Krugman Model NotesIssue is location of firms in conditions of economies of scale and fixed costs. When and why will core periphery patterns emerge. Anderson view:Explain when a firm will prefer to locate in a region, and when it would prefer to ship into the region. The key variables are the quantity of goods to be shipped and the transportation costs if the firm ships into the region, and the cost of building a factory if the firm locates in a region. One possibilityA firm in each region can attempt to produce for the entire market so that it builds only one factory and incurs the cost of shipping to the other region. It will do this if:Qship * t FC; it costs more to ship into the smaller region than to build a factory in it.In this case one firm may choose to supply East with a factory in East and West with a factory in West.Or if there are two independent producers Stanfields and Penmans textile mills, each can survive in its own market and will not ship into the other.Krugman approach:Krugman model:Determinants of distribution of manufacturing: Will a core form in one region, or will both share in manufacturing?East30 farmersWest30 farmersmountains40 workers locate where plants locate.Using Equations:Share of population in the West = SnShare of population in the East = (1- Sn)Share of workers (manufacturing) in West = SmShare of workers (manufacturing) in the East = (1- Sm)Share of farmers (agriculture) in West = SaShare of farmers (agriculture) in East = 1- SaIf Sm = 1, all the manufacturing is in West, so West is the core and East is a periphery with (1-Sm) = 0If Sm = 0, none of the manufacturing is in the West, so West is the periphery and East is a core with (1-Sm) = 1If Sm = .5, manufacturing is evenly distributed between the West and East. There is no core periphery pattern.The relationship between the share of the population and the share of manufacturing.Relationship between Sn and Sm can be expressed algebraically. The share of workers in the population and the division of farmers between the regions can be taken as parameters that can be changed. = share of population who are workersSa = share of farmers in West. (Krugman assumes Sa = )Sn = (1- )* Sa + * SmTHIS IS A KEY EQUATIONIn Words: Share of the population in West (Sn) equals the share of farmers in the total population (1- ) times the share of farmers in West (Sa) plus the share of workers in the total population () times the share of workers in West (Sm).Initial values: Sa = .5; = .4If West is the periphery Sm = 0 and given the initial values Sn = (1-.4)*.5 + .4 * 0 = .3West has .3 of the population.If West is the core, Sm = 1 and given the initial valuesSn = (1-.4)*.5 + .4 * 1 = .7No Core Periphery Pattern If manufacturing is evenly distributed, each unique product will be produced locally, so each firm will have a plant in each region and no goods will be shipped between regions. In that case each region manufactures for itself, and manufacturing must be in proportion to the population. (Remember in this simplified and artificial world we have assumed, all firms have identical fixed costs and transportation costs and consumption of goods is strictly proportional to population.)If West manufactures for itself only, then Sm = Sn. Manufacturing production must be geared to the size of Wests population.Sn = (1- )* Sa + * Sm, but Sm = Sn soSn = (1- )* Sa + * SnSn - ( * Sn) = (1- )* Sa(1- )* Sn = (1- ) * SaSn = SaThe number of workers is proportionate to the number of farmers. If each region manufactures only for itself, then the ratio of workers to farmers in the entire economy must be the ratio in each region.The share of the population in West equals its share in agriculture. Given Krugmans assumption of equal agricultural resources, Sn=.5.But if agriculture were not evenly divided, say West had 60% of the farmers, then Sn = .6 when manufacturing is equally divided. The Krugman BoxIf there is no manufacturing in West, Sm = 0, there are only farmers in West. The share of population in West is the farmers in West as a share of the total population. As the share of manufacturing increases in West, the share of population in West grows along the line P for population.When West has no manufacturing (Sm = 0) East has all the manufacturing. As Wests share of population grows, Easts share of population shrinks. When Sm = 1, East has only farmers and no manufacturing.2. Manufacturing depends on populationIf the population in West is small, because it has no manufacturing then it may be that it will cost less to ship into West from East than to build a factory there. Sm = 0 is possible, and West can be a periphery.If Sn F/tx then Qship to west * t FC it costs less to ship into West, Alberta, than to build a factory and produce there. Sm = 0 is possibleIf the population in East is small, because all the manufacturing is in West, than it may be that it will cost less to ship into East from West than to build a factory in East. West can continue to have all the manufacturing. Sm=1 is possible, and West can be the Core and east can be the periphery.If 1-Sn F/tx then Qship to east * t FC FC it costs less to ship into East , than to build a factory and produce there. Sm = 1 and 1-Sm = 0 is possible; East will have no manufacturing and West will have all of it. Sm = 1 is possible.1-Sn F/tx is the same as Sn 1-F/txIf even when West has no manufacturing, the share of population is large enough that it pays to build a factory in West rather than to ship into it, then Sm = 0 is not possible. West cannot be a periphery.If F/tx Sn then it costs less to build a factory than to ship to west. If that is true, if Sm = 0, firms would decide to locate in West rather than ship in and Sm would rise. Sm = 0, West is a periphery is not possible.The same is true for east if F/tx (1-Sn) (Easts share of population) when all the manufacturing is in West and non in East. That occurs when ever Sn Sn, West can be a peripheryIf Sm = 1, 1-F/tx Sn, West can be a core; East can be a peripheryIf Sm = Sn, East and West will each manufacture for themselves. A core periphery pattern is not possible.If any one of these equilibriums is established it will persist. Any one could be established. WHEN IS A CORE PERIPHERY PATTERN LIKELY?When will it be economical for a region to have all the manufacturing, to have none, or to have the same proportion of manufacturing as it has population?The answer depends (surprise, surprise!) on the relationship between fixed costs, transport costs, the total size of the market and Sa.Krugmans valuesFixed costs = F = 4transport costs per unit = t = 1units sold = x = 10Sa = .5 = .4We will use equations to see that with the given values, three types of worlds are possible. West could be the periphery and East the core, East could be the periphery and West the core, or manufacturing could be divided between the two.Can West be a periphery with no manufacturing?Do not build a plant in WestIf Sn * tx F that is if Sn F/txIt costs less to ship to West than to build a plant in West.Therefore, Sm = 0.In our Anderson notation, this is the situation when FC t * Qs; It costs more to ship the goods than to build a plant in West. In this situation, every firm will want to have a plant in West.Given the costs, if West has only farmers, Sm= 0.Sn = (1- )* Sa + * Sm; Sn = (1-.4) * .5 + .4 * 0 = .3 + 0 = .3 Sn = .3F/tx = 4/(1*10) = .4Sn F/tx because .3 .4Cost of building the factory is 4 (million dollars)Cost of shipping the goods into West is 3 (million dollars). That is $1 per unit, times 30 (million) units shipped. West has .3 of the population so by our assumption of per capital consumption, it will buy .3 of the 10 (million) units produced.Therefore, it is consistent for West to have no manufacturing and to be populated only by farmers. Building a plant would cost more than shipping from West to East if West had no manufacturing, Sn would be low enough that it paid to continue to have no manufacturing. If West began as a periphery, it would stay a periphery. West as a periphery is one equilibrium.Can West be a core, with all the manufacturing in West and none in East?Then East must be a periphery , and firms must be unwilling to locate in East.If West is a core, than Sm = 1; Sn = (1- )* Sa + * Sm; Sn = (1-.4) * .5 + .4 * 1 = .3 + 0 = .7 Sn = .7(1-Sn) = .3; East has only .3 share of the populationF/tx = 4/(1*10) = .4(1-Sn) F/tx because .3 t*Qs; Building a plant in East costs more than shipping from West to East.Do not build a plant in EastIf (1-Sn) * tx F that is if (1-Sn) 1-F/txIf it costs less to ship to East than to build a plant in East then no firm will want to locate in East. West will have all the manufacturing, and West will be the core. Therefore, Sm = 1. West has all the manufacturing, East has none.Therefore, it is consistent for West to have all the manufacturing and all the workers. Building a plant in East would then cost more than shipping from West to East. If West had all the manufacturing, 1-Sn would be low enough that it paid to continue to have no manufacturing in East and all the manufacturing in West.If West began as a core would stay a core. West as a core is one equilibrium.Can manufacturing be divided between West and East?West: If Sn * tx F, it costs more to ship into West than to build a plant in West, there will be manufacturing in West.East: If (1-Sn) * tx F then it costs more to ship into East than to build a plant in East, there will be manufacturing in East.West: Sn * tx F is the same as F/tx F is the same as (1-Sn) F/tx which is the same as Sn 1-F/tx) Therefore:If F/tx Sn F and (1-Sn) * tx F ; Both West and East are big enough to have factories built to serve there own markets. There will be no core periphery pattern and manufacturing will be dispersed between the two regions.For example, F/tx = .4, 1-F/tx = .6. So if the Sn is between .4 and .6, both regions can have manufacturing. When the manufacturing is equally divided, Sn = .5, so sharing manufacturing is also possible. If each region has half the manufacturing, then firms will find it cheaper to build a plant in each region rather than to build only one plant and ship into the other region. The costs of building the plant is 4 (million), but shipping in to either region will cost 5 (million) $1 per unit times half the 10 million units produced. Note: if F tx * .5 or F/tx .5 then it always costs more to build a plant than to ship the goods to the smaller region. The choice will always be to build one plant in the larger region and ship output to the smaller region.Note: if t = 0, transport costs have no influence on location decisions. Its a whole new ball game. However, even if the cost of shipping goods is zero (computer software) or very, very low (pharmaceuticals) there may still be costs of doing business at a distance from the market. Sometimes face to face meetings are very important to business, or firms need to locate where there is a large pool of labour with the skills they use.Situations with alternate fixed costs and transportation costs.Three equations show how the share of population depends on the share of manufacturing:Sn 1-F/tx Sm = 1F/tx Sn 1-F/tx Sm = Sn One equation shows how the share of population depends on the share of manufacturing.Sn = (1- )* Sa + * SmKeep the share of workers = .4 and agriculture is evenly divided. ThenSn = (1- .4)* .5 + .4 * SmIf Sm = 0, then Sn equals .3If Sm = 1, then Sn equals .7If Sm = .5 then Sn equals .5.3 1-F/tx Sm = 1F/tx .5 1-F/tx Sm = Sn Given F = 4; t = 1; and x = 10 then F/tx = .4.3 1-.4 = .6 Sm = 1 is a possible equilibrium.4 .5 .4 Sm = Sn is a possible equilibriumSuppose economic conditions were different F = 2; t = 2; and x = 5; F/tx = .2.3 1-.2 = .8; not true Sm = 1 is not a pos