von Thunen s Model Industrial Land Use the von Thunen Model Moving Forward von Thunen s Model Results

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1 von Thunen Model Indutrial Land Ue the von Thunen Model Philip A. Viton September 17, 2014 In 1826, Johann von Thunen, in Der iolierte Stadt (The iolated city) conidered the location of agricultural activitie around an iolated city with no interregional trade. Thi reflected the etting for many early-19th-century German citie, before the development of an extenive railway network. All agricultural output needed to be tranported (by hore and cart) to the city to be old. von Thunen work contained not only a ophiticated theoretical model but alo a careful empirical analyi of land productivity, carrying capacity (horepower) of the available tranport, and on the need to carry additional weight (fuel) to feed the hore. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 von Thunen Model Reult Moving Forward Rough pattern of von Thunen reult for 19-th century Germany: Perihable (fruit/vegetable/milk) 2. Lumber (heavy: you want to minimize tranport cot) 3. Other crop 4. Livetock, which are their own (weight-loing) tranport (von Thunen actual reult were more detailed than thi). Of coure, von Thunen model wa quite pecialized, both a to it time and it technology. Can we ue it to get inight into contemporary land ue? Let try to build on the pirit of von Thunen model. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

2 Space The Land Market (I) We will aume that all activity take place on a von Thunen plain : a flat featurele plain. Geographical feature (topography, etc) are ignored. We aume that there i a city located in the middle of thi plain. We will focu on land ue outide the city, o we repreent the city a a point in pace. Our tory i about competition for land by the repreentative of the variou indutrial ector. At any ditance from the city, land could be occupied by any of the ector. Which ector will actually get land at that ditance? We will make a natural aumption here: land goe to the highet bidder, ie whoever i prepared to pay mot for it. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 The Land Market (II) Accounting v Economic Profit (I) So we need to ak: how much i a repreentative of each indutry willing to pay (bid) for a plot of land located mile from the city? To dicu thi, we need to ay omething about why the firm do what they do, ie their behavioral motivation. Our anwer here i conventional: firm make their deciion in order to maximize profit. Thu, the amount a firm would be willing to bid for land at ome ditance from the city will depend on the level of it profit. So we begin with a brief dicuion of how profit are meaured. We will ditinguih two way of meauring profit. Accounting Profit: The definition of accounting profit i exactly what you would expect: accounting profit are the difference tetween total firm revenue, and the cot incurred to realize thoe revenue: Accounting Profit = Total Revenue Total Cot Economic Profit: Economic profit take accounting profit and ubtract out the profit that could be achieved, if reource were deployed in their next-bet poible way. So Economic Profit = Accounting Profit Next-Bet Profit Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

3 Accounting v Economic Profit (II) Accounting v Economic Profit (III) Before we ee why thi i a ueful ditinction, let look at an example. Suppoe there are two opportunitie, called A and B, decribed a follow: A B Total Revenue Total Cot Accounting Profit Accounting profit i jut the difference between total revenue and total cot, o = 18 for opportunity A, and = 25 for opportunity B. To calculate A economic profit, we note that if reource devoted to opportunity A had been devoted intead to opportunity B, they would have yielded an accounting profit of 25. And ince there are only two opportunitie, thi i the next-bet profit for A. To arrive at the economic profit for A we ubtract thi next-bet profit (25) from A accounting profit: the reult i = 7. Similarly for B: it next-bet opportunity i the profit offered by A, o economic profit i = 7 The uphot i: A B Total Revenue Total Cot Accounting Profit Economic Profit 7 7 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Accounting v Economic Profit (IV) Indutrie What doe thi tell u? The important thing here i the ign of the economic profit. Opportunity A ha a negative economic profit. Thi tell u that if we re out to maximize our return, that we can do better (we hould hift reource to opportunity B). On the other hand, opportunity B ha a poitive economic profit. Thi tell u that there i no better way on offer to make profit (the next-bet opportunity yield le profit). We now apply thee idea to our indutrial land-ue problem. We think of an indutry a made up of many mall identical firm. In the jargon, we will conider competitive indutrie. Therefore we can conider a repreentative firm in each indutry (indutrial ector). Each indutry manufacture a ingle product, and ell that product only in the city. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

4 Entry and Exit (I) We have aid that the willingne of firm to bid for land depend on their profit level. But what will thoe profit level be? The key to undertanding thi i that if a ector ha poitive economic profit, entrepreneur will enter that ector; if economic profit are negative, then entrepreneur will leave that ector (and migrate to another). Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Entry and Exit (II) We will aume that nothing prevent a firm from entering a profitable ector, or leaving an unprofitable one. In the jargon, there are no barrier to entry (or exit) for any indutrial ector. The mot important barrier to entry found in practice i regulatory: ome governmental agency retrict the ability of firm to enter (typically thi quetion doe not arie for leaving) an indutrial ector. Example: in many citie, entry to the taxicab ector i regulated; before the Air Deregulation Act of 1976, entry into the paenger air tranport market wa regulated (by the Civil Aeronautic Board). Some indutrie alo experience barrier to entry in the form of very high tart-up cot (think of the cot of electricity generation, or of contructing a ubway ytem) that dicourage entry. We are auming that thee do not apply, ie that there i Free Entry and Free Exit for all the indutrie that might locate on the plain. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Long-Run Profit Firm in an Indutry (I) If a ector ha poitive economic profit, entrepreneur will be induced to enter the ector. Thi will reduce the profitability of the ector (more competition). If the ector ha negative economic profit, entrepreneur will leave the ector. Thi will enhance the profitability of the remaining firm (le competition). Our aumption of free entry and exit mean that there are no reaon for thee change not to occur. The reult i that entry and exit will eventually (in the long run) drive economic profit (but not necearily accounting profit) to zero. Thi zero-profit outcome i the long-run indutry equilibrium for a competitive indutry. We will focu only on the long run. We will not be dicuing the path to equilibrium, ie how we actually get there. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Let return now to our dicuion of economic activity on a von Thunen plain. All input and output will be meaured in weight unit. Input will be aumed to be available everywhere (unlike in our Weber model): they are ubiquitou. Production will be according to fixed factor proportion and contant return to cale (jut a in our Weber model). Each firm act on the aumption that if it change i production deciion (how much to produce), thi will not change either the price it receive for it product in the city, or the price it mut pay for input. In the jargon, it i a price-taker in the input and output market. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

5 Firm in an Indutry (II) The Tranport Sector a b Becaue we are auming ubiquitou input, direction i alo irrelevant. If we know that an indutry i located between a and b in one direction from the city, we know that it will locate between a and b in all direction. Thi give rie to von Thunen ring pattern of indutrial location. We will aume that tranportation to and from the city i available everywhere. Becaue direction i irrelevant, we can conider a 1-dimenional patial tructure, a if the city and it hinterland were on a line. We will meaure ditance in any direction from the city a. Tranport cot will be k per ton-mile. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 A Modern von Thunen Model (I) We can now pull all thee trand together. Conider a repreentative firm in the j-th indutrial ector. The output of thi firm i old at a fixed price per ton p j in the city. Land devoted to thi product (indutry) ha a yield (productivity) of α j ton per acre. Production cot, excluive of land and tranportation cot, but including the next-bet attainable profit, are β j per acre. There are contant return to cale. Land at ditance from the city cot (rent for) R() per acre. A Modern von Thunen Model (II) We can now figure out (economic) profit per acre in indutry j occupying land at ditance from the city. Total revenue per acre will be the price per ton (p j ) multiplied by the yield α j per acre (ton per acre). So total revenue per acre in ector j will be α j p j Total economic cot per acre will be the um of Direct production cot (including next-bet profit): β j Tranport cot: α j k Land cot at : R() So total economic profit per acre, if indutry j locate mile from the city, will be: Π j () = p j α j (β j + α j k + R()) Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

6 Bid Rent (I) Bid Rent (II) We have aid that land at ditance goe to the indutry that i prepared to bid mot for it. So our crucial quetion i: how much i indutry j prepared to bid for a plot of land at ditance? We will refer to thi highet poible bid for land a indutry j Bid-Rent for land (at ). So our quetion i: how to determine indutry j bid-rent for land at ditance from the city? But thi i now eay to anwer: in the long run, given free entry, the mot that each firm in an indutry could bid for land i the amount conitent with zero economic profit in the long run. So we can find that bid by etting economic profit equal to zero and then olving for the maximum bid for land (land rent) that i conitent with zero economic profit. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Bid Rent in Indutry j (I) Bid Rent in Indutry j (II) Let do that. We have Π j = p j α j (β j + α j k + R()) We now et thi to zero, to be conitent with long-run indutry equilibrium: 0 = p j α j (β j + α j k + R()) and olve for the rent R() which we ll call Rj (), to remind u that i the maximum land rent (at ) that indutry j can pay, conitent with long-run (zero economic profit) indutry equilibrium. So: R j () = p j α j β j α j k Thi i called indutry j bid-rent at ditance from the city. The maximum that a firm in indutry j can pay for land (it bid-rent) at ditance from the city i R j () = (p j α j β j ) (α j k) Let clean thi up, and write, purely a notation: c j = p j α j β j g j = α j k Then we ee that indutry j bid-rent can be written a: R j () = c j g j Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

7 Bid Rent in Indutry j (III) Bid Rent in Indutry j (IV) How doe indutry j bid-rent vary over pace (ie, with ditance)? Clearly it i linear in ditance. At ditance = 0 we have R j () = c j. If thi indutry i to be viable, then clearly we mut have c j > 0 (check the definition of c j from the previou lide). The lope of the line i g j. Since both the yield per acre in indutry j (α j ) and the tranport cot per ton-mile (k) are poitive, we conclude that g j = α j k i poitive, and hence g j i negative. c j 0 lope = g j R j* () Bid Rent in indutry j lope downward in ditance from the city. The further away from the city the indutry i located, the le it i willing to bid for land. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Indutrial Land Ue (I) Indutrial Land Ue (II) We are now in a poition to tudy indutrial land ue over pace. Each indutry j will generate a bid-rent Rj () for land at each ditance from the city. That bid rent repreent the mot that it i prepared to bid for land at ditance, conitent with long-run equilibrium. We have aid that land goe to the highet bidder. So at any ditance the land goe to that indutry whoe Rj () i greatet. We can decribe thi a aying that the actual (oberved) pattern of land rent over pace i the upper envelope of the individual bid-rent. c 1 c 2 0 * * R2 () R 1* () Suppoe we have two indutrie competing for land outide the city. Their bid-rent are hown a R 1 () and R 2 (). A drawn, indutry 1 i prepared to bid more for land near the city. So it locate near the city. Indutry 2 locate further out. The point mark the boundary between the two indutrie (land ue). Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

8 Indutrial Land Ue (III) Indutrial Land Ue (IV) c 1 c 2 0 * For thi two-indutry etting, the oberved pattern of land-rent over pace will be the upper envelope of the individual bid-rent, hown by the heavy line. Note that it i alo downward-loping over pace, though it i not a moothly falling line (there i a kink at ). R 1* () R 2* () R 4* () R 3* () For more than 2 poible indutrie, the analyi i exactly the ame. Each indutry ha a linear downward-loping bid-rent function. At any point in pace, the indutry with the highet bid-rent get the land. Thi i the upper envelope (heavy line) of the bid-rent of the variou indutrie. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Indutrial Land Ue (V) Land Ue and Tranportation (I) R 1* () R 2* () R 4* () R 3* () Note what can happen: technological or economic condition can reult in indutrie being crowded out of the land market entirely. Thi i what happen in the figure to indutry 4. It i never able to outbid the other indutrie So it i not preent in the land-ue pattern around thi city. c j 0 What happen if tranportation cot decreae? Recall that R j () = c j g j. The term c j doe not involve the tranport cot k, and only the lope g j = α j k doe. So if tranport cot decline, g j fall, ie the lope of the bid-rent function get le teep. Since c j i unaffected, the decreae in tranportation cot caue indutry j bid-rent to pivot outward around c j. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

9 Land Ue and Tranportation (II) Agglomeration c 1 c 2 0 * R 2* () R 1* () The figure how what happen in the 2-indutry etting when tranport cot decreae. Bid-rent pivot outward. Each indutry occupie more land: the croover-point move outward from the city. Compare thi to the figure on lide 29. Agglomeration refer to the tendency of certain indutrie to cluter together. Let try to contruct a imple model of thi. A before, we ll concentrate in a repreentative firm in each indutry. We need a reaon for indutrie to cluter they have to have omething to do with one another. Let retrict our attention to a two-indutry model, and we ll uppoe that the output of indutry 2 i ued a an input to production in indutry 1. That will be it only ue: indutry 2 output will not be old at the city at all. So the reaon to cluter, if it occur, i to be near your cutomer (for indutry 2) or your input upplier (indutry 1). Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Indutry 1 Indutry 1 will be a before, except that (a) every ton of output require 1 ton of the output of indutry 2; and (b) we will let indutry 1 have it own tranportation cot per ton-mile, k 1. Note that becaue all of 1 output i old at the city, we don t need to worry about it price varying over pace and we write it a p 1. But indutry 2 output i old wherever indutry 1 locate, o it price will vary with pace. If p2 () i the elling price per ton of good 2 at location, then indutry 1 cot per acre from 2 input i p2 ()α 1, where a before α 1 i the productivity of land for indutry 1. So indutry 1 profit per acre at i Π 1 () = p 1 α 1 (β 1 + k 1 α 1 + R 1 () + p 2 ()α 1 ) And, impoing the zero (economic) profit condition, we have, for it bid-rent at : R 1 () = p 1 α 1 β 1 p 2 ()α 1 k 1 α 1 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Indutry 2 Aume that agglomeration doe actually take place. Then indutry 2 will have no tranport cot. So indutry 2 profit per acre at i: Π 2 () = p 2 ()α 2 (β 2 + R 2 ()) where a before α 2 i indutry 2 productivity per acre, and β 2 i it other cot per acre. Then under free entry, we can find indutry 2 bid-rent by equating profit per acre to zero. The reult i: R 2 () = p 2 ()α 2 β 2 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

10 Condition for Agglomeration (I) Condition for Agglomeration A to the firt condition, we equate the bid rent of the two indutrie and find When will the two indutrie locate in the ame place? There are two condition: 1. They mut offer the ame bid-rent at. 2. It mut be unprofitable for indutry 2 to hip it product anywhere. R 1 () = R 2 () p 1 α 1 β 1 p 2 ()α 1 k 1 α 1 = p 2 ()α 2 β 2 which give u an expreion for the equilibrium price for indutry 2 output at any location: p 1 α 1 β 1 k 1 α 1 + β 2 = p2 ()α 2 + p2 ()α 1 p 1 α 1 β 1 k 1 α 1 + β 2 = p2 ()(α 1 + α 2 ) p 2 () = p 1α 1 β 1 k 1 α 1 + β 2 (α 1 + α 2 ) Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Condition for Agglomeration (II) A to the econd condition, it mut be unprofitable for indutry 2 to hip any product. Now, p2 () i the price indutry 2 receive per ton, ie it total revenue per ton at, which we can write a p 2 () = p 1α 1 β 1 + β 2 (α 1 + α 2 ) k 1α 1 (α 1 + α 2 ) If indutry 2 were to hip 1 ton of output 1 mile, the change in it revenue per ton i k 1α 1 (α 1 + α 2 ) multiplied by whether we add (+1) or ubtract ( 1) from hipment. If k 2 i the hipping cot per ton-mile, the hipment will add k 2 to cot. Condition for Agglomeration (III) In an agglomeration equilibrium, a change in any direction mut be unprofitable, o indutry 2 tranportation cot k 2 per ton mut be greater than thi, o k 2 k 1 α 1 (α 1 + α 2 ) Then we have, remembering that everything i poitive, k 2 k 1 α 1 (α 1 + α 2 ) The ratio of the tranportation cot per ton-mile in the two indutrie mut exceed the proportional productivity of the two indutrie. When thi condition i not atified, then the two indutrie will locate at different point in pace (jut like our firt model), except that indutry 2 will hip product only to indutry 1 location. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43

11 An Urban von Thunen Model (I) An Urban von Thunen Model (II) In 1925 E.W. Burge publihed a von Thunen-type model of urban form. Hi reult were that land ue were arranged in ring around a CBD: moving outward in pace from the city center, he had: 1. Downtown 2. Indutrial + warehouing 3. Tranition (mixed indutrial + reidential ue) 4. Blue collar reidential 5. White collar reidential 6. Executive reidential The peculiar reult, where the indutrial + warehouing ector wa located cloe to the downtown wa baed on 1920 tranport technology. The truck ha barely entered the tranportation ytem, and indutry relied almot completely on railway tranportation. Railroad tended to group their facilitie in centrally located witching yard to take advantage of economie of cale; in Burge cae, indutrial location jut followed the tranport technology. Nowaday, with more decentralized tranport, indutrial location are more decentralized, too. Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43 Advanced Reading If you re comfortable with mathematical reaoning, there in intereting paper in thie vein that you might want to look at: Y-N Shieh, Launhardt on von Thunen Ring: A Note, Regional Science and Urban Economic 22 (1992), pp The model of agglomeration i a light re-write of a model in chapter 3 of the advanced book : Maahia Fujita and Jean-Françoi Thie, Economic of Agglomeration, Cambridge Univerity Pre, Philip A. Viton CRP 6600 von () Thunen Model September 17, / 43