Intertemporal Effects of Consumption and Their Implications for Demand Elasticity Estimates

Transcription

1 Intertemporal Effects of Consumption and Their Implications for Demand Elasticity Estimates Wesley R. Hartmann* Graduate School of Business Stanford University 58 Memorial Way Stanford, CA May 2006 Abstract Consumption of a good typically diminishes the marginal utility of consuming more, but for how long? This paper adapts a model of consumption capital to allow consumption to have a lasting effect that diminishes the marginal utility of future consumption. Estimates of the model find that it takes the 25 th, median and 75 th percentile of consumers 9, 32 and 43 days for their marginal utilities to return to pre-consumption levels, and they are forward-looking with respect to these effects. This generates intertemporal substitution of consumption that leads to an overestimate of the own-price elasticity of demand of ten percent when it is estimated using temporary price changes. In addition to these implications consumption effects share with those of durable and storable goods, consumption effects also raise concerns for capacity constrained industries because the timing of consumption affects capacity utilization. In the empirical application in this paper, price variation in one time period generates substantial changes in capacity utilization in that period, but minimal changes in other periods because the intertemporal substitution is spread over many time periods. Keywords: consumption, discrete choice, dynamic programming, random coefficients JEL: L0, D2, C6 * I am grateful to Dan Ackerberg, Phillip Leslie, Andrew Ainslie, Lanier Benkard, Latika Chaudhary, Harold Demsetz, Michaela Draganska, JP Dube, Joe Hotz, Matt Neidell, Aviv Nevo, two anonymous reviewers and seminar participants at UCLA, Stanford GSB, Penn State, UBC, UC Berkeley Haas School of Business, and University of Chicago GSB 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.

2 Introduction One would expect the marginal utility for nearly all goods to be diminished immediately following consumption. However, the length of time that it is diminished is likely to vary across both goods and consumers. When eating, diminishing marginal utility eventually leads people to stop eating until the utility from the meal (i.e. the satiation of hunger) has faded. When people visit a tourist attraction, the experience may provide utility for a lifetime to some people, rendering a return visit unnecessary. However, for others, the experience might provide utility for a shorter time, leading to a greater likelihood of a return visit. In this paper, I specify a dynamic discrete choice model of demand that allows consumption to affect utility for varying amounts of time. The common assumption of additively-separable utility in consumption is relaxed by modeling a consumption stock similar to that described by Stigler and Becker (977). Consumption of a good produces the stock and estimates of the model indicate that the stock provides utility over time, thereby leading to diminished marginal utility until the stock has depreciated. Consumption in one period can therefore substitute for consumption in another (i.e. there is intertemporal substitutability). The model allows quantification of these consumption dynamics, but it also can assist policy-makers and firms by yielding a set of parameters characterizing all crossprice and own-price elasticities. These elasticities share patterns with those in recent work on storable and durable goods. Intertemporal substitution, whether caused by enduring effects of consumption, storability, or durability, results in positive cross-time

3 price elasticities. Depending on whether consumers expect future prices to be lower or higher, they may either postpone or hasten their purchases. Researchers failing to account for dynamics when estimating demand will either miss or underestimate this type of substitution. In the case of storable goods, Erdem, Imai, and Keane (2003) and Hendel and Nevo (2002) have shown this can lead to large overestimates of long-run own-price elasticities. For durable goods where firms face intertemporal pricing problems, crosstime price elasticities themselves are important (e.g. Erdem, Keane and Strebel (2004) and Nair (2005)). I find similar patterns deriving from an entirely different source of dynamics where the depreciated value of the stock of past consumption creates the intertemporal substitutability. Intertemporal effects specific to consumption also provide a set of normative implications beyond those previously identified for storable goods. Specifically, for nondurable non-storable goods sold under capacity constraints (such as the rounds of golf analyzed in this paper), the timing of consumption is particularly important. Temporary price changes can shift demand either to or from capacity constrained time periods. The estimates of the model in this paper suggest that ten percent of the quantity increase from a price change is attributable to demand substituted from other time periods. In the present case this is spread over seven days suggesting that the effects on capacity utilization may be minimal. For industries such as electricity or airlines, where consumers may not substitute over such long time horizons, there may be greater effects on capacity utilization. The demand model specified in this paper is indirectly related to the literature on variety-seeking. Meyer (976) considered utility specifications similar to the 2

4 specification in this paper, but an operational version of the theory was not developed at the time. Jeuland (978) extended this work by recognizing that the concept of diminishing marginal utility could help explain variety-seeking behavior. McAlister (982) then refined the model of variety-seeking by focusing on diminishing marginal utility over attributes rather than goods. This spawned an extensive literature on varietyseeking which focuses on brand-switching. The present paper differs from the variety-seeking literature along two dimensions. First, the focus of the paper is not brand-switching. The demand model specified here could be adapted to consider intertemporal consumption of electricity from a public utility that has no obvious substitutes. Second, consistent with rational utility maximization, consumers are modeled to be forward-looking. This implies that consumers will optimally time purchases. In this paper, consumers are found to postpone consumption when they are aware of a future price discount. Ignoring the future in these types of models leads to an omitted variable bias, which affects the sign of the statedependence in this paper. In addition, a static demand model would not allow for prediction of the full-set of cross-time price elasticities because it would ignore the effect of a price change on previous demand. Furthermore, the dynamic model allows the researcher to infer the implications of permanent pricing policies through observed temporary price variation. The empirical application of golf in this paper provides two primary advantages for identifying these consumption effects. First, golf is both non-durable and nonstorable such that observations of purchases in scanner panel data actually reveal consumption, rather than leaving the researcher to derive an optimal consumption choice The literature on variety-seeking has not yet been extended to consider forward-looking consumers. 3

5 for analysis (as in Sun (2005)). Second, during the panel data at this course, consumers were informed in advance of future price discounts, such that the identification of forward-looking behavior is less reliant on the econometrician s assumptions regarding consumers beliefs about the future. 2 Another issue affecting estimation of these dynamic models and others (e.g. Erdem and Keane, 996 and Ackerberg, 2003) concerns the inclusion of sufficient heterogeneity to separately identify heterogeneity and state-dependence. Typically the dynamic programming problem must be solved for every possible consumer type at every value of the parameters considered in the optimization procedure. Rich heterogeneity implies many such consumer types, and thus a high degree of computational burden. One approach has been to assume a small number (e.g. 3 or 4) of discrete consumer types (e.g. Keane and Wolpin, 997 and Crawford and Shum, 2003) reducing the computational burden. In contrast, using a recently developed technique involving importance sampling and a change of variables (Ackerberg, 200), I am able to allow for a rich, multidimensional, continuous distribution of consumer heterogeneity. This is far more heterogeneity than has been accommodated in prior dynamic models in the literature. The estimates of my dynamic model of consumer choice find consumption to have a lasting effect on utility that induces substitutability across time. Specifically, the duration since the last consumption occasion is found to be positively related to the 2 Common specifications of consumers beliefs about the future in dynamic demand models imply that consumers are quite knowledgeable regarding both statistics and the prices they have observed in the past. The advance notice of future price changes reduces the extent to which I rely on such assumptions. A very different and interesting approach towards addressing these expectations issues is taken in Erdem, Keane and Strebel (2004). They observe individuals expectations about future prices. This allows them to use perceptions about the future that differ on a dimension reported by the decision-makers themselves. 4

6 consumption probabilities, and forward-looking consumers take this into account when choosing their future states. The time it takes marginal utility to return to preconsumption levels is approximately 9, 32 and 43 days for the 25 th percentile, median, and 75 th percentile of consumers. A simulated hypothetical price change in the model is used to estimate the effect of these consumption effects on elasticities. Cross-time price elasticities are found to be positive, suggesting that rounds of golf on the days before and after a price change are substitutes for the round experiencing the price change. The intertemporal substitutability further results in an own-price elasticity of demand that would be overestimated by ten percent if based on a temporary price change instead of a permanent price change. Other related counterfactuals demonstrate how both temporary and permanent price changes affect the distribution of demand across time. As described above, this is important for firms facing capacity constraints. The paper proceeds as follows. The following section describes the golf data. Section 3 defines the model of demand. Section 4 contains the empirical implementation of the model. The estimates of the model and elasticities are in Section 5. Section 6 concludes. 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 304 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 (8-Hole, Twilight which is typically between 9 and 8 holes, and 9-Hole or less) were recorded by swiping their membership card at purchase. The purchases during the 98-day period represent the choices that will 5

7 be analyzed, but I will also use their purchases in the months before to determine the initial duration since the last 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 technology to decrease the rigidity of their price menu. The course studied here has fixed prices for most days and times, except for Sundays, for which they occasionally e- mailed coupons to their members. Table depicts the prices by day of week and type of round. The Sunday 8-Hole round (top row) is the only price that varies. $70 is the base price for weekend 8-Hole rounds but prices are as low as $42, with a mean of $60.30, depending on the coupon sent. Other weekend rounds are priced at $44 and $22 for Twilight and 9-Hole rounds respectively. Weekday rounds are cheaper for all types: $22, $34, and $50 for 9-Hole, Twilight, and 8-Hole rounds respectively. This course, unlike many others, experienced a regularly low demand for Sunday rounds before Twilight. To fill up the excess capacity, it therefore began sending 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. 6

8 The coupon was ed to consumers three days prior to the effective date. I observe these coupon mailings for a 98-day period during the summer of 200. The panel consists of American Golf Players Association (AGPA) frequent golf club members. These golfers swipe a membership card whenever they purchase a round at the course. 3 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. While the AGPA records golfers swipes at other courses, the data set used in this analysis only contains purchases at this course. Summary statistics related to the panel members purchases are presented in Table 2. The golfers daily purchase probabilities (just over % for each type) indicate that they play approximately 5 rounds of each type per year. On average, they have not played in 28 days and 23% of all observations represent a golfer who has not played in 60 days or more. The difficulty of distinguishing between heterogeneity and state dependence can be illustrated by looking at how the average number of purchases individuals made over the panel differs by their history at the start of the panel. Figure shows that an individual who golfed the day before the 98-day panel began, on average, golfed more than 8 times over the 98 days; while those who had not golfed in 60 days at the beginning of the panel golfed an average of only 2.6 times over the 98 days. There are two possible explanations for this behavior. Either individuals like to golf less as the duration since 3 If the golfer does not have his card with him, there are other means by which he can get credit for the round. 7

9 the last purchase increases (positive state dependence), or there exists a group of lowdemand golfers who were more likely to be at a starting history of 60 (heterogeneity). The results from the demand estimation, reported in section 5, support the existence of extensive heterogeneity and show that the state-dependence is actually negative. The future price variation in the data could be used to identify cross-time price elasticities in descriptive regressions (e.g. regressing demand on Thursday on the price for the coming Sunday). Unfortunately there are not enough observations to identify each of the daily cross-time price elasticities. However, Table 3 reports regressions of the total rounds by each golfer on the three days before and after Sunday on the Sunday price. These results show evidence of intertemporal substitution. The substitutability with the three days after Sunday is significant, but the substitutability with the three days prior is not (though the sign is consistent with substitutability). The weakness of the latter may be due to the fact that some golfers did not read their s early enough to significantly affect their pre-sunday purchases. To account for this in the structural model, I will include a parameter that allows for heterogeneous probabilities of responding to future prices. The findings reported in section 5 suggest that there are distinct segments of consumers that either do or do not respond to future prices. Regressions of Sunday 8-Hole demand on the Sunday 8-Hole price are also included and the expected sign is found and significant. 4 The structure of the behavioral model in this paper (defined in the following section) should facilitate the estimation of the elasticities. Note that intertemporal 4 There are only 42 individuals in these regressions because the remainder of the golfers did not purchase an 8-Hole round on Sunday during the 98 days. However, their purchases on preceding and subsequent days were affected by this price. 8

10 substitution should be identified by two sources of variation in the data. First, as described above, one can examine the effect of past and future prices on current demand. Second, there is the effect of prior purchases (or alternatively prior prices) on current demand (i.e. the state dependence). The structural model used here is able to efficiently combine both these sources of variation to identify intertemporal substitution. This greatly reduces the demands on the data relative to a reduced-form approach, which would likely require one to estimate separate coefficients measuring the effects of the future and past prices (or purchases) on current demand. 5 3 Model of Golf Demand Defining the model of golf demand captures the two primary objectives of the paper. First, non-time separable preferences are allowed through a stock of past consumption affecting the marginal value of additional consumption. Estimates and analysis in section 5 show marginal utility is diminished most immediately following consumption and significant heterogeneity exists in how long it takes marginal utility to return to pre-consumption levels. Diminishing marginal utility in the presence of these preferences indicates intertemporal substitutability. Second, the set of preference parameters defined in the model determine all cross-time price and own-price elasticities. Time inseparability of preferences requires a utility specification that can incorporate past choices into current utility. A useful approach used in the economics literature is to model market choices or consumption to produce a consumption commodity. This literature once termed the New Theory of Consumer Behavior 5 In fact, one might need to estimate separate elasticities with respect to each future (and past) price. For example, the effect of a future price change on Sunday might have different implications on Thursday vs. Friday behavior. Again, in the dynamic structural model, this is all encompassed by the structural state dependence parameter and the price coefficient. 9

11 derived following Becker (965) has been implemented specifically for consumption by papers such as Stigler and Becker (977) and Becker and Murphy (988). In these applications, the focus of the consumption commodity is in its complementarity with current consumption, but the focus of the consumption commodity in this paper is in its substitutability with current consumption. 6 This substitutability comes through the negative state-dependence of current consumption with respect to past consumption. 3. The Current Period Utility I begin expressing the current period utilities as the indirect utility from each of the four daily choices a golfer can make. The outside option (i.e. not golfing) is the most common choice made by golfers. While the discrete choice model only allows us to estimate relative utilities which typically leaves the outside alternative normalized to zero, I will specify its utility as a function of the depreciated stock of past consumption to maintain a structural interpretation of how past choices affect current utility. Differences in opportunity costs of time on weekends relative to weekdays will also be included in the outside alternative. The utility of the outside alternative is specified: (, ) ( ; ) ( ) u H ε = ϕ H γ + γ Weekend + ε. () 0 it i0t it i0 id i0t γ id is a measure of potentially reduced opportunity costs of time facing a golfer during weekends. Ideally we would like to observe the stock of past consumption and include it in the estimation, but it is intangible and hence unobservable. So, I am forced 6 The notion of consumption capital and complementarities could be a feature of a model of golf demand. Specifically, new golfers that learn to play may develop an appreciation for golf that changes their willingness to play. The Stigler and Becker (977) model could accommodate this while retaining stable preference parameters. In my specification, I do not allow for consumption capital to accommodate changing preferences because I do not allow for all cumulative previous consumption to affect demand. This is a reasonable assumption in this case because the golf course had a separate membership for new golfers, while the golfers observed in this data already self-selected as frequent golfers. 0

12 to infer it from past choices and estimated preference parameters using the function ϕ ( ). H is the number of days since the last consumption occasion and γ i0 is a parameter vector that will determine whether this stock will positively or negatively affect the utility of the outside option. There are a few assumptions involved in this specification of the consumption stock which deserve elaboration. First, only the consumption stock acquired during the last consumption occasion is relevant to current purchases. Stated differently, all past consumption fully depreciates after a new consumption occasion. In this specification, the entire stock of past consumption can be summarized by stable preference parameters, γ i0, and a single state variable, H. This provides a clear and simple empirical implementation because H is observable, discrete, and evolves deterministically. Allowing the stock to accumulate across multiple consumption occasions would require the state variable to be all past choices or the stock itself, rather than H. The continuous stock as a state would still require a discretization and would require inference of the initial stock. By specifying past consumption to fully depreciate at a new consumption occasion, observing the initial H only requires searching through purchase data prior to the beginning of the price series for the most recent purchase. These advantages come at little cost in this empirical application. The temporary price decreases in the data are only valid for a single purchase, so consumers opportunities to stock up on consumption at low prices are very limited. In addition, the data show that most golfers space out their purchases, so attempts to compile a consumption stock of more than one round would be rare. Furthermore, the limited

13 consumption stock would tend to underestimate the incidence of intertemporal substitution, rather than create bias that would falsely find intertemporal substitution. The second assumption in this specification is that all consumption choices affect the stock of the consumption commodity the same. The primary reason for this is computational simplicity. Allowing each type of round of golf to have a different effect on the stock would triple the size of the state space. 7 Once again, this assumption does not bias in favor of finding intertemporal substitution because the ability to compile greater stocks of consumption for certain types of rounds would exaggerate the ability of consumers to intertemporally substitute. Finally, in estimation, I define ϕ( H; γ ) γ log ( H) γ ( H 59) = + >. i0 i0 i02 Depreciation of a consumption stock requires that as the time since the last purchase increases, the utility of the outside good should decrease at a decreasing rate. When γ i0 < 0, the above function will provide exactly this non-linear pattern and indicate the negative state-dependence necessary for intertemporal substitution. Notice that the specification does not force the result as γ i0 could be greater than zero in estimation. The latter term in the function accounts for the fact that the days since the last purchase has been capped at 60. We can now move on to specify the indirect utility from choosing each of the rounds of golf. The utility to individual i in time period t from choosing a type j round of golf is: 7 This would unnecessarily increase the computational burden and is similar to assumptions made in the literature. 2

14 (, ) u = γ + γ p D ξ + ε (2) ijt ij ip j t t ijt γ ij is the utility net of price and an individual, time and choice specific shock to preferences, ε ijt. γ ip is the individual s sensitivity to the price of the round, which is a function of the day of the week, D t, and the value of the weekly coupon, ξ w, which may apply on the given day. The ε s for each type of round of golf and the outside good are assumed to be logit errors distributed i.i.d. over individuals and time Dynamic Optimization Problem Dynamics in the choices arise through the state S { H, D, ξ}. The day of the week, D, and the price shocks, ξ, evolve exogenously to the consumer. The stock of past consumption as measured through the time since the last purchase, H, is therefore the state creating the dynamics in the consumer s problem, though the other two states certainly affect the dynamic incentives through H. After defining the consumer s maximization problem, I will discuss the effect of each of these variables. To begin defining the dynamics, the discounted present value of current and future utility to the consumer is: τ t V( Sit; γ i ) = max E β u Si, yi, ε ; γi Sit, Πi; γi Πi τ = t ( τ τ τ ) (3) 8 One might expect serial correlation in the logit errors given that these are daily decisions and vacations or other personal engagements may increase or decrease the value of the outside alternative on successive days. However, these types of positive serial correlation would actually bias in favor of positive statedependence, rather than the negative state-dependence found in this paper. 3

15 where Π i is a set of decision rules mapping states, Sit s, to choices, yit s. At every state, S, the consumer faces the same infinite-horizon maximization problem. The value function V( S ; γ ) defined in equation (3) above is the solution to the following Bellman s equation : it i ( ) + V( Sit; γ i ) = max u Sit, yit, εit; γi + βev( Sit ; γi ). (4) yit The approach used to solve this Bellman s equation with maximization over a discrete number of choices is based on Rust (987). Rather than iteratively solving for the value function itself, I define a value function that is conditional on the choice yt does not consider the taste shock ε jt : = j and that V ( S γ ) = v + βev( S, ε S, y = j, γ ) (5) j t i j t+ t+ t t i where v j is the non-stochastic portion of the current period utility defined in equation () or (2) above. By combining equations (4) and (5), we get the following set of equations that must be satisfied: 4

16 ( max ( +, +, + ) +, 0, y ) αie Vj Ht Dt ξt γi ε jt St yt γ + = i + V0( St γi) = v0 + β ( ) max (,, = 0 ) +, = 0, αie Vj Ht Dt ξt γi ε jt St yt γi V( St γi) = v+ β ( ) ( ) (6) αie Vj Ht Dt ξt γi ε jt St yt γ i V2( St γi) = v2 + β ( ) max (,, = 0 ) +, = 2, t+ αi E( j t t i t jt t t i y V H D γ ξ ε S y γ ) t+ ( max ( +, +, + ) + +, =, y ) + t+ αi E( max Vj Ht, Dt γi, ξt 0 ε jt St, yt, γ i y = + + = ) t+ ( max (,, ), 2, y = ) + t+ αi E( Vj Ht+ Dt+ γi ξt+ ε jt+ St yt γi y ) t+ ( max (,, ), 3, y = ) + t+ αi E( Vj Ht+ Dt+ γi ξt+ ε jt+ St yt γi y ) αie Vj Ht Dt ξt γi ε jt St yt γ i V3 ( St γ i ) = v3 + β ( ) max (,, = 0 ) +, = 3, t+ The effects of the state variables now become clear. An individual choosing not to golf receives a discounted present value of utility, V 0, including current utility v 0, which is a function of the existing stock of past consumption as determined by discounted future utility corresponding to H = H + such that the stock of past t+ t H t, and a consumption is reduced further. Given that the eventual estimate of γ i0 is negative, the value of the outside alternative will be lower and consequently the marginal value of golfing will be greater. If an individual chooses to golf one of the three types of rounds, a current period utility v j is received and the discounted future utility includes a restored stock of consumption corresponding to H it + =. The day of the week affects choices through H. Specifically, an individual with a higher opportunity cost of time during the week, indicated by γ id < 0, prefers to golf on the weekend, but also can increases the value of the outside alternative during the week 5

17 by insuring a maximal stock of past consumption, H =, on Monday. On the other hand, golfers without higher opportunity costs on weekends can avoid high weekend prices by purchasing and restoring their stock of past consumption during the week. Just as price variation across the days of the week can lead to intertemporal substitution, price variation across weeks on a given day can also affect incentives. ξ w is defined to be the weekly coupon schedule for the course. In the data, this coupon is sent out after golf on Wednesdays and is only valid for 8-hole (type ) rounds on Sundays. ξw = ξw if Dt 3 ξ = ξ {0, 0, 2, 7, 22, 25, 28} if D = 3 w+ t where ξ is a discrete random variable with the following probability function, reflecting the frequency with which the discounts were sent in the data 9 : On Sunday, w f ξ ( ξ) 3 if ξ = 0 7 if ξ = 0 7 if ξ = 2 7 = if ξ = 7. (7) 4 if ξ = 22 4 if ξ = 25 4 if ξ = 28 4 ξ affects the current period utility through p (, ) j D ξ. On Thursday through Saturday, greater Sunday discounts will lead to lower purchase incentives when γ i0 < 0 9 I assume that consumers knew this distribution ex-ante and their beliefs about it did not change over the 98-day panel. 6

18 through ξw s effect on the expected future value. On Monday through Wednesday, ξ w only has an indirect effect on demand through its past effect on H. Notice the parameter α i in equation (6). This is an individual-specific probability of taking the coupon (i.e., future price shock) into account when making current decisions. Specifically, it allows for unobserved factors that may have prevented the golfer from using this future information to make current decisions. Such unobserved factors include the probability that the golfer did not check before Sunday, or that there were constraints preventing the golfer from changing plans on short notice. 0 Intuitively, this parameter will be identified by the amount of variation in Thursday, Friday and Saturday choices that is induced by the golfers advance notification of the varying Sunday price. Because this is the same variation that would best identify the discount factor, I fix the discount factor in estimation. If instead we assumed α i = and estimated the discount factor, the behaviors described above would likely generate an unreasonably low estimate of the discount factor. The expectations in equation (6) are over ε t + every day. By defining the ε s to be distributed extreme value, we get the following simple form for the expectation: 3 V j E( max V0 + ε0,..., V3 + ε3 V0,..., V3) = ln e. (8) j= 0 0 It may be argued that i α should be included in the current period utility as well, but this specification would only affect Sunday choices. I assume that by Sunday, sufficient time has passed to ease these concerns. The definition and implications of α is also consistent with Israel (2005), which finds evidence that information can help explain an apparent unresponsiveness of consumers to future price discounts. 7

19 On Wednesdays, the above equation is solved for every possible value of ξ w + to form the correct expectation integrating out ξ w +. Substituting this into the problem above, we have: 3 V k αie ln e St, y = j, γi + k = 0 Vj( St γi) = vj + β 3 c= 0 V k ( αi) E.5772 ln e St, y j, γ + = i k = 0 (9) 2 Rust (987) proves that the iterative application of this mapping, starting at an arbitrary set of V s, eventually converges to the actual set. 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 purchase might be greater than the surplus of purchasing today. In this model, the 2 As this equation has already integrated out ε, the expectations are over ξ. 8

20 consumer knows the Sunday discounts three days in advance, with certainty. Given that a discounted Sunday is cheaper than a Saturday, yet 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. The choice-specific value functions defined above fit nicely into a discrete choice model to form the probabilities associated with each choice. 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. Discrete Choice The distribution of the ε s implies standard logit choice probabilities, which are a function of the choice-specific value functions defined above: Pr( y S ; γ ) J { yit = j}exp( Vjit ) = (0) j= 0 exp( Vkit ) it it i J k = 0 9

21 Given these defined probabilities of observing each choice, a likelihood across all time periods for a single individual can be defined: T L( S,..., S, y,..., y S, y, γ ) Pr( y S ; γ ) Pr( S S, y ) = () i i it i it 0 0 i it it i it it it t= To this point, the focus has been on defining a likelihood conditional on the preferences of individual i, i.e. γ i. We allow these (time-invariant) preferences to vary across individuals using a random effects specification. The following set of normally distributed random coefficients is defined to represent these preferences: γ = γ +Γ η (2) i i where η are distributed i.i.d. standard normal. Thus, the vector γ represents the means of the various coefficients, while Γ is the Cholesky decomposition of the variancecovariance matrix, Σ. To ease the computational burden, I assume that Σ is diagonal. An additional random coefficient that is used to estimate the parameters for α i, which is bounded between 0 and, is γ, where: αi ( γ αi ) ( γ ) exp αi =. (3) + exp αi Introducing the random coefficients allows the likelihood to now be expressed in terms of the model parameters θ = { γ, Σ }: T (, ) = Pr( ;,, Σ) Pr(, ) ( ) L S y θ y S η γ S S y f η dη (4) i i i it it i it it it i t= 20

22 The deterministic evolution of the state variables, Pr( Sit Sit, yit ) =, simplifies the likelihood to: 4.2 Simulation T (, ) = Pr( ;,, Σ) ( ) L S y θ y S η γ f η dη. (5) i i i it it i i t= 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 function evaluations required for convergence of the maximization routine. Ackerberg (200) 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. Applying the Ackerberg (200) technique, we get the following simulated likelihood for an individual: where ui γ ηi ( ns γ, Σ) g ( u ) NS T f u Li( Si, yi θ ) = Pr( yit Sit; uns) NS ns= t= ns = +Γ is the change of variables, and ( ) (6) g u is the importance sampling distribution specified with a non-zero density over the support of u : 2

23 ( ) (, ) g u = f u γ Σ (7) U γ and Σ are arbitrarily defined starting values for the true γ and Σ. Taking the product of this likelihood function across individuals yields the following simulated likelihood function to be maximized: N L= L (8) i= i * A maximization algorithm will find θ by iteratively searching over the possible values of θ. The computational advantage of the algorithm is that as θ changes, only f () in (6) changes. Pr( yit Sit; u ns) does not change during the maximization algorithm, so we do not have to continually re-solve the dynamic programming problem as we search over θ. 3 The parameter values that maximize this likelihood are consistent as NS approaches infinity (Gourieroux and Monfront, 996). Standard errors of the model parameters are calculated using the outer-product of a numerical gradient of the likelihood function. Simulation error is included in the standard errors by repeating the estimation for multiple sets of draws from g( u ). 5 Results The primary goal of this paper is to establish that consumption diminishes marginal utility for an extended period of time and that this has implications for both cross-time and own-price elasticities. The estimated model parameters show that utility 3 In practice, it is useful to change the importance sampling distribution occasionally. 22

24 from consumption is lowest immediately following consumption and that it gradually increases as the consumption experience fades. This implies positive cross-time price elasticities. I present the estimates of these cross-time price elasticities within a week as well as estimates of the own-price elasticity of demand and a cross-product price elasticity. 5. Model Estimates Three 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 and no random coefficients). Then I estimate a logit with random coefficients, but without forward-looking behavior. Finally, I estimate the version of the model specified in the previous sections. It is noticeable that the addition of heterogeneity and dynamics changes the state dependence from positive to negative. The simple logit model suggests habit persistence and complementarities across time, the heterogeneity in the random coefficients model removes the habit persistence, while the full model with heterogeneity and dynamics suggests intertemporal substitution. Table 4 presents the estimates of each of the models. The estimates for most parameters are reasonably stable across specifications. 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 value of the outside alternative is greater on weekdays, when some golfers may have to work. The parameter estimates that 23

25 change across models are the price coefficient (which becomes more negative) and those relating to state dependence. The parameters excluded from the static logit models are the discount factor, which is fixed at 0.99 in estimation, and α, the probability that a golfer incorporates future price variation in current decisions. The estimate of the normally distributed random coefficient γ αi has a mean insignificant from zero with a standard deviation of Transforming this variable to α, which is constrained to be between 0 and, implies a bimodal distribution of golfers probabilities of incorporating future price variation in current decisions. The modes are near 0 and, with approximately 33% of golfers having a 0 to % probability of incorporating the information, while 3% have a 99 to 00 % probability of taking the future information into account when purchasing. The other third of golfers occasionally take the future price information into account, though the majority either do this frequently or infrequently. The price coefficient is one parameter that substantively changes between the static random coefficients and dynamic random coefficients model. In the dynamic model, the parameter is more negative. The likely reason for this change is that in a static model, the price parameter is only identified from variation in the willingness to purchase on the day the price is changing. Table 3 notes that only 42 of the 304 golfers ever actually played on a Sunday. However, the dynamic model identifies the price parameter using choices on Sunday as well as Thursday through Saturday. The parameters estimating state dependence change as additional heterogeneity is incorporated. The parameter for the log of days since last round is positive in the simple logit, suggesting habit persistence. However, if heterogeneity is not accurately 24

26 controlled for, as is the case for the simple logit, the positive coefficient is likely picking up the fact that avid golfers are likely to have fewer days between rounds. The static random coefficients specification adds the heterogeneity necessary to remove its effect from the estimates of the state-dependence. Specifically, the coefficient on the log of days since last round shrinks to a statistically insignificant value of 0.05, while the indicator for 60 days since the last round changes signs to be -0.72, also insignificant. As expected, the addition of heterogeneity has removed the habit persistence suggested by the simple logit. Of note however is that the estimates of this model suggest no state-dependence, which would imply no intertemporal substitution. The addition of the forward-looking behavior in the full dynamic model specified above, controls for an additional bias that favors habit persistence. Specifically, the static specification omits future prices from the current period utility specification, which biases toward positive state-dependence. The intuition is as follows. In weeks in which the Sunday price is high (low), there will be an increased (a decrease) incentive to purchase on the preceding Thursday, Friday and Saturday. This creates a correlation in the purchase probabilities on these days. Accounting for this omitted variable bias, results in the estimated negative state-dependence in demand. The estimate of the coefficient on the log of days since last round in this model is -0.22, while the coefficient on 60 days since the last purchase is -0.27, both of which are statistically significant. It is this negative state dependence from consumption that suggests marginal utility is diminished immediately after consumption, then gradually increases as the experience fades. 25

27 5.2 Non-Time Separable Preferences and Marginal Utility The non-time separable preferences specified in section 3 incorporate utility from past consumption through a stock of past consumption that affects the value of the outside alternative. These coefficient estimates of the log of days since last round indicate that when the consumption stock is high (i.e. the days since the last round are low), the value of playing an additional round of golf will be low. This diminishing marginal utility creates intertemporal substitution because the choice to play a round of golf increases the consumption stock and reduces the utility to play on subsequent days. A useful way to quantify these effects is to revisit the question of how long it takes utility to return to pre-consumption levels. Recall from the introduction that while the answer to this question may be obvious for something like food, it is much less clear for other consumption goods. For golf, there is significant heterogeneity in how long it takes marginal utility to replenish. The model estimates imply that the time it takes marginal utility to reach pre-consumption levels is approximately 9, 32 and 43 days for the 25 th percentile, median, and 75 th percentile of consumers Effects of Temporary Price Changes The fact that marginal utility is diminished for an extended period of time implies that cross-time price elasticities will indicate intertemporal substitution. In this section, I estimate the elasticities in a scenario like that observed in the data in which there are temporary price changes. I measure the own-price elasticity and the cross-time price 4 In my specification, the number of days it takes an individual to return to his pre-consumption level of utility is actually the number of days it had been since the last consumption occasion. The calculation therefore involves weighting the various possible days it may have been since purchasing by the probability that a purchase would be made that many days after consumption by the particular individual. 26

28 elasticities for the days leading up to and following the price change. After estimating the elasticities I quantify the effects of the intertemporal substitution on revenue and demand, which is informative of the implications for capacity utilization. To estimate these elasticities, it is useful to think of a price change on a given day. This directly affects the consumption on the given day (implying an own-price elasticity) and also affects the state of consumption for subsequent days. The negative state dependence implies that the subsequent consumption will change in a direction opposite to that of the consumption change on the day of the price change. In addition, consumption prior to the day of the price change will also be affected because the price changes are announced in advance and consumers are forward looking. I estimate these cross-time price elasticities, an own-price elasticity for the Sunday 8-Hole round of golf (i.e. the one experiencing the price variation in the data), and a cross-product elasticity on Sunday by introducing a hypothetical price change into the model estimated above. The elasticities are estimated for a simulated sample of consumers drawn from the distribution of the random coefficients. The hypothetical price change involves estimating demand for seven consecutive days following a coupon offering a price of p 0 for the coming Sunday 8-Hole round of golf. Expected demands given this coupon are calculated for the first three days after the coupon ( q 3, q 2, q ), the effective date ( q 0 ), and three days following the effective date ( q,..., q 3 ). Then an alternative price, p ', is 0 considered, with the respective demands given this price calculated as q 3',... q0',..., q3' Elasticities are then calculated using the following midpoint method:. 27

29 η = t ( qt + qt) ( p 0 + p0) q t qt 2 p p (9) For all t, except t = 0, the quantities represent the quantities across all three types of rounds. When t = 0, I separate out the quantities for the Sunday 8-Hole round from the other two types of rounds. The model parameters also imply elasticities much further into the future than t = 3 defined above, however these would be estimated best using simulation. 5 These elasticities are reported in Table 5. The standard errors for the elasticities are calculated using bootstrapping. Each of the elasticities has the expected sign. The cross-time price elasticities are all positive, suggesting that rounds of golf on surrounding days are substitutes. The elasticities for the three days following a price change begin at and decrease to The elasticities for the three days prior to the effective date of the price change increase from to Both patterns suggests that substitutability decreases with the distance in time. The own-price elasticity of the Sunday 8-Hole round is estimated to be The cross-price elasticity with respect to the other two types of rounds sold on the effective date of the Sunday 8-Hole price change is All elasticities are significant except for the cross-price elasticity of Saturday with respect to Sunday. The insignificance arises because of the large standard error. This may be the result of only drawing 40 sets of parameters to use in bootstrapping. All 40 estimates of the Saturday elasticity are positive, with the minimum being It is 5 I do not simulate the elasticities further because the number of simulated logit errors required to estimate standard errors on the elasticities would become excessively large. 28

30 computationally difficult to increase the number of draws in the bootstrapping because for each draw of parameters, a simulated set of consumers must be drawn to estimate the elasticity. I currently use 00 simulated consumers for each of the 40 sets of parameters considered in the bootstrapping. This is already computationally demanding because the dynamic programming problem must be solved for each of these 4000 simulated consumers and each of their demands over seven days and four possible pricing policies must be derived. To quantify the implications of these elasticities and the intertemporal substitution for the firm, it is necessary to extrapolate from the purchase probabilities to a baseline demand, q 0. I set q 0 = 240, which is the demand at the course when it is at capacity. Prices vary between $42 and $70, so I assume that the capacity constraint binds at $42, but not at $70. When the course drops its price to $28 below its baseline price, its demand increases from just over 32 rounds of golf to 240 rounds. This is clearly a substantial increase in demand. While much of this demand increase is new demand, about 0 percent (just over 20 rounds) is substituted away from other time periods. Half comes from other time periods in the same day. The other half comes from days either before or after the Sunday price drop. As one would expect, the effect is largest on the days just before or just after the price change. 3 of the 0 rounds would have been purchased on Saturday, while 4.6 would have been purchased on Monday. This reveals that the intertemporal substitution is spread across many days such that there is not a strong shift in demand, and hence not a large change in capacity utilization, on any given day. 29