Urban Land Markets: Notes on Chap. 6 of O’Sullivan Urban Economics Text 1. Land-Market Equilibrium: Office Bid Rent with Factor Substitution In what follows here, we will elaborate on the analysis of land-market equilibrium with factor substitution, as outlined in the text. Starting with a model in the text, the key objective in this section is to derive the model’s endogenous variables (examples include land rent at each location, building height at each location) from the exogenous variables (examples include revenue received by office firms and “travel cost” – a variable capturing the additional cost faced by firms operating at a distance from the median location1). Before reading on, it is suggested that students read pages 104-112 and 127129 in the text (6th edition (2007) – earlier editions should not be used). Those pages should be kept handy since they will be referred to frequently. In what follows here, hyphenated numbers for tables or figures (for example Table 6-7) refer to tables or diagrams in the text. Table or figure numbers without a hyphen (for example Table 1) refer to tables or diagrams included in these notes (the diagrams in these notes appear at the end). We begin with Table 6-7 on page 111 of the text. We will re-order the table so that exogenous variables (given from outside the model) are grouped in columns 1-4; the endogenous variables (determined in the model) will be grouped in columns 5-9. The resulting table appears here as Table 1. Table 1 also includes values for variables on rows omitted in Table 6-7: the rows added here show all data for locations at 0, 4 and 6 blocks from the median location, while travel cost numbers have been added for locations at 2 and 3 blocks. The blank spaces for variables at 2 and 3 blocks are not filled in since the omitted data will not be needed in the discussion below.
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It is at the median location – that is, at the centre of the office market – that information services such as legal or financial advice are assumed to be provided at lowest cost. However, firms at less central locations can remain competitive because land rent will adjust as required to keep them competitive. This is the key point made in the text pages cited above and in these notes.
Table 1: Expanded / Re-ordered Version of Table 6-7 in Text 1.Distance in blocks (x) 0 1 2 3 4 5 6
2.Travel cost / day $0 $36 $74 $114 $156 $200 $246
3.Total revenue / day $500 $500 $500 $500 $500 $500 $500
4.Other non-land cost /day $150 $150 $150 $150 $150 $150 $150
5.Product’n site (hectares) 0.02 0.04
0.125 0.25 0.50
6.Bldg. height (floors) 50 25
8 4 2
7.Total rent paid /day $70 $64
$54 $50 $29
8.Capital cost of bldg./day $280 $250
$140 $100 $75
9.Bid rent per ha / day $3500 $1600
$432 $200 $58
The numbers in Table 1 could be interpreted as applying to seven firms, operating at seven different locations as indicated in Column 1. Alternatively, we could interpret Table 1 as applying to a single firm that moves from one location to another. Either way, the firm or firms will compete with large numbers of identical firms. The conditions for perfect competition are assumed to be met here, including zero economic profit in equilibrium (firms make only the profit required to keep them in business). Land owners are also assumed to meet conditions for perfect competition: in particular, each land owner controls only a small fraction of the total land supply in the market. Land owners at each location are also assumed to have zero supply elasticity since (1) the total land supply at each location is fixed; and (2) nothing land owners can do with their land is preferable to renting it to the highest bidder – even if the highest bidder offers a very low rent. For example, if there are 20 hectares of land at x = 1 block (a ring-shaped area of land at approximately 1 block radius centred on the median location), all 20 hectares will be supplied on a vertical supply curve. The actions of landowners in the x = 1 submarket (or any other submarket) will have no impact on land rent ($1600 in the case of the x = 1 sub-market). We will return to the supply side of the land market later in these notes. Exogenous Variables Since travel cost is an exogenous variable, any numbers could be assumed for Column 2. The assumption in the text is that for information industries (the industries occupying offices in this model), travel cost increases at an increasing rate moving away from the median location. In the example of Table 1, a move from 0 blocks (the median location itself) to one block away increases travel cost by $36 - $0 = $36 per day; a move from 1 block to 2 blocks increases travel cost by $74 - $36 = $38 per day, and so on.
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The text provides some of the numbers for travel cost, and in these notes we have added numbers for other locations. These added travel cost numbers are generally consistent with numbers in the text. The only inconsistency involves a reference on page 107 (2nd last line), where it says that moving from 4 blocks to 5 blocks increases travel costs by $50 – i.e. implicitly the travel cost number at x = 4 blocks must be $150 since travel cost at x = 5 blocks is $200. We have substituted travel cost = $156 at 4 blocks in these notes, since that number is an easier fit with other rows once they are filled in.2 In the table above, all variables with a time dimension have a per-day time period noted. The text indicates on page 106 that an office firm’s revenue ($500) is per day, so all other time-specific variables must also be measured on a per-day basis. (The time period could alternatively be a month, a year or any other interval, as long as it is applied consistently to all variables.) Not shown in the table is another exogenous variable assumed to apply equally to each firm. Each firm is assumed to require10,000 sq. metres (1 hectare) of office floor area to produce its output. The firm could rent that floor area from a separate firm providing office space for rent; however, in the model here and in the text the firm provides its own office space, renting land and capital for that purpose. It then uses its office space to produce output. We are not told what output the firm produces, but it is unnecessary to know this. Implicitly, we have an exogenous output level per firm and an exogenous output price (as with the bicycle industry discussed earlier in Chapter 6 – five bicycles a day per firm selling for $50 each). All we actually need with the office firm is the firm’s total revenue per day (output level per day times price), and we are given this revenue exogenously at $500 per day (Column 3). The firm’s product might be legal advice or financial advice – some type of “information” output that is more costly to produce (as a function of “travel cost”) if a firm locates away from the median location. Column 4 shows payment to inputs other than capital and land: for example, labour cost would be included here along with the profit required by the firm to stay in business. The assumption here is that the inputs included in Col. 4 cannot be substituted for land and / or capital: their cost is a constant $150 / day given exogenously.
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The point made at the end of page 107 remains valid whether travel cost at 4 blocks = $156 or $150. If $156 is used, rent at 4 blocks would be $376 instead of $400. The change in rent from 4 blocks to 5 would be $376 - $200 = $176 instead of $400 - $200 = $200 as in the text. Either way, the four-to-five block rent change ($200 or $176) would exceed the zero-to-one block rent change ($144). In other words, with no factor substitution and travel cost increasing at an increasing rate, the bid rent function in Figure 6-3 is “concave” – the point made on p. 107.
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Endogenous Variables Columns 5 and 6 are effectively the same variable. Column 5 shows the land area in hectares rented by a firm at each location, and Column 6 shows that land area divided into the firm’s fixed floor area (1 hectare). Thus Column 6 is the ratio of floor area to land area. This ratio can be referred to as “building height” (in floors) on the assumption that each floor covers all of the land (site). For example on the second row from the bottom of Table 1, the firm rents one quarter hectare of land. If the ground floor covers the entire site (no outdoor parking, no landscaping), and if every floor has the same area as the ground floor, the building will be four floors high once its total floor area is one hectare: four floors at one quarter hectare each. In the real world, buildings are not usually constructed to cover the entire site. With a floor area / land area ratio of 4.0, there are many alternatives to a fourfloor building: for example an eight-floor building could be constructed covering half the site, or a sixteen-floor building covering one quarter of the site. In each of these examples there is a one hectare building on one quarter hectare of land – but the land area directly under the building (known as the building “floor plate” in the development industry) is less than the total land area rented by the firm. The land not directly under the building is still required or the firm would not rent it – for example outdoor parking or landscaping may be needed to attract customers. The points in the preceding paragraph being noted, for convenience we will continue using “building height” (in floors) when referring to the floor area / land area ratio.3 Column 7 (Total rent paid) is land rent (Column 9) times the amount of land the firm rents (Column 5). For example on the second row from the bottom, a firm rents one quarter hectare of land as its production site, at a rent of $200 / hectare / day. Multiplying these numbers together gives us $50 per day. This “total rent paid” amount will also be referred to below as a firm’s “land cost”: what it pays for land per day. “Land cost” is also used as a label for this variable in Table 6-6 in the text. In Table 6-4 (p. 106) land cost is also referred to as willingness to pay (WTP) for land. When all costs except land cost are subtracted from revenue, the residual “leftover” amount is (by definition) available for land cost and economic profit. However, if any of this amount actually does go to economic profit, it will attract 3
But note: “building height” can only represent the floor area / land area ratio if that ratio > 1.0. That is because the minimum building height is one floor high. For example with a floor area / land area ratio = 0.5 (1 hectare floor area on 2 hectares of land), we would not have a building that is one-half floor high; we would either have a building one floor high covering half the site or some taller building covering less than half the site.
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“wannabe” firms. The wannabe firms will need land to get into production; however, with all land already occupied, there will be excess demand for land and rent levels will be bid up until economic profit is zero. Thus the leftover amount noted above must all go to land cost in equilibrium, and none of it to economic profit. “Willingness to pay” for land means the maximum amount a firm is willing to pay – what it is willing to pay if it has to (see text p. 102). When the land market is in equilibrium, the firm must indeed pay this amount. Column 8 shows the firm’s capital input. While the capital used in buildings has many components (bricks, concrete, glass, steel, elevators, air conditioners etc.), all capital is lumped together in the model so that it can fit onto one axis in an isoquant diagram like those in the text on page 128. Thus the capital input is measured in dollars rather than physical units: it is the dollar amount required per day to rent the capital used in the firm’s building. With capital measured this way, its “price” is $1 per unit per day by definition. The text (on p. 109) defines the term “building cost” as the total of land cost and capital cost. We will use this term frequently in what follows. While building cost does not have its own column in Table 1, building cost on any row is always the total of Column 7 and Column 8. Deriving endogenous variables from the exogenous variables Once the production function for office space is added to the model, we will see that given the exogenous variables, there will be a unique value for each endogenous variable in Table 1. The production function is the relationship between output (office floor space in this case) and inputs (capital and land). Geometrically, it is represented by an isoquant map. Figure 6A-2 shows the isoquant S = 1 in both panels of the figure. That isoquant shows all combinations of land and capital that can be used to produce one hectare of office floor space. Firms at different locations will be operating at different points on the S = 1 isoquant. One point at which some firms will operate is point m in Figure 6A-2 A. Point m has coordinates L = 0.25 hectares and C = $100. The slope of the isoquant at point m (marginal rate of technical substitution) is - $200. All of these numbers come from the production function for office space – actually one particular point on that function. Figure 1 (below) reproduces Figure 6A-2 A with the tangent line at point m extended until it meets the axes. C* is the intercept point on the C axis. Three equilibrium conditions can be analyzed using Figure 1: (1) each firm occupies one hectare of office space;
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(2) Each firm produces its hectare of office space using the least-cost combination of land and capital; (3) firms earn zero economic profit at each location. Condition (1) is met as long as firms are on the S = 1 isoquant. Condition (2) is met when firms choose the least-cost input combination on that isoquant; this means operating at a point of tangency between the isoquant and an isocost line. Thus the tangent line touching point m will be the isocost line on which firms at that location are operating, assuming fulfillment of conditions (1) and (2). When the vertical-axis input is measured in dollars, the slope of an isocost line = the price of the input on the horizontal axis (this input price will be land rent in the present case). This point is noted in the text (first paragraph following Figure 6A2). Another way of illustrating the point is as follows. The isocost line’s slope is the same as the isoquant slope at the point of tangency, and the production function has already given us the value for this slope: - $200. If we start at point C* (with zero land cost) and move along the isocost line to a point (not shown in Fig. 1 or the text diagrams) where land input = 0.1 hectares, the - $200 slope tells us that $20 less per day will be spent on capital. Along an isocost line, whatever is not spent on capital is spent on land, so $20 / day is spent on the 0.1 hectare of land. If one tenth of a hectare costs $20 per day, one hectare must cost $200 / day, so $200 / hectare / day is the price of land (i.e. land rent). Thus when conditions (1) and (2) are met, firms operating at point m will face a land rent = $200 / hectare / day. Given this rent value, we know that these firms’ building cost (defined above as the total of land cost plus capital cost) will be $150 / day. We can see this in Figure 6A-2A: The firm rents 0.25 hectares of land at $200 / hectare / day (so land cost = $50) plus $100 / day in capital cost. Since all points on an isocost line have the same building cost, the intercept point C* must also have $150 building cost. A point such as C* (all capital and no land) would obviously never be chosen by a firm, but the C-axis intercept point of an isocost line is analytically useful as we will see below. We now turn to the third equilibrium condition noted above: zero economic profit. For this condition to be met, we need Total Revenue = Total Cost. In other words for each firm, Total Revenue must equal travel cost + other nonland cost + capital cost + land cost. Rearranging: Total Revenue – travel cost – other non-land cost = capital cost + land cost.
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The left side of this equation is made up of exogenous variables in Table 1. Since total revenue and other non-land cost are constants (respectively $500 / day and $150 / day) at all locations, the left side of the equation becomes: $350 – travel cost. On the right side, capital cost + land cost = building cost (by definition). Thus $350 – travel cost = building cost. To know equilibrium building cost at any given location (i.e. on any row of Table 1), this equation tells us that we only need to know travel cost at that location – given other exogenous variables that total to a constant number, and given fulfillment of the zero-economic-profit equilibrium condition. Given this relationship, we could take two different routes to deriving values for the endogenous variables in Table 1. With the first route, we would turn first to the isoquant diagrams (Fig. 6A-2, Fig. 1), and examine point after point on the S = 1 isoquant. At each point, we would have coordinates for L and C, as well as a slope value and thus a value for rent. As we have already done for point m, we can proceed to calculate building cost associated with each isoquant point. It is then a straightforward task to locate the row in Table 1 corresponding to any given isoquant point. Simply look for the row on which $350 – travel cost equals building cost as just determined at the given isoquant point, and that will be the appropriate row. Then fill in all of the endogenous variables on that row. For example, with point m we determined in the isoquant diagram that building cost = $150 for firms operating at that point. In Table 1, the row on which $350 – travel cost = $150 is the row for x = 5 blocks, where travel cost is $200. We can then fill in the endogenous variables on that row from our findings in the isoquant diagram: 0.25 hectares for Column 5, 4 floors for Column 6, $50 for Column 7, $100 for Column 8 and $200 for Column 9. Via the route just outlined, we can match up points on the S = 1 isoquant with specific locations in the city. With the second route, we start with Table 1. On each row we have a location and a travel cost number. By subtracting travel cost from $350 we determine what building cost must be for firms operating at that location. Again using the x = 5 blocks row as an example, firms operating at that location must have building cost = $150, given that travel cost = $200. We then go to Figure 1 and mark the point on the C axis equal to building cost as just determined ($150 = C* in our current example). Then draw the tangent line from the marked C-axis point to the S = 1 isoquant. There will be only one tangent line to the isoquant that can be drawn from a point on the C axis, and only one point of tangency. Once we
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have the point of tangency, and the isocost / isoquant slope at that point, we have all of the endogenous variables for any given row in Table 1. Likewise, values for endogenous variables could be derived at x = 1 block. In Table 1 for that location, it can be seen that $350 – travel cost = $350 - $36 = $314 = building cost. While not shown in Figure 1, building cost = $314 could be marked on the C axis, with a tangent line being drawn from that point to the S = 1 isoquant. That tangent line, which is the isocost line for firms at x = 1, is actually shown in Fig. 6A-2B. As drawn there it stops short of its intercept on the C axis, but if extended it would hit the C axis at C = $314. With L = 0.04 hectares, building height is more than 6 times higher at x = 1 than it was at x = 5 (25 floors rather than 4). The highest buildings of all will be at x = 0, where travel cost is zero. The tangent line to S = 1 would have its intercept on the C axis at C = $350. This tangent is not shown in the diagram here or in the text, but estimated values for endogenous variables at x = 0 are shown in Table 1. As estimated, building height would be 50 floors and land rent would be $3500 / hectare / day. To obtain exact numbers, we would need to know the isoquant’s slope at its tangency point with a tangent line drawn from C = $350, as well as the coordinates of that tangency point.4 This completes our discussion of the second route to matching up rows in Table 1 with isoquant points in Figure 1. The Bid rent Function If the equilibrium rents in Column 9 of Table 1 are graphed as a function of x, a bid rent function (also called a bid rent curve) is obtained. This function, abbreviated here as BRF, is usually drawn as a continuous curve – an approximation of observable rent / distance relationships. To determine what is observable, one must recognize that firms rent 2dimensional sites (for example 100 metres by 100 metres which would comprise 10,000 sq. metres = 1 hectare); a single per-hectare (or per-square-metre) rent number is observable for the entire site, not different rents for different parts of the site. It is relevant here that travel cost, which drives rent in the model, will presumably be at one level for a firm – not higher for operations on one part of its site and lower for operations on another part. Starting from a given firm’s site, the next site inward toward the median location can be expected to have a slightly higher rent per hectare, with the next site outward having a slightly lower rent per hectare. Unless the sites are vanishingly small, it is not possible to observe continuously variable rent. Instead, observable rent will step upward or downward at discrete intervals – for example at 10-metre or 100-metre intervals. 4
Values for endogenous variables at x = 4 and x = 6, which are in Table 1 and which will be discussed later, have also been estimated.
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All the same, a continuously variable BRF is a reasonable approximation of observable BRF’s. Alternatively, a step-function approximation could be taken directly from Table 1, with rent steps at one-block intervals. As the text demonstrates (pp. 106-7), the BRF is “concave” in the absence of factor substitution and if travel cost increases at an increasing rate. Concavity in this context refers to a function that becomes less steep approaching the median location. Fixed (rather than substitutable) inputs such as those in Table 6-4 – which result there in buildings four floors high at all locations – might be the result of zoning rules imposed by a city government. A zoning rule limiting the floor area / land area ratio to a maximum such as 4.0 is actually quite typical of zoning rules in many cities. If firms would choose higher ratios without the zoning rule, the output / land ratio will be fixed by zoning. Even though the firm’s isoquant looks like the isoquant in Figure 6A-2, it will be unable to move along that isoquant in a northwesterly direction past a certain point: the point at which the floor area / land area ratio reaches the legal maximum. Least-cost tangency with an isocost line will then not be possible, because it will not be legal. Firms will be stuck at points like point m in Figure 6A-2B, which in fact does correspond to a floor area / land area ratio of 4.0. Returning to the case of unconstrained factor substitution, consider land rents at 4, 5 and 6 blocks in Table 1. Moving from 5 blocks to 6 blocks, rent changes by $142 / hectare / day. Moving from 5 blocks to 4 blocks, rent changes by $232 / hectare / day. Thus we are on a convex BRF like the upper function in Fig. 6-5: its slope increases as firms move toward the median location. If factor substitution is ruled out, and if we assume that inputs are fixed at their x = 5 values (land area fixed at 0.25 hectares, capital fixed at $100 day), the change in land rent from 5 blocks to 6 blocks would be equal to the difference in travel cost ($46) divided by 0.25.5 That is, rent would fall by $184 / hectare / day. (Thus R (6) would be only $16 / hectare / day, given that R (5) = $200.) Moving from 5 blocks to 4 blocks, again with factor inputs fixed at their x = 5 values, the change in rent is again equal to the difference in travel cost (this time $44) divided by 0.25. That is, rent would increase by $176 (from $200 to $376) moving from 5 blocks to 4 blocks. Without factor substitution, then, we are on a concave BRF like the lower function in Fig. 6-5: its slope decreases as firms move toward the median location. 5
With factor substitution, Table 1 indicates that the floor area / land area ratio would fall from 4.0 at 5 blocks to 2.0 at 6 blocks. A zoning rule setting 4.0 as a maximum would not prevent factor substitution in the case of a firm moving from 5 blocks to 6 blocks, since 2.0 is less than the maximum. To analyze a move from 5 blocks to 6 without factor substitution, it could be assumed that the zoning rule sets 4.0 as both the maximum and the minimum for the floor area / land area ratio. With that rule, 4.0 would be the only permitted ratio of floor area to land area.
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Returning again to the case of unconstrained factor substitution, it is possible – at least theoretically – that the potential for substitution is very limited. Perhaps regardless of how high or low land rent might be, the lowest floor area / land area ratio firms would choose could be 3.9 (for example), and the highest 4.1. This would contrast with the wide range in Table 1, where the floor area / land area ratio goes all the way from 2 up to 50. With very limited potential for factor substitution, the BRF might not be convex but rather just less concave than the no-substitution BRF – since one case is an approximation of the other.6 Very limited factor substitution, as in the example cited in the preceding paragraph, is unlikely in the real world – unless it were to result from zoning rules. In downtown Toronto, where zoning rules have recently been loosened up, one recently approved (but not yet built) project has a floor area / land area ratio approximating 48.0. That is a high number by any city’s standards. Since the proposed building will cover less than the total site area, it will be more than 48 floors high (somewhere in the 70+ range). This project illustrates the reality of factor substitution in building construction: large amounts of floor area can be piled onto a small area of land (in this case a parking lot with capacity for 35 cars, with room for about 10 more if there were not a small presentation-centre building on the site). Concluding Comments on the Land Market with Factor Substitution When factor substitution is ruled out, it is clear why high land rents are observable at certain locations in the city. These are locations where firms discover a cost advantage, and to get that advantage – under competitive market conditions – they have to pay for it in land rent. The cost advantage is the obvious cause of relatively high land rents. The greater the cost advantage over less attractive locations, the greater the rent premium. In referring to cost advantage, we of course exclude land cost since it is land cost that adjusts upward where other costs are low. When factor substitution is not ruled out (i.e. it is allowed), the picture becomes more complex. As a starting point, the cost advantage at certain locations 6
An alternative possibility is that travel cost increases linearly with distance, rather than increasing at an increasing rate. In this case the BRF is linear without factor substitution and must be convex with factor substitution. To see this, repeat the 4,5 and 6 block example above on the assumption that travel cost changes by $44 for both moves (from 6 blocks to 5 and from 5 blocks to 4). Without factor substitution, rent changes by $44 / 0.25 = $176 for both moves. Rents would be $376 at 4 blocks, $200 at 5 blocks and $24 at 6 blocks. These rents lie along a straight line. If factor substitution is then allowed, economic profit will emerge at the 4-block and 6-block locations (as firms reduce cost by substituting factors). At the 5-block location, we are assuming that factor inputs are the least-cost inputs whether factor substitution is allowed or not, so there is no economic profit there. With rents originally on a straight line, and then increasing above that line at both ends (as will be required to eliminate the economic profit at both ends), a convex function is formed.
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causes higher rents at those locations, as in the no-substitution case. But firms then use less land and more capital, temporarily earning economic profit as a result. That economic profit leads to even higher rent, as wannabe firms compete for already occupied land. That higher rent in turn leads to another round of factor substitution, and so on. Fortunately, this process does not go on forever but converges to equilibrium values for rent and factor inputs. We have already seen earlier in these notes how these equilibrium rents and factor inputs can be determined, for any given values of the exogenous variables. Observers looking at high land rent and high buildings at certain city locations often question the direction of causality. Are the high buildings the result of high land rent or is it the other way around? In the adjustment process described two paragraphs above, both directions were operative. However, in equilibrium land rent and building height alike are endogenous variables – determined by exogenous variables in the model and the production function for buildings. The cost advantage attached to these prime locations remains a significant influence, as it was in the case of no factor substitution. Factor substitution adds another influence. In the example of Fig. 6-5, it can be seen that land rent at x = 1 is $1600 / hectare / day given the ability to construct a 25-floor building at that location, but only $856 if limited to 4 floors. The cost advantage of being at 1 block (as opposed to farther out) is the same either way, so it is factor substitution that accounts for the difference. At least in this model, we will never see tall buildings constructed on cheap land. This point is illustrated in Table 6-6 in the text. It is possible to build a 25-floor building on the lowest cost land in that table ($40 / hectare / day rent), but the resulting building cost is far higher (at $251.60 / day) than the building cost associated with lower buildings on larger sites ($110 / day or $90 / day). A firm that did construct a 25-floor building at that location would soon go out of business, since it would have to compete with firms constructing low-rise buildings with only $90 / day in building cost. Likewise Table 6-6 illustrates the rule that low buildings will not be built on expensive land. When considering high land rents – typically near the city centre – the question may arise: does the limited supply of land at these central locations have anything to do with the high rents there? The geometry of circles does imply that a limited land supply is available closer to the centre – for example in a ring approximating a one-block radius compared with a ring farther out (say at 5 blocks). However, in this model the land supply does not affect rent. This point can be illustrated with Figure 2 in these notes. In Figure 2, it is assumed that the land supply at x = 1 block is 20 hectares. This might correspond to the number of hectares in a ring-shaped area of land (referred to in
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class as a bagel) having inner radius 0.5 blocks and outer radius 1.5 blocks – centred on the median location. The zero-elastic (vertical) supply curve for land in this sub-market is shown in the figure, positioned at L = 20 hectares. Even at low rents, the entire 20 hectares will be available for rent, on the assumption that there is no opportunity cost to offering the land for rent; that is, there is no personal use to which the land could be put for the owners’ benefit. Even at high rents, no additional land can be supplied within this fixed geographic area. The demand curve in the figure is horizontal (infinitely elastic) at the rent corresponding to zero economic profit ($1600 / hectare / day). At any higher rent, the quantity of land demanded in the x = 1 sub-market is zero since firms would earn negative economic profit at any rent higher than the zero-economicprofit rent. With a quantity of land demanded = zero, there would be excess supply and rent would fall to $1600. If hypothetically the city introduced rent control for land and set a minimum for this sub-market higher than $1600, all land in the sub-market would go vacant and remain vacant. Land owners would have no tenants but could not legally reduce rent as required to attract them. Firms would simply skip over this sub-market while continuing to produce elsewhere (assuming the rent control does not apply there as well). At any lower rent than $1600, there would be an indefinitely large excess demand due to positive economic profit (exactly how much excess demand does not matter). This excess demand is the result of wannabe firms attracted by economic profit; the wannabe firms find the entire 20 hectares on which this economic profit is earned already occupied, and have to offer higher rent to get land away from existing firms. Thus in any case where rent in the x = 1 submarket is below $1600, it will move up to $1600 which is the only possible equilibrium rent. To summarize on the demand curve for land, it shows the quantity of land that firms demand at each possible rent, given the zero-economic-profit requirement. It coincides with the rent axis at rents above $1600 – in that range the quantity demanded is zero. At rents below $1600, there is an indefinitely large number of hectares demanded – a large enough number that it is off the figure with the exact number irrelevant. At rent = $1600 the quantity demanded is sufficient to take all 20 hectares in the supply, and more hectares would be demanded if available at that rent; however, no one is willing to bid a higher rent to get land in the x = 1 sub-market when rent = $1600. To confirm that supply does not affect rent, imagine that the city buys half the land in the x = 1 submarket and uses it for a park. The supply curve, while remaining vertical, would move over so that it is positioned at 10 hectares. However, the equilibrium rent would still be $1600, given the horizontal demand curve for land. The quantity of output in the submarket would be reduced by half, but there would be no increase in rent.
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Later in the text (p. 345), consideration is given to a model with downwardsloping demand for land. In this model, the land supply is not divided into submarkets with different locational characteristics, and the output price is endogenous. Notes elaborating on this model will be posted on the course web site in July or August / 06. 2. Other Points re Chapter 6 in Text All of the land markets considered in Chapter 6 have land owned by one set of individuals while firms renting the land are owned by other individuals. However, in the real world, firms typically own the land on which they operate. In many cases where there is tenant occupancy, one firm owns both the land and building while another firm rents space in the building. However, cases where the land owner has no other involvement with the property do occur in the real world. Much of the farmland in Scotland, for example, is not owned by farmers working the land. The Scottish Parliament has been attempting to make it easier for farmers to buy the land they now rent. In the suburban fringe area of Toronto, many of the remaining farms are rented. The owners are firms intending eventually to develop housing, with the farm use being temporary. Even in downtown Toronto, there are properties where the landowner does not own the building. Buildings are typically built on rented land only if there is a long-term lease (for example 99 years). The long-term lease is designed to protect the building owner from increases in land rent that would make the investment unprofitable. Long-term leases are not required in the models of Chapter 6, because we assume that capital is mobile in those models. If land rent were to increase beyond what a firm owner is willing to pay, that owner would stop renting the capital in the building – at which point the capital owner would move it somewhere else. Again in the real world, buildings rarely move from one location to another because of the cost required to do this. (However, as was pointed out in class there are cases where buildings have been moved in Toronto, one of them – the University of Toronto Department of Physical Geography – four floors high.) Owner occupancy of land can readily be incorporated into the models of Chapter 6. Essentially, a firm that decides to own land rather than renting it pays its rent up front. To buy land, whether to rent out to someone else or use for one’s own firm, a bidder has to pay the market value. This market value is the present value of expected future land rents. Buyers and sellers have to estimate what future land rents will be – either on the basis of observed land rents or on the basis of “leftover amounts” (revenue that can be earned on the land minus the non-land costs required to earn that revenue).
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A discount rate is then applied to the expected future stream of land rents to calculate the market value. For purposes of this course, we will assume that market value is the present value of a perpetual and constant rental stream paid at annual intervals beginning in one year. Thus if R is the annual land rent and i is the discount rate, the market value = R / i. For example if R = $1000 per hectare annually and the discount rate is 0.05 (5 percent) then the market value per hectare = $1000 / 0.05 = $20,000. Assuming that future rents are as expected when land was purchased, owner occupant firms and tenant firms will compete on equal terms. The owner occupant paid $20,000 to own a hectare of land, paying the rent up front as a present value so that annual rent does not have to be paid. The tenant firm can take that same $20,000, invest it at 5 percent, and use the $1000 annual receipts to pay the annual land rent. In both cases they earn zero economic profit. Finally we turn to Fig. 6-10 in the text. This diagrams shows how the land market allocates land among four sectors: office, manufacturing, residential and agriculture. The horizontal axis is omni-directional, meaning that rents will vary with x as we move out from the centre in any direction: east, south, west, north or any points in between. This omni-directionality can be visualized by looking at Fig. 6-9 C on page 121 (drawn with 3-dimensional imaging software). In moving from Fig. 6-9 C to Fig. 6-10, the linear road crossing the city is eliminated. Also, in Fig. 6-10 housing occupies a ring-shaped area of land inside the bowl shown in Fig. 6-9 C; this ring-shaped area is between the rim of the bowl (representing the manufacturing rent peak at the circular beltway road) and the central rent cone (representing office bid rent peaking at the city centre). Fig. 6-9 C shows only office and manufacturing bid rent, not residential – also the case with other panels of Fig. 69. Also added in Fig. 6-10 is a residential ring on the outer rent slope outside the beltway. Finally, agricultural rent at the edge of the city is not shown in Fig. 6-9. Residential bid rent at each location is at the level required for zero economic profit earned by housing firms. Housing firms face an output price P (x) which varies by location. The relationship between P (x) and distance from the city centre x is called the housing price function. The housing price (monthly rental per sq. foot of residential floor area) goes up as commuting distance goes down, as is required to keep households on the same indifference curve wherever they live. Which indifference curve that will be depends on the utility households can get by moving to other cities. The utility level of households is exogenous in this model: costless moves in and out of town are assumed to be possible (“open-city model� assumptions). In most of the residential land market, P (x) slopes downward to the right, indicating longer commutes at housing locations farther from the city centre. However, in Fig. 6-10 there is a residential area (between x2 and x3) inhabited by
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outcommuters. These are commuters who travel away from the city centre on their way to work, towards it on their way home. They work for manufacturing firms close to the beltway but live inside the beltway. In this residential area the housing price function P (x) slopes upward to the right, so the residential bid rent function shown in this part of Fig. 6-10 also slopes upward to the right. Earlier in Chapter 6, when residential, office and manufacturing land markets were considered separately, x was measured in blocks for manufacturing and office firms but in miles for housing firms. Now that their bid rents are in the same diagram, a common unit of distance x must be used. Fig. 6-10 does not indicate what that unit is, but we will assume it is miles. [“Blocks” are not intended to be a standard unit of linear distance, but have been used as a convenient measure of linear distance applicable to business sector land markets. Metres (or yards) are too short a distance unit for measurable changes in rent to be evident, while kilometres (or miles) would represent such a large distance unit that land rent might decrease from its maximum to zero before reaching x = 1 unit. There is a linear distance unit called the hectometre (100 metres) which could approximate a block, but few people are familiar with it.] Likewise on the vertical axis of Fig. 6-10 a standardized rent unit is required. For office and manufacturing firms, per-hectare land rents have been specified on a daily basis while residential per-hectare land rents have been specified on a monthly basis. We will assume that all rents are now monthly. Note that outcommuters will never meet incommuters while travelling to or from work in this model. All commuting is by car, and all residential land is inhabited either by outcommuters or incommuters – not both. We assume that commuting travel inside business districts – either within the central business district (CBD) or within the manufacturing district – is short enough to ignore. Thus firms located farther from residents will not have to pay a higher wage than the wage paid by firms closer to residents. It is only within residential districts that commuting distances are significant. As Fig. 6-10 is drawn, x2 is approximately midway between x1 and x3. That implies equal commuting distance to either the CBD or manufacturing district from housing at at the point where the downward sloping and upward sloping residential BRF’s intersect. At that point of intersection, land rent is equal for housing occupied by outcommuters and incommuters, so housing prices must be the same for both. (x2 is the only residential location where both incommuters and outcommuters can live.) Given these conditions, office and manufacturing wages must be equal. If these wages were not equal, either incommuters or outcommuters would obtain higher utility since everything else affecting utility (commuting cost and housing price) would be equal. We are assuming here that all households, wherever they work,
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are identical and must obtain the same utility in equilibrium. There is no specialized labour that works only in office or manufacturing sectors. In equilibrium no one will be able to increase utility either by changing residential location or by changing jobs. However, we can consider the possibility that office firms have excess demand for labour, given that their workers all live in the residential district bounded by x1 and by x2. In that case, the office wage will go up, causing office bid rent to shift down and the “CBD workers” bid rent to shift up. This will move the boundary between CBD workers and manufacturing workers (x2) to the right, while the x1 boundary shifts left. With the land area accommodating CBD workers having increased, the office labour shortage is corrected. However, there will be impacts on the manufacturing sector as a result of the reduction in land area inhabited by outcommuting manufacturing workers, with further impacts back onto the office sector. We will not, at least in this course explore these labour market impacts further – limiting ourselves instead to the observation that office and manufacturing wages will generally need to be independently variable. In the appendix to Chapter 7, we will analyze labour market equilibrium, but only in the context of a one-industry labour market rather than a two-industry labour market. Another point to be noted with Fig. 6-10 is that land rent at x3 must be equal to land rent at x5. At both of these locations, there are households with zero commuting distance to their manufacturing jobs. Given that both groups of households are manufacturing workers, their wage levels are equal. Thus to obtain the same utility their housing prices must be the same, implying equal residential land rents. Since residential land rents are equal at these two locations, and since residential land rents intersect manufacturing land rents at these two locations, manufacturing land rents must also be equal at x3 and x5. Since manufacturing land rents depend on freight cost, which in turn depends on distance from the beltway, we know that points x3 and x5 are equidistant from the beltway. Note also that the radius of the beltway x4 is obviously exogenous to the model. It depends on where the city decided to build it. If the radius is small enough, it may not be possible for residents to live inside the beltway. This would occur if the office BRF intersected the upward-sloping manufacturing BRF at a rent high enough to eliminate housing at that intersection. That is, a housing firm at his intersection, even with output price P(x) as high as possible based on a zero commute, could not pay the land rent set by office and manufacturing firms at that intersection. If it did, it would have negative economic profit. In that case, the model would revert to “monocentric” status.
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In a monocentric model, as will be noted in the appendix to Chapter 7, there is only one employment district. It will be located at the centre of the city. In Fig. 610 as drawn, there are two separate employment districts separated by residents. This “multicentric� configuration is more representative of contemporary urban land markets than the monocentric configuration – which was driven by historical transportation systems (manufacturing output shipped out of town through a central rail terminal rather than by road).
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