J. Pack. Sci. Technol. Vol.5 No.4 (1996)

A Theoretical Model of Redemption System
when Consumer Inconvenience can be Described as a Linear
Function of Distance
Masanobu ISHIKAWA

Waste reduction policy by promotion of recycling has been introduced in several countries. "Redemption System" in the state of California (U.S.A.) has achieved its goal at least cost. This study illustrates the optimal features of this system in achieving the target recycling rate within budget constraints. We take into account for consumer inconvenience as cost and we consider the case when
the inconvenience of the consumers to return can be described as a linear function of the distancefrom the collection stations. The theoretical analysis revealed that the optimum refund size does
not depend on the target recycle rate. Thus, a policy of dispersing the collection stations and not
raising the refund rate resulted in the optimum performance.
Keyword s: Recycling, Redemption syste m, Post-consu mer waste, Inconvenience

1. Introduction

many municipalities have adopted a curb-sidecollection system. In European countries a variety

1.1. Backgroundof economic instruments for recycling were intro-Municipal waste management is now one of

duced in the 1980's. Recycling has become athe major policy issues in developed countries.

mo re polit ica l and inte rnat ional i ssu e sin ce

They all are having difficulties constructing new

Ger man y and France decided to adopt more

facilities, such as incinerators and land fills, due

strict mandatory deposit systems at the begin-

to environmental protection concerns, and theyning of the 1990's. These may exert some influ-also face increasing waste management costs.

ence on international economic and trade system.

To reduce waste and avoid land fill, these

Germany's new law, in particular, requires packag-countries are being obliged to adopt recycling

ing-related industries to recycle and reuse theirpackaging.

policies. In the USA, ten states have introduced
a deposit refund system since the 1980's and*Tokyo University of Fisheries, Department of Food Science and Technology, 4-5-7 Konan, Minato-ku, Tokyo,
108Japan-267-A Theoretical Model of Redemption System

1.2. Redemptio n sys t emtime, total revenue, determined by the recycling

Of all the systems so far, a unique deposit

rate and deposit value, must balance total cost.

refund system called the "Redemption system"

Total cost consists of the capital and operatingin California has achieved its goals at least cost

cost of the system and the total redemption paid
for returned containers.

and accomplished this over the shortest timeperiod. This system is unique in several aspects.

2. Theoretical modelFirst, the state originates and handles deposits
and retailers pay deposits only to the state. This

This study is intended to illustrate the optimal

makes the system more integrated and efficient,features subject to achieving a given recycling

since it eliminates the handling of returned

rate and within budget constraints.

containers by retailers or brand holders. Second,

In the theoretical mod el, the total system

the system is cheaper than other deposit-refund

costs plus the inconvenience to consumers are

systems because the returned containers areminimized for a given recycling rate. We hypoth-redeemed at "recycling centers" which are widelydispersed and close to residential areas. In

esized that inconvenience is an increasing functionof the distance that consumers have to travel tocontrast, other deposit refund systems havereturn containers to a recycling center. The

higher costs because all retailers must redeem

analysis seeks to determine which costs would

containers on their own premises. Third, thisbe less: setting a higher redemption rate orsystem provides consumers with the choice ofdisp ersing many recycling centers.

returning containers to recycling centers if they

Figure 1 illustrates a process flow of thewant refunds or to curb-side locations if they"Redemption system." Among the variables used,prefer such convenience.

number of stations, refund size, and deposit size

The performance of this Redemption system

are policy variables constrained in that income

provides some very interesting data about

should balance with expenditure.

c o n su me r p r e fe r en ce b et w ee n t h e mo n et a r y

In the present model we assume:refund (economic incentives) and the inconve-1) The target recycling rate is given.

n i e n ce o f r e t u r n i n g th e be v e r a ge c on t a in e r2) The in convenience of a consumer is alinear function of the distance to the collection center.

(distance to the collection point).
For higher recycling rates, policy alternatives
are limited to providing many convenient recycling3) The operation cost is a sum of the fixed
cost, in turn, is proportional to the number

centers or increasing the redemption size (value).
Both increase total system cost. At the same- 268 -J. Pack. Sci. Technol. Vol.5 No.4 (1996)

If we have a social consensus on the quan-of stations and the variable cost. This intitative evaluation of DE, the optimal recycl-

turn, is proportional to the amount of theing rate would be obtained from the conditioncollected containers.

of minimization of the social cost. However,4) The effect of deposit size on the purchas-

the DE consensus is very difficult to reach.

ing behavior of consumers is small.

Accordingly, the practical target is the mini-

Let us consider the social cost (SC) to be themization of the social cost under a given

sum of the decrease of total surplus including

recycling rate. When we consider the possi-

the total refund (TS), the inconvenience ofbility of social consensus, financial balance ofconsumers (TI) and the externalities (DE), suchas decrease of capacity of land fill site. Thus,
SC=TS+TI+DE

the redemp tion system would of course becritical.
The problem can be described as:(1)- 2 6 9 -

A Theoretical Model of Redemption SystemMin

Min
SC(r,d,ρ)=TS+TI+DE

(2)

(7)

s.t.

s.t.
R(r,ρ)=R0

SC(r,d,ρ)=OC＋TI(3)

OC+TR=TD+RV(4)

In the present model the cost for processingof the collected containers is not treated explic-R(r,ρ)=R0(3)

OC＋TR=TR＋RV(4)

We next assume the following to formulate
the inconvenience of consumers:5) Collection stations are located on the nodes

itly. This is because processing cost would bereflected in the price of the collected containers.

6) Consumers would decide to refund or not

Given that change in the sale of beverages

by comparing the refund price and thedue to the introduction of the redemption systeminconvenience after the consumption ofis small, the decrease of total surplus, includ-

the beverage.

ing the total refund (TS) can be approximatedas (Appendix A1)):of a normal grid.

From assumptions 2 and 6, there should bea critical distance (x c ) at which inconvenienceis equal to refund size. Consumers in the area

T S = T D - T R(5)where the distance to the recycling station isless than x c would choose to refund; the rest

The problem can then be rewritten as:
Minwould discard the containers (Figure 2).
The recycling rate can be expressed as theTSC(r,d,ρ)=OC-RV+TI+DE(6)

ratio of the hatched area of Figure 2.
A simple geometrical calculation (AppendixB) yields;s.t.
R ( r , ρ ) = R 0(3)

O C + T R = T D + R V(4)

Si nc e th e c h a ng e o f t ot a l b e v e r a g e sa le s issmall and the target recycling rate is given asa constant, sales of the recycled containers (RV)
a nd e xt e r n a l i t i e s ( D E ) a r e n ea r ly c o n st an t .
We can then simplify:- 2 7 0 -(9)J. Pack. Sci. Technol. Vol.5 No.4 (1996)

whereθ = 4 ρ x2C(10)The function f1(θ) exhibits the correction of theeffect of accumulating refund circles shown inFigure 2. From assumption 2, the inconvenience ofa consumer at the edge of the refund circles (I(xC)) isequal to the refund size:I ( xC) = r(ll)Therefore, the critical distance can be describedby refund size as:
x C = a 1 r + a 2(12)The relation between refund size and amountof recycled containers is depicted in Figure 3.
In Figure3, the sum of S1 and S2(RrQ)
indicates the total amount which is refunded toconsumers. The increasing curve in Figure 3
can be interpreted as the supply curve of theused containers, thus the area below the supply
curve, S 2 , is the total consumer inconvenience.

the nu mb er o f stations and the variable cost

R can be expressed as a function of r by using(CV), which, in turn, is proportional to the amountof recycled containers.

equations (8), (10), and (12). Thus,
OC=CS+R・Q・CV
(13)

(15)

and

TI is;(16)(14)

According to assumption 3, operation cost(OC) is the sum of the fixed cost (C S ) and thevariable cost. The fixed cost is proportional to- 271 -Substitution of equation (16) into (15) yields

A Theoretical Model of Redemption System

the following equation,t h e o p t i mu m s o l u t i o n b e c o me s

By substitution of equations (14), (17), (8),and (10) into (7) and (3), we can derive thisexpression.
Min
From equations (24) and (25), it is evident
that the optimum refund size is independent
from the target recycling rate. It only depends
on consumer behavior (I(x), γ)> fixed cost ofcollection station (C f ), and population density(ρ). Station density is proportional to the targetrecycling rate (Equation 26).
Let us consider the most simple case whena2 is null and the target recycle rate is less thanThe solution for r is;

π/4. The optimum refund size, the optimum
station density are:w h e r e

It is remarkable that the optimum refund size
is insensitive to the fixed cost of collection station(C S ). On the other hand, consumer behavior (a 1 )affect much on the optimum refund size.
The social cost is:
Station density and deposit size are given by,

The social cost described by equation (29)In the case when individual refund circles are
set apart,f 1 (θ), is unity from equation (9). Then- 2 7 2 -J. Pack. Sci. Technol. Vol.5 No.4 (1996)

does not include RV and DE. The total social
cost is:the fixed cost of collection stations.

Acknowledgment
The author would like to appreciate Professor
Yoshifusa Kitabatake of Graduate School of Kyoto
University, Professor Akinori Kumagaya ofAoyama Gakuin University, and Dr. Hiroshi
In equation (30), quantitative description of

Asano and Dr. Isamu Matsukawa of CRIEPI,

the externality term DE(R 0 ) is very difficult.
However, from this equation, we can obtain theProfessor Tsutomu Tanaka of Chuo University,
Mr. Isoroku Sawada and Mr. Masaru Eto ofnecessary condition for the rationalization of

Economic Planning Agency, Mr. Hiroshi Tsubouchi

the enhancement of recycling within budget
constraints as:of National Land Agency for valuable comment.
This study was done as a part ofa project of
Economic Research Institute in which the author
collaborated with Professor Yoshifumi Fujii of
Bunkyo University, Professor Kazuhiro Ueta of
Kyoto University, Professor Eiji Hosoda of Keio

(31)

University, Miss Mihoko Shimamoto of Hosei
The left-hand side of equation (31) is a margin-

University, Mr. Kazuhiro Noma, Mr. Toshiyuki

al decrease of externalities per a recycled contain-

Suzuki, and Mr. Hiroyoshi Nakayama of Econom-er. The first and the second term of the right-

ic Planning Agency: the author appreciate muchfor sincere discussion and helpful comments.

hand side are the marginal income per a recycledcontainer, the third term of the right-hand side

Appendices

implies the consumer inconvenience per a recycledcontainer.

Appendix A Total surplus
A supply curve and a demand curve are depict-

3. Conclusion

ed in Figure A-1. Since the price of beverages
Theoretical analysis proved that when weincrease by deposit size d after the introduction

minimize the sum of operation cost of a redemp-of redemption system, the supply curve (DJC)
would change vertically (EB).

tion system and inconvenience of consumers
under given target recycling rate, the optimal
refund size is not depend on the target recycling
rate up to π/4.
The optimum refund size is insensitive with-273-

A Theoretical Model of Redemption System

Thus the decrease of total surplus TS' (GBCJIG)
is:

(A-4)
In equation (A-4), because η S and η ｄ areusually less than unity, and d is much smaller
than p O , the second term is much smaller than
the first term. According to the study of Porteron the cost benefit analysis of deposit-refund
system, the second term of equation (A-4) isless than 2% of the first term 2) . Therefore, theQuantity
Fig. A-1 Change of total surplus by introductionof redemption systemchange of total surplus including refund (TS)is:The equilibrium point shift from C to B, thus
the total surplus not considering the refundchange from the area ACDA to the area ABEA.
The supply elasticity (η S ) and demand elasticity (ηd) are defined as:

T S = Q 0 { d - R r }= T D - T R( A - 5 )Appendix B Recycling rate and critical distance
When collection stations are located in thenodes of normal grids of size L, consumers
within the critical distance from collection stations
would return their containers(Figure B-1). The( A - l )andPB-PJ=d( A - 2 )We obtain pB and pJ as:( A - 3 )- 274 -recycling rate is the ratio of area of the refund
circles to th e to tal area.
The recycling rate (R) is:J. Pack. Sci. Technol. Vol.5 No.4 (1996)

Collection station
Fig. B-1 Collection station and critical distance

References
1) Fujii, Yoshifumi, Masanobu Ishikawa, Kazutaka
Noma and Toshiyuki Suzuki, Keizai Bunseki.; inprint 1996 (InJapanese)
2) Porter, R. C., Land Economics, 59, 177-194,(1983)
3) Ishikawa, Masanobu: A study of social cost of recycling by deposit-refund system; Shokuhinn
Youki Kannkyou Bika Kyoukai/ Institute for Policy Sciences, 1994 (In Japanese)
4) Saito, Ayako and Masanobu Ishikawa: An optimum
design of redemption system considering theinconvenience of consumers, 4th Annual Meetingof Society of Science and Technology of Packaging, Japan, 1995

Using station density of collection stations(ρ), equation (B-l) can be rewritten as:- 2 7 5 -A Theoretical Model of Redemption System-276-