Compare methods

	Urban Density Gradient Model	Bid-Rent Analysis
Field	Human Geography	Human Geography
Family	Regression model	Regression model
Year of origin	1951	1964
Originator	Colin Clark; Edwin Mills & Richard Muth (theory); Bruce Newling (quadratic form)	William Alonso
Type	Family of functional models of urban population density as a function of distance from the centre	Theory of urban land rent and land-use allocation by distance to the centre
Seminal source	Clark, C. (1951). Urban population densities. Journal of the Royal Statistical Society. Series A (General), 114(4), 490–496.	Alonso, W. (1964). Location and Land Use: Toward a General Theory of Land Rent. Harvard University Press, Cambridge, MA. ISBN: 9780674537019
Aliases	Urban Density Function, Population Density Gradient, Density-Distance Function, Monocentric Density Model	Bid-Rent Theory, Alonso Bid-Rent Model, Urban Land-Rent Model, Bid-Rent Curve Analysis
Related	4	4
Summary	The urban density gradient model is the broad family of functional relationships that describe how population density varies with distance from a city's centre. Its canonical member is Colin Clark's 1951 negative-exponential form, but the family also includes Bruce Newling's quadratic-exponential function that permits a density crater at the core, simpler linear and Smeed forms, and the economic micro-foundation supplied by the Muth-Mills monocentric city model. Together these give planners and economists a compact, comparable language for urban spatial structure.	Bid-rent analysis is the urban-economics theory that explains how land rent and land use are organized by distance to the city centre. Developed by William Alonso in 1964, it represents each land use — commerce, industry, housing — by a bid-rent curve giving the maximum rent that use is willing to pay at each distance from the central business district. Because uses with steeper curves outbid others for central land, the observed rent is the upper envelope of all the curves, and the city sorts into concentric zones with the highest bidder winning each ring.