$
#
$
$
#
$
$
where
1
CORE
P
denotes the
nominal
price of core goods in period 1,
1
NONCORE
P
denotes the
nominal
price of non-core goods in period 1, and similarly for
2
CORE
P
and
2
NONCORE
P
.
As usual,
1
Y
and
2
Y
denote
nominal
income in periods 1 and 2, respectively, and
i
is the nominal interest rate.
Finally, we can construct as usual the representative consumer’s nominal lifetime budget
constraint, which here is:
2
2
2
2
2
1
1
1
1
1
1
1
1
CORE
CORE
NONCORE
NONCORE
CORE
CORE
NONCORE
NONCORE
P
c
P
c
Y
P
c
P
c
Y
i
i
i
$
$
$
#
$
$
$$
This nominal LBC has the same interpretation as always:
the PDV of all lifetime nominal
consumption (which here takes into account both consumption of core and non-core goods) is
equal to the PDV of all lifetime nominal income. That is, in a lifetime sense, all consumption
spending equals all income regardless of “how many” goods there are to purchase.