Urban-rural imbalances: a model to evaluate how cooperation can efficiently
provide goods and services in peripheral areas fostering human and economic
development.
Andrea Salustri
Fondazione Economia‐Università Tor Vergata
via Columbia 2, 00133 Roma
E‐mail: [email protected]
Michele Mosca
Universitá Federico II Napoli
Via Rodinó 22, 80133 Napoli
E-mail: [email protected]
Federica Viganò
Libera Universitá di Bolzano‐ Free University Bozen‐ Freie Universität Bozen
Regensburger Allee 16 ‐ viale Ratisbona, 16 , 39042 Brixen‐Bressanone Italy
E‐mail: [email protected]
Abstract
Cities are an important driver for social and political change and for rural development, attracting
investments and providing infrastructural services. Nevertheless, the urban environment has been
addressed as the major cause of several problems affecting peripheral areas (suburbs, rural areas),
as cities polarize the space, marginalizing the rural areas, and “sprawling” in the neighbourhoods.
Indeed, in rural areas the lower “density” of the public and private supply of goods and services
has negative effects in terms of efficiency, as rent-seeking behaviors are not automatically
countered due to a lower presence of competitors. Laying on these premises, we propose a
theoretical model to show how in a spatial framework characterized by urban-rural imbalances,
cooperatives and nonprofit organizations, can exert a distributive function fostering a fair and
equal treatment among residents. Specifically, the market and the public sector might be unable
to satisfy the needs of rural or marginalized urban residents due to higher transport costs or lack
of financial resources, even if, considering both the accounting and the economic profit, there
might be incentives to serve also marginal areas in order to avoid environmental hazards, urban
sprawl and the development of the illegal economy.
Keywords: cooperative economics; human development; illegal systems; urban-rural
development; size and spatial distributions of regional economic activity; social capital.
JEL Classification: J54; O15; K 4; R11; R12; Z13
1
1.
Introduction
Cities are an important driver for social and political change and for rural development, attracting
investments and providing infrastructural services. Nevertheless, the urban environment has been
addressed as the major cause of several problems affecting peripheral areas (suburbs, rural areas),
as cities polarize the space, marginalizing the rural areas, and “sprawling” in the neighborhoods.
Indeed, in rural areas the lower provision of public and private goods and services could generate
lower level of efficiency, diminished income of residents and consequently a negative impact on
wellbeing of rural communities. Specifically, it is widely recognized the need of a more balanced
redistribution of urban growth, aimed at discouraging concentration in few and large urban
agglomerations, and promoting the development of peripheral areas. More in general, when
policy actions are inefficient, ineffective or absent, the process of urbanization determines the rise
of spatial inequalities, marginalization and social exclusion. Consequently, a by-product of rapid
urbanization is the increase of extreme poverty in rural areas, and the persistence of gaps in
urban-rural connectivity, despite the growing overall level of interdependence.
In times of recession, the action of private and public institutions (firms, public administrations,
banks…) is limited to what it can be considered “financially accountable”. Therefore, there might
be empty places in rural areas in a broader sense (countryside, natural areas) and in urban
peripheries, where “emptiness” can be measured in terms of lack of resources, welfare,
development and growth. In these not necessarily scarcely populated places, people might
repolarize the space creating networks able to overcome the lack of agglomeration and generate a
subsidiary welfare supply, while cooperatives and nonprofit organizations, by collectivizing the
services through the constitution of members-owned organizations, are a good response to several
societal challenges.
Laying on these premises, we propose a theoretical model to show how in a spatial framework
characterized by urban-rural imbalances, cooperatives and nonprofit organizations can
compensate or substitute this lack of supply by producing themselves goods and services,
fostering a fair and equal treatment among residents. Specifically, the market and the public
sector might be unable to satisfy the needs of rural or marginalized urban residents due to higher
distance costs or lack of financial resources, even if, considering both the accounting and the
economic profit, there might be incentives to serve also peripheral areas in order to avoid
environmental hazards, urban sprawl and the potential development of the illegal economy.
In this paper we illustrate how in areas where rural communities are in the need of goods and
services generally provided by the public or the private, there are two potential paths, the first
leading to reinforce the social capital boundaries and networks, revitalizing growth and fostering
human development and social cohesion. In this first hypothesis cooperatives and the nonprofit
sector can provide an additional supply at lower costs by repolarizing the space according to
people's needs, particularly in presence of public expenditure contraction or during crisis. The
second path could bring to a twofold vicious cycle which goes from the social and cultural
impoverishment to the increase of the social insecurity and the incentives to illegal activity, from
the spread of distrust between citizens or between them and the legal institutions to the
strengthening of a development model sustained by crime which inexorably undermines the
chances of economic and social growth of the territories.
2
2.
Literature review
We heavily lie the foundations of this analysis on the seminal research of Limao and Venables
(1999), about the relevance of transport and distance costs and remoteness and isolation of
countries as determinants for enhancing participation of territorial areas to the economic
production networks. A second body of literature considers the lack of a competitive market in
the rural areas and consequently, the retraction of the private sector, generally discouraged to
invest. Non-competitive markets, such as rural areas, can compromise efficiency gains: on the
one hand, transaction costs tend to rise in non-competitive markets because governments should
exercise greater oversight, because of the lack of discipline instilled by competition. On the other
hand, transport costs increase for both providers (public or private) and services beneficiaries (i.e.
citizens living in marginalized areas). There is a wide evidence that rural markets are
characterized by low levels of competition, thus less attractive also for the profit companies
(Kodrzycki 1994; Warner 2009; Bel and Fageda 2010; Warner and Hefetz 2003, Warner and
Hefetz 2008). Although linkages between rural and urban areas are intense, rural/marginalized
areas due to isolation and remoteness risk to not be served by both the public and the private
actors (Dijkgraaf & Gradus 2003; Levin & Tadelis 2012).
Coming to the concept of social capital, there is a long intellectual and empirical research. We
aim to recall the fundamental and seminal studies of Putnam (1993, 2000) and Coleman (1988,
1990), and we look in the perspective of social capital interactions and involvement towards
positive outcomes or discovering the potential “dark side of social capital” (Putnam 2000). The
notion of social capital illustrates how a social structure facilitate action between the members or
the subjects within the structure. Different patterns of relations can be distinguished: Putnam
distinguishes between bonding and bridging social capital. Bonding social capital is directed to
the inside of the group and it leads to exclusive identities and tends to reinforce homogeneous
groups. Examples of this type are family ties, ethic/religious fraternal organizations. Bridging
social capital is directed to the outside of a group and bridges people of different social classes.
Bridging social capital exists, for example, in civic movements associative organizations,
nonprofit and cooperative. The former type is likely to have illiberal effects because it seeks to
build networks of the already like-minded to the exclusion of others (Putnam 2000: 358).
Bridging social capital builds connections across groups and is inclusive in nature, offers the
potential to solve some of society’s most intractable problems (Putnam 2000: 363). Social capital
is thus a relational theory of social interaction which understands actors and their purposeful
actions as inter-dependent (Castiglione et al. 2008; Ayios et al. 2014).
In peripheral/marginalized areas, there are two possible evolution of social capital: the first
reflects the positive relation between actors, leading to collaborative and strong social ties,
facilitating community based solutions for the inefficient supply of public and market goods
coherent with people needs (Putnam 2000). The self-production and localization of such goods,
as our analysis proves, reduces transaction and transport costs for the public actor. A second
evolution of social capital discover one of the potential dark side of it. Specifically when public
policies are ineffective, public goods and services are not sufficiently delivered and citizens
perceive to be unfairly threatened as local resources are additionally subtracted through taxes,
there is a ground for the proliferation of criminal organizations. Because of the lower provision of
public goods, criminal organizations could take over (or distort) social capital. Criminal
organizations, in fact, use relationships and nets, which are built in the territories among
individuals and between these individuals and institutions. These nets arouse and display strength
and power and destroy the myth of invincibility on the public institutions. Moreover, criminal
3
organizations by exerting a strong and perverse control over social capital reduce individuals´
space of freedom and affect irreversibly the legal productive activities (Masciandaro 2000). As a
final result, even if apparently criminal organizations provide illegal services to communities,
they deeply influence the entrepreneur's behavior and produce a social and economic
impoverishment (Mosca and Villani 2013).
A remedial action of the policy maker at this regard is crucial to discourage people to choose
illegal activities. The promotion of policies sustaining the cooperative and nonprofit sector could
favour the accumulation of the “pure” social capital by activating the mechanisms, which push
individuals to prefer legal to illegal activity. These questions, however, have received little
attention in the literature. In particular, “civic norms may attach guilt and shame to criminal
behaviour and may also stimulate trust in others (...)” (Buonanno P., Montolio D. and Vanin P.,
2009).
3.
A model for evaluating distance costs between urban and rural areas
By developing a theoretical framework, we illustrates how land matters in determining the
localization and the degree of development of the market and of the public sector along a
continuous that goes from urban to rural areas. Moreover, we focus on how distance costs
determine the production and delivery of goods and services in the perspective of different actors,
the private, the public, the cooperatives´ and nonprofit organizations.
The logical framework, as reported in figure 1 in appendix, is made of three vertexes (“Urban”,
“Rural” and “Institutions”), and three sides (“People”, “Resources” and “Land”). We discuss the
relation among two vertexes (Urban, Rural) and a side (Land)1.
In Appendix A we provide a formal model. We assume that Land is a continuum that goes from
Urban to Rural characterized by a heterogeneous level of distance costs, population density and
per capita income. We assume also that there are no production costs, so that the only costs that
matter are “distance costs”. Specifically, distance costs increase as far as one moves from U
toward R due to several reasons (transport costs, corruption, less availability of infrastructures,
etc.). On the other hand, as far as one moves from U to R, population density and per capita
income are decreasing.
3.1 Market equilibria and profit maximization choices
Figure 1 illustrates how the market equilibria (the solution of firms’ profit maximization
algorithm) changes moving across land. Specifically, we illustrate how the aggregate demand and
the aggregate supply functions for a fixed level of Q vary across land. We notice how, as far as
we move from U to R, the aggregate demand is decreasing, as:
consumers, having a lower income, have a lower willingness (capability would be more
appropriate) to pay the same amount of goods;
population density decreases, therefore the aggregate demand becomes more inelastic.
On the supply side, as far as we move from U to R, in the absence of production costs and of
1
Notice how “Institutions”, “Resources” and “People” might recall the production function Y = f(K, L).
4
fixed distance costs, the marginal distance costs2 tend to increase. Therefore, due to a decreasing
“capability” to pay, more rigid aggregate demands, and higher distance costs, firms have a
convenience to serve only a certain share of land. Moreover, in order to make extra-profits, they
have an incentive to serve a place as much as it is closer to U.
The share of land served by the market depends on the slope of the demand and supply curve, but
it might be influenced also by the market power of the economic agents. In order to keep the
analysis as simple as possible, we analyze only the case with linear demand (with respect to Q)
and supply curve, when the market is perfectly competitive.
§ An example
Assume for example that an entrepreneur is looking for the optimal place to sell the level Qk of
output produced (recall that production costs are equal to zero). Moving from the headquarters of
the firm, located in the city, toward the rural areas, he selects three places (A, B, C), characterized,
respectively, by marginal distance costs that are:
- constant with respect to quantity;
- increasing in the distance from U (MCA < MCB < MCC);
- equal to the average distance costs (there are no fixed distance costs).
Moreover, the entrepreneur estimates in each place a linear aggregate demand, such that
𝑝𝐴 (𝑄) = 𝑎 − 𝑏𝑄, 𝑝𝐵 (𝑄) = 𝑎′ − 𝑏 ′ 𝑄, 𝑝𝐶 (𝑄) = 𝑎′′ − 𝑏 ′′ 𝑄,
With a > a’ > a’’, and b < b’ < b’’. Finally, he observes that MCC > a”. The results of the analysis
are summarized in Figure 1.
p
p
p
a
B
A
C
a’
MCC
a
”
MCB
MCA
O
b
Q
k
b’’
b’
Q
O
Fig. 1. The exchange process in the (p, Q) plan.
Q
k
Q
O
Q
Q
k
In Figure 2, we represent the same issues in the (p, L) plan, where L denotes “Land”, and goes
from U to R. Notice how in A, the quantity Qk is demanded and by selling it the entrepreneur
2
We draw the marginal distance cost line to highlight the equilibrium condition between marginal distance costs and
prices. Indeed, under the assumption of zero fixed costs, each point of the marginal distance cost line coincides with
the average costs that the entrepreneur pays in that place to supply the quantity Q.
5
obtains a positive profit. In B, the quantity Qk is still demanded, but due higher marginal distance
costs the entrepreneur would suffer a loss. In C, the quantity Qk is not demanded, both because of
lower incomes and lower population density. Therefore, the entrepreneur sells the output
produced in A, obtaining a positive profit.
MC(L, Qk)
p(L, Qk)
U
A
E
B
C
R
Fig. 2 – The exchange process in the (p, L) plan.
3.2 Public expenditure
Distance costs might play an important role also in determining the effective level of public
expenditure per individual g = (Public Expenditure)/(Population). Specifically, assume that the
public sector fixes a homogeneous level of public expenditure per individual across land, and
assume for simplicity that the level of taxation is equal to zero. When distance costs are positive
and increasing as far as one moves from U to R, the level of public expenditure per individual
decreases from gmax to gmin, raising a specific kind of spatial inequality (Fig. 3).
Moreover, in certain cases distance costs might determine perverse effects on land. Specifically,
if policy makers fix a level of g that on a certain share of land is not sufficient to cover the
distance costs, there might be places where people receive a negative contribution to their
wellbeing from the public sector. We observe how, while inequality might be a necessary cause
(but not sufficient), a detriment to wellbeing might be a sufficient cause, but not a legitimation,
for illegal and criminal activities.
6
gmax
0
gmin
U
R
Fig. 3 – Net contribution across land of a constant per capita public expenditure.
3.3 Household production and nonprofit activities
Distance costs might play an important role in determining households’ preferences for free time,
the amount of household production and the level of involvement in nonprofit activities.
Specifically, as we assumed that by construction land is characterized by a decreasing level of
income as far as one moves from U toward R, there is a decreasing opportunity cost for leisure (in
this case, we include in leisure also household production and non profit activities). Moreover, as
far as one goes from U to R, we assume that that the utility of leisure increases, as during the free
time an individual must satisfy more urgent needs that are neither provided from the market,
neither from the public sector. Therefore, there is a share of land in which it is convenient to
engage in household production and non profit activities (from R to E) and a share of land in
which it is not (from U to E) (Fig. 4).
̅ , L)=w
MC(𝑸
̅ , L)
MU(𝑸
E
U
Fig. 4 – Incentives for non profit activities across land.
R
7
Clearly, the slope of the marginal cost and of the marginal utility functions might affect the
amplitude of the share of land in which nonprofit activities and household production are
convenient. It is worth noticing how nonprofit activities contribute to reduce spatial inequality, as
their economic convenience is inversely proportional to the level of income that people can earn
and to the level of welfare provided by the institutions (public and private).
4.
Conclusions
The model shows that in peripheral/marginalized areas there might be conditions (lack of public
and private services, low income…) for citizens to act criminally, especially when public policies
pretend to provide a positive contribution to welfare, while indeed they are subtracting local
resources. Because of the lower (or negative) provision of public goods, criminal organizations
could proliferate taking over (or distorting) the social capital, whose endowment and
accumulation is essential for economic development of the territories. It emerges that criminal
behaviour is a complex phenomenon which depends on different socioeconomic factors where
social capital is a central element with which to build preventive policies to combat organized
crime complementary with repressive measures implemented by the police and the judiciary. In
fact, the creation and intensification of an adequate process of accumulation of “pure” social
capital may contribute to the liberation from the yoke imposed by organized crime in the areas
where its presence is massive, distorting the rules of functioning of markets and jeopardizing
social development and economic.
To conclude, we believe that it is possible to break the twofold vicious cycle which goes from:
- the social and cultural impoverishment to the increase of the social insecurity and the incentives
to illegal activity;
- the spread of distrust among citizens or among them and the legal institutions to the
strengthening of a development model sustained by crime which inexorably undermines the
chances of economic and social growth of the territories.
Assuming what has been described in our model, i.e. there is a lower provision of welfare services as
far as we move from urban to rural areas, according to the construct of the efficiency-equity trade-off
and the market conditions, we believe that cooperatives and more in general non profit institutions
might compensate the underprovision of public and private goods and services by generating a
complementary supply. Moreover, the private and the public sector might decide to facilitate this
process by contracting out one or more parts of the production process.
Referring to Hansmann (1996) if, e.g. the public government has lower transaction costs in
bringing stakeholders into ownership rather than having a contractual relationship with them, this
is the case to choose the cooperative solution. Second, Hansmann brings in market failures to
explain why people sometimes choose cooperatives even though the costs of ownership are
relatively high; it is because the alternatives – such as entering into market relations with a
private monopoly– are even more costly. Finally, there is a consistent literature on the role of
cooperative and of the non profit sector in reinforcing the social capital boundaries. The existing
views of social capital, share a fundamentally rational and instrumental vision of the role that
social capital plays in economic development. Social capital is explicitly regarded as a productive
resource just as much as financial, environmental, or human capital, which requires investments
and generates returns in the form of better access to information, better communication and
coordination, reduction of opportunistic behavior (Valentinov 2004).
Cooperatives and the non profit sector in this perspective are seen as potential tools for economic
8
development in that they make feasible that which could not be achieved by people operating
individually: they can build stock of capital, give members a “voice” to advocate change in
government policies, promote local ownership, create jobs, promote local control of capital, and
fight inequality (racism and segregation). When individuals interact regularly and trust, social
transactions are more efficient and common problems are more easily resolved. This, in turn,
facilitates community development.
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Mathematical appendix
Define 𝑄 ∈ [0, ∞] as the level of output Y and 𝐿 ∈ [0,1] as “Land”, where L = 0 represents
“Urban” and L = 1 represents “Rural”. The output Y can be distributed and consumed across land,
therefore Y is a heterogeneous good, as consuming the level QY in two different places does not
mean consuming the same level of Y. Specifically, we are assuming that by construction
Y = f(Q, L).
Now, assume that on the production side operates a profit maximizing firm (F) characterized by a
total cost function equal to
𝑇𝐶(𝑌) = 𝑇𝐶(𝑄, 𝐿) = 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡𝑠 (𝑄, 𝐿) + 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡𝑠 (𝑄, 𝐿)
For any fixed Q, assume that production costs are equal to zero and rename distribution costs as
“distance costs” (henceforth, DC), obtaining
𝑇𝐶(𝑄̅ , 𝐿) = 𝐷𝐶(𝑄̅ , 𝐿)
Moreover, assume that DC are constant in Q and increasing in L. Specifically,
10
𝜕𝐷𝐶(𝑄, 𝐿)
𝜕𝐷𝐶(𝑄, 𝐿)
𝜕𝐷𝐶(𝑄, 𝐿)
𝜕𝐷𝐶
𝜕𝐷𝐶
𝑑𝑄
𝑑𝐿 =
𝑑𝐿
=0
> 0 → 𝑑𝐷𝐶(𝑄, 𝐿) =
⏟+
𝜕𝑄
𝜕𝐿
𝜕𝐿
𝜕𝑄
𝜕𝐿
0
It follows that, for a given Q,
𝑑𝐷𝐶(𝑌) 𝑑𝐷𝐶(𝑄̅ , 𝐿)
=
= 𝑀𝐷𝐶𝐿 (𝑄̅ , 𝐿) = 𝑀𝐷𝐶𝐿 (𝐿)
𝑑𝑌
𝑑𝐿
On the other hand, F’s total revenues (TR) are defined as
𝑇𝑅(𝑌) = 𝑝(𝑌)𝑌 → 𝑇𝑅(𝑄, 𝐿) = 𝑝(𝑄, 𝐿)𝑓(𝑄, 𝐿)
For a given Q, the previous equation can be rewritten as
𝑇𝑅[𝑄̅ , 𝐿] = 𝑝(𝑄̅ , 𝐿)𝑓(𝑄̅ , 𝐿) = 𝑝(𝐿)𝑓(𝐿) = 𝑇𝑅(𝐿)
Therefore, for a given Q it must be that
𝑑𝑇𝑅 =
𝜕𝑇𝑅(𝑄̅ , 𝐿)
𝜕𝑇𝑅(𝑄̅ , 𝐿)
𝜕𝑇𝑅(𝑄̅ , 𝐿)
𝑑𝑄 +
𝑑𝐿 =
𝑑𝐿
𝜕𝐿
𝜕𝐿
⏟ 𝜕𝑄
It follows that, for a given Q,
0
𝑑𝑇𝑅(𝑌) 𝑑𝑇𝑅(𝑄̅ , 𝐿)
=
= 𝑀𝑅𝐿 (𝑄̅ , 𝐿) = 𝑀𝑅𝐿 (𝐿)
𝑑𝑌
𝑑𝐿
Then, for any given Q, F solves the following algorithm:
max 𝜋𝐹 (𝑌) = max 𝜋𝐹 (𝐿) = 𝑇𝑅(𝑄̅ , 𝐿) − 𝑇𝐶(𝑄̅ , 𝐿)
𝑌
𝐿
F.O.C. is
𝜕𝜋𝐹
= 𝑀𝑅(𝑌) − 𝑀𝐶(𝑌) = 𝑀𝑅𝐿 (𝐿) − 𝑀𝐷𝐶𝐿 (𝐿) = 0
𝜕𝑌
Finally, under perfect competition, F.O.C. becomes
{
𝑀𝑅(𝑌) = 𝑝(𝑌) = 𝑝(𝑄̅ , 𝐿)
→ 𝑀𝑅𝐿 (𝐿) = 𝑝(𝑄̅ , 𝐿) = 𝑀𝐷𝐶𝐿 (𝐿)
𝑀𝐶(𝑌) = 𝑀𝐶𝐷𝐿 (𝐿)
Cost analysis
Given the previous assumptions, in order to obtain Y, assume that a relation of imperfect
substitution relates Q to L. Moreover, F is characterized by a quadratic cost function in Y, so that
11
𝑌 = 𝑓(𝑄, 𝐿) = 𝑄𝐿
→
𝑇𝐶(𝑌) = 𝑇𝐶(𝑄, 𝐿) = 𝐹𝐶 + 𝑉𝐶(𝑄, 𝐿) = 𝐹𝐶 + 𝑐𝐿2 𝑄 2 .
Now, assume that Q is fixed, and that FC = 0. Therefore,
𝑇𝐶(𝑄̅ , 𝐿) = (𝑐𝑄̅ 2 )𝐿2 = 𝑐′𝐿2
It follows that
𝑀𝐶(𝑌) = 𝑀𝐷𝐶𝐿 (𝐿) = (2𝑐𝑄̅ )𝐿 = 𝐶𝐿
Finally, given (𝐿𝐴 , 𝐿𝐵 , 𝐿𝐶 ) ∈ 𝐿, where 𝐿𝐴 < 𝐿𝐵 < 𝐿𝐶 , it must be that
𝑀𝐶(𝑄̅ , 𝐿𝐴 ) = 𝐶𝐿𝐴 < 𝑀𝐶(𝑄̅ , 𝐿𝐵 ) = 𝐶𝐿𝐵 < 𝑀𝐶(𝑄̅ , 𝐿𝐶 ) = 𝐶𝐿𝐶
Market analysis
Assume that all consumers are identical and are characterized by a linear demand function for
good Y equal to
𝑝(𝑌) = 𝑎 − 𝑏𝑌, 𝑎, 𝑏 ≥ 0
Therefore, the aggregate demand function of n individuals is equal to
𝑏
𝑝(𝑌) = 𝑎 − 𝑌, 𝑎, 𝑏 ≥ 0,
𝑛
𝑛>0
Recalling that Y is a heterogeneous good in L, consider n as a measure of population density, and
assume that n = max {(d – L),0}, with 0 < d < 1. If every individual has the same income, the
aggregate demand of Y should vary across land according to the following function
𝑝(𝑌) = 𝑝(𝑄, 𝐿) = max {𝑎 −
𝑏𝐿
𝑄, 0} , 𝑎, 𝑏 ≥ 0
(𝑑 − 𝐿)
By construction, we assumed Land as characterized by a decreasing level of income, and that the
willingness to pay is bounded by the capability to pay of an individual. Therefore, we assume that
a = w(d’ – L), with d’ > 0. The aggregate demand function w.r.t. L becomes
𝑝(𝑌) = 𝑝(𝑄, 𝐿) = max {(𝑤𝑑′ − 𝑤𝐿) −
𝑏𝐿
𝑄, 0} , 𝑎, 𝑏 ≥ 0,
(𝑑 − 𝐿)
Its derivative w.r.t. L is
𝜕𝑝(𝑄̅ , 𝐿)
𝑏𝑄
=
−𝑤
𝜕𝐿
(𝑑 − 𝐿)2
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As
𝜕𝑝(𝑄̅ , 𝐿)
→ −𝑤
𝐿→∞
𝜕𝐿
lim
The aggregate demand function in the plan (p, L) is decreasing and convex w.r.t. the origin (U).
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