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13. Shech, J. S. and P. L. Wright (eds.) (1974), Marketing Analysis for 
 Societal Problems , Bureau of Economic and Business Research, Univ- 
 ersity of Illinois. 
 
 14. Zaltman, Gerald (1974), "Strategies for Diffusing Innovations," in 
 Sheth and Wright (eds.) Marketing Analysis for Societal Problems , 
 Bureau of Economic and Business Research, University of Illinois. 
 
Faculty Working Papers 
 
 College of Commerce and Business Administration 
 
 University of Illinois at U r b a n a - C h a m p a I g n 
 
FACULTY WORKING PAPERS 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 October 24, 1979 
 
 MODAL CHOICE AND WORK TRIPS 
 
 Peter F. Colwell, Associate Professor, 
 
 Department of Finance 
 Bruce S. Vanderporten, Loyola University 
 
 #623 
 
 Summary: 
 
 Theoretically, the choice of mode for the work trip is determined by the 
 abstract characteristics of the modes, the technology with which modes are com- 
 bined, labor market opportunities, and preferences. Institutional constraints 
 on labor markets have important effects. For example, the commuter who must 
 conform to a standard work day or who receives higher way rates from working 
 full-time than part-time may optimally choose to mix modes over time. However, 
 the attractiveness of modal mixing is diminished when economies of scale exist 
 in taking certain modes. With flexible hours at a part-time wage rate and 
 inflexible hours at a higher full-time wage rate, upper and lower bounds can 
 be established for valuing a transport improvement. Finally, the theory of 
 modal choice suggests structural characteristics for empirical models. 
 
MODAL CHOICE AND WORK TRIPS 
 Bruce S. Vanderporten and Peter F. Colwell* 
 
 Introduction 
 
 As new modes of transportation are introduced and existing modes 
 are modified or discontinued, the total number of trips taken as well 
 as the distribution of trips among modes changes. Further effects on 
 travel choice arise from changes in individual preferences and labor 
 market opportunities. These changes in travel decisions have interesting 
 welfare implications. 
 
 Three landmark articles in urban transportation economics have 
 examined these issues or closely related issues from different per- 
 spectives. These are Moses and Williamson [4], Quandt and Baumol [5], 
 and Gronau [2], The purpose of this paper is to provide a common frame- 
 work for examining these seemingly disparate transport studies. The 
 often implicit assumptions of these studies will be identified in rela- 
 tion to how they influence modal choice and how relaxation of these 
 assumptions may lead to other choices. The paper will also focus on the 
 subsidy issue in urban transport and attempt to establish more narrow 
 limits in valuing transport improvements. Finally, the paper will 
 attempt to explain how the theory of modal choice can be useful in 
 specifying empirical transport demand models. 
 
 It is advantageous to define modes by their abstract characteristics, 
 That is, rather than identifying actual modes, transit forms are defined 
 
 *The authors are Assistant Professor of Economics at Loyola University 
 of Chicago and Associate Professor of Finance at the University of 
 Illinois - Urbana, respectively. 
 
-2- 
 
 by characteristics such as cost, travel time, comfort, frequency of 
 service, etc. Thus, theoretical and empirical predictions of demand 
 for nonexistent, but conceivable modes are possible. Cost and travel 
 time are the characteristics given primary emphasis in all three 
 articles. This study will follow the emphasis of these articles and 
 focus on these two characteristics. 
 
 The Gronau Model 
 
 The simplest and least general of the articles is by Gronau. He 
 considered alternative modes of travel by their cost (C) and time (T) 
 
 characteristics. With two modes, A and B, where C. > CL, both modes 
 
 A b 
 
 will be relevant if T. < T„. If T A > T.,, mode A is an irrelevant 
 
 A B A B 
 
 mode, since with its higher cost and time characteristics, it would never 
 be rationally preferred to mode B. 
 
 Of the two relevant modes shown in Figure 1, mode A is the faster 
 and costlier while mode B is the slower and cheaper. The marginal cost 
 of time (MC) is the additional cost the traveler incurs per unit of time 
 when taking mode A rather than mode B. 
 
 MC AB = (C A- C B )/(T B-V- 
 
 In Figure 1, MC „ is the negative slope of line sement AB. In Gronau 's 
 analysis, a traveler's choice of mode depends on the relation between 
 his unit value of time (V) and the marginal cost of time. This may be 
 represented by considering two iso expenditure lines, E.. and E„, where 
 the negative slope of each equals the traveler's unit value of time. 
 Since lower isoexpenditure lines indicate smaller expenditures, a 
 
Trave' 
 
 Cost 
 
 e j 
 
 \ 2 
 
 
 
 C A 
 
 
 
 sL 
 
 
 
 Cn 
 
 
 
 
 ^_ -^>B 
 
 
 B 
 
 
 
 
 
 
 
 
 
 \ ' X^ 
 
 \ 1 
 
 N. r I 
 
 2 
 
 
 
 
 
 1 
 
 1 
 1 
 I 
 
 1 
 
 ^ E i 
 
 T A T B 
 
 Travel Time 
 
 Figure 1 
 
-3- 
 
 traveler prefers that mode which places him on the isoexpenditure line 
 closest to the origin. With isoexpenditure lines E- and E2, the mode 
 of choice would be the faster and costlier mode, A. Here the traveler's unit 
 value of time exceeds the marginal cost of time in taking mode A (V > MC » B ) . 
 For the traveler who places less value on his time such that V < MC. Rf , 
 mode B would be the preferred choice. This case corresponds to isoex- 
 penditure lines El and El, in Figure 1. 
 
 While Gronau's analysis provides some insight into modal choice, 
 its restrictive assumptions seriously limit the range of that choice. 
 In particular, Gronau assumes a constant unit value of time which is 
 presumably related to preferences and opportunities but is not made 
 explicit. The difficulty with this assumption is that it rules out 
 the existence of certain modes and combinations of modes which would 
 not be ruled out in the framework of conventional consumer choice theory. 
 A further unfortunate result of the Gronau approach is that a point 
 other than a modal point may never be though of as uniquely optimal. 
 
 One way in which the Gronau model restricts travel choice can be 
 seen by considering a new mode D such that C^ > C D > C„ and T^ < T_< T„. 
 Since D is not dominated by A or B in all of its characteristics, it is 
 not an irrelevant mode. Yet D can never be chosen if MC. n < MC™. 
 Figure 2 illustrates this situation. It is impossible to construct an 
 isoexpenditure line through D such that a lower isoexpenditure line 
 could not be attained by selecting either mode A or B. This implies 
 that for a new mode such as F to exist alongside A and B, mode F would 
 
 need to possess characteristics which would cause AFB to be convex to 
 
 2 
 the origin. 
 
Figure 2 
 
-4- 
 
 The usefulness of the line AFB to the Gronau analysis is that it 
 suggests that any modes to the northeast of it would never be chosen. 
 While Gronau refers to lines connecting relevant modes such as AFB as 
 isoquants, the terminology is somewhat unusual, since such lines do not 
 represent the actual opportunities available to a traveler. Rather, 
 the line AFB just connects opportunity points, A, F, and B in Figure 2. 
 For this line to represent an isoquant would require some knowledge 
 about how the modes, A, F, and B may be combined. 
 
 A Modification 
 
 The first modification of the Gronau model will be to allow for 
 mixing of modes. That is, rather than restricting the choices confronting 
 a traveler to discrete modes A or B in Figure 1, a traveler is able to 
 choose linear combinations of the two modes. For example, if the modal 
 choice is between an automobile with monetary outlay of $2.00 and travel 
 time of 30 minutes and a bus with a fare of $.50 and travel time of 60 
 minutes, the traveler might select two auto trips for each bus trip to 
 give an average trip cost of $1.50 and travel time of 40 minutes. 
 
 The actual use of linear combinations assumes certain conditions 
 
 3 
 are present. In particular, for straight line AB in Figure 1 to 
 
 represent the opportunities open to a traveler requires that constant 
 returns to scale be available in both modes A and B and that no indivi- 
 sibilities be present. When just one trip is involved, the assumption 
 is hardly realistic. That is, if the journey from Baltimore to Houston 
 is arranged so that half the trip is made by airplane and half by bus, 
 it is not likely that the cost-time expenditure would fall midway between 
 
-5- 
 
 the cost-time expenditures from exclusive use of either mode. But where 
 there is a round trip in which one leg is by airplane and the other by 
 bus, the combined cost- time expenditure would fall close to equally be- 
 tween the cost-time expenditures derived from using either mode alone. 
 The presence of indivisibilities rules out the existence of all but one 
 opportunity between A and B for one round trip. As the number of trips 
 increases, the opportunity expands for linear combinations along AB. 
 If both the automobile and bus are relevant modes for the work trip, 
 four combinations involving some use of each mode are possible over a 
 work week, assuming the same mode is used for both trips in one day. 
 Even more combinations exist if the time duration is longer or where it 
 is practical to use different modes for each direction of a round trip 
 such as bus one way and taxicab the other. 
 
 The potential to combine modes extends to situations where the 
 automobile is the only practical form of transporation. Since modes 
 are defined by their abstract characteristics, an auto trip along a 
 toll road that provides a fast journey is a different mode from the 
 same auto trip along free but slower surface streets. Motorists desir- 
 ing to mix modes might select the toll road for some trips and surface 
 streets for others. Greater modal mixing would likely occur if user 
 charges were to be imposed along heavily traveled highways to promote 
 economy in use [7], Faced with the choice of a faster-costlier highway 
 trip or a slower-cheaper trip along surface streets many motorists would 
 choose to combine modes. 
 
 The opportunity to mix modes in varying proportions is thus most 
 applicable to commuting where many trips are taken over an extended 
 
-6- 
 
 period of time. Here the line AB becomes an isoquant in the traditional 
 sense of representing a schedule of opportunities available to the pro- 
 ducer of a trip. For isoquant AB to be a straight line requires that 
 there be no fixed costs or associated cost or time advantage obtained 
 from modal specialization that would yield economies of scale. Factors 
 which lead to economies of scale include parking stickers, quantity dis- 
 counts for commutation tickets, and experience in commuting (e.g., 
 knowledge of schedules, road conditions, etc.). Where economies of scale 
 prevail, the isoquant would be concave to the origin or scalloped as 
 continuous line AB in Figure 3. This isoquant position is less preferred 
 to the one arising from the linear combination of the modes (the dashed 
 line AB). 
 
 In this context, it is interesting to reconsider mode D where 
 MC AD < MC D g. In the Gronau analysis, this mode could not be selected 
 because of the assumption of a constant value of time. Yet one need 
 not rule out mode D if linear combination of modes A and B are impossible. 
 If, however, it is found that a traveler selects mode D over either A 
 or B or a combination of the two, this would clearly indicate that there 
 must be economies of scale in travel. 
 
 While this section analyzes the production of modal mixing, the 
 next section provides a rationale for modal mixing. It does this by 
 integrating the Gronau model with the Moses and Williamson model. Modal 
 mixing will be shown to be capable of providing the optimal choice even 
 where economies of scale exist. 
 
Figure 3 
 
-7- 
 
 The Moses and Williamson Model 
 
 The Moses and Williams (M&W) study preceded Gronau's work by seven 
 years. Nevertheless, the M&W article is far more general, providing 
 sounder insights into the modal choice process, and suggesting the means 
 for analyzing the benefits from transport improvements. 
 
 Figure 4 can be viewed as a synthesis of the Gronau and M&W approaches. 
 The lower section of the Figure is an inverted Gronau diagram where the 
 vertical axis extending downward measures commuting cost and the hori- 
 zontal axis extending to the left measures commuting time. Points A and 
 B represent the cost and time characteristics of the available modes. 
 Isoquant AB represents achievable linear combinations of the two modes. 
 An individual selecting the faster- costlier mode, A, would spend KL time 
 commuting and ML in money cost. 
 
 The upper section of Figure 4 is developed in M&W space. Here the 
 vertical axis represents income (net of commuting costs). The horizontal 
 axis measures leisure time (time not spent commuting or working). The 
 combinations of income and leisure available to an individual are affected 
 by labor market constraints. M&W consider two possibilities. Either 
 the individual has the freedom to work flexible hours at a constant 
 wage rate or must conform to a standard work day. For the individual 
 who works flexible hours at a given wage rate, income- leisure opportun- 
 ities associated with taking mode A are represented by line AE. The 
 
 4 
 negative slope of AE is the constant wage rate. The situation changes 
 
 when a fixed work day constraint, JK, is imposed. Then only opportunity 
 
 point A on AE is attainable, and the individual spends KL time traveling, 
 
 JK working, OJ in leisure, and has income net of commuting costs of OH. 
 
Net Income 
 
 M&W Space 
 
 Travel Cost 
 
 Figure 4 
 
-8- 
 
 Similarly, BG, parallel to AD, is the opportunity line associated with 
 
 mode B in the flexible hours case and B' is the opportunity point for 
 
 mode B given the same restriction of work time, NQ = JK. Finally, line 
 
 segment A'B' in M&W space corresponds to isoquant AB in Gronau space, 
 
 both with the same slope. 
 
 Individuals are assumed to derive satisfaction from both income 
 
 and leisure. Their objective is to maximize their satisfaction from 
 
 these quantities subject to an income- leisure opportunity constraint. 
 
 On this basis they select their preferred mode. Their choice of mode 
 
 may differ depending on whether they work flexible hours or are required 
 
 to work a standard number of hours. With flexible hours, only the wage 
 
 rate and MC. D are necessary for establishing modal choices. As Figure 
 
 4 is constructed, w > MC AT , causes AE to be to the northeast of BG. 
 
 AB 
 
 Since income-leisure combinations along AE can always be found that are 
 superior to those along BG, mode A will clearly be the preferred mode. 
 Thus for the individual depicted in Figure 4 with indifference curves 
 I., and I 9 , point R along opportunity line AE would yield the optimum 
 mix of income and leisure. If w < MC,,,, BG would lie northeast of AE 
 
 AU 
 
 and mode B would be preferred. The flexible hours case resembles the 
 Gronau analysis in that nothing need be known about consumer preferences 
 to identify the preferred mode. A further consequence of flexible 
 hours is that linear combinations of modes are generally not relevant. 
 This is because opportunity lines AE and BG would coincide only in the 
 special case where w = MC, R . Even in this case there would be no 
 advantage to combining modes over specialization. 
 
-9- 
 
 If an individual must work standard hours, knowledge of his wage 
 rate together with the characteristics of available modes will not be 
 sufficient to establish his modal choice. Information on preferences, 
 how an individual values income relative to leisure, is also essential. 
 We shall concentrate here on circumstances where an individual chooses 
 to combine modes rather than use one exclusively. Modal mixing is more 
 likely to occur when there are substantial differences in the abstract 
 characteristics of available modes. 
 
 Figure 4 illustrates a situation where modal mixing at point S 
 would be preferred by the individual whose indifference curves are Ii 
 and I~ and who must work standard hours. It is assumed here that con- 
 stant returns to scale permit linear combinations of modes A and B. 
 
 Figure 5 is developed assuming a scalloped isoquant in Gronau space 
 which projects into a similarly shaped opportunity line connecting A' 
 and B' in M&W space. Maintaining the fixed hour assumption, Figure 5 
 shows that modal mixing is not limited to situations where constant 
 returns to scale apply. Line A'TB' represents opportunities available 
 to the commuter. With indifference curves I-j and IA, the commuter 
 would prefer position T which is achieved by combining modes A and B. 
 Since scalloped isoquant s involve a lower income-leisure position than 
 linear combinations, mixing is less frequent. Indeed, the significance 
 of scalloped isoquants may lie in explaining the infrequency of mixing. 
 
 From this analysis, more can be understood about the competitive 
 position of mode D in Gronau space (see Figures 2 & 3) . Recall that this 
 mode could never be selected in the Gronau analysis, because together 
 with proximate modes A and B, it formed an isoquant that was concave to 
 
Net Income 
 
 a 1 n 
 
 Leisure 
 
 Figure 5 
 
-10- 
 
 to the origin. Under the standard work hours condition, opportunity 
 point D' associated with mode D may be the preferred income-leisure 
 combination. This is the case in Figure 5 where the indifference curve 
 passing through D' is higher than those passing through A' and B' (not 
 shown) . Thus mode D is preferred. 
 
 Part Time and Full Time Work 
 
 This section and the following section utilize an assumption about 
 labor market opportunities which falls between the extreme assumptions 
 of Gronau and M&W. Gronau claimed that his approach avoided utility 
 analysis. As we have seen, the only labor market characteristics which 
 make utility irrelevant for modal choice is flexible hours with a con- 
 stant wage rate. M&W explicitly consider this case as well as the oppo- 
 site extreme, completely inflexible work hours. The intermediate case 
 considered here assumes flexible work hours at a part-time wage rate and 
 completely inflexible hours at a higher full-time wage rate. The assump- 
 tion of flexible hours at a part-time wage rate does not require that 
 there be a single employer who would offer flexible hours. It only re- 
 quires that an individual with one or more part-time jobs have only one 
 commute of any substance. 
 
 Consider the discontinuous opportunity curve in Figure 6. Beginning 
 with the cost and time of commuting indicated by point A, opportunities 
 for less than full-time employment extend to point A'. The negative 
 slope of AA' is the part-time wage rate. Point A' can be achieved by 
 working full time at the part-time wage rate, whereas point A" can be 
 achieved by working full time at the full-time wage rate. Opportunities 
 
Net Income 
 
 Time 
 
 Travel Cost 
 
 Figure 6 
 
-11- 
 
 from A" to E are obtained by working full time and moonlighting at part- 
 time jobs. So the negative slope of A"E is the part-time wage rate. 
 
 Figure 7 illustrates the opportunities available to an individual 
 having a choice of two relevant modes. In particular, Figure 7 shows a 
 situation in which the marginal cost of time exceeds the part-time wage 
 rate. In this situation, a part-time worker or moonlighter would take 
 mode B, the slower-cheaper mode. This is because the opportunity line 
 associated with mode B is above the line associated with mode A in the 
 range of part-time work. There is a minor exception to the rule that 
 the part-time worker takes the slower-cheaper mode. If the scallop 
 dips below the line A"E just to the northwest of A", it would be possible 
 for a moonlighter to take mode A, the faster costlier mode. 
 
 A full-time worker might take either mode or mix modes depending 
 on preferences. There is also a minor exception to this rule. Consider 
 the net cost saving from taking the slower-cheaper mode to be the direct 
 saving in cost minus the extra time in commuting valued at the part-time 
 wage rate. If the premium for working full time (i.e., the size of the 
 discontinuity) is less than the net cost saving, a full-time worker will 
 never specialize in the faster-costlier mode. 
 
 Modal choice is more clearcut if the marginal cost of time is less 
 than the part-time wage rate (not shown). Regardless of full-time or 
 part-time status, only the faster costlier mode will be selected. 
 Information on preferences is not required to determine modal choice. 
 
 The Subsidy Issue 
 
 The oft-quoted diversion prices from the M&W study assume flexible 
 hours at the individuals' wage rate. In the event that hours are completely 
 
Net Income 
 
 G 
 E 
 
 Travel Cost 
 
 Figure 7 
 
-12- 
 
 inflexible, M&W reveal the direction of bias in their diversion price 
 estimates. However, we have no way of knowing the extent of bias. In 
 this section, the intermediate case of flexible hours at a part-time 
 wage rate and inflexible hours at a full-time wage is used to provide a 
 range within which the diversion price must fall. 
 
 Our focus is on the interesting case of a full-time worker before 
 and after a transport improvement. The transport improvement under con- 
 sideration consists only of a reduction in time. The same two criteria 
 used by M&W, the income supplement and the income levey, will be used 
 to define the range of the diversion price. The income supplement is 
 the subsidy necessary to make the individual just as well off as if he 
 received the transport improvement. The income levy is the tax on the 
 transport improvement that would leave the individual no better off than 
 before the improvement. 
 
 Figure 8 illustrates the individual's opportunities before and after 
 the transport improvement. The income supplement can be thought of as 
 the necessary upward shift in the "before" opportunity line to reach an 
 indifference curve, like L. or I_, which goes through point A' but does 
 not otherwise touch the "after" opportunity line. If the indifference 
 curve is similar to I_, very steep as it passes through A' and falling 
 infinites imally close to the "after" line in the part-time region, the 
 upward shift in the "before" line is 
 
 z = (T., - Tjw 
 p B A p 
 
 where z = the minimum income supplement, and 
 
 w = the part-time wage rate. 
 
Jet Income 
 
 Time 
 
 Travel Cost 
 
 Figure 8 
 
-13- 
 
 A recipient of the minimum income supplement would work part-time and 
 end up at income-leisure position H. 
 
 For the other extreme, suppose that the indifference curve, like 
 Io in Figure 8, has a slope of -w at A'. In this case, the income 
 supplement must be equal to z plus the length of the discontinuity 
 in an opportunity line. That is, 
 
 the maximum income supplement = z + (w,. - w )h £ 
 
 *^ p f p f 
 
 where w, = the full-time wage rate, 
 
 and 
 
 h f = full-time work hours. 
 
 A recipient of the maximium income supplement would work part-time to 
 receive JG which together with the income supplement would give him a 
 final income of JA' . 
 
 Turning to the income levy criterion, we need to know the amount of 
 the downward shift in the "after" curve needed to just allow the individual 
 to reach the indifference curve, like Ij or I^t which passes through B' 
 but does not otherwise touch the "before" line. If this indifference 
 curve, IjL only barely falls above the "before" line in the part-time 
 region to the left of point G, the tax would have to be as much as 
 
 the maximum income levy = z + (w, - w )h-. 
 
 P f P f 
 
 An individual paying the maximum income levy would work full-time or 
 
 moonlight. 
 
 If, on the other hand, the difference curve has a slope of -w at 
 
 P 
 
 B*, the downward shift in the "after" line need only be 
 
 minimum income levy = z 
 
 P 
 
-14- 
 
 Any less of a shift would allow the individual to reach a higher indif- 
 ference curve. An individual paying the minimum income levy would work 
 overtime, receive income of NM, and after-levy income of NB'. Thus the 
 range of diversion prices is the same for the income levy criterion as 
 it is for the income supplement criterion. Note that if w = w,, the 
 diversion price is the same as in the M&W flexible hours case. This 
 result shows that the individual's diversion price can be found within 
 a rather narrow range if full-time and part-time wage rates are known. 
 
 Toward an Empirical Model 
 
 The theoretical structure developed in this paper can be used to 
 formulate econometric models of travel demand. As an illustration of 
 this capability, we consider the econometric treatment of competing 
 modes. Defects of existing approaches are examined and some remedies 
 are suggested based on the theory. 
 
 The Quandt and Baumol model estimated the demand for travel by a 
 mode k along a given route as a function of the best characteristics 
 of the modes serving the given route and the characteristics of the mode 
 k relative to the best characteristics. The model suffers from a number 
 of deficiencies [3, 6], In particular, the model is deficient in cap- 
 turing the impact of competing modes. Travel demand for mode k is not 
 affected by changes in the characteristics of a competing mode unless 
 those changes establish a new value for some best characteristics. 
 
 Young has attempted to remedy this deficiency by incorporating two 
 
 new variables representing characteristics of proximate modes [8]. The 
 
 proximate modes to mode k are the modes that are the next more expensive 
 
 and faster and the next cheaper and slower. If T, .. is the travel time 
 
 r k-1 
 
-15- 
 
 of the slower proximate mode and C, ... is the travel cost of the more 
 expensive proximate mode, Young's new relative performance variables are 
 T, /T, . and C, /C, . These variables can be added to the original Quandt 
 and Baumol equation. 
 
 These new variables give some indication of the impact proximate 
 modes have on mode k. Unfortunately, their usefulness is limited to 
 changes in the inferior characteristics of the proximate mode. Where 
 the superior characteristic undergoes change, there is no impact on mode 
 k according to Young's model. That is, if the faster proximate mode 
 becomes even faster, though not the absolute fastest, or the cheaper 
 proximate mode becomes even cheaper, though not the absolute cheapest, 
 these improvements will not reduce the demand for k. This problem can- 
 not be corrected by introducing new relative performance variables based 
 on the superior characteristic without risking pure multicollinearity. 
 Thus, while the Young analysis suggests the importance of examining com- 
 peting or proximate modes, it does not fully remove the deficiency in 
 the Quandt and Baumol formulation. 
 
 The theory of modal choice suggests improvements in capturing the 
 impact of competing modes on the demand for a particular mode. With two 
 relevant modes, A and B, in Gronau space as shown in Figure 9, assume a 
 hitherto unknown mode D is introduced. Suppose D is on the line segment 
 D.D-. We would like to construct a travel demand function with the 
 following desirable properties: 
 
 (a) If D is at D. , it will eliminate travel on modes A and B. 
 
 If D is at Dr, travel on it will be eliminated by diversion 
 to modes A and B. 
 
Figure 9 
 
-16- 
 
 (b) As D moves northeast from D^ (i.e., becomes more costly and 
 time-consuming) , the proportion of travelers taking competing 
 modes A and B increases. 
 
 (c) If linear combinations of modes are possible, mode D between 
 Do and Dr would never be selected. Alternatively, economies 
 of scale suggest that mode D would still attract travel demand 
 if it falls within some distance to the northeast of D,. 
 
 The number of travelers mode D takes away from modes A and B depends 
 on how extreme is the "convexity" to the origin of isoquant ADB. The 
 extent of "convexity" varies from a maximum at D.. to a minimum at Dr 
 (i.e., with negative convexity or concavity occurring between D_ and D-). 
 Figure 10 shows that a measure of convexity for mode D may be found by 
 forming the variable R = (90° - 3 - y) where 3 and y represent the angle 
 
 deflections from the maximum convexity positions to proximate modes A 
 
 — 1 DF — 1 — 1 BF — 1 
 
 and B. Since 3 = cot -rrr = cot " mc ^jj and y = tan — ■ tan ^DB» 
 
 the marginal costs of time can be incorporated by rewriting R as 
 
 R = 90° - cot" 1 MC^ - tan" 1 MC^g. 
 
 At maximum convexity where 3 = Y = 0» M C AQ = °° an ^ m ^db = ®* C ^ e 
 commuter always selects mode D since the marginal rate of substitution 
 of income for lesiure would always be between MC__ and MC. . R would 
 then take on its maximum value of 90°. With zero convexity where 
 3 + Y = 90° and MC^ = MC DB , R would be 0°. In this instance, there is 
 no advantage to the commuter of selecting mode D under the restrictive 
 assumption of a constant unit value of t ime . Under less restrictive 
 assumptions, however, D may be preferred. If ADB is concave to the origin 
 
-17- 
 
 (i.e., mode D> in Figure 10), then 6 + y > 90°, and R is negative. For it 
 to be possible for mode D to be selected where R is negative requires 
 economies of scale in modal mixing as well as diminishing marginal rates 
 of substitution of income for leisure. In summary, the competitive 
 position of mode D declines as R becomes smaller. 
 
 The drawback in using R as an index of the competitiveness of a 
 mode is that R is sensitive to the untis in which cost and time are 
 measured. It is therefore necessary to measure the angles of displace- 
 ment in standardized units. One approach, in the spirit of Quandt and 
 Baumol, would be to develop cost-time characteristics for each mode 
 relative to the best cost (C, ) and best time (T, ) characteristics. 
 Thus, if we define relative cost and time characteristics for mode k as 
 
 c £ = c k /c b and T k - W *«■ 
 
 . C £- C f MC AB 
 
 R* = 90° - cot" 1 MC£ D - tan" 1 MC£ B 
 
 MC AD 
 = 90° - cot rs ,„ t - tan 
 
 _-l AD ___-! M 
 
 The standardization of the marginal cost of time through diversion by 
 
 (C u/T N 
 
 b b ; represents just one approach. In lieu of (C /t ), suitable 
 divisers would be selected on the basis of how closely they approximate 
 the unobservable marginal value of time for the median traveler. This 
 hypothesis can be most easily understood by assuming that there are only 
 two modes, A and B. If the marginal value of time for the median traveler 
 
Figure 10 
 
-18- 
 
 equals MC» R , we would expect half the commuters to take mode A and half 
 to take mode B. With the diviser equal to the marginal value of time, 
 MCJ„ = 1, and therefore R* = 45° for both mode A and mode B. Giving 
 empirical life to this concept, one might use the median marginal cost 
 of time for the divisor. 
 
 Conclusion 
 
 Theoretically, the choice of mode for the work trip is determined 
 by the abstract characteristics of the modes, the technology with which 
 modes are combined, labor market opportunities, and preferences. Insti- 
 tutional constraints on labor markets have important effects. For 
 example, the commuter who must conform to a standard work day or who 
 receives higher wage rates from working full-time than part-time may 
 optimally choose to mix modes over time. However, the attractiveness 
 of modal mixing is diminished when economies of scale exist in taking 
 certain modes. With flexible hours at a part-time wage rate and in- 
 flexible hours at a higher full-time wage rate, upper and lower bounds 
 can be established for valuing a transport improvement. Finally, the 
 theory of modal choice suggests structural characteristics for empirical 
 models. 
 
-19- 
 
 FOOTNOTES 
 
 Consider mode D such that MC^ < MC DB . Whatever the traveler's 
 unit value of time, V, mode D will not be chosen. If V > MC^q, mode A 
 will be chosen over D, and if V < MCpg, mode B will be chosen over D. 
 Since MCaq < MC_, R , there is no value of V consistent with the choice 
 of mode D. 
 
 2 
 The limiting case for a relevant mode would be one on the line 
 
 AB. In that case, only if V = MC,r, could this mode share some travel 
 
 demand with modes A and B. 
 
 3 
 Linear combinations have been used to identify the benefits of 
 
 mixing trips. See [1], 
 
 4 
 Travelers are assumed to view work and travel time as equally 
 
 disagreeable so that time added to leisure can be considered a single 
 
 characteristic regardless of whether the time is taken from work time 
 
 or from travel time. 
 
 Modal choice with flexible hours yields identical results to 
 
 the Gronau analysis when the unit value of time (V) is defined as the 
 
 wage rate. Mode A will always be selected when V > MC^g and mode B 
 
 when V < MC AT ,. This is because with flexible hours the commuter will 
 AB 
 
 always re-negotiate his work day to ensure the equality of his wage rate 
 with his marginal value of time, his marginal rate of substitution of 
 income for leisure. 
 
-20- 
 
 REFERENCES 
 
 [1] W. B. Allen, "An Economic Derivation of the 'Gravity Law* of 
 Spatial Interaction: A Comment on the Reply." Journal of 
 Regional Science , 12 (1972), pp. 119-126. 
 
 [2] R. Gronau, "The Effect of Traveling Time on the Demand for 
 Passenger Transportation." Journal of Political Economy, 78 
 (1970), pp. 377-394. 
 
 [3] R. Gronau and R. E. Alcaly, "The Demand for Abstract Transport 
 
 Modes: Some Misgivings." Journal of Regional Science , 9 (1969), 
 pp. 153-157. 
 
 [4] L. N. Moses and H. F. Williamson, Jr., "Value of Time, Choice of 
 Mode, and the Subsidy Issue in Urban Transportation." Journal of 
 Political Economy, 71 (1963), pp. 247-264. 
 
 [5] R. E. Quandt and W. J. Eaumol, "The Demand for Abstract Transport 
 Modes: Theory and Measurement." Journal of Regional Science , 6 
 (1966), pp. 13-26. 
 
 [6] R. E. Quandt and W. J. Baurool, "The Demand for Abstract Transport 
 Modes: Some Hopes." Journal of Regional Science , 9 (1969), pp. 
 159-162. 
 
 [7] W. S. Vickrey, "Pricing in Urban and Suburban Transport." 
 American Economic Review, 53 (1963), pp. 452-465. 
 
 [8] K. H. Young, "The Abstract Mode Travel Demand Models: Estimation 
 and Forecasting." in Studies in Travel Demand , Vol. IV (1968), 
 pp. 1-36, Mathematica. 
 
 M/E/178