2nd World Congress of Environmental and Resource Economists
Monterey, California
24-27 June 2002
Tradable compliance credits for extended producer
responsibility: market power and the allocation of
initial property rights
Roger Salmons 1
Department of Economics, University College London
Abstract
This paper considers how the allocation of initial property rights can affect the performance of
a credit-trading scheme used to implement a recycling target under extended producer
responsibility (EPR). The analysis is based on a simple, stylised model of the credit-trading
scheme that has evolved in relation to extended producer responsibility for waste packaging in
the United Kingdom. It is demonstrated that, when firms act strategically in one of the
participating sectors, the choice of sector to receive the rights to the credits that are generated
when waste material is reprocessed can affect the scheme’s cost efficiency and its
environmental effectiveness. In doing so, the paper offers some new perspectives on the issue
of market power in permit-trading systems, and provides guidance to waste regulators who
may be considering the use of trading schemes in this policy area.
Keywords
Extended producer responsibility; Credit trading; Market power
1
Postal address: Department of Economics, University College London, Gower Street, London WC1E 6BT,
United Kingdom. Email address: [email protected]
1
1.
Introduction
Waste management systems are typically subject to a number of market failures. In
particular, municipal waste collection is often funded out of general taxation rather than by
charging households directly for the service, while the gate-fees charged by landfill and
incinerator operators do not properly incorporate the external costs of disposal. Together,
these failures lead to inefficient levels for the amount of waste that is generated (too high),
and for the amount that of material that is recovered from the waste stream (too low). In
theory, these failures can be addressed directly – and efficiency restored – by a combination
of externality charges applied to waste disposal, and unit-based pricing for household waste
collection (Fullerton & Wu, 1998). However, there are a number of problems with this
approach in practice. First, it may lead to an increase in the illegal dumping of waste by
households (Fullerton & Kinnaman, 1996). Second, efficiency would require that the
externality charges – and hence the collection charge – be differentiated between the various
components of the waste stream, to reflect their respective toxicities (Dinan, 1993). Even if
such as scheme were technically feasible, it would be prohibitively costly to administer.2
The potential limitations of the direct approach have led to various alternative policy
responses being proposed, with the focus being shifted upstream to producers. These can be
grouped under the collective heading of extended producer responsibility (EPR). Under EPR,
producers are required to assume responsibility for the life-cycle environmental impacts of
their products, with particular emphasis on disposal. Examples include the German
Packaging Ordinance of 1991, and the 1993 Producer Responsibility initiative in the United
Kingdom for six priority waste streams.3 More recently, the EU Directive on End of Life
Vehicles (2000/53/EC) has set recycling targets, and stipulated that producers must pay “all or
a significant part” of the costs of take-back and treatment from January 2007.
Within the overall umbrella of EPR, a distinction can be made between price-based responses
that combine some form of advanced disposal fee (including appropriate external costs) with a
subsidy to recycling (Dinan, 1993; Palmer & Walls 1999), and quantity-based responses.
2
Notwithstanding these problems, the use of unit-based pricing is growing. It is in widespread use in seven
European countries (ACR, 1999), and has been adopted by over 4000 communities in the USA, accounting for
around 12% of the population (Miranda et al, 1998).
3
The six priority waste streams were packaging, newspapers, tyres, batteries, vehicles and electronic equipment.
2
Quantity-based approaches can take several forms, but there is commonly a requirement to
recover (i.e. divert from the waste stream going to landfill) a given proportion of “end of life”
products, or component materials.4
Whatever form they take, a common characteristic of these initiatives is a recognition that
while individual producers should have an obligation to meet the cost of recovery, it is not
economically efficient to require them to undertake the necessary recovery themselves. One
common approach is to allow producers to discharge their individual obligations by joining a
collective compliance scheme (or producer responsibility organisation). The scheme then
assumes responsibility for ensuring that sufficient waste is collected and recovered to satisfy
the aggregate obligation; with the cost of these activities funded by the membership fees. A
good example of this approach is the Duales System Deutschland (DSD) for packaging in
Germany. However, this is not the only approach that can be used. An alternative, which
maintains the concept of individual responsibility, is to introduce a compliance credit trading
scheme. Under this approach, “recovery credits” are created (on a one-for-one basis) for
every product, or tonne of material, that is delivered to reprocessors, and producers are
required to purchase sufficient credits to satisfy their individual obligations. An example of
this trading approach has evolved over recent years in relation to waste packaging in the
United Kingdom (Salmons, 2002a).5
This type of credit trading scheme has a number of important differences compared to
traditional permit trading applications such as air pollution. In most traditional applications,
the target market (i.e. the market in which the regulator wishes to intervene) is incomplete –
i.e. the supply side of the market is missing. Furthermore, the underlying objective is usually
expressed in absolute terms, which is translated into a fixed, ex ante “cap” on the total
number of permits. In contrast, in a compliance credit trading scheme for EPR, the target
market is complete (i.e. both the demand and supply sides exist), and the target is expressed in
relative terms (i.e. as a recovery percentage). Consequently, the total number of credits is
only determined ex post, as part of the market equilibrium.
4
The recovery target may be combined with a requirement to take back end-of-life products, or component
materials, from consumers.
3
The differences between the two types of application are illustrated in Figure 1, which
compares the trading system structure of a typical traditional application of trading – SO2
emissions by electricity generators, with that of a typical EPR application – waste packaging.
In the first case, the obligation to purchase permits is imposed on one of the sectors
participating in the target market – i.e. the “market” for environmental services provided by
the atmosphere.6 Of course, in this application the supply sector is missing, and so the only
option is to place the obligation on the electricity generators. However, there is a choice
regarding the initial property rights to the permits. They can be granted either to the
generators under some form of grandfathering (as is shown in Figure 1), or to the regulator for
them to auction off. In contrast, in the EPR application, the obligation is imposed on a sector
– the producers – that does not participate in the target market. Furthermore, because this
market is complete, there is a choice over which sector is granted the initial property rights
when the waste packaging is received by the reprocessors – i.e. the credits can be given either
to the demand side (the reprocessors), or to the supply side (the waste collectors).
Salmons (2002b) has shown that compliance credit trading provides a cost-efficient
implementation mechanism for a recovery target under EPR, provided that households (or
producers) are charged for the private cost of disposal.7 However, this conclusion relies on
the assumption that all markets are perfectly competitive. The objective of this paper is to
investigate the implications of relaxing this assumption. In particular, it considers whether
the presence of market power in the target market has any implications for the choice of
sector to be granted the initial property rights to the credits that are created.
There is a growing literature on the issue of market power in permit trading systems. Hahn
(1984) has considered the implications of monopoly power in the permit market; while
Mauleg (1990) and Sartzetakis (1997a) have investigated the interactions between a
competitive permit market and an oligopolistic product market. Finally, Sartzetakis (1997b)
5
The UK scheme is actually a hybrid of the two approaches. Obligated producers can either join a collective
compliance scheme (of which there are seventeen); undertake the necessary recovery themselves; or purchase
Packaging Waste Recovery Notes (PRNs) which are created when waste packaging is received by reprocessors.
6
The output of emissions to the atmosphere can be interpreted as an input of environmental services provided by
the atmosphere.
7
The cost-efficient implementation of a given recovery target will not in general correspond to the social
optimum, even if the target is set equal to the socially optimal rate. It will only be so if optimality requires 100%
of the waste product or material to be recovered.
4
and von der Fehr (1993) have analysed the situation in which there is strategic interaction in
both the permit and the product markets. These all conclude that cost efficiency is likely to be
undermined if firms can act strategically in the permit or product markets. However, these
analyses have all been conducted in the context of traditional permit trading applications, in
which the issue of market power in the target market does not arise, and in which the
objective is absolute. Consequently, this analysis provides some new perspectives on the
topic.
The issue is analysed using a simple, stylised model of EPR for a single packaging material,
in which there is only one reprocessor, that can exercise monopoly (or monopsony) power in
any market in which it operates. The model captures some of the salient features of the
trading scheme that has evolved in the United Kingdom for waste packaging.8 A detailed
description of the model is provided in the next section. While the focus of the paper is on the
implications of market power, it is instructive to compare this situation with the perfectly
competitive case, in which all participants are price-takers. Consequently, the perfectly
competitive case is considered briefly in section 3, before turning to the monopolistic case in
section 4. In each case, the market equilibrium conditions are derived under the alternative
property rights regimes, and these are then used to draw some conclusions regarding the
relative values of the system variables. In section 5, a specific numerical example is used to
illustrate the results, and to investigate some issues that cannot be determined analytically.
Finally, the implications of the analysis are discussed in section 6.
2.
Stylised model of credit trading system for EPR
In this simple, stylised model of a credit trading system used to implement EPR for a single
packaging material, there are three sectors: a waste reprocessing sector; a waste collection
sector; and a production sector – i.e. the producers of packaged goods. It is assumed that
there is complete separation between the three sectors, i.e. the production sector does not
collect any waste packaging, and the waste reprocessing sector has no obligations under the
EPR regulations.
8
However, it should be noted that the UK trading scheme is much more complex than the model, with multiple
materials, multiple targets (i.e. an aggregate target and minimum thresholds for individual materials), and
oligopolistic market structures rather than a single monopoly.
5
Figure 2 provides a schematic representation of the model, which is developed at the sectorlevel.9 The tonnage of waste packaging diverted from the waste stream (w) is a function of
the amount of capital devoted to waste diversion activities (kd). The resultant capital
requirement function kd = g(w) is assumed to be convex, with g//(w) + w g///(w)  0.10
Diverted waste packaging is transformed into reprocessed material according to the
production function y = f(w,kr), where kr is the level of reprocessing capital. It is assumed that
f(w,kr) is strictly concave with fkw = fwk > 0. The production sector’s inverse demand function
for packaging (z) is denoted by Pz(z). It is assumed that Pz(z) is continuously differentiable,
with Pz/(z) < 0, 2 Pz/(z) + z Pz//(z) < 0, and lim ( Pz(z) + z Pz/(z) ) > pz.11
z 0
The target recovery rate for waste packaging is denoted by  [0,1], and this is used to
define a “recovery obligation” (in tonnes) for the production sector – based on the amount of
packaging produced.12 However, the sector is not required to undertake the necessary
diversion of waste packaging itself (i.e. incur the cost of diversion directly). Rather it is
required to purchase sufficient recovery credits (n) to satisfy its obligation, i.e. n  z.
Recovery credits may only be created – on a tonne-for-tonne basis – when the diverted waste
packaging is received by the reprocessor, i.e. n  w. Two alternative scenarios are considered
regarding the assignment of the initial property rights to these credits. In the first scenario (A)
the rights are assigned to the reprocessing sector; in the second scenario (B) they are granted
to the waste collection sector.
The market prices of delivered waste (qw) and recovery credits (qn) are determined
endogenously within the model, while the prices of capital (pk), packaging (pz), reprocessed
material (py) and landfill (pl) are all assumed to be exogenous (i.e. fixed).
9
Because the focus is on the interaction between the three sectors, the firm-level detail has been abstracted. This
has the advantage of simplifying the notation, and has no impact on the generality of the results.
10
This condition is satisfied for common single-variable convex functions – for example kd = a wb (a>0, b>1),
and kd = eaw – 1 (a>0).
11
The last two assumptions ensure that the resultant marginal revenue function for packaging MRz(z) is
downward sloping, and that MRz(0) > pz.
12
It should be noted that the obligations are imposed on individual firms, based on the amount of packaging that
they produce.
6
The model is static, with the values of all variables being determined simultaneously. Thus,
the amount of waste packaging that is sent to landfill (l) by the waste collectors is equal to the
difference between the amount that is produced and the amount that is diverted, i.e. l = z – w.
The model is also open, in that it is assumed that the reprocessed material is not used for
packaging.13
3.
Market equilibrium under perfect competition
In this section the outcome is determined under the assumption of perfect competition in all
markets – i.e. none of the sectors can exercise market power in any of the markets in which
they operate.
Scenario A:
Property rights assigned to the reprocessing sector
Given exogenous prices py, pk, pl, and pz, a simultaneous market equilibrium comprises prices
qw* and qn*, and non-negative quantities y*, w*, n*, z*, kr* and kc*, such that these values solve
the three sectors’ optimisation problems, and satisfy the respective market clearing conditions
for diverted waste packaging and recovery credits. When the initial property rights are
allocated to the reprocessing sector, the optimisation problems for the three sectors are:
Reprocessors
Collectors
Max
py f(wr,kr) + qn nr  qw wr  pk kr
s.t.
nr - wr  0
Max
qw wc  pk kd + pl wc
s.t.
g(wc)  kd  0
13
The validity of these two assumptions depends on the characteristics of the material / product to which EPR is
being applied. They are reasonable in the case of plastic packaging for fast-moving consumer goods (FMCGs),
which have a relatively short life-span, and where the recovered material is used in a wide range of products.
However, in other applications it may be more appropriate to use a dynamic model (e.g. end-of-life vehicles),
and / or a closed model (e.g. aluminium beverage cans).
7
z
Producers
14
Max
 P( ) d
 pz z  qn np
0
s.t.
z  np  0
and the market clearing conditions for the two markets are:
wc  wr
=
0
np  nr
=
0
The Kuhn-Tucker sufficiency conditions are satisfied for all three optimization problems.
Consequently, the Kuhn-Tucker first order conditions for the three problems, together with
the two market clearing conditions, are necessary and sufficient for a unique simultaneous
market equilibrium. Assuming an internal solution, it is straightforward to derive a set of
market equilibrium conditions expressed solely in terms of the real variables and market
prices (i.e. eliminate all shadow prices). These are given in the first column of Table 1.
There are several points to note from the market equilibrium conditions. First, the
reprocessors equate the value of the marginal product of diverted waste packaging to the net
price of diverted waste (i.e. after subtracting the value of the credit that is generated). Put
another way, the waste packaging has value to the reprocessors both in terms of the material
that they can recover, and in terms of the credit that they can sell. Second, all of the credits
that are generated when the waste is received are sold to the producers (i.e. w* = n*).15
Finally, the requirement for producers to purchase credits in order to meets their obligations
has the effect of raising the price of packaging material, by an amount equal to the credit price
(qn) multiplied by the target recovery rate (). Thus, from the producers’ perspective, the
trading scheme is equivalent to a tax on packaging. Indeed the same overall outcome could
If the price in the product market is held constant, then the “consumer surplus” in the market for any essential
factor input is equal to the producer surplus in the product market (i.e. operating profit before fixed costs). If the
product price is allowed to adjust to changes in the factor price, then the consumer surplus in the factor market is
equal to the sum of the producer and consumer surpluses in the product market (provided that the latter is
defined).
15
The only situation in which this will not be the case is if the target recovery rate is lower than the actual
performance prior to the introduction of the regulation. Of course, in this trivial case, the price of permits will be
zero, and – assuming that the producers do not purchase more credits than they require – some credits will
remain unsold, i.e. w* > n*.
14
8
be achieved if the government were to levy a tax on packaging material (z = qn), and use the
revenues to subsidise the price of diverted waste packaging (w = -qn).16
The market equilibrium conditions yield the following expression for the inverse demand
curve for credits; giving the amount that the producer would be willing to pay for a credit,
conditional on particular values of  and pz.
Qn(n; , pz)
=
1   n
P 
  z   


pz 

From this expression it is clear that the demand curve for credits will be downward sloping,
and that if the performance target is set at the rate achieved pre-regulation (i.e. 0 = w0/z0)
then qn = 0 when n = n0 = w0. It is also clear that the impact of a change in the target recovery
rate () on the demand curve for credits will depend critically on the shape of the packaging
demand curve. This is illustrated in Figure 3, which shows the derived demand curves for
credits for two different functional forms for Pz(z), together with two possible supply curves
for credits – a “high cost” curve S1, and a “low cost” curve S2. With a constant-elasticity
demand curve for packaging, an increase in the performance target causes the demand curve
for credits to shift upwards. Consequently, the equilibrium price of credits increases
irrespective of the slope of the supply curve. In contrast, when the demand curve for
packaging is linear, an increase in the target recovery rate causes the demand curve for credits
to rotate. The impact on the price of credits now depends on the slope of the supply curve.
When the supply curve is relatively flat (S2), the price of credits increases. However, when it
is relatively steep (S1), the price of credits decreases as the performance target becomes more
stringent!
Scenario B:
Property rights assigned to the waste collection sector
When the initial property rights are allocated to the waste collection sector, the optimisation
problem for the producers and the market clearing conditions are unchanged.17 However, the
optimisation problems for the reprocessors and the waste collectors are now:
16
17
The tax-subsidy scheme would be revenue neutral for the government.
Apart from a change to the superscript for the supply of n.
9
Reprocessors
Collectors
Max
py y  qw wr  pk kr
s.t.
y  f(wr,kr)  0
Max
qw wc + qn nc pk kd + pl wc
s.t.
g(wc)  kd  0
nc  wc  0
The derived market equilibrium conditions are given in the second column of Table 1.
Comparing these equilibrium conditions with those derived for scenario A, it is clear that the
equilibrium values of the physical variables and the price of credits will be the same in each
case. However, the price of diverted waste packaging will differ between the two scenarios,
with qw** = qw*  qn*.18 Thus, the difference between the price of delivered waste in the two
cases is exactly equal to the credit price. The total cost of meeting the target recovery rate is
the same in each case, as is the distribution of costs and benefits.
4.
Market equilibrium with monopoly power
In this section it is assumed that, while waste collectors and producers are price-takers in any
market in which they operate, there is a single reprocessor, which acts as a monopsonist in the
market for diverted waste, and as a monopolist in market for recovery credits.
Scenario A:
Property rights assigned to the reprocessing sector
When the initial property rights are allocated to the reprocessor, the optimisation problems for
the waste collectors and the producers are the same as for scenario A under perfect
competition, as are the market clearing conditions. However, the optimisation problem for
the reprocessor is now:
10
Reprocessor
Max
py y + Qn(nr) nr  Qw(wr) wr  pk kr
s.t.
y  f(wr,kr)  0
nr - wr  0
Unlike the case of perfect competition, the second constraint may not always be binding in the
solution, even with a positive price for recovery credits (i.e. it is possible to have n* < w*). As
can be seen in Figure 4, for values of  below a particular threshold value (  ), the
reprocessor’s marginal revenue curve for credits will intersect the marginal cost curve to the
left of the point n0 = w0 (i.e. the number of credits generated at the pre-regulation level of
delivered waste). If this is the case, then the reprocessor can divorce its two decisions, and set
a monopoly price for credits based on its zero marginal cost. In contrast, when the target
recovery rate exceeds this threshold value, the reprocessor must make a joint decision. The
value of  will depend on the functional form of Qn(n), which in turn depends on the
functional form of Pz(z), but it will always be strictly greater than the pre-regulation recovery
rate (0).
Proposition 1:
For any functional form of Pz(z) that satisfies the assumptions set out in
section 2, there exists a threshold value  > 0 such that n* < w* = w0 for
all values of   .
Proof: see appendix A1.1
The derived market equilibrium conditions are given in Table 2 for values of  below the
threshold, and above the threshold. There are two main points to note from these conditions.
First, the value of the marginal product of diverted waste is now set equal to the difference
between the marginal cost of purchasing the waste packaging and the marginal revenue
resulting from the sale of the credits.19 Second, when the target recovery rate is less than the
threshold value the actual recovery rate is greater than the target (i.e. the target is overachieved).
18
By comparing the two sets of market equilibrium conditions it is clear that the conditions for case B will be
satisfied by y** = y*, n** = n*, etc. ), with qn** = qn* and qw** = qw*  qn*
19
When the target recovery rate is less than the threshold value, the marginal revenue from the sale of credits is
equal to zero.
11
Proposition 2:
If the target recovery rate is less than the threshold rate (  ), the actual
recovery rate is equal to the threshold rate, i.e.


w*
z*

n*
z*
 
for all  
Proof: see appendix A1.2
Scenario B:
Property rights assigned to the waste collection sector
When the initial property rights are allocated to the waste collectors, the optimisation problem
for the producers and the market clearing conditions are unchanged. However, the
optimisation problems for the reprocessor and the waste collectors are now:
Reprocessor
Collectors
Max
py y  Qw(wr) w  pk kr
s.t.
y  f(wr,kr)  0
Max
qw wc + qn nc pk kd + pl wc
s.t.
g(wc)  kd  0
nc - wc  0
In order to derive the market equilibrium conditions it is necessary to make an assumption
about the rationality of the reprocessor. Table 3 gives the market equilibrium conditions for
two different assumptions. In the first case, the reprocessor is assumed to be completely
rational, and understands that its decisions regarding the amount of diverted waste that it
accepts will affect the purchase price in two ways. First, it will affect the waste collectors’
marginal cost. Second, it will affect the price that the waste collectors receive for the credit
that they can sell. Thus, even though the individual waste collectors cannot affect the price of
credits (i.e. they act as a price-takers), the reprocessor can do so by manipulating the variable
to which they are directly linked. In the second case, it is assumed that while the reprocessor
realises that the supply curve for delivered waste is shifted by an amount equal to the market
price of credits, it fails to understand that it can influence that price by its actions in the
diverted waste market. The plausibility of these two assumptions is discussed in section 6.
12
Analysis of the market equilibrium conditions in Table 2 and Table 3 yields the following two
propositions.
Proposition 3:
If the reprocessor is rational, and the target recovery rate is greater than the
threshold value (  ), then the outcome is the same irrespective of whether
the initial property rights are granted to the reprocessor or to the waste
collectors. However, if the target rate is less than the threshold value, then
the outcomes will differ. In this case, if the initial property rights are
granted to the waste collectors, then the market price of:

recovery credits is lower (i.e. qn** < qn*)

diverted waste is lower (i.e. qw** < qw*)
and the quantity of:

diverted waste packaging is lower (i.e. . w** < w*)

recovery credits is higher (i.e. n** > n*)

packaging produced is higher (i.e. z** > z*)

waste packaging sent to landfill is higher (i.e. l** > l*)

reprocessed material is lower (i.e. . y** < y*)
Proof: see appendix A1.3
Proposition 4:
If the initial property rights are granted to the waste collectors then, for all
values of the target recovery rate, the outcomes will differ depending on the
assumption that is made regarding the rationality of the reprocessor. If the
reprocessor is myopic, then the market price of:

recovery credits is lower (i.e. qn# < qn**)

diverted waste is higher (i.e. qw# > qw**)
and the quantity of:

diverted waste packaging is higher (i.e. . w# > w**)

recovery credits is higher (i.e. n# > n**)

packaging produced is higher (i.e. z# > z**)
13

waste packaging sent to landfill is higher (i.e. l# > l**)

reprocessed material is higher (i.e. y# > y**)
Proof: see appendix A1.4
Thus, it is clear that the differences between the outcomes under the two alternative property
rights regimes depends critically on the assumption that is made regarding the rationality of
the reprocessor, and on the value of the target recovery rate (). If the reprocessor is rational,
and the target rate is above the threshold value, then it makes no difference whether the initial
property rights are granted to the reprocessor or to the waste collectors. The outcome, and the
distribution of costs and benefits will be exactly the same in each case. However, if the
reprocessor is myopic, or if target rate is below the threshold level then the outcomes will
depend on which sector is granted the initial property rights.
By combining propositions 3 and 4 it is possible to compare the equilibrium values of the
various model variables for the three alternative combinations of property right assignments
and rationality assumptions. These rankings are shown in Table 4, from which it can be seen
that the environmental outcome is always worse (or at best no better) when the initial property
rights are granted to the waste collectors. Under either rationality assumption, and for any
recovery rate, the amount of packaging used by the producers, and the amount of waste
packaging sent to landfill, is greatest when the waste collectors have the property rights.
5.
Illustrative Example
In order to gain a clearer understanding of the relative outcomes under the different scenarios
analysed in section 4, the various market equilibria will be derived for an illustrative example,
using specific functional forms and parameter values (see Table 5). The functional forms and
parameter values have been chosen so as to allow an analytical solution to of the market
equilibrium conditions to be derived, and therefore the results should be taken as illustrative
rather than general.
14
Using these assumptions it is straightforward to derive algebraic expressions for the physical
variables and the endogenous prices in terms of the target level of recovery ().20 The preregulation rate of recovery is 25%, and the threshold recovery rate for scenario A is 50%. It
follows from the assumption of a linear demand curve for packaging material that the demand
curve for credits and the marginal revenue curve are both linear in n, taking the forms:
Qn(n;,1)
=
5


n
2
and
MRn(n;,1)
=
5


2n
2
Thus, as  increases, the curves rotate anti-clockwise, becoming flatter at higher target
recovery rates (see Figure 3).
Figure 5 shows the equilibrium values of the main system variables for all values of the
target recovery rate above the pre-regulation value. As can be seen, the relative values of the
variables are the same as those predicted in Table 4. However, the various panels also exhibit
some interesting features (particularly the responses to increases in the target recovery rate)
that will be discussed briefly.
a)
Price of recovery credits
When the property rights are granted to the reprocessor (or when the waste collectors have the
rights and the reprocessor is rational), the price of credits declines as the target recovery rate
increases. In contrast, the price rises when the waste collector has the rights and the
reprocessor is myopic. As has been noted previously, the negative relationship between the
price of credits and the target recovery rate reflects the assumption of a linear demand curve
for packaging, and is not a consequence of market power per se. However, the difference
between the two cases implies that in the presence of market power, the slope of the supply
curve for credits (i.e. the marginal cost to the holder of creating them) will depend on the
allocation of the initial property rights; being steeper when they are granted to the reprocessor.
20
These are available on request from the author.
15
b)
Diverted waste packaging
When the reprocessor has the property rights, the introduction of the EPR regulations has no
impact on the market for diverted waste packaging if the target recovery rate is below the
50% threshold. This reflects the fact that the reprocessor can divorce its decisions in the two
markets, and has no reason to deviate from its pre-regulation intake of diverted waste.
However, as the target rate increases above the threshold, the reprocessor loses this freedom
of action, and must increase the quantity of diverted waste packaging that it accepts if it
wishes to increase the number of credits that it can sell.
When the rights are granted to the waste collectors, the impact on the target market is very
different under the two alternative assumptions regarding the rationality of the reprocessor. If
the reprocessor is myopic, then as the target becomes more stringent, the quantity of diverted
waste rises from its pre-regulation level, while the price falls. This is in marked contrast to
the impact if the reprocessor is rational. In this case the reprocessor realises that it can
manipulate both the price of diverted waste and the price of credits by changing the amount of
diverted waste that it accepts. Consequently, it reduces its intake of diverted waste, and
drives up the price of credits to a level at which the waste collectors are willing to pay the
reprocessor to take additional waste packaging in order that they can sell additional credits to
the producers. However, it should be noted (from panels a) and b) of Figure 5) that the
combined price of diverted waste and recovery credits is always positive – i.e. the net revenue
to the waste collectors is positive.
c)
Packaging produced, and waste packaging sent to landfill
As was noted in section 3, the imposition of the obligation on the producers has the effect of
increasing the price of packaging – by an amount equal to the credit price multiplied by the
target recovery rate. Consequently, the higher the price of credits, the smaller the quantity of
packaging that is produced (i.e. the greater the level of source reduction). When the
reprocessor has the property rights and the target recovery rate is below the 50% threshold,
the price of credits is inversely proportional to the recovery rate.21 Consequently, the value of
the “packaging tax” is constant; as is the amount of packaging used (albeit at half the pre-
21
This follows directly from the expressions for Qn(n) and MRn(n), noting that n* = MRn-1(0).
16
regulation level). As expected, the actual recovery rate is equal to the threshold value of 50%.
However, when the target recovery rate is above the threshold, or when the rights are granted
to the waste collectors, the value of the “packaging tax” is positively related to the target
recovery rate, and hence the amount of packaging used declines as the target recovery rate
rises.22
With regard to the quantity of waste packaging that is landfilled, a very similar picture can be
seen. When the target recovery rate is below the threshold and the reprocessor has the
property rights, then both the amount of packaging produced and the amount diverted is
constant, and hence so to is the amount going to landfill. In all of the other cases the amount
of diverted waste packaging is equal to the number of credits purchased by the producers, and
consequently landfilled waste is equal to the (1-)% of the amount produced. Consequently,
as the amount produced declines, so too does the amount going to landfill.
Figure 6 shows the total cost under the alternative property rights regimes for different values
of the target recovery rate (i.e. the aggregate reduction in profits across the three sectors).23 It
is clear from this that the relative efficiency of the two property rights regimes depends
critically on the assumption that is made about the rationality of the reprocessor. If the
reprocessor is rational then, for target rates below the 50% threshold, the aggregate cost is
higher when the property rights are granted to the waste collectors. However, if the
reprocessor is myopic then the aggregate cost is always lower when the waste collectors have
the rights. Indeed, for recovery rates below 60%, aggregate profits actually increase (i.e. the
aggregate cost is negative). This is because at these rates, the scheme is also effectively
acting as an instrument of market regulation – reducing the deadweight loss in the target
market arising from the monsopony power of the reprocessor (with the producers rather than
the government providing the subsidy). For relatively small increases in the target recovery
rate (versus the pre-regulation value of 25%), and hence relatively small increases in the
quantity of diverted waste packaging, this gain is sufficiently large to outweigh the cost
imposed on the producers.
22
When the reprocessor has the property rights, the decline in the price of credits is more than offset by the
increase in the recovery target (see panel a)).
23
This has been calculated under the assumption that households are charged for the waste that they generate,
and hence that source reduction by producers has no financial impact on waste collectors.
17
Given that the environmental outcomes can differ under the alternative property rights
regimes (see panel e) and f) of Figure 5), it may be more appropriate to compare efficiency in
terms of the cost per tonne of abatement (i.e. the reduction in waste packaging that is sent to
landfill 24). However, as can be seen from panel b) of Figure 6, this makes no difference to
the conclusions regarding the relative costs of the two property rights regimes.
Finally, Figure 7 considers the distribution of costs and benefits for two particular target
recovery rates – one below the 50% threshold, and one above. There are two main points to
note. First, the cost to the production sector is always greater than the aggregate cost across
the all three sectors. Thus, even in the case when there is an aggregate benefit, the producers
suffer a reduction in profits. Second, while the reprocessor always benefits from the
introduction of the scheme, the impact on the waste collection sector depends on the level of
the target, the allocation of the property rights, and the assumption that is made about the
rationality of the reprocessor. Interestingly, when the target recovery rate is 35% and the
reprocessor is rational, the waste collectors are worse off if they have the property rights.
6.
Conclusions
This analysis has been based on a highly stylised model of EPR with a single packaging
material, and the simplest case of market power – i.e. monopoly power. Nevertheless, it
provides some useful insights that have important implications for the design of credit trading
schemes for EPR (and potentially for other applications where the target market is complete,
and the target is expressed in relative terms).
The generic analysis in section 4, and the illustrative example in section 5, demonstrate the
potential sensitivity of the economic and environmental outcomes to the assignment of initial
property rights when there is asymmetric market power in the target market. However, it is
clear that the extent of this sensitivity depends on the value of the target recovery rate, and on
the assumption that is made about the rationality of the reprocessor. In particular, if the
reprocessor is rational and the target recovery rate is above the threshold level, then the
24
The reduction is equal to the reduction in the amount of packaging produced, plus the increase in the amount
of waste packaging diverted.
18
environmental and economic outcomes are completely unaffected by the choice of sector to
receive the initial property rights.
It is important therefore to consider the likely magnitude of the threshold recovery rate. In the
illustrative example the threshold was equal to twice the pre-regulation rate (i.e. 50% versus
25%). However, this ratio is an artefact of the assumption of a linear demand curve for
packaging. If an iso-elastic demand curve had been assumed with elasticity = - 0.5, then the
threshold rate would have been four times greater than the pre-regulation rate (i.e. 100%).
Consequently, it is quite possible that the threshold rate will exceed any feasible target
recovery rate, and therefore that the outcomes will depend on the which sector is granted the
initial property rights.
This leads to the question of which assumption is most realistic regarding the rationality of
the reprocessor – i.e. is it more likely to be myopic or rational. Unfortunately, neither
assumption seems entirely plausible. The rational assumption implies that the reprocessor
fully understands the demand for credits, so that it can correctly forecast how its actions in the
diverted waste packaging market will affect the price of credits. However, this seems
unlikely, given that it does not itself participate in the market. On the other hand, the myopic
assumption implies that the reprocessor can correctly predict the equilibrium price in the
market for credits, but does not have any information about the shape of the demand curve.
One possible explanation for this apparent anomaly would be that the price is correctly
forecast by outside experts such as market analysts, who then make the information public
without revealing their underlying assumptions. A more realistic assumption might be that
the reprocessor is partially rational – i.e. it understands that it can influence the price of
credits, but it has imperfect information about the shape of the demand curve. This would
provide a useful extension to the relatively simple model considered here.
If there is reason to believe that the reprocessor is rational, then it would be better to grant the
property rights to the reprocessor. The environmental outcome is better (i.e. less packaging is
produced, and less waste packaging is sent to landfill), and the aggregate cost is lower.25 The
only caveat is that the lower cost hides a greater financial transfer to the reprocessor, and the
gross cost to the producer is higher. If on the other hand, it is felt that the myopic assumption
19
is more appropriate, then the regulator is faced with a trade-off. The aggregate cost will be
lower if the property rights are granted to the waste collectors, but the environmental outcome
will be worse. One approach to reconciling this trade-off is to base the decision on the
average cost per tonne of abatement (i.e. reduction in landfilled waste packaging). On this
basis it is preferable to assign the property rights to the waste collectors. However, in order to
get the same environmental outcome it will be necessary to set a higher target recovery rate.
References
ACR (1999). The Application of Local Taxes and Fees for the Collection of Household Waste: Local Authority
Jurisdiction and Practice in Europe, Technical Report, Association of Cities for Recycling, Brussels
Dinan, T.M. (1993). “Economic Efficiency Effects of Alternative Policies for Reducing Waste Disposal”,
Journal of Environmental Economics and Management, 25: 242-256
Fullerton, D & Kinnaman, T.C. (1996). “Household Responses to Pricing Garbage by the Bag”, American
Economic Review, 86(4): 971-984
Fullerton, D & Wu, W. (1998). “Policies for Green Design”, Journal of Environmental Economics and
Management, 36: 131-148
Hahn, R. W. (1984). "Market Power and Transferable Property Rights." Quarterly Journal of Economics 99(4):
753-765.
Mauleg, D.A. (1990). “Welfare Consequences of Emission Credit Trading Programs”, Journal of Environmental
Economics and Management, 18 (1): 66-77
Miranda, M.L., LaPalme, S. & Bynum, D.Z. (1998), Unit-based pricing in the United States: a tally of
communities, Report to the USEPA, July 1998
Palmer, K. & Walls, M. (1999). Extended Product Responsibility: An Economic Assessment of Alternative
Policies, Discussion Paper 99-12, Resources for the Future, Washington
Salmons, R. (2002a). “Solid waste management”, in New Areas for Application of Tradable Permits, OECD,
Paris (forthcoming).
25
The conclusions regarding the relative costs of the alternative property rights regimes are based on the
simulation results in section 5, and have not been proved analytically.
20
Salmons, R. (2002b). Cost efficiency of tradable compliance credits for extended producer responsibility,
Discussion Paper, CSERGE, University College London
Sartzetakis, E. S. (1997a). "Tradeable Emission Permits Regulations in the Presences of Imperfectly Competitive
Product Markets: Welfare Implications." Environmental and Resource Economics 9(1): 65-81.
Sartzetakis, E. S. (1997b). "Raising Rivals' Costs Strategies via Emission Permits Markets." Review of Industrial
Organization 12(5-6): 751-765.
von der Fehr, N.-h. (1993). "Tradable Emission Rights and Strategic Interaction." Environmental and Resource
Economics 3: 129-151.
21
Appendix 1: Proofs of propositions
A1.1
Proposition 1
If the pre-regulation recovery rate is 0, then w0 = 0z0
By construction, if n0 = w0 then Qn(n0; 0, pz) =

MRn(n0; 0, pz) < 0
since
and
P(z0) = pz

1   n0 
P 0   pz  = 0
0 
   

MRn(n) < Qn(n)
for all n > 0
By assumption MRz(0) > pz

MRn(0; 0, pz)
=

1 
0
 0   pz 
0  MR z
 
 

> 0
By assumption, MRz(z) is continuous and MRz/(z) < 0 for all z. Therefore, by the intermediate
value theorem, there exists a unique value n̂ (0)  (0, n0) such that MRn( n̂ (0), 0; pz) = 0.
For   0, let n̂ solve the implicit function MRn( n̂ , ; pz) = 0. By the implicit function
theorem
dn̂
d

n̂


n̂ = k 
(for some constant k > 0)
Therefore, there exists a finite value  > 0, such that n̂ = n0 = w0, and for all   [0,  ),
n̂ < n0 = w0. However, for all n  w0, the marginal cost of creating credits is zero. Thus, for
all   [0,  ), the reprocessor will maximise profits by setting w* = w0 and n* = n̂ < w0.
QED
22
A1.2
Proposition 2
From the proof of Proposition 1
k
=
n*
<
w*
=
w0
=
n0
=
k
=
w0
z*
=
n0
z*
=
k

z*
Therefore, since z* > 0 it follows that
k

z*
=
n*
z*
<
w*
z*
But from the market equilibrium conditions

k
= 1
z*

w*
= 
z*
n*
=  
z*
QED
23
A1.3
Proposition 3
Denote the market equilibrium values when the reprocessor has the initial property rights by
qn*, qw*, w*, n*, z*, kc* and kr*; and denote the market equilibrium values when the waste
collector has the rights and the reprocessor is rational by qn**, qw**, w**, n**, z**, kc** and kr**.
These values must satisfy the respective necessary conditions for a simultaneous market
equilibrium.
a)
Target recovery rate above threshold (i.e.    )
Proof is by contradiction. Assume that qn** + qw** > qw*

g/(w**) > g/(w*)

w** >
w*

n**
>
n*
z**
>
z*

qn** <
qn*
and
kc** >
since
Qn/(n) < 0
qn** + Qn/(n**) n**
<
kc *
qn* + Qn/(n*) n*
since

 w** 
P

  
2
Noting that
Qw/(w**)
pk g (w ) 
//
=
g//(w) > 0
since
**
1

MRn/(n) < 0
/
it follows that

2
Qw/(w**)
**
//
**
**
+ qn
**
 qn
**

w
=
pk g (w ) w
qw** + Qw/(w**) w**
=
pk g//(w**) w** + qw** + qn** 
1

 n**  **
P  n

/
[ qn**  Qn/(n**) n** ]
24
Consequently, using the original assumption and the derived implications, it follows that
qw** + Qw/(w**) w**
>
pk g//(w*) w* + qw*  [ qn*  Qn/(n*) n* ]
=
[ qw*  Qw/(w*) w* ]
=
fw(w*, kr*)

[ qn*  Qn/(n*) n* ]
However, the concavity of f(w, kr) implies that
[ fw(w*, kr*)  fw(w**, kr**) ] (w**  w*)
But since
fw(w**, kr**)
py fk(w**, kr**)

=
pk
fw(w*, kr*)

[ fk(w*, kr*)  fk(w**, kr**) ] (kr**  kr*)
=
py fk(w*, kr*)
if
w** > w*
it follows that
Therefore, we have shown that qw** + Qw/(w**) w** > fw(w**, kr**), which contradicts one
of the necessary conditions for an equilibrium. Consequently, the assumption cannot be true.
Assume instead that qn** + qw** < qw*. Following a similar logic, it is straightforward to
show that this implies qw** + Qw/(w**) w** < fw(w**, kr**), which again contradicts one of
the necessary conditions for an equilibrium.
Consequently, it must be that qn** + qw** = qw* (i.e. qw** < qw*), in which case it follows
from the above logic that w** = w*, n** = n*, qn** = qn*, kc** = kc*, z** = z*, and hence that l** =
l*. Since fk(w**, kr**) = fk(w*, kr*), the assumption that fkw > 0 and fkk < 0 implies that kr** =
kr*, and hence that y** = y*.
QED
25
b)
Target recovery rate below threshold (i.e.  <  )
Proof is by contradiction. Assume that qn** + qw**  qw*

g/(w**)  g/(w*)

w** 
w*

n**
>
n*
z**
>
z*

qn** <
qn*
and
kc** 
since
Qn/(n) < 0
qn** + Qn/(n**) n**
<
kc *
qn* + Qn/(n*) n*
since

 w** 
P

  
2
Noting that
Qw/(w**)
pk g (w ) 
//
=
g//(w) > 0
since
**
1

MRn/(n) < 0
/
it follows that

2
Qw/(w**)
**
//
**
**
+ qn
**
 qn
**

w
=
pk g (w ) w
qw** + Qw/(w**) w**
=
pk g//(w**) w** + qw** + qn** 
1

 n**  **
P  n

/
[ qn**  Qn/(n**) n** ]
Consequently, using the original assumption and the derived implications, it follows that
qw** + Qw/(w**) w**
>
pk g//(w*) w* + qw*  [ qn*  Qn/(n*) n* ]
=
[ qw*  Qw/(w*) w* ]
=
fw(w*, kr*)

[ qn*  Qn/(n*) n* ]
26
However, the concavity of f(w, kr) implies that fw(w**, kr**)  fw(w*, kr*) if
w** > w*
(see proof of part a)
Therefore, we have shown that qw** + Qw/(w**) w** > fw(w**, kr**), which contradicts one of
the necessary conditions for an equilibrium. Consequently, the assumption cannot be true,
and it must be that qn** + qw** < qw* (i.e. qw** < qw*), in which case it follows from the
above logic that w** < w* and kc** < kc*. Since fk(w**, kr**) = fk(w*, kr*), the assumption that
fkw > 0 and fkk < 0 implies that kr** < kr*, and hence that y** < y*.
Because of the strict inequality w* > n*, it is not possible to make any inferences about the
other variables using the foregoing logic (i.e. one cannot infer the relative magnitudes of n*
and n** from w* and w**). However, if w** < w*, then fw(w**, kr**)  fw(w*, kr*)

qw** + Qw/(w**) w**

qw* + Qw/(w*) w*

Qw/(w**) w**  qn**
>
Qw/(w*) w*  qn*  Qn/ n*
qn*  Qn/ n* = 0
since qn** + qw** < qw* and

2

//
**

**
pk g (w ) w
1

 w**  **
 qn**
P/ 
 w
  

2
>

//
*
*
pk g (w ) w

1

 n* 
P/   n*  qn*
 
pk [ g//(w**) w**  g//(w*) w* ]
>
 **
 qn 


But [ g//(w**) w**  g//(w*) w* ]  0
2
1


 n** 
P   n** 


/
 *
  qn

since, by assumption

2

1

 n*  
P /   n* 
   
g//(w) + w g///(w)  0
27

MRn(n*) > MRn(n**)

n** > n*
since MRn/(n) < 0

qn** < qn*
since Qn/(n; ) < 0
z** > z*
Finally, since z** > z* and w** < w* it follows that l**  l* .
QED
28
A1.4
Proposition 4
Denote the market equilibrium values when the reprocessor is myopic qn#, qw#, w#, n#, z#, kc#
and kr#; and denote the market equilibrium values when the reprocessor is rational by qn**,
qw**, w**, n**, z**, kc** and kr**. These values must satisfy the respective necessary conditions
for a simultaneous market equilibrium.
Proof is by contradiction. Assume that qn** + qw**  qn# + qw#

g/(w**)  g/(w#)

w** 
w#

n**
>
n#
z**
>
z#
l**
>

qn** <

qw#

and
kc** 
l#
since
(1 -  z** = l** and (1 -  z# = l#
qn#
since
Qn/(n) < 0
qw**
since
kc#
since
qn**  qn# 
g/(w) > 0 and g//(w) > 0
qw#  qw**
The respective equilibrium conditions

Qw/(w**) w**  Qw/(w#) w#
=
pk [ g//(w**) w**  g//(w#) w# ]

2

1

 w**  **
P/ 
 w
  
29
But [ g//(w**) w**  g//(w#) w# ]  0 since, by assumption
g//(w) + w g///(w)  0 and
P/() < 0.

Qw/(w**) w**  Qw/(w#) w#
> 0

qw** + Qw/(w**) w**
qw# + Qw/(w#) w#
>
However, the concavity of f(w, kr) implies that fw(w**, kr**)  fw(w#, kr#) if w** > w# (see
proof of proposition 2)
Therefore, we have shown that
fw(w#, kr#)

fw(w**, kr**)
=
qw** + Qw/(w**) w**
>
qw# + Qw/(w#) w#
which contradicts one of the necessary conditions for an equilibrium. Consequently, the
assumption cannot be true, and it must be that qn** + qw** < qn# + qw#, in which case it
follows from the above logic that w** <
w#, n** < n#, z** < z#, l** < l#, kc** < kc#, qn** > qn#,
and qw** < qw#. Since fk(w**, kr**) = fk(w#, kr#), the assumption that fkw > 0 and fkk < 0
implies that kr** < kr#, and hence that y** < y#.
QED
30
Figures
Figure 1:
a)
Trading system structures
SO2 emissions by electricity generators
Sector
B
Electricity
Sector
A
Emissions
Disposal
Services
||
Sector A:
Sector B:
Electricity generators
Electricity users
Permits
Sector
A
Waste
Packaging
Sector
B
Service / permit flows
Financial flows
b)
Extended producer responsibility for packaging
Sector
D
Reprocessed
Material
Sector
A
||
Sector A:
Sector B:
Sector C:
Sector D:
Material reprocessors
Waste collectors
Obligated producers
Producers
Credits
Sector
C
Material / credit flows
Financial flows
31
Figure 2:
Model structure
Waste
Collection
Sector
kc
(Case B)
n
n
Obligated
Production
Sector
z
w
Waste
Reprocessing
Sector
(Case A)
kr
y
Figure 3:
Inverse demand curves for credits Qn(n)
when P(z) = A – bz
P(z) = A z-1
when
30
30

S1
S1


20
20
10
10
S2
0
S2
0
0
20
40
60
0
20
40
60
32
Figure 4:
Threshold recovery rate (  )
(i)  < 
qn
(ii)  > 
qn
S(n)
S(n)
qn*
qn*
Q(n)
Q(n)
MR(n)
n0
MR(n)
n0
33
Figure 5:
a)
System variables
Price of credits (qn)
12
2B myopic
2B rational
2A
10
euro
8
6
4
2
0
25%
50%
75%
100%
Recovery target
b)
Price of diverted waste packaging (qw)
4
2B myopic
2B rational
2A
2
euro
0
-2
-4
-6
-8
-10
25%
50%
75%
100%
Recovery target
34
Figure 5:
c)
System variables (continued)
Quantity of diverted waste packaging (w)
3
2B myopic
2B rational
2A
tonnes
2
1
0
25%
50%
75%
100%
Recovery target
d)
Quantity of recovery credits (n)
3
2B myopic
2B rational
2A
tonnes
2
1
0
25%
50%
75%
100%
Recovery target
35
Figure 5:
e)
System variables (continued)
Packaging used by producers (z)
6
2B myopic
2B rational
2A
tonnes
4
2
0
25%
50%
75%
100%
Recovery target
f)
Waste packaging sent to landfill (l)
tonnes
4
2B myopic
2B rational
2A
2
0
25%
50%
75%
100%
Recovery target
36
Figure 6:
a)
Economic efficiency
Total cost
6
2B myopic
2B rational
2A
euro
4
2
0
-2
25%
50%
75%
100%
Recovery target
b)
Cost per tonne of reduction of landfilled packaging
3
2B myopic
2B rational
2A
euro per tonne
2
1
0
-1
-2
25%
50%
75%
100%
Recovery target
37
Figure 7:
a)
Distribution of costs and benefits
Recovery target = 35%
12
Reprocessor
Collectors
Producers
8
euro
4
0
-4
-8
2B myopic
b)
2B rational
2A
Recovery target = 70%
12
Reprocessor
Collectors
Producers
8
euro
4
0
-4
-8
2B myopic
2B rational
2A
38
Table 1: Necessary and sufficient conditions for a market equilibrium conditions under perfect
competition
Scenario A
Scenario B
py fw(w*,kr*)
=
qw*  qn*
py fw(w**,kr**)
=
qw**
py fk(w*,kr*)
=
pk
py fk(w**,kr**)
=
pk
pk g/(w*)
=
q w* + p l
pk g/(w**)
=
qw** + qn** + pl
P(z*)
=
pz + qn*
P(z**)
=
pz + qn**
g(w*)
=
kd*
g(w**)
=
kd**
w*
=
n*
w**
=
n**
=
 z*
=
 z**
39
Table 2: Necessary and sufficient conditions for a market equilibrium under monopoly power when
initial property rights are assigned to the reprocessing sector
Target below threshold ( i.e.  <  )
Target above threshold (    )
py fw(w*,kr*) = (qw* + Qw/ w*)
py fw(w*,kr*) = (qw* + Qw/ w*)  (qn* + Qn/ n*)
= (qn* + Qn/ n*)
0
py fk(w*,kr*) = pk
py fk(w*,kr*) =
pk
pk g/(w*)
=
pk g/(w*)
=
P(z*)
= pz + qn*
P(z*)
=
pz + qn*
g(w*)
= kd*
g(w**)
=
kd**
w*
=
n*
Qw/(w*)
=
pk* g//(w*)
Qn/(n*)
=

w*
>
Qw/(w*)
q w* + p l
=
=
pk* g//(w*)

2
Qn/(n*)
=
 z*
n*
1

*
/ n 
P 
 
q w* + p l
=
2
1

 z*
*
/ n 
P 
 
40
Table 3: Necessary and sufficient conditions for a market equilibrium under monopoly power when the
initial property rights are assigned to the waste collection sector
Rational reprocessor
Myopic reprocessor
py fw(w**,kr**)
=
(qw** + Qw/ w**)
py fw(w#,kr#)
=
(qw# + Qw/ w#)
py fk(w**,kr**)
=
pk
py fk(w#,kr#)
=
pk
pk g/(w**)
=
qw** + pl
pk g/(w#)
=
q w# + p l
P(z**)
=
pz + qn**
P(z#)
=
pz + qn#
g(w**)
=
kd**
g(w#)
=
kd#
w**
=
n**
w#
=
n#
Qw/(w#)
=
pk g//(w#)
=
 z**

2
Qw/(w**) =
pk g//(w**) 
1

 w** 
P

  
=
 z#
/
41
Table 4: Relative values of system variables
Below threshold ( i.e.  <  )
Above threshold (    )
qn
2B(M)
<
2B(R)
<
2A
2B(M)
<
2B(R)
=
2A
w
2B(M)
>
2A
>
2B(R)
2B(M)
>
2A
=
2B(R)
y
2B(M)
>
2A
>
2B(R)
2B(M)
>
2A
=
2B(R)
n
2B(M)
>
2B(R)
>
2A
2B(M)
>
2B(R)
=
2A
z
2B(M)
>
2B(R)
>
2A
2B(M)
>
2B(R)
=
2A
l
2B(M)
>
2B(R)
>
2A
2B(M)
>
2B(R)
=
2A
Table 5:
Parameter values and functional forms
Exogenous price parameters
py
=
4
pk
=
1
pz
=
2
pl
=
1
Reprocessing production function
y
=
f(w,kr)
=
w½ kr½
Capital requirement function
kd
=
g(w)
=
w2
Inv. Demand for packaging material
P(z)
=
7 - z
42