efficiency could be achieved by either assignment of management responsibilities and knowledge.
For dynamic resource problems, government involvement in management may be necessary at
least as a repository of knowledge about the resource see Hurwicz and Weinberger, 1990, for
discussion about the transversality condition as a reason for government involvement in dynamic
problems. Therefore, full privatization may not be possible. Below we develop this argument for
the soil erosionpollution framework.
3. Social efficiency in the dynamic system
The basis for policy design is the First Funda- mental Theorem of welfare economics extended to
a dynamic setting with externalities. The First Theorem of welfare economics in the static case
states the correspondence of a decentralized equi- librium with private decision-making to social
efficiency, given certain conditions about prefer- ences and production sets. For an externality, a
policy — such as a tax or subsidy — is added to the equilibrium system to bring about the correspon-
dence. See Negishi 1960 and Laffont 1989 for the static correspondence. See Kemp and Long
1982 for discussion of the dynamic extension without externalities. Eichhorn and Spremann
1982 is also relevant for discussion of pricing when there are intertemporal externalities.
Two types of mathematical models for our two-sector dynamic externality setting are given in
Appendix A: Section A.1 social efficiency; Sec- tion A.2 decentralized equilibrium under alterna-
tive policies Sections A.2.1, A.2.2 and A.2.3 to achieve efficiency. For each of Sections A.2.1,
A.2.2 and A.2.3, first-order conditions for social efficiency are satisfied by a decentralized equi-
librium
corresponding to
private decisions
modified by a given policy. Each alternative type of decentralization has different implications
about the role of government. Below, we describe these policy alternatives.
3.1.
Basic relationships Our extension of Larson and Bromley’s model
embeds the rural agricultural household that performs agricultural production in a two-sector
economy that also includes urban manufactur- ing households. This simplified economic system
is the basis for defining social efficiency and iden- tifying policies that could lead to social efficiency.
Fig. 1 describes the basic flows of labor and capital inputs, and food, manufacturing and pol-
lution outputs among rural and urban sectors in the economy. Both rural and urban households
consume manufactured goods M and food F, and supply labor L for production. Utility or
satisfaction levels for urban households are deter- mined by pollution, consumption of food and
manufactured items, and labor supplied. Simi- larly, rural household preferences are in terms of
consumption of food and manufactured items and labor supplied; pollution is not assumed to affect
Fig. 1. Relations between urban and rural sectors.
rural households directly. Declining marginal util- ity
of consumption
and declining
marginal product of inputs are assumed. A constraint on
resource allocation is providing a minimum food consumption for each type of household to avoid
starvation in any time period.
On the production side, agricultural households use labor and purchase capital to produce food
for both themselves and urban households; agri- cultural labor can also be used for soil resource
improvementpollution reduction. Urban house- holds supply labor to produce capital that is used
for manufactured goods and for agricultural production.
Intensive capital used for food production by rural households is assumed to cause soil erosion.
Soil erosion is a stock externality because it re- duces food production over time. At the same
time soil erosion causes pollution, with soil parti- cles carrying pollutants such as fertilizer and pes-
ticides to water bodies. Pollutants accumulate in streams in sedimentary layers over time; thus as-
sociated pollution is also a stock externality. These pollutants can affect the welfare of urban
households, e.g. urban drinking water quality. Rural residents may get their drinking water from
wells that are less affected by pollutants. Rural labor can simultaneously reduce soil erosion and
pollution through improved practices. For exam- ple, labor for terracing and buffer strips can im-
prove soil loss and reduce pollution.
Production is described not only by the familiar production functions of classical economics but
also by dynamic relationships. In this simplified economy, there are three dynamic production re-
lationships for capital, pollution, and soil quality. The dynamic relationship describing capital is
similar to that used in macroeconomic models: capital is a stock which grows in proportion to
labor input and declines with depreciation. Both pollution and soil quality are described by stock
relationships similar to the capital relation. In the dynamic relationship describing soil, its quality is
reduced by use of intensive capital, and it can be improved either through the reduction of inten-
sive capital or through increased labor used for pollution reduction.
3.2.
Social efficiency Following the paradigm of welfare economics,
the conditions for socially efficient Pareto opti- mal resource allocation for this dynamic context
are described. Social efficiency see Appendix A, Section A.1, as in the static case, concerns the
allocation of capital and labor inputs among food and manufacturing and the allocation of final
goods between the two sectors, considering the impacts of pollution and soil erosion.
The social objective function used to describe the social efficiency problem Section A.1 is the
weighted sum of discounted utility for all persons in the society, here represented by the two sectors
C and A. An allocation of a consumption good x in limited supply satisfies — from the slope of
the social objective function — the following rela- tion between the marginal utilities for each sector:
u
C
x
C
u
A
x
A
= − a
2
a
1
. That is, given weights a
i
on sectors, each con- sumption good produced x is allocated so that
marginal utility of consumption for each sector is inversely proportional to its social weight. Then,
because of declining marginal utility, relatively more is given to a sector with a larger weight.
Extending the static case Negishi, 1960, we do not assume there is a social planner who defines
these weights, but rather that any efficient solu- tion can be associated with some weights. By
varying the weights over the set of possible weightings, all possible socially efficient solutions
can be generated. We assume that society would rather be at an efficient solution than otherwise;
that is, our arguments do not require determining which of the efficient outcomes should be selected.
The first-order conditions necessary for social effi- ciency apply for any set of weights.
An important issue for problems with time dimensions is the discount factor r commonly
used to put events over time into common value units. Its selection is highly controversial see
Kula, 1992 since the use of such a discount factor implies that ‘a util tomorrow is worth less than a
util today’. With a positive discount factor, future generations’ utilities receive lower weights than
today’s generation. A discount rate of zero would imply equal weights over time. The same
form for efficiency conditions holds for any dis- count rate: by varying the rate, alternative effi-
cient solutions can be generated. Below, we assume that the discount rate used by each sec-
tor in equilibrium formulations is the same as in the social efficiency problem.
In determining policy implications below, we focus particularly on the dynamic first-order
conditions for soil quality and pollution and on the marginal value conditions for labor and cap-
ital. See Kamien and Schwartz, 1982, for the general method of dynamic optimization. The
first-order conditions given in the appendix are necessary conditions formulated in terms of
shadow prices for pollution, soil, and capital. The shadow price for capital reflects the long-
term social opportunity cost of current capital use decisions. Similarly, the opportunity cost of
soil
quality reflects
the marginal
long-term benefits of maintaining soil quality. The shadow
price for pollution indicates the long-term mar- ginal social damage from pollution; it is nega-
tive. First-order conditions given in Appendix A show that the opportunity cost for pollution at
the optimal level should reflect the long-term ef- fects of pollution on the welfare of urban house-
holds. Similarly, the opportunity cost for soil quality relates to its long-term productivity.
The optimal time path that solves the set of first-order conditions and constraints depends on
the initial conditions in an economy: societies with different initial conditions here including
pollution and soil quality will have different op- timal time paths. Regardless, from the supply –
demand constraints for labor and capital, the marginal value for each input should be the
same in each of its uses.
3.3.
Decentralized equilibrium and pri6ate decision-making
Decentralized equilibrium models see Ap- pendix A, Sections A.2.1, A.2.2 and A.2.3 rep-
resent different organizations of consumption and production through private decision-making
by actors in the economic system. Generally, manufacturing decisions are made in the urban
area, with urban households determining the labor supplied for manufacturing. Ownership
and production of capital are assigned to the urban sector. The rural households make deci-
sions about food production and allocate labor for food and pollution reduction production.
Consumption decisions are made separately by urban and rural sectors. Manufacturing is as-
sumed to operate in a ‘spot’ market, so that production decisions and demand for labor are
made period by period given relative prices.
By dynamic equilibrium we mean a simulta- neous set of decisions across time for each actor
in the dynamic system, such that no actor would change her decisions given the equilibrium deci-
sions by the other actors. Necessary conditions that determine equilibrium are the first-order
conditions for each sector’s decentralized prob- lem, the constraints for each sector’s decentral-
ized
decision problem,
and supply – demand
equilibrium conditions. The dynamic equilibrium includes prices for each period; equilibrium
prices are a time vector for each commodity. Given the equilibrium vector of prices over time,
households make decisions about consumption and labor supply for the whole time period.
The existing situation in many countries with soil degradation may correspond to a social so-
lution in which the social costs due to pollution are ignored. In a decentralized private decision-
making system without a social policy, for rural people to meet the ‘no starvation’ requirement
with low initial income may result in zero labor allocated to pollution reduction, producing soil
degradation and decreased productivity of agri- culture in the long run. Given social costs due
to pollution and soil erosion, such an equi- librium outcome would not achieve social effi-
ciency. Even though increasing rural income equivalent to increasing the weight on rural
households relative to urban households could reduce soil degradation, social efficiency condi-
tions could still not be satisfied.
Fig. 2. Alternative decentralized policy models to alleviate soil erosion.
3.4.
Alternati6e policy designs Alternative policy designs below differ in terms
of who is assigned the responsibility for managing dynamic production relationships for soil and pol-
lution. Each policy brings about the correspon- dence of private decisions with social efficiency.
Policies could involve different price and income effects in the economy as a whole. See Fig. 2 for
a comparison of policy types.
3.4.1.
Pigou6ian solution Section A.
2
.
1
This policy employs both a tax on polluting capital and a subsidy for labor to reduce pollu-
tion. Extending the static paradigm, the Pigouvian case requires that an external resource manager a
role that can be carried out by government to know all demand and supply relationships in the
economy including the dynamic relationships, to solve the social optimality problem, and to deter-
mine and apply the appropriate tax and subsidy package accordingly. Urban and rural household
private decisions are affected only through price levels. The rural household would not have to
know the dynamic environmental relationships in order for the efficiency conditions to be satisfied.
3.4.2.
Bargaining solution Section A.
2
.
2
In this case, an entitlement level P
e
for pollution is first defined such that pollution should never be
greater than this level. The urban household may then pay the rural household to obtain pollution
below the entitlement level. In this model, it is assumed that rural producers know the dynamic
production relationships for soil and pollution and manage them in response to payment levels
from urban households. Rural producers then choose intensive capital and labor for pollution
reduction to satisfy both the urban demand for pollution reduction and for food.
While the first-order efficiency conditions are satisfied at any equilibrium, the nature of the
equilibrium would generally depend on the enti- tlement level. The role of government in this case
would be to set the entitlement level, e.g. based on equity considerations.
3.4.3.
Go6ernment as co-producer of en6ironmental goods
Section A.
2
.
3
This model differs from the Pigouvian model in that the government is required to know only the
dynamic environmental relationships. It also dif- fers from the bargaining model in that the govern-
ment resource manager becomes a part of the equilibrium system as a co-producer of environ-
mental goods. The resource manager purchases the labor input for pollution reduction from agri-
cultural labor to meet the urban demand for pollution reduction; urban payment p
p
per unit pollution reduced is paid to the resource manager.
At the same time, the government resource man- ager receives a tax t on intensive capital from
rural producers, pays a subsidy s per unit labor for pollution reduction, and also pays rural pro-
ducers p
s
per unit soil improvement. The tax, subsidy, and the prices for soil improvement and
pollution reduction are then determined through operation of the equilibrium system. The combi-
nation of these four price instruments is required to satisfy social efficiency conditions. Prior to any
market exchange, the pollution entitlement is set at P
e
. The initial soil level is S
o
. Equilibrium prices and outcomes would generally be affected by the
entitlement and initial soil levels. Knowledge is required about the dynamic envi-
ronmental relations for soil and pollution for the resource manager’s decision problem. Rural pro-
ducers operate in a spot market without require- ments for knowledge of dynamic environmental
relations.
4. Conclusions
