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The demand for vanity (plates): Elasticities, net revenue maximization, and deadweight loss.

ERIK D. CRAFT (*)

I. INTRODUCTION

Each state is a monopolist in the production of personalized license plates, otherwise known as vanity license plates. (1) State policymakers, therefore, must answer the same questions when setting the price of personalized plates as they do when determining income tax rates, excise tax rates, lottery payout rates, and other user fees. How will revenue change as the price (tax rate) is increased? At what price (tax rate) will revenue be maximized? What is the ratio of the marginal excess burden (deadweight loss) to marginal net revenue at a given price (tax rate)? Any state charging a price that exceeds the monopoly price not only causes a socially suboptimal level of consumption but forgoes revenue as well. In the absence of distributional concerns, the marginal dollar of net revenue raised should have the same deadweight or efficiency cost across all revenue sources. The socially efficient price for a personalized license plate is likely below the net revenue-maximizing price, because social surplus is usu ally maximized when products are priced at marginal cost.

This article offers answers to the preceding questions and others using a new data set gathered by the Virginia Department of Motor Vehicles on the prevalence of and fees for personalized passenger license plates in 1997. The article also will argue that special background license plates are a complement to vanity license plates and will extend previous conclusions regarding the optimal pricing of vanity license plates.

II. PAST RESEARCH

Alper et al. (1987) used 1983 data to present the first published estimates of the elasticity of demand for personalized license plates. Their key price variable was the five-year average annualized cost of owning personalized plates, which they used along with other variables to explain the percentage of license plates that were personalized. Elasticity estimates from their analysis ranged from -0.17 for Alaska to -2.64 for Ohio. If the marginal cost of vanity license plates is close to zero, then the net revenue-maximizing elasticity should be near one. Finding only 12 of 44 states with elasticities in the range between 0.75 and 1.24 in absolute value, the authors concluded that "vanity plated are substantially too cheap in some states and substantially too expensive in other" (p. 108). Harrington and Krynski (1989) extended the set of independent variables used by Alper et al. (1987) to explain the purchase of personalized plates. They also corroborate Alper et al. in showing that many states charge price s too high to maximize net revenues. In addition, Harrington and Krynski (1989) analyzed the decision to purchase personalized plates by including both the initial fee and renewal fee in their regressions. They estimated upper-bound estimates of the initial net revenue-maximizing price and concluded that many states were setting the initial fees too high.

Previous research proposed that personalized license plates were normal goods. Researchers also assumed that the demand would be higher in states in which both front and back license plates were required, because personalized plates will have greater visibility. A potential bandwagon effect suggested that the longer personalized plates had been available in a state, the greater the demand. The greater the number of potential combinations, as determined by the number of spaces and extra characters available, the higher should be the demand. Though not justified explicitly, the percentage of the population aged 25-44 years was also included in the set of independent variables. One hypothesis is that such persons are more likely to seek to create an identity or attract a mate. Craft and Schmidt (1999) find evidence that relative increases in this demographic group raise the average level of vehicular capital, holding other factors constant. (2) Harrington and Krynski (1989) added the average registration fee and a dummy variable for earmarked funds. Vehicle owners might be more likely to purchase vanity plates if the funds support a favored program and a residual claimant is created.

Biddle (1991) used an 11-state 100-point panel data set to test for the existence of a bandwagon effect. He found some evidence, but some of his results were consistent with and supported an information diffusion model. In the process of developing a test for the bandwagon effect, Biddle estimated the log of the number of personalized plates directly as a function of the logs of price, income, and number of cars; a dummy for the presence of promotional activity; the number of potential characters; the age of the vanity plate program; and state dummies. His standard model yielded a price elasticity of demand estimate of -1.30, and his bandwagon demand model using a lagged dependent variable estimated the price elasticity of demand to be -0.62. Unfortunately, his demand specification provides no guidance to states on the influence of factors they control, such as the influence of the number of alternative background plates or two plate requirements. Biddle's careful analysis also limited him to using data from states whose initial and renewal fees were identical.

III. DATA AND SPECIFICATION

The Virginia Department of Motor Vehicles surveyed all states in early 1999, requesting data on the fees and numbers of various categories of active passenger license plates as of December 31, 1997. Descriptive statistics of this and additional data are found in Table 1. Table 2 presents the correlation matrix.

The percentage of passenger plates personalized has increased from 1.78% to 3.54% (3.11% when the latter sample is weighted) in the 15 years from 1983 to 1997 (Alper et al., 1987, p. 105). The average four-year annualized price of a personalized license plate in the 1997 sample is $24.13 ($25.07). When calculated as five-year annualized price (per previous research), the amount is $23.62 ($24.44), compared to $22.29 in 1983. The unweighted average percentage of plates with special backgrounds in this 31 state sample is 5 (4.28 when weighted). Nearly two-thirds of the states require license plates in both the front and back of the vehicle. Four-fifths of the states in the sample earmark revenues raised from the sale of vanity plates for specific programs. The data showing the 1997 and 1983 annualized price and prevalence of vanity plates by state can be viewed in Table 3. (3)

Past research, although not explicitly addressing demographic factors, has shown that the higher the proportion of inhabitants aged 25-44 in a state, the greater is the demand for personalized license plates. If there is a life-cycle effect whereby young adults seek to attract attention as part of their social or mating activities, then this demographic variable can be refined further to explain the demand for vanity plates even better. The percentage of the population aged 25-34 may identify even more accurately single persons seeking to differentiate themselves from others. This age group consists of a high percentage of single persons. These individuals are also at an early stage in their careers. Demographic estimates on the distribution of the population by age categories are found in the U.S. Bureau of Census (2000) data.

Note that the percentage of special background license plates exceeds that of vanity license plates in the sample. Anecdotal evidence suggests that the number of license plate special background options has proliferated in recent years. For example, in Virginia one can show support for the Chesapeake Bay (one of 51 special-interest plates), identify oneself as an alumnus/alumna at the University of Richmond (2 choices out of 65 university or college plates), or indicate past or present military service (45 varieties). (4) If a vehicle owner purchases special background plates, the opportunity to personalize one's vehicle is further enhanced. Not only do special background plates effectively extend the range of potential statements using personalized license plates, but the marginal time cost of ordering personalized plates is lower if one is also buying special background plates. Therefore, not only may personalized license plates and special background plates be complements--as suggested by the negative corr elation between the price of vanity plates and the percentage of special background plates--but the purchase of one type of plate may lower the full cost of purchasing the other type.

The Virginia Department of Motor Vehicles' survey attempted to collect extensive data on background plate choices and prices, but different interpretations of some categories of background plates and the wide variety of prices for different backgrounds within each state make consistent comparisons across states impossible. Many states, however, were able to provide the total number of special background plates active on December 31, 1997. This data along with the total number of active passenger plates were used to calculate the percentage of special background passenger plates.

The complete demand specification for vanity license plates would include the prices of related substitutes or complements, such as special background plates (standard determinant of demand), the number of choices of special background plates (more choices increase the number of complements and/or substitutes), and the total number of special background plates (whenever the extra effort is made to purchase a special background plate by visiting the Department of Motor Vehicles, or more recently by going online, the combined marginal monetary and time cost of ordering a vanity plate is reduced). But this ideal strategy is not feasible given the absence of a single or an average price for special background plates in each state and consistent data on overall background plate options. Previous research did not include any information related to special background plates, and excluding correct regressors can lead to biased estimators, Therefore, demand specifications in this article will combine the three precedi ng theoretical demand factors into one proxy variable, the percentage of special background plates. If special background plates are complements to vanity plates, then the three combined factors affect the demand for vanity plates positively.

The size of the license plate data set collected by the Virginia Department of Transportation differs by variable. Five states (North Dakota, Ohio, New Hampshire, Indiana, and Hawaii) and the District of Columbia did not return the survey form. Six states (Connecticut, Georgia, Iowa, Pennsylvania, New Jersey, and Texas) may have provided flow data on personalized license plates but could not provide stock data for the end of calendar year 1997. Kansas and Rhode Island price vanity plates in an incomparable manner. Data on the number of special background plates or year of program inception were also missing for some states, preventing their use in the specification of demand. If necessary, states received two follow-up letters requesting the return of the survey and two telephone calls inquiring into the availability of missing or inconsistent data. The descriptive statistics are calculated for the 31 states that reported all variables used in the subsequent regression analyses. (5)

Previous vanity plate research employed both linear and logit regression specifications but emphasized the latter. There are, however, good reasons for employing a linear probability model with this new data set. There are trade-offs between estimation techniques. The primary advantages of the logit technique are (1) a reasonable functional that captures the nonlinearities of probability estimates; (2) the bounding of probability estimates between zero and one; and (3) the amelioration of heteroscedasticity, which can arise from applying ordinary least squares inappropriately to a dichotomous dependent variable. None of these advantages are important in the present estimation. (1) The dependent variable represents statewide proportions with an observed range from 0.005 to 0.098, rather than from 0 to 1. The logit and probit functional forms are essentially linear over this particular range. Thus, neither offers any real advantage over least-squares estimation. (2) None of the least-squares probability estima tes fall outside the acceptable range of 0 to 1. (3) Because the data lie within a linear portion of the logit function, the heteroskedasticity issue is moot. There is, however, another heteroskedasticity issue, that is important in this instance. The states are of widely varying population sizes. Consequently, weighted least squares is employed, with the number of passenger vehicle registrations providing the weights.

On the other hand, least-squares estimation offers advantages of its own. Coefficients are readily interpretable. It is more robust to model misspecification than are maximum-likelihood techniques such as logit. (6) Most important, purchasing a vanity license plate may increase the probability of purchasing a special background plate, or a third variable may influence the probability of purchasing both variations of the standard license plate. Logit estimation using instrumental variables would be unnecessarily complicated in this case. For these and other reasons, researchers continue to defend and apply linear probability models under appropriate circumstances. (7)

IV. REGRESSION RESULTS

The cost of owning personalized license plates is modeled in two different ways. First, both initial and renewal fees will be used concurrently in the estimation of the demand schedule for personalized license plates. Second, an undiscounted four-year annualized price variable will be constructed. Adding the initial additional fee for personalized license plates to the renewal fee multiplied three times and then dividing the combined sum by four gives the undiscounted four-year annualized price. (8) The arithmetic mean age of license plates in Virginia was 3.8 years in 1998 and has been at that level since the early 1990s (Virginia Department of Motor Vehicles, 1998, p. 6). The four-year annualized price variable also fits the data better than a five-year annualized price variable used by Alper et al. (1987).

Table 4 presents the regression analysis results of the percentage of passenger plates that are personalized first using the initial fee and renewal fee variables, followed by the four-year annualized price variable. The initial fee, renewal fee, percentage of the population aged 25-34, percentage of special background plates, two-plate dummy variable, and personal income variable are each statistically significant at the 1% level in Table 4. Each of these variables is economically significant as well. Each increase of 1% of the population aged 25-34 raises the percentage of vanity plates by about 0.8. When reporting how independent variables affect the dependent variable, the percentage of passenger license plates that are personalized, all percentage changes in this article refer to changes of the final absolute percentage, not changes relative to some initial or prior level. Each increase of 1% in the percentage of all license plates that have special backgrounds is associated with a 0.13 increase in the final percentage of vanity plates. Living in a state that requires license plates on both the front and rear of a vehicle increases the level of personalized plates by over 3%. A $1000 increase in the per capita income lowers the percentage by over 0.3 of one percentage point.

The remaining variables generally have the expected signs but are statistically insignificant. The statistical insignificance of the possible vanity plate combinations is not surprising if the available choices are so numerous that the constraint is not binding. (9) In previous research, the number of years that vanity plates had been available was sometimes statistically significant. Given the many years since the inception of states' vanity plate programs, this bandwagon/information diffusion effect may now be less consequential. In contrast to research by Harrington and Krynski (1989) there is little evidence that earmarking funds from personalized plates sales has an effect on demand.

The personal income coefficients may appear to imply that vanity plates are inferior goods. This need not be the case. The income coefficients are estimating the effect on a ratio (the number of personal vehicles with vanity plates over the number of vehicles). If both personal vehicles and vanity plates are normal goods, but the income elasticity of demand for personal vehicles is higher, then this ratio will drop as income increases.

A full model would estimate the expected length of time one owns personalized plates or probability of renewing them as a function of both the initial fee and renewal fee. The regression equations in which both the initial fee and renewal fee for personalized plates are present contain more information in the sense that an annualized price variable is a function of the two other fees. Ideally, elasticities, revenue effects, and deadweight loss estimates would be derived from the direct use of the initial fee and renewal fee variables in which no information is wasted.

As Harrington and Krynski (1989) showed, maximizing total profits from the sale of personalized plates requires estimates of the probabilities of purchase and renewal of personalized plates. (10) They estimated the upper bound of the initial price that maximizes net revenues for each state. Although their published article did not explore the issues further, in theory it would be relatively straight-forward to estimate both probabilities. If it is assumed the number of personalized license plates is in a steady state and the number of new vanity plates is known each year, the probabilities of purchasing vanity plates and renewing them can be calculated. Each probability could then be regressed on both the initial fee and renewal fee and other factors to determine the effect of each on the decision to buy and to renew. Once each probability is known as a function of both fees, one optimizes a two-variable function. The recent Virginia Department of Motor Vehicles (1999) survey attempted to distinguish between new personalized plates and renewals, but most states could not answer the more detailed survey questions or did so inconsistently. This ideal strategy, therefore, was not feasible.

Note the relative size of the coefficients for the initial fee and renewal fee variables in equations (1) and (2). Each $1 increase in the initial fee lowers the percentage of plates that are personalized by 0.04%. If the average license plate owner possesses the plates for four years and the initial fee and renewal fees were equal, a 6% discount rate would then imply that the initial fee represents 27.2% of the present discounted value of the total costs. The ratio of initial fee coefficient to renewal fee coefficient according to these assumptions is expected to be 0.374. Inspection of the estimated coefficients in equations (1) and (2) indicates higher ratios. This is evidence that buyers are unusually sensitive to the initial cost of vanity plates or owners do not expect to keep their plates four years on average. If the former alternative is true, the difference between the renewal fees (marginal cost of zero) and the initial fees should be less than half of the marginal production and administrative co st of providing personalized license plates. Recall that with a linear demand curve and constant marginal costs, each $1 change in a monopolist's marginal cost leads to a 50-cent change in the revenue-maximizing price.

Table 4 also estimates the quantitative significance of factors shifting the demand curve for vanity license plates using a four-year annualized price variable to measure movements along the demand curve. The annualized price variable, demographic variable, extent of special background plates variable, two-plate dummy variable, and income remain highly statistically and economically significant. Each $1 increase in the average annualized price of a personalized license plate decreases the percentage of passenger plates that are personalized by slightly less than 0.1 of 1%.

When the percentage of the population aged 25-34 rises by one, the percentage of vanity license plates rises by 0.8. Each percentage point increase in the number of plates with special backgrounds is associated with a 0.12-0.23 rise in the number of plates that are personalized. The requirement of front and back plates is expected to raise the number of personalized plates by over 3%. Surprisingly, each increase in per capita income of $1000 is estimated to decrease the percentage of vanity license plates by nearly 0.4%.

Regression equation (5) tests one alternative to the hypothesis that the demand curve is linear over the range of annualized prices. Neither of the two quadratic price coefficients is statistically significant.

Given the potential endogeneity of vanity license plates and special background plates (purchase of either may increase complementarity options or lower the marginal cost of the other), equation (7) applies a two-stage least squares procedure. (11) Recall that the prices for special background plates vary within each state and are unavailable. Although background plate choices differ across states, plates indicating past or present military service or support appear to be common options. In Virginia there were 42 such plate options in 2000. Therefore, equation (6) uses the percentage of the population with veteran status and the percentage of the population on active duty as instruments. A population with many individuals who are on active duty or who are veterans is likely to be associated with the purchase of more special background plates but increases in the percentage of vanity license plates should have no effect on the number of veterans or military personnel. Even though license plates with universit y and college backgrounds are ubiquitous, the percentage of the population that is college-educated was a poor instrument. The two-stage least squares results are not sensitive to the inclusion in the sample of Virginia, a state with low vanity plate prices, high vanity plate sales, and a large military population. The two-stage least squares procedure indicates a stronger effect from the presence of special background plates and a weaker influence of the other variables.

To the extent that there is inertia in vehicle owners' responses to vanity plate price changes or that present demand depends on past demand, vanity plate price and quantity combinations do not necessarily correspond to their long-run equilibrium. (12) Because some vehicle owners likely are still adjusting to changes in prices at any given time, the effect of price on demand is underestimated, and the estimated revenue-maximizing prices in Table 3 are even lower. The Breusch-Pagan test for heteroskedasticity using the independent variables as explanatory variables for the variance of the error terms cannot reject the null hypothesis of homoskedasticity in the errors for the specifications found in equations (3), (4), and (5).

V. REVENUE AND EFFICIENCY EFFECTS

Table 3 presents estimates of the price elasticity of demand, net revenue-maximizing price, and social welfare loss per marginal dollar of net revenue, calculated at the 1997 annualized price and percentage vanity plate levels using equation (6), the specification indicating the weakest relationship between price and quantity demanded. After transforming the percentage of vanity plates data point and demand slope estimate into a standard demand curve, one easily can calculate marginal effects and the net revenue-maximizing price. One calculates the welfare loss per marginal dollar of net revenue by creating a ratio of the difference between the price and social cost (numerator) to the marginal net revenue at the given price (denominator). (13) No welfare loss entry in Table 3 implies an annualized price already exceeding the net revenue-maximizing price. Elasticities and revenue effects cannot be estimated for states not providing data on personalized plate fees or total active personalized plates, although ordinary least squares-based elasticities can be calculated for states whose lack of other variables precluded their use in the regression analysis.

The Virginia Department of Motor Vehicles (1998, pp. 10-12) estimated its average cost of producing special or personalized license plates as $6.35 per pair. The Virginia Department of Motor Vehicles also cites evidence that its costs are above the average costs of other states. If there is a slightly higher administrative cost of providing personalized plates of over $1 relative to standard plates, the marginal cost of providing vanity plates may approach $8. This one-time $8 cost is spread over four years. For simplicity's sake, assume the annualized marginal cost to be $2. Because equation (1-4) and (6) are linear demand curves, each $1 change in the marginal annual cost changes the estimate of the profit-maximizing price by only 50 cents.

The data and elasticity estimates for 1983 are listed in parentheses (Alper et al., 1987) in Table 3. Recall Alper et al. use a five-year annualized price variable. Elasticity estimates using the logit procedure and the variables used in equations (4) and (6) are also presented, as well as the logit elasticity estimates from Alper et al. to offer a benchmark with past research. The new logit elasticity estimates are generally compressed in comparison with the two-stage least squares estimates.

Least-squares price elasticity of demand estimates in Table 3 range from a low of -0.09 in Virginia to a high of -4.03 in South Dakota. Given the effects of other variables, which shift the demand curves, net revenue-maximizing prices range from a low of $15.79 in Florida to a high of $66.91 in Nevada. Ten of the 37 states listed in Table 3 charge prices above the profit-maximizing level. If Massachusetts lowered the annualized price of owning vanity plates from $50 to $33, the estimates imply that its vanity plates sales would rise by nearly 56,000 in equilibrium. Revenue would increase by over $1 million per year, whereas plate production costs go up by only $112,000 per year in the steady state. A further 13 states could raise vanity plate prices and increase net revenues, but the marginal deadweight loss per marginal dollar raised exceeds 100%.

The intersection of elasticity estimates in Alper et al. (1987) and Table 3 includes 27 states. Fourteen states in this subset could have raised total revenue by lowering the annual cost in 1983. When using the five-year annualized price of Alper et al., 8 of these 14 states actually lowered the annualized price between 1983 and 1997. Perhaps economic research does affect policy. Estimates imply that between four (logit) and seven (ordinary least squares) states in the subset have had their price elasticities of demand exceeding one rise even further since 1983.

The correlation matrix in Table 2 and the regression results in Table 4 suggest that personalized license plates and special background license plates are complements in consumption or that the purchase of a special background plate lower the effective cost (including time) of a personalized plate. This observation implies that the optimal prices of both personalized plates and special background plates will be even lower when analyzed in the broader context of interrelated demand functions. The use of the price coefficient in equation (6), rather than from the other specifications, also biases downward (upward) estimates of the price elasticity of demand (net revenue-maximizing price) and welfare losses per marginal dollar of net revenue.

Note that there is not one number representing the price elasticity of demand or the optimal price for personalized license plates in all states. Policy makers should remember that there is not necessarily one elasticity estimate applicable to all states or to all groups when choosing among different revenue sources. This implies a fundamental point: the optimal tax rates and mix of revenue sources can be expected to vary across states. (14)

VI. CONCLUSIONS

The level of initial fees relative to renewal fees may have a disproportionate effect on the decision to own vanity license plates. This result implies that states seeking to maximize revenues or minimize the deadweight loss per dollar revenue should consider keeping renewal fees at a level equal to or higher than initial fees. Yet the marginal cost to the state of renewing personalized plates is essentially zero. This fact works to lower the optimal renewal fee below the level of the initial fee. Without estimates of the probability of renewal as a function of both fees, one can argue only that the difference between the initial fee and renewal fee should be less than half of the marginal production and administrative cost of the personalized license plates.

There is evidence that at least ten states charge such a high price for personalized license plates that net revenues would rise by lowering the fees. Optimal pricing, however, should be analyzed in the context of alternative revenue sources, not just revenue maximization. An additional 13 or more states raise marginal funds at a deadweight cost of more than $1 per dollar revenue.

In addition, there is evidence that special background plates and personalized license plates are complements and/or the purchase of special background plates lowers the marginal cost (including time and effort) of personalizing one's plates. States will raise even more revenues from vanity plates by selling more special background plates and requiring front and back plates for vehicles. Contrary to past research, there is no longer evidence that age of the program or the earmarking of revenues from the sale of personalized plates has any effect on sales.

Further research requires better data on the distribution of all personalized license plates between the categories of new plates and renewed plates. This will permit joint estimation of the optimal initial fee and renewal fee. Future results also may be improved with better data on the precise distribution of vanity plate revenues. Although the extreme variety of special background plates and their prices, even within states, makes comparisons across states difficult, integrating the analysis of the demand for personalized plates and for special background plates is necessary to estimate the optimal pricing structure of both license plate options.

Finally, the profit-maximizing prices and marginal profit estimates vary considerably across states. These results are a strong reminder that optimal fiscal policy does not imply that each state, county, or city will choose the same mix of revenue sources when seeking to raise funds with the lowest possible deadweight loss. The literature on the social cost of public funds raised from alternative sources at the state and local level remains thin or nonexistent. These levels of government do not raise inconsiderable sums of revenue. (15) Further research on the social cost of revenue raised from property taxes (both real estate and vehicle), lotteries, sales taxes, and income taxes at the state and local level is necessary if state and local governments are to finance programs efficiently.
TABLE 1

Descriptive Statistics (1997)

 Mean Standard
Variable (Weighted) Deviation

Vanity plates as percentage of 3.54 2.58
 all passenger plates (Dec. 31, 1997) (3.11)
Four-year annualized price 24.13 11.31
 (25.07)
Initial fee 31.72 19.86
 (34.55)
Renewal fee 21.60 12.37
 (21.92)
Percentage of state population age 25-34 14.09 1.35
 (14.70)
Special background plates as 5.02 6.04
 percentage of all passenger plates (4.28)
Two-plate dummy variable 0.65 0.49
 (0.63)
Earmark dummy variable 0.81 0.40
 (0.80)
Personal income per capita 23.547 3.373
 (in thousands) (25.016)
Age of vanity plate program 25.0 6.6
 (26.6)
Possible vanity plate 2,359,524 9,763,127
 combinations (in millions) (3,898,164)
Percentage of population 0.51 0.50
 that is on active military duty (0.54)
Percentage of population 9.82 1.11
 that has veteran status (9.61)


Variable Minimum Maximum

Vanity plates as percentage of 0.51 9.75
 all passenger plates (Dec. 31, 1997)
Four-year annualized price 7.50 50.75

Initial fee 10 100

Renewal fee 0 50

Percentage of state population age 25-34 11.15 16.57

Special background plates as 0.16 28.90
 percentage of all passenger plates
Two-plate dummy variable 0 1

Earmark dummy variable 0 1

Personal income per capita 18.272 31.239
 (in thousands)
Age of vanity plate program 15 43

Possible vanity plate 1.68 54,507,959
 combinations (in millions)
Percentage of population 0 2.40
 that is on active military duty
Percentage of population 6.49 12.13
 that has veteran status

Source: Virginia Department of Motor Vehicles (1998); U.S. Bureau of the
Census (1999-2000); U.S. Department of Commerce (1999); Archibald
(1999). Descriptive statistics are provided for the 31 states included
in the subsequent analyses.
TABLE 2

Correlation Matrix

 Percent Percent
 Percent Annualized Aged Special Two-Plate
 Vanity Price 25-34 Plates Dummy

Percent vanity 1.00
Annualized price -0.34 1.00
Percent aged 25-34 0.01 0.31 1.00
Percent special plates 0.53 -0.17 0.02 1.00
Two-plate dummy 0.47 0.06 -0.02 0.16 1.00
Per capita personal 0.02 -0.03 0.65 -0.04 0.42
 income
Age of program 0.11 -0.21 0.30 -0.11 0.16
Percent active duty 0.16 -0.21 0.06 0.38 -0.01
Percent veteran 0.26 -0.35 -0.13 0.25 0.01

 Per Capita Percent
 Personal Age of Active Percent
 Income Program Duty Veteran

Percent vanity
Annualized price
Percent aged 25-34
Percent special plates
Two-plate dummy
Per capita personal 1.00
 income
Age of program 0.49 1.00
Percent active duty 0.04 0.16 1.00
Percent veteran 0.04 0.16 -0.04 1.00

Note: See Table 1 for data sources.
TABLE 3

Elasticity, Net Revenue- Maximizing Price, and Welfare Loss Estimates
(Equation [7])

 Four-Year Elasticity:
 Percent Annualized Price Logit
 Vanity Price Elasticity Estimates

Alabama 1.41 (0.17) 50.00 (50.00) -2.94 -1.09 (-2.20)
Alaska 2.72 (0.84) 7.50 (4.00) -0.23 (-0.17)
Arizona 3.74 25.00 -0.56 -0.36
Arkansas 1.23 (0.50) 25.00 (25.00) -1.69 -1.18 (-1.09)
California 4.47 (3.40) 28.75 (23.00) -0.53 -0.74 (-0.98)
Colorado 1.81 (0.46) 27.50 (27.00) -1.26 -0.88 (-1.18)
Delaware 2.54 (2.03) 20.00 (25.00) -0.65 -0.33 (-0.31)
Florida 1.46 (0.70) 12.00 (12.00) -0.68 -0.31 (-0.52)
Idaho 3.50 (0.76) 17.50 (25.00) -0.42 -0.66 (-1.09)
Illinois 2.69 (1.30) 26.25 (23.00) -0.81 -0.64 (-1.00)
Kentucky 0.99 (0.20) 25.00 (38.50) -2.10 -1.32 (-1.69)
Louisiana 0.55 (0.27) 25.00 (32.40) -3.78 -2.24 (-1.42)
Maine 8.00 (0.44) 15.00 (15.00) -0.16 -0.38 (-0.66)
Maryland 2.13 (1.10) 25.00 (25.00) -0.97 (-1.09)
Massachusetts 1.18 (1.70) 50.00 (29.60) -3.52 -1.82 (-1.28)
Michigan 1.56 20.00 -1.06 --
Minnesota 1.05 (0.57) 25.00 (20.00) -1.97 -1.77 (-0.88)
Mississippi 2.63 (0.16) 30.00 (55.00) -0.95 (-2.42)
Missouri 4.08 (2.30) 15.00 (12.00) -0.31 -0.41 (-0.52)
Montana 8.69 (3.70) 13.75 (8.00) -0.13 -0.19 (-0.34)
Nebraska 4.08 30.00 -0.61 -0.52
Nevada 8.95 24.00 -0.22 -0.27
New Mexico 2.09 (1.10) 15.00 (15.00) -0.60 -0.47 (-0.65)
New York 2.88 (3.10) 28.06 (16.05) -0.81 -0.76 (-0.68)
North Carolina 2.81 (2.10) 20.00 (10.00) -0.59 -0.45 (-0.43)
Oklahoma 2.46 (0.25) 13.25 (10.35) -0.45 -0.35 (-0.45)
Oregon 2.23 (0.09) 50.75 (50.00) -1.89 -1.14 (-2.20)
South Carolina 2.36 (0.73) 27.00 (15.00) -0.95 (-0.66)
South Dakota 0.51 (0.11) 25.00 (38.00) -4.03 -4.20 (-1.67)
Tennessee 1.18 (0.08) 25.00 (50.00) -1.76 (-2.20)
Utah 4.31 (0.83) 39.63 (10.00) -0.76 -0.92 (-0.44)
Vermont 7.58 (4.20) 20.00 (15.00) -0.22 -0.46 (-0.63)
Virginia 9.75 (7.40) 10.00 (10.00) -0.09 -0.26 (-0.41)
Washington 2.60 (0.80) 34.00 (22.00) -1.08 -0.83 (-0.96)
West Virginia 2.59 (0.50) 15.00 (25.00) -0.48 -0.32 (-1.09)
Wisconsin 5.40 (3.20) 15.00 (10.00) -0.23 -0.36 (-0.43)
Wyoming 4.19 (0.40) 7.50 (30.00) -0.15 -0.18 (-1.32)

 Net Revenue Welfare Loss
 Maximizing per Marginal
 Price $ Net Revenue

Alabama 34.51 --
Alaska 21.16 0.20
Arizona 36.00 1.05
Arkansas 20.88 --
California 42.31 0.99
Colorado 25.65 --
Delaware 26.31 1.42
Florida 15.79 1.32
Idaho 30.83 0.58
Illinois 30.33 2.97
Kentucky 19.46 --
Louisiana 16.81 --
Maine 56.71 0.16
Maryland 26.37 8.39
Massachusetts 33.10 --
Michigan 20.40 22.5
Minnesota 19.85 --
Mississippi 31.85 7.55
Missouri 33.06 0.36
Montana 60.23 0.13
Nebraska 40.55 1.33
Nevada 66.91 0.26
New Mexico 21.09 1.07
New York 32.37 3.02
North Carolina 27.95 1.13
Oklahoma 22.45 0.61
Oregon 39.79 --
South Carolina 28.71 7.29
South Dakota 16.60 --
Tennessee 20.60 --
Utah 46.80 2.62
Vermont 56.65 0.25
Virginia 64.73 0.07
Washington 33.68 --
West Virginia 24.11 0.71
Wisconsin 41.06 0.25
Wyoming 30.00 0.12

Notes: The data and Logit elasticity estimates for 1983 are in
parentheses (Alper et al., 1987 Recall Alper et al. used a five-year
annaulized price variable. The last two columns are derived using the
linear probability model estimates in Equation VI. Welfare
loss per marginal dollar of net revenue is the social welfare
loss in dollars caused by the marginal net dollar of revenue when
raising price and lowering output at the present annualized price. No
welfare loss number implies an annualized price already exceeding the
net revenue-maximizing price.
TABLE 4

Regression Results

 (1) (2) (3)

Intercept -1.71 -1.38 -1.81
 (0.57) (0.53) (0.60)
Initial fee -0.038 (**) -0.042 (***) --
 (2.60) (3.27)
Renewal fee -0.054 (**) -0.054 (**) --
 (2.48) (2.76)
Annualized price -- -- -0.091 (***)
 (3.71)
[(Annualized price).sup.2] -- -- --

Percent aged 25-34 0.79 (**) 0.85 (***) 0.82 (**)
 (2.62) (3.24) (2.71)
Percent special plates 0.13 (***) 0.13 (***) 0.15 (***)
 (3.05) (3.15) (3.38)
Two-plate dummy 3.29 (***) 3.25 (***) 3.10 (***)
 (4.69) (5.41) (4.51)
Personal income -0.34 (**) -0.31 (**) -0.38 (**)
 (2.53) (2.76) (2.91)
Age of program 0.046 -- 0.065
 (0.91) (1.36)
Earmark dummy 0.42 -- 0.40
 (0.59) (0.55)
Possible combinations 1.35E-9 -- 9.74E-10
 (0.06) (0.04)
F-value 7.70 13.25 8.35
Adjusted [R.sup.2] 0.67 0.70 0.66

 (4) (5) (6) (*)

Intercept -1.51 -0.76 -1.68
 (0.55) (0.27) (0.57)
Initial fee -- -- --

Renewal fee -- -- --

Annualized price -0.093 (***) -0.20 -0.083 (***)
 (4.06) (1.63) (3.11)
[(Annualized price).sup.2] -- 1.71E-3 --
 (0.89)
Percent aged 25-34 0.87 (***) 0.96 (***) 0.75 (**)
 (3.20) (3.30) (2.37)
Percent special plates 0.14 (***) 0.12 (**) 0.23 (**)
 (3.48) (2.37) (2.35)
Two-plate dummy 3.16 (***) 3.34 (***) 2.80 (***)
 (4.93) (4.95) (3.57)
Personal income -0.39 (***) -0.41 (***) -0.35 (**)
 (3.21) (3.31) (2.37)
Age of program 0.054 0.053 0.059
 (1.32) (1.29) (1.31)
Earmark dummy -- -- --

Possible combinations -- -- --

F-value 11.92 10.24 9.33
Adjusted [R.sup.2] 0.69 0.68 0.62

Notes: The dependent variable is the percentage of all passenger license
plates that are personalized. Absolute values of t-Statistics are in
parentheses. Data sources are described in Table 1. There are 31 data
points.

(a)Equation (6) is two-stage least squares.

(*)Significant at the 10% level of confidence.

(**)Significant at the 5% level of confidence.

(***)Significant at the 1% level of confidence.


(*.) This is a revision of a paper presented at the Western Economic Association International 74th annual conference in San Diego on July 9, 1999. I wish to thank James Kimball, Marcy Kaplan, and Sylvia Williams at the Virginia Department of Motor Vehicles for their assistance with the license plate data and James Kimball, Kevin Beam, Robert Nicholson, Robert Schmidt, Bill Ross, Robert Archibald, and anonymous referees for their comments and suggestions in this research. Summer research support from the University of Richmond is gratefully acknowledged.

(1.) A vanity or personalized license plate is one whose combination of letter, numbers, and other characters is determined by the owner of the vehicle. Special background license plates are distinguished from normal plates in that they allow the motorist to choose the license plate background on which the vehicle license identification is written. Persons can own plates for a vehicle that are both personalized and possess special backgrounds at the same time.

(2.) A colleague of the author has the name of a sensuous Swedish folk dance (SVEV) on the license plate of his Saab. An acquaintance has plates that read ICY COOL; her original choice was 2SPYC4U, but she feared certain interpretations.

(3.) After the data were collected, the Virginia Department of Motor Vehicles encouraged its personnel to ask owners of newly registered vehicles if they desired personalized license plates. Subsequently, the percentage of all personal vehicles with vanity plates rose even further, from 9.8% in 1997 to nearly 13% in 2000. This is one example of an excluded factor that affects both the demand for vanity and special background plates, promotion efforts by departments of Motor Vehicles. I thank a referee for this insight. Although Connecticut could not identify the number of personalized plates in 1997, today one-quarter of its plates allegedly are personalized on account of its one-time-only fee. Then the fee was reported to be $7; today it is $81 (Gross, 2001, personal communication with Gross, Virginia Department of Motor Vehicles unpublished survey).

(4.) Virginia may have been the first state (August 1997) in which one could use the Internet to determine if one's desired plate is available. One can even view it on the special background plate of one's choice.

(5.) The average registration fee for a vehicle was used as an independent variable by Harrington and Krynski (1989) to estimate the demand for vanity license plates. The registration fee for a vehicle, however, affects the demand for owning and registering a vehicle, not the demand for personalizing its license plates.

(6.) Application of the logit procedure leads to over 100 million events, making essentially any hypothetical independent variable statistical significant.

(7.) For recent applications see Hu (1998) on welfare program participation; Bassett et al. (1998) on 401(k) participation decisions; Gruber et al. (1999) on welfare program participation, probability of death during a year, and low birth weights; Barrows (1999) on the probability of working after giving birth; Yelowitz (2000) on participation in SSI; and Goldin and Rouse (2000) on advancing to subsequent audition stages. Greene (1997, p. 874) cites numerous other authors using or defending the use of the linear probability model.

(8.) Creating a discounted annualized four-year cost variable has little effect over such a short time period, because most of the effect would show up as a change in the factor of proportionality. Such modifications have a first-degree effect similar to increasing the assumption of average vanity license plate ownership length by a few months.

(9.) An alternative specification used the number of spaces available for a personalized license plate. This variable was statistically insignificant as well.

(10.) In a two-period world, they show that states would maximize the function ([P.sub.N] + ([P.sub.R]/[1 + r])[[pi].sub.R] - MC)[[pi].sub.N] where [P.sub.N] is the initial fee for a new personalized plate, [P.sub.R] is the renewal fee, r is the interest rate, MC is the marginal cost of new personalized plates, [[pi].sub.N] is the probability of purchasing new personalized plates, and [[pi].sub.R] is the probability of renewing such plates.

(11.) If the percentage of special background plates is dropped from equation 4, the price coefficient rise to 0.10 and remains statistically significant with a t-value of 4. This is consistent with upward bias due to the absence of a negatively correlated variable of opposite effect.

(12.) There were 31 states whose data were used to estimate equations 1-6. Of those states, the annualized costs of 28 states can be compared with the 1983 annualized costs reported in Alper et al. (1987). In nominal terms, 8 states did not change their prices, 12 states raised their prices an average of $8.84, and 8 states lowered their prices an average of $10.83.

(13.) Marginal net revenue is simply the derivative of the total net revenue function with respect to output (how a state government's profit changes as price is lowered to sell one more unit). For any price and quantity combination, the deadweight loss of the marginal unit that could have been sold is the difference between its price and the social cost.

(14.) This is a generalization of the basic observation that raising the sales tax rate in a small state, ceteris paribus, will have greater revenue and deadweight loss effects that in a large state, because a larger proportion of inhabitants can shop more easily in another state.

(15.) Over one-third of all government receipts are collected at the state and local level (Economic Report of the President, 2000, p. 401).

REFERENCES

Alper, N. O., R. B. Archibald, and E. Jense. "At What Price Vanity?: An Econometric Model for Personalized License Plates." National Tax Journal, 40(1), 1987, 103-09.

Archibald, R. B. "Memo on inception dates of state's personalized license plate programs." Memo, 1999.

Barrows, L. "An Analysis of Women's Return-to-Work Decisions." Economics Inquiry, 37(3), 1999, 432-51.

Bassett, W. F., M. J. Fleming, and A. P. Rodrigues. "How Workers Use 401(k) Plans: The Participations, Contribution, and Withdrawal Decisions." National Tax Journal, 51(2), 1998, 263-89.

Biddle, J. "A Bandwagon Effect in Personalized License Plates." Economic Inquiry, 29(2), 1991, 375-88.

Craft, E., and R. Schmidt. "A Preliminary Analysis of the Effects of Vehicle Taxes on Vehicle Demand." University of Richmond Working Paper, 1999.

Economic Report of the President. Washington, DC: U.S. Government Printing Office, 2000.

Goldin, C. and C. Rouse. "Orchestrating Impartiality: The Impact of 'Blind' Auditions on Female Musicians." American Economic Review, 90(4), 2000, 715-41.

Greene, W. H. Econometric Analysis. Upper Saddle River, NJ: Simon and Schuster, 1997.

Gross, E. "VANI-T PL8." Fredericksburg Free Lance-Star, July 22, 2001, B-1.

Gruber, J., P. Levine, and D. Staiger. "Abortion Legalization and Child Living Circumstances: Who Is the Marginal Child?" Quarterly Journal of Economics, 64(1), 1999, 263-91.

Harrington, D. E., and K. Krynski. "State Pricing of Vanity License Plates." National Tax Journal, 42(1), 1989, 95-99.

Hu, W.-J. "Elderly Immigrants on Welfare." Journal of Human Resources, 33(1), 1998, 711-41.

U.S. Bureau of the Census. "Population Estimates for the U.S., Regions, Divisions, and States by 5-year Age Groups and Sex: Times Series Estimates, July 1, 1990 to July 1, 1999 and April 1, 1990 Census Population Counts." 2000.

-----. Statistical Abstract of the United States, 1999.

U.S. Department of Commerce. Survey of Current Business, May 1998.

Virginia Department of Motor Vehicles. "Study of the Costs and Benefits of a Regular License Plate Replacement Program." 1998.

Virginia Department of Motor Vehicles. Unpublished survey on license plate registrations, license plate prices, and license plate options, 1999.

Yelowitz, A. S., "Using the Medicare Buy-in Program to Estimate the Effect of Medicaid on SSI Participation." Economic Inquiry, 38(3), 2000, 419-41.

Craft: Associate Professor, Department of Economics, Robins School of Business, University of Richmond, Richmond, VA 23173. Phone 1-804-287-6573, Fax 1-804-289-8878, E-mail ecraft@richmond.edu
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