Who gets the urban surplus?

Transcription

1 8/11/17 Wo gets te urban surplus? Paul Collier Antony J. Venables, University of Oxford and International Growt Centre Abstract Hig productivity in cities creates an economic surplus relative to oter areas. How is tis divided between workers and land-owners? Simple models wit omogenous labour suggest tat it accrues largely or entirely in te form of land-rents. Tis paper sows tat eterogeneity of labour in two main dimensions (productivity differentials and ousing demands) radically canges tis result. Even a modest amount of eterogeneity can drive te land sare of surplus down to 2/3rds or lower, as ig productivity and/or low ousing demand individuals receive large utility gains. It follows tat land value appreciation understates te value of urban amenities and infrastructure. In a system of cities te sorting of workers across cities means tat, wile total rent is igest, te rent sare of surplus is lowest in te largest and most productive cities. Keywords: Land rent, cities, productivity, wages, sorting JEL classification: R1, R1, R2 Acknowledgements: Tanks to Gilles Duranton, Vernon Henderson, Guy Micaels, referees and an editor for elpful comments. Autors addresses: Paul Collier, Blavatnik Scool of Government, Walton Street, Oxford, paul.collier@bsg.ox.ac.uk A. J. Venables Department of Economics Manor Road Oxford OX1 3UQ UK tony.venables@economics.ox.ac.uk

2 1. Introduction Wo benefits from te ig productivity levels of workers in cities? A standard answer is tat te surplus goes principally to land owners. Te limiting case is tat tere is an infinitely elastic supply of omogenous workers so wit tis assumption fixing teir reservation utility levels and wit no oter claimants on te surplus land owners must take 1%. Tis paper is an exploration of wat appens wen workers are eterogeneous, focusing on two salient dimensions of eterogeneity. One is te productivity differential te worker receives by locating in te city; te oter is te amount of ousing te worker consumes. Te idea is simple. Workers wit a large productivity differential and small ousing demand will do better tan tose wit te opposite caracteristics, and will terefore capture some of te urban surplus. Fundamentally, te presence of sufficient eterogeneity means tat te assumption of an infinitely elastic supply of workers becomes untenable. If tere were infinitely elastic supply te city would be occupied entirely by te type of worker tat gets te greatest return (bankers wit a preference for small apartments). We know tat wile sorting takes place tis is not te case. Te questions ten are: moving away from omogeneity, ow muc of te surplus still goes to landlords? Is te cange in teir sare large or small, and wat does it depend on? We find tat te effect is large. Modest amounts of eterogeneity appear to reduce te sare received by landlords to between two-tirds and one-alf. Te relationsip is convex, so tat even a small amount of eterogeneity brings a relatively large reduction in rent sare. Heterogeneity in eiter te productivity differential or ousing demand reduces te rent sare, and te effect is largest wen tere is a negative correlation between te two attributes of individuals. Looking at systems of cities, cities tat offer relatively large productivity differentials ave te igest absolute levels of rent, but te lowest sare of rent in surplus; tese are te largest cities, and tose in wic sorting as te most extreme effect. Tese answers are important for a number of issues. Analytically, tey matter for our understanding of urban models. Empirically, muc work rests on te assumed omogeneity of workers, for example recent work by Albouy (216) deriving te value of cities from data on urban wages. Evidently, answers matter for income distribution. Tey also matter for public finance and te extent to wic land taxation can capture te value created by urban agglomerations, as studied in Henry George teorems. Tey matter for project appraisal, since improvements in urban infrastructure increase rents and land values by less tan te full value of benefits created. Existing literature recognises te critical nature of te omogeneity assumption, as succinctly stated by Arnott and Stiglitz (1979) wo state te conceptual basis of capitalization studies is sound only wen marginal individuals are very similar to infra-marginal individuals... (p496). Duranton and Puga (215) discuss te issues tat arise as eterogeneous individuals 1

3 sort into different locations, and te tecnical complexities tat follow. Te recent literature on eterogeneity focuses on constructing large scale models tat analyse bot te sources of eterogeneity and teir implication for city systems, as exemplified by Berens et al. (214) (see also te survey by Berens and Robert-Nicoud 215). Te approac of tis paper is, by contrast, to focus tigtly on te question posed in te title and derive a quantitative sense of te answer to te question. We take spatial productivity differentials and ousing preferences as exogenous, drawing out teir implications rater tan modelling teir causes. Te remainder of te paper is as follows. Section 2 looks at an economy containing a single city, and at individuals wo decide weter or not to enter te city and were to live in te city. Section 2.1 presents some analytical results (solving te assignment problem for two simple cases), and section 2.2 presents te core quantitative results on te division of urban surplus between landowners and individuals, based on numerical analysis of te model. In section 2.3 we illustrate ow te benefits of an urban transport improvement, affecting just part of te city, are divided between rents and residents utility. Section 3 moves to a multicity variant of te model in wic individuals sort between cities, and demonstrates ow te sare of surplus tat is captured by rent varies across city types, and is lowest in te largest and most productive cities. 2. Model 1: A single city We start by considering an economy wit a single city, an outside location, and a given total population. Some individuals coose to live and work in te city, and oters outside. Eac worker as a productivity differential between working in te city and outside, denoted and a ousing preference parameter. Sources of tese productivity differentials ave been extensively analysed elsewere. 1 In te baseline version we outline in tis section is simply te quantity of ousing te individual consumes, a more general case being set out in te appendix and utilised in simulations. For eac worker te attributes q and are exogenous, as is te density function f ( ) over { } space tat gives te distribution of tese attributes over te population. Urban workers commute to jobs in te CBD, and a worker living at distance x from te CBD faces commuting costs tx and pays price differential (relative to living outside te city) p( for a unit of ousing. 2 Te utility an individual wit attributes, derives from living in te city at distance x from te CBD, as compared to living outside te city is terefore te productivity increment minus incremental ousing costs and commuting costs, 1 See Duranton and Puga (24) for te definitive survey. We assume constant productivity and output prices, so productivity differentials q are constant parameters. 2 We use te simplest form of te Alonso-Mills-Mut open city model, see Duranton & Puga (214). Te price function p( is exogenous to eac individual and endogenous to te equilibrium. 2

4 u( ; q p( tx. (1) Individuals coose to reside in te place tat yields igest utility. Given te ouse price function p(, an individual wit caracteristics { } makes coices: live in city at x *( ) arg max u( ; if u ( ; x *( )), (2) live outside city if u ( ; x *( )). x We define ( ; as te indicator function, equal to unity if type { } lives in te city at distance x and equal to zero oterwise, so te number of type { } people coosing to live at x is N( ; ( ; f ( ). (3) Te demand for ousing at eac distance x is ten N( ; dqd. Housing at eac distance is initially assumed to be in fixed supply, s(, so market clearing for ousing at x is N( ; dqd s(. (4) q Te simplest case is a linear city in wic land area and te number of ouses at eac distance is a constant, s( = s. Te ouse price differential, p(, adjusts to clear te market, and te city edge (denoted x ~ ) is were tis price differential equals zero, p ( ~ x ). (5) Equilibrium is caracterised by tese five equations giving utility, location coice, numbers of individuals of eac type at eac place, te ouse price function p(, and te city edge. 3 From tis we derive te variables we are interested in. Te total number of people living at x: N ( N(, ddq. (6) q Te total utility differential accruing to urban workers: q U ~ x q u( ; N( ; ddqdx. (7) Wit fixed supply of ousing and no construction costs, te ouse price differential p( is equal to land rent so total differential rent is: 3 We will often refer to differential utility, ouse prices, rent and productivity simply as utility, ouse prices, rent, and productivity. Trougout, we maintain te assumption tat real incomes of all individuals outside te city are constant, unaffected by city size. 3

5 ~ R x s( p( dx. (8) Te total urban surplus is differential rent and utility, R + U, and our primary question is, ow is tis surplus divided between tese two components? Before answering tese questions, we note tat te expressions above give te simplest form of te model. Tis is used in te analysis of next subsection and is te base case trougout te paper. In simulation we also use a version tat is more general in two respects. First, te demand for ousing of individuals of eac type also depends on price, wit elasticity ε; wilst adding a price elasticity we retain zero income elasticity, a consequence of our working wit productivity and income differentials not levels. 4 Second, ousing supply is endogenised, adding a construction sector tat cooses supply of ousing per unit land, tis creating a price elasticity η of ousing supply at eac distance, x. Te preference and tecnology apparatus tat supports tese generalisations is set out in appendix 1, as are te more general forms of eqns. (1) (8). 2.1 Analysis Drawing out answers to te question posed in tis paper, for a wide range of different parameters, city sapes, distributions of individual caracteristics, and specifications of preferences and tecnology, requires numerical simulations. Tese are presented and discussed in following sections of te paper, wile in tis section we briefly derive analytical solutions for some special cases. We take eac of our two dimensions of eterogeneity separately, in line wit te one-dimensional ouseold eterogeneity tat as been studied by various autors. 5 Analytical approaces ave not yet provided closed form solutions for te multi-dimensional case. 6 Homogenous population: Te bencmark is wen workers are omogenous and te common values of teir productivity differentials and ousing demands are q,. Te utility differential is zero, i.e. u ( ; q p( tx, giving ouse price scedule p ( q tx/ and city edge ~ x q / t. Values of oter variables are 4 For example, deriving demand from Cobb-Douglas preferences would require specification of individuals income levels, but our structure imposes no relationsip between te urban income (productivity) differential and te absolute income level of te individual. 5 Te literature focuses on income eterogeneity and dates back to Beckman (1969) and Fujita (1989); see Maatttanen and Tervio (214) for recent work. 6 For recent progress on multi-dimensional matcing problems see Ciappori et al. (216) and, in an urban context, Gaigne et al. (217). 4

6 2 R sq / 2t, N ~ xs / sq / t, U. (9) Te entire urban surplus goes in rent, R/(R + U) = 1. Heterogeneous productivity differential: If individuals all ave te same demand for ousing, ten tose wo enter te city are indifferent as to were tey live. Heterogeneous productivity differentials means tat individuals wit productivity greater tan or equal to some (endogenous) cut-off value q ~ will enter te city, and te marginal worker as u ( q~, ; q~ p( tx for all x ~ x. Tis sets te ouse price scedule as p ( q ~ tx/, wit boundary value p ( ~ x ), so ~ x q ~ / t. Te marginal worker is suc tat te city is filled wen occupied by all individuals wit q q~, i.e. q ~ satisfies f q dq sx ~ (, ). (1) q ~ For present purposes, te point is simply tat te level of rent is set by te productivity differential of te marginal worker, q ~, and intra-marginal workers eac capture surplus q q ~. Te total size of tis, relative to city rents, is greater te larger is te dispersion of as will be sown in section 2.2. Heterogeneous ousing demand: Te converse case is were individuals ave different ousing demands but te same productivity differentials, q q 1. Tey coose different locations witin te city, wit lower individuals coosing iger rent (closer to te CBD) places. Eac type s location coice, x *( ), is given by te first order condition for maximisation of eqn. (1), p x* ( ) / x t / 5. Caracterisation of te equilibrium involves constructing te price function p( tat satisfies tis equation across a population of individuals wit different values of. To do tis, note tat te city out to distance x from te CBD will be occupied by individuals wit ousing demand less tan or equal to some value, (, defined by equality of ousing demand and supply up to x, ( f ( ) d sx. Tis gives a relationsip between and x wic can be used to turn te first order condition into a simple differential equation in x. Tis, togeter wit te boundary condition tat at te city edge (were p ( ~ x ) ) marginal workers ave zero utility differential, u ( ( ~ ; x *( ( ~ )), defines te price function. If as uniform distribution wit lower support and density f ten te price function takes te form q tx2/ 2 sq / tf 1/ 2 sx f p(. (11) / (see appendix 2). Tis decreases wit distance, is convex, and takes value zero at te city edge. However, rent does not capture te entire city surplus, except in te limiting case 1/ 2

7 were te distribution collapses to a point. Te division of surplus is illustrated in Fig. 1 (parameters given in appendix 3). Panel (a) gives te rent (ouse price) function, eqn. (11), and population density as a function of distance x from te CBD; in tis example equilibrium city size is unity and population density varies by a factor of 4:1, reflecting eterogeneity of. Panel (b) gives te distribution of te surplus. All workers ave te same productivity differential, q 1, but its division between commuting costs, rent and utility depends on teir ousing demand and cosen location. Working down from te top of Fig 1(b), part goes to commuting costs, tx, part goes to rent, p, and te remainder is differential utility. Low individuals live near te CBD and pay low commuting costs and ig rent, but since tey consume little ousing tey receive a net utility benefit. Integrating across te population of te city gives total differential rent R and utility U (areas p and u, eac weigted by population). In te example of Fig. 1 two-tirds of te urban surplus is captured by land rent, R/(R + U) = Figure 1: Heterogeneous ousing demand (a) Rent and population by distance (b) Commuting cost, ousing expenditure, utility, per person by distance q 1 tx Population Rent, p( u u(: p 2.2 Numerical exploration: We now move to te general case were individuals vary in bot caracteristics, q and, and assess te sares of te urban surplus captured by landowners and by workers. We look at different joint distributions of worker attributes, f ( ), and extend te model to include price elastic individual ousing demand and ouse supply. Answers are derived numerically, centred around te case in wic mean values of te productivity differential and ousing 7 Te dased line on Fig. 1b gives te utility of a worker of given type as a function of x, its maximum indicating te worker s location coice. Te locus of suc maxima is te curve u. 6

8 demand are set to unity, q 1, 1, total population (urban and non-urban) is 1, t = 1, te city is linear wit s = 5, and price elasticities ε, η, are initially equal to zero. If workers are omogenous ten values of endogenous variables come from eqns. (9). Te city edge is ~ x q / t 1 and urban population is N ~ xs / 5, i.e. ½ of te total population occupy te city. R =25, U =, so te entire city surplus goes to landowners. We start exploring eterogeneity assuming tat te distribution of caracteristics f ( ) is bivariate normal on te support q [,2], [.1,2], wit means q 1, 1, and variances and covariance q,, q. For computational reasons we smoot location coice using te logit function so te probability of individual of type { } living at x is ~ x ( ; exp[ u( : ] / exp[ u( : ] dx. (12) Te logit parameter μ is set at a ig value, so te function focuses probability quite tigtly on te location tat gives te igest utility, as will be clear from te figures tat follow. Tis and oter details of computation are given in appendix 3. Individuals sort into different locations, and visualising tis is assisted by Fig. 2, constructed wit.1,. Te orizontal plane is { } space, te vertical axis gives te q q density of individuals and te volume under te surface is number of people. Te entire population is te full bivariate normal, and panel (a) of te figure gives te subset of population resident in te city; it is te part of te normal density containing individuals wo gain utility from being in te city, i.e. tose wit relatively low and/or ig q. 8 Oter panels give population at different points in te city. Panel (b) is population in a range of locations close to te CBD. As would be expected given te urban rent gradient, tese are people wit very low, altoug covering a wide range of productivity differentials. Panel (c) is mid-city and (d) is near te edge, tese areas picking up individuals wit greater demand for ousing. Panels (b) (d) eac appear like a fin across te surface. Wit an infinitely fine grid and exact location coice tey would be lines giving te combinations { } for wic a particular x maximises, u ( ;, given equilibrium ouse prices. 8 I.e. te set { } for wic u ( ; x *( )). Te intuition beind te curvature of te edge of tis set is tat low people live near te CBD. An increment d requires a larger dq to compensate near te CBD were te cost of, p(, is ig. 7

9 Figure 2: Te urban population at different locations a) Entire city (b) Near CBD c) Mid-city d) Near edge Figure 3: (a) Rent and population by distance. (b) Average and q by distance. E(q Population, N( E( Rent, p( ~ x 8

10 Te intra-city distribution of population is summarised in Fig. 3. Panel (a) gives city population N( and ouse prices p( by distance from te CBD. House prices (and rent) are iger near te CBD, as expected, and so too is population density since low individuals coose to live tere. Panel (b) gives average ousing demand per person and productivity differential at eac distance (i.e. averaging over te caracteristics of individuals living at eac. Sorting means tat ousing demand increases wit distance, and so too does average productivity differential; even at te lower rents near te city edge ig individuals only coose to live in te city if tey are also ig q. Wat of our main question; wo captures te urban surplus? Answers are given in Figs. 4(a) and 4(b), eac of wic as te rent sare of city surplus, R/(R + U), on te vertical axis, and a measure of eterogeneity on te orizontal axis. Te lowest line on Fig. 4(a) as equal standard deviation of q and wit value given by te orizontal axis, σq = σ = σ; covariance is zero. Te line illustrates tat as σ goes to zero so te rent sare goes to unity, as it must. Higer values of σ lead to a muc lower rent sare, and convexity of te relationsip indicates tat even a small amount of eterogeneity as a large impact on te rent sare. Te sare drops below 5% if σ >.17. To interpret tis, σ >.17 means tat te proportion of total population wit caracteristics { } outside a circle wit radius.5 around te mean of {1, 1} is greater tan 45%. 9 Tis does not seem an excessive amount of eterogeneity, yet it alves te rent sare. We use tis as our reference case, and look at variations in oter parameters in turn. Te upper two lines in Fig. 4a switc off variance of q and in turn (or more precisely, set eac equal to σ/1). Tis approximately alves te effect of eterogeneity, te effect being similar wicever caracteristic remains eterogeneous. Te main message is tat eterogeneity in a single dimension is sufficient to reduce te rent sare, in line wit te analysis of section 2.1. Figure 4b looks at covariance. Te central line reproduces te reference case of Fig. 4a (zero covariance) and te bottom line assumes negative covariance, σq = -.95σ. Tis reduces te rent sare, te intuition being tat individuals wit te igest productivity differential also demand te least ousing, and tese are te individuals wo capture te most surplus. Conversely, positive covariance (top line) raises te rent sare because tere is little variation in ousing demand per unit productivity differential, and it is tis relativity tat is crucial. 9 From numerical investigation of te bivariate normal σ q = σ =.17, σ q =. 9

11 Figure 4: (a) Rent sare of surplus: by variance σq = σ/1, σ =σ, σq =. σq = σ, σ = σ/1, σq =. σq = σ =σ, σq =. (b) Rent sare of surplus: by variance and covariance σq = σ =σ, σq =.95σ. σq = σ =σ, σq =. σq = σ =σ, σq =.95σ. Table 1 reports te way tat outcomes vary wit oter parameters, and also adds price elasticity to individuals ousing demand and to ousing supply, as described in appendix 1. Eac row varies one or more features of te model from te base case (row 1) wic is as in Figs. 4. Making te city circular reduces its population (given our parameters) and as virtually no effect on te rent sare of surplus. Log-normality makes very little difference. A larger total (i.e. national) population increases city size and reduces te sare of population in te city, wit te effect of sifting te slice across te normal distribution in Fig. 2a, clockwise around te origin. Te rent sare rises, consistent wit te idea tat labour supply to te city is more elastic in a larger economy. 1

12 Remaining rows allow for positive price elasticity of ousing supply, η, and of individual s demand for ousing, ε. As outlined in appendix 1, ousing supply becomes sp ( and ousing demand becomes p (. 1 As expected, positive elasticities increase city population and reduce te sare of rent in surplus, tis going to less tan 4% in te more extreme case reported. Table 1: Sensitivity: Rent/city surplus R/(R + U) City population, (% of total population) Base: σq = σ =.1, σq = ; linear city, normal distribution, total population = 1, ε =, η =. Circular city Lognormal distribution Total population = (37.5%) Total population = (83.2%) Elasticity of demand for ousing, ε = Elasticity of supply of ousing, η = Bot; ε = η = Elasticity of demand for ousing, ε = Total population =2, ε =1.5, η = (74%) Total population =1, ε =1.5, η = (45%) 2.3 Partial capitalisation of local amenity improvements. An improvement in local amenities at a point in te city (suc as a transport improvement) leads to an increase in land values around te improvement, as would be expected and as is confirmed by empirical studies (e.g. Gibbons and Macin 25). Tis as led policy work to argue tat te increase in land values sould be used in cost-benefit analysis as a summary measure of te benefits of te improvement. 11 Wile correct in te limiting case (infinitely elastic supply of omogenous workers), it generally only captures a part of te benefit, some accruing to workers. How does tis work out in our more general framework? 1 Appendix 1 gives te and individual utility functions and construction tecnology from wic tese are derived. Note tat tis is te elasticity of supply per unit land and elasticity of eac individual s demand. Total supply of ousing in te city is assumed endogenous trougout, since we use an open city model wit endogenous city edge. 11 See DCLG (216) for use of tis in project appraisal. 11

13 An example is sown in fig. 5, were te cost of commuting is alved for locations at distance x [.4,.5], and uncanged elsewere (so giving x [.4,.5] better access to te CBD tan some places closer in). If tere were to be no population movement and utility levels of all individuals were eld constant, ten all te benefit of lower commuting cost would be captured in rent. However, tis is not te equilibrium. Individuals in te interval x [.4,.5] consume relatively muc ousing and now face iger rent, so tey will move. Tey are replaced by individuals from closer to te CBD wit lower ousing demand and terefore willing to pay te iger rent. Population density increases in te affected interval and decreases in areas closer to te CBD. Tis is illustrated by te solid line in fig. 5. Te cange in rent in eac place is given by te iger dased line, and te cange in utility (tat of all inabitants in eac place) by te lower dased line. Tere is a net utility increase and, across te urban population as a wole, tis experiment yields increase in surplus of wic 82% accrues in rent and 18% in ouseold utility. Tese numbers are particular to tis example and are sensitive to te exact magnitude and location of te transport improvement. However, tey make te point tat use of te rent increment as a measure of benefit sould take into account te equilibrium suffling of urban population tat, in te long run, a transport improvement will bring. Te suffling is associated wit a transfer of some of te benefit from rent to ouseold utility. Figure 5: A reduction in commuting costs; canges in population, rent and utility Cange in rent Cange in population Δ utility Cange in utility.5 distance, x 1 12

14 3. Model 2: a system of cities Do tese or similar results obtain wit a system of cities? We investigate tis in a restatement of te model used above. We now assume tat cities ave no internal geograpy and reinterpret x as an index over a set of cities, x (,1]. Te utility derived from living and working in city x (relative to being outside, i.e. not in any city) is u( ; qx p(. (1 ) Comparing tis wit eqn. 1, tere are no commuting costs but eterogeneity across cities is introduced by supposing tat te productivity differential of a type q worker is qx, i.e. low x cities deliver little additional productivity, by construction. Many mecanisms could be posited to generate tis cross-city eterogeneity but, in te spirit of tis paper, we give it te simplest possible form and investigate its implications. Eqns. (2), (3) and (4) are as before, wit N( ; ( ; f ( ) now interpreted as te number of people of type { } living in city x. In equilibrium all cities will operate, since tere is a positive (if small) productivity differential for all cities relative to outside. Since tere is no internal geograpy and p( is te price of all ouses in city-index x, eqn. (5) for te city edge no longer exists, and te population of eac city is N(, as given by eqn. (6). Eqns. (7) and (8) still old, now wit ~ x 1, i.e. adding over all populated cities. We report results for te case in wic tere are 2 cities,.1, and tere is a positive elasticity of ousing supply, η = Fig. 6a illustrates te equilibrium distribution of population, analogous to te intra-city distribution of Fig. 3a. As expected, ig index (so ig productivity-differential) cities are larger and ave iger ouse prices. Fig. 6b illustrates tat tis is driven by ig productivity-differential individuals sorting very strongly to cities tat lever teir productivity advantage, ig x cities. Te consequent ig ouse prices in tese cities induce furter sorting according to individuals ousing demand. Wat about te sare of urban surplus going to rents? Wit.1, te sare across all cities is 59%, virtually identical to tat reported for model 1 in Fig. 4 and Table 1. Te sare falls wit variance, as before. More interesting, is to see ow te sare varies wit x, te city productivity increment, and ence city size. Tis is illustrated in Fig.7. Te bars give te rent sare by city and indicate tat te sare of rent in total surplus is smaller for larger (ig city-inde cities. 13 Te ig productivity differential gives tese cities an q q q q 12 η =.33 implies tat 25% of ousing costs are construction, and 75% land rent, see appendix. In tis example te total urban population is 6% of te entire population. Te example as ε =, and raising tis elasticity reduces te rent sare, as in Table Total land rent is iger in tese cities, altoug it is a smaller sare of total surplus. 13

15 absolute advantage for all individuals and tis bids up ouse prices. Comparative advantage makes tese cities attractive for ig q and low individuals (as illustrated in Fig. 6b), and tese are precisely te individuals wo capture a larger sare of te surplus. Te city wit te igest index and ence te largest city captures just 39% of te surplus it generates in te form of land rent. Figure 6: Heterogeneous cities: (a) Rent and population by city index. (b) Average and q by city index E( Population House price E(q Figure 7: Rent sare of surplus: by city index. σq = σ =.1, σq =, η = Concluding comments Homogenous labour is a useful simplifying assumption in city modelling, but we know tat a key feature of cities is te distinctive skill mix of teir population (e.g. Glaeser and Gottlieb 14

16 29). Tis paper as argued tat te assumption of omogenous labour is particularly misleading wen it comes to tinking about te distribution of urban surplus between land and labour. Wit omogenous labour te sare of land-rent in surplus takes te extreme boundary value, of unity. Heterogeneity can only move tis in one direction, so te question is, by ow muc? Tis paper suggests tat te answer is, a lot. Heterogeneous productivity differentials or ousing demands pull te sare down, in a convex relationsip so tat even small amounts of eterogeneity matter. Combining bot dimensions of eterogeneity suggests tat, for modest degrees of population eterogeneity, rent migt capture between one and two-tirds of total surplus, te rest accruing to individuals. Te result extends from a single city context to multiple cities wic differ in te productivity differential tey offer to workers; in tis case it is te largest and most productive cities tat ave te lowest sare of rent in te urban surplus tat tey create. Tese findings matter for our tinking about te income distributional implications of urbanisation, and about urban public finance. Tey may also set te stage for future empirical and policy work tat seeks to understand and quantify te various impacts of urban development. References: Albouy, D. (216), Wat are cities wort? Land rents, local productivity and te total value of amenities, Review of Economic Statistics 98, Arnott R. J. and J. E. Stiglitz (1979) Aggregate Land Rents, Expenditure on Public Goods, and Optimal City Size, Quarterly Journal of Economics. 93 (4): Beckmann, M.J. (1969) On te distribution of urban rent and residential density. Journal of Economic Teory 1: Berens, K. And F. Robert-Nicoud (215), Agglomeration Teory wit Heterogeneous Agents in G. Duranton, J. V. Henderson, and William C. Strange (eds.) Handbook of Regional and Urban Economics, vol. 5, Elsevier: Nort-Holland, pp Berens, K. G. Duranton, and F. Robert-Nicoud, (214) Productive Cities: Sorting, Selection, and Agglomeration, Journal of Political Economy 122, Ciappori, P.-A., R. McCann and B. Pass (216) Multidimensional matcing, Columbia University, NY. DCLG (216) Te DCLG appraisal guide, Dept. for Communities and Local Government, London, ttps:// Duranton, G., and D. Puga (215) Urban land use in G. Duranton, J. V. Henderson, and William C. Strange (eds.) Handbook of Regional and Urban Economics, vol. 5, Elsevier: Nort-Holland, pp

17 Duranton, G., and D. Puga (24). Micro-foundations of urban agglomeration economies in J. V. Henderson and J.-F. Tisse (eds.) Handbook of Regional and Urban Economics, vol. 4, Elsevier: Nort-Holland, pp Fujita, M. (1989) Urban Economic Teory. Cambridge: Cambridge University Press. Gaigne, C., H. Koster, F. Moizeau and J-F. Tisse (217) Amenities and te social structure of cities, CEPR dp Gibbons, S., and Macin, S. (25) Valuing rail access using transport innovations. Journal of Urban Economics, 57(1): Glaeser, E.L., and J.D. Gottlieb (29). Te wealt of cities: agglomeration economies and spatial equilibrium in te United States, Journal of Economic Literature, 47, Maattanen, N. and M. Tervio (214). Income distribution and ousing prices; an assignment model approac, Journal of Economic Teory, 11, Appendix 1: Housing demand and supply. Price elastic individual ousing demand: Housing demand is generalised by supposing tat te net utility an individual of type derives from consuming H units of ousing at price p( is 1/ ( 1) / v H /( 1) p( H. Utility maximising coice of H is 16 H p( 1 and maximised utility is given by (indirect) utility function v * ( p(, ) p( /(1 ). Te price of outside ousing is denoted p, so te more general form of eqn (1) is u( ; q 1 1 p( p /(1 ) tx. (1A) Te term in square brackets corresponds to te differential ouse price of te text. In section 3 eqn. (1 ) becomes 1 1 p( p /(1 ) u ( ; qx. (1 A) Price elastic ousing supply: Te cost of building S units of ousing on a unit of land is (1 ) / S /( 1), increasing and convex so η >. Te profit of a developer is (1 ) / ( x ) p( S S /( 1) and optimal coice gives S p(. Maximised profit per (1 ) unit land is * ( p ( ) p( /( 1). Land rents are equal to maximised profits. Notice also tat rents are fraction 1/(η +1) of gross revenue (= ouseold expenditure on ousing). Tus, η =.5 implies tat land rents are 2/3rds of ousing expenditure, te rest being construction cost. Differential rent at place x wic as s( units of land is (1 ) (1 ) p( p /( 1) r *( p( ) s(.

18 Equilibrium: Market clearing for ousing, eqn. (4) is p ( N( ; dqd s( p( q Te city edge, eqn. (5), is p( ~ x ) p. Total differential utility is U ~ x q Total differential rent is R ~ x. (4A) 1 1 q p( p /(1 ) tx N( ; ddqdx. (7A) ~ x (1 ) (1 ) r *( p( ) dx s( p( p /( 1) dx (8A) Analytical results and te base case are derived wit ε = η = and p, and equations above collapse to tose of te text. Simulation generalises to allow ε, η >, and renormalizes using p 1 (tis positive lower bound required simply because an iso-elastic demand function gives infinite demand for ousing at p = ). Appendix 2: Te assignment problem wit eterogeneous ousing demand: Te first order condition for coice of location is p x* ( ) / x t / distribution of eqn. (6) gives ( f d sx 2 1/ 2 first order condition becomes / x t2 sx / f 2 p( K tf / s2 sx / f 1/ 2. Wit uniform 2 implying tat ( 2 sx / f 1/ 2. Te p. Integrating, tis gives were K is a constant of integration. At te city edge te land price and differential utility are bot equal to zero, ( ~ 2 p K tf / s2 sx f and u ( ; q t~ x. Togeter tese give te constant of integration 2 2 sq / tf 1/ 2 K tf / s and ence te ouse price function, eqn. (8), sq / tf 1/ 2 sx f / 1/ 2 p( tf / s. Tis can be rearranged, denoting te terms in square brackets a, b, and noting tat a 1/ 2 1/ / 1/ 2 2 b a b / a b, so p( ~ x ) 2q tx/ a 1/ b 1/ q tx2/ 2 sq / tf 1/ 2 sx f / ~ /. Tis gives 1/ 2 p(. Hence, if and f ten p( x q tx/ ~ ). Fig. 1 is constructed wit q 1, t = 1, s = 1,. 5 and f. 5. 1/ 2 17

19 Appendix 3: Numerical procedures: Simulations are derived from Matlab, using a 9x9 grid of { }, ranging from q [,2], [.1,2]. In section 2 tere are 75 equal spaced distances tat may be occupied by te city. In section 3 tere are 2 cities. Te logit parameter is μ = 25. All simulations used in Fig.2 and beyond use te general structure outlined in appendix 1 (ousing demand and supply), wic reduces to tat of te text if elasticities ε and η are equal to zero and p =. 18