Consumption Ratcheting on
Financial Markets
Simon Krsnik
Bielefeld University
IMW - Institut für Mathematische Wirtschaftsforschung
EBIM - Economic Behavior and Interaction Models
Abstract
In financial economics the problem of optimal portfolio choice and utility maximization under budget constraints (Merton, 1969) has been studied under several
aspects of market dynamics and investor behaviors. It has been elaborately discussed (Ryder and Heal, 1973; Constantinides, 1990) why loosening the assumption of investor preferences to be non time-additive is reasonable. This advisement leads to a broad literature of habit-formation Models examing Mertons Problem
(e.g. Ingersoll (1992)). We consider a habit-formation framework (Dybvig, 1995) that models the behavior of an investor who claims consumption ratcheting.
1. Motivation
It can be widely observed that previous consumption (i.e. standard of living) affects current consumption decisions. Standard habit-formation literature often catches this idea with punishing any
deviation from an endogenously or exogenously
given consumption process or any decrease in
consumption. This raises different questions:
Is it reasonable to punish reaching a higher and
everlasting standard of living? Further, is every
small decline in living standards worth to be punished?
Dybvig (1995) introduced a model that covers all
this. Using the preferences given there we are
able to model the behavior of two types of investors. The first one forgets slowly about previous living standards and therefore tolerates a
moderate decrease in consumption and the second one desires a safe non-decreasing consumption process.
2. The Model
Let Ω, F, P, {Ft}t≥0 be a filtered probability
space (with infinite time horizon).
We call C the set of all positive rate of consumption processes c = {ct}t≥0. Since we assume an
initial consumption rate c− ≥ 0, we are looking
at investors who set a high value on rate of consumption processes of the following type.
c0 ≥ c− and ct ≥ e
−α(t−s)
cs ∀s ≤ t
(1)
Where α ∈ R reflects the memory or requirement
of the investor. The following Figure explaines
possible rate of consumption processes for c− = 1
and various α.
It can easily be shown that the problem for α ∈ R
reduces to the problem of a RI. To see the
strength ratcheting behavior Dybvig proposes.
Example (Dybvig 1995). Consider Ω = {ω1, ω2}
with 2 equally probable states and the lotteries:
lottery A [0, 1) [1, ∞)
ω1
1e 1e
ω2
2e 2e
lottery B [0, 1) [1, ∞)
ω1
1e 2e
ω2
2e 1e
Although the AI is indifferent between this two lotteries the profound aversion against a decrease
in his consumption rate makes the RI prefer lottery A. His immense loss aversion assignes the
value −∞ to lottery B.
We contemplate Problem (2) for a RI (with u =
log) on a complete financial market with 2 assets.
S0(t) = ert
for a riskless asset
S1(t) = eµt+σWt for a risky asset
where Wt is a standard Brownian Motion and
r, µ, σ > 0 are constants. Therefore the market
price of risk is ϑ := µ−r
σ .
Since an AI is not averse to decreases in consumption, his initial consumption rate c− plays no
role. Under these circumstances he behaves optimally if his wealth develops as follows.
XtA = w exp (r − δ + 21 ϑ2)t + ϑWt
For α ∈ R investors will be assumed to evaluate
utility as a time-additive investor (AI) as long as a
rate of consumption process satisfies (1). Their
preferences will respect the following utility function
 hR
i
E ∞ e−δtu(c ) dt
if c satisfies (1)
t
0
Vu,α(c) = 
−∞
else
Optimal consumption processes cA and optimal
portfolios (investment in risky asstes) π A can be
described by
cAt = δXtA
and πtA = ϑXtA
Therefore he should always spend a constant
fraction of his wealth in consumption and in the
risky asset (see Merton (1969)).
For the RI problem (2) is feasible only if he is able
to consume c− all the time (i.e. w ≥ cr− ). If the
problem is feasible an optimum exists (see Dybvig (1995)) and his optimal consumption cR is
cR
t := max c−, λ exp
For given α• the investor wants a rate of
consumption process above the corresponding line. Obviously αgreen (α = 0) reflects a
consumption ratcheting investors (RI).
!)
sup r − δ + 21 ϑ2 s + ϑWs
0≤s≤t
max Vu,α(c)
c∈C
s.t. E
0
e
−rt
γ=q
(δ − r +
2δ
ϑ2 2
2)
− 2rϑ − (δ − r +
ϑ2
2)

rw(1 − γ1 )
1
and λ =
c (γ − 1) rw − 1 γ
−
c−
if rw(1 − γ1 ) ≥ c−
else
Therefore both investors
base their decisions
on
the same process exp (r − δ + 21 ϑ2)t + ϑWt . The
behavior of the RI can be interpreted as follows.
cR
R
If he consumes ct today, he needs rt to hold his
consumption constant. Whenever the process
above reaches a new supremum the RI plans to
finance a higher consumption level for the rest of
his lifetime (see the figure below).
(2)
λ
1 2
exp
(
2 ϑ − δ)t + ϑWt
R
ct
and the optimal portfolio process is given by
πtR = γ σϑ
XtR −
cR
t
!
r
Since the RI always has to think about his future
he only invests a fraction of the money he does
not need to finance his current consumption level
over time in the risky asset.
Optimal consumption, wealth and investment boundary
As soon as the optimal wealth process (blue in
figure above) hits the lower boundary (green, i.e.
cR
R
Xt = rt ) the RI has no money left for speculating.
From that time on he leaves all of his money in
the riskless asset. Notice that optimal consumption starts with c− = 5, which is above the optimal
wealth level. The optimal consumption process
develops as described before.
4. Outlook
Up to here we considered a complete financial
market. Riedel (2007) used other techniques and
generalized Dybvigs work, which allows us to look
at Lévy processes. This enables us to look at
incomplete markets and generalize Dybvigs results. Further the question rises if important results on incomplete markets like the theorem of
Cvitanić and Karatzas (1992, 10.1 Theorem) still
hold for investors of this kind? Looking at other financial markets driven by Lévy processes is also
planned.
References
Constantinides, George M. 1990. Habit Formation: A Resolution of the
Equity Premium Puzzle, The Journal of Political Economy 98, no. 3,
519–543.
Cvitanić, Jakša and Ioannis Karatzas. 1992. CONVEX DUALITY IN
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Probability 2, no. 4, 767–818.
Dybvig, Philip H. 1995. Duesenberry’s Ratcheting of Consumption:
Optimal Dynamic Consumption and Investment Given Intolerance for
any Decline in Standard of Living, Review of Economic Studies 62,
287–313.
Ingersoll, Jr., Jonathan E. 1992. Optimal cunsumption and portfolio
rules with intertemporally dependent utility of consumption, Journal of
Economic Dynamics and Control 16, 681–712.
Merton, Robert C. 1969. Lifetime Portfolio Selection under uncertainty:
The Continuous-Time Case, The Review of Economics and Statistics
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Riedel, Frank. 2007. Optimal Consumption Choice with intolerance for
Declining Standard of Living, IMW Workingpaper 394.
ct dt ≤ w
γ
R
Xt =
γ−1 r
!γ
where
where δ > 0 is the discount factor and u : R+ → R
the intertemporal utility function (strictly concave,
strictly increasing,limx→∞ u0(x) = 0).
Given an initial endowment w > 0 the investors
problem is the following
Z ∞
cR
t
3. Consumption Ratcheting on a complete
financial market
(
αred = −1, αgreen = 0, αyellow = 0.1
Further the wealth process is given by
The process exp (r − δ + 12 ϑ2)t + ϑWt
Ryder, Jr., Harl E. and Geoffrey Heal M. 1973. Optimal Growth with Intertemporally Dependent Preferences, The Review of Economic Studies Ldt. 40, no. 1, 1–31.