# A bandwagon effect in personalized license plates?

A BANDWAGON EFFECT IN PERSONALIZED LICENSE PLATES?

The bandwagon effect is a consumption externality that exists when an individual's demand for a good is increased by his observation of other consumers using that good. This paper models a product demand curve with a bandwagon effect and, using data on sales of personalized license plates, estimates such a demand curve. Certain more conventional models of product demand, including information diffusion and habit formation models, are observationally similar to the bandwagon model, despite being conceptually different from it. I attempt to use the license plate data to discriminate between the bandwagon model and these other models.

In 1950, Harvey Leibenstein described the impact on product demand curves of what he called a "bandwagon effect." The bandwagon effect was said to exist if peoples' valuations of a good (and thus demand for that good) increased when they observed others consuming the good. No one to my knowledge has attempted to empirically verify the existence of or estimate the magnitude of a bandwagon effect in any product market. In this paper I attempt to do both by looking at the demand for personalized license plates, a good that could plausibly be subject to a bandwagon effect.

In the first section of the paper I develop a simple model of product demand in the presence of a bandwagon effect. The model is based on Leibenstein's treatment but is modified to fit the case of personalized license plates. In the second section, data on the sales of personalized plates are described and used to estimate a demand function. The results of this estimation are consistent with the theoretical model of the bandwagon effect.

The search for the bandwagon effect cannot end there, however. The bandwagon model, in which consumers are moved by considerations of fashion and status emulation, is observationally similar to other well-known but conceptually quite different models of product demand in which consumers are more sober and level-headed. In particular, models of information diffusion or habit formation imply patterns in the data very much like those that follow from the bandwagon model. Section III discusses this matter in more detail and presents statistical tests designed to differentiate between the bandwagon model and alternative models of product demand. In the final section I draw some conclusions from the empirical evidence.

I. THE BANDWAGON EFFECT

Consider a product that lasts for a single period of well-defined length. Each period, a consumer may purchase at most one of the product. The personalized license plate is a product of this sort, as are subscriptions to a particular magazine, hairstyles, club memberships, vacations to chic resorts, etc. Assume first that there is no bandwagon effect and let (1) [V.sub.it]([X.sub.it], [Z.sub.t]) represent the value of the product to individual i in period t, where [X.sub.it] is a vector of individual characteristics such as age or income, and [Z.sub.t] is a vector of product characteristics in period t. The function [V.sub.it] is a random variable defined over the population of potential buyers. Let [F.sub.t](V) = prob([V.sub.it] < V) in period t. Then, if [M.sub.t] is the population of potential consumers in period t, market level demand in period t is (2) [Q.sub.t] = [M.sub.t][1 - [F.sub.t]([p.sub.t])] where [p.sub.t] is the product price. Demand is a decreasing function of price. It also depends on product characteristics and the distribution of personal characteristics in the population.

Adding a bandwagon effect to this model involves allowing individuals' preferences to depend on aggregate behavior. Liebenstein assumed that in the presence of a bandwagon effect the quantity of a product demanded by each consumer at a given price depended on market level demand at that price. This creates a problem when moving from individual to market level demand functions, for the market level demand curve is no longer a lateral summation of individual demand curves. The problem can be avoided if we replace Leibenstein's characterization of the bandwagon effect with one that is arguably more realistic by assuming that consumer demand in period t depends on the market level demand in the preceding period.(1) In the case of personalized license plates, this amounts to assuming that the likelihood that a person buys or renews a personalized plate at any given price depends on the number of those plates he has recently seen on other peoples' cars. The assumption can be incorporated into the model by changing the individuals' valuation functions to (1') [V.sub.it] = [V.sub.it]([X.sub.it],[Z.sub.t],[Q.sub.t-1]) where [Q.sub.t-1] is market level demand for the product in the previous period. Then, aggregating in the manner of equation (2), market level demand can be written (3) [Q.sub.t] = f([p.sub.t], [Q.sub.t-1]) where [f.sub.1] < 0 and [f.sub.2] > 0. The dependence of demand on product characteristics and the distribution of population characteristics has been suppressed.

In this model of the bandwagon effect demand can grow over time when price and other demand shifters remain constant, growing explosively or reaching a steady state in which f([p.sub.t], [Q.sub.t-1]) = [Q.sub.t-1]. A steady state is stable if [f.sub.2]([p.sub.s],[Q.sub.s]) < 1, where [p.sub.s] and [Q.sub.s] are steady state price and quantity. Leibenstein insured the existence of a stable steady state by assuming at the individual level a "diminishing marginal effect of external consumption" which translates into a negative second derivative of [V.sub.it] with respect to [Q.sub.t-1] in equation (1'). Leibenstein's solution is not appropriate for the sort of product considered here, as each person is assumed to purchase at most one product per period. However, a case for the existence of a stable steady state can still be made by assuming a heterogeneous population in which, at any level of demand, some peoples' valuation functions are decreasing in [Q.sub.t-1], a phenomenon Leibenstein called the "snob effect." As the level of total consumption grows, people whose tastes are subject to the snob effect experience a fall in their [V.sub.it] functions. Those who have been in the market in previous periods will drop out, and those who have never entered the market will continue to stay out. If a sufficient number of people are "snobbish," there will be a stable steady state level of total demand, even if many individuals do not experience a diminishing marginal effect of external consumption.

Liebenstein demonstrates that the presence of a bandwagon effect makes demand more price elastic because the quantity change brought forth by a price change leads to further quantity change. The model just outlined also has this feature, although the augmented effect of a price change works itself out over several periods: a fall in the current price leads to an increase in current demand, which in turn causes an increase in next period's demand through the bandwagon effect, and so on. Figure 1 shows the steady state demand curve (relating price to steady state demand) in a market with a bandwagon effect. The top panel shows the relationship between [Q.sub.t] and [Q.sub.t-1] at three different prices. The bottom panel derives from the top panel the relationship between price and steady state demand. The top panel also shows the multi-period process of adjustment following a fall in price from [p.sub.3] to [p.sub.2], with [Delta][Q.sub.1] and [Delta][Q.sub.2] being the quantity adjustments in the first and second periods.

II. THE BANDWAGON EFFECT IN THE MARKET FOR PERSONALIZED LICENSE PLATES

My attempt to estimate a bandwagon effect involves a product that could plausibly generate such an effect: the personalized or "vanity" license plate, a commodity whose very name suggests the sort of status conscious or emulative tendencies supposed to lie behind the bandwagon effect. All fifty states and the District of Columbia currently sell vanity plates, allowing vehicle owners who pay a special fee to choose the sequence of symbols that will appear on their license plate. In this section I will use data on the sales of vanity plates to estimate a market demand function, first using a standard specification, then using a specification adjusted to allow for the presence of a bandwagon effect.(2)

Officials from twenty-one states were kind enough to supply me with data on personalized plate sales in their states and to describe the regulations governing their plate programs. Additional information on personalized plate programs was taken from Kripaitis-Neely [1985]. The annual sales data from the states cover periods of time ranging from three to eighteen years in length. Unfortunately, differences in pricing policies and record keeping create some noncomparabilities between data from different states. The estimation discussed in this section involves data from eleven states that use the most prevalent pricing system, one which is consistent with the assumptions of the model developed above. Under this system, vehicle owners decide once each year whether or not they want a personalized plate for the upcoming year. If they do, they pay a fixed fee over and above their basic registration fee. The fee is the same whether they are buying a new personalized plate or renewing one from a previous year. These states keep track of the number of valid personalized plates on the road each year, that is, the number of personalized plates issued for the first time that year plus the number of plates renewed from the previous year. The eleven states taken together provide 100 observations on annual sales. (Details on these data are in the appendix.)

Estimating equations (2) and (3) requires specifications of [Q.sub.t], [M.sub.t], [Z.sub.t], and [p.sub.t] appropriate to the market for personalized license plates. [Q.sub.t] is defined as the number of valid plates on the road at a particular point in year t. Since the potential market for personalized plates in a state is limited by the number of vehicles in the state eligible to bear such plates, the [M.sub.t] variable is constructed using Department of Transportation information on state vehicle registration in conjunction with each state's rules regarding eligibility for personalized plates. The [Z.sub.t] vector includes variables specific to a state's plate program in a year: the number of characters permitted on the plate, the age of the state's program, and a variable indicating whether or not a state promoted its program that year.(3) Data on state per-capita incomes and unemployment rates have been collected to measure differences across states in consumers' incomes and economic situations. Finally, the price of a plate is measured by the amount consumers must pay for a personalized plate over and above regular license and registration fees. Nominal price and income data have been deflated using the Consumer Price Index.(4)

The data were first used to estimate a standard demand function corresponding to equation (2), and the results are reported in column A of Table I. The dependent variable was the log of valid plates; independent variables included the logs of price, per-capita income, and the number of eligible vehicles; a dummy variable for promotion activity, the number of characters permitted on the plate, and the age of the state's program.(5) State dummy variables were also added to the regression to capture state by state differences in demand arising from excluded [Z.sub.t] variables and differences across states in the distribution of consumer characteristics.(6,7) The form of the error term was assumed to be [e.sub.jt] = [[Rho]e.sub.jt-1] + [u.sub.jt] where j indexes states, t indexes time periods, and [Mathematical expression omitted] thus allowing for both serial correlation and heteroskedasticity. A Durbin-Watson test suggested the presence of serial correlation, and there was also evidence that [Mathematical Expressin Omitted] might not be constant across states. The estimated standard errors of the OLS coefficient estimates reported in column A of Table I have been adjusted to reflect these findings.(8)

The column A estimates suggest that the demand for personalized plates is price elastic, and the coefficient of the income variable is positive but not significant. The coefficient on eligible vehicles is negative but smaller than its standard error, presumably because controlling for interstate differences in vehicle population with state dummy variables leaves little leverage in the data for precisely estimating the impact on plate sales of changes in the number of eligible vehicles. States can increase sales with promotional campaigns or by increasing the number of characters allowed on the plate. Plate sales grow by about 4 percent a year as a state's program ages, holding constant the contemporaneous demand shifters included in the regression. This, of course, is one implication of the bandwagon model.

To estimate equation (3), the bandwagon demand function, I adjusted the specification used to generate the estimates of Table I, column A to allow the log of current demand to depend on the log of the previous period's demand. Once again, state dummies were included in the regression. The results of estimating this equation are reported in column B of Table I. Durbin's h statistic, designed to detect first-order autocorrelation in the presence of a lagged dependent variable (see Judge et al. [1985, 326]) failed to reject the hypothesis of no autocorrelation. The test for heteroskedasticity used on the column A estimates was also applied to the bandwagon specification, and the hypothesis of homoskedasticity could not be rejected.(9)

The column B results are in accordance with the predictions of the bandwagon model. The coefficient of the lag of plate sales is positive, significant, and less than one. The coefficient of the price variable is roughly half the size of its column A counterpart, but this too is consistent with the bandwagon model. Consider a state that lowers its plate price in the fifth year of its program and keeps it low in the ensuing years. Other things remaining equal, the demand in the sixth year will be higher than it was in the fourth year for two reasons: the lower price in the sixth year and the higher demand in the fifth year engendered by the lower price in the fifth year. The price coefficient in column A captures both of these effects, while in column B the two effects are separated out. Experimentation with other functional forms for the relationship between plate sales, price, income, and previous period's plate sales produced results that were essentially the same as those in column B.(10)

One puzzling aspect of the column B results is that the estimated effect of the age of the program on sales is basically unaffected by the introduction of lagged sales into the demand equation. Further, when a second lag of plate sales is also added to the equation, its coefficient is insignificant and the coefficient of the age-of-program variable remains unchanged. In effect, as a state's plate program gets older demand grows, holding constant price, income, number of characters allowed on the plate, promotional activity, and any effect on demand that comes from past plate sales. A possible explanation for this that is in the spirit of the bandwagon model involves the possibility that "word of mouth" can augment the actual display of personalized plates as fuel for the bandwagon. As time passes following the introduction of a plate program by a state, a typical resident will repeatedly become involved in or overhear discussions of interesting personalized plates. Each of these experiences will add to his impression that a bandwagon is rolling and raise his valuation of the personalized plate.

Clearly, if one is willing to accept this story as well as the model of section I as an appropriate characterization of the market for personalized plates, then we can conclude from the column B estimates that the bandwagon effect is an important factor in that market. Consumers are more likely to purchase plates if they have seen others displaying the plates, and thus increases in sales lead, in and of themselves, to further increases in sales.

III. ALTERNATIVE MODELS OF PRODUCT DEMAND

It was taken as given in discussing the Table I results that they reflected the presence of a bandwagon effect, but the bandwagon interpretation is not the only possible one. Other models of product demand can lead to similar patterns in the data; more conventional models that require no assumption of faddish, frivolous, or fashion-conscious behavior on the part of consumers. There are models of information diffusion in which it takes time for consumers to receive and process information about new products, and models of habit formation in which a person's purchase of a product in one period can alter the chance that he will purchase the product in subsequent periods. This section explores the problem of discriminating between these models and the bandwagon model of demand.

New Product-Information Diffusion

Many researchers have outlined models in which demand for a new product grows as information about or awareness of the new product spreads.(11) A model of this sort could be applied to personalized license plates by assuming that when a personalized plate program is first introduced into a state, many who would otherwise buy a plate do not because they are not fully informed about the program. As time passes, information about the program is diffused and previously uninformed buyers enter the market, leading to a growth in plate sales over time.

Though both the bandwagon model and this diffusion model imply a growth in sales over time even if conventional demand shifters remain constant, the causes of that growth are conceptually different in the two models. In the bandwagon model everyone is aware of the existence of the plate program. The purchase and display of plates in one period influences demand in the next period by changing peoples' tastes and raising their willingness to pay for a plate. In the diffusion model each individual's willingness to pay for a plate is independent of the total number of plates sold. However, not all individuals are aware of the program, and the uninformed do not purchase plates even though they would value a plate at more than its price. The mere passage of time leads to growth in plate sales as more potential buyers become aware and enter the market. If one also assumes that the display of personalized plates spreads information about the existence of the plate program, the number sold in a given period will influence the number of plates sold in subsequent periods.

Despite the conceptual differences between the two models, they are observationally very similar. As already noted, in both models demand tends to grow over time, and in both models this period's demand can depend on last period's sales as well as this period's demand shifters. Another similarity is that a fall in price in the present period leads to growth in demand beyond that caused by a pure price effect, since the increased sales in the present cause further sales growth in the future. The positive coefficient on the promotion variable in Table I is consistent with the information diffusion story, because promotion is a way of spreading information. However, it can also be argued that promotion actually changes peoples' preferences and perhaps fuels the bandwagon. Obviously, discriminating between the bandwagon and diffusion models can be difficult.(12)

There is at least one important difference between the two models. An information diffusion process can only affect a product demand curve for a limited period of time, i.e., until information about a product is fully diffused. On the other hand, the bandwagon effect modeled above is permanent, in that increases or decreases in sales always affect future demand. This difference between the models suggests ways of gauging the importance of information diffusion as an explanation of the growth in vanity plate sales.

For example, consider the relationship between plate sales and the age of a plate program. If we accept the information diffusion story, demand increases as time passes because each year more people become informed about the existence of the plate program. However, these increases can continue only so long as there remain potential consumers who are unaware of the plate program, and they will diminish over time as this pool of uninformed people dries up. In the sample used in section II, the program age variable ranges from 1 to 23, with a mean of 9.5 years. It would seem reasonable to assume that 9.5 years would be long enough for the vast majority of a state's motorists to learn of the existence of an ongoing personalized plate program in their state, yet the estimates of Table I imply that total sales increase at an increasing rate (i.e., a constant percentage rate) as time passes. To determine whether this result was simply a product of misspecification, the square of program age was added to the bandwagon equation. Its coefficient was small, positive, and insignificant, whereas the information diffusion model would predict a negative coefficient.

TABLE I

Dependent Variable: Log of Personalized Plate Sales 10 State Dummies included Standard errors are in parentheses under estimates

(a)Durbin-Watson statistic calculated as: [Mathematical Expression Omitted] Where [e.sub.jt] is the estimated OLS regression residual, and [T.sub.j] is the number of observations for state j.

The same logic provides the basis for a second test. As mentioned above, both the bandwagon and the diffusion model imply that sales are related negatively to both current and lagged price. Current price is directly related to current quantity, and lagged price is indirectly related to current quantity through its effect on lagged quantity. In the new product diffusion model, this indirect effect exists because price influences the rate of diffusion of information. As time passes and information spreads, the indirect effect of lagged price on current sales should diminish and eventually disappear. In the bandwagon model, on the other hand, the indirect effect never disappears. A price change will always have an impact on future sales beyond the standard price effect. This line of reasoning led me to add the log of lagged price and an interaction between it and program age to the basic demand equation. To be consistent with the new product diffusion model, the coefficient of the log of lagged price should be negative, and that of the interaction term should be positive. The presence of this pattern does not necessarily imply the existence of a pure diffusion process, as it would be consistent with a hybrid bandwagon-diffusion demand situation (see note 12) or a bandwagon effect that diminishes as total consumption grows. However, the absence of a diminishing impact from lagged price would be evidence against the importance of information diffusion in the market for personalized license plates.(13)

Table II presents the results of the test. The coefficients on lagged price and the interaction term correspond to the pattern suggested by the diffusion model, and the coefficient of the interaction term is significant. Thus, we cannot rule out information diffusion as one source of demand growth. However, the coefficient of -.8 on the lagged price variable and the coefficient of .033 on the interaction term taken together imply that it takes about twenty-four years for the influence of lagged price on present demand to disappear. This is puzzling since it seems reasonable to suppose that information about a personalized plate program would become fully diffused more quickly than that.

TABLE II

Test For Diffusion Process Dependent variable: Log of personalized plate sales n = 89 Regression includes 10 dummy variables and regressors of Table IA in addition to reported variables. Standard errors are in parentheses under estimates.

The lingering impact of lagged price on sales can be reconciled with the information diffusion model by considering the impact of consumer mobility on the level of awareness in a state. It seems safe to assume that the yearly flow of vehicle owners out of a state will have little effect on the level of awareness in that state, as there is little reason to believe that those who leave are drawn disproportionately from the informed or the uninformed segment of the state's population. Vehicle owners who have recently moved into the state, however, are much less likely than the average state resident to be informed about the existence of a personalized plate program in their new state. This reasoning suggests that as long as there is interstate mobility, there will be at least some segment of a state's car-owning population that is unaware of the state's plate program. It also suggests that, other things being equal, the proportion of potential plate consumers who are uninformed will be larger in a state in which a higher percentage of residents are recent arrivals. And, if lack of information is a barrier to the purchase of a personalized license plate, plate sales will thus be lower in states with a higher proportion of new residents. The 1980 census included information on the proportion of each state's population that had moved into that state in the previous five years. To test for an effect of interstate migration on plate sales, the regression reported in column B of Table I was re-estimated with this variable on the left hand side instead of the state dummy variables. Its coefficient was negative and significant, indicating that an increase of one percentage point in the migration variable decreased plate sales by 1.6 percent.(14)

With this result it becomes easier to square the estimates in Table III with the information diffusion story and to conclude that information diffusion is at least one of the processes generating the observed growth in sales of personalized license plates. It remains difficult to reconcile the effect of the age-of-program variable with the diffusion model. Perhaps a hybrid bandwagon-information diffusion model is the more appropriate one in this market.

TABLE III

Interstate Bandwagon Test n = 9 (NC, MN, WI, WY, WV, VA, IL, ID, CO) Standard errors are in parentheses under estimates.

Habit Formation and State-Dependent Behavior

In the bandwagon model an individual's preferences are affected by other consumers' behavior. A model of habit formation or state dependence would propose that an individual's purchase of a plate in the present period affects his likelihood of buying a plate in future periods. It could be that the experience of owning a plate changes or helps him discover the true nature of his preferences, or that the renewal of an existing plate has lower transaction costs than the purchase of a new plate.(15) Either of these assumptions can be captured by rewriting the value function of equation (1) as (1") [V.sub.it] = [V.sub.it]([X.sub.it], [Z.sub.t], [q.sub.t-1]); [V.sub.q] > 0 where [q.sub.t-1] = 1 if the individual owned a personalized plate in the previous period and zero if not.

Equation (1") will lead to a market level demand curve identical in form to the bandwagon demand curve of equation (3). Further, suppose that one element of [X.sub.it] is a zero-mean random shock to preferences, so that the [V.sub.it] function in equation (1") can be written as [Mathematical Expression Omitted]. Then, market level demand will grow over time even if the conventional demand shifters are stable. Consumers who buy a plate due to a positive [U.sub.it] in one period are likely to continue purchasing in future periods even in the absence of a positive [U.sub.it]. There will also be a positive relationship between present demand and demand in the previous period, even if one controls for present demand shifters. Thus, with only market level time series data on total sales it is difficult to discriminate between a bandwagon model and a habit formation or state dependence model.(16)

There is, fortunately, another approach to making such a discrimination. In the first year of a plate program, everyone who purchases a plate is doing so for the first time, and the previous purchase of a plate cannot affect behavior. However, a bandwagon effect, if it is assumed to operate across state lines, could have an impact on the demand for plates in a state just introducing the program. The willingness to pay of individuals in a state introducing a program might be influenced by the observation of out-of-state cars with personalized plates.

Evidence that the demand for personalized license plates in one state might be affected by the behavior of residents in neighboring states is provided by an examination of the order in which personalized plate programs were adopted in the lower forty-eight states. The initial programs were in New England, and as of 1965, all but two of the ten existing programs were in the Northeast. Ten more programs were established between 1966 and 1970; eight of these in states that bordered on a state already having a program. Of the first twenty states that introduced programs, thirteen or 65 percent were bordered by states that already had programs, and one more was within close proximity of a state with a program. To put this 65 percent figure into perspective, I performed an experiment in which twenty states were chosen at random, with each chosen state being checked to see if it was bordered by a previously chosen state. The experiment was repeated fifty times, and in only five cases were thirteen or more states bordered by previously chosen states.(17)

One possible explanation for this result is that residents of a state with no program see personalized plates on out-of-state cars, grow envious of these plates (via a bandwagon effect) and urge their lawmakers to establish a personalized license plate program in their state. (Some state officials volunteered that this was the case in their state.) Of course, it could be that a bandwagon effect of a different sort operates among politicians--seeing a successful revenue-generating program in a neighboring state, they are eager to adopt a similar program in their state.

There is another test that may shed light on whether an interstate bandwagon effect operates on the (potential) consumers of personalized plates, and that avoids the possible confusion of state dependence and bandwagon behavior. If the bandwagon effect operates across state lines, then among states instituting new programs plate sales should be higher, other things being equal, in states where citizens have been exposed to more out-of-state personalized plates. Table III reports the results of two regressions designed to check for this situation. Data on first-year sales are available for seven states; for two more long time series starting in the third year of the program were used to estimate first-year sales. The log of first-year sales was regressed on log price, log of eligible vehicles, log income, and one of two variables designed to proxy the exposure of citizens to out-of-state personalized plates: either the percentage of the state's border touching a state with a program already in place or the number of paved roads crossing into the state from states that already have programs (with multilane highways counting as two roads). The estimated coefficients of both variables have the positive signs predicted by an interstate bandwagon model, but they are not statistically significant, which is not surprising given the few degrees of freedom available. A more decisive test would require data on first-year sales from more states, or better still, individual level data on personalized plate demand.(18)

IV. CONCLUSION

In the years since Leibenstein first modeled the bandwagon effect, few if any economists have tried to find one in the real world. This may be because the bandwagon effect is unlikely to be important in many markets, or because, as this paper has shown, it is easily confused with more widely accepted and commonly modeled demand phenomena. However, if you accept the bandwagon effect as a possibility and wish to look for one, the market for vanity plates is a logical place to search. After all, one purpose of the plates is to communicate with other consumers, and communications have a way of eliciting responses.

My own search for the bandwagon effect has not resulted in an indisputable sighting of the quarry, but this paper has offered some positive evidence of a bandwagon effect in the market for personalized plates. The estimated demand curve in that market is consistent with the bandwagon model; particularly the dependence of present demand on past demand. This pattern in the data is also consistent with a product diffusion model, and the existence of a negative partial correlation between plate sales and the proportion of new residents in a state supports such an interpretation of the data. However, the fact that plate sales increase yearly at an increasing rate even when other demand shifters are held constant suggests that information diffusion is not the whole story. An attempt to determine whether the effect of past purchases on present demand has an interpersonal or an intrapersonal basis led to the question of whether a bandwagon effect might be operating across states lines, and the historical pattern of adoption of personalized plate programs supports the proposition that it is. Based on the available evidence, I would conclude that information diffusion is one factor underlying the pattern of growth in the sales of personalized places, but I would not rule out the existence of a bandwagon effect in this market. [Appendix Omitted]

(1)Pollak [1970, 760] also suggests this approach to modeling demand in the presence of consumption externalities. (2)Alper, Archibald, and Jensen [1987] also estimate a demand function for personalized plates, using 1983 sales data from forty-three states. (3)Promotional efforts have varied from state to state, ranging from attempts to make purchasing plates easier for consumers, to occasional public relation campaigns, to Virginia's current use of aggressive marketing techniques. In the empirical analysis, a dummy variable is used to signal the occurrence of any sort of extraordinary marketing activity. (4)Personal income data comes from the U.S. Department of Commerce's estimates of state personal income, 1969-1985. (See the Survey of Current Business, August 1986). State unemployment rates are from annual issues of the Handbook of Labor Statistics, 1975-1985, Department of Labor. Motor vehicle registration information comes from annual issues of Highway Statistics (U.S. Department of Transportation). All data used are available from the author upon request. (5)The state unemployment rate is not included, as it is not available before 1975 and using it cuts the sample size to eight-five. Regressions with this smaller smaple that include the state unemployment rate do not suggest that it is a determinant of personalized plate sales. (6)I assume that the supply curve for personalized plates in any state is perfectly elastic; that is, the state supplies plates to all demanders at the legislated price. If states behaved as profit-maximizing monopolists, there could be some simultaneity between quantity and price, but conversations with state officials leave me confident that this is not the case. Revenue maximization is not the goal of the legislatures that set the nominal price, and much of the variation in real prices is due to inflation. (See also Alper, Archibald, and Jensen [1987].) (7)The dummy variable approach is potentially less efficient than an error components--generalized least squares (GLS) approach, which would use both within--and across state variation to estimate the parameters of the demand function. However, the small number of states used in estimation raises the possibility of a sample, if not population, level correlation between excluded state specific effects and included variables, and such correlation would bias the GLS estimates. A Hausman test comparing the GLS and dummy variable estimates rejects the consistency of the GLS estimates. Estimation methods and tests used are described in Judge et al. [1985, 521-27]. (8)To test for heteroskedasticity, [Mathematical Expression Omitted] was regressed on the state dummy variables, and an F-test for the joint significance of the dummy variables was conducted. The F-statistic had a prob-value of .08. To correct standard errors, the OLS residuals were used to estimate the autocorrelation coefficient and the state specific variances. These estimates were used to construct the estimated error covariance matrices needed to calculate the adjusted standard errors for the OLS estimates. (9)The alpha level for each of these tests was .1. The improper exclusion of lagged sales from the column A estimates is one possible reason for the presence of autocorrelation and heteroskedasticity in the column A estimates, and their absence from column B estimates. (10)The results of these and all other regressions discussed but not reported in the text are available upon request. (11)New product diffusion models and related models of the diffusion of technological innovations abound in the literatures of economics, marketing, and management science. Rogers [1983] provides a good survey. (12)The information diffusion model and the bandwagon model can be seen as portraying two aspects of a broadly defined diffusion process: the spread of information and the influence exercised by trendsetting consumers on the rest of the population. See Gatignon and Robertson [1985]. (13)The diffusion model contemplated in this section concerns the spread of information about the existence of the plate program. One could also consider a model in which information about price changes takes time to spread. In that case, there would be a direct effect of lagged prices on sales following any price decline, and the test proposed above would not be useful. However, in the data used herein, real price changes come from two sources: changes in the price level and legislated changes in the nominal price charged for the plate. Changes in the price level arguably become apparent to all consumers at about the same time. The legislated price changes have (with one exception) been increases. A price increase affects plate sales only through driving current plate holders out of the market. Those not in the market, regardless of when or whether they learn about the price increase, will not enter the market. Current plate holders receive information on the price increase in the period it occurs, when their renewal notice arrives. Thus, increases in sales caused by a delayed realization of a price decline by some consumers are not a plausible explanation of Table I results. (14)The migration variable ranged from 5.1 to 28.3 with a mean of 13.2. Its estimated coefficient was -.016 (s.e. = .005). The coefficient on lagged plate sales was almost twice as large as its Table I counterpart, and the coefficient on price fell in absolute value. The coefficient on eligible vehicles became positive and significant. These changes were more due to the necessary exclusion of the state dummies from the regression than the inclusion of the migration variable. Indeed, the significant coefficient on the migration variable may be a product of the sort of spurious correlation discussed in note 7, but it does hold up even when the observations from Wyoming (the state with the highest proportion of new residents) are excluded from the sample. (15)Some pertinent general models of habit formation are presented in Pollak [1970], Becker and Murphy [1988], and Michaels [1988]. (16)A time series on annual sales to first time buyers could be used to discriminate between the two models, as the positive partial correlation between this measure and the lagged growth rate of total sales predicted by the bandwagon model could not be attributed to habit formation. Some states provide data on annual sales of new plates (i.e., new combinations as opposed to renewals), and data from these states yield significant evidence of the above-mentioned correlation. However, because buyers of new plates are not always first-time plate buyers and because of problems in the data on total sales from these states, this evidence in support of the bandwagon effect should be discounted somewhat. (17)The first twenty states to adopt programs, and the year in which they adopted them, are as follows: 1937-Connecticut; 1955-New York; 1957 New Hampshire; 1958-Vermont; 1961-Rhode Island; 1962-Maine; 1963-North Dakota; 1964-Pennsylvania; 1965-Texas; 1966-Massachusetts; 1967-New Mexico; 1968-Arkansas, Delaware, North Carolina, Oklahoma, 1968-Nevada; 1970-California, Oregon, Wyoming. (18)A third model leading to an equation like (3) is the standard partial adjustment model. This model is often applied to durables, and has been successfully applied to nondurables such as electricity or beer. However, I see no compelling reason to interpret the Table I estimates in a partial adjustment framework. A commodity like electricity is nondurable but complementary with certain durable goods, so that adjustment in electricity use follows adjustment in the stocks of complementary durables. With a good like beer, it can be argued that aggregate consumption increases through gradual marginal increases in consumption by individuals. There are no equally plausible arguments for applying a partial adjustment model to personalized plates. The personalized plate is distinct from most goods in that individuals generally purchase only one, once a year. Changes in the stock of personalized plates come almost entirely through entry into or exit from the market, not from alterations in the consumption flows of those in the market. If a fall in price is to have an impact on demand more than a year after its occurrence, two possible explanations suggest themselves: either the price decrease somehow alters the tastes of some individuals over a year after its occurrence, or some individuals take more than a year to find out about the price decrease. The bandwagon model is a specific version of the first explanation, while the second suggests the sort of information diffusion model discussed in note 12. A third justification for applying a partial adjustment interpretation to the table I results is not readily apparent.

REFERENCES

Alper, Neil, Robert B. Archibald, and Eric Jensen. "At What Price Vanity?: An Econometric Model of

the Demand for Personalized License Plates." National Tax Journal, March 1987, 103-10. Becker, Gary S. and Kevin M. Murphy. "A Theory of Rational Addiction." Journal of Political Economy,

August 1988, 675-700. Gatignon, Hubert and Thomas S. Robertson. "A Propositional Inventory for New Diffusion Research."

Journal of Consumer Research, March 1985, 849-67. Judge, George G. et al. The Theory and Practice of Econometrics, 2nd ed. New York: Wiley and Sons,

1985. Kripaitis-Neely, Paula. "Personalized License Plates in the United States: 1983 Survey Results."

Mimeo, Virginia Division of Motor Vehicles, 1985. Leibenstein, Harvey. "Bandwagon, Snob, and Veblen Effects in the Theory of Consumer Demand." The

Quarterly Journal of Economics, August 1950, 183-207. Michaels, Robert J. "Addiction, Compulsion, and the Technology of Consumption." Economic Inquiry,

January 1988, 75-88. Pollak, Robert A. "Habit Formation and Dynamic Demand Functions." Journal of Political Economy,

August 1970, 745-63. Rogers, Everett M. Diffusion of Innovations, 3rd ed. New York: The Free Press, 1983.

JEFF BIDDLE, Assistant Professor of Economics, Michigan State University. I would like to thank the many state officials who so willingly provided the information used in this study, and to acknowledge the helpful comments of Charles Clotfelter, Dan Hamermesh, Tim Lane, and Dan Suits.

The bandwagon effect is a consumption externality that exists when an individual's demand for a good is increased by his observation of other consumers using that good. This paper models a product demand curve with a bandwagon effect and, using data on sales of personalized license plates, estimates such a demand curve. Certain more conventional models of product demand, including information diffusion and habit formation models, are observationally similar to the bandwagon model, despite being conceptually different from it. I attempt to use the license plate data to discriminate between the bandwagon model and these other models.

In 1950, Harvey Leibenstein described the impact on product demand curves of what he called a "bandwagon effect." The bandwagon effect was said to exist if peoples' valuations of a good (and thus demand for that good) increased when they observed others consuming the good. No one to my knowledge has attempted to empirically verify the existence of or estimate the magnitude of a bandwagon effect in any product market. In this paper I attempt to do both by looking at the demand for personalized license plates, a good that could plausibly be subject to a bandwagon effect.

In the first section of the paper I develop a simple model of product demand in the presence of a bandwagon effect. The model is based on Leibenstein's treatment but is modified to fit the case of personalized license plates. In the second section, data on the sales of personalized plates are described and used to estimate a demand function. The results of this estimation are consistent with the theoretical model of the bandwagon effect.

The search for the bandwagon effect cannot end there, however. The bandwagon model, in which consumers are moved by considerations of fashion and status emulation, is observationally similar to other well-known but conceptually quite different models of product demand in which consumers are more sober and level-headed. In particular, models of information diffusion or habit formation imply patterns in the data very much like those that follow from the bandwagon model. Section III discusses this matter in more detail and presents statistical tests designed to differentiate between the bandwagon model and alternative models of product demand. In the final section I draw some conclusions from the empirical evidence.

I. THE BANDWAGON EFFECT

Consider a product that lasts for a single period of well-defined length. Each period, a consumer may purchase at most one of the product. The personalized license plate is a product of this sort, as are subscriptions to a particular magazine, hairstyles, club memberships, vacations to chic resorts, etc. Assume first that there is no bandwagon effect and let (1) [V.sub.it]([X.sub.it], [Z.sub.t]) represent the value of the product to individual i in period t, where [X.sub.it] is a vector of individual characteristics such as age or income, and [Z.sub.t] is a vector of product characteristics in period t. The function [V.sub.it] is a random variable defined over the population of potential buyers. Let [F.sub.t](V) = prob([V.sub.it] < V) in period t. Then, if [M.sub.t] is the population of potential consumers in period t, market level demand in period t is (2) [Q.sub.t] = [M.sub.t][1 - [F.sub.t]([p.sub.t])] where [p.sub.t] is the product price. Demand is a decreasing function of price. It also depends on product characteristics and the distribution of personal characteristics in the population.

Adding a bandwagon effect to this model involves allowing individuals' preferences to depend on aggregate behavior. Liebenstein assumed that in the presence of a bandwagon effect the quantity of a product demanded by each consumer at a given price depended on market level demand at that price. This creates a problem when moving from individual to market level demand functions, for the market level demand curve is no longer a lateral summation of individual demand curves. The problem can be avoided if we replace Leibenstein's characterization of the bandwagon effect with one that is arguably more realistic by assuming that consumer demand in period t depends on the market level demand in the preceding period.(1) In the case of personalized license plates, this amounts to assuming that the likelihood that a person buys or renews a personalized plate at any given price depends on the number of those plates he has recently seen on other peoples' cars. The assumption can be incorporated into the model by changing the individuals' valuation functions to (1') [V.sub.it] = [V.sub.it]([X.sub.it],[Z.sub.t],[Q.sub.t-1]) where [Q.sub.t-1] is market level demand for the product in the previous period. Then, aggregating in the manner of equation (2), market level demand can be written (3) [Q.sub.t] = f([p.sub.t], [Q.sub.t-1]) where [f.sub.1] < 0 and [f.sub.2] > 0. The dependence of demand on product characteristics and the distribution of population characteristics has been suppressed.

In this model of the bandwagon effect demand can grow over time when price and other demand shifters remain constant, growing explosively or reaching a steady state in which f([p.sub.t], [Q.sub.t-1]) = [Q.sub.t-1]. A steady state is stable if [f.sub.2]([p.sub.s],[Q.sub.s]) < 1, where [p.sub.s] and [Q.sub.s] are steady state price and quantity. Leibenstein insured the existence of a stable steady state by assuming at the individual level a "diminishing marginal effect of external consumption" which translates into a negative second derivative of [V.sub.it] with respect to [Q.sub.t-1] in equation (1'). Leibenstein's solution is not appropriate for the sort of product considered here, as each person is assumed to purchase at most one product per period. However, a case for the existence of a stable steady state can still be made by assuming a heterogeneous population in which, at any level of demand, some peoples' valuation functions are decreasing in [Q.sub.t-1], a phenomenon Leibenstein called the "snob effect." As the level of total consumption grows, people whose tastes are subject to the snob effect experience a fall in their [V.sub.it] functions. Those who have been in the market in previous periods will drop out, and those who have never entered the market will continue to stay out. If a sufficient number of people are "snobbish," there will be a stable steady state level of total demand, even if many individuals do not experience a diminishing marginal effect of external consumption.

Liebenstein demonstrates that the presence of a bandwagon effect makes demand more price elastic because the quantity change brought forth by a price change leads to further quantity change. The model just outlined also has this feature, although the augmented effect of a price change works itself out over several periods: a fall in the current price leads to an increase in current demand, which in turn causes an increase in next period's demand through the bandwagon effect, and so on. Figure 1 shows the steady state demand curve (relating price to steady state demand) in a market with a bandwagon effect. The top panel shows the relationship between [Q.sub.t] and [Q.sub.t-1] at three different prices. The bottom panel derives from the top panel the relationship between price and steady state demand. The top panel also shows the multi-period process of adjustment following a fall in price from [p.sub.3] to [p.sub.2], with [Delta][Q.sub.1] and [Delta][Q.sub.2] being the quantity adjustments in the first and second periods.

II. THE BANDWAGON EFFECT IN THE MARKET FOR PERSONALIZED LICENSE PLATES

My attempt to estimate a bandwagon effect involves a product that could plausibly generate such an effect: the personalized or "vanity" license plate, a commodity whose very name suggests the sort of status conscious or emulative tendencies supposed to lie behind the bandwagon effect. All fifty states and the District of Columbia currently sell vanity plates, allowing vehicle owners who pay a special fee to choose the sequence of symbols that will appear on their license plate. In this section I will use data on the sales of vanity plates to estimate a market demand function, first using a standard specification, then using a specification adjusted to allow for the presence of a bandwagon effect.(2)

Officials from twenty-one states were kind enough to supply me with data on personalized plate sales in their states and to describe the regulations governing their plate programs. Additional information on personalized plate programs was taken from Kripaitis-Neely [1985]. The annual sales data from the states cover periods of time ranging from three to eighteen years in length. Unfortunately, differences in pricing policies and record keeping create some noncomparabilities between data from different states. The estimation discussed in this section involves data from eleven states that use the most prevalent pricing system, one which is consistent with the assumptions of the model developed above. Under this system, vehicle owners decide once each year whether or not they want a personalized plate for the upcoming year. If they do, they pay a fixed fee over and above their basic registration fee. The fee is the same whether they are buying a new personalized plate or renewing one from a previous year. These states keep track of the number of valid personalized plates on the road each year, that is, the number of personalized plates issued for the first time that year plus the number of plates renewed from the previous year. The eleven states taken together provide 100 observations on annual sales. (Details on these data are in the appendix.)

Estimating equations (2) and (3) requires specifications of [Q.sub.t], [M.sub.t], [Z.sub.t], and [p.sub.t] appropriate to the market for personalized license plates. [Q.sub.t] is defined as the number of valid plates on the road at a particular point in year t. Since the potential market for personalized plates in a state is limited by the number of vehicles in the state eligible to bear such plates, the [M.sub.t] variable is constructed using Department of Transportation information on state vehicle registration in conjunction with each state's rules regarding eligibility for personalized plates. The [Z.sub.t] vector includes variables specific to a state's plate program in a year: the number of characters permitted on the plate, the age of the state's program, and a variable indicating whether or not a state promoted its program that year.(3) Data on state per-capita incomes and unemployment rates have been collected to measure differences across states in consumers' incomes and economic situations. Finally, the price of a plate is measured by the amount consumers must pay for a personalized plate over and above regular license and registration fees. Nominal price and income data have been deflated using the Consumer Price Index.(4)

The data were first used to estimate a standard demand function corresponding to equation (2), and the results are reported in column A of Table I. The dependent variable was the log of valid plates; independent variables included the logs of price, per-capita income, and the number of eligible vehicles; a dummy variable for promotion activity, the number of characters permitted on the plate, and the age of the state's program.(5) State dummy variables were also added to the regression to capture state by state differences in demand arising from excluded [Z.sub.t] variables and differences across states in the distribution of consumer characteristics.(6,7) The form of the error term was assumed to be [e.sub.jt] = [[Rho]e.sub.jt-1] + [u.sub.jt] where j indexes states, t indexes time periods, and [Mathematical expression omitted] thus allowing for both serial correlation and heteroskedasticity. A Durbin-Watson test suggested the presence of serial correlation, and there was also evidence that [Mathematical Expressin Omitted] might not be constant across states. The estimated standard errors of the OLS coefficient estimates reported in column A of Table I have been adjusted to reflect these findings.(8)

The column A estimates suggest that the demand for personalized plates is price elastic, and the coefficient of the income variable is positive but not significant. The coefficient on eligible vehicles is negative but smaller than its standard error, presumably because controlling for interstate differences in vehicle population with state dummy variables leaves little leverage in the data for precisely estimating the impact on plate sales of changes in the number of eligible vehicles. States can increase sales with promotional campaigns or by increasing the number of characters allowed on the plate. Plate sales grow by about 4 percent a year as a state's program ages, holding constant the contemporaneous demand shifters included in the regression. This, of course, is one implication of the bandwagon model.

To estimate equation (3), the bandwagon demand function, I adjusted the specification used to generate the estimates of Table I, column A to allow the log of current demand to depend on the log of the previous period's demand. Once again, state dummies were included in the regression. The results of estimating this equation are reported in column B of Table I. Durbin's h statistic, designed to detect first-order autocorrelation in the presence of a lagged dependent variable (see Judge et al. [1985, 326]) failed to reject the hypothesis of no autocorrelation. The test for heteroskedasticity used on the column A estimates was also applied to the bandwagon specification, and the hypothesis of homoskedasticity could not be rejected.(9)

The column B results are in accordance with the predictions of the bandwagon model. The coefficient of the lag of plate sales is positive, significant, and less than one. The coefficient of the price variable is roughly half the size of its column A counterpart, but this too is consistent with the bandwagon model. Consider a state that lowers its plate price in the fifth year of its program and keeps it low in the ensuing years. Other things remaining equal, the demand in the sixth year will be higher than it was in the fourth year for two reasons: the lower price in the sixth year and the higher demand in the fifth year engendered by the lower price in the fifth year. The price coefficient in column A captures both of these effects, while in column B the two effects are separated out. Experimentation with other functional forms for the relationship between plate sales, price, income, and previous period's plate sales produced results that were essentially the same as those in column B.(10)

One puzzling aspect of the column B results is that the estimated effect of the age of the program on sales is basically unaffected by the introduction of lagged sales into the demand equation. Further, when a second lag of plate sales is also added to the equation, its coefficient is insignificant and the coefficient of the age-of-program variable remains unchanged. In effect, as a state's plate program gets older demand grows, holding constant price, income, number of characters allowed on the plate, promotional activity, and any effect on demand that comes from past plate sales. A possible explanation for this that is in the spirit of the bandwagon model involves the possibility that "word of mouth" can augment the actual display of personalized plates as fuel for the bandwagon. As time passes following the introduction of a plate program by a state, a typical resident will repeatedly become involved in or overhear discussions of interesting personalized plates. Each of these experiences will add to his impression that a bandwagon is rolling and raise his valuation of the personalized plate.

Clearly, if one is willing to accept this story as well as the model of section I as an appropriate characterization of the market for personalized plates, then we can conclude from the column B estimates that the bandwagon effect is an important factor in that market. Consumers are more likely to purchase plates if they have seen others displaying the plates, and thus increases in sales lead, in and of themselves, to further increases in sales.

III. ALTERNATIVE MODELS OF PRODUCT DEMAND

It was taken as given in discussing the Table I results that they reflected the presence of a bandwagon effect, but the bandwagon interpretation is not the only possible one. Other models of product demand can lead to similar patterns in the data; more conventional models that require no assumption of faddish, frivolous, or fashion-conscious behavior on the part of consumers. There are models of information diffusion in which it takes time for consumers to receive and process information about new products, and models of habit formation in which a person's purchase of a product in one period can alter the chance that he will purchase the product in subsequent periods. This section explores the problem of discriminating between these models and the bandwagon model of demand.

New Product-Information Diffusion

Many researchers have outlined models in which demand for a new product grows as information about or awareness of the new product spreads.(11) A model of this sort could be applied to personalized license plates by assuming that when a personalized plate program is first introduced into a state, many who would otherwise buy a plate do not because they are not fully informed about the program. As time passes, information about the program is diffused and previously uninformed buyers enter the market, leading to a growth in plate sales over time.

Though both the bandwagon model and this diffusion model imply a growth in sales over time even if conventional demand shifters remain constant, the causes of that growth are conceptually different in the two models. In the bandwagon model everyone is aware of the existence of the plate program. The purchase and display of plates in one period influences demand in the next period by changing peoples' tastes and raising their willingness to pay for a plate. In the diffusion model each individual's willingness to pay for a plate is independent of the total number of plates sold. However, not all individuals are aware of the program, and the uninformed do not purchase plates even though they would value a plate at more than its price. The mere passage of time leads to growth in plate sales as more potential buyers become aware and enter the market. If one also assumes that the display of personalized plates spreads information about the existence of the plate program, the number sold in a given period will influence the number of plates sold in subsequent periods.

Despite the conceptual differences between the two models, they are observationally very similar. As already noted, in both models demand tends to grow over time, and in both models this period's demand can depend on last period's sales as well as this period's demand shifters. Another similarity is that a fall in price in the present period leads to growth in demand beyond that caused by a pure price effect, since the increased sales in the present cause further sales growth in the future. The positive coefficient on the promotion variable in Table I is consistent with the information diffusion story, because promotion is a way of spreading information. However, it can also be argued that promotion actually changes peoples' preferences and perhaps fuels the bandwagon. Obviously, discriminating between the bandwagon and diffusion models can be difficult.(12)

There is at least one important difference between the two models. An information diffusion process can only affect a product demand curve for a limited period of time, i.e., until information about a product is fully diffused. On the other hand, the bandwagon effect modeled above is permanent, in that increases or decreases in sales always affect future demand. This difference between the models suggests ways of gauging the importance of information diffusion as an explanation of the growth in vanity plate sales.

For example, consider the relationship between plate sales and the age of a plate program. If we accept the information diffusion story, demand increases as time passes because each year more people become informed about the existence of the plate program. However, these increases can continue only so long as there remain potential consumers who are unaware of the plate program, and they will diminish over time as this pool of uninformed people dries up. In the sample used in section II, the program age variable ranges from 1 to 23, with a mean of 9.5 years. It would seem reasonable to assume that 9.5 years would be long enough for the vast majority of a state's motorists to learn of the existence of an ongoing personalized plate program in their state, yet the estimates of Table I imply that total sales increase at an increasing rate (i.e., a constant percentage rate) as time passes. To determine whether this result was simply a product of misspecification, the square of program age was added to the bandwagon equation. Its coefficient was small, positive, and insignificant, whereas the information diffusion model would predict a negative coefficient.

TABLE I

Dependent Variable: Log of Personalized Plate Sales 10 State Dummies included Standard errors are in parentheses under estimates

A B Standard Demand Bandwagon Demand n = 100 n = 89 Log (Price) -1.30 Log (Price) -.62 (.18) (.19) Log (Lag (Plates)) .35 (.07) Log (Income) .57 Log (Income) .03 (.45) (.30) Log (Cars) -.30 Log (Cars) -.45 (.44) (.34) Promotional .36 Promotional .63 Activity (.12) Activity (.16) Number of .12 Number of -.02 Characters (.057) Characters (.06) Age of .036 Age of .05 Program (.018) Program (.02) [R.sup.2] .98 [R.sup.2] .99 DW(a) 1.57

(a)Durbin-Watson statistic calculated as: [Mathematical Expression Omitted] Where [e.sub.jt] is the estimated OLS regression residual, and [T.sub.j] is the number of observations for state j.

The same logic provides the basis for a second test. As mentioned above, both the bandwagon and the diffusion model imply that sales are related negatively to both current and lagged price. Current price is directly related to current quantity, and lagged price is indirectly related to current quantity through its effect on lagged quantity. In the new product diffusion model, this indirect effect exists because price influences the rate of diffusion of information. As time passes and information spreads, the indirect effect of lagged price on current sales should diminish and eventually disappear. In the bandwagon model, on the other hand, the indirect effect never disappears. A price change will always have an impact on future sales beyond the standard price effect. This line of reasoning led me to add the log of lagged price and an interaction between it and program age to the basic demand equation. To be consistent with the new product diffusion model, the coefficient of the log of lagged price should be negative, and that of the interaction term should be positive. The presence of this pattern does not necessarily imply the existence of a pure diffusion process, as it would be consistent with a hybrid bandwagon-diffusion demand situation (see note 12) or a bandwagon effect that diminishes as total consumption grows. However, the absence of a diminishing impact from lagged price would be evidence against the importance of information diffusion in the market for personalized license plates.(13)

Table II presents the results of the test. The coefficients on lagged price and the interaction term correspond to the pattern suggested by the diffusion model, and the coefficient of the interaction term is significant. Thus, we cannot rule out information diffusion as one source of demand growth. However, the coefficient of -.8 on the lagged price variable and the coefficient of .033 on the interaction term taken together imply that it takes about twenty-four years for the influence of lagged price on present demand to disappear. This is puzzling since it seems reasonable to suppose that information about a personalized plate program would become fully diffused more quickly than that.

TABLE II

Test For Diffusion Process Dependent variable: Log of personalized plate sales n = 89 Regression includes 10 dummy variables and regressors of Table IA in addition to reported variables. Standard errors are in parentheses under estimates.

Log (Price) -.61 (.20) Lag (Log (Price)) -.80 (.18) Age of Program -.006 (.027) (Age of Program) X [Lag (Log (Price))] .033 (.011)

The lingering impact of lagged price on sales can be reconciled with the information diffusion model by considering the impact of consumer mobility on the level of awareness in a state. It seems safe to assume that the yearly flow of vehicle owners out of a state will have little effect on the level of awareness in that state, as there is little reason to believe that those who leave are drawn disproportionately from the informed or the uninformed segment of the state's population. Vehicle owners who have recently moved into the state, however, are much less likely than the average state resident to be informed about the existence of a personalized plate program in their new state. This reasoning suggests that as long as there is interstate mobility, there will be at least some segment of a state's car-owning population that is unaware of the state's plate program. It also suggests that, other things being equal, the proportion of potential plate consumers who are uninformed will be larger in a state in which a higher percentage of residents are recent arrivals. And, if lack of information is a barrier to the purchase of a personalized license plate, plate sales will thus be lower in states with a higher proportion of new residents. The 1980 census included information on the proportion of each state's population that had moved into that state in the previous five years. To test for an effect of interstate migration on plate sales, the regression reported in column B of Table I was re-estimated with this variable on the left hand side instead of the state dummy variables. Its coefficient was negative and significant, indicating that an increase of one percentage point in the migration variable decreased plate sales by 1.6 percent.(14)

With this result it becomes easier to square the estimates in Table III with the information diffusion story and to conclude that information diffusion is at least one of the processes generating the observed growth in sales of personalized license plates. It remains difficult to reconcile the effect of the age-of-program variable with the diffusion model. Perhaps a hybrid bandwagon-information diffusion model is the more appropriate one in this market.

TABLE III

Interstate Bandwagon Test n = 9 (NC, MN, WI, WY, WV, VA, IL, ID, CO) Standard errors are in parentheses under estimates.

Log (Price) -1.41 -1.31 (.82) (.63) Log (Income) 2.22 2.63 (2.62) (1.82) Log (Cars) .77 .73 (.36) (.39) Border Variable .51 -- (1.57) Roads Variable -- .0045 (.0130) [R.sup.2] .81 .82

Habit Formation and State-Dependent Behavior

In the bandwagon model an individual's preferences are affected by other consumers' behavior. A model of habit formation or state dependence would propose that an individual's purchase of a plate in the present period affects his likelihood of buying a plate in future periods. It could be that the experience of owning a plate changes or helps him discover the true nature of his preferences, or that the renewal of an existing plate has lower transaction costs than the purchase of a new plate.(15) Either of these assumptions can be captured by rewriting the value function of equation (1) as (1") [V.sub.it] = [V.sub.it]([X.sub.it], [Z.sub.t], [q.sub.t-1]); [V.sub.q] > 0 where [q.sub.t-1] = 1 if the individual owned a personalized plate in the previous period and zero if not.

Equation (1") will lead to a market level demand curve identical in form to the bandwagon demand curve of equation (3). Further, suppose that one element of [X.sub.it] is a zero-mean random shock to preferences, so that the [V.sub.it] function in equation (1") can be written as [Mathematical Expression Omitted]. Then, market level demand will grow over time even if the conventional demand shifters are stable. Consumers who buy a plate due to a positive [U.sub.it] in one period are likely to continue purchasing in future periods even in the absence of a positive [U.sub.it]. There will also be a positive relationship between present demand and demand in the previous period, even if one controls for present demand shifters. Thus, with only market level time series data on total sales it is difficult to discriminate between a bandwagon model and a habit formation or state dependence model.(16)

There is, fortunately, another approach to making such a discrimination. In the first year of a plate program, everyone who purchases a plate is doing so for the first time, and the previous purchase of a plate cannot affect behavior. However, a bandwagon effect, if it is assumed to operate across state lines, could have an impact on the demand for plates in a state just introducing the program. The willingness to pay of individuals in a state introducing a program might be influenced by the observation of out-of-state cars with personalized plates.

Evidence that the demand for personalized license plates in one state might be affected by the behavior of residents in neighboring states is provided by an examination of the order in which personalized plate programs were adopted in the lower forty-eight states. The initial programs were in New England, and as of 1965, all but two of the ten existing programs were in the Northeast. Ten more programs were established between 1966 and 1970; eight of these in states that bordered on a state already having a program. Of the first twenty states that introduced programs, thirteen or 65 percent were bordered by states that already had programs, and one more was within close proximity of a state with a program. To put this 65 percent figure into perspective, I performed an experiment in which twenty states were chosen at random, with each chosen state being checked to see if it was bordered by a previously chosen state. The experiment was repeated fifty times, and in only five cases were thirteen or more states bordered by previously chosen states.(17)

One possible explanation for this result is that residents of a state with no program see personalized plates on out-of-state cars, grow envious of these plates (via a bandwagon effect) and urge their lawmakers to establish a personalized license plate program in their state. (Some state officials volunteered that this was the case in their state.) Of course, it could be that a bandwagon effect of a different sort operates among politicians--seeing a successful revenue-generating program in a neighboring state, they are eager to adopt a similar program in their state.

There is another test that may shed light on whether an interstate bandwagon effect operates on the (potential) consumers of personalized plates, and that avoids the possible confusion of state dependence and bandwagon behavior. If the bandwagon effect operates across state lines, then among states instituting new programs plate sales should be higher, other things being equal, in states where citizens have been exposed to more out-of-state personalized plates. Table III reports the results of two regressions designed to check for this situation. Data on first-year sales are available for seven states; for two more long time series starting in the third year of the program were used to estimate first-year sales. The log of first-year sales was regressed on log price, log of eligible vehicles, log income, and one of two variables designed to proxy the exposure of citizens to out-of-state personalized plates: either the percentage of the state's border touching a state with a program already in place or the number of paved roads crossing into the state from states that already have programs (with multilane highways counting as two roads). The estimated coefficients of both variables have the positive signs predicted by an interstate bandwagon model, but they are not statistically significant, which is not surprising given the few degrees of freedom available. A more decisive test would require data on first-year sales from more states, or better still, individual level data on personalized plate demand.(18)

IV. CONCLUSION

In the years since Leibenstein first modeled the bandwagon effect, few if any economists have tried to find one in the real world. This may be because the bandwagon effect is unlikely to be important in many markets, or because, as this paper has shown, it is easily confused with more widely accepted and commonly modeled demand phenomena. However, if you accept the bandwagon effect as a possibility and wish to look for one, the market for vanity plates is a logical place to search. After all, one purpose of the plates is to communicate with other consumers, and communications have a way of eliciting responses.

My own search for the bandwagon effect has not resulted in an indisputable sighting of the quarry, but this paper has offered some positive evidence of a bandwagon effect in the market for personalized plates. The estimated demand curve in that market is consistent with the bandwagon model; particularly the dependence of present demand on past demand. This pattern in the data is also consistent with a product diffusion model, and the existence of a negative partial correlation between plate sales and the proportion of new residents in a state supports such an interpretation of the data. However, the fact that plate sales increase yearly at an increasing rate even when other demand shifters are held constant suggests that information diffusion is not the whole story. An attempt to determine whether the effect of past purchases on present demand has an interpersonal or an intrapersonal basis led to the question of whether a bandwagon effect might be operating across states lines, and the historical pattern of adoption of personalized plate programs supports the proposition that it is. Based on the available evidence, I would conclude that information diffusion is one factor underlying the pattern of growth in the sales of personalized places, but I would not rule out the existence of a bandwagon effect in this market. [Appendix Omitted]

(1)Pollak [1970, 760] also suggests this approach to modeling demand in the presence of consumption externalities. (2)Alper, Archibald, and Jensen [1987] also estimate a demand function for personalized plates, using 1983 sales data from forty-three states. (3)Promotional efforts have varied from state to state, ranging from attempts to make purchasing plates easier for consumers, to occasional public relation campaigns, to Virginia's current use of aggressive marketing techniques. In the empirical analysis, a dummy variable is used to signal the occurrence of any sort of extraordinary marketing activity. (4)Personal income data comes from the U.S. Department of Commerce's estimates of state personal income, 1969-1985. (See the Survey of Current Business, August 1986). State unemployment rates are from annual issues of the Handbook of Labor Statistics, 1975-1985, Department of Labor. Motor vehicle registration information comes from annual issues of Highway Statistics (U.S. Department of Transportation). All data used are available from the author upon request. (5)The state unemployment rate is not included, as it is not available before 1975 and using it cuts the sample size to eight-five. Regressions with this smaller smaple that include the state unemployment rate do not suggest that it is a determinant of personalized plate sales. (6)I assume that the supply curve for personalized plates in any state is perfectly elastic; that is, the state supplies plates to all demanders at the legislated price. If states behaved as profit-maximizing monopolists, there could be some simultaneity between quantity and price, but conversations with state officials leave me confident that this is not the case. Revenue maximization is not the goal of the legislatures that set the nominal price, and much of the variation in real prices is due to inflation. (See also Alper, Archibald, and Jensen [1987].) (7)The dummy variable approach is potentially less efficient than an error components--generalized least squares (GLS) approach, which would use both within--and across state variation to estimate the parameters of the demand function. However, the small number of states used in estimation raises the possibility of a sample, if not population, level correlation between excluded state specific effects and included variables, and such correlation would bias the GLS estimates. A Hausman test comparing the GLS and dummy variable estimates rejects the consistency of the GLS estimates. Estimation methods and tests used are described in Judge et al. [1985, 521-27]. (8)To test for heteroskedasticity, [Mathematical Expression Omitted] was regressed on the state dummy variables, and an F-test for the joint significance of the dummy variables was conducted. The F-statistic had a prob-value of .08. To correct standard errors, the OLS residuals were used to estimate the autocorrelation coefficient and the state specific variances. These estimates were used to construct the estimated error covariance matrices needed to calculate the adjusted standard errors for the OLS estimates. (9)The alpha level for each of these tests was .1. The improper exclusion of lagged sales from the column A estimates is one possible reason for the presence of autocorrelation and heteroskedasticity in the column A estimates, and their absence from column B estimates. (10)The results of these and all other regressions discussed but not reported in the text are available upon request. (11)New product diffusion models and related models of the diffusion of technological innovations abound in the literatures of economics, marketing, and management science. Rogers [1983] provides a good survey. (12)The information diffusion model and the bandwagon model can be seen as portraying two aspects of a broadly defined diffusion process: the spread of information and the influence exercised by trendsetting consumers on the rest of the population. See Gatignon and Robertson [1985]. (13)The diffusion model contemplated in this section concerns the spread of information about the existence of the plate program. One could also consider a model in which information about price changes takes time to spread. In that case, there would be a direct effect of lagged prices on sales following any price decline, and the test proposed above would not be useful. However, in the data used herein, real price changes come from two sources: changes in the price level and legislated changes in the nominal price charged for the plate. Changes in the price level arguably become apparent to all consumers at about the same time. The legislated price changes have (with one exception) been increases. A price increase affects plate sales only through driving current plate holders out of the market. Those not in the market, regardless of when or whether they learn about the price increase, will not enter the market. Current plate holders receive information on the price increase in the period it occurs, when their renewal notice arrives. Thus, increases in sales caused by a delayed realization of a price decline by some consumers are not a plausible explanation of Table I results. (14)The migration variable ranged from 5.1 to 28.3 with a mean of 13.2. Its estimated coefficient was -.016 (s.e. = .005). The coefficient on lagged plate sales was almost twice as large as its Table I counterpart, and the coefficient on price fell in absolute value. The coefficient on eligible vehicles became positive and significant. These changes were more due to the necessary exclusion of the state dummies from the regression than the inclusion of the migration variable. Indeed, the significant coefficient on the migration variable may be a product of the sort of spurious correlation discussed in note 7, but it does hold up even when the observations from Wyoming (the state with the highest proportion of new residents) are excluded from the sample. (15)Some pertinent general models of habit formation are presented in Pollak [1970], Becker and Murphy [1988], and Michaels [1988]. (16)A time series on annual sales to first time buyers could be used to discriminate between the two models, as the positive partial correlation between this measure and the lagged growth rate of total sales predicted by the bandwagon model could not be attributed to habit formation. Some states provide data on annual sales of new plates (i.e., new combinations as opposed to renewals), and data from these states yield significant evidence of the above-mentioned correlation. However, because buyers of new plates are not always first-time plate buyers and because of problems in the data on total sales from these states, this evidence in support of the bandwagon effect should be discounted somewhat. (17)The first twenty states to adopt programs, and the year in which they adopted them, are as follows: 1937-Connecticut; 1955-New York; 1957 New Hampshire; 1958-Vermont; 1961-Rhode Island; 1962-Maine; 1963-North Dakota; 1964-Pennsylvania; 1965-Texas; 1966-Massachusetts; 1967-New Mexico; 1968-Arkansas, Delaware, North Carolina, Oklahoma, 1968-Nevada; 1970-California, Oregon, Wyoming. (18)A third model leading to an equation like (3) is the standard partial adjustment model. This model is often applied to durables, and has been successfully applied to nondurables such as electricity or beer. However, I see no compelling reason to interpret the Table I estimates in a partial adjustment framework. A commodity like electricity is nondurable but complementary with certain durable goods, so that adjustment in electricity use follows adjustment in the stocks of complementary durables. With a good like beer, it can be argued that aggregate consumption increases through gradual marginal increases in consumption by individuals. There are no equally plausible arguments for applying a partial adjustment model to personalized plates. The personalized plate is distinct from most goods in that individuals generally purchase only one, once a year. Changes in the stock of personalized plates come almost entirely through entry into or exit from the market, not from alterations in the consumption flows of those in the market. If a fall in price is to have an impact on demand more than a year after its occurrence, two possible explanations suggest themselves: either the price decrease somehow alters the tastes of some individuals over a year after its occurrence, or some individuals take more than a year to find out about the price decrease. The bandwagon model is a specific version of the first explanation, while the second suggests the sort of information diffusion model discussed in note 12. A third justification for applying a partial adjustment interpretation to the table I results is not readily apparent.

REFERENCES

Alper, Neil, Robert B. Archibald, and Eric Jensen. "At What Price Vanity?: An Econometric Model of

the Demand for Personalized License Plates." National Tax Journal, March 1987, 103-10. Becker, Gary S. and Kevin M. Murphy. "A Theory of Rational Addiction." Journal of Political Economy,

August 1988, 675-700. Gatignon, Hubert and Thomas S. Robertson. "A Propositional Inventory for New Diffusion Research."

Journal of Consumer Research, March 1985, 849-67. Judge, George G. et al. The Theory and Practice of Econometrics, 2nd ed. New York: Wiley and Sons,

1985. Kripaitis-Neely, Paula. "Personalized License Plates in the United States: 1983 Survey Results."

Mimeo, Virginia Division of Motor Vehicles, 1985. Leibenstein, Harvey. "Bandwagon, Snob, and Veblen Effects in the Theory of Consumer Demand." The

Quarterly Journal of Economics, August 1950, 183-207. Michaels, Robert J. "Addiction, Compulsion, and the Technology of Consumption." Economic Inquiry,

January 1988, 75-88. Pollak, Robert A. "Habit Formation and Dynamic Demand Functions." Journal of Political Economy,

August 1970, 745-63. Rogers, Everett M. Diffusion of Innovations, 3rd ed. New York: The Free Press, 1983.

JEFF BIDDLE, Assistant Professor of Economics, Michigan State University. I would like to thank the many state officials who so willingly provided the information used in this study, and to acknowledge the helpful comments of Charles Clotfelter, Dan Hamermesh, Tim Lane, and Dan Suits.

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Publication: | Economic Inquiry |

Date: | Apr 1, 1991 |

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