CANADIAN APPLIED
MATHEMATICS QUARTERLY
Volume 17, Number 4, Winter 2009
DEVELOPING UTILITY FUNCTIONS FOR
OPTIMAL CONSUMPTION IN MODELS WITH
HABIT FORMATION AND CATCHING UP
WITH THE JONESES
ROMAN NARYSHKIN AND MATT DAVISON
ABSTRACT. This paper analyzes popular time-nonseparable
utility functions that describe “habit formation” consumer preferences comparing current consumption with the time averaged
past consumption of the same individual and “catching up with
the Joneses” (CuJ) models comparing individual consumption
with a cross-sectional average consumption level. Few of these
models give reasonable optimum consumption time series. We
introduce theoretically justified utility specifications leading to
a plausible consumption behavior to show that habit formation preferences must be described by a power CRRA utility
function different from the exponential CARA used for CuJ.
We also introduce a continuous time formulation for the habit
formation problem which yields a nonlinear ordinary differential equation (ODE). The numerical solutions of this ODE are
presented and analyzed.
1 Introduction Mathematically, the decision criteria for human
beings may be based on their personal utility function U, which is a measure of the relative satisfaction they derive from wealth W or consumption of goods C. Utility functions were introduced by Daniel Bernoulli
[3] in order to solve the St. Petersburg paradox. Bernoulli supposed an
individual’s utility to be strictly convex rather than linear, meaning that
having twice as much wealth does not make someone twice as happy. A
general decision-making framework for economics optimization problems
was developed by John von Neumann and Oskar Morgenstern two centuries later [15]. From that time on, the maximization of the expected
value of a utility function E{U (x)} defined over outcomes x has been
widely accepted as a good criterion for finding optimal economic strategies. The main properties of the Morgenstern-von Neumann utility function u(ct ) are convexity (u00 < 0) and time-separability, meaning that
utility is derived from current consumption ct independent of the conc
Copyright Applied
Mathematics Institute, University of Alberta.
703
704
R. NARYSHKIN AND M. DAVISON
sumption history. These simplifying assumptions have faced repeated
theoretical criticism [12, 19] and lead to paradoxes when compared to
empirical data [7, 14].
Numerous attempts to resolve these problems within the expected
utility approach can be classified (following [2]) into two categories by
the ways in which the time-nonseparable utility function is constructed.
The first category considers the averaged consumption of the economy
as a standard of living benchmark compared to which the current individual’s consumption preferences are computed. This “catching up with
the Joneses” (CuJ) approach was introduced in [6] and more recently
developed by Abel and Galı́. The resulting CuJ utility function typically
has the form u(c/C D ), where C is the per capita consumption in the
economy [1, 10]. The second category uses the individual’s own past
consumption as a benchmark for the current consumption. This “habit
formation” approach was initially proposed in [18] and usually assumes
that Rthe current consumption ct is compared to the weighted aggregate
t
zt = −∞ e−a(t−τ ) cτ dτ or recent past ct−1 consumption either additively
u(ct − bzt ) [5, 20, 21] or multiplicatively via u(ct /cdt−1 ) [1, 8] in the
utility function. The papers [1, 2] and [11] attempt to combine both
habit formation and CuJ in a single utility function.
The present work examines some existing time-nonseparable utility
specifications and discuss their feasibility. Qualitative analysis is also
used to construct utility functions which satisfy both analytical requirements and empirical data.
2 Habit formation
2.1 Critique of the short-memory utility function Consider a
short-memory utility function in which the current consumption is compared to the past consumption during the last-period only, namely u =
u(ct , ct−1
). The typical form used in the literature (e.g., [1, 8]) is u =
u cdct , where d is a constant and u has an isoelastic form uCRRA (x) =
t−1
xγ /γ with a constant relative risk aversion −xu00 /u0 = 1 − γ. To see
whether such a utility function leads to a plausible optimum policy for
consumption, we maximize the life-time discounted utility
max
c1 ,...,cN
N
X
t=1
e
−ρt
uCRRA
ct
cdt−1
DEVELOPING UTILITY FUNCTIONS
705
subject to
N
X
ct = W 0 .
t=1
The maximization of the habit-forming objective functions in discrete time is done in Maple 12 using the function Maximize from the
Optimization package with parameter assume = nonnegative which accounts for non-negativity of consumption. The optimization algorithms
take advantage of built-in library routines provided by the Numerical
Algorithms Group (NAG). For nonconvex objective functions, Maple 12
searches for a local maximum. In principle, the optimization could be
carried out using the Lagrange method which would, in this case, lead
to a system of nonlinear algebraic equations. Numerical experimentation with the general case shows that the optimal solution exists and
smoothly transforms to the non-habit-forming solution when β → 0.
The numerical optimization was carried out using the following values for risk aversion 1 − γ = 0.5, consumption period N = 20 years,
discount factor ρ = 3%, initial wealth W0 = $1, 000, 000 and inherited
consumption c0 =$100,000 are shown in Figure 1. If consumption memory persists for one year only, consumption does grow with time for large
values of d. However, this choice also creates huge jumps in consumption
at the first and at the last year. In a particular case of the logarithmic
utility U (x) = log x (CRRA with γ → 0) with d = 1 and ρN 1, the
140000
Consumption
120000
100000
80000
60000
40000
20000
0
2
4
6
8
10
Time
12
14
16
18
20
d=0
d=0.1
d=0.8
d=1.5
d=2.5
FIGURE 1: Optimum consumption in a 20-year period under habit
formation with short (1-year) memory.
706
R. NARYSHKIN AND M. DAVISON
objective function simplifies to
max
c1 ,...,cN
N
X
e−ρt log
t=1
ct
ct−1
= max {(1 − ρN ) log cN
c1 ,...,cN
− log c0 + ρ(log c0 + · · · + log cN −1 ) + O(ρ2 )},
suggesting that it becomes optimal to wait until the end and consume almost all the wealth in the last year (c1 = c2 = · · · = cN −1 = O(ρ), cN =
W0 [1 − O(ρ)]), which seems unrealistic, because there is a trade-off between habit formation and the consumer’s time preferences described by
the discount factor.
A similar phenomenon is observed with an M -period utility for which
the objective function takes the form (d = 1):
max
c1 ,...,cN
N
X
t=1
e
−ρt
ct
uCRRA
c0 +
1
M
t−1
P
c t0
!
.
t0 =t−M
Numerical results for various memory lengths in Figure 2 show that the
length of the consumer’s memory M drastically changes his consumption in the first M and last M years, leaving intermediate consumption
almost unchanged.
Avoiding abrupt changes in the optimum consumption behavior requires memory of every past period. Only in this case does the consumption become smooth and give plausible habit forming behavior—
consumption increases as habits form, then becomes saturated and afterwards slowly decreases due to the discount factor ρ, displaying the
tradeoff between habit formation and the consumer’s time preferences.
The habit formation’s strength can be varied by introducing
a pa
ct
,
where
c̄t is
rameter β such that the utility function becomes u c0 +βc̄
t
1 Pt−1
the averaged past consumption c̄t = t−1 t0 =1 ct0 .
2.2 Critique of the additive utility function Another class of habit
formation utility functions includes full memory about past consumption. Many authors (e.g., [5, 20, 21]) consider the utility of the additive
Rt
form u(ct − bzt ), where zt = −∞ e−a(t−τ ) cτ dτ is the weighted aggregate
past consumption. For the isoelastic utility uCRRA (x) = xγ /γ, this specification implies that consumption ct cannot fall below bzt . As a result
of this “additive = addictive” utility, the optimum consumption suffers
DEVELOPING UTILITY FUNCTIONS
707
65000
Consumption
60000
55000
50000
45000
40000
0
5
10
Time
15
20
No-Memory Case (M=0)
Short Memory Case (M=3)
Medium Memory Case (M=N/5)
Long Memory Case (M=N/2)
Full-Memory Case (M=N)
FIGURE 2: Optimum consumption under habit formation with various
lengths of memory.
from the pathological solution allowing “bankruptcy” ct = 0 for some
t > tb . This pathology remains, even if it is slightly
R t reduced by replacing zt by the averaged consumption c̄t = c̄0 + 1t 0 cτ dτ, or its weighted
analogue, and simply stops working when b becomes sufficiently large.
One remedy might be to replace the CRRA utility by the constant
absolute risk aversion (CARA) utility uCARA (x) = −e−ηx /η therefore
allowing negative values of x. To see whether this is the case, consider
the problem of optimizing the objective function
max
c1 ,...,cN
N
X
e−ρt uCARA (ct − bc̄t ).
t=1
The optimum solution is shown in Figure 3. This consumption path
begins with an intuitively challenging initial trough. A better way to
fix the problem was already mentioned in the previous section. The
ct
is a good candidate for the
“multiplicative” utility u(ct , c̄t ) = u c̄0 +βc̄
t
habit formation both from a theoretical perspective as well as from the
point of view of the optimum solution behavior. Numerical solutions for
several parameters β are shown in Figure 4.
2.3 Utility function for habit formation Another type of utility
function of the form u(ct , c̄t ) = u1 (ct ) + βu2 (c̄t ) was also examined and
found to be unsuitable for the description of habit formation because
708
R. NARYSHKIN AND M. DAVISON
60000
Consumption
55000
50000
45000
40000
35000
30000
0
5
10
Time
15
20
b=0
b=0.3
b=0.6
b=1
FIGURE 3: Optimum consumption under habit formation with additive
utility.
Consumption
80000
60000
40000
20000
0
5
10
Time
15
20
beta=0
beta=0.1
beta=0.3
beta=0.5
beta=1
FIGURE 4: Optimum consumption under habit formation with multiplicative utility.
it led to decreasing consumption paths. In summary, just two simple
types of utility functions appear to give feasible optimum solutions for
consumption:
DEVELOPING UTILITY FUNCTIONS
•
•
709
ct
u(ct , c̄t ) = uCRRA
,
+ βc̄t
c̄0 ct
u(ct , c̄t ) = uCRRA d .
c̄t
The latter form of utility is used by Alvarez-Cuadrado et al. [2] and
Gómez [11]. However, they also include a second part that represents
CuJ and seems to be misused for that purpose, as explained in the next
section. Moreover, the power function of c̄t requires further adjustment
to make the numerator’s dimension ($/year) match the denominator’s.
3 Catching up with the Joneses
3.1 Critique of the multiplicative utility function The “Catching up with the Joneses” (CuJ) idea introduces external preferences into
the consumer’s behavior. An individual’s level of consumption ct is
directly related to the whole economy through the per-capita consumption Ct : if the economy grows, an individual will feel a need to increase
his consumption to maintain his happiness. Therefore, one expects a
positive change in the optimum consumption (relative to the standard
time-separable solution) with a positive change in per-capita consumption. Unfortunately, we will see that some of the popular models of
utility functions for CuJ lead to the opposite behavior.
Following Abel [1], Alvarez-Cuadrado et al. [2] and Gómez [11], we
consider the utility function of the form
ct
,
u(ct , Ct ) = u
CtD
where D is a tunable constant. We also assume, with Galı́, that the percapita consumption is a linear function approximately doubling every
30 years Ct ≈ C0 (1 + t/30) [9]. The numerical results are shown in
Figure 5. We can see that the consumption decreases even faster with
time than in the standard D = 0 model, opposite to the assumption of
growing economy. This multiplicative utility type is therefore unsuitable
for modeling CuJ. This result is easily extended to any growing function
Ct and is particularly obvious for exponentially increasing Ct = C0 eλt
which imply, for any positive D and isoelastic preferences, decreasing in
time utility
uCRRA (ct , Ct ) =
1 cγt
1 cγt −γλDt
=
e
γ CtγD
γ C0γD
710
R. NARYSHKIN AND M. DAVISON
100000
Consumption
80000
60000
40000
20000
0
5
10
Time
15
20
D=0
D=0.1
D=0.4
D=0.8
D=1
FIGURE 5: Optimum consumption under catching up with the Joneses
with multiplicative utility.
equivalent to the utility without CuJ discounted; however, with a bigger
factor ρ0 = ρ + γλD. Moreover, even the case where D is negative,
|D|
equivalent to the utility function u(ct , Ct ) = u(ct Ct ) considered by
Galı́ [10], fails to predict increasing consumption.
3.2 Utility function for catching up with the Joneses A hint
on modelling CuJ comes from the empirical “Easterlin Paradox” [4, 7]
which shows that, despite sharp rises in per-capita GDP, average happiness has remained constant over time (see Figure 6). Mathematically,
this behavior can be expressed as u(ct , Ct ) = const or
∂u dc
∂u dC
du(ct , Ct )
=
+
= 0.
dt
∂c dt
∂C dt
Empirical data [9] suggest that Ct ≈ C0 (1 + t/30) which gives dC/dt ≈
C0 /30 = const, and assuming ct to be approximately linear (suggested
by Figure 7) yields a simple partial differential equation of the form
∂u
∂u
+α
= 0,
∂c
∂C
with a constant α, which is easily integrated giving
u = u(ct − αCt ).
DEVELOPING UTILITY FUNCTIONS
Real per−capita consumption in USA ($)
4
4
x 10
3
711
Source: Penn World Table
http://pwt.econ.upenn.edu/
2
1
0
1970
1975
1980
1985
1990
1995
2000
2005
2000
2005
Average life satisfaction in USA (%)
100
80
60
40
20
0
1970
Source: World Database of Happiness
http://worlddatabaseofhappiness.eur.nl
1975
1980
1985
1990
1995
FIGURE 6: Real per-capita consumption and average happiness in the
USA, 1970–2006.
So the utility function for CuJ must be additive.
Further numerical analysis shows that, similar to the case of habit
formation, the additive utility cannot be assumed isoelastic (as in [13])
because of possible negative values of ct − αCt , so must be of the CARA
type. In Figure 7 we present optimal consumption for
u = uCARA (ct − αCt ).
dc
Note that the above equations imply that dC
dt = −α dt , where C is
the per capita income in the economy and c is the income of the given
agent we are studying. When α = 0 this implies that C = const. While
it is tempting to conclude from this that Catching up with the Joneses is
the impetus behind all economic growth, the reality is of course that our
model isn’t suited for making statements about the exogenous variable
C; it is focused on the endogenous variable c. The special case α = 0
places no additional restrictions on the optimal
consumption c.
ct
Two other types of utilities: u C0 +αC
and
u1 (ct ) + αu2 (Ct ) were
t
also considered and found unsuitable for CuJ because the former led to
decreasing consumption paths and the latter didn’t influence the optimum consumption at all (because Ct separates from ct ).
We conclude that empirical consumption growth evidence suggests
712
R. NARYSHKIN AND M. DAVISON
60000
Consumption
55000
50000
45000
40000
0
5
10
Time
15
20
alpha=0
alpha=0.1
alpha=0.2
alpha=0.3
alpha=0.4
FIGURE 7: Optimum consumption with catching up with the Joneses,
additive CARA utility.
a simple utility form which best describes consumption externalities
should be exponential CARA type u = uCARA (ct − αCt ).
4 Optimum consumption under habit formation
4.1 Differential equation that governs optimum consumption
Consider the deterministic optimum consumption problem under habit
formation in continuous time. We will use a habit formation utility
function that has no constraints on the current consumption c(t) (i.e.,
investor’s preferences are not addictive) in the form suggested in Section 2:
c(t)
U [c(t), c̄(t)] = U
,
c0 + βc̄(t)
where c0 is the inherited level of consumption at the moment t = 0, the
positive constant β is the memory parameter measuring the influence
of the past consumption on the present one (β = 0 corresponds to the
standard Merton formulation), and c̄(t) is the averaged past consumption until the present moment t :
c̄(t) =
1
t
Z
t
c(s) ds.
0
DEVELOPING UTILITY FUNCTIONS
713
Continuous time problems are more easily solved using a new variable
Z t
L(t) ≡
c(s) ds,
0
which represents the aggregate consumption over the life time. After the
change of variables, the objective function of the optimization problem
can be written as
Z T
L̇(t)
−ρt
e
U
(1)
max
dt + B[W (T ), T ] ,
c0 + βL(t)/t
L(s)
0
where the bequest function is
B[W (T ), T ] = bT e−ρT U
WT
,
W0
and wealth W is related to L via
L̇(t) = −Ẇ (t) = c(t).
The variational principle (Lagrange method) gives the set of equations
for determining the optimal solution:
d −ρt ∂U
∂U
,
e
= e−ρt
dt
∂L
∂ L̇
∂B
∂U
(T ) = e−ρT
(T ),
∂WT
∂ L̇
and the boundary conditions
L(0) = 0,
L(T ) = W0 − WT .
The case of the simplest logarithmic utility function (U (x) = log x)
gives the nonlinear second order ordinary differential equation (ODE)
L̈ + ρL̇ −
β L̇2
= 0,
c0 t + βL
L(0) = 0,
L(T ) + bT L̇(T ) = W0 .
For the CRRA utility function we obtain a slightly more complicated
equation
γ L̇L
ρ
β
L̇2 +
= 0,
(2)
L̈ +
L̇ −
1−γ
c0 t + βL
1−γ t
714
R. NARYSHKIN AND M. DAVISON
with boundary conditions
(3)
L(0) = 0,
L(T ) + b
1
1−γ
γ
L(T ) 1−γ
L̇(T ) = W0 .
T 1+β
W0
The differential equations studied here are very nonlinear boundary
value problems, with one boundary condition of Dirichlet type and the
other of Robin type. What is worse, since L(0) = 0, both equations
contain a singularity at the t = 0 boundary. It is natural to ask how
one could know that such an equation has a solution at all, let alone
a unique one. It is remarkably difficult to obtain results of this nature from the equations themselves, but we have additional information
about them. Solution existence comes from the fact that these equations are the Euler-Lagrange equations corresponding to a simple utility
maximization problem. While such problems are not in general unique
(think of finding geodesics on arbitrary surfaces), in this case we have
a great deal of additional structure because of the convexity of the utility function to be optimized which strongly implies uniqueness. These
equations would, however, repay more detailed scrutiny in their own
right. For the purposes of this paper, we simply take these heuristic
arguments, together with the stability of our numerical methods, and
proceed to draw economic conclusions.
In terms of numerical evidence, the existence of a solution to (2)–(3)
can be intuitively seen from the discrete version of the problem analyzed
earlier. As the number of consumption points Ci increases, the graph
containing the optimum solution becomes more dense while does not
change its form, so that in the limit N → ∞ it will converge to the
optimal solution which will be the solution of the derived ODE.
4.2 Numerical results The nonlinear differential equation (2) was
solved numerically in Maple 12 with dsolve. The dsolve routine uses a
midpoint method enhanced by deferred corrections (method = bvp[middefer]). Figure 8 shows the optimum consumption for various memory parameters β. We can see that habit formation radically changes
consumer behavior. The optimum consumption for a habit-forming consumer is qualitatively different from Merton’s consumer (who has β = 0).
Moreover, habit formation (measured by the memory parameter β)
and time preferences (measured by the discount factor ρ) are competitive
forces. Optimum consumption for various discount factors is shown in
Figure 9. While time preferences require higher consumption at initial
and lower consumption at later times, habit formation requires lower
DEVELOPING UTILITY FUNCTIONS
715
180000
Optimum Consumption
160000
140000
120000
100000
80000
60000
40000
20000
0
5
10
Time
15
20
beta=0
beta=0.01
beta=0.1
beta=1
beta=10
FIGURE 8: Optimum consumption under habit formation for various
memory parameters β.
consumption in the beginning and higher consumption later1 . The combination of the two effects results in the humped optimum consumption,
whose maximum decreases and shifts to the right for consumers with
stronger levels of habit formation. This represents the “pure” effect
of habit formation on optimum consumption without obscuring it with
interest income and the randomness of investment returns.
Another effect, already indirectly evident in Figure 8, is that the bequest (terminal wealth) increases with the strength of habit formation.
The exact relationship between the bequest WT and β is shown in Figure 10. We see that the lifetime savings of a habit-forming investor are
larger that those of the Merton investor.
The increased bequest under habit formation may be interpreted as
follows. Habit-forming investors prefer to consume less at initial times as
seen from Figure 9 and more or less comparable amounts at later times.
Those initial savings, together with the earned returns on the saved
amount (in the presence of investment opportunities), lead to the larger
bequest. From the analytical point of view, the utility of consumption in
(1) has a lower “weight” with higher β comparing to the β-independent
utility of bequest B. Therefore, as habits become stronger, utility is
more easily gained from bequest rather than from consumption favoring
larger bequests.
1 It is worth emphasizing that the initial optimum consumption c(0) equals neither
0 nor c0 and depends on β.
716
R. NARYSHKIN AND M. DAVISON
Optimum Consumption
50000
40000
30000
20000
10000
0
5
10
Time
15
20
beta=1, rho=1%
beta=1, rho=5%
beta=1, rho=10%
FIGURE 9: Optimum consumption under habit formation for various
discount factors ρ.
700000
600000
Bequest
500000
400000
300000
200000
100000
0
2
4
6
8
10
Beta
FIGURE 10: Optimum bequest under habit formation.
5 Conclusions This paper examines several widely-used utility
specifications designed to describe internal consumer preferences specifying consumption path based on past consumption (habit formation),
and preference externalities (catching up with the Joneses). Few of
these models lead to feasible patterns in the consumption path of the
consumer. Based on discrete time arguments, we recommend how to
DEVELOPING UTILITY FUNCTIONS
717
change the utility function to correctly reflect the consumer’s theoretical and empirical behavior.
80000
Consumption
70000
60000
50000
40000
30000
20000
0
5
10
Time
15
20
beta=0, alpha=0.3
beta=0.1, alpha=0.3
beta=0.3, alpha=0.3
beta=0.5, alpha=0.3
beta=1, alpha=0.3
FIGURE 11: Optimum consumption under habit formation and catching up with the Joneses.
The paper concludes that habit formation should be described by the
isoelastic CRRA utility of the form
γ
ct
1
ct
u(ct , c̄t ) = uCRRA
,
=
c̄0 + βc̄t
γ c̄0 + βc̄t
and that catching up with the Joneses should be described by the exponential CARA utility
1
u(ct , Ct ) = uCARA (ct − αCt ) = − e−η(ct −αCt ) .
η
These may be combined in a single utility function via the linear
combination
ct
u(ct , c̄t , Ct ) = AuCRRA
+ (1 − A)uCARA (ct − αCt ).
c̄0 + βc̄t
The case A = 1 corresponds to pure habit formation while A = 0 corresponds to pure catching up with the Joneses. It is easy to solve any value
of A. The optimum consumption for A = 1/2 is displayed in Figure 11.
718
R. NARYSHKIN AND M. DAVISON
Note the consumption behavior here is very similar to that displayed in
Figure 8.
In the final section of the paper we presented a continuous time formulation for the habit formation problem. This yields a nonlinear ordinary
differential equation the numerical solutions of which are qualitatively
consistent with the discrete time results in Section 3. In [16] we present
some analysis of this habit formation utility when used to modify the
standard Merton consumption-investment problem. A comprehensive
analysis of both analytical and numerical methods can be found in [17].
REFERENCES
1. A. Abel, Asset prices under habit formation and catching up with the Joneses,
Amer. Econom. Rev. 80 (1990), 38–42.
2. F. Alvarez-Cuadrado, G. Monteiro and S. Turnovsky, Habit formation, catching up with the Joneses, and economic growth, J. Econom. Growth 9 (2004),
47–80.
3. D. Bernoulli, 1954, Exposition of a new theory on the measurement of risk,
Econometrica 22(1) (1954), 23–36; Translated from Bernoulli, D., 1738, Specimen theoriae novae de mensura sortis, Commentarii Academiae Scientarium
Imperalis Petropolitanae, V, 175–192.
4. Clark, A. P. Frijters and M. Shields, Relative income, happiness and utility:
An explanation for the Easterlin paradox and other puzzles, J. Economic Literature XLVI (2008), 95–144.
5. G. Constantinides, 1990, Habit formation: A resolution of the equity premium
puzzle, J. Political Economy 98 (1990), 519–543.
6. J. Duesenberry, Income, Saving and the Theory of Consumer Behavior, Harward University Press, Cambridge, MA, 1949.
7. R. Easterlin, Does economic growth improve the human lot? Some empirical
evidence, in R. David and R. Reder (eds.), 89–125, Nations and households
in economic growth: Essays in honor of Moses Abramovitz, Academic Press,
New York, 1974.
8. J. Fuhrer, Habit formation in consumption and its implications for monetary
policy models, Amer. Econom. Rev. 90 (2000), 367–390.
9. J. Galı́, Finite horizons, life-cycle savings, and time-series evidence on consumption, J. Monetary Econom. 26 (1990), 433–452.
10. J. Galı́, Keeping up with the Joneses: Consumption externalities, portfolio
choice, and asset prices, J. Money, Credit and Banking 26 (1994), 1–8.
11. M. Gómez, Equilibrium efficiency in the Ramsey model with habit formation,
Berkeley Electronic Press, Stud. Nonlinear Dyn. Econometrics 11 (2007), 1–32.
12. D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under
risk, Econometrica 47 (1979), 263–291.
13. L. Ljungqvist and H. Uhlig, Tax policy and aggregate demand management
under catching up with the Joneses, Amer. Econom. Rev. 90 (2000), 356–366.
14. R. Mehra and E. Prescott, The equity premium: A puzzle, J. Monetary Econom.
15 (1985), 145–161.
15. O. Morgenstern and J. von Neumann, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944.
DEVELOPING UTILITY FUNCTIONS
719
16. R. Naryshkin and M. Davison, Numerical methods for portfolio optimization
under habit formation and transaction costs, Proc. China-Canada Industry
Workshop on Enterprise Risk Management (Wuhan, China, Nov. 25–30, 2008),
1–5.
17. R. Naryshkin, Portfolio optimization under habit formation and transaction
costs, PhD Thesis, University of Western Ontario, 2010.
18. H. Ryder and G. Heal, Optimal growth with intertemporally dependent preferences, Rev. Economic Stud. 40 (1973), 1–31.
19. P. Shoemaker, The expected utility model: Its variants, purposes, evidence and
limitations, J. Econom. Literature 20 (1982), 529–563.
20. W. Smith, Consumption and saving with habit formation and durability, Economic Lett. 75 (2002), 369–375.
21. S. Sundaresan, Intertemporally dependent preferences and the volatility of consumption and wealth, Rev. Finan. Stud. 2 (1989), 73–89.
Department of Applied Mathematics, University of Western Ontario,
1151 Richmond Street, London, ON, Canada.
E-mail address: [email protected]
Department of Statistical and Actuarial Sciences,
University of Western Ontario, 1151 Richmond Steet, London, ON, Canada.