Douglass B. Lee, Jr.1
The criterion for making good investments is to select projects for which the net benefits are positive, i.e., incremental benefits exceed incremental costs. The major analytic steps are: define alternatives, evaluate impacts, and select the project with highest net benefits. If pricing is determined independently of marginal cost, however, pricing is exogenous and the investment evaluation is necessarily in a second-best mode.
The HERS model incorporates demand elasticity and benefit-cost evaluation principles that are specific to the investment and policy alternatives typically considered at national and local levels. As highway investment concerns shift, the HERS model attempts to adapt by making explicit the variables and relationships that will permit the model to realistically address the new concerns. In doing so, the model becomes more general in its scope and more flexible in its application to questions of interest.
For the 1997 Conditions and Performance report to Congress, the HERS model was extended to utilize demand elasticities, such that highway improvements that lowered or raised user costs could lead to changes in travel volumes, and that over the long run the effects of improvements could be to shift the demand curve from where it might have been placed in initial forecasts (see Appendix B, "Induced Traffic and Induced Demand" and Appendix C, "Demand Elasticities for Highway Travel"). Also, HERS was modified to estimate and take into account emissions of air pollutants, although the feature was not turned on until the 1999 report to Congress. More recently, HERS has been modified to incorporate a money price that is separate from user time and operating costs. Fuel taxes and tolls are therefore recognized, both in estimating demand and as potential policy instruments such as for congestion pricing. Thus the principles outlined in this Appendix have served as a guide in developing the economic portions of the model, and the model now embodies them to a large degree.
A physical facility can be represented for evaluation purposes by its unit costs with respect to traffic volume, measured as vehicle trips per hour. Three functions of volume provide the information necessary for calculating net operating benefits: average variable cost (AVC), marginal cost (MC), and price. Assuming a base alternative and one project alternative, the physical characteristics of each alternative are given by their variable cost curves, while the price curve constitutes the policies affecting how the facility is operated. All variable costs, whether monetized or not, are included. Cost and price components are assumed to be converted into a common numeraire (dollars), referred to as generalized cost or generalized price, meaning that it combines money and in-kind components on the same scale. Neither fixed costs nor fixed charges (e.g., annual vehicle license) are represented in the diagrams.
In general, at any given volume, marginal cost, average variable cost, and price to the user are all different. MC and AVC are mathematically related, and will diverge if any component of cost varies with volume (or v/c), i.e., MC is unequal to AVC if unit cost (AVC) goes up or down with volume. Because unit travel time costs rise with congestion, for most volume levels marginal cost lies above average cost. Price includes user charges, which are transfers and not costs, and excludes externalities and agency costs (facility wear, maintenance, and operation), which are costs that are not part of the price.
The marginal social cost curve is the guide for efficient pricing, so if p = MC at all volumes, then net benefits in the short run are maximized for the facility. In this special (first-best) case, price and marginal cost are the same. As shown more generally in Figure D-1, price, represented by the price function, is not directly tied to marginal cost, labeled MC. Since the price function determines the quantity demanded by its intersection with the demand curve, actual volume is q0 at a price of p, with a marginal cost of mc and an average cost of ac. The inefficiency from not pricing at marginal cost is given by the triangular area bounded by pmc, mc, and p.2
2. The welfare loss from inefficient pricing does not enter in to the measurement of second-best benefits, but recognition of the inefficiency provides some insights when comparing second-best benefits to first-best pricing.
Variable social costs include travel time, fuel, accidents, other vehicle wear and operating costs, damage from emissions and noise, and facility wear, maintenance, and operation (agency costs), but not fuel taxes or tolls. They are variable because they increase with vehicle miles traveled. Their combined average unit cost per vehicle mile (AVC) might rise, decline, or remain constant with volume, which is a rate of flow. In fact, most of the components of variable cost vary slightly with volume, due to congestion, but the one that varies by far the most is travel time.
Because users are faced with the average rather than the marginal cost of travel time, it is frequently assumed that price and average cost are the same, but this usually is not true because of user charges, agency costs, and externalities. The AVC and MC functions are mathematically related, such that either one could be derived from the other, but it is the components of average cost that can be observed empirically.
Total variable cost can be measured either as the area under the marginal cost curve (e.g., up to q0) or as the average variable cost (ac) times the volume (q0), the latter being a rectangle, as shown in Figure D-1. This relationship will be used later.
Price is the cost to the user, and includes travel time, accidents, and operating costs as well as money payments that vary with usage. The price function in Figure D-1 assumes that travel time is the main reason the generalized price varies with volume; user charges are approximately constant per vehicle mile, such as through a fuel tax.
The price function in this diagram is shown as lying above average cost. This might be the case if variable user charges exceed variable externalities and agency costs. If the reverse is true, then the price function lies below AVC, as shown in Figure D-2. The same relationships hold as before, although the inefficiency triangle is relatively larger. For congested conditions, it is unlikely that price will be above MC without a congestion-related toll, but price could be above AVC. Whether price is above or below AC depends upon the magnitude and valuation of externalities and agency costs relative to user charges.
Thus volume could be determined by any of the three functions, shown at the circled points in Figure D-1 or Figure D-2: by marginal cost at pmc, for efficient pricing and first-best evaluation; by average variable cost at pac, which ignores actual user charges, agency costs, and externalities; or by the price function at p, which is the most general case.
The demand curve shows the quantity that will be taken by consumers across a range of prices, generalized to include time and running costs. In reality, this demand curve is constantly shifting, affected by user preferences as well as their knowledge of what conditions will actually pertain at their time of usage. For analytic purposes, the demand curve is assumed to be fixed - or represented by some average - for some period of time. An example of a demand period is the AM peak period, lasting several hours in larger urban areas. A single demand curve might represent the AM peak, or the AM and PM peaks combined; time periods do not need to be contiguous to be treated as a single demand period.
The minimum information needed to construct a demand curve is a price, a quantity, and an elasticity at that point. The price could be the average generalized price during peak periods, namely, the sum of travel time, running costs, and user fees representative of peak times.3 The quantity is the average traffic volume for the peak, in vehicles per hour. An elasticity can be selected by comparing the project to other facilities with respect to the mix of components of price, the substitutes available, and the types of vehicles and trip purposes. Ideally, the elasticity should be compatible with the nature of the project, e.g., include diversions if the project is a single facility. The demand curve could be disaggregated into separate analyses for each market segment (e.g., trucks, commuters, recreational travelers), but some averaging is always necessary. A single demand curve for all users is assumed here.
With a single elasticity value, the functional form of the demand curve can be either straight line or constant elasticity. Given the demand point (p, q0) in Figure D-2, only one demand curve of each type passes through that point with the given elasticity at that point. A straight line is used here.
To evaluate a project, a set of curves is needed for each of the base and the project alternatives, as shown in Figure D-3. Data for the base alternative are designated with a "0" subscript, and with a "1" for the project alternative.
The marginal and average cost functions are characteristics of the facility, resulting from its capacity, geometrics, terrain, pavement condition, and so forth. The price function is partly endogenous to the facility in that it includes some variable costs, and partly exogenous in the form of user charges and regulation. The price function could be made to go through the point pmc by the correct congestion toll, in which case the facility would be operating efficiently. Another possibility is that externalities (except delay) and agency costs are exactly offset by user charges, such that the price function follows the AC curve. These are special cases of the general case presented here.
3. See Table D-1 below, page D-10.
A highway improvement - resurfacing, reconstruction, additional lanes - will change user costs by some amount, resulting in operating benefits. Reductions in running costs, travel time, and accidents are both reductions in price and real benefits. Savings in agency costs and externalities are real benefits but not included in the price, whereas savings in user fees are not real benefits. The impacts of each improvement can be estimated from its induced traffic volume (based on the price and demand curve) and variable cost savings. These net operating benefits (NOB) are estimated for the current period, and subsequent periods, over the lifetime of the improvements. Any improvement whose NOB over its lifetime exceeds its capital costs is considered feasible; among feasible improvement projects, the one generating the highest net benefits is preferred.
Highway improvements that reduce congestion (by expanding capacity), or reduce vehicle wear and fuel consumption, or reduce accidents, have the effect of lowering the price to the user and stimulating greater volumes, depending upon the elasticity of demand. If the short-run price elasticity is non-zero, changes in the generalized price will cause changes in volume, within the same period, by movement along the demand curve.
To some extent, capacity expansions are self-limiting, in that induced traffic reintroduces congestion, which offsets some of the initial time savings from expansion. This supply-demand equilibrium may not result in as high a volume as would be the case if there were no congestion, but congestion will remain below the original congestion level before the capacity expansion. It is not possible for the same level of congestion to return after the expansion as before, because the short-run demand curve slopes downward to the right, and demand in the short run stays on the same demand curve. In subsequent demand periods, shifts in the demand curve might lead to higher congestion than in the current period, but such demand growth would be at least partly exogenous.4
Benefits of the project are generally a combination of cost savings and additional travel. The net of such operating benefits is compared to the net or sum or fixed costs, with all values discounted or annualized as appropriate. Figure D-3 can be used to illustrate the net operating benefit (NOB) of the project. This diagram assumes price lies above AVC; the (minor) consequences of changing this assumption will be shown subsequently. The measurement of NOB can be defined in two ways, using different combinations of the variable cost and pricing functions and the demand curve.
Total variable costs for the base alternative are represented by the area under the MC curve up to the existing volume q0. For the project alternative, the corresponding area is lower but extends out to q1. The cost difference is an area of cost savings between the two curves up to q0, and an area of additional costs under MC1 from q0 to q1. The latter is offset by the (not necessarily equal) incremental benefits from the additional trips, represented by the area under the demand curve from q0 to q1. The resulting NOB is the area outlined by the dot-dash line. It can be described as the area between the two MC curves and under the demand curve.
Where MC crosses above the demand curve, the area - marked "additional costs" - is negative; these disbenefits are a consequence of underpricing the project alternative, relative to marginal cost pricing. NOB could be increased by this amount if the new project were efficiently priced, but this is not an option with exogenous pricing. Correspondingly, NOB would be smaller if it did not include the inefficiency from underpricing the base alternative.
Because areas under the marginal cost curve can also be represented by rectangles constructed from the AVC curve, using the relation,
the area under MC0 up to q0 is equal to the rectangle whose length is q0 and whose height is ac0 (read from AVC0), as shown in Figure D-4. Similarly, the area under MC1 up to q1 is equal to the rectangle q1 by ac1. The difference between these two rectangles is the shaded area labeled "delay and cost savings," minus the additional costs from q0 to q1, plus the area under the demand curve from q0 to q1. This shaded area is exactly equal to the outlined area derived from the MC curves.
In practice, a distinction is made between trips that are already being made in the base case (up to q0), or "old" trips, on the one hand, and additional trips (from q0 up to q1) generated by the reduction in price (from p0 to p1), or "new" trips, on the other.5 A reason for making this distinction is the nature of the benefits to the two groups: old users have "demonstrated" or "revealed" (even if the demand curve is estimated or forecast) their willingness to pay for their travel, and so the benefits to them are the cost savings over their previous generalized cost. New trip makers on this facility, in contrast, have not shown any willingness-to-pay, so their benefits must be estimated from the demand curve as consumer surplus over what they actually pay when using the new project.
5. It is likely, though not necessary, that most previous users of the base facility remain on the new facility to become "old" users, since they obtain what they had previously but now at a lower generalized price. The old-new distinction, however, is heuristic, rather than defining a fixed set of vehicles.
Although the total NOB areas are the same whether defined by MC or AVC, the way they partition the benefits between old and new users is not. The area under MC1 up to q0 indicates the total average cost if volume on the new facility were held at the volume on the base facility, but that is not what will happen; the new volume will be q1, and costs for "old" users will be higher than if volume were held to q0. At a volume of q0, ac1 would occur where AVC1 crosses the vertical at q0, but none of the old users actually faces this hypothetical cost on the new facility; instead they all pay the actual ac1 at q1. The shaded area representation based on AVC provides a more useful interpretation with respect to old and new users, and also allows for direct empirical estimation of the benefit components.
With price above AVC, the NOB diagram looks similar in shape to the first best case. This is because with price being above AVC there is at least a partial "toll" even if it is below marginal cost. If it is assumed that p < AVC, the diagram is slightly different. Figure D-5 shows such a situation, in which price is below AVC for both the base and project alternatives.
This diagram can be compared to Figure D-4. The outline of NOB based on MC is essentially the same, but the area defined by AVC curves has a somewhat different shape. Savings on old trips start above the current price, because the elimination of externalities in the base case is a benefit. Correspondingly, the benefits stop farther up, because some of the travel time and cost savings are offset by agency costs or externalities in the project alternative; cost savings would come down to p1 were it not for the new externalities. Consumer surplus is the same in both diagrams, but a share of incremental consumer surplus in Figure D-5 is offset by additional agency costs or externalities from new trips, which might be thought of as negative producer surplus.
Breaking NOB into components is useful for several reasons:
Most of the components can be disaggregated further than the major categories shown in Figures D-4 and D-5 and described below. A more detailed breakdown is provided in Table D-1. The first three columns refer to the three functions of volume previously described. Travel time is divided into normal travel time and delay, which can be anything above free-flow speed or above a v/c of, say, 0.8 of capacity. Some operating and infrastructure costs (vehicle ownership and pavement) are divided into variable and fixed as well as internal (paid by the user) and external (paid or suffered by others). Parking cost is excluded for purposes of project evaluation of highway facilities, although it may be relevant to other purposes such as price elasticities.
The fourth column notes those components included in the HERS model. Travel time and operating cost components are included in both price and cost, while infrastructure costs (agency costs) and externalities are included in cost only. HERS does not include user fees such as fuel taxes or tolls. Neither HERS nor the general theory deal with fixed user fees, such as annual fees not based on miles traveled.
|Marginal Cost||Average Cost||Price||HERS|
|y = cost component is included in the total for the column category|
|MT = marginal time cost is included|
|AD = average delay cost is included|
|Maintenance and Operation||y||y||y|
|Noise and Vibration||y||y|
|Other Variable Fees||y|
Trips being taken on the base facility before the improvement and remaining on the new facility receive benefits in the form of reduced delay and operating costs. In Figure D-6, the average cost with no improvement is AVC0 and yields an average cost of ac0 for the base or "unimproved" volume q0. With the improvement, ac1 is the cost as read off the AVC1 curve.
Savings on old trips, then, is the difference in average cost (a0−a1) times the unimproved volume q0, indicated by the shaded rectangle. All costs listed in the "Average Cost" column of Table D-1 are included. For example, fuel savings, highway maintenance cost savings, and pollution reduction are included, but fuel tax savings are ignored.
Consumer surplus is the amount users would be willing to pay above what they actually pay; it is measured as an area under the demand curve between the "with" and "without" volumes, and above the price. Because incremental consumer surplus applies to induced or "new" trips, the relevant volumes in Figure D-6 are q0 (with no improvement) and q1 (with improvement). Data required consist of two prices and the demand curve. This consumer surplus is a triangular area whose hypotenuse is the demand curve between p0 and p1, and whose legs are formed by q0 and p1 (only the top of this triangle is shown, because the bottom is offset by external costs). Both fuel costs and fuel taxes, for example, are included in the measurement.
Producer surplus is an area under the demand curve that is below what users pay but above short-run variable cost. Normally, user fees are regarded as transfers and therefore ignored in estimating benefits, but here it is simply a part of the means for valuing induced travel. Like consumer surplus, it indicates a willingness to pay for new trips. A congestion toll generates producer surplus, but only the portion on new trips is counted as a benefit; the portion applying to old trips is already counted in the time and cost savings on old trips. The net of revenues above incremental agency costs and externalities is producer surplus.
Producer surplus can be negative if payments are less than average cost. Although not comprised of revenues, users create negative externalities that are omitted from the price, so these costs can be treated symmetrically to positive producer surplus. Negative producer surplus is shown in Figures D-5 and D-6, while Figure D-4 shows positive producer surplus. In Figure D-7, the surplus of revenues over short-run cost is the rectangle with a height of (p0−ac0) and a length of q0; this area is excluded from NOB of the project because it occurs in the base case.
Negative externalities shift the marginal and average social cost curves upward, but not the price function. External costs are included in the average variable cost (AVC) curves in Figure D-6. Thus the MC and AVC curves in the diagram include both externality as well as time costs.
A negative externality has the opposite effect as a user charge. If the user charge and the externality happened to be equal in value (at all volume levels), then there would be no externality. In outlining the incremental NOB area, external costs are rectangles taken from the average cost curves, and are negative in sign. In Figure D-7, for example, the net of external and agency costs over user payments in the project alternative has a height of (ac1−p1) and a length of q1. Subtracting this area from consumer surplus (for new trips) and user cost savings (on old trips) leaves the small triangle above the demand curve at the outer end in Figure D-6 as a negative benefit (i.e., traffic induced by underpricing produces costs that exceed internal benefits).
Externalities caused by induced trips diverted from other facilities may not be adding to the total emissions of pollutants, but this is irrelevant to the present project. The only way to incorporate changes in externalities in related markets (e.g, parallel facilities) is to measure the difference in the total inefficiency with and without the improvement project; it cannot be done one item (e.g., pollution or congestion) at a time.
It seems conceptually plausible to sum up the net change in air pollution caused by a project for a region, say, and count that as the project's pollution benefit or disbenefit; then do the same for travel time, accidents, and running costs. While certainly a chore to detect the thousands of microscopic impacts occurring throughout the region, the task might be accomplished with regional simulation models.
The real problem, however, is that all of these calculations are meaningless without also calculating and summing all of the changes in consumer utility occurring at the same time, each of them the result of a shift in the demand curve in the relevant market. The likely error in such an estimate would greatly exceed the magnitude of the impact being estimated. Total air pollution in the region may be a performance measure of important policy concern, but the change in that index is not a basis for evaluating individual projects. The practical solution is to ignore what happens in related markets, except perhaps to trace out the efficiency changes for a few externalities in a few closely-related markets. The magnitude of such differences in related markets is generally small relative to benefits in the primary market.
The above steps describe a static equilibrium analysis conducted within a single short-run demand period. For each improvement alternative, the steps are repeated for each demand period over the lifetime of the improvement. Once the lifetime NOB is accumulated for each alternative and compared to costs, the investment choice can be made for that project.
The demand curve shifts over time in two primary patterns:
There are also periodic fluctuations over days of the week, and days or seasons of the year. Daily commuting peaks may be unimportant on some facilities. For evaluation, however, it is usually sufficient to recognize 1-3 daily demand period "types" and 1-4 demand periods over the investment lifetime, depending upon the rate of traffic growth.
The overall analysis period (e.g., twenty years) can be broken into shorter demand periods (e.g. 1-5 years), depending upon how rapidly exogenous demand factors are changing. Each demand period embodies a short run during which demand is assumed to be fixed, meaning that a single short-run demand curve applies for the duration of the period. This "single" demand can still be composed of several periodic demand curves, such as peak and off-peak, or it could be a daily average.
Once the overall analysis period is broken into demand periods, the secular or trend forecast becomes a series of discrete points representing the midpoints of demand periods. These points provide the origins or calibration points for the associated short-run demand curves.
Even in the short run, demand stimulated by reduction in the generalized price generates enough traffic to partly offset the gains from increased capacity. In the long run, this effect can be exaggerated, when general growth in demand and highway improvements reinforce each other to increase traffic volumes. A casual observer of this process can easily come to the conclusion that building more road capacity is self-defeating, because congestion is soon back to where it was. For those trying to carry out benefit-cost analysis, the benefits seem to disappear. The reality, however, is a bit different.
The simplest case is shown in Figure D-8, in which a base alternative and a project alternative are represented by their AVC curves, and these also give the price to the user under each alternative. Two demand curves are included, D1 for the first period of time and D2 for the next period. The curves are drawn such that, by coincidence, the cost to the user in the first demand period under the no-build alternative is the same as the price in the second period under the project alternative. In other words, users are individually no better off after the improvement than before.
This does not mean, however, that there are no benefits. First, there is more travel than was the case under the base alternative. Second, the relevant comparison is not to the price and volume in period one, but to the period two base case - i.e., the conditions that would have occurred in period two if the improvement had not been made. With an exogenous growth in demand, congestion would have been much worse without the improvement, and less travel would have been served. Hence, there is positive incremental NOB in the first period, and additional (and larger) NOB in the second demand period. Together (assuming only two periods) these account for project benefits, to be compared against incremental capital costs.
A more general case is illustrated in Figure D-9, in which user price does not follow the average cost curve. In this case, price is above average cost (compare this to Figure D-4, the single-period case with price above AVC). The MC curve is omitted because pricing inefficiency is ignored, i.e., a second-best investment comparison is assumed. Price is again arbitrarily set so as to equate base alternative price in the first period to the price with the improvement in the second period. Again, the areas of NOB are outlined and shaded, and together form the NOB for the project alternative. Because price is above AVC for the project alternative, NOB includes some producer surplus on new trips, in addition to consumer's surplus.
A final configuration, shown in Figure D-10 illustrates the case when price is below social cost for both alternatives. The two demand periods are independent of each other. Each of these periods is similar to Figure D-5.
In summary, each demand period is handled as a single period, in which the short-run demand curve is fixed to a point based on the actual or forecast traffic at an associated price. Within the demand period, volume can move along the demand curve depending upon reductions in generalized price resulting from improvements being evaluated. Between demand periods, demand can shift and facilities wear out, resulting in a new set of cost and demand curves. The sum of the discounted benefits in each demand period is the present value of project benefits. This is true whether pricing is efficient at p = MC, price follows AC, or pricing follows neither of the above.
The HERS model produces results like those shown in Figure D-10 because agency costs and air pollution (if the module is enabled) are included. If other negative externalities (e.g., noise, water pollution, external costs of accidents) were modeled in HERS, the gap between average variable cost and the price function would be wider. The model cannot produce results of the type shown in Figure D-9 because there is no price in the model that is separate from other user costs.
These concepts are readily made operational, and can be implemented in spreadsheet or other models.
Table D-2 shows some hypothetical data for a single demand period for one project alternative versus the base case. All of the data are converted from whatever natural units (e.g., minutes, crashes, grams) they might have been generated in to dollars per vehicle trip over the facility. All of the bolded numbers are required input data about costs and pricing that must be estimated for the specific conditions of the project, including the volumes that will occur at the relevant prices. Capacities of the existing and expanded facilities are also required.
Calculations are done on the basis of two contrasting assumptions:
Total NOB as well as the major components under each of the two assumptions are shown in Table D-3. Areas in the diagrams correspond to components of NOB in the table. For example, savings on old trips in the first-best evaluation are measured by the rectangle
|SOT||=||(6.20 ∠ 4.65) × 4000|
which is composed of normal running time savings = (1.80−1.40)×4000=1,600, plus running cost savings = (2.80∠1.75)×4,200, plus delay savings = zero with efficient pricing, plus highway operating cost savings (0.50∠0.40)×4,000=400.
|$ per trip||BASE||PROJECT||Marginal Cost||Price||Average Cost|
|TRAVEL TIME COST||6.80||3.40|
|OTHER DATA AND PARAMETERS|
Numbers for the "Existing" facility show the net benefits of operating the facility efficiently (i.e., correctly priced) or inefficiently, and do not enter into the benefit-cost evaluation of the expansion project.
|NET OPERATING BENEFITS||Existing||Expansion|
|normal running time savings on old trips||1,600||2,000|
|running cost savings on old trips||4,200||5,250|
|delay savings on old trips||15,000|
|externality cost savings on old trips||0||0|
|highway operating cost savings on old trips||400||500|
|incremental consumer surplus on new trips||19,500||30,469||4,875||4,062|
|producer surplus on new trips||23,950||750||5,325||274|
|externality costs on new trips||(5,500)||(2,008)|
|highway operating costs on new trips||(2,500)||(730)|
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