Fall 2022 Nicolas M¨ader
Problem Set 5: Inter temporal optimization:
Consumption-savings
0 Preface (0 points)
We continue to assume that the flow of utility associated with by consumption c is captured by
a function of the type,
u(c) = c
1−γ − 1
1 − γ
(CRRA)
To motivate the household’s savings decision, we appeal to the fact that people live for multiple
periods. Specifically, we will assume that households either live for two periods, although virtually
all two-period results extend to the more general finite and infinite horizon setups. Moreover, we
assume that ‘lifetime utility’ is additively separable, meaning that it takes the form of a sum, and
that the flow of utility from future consumption is discounted by a discount factor β ∈ (0, 1).
1 Two-period model (20 points)
In the the two-period model, household ‘lifetime utility’ takes the following form,
UHH =
c
1−γ
0 − 1
1 − γ
+ β
c
1−γ
1 − 1
1 − γ
!
where c0 denotes contemporaneous consumption, and c1 denotes future consumption. We fur ther assume that lifetime wealth, or permanent income, is predetermined and given by mP = m0
meaning that all income is earned today such that the income model collapses to the wealth model.
In this case, we can write our household’s optimization problem as follows,
1
max
s
c
1−γ
0 − 1
1 − γ
+β
c
1−γ
1 − 1
1 − γ
!
s.t. c0 ≡ mP − s
c1 ≡ (1 + r)s
where s is saving.
a) (2 points) To get a sense of how our household’s lifetime utility UHH looks like, assume γ = 0.5,
β = 0.97, r = 0.05, mP = 1, and visit WolframAlpha to plot UHH as a function of s (i.e. copy-paste
“plot ((1-s)ˆ(1-0.5)-1)/(1-0.5) + 0.97((1.05*s)ˆ(1-0.5)-1)/(1-0.5) between 0 and 1”). Like in class,
use the max operator to find the optimal amount of savings s
?
.
Using calculus, it is possible to show that our household’s optimal plan must satisfy the following
optimality condition,
c
?
1
c
?
0
≡ [β(1 + r)]
1
γ (OCG)
Combining (OCG) with the two budget constraint allows us to solve for the following general
— in the sense that γ, β, r, and mP are left unspecified — consumption-savings solution,
c
?
0 ≡
1
1 + κ
mP
c
?
1 ≡
κ(1 + r)
1 + κ
mP
s
? = m0 − c
?
0
where κ = β
1
γ (1 + r)
1−γ
γ .
b) (3 points) Calculate the values of (c
?
0
, c?
1
, s?
) by either plugging γ = 0.5, β = 0.97, r = 0.05,
mP = 1 into the above equations or, alternatively, by plugging them into the corresponding Excel
sheet. Does your solution match the result you found in a)?
c) (1 point) Using equation (OCG) and/or the provided Excel sheet, how does the optimal ratio
c
?
1
c
?
0
respond to an increase in permanent income mP ? Are (c
?
1
, c?
0
) invariant to changes in mP ?
2
d) (2 points) The parameter β represents the extent to which our households cares about future
consumption relative to contemporaneous consumption. Using equation (OCG) and/or the pro vided Excel sheet, how does the optimal ratio c
?
1
c
?
0
respond to an increase in β? What is the economic
intuition underlying this result?
e) (2 points) The real interest rate r represents a physical exchange rate between current consump tion and future consumption.1 Holding permanent income mP fixed, how would you expect the
optimal ratio c
?
1
c
?
0
to respond to an increase in r? (Hint: Verify your solution using equation (OCG)
and/or the provided Excel sheet.) What is the economic intuition underlying this result?
The effect of rising interest rates on c
?
1
c
?
0
reflects a substitution effect and is thus, given its
intertemporal nature, often referred to as ‘intertemporal substitution’. Moreover, we often call the
partial derivative of ln(c
?
1
/c?
0
) with respect to r,
ε ≡
∂ ln(c
?
1
/c?
0
)
∂r
≈
1
γ
(EIS)
the elasticity of intertemporal substitution. Even though you do not need to be able to derive
equation (EIS), I would like you to understand its implication, namely that larger values of γ
reflect lower levels of intertemporal substitutability of consumption. Specifically, γ represents a
household’s desire to smooth consumption over time.
f) (3 points) Using equation (EIS), what happens to ε as γ goes to infinity? Using equation (OCG)
or the corresponding Excel sheet, what happens to c
?
1
c
?
0
as γ goes to infinity? How do these two
results relate?
Recall from the consumption-leisure decision that we can decompose an agent’s optimal response
to a change in prices — such as wages or interest rates — into a substitution effect and a wealth
effect. By examining the effect of r on
c
?
1
c
?
0
, we only studied intertemporal substitution without
considering the effects on c
?
0
and c
?
1
separately.
g) (2 points) Using the consumption-savings solution or the corresponding Excel sheet, continue to
assume β = 0.97, mP = 1 as before, and examine what happens to c
?
0
as you vary the real interest
rate r for γ = 0.5 and then repeat the same exercise for γ = 2. What do you observe? How can
1
“How much could I consume more in the future if I consumed a little bit less today?”
3
you explain the fact that c
?
0 may increase in r?
h) (5 points) In all likelihood, you will someday find yourself in the position, in which you will be
debating how much of your monthly income to save for retirement. Based on the discussion above,
do you expect to consume more, less, or the same as interest rates increase? In other words, how
much do you care about consumption smoothing? (Hint: Once again, there is no correct answer
here. Show me that you did some thinking.)
2 Finite horizon model (0 points)
While the two-period model helps us foster economic intuition regarding intertemporal substitu tion, the assumption that our agents only live for two periods seems awfully artificial. This is why
economists typically allow households to live for an arbitrary number of periods T. For example,
the T-period analogue of our two-period consumption-savings household would look as follows,
max
{ct}
T
t=0
X
T
t=0
β
t
”
c
1−γ
t − 1
1 − γ
#
s.t. X
T
t=0
ct
(1 + r)
t
| {z }
P Vc
= mP
where permanent income mP is taken as given and we are looking for an optimal sequence
of consumption {c
?
t }
T
t=0. Somewhat unsurprisingly, the optimality condition of the finite horizon
model generalizes the optimality condition from the two-period case as follows,
c
?
t+1
c
?
t
≡ [β(1 + r)]
1
γ (OCG’)
which simply means that the optimal growth rate of consumption is time-invariant (and equal
to the same value as in the two-period case). Combining (OCG’) with the above budget constraint
helps us find the following general solution of the finite horizon consumption-savings problem,
4
c
?
0 ≡
1 − κ
1 − κ
T +1
mP
c
?
t = [β(1 + r)]
t
γ c
?
0
where κ = β
1
γ (1 + r)
1−γ
γ as before. We can use this solution to extend our analysis of intertem poral substitution — the notion that people shift consumption from today into the future as interest
rates rise — from the original two-period setup to one with an arbitrary number of periods.
Figure 1. Optimal consumption paths {c
?
t }
50
t=0 for two different values of r (and various γ)
5
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