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The consumer's micro-micro gasoline buying decision.

Abstract A "micro-micro" consumer problem of gasoline purchases is examined using daily price data. Comparing the optimizing consumer with one who buys gasoline at random, the paper finds optimizers save about 4% of their annual gasoline bill. The paper also provides some evidence about the costs of non-optimal gasoline buying strategies.

Keywords Gasoline * Consumer * Value of information

JEL Classification Q40

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This paper investigates the "micro-micro" consumer problem of gasoline purchases during a given planning period--that is, how much an idealized consumer (i.e., an "optimizer" or cost minimizer) who buys gasoline from one of four stations in Peoria, Illinois between November 18, 2003 and October 25, 2005 would pay for gasoline as compared to various other ad hoc gasoline buying strategies. Buying gasoline requires that consumers track gasoline usage and plan the next refueling before running out of gas. On the other hand, a finite tank size limits stocking up to take advantage of low prices. Refueling is time consuming so consumers will not refuel any more often than necessary. Finally, gas prices are prominently posted, visibly changing from day to day. A consumer may buy gas today and then be chagrined to see tomorrow that the price has fallen.

The "idealized" consumer assumed herein is allowed information that real consumers do not have--the price of gasoline tomorrow and the day after. Thus, the paper could also be seen as a test of the value of that information under various circumstances. The paper also measures the amount, on average, an idealized consumer can save by refueling more than once per planning period. An individual who has a good sense of the value of his or her time could thus use the experiments of this paper to provide some evidence about whether taking the time and trouble to buy gas as cheaply as possible is in fact optimal for him or her. In general, it is found that the optimizer saves at most about 8% on gasoline expenditures; often savings are about half that, around 4%, and under a considerable number of circumstances the savings are in the 2-3% range.

[FIGURE 1 OMITTED]

Although the larger problem of consumer demand for gasoline is well studied (gasoline demand being inelastic) Nicol (2003), Cheung and Thomson (2004), Ramanathan and Subramanian (2003), and Espey (1998), it seems that the micro-micro problem using daily prices in this paper has not been considered in the literature.

Data and Parameters

The gasoline prices were obtained from the (lowest grade) postings outside four stations in Peoria Illinois each day that an author was able to view them for the study period of November 18, 2003 to October 25, 2005. The stations, along a 3-mile length of a single major thoroughfare, included major brands, local brands, and stations with and without repair facilities. The maximum and minimum price per planning period is shown for the 52 12-day (non-overlapping) planning periods for which full data exists. The average range per planning period is about $0.20, and the range (of ranges) is $0.090 to $0.46 (Fig. 1).

The "micro-micro" problem requires consumer data on daily fuel use, tank size, and the length of the planning period. Our parameters are given in Table 1 and explained next.

Vehicle classifications are derived from cars.com ("large cars" being midsize and full size) whose selections include domestic and foreign, low priced and high priced cars. The parameter values are averages for the 2004 model year (the middle year of the gas price data). Daily fuel usage is calculated by

daily fuel use = [(miles per year)/(miles per gallon)]/(365 days per year) = gallons/day

where miles per year is from Federal Highway Administration data and average miles per gallon is from cars.com. Average tank size is from cars.com as well. The planning periods of 12 and 10 days are the longest feasible given the preceding parameters and the assumption that consumers may refuel only once but will not refuel more than twice per planning period. (This assumption will be reconsidered in a later section.) Finally, the number of planning periods in the study is determined by the number of (non-overlapping) periods of 12 and 10 consecutive days in the daily gasoline price data that was collected.

The Consumer's Problem

It is assumed that the consumer knows the gas prices during the current planning period, but knows nothing about prices in the next planning period; thus he or she begins and ends each planning period with a half a tank of gas. A "feasible" gasoline buying strategy requires refueling before running out of gas, and having enough gasoline in the tank by the last refueling to have half a tank of gasoline at the end of the planning period. The solution of the optimization problem is to compute all feasible refueling possibilities, find the cost of each, and then pick the minimum.

For a closer look at the consumer's problem consider a 12-day planning period for an owner of a large car. Suppose the consumer refuels either once or twice. Since a large car uses 1.3 gal a day, the consumer must buy 1.3 x 12 = 15.6 gal each planning period. The first refueling must be on or before day 6 (subtract 1.3 gal per day from the amount in the tank on day 0). Similarly, the second refueling must be on day 6 or later in order for the tank to be exactly half-full on the last day of the planning period (add 1.3 gal per day to the amounted needed in the tank on the last day of the planning period). It is possible for consumer to refuel just once but only if refueling is done on day 6.

Denote the quantity of gasoline purchased on day i by [q.sub.i] with at most two [q.sub.i]'s nonzero. An example of a feasible pair of refueling amounts is [q.sub.1] = 9.8 and [q.sub.7] = 5.8 (and the other [q.sub.i] = 0). To check feasibility suppose that prices on days 1 and 7 satisfy [q.sub.1] < [q.sub.7] so that the maximum amount is purchased on day 1. This maximum amount is the amount of empty space in the tank after using 1.3 gal on day 1: 8.5-1.3=9.8. This amount is less than 15.6, the total needed. On day 7 the rest of the gasoline is purchased: 15.6-9.8=5.6. To complete the feasibility check, note that there is sufficient room in the tank to accommodate this purchase and there will be 8.5 gal in the tank at the end day 12. So it has been determined that [q.sub.1]=9.8 and [q.sub.7]=5.8 is a feasible pair of refueling amounts. (If [p.sub.1]>[p.sub.7], that the maximum amount is purchased on day 1.) For a 12-day planning period with at most two refuelings, 42 out of the possible 78 pairing are feasible.

A two-refueling solution will often use the lowest price of the planning period for one of the refuelings, and the relatively lowest price (for the interval 1-6 days or the interval 6-12 days) for the other refueling. But this strategy of just looking for the lowest price is not necessarily optimal for all price vectors. As an example, for Station 3, October 10-17, 2005 the prices were {2.949, 2.899, 2.899, 2.869, 2.869, 2.779, 2.759, 2.749, 2.679, 2.679, 2.679, 2.649}. The price vector is decreasing (actually, nonincreasing) so the minimum price occurs on day 12. Using the strategy of the first sentence of the paragraph, buy 7.1 gal on day 6 and 8.5 gal on the price minimizing 12th day gives a cost of $42.25. However the consumer is better off buying less on day 6 when prices are very high, than buying the additional amount needed to get to day 12 rather than day 9. Specifically, the cost of buying 3.2 gal on day 6 and 12.4 gal on day 9 is $42.11, slightly smaller than the first strategy and becomes the minimal cost for the planning period.

Table 2 gives the average refueling costs for the vehicle-planning period combinations summarized in Table 1. The sub-row labeled "1 refueling" in each row gives the average over all experiments of the stated kind (e.g., Large Car-Homogeneous) of the minimized cost of buying gasoline, given that the consumer refuels just once. The stations are considered "homogeneous" when the consumer chooses the station with the lowest price on any given day and without regard to brand or services available. For example, for Large Car-single refueling-Homogeneous Stations, the average refueling cost is $30.52. The other entries are analogous; the case of three refuelings will be discussed in the next section.

Notice first that there is only one slight difference between the Large Car column and the SUV/Van column. Not surprisingly, in general, the per planning period average refueling cost for Compacts is lower than for the other vehicles. (But note that to calculate annual costs a 12-day planning period implies the average cost is multiplied by 365/12 while a 10-day planning period implies the average cost is multiplied by 365/10).) Comparing stations, Station 2 is the cheapest. A person willing to hunt for the lowest price, i.e., the Homogeneous row, saves money, but never more on average than 14 cents per planning period. (The case of the Large Car and SUV/Van three refuelings is anomalous--the average cost for Station 2 is lower than for the Homogeneous row, but the reason is that all experiments in these columns have three refueling Homogeneous solutions, while two three refueling solutions failed to exist for Station 2.) Station 3 is at least on average, the most expensive station.

The next section will consider the question of the number of refuelings in more detail and the final section will consider the larger question of how much a consumer could save with perfect information on gasoline prices.

The Number of Refuelings

The paper makes no explicit assumptions about the value of the consumer's time so it is not possible to find the optimal number of refuelings per planning period. However this section provides some evidence on the relation between the opportunity cost of time and the number of refuelings and in particular on the tradeoffs between number of refuelings and monetary cost of gasoline. As seen in Table 2, the cost of buying gasoline for a given a number of refuelings is easily computed. When the monetary cost of a single refueling is the same as for two refuelings, a single refueling is clearly optimal (unless the value of time is zero) and so these costs can be compared. (There are a few anomalous looking results reported in Table 2 for which it appears that using three refuelings actually is more expensive than two refuelings. Briefly, the explanation is that if there is a case where refueling any three times raises the planning period cost of gasoline as compared to refueling twice or once, then we will say a "3 refueling" solution fails to exist. Ties, however, are allowed; if a 3 refueling costs the same (to within a penny) as a 2 refueling or a 1 refueling, then we will say the 3 refueling exists.)

See Table 3. For Large Cars and Homogeneous Stations, two refuelings were optimal for exactly half of the 52 two 12-day planning periods in the data. ("Optimal" here means that the cost of the 2 refueling solution is lower than the best 1 or 3 refueling solution. In the "Count" data, ties are assigned to the lower number of refuelings. That is, for an experiment where the minimum 1 refueling solution is the same as the minimum 2 refueling, the experiment is counted as a 1 refueling solution.) The average cost for Large Cars and Homogeneous Stations is $28.63, and the average savings over a single refueling is $0.86. This saving between refueling twice and once is denoted in the table by "Savings (2-1)."

The proportion of times two refuelings is optimal is higher for Stations 1 and 4, and lower for Compacts because the data has 63 10-day planning periods. The range of average savings comparing one and two refuelings--when two are optimal--is from $0.67 to $1.26. So consumers for whom the opportunity cost of the time it takes to refuel (and/or distaste for refueling) is less than $0.67 should certainly refuel twice (for those planning periods when refueling twice is optimal). Consumers for whom the opportunity cost of a refueling is greater than $1.26 should simply refuel once regardless since the benefit in gasoline savings of an extra refueling does not outweigh the cost.

Table 4 reports similar results for the planning periods when three refuelings is optimal. The counts are smaller, but still comprise at least 15% of the experiments. The savings for these cases between three refuelings and one refueling ranges from $0.88 to $1.95, but by observing Savings (3-1) and Savings (3-2) in the table notice that most of these benefits come from the second refueling. The amount saved comparing two refuelings with three--given that three is optimal--ranges from $0.09 to $0.22. There may be people with an opportunity cost of a refueling below $0.22 (and don't mind the smell of gasoline on their hands), but the paper will proceed on the assumption that consumers will do one or two refuelings only. Limiting the refuelings to one or two, given the other parameters of the model, implies the maximum-length planning period given in Table 1.

The Value of Limited Information on Prices

The previous section concerns only the minimal costs incurred per planning period. This section investigates how much an idealized consumer (i.e., the "optimizer") would benefit from perfect information about gas prices for a given planning period, given that the consumer knows nothing about prices in the next planning period. This is the sense in which the consumer has "limited information on prices." Each of the 52 (non-overlapping) 12-day planning periods for which price data exists is treated as a separate experiment for Large Cars and SUV/Vans; similarly, each of the 63 (non-overlapping) 10-day planning periods is a separate experiment for Compacts. The results reported are the average over all relevant experiments. In this section, it is assumed that the consumer refuels either once or twice, choosing the number of refuelings which yields the lowest planning period cost of gasoline.

Consider Large Cars first. In Table 5 "Min Cost" refers to the average optimal solution (to the problem of minimizing planning period cost of gasoline) for the various stations or choosing from all the stations. So in the first section titled "Planning Period Average Costs," the first row in Table 5 gives the minimum cost of buying gasoline in the various cases. The "Max Cost" gives a maximum cost (given one or two refuelings), calculated by selecting the prices and the feasible quantities that gives the highest total cost in the various cases.

The "Random Cost" is calculated as follows. In the homogeneous case, for each planning period, randomly select: a feasible pair of days, two stations, whether to buy more gas on the first day of the feasible pair or on the second. This information then produces a total cost for the particular planning period. This process was repeated 1000 times for each planning period as if 1000 consumers were randomly purchasing fuel from these stations during each period. The same procedure was followed for the other columns, except that the station was fixed.

The next section of the table gives the same costs on an annual basis. The "Yearly Avg Savings" give the savings of the optimal solution (i.e., "Min") gives the average annual amounts saved from the combinations shown, and the final section gives those results in percentage terms (biggest number-smallest number/biggest).

For example, a consumer who thinks of the stations as homogenous, would optimally spend $29.81*365/12=$906.71 in 1 year on gasoline, as compared to the consumer with the worst possible gas buying luck, who would pay $992.56. The optimizer would save $85.85, or 8.65% of his fuel cost in a year. This is the largest number in the saving section of the table, and can be considered as the upper bound on saving by optimizing. Comparing the optimizer to the random outcome, the equivalent savings (in the homogenous stations case) are $39.05 or 4.13% of fuel costs per year.

Comparing the homogenous case to those customers who have a preference for a given station because of brand or service, the annual cost for Station 2 is only 72 cents more than for the homogenous case. As noted previously, often Station 2 has the lowest prices. And notice that if a consumer were to simply decide to always buy from Station 2 (the cheapest on average), the optimizer saves only $26.03 per year or 2.79% compared to the random buyer. At the other extreme, the maximum cost for the homogeneous case is only $3.32 per year more than the maximum for Station 3, suggesting that Station 3 often has the highest prices. (Station 3 is a full service station with an active repair business and the shortest waiting time to buy gasoline.) A consumer who always chooses Station 3, but otherwise optimizes pays $33.85 more a year than an optimizer who treats the stations as homogeneous. Notice also from the percentage savings section of the table, which on average the cost of buying gas randomly is closer to the optimizer's cost than the maximizer's. Similar analysis is repeated for the other vehicles in Tables 6 and 7.

Finally, although the details are not reported here, an analysis was also done on the benefits of optimization as compared to the dispersion of gasoline prices (as measured by the range of gasoline prices in a given planning period). The benefits of optimizing were found to increase as the range of gas prices increased, however the typical results varied little from the preceding analysis. The most common price range was a spread of gasoline prices per planning period of about 5% above the minimum price. Assuming the stations are homogeneous and comparing the random outcome with the optimizer, the average savings would be about 4% of the gasoline bill--i.e., the same percentage savings shown in Table 5.

Briefly, consider consumers who think of all stations as being homogeneous. Comparing optimal behavior with random, for compact cars the savings are about $34.26 (or 3.93%) a year and for SUV/vans $48.05 (or 4.13%) a year. Comparing the optimizing consumer with one who inadvertently maximizes the cost of gasoline, for SUV/vans, the maximizer pays $105.78 and 8.66% more, while for compacts the maximizer pays $76.22 and 8.93% than the optimizer.

For one last comparison, the previous section argued that consumers who have an opportunity cost of the time for refueling of more than about $2 would not consider refueling more than once a planning period. But it seems likely that those with a very high opportunity cost of their time would prefer to spend as little time buying gas as possible, even ignoring the difference in prices between stations, or between days. They might even prefer the station that casual observation suggests will have the shortest refueling time (because it has the highest prices and the fewest customers)--in this data set, Station 3. To get the flavor of how much, on average, the "time saving" strategy of always refueling just once per planning period at Station 3, see Table 8.

It is seen that the average cost per planning period of such a time saving strategy is between $1.37 and $2.22 per planning period, and barely ever exceeds $5.50. This suggests that customers with opportunity costs of time to refuel in double digits should certainly just buy their gasoline so as not to run out, i.e. on day 6 for Large Cars and Vans (and day 5 for Compacts).

Conclusion

The primary conclusion of the current paper is that if one of the most vexing aspects of the consumer's gasoline buying problem--price uncertainty--disappeared and prices were completely predictable, the percentage savings for consumers would be measured in the single digits, and would often be below five percent (see Tables 5, 6, and 7). In dollar terms, the annual savings available by optimizing are in the $35-$65 range--enough to buy the family a meal or a night at the movies. A secondary finding is that even when the opportunity cost of the consumer's time is rather low, the consumer is often better off saving time rather than money in buying gasoline.

References

Cheung, K., & Thomson, E. (2004). The demand for gasoline in China: A cointegration analysis. Journal of Applied Statistics, 31(5), 533-544.

Espey, M. (1998). Gasoline demand revisited: An international meta-analysis of elasticities. Energy Economics, 20(3), 273-295.

Nicol, C. J. (2003). Elasticities of demand for gasoline in Canada and the United States. Energy Economics, 25(2), 201-214.

Ramanathan, R., & Subramanian, G. (2003). Elasticities of gasoline demand in the sultanate of Oman. Pacific and Asian Journal of Energy, 13(2), 105-113.

Published online: 15 August 2007

[c] International Atlantic Economic Society 2007

J. Highfill ([mailing address])

Department of Economics, Bradley University, Peoria, IL 61625, USA

e-mail: highfill@bradley.edu

M. McAsey

Department of Mathematics, Bradley University, Peoria, IL 61625, USA
Table 1 Parameter values

                                         Planning  Number of
   Vehicle    Daily Fuel Use  Tank Size  Period    Planning Periods

1  Compact    1.2             14         10        63
2  Large car  1.3             17         12        52
3  SUV/van    1.6             21         12        52

Table 2 Average refueling costs

                                                             Compact
                           Large Car 12-day  SUV/Van 12-day  10-day

Homogeneous  1 Refueling   30.52             30.52           23.52
  stations   2 Refuelings  29.81             29.81           22.99
             3 Refuelings  29.90             29.90           23.02
Station 1    1 Refueling   31.00             31.00           23.95
             2 Refuelings  30.24             30.24           23.30
             3 Refuelings  30.33             30.33           23.28
Station 2    1 Refueling   30.66             30.67           23.57
             2 Refuelings  29.83             29.83           23.01
             3 Refuelings  29.83             29.83           23.10
Station 3    1 Refueling   31.58             31.58           24.32
             2 Refuelings  30.92             30.92           23.81
             3 Refuelings  31.01             31.01           23.99
Station 4    1 Refueling   30.93             30.93           23.82
             2 Refuelings  30.07             30.07           23.19
             3 Refuelings  30.19             30.19           23.36

Table 3 Average savings when 2 refuelings are optimal

                                                              Compact
                            Large Car 12-day  SUV/Van 12-day  10-day

Homogeneous  Count          26                26              26
  stations   2-Refuel Cost  28.63             35.24           22.25
             Savings (2-1)   0.86              1.05            0.81
Station 1    Count          31                31              38
             2-Refuel Cost  29.34             36.12           23.18
             Savings (2-1)   0.91              1.13            0.87
Station 2    Count          27                27              27
             2-Refuel Cost  28.36             34.90           22.01
             Savings (2-1)   1.02              1.26            0.88
Station 3    Count          27                27              34
             2-Refuel Cost  30.22             37.20           23.55
             Savings (2-1)   0.84              1.04             .67
Station 4    Count          30                30              36
             2-Refuel Cost  29.52             36.32           22.05
             Savings (2-1)   0.97              1.19            0.83

Table 4 Average savings when 3 refuelings are optimal

                                                              Compact
                            Large Car 12-day  SUV/Van 12-day  10-day

Homogeneous  Count          15                15              17
  stations   3-Refuel Cost  33.51             41.24           25.82
             Savings (3-1)   1.09              1.34            0.88
             Savings (3-2)   0.10              0.12            0.09
Station 1    Count           9                 9              10
             3-Refuel Cost  33.29             40.97           25.00
             Savings (3-1)   1.47              1.81            0.91
             Savings (3-2)   0.18              0.22            0.13
Station 2    Count          13                13               0.10
             3-Refuel Cost  33.98             41.81           16
             Savings (3-1)   1.33              1.63           26.08
             Savings (3-2)   0.11              0.14            0.85
Station 3    Count          12                12              10
             3-Refuel Cost  32.67             40.21           26.83
             Savings (3-1)   1.13              1.39            1.05
             Savings (3-2)   0.17              0.20            0.13
Station 4    Count          11                11              12
             3-Refuel Cost  33.22             40.88           26.88
             Savings (3-1)   1.58              1.95            0.96
             Savings (3-2)   0.11              0.14            0.10

Table 5 Large cars: planning period average costs and savings

Planning Period     Homog
Avg Costs           Stations  Station 1  Station 2  Station 3  Station 4

  Min cost           29.81     30.23      29.83      30.92      30.06
  Max cost           32.63     32.11      31.87      32.52      32.31
  Random             31.09     31.05      30.69      31.62      31.02
Yearly avg cost
  Min cost          906.71    919.50     907.43     940.56     914.30
  Max cost          992.56    976.76     969.46     989.24     982.69
  Random            945.75    944.45     933.46     961.88     943.51
Yearly avg savings
  Min v max          85.85     57.25      62.03      48.68      68.38
  Min v random       39.05     24.95      26.03      21.32      29.20
  Max v random       46.81     32.30      36.00      27.36      39.18
Percent of yearly
  av sav
  Min v max           8.65      5.86       6.40       4.92       6.96
  Min v random        4.13      2.64       2.79       2.22       3.10
  Max v random        4.72      3.31       3.71       2.77       3.99

Table 6 SUV/Van: planning period average costs and savings

Planning Period     Homog
Avg Costs           Stations  Station 1  Station 2  Station 3  Station 4

Min cost               36.69     37.21      36.72      38.06      36.99
Max cost               40.16     39.53      39.23      40.03      39.77
Random                 38.27     38.21      37.77      38.92      38.18
Yearly avg cost
  Min cost          1,115.90  1,131.65   1,116.79   1,157.56   1,125.25
  Max cost          1,221.68  1,202.24   1,193.26   1,217.59   1,209.55
  Random            1,163.95  1,162.18   1,148.71   1,183.91   1,161.24
Yearly avg savings
  Min v max           105.78     70.59      76.47      60.03      84.30
  Min v random         48.05     30.53      31.92      26.35      35.99
  Max v random         57.73     40.06      44.55      33.68      48.31
Percent of yearly
  av sav
  Min v max            8.66       5.87       6.41       4.93       6.97
  Min v random         4.13       2.63       2.78       2.23       3.10
  Max v random         4.73       3.33       3.73       2.77       3.99

Table 7 Compacts: planning period average costs and savings

Planning Period     Homog
Avg Costs           Stations  Station 1  Station 2  Station 3  Station 4

Min cost             22.97     23.30      23.00      23.81      23.18
Max cost             25.06     24.63      24.39      24.95      24.76
Random               23.91     23.85      23.57      24.30      23.83
Yearly avg cost
  Min cost          838.58    850.42     839.41     869.07     846.12
  Max cost          914.80    899.05     890.10     910.51     903.69
  Random            872.84    870.64     860.44     887.00     869.72
Yearly avg savings
  Min v max          76.22     48.63      50.69      41.45      57.56
  Min v random       34.26     20.22      21.03      17.93      23.59
  Max v random       41.96     28.41      29.66      23.52      33.97
Percent of yearly
  av sav
  Min v max           8.33      5.41       5.70       4.55       6.37
  Min v random        3.93      2.32       2.44       2.02       2.71
  Max v random        4.59      3.16       3.33       2.58       3.76

Table 8 Station 3 one refueling vs. homogeneous stations optimum

         Large Car 12-day  SUV/Van 12-day  Compact 10-day

Average  $1.80             $2.22           $1.37
Std dev   0.77              0.95            0.58
Range     0.61-4.49         0.76-5.54       0.24-2.64
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Title Annotation:purchasing behaviour of gasoline by consumers
Comment:The consumer's micro-micro gasoline buying decision.(purchasing behaviour of gasoline by consumers)
Author:Highfill, Jannett; McAsey, Michael
Publication:International Advances in Economic Research
Geographic Code:1USA
Date:Nov 1, 2007
Words:4718
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