Breaking into Tradables: Urban Form and Urban Function in a Developing City

Many cities in developing economies, particularly in Africa, are experiencing urbanization without industrialization. This paper conceptualizes this in a framework in which a city can produce non-tradable goods and -- if it is sufficiently competitive -- also internationally tradable goods, potentially subject to increasing returns to scale. A city is unlikely to produce tradables if it faces high urban and hinterland demand for non-tradables, or high costs of urban infrastructure and construction. The paper shows that, if there are increasing returns in tradable production, there may be multiple equilibria. The same initial conditions can support dichotomous outcomes, with cities either in a low-level (non-tradable only) equilibrium, or diversified in tradable and non-tradable production. The paper demonstrates the importance of history and expectations in determining outcomes. Essentially, a city can be built in a manner that makes it difficult to attract tradable production. This situation might be a consequence of low (and self-fulfilling) expectations or history. The predictions of the model are consistent with several observed features of African cities.

Many cities in developing economies, particularly in Africa, are experiencing urbanization without industrialization. This paper conceptualizes this in a framework in which a city can produce non-tradable goods and-if it is sufficiently competitive-also internationally tradable goods, potentially subject to increasing returns to scale. A city is unlikely to produce tradables if it faces high urban and hinterland demand for non-tradables, or high costs of urban infrastructure and construction. The paper shows that, if there are increasing returns in tradable production, there may be multiple equilibria. The same initial conditions can support dichotomous outcomes, with cities either in a low-level (non-tradable only) equilibrium, or diversified in tradable and non-tradable production. The paper demonstrates the importance of history and expectations in determining outcomes. Essentially, a city can be built in a manner that makes it difficult to attract tradable production. This situation might be a consequence of low (and self-fulfilling) expectations or history. The predictions of the model are consistent with several observed features of African cities.

Introduction
Cities in the developing world face the challenge of accommodating a predicted 2.5 billion more people by 2050, and the fastest urbanizing region will be Sub-Saharan Africa, where urban population is predicted to treble to more than 1 billion. Urbanization is occurring at lower per capita income levels than was historically the case, sometimes referred to as 'urbanization without growth' (Fay and Opal, 2000;Jedwab and Vollrath 2015). 1 The performance of developing cities is heterogeneous, and in one area has been sharply dichotomous. Many Asian cities have been able to create jobs in tradable goods sectors and have become internationally competitive, producing large volumes of exports. African cities have failed to do this, and have instead grown on the basis of supplying local and perhaps regional markets. The phrase 'urbanization without industrialization' has gained currency, and Gollin et al. (2016) point to the prevalence of this phenomenon in resource rich developing countries.
The present paper analyzes the factors that shape this aspect of performance and that determine the extent to which developing cities are able to succeed in attracting high productivity tradable goods (or service) sectors, or instead remain specialized in producing non-tradables for local markets. The paper is primarily theoretical, and is based on interactions between 'urban function' -the economic activity that takes place in the city -and 'urban form' -the way in which the city is constructed and the efficiency with which it operates. To capture this, the model that is developed in the paper has several key ingredients. On the production side, we distinguish between non-tradable and tradable sectors of production. The former is likely to encounter diminishing returns because it is limited by the size of local markets, while the latter offers the prospect of increasing returns and agglomeration economies. On the residential side, urban form is captured by a standard urban model in which buildings are durable and density of construction is endogenous. The residential capital stock -and hence the size and density of the city -therefore depend on both past history and expectations of future returns.
Three sets of results are established. First, we establish conditions which are likely to lead to a city being specialized in non-tradables, as opposed to diversified into both non-tradables and tradables. Conditions include the presence of high urban and hinterland demand for nontradables, and high costs of urban infrastructure and construction. Second, we show how, if there are increasing returns in tradable production, there may be multiple equilibria. The same initial conditions can support dichotomous outcomes, with cities either in a low-level (nontradable only) equilibrium, or diversified in both tradable and non-tradable production. Third, 3 we demonstrate the importance of history and of expectations in determining outcomes. Essentially, a city can be built in a manner which makes it difficult to attract tradable production. This might be a consequence of low (and self-fulfilling) expectations or of history.
While the paper is primarily theoretical, we start by outlining three features of African cities that are illuminated by the model. The first is African cities' failure to create jobs in internationally tradable goods or service sectors. Gollin et al. (2016) investigate this, principally at the national level, establishing the adverse effect of natural resource sectors on manufacturing employment. Natural resource dependence is only part of the story, and there are also significant regional differences. Focusing on Africa, Jones (2016) compares manufacturing shares of GDP in African and non-African economies at different stages of urbanization. In non-African economies, the manufacturing share of GDP rises from 10% to nearly 20% as the urban population share rises to 60%, above which it falls back. In Africa, the manufacturing share remains flat (or somewhat falling) at around 10% of GDP through a cross-section of urbanization rates ranging from 10% to 70%. 2 A more urban focus can be derived by using spatially disaggregated IPUMS data. 3 These are sample data collected at the individual level, with selfdeclared sector of employment. Table 1 reports the share of employment declaring to be in manufacturing in areas classified as urban. The data are presented by city (selected by size within country) and by country (in India this is urban area by state, with separate city level data unavailable). While manufacturing is not synonymous with tradable goods, the data indicate clearly the extent to which Africa is different from other regions. Simple averages across countries suggest manufacturing shares of urban employment nearly three times greater in Asia than in Africa, and increasing through time, whereas Africa's shares have declined slightly. Accra appears as the African city with the highest share of employment in manufacturing, at 14.9%, whereas shares in Asian cities rise well in excess of 30%. 4 The second feature is the problematic 'urban form' of African cities. The weakness of African urban form has numerous manifestations, showing up most obviously in low stocks of key capital assets, including housing and infrastructure. While low stocks of these assets is partly a function of low income (urbanization at relatively low levels of per capita income) it is also due to market and governance failures. Lack of clarity in land-tenure is widespread, disputes between multiple claimants on land are frequent, and invasions to seize urban land are a problem. Private investment is also discouraged by inappropriately high building and land use regulations, and mortgage finance is hard to obtain (Collier and Venables 2015). Provision of public capital is low, with estimates of the infrastructure gap suggesting that as much as 20% of urban GVA needs to be spent over a period of decades to fill the gap (Foster et al. 2010). These features are captured in our modeling as high costs of building ('costs' including non-monetary obstacles) and of urban transport. Within the model, the consequences are low levels of residential investment and high costs of accessing jobs. This corresponds to the reality of widespread informal settlement, generally single story and constructed of mud and sheet metal.
Some 62% of Africa's urban population lives in slums (UN-Habitat, 2010) and there is often a hodgepodge of land use, with slum areas persisting next to modern developments near city centers. Inefficient land use and sub-optimal stocks of residential and infrastructure capital have costs for the functioning of the city as a whole. There are direct costs as, for example, data for Nairobi suggest that land near the city center that is currently occupied by slums foregoes around two-thirds of its value compared to neighboring formal developments ).
There are wider costs of loss of connectivity between economic agents, imposing costs on firms and manifest in some of the longest commuting times in the world.
The third feature of African cities is the apparent paradox of relatively high nominal wages and prices in many cities, despite their low real income and lack of modern sector employment. This is a feature that, as we will see, can emerge as an outcome in the model. The evidence for it comes from several sources. Jones (2016) reports that firms in African cities pay wages (at official exchange rates) about 15% higher than in non-African cities, conditional on national real GDP pc. Labor costs are estimated at up to 50% higher. Corresponding to this, sales per worker are about 25% higher than in comparable non-African cities, but this is largely in the non-traded sectors and appears to simply reflect higher prices not physical productivity. 5 High nominal wages are matched by high prices of goods and services.  use ICP data to study the cost of living of urban households across countries and find that it is some 20-30% higher in African countries than in other countries at similar income levels. 6 Part of this is due to high rents and urban transport costs (respectively 55% and 42% higher than in comparable places elsewhere) although it extends to other commodities. Prices of food and other goods are also relatively high. Henderson and Nigmatulina (2016) show that, across developing country cities, high prices are negatively associated with a measure of connectivity between people in the city.
Our model captures these features, and is developed in a series of stages. Section 2 lays out the ingredients, characterizes equilibrium and undertakes comparative statics to establish the determinants of city specialization, as well as of city size, density and rent levels. While Section 2 maintains the assumption of constant returns to scale in tradable production, Section 3 moves to environments with increasing returns, establishing the possibility of multiple equilibria and the roles of history and expectations in shaping outcomes. Section 4 concludes and offers policy implications. 6

The model
The model focuses on a single city which sells goods within and outside the city, and which is able to draw labor from the wider economy. Labor is the only input to production, and land is used for housing urban workers. Analysis is based around labor demand and labor supply. The former gives the relationship between the wage and the level of employment, depending on productive activity in the city ('urban function'). The latter is the relationship between the wage and city population, this depending on migration and on the cost of living in the city, including costs of construction and of commuting ('urban form').

Production and labor demand:
Labor is demanded by the production side of the city economy in which there are (potentially) two produced goods, tradables and non-tradables, with employment levels T L and N L giving total city employment . The wage is w, the same in both sectors, and since labor is the only input the value of output produced in the city is wL . 7 To derive the city's labor demand function we look first at demand for labor in nontradable production, and then in tradables.
Non-tradable goods meet demands from the local market and have price N p , determined endogenously by supply and demand. 8 They are produced under constant returns to scale so, choosing units such that one unit of labor produces one unit of output, w p N  and the value of supply is N wL . The value of demand for non-tradables is In the first term, wL is the city wage bill and this is spent on a composite good which is a Cobb-Douglas aggregate of tradables with share  , and non-tradables with share ) 1 (   . As we will see below, this spending takes different forms -final consumption, commuting and construction costs -but all demand the same composite, so the city's income generates demand for non-tradables , is spending on non-tradables from income generated outside the city. Such spending consists of several elements. One is spending from transfer payments to the city, such as natural resource revenues, taxes, or foreign aid. Another is 7 hinterland spending on non-tradables produced in the city. We refer to   N p h as the hinterland demand function, and take it as exogenous and decreasing in price; it is shifted upwards by higher demand from these extra-city sources. Using w p N  and , the equality of supply and demand for non-tradables is  (1) This is the wage at which net urban supply of non-tradables equals hinterland demand, where is the inverse hinterland demand curve. Crucially, its downwards slope captures the fact that expanding urban employment in non-tradables reduces the wage paid, since increased supply reduces the price of non-tradables.
In contrast, tradable goods face perfectly elastic world demands, have fixed world price, and will be taken as numeraire. Labor productivity in the tradable sector is   T L a , and this is the wage offered in the sector. We assume that productivity is either constant or increasing in T L , increasing returns arising because of agglomeration economies external to the firm but internal to the tradable sector. The sector operates if labor productivity is greater than or equal to the market wage, w, so the following relationships must hold, Together, Eqns. (1) and (2) implicitly define the city's (inverse) labor demand schedule ) (L w D , i.e. they give the value of w at which urban employment (in non-tradables and tradables) fully employs a labor force of size L. This labor demand curve generally has a kink in it at the 'trigger wage', ) 0 ( 0 a a  , at which tradable production commences. We summarize it as follows: If the tradable sector is inactive, then 0  T L and, from (1), If the tradable sector is active, then 0  T L and, from (1) and (2), The wage offered is the maximum, This is illustrated by the bold curve in Fig. 1 which has city employment (= population) on the horizontal axis and the wage on the vertical. The figure is drawn with constant returns to scale in tradable production so labor productivity in tradables is a constant   0 a L a T  . On the downward sloping segment the entire labor force is employed in the non-tradable sector, , downward sloping as more employment increases supply of the nontraded good, reducing price and wages. If the ensuing wage is less than 0 a then tradable production is profitable, giving the horizontal segment, Urban costs and labor supply: Labor supply comes from the location decisions of mobile workers. It depends on utility outside the city, set as exogenous constant 0 u , and utility inside.
This depends on the wage, the prices of goods consumed, and any further costs associated with urban living. These include direct utility costs (and benefits) from crowding, congestion, and provision of public services. And financial costs, arising from land and house prices and from commuting costs.
Our modeling of urban costs is in the tradition Alonso (1964) and others. 9 Workers are employed in the central business district (CBD) and incur commuting costs. The city is linear, and x denotes distance from the CBD. 10 Each worker occupies a unit of housing, and housing density (hence the amount of land occupied per unit housing) is chosen by profit maximizing developers. The utility of a worker living at distance x from the CBD is v(x), (4) The term in square brackets is the wage net of housing and commuting costs. The cost of housing at distance x is p(x) and commuting incurs t units of the composite good per unit distance, so a worker living at distance x from the CBD pays commuting costs w is the price index of the composite good. 11 Income net of housing and commuting costs (i.e. the term in square brackets) is spent on the composite good, so utility is derived by deflating by the price index. 12 9 See Duranton and Puga (2015) for full exposition and analysis of the Alonso-Muth-Mills urban model. 10 To minimize notation we work with a linear city with a single spoke from the CBD. 11 As noted above, the composite is Cobb-Douglas with tradable share θ, non-tradable share 1 -θ. Since tradables are the numeraire and the price of non-tradables is w, the price index is Workers are perfectly mobile, choosing between living outside the city with utility 0 u , or at locations x in the city. For all occupied urban locations, it must therefore be the case that The supply of housing depends on residential construction decisions, taken to maximize land rent at each point. The number of housing units -or density -built at x is N(x), and rent, ) (x r , is revenue from housing minus construction costs, i.e.
Construction costs per unit land are , an increasing and convex function of density (e.g. the cost of building taller). 13 We assume that this relationship is iso-elastic with parameter γ, and that costs are incurred in units of the composite good, i.e. have price   1 w . Developers choose density to maximize rent, giving first order condition and maximized rent, The edge of the city is at distance x where rent equals the exogenous outside rent 0 r , i.e.
Housing capacity and total city population up to this edge are Eqns. (5) - (9) give values of three variables that vary with distance, N(x), p(x), and r * (x) and two scalars, L and x , as functions of parameters and the wage. They implicitly define the residential structure of the city, its built density, house prices, land rents, and total population and city edge. They also define the labor supply function -the relationship between w and L. An explicit form of this relationship is useful for expositional purposes and can be derived with two simplifying they create demand for non-tradables, they do not provide utility for urban residents.
10 assumptions. 14 The first is that outside rent is zero, 0 ) ( ; this implies that the edge of the (from (6') and (7)), i.e. that the city converges to zero density at its edge. It follows from (5) that the city edge is given by: Second, we assume that construction costs increase with the square of density, γ = 2, giving population This uses (7), (5) and (10) in (9); the appendix gives the case for general values of γ. Inverting, the urban wage required to attract and accommodate population L is This is the inverse labor supply curve, ) (L w S illustrated by the upwards sloping curves on Fig. 1, the upper one drawn for a city with higher urban costs. 15 Several remarks are in order. First, concavity is a general property of this supply function. Intuitively, if the wage (and hence land rent) is low the city is built at low density, so accommodates a small population. 16 Higher wages increase population through two margins: the city becomes larger and is built at higher density, the interaction between the two giving concavity. Second, this supply curve is derived from the fundamentals of commuting costs, construction, and land prices, but these are only part of a more general urban cost relationship. If other factors -congestion, loss of amenity -increase with city size, this too will create the upward sloping relationship as a larger city requires higher wages as compensating differential to offset these costs (see e.g. Duranton's 2008 discussion of the 'cost of living curve').

Equilibrium with constant returns in tradable production
The urban labor demand and supply curves, Eqs. (3) and (12) (11) and (3b)). The message is simple. Both cases have the same real wage, 0 u . However, the equilibrium without tradable production, E + , has higher nominal wages and higher non-tradable prices. High urban costs can be passed on to consumers of non-tradables, but cannot be passed on in tradable sector, meaning that the city is uncompetitive in the production of these goods.
What factors shape the likelihood of each of these outcomes occurring? The dividing line between cases is where wages and employment levels satisfy This is more likely if, from the left-hand side of the inequality, 0 a is small, i.e. the productivity of workers in the tradable sector is low. This could arise because of low productivity in the sector, or transport barriers imposing additional costs on exporting. On the right hand side of the inequality four parameters enter multiplicatively, c, t, H, 1/θ, where H represents a multiplicative shift parameter in the hinterland demand function. 17 Before discussing the impact of these parameters, it is useful to establish their impact on other aspects of the equilibrium, in particular the area of the city, its average density, and urban land rents. Area is captured by x , Eqn. (10), and average density is x L/ . Total rents, R, are derived by integrating over rents at each point in the city (Eqn. 6'), where the last equation uses γ = 2 (see appendix for general form). The effect of parameters on endogenous variables is found by log-linearizing the equilibrium conditions, and full expressions are given in the appendix. Here we simply note the comparative static signs, with Table 2 giving the sign of an increase in each of the parameters on specialization and other endogenous variables.
The city is more likely to produce non-tradables only, LT = 0, the higher are c, t, H, and 1/θ. The last two of these capture high demand for non-tradables, either from the hinterland or from a high share of city income being spent on non-tradables. The first two capture high urban costs, of construction and commuting, respectively. While demand and costs have the same qualitative impact on city specialization and on nominal wages, they have opposite effects on city population. If LT = 0, higher demand is associated with larger population, and higher costs with a smaller population. Higher values of each of these parameters increase nominal wages and the ratio of rent to wage bill. Notice however that the two cost parameters have opposite implications for city density and geographic size. High construction costs reduce density, and city size is greater despite lower population. High transport costs reduce size, raising rents and hence city density. 17 I.e., the demand function becomes If the city is diversified, LT > 0, then nominal wages are set by productivity in tradables, a0. Demand for non-tradables has no effect on city population -high demand is accommodated by lower tradable production. High costs do however reduce city population, because of their direct effect on density of building (for construction costs) or the extent of the city (for commuting costs).
Notes: Δ denotes a proportional change in parameter. ^ denotes a proportional change in a variable. For each variable, the first column gives comparative statics when equilibrium has LT = 0 and the second when LT > 0.
In summary, characteristics of the city equilibrium are determined by multiple factors, and the model gives clear predictions about the mapping between parameters and outcome variables. Of course, the model parameters are themselves just summaries of complex realities, some elements of which can be changed by policy, others not. Thus, building costs include the ad valorem equivalent of the multiple obstacles to private construction that were outlined in the introduction. Commuting costs are to do with transport investment, and also reflect income levels -cities in which most people have to walk to work. Similarly, demand for non-tradables may be driven by the distribution of tax revenues, foreign aid or natural resource revenues which may themselves be the outcome of a political economy of urban bias (Bates 1981, Lipton 1977, Ades and Glaeser 1991. Hinterland demand for non-tradables is also a function of the income and economic geography of the region in which the city is located; for example, demand will be high if there are no nearby cities offering alternative sources of supply.

Increasing returns, expectations, and coordination failure
We now open up two central features of the model. The first is increasing returns to scaleagglomeration economies -in tradable production, this creating the possibility of multiple equilibria. 18 We show that it is possible that there is a low equilibrium in which the city produces tradables only, and also a high equilibrium in which the city is active in both sectors. The possibility of being trapped in the low-equilibrium arises because of coordination failure. 19 In the simplest case (section 3.1) this is an inter-firm coordination failure; potential producers of tradables do not coordinate to internalize the external economies of scale associated with agglomeration. However, given endogenous choice of the way the city is built -its size and density -the coordination failure goes deeper. Section 3.2 turns to the role of sunk costs and expectations in shaping construction decisions, and shows how (self-fulfilling) expectations may be such that the city is constructed in a way that locks it into the low equilibrium. It is possible that -even if the inter-firm coordination failure between produces of tradables were somehow resolved -the built urban form is incompatible with expanding into tradable goods production.

Increasing returns and multiple equilibria
Suppose that tradable production is subject to agglomeration economies so that productivity in tradable production is increasing with the size of the sector. productivity. This is associated with larger city size and employment and with higher nominal wages (although real wages are held to 0 u by migration and labor supply).

Figure 2: Increasing returns and multiple equilibria
What are the conditions that support this configuration? The appendix gives conditions on parameters, and here we just make three remarks. First, point M exists if , exactly as discussed in section 2.2. Second, city size must (we assume) be bounded above, so it must be the case that as L becomes very large so  T . This is larger the fewer non-tradable jobs are created by demand from additional urban population (high θ), and the more that the increasing wage chokes off hinterland demand for non-tradables (more elastic hinterland demand for non-tradables).
There are two further comments. First it can be shown that aggregate welfare is higher in the equilibrium M' than M. Comparing the two, workers' utility is, by construction, unchanged. Total city rent is higher, since along the labor supply curve it is increasing in wage and city size. There is a third element of welfare change, which is that of 'hinterland' consumers of the nontradable good, whose welfare depends on the price of non-tradables. We show in the appendix that the sum of these elements is increasing in the wage and hence, comparing points on the labor supply curve, welfare is higher at M' than M. However, the distributional impact, given constant real wages in the city, is no change for workers, gain for urban landowners, and loss for outsiders consuming non-traded goods produced in the city.
Second, we suggested in the introduction that cities' experience seemed to be dichotomous, some trapped in non-tradable production, while others have been successful in attracting tradables and have grown large clusters of tradable activities. The possibility of multiple equilibria captures this dichotomous response, showing how small initial differences can lead to quite different outcomes. However, the low equilibrium is supported by a simple inter-firm coordination failure between tradable sector producers. We now enrich this, adding interaction between the tradable sector and the way in which the city is constructed.

Expectations, sunk costs, and construction
Buildings are long lived and most of the costs incurred in constructing them are sunk. This has two implications. One is that construction decisions involve forward looking expectations, and the other is that a historical legacy of urban form shapes the present urban equilibrium. It is therefore possible that cities are 'locked-in' to outcomes not simply because of coordination failure between firms, as we saw in the previous section, but because of a coordination failure spanning time periods and the actions of builders and firms. To capture this, a time dimension must be added to the model, and this is done simply by adding a second time period. We will show how low expectations mean that the city may be constructed at a scale and density that precludes attracting tradable sector production. High expectations can be self-fulfilling as they support construction at greater scale and density, an urban form that lowers costs, attracts tradables, and generates an outcome with higher welfare.
The extension to two periods is straightforward. Production and labor demand are as above, but residential building is durable. Buildings constructed in the first period last into the second, meaning that first period construction decisions depend on prices in period 1 and on expected prices in period 2. Time periods are indicated by subscripts 1, 2 and expectations of future values by superscript E, so period 1 expectations of period 2 prices of houses and labor are, . Worker mobility means that house prices are expected to satisfy indifference condition (5) in both periods, so The expected present value of a house built at x in period 1 is , where δ and 1 -δ are weights attached to each period, and it is this that guides period 1 construction decisions. 20 The expected present value of construction at density ) ( 1 x N on land at x in period 1, and the optimal choice of density are The edge of the city is at distance 1 x where rent equals the exogenous outside rent 0 r , and city population follows, so These equations are analogous to Eqs (5) -(9), except that future expected prices influence construction decisions.
In the second and final period house prices satisfy the indifference condition Building takes place on land that is not previously developed so, for 1 x x  , rent, construction decisions and population are given by and Once again, these are analogous to Eqs. (5)-(9), except that they now apply only to land that is not previously developed, 1 x x  , and the housing stock (and hence population) includes that inherited from period 1.
The full equilibrium comes from the labor demand functions (as previously) and from noting that Eqs. (5a) -(9a) and (5b) -(9b), respectively, define first and second period inverse labor supply functions, ) : ( . Period 1 labor supply now depends also on expectations about period 2: and period 2 labor supply depends also on inherited residential stock and hence population 1 L . It is not generally possible to derive explicit expressions for these supply functions, so we proceed by demonstrating some special cases, and then going to numerical simulation.
First, notice that, by construction, the two equilibria of sub-section 3.1 are stationary perfect foresight equilibria of the two-period model. Thus, if and γ = 2, in which case the edge of the city in period 1 is If the same assumptions hold in period 2 then (5b) -(9b) give . This enables the integral giving city population in period 2 to be evaluated as Rearranging, the inverse labor supply curve is  (14) and reflects that fact that, if w2 is only slightly greater than w1, then the only new building undertaken will be close to the edge of the period 1 city, and low price, rent, and density, therefore accommodating few workers. 22 Quite generally, ) : ( 1 2 2 L L w S must lie on or above ) (L w S . The latter optimizes construction for each value of L; the former inherits a housing stock optimized for first period value of 1 L , and then builds beyond. Some of the city building stock is therefore optimized for the relatively small population at M, a building stock that is not optimal for the larger population.The consequences are clear, from the fact thatas illustrated -labor supply ) : ( does not intersect with labor demand except at M. Thus, even if inter-firm coordination failure were to be resolved, the high equilibrium is unattainable. Starting from period 1 equilibrium at M, the only equilibrium is the stationary one, remaining at M. The previous case worked with period 1 equilibrium at M -as would be the case with stationary (and self-fulfilling) expectations. What if expectations are more positive, expecting tradable production in period 2? We have already seen that {M', M'} is an equilibrium, but it is more interesting to study the transition from non-tradable specialization to tradables. We therefore assume that tradable production is technologically impossible in the first period (e.g. city tradable productivity is very low), but becomes possible in the second period. The first period equilibrium therefore lies on , but in {F1, F2} expectations are 'optimistic', and perfectly foresee the second period outcome with tradable production. This gives higher present value rents, more first period construction, and hence greater first period housing capacity which reduces house prices and first period nominal wages. Importantly if point F1 lies below a0, then the inter-firm coordination problem is resolved. City population and the supply of non-tradables is large enough at F1 for the wage (absent tradable production) to be less than the trigger wage, ensuring that tradable production will take place in the second period. Once again, the built structure of the city overcomes a potential inter-firm coordination failure.

The urban profile
Finally, we illustrate results by developing a numerical example. This puts numbers on the arguments above and demonstrates the existence of the equilibria discussed. It also enables us to relax assumptions (in particular letting outside land rent be positive), establish welfare effects and undertake comparative statics. Perhaps most importantly, it generates illustrations of the city profile -the way in which land rent, house prices and building density varies across the city and between equilibria. The simulations use the same parameter values as underlie earlier figures, except that 0 0  r . Values are given in the appendix; to scale results we note that outside utility is set at unity and the two time-periods have equal weight, δ = ½.

21
Outcomes are illustrated in the panels of Fig. 4. Values of city-wide variables are given in the table and intra-city profiles of rent, house prices, and density are given in the three plots which have distance from the CBD on the x-axis. The , with a 65% premium over outside utility going to meet the urban cost of living, including commuting and housing costs. This wage is consistent with no tradable production occurring if it is greater than the trigger wage, i.e. House prices fall linearly, since commuting costs are assumed linear in distance. Density and rents decline with distance, as expected.
The solid lines, subscripted by period, are equilibrium {F1,F2} in which there is tradable production in the second period, but not the first. The example has 2  m a , and this is the second period wage. This generates high second period house prices and land rents, and also therefore relatively high expected returns to period 1 building, does not occur in period 1, but will surely do so in period 2.

Figure 4: City profiles
The table in Fig. 4 also gives population, real incomes and welfare measures. Equilibrium {F1, F2} supports nearly twice the first period population of equilibrium {M, M}, and more than three times the second period population. Real income gains accrue in the form of total land rents, R, and the present value of these are nearly four times larger in equilibrium {F1,F2} than {M, M} (and are also larger relative to the wage bill); deflating by the price index, real rents are around 3.7 times higher The growth of the city and expansion of tradable production is associated with varying supply of non-tradables to the hinterland, this increasing hinterland consumer surplus in period 1 but reducing it in period 2. The bottom row of the

House prices
Density (=population) increasing city population. Higher construction and commuting costs shift points {F1,F2} to the left, raising the first period wage and reducing population in both periods. The importance of future expectations can be seen by changing δ, the weight on the first period relative to the second. A higher δ has no effect on equilibrium {M, M} and again moves points {F1,F2} to the left, raising w1 and reducing population. Each of these parameter changes increases the likelihood that w1 exceeds 0 a , in which case the city fails to attract tradable production and equilibrium of type {F1,F2} does not exist.

Concluding comments
The introduction to this paper sets out stylized facts about developing country cities, particularly those in Africa, indicating the presence of relatively high urban costs, a high cost of living, high nominal wages (alongside a low real wage), and failure to attract investment in tradable sectors of activity. Each of these is captured in the model developed in this paper. The analysis shows how the combination of increasing returns to scale, durable capital and sunk costs make for multiple equilibria, and the possibility -depending on parameters -that the city is built in a way that precludes establishment of the tradable goods sector.
Several broad policy messages follow. The first is the need to see the city as a whole. Policy has often been siloed, while our analysis highlights the interaction between all aspects of a city's urban form -residential construction, infrastructure, transport -and its economic performance. The second is the high cost of policy failure. Small differences in initial conditions can -with increasing returns, sunk costs, and expectations -set cities on quite different development paths.
In terms of specific policy instruments, the paper points to the importance of efficient land use and infrastructure provision. The consequences of high building and commuting costs go far beyond their direct effects as they shape the sort of economic activity that takes place in the city. The many obstacles to residential construction that were noted in the introduction are all sources of inefficiency, which necessarily raise urban costs and thereby make the city a less attractive place for tradable goods production. Reducing these obstacles has direct benefits, and also increases the likelihood that the city will develop new sectors of activity. Expectations also matter, as the form and extent of investment in durable structures depend on the expected future prosperity of the city. Expectations need to be coordinated in some way, so that investors have confidence that the city -or a particular area within it -is likely to grow. Setting these expectations in a credible way is difficult and may require commitment in the form of investment in public infrastructure.
Stepping outside the confines of the model, two further points can be made. While the model focuses on investment in physical capital, investment in human capital is at least as important. Acquisition of the specialist skills needed to run modern production -and to run the city -will take place only if the costs of the investment are not too high and there are expectations of positive returns. Thus, the arguments that the paper has made with respect to physical capital apply with at least equal force to human capital. The model also draws out the possibility of vicious or virtuous circles leading to multiple equilibria, and here too further forces might be at play. In particular, a fiscal feedback -not present in the model -may be important. A weak tax base is likely to lead to poor infrastructure and public services. This will raise urban costs, directly in the form of high transport and congestion costs and limited availability of power and other utilities, and indirectly via reducing the well-being of workers who require compensating wage payments. As we have seen, higher urban costs will undermine the city's economic performance and hence its tax base (e.g. lowering rents and land values). This in turn reduces the city's ability to provide such services, completing the vicious circle.
Finally -and for future research -this urban model needs to be placed in a wider model of urban hierarchy, within which different cities perform different functions. Some cities will specialize in non-tradables but, in all but the most natural resource abundant countries, some cities that are able to compete in non-resource tradable activities will surely be needed.