We extend the Ravn, Schmitt-Grohe and Uribe (Review of Economic Studies, 2006) model of external deep-habits with the idea that some product varieties are more prone to habit formation than others. This s uncertainty in habit formation which affects firm's pricing. Provided that uncertainty is strong, a profound implication is that role of market frictions, such as habit formation, and its consequences for the dynamic variations in the markups can be reversed.

Ali Choudhary

Young-Bae Kim

English

Research Papers in Economics

Habit Formation: Deep and Uncertain 
Ali Choudhary Young-Bae Kim
University of Surrey www.econ.bbk.ac.uk
Abstract
We extend the Ravn, Schmitt-Grohe and Uribe (Review of Economic Studies, 2006) model
of external deep-habits with the idea that some product varieties are more prone to habit
formation than others. This creates uncertainty in habit formation which affects firm's
pricing. Provided that uncertainty is strong, a profound implication is that role of market
frictions, such as habit formation, and its consequences for the dynamic variations in the
markups can be reversed.
Citation: Choudhary, Ali and Young-Bae Kim, (2007) "Habit Formation: Deep and Uncertain." Economics Bulletin, Vol. 4, No.
5 pp. 1-7
Submitted: November 2, 2006.  Accepted: February 18, 2007.
URL: http://economicsbulletin.vanderbilt.edu/2007/volume4/EB-06D10002A.pdf1
1. Introduction
The important contribution by Ravn, Schmitt-Grohé and Uribe (2006, RSU henceforth) presents
a model where customers develop external habits as in Abel (1990) over varieties of consumer goods.
In such a framework, the price mark-up policy is countercyclical in that a rise in current demand
increases the importance of short-run elasticity relative to long-run elasticity- in the form of habits
over varieties. Moreover, current pricing also aﬀects future business as ﬁrms expect habit formation
to take place through time. Therefore, when ﬁrms expect high future return they encourage habit
formation by lowering their markups today; an idea reminiscent of the customer-market theory of
Phelps and Winter (1970).
However, this picture ignores the possibility that at any given point in time households are not
always certain about the varieties their neighbors are buying, unless of course there is a complete
ﬂow of information on the consumption of goods between neighbors. While some varieties may be
infectious in the sense of ‘catching-up with the Jonses,’ others would fail to generate such desires.
This is perhaps because some varieties are easily observable and distinguishable (i.e. conspicuous)
than others ; for example cars and mobile phones against jewelry sets and evening dresses. Therefore,
it is plausible that varieties are not equally habit-forming which in turn introduces some uncertainty in
the process of habit formation. The RSU results overlook the relative importance of such uncertainty
and its potential impact on ﬁrms’ pricing policy.
In this paper we extend the RSU framework to include uncertainty in habit-formation. In doing
so we ﬁnd two important results: a) the intertemporal eﬀe c t so fh a b i t sm a t t e rl e s sf o rp r i c i n ga n db )
markups are not necessarily countercyclical.
The following section lays down the model. Section 3 solves for a general equilibrium followed by
a concluding section.
2. The Model
Assume an economy where there is a continuum of j representative households of measure one
so that j [0,1].A t t i m e t each household consumes a variety of diﬀerentiated goods indexed by
i [0,1]. In the ﬁrst instance, the household seeks to minimize total expenditure over consumption
subject to an aggregation constraint so that2
Ptx
j
t =
1 Z
0
Pitc
j
itdi, (1a)
s.t x
j
t =
⎡
⎣
1 Z
0
³
c
j
it − αθcit−1 +( 1− α)zt
´1− 1
η
di
⎤
⎦
1
(1− 1
η)
, (1b)
cit−1 =
1 Z
0
c
j
it−1dj, zt = ρzt−1 + εt, and εt ∼ iid
¡
µ,σ2¢
. (1c)
where Pt =
∙Z 1
0
P
1−η
it di
¸ 1
1−η
is the aggregate Dixit-Stiglitz price index, x
j
t denotes an aggregation
of jth agent’s consumption of i varieties, Pit is the nominal price of the ith variety, c
j
it is jth agent’s
consumption of variety i at time t. The agent displays a tendency for external habit formation in
varieties where cit−1 denotes an exogenously given average consumption over j agents of variety i in
period t−1.T h e t e r m θ captures the strength of the externality-eﬀect of habit formation. The process
of habit formation is also subject to uncertainty captured by the auto-regressive process zt where εt
is iid. The term α (0,1) captures the extent to which a given variety is prone to habit-formation
without uncertainty. For example, α =0implies a habit-forming process dominated by uncertainty.
The term η is the immediate demand elasticity.
Using standard methods, the optimal level of c
j
it is
c
j
it =
µ
Pit
Pt
¶−η
x
j
t + αθcit−1 − (1 − α)zt. (2)
Eq. (2) is jth agent’s demand function for variety i that is adjusted for habit formation and
uncertainty. Thus, demand is falling in relative prices but increasing in habit-adjusted consumption.
Note that a rise in x
j
t raises the importance of short-run elasticity; hence elasticity is procyclical.
Demand positively depends of past average consumption through habits but is negatively aﬀected by
our random variable; the last two terms on the RHS of (2) respectively. For example when α =1 ,
habit formation over each variety occurs with certainty and its eﬀects would be fully reﬂected on the
demand function.
The cost-minimization problem set out above is secondary, the primal issue faced by the represen-
tative household is to maximize the utility over x
j
t and h
j
t (hours worked by household j)s u b j e c tt o
his/her period-by-period budget constraint. In this paper, the focus is on the role of uncertainty in
the habit formation process. Nonetheless, the introduction of uncertainty does not directly aﬀect the3
utility function and the budget-constraint of the households. Thus, the typical ﬁrst-order conditions
associated with the household’s utility maximization problem in RSU are used to generate the equi-
librium conditions for marginal substitutions and labour-leisure choice as shown below (see equations
8b and 8c).
2.1. Firms
Each diﬀerentiated good i is produced by ﬁrms in a monoplistiscally competitive environment.
The aggregate demand for each variety i is simply cit ≡
Z 1
0
c
j
itdj, which is the sum of (2) over j
households so that
cit =
µ
Pit
Pt
¶−η
xt + αθcit−1 − (1 − α)zt,x t ≡
1 Z
0
x
j
tdj (3)
Assume that each ﬁrm manufacturing variety i is endowed with a linear technology and employs
labour as the only factor of production with wage wt. Then, the proﬁto fﬁrm i in period t can be
expressed as1
Φit =
µ
µit − 1
µt
¶
cit, (4)
where µit and µt denote the markup (price-cost margin) of variety i and the average economy-wide
markup respectively.
The objective of the ﬁrm is to maximize the expected present value of proﬁts given by
Max
µit,cit
Et
∞ X
j=0
rt,t+1Φit+j (5)
subject to a process for the discount rate r, (3) and µt.
The Lagrangian for the ﬁrm’s optimization problem is
L = E0
∞ X
j=0
r0,t{cit
µ
µit − 1
µt
¶
(6)
+vit
"µ
µit
µt
¶−η
xt + αθcit−1 − (1 − α)zt − cit
#
},
where vit is the shadow value of selling a marginal unit of the variety i.T h e ﬁrst-order conditions
associated with (6) are
vit =
µ
µit − 1
µt
¶
+ αθEtrt,t+1vit+1, (7a)
cit = ηvit
µ
µit
µt
¶−η−1
xt. (7b)4
Eq. (7a) says that the value of an extra unit of sale is composed of the unit proﬁt, the ﬁrst term on
the right-hand-side, and the uncertainty-adjusted expected value of future sales, the second term. For
example when α =1 /2,t h eﬁrm values future sales half as much due to the uncertain nature of habits
for its variety. The second condition (7b) says that at the optimum the marginal beneﬁtf r o mr a i s i n g
the markup (the left-hand-side) —arising from selling all output at a higher prices— must equate to its
marginal cost in the form of lower demand evaluated at its shadow value. The ﬁrst-order-condition
collapses to the text-book monopoly markup, η/(η − 1), without habit formation (i.e., θ =0 ) .
3. Equilibrium
By restricting ourselves to symmetry across ﬁrms and households (i.e. dropping both i and j)w e
a r ea b l et od e ﬁne the steady-state using the following set of equations
Demand; xt = ct − αθct−1 +( 1− α)zt, (8a)
Marginal Substitution; 0=βUx(xt+1,h t+1) − Ux(xt,h t)rt,t+1, (8b)
Labour-Leisure Choice; wt = −
Uh(xt,h t)
Ux(xt,h t)
, (8c)
Markup; µt = At/wt, (8d)
Shadow Value of Marginal Unit; vt = αθEtrt,t+1vt+1 +1−
1
µt
, (8e)
Combining (8a) and (7b) ct = ηvt(ct − αθct−1 +( 1− α)zt). (8f)
What sets these conditions apart from those presented in RSU is the presence of uncertainty in
(8a) (8e) and (8f). Indeed with high uncertainty (e.g. α =0 )the ﬁrm cares less about the future
in (8e) and together with (8f) it will charge a markup with a premium on the text-book monopoly
markup to correct for the presence of uncertainty as we show below. Let us consider the mark-up for
the more general case. Manipulating (8f) together with (8e) and using standard methods we obtain
the general time-varying markup equation
µt =
∙
1 −
1
η(1 − (αθct−1 − (1 − α)zt)/ct)
+ αθEtrt,t+1vt+1
¸−1
(9)
This expression is convenient as it deﬁnes the markup in terms of short-run and long-run elasticities
adjusted for the presence of uncertainty habit formation. Consider the simplest case where α =1
and θ =0so that there are no uncertainty and habit formation. In this case, the markup collapses
to the constant text-book monopoly markup
η
η−1.5
Now consider the case where there is no uncertainty (α =1 )but habits accrue with strength θ>0.
In this case the general markup collapses to that of RSU,
µt =
"
1 −
1
η(1 − θ
ct−1
ct )
+ θEtrt,t+1vt+1
#−1
(10)
The term η(1−αθ
ct−1
ct ) captures the short-run elasticity and it rises with current consumption ct,
and as a result markups fall. Hence, markups appear countercyclical. Furthermore, a rise in the
expected value of sales of the marginal unit, through the creation of habits θ evaluated at its shadow-
value vt+1, induces ﬁrms to reduce markups today; this is the famous long-run customer-market eﬀect
which is responsible for the intertemporal dynamics in the markup.
The eﬃcacy of both the eﬀects discussed above is reduced in the presence of uncertainty. Indeed,
from (9), the short-term elasticity is η(1 − (αθct−1 − (1 − α)zt)/ct) and its procylicality very much
depends on α, the certainty with which customers form habits. A suﬃciently low α will dampen the
role of countercyclicality and in the limit it may in fact raise the markup to correct for the fact that
ﬁrms have to bear sales uncertainty; reversing the RSU result of procyclical elasticities. Moreover, a
low α implies that ﬁrms care less about the future, as habit-formation is less likely to occur, and thus
invest less on the expansion of their customer-base by trimming markups.
4. Conclusion
In this paper we revisited the Ravn, Schmitt-Grohe and Uribe (2006) results that in the presence
of external habit formation: a) short-run elasticities are procyclical hence markups are countercyclical
and b) ﬁrms invest on their future customer-base by lowering their current markups. We ﬁnd that
these results qualitatively change when there is uncertainty in habit formation as some products may
fail to generate suﬃcient public interest. The context in which we develop our results is simple,
but extending it to include other sectors such as the government will make a quantitative diﬀerence.
A profound implication is that shifts in aggregate demand will induce economy-wide countercyclical
markups if-and-only-if the proportion of products that create market frictions in the economy, in the
form of habit formation, outweigh those which constrain the generation of such frictions.
Acknowledgments
We would thank Paul Levine, Gylﬁ Zoega, Vasco Gabriel, Simra Khan and Joohee Chung for
helpful discussions. The usual disclaimer applies.6
References
Abel, A, B., 1990, Asset Price Under Habit Formation and Catching Up with the Joneses, The
American Economic Review Papers and Proceedings, 80, 38-42.
Ravn, M., Schmitt-Grohe, S., and Uribe, M., 2006, Deep Habits, Review of Economic Studies, 73(1),
Phelps, E. S., Winter, S. G., 1970, Optimal Price Policy Under Atomistic Competition, in E. Phelps,
Ed., Microfoundations of Employment and Inﬂation Theory, New York, NY: W.W Norton.
Notes
1The linear production technology for ﬁrm i is given by yit = Athit where yit denotes output of
variety i, At denotes an aggregate technology shock and hit denotes labour input. Firm i must satisfy
yit ≥ cit, then we can rewrite the aggregate demand as cit = Athit or cit
At = hit. Nomainal marginal
costs are given by MCt =( Ptwt/At) where wt/At is real marginal costs and wt is the real wage rate.
The proﬁto fﬁrm i in period t is given by Φ
j
t =
³
Pit
Pt
´
cit −wthit. The markup of prices over marginal
costs charged by ﬁrm i is given by µit = Pit/MCt and the average markup charged in the economy is
given by µt = Pt/MCt.T h e nw ec a nr e w r i t eﬁrm i’s proﬁts as
Φ
j
t =
³
µitMCt
µtMCt
´
cit − wt
³
cit
At
´
=
³
µit
µt
´
cit −
³
MCt
Pt
´
cit =
³
µit
µt
´
cit −
³
MCt
µtMCt
´
cit
and thus
Φ
j
t =
³
µit−1
µt
´
cit.

Asset Price Under Habit Formation and Catching Up with the Joneses, The American Economic Review Papers and Proceedings,

Deep Habits,

Habit Formation: Deep and Uncertain

http://www.accessecon.com/pubs/EB/2007/Volume4/EB-06D10002A.pdf

Habit Formation: Deep and Uncertain

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