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Lecture 3

ECO333H1 Lecture 3: Factor Substitution

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Peter Tomlinson

Urban Land Markets with Factor Substitution January 2014 In what follows here, we will elaborate on the analysis of land-market equilibrium with factor substitution, as outlined in O’Sullivan’s text. Starting with a model in Chapter 6 of the text, we derive the model’s endogenous variables from its exogenous variables. Factor substitution refers to firms’ ability to substitute capital for land at locations where land rent is relatively high – rather than using fixed land and capital inputs regardless of land rent. The factor-substitution model in the text considers firms that occupy office buildings. However, a factor substitution model can apply to any type of firm. In these notes, factor substitution can be switched on or off so that its impact on endogenous variables can be isolated. A zoning law controlling firms’ land input is the tool used to switch factor substitution on or off. Firms in the textbook model sell information products to their customers. Customers might be paying for financial advice or legal advice – two examples of information products. Firms need data – such as data on financial markets or data on matters involved in contract law – in order to develop saleable information products. The location variable in the model is distance from the office land market’s median location (the location at the centre of a circular land market). We assume that the cost of obtaining required data increases with distance from the median location. Required data is obtained at meetings, and “travel cost” (an exogenous variable) measures the cost of attending these meetings. 1 We begin with Table 6-7 in 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 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 t he 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 data at these locations will not be needed in the discussion below. The numbers in Table 1 could be interpreted as applying to seven firms at the seven locations indicated in Column 1. Alternatively, we could interpret Table 1 1In what follows here, hyphenated numbers for tables or figures (for example Table 6-7) refer to tables or figures in the text. These numbers are the same in both the 6th and 7th editions of the text. Table or figure numbers without a hyphen (for example Table 1) refer to tables or figures included in these notes. These notes cannot be read without the text book at hand, since there will be numerous references to tables and diagrams in the text that are not reproduced in the notes. 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). The profit required to stay in business is included in the “other non- land cost” variable. Table 1: Expanded / Re-ordered Version of Table 6-7 in Text 1. Distance 2. Travel 3. Total 4. Other 5. Product’n 6. Bldg. 7. Total 8. Capital9. Bid in blocks cost revenue non-land site height rent paid cost of rent per ( x ) cost (hectares) (floors) bldg. ha 0 $0 $500 $150 0.02 50 $70 $280 $3500 1 $36 $500 $150 0.04 25 $64 $250 $1600 2 $74 $500 $150 3 $114 $500 $150 4 $156 $500 $150 0.125 8 $54 $140 $432 5 $200 $500 $150 0.25 4 $50 $100 $200 6 $246 $500 $150 0.50 2 $29 $75 $58 Exogenous Variables Column 1 in Table 1 shows distance from the median location, measured in “blocks”. A block is treated as a standard distance measure, comparable to kilometres or miles. This variable measures distance from the median location in any direction – whether east, south, west, north (or any other compass direction) makes no difference. All variables in the model will have equal values at any given x, regardless of direction. Travel cost is shown in Column 2. Since travel cost is an exogenous variable, any numbers could be assumed for Column 2. The assumption in the text is that 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. The text provides some of the numbers for travel cost, and in Table 1 numbers have been filled in for other locations. All variables with a time dimension (Columns 2, 3, 4, 7, 8, 9) are specified on a per-day basis. The text indicates that an office firm’s revenue ($500) is per day, so all other time-specific variables must be measured on a per-day basis. (The time period could alternatively be a month, a year or any other time interval, as long as it is specified consistently for all variables.) The product that is sold for $500 per day is not identified in the text, but identifying it is unnecessary. Whatever the product is, both the output level produced by a firm and the output price are exogenous; price times quantity of output is fixed at $500 per day per firm. Not shown in Table 1 is another exogenous variable assumed to apply equally to each firm. Each firm is assumed to require an office building with 10,000 sq. metres (1 hectare) of floor area to produce its output. The firm could rent that 2 floor area from another firm producing office space for rent. However, in the model here and in the text the firm constructs its own office building, renting land and capital for that purpose. It then uses its office building to produce the information product that sells for $500 per day. Column 4 (“other non-land cost”) shows a firm’s cost of inputs other than travel cost, capital cost and land cost. ONLC includ2s labour cost and the profit required by the firm to stay in business. The assumption is that inputs included in ONLC cannot be substituted for land or capital: their cost is a constant $150 / day given exogenously. Endogenous Variables Columns 5 and 6 in Table 1 are effectively the same variable. Column 5 shows the land input in hectares rented by a firm at each location (x-value), and Column 6 shows that same land input divided into the firm’s fixed floor area (1 hectare). Thus Column 6, the ratio of floor area to land area, is the reciprocal of Column 5. For example firms at x = 5 rent 0.25 hectares of land on which they construct 1 hectare of floor area, so the floor area / land area ratio is 1 / 0.25 = 4. In these notes we will refer to “land input” rather the equivalent term “production site” used in the text. We assume that a firm’s land input is limited to land under the building (no additional land for landscaping, parking, etc.). We also assume that every floor has the same area as the ground floor, so a building on 0.25 hectares of land has four floors of 0.25 hectares each. Given these assumptions, the floor area / land area ratio is also “building height” measured in numbers of floors. 3 The variable in Column 7 is a firm’s land cost. Land cost is a firm’s land input in hectares (Column 5) times land rent per hectare (Column 9). For example at x = 5, a firm rents one quarter hectare of land at a rent of $200 / hectare. Multiplying these numbers together yields land cost of $50. The text uses an equivalent term for land cost: “total rent paid”. In equilibrium, land cost can also be referred to as “willingness to pay” for land (“WTP” – see text Table 6-4). Willingness to pay for land means the maximum amount a firm is willing to pay – paying more would reduce profit below the level required to keep the firm in business. 2 Other non-land cost may also include travel cost at x = 0. In Table 1 and in the text, travel cost at x = 0 is set equal to zero. However, that is consistent with firms at x = 0 actually incurring travel cost. To achieve that consistency, define travel cost as the difference between travel cost at each x and travel cost at x = 0. Travel cost at x = 0 is then included with the other constants in ONLC. 3Note: “building height” is equivalent to the floor area / land area ratio only if that ratio ≥ 1.0. The minimum building height is obviously one floor. For example with a floor area / land area ratio = 0.5 (1 hectare floor area on 2 hectares of land), we would either have a building one floor high covering half the site or some taller building covering less than half the site. A building 0.5 floors high covering the entire site is not an option. 3 When costs in Columns 2, 4 and 8 are subtracted from revenue (Column 3), the residual amount is available for land cost and (potentially) for economic profit. However, if any of this residual actually does go to economic profit, “wannabe” firms will be attracted. These wannabe firms will need land to get into production. With all land already occupied, there will be excess demand for land so rent levels will be bid up until economic profit is zero. At that point, firms will pay land cost = the maximum they are willing to pay for land. In these notes we will refer to “land cost” rather than the equivalent terms “total rent paid” or “willingness to pay” – thus assuming land rent and land input are at their equilibrium values. Column 8 shows the firm’s capital input. While the capital used in buildings has multiple components (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 / isocost diagram – for example Figure 1 below, or Figure 6A-2 in the text. Thus the capital input is measured in dollars rather than physical units: it is the dollar amount per day required to rent the capital used in the firm’s building. With capital measured in dollar units, its unit price is $1 by definition. The text defines the term “building cost” as the total of a firm’s capital cost plus land cost. However, in these notes we will refer to “capital cost plus land cost” rather than using a separate label for this total. Deriving endogenous variables from the exogenous variables In this section we will use Figure 1 to derive endogenous variables graphically from exogenous variables. Figure 1 is modeled on Figure 6A-2 in the text, with a key modification. The modification is to extend the tangent isocost line shown in 6A-2 to the vertical axis, with intercept point Cishown in Figure 1. Figure 1 then resembles Figure 6A-1 in the text. Although Figure 6A-1 illustrates a different model – a model of residential land market equilibrium – the relationship between exogenous and endogenous variables is similar in both Figures 6A-1 and 6A-2. The vertical axis in both Figures 6A-1 and Figure 6A-2 is measured in dollar units. As was noted above, Figure 6A-2’s vertical axis is a firm’s capital input. Fig. 6A-1’s vertical axis is a resident’s monthly spending on non-housing goods. Given an axis measured in dollars, other variables measured in dollars can be displayed on that axis. In Figure 6A-1, values for monthly income net of commuting cost can be displayed. Point A is income net of commuting cost for a resident with a 10 mile commute, assuming commuting cost t is $50 / month / mile. All residents are assumed to have $2000 monthly income, so a resident with (10 X $50) = $500 / month commuting cost will have $1500 / month available to spend on housing and other goods. Because residents are assumed to spend all of their income, $1500 / month will in fact be spent on housing and other goods where commuting distance x is 10 miles. 4 Figure 1 Capital Input ($) 150 Ci 100 m S =1 0.25 Land Input (Hectares) Still referring to Figure 6A-1 in the text, point A = $1500 is one point on the equilibrium budget line for a resident at x = 10 miles. One other point is required to determine the position of that line. Tangency point i on the equilibrium indifference curve U is1the required point. No line through A other than Ai can be the equilibrium budget line. A line steeper than Ai will not allow residents to reach the equilibrium indifference curve U . 1 The slope of a budget line is equal to the price of housing, so a slope that is too 4 steep means the price of housing is temporarily above its equilibrium level. If the price of housing at x = 10 miles will not allow residents to obtain the equilibrium utility level U1, residents will not pay that price. They will offer a lower price and move to another location if housing firms do not accept the offer. Thus a budget line through A that is too steep will rotate counter-clockwise on pivot point A until it coincides with the tangent line Ai. A line through A flatter than Ai will allow residents to reach indifference curves above U , If residents can reach Indifference curves above U , their utility would 1 1 be temporarily above the equilibrium level. Other residents would be attracted to the location x = 10 miles, causing the budget line to rotate clockwise on pivot point A, until it coincides with Ai. After that, residents can no longer reach higher . indifference curves than the equilibrium indifference curve U 1 Thus the tangent line Ai is the resident’s equilibrium budget line. As was noted above, Its slope is the price of housing at x = 10 miles: $0.30 per square foot per 4A budget line’s slope (in absolute value) is the ratio of two prices: the price of the good on the horizontal axis divided by the price of the good on the vertical axis. The latter price equals $1, so the budget-line slope is $p / $1 = $p: the equilibrium housing price at x = 10 miles. 5 month as shown in Figure 6A-1. The coordinates of the tangency point i are G (non housing goods consumption) = $1200 / month; and housing consumption h = 1000 square feet. The variables in this paragraph are equilibrium values for the endogenous variables at x = 10 miles. The exogenous variables from which these values are derived are income (w) = $2000 per month; commuting cost (t) = $50 per mile per month; and utility level U = U . 1 Returning to the isoquant / isocost diagram (Figure 1), derivation of equilibrium endogenous variables at any given location (for example x = 5 blocks) would proceed via these steps: Since the zero economic profit condition requires equality of total revenue and total cost for each firm, total revenue minus other non-land cost minus travel cost = (capital cost plus land cost). Using the relevant values from Table 1, we have $500 (total revenue) minus $150 (other non-land cost) minus $200 (travel cost) = $150 = total of capital cost plus land cost. $150 is then the C-axis intercept of the equilibrium isocost line at x = 5 blocks. This point is represented in Figure 1 as C = $i50. Firms at x = 5 must spend $150 per day on capital and land to build an office building with 1 hectare of floor area while earning zero economic profit. C is one point on the equilibrium isocost line facing firms at x = 5. One other point is required to determine the position of the equilibrium isocost line. The required point is tangency point m, on the isoquant for an office building with 1 hectare floor area (labeled the S = 1 isoquant). This proposition is demonstrated via steps similar to those already taken with the residential model discussed above. The slope of an isocost line (in absolute value) is land rent [R (x)] divided by the price of capital ($1 by definition). For example if an isocost line has slope = minus $200, land rent is $200 / hectare / day. If the x = 5 isocost line through C iis temporarily steeper than Cm, firmi will not be able to reach the isoquant required for a 1-hectare office building. No firm will pay a land rent so high that it cannot earn the profit required to stay in business while also occupying the required 1 hectare of floor area. In that case land owners will have to reduce rents. That will make the isocost line through C iless steep, rotating counter-clockwise on pivot point C until it coincides with Cm. i i If the x = 5 isocost line through C is timporarily flatter than Cm, firms iill be able to reach the S = 1 isoquant while paying less than $150 / day, thus earning economic profit. In this case wannabe firms will be attracted by the economic profit. Requiring land to get into production, and with all land already occupied, the wannabe firms will bid up land rent. This will rotate the isocost line clockwise 5An isocost line through A that is temporarily flatter ihan Cm will cut through the S = 1 isoquant. Points on the isoquant under the isocost line will be on lower isocost lines, so firms can (temporarily) pay for their required 1-hectare office building and have money left over. 6 on the pivot point C uniil it coincides with Cm. i Thus the tangent line Cm is ihe equilibrium isocost line for firms at x = 5. Its slope (in absolute value) is $200 / hectare. Thus $200 per hectare is the equilibrium land rent entry in Column 9 of Table 1. The C-axis coordinate of point m is the Column 8 entry ($100). The L-axis coordinate of point m is the Column 5 entry (0.25 hectares). Multiplying the 0.25 hectare land input by the $200 / hectare land rent yields the $50 land cost entry (column 7). The 0.25 hectare land input divided into one hectare yields the building height entry in Column 6: building height = 1 / 0.25 = 4 floors. Similar derivations of endogenous variables can be carried out at other locations in the model. At any given value of x, subtract other non-land cost and travel cost from total revenue. Plot the result of this subtraction, which will be different at each x (since travel cost is different at each x), on the vertical axis of the isoquant diagram to obtain C(x) i the equilibrium isocost line’s vertical-axis intercept at the given location. Finally, draw the tangent from C ix) to the S = 1 isoquant. The tangency point and tangent slope will yield equilibrium values for all endogenous variables at the given value of x. At central locations where travel cost is relatively low (for example at x = 1), C ix) will be relatively high up on the C axis. On the x = 1 line of Table 1, C will i equal [$500 (total revenue) minus $150 (other non-land cost) minus $36 (travel cost)] = $314. A tangent to the S = 1 isoquant from C = $314 will be relatively i steep (i.e. land rent will be relatively high). Als
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