Preliminary –Comments and Suggestions are Welcome
Intertemporal Effects of Consumption and Their
Implications for Demand Elasticity Estimates
Wesley R. Hartmann*
Graduate School of Business
Stanford University
[email protected]
November 2003
Abstract
The advent of scanner data has expanded the number of goods for which firms and
economists can estimate elasticities to assess pricing strategies, the potential for new
products, and the competitiveness of markets. An important caveat of such data is that
temporary price changes, which are often the source of identification in these studies, can
have demand effects in both preceding and subsequent periods. Recent work studying
storable goods has shown that static demand analyses ignoring these effects can yield
inaccurate elasticity estimates. Storability represents one source of these intertemporal
demand effects, however, consumption substitutability or complementarity across time is
another source. To empirically document the importance of intertemporal effects of
consumption and to demonstrate that these elasticity concerns spread beyond the realm of
durable and storable goods, this paper analyzes a good that is clearly non-storable: rounds
of golf. A dynamic model of consumer choice that allows for extensive heterogeneity is
estimated using recently developed techniques involving importance sampling.
Estimated model parameters suggest that consumption in one period significantly affects
past and future consumption choices, and that a static demand analysis ignoring these
consumption effects would overestimate the firm’s own-price elasticity by thirteen
percent.
* I am grateful to Dan Ackerberg, Phillip Leslie, Andrew Ainslie, Lanier Benkard, Harold Demsetz, Joe
Hotz, Matt Neidell, and participants of the Industrial Organization Proseminar at UCLA for their helpful
comments. I would also like to thank Ken Guerra and Steve Fendrick at American Golf for providing the
data. All errors are mine.
Accurate elasticity estimates form the basis for firms’ and economists’
assessments of pricing strategies, the potential for new products, and the competitiveness
of markets. Recent research facilitates these estimates by developing new techniques to
more precisely characterize substitution across both products (e.g. Berry, Levinsohn and
Pakes, 1999) and time (Hendel and Nevo, 2002 and Erdem, Imai and Keane, 2001).
Until now, empirical studies of substitution across time have focused on goods that can
be stored and consumed post-purchase. While storage constitutes an obvious source of
intertemporal substitution, consumption itself, say of a blockbuster movie, can also have
demand effects in subsequent time periods. An individual’s reduced (or increased)
probability of seeing the movie in the future should be incorporated in own-price
elasticity estimates. To isolate these types of consumption effects and to demonstrate
their importance to elasticity estimates, this paper analyzes a non-storable good (rounds
of golf) with data uniquely suited to this empirical application.
A consumer’s willingness to substitute a good across time affects elasticities in a
manner similar to a consumer’s willingness to substitute across the products offered by a
firm in a single time period. It is useful to think of a price decrease that yields a quantity
increase composed of both newly generated sales and sales taken from substitutes. The
substitutes may be other firms’ products, other products sold by the firm, or the same
products sold in different time periods. Just as a firm must know the magnitude of sales
stolen from its other products, it must also know the magnitude of sales stolen from other
time periods in which it sells the products.1
1
The degree of complementarity between a firm’s products, or between purchases of the same product over
time, are equally as important to the estimation of elasticities.
1
Though cross time and cross product elasticities are similar, a unique set of
technical difficulties is associated with the estimation of demand effects across time. In
particular, the econometrician does not know consumers' expectations of future prices.
Dynamic models can incorporate the effects of these expectations, however, the models
typically rely on strong assumptions about the formation of expectations. As will be
described in more detail below, the pricing strategy of the firm analyzed in this paper
includes future prices, suggesting that the model will be less reliant on assumptions about
consumers’ expectations.
The types of products exhibiting intertemporal demand effects often share the
feature that a purchase in a given time period has the ability to affect utility in later
periods.2 This allows an individual to purchase future utility in the present. In the case
of storable goods, a good purchased today can be “carried over” into future periods for
future consumption (and hence future utility). While the ability of the good to be
physically present in multiple time periods can obviously cause intertemporal
substitution, the “purchase carryover” need not be physical. The utility from
consumption can similarly carry over into future periods, resulting in a good purchased
and consumed today providing utility in both the present and future.
To understand how the utility from consumption may carry over into future
periods, consider a purchase of a round of golf. On Tuesday, a golfer purchases and
plays a round of golf. On Wednesday, Thursday, Friday, etc., the experience of Tuesdays
round remains with the golfer. The round may serve as a topic of conversation with
friends, or perhaps it provided a break in an otherwise stressful workweek. Whatever the
2
Intertemporal substitution may also arise when a purchase indirectly affects utility in future periods
through a budget constraint.
2
reason may be, the golfer may not find it as useful to play again until this experience
fades and needs to be replaced. Intertemporal substitution can arise when a future price
decrease leads the golfer to postpone the replacement of the experience until a day in
which the cost is significantly cheaper.
The idea that past consumption can affect current utility is not new, and in fact,
the existing literature highlights the possibility that recent consumption may actually
increase, or complement, current consumption. Stigler and Becker (1977) describe a
consumption capital through which past consumption affects current utility and hence
current choices. The behavioral economics literature also suggests that consumption
experiences in the past affect current utility (Elster and Loewenstein, 1992). Both pay
considerable attention to the concept of addiction or habit persistence, through which past
or recent consumption experiences can increase current and future purchases. The reason
a history of extensive music listening is associated with more listening is described as
either increasing ones tastes for music, or as developing a stock of music knowledge
upon which other music becomes more valuable. Such a force leads to
complementarities across time that may either outweigh or be dominated by the utility
carryovers described above as causing intertemporal substitution.
Empirically evaluating the intertemporal effects of consumption and their
implications for demand elasticities can be difficult because consumption is rarely
observed. However, the case of golf avoids this problem because the good is nonstorable, implying consumption at the time of purchase, which is observable. The
intervals between golf purchases therefore allow for the identification of intertemporal
effects of consumption. A positive relationship between the length of this interval and
3
the purchase probability suggests a fading experience that increases the individual’s
marginal utility over time: implying substitutability. However, a negative relationship is
indicative of habit persistence and suggests complementarity in that the marginal utility
of purchasing declines as the time since the last purchase increases.
I estimate a dynamic discrete choice model that allows for intertemporal effects
both forward and backward in time. The state variable, the time since the last purchase,
is a function of lagged dependent variables and therefore characterizes how a
consumption choice affects choices in subsequent periods. The effects of a choice on
preceding periods is captured through the dynamic programming problem in which
consumers choose their future lagged dependent variables by adjusting purchases in
anticipation of future price changes.
The effects of the state variable are not, however, easy to estimate in the presence
of heterogeneous consumers. In such cases, it is widely recognized that rich
heterogeneity must be included in estimation to properly identify the state dependence
(Heckman, 1981 , Keane, 1997). This paper introduces heterogeneity through a set of
random coefficients, estimable in this dynamic model by use of recently developed
methods involving importance sampling and a change of variables (Ackerberg, 2001).
Another valuable feature of the case studied here is the type of price variation at
the golf course. In particular, while experiencing a persistent excess supply on Sundays,
the course occasionally sent its consumers an email coupon three days in advance. The
course “mixed up” both the weeks in which it sent the coupon and its value so as not to
lead consumers to permanently adjust their playing patterns. Exogeneity of the price is a
result, but its advance notice also assists estimation in that there is observed variation in
4
price expectations. Specifically, while long-run price expectations will be left to
assumptions similar to those in the existing literature, consumers' expectations in the
three days prior to a price change are known with certainty.
The estimates of this dynamic model of consumer choice support the existence of
intertemporal effects of consumption. The duration since the last purchase is related to
the purchase probabilities, and forward-looking consumers will take this into account
when choosing their future states. In addition, elasticity estimates suggest that rounds on
the three days prior and three days following a price change are substitutes for the round
experiencing the price change. The change in sales on these six days cumulatively
represents approximately 8 percent of the change in sales on the day of the price change.
Accounting for this intertemporal substitution reduces the estimate of the own-price
elasticity by approximately 11 percent.
The paper proceeds as follows. The following section reviews the literature.
Section 2 describes the data. Section 3 presents the dynamic model of demand. Section
4 empirically implements the model. Section 5 presents results and Section 6 concludes.
1 Literature Review
Estimating intertemporal effects of consumption involves modeling similar to
studies of storable goods, but the nature of the state dependence is different. I begin by
discussing the literature on state dependence in demand generally (purchase carryovers),
then move on to address state dependence in consumption specifically. A discussion of
modeling issues involved in recent studies of dynamically optimizing consumers of
storable goods follows.
5
1.1 State Dependence in Demand
There are a variety of economic phenomena that can lead to state dependence in
demand. Consumers may store products (e.g. Hendel and Nevo, 2002, Erdem, Imai and
Keane, 2001). Their knowledge of the product may be based on past experiences with
the product (e.g. Allenby and Lenk, 1994, Erdem and Keane, 1996, Ackerberg, 1998).
Consumers may exhibit habit persistence, or variety seeking behavior (e.g. McAlister,
1982, Moeltner and Englin, 2001). In each of these cases, consumer decisions are based
in part on past purchases.
State dependence specific to consumption has been addressed by Stigler and
Becker (1977), which introduces the concept of consumption capital. A stock of past
consumption that can depreciate over time is described to affect current choices. Focuses
are the issues of addiction and habit persistence through which past consumption can
increase the marginal utility of current consumption. Related empirical work in Becker,
Grossman, and Murphy (1994) uses aggregate cigarette data to demonstrate the
implications of addiction for long-run and short-run elasticities, showing that both past
and future price changes impact current consumption. Though the aim of my paper is
similar, I use individual level data to identify state dependence in consumption causing
substitutability across time that may outweigh the complementarities associated with
habit persistence or addiction.
The issue of intertemporally substituting consumption of a good is somewhat
similar to studies involving life-cycle labor supply. In these studies, consumption
(generally) and leisure can be shifted across an individual’s lifetime in response to
6
changes in either the wage rate or interest rate.3 Intertemporal substitution of a single
good may involve similar shifting of consumption across time, such as from weekdays to
weekends when wage opportunities may be lower. However, the good relevant to any
firm or market is only one form of consumption and may involve these wage related
dynamics or a separate set of dynamics derived from the diminishing marginal utility of
consumption of the good itself, or other purchase carryovers described above.
1.2 Estimating State Dependence
The primary difficulty in identifying state dependence in individual level data is
insuring that the significance of the past purchases is not caused by heterogeneity. If
heterogeneity is not controlled for, frequent past purchases will be associated with greater
purchase probabilities because high value consumers purchase more frequently. If state
dependence exists, a recent purchase may actually cause a high or low purchase
probability. Keane (1997) describes the problem in detail and demonstrates the existence
of both state dependence and heterogeneity in ketchup purchases by analyzing a panel of
consumers and allowing for rich heterogeneity. By observing individuals over time,
patterns consistent with either high or low values for goods can be picked up.
The effect of heterogeneity on parameters intended to identify state dependence in
this paper is quite similar to that in ketchup. High value golfers tend to purchase more
frequently. If heterogeneity is not controlled for, the probability of purchase will appear
to be greater for those who have purchased recently. For goods such as ketchup where
habit persistence is the likely form of state dependence, the parameter describing past
3
For discussion of this literature see Blundell and MaCurdy (1999), or see MaCurdy (1981) for an early
empirical application.
7
choices will have the same sign, regardless of whether the state dependence or
heterogeneity is being described. Golf, and other goods involving substitutability across
time differ in that state dependence implies a different sign than heterogeneity. In such a
case, the inability to adequately account for heterogeneity can result in not realizing the
proper sign of the state dependence. The results presented in section 6 illustrate this
phenomenon in that, as additional heterogeneity is added, the sign of the state
dependence actually changes from negative to positive.
1.3 Dynamic Models of Consumer Choice
As stated previously, the papers modeling dynamic price effects, Hendel and
Nevo (2002) and Erdem, Imai and Keane (2001), focus on storable goods. While
storability is the basis for modeling the state dependence in both, the two papers differ in
their modeling because of the computational complexities associated with brand and
quantity choices in a dynamic framework.
In a model of dynamic behavior based on storability, there is a stock of the
product that can carryover across periods. Each period, the consumption choice depletes
the stock and the consumer has the ability to augment that stock through additional
purchases. Consumption itself is not observed, resulting in models that restrict
individuals to consume either a constant or stochastic amount each period.
Computational complexities arise because the state variable must summarize the
stock. In the case of many brands, the dimension of the state variable can be quite large.
Erdem, Imai and Keane (2001) choose ketchup because the number of brands and sizes is
small, thereby decreasing the size of the state space. Hendel and Nevo (2002) simplify
8
the analysis by assuming that all brands are identical after purchases. This reduces the
state space to only a quantity rather than multiple brands and quantities.
In this paper, the state space is quite small because of the nature of the data. First,
golfers have only three types of rounds from which to choose on a given day.4 Second,
the quantity choice is binary, because a golfer chooses to either play or not on a given
day. While there are a variety of possible specifications of the state variable, I choose the
number of days since the last round. It clearly picks up the intertemporal nature of the
data and has the nice feature that it is an integer that can be capped at a level after which
all golfers can be assumed inactive.
2 Data
The data consist of purchases of rounds of golf by a panel of consumers at a
single golf course. The panel is composed of 487 individuals who were members of a
frequent golf program during a 98-day period over which I have price data. Their
purchases of the three different types of rounds (18 holes, twilight which is typically
between 9 and 18 holes, and 9 holes or less) were recorded by swiping their membership
card at purchase. In all, the data provide observation of the type of round purchased, the
weekly price menu at time of purchase, and the duration since the last purchase.
Prices for golf are typically fixed for a given day and time, but have experienced
increasing variation in recent years. The industry has taken advantage of email
technology to decrease the rigidity of their price menu. The course studied here has
fixed prices for most days and times, except for Sundays (see Table 2-1 for summary
4
While other golf courses could be included to provide additional choices, the focus here is on a single
course. This implies that the carry over will be for golf at this course, rather than golf in general. This is
the primary concern of the firm and is a reasonable focal point for a highly differentiated product.
9
statistics of the prices by day of week and type of round). This course, unlike publicly
owned courses, experienced a regularly low demand for Sunday rounds before twilight.
To fill up the excess capacity, it therefore began sending email coupons for this time slot
to members of its frequent golf program. The value of the coupon and the weeks in
which it was sent were “mixed up” so that consumers would not permanently adjust their
play patterns. Though discussions with management indicated that a literal pricing
experiment was not conducted, the discussions did suggest that treatment of the price
variation as random is reasonable. This provides exogenous variation of a coupon value
that ranges between $0 (when there is no coupon) and $28. The coupon was also emailed
to consumers three days prior to the effective date. I observe these coupon mailings for a
98-day period during the summer of 2001.
10
Table 2-1
Prices by Day of Week and Type of Round
Obs
Mean Std. Dev.
Min
Max
11
Sundays
18-hole Round
Twilight Round
9-hole Round
14
14
14
60.3
44.0
22.0
10.2
0.0
0.0
42
44
22
70
44
22
Saturdays
18-hole Round
Twilight Round
9-hole Round
14
14
14
70.0
44.0
22.0
0.0
0.0
0.0
70
44
22
70
44
22
Weekdays
18-hole Round
Twilight Round
9-hole Round
70
70
70
50.0
34.0
22.0
0.0
0.0
0.0
50
34
22
50
34
22
The panel consists of American Golf Players Association (AGPA) frequent golf
club members. The members provide their address, birthdate and some additional
information when signing up. By geocoding the addresses, I am able to calculate the
distance from the course to the home of each golfer. This and the age of the individual
are characteristics that I will use in future versions of this paper to account for some of
the heterogeneity in purchase behavior.
These golfers swipe a membership card whenever they purchase a round at the
course.5 The swipe becomes a record in the AGPA database containing the member’s
identification number, the type of round purchased and the exact time of purchase.
Golfers reveal this data because every tenth round purchased is rewarded with a discount
voucher to be used at one of the courses. The data set used in this analysis only contains
purchases at the course in which the prices are observed. Summary statistics related to
the panel members’ purchases are presented in Table 2-2.
5
If the golfer does not have their card with them, there are other means by which they can get credit for the
round.
12
Table 2-2
Summary Statistics
Individual- and Course-Specific Variables
(Approximate Standard
Mean / Year) Deviation
(3.10)
0.0917
13
18 Hole Purchase
N
47,726
Mean
0.0085
Twilight Purchase
47,726
0.0087
(3.18)
9 Hole Purchase
47,726
0.0093
(3.41)
Days Since Last Round
Days Since Last Round Sq
60 Days Since Last Round
47,726
47,726
47,726
28.9
1287.1
0.1900
Minimum
0
Maximum
1
0.0930
0
1
0.0962
0
1
21.2
1392.8
0.3923
1
1
0
60
3600
1
A preliminary reduced form analysis of the data suggests that intertemporal
substitution may be present. An examination of the aggregate sales on the days
surrounding the 14 Sundays in which prices vary is presented in Table 2-3. The reduced
form equation that is estimated is as follows:
ydw = δ 0 d + δ1d pw + δ 2 X dw
where w represents a Thursday to Wednesday week, and d is a single day, or a group of
days. X dw is week and day, or day group, specific covariates, such as a time trend or
indicator for the 4th of July holiday. In the table, there are either 3 or 7 day group
controls. When the effects are aggregated, there are 3: days before, Sunday, and days
after. When the effects are estimated daily, there are 7 groups: Thursday through
Wednesday. The first group is that which is always left out of the regression. The four
price effects under the 4th of July are interactions with the 4th of July to allow the effect of
the 4th of July holiday to be different for different days of the week (or groups of days).
The first set of results in the column titled “Aggregated Effects” suggest that
rounds on the 3 days prior and 3 days after a Sunday price change are substitutes in that a
dollar change in the Sunday price suggests a 0.03 increase in course demand on the days
before and a 0.26 increase in the days after. As expected, the price effect on Sunday is
negative at -0.19, however this is less in absolute value than the cumulative change on the
other days. This suggests that any increased sales that may be caused by a Sunday price
decrease would be more than offset by lost sales on the surrounding days. However, the
P-Values note that not a single estimate is significant at more than the 85% level.
14
Table 2-3
Reduced Form Estimates of Price Effects
Aggregated Effects
Daily Effects
Coef. Std. Err. P-Value
Coef. Std. Err. P-Value
Price Effect on
Thursday
-0.11
0.10
0.26
Friday
0.08
0.10
0.42
Saturday
0.04
0.10
0.73
-0.15
0.10
0.15
Monday
-0.11
0.11
0.33
Tuesday
0.19
0.11
0.10
Wednesday
0.17
0.11
0.15
Rounds 3 Days Prior
Sunday Rounds
Rounds 3 Days After
0.03
0.16
0.85
-0.19
0.16
0.24
0.26
0.18
0.15
Controls
Day Group 2
-6.13
13.06
0.64
-17.88
8.44
0.04
Day Group 3
-14.89
14.24
0.30
-12.19
8.44
0.15
Day Group 4
3.59
8.44
0.67
Day Group 5
-3.61
9.16
0.70
Day Group 6
-19.92
9.16
0.03
Day Group 7
-22.17
9.16
0.02
0.36
1.95
0.86
*Price Effect Monday
0.00
0.06
0.96
*Price Effect Tuesday
0.21
0.06
0.00
4th of July Week
0.61
4.26
0.89
*Price Effect 3 Days After
0.17
0.10
0.12
To better understand the price effects, the second column of results titled “Daily
Effects” breaks out the results by the individual day. Significance is still quite limited
(e.g. the highest levels are at 85% for the day of the price change and 90% for the
following Tuesday) in this analysis. In addition, some odd substitution patterns arise in
that the Thursday prior to a price decrease, and the Monday after are actually considered
complements, while the Tuesday and Wednesday after are significant substitutes.
15
While some intertemporal substitution is suggested, I believe the limited number
of data points (only 14 observed price-quantity pairs for each day or group of days)
prevents a conclusive reduced form analysis. The behavioral model defined in the next
sections uses individual variation in the duration between purchases to help identify the
cross-price elasticities. In particular, the relationship between a purchase decision and
the number of days since the last purchase defines the intertemporal link between
purchases on different days. This, when considered together with the own-price effect on
Sundays, will identify the price effects across days.
3 Demand Model
The primary purpose of the demand model is to define a set of parameters that can
characterize substitution both backward and forward in time. The latter type of
substitution involving forward-looking purchase behavior necessitates that the modeled
consumers dynamically optimize. In addition to the intertemporal choices, the individual
also makes a discrete choice between the set of goods offered in a given time period. The
model specified below is therefore a dynamic discrete choice model of demand.
3.1 The Current Period Utility
The utility individual i receives each period conditional on the state,
S = {H , D, c} , and preferences, γ , is:
⎧⎪v0 + ε 0t
u ( Sit , yit , pit ;θ ) = ⎨
⎪⎩vijt + ε ijt
if yit = 0
if yit = j ∀
j ∈ {1,..., J }
(1)
vijt = γ ij 0 + γ i1 ⎡⎣ p jt ( Dt ) − α i c jt ⎤⎦ + ϕ ( H it ; γ ij 2 )
16
where vijt represents the utility associated with choice j at time t net of ε ijt , an
individual and time specific shock to preferences distributed extreme value. H it is the
number of days since the last purchase and is incorporated in the current period utility
through the non-linear function ϕ . The price of product j depends on Dt , the day of the
week, and ct , a Sunday type 1 round specific shock to prices (i.e. the coupon) that affects
an individual’s choice with probability α i . α i primarily accounts for cases in which
consumers do not receive the coupon (e.g. they fail to check their email). In a reduced
form sense, α i also may pick up cases in which consumers ignore the coupon because
they do not want the hassle, or because they made plans before the coupon was sent. v0
represents the utility of the outside good and is normalized to 0 , such that the utility of
each product will be measured as the difference in utility from consuming the particular
good relative to the utility of consuming the outside good.
3.2 Dynamic Optimization Problem
Dynamics arise in this model through the state variable H it . Specifically, the
consumer is modeled to recognize the implications the choice at time t has on H it +1 and
hence the utility received in period t + 1 . The present discounted value of future utility to
the individual is therefore:
⎡ ∞ τ −t
⎤
Vi ( Sit ; γ i ) = max E ⎢ ∑ β u ( Siτ , yiτ , pτ ; γ i ) | Sit , Π i ; γ i ⎥
Πi
⎣⎢ τ =t
⎦⎥
where Π is a set of decision rules mapping states, S s, to choices, y s. This value
function V ( Sit ; γ i ) is the solution to the following “Bellman’s equation”:
17
(2)
Vi ( Sit ; γ i ) = max ⎡⎣u ( Sit , yit , pt ; γ i ) + β EV ( Sit +1 ; γ i ) ⎤⎦
(3)
⎡α i V ( H it +1 , Dt +1 , ct +1 , ε t +1 ) pε ( d ε )
⎤
⎢ ∫ε
⎥
EV ( St +1 ; γ i ) = ∫ ⎢
⎥ pc ( dct +1 | St ) .
c ⎢ + (1 − α i ) ∫ V ( H it +1 , Dt +1 , ε t +1 | ct +1 = 0 ) pε ( d ε ) ⎥
ε
⎣
⎦
(4)
yit
where
The laws of motion of the state variables are quite clear from their definition. The
length of time since the last round, H it , increases by one whenever the outside good is
chosen, and goes to 1 whenever the individual chooses one of the three types of rounds.
⎧ H it + 1
⎪
H it +1 = ⎨ H it
⎪1
⎩
if
yit = 0 and
if
yit = 0 and
if
yit ∈ {1, 2,3}
H it < 60 ⎫
⎪
H it = 60 ⎬
⎪
⎭
H it is bounded above at 60 because consumers are assumed inactive once they have not
purchased in the past 60 days. The days of the week are numbered 1 to 7, beginning with
Monday.
⎧ Dt + 1
Dt +1 = ⎨
⎩1
if
if
Dt < 7 ⎫
⎬
Dt = 7 ⎭
The shock to prices is the way in which the model incorporates the coupon. Every
Thursday, as was the case with the emailed coupon, the golfer receives the price discount
for Sunday. The golfer therefore knows the Sunday price until the end of Wednesday,
day 3, after which a new price shock for the coming Sunday will be revealed on
Thursday. The Sunday price shock, c , therefore evolves as follows:
18
⎧ct
ct +1 = ⎨
⎩c ∈ {0, −10, −12, −17, −22, −25, −28}
if
if
Dt ≠ 3⎫
⎬
Dt = 3 ⎭
where c is a discrete random variable with the following probability function, reflecting
the frequency with which the discounts were sent in the data:
⎧3
⎪ 7
⎪1
⎪ 7
⎪1
⎪ 7
⎪
f c ( c ) = ⎨ 1
14
⎪
⎪ 114
⎪
⎪1
⎪ 14
⎪1
⎩ 14
if c = 0
if c = −10
if c = −12
if c = −17
(5)
if c = −22
if c = −25
if c = −28
The model of demand specified above is a frequency of purchase model.
Typically, one might think of such a problem as primarily having a historical component.
That is, the value an individual receives from purchasing in a given period t is related to
how long it has been since the last purchase. A purchase is made if there is a positive
surplus from purchasing, given the time since the last purchase.
A forward-looking component to the frequency of purchase problem is less
obvious. It implies that a consumer might wait to purchase, even if a purchase would
bring positive surplus. An obvious necessary condition for waiting is that the present
discounted value from waiting and purchasing in the future is greater than that of
purchasing today. This might seem unlikely because the mere presence of a discount
factor implies that future utility receives less weight in decision-making. However, if
future prices are lower, then the discounted future surplus associated with a delayed
19
purchase might be greater than the surplus of purchasing today. In this model, the
consumer knows the Sunday discounts three days in advance, with certainty. Given that
a discounted Sunday is cheaper than a Saturday, but still a weekend, it is quite possible a
golfer may delay a purchase until Sunday to take advantage of the lower price.
4 Empirical Implementation
In this section I specify the details of the demand model such that an estimable
likelihood function results. Following other dynamic discrete choice models (Rust,
1987), a joint contraction mapping is used to compute the discounted present value of
each choice, net of the time specific shock to preferences and conditional on the state and
preferences of individual i . Given these choice specific value functions, a discrete
choice model and the probabilities associated with each choice are defined. These
probabilities are then used to define a likelihood function conditional on the preferences
of a single individual. To allow for the identification of demand for a heterogeneous
group of consumers I then define a set of random coefficients. Integration over the
distribution of the random coefficients is shown to produce a new likelihood function
conditional on the distribution of the random coefficients. A computationally efficient
simulation method is then used to estimate the parameters that maximize this simulated
likelihood.
4.1 Value Function
Suppressing the subscript i and setting J = 3 , as is the case in the data, the
Bellman’s equation defined in the previous section can be written:
V ( St , ε t | γ ) = max ⎡⎣v jt + ε jt + β EV ( St +1 , ε t +1 | St , yt = j , γ ) ⎤⎦
yt
20
(6)
The approach used to solve this Bellman’s equation that involves the maximization over
a discrete number of choices is based on Rust (1987). Rather than iteratively solving for
the value function itself, I define a value function that is conditional on the choice yt = j
and that does not consider the taste shock ε jt :
V j ( St | γ ) = v j + β EV ( St +1 , ε t +1 | St , yt = j , γ )
(7)
This choice specific value function can then be solved for using a joint contraction
mapping.
4.1.1 Joint Contraction Mapping
In order to solve for the V s, the following two-period problem must be
considered:
(
)
⎛ α E max ⎡V S | γ + ε ⎤ | S , y = 0, γ +
⎞
) jt +1 ⎦ t t
j ( t +1
⎜ i
⎟
yt +1 ⎣
V0 ( St | γ ) = v0 + β ⎜
⎟
⎜⎜ (1 − α i ) E max ⎡V j ( H t +1 , Dt +1 | γ , ct +1 = 0 ) + ε jt +1 ⎤ | St , yt = 0, γ ⎟⎟
⎦
yt +1 ⎣
⎝
⎠
⎛ α E max ⎡V S | γ + ε ⎤ | S , y = 1, γ +
⎞
) jt +1 ⎦ t t
j ( t +1
⎜ i
⎟
yt +1 ⎣
V1 ( St | γ ) = v1 + β ⎜
⎟
⎜⎜ (1 − α i ) E max ⎡V j ( H t +1 , Dt +1 | γ , ct +1 = 0 ) + ε jt +1 ⎤ | St , yt = 1, γ ⎟⎟
⎦
yt +1 ⎣
⎝
⎠
(8)
⎛ α E max ⎡V S | γ + ε ⎤ | S , y = 2, γ +
⎞
) jt +1 ⎦ t t
j ( t +1
⎜ i
⎟
yt +1 ⎣
V2 ( St | γ ) = v2 + β ⎜
⎟
⎜⎜ (1 − α i ) E max ⎡V j ( H t +1 , Dt +1 | γ , ct +1 = 0 ) + ε jt +1 ⎤ | St , yt = 2, γ ⎟⎟
⎦
yt +1 ⎣
⎝
⎠
⎛ α E max ⎡V S | γ + ε ⎤ | S , y = 3, γ +
⎞
)
j ( t +1
jt +1 ⎦
t
t
⎣
⎜ i
⎟
yt +1
V3 ( St | γ ) = v3 + β ⎜
⎟
⎜⎜ (1 − α i ) E max ⎡V j ( H t +1 , Dt +1 | γ , ct +1 = 0 ) + ε jt +1 ⎤ | St , yt = 3, γ ⎟⎟
⎦
yt +1 ⎣
⎝
⎠
(
(
(
(
)
(
)
(
)
(
Given that ε is distributed extreme value,
21
)
)
)
)
⎛ 3 V ⎞
E max ⎡⎣V0 + ε 0 ,...,V3 + ε 3 ⎤⎦ | V0 ,..., V3 = .5772 + ln ⎜ ∑ e j ⎟
⎝ j =0 ⎠
(
)
(9)
Substituting this into the problem above, we have:
⎛
⎞
⎛
⎞
⎛ 3 V ⎞
⎜ α i E ⎜ .5772 + ln ⎜ ∑ e k ⎟ | St , y = j , γ ⎟ +
⎟
⎝ k =0 ⎠
⎝
⎠
⎜
⎟
V j ( St | γ ) = v j + β ⎜
⎟
3
c =0
⎜ (1 − α i ) E ⎛⎜ .5772 + ln ⎛ ∑ eVk ⎞ | St , y = j , γ ⎞⎟ ⎟
⎜
⎟
⎜
⎟
⎝ k =0
⎠
⎝
⎠⎠
⎝
(10)
The solution to this set of choice specific value functions is found by iteratively solving
for the set of equations, until the set converges.
4.2 Discrete Choice
The set of choice specific value functions defined above are used to define the
discrete choice probabilities:
J
Pr( yit | Sit ; γ i ) = ∑
{ yit = j}exp (V jit )
J
∑ exp (V )
j =0
k =0
(11)
kit
Given these defined probabilities of observing each choice, a likelihood across all
time periods for a single individual can be defined:
T
Li ( Si1 ,..., SiT , yi1 ,..., yiT | S0 , y0 , γ i ) = ∏ Pr( yit | Sit ; γ i ) p( Sit | Sit −1 , yit −1 )
(12)
t =1
This likelihood incorporates a conditional independence assumption:
p( Sit +1 , ε it +1 | Sit , ε it , y, γ ) = q(ε it +1 | Sit +1 ) p( Sit +1 | Sit , yit )
To this point, the focus has been on defining a likelihood conditional on the
preferences of individual i . However, the data include a heterogeneous group of
individuals for which only the means and variances of their preferences will be
22
(13)
identifiable. Therefore, the following set of random coefficients is defined to represent
these preferences:
γ i = γ + Γηi
(14)
where Γ is the Cholesky decomposition of the variance-covariance matrix, Σ . To ease
the computational burden, I assume the elements in η are distributed i.i.d. normal such
that Σ is diagonal. The key drawback of this assumption is that if the current period
utility of each choice has its own specific intercept, as in equation (1), then utilities for
the three types of rounds of golf will be uncorrelated. In such a case, heterogeneity in
consumers’ overall tastes for golf will not be included. However, such tastes probably do
exist, so I impose correlation by redefining the current period utilities as follows:
vi1t = γ i10 + γ i1 p1t + γ i 2 H it + γ i 3 H it2 + γ i 4 ( H it = 60 ) + γ i 5 ( D > 5 )
vi 2t = γ i10 + γ i 20 + γ i1 p2t + γ i 2 H it + γ i 3 H it2 + γ i 4 ( H it = 60 ) + γ i 5 ( D > 5 )
vi 3t = γ i10 + γ i 30 + γ i1 p3t + γ i 2 H it + γ i 3 H it2 + γ i 4 ( H it = 60 ) + γ i 5 ( D > 5 )
(15)
p1t = p1t ( Dt ) − α i ct
Notice that the intercept of the first type of round, γ i10 , enters into the current
period utility for each of the other types of rounds as well. This term may therefore be
considered to represent consumer tastes for 18 hole rounds of golf, while the terms γ i 20
and γ i 30 represent the deviations from that value for the twilight and 9-hole golf rounds.
This set of current period utilities also highlights the empirical specification of the
other parameters to be estimated. In particular, the days since last round, H it , are
assumed to enter in a quadratic form, with an indicator for having reached the maximum
23
of 60. Preferences for weekend golf are captured in the term γ i 5 , and the effect of prices
on choice utilities is γ i1 .
Introducing the random coefficients allows the likelihood to now be expressed in
terms of the model parameters θ = {γ , σ } :
T
Li ( Si , yi | θ ) = ∫ ∏ Pr( yit | Sit ;ηi , γ , Σ) p ( Sit | Sit −1 , yit −1 ) f (ηi ) dη
(16)
t =1
This expression simplifies to:
T
Li ( Si , yi | θ ) = ∫ ∏ Pr( yit | Sit ;ηi , γ , Σ) f (ηi ) dη
(17)
t =1
because p( Sit | Sit −1 , yit −1 ) = 1 due to the deterministic evolution of S .
4.3 Simulation
The way one would initially think about estimating a problem such as that defined
above would be to simulate the integral over η and use a maximization algorithm to
search for the θ that maximizes the probability of observing the actual purchases. The
problem with such an approach in the model specified above is that the dynamic
programming problem will need to be solved N × NS × R times, where NS is the number
of sample draws for each individual and R is the number of iterations required for θ to
converge in the maximization routine. Ackerberg (2001) describes a change of variables
and importance sampling technique that can be used to reduce the computational burden
such that the dynamic programming problem only needs to be solved N × NS or N
times.
24
4.3.1 Importance Sampling and a Change of Variables
The integral in equation (17) will still be simulated, but over a newly defined, or
changed variable, γ i . The maximization algorithm will differ in that the search for θ
will be to weight a set of simulated individuals, rather than find the set of simulated
individuals, such that the probability of observing the actual purchases is maximized.
4.3.1.1 Change Of Variables
I will define the following change of variables:
ui = γ + Γηi
(18)
such that ui is distributed g (ui ) . A draw from g (ui ) will therefore constitute a
simulated individual.
4.3.1.2 Importance Sampling
As stated above, I will use importance sampling to simulate the integral in (17).
Dividing and multiplying (17) by g (ui ) :
T
Li ( Si , yi | θ ) = ∫ ∏ Pr( yit | Sit ;ηi , γ , Σ)
t =1
f (ηi )
g ( ui ) dη
g ( ui )
(19)
Simulating the integral by taking draws, uns , we get:
Li ( Si , yi | θ ) =
1
NS
NS
⎡
∑ ⎢π ( S , y | η
ns =1
⎣
i
i
ns
,γ , Σ)
f ( uns | γ , Σ ) ⎤
⎥
g ( uns ) ⎦
(20)
T
where π ( Si , yi ,| ηns , γ , Σ ) = ∏ Pr( yit | Sit ;ηns , γ , Σ) . For the first iteration with the initial
t =1
set of parameters, θ1 , the expression
f ( uns | γ , Σ )
= 1 and is therefore identical to how
g ( uns )
25
one would typically simulate (17). However, θ r for each successive iteration of the
maximization algorithm will imply a different weight,
f ( uns | γ r , Σ r )
. Intuitively, we
g ( uns )
can think of the γ r implying an η for each simulated individual, uns . The maximization
algorithm will seek the γ that implies smaller η s for simulated individuals who better
reflect the observed purchases. A smaller η implies a larger numerator in the weighting
function and hence a greater weight for the simulated individual.
4.3.2 Construction of the Likelihood Function
Given the simulation in (20), the likelihood function is:
N
L = ∏ Li
(21)
i =1
The parameter values that maximize this likelihood are consistent as NS approaches
infinity (Gourieroux and Monfort, 1996). Standard errors of the model parameters are
calculated using the outer-product of a numerical gradient of the likelihood function. It
should also be noted that current estimates of the standard errors reported in the following
section do not account for simulation error.
5 Results
The research strategy implemented to accurately characterize the elasticities of
this non-storable good involves the use of estimated behavioral parameters from the
model to calculate a set of elasticities over time. The following subsections present the
estimates of the behavioral model and describe the methodology and estimates of the
elasticities.
26
5.1 Model Estimates
Two versions of the discrete choice model specified above are estimated. First, I
estimate a simple logit model that does not allow for heterogeneity or forward-looking
behavior (i.e. no dynamic programming problem). Then I estimate the version of the
model specified in the previous sections. It is noticeable that as heterogeneity and
dynamics are added into the model, the state dependence evolves. The simple logit
model, with biased state dependence because it does not account for heterogeneity,
suggests habit persistence and complementarities across time, while the full model with
heterogeneity (i.e. random coefficients) and dynamics suggests intertemporal
substitution.
Table 5-1 presents the estimates of each of the models. The estimates for most
parameters are stable throughout. The intercepts and indicators for different types of
rounds maintain the same ordering in each of the models estimated. Those rounds with
more holes are preferred. The price coefficient is significantly negative. Weekend
rounds are always preferred to weekday rounds. The only parameter estimates that
change across models are those relating to state dependence.
27
Table 5-1
Summary of Results
Simple Logit
Coeff
28
Current Period Utility
Intercept
Twilight Indicator
9 Hole Indicator
Price
Days Since Last
Days Since Last Sq /1K
60 Days Since Last
Weekend
alpha
-1.119
-0.805
-1.497
-0.049
-0.037
-0.047
-0.002
0.542
SE
Dynamic Logit with RC
Tstat
0.277
0.091
0.171
0.005
0.006
0.131
0.224
0.053
Discount Factor
Standard Deviations of Random Coefficients
Intercept
Twilight Indicator
9 Hole Indicator
Price
Days Since Last
Days Since Last Sq
60 Days Since Last
Weekend
alpha
-4.0
-9.9
-23.7
-9.3
-6.7
-0.4
0.0
10.2
Coeff
-1.366
-1.641
-2.317
-0.066
0.026
-0.464
0.863
0.376
0.675
SE
Tstat
0.045
0.109
0.087
0.002
0.002
0.037
0.176
0.091
0.257
-30.3
-15.0
-26.8
-36.7
11.5
-12.5
4.9
4.1
2.6
0.0231
0.0842
0.0678
0.0010
0.0010
0.0143
0.0672
0.0531
0.0048
16.4
15.2
13.3
15.4
20.2
23.5
23.1
18.0
64.8
0.990
0.3793
1.2829
0.8990
0.0154
0.0202
0.3358
1.5496
0.9533
0.3112
Parameters estimating state dependence should be expected to change as
additional heterogeneity is incorporated. The parameter estimating how “days since last
round” relates to the value of a particular choice is negative in the simple logit. The
negative coefficient would tend to suggest habit persistence. However, if heterogeneity is
not accurately controlled for, as is the case for the simple logit, the negative coefficient is
likely picking up the fact that avid golfers are likely to have fewer days between rounds.
The full model estimates involving heterogeneity introduced through random
coefficients suggest there is a large degree of heterogeneity. As expected, when
controlling for heterogeneity, the coefficient on days since last round is positive.
The parameter estimates also support the presence of forward-looking behavior.
The state dependence parameters in the models with random coefficients are consistent
with the intertemporal linkage of utility. The positive first order effects suggest
substitutability across time.
5.2 Elasticities
The significant link between demands across periods is the basis for intertemporal
substitution that can be characterized by dynamic price elasticities. In fact, the set of
behavioral parameters imply a set of dynamic price elasticities that can be analytically
solved for the three periods before and after a price change. The estimated elasticities
will be for a set of individuals simulated from the set of random coefficients and there
standard deviations..
Using a decision tree starting on a Thursday in which consumers receive the price
for an 18 hole round of golf on the following Sunday, I am able to calculate the expected
29
quantity of rounds on each day for seven consecutive days.6 I do this for two different
Sunday 18-hole prices and calculate the elasticity using the following midpoint method:
ηt =
1
qt′ − qt
1
2
(
qt′ + qt
)
2
(p′+ p )
0
p0′ − p0
0
(22)
for t between –3 and 3. The q s are the expected number of rounds (of all types for all
days, except day 0 for which only the 18 hole rounds are included) for the individual on a
given day. The elasticities are reported in Table 5-2.
6
I do not calculate the elasticities for more than a week because the seven possible prices in the following
week drastically increases the number of branches on the decision tree.
30
Table 5-2
Dynamic Elasticity Estimates
For a Change in the Sunday 18-Hole Price
31
Days Relative to
Price Change
Day of the
Week
Elasticity for
Playing Any Type
-3
-2
-1
0
1
2
3
Thursday
Friday
Saturday
Sunday
Monday
Tuesday
Wednesday
0.0052
0.0054
0.0084
-3.8431
0.0083
0.0032
0.0038
Each of the elasticities has the appropriate sign. The cross-price elasticities (i.e.
those with respect to other time periods) are all positive suggesting that rounds of golf on
surrounding days are substitutes. The elasticities decrease as the time period increases or
decreases from the time of the price change.
The elasticities on the three days following a price change begin at 0.0083 and
decrease to about 0.0035. The forward-looking elasticities range from 0.0052 to 0.0084.
Together, the rounds lost on the three days preceding and three days following a Sunday
price change represent 8% of the total quantity change on Sundays.
At first glance, the price elasticity of the Sunday rounds of golf appears to be –
3.8431. However, if the effects on surrounding days are accounted for, the price
elasticity net of both cross-product and cross-time substitution is more appropriately –
2.9860, after accounting for the 8% of the Sunday sales that were cannibalized from the 3
days before and after. Had the adjustment only included the cross-product effects, the
elasticity would have only been -3.3616. Therefore, the elasticity drops an additional
11% when the dynamic effects are included. The elasticity could be reduced even further
if substitutability beyond Wednesday were included. Unfortunately, the analytical
methods used to this point cannot extend beyond a seven-day period because of
computational issues.
6 Conclusion
This paper demonstrates that an individual's willingness to substitute consumption
across time can result in misleading estimates of demand elasticities. Estimated
parameters from a dynamic model of consumer choice show that consumption of a good
32
can affect choices in both past and future periods. These parameters imply a set of
dynamic price elasticities that suggest purchases in the three days prior and three days
following a price change are substitutes for the round with the changing price. Failure to
account for such intertemporal substitution results in a 13 percent overestimate of the
own-price elasticity of the firm.
The existence and degree of intertemporal substitution of consumption choices
also has important implications for the nature of consumer choice dynamics and the types
of goods analyzed in a dynamic context. First, utility carryovers may be relevant for a
variety of goods, including storable goods. The existence of multiple sources of
intertemporal effects makes it difficult to understand the nature of the effects and presents
difficulties in empirically distinguishing them. As an example, future choices may be
affected because the good is in storage and ready for future consumption, or the good
may have been consumed at purchase and the effects of that consumption may remain
into the future. Second, intertemporal effects of consumption raise dynamic concerns for
non-durable and non-storable goods that have traditionally been analyzed statically. As a
result, a wider scope of goods should be modeled with dynamic optimization in mind.
The empirical evidence presented in this paper analyzes a single good to highlight the
importance of these broader issues.
33
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