Who gets the urban surplus?

Who gets the urban surplus? Abstract High productivity in cities creates an economic surplus relative to other areas. How is this divided between workers and landowners? Simple models with homogenous labour suggest that it accrues largely—or entirely—in the form of land rents. This article shows that heterogeneity of labour in two main dimensions (productivity differentials and housing demands) radically changes this result. Even a modest amount of heterogeneity can drive the land share of surplus down to two-thirds or lower, as high productivity and/or low housing demand individuals receive large utility gains. It follows that land value appreciation understates the value of urban amenities and infrastructure. In a system of cities the sorting of workers across cities means that, while total rent is highest, the rent share of surplus is lowest in the largest and most productive cities. 1. Introduction Who benefits from the high productivity levels of workers in cities? A standard answer is that the surplus goes principally to landowners. The limiting case is that there is an infinitely elastic supply of homogenous workers so—with this assumption fixing their reservation utility levels and with no other claimants on the surplus—landowners must take 100%. This article is an exploration of what happens when workers are heterogeneous, focusing on two salient dimensions of heterogeneity. One is the productivity differential the worker receives by locating in the city; the other is the amount of housing the worker consumes. The idea is simple. Workers with a large productivity differential and small housing demand will do better than those with the opposite characteristics, and will, therefore, capture some of the urban surplus. Fundamentally, the presence of sufficient heterogeneity means that the assumption of an infinitely elastic supply of workers becomes untenable. If there were infinitely elastic supply the city would be occupied entirely by the type of worker that gets the greatest return (bankers with a preference for small apartments). We know that—while sorting takes place—this is not the case. The questions then are: moving away from homogeneity, how much of the surplus still goes to landlords? Is the change in their share large or small, and what does it depend on? We find that the effect is large. Modest amounts of heterogeneity appear to reduce the share received by landlords to between two-thirds and one-half. The relationship is convex, so that even a small amount of heterogeneity brings a relatively large reduction in rent share. Heterogeneity in either the productivity differential or housing demand reduces the rent share, and the effect is largest when there is a negative correlation between the two attributes of individuals. Looking at systems of cities, cities that offer relatively large productivity differentials have the highest absolute levels of rent, but the lowest share of rent in surplus; these are the largest cities, and those in which sorting has the most extreme effect. These answers are important for a number of issues. Analytically, they matter for our understanding of urban models. Empirically, much work rests on the assumed homogeneity of workers, for example, recent work by Albouy (2016) deriving the value of cities from data on urban wages. Evidently, answers matter for income distribution. They also matter for public finance and the extent to which land taxation can capture the value created by urban agglomerations, as studied in ‘Henry George’ theorems. They matter for project appraisal, since improvements in urban infrastructure increase rents and land values by less than the full value of benefits created. Existing literature recognises the critical nature of the homogeneity assumption, as succinctly stated by Arnott and Stiglitz (1979) who state ‘the conceptual basis of capitalization studies is sound only when marginal individuals are very similar to infra-marginal individuals …’ (496). Duranton and Puga (2015) discuss the issues that arise as heterogeneous individuals sort into different locations, and the technical complexities that follow. The recent literature on heterogeneity focuses on constructing large-scale models that analyse both the sources of heterogeneity and their implication for city systems, as exemplified by Behrens et al. (2014) (see also the survey by Behrens and Robert-Nicoud (2015)). The approach of this article is, in contrast, to focus tightly on the question posed in the title and derive a quantitative sense of the answer to the question. We take spatial productivity differentials and housing preferences as exogenous, drawing out their implications rather than modelling their causes. The remainder of the article is as follows. Section 2 looks at an economy containing a single city, and at individuals who decide whether or not to enter the city and where to live in the city. Section 2.1 presents some analytical results (solving the assignment problem for two simple cases), and Section 2.2 presents the core quantitative results on the division of urban surplus between landowners and individuals, based on numerical analysis of the model. In Section 2.3 we illustrate how the benefits of an urban transport improvement, affecting just part of the city, are divided between rents and residents’ utility. Section 3 moves to a multi-city variant of the model in which individuals sort between cities, and demonstrates how the share of surplus that is captured by rent varies across city types, which is lowest in the largest and most productive cities. 2. Model 1: a single city We start by considering an economy with a single city, an ‘outside’ location and a given total population. Some individuals choose to live and work in the city, and others outside. Each worker has a productivity differential between working in the city and outside, denoted q, and a housing preference parameter h. Sources of these productivity differentials have been extensively analysed elsewhere.1 In the baseline version we outline in this section h is simply the quantity of housing the individual consumes, a more general case being set out in the Appendix and utilised in simulations. For each worker the attributes q and h are exogenous, as is the density function f(q,h) over {q, h} space that gives the distribution of these attributes over the population. Urban workers commute to jobs in the central business district (CBD), and a worker living at distance x from the CBD faces commuting costs tx and pays price differential (relative to living outside the city) p(x) for a unit of housing.2 The utility an individual with attributes q, h, derives from living in the city at distance x from the CBD, as compared to living outside the city is, therefore, the productivity increment minus incremental housing costs and commuting costs, u(q,h;x)=q−hp(x)−tx. (1) Individuals choose to reside in the place that yields highest utility. Given the house price function p(x), an individual with characteristics {q, h} makes choices: live in city atx*(q,h)=arg maxx u(q,h;x) if u(q,h;x*(q,h))≥0,live outside city ifu(q,h;x*(q,h))<0. (2) We define π(q,h;x) as the indicator function, equal to unity if type {q, h} lives in the city at distance x and equal to zero otherwise, so the number of type {q, h} people choosing to live at x is N(q,h;x)=π(q,h;x)f(q,h). (3) The demand for housing at each distance x is then ∫hh∫qN(q,h;x)dqdh. Housing at each distance is initially assumed to be in fixed supply, s(x), so market clearing for housing at x is ∫hh∫qN(q,h;x)dqdh=s(x). (4) The simplest case is a linear city in which land area and the number of houses at each distance is a constant, s(x) = s. The house price differential, p(x), adjusts to clear the market, and the city edge (denoted x˜) is where this price differential equals zero, p(x˜)=0. (5) Equilibrium is characterised by these five equations giving utility, location choice, numbers of individuals of each type at each place, the house price function p(x) and the city edge.3 From this we derive the variables we are interested in. The total number of people living at x: N(x)=∫q∫hN(q,h,x)dhdq. (6) The total utility differential accruing to urban workers: U=∫0x˜∫q∫hu(q,h;x)N(q,h;x)dhdqdx. (7) With fixed supply of housing and no construction costs, the house price differential p(x) is equal to land rent so total differential rent is: R=∫0x˜s(x)p(x)dx. (8) The total urban surplus is differential rent and utility, R + U, and our primary question is, how is this surplus divided between these two components? Before answering these questions, we note that the expressions above give the simplest form of the model. This is used in the analysis of next subsection and is the base case throughout the article. In simulation we also use a version that is more general in two respects. First, the demand for housing of individuals of each type h also depends on price, with elasticity ɛ; whilst adding a price elasticity we retain zero income elasticity, a consequence of our working with productivity and income differentials not levels.4 Secondly, housing supply is endogenised, adding a construction sector that chooses supply of housing per unit land, this creating a price elasticity η of housing supply at each distance, x. The preference and technology apparatus that supports these generalisations is set out in Appendix A, as are the more general form of Equations (1–8). 2.1. Analysis Drawing out answers to the question posed in this article, for a wide range of different parameters, city shapes, distributions of individual characteristics and specifications of preferences and technology, requires numerical simulations. These are presented and discussed in following sections of the article, while in this section we briefly derive analytical solutions for some special cases. We take each of our two dimensions of heterogeneity separately, in line with the one-dimensional household heterogeneity that has been studied by various authors.5 Analytical approaches have not yet provided closed-form solutions for the multidimensional case.6 2.1.1. Homogenous population The benchmark is when workers are homogenous and the common values of their productivity differentials and housing demands are q¯, h¯. The utility differential is zero, i.e. u(q¯,h¯;x)=q¯−h¯p(x)−tx=0, giving house price schedule p(x)=(q¯−tx)/h¯ and city edge x˜=q¯/t. Values of other variables are R=sq¯2/2h¯t,N=x˜s/h¯=sq¯/h¯t,U=0. (9) The entire urban surplus goes in rent, R/(R + U) = 1. 2.1.2. Heterogeneous productivity differential If individuals all have the same demand for housing h¯, then those who enter the city are indifferent as to where they live. Heterogeneous productivity differentials means that individuals with productivity greater than or equal to some (endogenous) cut-off value q˜ will enter the city, and the marginal worker has u(q˜,h¯;x)=q˜−h¯p(x)−tx=0 for all x≥x˜. This sets the house price schedule as p(x)=(q˜−tx)/h¯, with boundary value p(x˜)=0, so x˜=q˜/t. The marginal worker is such that the city is filled when occupied by all individuals with q≥q˜, i.e. q˜ satisfies h¯∫q˜∞f(q,h¯)dq=sx˜. (10) For present purposes, the point is simply that the level of rent is set by the productivity differential of the marginal worker, q˜, and intramarginal workers each capture surplus q−q˜>0. The total size of this, relative to city rents, if greater the larger is the dispersion of q, as will be shown in Section 2.2. 2.1.3. Heterogeneous housing demand The converse case is where individuals have different housing demands h but the same productivity differentials, q=q¯=1. They choose different locations within the city, with lower h individuals choosing higher rent (closer to the CBD) places. Each type’s location choice, x*(q¯,h), is given by the first-order condition for maximisation of Equation (1), ∂p(x*(q¯,h))/∂x=−t/h. Characterisation of the equilibrium involves constructing the price function p(x) that satisfies this equation across a population of individuals with different values of h. To do this, note that the city out to distance x from the CBD will be occupied by individuals with housing demand less than or equal to some value, h(x), defined by equality of housing demand and supply up to x, ∫0h(x)hf(q¯,h)dh=sx. This gives a relationship between h and x which can be used to turn the first-order condition into a simple differential equation in x. This, together with the boundary condition that at the city edge (where p(x˜)=0) marginal workers have zero utility differential, u(q¯,h(x˜);x*(q¯,h(x˜)))=0, defines the price function. If h has uniform distribution with lower support h0 and density fh then the price function takes the form p(x)=(q¯−tx)2/{[2sq¯/tfh+h02]1/2+[2sx/fh+h02]1/2} (11) (see Appendix B). This decreases with distance, is convex, and takes value zero at the city edge. However, rent does not capture the entire city surplus, except in the limiting case where the distribution collapses to a point. The division of surplus is illustrated in Figure 1 (parameters given in Appendix C). Panel (a) gives the rent (house price) function, Equation (11), and population density as a function of distance x from the CBD; in this example equilibrium city size is unity and population density varies by a factor of 4: 1, reflecting heterogeneity of h. Panel (b) gives the distribution of the surplus. All workers have the same productivity differential, q¯=1, but its division between commuting costs, rent and utility depends on their housing demand and chosen location. Working down from the top of Figure 1(b), part goes to commuting costs, tx, part goes to rent, ph, and the remainder is differential utility. Low h individuals live near the CBD and pay low commuting costs and high rent, but since they consume little housing they receive a net utility benefit. Integrating across the population of the city gives total differential rent R and utility U (areas ph and u, each weighted by population). In the example of Figure 1 two-thirds of the urban surplus is captured by land rent, R/(R + U) = 0.66.7 Figure 1 View largeDownload slide Heterogeneous housing demand: (a) Rent and population by distance. (b) Commuting cost, housing expenditure, utility, per person by distance. Figure 1 View largeDownload slide Heterogeneous housing demand: (a) Rent and population by distance. (b) Commuting cost, housing expenditure, utility, per person by distance. 2.2. Numerical exploration We now move to the general case where individuals vary in both characteristics, q and h, and assess the shares of the urban surplus captured by landowners and by workers. We look at different joint distributions of worker attributes, f(q,h), and extend the model to include price elastic individual housing demand and house supply. Answers are derived numerically, centred around the case in which mean values of the productivity differential and housing demand are set to unity, q¯=1, h¯=1, total population (urban and non-urban) is 100, t = 1, the city is linear with s = 50, and price elasticities ɛ, η, are initially equal to zero. If workers are homogenous then values of endogenous variables come from Equation (9). The city edge is x˜=q¯/t=1 and urban population is N=x˜s/h¯=50, i.e. ½ of the total population occupy the city. R =25, U = 0, so the entire city surplus goes to landowners. We start exploring heterogeneity assuming that the distribution of characteristics f(q,h) is bivariate normal on the support q∈[0,2], h∈[0.1,2], with modes q¯=1, h¯=1, and variances and covariance σq2,σh2,σqh2. For computational reasons we smooth location choice using the logit function so the probability of individual of type {q, h} living at x is π(q,h;x)=exp [μu(q,h:x)]/∫0x˜exp [μu(q,h:x)]dx. (12) The logit parameter μ is set at a high value, so the function focuses probability quite tightly on the location that gives the highest utility, as will be clear from the figures that follow. This and other details of computation are given in Appendix C. Individuals sort into different locations, and visualising this is assisted by Figure 2, constructed with σq=σh=0.1,σqh=0. The horizontal plane is {q, h} space, the vertical axis gives the density of individuals and the volume under the surface is number of people. The entire population is the full bivariate normal, and Panel (a) of the figure gives the subset of population resident in the city; it is the part of the normal density containing individuals who gain utility from being in the city, i.e. those with relatively low h and/or high q.8 Other panels give population at different points in the city. Panel (b) is population in a range of locations close to the CBD. As would be expected given the urban rent gradient, these are people with very low h, although covering a wide range of productivity differentials. Panel (c) is mid-city and (d) is near the edge, these areas picking up individuals with greater demand for housing. Panels (b)–(d) each appear like a fin across the surface. With an infinitely fine grid and exact location choice they would be lines giving the combinations {q, h} for which a particular x maximises, u(q,h;x), given equilibrium house prices. Figure 2 View largeDownload slide The urban population at different locations: (a) Entire city. (b) Near CBD. (c) Mid-city. (d) Near edge. Figure 2 View largeDownload slide The urban population at different locations: (a) Entire city. (b) Near CBD. (c) Mid-city. (d) Near edge. The intracity distribution of population is summarised in Figure 3. Panel (a) gives city population N(x) and house prices p(x) by distance from the CBD. House prices (and rent) are higher near the CBD, as expected, and so too is population density since low h individuals choose to live there. Panel (b) gives average housing demand per person and productivity differential at each distance (i.e. averaging over the characteristics of individuals living at each x). Sorting means that housing demand increases with distance, and so too does average productivity differential; even at the lower rents near the city edge high h individuals only choose to live in the city if they are also high q. Figure 3 View largeDownload slide (a) Rent and population by distance. (b) Average h and q by distance. Figure 3 View largeDownload slide (a) Rent and population by distance. (b) Average h and q by distance. What about our main question; who captures the urban surplus? Answers are given in Figures 4(a) and 4(b), each of which has the rent share of city surplus, R/(R + U), on the vertical axis, and a measure of heterogeneity on the horizontal axis. The lowest line on Figure 4(a) has equal standard deviation of q and h with value given by the horizontal axis, σq = σh= σ; covariance is zero. The line illustrates that as σ goes to zero so the rent share goes to unity, as it must. Higher values of σ lead to a much lower rent share, and convexity of the relationship indicates that even a small amount of heterogeneity has a large impact on the rent share. The share drops below 50% if σ > 0.17. To interpret this, σ > 0.17 means that the proportion of total population with characteristics {q, h} outside a circle with radius 0.5 around the mean of {1, 1} is greater than 45%.9 This does not seem an excessive amount of heterogeneity, yet it halves the rent share. Figure 4 View largeDownload slide (a) Rent share of surplus: by variance. (b) Rent share of surplus: by variance and covariance. Figure 4 View largeDownload slide (a) Rent share of surplus: by variance. (b) Rent share of surplus: by variance and covariance. We use this as our reference case, and look at variations in other parameters in turn. The upper two lines in Figure 4(a) switch off variance of q and h in turn (or more precisely, set each equal to σ/100). This approximately halves the effect of heterogeneity, the effect being similar whichever characteristic remains heterogeneous. The main message is that heterogeneity in a single dimension is sufficient to reduce the rent share, in line with the analysis of Section 2.1. Figure 4(b) looks at covariance. The central line reproduces the reference case of Figure 4(a) (zero covariance) and the bottom line assumes negative covariance, σhq= −0.95σ. This reduces the rent share, the intuition being that individuals with the highest productivity differential also demand the least housing, and these are the individuals who capture the most surplus. Conversely, positive covariance (top line) raises the rent share because there is little variation in housing demand per unit productivity differential, and it is this relativity that is crucial. Table 1 reports the way that outcomes vary with other parameters, and also adds price elasticity to individuals’ housing demand and to housing supply, as described in Appendix A. Each row varies one or more features of the model from the base case (row 1) which is as in Figure 4. Making the city circular reduces its population (given our parameters) and has virtually no effect on the rent share of surplus. Lognormality makes very little difference. A larger total (i.e. national) population increases city size and reduces the share of population in the city, with the effect of shifting the slice across the normal distribution in Figure 2(a), clockwise around the origin. The rent share rises, consistent with the idea that labour supply to the city is more elastic in a larger economy. Table 1 Sensitivity Rent/city surplus R/(R + U) City population (% of total population) Base: σq = σh = 0.1, σhq= 0; linear city, normal distribution, total population = 100, ɛ =0, η = 0. 0.583 58.8 Circular city 0.583 40.2 Lognormal distribution 0.589 57.3 Total population = 200 0.659 75.1 (37.5) Total population = 50 0.441 41.6 (83.2) Elasticity of demand for housing, ɛ = 0.5 0.529 66.3 Elasticity of supply of housing, η = 0.5 0.536 65.5 Both; ɛ = η = 0.5 0.476 73.0 Elasticity of demand for housing, ɛ = 1.5 0.375 83.0 Total population =200, ɛ =1.5, η = 0.5 0.390 149 (74) Total population =1000, ɛ =1.5, η = 0.5 0.432 447 (45) Rent/city surplus R/(R + U) City population (% of total population) Base: σq = σh = 0.1, σhq= 0; linear city, normal distribution, total population = 100, ɛ =0, η = 0. 0.583 58.8 Circular city 0.583 40.2 Lognormal distribution 0.589 57.3 Total population = 200 0.659 75.1 (37.5) Total population = 50 0.441 41.6 (83.2) Elasticity of demand for housing, ɛ = 0.5 0.529 66.3 Elasticity of supply of housing, η = 0.5 0.536 65.5 Both; ɛ = η = 0.5 0.476 73.0 Elasticity of demand for housing, ɛ = 1.5 0.375 83.0 Total population =200, ɛ =1.5, η = 0.5 0.390 149 (74) Total population =1000, ɛ =1.5, η = 0.5 0.432 447 (45) Table 1 Sensitivity Rent/city surplus R/(R + U) City population (% of total population) Base: σq = σh = 0.1, σhq= 0; linear city, normal distribution, total population = 100, ɛ =0, η = 0. 0.583 58.8 Circular city 0.583 40.2 Lognormal distribution 0.589 57.3 Total population = 200 0.659 75.1 (37.5) Total population = 50 0.441 41.6 (83.2) Elasticity of demand for housing, ɛ = 0.5 0.529 66.3 Elasticity of supply of housing, η = 0.5 0.536 65.5 Both; ɛ = η = 0.5 0.476 73.0 Elasticity of demand for housing, ɛ = 1.5 0.375 83.0 Total population =200, ɛ =1.5, η = 0.5 0.390 149 (74) Total population =1000, ɛ =1.5, η = 0.5 0.432 447 (45) Rent/city surplus R/(R + U) City population (% of total population) Base: σq = σh = 0.1, σhq= 0; linear city, normal distribution, total population = 100, ɛ =0, η = 0. 0.583 58.8 Circular city 0.583 40.2 Lognormal distribution 0.589 57.3 Total population = 200 0.659 75.1 (37.5) Total population = 50 0.441 41.6 (83.2) Elasticity of demand for housing, ɛ = 0.5 0.529 66.3 Elasticity of supply of housing, η = 0.5 0.536 65.5 Both; ɛ = η = 0.5 0.476 73.0 Elasticity of demand for housing, ɛ = 1.5 0.375 83.0 Total population =200, ɛ =1.5, η = 0.5 0.390 149 (74) Total population =1000, ɛ =1.5, η = 0.5 0.432 447 (45) Remaining rows allow for positive price elasticity of housing supply, η, and of individual’s demand for housing, ɛ. As outlined in Appendix A, housing supply becomes sp(x)η and housing demand becomes hp(x)−ɛ.10 As expected, positive elasticities increase city population and reduce the share of rent in surplus, this going to less than 40% in the more extreme case reported. 2.3. Partial capitalisation of local amenity improvements An improvement in local amenities at a point in the city (such as a transport improvement) leads to an increase in land values around the improvement, as would be expected and as is confirmed by empirical studies (e.g. Gibbons and Machin, 2005). This has led policy work to argue that the increase in land values should be used in cost-benefit analysis as a summary measure of the benefits of the improvement.11 While correct in the limiting case (infinitely elastic supply of homogenous workers), it generally only captures a part of the benefit, some accruing to workers. How does this work out in our more general framework? An example is shown in Figure 5, where the cost of commuting is halved for locations at distance x∈[0.4,0.5], and unchanged elsewhere (so giving x∈[0.4,0.5] better access to the CBD than some places closer in). If there were to be no population movement and utility levels of all individuals were held constant, then all the benefits of lower commuting costs would be captured in rent. However, this is not the equilibrium. Individuals in the interval x∈[0.4,0.5] consume relatively much housing and now face higher rent, so they will move. They are replaced by individuals from closer to the CBD with lower housing demand and, therefore, willing to pay the higher rent. Population density increases in the affected interval and decreases in areas closer to the CBD. This is illustrated by the solid line in Figure 5. The change in rent in each place is given by the higher dashed line, and the change in utility (that of all inhabitants in each place) by the lower dashed line. There is a net utility increase and, across the urban population as a whole, this experiment yields increase in surplus of which 82% accrues in rent and 18% in household utility. Figure 5 View largeDownload slide A reduction in commuting costs; changes in population, rent and utility. Figure 5 View largeDownload slide A reduction in commuting costs; changes in population, rent and utility. These numbers are particular to this example and are sensitive to the exact magnitude and location of the transport improvement. However, they make the point that use of the rent increment as a measure of benefit should take into account the equilibrium shuffling of urban population that, in the long run, a transport improvement will bring. The shuffling is associated with a transfer of some of the benefit from rent to household utility. 3. Model 2: a system of cities Do these or similar results obtain with a system of cities? We investigate this in a restatement of the model used above. We now assume that cities have no internal geography and reinterpret x as an index over a set of cities, x∈(0,1]. The utility derived from living and working in city x (relative to being ‘outside’, i.e. not in any city) is u(q,h;x)=qx−hp(x). (13) Comparing this with Equation (1), there are no commuting costs but heterogeneity across cities is introduced by supposing that the productivity differential of a type q worker is qx, i.e. low x cities deliver little additional productivity, by construction. Many mechanisms could be posited to generate this cross-city heterogeneity but, in the spirit of this article, we give it the simplest possible form and investigate its implications. Equations (2), (3) and (4) are as before, with N(q,h;x)=π(q,h;x)f(q,h) now interpreted as the number of people of type {q, h} living in city x. In equilibrium all cities will operate, since there is a positive (if small) productivity differential for all cities relative to ‘outside’. Since there is no internal geography and p(x) is the price of all houses in city index x, Equation (5) for the city edge no longer exists, and the population of each city is N(x), as given by Equation (6). Equations (7) and (8) still hold, now with x˜=1, i.e. adding over all populated cities. We report results for the case in which there are 20 cities, σq=σh=0.1,σqh=0 and there is a positive elasticity of housing supply, η = 0.33.12Figure 6(a) illustrates the equilibrium distribution of population, analogous to the intracity distribution of Figure 3(a). As expected, high index (so high productivity-differential) cities are larger and have higher house prices. Figure 6(b) illustrates that this is driven by high-productivity differential individuals sorting very strongly to cities that lever their productivity advantage, high x cities. The consequent high house prices in these cities induce further sorting according to individuals’ housing demand. Figure 6 View largeDownload slide Heterogeneous cities: (a) Rent and population by city index. (b) Average h and q by city index. Figure 6 View largeDownload slide Heterogeneous cities: (a) Rent and population by city index. (b) Average h and q by city index. What about the share of urban surplus going to rents? With σq=σh=0.1,σqh=0 the share across all cities is 59%, virtually identical to that reported for Model 1 in Figure 4 and Table 1. The share falls with variance, as before. More interesting, is to see how the share varies with x, the city productivity increment, and hence city size. This is illustrated in Figure 7. The bars give the rent share by city and indicate that the share of rent in total surplus is smaller for larger (high city index) cities.13 The high-productivity differential gives these cities an absolute advantage for all individuals and this bids up house prices. Comparative advantage makes these cities attractive for high q and low h individuals (as illustrated in Figure 6(b)), and these are precisely the individuals who capture a larger share of the surplus. The city with the highest index—and hence the largest city—captures just 39% of the surplus it generates in the form of land rent. Figure 7 View largeDownload slide Rent share of surplus: by city index. σq= σh= 0.1, σhq= 0, η = 0.33. Figure 7 View largeDownload slide Rent share of surplus: by city index. σq= σh= 0.1, σhq= 0, η = 0.33. 4. Concluding comments Homogenous labour is a useful simplifying assumption in city modelling, but we know that a key feature of cities is the distinctive skill mix of their population (e.g. Glaeser and Gottlieb, 2009). This article has argued that the assumption of homogenous labour is particularly misleading when it comes to thinking about the distribution of urban surplus between land and labour. With homogenous labour the share of land rent in surplus takes the extreme boundary value, of unity. Heterogeneity can only move this in one direction, so the question is, by how much? This article suggests that the answer is, a lot. Heterogeneous productivity differentials or housing demands pull the share down, in a convex relationship so that even small amounts of heterogeneity matter. Combining both dimensions of heterogeneity suggests that, for modest degrees of population heterogeneity, rent might capture between one- and two-thirds of total surplus, the rest accruing to individuals. The result extends from a single city context to multiple cities which differ in the productivity differential they offer to workers; in this case it is the largest and most productive cities that have the lowest share of rent in the urban surplus that they create. These findings matter for our thinking about the income distributional implications of urbanisation, and about urban public finance. They may also set the stage for future empirical and policy work that seeks to understand and quantify the various impacts of urban development. Footnotes 1 See Duranton and Puga (2004) for the definitive survey. We assume constant productivity and output prices, so productivity differentials q are constant parameters. 2 We use the simplest form of the Alonso–Mills–Muth open city model, see Duranton and Puga (2015). The price function p(x) is exogenous to each individual and endogenous to the equilibrium. 3 We will often refer to differential utility, house prices, rent and productivity simply as utility, house prices, rent and productivity. Throughout, we maintain the assumption that real incomes of all individuals outside the city are constant, unaffected by city size. 4 For example, deriving demand from Cobb–Douglas preferences would require specification of individuals’ income levels, but our structure imposes no relationship between the urban income (productivity) differential and the absolute income level of the individual. 5 The literature focuses on income heterogeneity and dates back to Beckmann (1969) and Fujita (1989); see Maattanen and Tervio (2014) for a recent work. 6 For recent progress on multidimensional matching problems, see Chiappori et al. (2016) and, in an urban context, Gaigne et al. (2017). 7 The dashed line on Figure 1(b) gives the utility of a worker of given type as a function of x, its maximum indicating the worker’s location choice. The locus of such maxima is the curve u. 8 That is, the set {q, h} for which u(q,h;x*(q,h))≥0. The intuition behind the curvature of the edge of this set is that low h people live near the CBD. An increment dh requires a larger dq to compensate near the CBD where the cost of h, p(x), is high. 9 From numerical investigation of the bivariate normal σq = σh = 0.17, σhq= 0. 10 Appendix 1 gives the individual utility functions and construction technology from which these are derived. Note that this is the elasticity of supply per unit land and elasticity of each individual’s demand. Total supply of housing in the city is assumed endogenous throughout, since we use an open city model with endogenous city edge. 11 See DCLG (2016) for use of this in project appraisal. 12 η = 0.33 implies that 25% of housing costs are construction, and 75% land rent, see Appendix. In this example the total urban population is 60% of the entire population. The example has ɛ = 0, and raising this elasticity reduces the rent share, as in Table 1. 13 Total land rent is higher in these cities, although it is a smaller share of total surplus. References Albouy D. ( 2016 ) What are cities worth? Land rents, local productivity and the total value of amenities, Review of Economic Statistics , 98 : 477 – 487 . Google Scholar CrossRef Search ADS Arnott R. J. , Stiglitz J. E. ( 1979 ) Aggregate land rents, expenditure on public goods, and optimal city size, Quarterly Journal of Economics , 93 : 471 – 500 . Google Scholar CrossRef Search ADS Beckmann M. J. ( 1969 ) On the distribution of urban rent and residential density . Journal of Economic Theory , 1 : 60 – 67 . Google Scholar CrossRef Search ADS Behrens K. , Robert-Nicoud F. ( 2015 ) Agglomeration theory with heterogeneous agents. In Duranton G. , Henderson J. V. , Strange William C. (eds) Handbook of Regional and Urban Economics , pp. 171 – 245 . North-Holland : Elsevier . Behrens K. , Duranton G. , Robert-Nicoud F. ( 2014 ) Productive cities: sorting, selection, and agglomeration . Journal of Political Economy , 122 : 507 – 553 . Google Scholar CrossRef Search ADS Chiappori P. -A. , McCann R. , Pass B. ( 2016 ) Multidimensional Matching . New York : Columbia University . DCLG ( 2016 ) The DCLG Appraisal Guide. London: Department for Communities and Local Government. https://www.gov.uk/government/publications/department-for-communities-and-local-government-appraisal-guide. Duranton G. , Puga D. ( 2015 ) Urban land use. In Duranton G. J. , Henderson V. , Strange W. C. (eds) Handbook of Regional and Urban Economics , pp. 467 – 560 . North-Holland : Elsevier . Duranton G. , Puga D. ( 2004 ) Micro-foundations of urban agglomeration economies. In Henderson J. V. , Thisse J. -F. (eds) Handbook of Regional and Urban Economics , pp. 2063 – 2117 . North-Holland : Elsevier . Fujita M. ( 1989 ) Urban Economic Theory . Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Gaigne C. , Koster H. , Moizeau F. , Thisse J-F. ( 2017 ) ‘Amenities and the social structure of cities’, CEPR Discussion Paper 11958. Gibbons S. , Machin S. ( 2005 ) Valuing rail access using transport innovations . Journal of Urban Economics , 57 : 148 – 169 . Google Scholar CrossRef Search ADS Glaeser E. L. , Gottlieb J. D. ( 2009 ) The wealth of cities: agglomeration economies and spatial equilibrium in the United States . Journal of Economic Literature , 47 : 983 – 1028 . Google Scholar CrossRef Search ADS Maattanen N. , Tervio M. ( 2014 ) Income distribution and housing prices; an assignment model approach . Journal of Economic Theory , 11 : 381 – 410 . Google Scholar CrossRef Search ADS Appendix A. Housing demand and supply A.1. Price elastic individual housing demand Housing demand is generalised by supposing that the net utility an individual of type h derives from consuming H units of housing at price p(x) is v=h1/ɛH(ɛ−1)/ɛɛ/(ɛ−1)−p(x)H. Utility maximising choice of H is H=hp(x)−ɛ and maximised utility is given by (indirect) utility function v*(p(x),h)=−hp(x)1−ɛ/(1−ɛ). The price of outside housing is denoted p0, so the more general form of Equation (1) is u(q,h;x)=q−h[p(x)1−ɛ−p01−ɛ]/(1−ɛ)−tx. (A.1) The term in square brackets corresponds to the differential house price of the text. In Section 3 Equation (1) becomes u(q,h;x)=qx−h[p(x)1−ɛ−p01−ɛ]/(1−ɛ). A.2 Price elastic housing supply The cost of building S units of housing on a unit of land is S(1+η)/ηη/(η+1), increasing and convex so η > 0. The profit of a developer is Π(x)=p(x)S−S(1+η)/ηη/(η+1) and optimal choice gives S=p(x)η. Maximised profit per unit land is Π*(p(x))=p(x)(1+η)/(η+1). Land rents are equal to maximised profits. Notice also that rents are fraction 1/(η + 1) of gross revenue (= household expenditure on housing). Thus, η = 0.5 implies that land rents are two-thirds of housing expenditure, the rest being construction cost. Differential rent at place x which has s(x) units of land is r*(p(x))=s(x)[p(x)(1+η)−p0(1+η)]/(η+1). A.3 Equilibrium Market clearing for housing, Equation (4) is ∫hhp(x)−ɛ∫qN(q,h;x)dqdh=s(x)p(x)η. (A.2) The city edge, Equation (5), is p(x˜)=p0. Total differential utility is U=∫0x˜∫q∫h{q−h[p(x)1−ɛ−p01−ɛ]/(1−ɛ)−tx}N(q,h;x)dhdqdx. (A.3) Total differential rent is (A.4) Analytical results and the base case are derived with ɛ = η = 0 and p0=0, and equations above collapse to those of the text. Simulation generalises to allow ɛ, η > 0, and renormalises using p0=1 (this positive lower bound required simply because an isoelastic demand function gives infinite demand for housing at p = 0). B. The assignment problem with heterogeneous housing demand The first-order condition for choice of location is ∂p(x*(q¯,h))/∂x=−t/h. With uniform distribution of hEquation (6) gives ∫h0h(x)hfhdh=sx implying that h(x)=[2sx/fh+h02]1/2. The first-order condition becomes ∂p/∂x=−t[2sx/fh+h02]−1/2. Integrating, this gives p(x)=K−tfh/s[2sx/fh+h02]1/2 where K is a constant of integration. At the city edge the land price and differential utility are both equal to zero, p(x˜)=K−tfh/s[2sx˜/fh+h02]1/2=0 and u(q,h;x)=q¯−tx˜=0. Together these give the constant of integration K=tfh/s[2sq¯/tfh+h02]1/2 and hence the house price function, Equation (8), p(x)=tfh/s{[2sq¯/tfh+h02]1/2−[2sx/fh+h02]1/2}. This can be rearranged, denoting the terms in square brackets a, b and noting that a1/2−b1/2=(a−b)/(a1/2+b1/2), so p(x˜)=2(q¯−tx)/(a1/2+b1/2). This gives p(x)=(q¯−tx)2/{[2sq¯/tfh+h02]1/2+[2sx/fh+h02]1/2}. Hence, if h0=h¯ and fh→∞ then p(x˜)=(q¯−tx)/h0. Figure 1 is constructed with q¯=1, t = 1, s = 1, h0=0.5 and fh=0.5. C. Numerical procedures Simulations are derived from Matlab, using a 90 × 90 grid of {q, h}, ranging from q∈[0,2], h∈[0.1,2]. In Section 2 there are 75 equal spaced distances that may be occupied by the city. In Section 3 there are 20 cities. The logit parameter is μ = 250. All simulations used in Figure 2 and beyond use the general structure outlined in Appendix A (housing demand and supply), which reduces to that of the text if elasticities ɛ and η are equal to zero and p0 = 0. © The Author (2017). Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Economic Geography Oxford University Press

Who gets the urban surplus?

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Abstract

Abstract High productivity in cities creates an economic surplus relative to other areas. How is this divided between workers and landowners? Simple models with homogenous labour suggest that it accrues largely—or entirely—in the form of land rents. This article shows that heterogeneity of labour in two main dimensions (productivity differentials and housing demands) radically changes this result. Even a modest amount of heterogeneity can drive the land share of surplus down to two-thirds or lower, as high productivity and/or low housing demand individuals receive large utility gains. It follows that land value appreciation understates the value of urban amenities and infrastructure. In a system of cities the sorting of workers across cities means that, while total rent is highest, the rent share of surplus is lowest in the largest and most productive cities. 1. Introduction Who benefits from the high productivity levels of workers in cities? A standard answer is that the surplus goes principally to landowners. The limiting case is that there is an infinitely elastic supply of homogenous workers so—with this assumption fixing their reservation utility levels and with no other claimants on the surplus—landowners must take 100%. This article is an exploration of what happens when workers are heterogeneous, focusing on two salient dimensions of heterogeneity. One is the productivity differential the worker receives by locating in the city; the other is the amount of housing the worker consumes. The idea is simple. Workers with a large productivity differential and small housing demand will do better than those with the opposite characteristics, and will, therefore, capture some of the urban surplus. Fundamentally, the presence of sufficient heterogeneity means that the assumption of an infinitely elastic supply of workers becomes untenable. If there were infinitely elastic supply the city would be occupied entirely by the type of worker that gets the greatest return (bankers with a preference for small apartments). We know that—while sorting takes place—this is not the case. The questions then are: moving away from homogeneity, how much of the surplus still goes to landlords? Is the change in their share large or small, and what does it depend on? We find that the effect is large. Modest amounts of heterogeneity appear to reduce the share received by landlords to between two-thirds and one-half. The relationship is convex, so that even a small amount of heterogeneity brings a relatively large reduction in rent share. Heterogeneity in either the productivity differential or housing demand reduces the rent share, and the effect is largest when there is a negative correlation between the two attributes of individuals. Looking at systems of cities, cities that offer relatively large productivity differentials have the highest absolute levels of rent, but the lowest share of rent in surplus; these are the largest cities, and those in which sorting has the most extreme effect. These answers are important for a number of issues. Analytically, they matter for our understanding of urban models. Empirically, much work rests on the assumed homogeneity of workers, for example, recent work by Albouy (2016) deriving the value of cities from data on urban wages. Evidently, answers matter for income distribution. They also matter for public finance and the extent to which land taxation can capture the value created by urban agglomerations, as studied in ‘Henry George’ theorems. They matter for project appraisal, since improvements in urban infrastructure increase rents and land values by less than the full value of benefits created. Existing literature recognises the critical nature of the homogeneity assumption, as succinctly stated by Arnott and Stiglitz (1979) who state ‘the conceptual basis of capitalization studies is sound only when marginal individuals are very similar to infra-marginal individuals …’ (496). Duranton and Puga (2015) discuss the issues that arise as heterogeneous individuals sort into different locations, and the technical complexities that follow. The recent literature on heterogeneity focuses on constructing large-scale models that analyse both the sources of heterogeneity and their implication for city systems, as exemplified by Behrens et al. (2014) (see also the survey by Behrens and Robert-Nicoud (2015)). The approach of this article is, in contrast, to focus tightly on the question posed in the title and derive a quantitative sense of the answer to the question. We take spatial productivity differentials and housing preferences as exogenous, drawing out their implications rather than modelling their causes. The remainder of the article is as follows. Section 2 looks at an economy containing a single city, and at individuals who decide whether or not to enter the city and where to live in the city. Section 2.1 presents some analytical results (solving the assignment problem for two simple cases), and Section 2.2 presents the core quantitative results on the division of urban surplus between landowners and individuals, based on numerical analysis of the model. In Section 2.3 we illustrate how the benefits of an urban transport improvement, affecting just part of the city, are divided between rents and residents’ utility. Section 3 moves to a multi-city variant of the model in which individuals sort between cities, and demonstrates how the share of surplus that is captured by rent varies across city types, which is lowest in the largest and most productive cities. 2. Model 1: a single city We start by considering an economy with a single city, an ‘outside’ location and a given total population. Some individuals choose to live and work in the city, and others outside. Each worker has a productivity differential between working in the city and outside, denoted q, and a housing preference parameter h. Sources of these productivity differentials have been extensively analysed elsewhere.1 In the baseline version we outline in this section h is simply the quantity of housing the individual consumes, a more general case being set out in the Appendix and utilised in simulations. For each worker the attributes q and h are exogenous, as is the density function f(q,h) over {q, h} space that gives the distribution of these attributes over the population. Urban workers commute to jobs in the central business district (CBD), and a worker living at distance x from the CBD faces commuting costs tx and pays price differential (relative to living outside the city) p(x) for a unit of housing.2 The utility an individual with attributes q, h, derives from living in the city at distance x from the CBD, as compared to living outside the city is, therefore, the productivity increment minus incremental housing costs and commuting costs, u(q,h;x)=q−hp(x)−tx. (1) Individuals choose to reside in the place that yields highest utility. Given the house price function p(x), an individual with characteristics {q, h} makes choices: live in city atx*(q,h)=arg maxx u(q,h;x) if u(q,h;x*(q,h))≥0,live outside city ifu(q,h;x*(q,h))<0. (2) We define π(q,h;x) as the indicator function, equal to unity if type {q, h} lives in the city at distance x and equal to zero otherwise, so the number of type {q, h} people choosing to live at x is N(q,h;x)=π(q,h;x)f(q,h). (3) The demand for housing at each distance x is then ∫hh∫qN(q,h;x)dqdh. Housing at each distance is initially assumed to be in fixed supply, s(x), so market clearing for housing at x is ∫hh∫qN(q,h;x)dqdh=s(x). (4) The simplest case is a linear city in which land area and the number of houses at each distance is a constant, s(x) = s. The house price differential, p(x), adjusts to clear the market, and the city edge (denoted x˜) is where this price differential equals zero, p(x˜)=0. (5) Equilibrium is characterised by these five equations giving utility, location choice, numbers of individuals of each type at each place, the house price function p(x) and the city edge.3 From this we derive the variables we are interested in. The total number of people living at x: N(x)=∫q∫hN(q,h,x)dhdq. (6) The total utility differential accruing to urban workers: U=∫0x˜∫q∫hu(q,h;x)N(q,h;x)dhdqdx. (7) With fixed supply of housing and no construction costs, the house price differential p(x) is equal to land rent so total differential rent is: R=∫0x˜s(x)p(x)dx. (8) The total urban surplus is differential rent and utility, R + U, and our primary question is, how is this surplus divided between these two components? Before answering these questions, we note that the expressions above give the simplest form of the model. This is used in the analysis of next subsection and is the base case throughout the article. In simulation we also use a version that is more general in two respects. First, the demand for housing of individuals of each type h also depends on price, with elasticity ɛ; whilst adding a price elasticity we retain zero income elasticity, a consequence of our working with productivity and income differentials not levels.4 Secondly, housing supply is endogenised, adding a construction sector that chooses supply of housing per unit land, this creating a price elasticity η of housing supply at each distance, x. The preference and technology apparatus that supports these generalisations is set out in Appendix A, as are the more general form of Equations (1–8). 2.1. Analysis Drawing out answers to the question posed in this article, for a wide range of different parameters, city shapes, distributions of individual characteristics and specifications of preferences and technology, requires numerical simulations. These are presented and discussed in following sections of the article, while in this section we briefly derive analytical solutions for some special cases. We take each of our two dimensions of heterogeneity separately, in line with the one-dimensional household heterogeneity that has been studied by various authors.5 Analytical approaches have not yet provided closed-form solutions for the multidimensional case.6 2.1.1. Homogenous population The benchmark is when workers are homogenous and the common values of their productivity differentials and housing demands are q¯, h¯. The utility differential is zero, i.e. u(q¯,h¯;x)=q¯−h¯p(x)−tx=0, giving house price schedule p(x)=(q¯−tx)/h¯ and city edge x˜=q¯/t. Values of other variables are R=sq¯2/2h¯t,N=x˜s/h¯=sq¯/h¯t,U=0. (9) The entire urban surplus goes in rent, R/(R + U) = 1. 2.1.2. Heterogeneous productivity differential If individuals all have the same demand for housing h¯, then those who enter the city are indifferent as to where they live. Heterogeneous productivity differentials means that individuals with productivity greater than or equal to some (endogenous) cut-off value q˜ will enter the city, and the marginal worker has u(q˜,h¯;x)=q˜−h¯p(x)−tx=0 for all x≥x˜. This sets the house price schedule as p(x)=(q˜−tx)/h¯, with boundary value p(x˜)=0, so x˜=q˜/t. The marginal worker is such that the city is filled when occupied by all individuals with q≥q˜, i.e. q˜ satisfies h¯∫q˜∞f(q,h¯)dq=sx˜. (10) For present purposes, the point is simply that the level of rent is set by the productivity differential of the marginal worker, q˜, and intramarginal workers each capture surplus q−q˜>0. The total size of this, relative to city rents, if greater the larger is the dispersion of q, as will be shown in Section 2.2. 2.1.3. Heterogeneous housing demand The converse case is where individuals have different housing demands h but the same productivity differentials, q=q¯=1. They choose different locations within the city, with lower h individuals choosing higher rent (closer to the CBD) places. Each type’s location choice, x*(q¯,h), is given by the first-order condition for maximisation of Equation (1), ∂p(x*(q¯,h))/∂x=−t/h. Characterisation of the equilibrium involves constructing the price function p(x) that satisfies this equation across a population of individuals with different values of h. To do this, note that the city out to distance x from the CBD will be occupied by individuals with housing demand less than or equal to some value, h(x), defined by equality of housing demand and supply up to x, ∫0h(x)hf(q¯,h)dh=sx. This gives a relationship between h and x which can be used to turn the first-order condition into a simple differential equation in x. This, together with the boundary condition that at the city edge (where p(x˜)=0) marginal workers have zero utility differential, u(q¯,h(x˜);x*(q¯,h(x˜)))=0, defines the price function. If h has uniform distribution with lower support h0 and density fh then the price function takes the form p(x)=(q¯−tx)2/{[2sq¯/tfh+h02]1/2+[2sx/fh+h02]1/2} (11) (see Appendix B). This decreases with distance, is convex, and takes value zero at the city edge. However, rent does not capture the entire city surplus, except in the limiting case where the distribution collapses to a point. The division of surplus is illustrated in Figure 1 (parameters given in Appendix C). Panel (a) gives the rent (house price) function, Equation (11), and population density as a function of distance x from the CBD; in this example equilibrium city size is unity and population density varies by a factor of 4: 1, reflecting heterogeneity of h. Panel (b) gives the distribution of the surplus. All workers have the same productivity differential, q¯=1, but its division between commuting costs, rent and utility depends on their housing demand and chosen location. Working down from the top of Figure 1(b), part goes to commuting costs, tx, part goes to rent, ph, and the remainder is differential utility. Low h individuals live near the CBD and pay low commuting costs and high rent, but since they consume little housing they receive a net utility benefit. Integrating across the population of the city gives total differential rent R and utility U (areas ph and u, each weighted by population). In the example of Figure 1 two-thirds of the urban surplus is captured by land rent, R/(R + U) = 0.66.7 Figure 1 View largeDownload slide Heterogeneous housing demand: (a) Rent and population by distance. (b) Commuting cost, housing expenditure, utility, per person by distance. Figure 1 View largeDownload slide Heterogeneous housing demand: (a) Rent and population by distance. (b) Commuting cost, housing expenditure, utility, per person by distance. 2.2. Numerical exploration We now move to the general case where individuals vary in both characteristics, q and h, and assess the shares of the urban surplus captured by landowners and by workers. We look at different joint distributions of worker attributes, f(q,h), and extend the model to include price elastic individual housing demand and house supply. Answers are derived numerically, centred around the case in which mean values of the productivity differential and housing demand are set to unity, q¯=1, h¯=1, total population (urban and non-urban) is 100, t = 1, the city is linear with s = 50, and price elasticities ɛ, η, are initially equal to zero. If workers are homogenous then values of endogenous variables come from Equation (9). The city edge is x˜=q¯/t=1 and urban population is N=x˜s/h¯=50, i.e. ½ of the total population occupy the city. R =25, U = 0, so the entire city surplus goes to landowners. We start exploring heterogeneity assuming that the distribution of characteristics f(q,h) is bivariate normal on the support q∈[0,2], h∈[0.1,2], with modes q¯=1, h¯=1, and variances and covariance σq2,σh2,σqh2. For computational reasons we smooth location choice using the logit function so the probability of individual of type {q, h} living at x is π(q,h;x)=exp [μu(q,h:x)]/∫0x˜exp [μu(q,h:x)]dx. (12) The logit parameter μ is set at a high value, so the function focuses probability quite tightly on the location that gives the highest utility, as will be clear from the figures that follow. This and other details of computation are given in Appendix C. Individuals sort into different locations, and visualising this is assisted by Figure 2, constructed with σq=σh=0.1,σqh=0. The horizontal plane is {q, h} space, the vertical axis gives the density of individuals and the volume under the surface is number of people. The entire population is the full bivariate normal, and Panel (a) of the figure gives the subset of population resident in the city; it is the part of the normal density containing individuals who gain utility from being in the city, i.e. those with relatively low h and/or high q.8 Other panels give population at different points in the city. Panel (b) is population in a range of locations close to the CBD. As would be expected given the urban rent gradient, these are people with very low h, although covering a wide range of productivity differentials. Panel (c) is mid-city and (d) is near the edge, these areas picking up individuals with greater demand for housing. Panels (b)–(d) each appear like a fin across the surface. With an infinitely fine grid and exact location choice they would be lines giving the combinations {q, h} for which a particular x maximises, u(q,h;x), given equilibrium house prices. Figure 2 View largeDownload slide The urban population at different locations: (a) Entire city. (b) Near CBD. (c) Mid-city. (d) Near edge. Figure 2 View largeDownload slide The urban population at different locations: (a) Entire city. (b) Near CBD. (c) Mid-city. (d) Near edge. The intracity distribution of population is summarised in Figure 3. Panel (a) gives city population N(x) and house prices p(x) by distance from the CBD. House prices (and rent) are higher near the CBD, as expected, and so too is population density since low h individuals choose to live there. Panel (b) gives average housing demand per person and productivity differential at each distance (i.e. averaging over the characteristics of individuals living at each x). Sorting means that housing demand increases with distance, and so too does average productivity differential; even at the lower rents near the city edge high h individuals only choose to live in the city if they are also high q. Figure 3 View largeDownload slide (a) Rent and population by distance. (b) Average h and q by distance. Figure 3 View largeDownload slide (a) Rent and population by distance. (b) Average h and q by distance. What about our main question; who captures the urban surplus? Answers are given in Figures 4(a) and 4(b), each of which has the rent share of city surplus, R/(R + U), on the vertical axis, and a measure of heterogeneity on the horizontal axis. The lowest line on Figure 4(a) has equal standard deviation of q and h with value given by the horizontal axis, σq = σh= σ; covariance is zero. The line illustrates that as σ goes to zero so the rent share goes to unity, as it must. Higher values of σ lead to a much lower rent share, and convexity of the relationship indicates that even a small amount of heterogeneity has a large impact on the rent share. The share drops below 50% if σ > 0.17. To interpret this, σ > 0.17 means that the proportion of total population with characteristics {q, h} outside a circle with radius 0.5 around the mean of {1, 1} is greater than 45%.9 This does not seem an excessive amount of heterogeneity, yet it halves the rent share. Figure 4 View largeDownload slide (a) Rent share of surplus: by variance. (b) Rent share of surplus: by variance and covariance. Figure 4 View largeDownload slide (a) Rent share of surplus: by variance. (b) Rent share of surplus: by variance and covariance. We use this as our reference case, and look at variations in other parameters in turn. The upper two lines in Figure 4(a) switch off variance of q and h in turn (or more precisely, set each equal to σ/100). This approximately halves the effect of heterogeneity, the effect being similar whichever characteristic remains heterogeneous. The main message is that heterogeneity in a single dimension is sufficient to reduce the rent share, in line with the analysis of Section 2.1. Figure 4(b) looks at covariance. The central line reproduces the reference case of Figure 4(a) (zero covariance) and the bottom line assumes negative covariance, σhq= −0.95σ. This reduces the rent share, the intuition being that individuals with the highest productivity differential also demand the least housing, and these are the individuals who capture the most surplus. Conversely, positive covariance (top line) raises the rent share because there is little variation in housing demand per unit productivity differential, and it is this relativity that is crucial. Table 1 reports the way that outcomes vary with other parameters, and also adds price elasticity to individuals’ housing demand and to housing supply, as described in Appendix A. Each row varies one or more features of the model from the base case (row 1) which is as in Figure 4. Making the city circular reduces its population (given our parameters) and has virtually no effect on the rent share of surplus. Lognormality makes very little difference. A larger total (i.e. national) population increases city size and reduces the share of population in the city, with the effect of shifting the slice across the normal distribution in Figure 2(a), clockwise around the origin. The rent share rises, consistent with the idea that labour supply to the city is more elastic in a larger economy. Table 1 Sensitivity Rent/city surplus R/(R + U) City population (% of total population) Base: σq = σh = 0.1, σhq= 0; linear city, normal distribution, total population = 100, ɛ =0, η = 0. 0.583 58.8 Circular city 0.583 40.2 Lognormal distribution 0.589 57.3 Total population = 200 0.659 75.1 (37.5) Total population = 50 0.441 41.6 (83.2) Elasticity of demand for housing, ɛ = 0.5 0.529 66.3 Elasticity of supply of housing, η = 0.5 0.536 65.5 Both; ɛ = η = 0.5 0.476 73.0 Elasticity of demand for housing, ɛ = 1.5 0.375 83.0 Total population =200, ɛ =1.5, η = 0.5 0.390 149 (74) Total population =1000, ɛ =1.5, η = 0.5 0.432 447 (45) Rent/city surplus R/(R + U) City population (% of total population) Base: σq = σh = 0.1, σhq= 0; linear city, normal distribution, total population = 100, ɛ =0, η = 0. 0.583 58.8 Circular city 0.583 40.2 Lognormal distribution 0.589 57.3 Total population = 200 0.659 75.1 (37.5) Total population = 50 0.441 41.6 (83.2) Elasticity of demand for housing, ɛ = 0.5 0.529 66.3 Elasticity of supply of housing, η = 0.5 0.536 65.5 Both; ɛ = η = 0.5 0.476 73.0 Elasticity of demand for housing, ɛ = 1.5 0.375 83.0 Total population =200, ɛ =1.5, η = 0.5 0.390 149 (74) Total population =1000, ɛ =1.5, η = 0.5 0.432 447 (45) Table 1 Sensitivity Rent/city surplus R/(R + U) City population (% of total population) Base: σq = σh = 0.1, σhq= 0; linear city, normal distribution, total population = 100, ɛ =0, η = 0. 0.583 58.8 Circular city 0.583 40.2 Lognormal distribution 0.589 57.3 Total population = 200 0.659 75.1 (37.5) Total population = 50 0.441 41.6 (83.2) Elasticity of demand for housing, ɛ = 0.5 0.529 66.3 Elasticity of supply of housing, η = 0.5 0.536 65.5 Both; ɛ = η = 0.5 0.476 73.0 Elasticity of demand for housing, ɛ = 1.5 0.375 83.0 Total population =200, ɛ =1.5, η = 0.5 0.390 149 (74) Total population =1000, ɛ =1.5, η = 0.5 0.432 447 (45) Rent/city surplus R/(R + U) City population (% of total population) Base: σq = σh = 0.1, σhq= 0; linear city, normal distribution, total population = 100, ɛ =0, η = 0. 0.583 58.8 Circular city 0.583 40.2 Lognormal distribution 0.589 57.3 Total population = 200 0.659 75.1 (37.5) Total population = 50 0.441 41.6 (83.2) Elasticity of demand for housing, ɛ = 0.5 0.529 66.3 Elasticity of supply of housing, η = 0.5 0.536 65.5 Both; ɛ = η = 0.5 0.476 73.0 Elasticity of demand for housing, ɛ = 1.5 0.375 83.0 Total population =200, ɛ =1.5, η = 0.5 0.390 149 (74) Total population =1000, ɛ =1.5, η = 0.5 0.432 447 (45) Remaining rows allow for positive price elasticity of housing supply, η, and of individual’s demand for housing, ɛ. As outlined in Appendix A, housing supply becomes sp(x)η and housing demand becomes hp(x)−ɛ.10 As expected, positive elasticities increase city population and reduce the share of rent in surplus, this going to less than 40% in the more extreme case reported. 2.3. Partial capitalisation of local amenity improvements An improvement in local amenities at a point in the city (such as a transport improvement) leads to an increase in land values around the improvement, as would be expected and as is confirmed by empirical studies (e.g. Gibbons and Machin, 2005). This has led policy work to argue that the increase in land values should be used in cost-benefit analysis as a summary measure of the benefits of the improvement.11 While correct in the limiting case (infinitely elastic supply of homogenous workers), it generally only captures a part of the benefit, some accruing to workers. How does this work out in our more general framework? An example is shown in Figure 5, where the cost of commuting is halved for locations at distance x∈[0.4,0.5], and unchanged elsewhere (so giving x∈[0.4,0.5] better access to the CBD than some places closer in). If there were to be no population movement and utility levels of all individuals were held constant, then all the benefits of lower commuting costs would be captured in rent. However, this is not the equilibrium. Individuals in the interval x∈[0.4,0.5] consume relatively much housing and now face higher rent, so they will move. They are replaced by individuals from closer to the CBD with lower housing demand and, therefore, willing to pay the higher rent. Population density increases in the affected interval and decreases in areas closer to the CBD. This is illustrated by the solid line in Figure 5. The change in rent in each place is given by the higher dashed line, and the change in utility (that of all inhabitants in each place) by the lower dashed line. There is a net utility increase and, across the urban population as a whole, this experiment yields increase in surplus of which 82% accrues in rent and 18% in household utility. Figure 5 View largeDownload slide A reduction in commuting costs; changes in population, rent and utility. Figure 5 View largeDownload slide A reduction in commuting costs; changes in population, rent and utility. These numbers are particular to this example and are sensitive to the exact magnitude and location of the transport improvement. However, they make the point that use of the rent increment as a measure of benefit should take into account the equilibrium shuffling of urban population that, in the long run, a transport improvement will bring. The shuffling is associated with a transfer of some of the benefit from rent to household utility. 3. Model 2: a system of cities Do these or similar results obtain with a system of cities? We investigate this in a restatement of the model used above. We now assume that cities have no internal geography and reinterpret x as an index over a set of cities, x∈(0,1]. The utility derived from living and working in city x (relative to being ‘outside’, i.e. not in any city) is u(q,h;x)=qx−hp(x). (13) Comparing this with Equation (1), there are no commuting costs but heterogeneity across cities is introduced by supposing that the productivity differential of a type q worker is qx, i.e. low x cities deliver little additional productivity, by construction. Many mechanisms could be posited to generate this cross-city heterogeneity but, in the spirit of this article, we give it the simplest possible form and investigate its implications. Equations (2), (3) and (4) are as before, with N(q,h;x)=π(q,h;x)f(q,h) now interpreted as the number of people of type {q, h} living in city x. In equilibrium all cities will operate, since there is a positive (if small) productivity differential for all cities relative to ‘outside’. Since there is no internal geography and p(x) is the price of all houses in city index x, Equation (5) for the city edge no longer exists, and the population of each city is N(x), as given by Equation (6). Equations (7) and (8) still hold, now with x˜=1, i.e. adding over all populated cities. We report results for the case in which there are 20 cities, σq=σh=0.1,σqh=0 and there is a positive elasticity of housing supply, η = 0.33.12Figure 6(a) illustrates the equilibrium distribution of population, analogous to the intracity distribution of Figure 3(a). As expected, high index (so high productivity-differential) cities are larger and have higher house prices. Figure 6(b) illustrates that this is driven by high-productivity differential individuals sorting very strongly to cities that lever their productivity advantage, high x cities. The consequent high house prices in these cities induce further sorting according to individuals’ housing demand. Figure 6 View largeDownload slide Heterogeneous cities: (a) Rent and population by city index. (b) Average h and q by city index. Figure 6 View largeDownload slide Heterogeneous cities: (a) Rent and population by city index. (b) Average h and q by city index. What about the share of urban surplus going to rents? With σq=σh=0.1,σqh=0 the share across all cities is 59%, virtually identical to that reported for Model 1 in Figure 4 and Table 1. The share falls with variance, as before. More interesting, is to see how the share varies with x, the city productivity increment, and hence city size. This is illustrated in Figure 7. The bars give the rent share by city and indicate that the share of rent in total surplus is smaller for larger (high city index) cities.13 The high-productivity differential gives these cities an absolute advantage for all individuals and this bids up house prices. Comparative advantage makes these cities attractive for high q and low h individuals (as illustrated in Figure 6(b)), and these are precisely the individuals who capture a larger share of the surplus. The city with the highest index—and hence the largest city—captures just 39% of the surplus it generates in the form of land rent. Figure 7 View largeDownload slide Rent share of surplus: by city index. σq= σh= 0.1, σhq= 0, η = 0.33. Figure 7 View largeDownload slide Rent share of surplus: by city index. σq= σh= 0.1, σhq= 0, η = 0.33. 4. Concluding comments Homogenous labour is a useful simplifying assumption in city modelling, but we know that a key feature of cities is the distinctive skill mix of their population (e.g. Glaeser and Gottlieb, 2009). This article has argued that the assumption of homogenous labour is particularly misleading when it comes to thinking about the distribution of urban surplus between land and labour. With homogenous labour the share of land rent in surplus takes the extreme boundary value, of unity. Heterogeneity can only move this in one direction, so the question is, by how much? This article suggests that the answer is, a lot. Heterogeneous productivity differentials or housing demands pull the share down, in a convex relationship so that even small amounts of heterogeneity matter. Combining both dimensions of heterogeneity suggests that, for modest degrees of population heterogeneity, rent might capture between one- and two-thirds of total surplus, the rest accruing to individuals. The result extends from a single city context to multiple cities which differ in the productivity differential they offer to workers; in this case it is the largest and most productive cities that have the lowest share of rent in the urban surplus that they create. These findings matter for our thinking about the income distributional implications of urbanisation, and about urban public finance. They may also set the stage for future empirical and policy work that seeks to understand and quantify the various impacts of urban development. Footnotes 1 See Duranton and Puga (2004) for the definitive survey. We assume constant productivity and output prices, so productivity differentials q are constant parameters. 2 We use the simplest form of the Alonso–Mills–Muth open city model, see Duranton and Puga (2015). The price function p(x) is exogenous to each individual and endogenous to the equilibrium. 3 We will often refer to differential utility, house prices, rent and productivity simply as utility, house prices, rent and productivity. Throughout, we maintain the assumption that real incomes of all individuals outside the city are constant, unaffected by city size. 4 For example, deriving demand from Cobb–Douglas preferences would require specification of individuals’ income levels, but our structure imposes no relationship between the urban income (productivity) differential and the absolute income level of the individual. 5 The literature focuses on income heterogeneity and dates back to Beckmann (1969) and Fujita (1989); see Maattanen and Tervio (2014) for a recent work. 6 For recent progress on multidimensional matching problems, see Chiappori et al. (2016) and, in an urban context, Gaigne et al. (2017). 7 The dashed line on Figure 1(b) gives the utility of a worker of given type as a function of x, its maximum indicating the worker’s location choice. The locus of such maxima is the curve u. 8 That is, the set {q, h} for which u(q,h;x*(q,h))≥0. The intuition behind the curvature of the edge of this set is that low h people live near the CBD. An increment dh requires a larger dq to compensate near the CBD where the cost of h, p(x), is high. 9 From numerical investigation of the bivariate normal σq = σh = 0.17, σhq= 0. 10 Appendix 1 gives the individual utility functions and construction technology from which these are derived. Note that this is the elasticity of supply per unit land and elasticity of each individual’s demand. Total supply of housing in the city is assumed endogenous throughout, since we use an open city model with endogenous city edge. 11 See DCLG (2016) for use of this in project appraisal. 12 η = 0.33 implies that 25% of housing costs are construction, and 75% land rent, see Appendix. In this example the total urban population is 60% of the entire population. The example has ɛ = 0, and raising this elasticity reduces the rent share, as in Table 1. 13 Total land rent is higher in these cities, although it is a smaller share of total surplus. References Albouy D. ( 2016 ) What are cities worth? Land rents, local productivity and the total value of amenities, Review of Economic Statistics , 98 : 477 – 487 . Google Scholar CrossRef Search ADS Arnott R. J. , Stiglitz J. E. ( 1979 ) Aggregate land rents, expenditure on public goods, and optimal city size, Quarterly Journal of Economics , 93 : 471 – 500 . Google Scholar CrossRef Search ADS Beckmann M. J. ( 1969 ) On the distribution of urban rent and residential density . Journal of Economic Theory , 1 : 60 – 67 . Google Scholar CrossRef Search ADS Behrens K. , Robert-Nicoud F. ( 2015 ) Agglomeration theory with heterogeneous agents. In Duranton G. , Henderson J. V. , Strange William C. (eds) Handbook of Regional and Urban Economics , pp. 171 – 245 . North-Holland : Elsevier . Behrens K. , Duranton G. , Robert-Nicoud F. ( 2014 ) Productive cities: sorting, selection, and agglomeration . Journal of Political Economy , 122 : 507 – 553 . Google Scholar CrossRef Search ADS Chiappori P. -A. , McCann R. , Pass B. ( 2016 ) Multidimensional Matching . New York : Columbia University . DCLG ( 2016 ) The DCLG Appraisal Guide. London: Department for Communities and Local Government. https://www.gov.uk/government/publications/department-for-communities-and-local-government-appraisal-guide. Duranton G. , Puga D. ( 2015 ) Urban land use. In Duranton G. J. , Henderson V. , Strange W. C. (eds) Handbook of Regional and Urban Economics , pp. 467 – 560 . North-Holland : Elsevier . Duranton G. , Puga D. ( 2004 ) Micro-foundations of urban agglomeration economies. In Henderson J. V. , Thisse J. -F. (eds) Handbook of Regional and Urban Economics , pp. 2063 – 2117 . North-Holland : Elsevier . Fujita M. ( 1989 ) Urban Economic Theory . Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Gaigne C. , Koster H. , Moizeau F. , Thisse J-F. ( 2017 ) ‘Amenities and the social structure of cities’, CEPR Discussion Paper 11958. Gibbons S. , Machin S. ( 2005 ) Valuing rail access using transport innovations . Journal of Urban Economics , 57 : 148 – 169 . Google Scholar CrossRef Search ADS Glaeser E. L. , Gottlieb J. D. ( 2009 ) The wealth of cities: agglomeration economies and spatial equilibrium in the United States . Journal of Economic Literature , 47 : 983 – 1028 . Google Scholar CrossRef Search ADS Maattanen N. , Tervio M. ( 2014 ) Income distribution and housing prices; an assignment model approach . Journal of Economic Theory , 11 : 381 – 410 . Google Scholar CrossRef Search ADS Appendix A. Housing demand and supply A.1. Price elastic individual housing demand Housing demand is generalised by supposing that the net utility an individual of type h derives from consuming H units of housing at price p(x) is v=h1/ɛH(ɛ−1)/ɛɛ/(ɛ−1)−p(x)H. Utility maximising choice of H is H=hp(x)−ɛ and maximised utility is given by (indirect) utility function v*(p(x),h)=−hp(x)1−ɛ/(1−ɛ). The price of outside housing is denoted p0, so the more general form of Equation (1) is u(q,h;x)=q−h[p(x)1−ɛ−p01−ɛ]/(1−ɛ)−tx. (A.1) The term in square brackets corresponds to the differential house price of the text. In Section 3 Equation (1) becomes u(q,h;x)=qx−h[p(x)1−ɛ−p01−ɛ]/(1−ɛ). A.2 Price elastic housing supply The cost of building S units of housing on a unit of land is S(1+η)/ηη/(η+1), increasing and convex so η > 0. The profit of a developer is Π(x)=p(x)S−S(1+η)/ηη/(η+1) and optimal choice gives S=p(x)η. Maximised profit per unit land is Π*(p(x))=p(x)(1+η)/(η+1). Land rents are equal to maximised profits. Notice also that rents are fraction 1/(η + 1) of gross revenue (= household expenditure on housing). Thus, η = 0.5 implies that land rents are two-thirds of housing expenditure, the rest being construction cost. Differential rent at place x which has s(x) units of land is r*(p(x))=s(x)[p(x)(1+η)−p0(1+η)]/(η+1). A.3 Equilibrium Market clearing for housing, Equation (4) is ∫hhp(x)−ɛ∫qN(q,h;x)dqdh=s(x)p(x)η. (A.2) The city edge, Equation (5), is p(x˜)=p0. Total differential utility is U=∫0x˜∫q∫h{q−h[p(x)1−ɛ−p01−ɛ]/(1−ɛ)−tx}N(q,h;x)dhdqdx. (A.3) Total differential rent is (A.4) Analytical results and the base case are derived with ɛ = η = 0 and p0=0, and equations above collapse to those of the text. Simulation generalises to allow ɛ, η > 0, and renormalises using p0=1 (this positive lower bound required simply because an isoelastic demand function gives infinite demand for housing at p = 0). B. The assignment problem with heterogeneous housing demand The first-order condition for choice of location is ∂p(x*(q¯,h))/∂x=−t/h. With uniform distribution of hEquation (6) gives ∫h0h(x)hfhdh=sx implying that h(x)=[2sx/fh+h02]1/2. The first-order condition becomes ∂p/∂x=−t[2sx/fh+h02]−1/2. Integrating, this gives p(x)=K−tfh/s[2sx/fh+h02]1/2 where K is a constant of integration. At the city edge the land price and differential utility are both equal to zero, p(x˜)=K−tfh/s[2sx˜/fh+h02]1/2=0 and u(q,h;x)=q¯−tx˜=0. Together these give the constant of integration K=tfh/s[2sq¯/tfh+h02]1/2 and hence the house price function, Equation (8), p(x)=tfh/s{[2sq¯/tfh+h02]1/2−[2sx/fh+h02]1/2}. This can be rearranged, denoting the terms in square brackets a, b and noting that a1/2−b1/2=(a−b)/(a1/2+b1/2), so p(x˜)=2(q¯−tx)/(a1/2+b1/2). This gives p(x)=(q¯−tx)2/{[2sq¯/tfh+h02]1/2+[2sx/fh+h02]1/2}. Hence, if h0=h¯ and fh→∞ then p(x˜)=(q¯−tx)/h0. Figure 1 is constructed with q¯=1, t = 1, s = 1, h0=0.5 and fh=0.5. C. Numerical procedures Simulations are derived from Matlab, using a 90 × 90 grid of {q, h}, ranging from q∈[0,2], h∈[0.1,2]. In Section 2 there are 75 equal spaced distances that may be occupied by the city. In Section 3 there are 20 cities. The logit parameter is μ = 250. All simulations used in Figure 2 and beyond use the general structure outlined in Appendix A (housing demand and supply), which reduces to that of the text if elasticities ɛ and η are equal to zero and p0 = 0. © The Author (2017). Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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Journal of Economic GeographyOxford University Press

Published: Dec 14, 2007

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