Cost-Benefit Analysis of Convention Center PDF

# Choosing the Best Alternative: Cost-Benefit Analysis Your city is considering acquiring a parcel of vacant land near the center of the city to develop a new convention center with a hotel complex. Your job, as a policy analyst, is to evaluate the proposal and send your recommendation to the city...

# Choosing the Best Alternative: Cost-Benefit Analysis Your city is considering acquiring a parcel of vacant land near the center of the city to develop a new convention center with a hotel complex. Your job, as a policy analyst, is to evaluate the proposal and send your recommendation to the city council. The question is, of course, how to go about it. It is obvious that you will recommend the project if the benefits outweigh the costs. When we make a decision, any decision, we consciously or unconsciously evaluate its potential benefits and costs. In fact, this evaluation process is so fundamental to human cognition that many economists have equated it with the very notion of human rationality. Economists argue that when individuals make decisions, even emotional ones, they have a good sense of the benefits and costs of their actions. How do we choose our mates in marriage? Poets and authors of romantic novels claim that love is blind anyone can fall in love with anyone else without following any definite pattern of behavior. Yet statistical data show that marriage partners tend to match each other's "endowments" (age, looks, wealth, education, social standing, and so forth). That is, their actual preferences reveal a "rational choice" based loosely on the economic notion of cost-benefit analysis. Thus, you will probably not be surprised that this chapter on cost-benefit analysis mirrors chapter 5 on critical thinking. Cost-benefit analysis, which emphasizes monetary evaluation, provides critical thinking with a more formal structure. I mentioned that the techniques of statistics and operations research greatly aid the process of thinking critically. Hence, because cost-benefit analysis draws on all these tools, I am discussing it at the very end of our discussion of analytical techniques. ## In sum, the fundamental principles of cost-benefit analysis are the following: - When considering a single project, accept it if its benefits are greater than its costs. - When considering alternative projects, choose the one that gives you the highest benefits in relation to the costs. In our convention center example, we should go ahead with the project if its benefits are greater than its costs. Similarly, we should evaluate alternative uses for the land (for example, a new public library, a public park, or a shopping mall) with regard to their respective benefits and costs and choose the one that gives us the highest benefit relative to the costs. If you recommend building a new convention center, how much should the city pay for it? Those who derive their livelihoods from the tourist industry may benefit more than those who do not. Thus, those in the former group may be willing to pay more for it than others. If the total amount the city is willing to pay (reflecting the total utility of the project to its stakeholders) is greater than the cost of the project, it is worth undertaking. This total amount is known in economic terms as the consumer surplus (see chapter 3). Therefore, this question goes to the heart of cost-benefit analysis. However, the more we look into the issue, the more complicated it becomes. To fully appreciate the complexity of the problem, we start with the process of conducting a cost-benefit analysis. ## Similar to the familiar process of critical thinking, the process of conducting a cost-benefit analysis is as follows: 1. Define the goal(s) of the project. 2. Identify the alternatives. 3. Make an exhaustive list of all benefits and costs, present and future. 4. Estimate and express benefits and costs in monetary terms. 5. Forecast the future streams of benefits and costs, if needed. 6. Choose the alternative with the largest benefit in comparison with the cost. Let us now discuss in detail each of these six steps. This powerful analytical tool can be used to evaluate almost every social decision. And as you will see from our discussion, its thoroughness (which is directly linked to the cost of the study) must be matched against the importance of the project. If we want to evaluate a proposal for a neighborhood park, we may want to devote a modest amount of time and effort. In contrast, the evaluation for the construction of an atomic power plant, which may affect the health and welfare of hundreds of thousands of residents in the region, would require a much greater degree of precision. ## Social Versus Private Cost-Benefit Analysis Before getting to the steps mentioned above, we should briefly discuss the various ramifications for the public versus the private sector. An example may clarify the distinction. An analyst friend, who works for a small neighboring town, called me to say that he was disappointed. The town was negotiating with a national retail chain store to locate within its boundaries. After months of negotiation, the store decided to locate in a more affluent medium-sized city next door. Because my friend's town was coming out of a long fiscal slump and the store would have given it a significant boost, he complained, "Didn't the management of the retail chain know how much social good they could have achieved by choosing to locate here?" Indeed, if you considered the positive externalities of this relocation decision, you might have agreed with him. However, from the standpoint of the retail chain, the private net benefit was the only calculation that counted. This difference in perception explains the distinction between private and public cost-benefit analysis. Every project, large or small, contains social benefits and costs. Retail stores generate employment, produce sales and income tax, and may even increase property values in the surrounding areas. In contrast, they may also create traffic congestion and pollution. When conducting a cost-benefit analysis for a public project, we must take into account these positive externalities and negative externalities; because private firms are not rewarded for the positive externalities they generate in the community, nor penalized for negative externalities (unless their actions cause a lawsuit), they do not consider them in their cost-benefit calculations. ## Defining Goals The first task of a cost-benefit analysis is to identify the goals of the project. The clearer the goals are, the easier it will be for an analyst to select the best course of action for achieving them. In chapter 5, I demonstrated the importance of defining the goals of projects. When evaluating the convention center proposal, we must determine whose goals to maximize, the city's, the county's, or the convention center authorities. We also need to know the goals of the project. We may identify as a goal an increase in tax revenue, urban renewal (as a result of the new convention center), or the enhanced image of the city. Elected representatives may set the goals, or we may determine the wishes of the stakeholders through surveys and focus groups. ## Identifying Alternatives The second step in preparing a cost-benefit analysis is to identify the alternatives. Typically, when there are not many identifiable alternatives, we evaluate a project against the option of doing nothing. However, if there are other feasible uses of the site, such as a new public library, a park, or a shopping mall, we should consider them in our analysis. ## Listing the Costs and Benefits of the Alternatives After selecting the alternatives, one should make as exhaustive a list as possible of the costs and benefits of the various alternatives. It may be useful to use the scheme shown in Figure 14.1 to delineate costs and benefits. They can be broadly classified into two categories: direct and indirect. The direct costs and benefits are those that are associated directly with the project itself. The indirect costs and benefits are those that affect the surrounding community but do not show up in the ledger of the project. In our convention center example, the revenues from the new facilities are considered to be the direct benefits, and the construction costs are the direct costs of the project. However, beyond these benefits and costs are the externalities of the proposed project. Thus, the generation of economic activities, such as increased business for the surrounding areas, is to be counted as an indirect benefit. The increased activities, however, may create factors that are detrimental to the city, such as increased traffic, pollution, or crime. These are the social or indirect costs of the project. Not all costs and benefits are measurable in monetary terms. Many costs and benefits are primarily qualitative in nature and as such cannot be readily expressed in dollars and cents. For example, a beautiful convention center can produce the intangible benefit of newfound pride in the city. Its construction can bring about a change in attitude in the city's citizens. In contrast, such negative conditions as environmental hazards brought about by the construction of the hotel and convention center would constitute part of its social costs. These are the intangible costs and benefits of a project. Analysts should be as comprehensive as possible in enumerating the costs and benefits of a public project, both intangible and tangible. I present a list of possible tangible and intangible costs and benefits in Table 14.1. ## Estimation and Valuation of Benefits and Costs One of the most difficult problems of conducting a cost-benefit analysis is that many of the benefits and costs may not be measurable in monetary terms. Another problem is that if they are to be accrued in the future, they must be estimated. It is not enough to state that the construction of a new convention center would increase revenue for the city. We need to come up with a reasonable estimate of how much additional revenue there would be. A convention center would draw groups of people from outside the region for business meetings. The proposed center would provide large rooms for various gatherings as well as hotel space for convention participants and their guests. Suppose that city hotels can accommodate 300,000 guests per year. The new convention center would increase this capacity to 350,000. The average cost of renting a room in the city is $50, but the increased supply of rooms would reduce this cost to $45 (see Figure 14.2). You may recall our discussion of consumer surplus from chapter 3. We illustrate this concept in Figure 14.2, in which the demand curve for hotel rooms (D) is shown as a downward-sloping heavy line. Suppose that before construction of the convention center, the supply curve was vertical line S, which gave us an equilibrium occupancy rate of 300,000 rooms at an average room cost per room of $50. At that point, the triangle abc represented the total benefit, or consumer surplus. As a result of the convention center, the supply increased to S', with an equilibrium room occupancy rate of 350,000 and an average rental rate of $45. As you can see, this increase in supply and reduction in price have allowed 50,000 extra guests to visit the city. This new situation has enlarged the area of total consumer surplus to the triangle aef. Of the total consumer surplus represented by triangle aef, the part described by triangle abc is not new. However, the convention center project has in fact created two new areas of consumer surplus, the rectangle cbdf and the triangle bde. Rectangle cbdf is an added surplus to consumers. That is, as a result of the reduced price of lodging in the city, the gain to consumers is a dollar-for-dollar loss to producers-the hotel and motel owners who had to lower their rates to fill the additional capacity. Rectangle cbdf illustrates what microeconomists call the pecuniary effect, which occurs when a change in the welfare of one group of individuals comes at the expense of some other group. Since the gains of the gainers exactly match the losses of the losers, for society as a whole, there is no change in welfare, unless we want to value one group's gain differently from the other group's loss. In that case, society must make a value judgment about the redistribution of income. We will discuss this issue later in the chapter. Returning to Figure 14.2, we see that the true additional consumer surplus is represented by triangle bde, the area of net social benefit. Using the Pythagorean theorem, we can calculate the area of triangle bde as $\\frac{Height \times Width}{2} = \\frac{(50-45) * 50,000}{2} = $125,000 Thus, the net gain in consumer surplus to the city is $125,000. Let us suppose that the project has been financed with municipal bonds. The bonds cost city taxpayers $75,000 per year. Therefore, the net gain to the city is $50,000, the difference between the gain in consumer surplus and the cost of servicing the loan ($125,000-$75,000 = $50,000). The preceding problem was an easy one to solve. In real life, calculations of costs and benefits are rarely as simple as those in our example. Table 14.1 gives you an idea of the different kinds of costs and benefits that have to be estimated and translated into monetary terms. One of the most important aspects of conducting a cost-benefit analysis is the valuation of intangibles, which are not bought and sold in the market. Yet for the sake of comparison, an analyst must ascribe monetary values to these items. Let us consider a few examples. ## Can We Put a Price Tag on the Intangibles of Life? How would you value the life of a human being, or the risk of injury that could result in physical disfigurement, or irreparable damage to the environment? Putting a value on such matters evokes controversy, yet it has become a routine matter. For example, when a jury hands down an award for pain and suffering, loss of face, disability, or loss of life, it is imputing a value to the most intangible aspects of life. The case of State of Alaska v. Exxon Corporation, resulting from an oil spill in the pristine and ecologically fragile Prince William Sound, is an example of putting a specific monetary value on the loss of habitat. It is indeed legitimate and necessary to consider ways of imputing monetary value to these intangibles when conducting a cost-benefit analysis. For instance, our large construction project carries the possibility of accidents resulting in severe injuries and even death. In such circumstances, it is essential to include the costs of such accidents. There are several methods for calculating these costs: the face value of life insurance, discounted future earnings, and required compensation. The face value of life insurance measures the monetary worth of one's life by the amount of one's life insurance policy. However, people buy insurance for many different reasons (for example, to some, it is a form of forced saving), and as such, their purchase may have little to do with their perception of the value of their own lives. In the discounted future earnings approach, often used in court cases, a person's life is worth the discounted value of future income. Future earnings are discounted because a dollar in the future is worth less in today's money. But since it evaluates life by one's earning potential, this approach undervalues the lives of those individuals whose talents are not sold in the market or who have stopped earning money. Therefore, it will fail to place much value on homemakers, retirees, and people with disabilities who cannot work. The market valuation of life does not address the crucial question of the worker's perception of the extent to which the added risk is compensated by the income differential. The answer would imply how much an individual believes his or her life is worth. There are people whose jobs carry almost no risk of death (for example, elementary school teachers, bank clerks), whereas the jobs of others carry an inordinate amount of risk (for example, firefighters, members of a bomb squad). Therefore, if we take an individual's education, age, experience, and other relevant factors of earning determination as constant, we will arrive at a larger payment for the risky jobs to compensate those workers for their added risk. The calculation of this margin of compensation for risk is called the required compensation principle. This is an important issue, and many economists have attempted to estimate the size of this margin. From their studies it seems that this value varies from a lower boundary of $2.5 million to $5 million in 1988 constant dollars. This value turns out to be, on average, five to ten times the value of life calculated under the discounted future earnings principle. As you can imagine, the imputing of a very high value for human life would make many projects less than economically viable. Therefore, you may think of these numbers as quite excessive, until you realize that an individual who took such a job might not have been totally aware of the risk involved. This is particularly true in the field of high technology; for example, workers whose job was sealing radiation chambers in the construction of atomic power plants complained that they were not adequately apprised of the risk by management. Even individuals who are aware of risk may not have the bargaining power to gain adequate monetary protection against the loss of life. Finally, this measure does not take into account the externalities of such a loss. The death of an individual can destroy a family and cause irreparable damage to the welfare of those who were dependent on this person for financial and emotional security. In light of these kinds of externalities, the U.S. military often exempted only sons from the draft or ensured that two brothers did not serve on the same ship. As technology improves we come to realize the deleterious effects of substances such as asbestos, whose risks most people were not aware of until many years later. In December 2000 the Environmental Protection Agency (EPA) announced its decision to clean up a dangerous chemical, PCB, from the waters of the upper Hudson River by dredging 2.65 million cubic yards of sediment along a forty-mile stretch. General Electric had discharged an estimated 1.1 million pounds of PCBs into the river before 1977 from capacitor plants in Fort Edward and Hudson Falls, about forty miles north of Albany. As a result, the EPA claimed that a 200-mile stretch of the river, down to New York City, was contaminated. The project would cost GE an estimated $460 million. However, the effects of PCBs had not been previously known, and therefore they were not banned before 1977. An example may help you understand the differences among the three methods of valuing a life. Suppose an innocent thirty-five-year-old schoolteacher is killed by the police during a high-speed car chase. Pursuing a suspect, a police cruiser goes through a red light and crashes into the teacher's car. The city is asked to pay compensation for this loss of life. Assume that the young man was earning $35,000 a year and had purchased a life insurance policy worth $150,000. According to the face value of life insurance method, the city will be liable for $150,000. However, if we assume that the young man would have lived for another thirty years and earned his current salary, his lifetime earnings discounted at a 6 percent rate turn out to be $201,022.19 (see the discussion of present value later in this chapter). In contrast, since he was not in a hazardous job, the required compensation method would estimate the value of his life at several million dollars. Since valuation of most of the intangibles in life is often highly subjective, the numbers can vary, climbing to absurd amounts. In 1991, two years after the Exxon tanker Valdez had spilled oil, causing extensive environmental damage in Alaska, Exxon agreed to settle criminal and civil complaints brought by Alaska and the federal government for $1.25 billion. Yet within a relatively short period, a study commissioned by the state and federal governments put the damage to the ecology at $15 billion. Indeed, as a society we may at times place extremely high prices on projects. If a project threatens a species with extinction or destroys a place of national interest or veneration, then we may assume that the cost of its destruction is too high for any conceivable monetary compensation. To prevent the extinction of spotted owls, the U.S. government declared a moratorium on logging in Oregon in 1991. ## How Can We Measure Future Loss or Gain? The prospect of future loss poses one of the most difficult obstacles to public projects. In popular terminology, this is the dreaded NIMBY (not in my backyard) factor, which community groups can effectively use to stop the construction of projects that have widespread indirect benefits but impose specific costs on a certain community. Thus, while the construction of a new airport may prove to be a boon to a region's economy, the question remains as to which community will have to live with the noise and increased traffic. Although small in proportion to the total gain to the region, the cost of increased noise can have disastrous effects on property values in nearby neighborhoods. An analyst is often faced with estimating a loss of property value that has not yet occurred. This estimate can typically be carried out using a causal regression model, discussed in Appendix B. We can form a regression model in which the price of property will be a function of changes (D) in Price = fA (Noise, Pollution, Travel time, Other factors) In this case, we hypothesize that the price of property will depend on the altered levels of noise (measured in decibels) and pollution (measured by various standardized emission units), which will have a negative effect on the price. A decrease in travel time (measured in terms of minutes to the airport), in contrast, is likely to increase the price. Other factors include the property's size, location, view, and so forth. For the model we can take these factors as given because the construction of the airport will not change them. Taking a cross-section of city properties, we can estimate the relevant coefficients for noise, pollution, and travel time. The coefficients for each term measure the impact on the dependent variable of a one-unit change in the independent variable. By multiplying the estimated coefficients with the expected change in that variable as a result of airport construction, we can estimate the total loss to the property. Suppose our regression coefficient for the effect of noise on the price of property turns out to be $5,000. This would mean that a one-unit increase in decibel level would reduce the price of a piece of property by $5,000. Suppose the environmental impact statement estimates that the new airport would add five decibels to the already existing noise level of a particular neighborhood. We can estimate the loss of property value for that neighborhood to be -$5,000 x5-$25,000. Other coefficients can be used in a similar manner to measure the total impact the new airport would have on property values. This kind of estimation poses many problems. Property owners are likely to dispute the results, because the estimates certainly will not cover all the costs associated with increased noise and other kinds of pollution (such as the effect on the physical and psychological health of the residents). It is interesting to note that frequently there exists an asymmetry in information between the gainers and losers of large public projects. In some instances a small group of potential losers tends to know and care about its losses a lot more than the larger group of potential gainers. In such cases, well-organized groups are often able to stop a project through political protests or obstructive legal actions. Conversely, in other cases, in which the potential for individual gains is strong, a handful of powerful interest groups are able to get approval for a project that may inflict costs on a wide segment of society. It should be obvious by now that inferring the value of nonmarketable items is not an easy task and often creates controversy. Yet as an analyst you may have to estimate the value of time saved as a result of a traffic diversion or the emotional cost of destroying a community to build a freeway through it. You have to approach such matters boldly but with caution. For example, a recent report suggested that the construction of high-occupancy vehicle lanes (the highway lanes set aside for vehicles carrying more than a certain number of passengers) on the perennially clogged Atlanta freeways reduced commuting time by fifteen minutes. You may be tempted to put a value on the time saved by multiplying it by the average wages of the commuters multiplied by their number, until you realize that the time saved is not likely to increase the commuters' working hours. Instead the fifteen minutes that would have been spent sitting in a traffic jam will now be spent pursuing enjoyable activities that carry no commercial value. You may instead consider the amount of gasoline saved by having to run the car engine for fifteen minutes and then calculate the money saved by commuters. In addition, you may look for the environmental benefits of reduced auto exhaust emissions. In the previous pages we discussed the problems of putting monetary values on nonmarketable items. If you find items that are simply not translatable in money, you may do well not to overstretch your imagination. As we have seen, unless you are careful, the valuing of nonmarketable items can quickly veer toward the ridiculous. Therefore, in such cases, an analyst should report accurately the intangible effects of the proposed project so that political decision makers can make informed decisions. ## Introduction of Time: Present Value Analysis In the preceding example of the convention center, we have a relatively simple choice to make on the basis of a onetime, lump-sum net benefit. However, the benefits and costs of most projects do not occur at one time. Instead, they come in over a period of time. This inclusion of time adds one more dimension to our problem. To begin with, a dollar received a number of years down the road may not be worth as much as a dollar already in our pockets. I am always reminded of a local television commercial for an annuity program. The announcer asks viewers to join a "millionaires' club." If a young adult saves a certain amount of money per month, at the end of nearly thirty-five years, this individual will receive $1 million from the annuity. Of course, during the dreamy announcement part of this commercial, the camera lens pans over all the trappings that are commonly associated with the lives of millionaires a fancy home, a limousine parked in the driveway, and so on. Ask yourself, though, would $1 million thirty-five years from now be worth $1 million in today's money? Obviously, the answer is that the two amounts of money are not equal. But the question of the difference between a dollar in my pocket today and one in the future can be answered only if we understand the process of discounting. To explain the process of discounting, I must first explain the process of compounding. Suppose I have invested $100 in a certificate of deposit, maturing at the end of the year, at a 10 percent interest rate. At the end of the year, I will receive $110. Thus, $100 invested at 10 percent interest for a year will yield $100×(1+0.1), or $110. The preceding formulation is perfectly obvious. If I keep this investment one more year at the compounding interest rate of 10 percent, at the end of the second year, I will get back not another $10, but $11, because I will earn interest on the previous year's interest. Therefore, at the end of the second year, I will receive $110 x (1+0.1), or $121. By inserting into the preceding equation the formula by which we obtained the result of $110, we get $100 x (1+0.1) x (1+0.1), or $121, which can be rewritten as $100 x (1+0.1)2=$121. If you are observant, you will note that keeping the money for two years requires us to multiply the original amount of money invested by 1 plus the interest rate (10 percent, or 0.1 in this case), the quantity raised to the power of 2. Therefore, if I had kept the money for three years, the exponent of the term within the parentheses would have to be raised to 3. By extending this logic, we can generalize by stating that the original investment of Po at a rate of interest of r percent for 1 years will give us P, amount of money: $\\ P = P_0 * (1+r)^n$ where P, is the principal amount at the end of the nth period, Po is the original principal amount at period 0, and r is the rate of interest. Using a calculator, we can determine that $155, invested at a 6.5 percent rate for seventeen years, would yield $155 x (1.065)17, or $452.14. This is the formula for the computation of compound interest; it tells you how much a dollar invested today at a certain interest rate would be worth in the future. In contrast, a dollar in the future may not be worth its full face value in today's currency; the forces of inflation, uncertainty, risk, and the plain fact that you would rather have your money now than at a later date may eat away much of its value. In other words, I may pose the question from the opposite direction: how much would a dollar be worth to you in the #th year in the future? In such a case, without compounding your initial investment, you would have to discount your future income. Let us take a specific example. Suppose I were to receive $100 a year from today. Since I will be getting the money in the future, if I use a discount rate (which measures the intensity with which I want my money in the present) of 10 percent, the $100 will be equal to $\\frac{100}{1.1} = $90.91 As in our previous example, if we are considering n years in the future, our future gain will have to be discounted by 1 plus the rate of discount, raised to the number of years we have to wait for the money. Thus, we can generalize the formula as $\\ P = \\frac{P_0}{(1+r)^n}$ Going back to the example of the millionaires' club, we can see that if the discounting factor is 10 percent, then $1 million received thirty-five years from now will be equal to $\\frac{ $1,000,000}{(1+0.1)^{35}} = $35,584.10$ Alas, in light of this analysis, it appears that the dream of $1 million has to be curtailed; the present value of $1 million received thirty-five years in the future is only about $35,000. With this amount, one certainly cannot expect all the trappings required for membership in a millionaires' club. This process is called present value analysis, by which we calculate the current or present value of a dollar to be gained in the future. We can use this formula to calculate the present value of a stream of benefits and costs to arrive at the net present value of a project. Notice that the larger the discount rate, the lower the present value of future dollars. If we were to discount $1 million at a 15 percent rate, we would arrive at the paltry sum of $7,508.89 for the same time period. You can see that as we increase the discount rate, the future dollars look smaller and smaller. As a result, gains to be made in the future look increasingly less attractive; similarly, the prospect of losses in the distant future looks less ominous. Therefore, the discount rate captures the strength of the desire to have money now as opposed to sometime in the future. This desire is called time preference. The relationship between time preference and the discounted future value of a dollar is shown in Figure 14.3. In this figure, we have plotted this year's earnings on the horizontal axis and next year's earnings on the vertical axis. If we do not have a time preference, we will be indifferent between $10 today and $10 next year. The line connecting $10 on the two axes shows an indifference map (we are indifferent between, or prefer equally, any two points along this line) with no time preference. However, if we discount the future earnings at a 10 percent rate, then to be on the same utility plane, we must earn $11 next year. If we have an even higher time preference, equal to a 15 percent rate of discount, unless we earn $11.50 in the following year, we would prefer to have $10 today. Let us consider a concrete example. Suppose we are evaluating two projects with the streams of benefits and costs shown in Table 14.2. From this table, it is clear that project A provides us with double the net benefit ($80) provided by project B ($40). Should we automatically choose project A over project B? If we were to jump to this conclusion, we would be remiss, since we would fail to consider that the increased net benefits of project A come at a later stage in the project's life. Our choice between the two projects will depend on the strength of our time preference the willingness to wait for future returns. Therefore, to compare the two projects on level ground, we must translate these future streams of benefits and costs into their present values. The formula is written as $PV = Y \\sum_{t=0}^n\\frac{(B_t-C_t)}{(1+r)^{t}}$ where PV is the present value of the project, B, is the benefit, and C, is the cost at time t. We can expand this expression and write $PV = \\frac{(B_0-C_0)}{(1+r)^0} + \\frac{(B_1-C_1)}{(1+r)^1} + \\frac{(B_2-C_2)}{(1+r)^2} + .... + \\frac{(B_n-C_n)}{(1+r)^n}$ Since any number raised to the power 0 is equal to 1, the expression can be written as $PV = (B_0-C_0) + \\frac{(B_1-C_1)}{(1+r)} + \\frac{(B_2-C_2)}{(1+r)^2} + .... + \\frac{(B_n-C_n)}{(1+r)^n}$ You may notice that the 0th year's net benefits are not discounted. This makes eminent sense, because the current year's dollar is equal to its face value and hence does not need to be discounted. If we are willing to wait (that is, it does not matter to us whether we receive our payments today or tomorrow), our time preference is said to be nil. In such a situation, we discount the future stream of net benefits with a zero discount rate and therefore do not discount at all. Thus, if we do not have any time preference, the net benefits for projects A and B are $80 and $40. In contrast, suppose we do have a definite time preference, and we want to evaluate the future stream of net benefits for the two projects at a 10 percent discount rate. In such a case, we can write the present values of project A (PVA) and project B (PVB), discounted at 10 percent, as $PVA =(0-30) + \\frac{0-15}{(1+0.1)} + \\frac{0-10}{(1+0.1)^2} + \\frac{10-5}{(1+0.1)^3} + \\frac{20-5}{(1+0.1)^4} + \\frac{120-5}{(1+0.1)^5}$ = -30-13.64+8.26+3.76+10.25+71.41-33.52+(1+0.1) and $PVB =(15-5) + \\frac{15-5}{(1+0.1)} + \\frac{15-5}{(1+0.1)^2} +\\frac{15-10}{(1+0.1)^3} + \\frac{15-10}{(1+0.1)^4} + \\frac{15-15}{(1+0.1)^5}$ = 10+9.09+8.26+3.76+3.41+0+(1+0.1) From the preceding calculation, we can see that discounted at a 10 percent rate, project A is less preferable than project B because it carries a lower present value. We can also see that the present values of the two projects will depend on the rate of discount, which determines their relative desirability. I have plotted the present values as functions of the rate of discount in Figure 14.4. From this figure, it can be seen that the two projects become equally desirable at a discount rate slightly less than 10 percent. For discount rates below 10 percent, project A is preferable to project B, but the relative desirability changes for discount rates of 10 percent and above. This change reflects how project A's benefits come at the end of its life. In contrast, project B yields positive net benefits from its inception. Therefore, if we can afford to wait (and have a small discount rate), we would prefer project A. However, if we are in a hurry to get back the returns on investment (and therefore have a stronger time preference), we should choose project B. Finally, if the present value is negative (as project A is for a discount rate close to 24 percent), we should reject the project. ## Choice of Time Horizon The choice of an appropriate time horizon is of crucial importance for a cost-benefit analysis. The relative desirability of a project is intrinsically connected to when it ends. The length of a time period affects the desirability of long- and short-term projects. Many projects for drug interdiction are designed for short-term results. In these projects, drug use is considered primarily as a law-and-order problem, and efforts are made to lower the supply of illicit drugs by police action. In contrast, programs treating drug abuse are seen as long-term public health initiatives that require expenditures on education, rehabilitation, and employment opportunities. These demand-side efforts (trying to reduce the drug demand) offer longer term solutions. So unless they are given longer time horizons, their impacts on drug use will not be fully realized. Project desirability dependent on the choice of time horizon is shown in Figure 14.5. You can see that before the critical point in time, T,,, the short-term project (Project I) yields higher net present value. However, beyond this point, the long-term project (Project II) becomes