NOTES ON CONSUMPTION AND SAVING
I. Twoperiod model. We assume that are only two periods—the current
period and the future period. Consumers must decide how to allocate
their current and future income across current and future consumption.
Utility Function: U = C
1
a
C
2
1a
.
C
1
= real consumption in period 1 (current period)
C
2
= real consumption in period 2 (future consumption).
r = real rate of interest
Y
1
= real income in period one (current period)
Y
2
= real income in period two (future period)
S
1
= real private saving = Y
1
– C
1
. If S
1
> 0 this represents lending and if S
1
< 0 means
borrowing.
The “a” in the utility function
is a taste parameter.
To find the utilitymaximizing combination of C
1
and C
2
we would:
1. Find the MRS (the slope of the IC).
MRS= (MU
C1
/MU
C2
) = (
∂
U/
∂
C
1
)/ (
∂
U/
∂
C
2
) =
=
[aC
1
a1
C
2
1a
/1a C
1
a
C
2
a
]=
[a/(1a)] /
[C
2
/C
1
]
2. Set the MRS equal to the slope of the budget line (which is 1+r) and solve for C
2
:
[a/(1a)] /
[C
2
/C
1
] = (1+r)
b
C
2
=
[(1a)/a] C
1
(1+r)
3. Show the budget constraint:
C
1
+ C
2
/(1+r) =
Y
1
+ Y
2
/(1+r).
This equation tells us
that the
present value of lifetime consumption equals the present value of lifetime income.
4. Plug (2) into (3) and solve for C
1
:
C
1
+ [(1a)/a] C
1
=
Y
1
+ Y
2
/(1+r)
b
[a/a] C
1
+ [(1a)/a] C
1
=
Y
1
+ Y
2
/(1+r)
b
[1/a]C
1
= Y
1
+ Y
2
/(1+r)
b
C
1
= a [Y
1
+ Y
2
/(1+r)]
This equation tells us the consumer’s optimal C
1
for given values of a, Y
1
, Y
2
and r.
[Notice that
a = C
1
/[Y
1
+ Y
2
/(1+r)]
In other
words, the exponent “a” represents current
consumption’s share in total expenditures. This is useful to know because it provides a
convenient shortcut for finding (4)]
5. Plugging the result from (4) into (2):
C
2
=
[(1a)/a] C
1
(1+r)
b
C
2
=
(1a) [Y
1
(1+r) + Y
2
]
This equation tells us the consumer’s
optimal C
2
for given values of a, Y
1
, Y
2
and r.
[C
2
/(1+r) = (1a) [Y
1
+ Y
2
/(1+r)]
b
(1a) =
C
2
/(1+r) divided by [Y
1
+ Y
2
/(1+r)] . So it can be
seen that (1a) represents future consumption’s share in total expenditures. Again, this is
useful because it provides a shortcut for finding (5)]
Comparative statics:
1.dC
1
/dY
1
= a > 0 . An increase in current income results in an increase in current consumption,
although by some fraction “a” of the change in income. Intuition: Because the consumer desires
to smooth consumption over time, only part of the increase in current income results in an
increase in current consumption. The remaining fraction will be saved, thereby allowing the
consumer to increase consumption in the future.
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View Full Document2. dC
1
/dY
2
= a/(1+r) > 0. Increase in future income results in an increase in current consumption
(by some fraction). Again, because of the desire to smooth consumption over time, the consumer
will “bring forward” some of the increase in future income by saving less today (which could be
interpreted as borrowing more).
Note: Suppose dY
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 Spring '08
 Serra
 Macroeconomics, Utility, Consumption function

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