WPS7950
Policy Research Working Paper 7950
Breaking into Tradables
Urban Form and Urban Function in a Developing City
Anthony J. Venables
Social, Urban, Rural and Resilience Global Practice Group
January 2017
Policy Research Working Paper 7950
Abstract
Many cities in developing economies, particularly in there may be multiple equilibria. The same initial condi-
Africa, are experiencing urbanization without industrial- tions can support dichotomous outcomes, with cities either
ization. This paper conceptualizes this in a framework in in a low-level (non-tradable only) equilibrium, or diversi-
which a city can produce non-tradable goods and—if it fied in tradable and non-tradable production. The paper
is sufficiently competitive—also internationally tradable demonstrates the importance of history and expectations in
goods, potentially subject to increasing returns to scale. A determining outcomes. Essentially, a city can be built in a
city is unlikely to produce tradables if it faces high urban manner that makes it difficult to attract tradable production.
and hinterland demand for non-tradables, or high costs of This situation might be a consequence of low (and self-ful-
urban infrastructure and construction. The paper shows filling) expectations or history. The predictions of the model
that, if there are increasing returns in tradable production, are consistent with several observed features of African cities.
This paper is a product of the Social, Urban, Rural and Resilience Global Practice Group. It is part of a larger effort by
the World Bank to provide open access to its research and make a contribution to development policy discussions around
the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be
contacted at tony.venables@economics.ox.ac.uk.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Breaking into Tradables:
Urban Form and Urban Function in a Developing City*
Anthony J. Venables
University of Oxford,
CEPR and International Growth Centre
Keywords: city, urban, economic development, tradable goods, structural transformation.
JEL classification: O14, O18, R1, R3
* Forthcoming, Journal of Urban Economics
Acknowledgements: I gratefully acknowledge support from the Africa Research Program on
Spatial Development of Cities at LSE and Oxford, funded by the Multi Donor Trust Fund on
Sustainable Urbanization of the World Bank and supported by the UK Department for
International Development. Thanks also to referees, an editor, Paul Collier, Avinash Dixit,
Vernon Henderson, Patricia Jones, Somik Lall, and seminar participants for helpful comments,
and to Sebastian Kriticos for research assistance.
Author’s Address:
Department of Economics
Manor Road
Oxford OX1 3UQ, UK
tony.venables@economics.ox.ac.uk
1. 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 non-
tradables, 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 (non-
tradable only) equilibrium, or diversified in both tradable and non-tradable production. Third,
1
Jedwab and Vollrath (2015) discuss alternative reasons for this, including urban technologies, push from
agriculture, politics, and natural population increase.
2
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 self-
declared 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
2
Jones (2016). Manufacturing shares are predicted values from a quadratic function fitted to the cross
section data.
3
Integrated Public Use Microdata Series (iPums) available for download at: https://usa.ipums.org/usa-
action/variables/group
4
Ethiopian data dates from 1994; recent manufacturing investments suggest that it could now exceed this.
3
Table 1: Share of employment in manufacturing: Urban areas
Africa India Other Asia
1990s 2000s States 1999 2004 1990s 2000s
Cameroon - 10.2 Maharashtra 23.4 23.8 Thailand 10.1 12.2
Ethiopia 7.98 - West Bengal 24.6 27.5 Bangkok 25.2 20.3
Addis Ababa 18.9 - Delhi 23.1 25.5 Samut Prakan 24.9 48.9
Ghana 14.4 12.6 Tamil Nadu 27.0 30.7 Nonthaburi 18.8 15.0
Accra 17.5 14.9 Karnataka 23.8 21.9 Udon Thani 9.87 6.93
Liberia - 1.6 Andhra Pradesh 17.7 20.1 Vietnam 16.8 16.9
Malawi 7.19 7.09 Gujarat 25.11 37.2 Ho Chi Minh 37.5 35.1
Blantyre 10.7 11.4 Uttar Pradesh 22.9 26.8 Ha Noi 21.6 18.4
Mali 6.26 6.55 Rajasthan 20.6 23.3 Da Nang 21.4 20.5
Mozambique 5.3 5.1 Bihar &Jharkhand 16.2 12.6 Hai Phong 24.8 24.3
Maputo 9.46 6.8 Punjab 22.7 27.2 Malaysia 20.1 22.4
Rwanda - 2.58 Kerala 21.2 15.7 Kuala Lumpur 25.2 20.9
Kigali - 4.92 Haryana 20.3 25.9 Seberang 40.3 44.4
Sierra Leone - 0.74 Pondicherry 28.5 20.8 Indonesia 9.19 8.00
Sudan - 6.12 Chhattisgarh 17.6 19.0 Jakarta 18.9 15.5
Tanzania - 4.96 Orissa 16.8 14.3 Bandung 20.6 23.0
DaresSalaam - 8.85 Chandigarh 15.7 17.4 Cambodia 5.01 6.92
Uganda - 4.82 Daman, Diu & Goa 11.5 15.8 Phnom Penh 13.2 26.9
Kampala - 6.92 Himachal Pradesh 8.86 14.3 Takeo 2.43 6.84
Dadra &Nagar
Zambia 6.11 - 52.5 27.9 Sihanoukville 8.02 17.5
Haveli
Lusaka 10.4 - Battambang 8.25 8.16
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.
4
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 (Henderson et al. 2016).
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. Nakamura et al. (2016) 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.
5
These findings on labor costs and price levels are consistent with earlier work by Gelb et al. (2013,
2015), although this work does not have an urban focus.
6
Nakamura et al. also report findings from the Economist Intelligence Unit’s Worldwide Cost of Living
Survey. While this is produced principally for expatriates its findings are consistent, indicating African
cities are 30% more costly for households than cities in low- and middle-income countries elsewhere.
5
2. Production and urban form
2.1 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 LT and LN giving total city employment L LT LN . 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 non-
tradable production, and then in tradables.
Non-tradable goods meet demands from the local market and have price pN , 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, pN w and the value of
supply is wLN . The value of demand for non-tradables is (1 ) wL pN h pN . 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
(1 ) wL . The second term, pN h pN , 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
These assumptions imply full employment and an integrated labor market. It would be possible to add
labor market imperfections (for example, part of non-tradable production being undertaken by an informal
sector), but this is inessential for the arguments being made.
8
The definition of non-tradables is necessarily quite elastic. The crucial distinction is the extent to
which the price depends on supply from the city, or is set on a wider regional or international market.
6
hinterland spending on non-tradables produced in the city. We refer to h pN 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 pN w and LN L LT , the equality of
supply and demand for non-tradables is L LT w (1 ) wL whw . Rearranging,
w h 1 L LT . (1)
This is the wage at which net urban supply of non-tradables equals hinterland demand, where
h1 . 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 aLT , and this is the wage
offered in the sector. We assume that productivity is either constant or increasing in LT ,
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,
w aLT , LT 0 ; w aLT , LT 0 . (2)
Together, Eqns. (1) and (2) implicitly define the city’s (inverse) labor demand schedule w D ( L) ,
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’, a0 a(0) , at which tradable production commences. We summarize it as
follows:
If the tradable sector is inactive, then LT 0 and, from (1),
w D ( L : LT 0) h 1 L . (3a)
If the tradable sector is active, then LT 0 and, from (1) and (2),
wD ( L : LT 0) a( LT ) , with LT solving a( LT ) h1 L LT . (3b)
The wage offered is the maximum,
wD ( L) max wD ( L : LT 0), w D ( L : LT 0) . (3c)
7
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 aLT a0 . On the downward
sloping segment the entire labor force is employed in the non-tradable sector,
w D ( L : LT 0) h 1 L , downward sloping as more employment increases supply of the non-
traded good, reducing price and wages. If the ensuing wage is less than a0 then tradable
production is profitable, giving the horizontal segment, w D ( L : LT 0) a0 . The dividing line is
where the urban population is L ha0 / .
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 u0 , 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),
v( x) w p( x) xtw1 / w1 w p( x) w 1 xt . (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 xtw1 , where
w1 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
1
tradables are the numeraire and the price of non-tradables is w, the price index is w .
12
If there are transfer payments spent in the city (resource rents, taxes, foreign aid) we assume that, while
8
Workers are perfectly mobile, choosing between living outside the city with utility u0 , or at
locations x in the city. For all occupied urban locations, it must therefore be the case that
v( x) u0 , implying that house prices at each distance, p(x), satisfy the indifference condition
p( x) w u0 xt w1 . (5)
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, r ( x) , is
revenue from housing minus construction costs, i.e.
r ( x) p( x) N ( x) cN ( x) w1 , γ > 1. (6)
Construction costs per unit land are cN ( x) w1 , 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 w1 . Developers
choose density to maximize rent, giving first order condition and maximized rent,
N ( x) p( x)w 1 / c 1 /( 1)
, (7)
r * ( x) 1 1 / p( x) N ( x) 1 1 / p( x) /( 1) w 1 / c
1 /( 1)
. (6’)
The edge of the city is at distance ~
x where rent equals the exogenous outside rent r0 , i.e.
r * (~
x ) r0 . (8)
Housing capacity and total city population up to this edge are
~
x
L 0 N ( x)dx . (9)
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.
13
See Henderson, Tanner and Venables (2016) for a richer modelling of construction technologies that
differentiates between formal and informal housing. Qualitative results of the present paper would be
unchanged with this more general formulation.
9
assumptions.14 The first is that outside rent is zero, r * ( ~x ) 0 ; this implies that the edge of the
x ) N (~
city has p( ~ x ) 0 (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:
x ) 0 implies p( ~
r * (~ x ) 0 and hence ~
x w u0 / t . (10)
Second, we assume that construction costs increase with the square of density, γ = 2, giving
population
w u
~ ~
xt / 2c.dx w u0
x x 2
L N ( x)dx
0 / 4tc . (11)
0 0
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
wS ( L) u0 2tcL1 / 2 1/
. (12)
This is the inverse labor supply curve, wS ( L) 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’).
14
These assumptions are relaxed in simulations later in the paper.
15
This and subsequent Figs. are derived by numerical simulation, parameters given in the appendix.
16
If r0 = 0 the curve is vertical at L = 0, as illustrated.
10
Figure 1: Urban equilibrium with constant returns to scale
w
wD (L : LT 0) wS (L : )
w S ( L)
E+
E
a0
w D ( L : LT 0)
1/
u0
L ha0 /
L a0 u0
2
/ 4tc L
2.2 Equilibrium with constant returns in tradable production
The urban labor demand and supply curves, Eqs. (3) and (12), give equilibrium values of the
wage, city population and sectoral employment, as illustrated by points E and E+ on Fig. 1. Point
E+ gives the equilibrium if urban costs are high, wS ( L : ) , in which case the wage exceeds a0
and city produces only non-tradables. At lower costs equilibrium is at E, with both sectors
active, wage rate w a0 , and hence city population and employment in tradables respectively,
L a0
u0 / 4tc and LT L ha0 (from Eqs. (11) and (3b)). The message is simple. Both
2
cases have the same real wage, u0 . 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.
11
What factors shape the likelihood of each of these outcomes occurring? The dividing line
between cases is where wages and employment levels satisfy a0 wS ( L) wD ( L : LT 0) , so
the city is specialized in non-tradables if a0 u0 2cth(a0 ) /
1/ 2 1/
(using Eqs. 12 and 3a).
This is more likely if, from the left-hand side of the inequality, a0 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 L / ~x . Total rents, R, are derived
by integrating over rents at each point in the city (Eqn. 6’),
~
x 3
R (1 1 / ) 0 p( x) N ( x)dx w1 w u 0 / 12tc (13)
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 h( p) Hh ( p)
12
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).
Table 2: Comparative statics
Rent
Likelihood Population Wage Size Density ˆLˆ,
ˆ ~
ˆ ˆ~ˆ R
LT = 0 L ˆ
w x L x ˆL
R ˆw ˆ
LT = 0: LT > 0 LT = 0: LT > 0 LT = 0: LT > 0 LT = 0: LT > 0 LT = 0: LT > 0
ΔH > 0 + + 0 + 0 + 0 + 0 + 0
Δc > 0 + - - + 0 + 0 - - + 0
Δt > 0 + - - + 0 - - + 0 + 0
Δa0 > 0 -
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.
13
3. Increasing returns, expectations, and coordination failure
We now open up two central features of the model. The first is increasing returns to scale –
agglomeration 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.
3.1 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. The left-hand segment of the labor
demand curve, w D ( L : LT 0) is unchanged, and the segment with tradable sector active,
w D ( L : LT 0) , becomes upwards sloping. Fig. 2 illustrates for the case in which productivity is
linear in tradable employment between lower and upper bounds, a0 , am ,
so a( LT ) mina0 LT , am , 0, a0 am .
As illustrated, returns to scale are strong enough for labor demand to have three intersections
with labor supply. Points M and M’ are stable equilibria, while the intermediate intersection is
unstable (under a dynamic in which the tradable sector expands or contracts according to
whether profits are positive or negative). At the lower point, M, the city is specialized in non-
tradable production and wages are above the trigger point, a 0 , at which tradable production
commences. Entry of a small mass of tradable produces is not profitable; this point is an
equilibrium if coordinated entry of a sufficiently large mass of tradable producers (who would
reap the benefits of agglomeration economies) is not possible. At the upper equilibrium, M’, the
tradable sector is active and is large enough for agglomeration economies to have cut in, raising
18
For discussion of agglomeration economies in the context of developing economies see World Bank
(2009) and for evidence see Chauvin et al. (2016).
19
As in, for example Murphy et al. (1989), Henderson and Venables (2009).
14
productivity. This is associated with larger city size and employment and with higher nominal
wages (although real wages are held to u0 by migration and labor supply).
Figure 2: Increasing returns and multiple equilibria
w wD (L : LT 0)
M’ w D ( L : LT 0)
M
a0
wS ( L)
1/
u0
L
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
a0 w S L w D ( L : LT 0) , 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
wS L w D ( L : LT 0) , as illustrated. Third, a necessary condition for multiple intersections is
that economies of scale in tradables are large enough that, in some range, the slope of the labor
demand curve exceeds that of labor supply. Recall that, along w D ( L : LT 0), LT solves
dwD dL a '
a( LT ) h1 L LT , Eqn. (3b). The slope is therefore a' ( LT ) T . The slope
dL dL 1 h' a'
is steeper the greater are marginal returns to scale, a' ( LT ) , adjusted for the fact that as L increases
some extra jobs are created in the non-tradable sector so dLT / dL 1 . This is larger the fewer
non-tradable jobs are created by demand from additional urban population (high θ), and the more
15
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 non-
tradable 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.
3.2 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
16
values by superscript E, so period 1 expectations of period 2 prices of houses and labor
E E
are, p2 ( x), w2 . Worker mobility means that house prices are expected to satisfy indifference
condition (5) in both periods, so
p1 ( x) w1 u0 xt w1
1
, E
p2 E
( x) w2 u0 xt w2
E
.
1
(5a)
The expected present value of a house built at x in period 1 is p1 ( x) (1 ) p2
E
( x) , 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 N1 ( x) on land at x in period
1, and the optimal choice of density are
r1E ( x) p1 ( x) (1 ) p2
E
( x) N1 ( x) cN1 ( x) w1
1
, (6a)
N1 ( x) p1 ( x) (1 ) p2
E 1
( x) w1
/ c 1 /( 1)
, (7a)
The edge of the city is at distance ~
x1 where rent equals the exogenous outside rent r0 , and city
population follows, so
~
r1* ( ~
x
x1 ) r0 , L1 0 1 N1 ( x)dx . (8a), (9a)
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
p2 ( x) w2 u0 xt w1
2 . (5b)
Building takes place on land that is not previously developed so, for x ~
x1 , rent, construction
decisions and population are given by
r2 ( x) p2 ( x) N 2 ( x) cN2 ( x) w1
2
, (6b)
N 2 ( x) p2 ( x) w 1 / c
1 /( 1)
, (7b)
~
r2* ( ~
x
and x2 ) r0 , L2 L1 ~
x
2
N ( x)dx . (8b), (9b)
1
20
We continue to use the word house ‘price’ rather than rent, although payments are made in both
periods.
17
Once again, these are analogous to Eqs. (5)-(9), except that they now apply only to land that is
not previously developed, x ~
x , and the housing stock (and hence population) includes that
1
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
E
functions, w1S ( L1 : w2 S
) and w2 ( L2 : L1 ) . 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 L1 . 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 w1 w2
E
w2 so that
E
p1 ( x) p2 ( x) p2 ( x) then L2 L1 and the model gives stationary outcomes identical to those
in section 3.1.21 This is illustrated on Fig. 3, with points M, M’ and labor supply curve
wS ( L) exactly as in Fig. 2. We refer to these equilibria as {M, M} and {M’, M’}, the two
elements in brackets representing the outcome in the two time periods.
Second, suppose that the first period equilibrium is at M (so the city was built with stationary
E
expectations, w2 w1 ). What effect does this have on the set of opportunities attainable in
S
period 2 – or more formally, what does w2 ( L2 : L1 ) look like? To answer this we assume that
r * (~
x ) 0 and γ = 2, in which case the edge of the city in period 1 is ~ 1
x w u / t (Eqn. 10).
1 0
If the same assumptions hold in period 2 then (5b) – (9b) give ~
x2 w
2 u 0 / t . This enables
the integral giving city population in period 2 to be evaluated as
~ ~
u0 xt / 2c.dx L1 w
x2 x2 2
L2 L1 ~ N 2 ( x)dx L1 ~ w2
2 w1 / 4tc
x1 x1
Rearranging, the inverse labor supply curve is
S
w2
( L2 : L1 ) w1 2tcL2 L1
1/ 2
1/
. (14)
E
21
With w1 w2
E
w2 and p1 ( x) p2 ( x) p2 ( x) Eqns. 5a-8a, 5b-8b and 5-8 are identical. It follows
x1 ~
that ~ x2 and L1 L2 .
18
S
This is the dashed line w2 ( L2 : L1 ) through point M on Fig 3. The steepness of this curve close
to M is apparent from Eq. (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
S
price, rent, and density, therefore accommodating few workers.22 Quite generally, w2 ( L2 : L1 )
must lie on or above wS ( L) . The latter optimizes construction for each value of L; the former
inherits a housing stock optimized for first period value of L1 , 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 that –
S
as illustrated – labor supply w2 ( L2 : L1 ) 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.
Figure 3: Multiple equilibria with sunk costs
wD (L : LT 0) S
w2 ( L2 : L1 )
S
w2 ( L2 : L1 )
w
F2
M’ w D ( L : LT 0)
M
a0
F1
wS ( L)
1/
u0
L
22
See Eqn. 12 and following discussion for comparison.
19
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 w D ( L : LT 0) but, if expectations are more positive, will not be at
E
point M. If tradable production is expected in period 2 then w2 w2 aM . This raises the
returns to building in period 1 (Eqns. 5a-7a) , so there is more and denser period 1 building,
accommodating a larger city population and moving the first period equilibrium along
S
w D ( L : LT 0) to point F1. The second period labor supply curve through this point, w2 ( L2 : L1 ) ,
does not have a closed form solution, but is qualitatively similar to (14), as illustrated on Fig. 3,
and gives the perfect foresight equilibrium pair {F1, F2}.
The essential difference between the two equilibria, {M, M} and {F1, F2} lies in expectations.
Both start with LT 0 , 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.
3.3 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 r0 0 . 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.
20
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 table reports outcomes for three equilibria: the
stationary low and high equilibria, {M, M} and {M’, M’} respectively, and the equilibrium
{F1,F2}. For clarity, the plots just give {M, M} (dashed lines) and {F1,F2}, for which the two
time periods are subscripted.
Equilibrium {M, M} has wage in each period w 1.65 , 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.
a0 1.65 . The intra-city profiles are given by the dashed lines on the figure. The city edge is
the kink in the rent function (up to which r ( x) r0 and beyond which land is undeveloped).
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 am 2 , 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, r1E ( x) . The city is therefore built
both larger and denser in the first period than is the case in equilibrium {M, M}, as is clear from
the density profile N1 ( x) . Period 2 land rents are higher again, so that there is further
construction and this is at high density, curve N 2 ( x) . The relatively large amount of building in
period 1 accomodates a large population, as a consequence of which period 1 wage, w1 1.42 ,
and house prices, p1 ( x) , are lower than that in equilibrium {M, M}. If the wage is less than a0
enough (in this example, requiring a0 1.42 ) tradable production is triggered. By assumption, it
does not occur in period 1, but will surely do so in period 2.
21
Figure 4: City profiles
Rent
{M,M} {M’,M’} {F1, F2}
w1= w2= 1.65 w1= w2=2 w1 = 1.42 w2 = 2
L1= L2 = 84 L1= L2=221 L1 = 129 L2 = 186
R1= R2=5.1 R1= R2=29 R1 = 10.1 R2 = 29
r1E ( x) r2 ( x)
Hinterland Hinterland H-land H-land
CS1=CS2=28 CS1=CS2=19 CS1= 37 CS2 = 19 r ( x)
- CV1= CV2=13 CV1=16 CV2=13
House prices Density (=population)
N1 ( x)
N 2 ( x)
p 2 ( x)
p ( x) N ( x)
p1 ( x)
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 table gives the combined welfare
change from {M, M} to other cases as a compensating variation. The gain ranges from 9% to
12% of the wage bill in {M, M}.23
The effects of varying parameters of the model are as would be expected. Higher demand for
non-tradables, H, shifts labor demand w D ( L : LT 0) to the right, raising the period 1 wage and
23
The compensating variation is a discrete version of the marginal welfare change given in the appendix.
22
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 a0 , in which case the city fails to attract tradable production and
equilibrium of type {F1,F2} does not exist.
4. 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.
23
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.
24
Appendix
Parameters used in simulation
u0 = 1; θ = 0.4; t = 0.001; δ = 0.5; c = 0.1; γ = 2; h(w) = Hw-ε, H = 150, ε =3.
In Fig. 1 a0 = 1.45, am = 1.45, α = 0. In figures 2-4, a0 = 1.45, am = 2; α = 0.008.
In Fig. 1- 3 r0 = 0. In Fig. 4 r0 = 0.05.
Section 2.1
1) Derivation of city population, (11), with general γ.
1 /( 1) 1 /( 1)
w 1 w 1
w (u
~
x ~
x ~
x 1 /( 1)
L N ( x)dx p( x) 1 /( 1)
dx xt ) w1 dx
c c
0
0 0 0
1 /( 1)
1 1
t w
u0
/( 1)
.
c
2) Derivation of total land rent, (13), with general γ.
1 /( 1)
~
x 1 w 1 ~
x
R (1 1 / ) p( x) N ( x)dx
1
c p( x) /( 1) dx
0
0
1 /( 1)
( 1)2 1
t (2 1)
w1 w u0
( 2 1) /( 1)
.
c
R 1 w u0
.
wL 2 1 w
Section 2.2
3) Comparative statics:
Equilibrium conditions:
Labor supply (11):
L w u0 / 4tc . 2
Labor demand (3): If LT = 0, L h( w) H / , elasticity of demand wh' ( w) / h( w) .
If LT > 0, a(LT) = a0, elasticity of demand .
City area (10): ~
x w u / t . 0
Total rent (13):
R w1 w u0 / 12tc . 3
Define w / w u0 0 , and note that u0 / w u0 0 .
Totally differentiating and solving the log-linearized system gives comparative static responses:
25
ˆ H
w
ˆ t
ˆ / 2
ˆc
ˆ w
L ˆH
ˆ tˆ c
ˆ 2H
ˆ / 2
~
ˆ w
x ˆ tˆ Hˆ c ˆ tˆ/ 2
ˆ 1 3w
R ˆ tˆ c ˆ 1 3H ˆ 1 t ˆ / 2
ˆc .
It follows that:
Average density: ˆ~
L ˆ c
ˆ H
x ˆ / 2
ˆ t
Average rent: R ˆ 1 2 w
ˆ ~
x ˆ c
ˆ 1 2 H ˆ 1 c
ˆ t
ˆ / 2
Rent per person: ˆ 1 H
ˆL
R ˆ tˆcˆ / 2
Rent/ wage bill: ˆL
R ˆw ˆ Hˆ tˆcˆ / 2
Section 3.1: Multiple equilibria
Existence of equilibrium M (Fig. 2). Non-tradable production commences at L ha0 / so
existence of point M requires w S ( L) u0 2(tch(a0 ) / )1/ 2
1/
a0 .
Existence of equilibrium M’: (Fig. 2). If the productivity relationship is linear in tradable
employment between lower and upper bounds, a0 , am , then a( LT ) mina0 LT , am . The
maximum level of productivity is first attained at LT (am a0 ) / . From Eqn. (3b), at point M’,
h(am ) L LT , hence L am a0 / ham / . Point M’ exists if
w S ( L) u0 2tcam a0 / ham /
1/ 2 1/
am .
Section 3.1. Welfare analysis
We measure the change in welfare between situations 1 and 0 as the compensating variation
CV R1 CS1 CS0 / w1
1
R0 / w1
0
d ( R / w1 ) dCS / w1 where CS is hinterland
consumer surplus. The interpretation is the change in the real value of rents net of
‘compensation’ of hinterland consumers for a change in consumer surplus. Total rents are
3
R w1 w u0 / 12tc . Differentiating with respect to the wage and using L w u0 / 4tc ,
2
gives d ( R / w1 ) Lw 1dw. The change in hinterland consumer surplus is simply quantity
times price change, so dCS LT Ldw . Hence, CV LT w 1dw , i.e. the real value of the
productivity increase in tradables.
26
Section 3.2. Second period population
Using (5b) and (7b),
~ ~
u 0 xt / 2c.dx w2 u 0 t ~ x 2 / 2 ~
x1 ~ x1 / 2c which, with
x2 ~
x2 x2
N 2 ( x)dx ~ w2
~
x1 x1
i~
x1 w1
u 0 / t and ~
x 2 w
2 u 0 / t gives the expression in the text.
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