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Notes on General Equilibrium Model

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University of Toronto St. George
Peter Tomlinson

The General Equilibrium Rectangular City Model June 25, 2009 These notes summarize lecture presentations from June 18 and 23, dealing with the general equilibrium model (Figures 7A-3 and 7A-4 plus related discussion in the textbook). General equilibrium, as defined in this model, means that land and labour markets are both in equilibrium simultaneously. A new endogenous variable is introduced here: the wage paid to employees of manufacturing firms. These employees are also the model citys residents the customers of housing firms in the city. Land Market Equilibrium Land market equilibrium, as outlined in Chapter 6, requires that manufacturing firms and housing firms pay land rents giving them zero economic profit at all locations. For housing firms there is an additional equilibrium requirement discussed in Chapter 6: the prices that residents pay for the firms output must allow residents to obtain the equilibrium utility level again at all locations. The location variable for residents is commuting distance to the business area. Residents equilibrium utility level is exogenous; it is fixed in the model city because residents can move in or out of the model city to other cities at no cost (the open city model assumption). Residents of other cities are assumed to have a fixed utility level. If utility in the model city is temporarily lower, model city residents will move to other cities. A population outflow causes housing prices in the model city to fall, so its remaining residents eventually regain the equilibrium utility level. Likewise if utility in the model city is temporarily higher than in other cities, a population inflow results, causing housing prices to increase until the equilibrium utility level is regained. Since this model assumes no consumer substitution, residents obtain equilibrium utility if they can just afford a fixed (exogenous) house size and a fixed(exogenous) consumption level of non-housing goods. After paying the cost of commuting to the business area, residents must have just enough money left to afford this fixed consumption bundle. Equilibrium housing prices at all locations will allow this requirement to be met. The no-consumer-substitution assumption differs from the indifference curve analysis in the appendix to Chapter 6. If residents can move along an indifference curve giving them the equilibrium utility level, house size and non-housing consumption would be endogenous variables. To eliminate consumer substitution here, well assume the city has a zoning rule. 1 The rule allows only a fixed housing unit size to be produced by housing firms. That being so, residents can reach just one point on their equilibrium indifference curve. Commuting cost is a function of distance between a residents housing unit and the residential/business boundary. Commuting inside the business area is costles s. As in the text, both manufacturing and housing firms are assumed to produce a fixed output level. Also as in the text, we assume that factor substitution is not possible. In the case of manufacturing firms, their output is produced using fixed inputs of land, labour, and capital. In the case of housing firms, their output is produced using fixed inputs of land and capital. Manufacturing firms ship their output to a highway or port, incurring freight cost along local roads. The farther they are from the highway or port, the more freight cost they incur. In equilibrium, land rents will adjust so that firms earn zero economic profit at all locations. There are no office firms in the model. Manufacturing firms are the only employers. Their output is exported from the city, and is sold at an exogenous price. Farms are included in the model. Competing against each other for land, farmers pay an equilibrium land rent. We assume this agricultural rent is exogenous that is, independent of distance from the highway or port. Labour Market Equilibrium Given the absence of consumer and factor substitution, employment and residential densities will be fixed. The land area occupied by businesses multiplied by the fixed employment density equals the number of jobs in the city. The land area occupied by resident s multiplied by the fixed residential densityequals the number of residents in the city. For the labour market to be in equilibrium, every resident must have a job and every employer must have the number of employees required to produce its output. If there is either excess supply or excess demand in the labour market, the wage level will change. A change to the wage level will have impacts in the land market. Business and residential land areas will adjust until the number of jobs in the business area equals the number of residents in the residential area. Thus land and labour markets reach equilibrium simultaneously. Geography of the Rectangular City The city has northern and southern boundaries fixed by zoning rules. Manufacturing and housing are not permitted outside these boundaries. As with rectangles in general, there is a uniform north-south distance for the entire city. This distance is an exogenous variable determined by the zoning rules. 2 The western boundary is formed by a highway or port (well assume its a port), to which manufacturing firms ship their output on local roads. The eastern boundarys location is determined in the land market. It is where residential bid rent equals the exogenous agricultural rent. In the region farther east, all land is agricultural because housing firms in that region could not pay the agricultural land rent. Variables in the Model The following variables are applicable to each manufacturing firm: Exogenous output = q units/month Exogenous output price = $p/unit Exogenous freight cost =$T/unit of output/mile distance from port (x) Exogenous labour input = E employees Exogenous capital input = $C/ month(capital is defined here to include other non-land, non-labour costs such as profit needed to stay in business) Exogenous land input = L bectares Endogenous wage =$w/employee/month Endogenous land rent = $R (b,w)/hectare/month The following variables are applicable to each housing firm: Exogenous output = Q sq.ft. of floor area Exogenous land input = L hectares Exogenous capital input = $K/month (capital is defined here to include other non- land costs such as labour and profit needed to stay in business) Endogenous land rent = $R (h,w)/hectare/month Endogenous output price = $P(x,w) / sq. ft. Floor area/month The following variables are applicable to each resident: Exogenous housing-unit size = h sq. ft. Exogenous consumption of non-housing goods = $G/month Exogenous commuting cost = $t/mile/month Endogenous housing price = $P(x,w)/sq. ft./ month Endogenous income = $w/month Other exogenous variables: Agricultural rent = $Ra/hectare/month North-south distance between zoning boundaries = dmiles 3 Other endogenous variables: The business/residential boundary (also known as the zero-commute location) is x (w1 The residential/agricultural boundary (also known as the city limit) is2x (w) The city population is N The number of manufacturing firms in the city is M The aggregate manufacturing output in the city is B The total number of housing firms in the city is F The variable x is the number of miles from the citys western boundary to any given location in the city, measured along an east-west line. While the business area is at the western end of the city, it will sometimes be referred to as the CBD (short for central business district). The term CBD boundary refers to the boundary between business and residential areas at x (w)1 Solving the Model: a Numerical Example The following numbers apply to each manufacturing firm: Exogenous output q = 200 units/month Exogenous output price p = $190/unit Exogenous freight cost T = $10/unit output/mile Exogenous labour input E = 10 employees Exogenous capital input C = $2000/month Exogenous land input L =b1 hectare The following numbers apply to each housing firm: Exogenous output Q = 15,000 sq. ft. floor area Exogenous land input L =h3 hectares Exogenous capital input K = $6000/month The following numbers apply to each resident: Exogenous unit commuting cost t = $100/mile between the residents housing unit and the CBD boundary at x (1) Exogenous housing consumption h = 1000 sq. ft. Exogenous consumption of other goods G = $400/month The following additional numbers are used: Agricultural land rent R = $1000/ha/month a Distance between northern and southern zoning boundaries d = 3 miles 4
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