A mathematical challenge: the centre size distribution in a dynamic retail model. CASA Working Paper 180, University College London, Centre for Advanced Spatial Analysis

The standard retail model can be constructed as follows (cf. Wilson, 1970). Let {Sij} be a matrix of money flows from residents in each zone i to retail centres in each zone j. Let ei be the per capita expenditure in i and Pi the population, so that eiPi is the total retail expenditure leaving i. Let cij be transport costs as before with C as the total transport expenditure. Suppose residents gain a benefit from using centres of a particular size that is proportional to logWj, say, where Wj is the size in floor space of centre j. Let X be the total of such benefits. Then we can maximise an entropy function subject to the constraints that represent our knowledge of the system. Max S = -ΣSijlogSij (1) such that ΣjSij = eiPi (2) ΣijSijlogWj = X (3) ΣijSijcij = C (4) which, after the usual manipulations, leads to Sij = AieiPiWjαexp(-βcij) (5) with Ai = 1/∑kWkαexp(-βcik) (6) Because the attraction ’end’ is unconstrained, this allows us to calculate Dj = ΣiSij (7) which is the total amount of money attracted into a centre. We thus have a model that predicts a key locational variable, Dj.