File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 4/18 distance from the center are all revenues exhausted? Locating outside of that distance would produce negative revenue, an economic consequence that prevents a user from locating there. Notice that, given the inputs, the wheat farmer can afford to locate farther away. Stated differently, the pea farmer MUST locate closer in. Wheat farmer Pea farmer R ¼ pa À w Àtam ¼ 0 R ¼ 10 à 10 À 50 À :5 à 10m ¼ 0 R ¼ 15 à 10 À 75 À 1 à 10m ¼ 0 m ¼ 10 ¼ Maximum distance m ¼ 7:5 ¼ Maximum distance By assuming an arbitrary value for m and solving for t, we can determine the slope of each party’s bid rent curve. Notice that the pea farmer’s slope is greater. What does this mean to the way both parties will bid for land closer to the center of the city? Wheat farmer Pea farmer R ¼ 10 à 10 À 50 À t à 10 à 10 ¼ 0 R ¼ 10 à 10 À 50 À t à 10 à 7:5 ¼ 0 t ¼ :5 ¼ Slope of bid rent curve t ¼ 1 ¼ Slope of bid rent curve Placing them both on the same plot is useful at this stage, noting that the point where the curves cross is the point on the land where the bids are equal. Prior to that point, the pea farmer is willing to pay the most for the land; beyond that point, the wheat farmer bids more than the pea farmer. Setting the two rent equations equal to each other, inserting the fixed inputs, and solving for m tells us the location on the land of the crossover point. Figure 1-2 shows the point on the land where both parties bid an equal rent and the amount of that rent. 0246810 Distance 20 40 60 80 100 Rent Wheat Farmer 01234567 Distance 20 40 60 80 100 Rent Pea Farmer FIGURE 1-1 Wheat and pea farmers’ bid rent curve. 4 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 5/18 10 à 10 À 50 À :5 à 10m ¼ 15 à 10 À 75 À 1 à 10mm¼ 5 R ¼ 10 à 10 À 50 À :5 à 10 à 5 R ¼ 25 A little experimentation with different values for the fixed inputs leaves one with the insight that (in our stylized example) nothing matters but transportation cost. Mathematically, this can be verified by taking the first derivative of R with respect to m, with the quantity produced standardized to 1. dR dm ¼Àt From this, we again see that in our simple model rent is a negative function of transportation cost. EXAMPLE #2—SEVERAL COMPETING USERS IN DIFFERENT INDUSTRIES Building on this, let us model an entire city with multiple users, each having a different transportation cost. We assume that user classes locate in concentric rings radiating out from the center of the city. The innermost is the central business core of commercial users (com), followed by an interior light industrial ring (indI), then residential (res), a second industrial ring of heavy manufacturers (indII), and finally, agricultural users (agr). Note that transportation costs per unit decrease in the outward direction with each user, resulting in a flatter slope for each curve as we progress outward. The combination of all users on a single graph leads to what is known as the bid 5 Distance 25 Rent bids are equal pea wheat FIGURE 1-2 Rent at the point where bids are equal. Why Location Matters 5 File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 6/18 rent surface or rent gradient. Note in Figure 1-3 that the largest land mass is taken by residential. Why might that be so? Following our wheat/pea farmer procedure, we can solve for each cross- over point. Table 1-1 reflects these values. We can link the crossover points to the change in use on the land by connecting the points to the perimeters of the appropriate circle (Figure 1-4). A different perspective is provided by placing them all on the same plane (Figure 1-5). The amount of land devoted to each use is dependent upon the size of the circles conscribing it. We can compute the total area of each concentric ring, noting that in this example land mass devoted to each use generally increases as we move away from the center (Table 1-2). 1 10 20 30 40 50 Distance 140 104 90 30 10 Rent All Users Agricultural Industrial II Residential Industrial I Commercial FIGURE 1-3 Bid rent curves for a city with different land uses. TABLE 1-1 Cross points and rent where land use changes Distance Rent com 0 140 com-indI 3. 104 indI-res 5. 90 res-indII 25. 30 indII-agr 35. 10 1 It is, of course, possible to make a simple supply and demand argument for lower rent for sectors in which more acreage is available. 6 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 7/18 IS THE BID RENT CURVE LINEAR? Joining the crossover points creates a bid rent surface for the entire city (Figure 1-6). Note that for the aggregate of these user classes, the bid rent surface is non-linear. It is clear from the plot in Figure 1-6 that multiple classes of users with a sequence of crossover points produce a bid rent surface for the entire city that 525 35 Distanc e 140 104 90 30 10 Rent FIGURE 1-5 Land use mapped on a single plane. 35 25 35 Distanc e 140 104 90 30 10 Rent FIGURE 1-4 Change in land use on a map of the city. Why Location Matters 7 File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 8/18 is not strictly linear, but appears linear on a piecewise basis. The aggregation of various uses, each with a different transportation cost (and, therefore, a different slope), creates this shape. From this we may speculate that different individual users within any one sector each may also have slightly different transportation costs, and the aggregate of the linear bid rent curves of these different users produces a curve for any specific use that is also not a straight line (Figure 1-7). Under these conditions one might reasonably assume that the functional form of the bid rent curve for all individual users would be R ¼ e Àax , where x is distance from the center of the city, the exponent a is a decay rate that may be observed in the market as one moves away from the center, and e is the base of the natural logarithm. EMPIRICAL VERIFICATION Suppose we collect data on actual rent paid by users along a line in a certain direction moving away from the center of the city (or any high rent point), TABLE 1-2 Land Mass in Square Miles Allocated to Different Uses com area 28.27 indI area 50.27 res area 1884.96 indII area 1884.96 agr area 2513.27 52535 Distanc e 140 104 90 30 10 Rent FIGURE 1-6 Bid rent surface for the entire city. 8 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 9/18 such as reflected in Table 1-3. The first element in each pair is the distance from the center, the second is the rent paid at that point, and the third is the natural log of the rent, a useful conversion for further analysis. A plot of the distance and rent data in Figure 1-8 shows a nearly linear decay in rent as distance increases. We are interested in the relationship between distance and rent. A common method for investigating the relationship between two variables is linear regression analysis. For this, we use the natural log of rent as the dependent variable. Figure 1-9 shows a plot of the data in Table 1-3. Not surprisingly, it appears linear because taking the natural log of a curved function has the effect of ‘‘linearizing’’ the function. We then fit the regression model (Equation 1-3): Log R½¼Log ke Àxd Âà ¼ Log k½Àxd ð1-3Þ where k is the regression constant, x is the slope, and d is distance from the center. The intercept and slope terms are shown in the regression equation: Log R½¼6:71003 À0:0155191x (A complete regression analysis appears among the electronic files for this chapter.) Exponentiating 2 both sides of the regression equation produces the conclusion that one may estimate rent based on a fixed intercept multiplied 1234567 Distance 0.2 0.4 0.6 0.8 1 Rent R= e − ax FIGURE 1-7 A well-behaved, smooth bid rent curve. 2 There is some doubt that ‘‘exponentiating’’ is a word. The Oxford English Dictionary does not carry ‘‘exponent’’ as a verb. However, we need a word for the cumbersome statement ‘‘using each side of the entire equation, each, as an exponent for the base of the natural log ’’ For this we press ‘‘to exponentiate’’ into service. Why Location Matters 9 File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 10/18 TABLE 1-3 Rent Data Distance Rent LN (rent) 0 821 6.71052 1 808 6.69456 2 795 6.67834 3 783 6.66313 4 771 6.64769 5 759 6.632 6 748 6.6174 7 736 6.60123 8 725 6.58617 9 714 6.57088 10 703 6.55536 11 692 6.53959 12 681 6.52356 13 671 6.50877 14 660 6.49224 15 650 6.47697 16 640 6.46147 17 630 6.44572 18 621 6.43133 19 611 6.4151 20 602 6.40026 21 592 6.38351 5101520 Distanc e 650 700 750 800 Rent FIGURE 1-8 Plot of rent vs. distance. 10 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 11/18 times the base of the natural logarithm taken to an exponent that is composed of the product of the decay rate (as a negative number) and the distance. R ¼ 820:597e À0:0155191x Hence, if one is at the center, where distance is zero (x ¼ 0), the rent is the intercept. R ¼ 820:597 when x ¼ 0 On the other hand, if one is ten miles from the center (x ¼ 10), the rent is R ¼ 702:638 when x ¼ 10 Recall Figure 1-7 and its pronounced convexity to the origin. This noticeable convexity is because the decay rate (.5) was fairly large. Figure 1-10 reflects the decay rate derived from our regression. As the decay rate is quite small and the range of distance is short, the curve appears linear. The same curve is more pronounced over a longer distance (Figure 1-11). So we see that while the curve is a function of the decay rate, for small decay rates its curvature is only apparent over longer distances. 0 5 10 15 20 Distance 6.4 6.45 6.5 6.55 6.6 6.65 6.7 Log [Rent ] FIGURE 1-9 Plot of natural log of rent vs. distance. Why Location Matters 11 File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 12/18 AN ECONOMIC TOPOGRAPHICAL MAP The world is not flat and neither are its land economics. The story becomes more realistic when one considers the theory in three dimensions. After all, there are an infinite number of directions away from any particular high rent location. One would expect the decay rate to vary in different directions. A stylized version of this uses the trigonometry employed in topography. 3 1234567 Distance 760 780 800 820 Rent R=820.597e −ax FIGURE 1-10 Bid rent curve suggested by regression analysis. 50 100 150 200 Distance 200 400 600 800 Rent R=820.597 e −ax Distance 0–200 FIGURE 1-11 Regression bid rent curve over a longer distance. 3 A more complete elaboration of this process with interactive features may be found at www.mathestate.com. 12 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch001.3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22.12.2004/3:21pm Page: 13/18 The so-called ‘‘path of progress’’ is the direction in which the decline in rent is the slowest, thus the decay rate is the slowest because higher rent is persistent in that direction. In that direction the decline is relatively flat. The opposite case is that of the steepest decay rate. As rents decline fastest, the decay rate is larger in the direction people are not locating. The three-dimensional parametric plots in Figure 1-12 show the economic topography where a ¼ .1 (Figure 1-12a) or a ¼ .02 (Figure 1-12b) to simulate the way rent changes as one travels around the land. RELAXING THE ASSUMPTIONS All models are only approximations of reality. Unfortunately, we attempt better approximations at the expense of generality. Nonetheless, the exercise of testing the model under more realistic assumptions is useful. One way to move closer to what we actually observe is to relax some of the assumptions. The first might be the idea that the urban business environ- ment is monocentric. In Figure 1-13a we see the potential for two high rent areas in a given market. This representation suggests that the secondary point of high activity might be somewhat flat at the top, representing an econo- mic oasis of activity where rents are generally high in a small area. This is the relaxation of the assumption that the greatest activity takes place at the absolute center. Rotating Figure 1-13a to see the rear of it in Figure 1-13b reveals an area of depressed rent. Clearly, there are as many portrayals of this condition as there are different cities on earth. Figure 1-13 could also depict the relaxation of the no transaction costs assumption. Zoning, a constraint on freedom of choice in how one uses one’s land, is essentially a transaction cost. If government imposes zoning that prohibits land use in a certain area, the consequence can be higher rent for that use in the area where that use is permitted. Another explanation for a plot like Figure 1-13 might be non-uniform transportation costs in one direction caused by natural barriers such as a river or mountain that must be crossed. One might also see an impact on the rent gradient as transportation costs differ in directions served by mass transit. Whether these graphical depictions represent reality is an interesting debate. One can challenge the notion that the market is symmetrical around a point, calling into question whether the most intense activity takes place on a single spot. Clearly, over time ‘‘clusters’’ of similar businesses gather in certain areas. Particular areas become ‘‘attractors’’ for certain kinds of industries. The list of exceptions to the basic theory is long. The primary value of the sort of analysis undertaken in this chapter is to provide a logical framework for location decisions and guide the thoughtful land consumer to a rational Why Location Matters 13 [...]... the CEO of an REIT or real estate fund to visit a new city and investigate real estate opportunities there, an acquisition team may first consult data before landing in a market where local players dominate transactions A WINDOW TO THE FUTURE Table 1-3 shows rent data collected along a line stretching away from a high rent location Real estate data always has some location attribute In the past that attribute... location decision through data REFERENCES 1 Alonzo, W Location and Land Use Cambridge, MA: Harvard University Press 2 Geltner, D M., & Miller, N G Commercial Real Estate Analysis and Investments Upper Saddle River, NJ: Prentice Hall 3 Kline, M., Mathematics for the Non-Mathematician New York: Dover Publications, Inc 4 von Thunen, J H (1966) The Isolated State New York: Pergamon Press 5 www.mathestate.com... how land may be used The unanswered question is: Shall the choice be made by the landowner or the community in which the land is located? Tariffs and trade agreements govern how commerce crosses international boundaries Laws prohibiting collusive and coercive activities govern domestic trade at a national level Our interest lies in local government For the private real estate investor, local land use... finer and the picture more complete There are a number of excellent data gatherers and providers; some are independent firms, and some are in-house for major real estate companies It is to these industry support groups we direct a final appeal As real estate data becomes more plentiful, observations of rent across the land will become more compact, filling in the grids necessary to describe the actual... of data often In the case of market rents, one must be mindful of the fact that no dataset supplants a careful market survey in the local area of a target acquisition However, as real estate markets become more efficient and data is more robust, the sort of models developed here will assist buyers in ‘‘getting up to speed’’ in an unfamiliar market Having been instructed by the CEO of an REIT or real estate. .. operating as a governmental jurisdiction  Build and test a model that chooses the proper level of regulation that optimizes community satisfaction  Explore the consequences of over-regulation and its affect on other municipal services 19 20 Private Real Estate Investment  Review a case study using actual data in a real setting to illustrate how land users may deal with local government in the face... address Later, a zip code was added Recently, longitude and latitude points have been included Each of these steps moves us closer to a time when the theoretical graphs shown in this chapter can be displayed as actual data points and the economic topographical map will represent a real world situation Data represents reality, and there will be times when reality conflicts with theory In Figure 1-14a we see... investor, local land use regulation is a significant aspect of the decision making process In urban settings it is no overstatement to say that real estate investment success is, in large part, dependent on an understanding of the regulatory environment in which the local real estate market exists Whether zoning or rent control, real estate investors ignore local politics at their peril Several general... down There are hundreds, if not thousands, of examples from the residential field to draw from Rather than take one of those and its somewhat straightforward Land Use Regulation 23 analysis, the setting for the analysis here comes from the commercial area This presents additional challenges that deserve attention and at the same time illustrates how a somewhat esoteric land use conflict can be modeled THE... maximizes the function because the Log is monotonic and concave for all positive log bases 4 This ignores the interplay between taxes and the level of sales which is not our story 28 Private Real Estate Investment equation ab g À ag ¼0 þ A À A0 A ð2-8Þ Transferring the second term on the left of Equation (2-8) to the right-hand side produces Equation (2-9) and sets marginal cost equal to marginal benefit . iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22 . 12. 2004/3 :21 pm Page: 15/18 25 0 25 North–South (a) 25 0 25 East–West 0 .25 0.5 0.75 Rent 0 25 25 0 25 0 .25 0.5 0.75 Rent 25 North–South (b) East–West FIGURE. Date/Time: 22 . 12. 2004/3 :21 pm Page: 18/18 File: {Elsevier}Brown/Revises-II/3d/Brown-ch0 02. 3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22 . 12. 2004/3 :21 pm Page: 19/38 CHAPTER 2 Land Use. Economics, 3, 1–44. 20 Private Real Estate Investment File: {Elsevier}Brown/Revises-II/3d/Brown-ch0 02. 3d Creator: iruchan/cipl-un1-3b2-1.unit1.cepha.net Date/Time: 22 . 12. 2004/3 :21 pm Page: 21 /38 economic