Recycling of Non-Renewable Resources and

the Least-Cost-First Principle

John R. Boyce∗

Department of Economics

University of Calgary

May 2012

Abstract

This paper analyzes the economics of recycling a non-renewable resource. Recycling

is two separate activities: sorting of recyclables from the waste stream into recyclable

stocks, and producing final goods from recycled stocks. With constant marginal pro-

duction and sorting costs, when recycled stocks are the least-cost source, a blocked

interval occurs where all possible waste is recycled and a constant proportion of output

is from recycled sources. When virgin stocks are the least-cost source, sorting into

recycled stocks occurs as a speculative activity for use when mining stocks run out. In

either case, whether all of the waste is sorted into recycled stocks depends on the cost

of sorting relative to the value of the recycled stocks. The effects of landfill costs are

also examined.

Key Words: Recycling, Non-Renewable Resources, Least-Cost-First Principle

JEL Codes: Q53, Q30, Q31, Q58.

∗Professor of Economics, Department of Economics, University of Calgary, 2500 University Drive, N.W.,Calgary, Alberta, T2N 1N4, Canada. email: boyce@ucalgary.ca; telephone: 403-220-5860.

1 Introduction

When the twin towers of the World Trade Center were destroyed by al-Qaeda on

September 11, 2001, left behind were approximately 300,000 tons of iron and steel

scrap among the 1.2 million tons of debris. Some of this iron and steel is now in

museums and in memorials to the loss, but most of it met the same fate as other scrap

iron and steel, it was recycled: twenty-four tons became part of the naval warship, the

USS New York ; and thousands of tons of material were shipped to China and India to

be recycled.

There are at least two benefits to recycling a non-renewable resource. The first

is the simple and appealing notion that if proportion δ of the stock is recycled each

cycle, then a stock of size S0 can be used over and over, resulting in total potential

use of S0/(1 − δ) > S0. Recycling half of the flow doubles the stock; recycling ninety

percent increases the stock ten-fold; and as the recycling rate δ approaches one, the

stock effectively becomes unbounded. A second benefit of recycling a non-renewable

resource is lower landfill requirements for waste (Environmental Protection Agency,

1997).

But recycling uses real resources. Whether recycling is economic hinges upon the

marginal costs of recycling vis-a-vis other sources of production. When a non-renewable

resource is recyclable, there are three sources of raw materials for final production:

producing from virgin stocks, producing from the recycled stock of pre-sorted recyclable

materials, and producing recyclables directly from the waste stream.

In this paper, I show that recycling of exhaustible resources may take several forms.

I consider a simple partial equilibrium model of an exhaustible resource in which

marginal production and sorting costs are each constant and in which there exists

a backstop technology. In such a model, it is natural to consider whether the least-

cost-first principle of Herfindahl (1967) holds. That principle states that when there

exist two (or more) non-renewable resource stocks with different marginal production

costs, it is both optimal and a market equilibrium that the least-cost stock is fully ex-

hausted before the high-cost stock is utilized. With recycling, however, I find that the

set of possible equilibrium sequences are much richer than in the standard two-stock

exhaustible resource environment. This results from two features of the economics of

recycling. The first is that recycling is actually two distinct economic activities: the

sorting recyclable materials from the waste stream, and producing from the recycled

stocks. This simple insight reveals some surprising answers to the question of when is

1

recycling economic. Second, there are two key constraints which affect the recycling

equilibrium. The quantity that can be sorted from the waste stream is constrained by

the rate of flow into the waste stream, and, when recycled stocks are exhausted, pro-

duction from the recycled stock is constrained by the flow of recyclables from the waste

stream. These constraints cause there to exist “blocked intervals” in which the demand

for sorting recyclables from the waste stream or the demand for recycled production

(or both) exceeds supply. These constraints play an important role in understanding

recycling equilibria in non-renewable resource markets.

When the virgin-stock is the least-cost source, there may exist an interval in which

sorting of recyclables into recycled stocks occurs, while current production is entirely

from virgin stocks. Sorting occurs because even though recycled stocks cannot currently

compete with virgin stocks, they are expected to be able to compete once virgin stocks

are exhausted. Thus, the sorting of recyclables during this interval occurs entirely as

a rational speculative equilibrium activity. When the costs of sorting from the waste

stream are sufficiently high relative to the value of the recycled stocks, however, this

interval may be preceded by an interval in which neither form of recycling occurs.

Thus, when the virgin stock is the least-cost source, the answer to the question of

when is recycling economic hinges upon the value of the recycled stock vis-a-vis its

cost of accumulation. In either case, however, production from recycled stocks occurs

only after the least-cost source virgin stocks have been exhausted.

In contrast, when the recycled stock is the least-cost source, recycled stocks are

always used first if they exist. When the cost of sorting is sufficiently high relative

to the value of the recycled stock, however, there may exist an interval in which all

production comes from recycled stocks, but no sorting of waste into recyclables occurs.

Thus, no matter which stock is the least-cost source, when sorting costs are sufficiently

high, a portion of the waste stream is never sorted into recycled stocks. This again

highlights the importance of separating the two activities of recycling, and suggests

that the answer to the question of when recycling is economic may be more subtle

than expected. Furthermore, when the recycled stock is the least-cost source, I find

that there always exists an interval, just prior to switching to the backstop technology,

in which the recycled and virgin stocks are used simultaneously. This interval occurs

only after existing least-cost recyclable stocks are exhausted. Once the recyclable

stocks are exhausted, production must come from virgin stock sources. But since

recyclable stocks are the least-cost source, any waste that is able to be sorted into

recyclables is immediately put to use, causing both stocks to be used simultaneously.

2

With linear marginal production costs, this causes the proportion of production coming

from recycled stocks to be constant over time. This may explain why the rate of

recycling has remained roughly constant for many of the sixteen metals for which

recycling data is available in the United States over the past century.

Herfindahl’s least-cost-first model has been extended in various ways by Solow and

Wan (1976), Hartwick (1978), Kemp and Long (1980), Lewis (1982), Drury (1982),

Swierzbinski and Mendelsohn (1989), Chakravorty and Krulce (1994), Amigues, Favard,

Gaudet, and Moreaux (1998), Gaudet, Moreaux and Salant (2001), Holland (2003),

Chakravorty, Magne, and Moreaux (2006) and Chakravorty, Moreaux, and Tidball

(2008). With recycling, the two stocks are the virgin and recycled stocks. To my

knowledge, this is the first paper to apply Herfindahl’s idea to recycling.

The economic literature on recycling began with Smith (1972), who considered a

model in which waste is a social bad that can be alleviated by recycling or by lowering

consumption, both of which are costly. This has since been extended by Ready and

Ready (1995) and Fullerton and Kinnaman (1995), who each considered models in

which society cared about the cost of waste through landfill usage. There is now

a large literature on the appropriate pricing strategies to encourage recycling [e.g.,

Baumol (1977), Fullerton and Kinnaman (1995), Palmer and Walls (1997), Conrad

(1999), Walls and Palmer (2001), Calcott and Walls (2005), Takayoshi (2007)]. Slade

(1980), for copper, and Sigman (1995), for lead, considered the effects of different

policies for recycling in a static partial equilibrium model. None of these papers,

however, examined how recycling is affected by rising scarcity rents in an exhaustible

resource model nor attempted to characterize the equilibrium in an exhaustible resource

market.

In addition, several authors have considered the effects of recycling on economic

growth. In Di Vita (2001, 2006, 2007) recycled stocks and virgin stocks have identical

costs. In Pittel (2006) and Pittel, Amigues and Kuhn (2010) recycled and virgin sources

are both essential in production and are imperfect substitutes. In contrast, here virgin

and recycled stocks are perfect substitutes differentiated only by their relative costs

— since a metal of a particular grade is that metal of that grade no matter what its

source1 — and neither stock is essential, as there exists a backstop technology available

at a higher cost.

The remainder of the paper is organized as follows. Section 2 documents recycling

1For example, scrap iron and steel yields a higher price than raw iron because scrap iron and steel aregenerally alloys.

3

050

100150

050

100150

050

100150

050

100150

1900 1925 1950 1975 2000 1900 1925 1950 1975 2000 1900 1925 1950 1975 2000 1900 1925 1950 1975 2000

Aluminum Antimony Chromium Cobalt

Copper Gold Iron & Steel Lead

Magnesium Mercury Nickel Platinum

Silver Tin Tungsten Zinc

Recycled % of Consumption Recycled % of Production

Per

cent

Rec

ycle

d

Year

Graphs by Mineral

Figure 1: Recycling Share of U.S. Consumption and Production for 16 Minerals, 1900-2010.

statistics for the United States over the past century. Section 3 outlines the basic

assumptions, and derives the necessary conditions that must hold along five subpaths

which make up a competitive equilibrium. Section 4 derives the equilibria for the case

where the recycled stocks have lower costs of production than mining virgin stocks.

Section 5 derives the equilibria for the case where marginal mining costs are less than

the marginal costs of using recycled stocks. Section 6 briefly shows how the model can

be extended to include landfill costs and costs of storing recycled stocks. Section 7

concludes.

4

2 Recycling of Exhaustible Resources

Fig. 1 shows the share of domestic consumption (solid line) and domestic production

(dashed line) that is from recycled sources in the United States for sixteen minerals from

1900-2010.2 For minerals such as aluminum, copper, iron and steel,3 and lead, most

consumption is domestically produced, so these rates are very similar. For minerals

such as chromium, cobalt, tin and tungsten, there is no domestic primary production in

the United States, so recycling rates are 100% of domestic production, and for nickel,

tin, and platinum, most consumption is imported.

The total production that comes from recycling is affected by the products in which

the resource is used and decisions about when and how to retire durable goods. Iron, for

example, is used in tools, appliances, automobiles, buildings, ships and bridges, among

other things. Each use has a very different lifespan and even different likelihoods that

the product is recycled. The iron in most buildings and bridges are eventually recycled,

but many ships are either lost at sea or are sunk at the end of their useful life because

the costs of recycling are too high. As the 9/11 attack showed, the proportion of pro-

duction from recycling can be affected by shocks in retirement of durable goods. These

are driven in part by the business cycle, as shown by the spike in silver, aluminum,

and copper recycling percentage of consumption during the Great Depression and the

similar spike in 2009 in aluminum, chromium, and iron & steel. They can also be

driven by other factors such as technological change, war, and other factors. Copper,

for example, which had much use in telecommunications and plumbing in the early

Twentieth Century, has been largely replaced by fiber-optics and plastics, respectively.

The recycling rates in Fig. 1 show that while there have been variations over time,

the recycling of the major metals has remained relatively constant over the last century.

The exceptions, lead, mercury, and aluminum, have had changes in recycling largely

driven by government policy. The increase in the recycling rates of tungsten reflect a

decline in domestic production. While recycling rates for iron & steel, lead, and nickel

have been rising, recycling rates for antimony, copper, and platinum have been falling

in recent decades.2The data is from the U.S. Geological Survey (Kelly and Matos 2011).3For iron and steel, secondary production equals “scrap iron and steel production” and primary production

equals “iron ore production.” Apparent consumption is the sum of apparent consumption of iron ore andscrap iron and steel. This recycling rate is not directly comparable to the Steel Recycling Institute’s recyclingrate, although the numbers are similar, as they define the recycling rate to be the ratio of total scrap recoveredto total raw steel production. See Fenton (2003).

5

3 Theoretical Model

I now turn to the application of the Herfindahl two-stock model to recycling. This

section explains the model assumptions and characterizes the five subpaths of which

combinations of comprise any non-renewable resource recycling equilibrium.

3.1 Model Assumptions

There are three sources of consumption: virgin stocks (the stock found in nature)

from which production arises through mining; recycled stocks, which are stocks of raw

materials that have been sorted from the waste stream; and the waste stream itself,

which is a flow at each instant in time.

The virgin stock of a recyclable material remaining at time t is St. This stock is

depleted by primary production, qt, at rate

dStdt≡ St = −qt, S0 > 0. (1)

Let xt denote quantity of recycled materials that is sorted from the waste stream, and

let yt denote production from recycled stocks. Thus total production is qt + yt. The

recycled stock is Rt, and the equation of motion for this stock is given by

dRtdt≡ Rt = xt − yt, R0 ≥ 0. (2)

Unlike the virgin stock, there may initially be zero recycled stock. This distinction is

important when the recycled stock is the least-cost source.

The maximum proportion of production that can be recovered through recycling

is 0 < δ < 1. The loss of proportion 1 − δ of production can be due to rusting, to

contamination from mixing with other goods in manufacturing or when introduced

to the waste stream, or any other processes which renders the materials so that it is

uneconomic to recover recyclable materials from the waste. Thus the effective stock is

at most (R0 + S0)/(1 − δ), which is finite.4 Therefore, the marginal costs of sorting

recyclable materials from the waste stream is γ > 0, for quantities less than or equal

to proportion δ of production, and is infinite for quantities greater than proportion δ

4This ignores activities such as the painting of bridges and buildings to prevent rusting, or altering thedesign of goods to affect the proportion which is recoverable which may affect δ. While these are interestingissues, the assumption that δ < 1 embodies the view that it is economically, if not physically, impossible torecycle 100% of the waste stream.

6

of production.

For simplicity, assume that consumers use the resource only for an instant, so that

all production immediately becomes available for recycling.5 Thus the quantity that

can be sorted from the waste stream is constrained to satisfy

xt ≤ δ(qt + yt). (3)

Implicit in (3) is the assumption that once waste is deposited in landfills, it becomes

uneconomic to mine. Thus, recyclables can only be accumulated as they pass through

the waste stream.

When the recycled stock is positive, production from recycled stocks, yt, is bound

only by the constraint that Rt ≥ 0. When Rt = 0, however, production from the

recycled stock is constrained by the rate of inflow to the recycled stock:

yt ≤ xt when Rt = 0. (4)

When this constraint is slack, recycled stocks grow, causing yt to be once again un-

bounded.

These two constraints play a major role in determining the recycling equilibrium.

When these constraints bind, they “block” production or sorting which would be prof-

itable from occurring were the constraint not binding. Thus, these blocked intervals

are characterized by a premium which owners of the recycled stock are willing to pay

for the sorted waste.

The model is partial equilibrium.6 Gross utility from consumption of the resource

is given by u(qt + yt), where u(0) = 0, u′(.) > 0 and u′′(.) < 0. There exists a

perfect substitute (a ‘backstop’ technology) which may be produced without bound

at price p < ∞. Therefore, the choke price on the exhaustible resource is u′(0) = p.

The parameters α and β, each less than p, are the production costs from the virgin

and recycled stocks, respectively, and γ is the cost of sorting the recycled stock from

the waste flow. Recycling can never be profitable when β + γ ≥ p. Therefore, the

interesting case occurs when β + γ < p.7 The discount rate is r, which is taken to5This assumption fits for aluminum cans; less so for bridges and buildings. If durable goods, however, are

of the ‘one-hoss-shay’ variety, with a lifetime of s periods, then if total production in period t is zt, and shareδ can be recycled, then the quantity available for recycling in period t is δzt−s. With stationary demand,this results in greater waste flow at each instant, since production is declining.

6Lewis (1982) showed that as long as there exists an alternative asset that yields a positive rate of return,the least-cost-first principle holds in general equilibrium.

7If β < p < β+γ, then any initial recycled stock would be used in equilibrium, but no gross accumulation

7

be the return on all other assets in the economy. Together with the assumption that

0 < δ < 1, the assumptions that α, β, and β + γ are each less than p ensures that the

entire resource stock is depleted in finite time. Hence T < ∞. In addition, for now

landfill costs for unsorted wastes and storage costs for recycled stocks are each ignored.

These considerations are reintroduced below in Section 6.

3.2 Five Subpaths

Depending upon the initial conditions and the relative marginal production and sort-

ing costs, a sequence of combinations of up to five subpaths, together with constants

describing the shadow values of the stocks and the times demarcating the end of each

subpath in the sequence, makes up an exhaustible resource recycling equilibrium. Each

subpath is denoted with a bold-face font, with the stock being extracted indicated first

(S for the virgin stock and R for the recycled stock) and an x following the stock being

extracted indicating that sorting from the waste stream occurs, while the absence of an

x indicates no sorting occurs. In each subpath, I show that a Hotelling r% rule holds.

For all subpaths except subpath SRx, I find that the current value of the stock

rises exponentially at the rate of interest. Therefore, let λ0ert and µ0e

rt denote the

current-value in-situ prices of the virgin and recycled waste stocks of the resource,

respectively. In subpath SRx, the current value of the recycled waste stock is written

as µtert.

Subpath S: Production from Virgin Stocks; No Sorting of Waste

In subpath S, there is production from virgin stocks, but no sorting of waste into

recycled stocks. Thus,

pt = u′(qt) = α+ λ0ert, (5a)

pt = u′(qt) ≤ β + µ0ert, for t ∈ S, (5b)

and

γ ≥ µ0ert, (5c)

where total production is qt = u′−1(α+λ0ert). Equation (5a) ensures that virgin stock

producers are indifferent between producing anywhere along subpath S. The inequality

of recycled stocks would ever occur.

8

in (5b) ensures that recycled stock owners prefer to not produce during this interval.

The inequality in (5c) implies that the cost of sorting from the waste stream is greater

than the value of the recycled stocks. Each of these inequalities holds strictly within

any subpath, but may hold as an equality at the endpoints of the subpath. Over

subpath S, the virgin stock declines at rate St = −qt, while the recycled stock remains

constant.

The equilibrium price path in subpath S depends only on the extraction costs, α,

and the current scarcity rental value, λ0ert, of the virgin stocks. Thus, over subpath

S, the following Hotelling condition holds:

ddt [pt − α]pt − α

= r, for t ∈ S. (6)

Subpath R: Production from Recycled Stocks; No Sorting of Waste

In subpath R, production is from recycled stocks, and no sorting of waste occurs. Thus

subpath R is characterized by the following conditions:

pt = u′(yt) = β + µ0ert, (7a)

pt = u′(yt) ≤ α+ λ0ert, for t ∈ R, (7b)

and

γ ≥ µ0ert, (7c)

where total production is yt = u′−1(β + µ0ert). Equation (7a) ensures that recycled

stock producers are indifferent between producing anywhere along subpath R. The

inequality in (7b) ensures that virgin stock owners prefer to not produce during this

interval. The inequality in (7c) ensures that no gross accumulations to recycling stocks

occurs. In subpath R, the virgin stock remains unchanged, but the recycled stock

declines at rate Rt = −yt.The equilibrium price path in subpath R depends only on the extraction costs, β,

and the scarcity rents, µ0ert, of the recycled costs. Thus, over subpath R, the following

Hotelling condition holds:

ddt [pt − β]pt − β

= r, for t ∈ R. (8)

9

Subpath Sx: Production from Virgin Stocks; Sorting of Waste

Subpath Sx is characterized by the following conditions:

pt = u′(qt) = α+ γδ + (λ0 − δµ0)ert, (9a)

pt = u′(qt) ≤ β + γδ + (1− δ)µ0ert, for t ∈ Sx, (9b)

and

φtert = µ0e

rt − γ ≥ 0, (9c)

where total production is qt = u′−1(α+γδ+(λ0−δµ0)ert). Equation (9a) ensures that

virgin stock producers are indifferent between producing anywhere along subpath Sx.

The inequality in (9b) ensures that recycled stock owners prefer to not produce during

this interval. The inequality in (9c) ensures that gross accumulations to recycling stocks

occurs. For both virgin and recycled stock owners, the opportunity cost of producing

includes the direct production costs as in subpath S, plus a factor δ(µ0ert − γ), which

reflects payments earned from contributing proportion δ of production from the waste

stream to the accumulation of recycled stocks. Thus, over subpath Sx, the virgin

stock declines at rate St = −qt and the recycled stock increases at rate Rt = δqt, since

constraint (3) binds, implying that xt = δqt.

Over subpath Sx, the following Hotelling condition holds:

ddt [pt − α− γδ]pt − α− γδ

= r, for t ∈ Sx. (10)

Subpath Rx: Production from Recycled Stocks; Sorting of Waste

Subpath Rx is characterized by the following conditions:

pt = u′(yt) = β + γδ + (1− δ)µ0ert, (11a)

pt = u′(yt) ≤ α+ γδ + (λ0 − δµ0)ert, for t ∈ Rx, (11b)

and

φtert = µ0e

rt − γ ≥ 0, (11c)

where total production is yt = u′−1(β+γδ+(1−δ)µ0ert). Equation (11a) ensures that

recycled stock producers are indifferent between producing anywhere along subpath

Rx. The inequality in (11b) ensures that virgin stock owners prefer to not produce

10

during this interval. Equation (11c) ensures that no gross accumulations to recycling

stocks occurs. For both virgin and recycled stock owners, the opportunity cost of

producing includes the direct production costs as in subpath R, plus the payment

δ(µ0ert − γ), for each unit of proportion δ of production which can be sorted from

the waste stream to the accumulation of recycled stocks. Thus, over subpath Rx, the

following Hotelling condition holds:

ddt [pt − β − γδ]pt − β − γδ

= r, for t ∈ Rx. (12)

In subpath Rx, the virgin stock remains unchanged, St = 0, while the recycled

stock declines at rate Rt = −(1− δ)yt, since xt = δyt.

Subpath SRx: Simultaneous Production from Virgin and Recycled

Stocks; Sorting of Waste

Subpath SRx involves simultaneous production from both stocks plus sorting of waste.

This path can only occur when Rt = 0 over the interval and when marginal production

costs from recycled stocks are less than marginal production costs from virgin stocks.

Thus, both stocks are utilized simultaneously in this subpath, since all recycled stock

is immediately used, implying (4) binds, and since Rt = 0 implies that recycled pro-

duction must come from the waste stream. Let µtert denote the current value of the

recycled stock during this interval (where µt may not be constant).

Since sorting of recyclables occurs, it must be that

φtert = µte

rt − γ ≥ 0. (13)

For price pt, the current value of the recycled stock is the difference between the

price received and the marginal cost of production,

µtert = pt − β + δ(µtert − γ), for t ∈ SRx, (14)

where the term δ(µtert − γ) is the payment received from recycled stock owners to

acquire flow from the waste stream. Since virgin stock owners are also producing

during this interval, it follows that the price must obey

pt = α+ λ0ert − δ(µtert − γ), for t ∈ SRx. (15)

11

Substituting for µtert from (14) into (15) yields

pt = u′(qt + yt) = (1− δ)α+ δ(β + γ) + (1− δ)λ0ert, for t ∈ SRx, (16)

where total production is qt + yt = u′−1[(1− δ)α+ δ(β + γ) + (1− δ)λ0ert]. The price

is therefore equal to the weighted sum of the marginal production costs from each

source, where marginal production costs from the virgin stock are α and the marginal

production costs from the recycled stock are β + γ, plus the scarcity rental cost of the

share of production that comes from virgin stocks, which are the only stock that is

being depleted. Therefore, over subpath SRx, the following Hotelling condition holds:

ddt [pt − (1− δ)α− δ(β + γ)]pt − (1− δ)α− δ(β + γ)

= r, for t ∈ SRx. (17)

Equations (14) and (15) hold because both virgin stock holders and recycled stock

holders simultaneously wish to produce, given prices and costs. Because φtert =

δ(µtert − γ) > 0, it follows that xt = δ(qt + yt), and because µt > 0, it follows that

yt = xt. Thus, in subpath SRx, the shares of production coming from each source are

constant:

ytqt + yt

= δ, andqt

qt + yt= 1− δ, for t ∈ SRx. (18)

Therefore, in subpath SRx, the virgin stock declines at rate St = −qt = −(1− δ)(qt +

yt), while the recycled stock remains unchanged, Rt = xt − yt = 0, since yt = xt.

In the subpath SRx, the time paths of µt and φt are determined by the condition

that both qt and yt are consumed in positive proportions. Thus,

µtert = α− β + λ0e

rt and φtert = α− (β + γ) + λ0e

rt. (19)

The current value of additional recycled stock, µtert, in this interval is equal to the

current value of the virgin stock less the cost difference between the recycled and virgin

stocks, β − α. The current value of gross accumulation to recycled stocks is equal to

the value of the recycled stock less the cost of sorting.

12

4 If Recycled Stocks are the Least-Cost Source

When the marginal cost of extracting from the recycled stock is less than the marginal

cost of extracting from the virgin stock, i.e., β < α, all equilibria in which some of the

waste stream is recycled end in the blocked interval SRx where everything that can

be obtained from the waste stream is sorted and recycled into final products. Thus, in

this interval, both stocks must be used simultaneously, since the gross accumulations

from just the recycled stock are insufficient to maintain production. Conversely, when

the cost of sorting recyclable materials from the waste stream is sufficiently high, two

paradoxical equilibria can arise. When the initial recycled stock is positive, it may

occur that the equilibrium begins in subpath R where extraction is from the recycled

stock, but the waste from the recycled stock is not sorted into recycled stocks because

the cost of sorting is too high relative to the value of the stock. Alternatively, when the

initial recycled stock is zero, the equilibrium may begin in subpath S, where extraction

is from the virgin stock and no sorting occurs, even though the recycled stocks are

lower cost, if accumulated. These equilibria highlight the important difference between

production from recycled stocks and gross accumulation to recycled stocks. They also

imply that the size of the initial recycled stock matters.

4.1 Equilibria with Zero Initial Recycled Stocks

I first consider equilibria in which the initial recycled stock is zero.

Equilibrium Sequence S→ SRx

Equilibrium sequence S→ SRx occurs when sorting costs are sufficiently high and the

initial recycled stock is zero: R0 = 0. In the initial subpath S only the virgin stock

is exploited, and no sorting into recycled stocks recycling occurs. This is followed by

subpath SRx in which simultaneous production from recycled stocks and the virgin

stock occurs. Such an equilibrium is described by the the conditions (5) over subpath

S and (13)-(15) over subpath SRx.

This equilibrium has four constants, the endpoints of each subpath, TS and TSRx,

and the present values of the stocks, λ0 and µ0. These constants are found by solving

for the exhaustion and no-arbitrage conditions which must hold along an equilibrium

path.

13

At time TS , the conditions which must hold are

µ0erTS = γ, (20a)

and

pTS= α+ λerTS = (1− δ)α+ δ(β + γ) + (1− δ)λ0e

rTS . (20b)

The condition (20a) is what makes it possible to begin sorting into recycled stocks at

time TS . The condition (20b) implies that the price path is continuous at time TS . This

is necessary to ensure that no arbitrage opportunity remains, since the left-hand-side is

the marginal profit at an instant before time TS and the right-hand-side is the marginal

profit an instant after time TS . Neither a virgin stock producer who withheld a unit

of production from the interval [0, TS) can earn a capital gain by waiting to produce

after time TS , nor a virgin stock owner who moved a unit of production forward from

the interval [TS , TSRx) to a moment before time TS can earn a capital gain. The

no-arbitrage condition (20b) rules out either of these types of opportunities.

The second set of equalities occur at time TSRx:

pTSRx= (1− δ)α+ δ(β + γ) + (1− δ)λ0e

rTSRx = p, (21a)

and

(1− δ)∫ TSRx

TS

u′−1[(1− δ)α+ δ(β + γ) + (1− δ)λ0e

rt]

dt =

S0 −∫ TS

0u′−1

[α+ λ0e

rt]

dt, (21b)

where the integrands equal total production at each instant in each interval. Equation

(21a) ensures that demand is choked off at time TSRx, while equation (21b) ensures

that at the same moment supply is also exhausted. The right-hand-side of the ex-

haustion condition (21b) takes account of the fact that reserves of the virgin stock

extracted before sorting begins at time TS are not recoverable. Because sorting from

the recycled stream began at time TS > 0, total consumption is less than the potential

resource supply, S0/(1 − δ). These two equations are two sides of a no-arbitrage con-

dition. If demand were not choked off at time TSRx, i.e., if pTSRx< p but STSRx

= 0,

then a producer who withheld a unit of production from sometime within the interval

[TS , TSRx) could earn a capital gain of p− pTSRxat time TSRx by doing so. Similarly,

if STSRx> 0 at time TSRx while pTSRx

= p, then all producers holding stock at that

14

point in time TSRx would own an asset which earned a zero rate of return, since price

cannot rise above p, which is less than the return of r earned on other assets. Thus,

those producers would do better by selling their stock at an earlier date. When all of

these arbitrage opportunities have been eliminated, equations (21) hold.

This equilibrium is depicted in Fig. 2. The thick exponential curve is the equilib-

rium price path. The thick step-function is the relevant opportunity cost of extraction

at each instant. From (19), the current value of the recycled stocks in subpath SRx,

µtert, is the difference between the curve α+λ0e

rt and the marginal cost of production

from the recycled stocks, β. As can be seen, this is rising at less than the rate of

interest, since α > β. This is why those who accumulate recycled stocks wish to sell

it immediately. The value of gross additions to recycled stocks, φert is equal to the

difference between the curve α + λ0ert and β + γ. This is zero at TS and is rising at

a rate greater than r over subpath SRx, which is why recycled stock owners wish to

accumulate the maximum possible from the waste stream in subpath SRx, but not

before.

Equilibrium Sequence SRx

When sorting costs are sufficiently small relative to the values of the recycled stock,

sorting from the waste stream begins immediately. Thus, when R0 = 0, the equilibrium

sequence is simply subpath SRx.

Since the recycled stock is held at zero throughout this equilibrium, it is the deple-

tion of the virgin stock which dictates when the sequence ends. At time TSRx, physical

and economic exhaustion occurs:∫ TSRx

0u′−1

[(1− δ)α+ δ(β + γ) + (1− δ)λ0e

rt]

dt =S0

1− δ, (22a)

and

pTSRx= (1− δ)α+ δ(β + γ) + (1− δ)λ0e

rTSRx = p (22b)

where the integrand of (22a) is total production over subpath SRx. The constants

λ0 and TSRx are found by solving the implicit equations (22), and the equilibrium

proceeds with production satisfying (18) and price and total output satisfying (16).

In Fig. 2, this equilibrium occurs when virgin stocks are such that time zero occurs

somewhere between time TS and time TSRx, so that only subpath SRx occurs. Because

sorting from the recycled stream began at time 0, the entire potential resource supply,

S0/(1− δ), is consumed.

15

TSRxTS

Α

p

Β

Β+Γ

H1-∆LΑ+

∆HΒ+ΓL

Β+Μ0

Α+Λ0

H1-∆LΑ+∆HΒ+ΓL+H1-∆LΛ0ert

Α+Λ0ert

Β+Μ0ert

t

$

Figure 2: Equilibrium Sequence S→ SRx, When Recycling is Least-Cost, Initial Recycled Stocksare Zero, & Sorting Costs are High.

Impossibility of Equilibrium Sequence S→ Sx→ SRx

No equilibrium sequence S → Sx → SRx can occur. Because extraction costs are

lower for recycled stocks than for virgin stocks, if sorting to the recycled stocks were

to occur, that stock immediately becomes the least-cost alternative, so subpath S is

always followed by subpath SRx when β < α.

Hence, the only equilibria that can occur when β < α and R0 = 0 are equilibrium

sequences SRx and S → SRx. Whether equilibrium sequence SRx or equilibrium

sequence S → SRx occurs depends on the size of the virgin stock, S0, and the value

of the sorting costs, γ. In the appendix, it is shown that TS in equilibrium sequence

S → SRx is increasing in both S0 and γ. Thus, for small enough S0 or γ, sorting

begins immediately. The reason this occurs for an increase in S0 is that an increase in

S0 causes λ0 to decline, which causes TS to rise to satisfy (20b). The effect from an

increase in γ is more subtle, since an increase in γ causes both λ0 and TS to increase.

16

4.2 Equilibria with Positive Initial Recycled Stocks

Next, consider equilibria in which R0 > 0 when β < α.

Sequence Rx→ SRx

When R0 > 0 and β < α, if γ or S0 + R0 is sufficiently small, sorting from the waste

stream occurs from the beginning, implying equilibrium sequence Rx→ SRx. In this

equilibrium, there are three constants to be determined, the two ending times, TRx and

TSRx, and the present value of the virgin stock, λ0.

The conditions which must hold at time TRx are

pTRx= β + δγ + µ0(1− δ)erTRx = α(1− δ) + (β + γ)δ + (1− δ)λ0e

rTSRx , (23a)

and ∫ TRx

0u′−1

[β + δγ + µ0(1− δ)ert

]dt =

R0

1− δ, (23b)

where the integrand of (23b) is the production rate over interval [0, TRx), of which

share δ comes from recycling the waste stream. Equation (23a) is the no-arbitrage

condition which makes producers of the recycled stock indifferent between producing

in subpath Rx and subpath SRx. Equation (23b) is the exhaustion condition for the

recycled stocks.

The conditions which must hold at time TSRx are that

pTSRx= (1− δ)α+ δ(β + γ) + (1− δ)λ0e

rTSRx = p, (24a)

and ∫ TSRx

TRx

u′−1[(1− δ)α+ δ(β + γ) + (1− δ)λ0e

rt]

dt =S0

1− δ. (24b)

Equation (24a) ensures that demand is choked off at the same moment that supply is

exhausted, and (24b) is the supply exhaustion condition for the virgin stock. Because

sorting from the recycled stream began at time 0, the full potential resource supply,

(R0 + S0)/(1− δ), is consumed.

An example of this equilibrium is depicted in Fig. 3, where time t = 0 occurs

in the interval between time TR and time TRx. Again, the thick exponential growing

price path is the equilibrium price path and the thick step-function shows the relevant

extraction costs over each subpath.

17

Equilibrium Sequence R→ Rx→ SRx

When γ is sufficiently large, sorting from the waste stream does not occur at the

beginning, even though the recycled stock is the least-cost source. Thus, it is possible

to have an equilibrium sequence R → Rx → SRx where in the initial subpath R,

production occurs from the recycled stocks, but gross additions to the recycled stocks

do not occur. Over this equilibrium sequence, there are five constants to be determined,

the three end points of each subpath, TR, TRx, and TSRx, and the two present values

of the stocks, λ0 and µ0.

The conditions which must hold at time TR are

µ0erTR = γ, (25a)

and

pTR= β + µ0e

rTR = β + δγ + µ0(1− δ)erTR . (25b)

Condition (25a) ensures that it becomes feasible to begin sorting the waste stream into

recyclables at time TR. Equation (25b) is the no-arbitrage condition which implies

that marginal profits to recyclable stock holders are equal in in present value in each

subpath. The condition (25b), however, is implied by condition (25a).

The conditions which must hold at time TRx are

pTRx= β + δγ + µ0(1− δ)erTRx = α(1− δ) + (β + γ)δ + (1− δ)λ0e

rTRx , (26a)

and

(1− δ)∫ TRx

TR

u′−1[β + δγ + µ0(1− δ)ert

]dt = R0 −

∫ TR

0u′−1

[β + µ0e

rt]

dt. (26b)

Equation (26a) is the no-arbitrage condition which makes producers of the recycled

stock indifferent between producing in subpath Rx and in subpath SRx. Equation

(26b) is the condition ensuring exhaustion of the recycled stocks at time TRx.

The conditions which must hold at time TSRx are

pTSRx= (1− δ)α+ δ(β + γ) + (1− δ)λ0e

rTSRx = p, (27a)

∫ TSRx

TRx

u′−1[(1− δ)α+ δ(β + γ) + (1− δ)λ0e

rt]

dt =S0

1− δ. (27b)

Condition (27a) ensures demand is exhausted at time TSRx while condition (27b) en-

18

TSRxTR TRx

p

Α

Β

Β+Γ

Β+Μ0

Α+Λ0

Β+Γ∆

H1-∆LΑ+

∆HΒ+ΓL

Β+Μ0ert

Α+Λ0ert

Β+Γ∆+H1-∆LΜ0ert

H1-∆LΑ+∆HΒ+ΓL+H1-∆LΛ0ert

t

$

Figure 3: Equilibrium Sequence R→ Rx→ SRx, When Recycling is Least-Cost, Initial RecycledStocks are Positive, & Sorting Costs are High.

sures that supply is exhausted at time TSRx.

Because sorting from the recycled stream began at time TR > 0 rather than at

time 0, the potential resource supply, (R0 + S0)/(1− δ), is not fully consumed in this

equilibrium.

An example of the equilibrium sequence R → Rx → SRx is depicted in Fig.

3. Sorting of recyclables from the waste stream occurs in both intervals [TR, TRx) and

[TRx, TSRx]. In interval [TR, TRx), the net value additions to the recyclables stocks, φert,

is the difference between the curve β + µ0ert and β + γ, while in interval [TRx, TSRx]

it is the difference between the curve α + λert and β + γ. In interval [TR, TRx), the

value of the recycled stocks are µ0ert, which grows at the rate of interest. In interval

[TRx, TSRx], the gross value of the recycled stocks is α+ λ0ert − β, which grows at less

than the rate of interest. This is why these stocks are not rebuilt up during interval

[TRx, TSRx]. In the appendix, I show that TR is increasing in each of the stocks and in

sorting costs.

19

Impossibility of Equilibrium Sequence R→ S→ SRx

It might be thought that when R0 > 0 that there exists an equilibrium sequence R→S → SRx if sorting costs are sufficiently high. However, this is not so. Suppose that

such an equilibrium exists. Then over interval [0, TR), the recycled stock is exploited,

and at time TR, the economy switches to the higher cost virgin stock. Thus at time

TR, it must be that β + µerTR = α + λ0erTR . Then over the interval [TR, TS), the

virgin stock is exploited until it becomes profitable to sort the waste stream. Thus,

at time TS , µ0erTS = γ, which just makes it profitable to begin sorting the waste

stream into recycled stock and waste. But because β < α, once recycled stock becomes

available, it is immediately exploited, which implies that both stocks are exploited

simultaneously. However, in order that there be no possibility of arbitrage, it must be

that α + λerTS = (1 − δ)α + δ(β + γ) + (1 − δ)λ0erTS . Subtracting α + λ0e

rTS from

each side and dividing by δ leaves α+ λ0erTS = β + γ = β + µ0e

rTS , where the second

equality uses the fact that µ0erTS = γ, since recycling just begins at time TS . This,

however, requires that TS = TR, which contradicts the assumption that an interval

[TR, TS) exists in which the virgin stock is exploited following the consumption of the

recycled stock.

Thus, when the initial recycled stock is positive, R0 > 0, and when the recycled

stock is the least-cost source, β < α, only two possible equilibria exist, equilibrium

sequence Rx → SRx and equilibrium sequence R → Rx → SRx. Which of these

occurs depends upon the sizes of the two stocks and the sorting costs. The effect of

sorting costs is easiest to see. Since µ0 is constrained to lie between 0 and p − β, the

sorting condition (25a) implies that there exists a value of γ close enough to zero such

that TS = 0. Thus, as γ rises above this value, TS increases. The effects of R0 and S0

occur because an increase in either causes µ0 to decrease, which implies that TS must

increase.

4.3 Summary of Equilibria when β < α

Table 1 summarizes the possible equilibria when extracting from the recycled stock is

the least-cost alternative. Equilibria in which subpath SRx occur yield intervals in

which constant proportions of production come from recycled and virgin stocks. These

proportions are related to the proportion of the waste stream that is feasible to recover,

δ, since when the recycled stock has the lowest cost, it is always used immediately

during subpath SRx. This may explain why Fig. 1 shows that the proportions of

20

Table 1: Equilibrium Outcomes when the Recycled Stock is Least-Cost: β < α.

Recycled Stock SizeR0 > 0 R0 = 0

Sorting Cost / Stocks Equilibrium % Gain Equilibrium % Gainγ ‘low’ / ‘large’ Stocks Rx→ SRx 100% SRx 100%γ ‘high’ / ‘small’ Stocks R→ Rx→ SRx < 100% S → SRx < 100%γ + β > p R→ S 0% S 0%Notes: The column labeled ‘% Gain’ indicates the proportion of the differencebetween (S0 +R0)/(1− δ) and S0 +R0 which is gained by recycling.

consumption from recycled stocks in the U.S. have remained roughly constant over the

twentieth century.

5 If Virgin Stocks are the Least-Cost Source

Consider next the equilibria which occur when the virgin stock is the least-cost source,

i.e., when α < β. In these equilibria, since S0 > 0, the virgin stock is always exhausted

before the recyclable stock, and if sorting of recycled stocks occurs, it occurs for the

purpose of speculation, since it is not viable to produce from recycled stocks until after

the virgin stocks have been exhausted. Thus, there are no blocked intervals in which

the constraint (4) for production from the recycled stock binds, although there do exist

intervals in which the constraint for sorting from the waste stream (3) binds, so long

as it ever becomes feasible for sorting to occur, which happens when β + γ < p.

There are three types of equilibria which occur when the virgin stock is the least-

cost source. When γ is sufficiently low or stocks sufficiently low, equilibrium sequence

Sx → Rx occurs, where sorting occurs from the beginning, and total consumption

equals the full potential production of (R0 + S0)/(1 − δ). For higher levels of γ or

higher levels of the stocks, equilibrium sequence S → Sx → Rx occurs, where there

exists a subpath at the beginning of the extraction profile in which no sorting of the

waste stream into recyclables occurs. If so, then less than the full potential consumption

of (R0+S0)/(1−δ) occurs. And when γ is very high, or stocks very high, and if R0 > 0,

equilibrium sequence S → R → Rx occurs, where sorting from the waste stream into

the recycled stock occurs only after all of the virgin stock and some of the recycled

stock have been exhausted. Again, less than full potential use of the resource occurs.

21

These three sequences are shown to be the only possible sequences that can occur when

α < β and when γ + β < p.

Equilibrium Sequence S→ Sx→ Rx

Consider the equilibrium sequence S → Sx → Rx. When this equilibrium sequence

occurs, there are five constants to be solved for, the three ends of the subpaths, TS ,

TSx, and TRx, and the present values of the virgin and recycled stocks, λ0 and µ0.

Over the first interval, the equilibrium follows subpath S where only the virgin

stock is exploited at rate qt = u′−1(pt), with pt given by (5a), and no sorting of

waste into recycled stocks occurs. Therefore, total consumption is less than potential

consumption, (S0 +R0)/(1− δ).The conditions which must hold at time TS are

erTSµ0 = γ, (28a)

and

pTS= α+ λ0e

rTS = α+ γδ + (λ0 − δµ0)erTS . (28b)

Equation (28a) is necessary for accumulation of recycled stocks to commence at time

TS . Equation (28b) ensures that no arbitrage opportunities remain for those holding

the virgin stock, given that in interval [TS , TSx), the virgin stock has additional value

δ(µ0ert−γ), which is the equilibrium payment made by those producers who are accu-

mulating recycled stocks. The condition (28b), however, is redundant to the condition

(28a).

Over the interval [TS , TSx), the economy follows subpath Sx, with only the virgin

stock exploited at rate qt = u′−1(pt), with pt given by (9a), but with maximum accu-

mulation of recyclable stocks from the waste stream. This subpath continues until the

virgin stock is exhausted at some time TSx, at which point production switches to the

recycled stocks. Thus, at time TSx,∫ TS

0u′−1(α+ λ0e

rt)dt+∫ TSx

TS

u′−1[α+ γδ + (λ0 − δµ0)ert

]dt = S0, (29a)

and

pTSx= α+ γδ + (λ0 − δµ0)erTSx = β + γδ + (1− δ)µ0e

rTSx , (29b)

where the integrands in (29a) are the rates of production during subpaths S and Sx,

22

respectively. Equation (29a) is the exhaustion condition for the virgin stock. The

equality in (29b) holds because at time TSx, production from the recycled stocks begin,

and the price path must be continuous in order that there be no arbitrage opportunities.

Observe that (29b) implies that α+ λ0ert = β + µ0e

rt, since a common δ(γ − µ0erTSx)

can be canceled from each side. Again, this highlights the importance of the separation

of the recycled stock production decision from recycled stock accumulation decision.

Over interval [TSx, TRx], the equilibrium follows subpath Rx, with yt = u′−1(pt),

where pt follows (11a). This ends at time some time TRx, when the recycled stock is

exhausted. Thus at TRx,

R0 + δ

∫ TSx

TS

u′−1[α+ γδ + (λ0 − δµ0)ert

]dt

= (1− δ)∫ TRx

TSx

u′−1[β + γδ + (1− δ)µ0e

rt]

dt,(30a)

and

pTRx= β + γδ + (1− δ)µ0e

rTRx = p, (30b)

where the integrands in (30a) are the production during subpaths Sx and Rx, respec-

tively. The left-hand-side of (30a) is the initial recycled stock plus the proportion of

virgin stock that was recycled over interval [TS , TSx). This is equated with net reduc-

tions to the recycled stock over the interval [TSx, TRx]. While (30a) ensures exhaustion

of supply at time TRx, the (30b) ensures that economic exhaustion also occurs at time

TRx in the sense that demand is just choked off at the moment when supplies give out.

The equilibrium sequence S → Sx → Rx is depicted in Fig. 4. The thick contin-

uously rising lower-envelope of the price paths depicts the equilibrium price path, and

the thick discontinuous step-function shows which costs are relevant in each interval.

The vertical difference between the curve labeled α + λ0ert and the equilibrium price

path, pt = α+ δγ + (λ0 − δµ0)ert, in the interval [TS , TSx) is equal to δφtert, which is

the current net return to the owner of the recycled stock of being able to purchase from

the waste flow. Similarly, in the interval [TSx, TSRx), the vertical difference between

the curves β + µ0ert and the equilibrium price path pt = β + δγ + (1− δ)µ0e

rt equals

δφtert, which again is the net return an owner of the recycled stock earns per unit

waste flow. Thus, δφtert is the current price a recycled stock owner is willing to pay for

access to sorting the waste stream, and φtert is the current purchase price of recycled

waste. In interval [0, TS), this price is zero, since the cost of sorting is greater than the

future sales value of the recycled stock.

23

TS TSx TRx

Α

p

Β

Α+Γ∆

Β+Γ∆

Β+Γ

Β+Γ∆+H1-∆LΜ0ert

Β+Μ0ert

Α+Λ0ert

Α+Λ0

Β+Μ0

Α+Γ∆+HΛ0-∆Μ0Lert

t

$

Figure 4: Equilibrium Sequence S → Sx → Rx When Virgin Stock is Least-Cost and SortingCost Is Medium.

In this equilibrium sequence, recycling occurs as a speculative activity in subpath

Sx. The waste stream that is sorted into recycling stocks is not immediately used;

rather, it is stored for later use. During subpath Sx, holders of the recycled stock must

content themselves to earning only capital gains on their investment. Furthermore,

their cash-flow is negative during this interval since they must pay price φtert for each

unit of the waste stream that is sorted into recyclables. It is only in subpath Rx that

recycling turns into immediate production and positive cash-flow for the owners of the

recycled stock. In equilibrium, recycling firms earn normal rates of return, but only

over the whole sequence taken together.

Not all of the virgin stock is recycled in sequence S→ Sx→ Rx. It is economically

rational to allow some of the waste stream to be lost forever because the costs of sorting

are too high relative to the value of the sorted stock. As shown in the appendix, this

occurs either because sorting costs are very high or because there is great abundance

of the virgin and recycled stocks.

24

Equilibrium Sequence Sx→ Rx

When sorting costs are sufficiently small, equilibrium sequence Sx→ Rx occurs, where

sorting from the waste stream begins at time zero. The necessary condition to be in

this sequence is that the initial (endogenous) marginal value of an additional unit of

the recycled stock is greater than the cost of sorting: µ0 > γ.

In the sequence Sx → Rx, there are four constants to be determined, the two

endpoints of the subpaths, TSx and TRx, and the two values of the stocks, λ0 and µ0.

During the initial subpath Sx the virgin stock is utilized at rate qt = u′−1(pt), with

pt given by (9a), and sorting of the waste stream into recyclables occurs. Thus, at time

TSx, the following conditions must hold:∫ TSx

0u′−1

[α+ γδ + (λ0 − δµ0)ert

]dt = S0, (31a)

and

pTSx= α+ γδ + (λ0 − δµ0)erTSx = β + γδ + (1− δ)µ0e

rTSx , (31b)

where the integrand in (31a) is production in subpath Sx. Equation (31a) is the

exhaustion condition for the virgin stock. Equation (31b) is the no-arbitrage condition,

which ensures that the price at time TSx is continuous so that neither a virgin stock

producer who withheld a unit of production from the interval [0, TSx) would earn a

capital gain by waiting to produce at time TSx, nor a recycled stock owner who moved

a unit of production forward from the interval [TSx, TRx) to a moment before time TSxwould earn a capital gain.

Then over subpath Rx, the recycled stock is exhausted. Exhaustion must occur

because the quantity produced, yt, is greater than the quantity added to recycled

stocks, xt = δyt, so the recycled stocks decline at rate Rt = −(1 − δ)yt. Thus, the

conditions which must hold at time TRx are∫ TRx

TSx

u′−1[β + γδ + (1− δ)µ0e

rt]

dt =R0 + δS0

1− δ, (32a)

pTRx= β + γδ + (1− δ)µ0e

rTRx = p, (32b)

From (32a), total production during subpath Rx equals the total possible consumption

starting at time TSx, (R0+δS0)/(1−δ). Because stock S0 was consumed in subpath Sx,

total consumption is (S0 + R0)/(1 − δ), so the maximum total possible consumption

given the initial stocks and the feasibility of recycling occurs. From (32b), at the

25

moment of physical exhaustion, demand is also choked off. Again, these equations

have a no-arbitrage interpretation.

Unlike equilibrium sequence S→ Sx→ Rx, in equilibrium sequence Sx→ Rx the

potential stock is utilized to the maximum extent possible: none of the waste enters

landfills or is otherwise lost. But in subpath Sx, only the sorting part of recycling

occurs. The recycled stock itself is not used for production after time TSx. Thus, the

motivation for recycling is again a speculative motivation. In Fig. 4, this equilibrium

sequence occurs when the stocks are small enough or sorting costs are sufficiently small

so that time zero occurs between TS and TSx, so that subpath S does not exist.

Equilibrium Sequence S→ R→ Rx

A third possible equilibrium occurs when R0 > 0 and either γ is sufficiently high or

stocks are sufficiently high that accumulation to the recycled stock does not occur until

after the virgin stock is completely exhausted. Of course, when α < β, R0 would be

greater than zero only by mistaken actions of overly enthusiastic amateurs or due to

some misguided public policy.

In this equilibrium, the virgin stock is consumed over subpath S, which lasts from

[0, TS), and then the recycled stock is consumed in subpath R over interval [TS , TR),

followed finally by a subpath Rx over interval [TR, TRx), during which sorting from

the waste stream occurs and production is from the recycled stock, which is all that

remains.

Thus, at time TS , the virgin stock is exhausted, and production switches to the

recycled stock. Thus, at time TS ,∫ TS

0u′−1(α+ λ0e

rt)dt = S0, (33a)

pTS= α+ λ0e

rTS = β + µ0erTS . (33b)

The equality in (33a) is the exhaustion condition for the virgin stock. The equality in

(33b) holds because at time TS , production from the recycled stocks begin, and the

price path must be continuous in order that there be no arbitrage opportunities for

either virgin or recycled stock owners.

Over subpath R, which occurs in the interval [TS , TR), production occurs from the

recycled stock, but it still does not pay recycled stock owners to accumulate stock by

sorting from the waste stream. At time TR, it finally becomes profitable for recycled

26

stock owners to begin accumulating recycled stocks from the waste stream.

erTRµ0 = γ, (34a)

and

pTR= β + µ0e

rTR = β + γδ + (1− δ)µ0erTR . (34b)

Condition (34a) is what causes accumulation through sorting the recyclables from the

waste stream to finally be profitable for owners of the recycled stock. Equation (34b)

is the no-arbitrage condition, which ensures that recycled stock owners do not which

to move production from subpath R to subpath Rx or visa-versa. This equation,

however, is redundant to (34a).

In the final subpath Rx, extraction occurs from the recycled stock and all of the

waste stream that can be sorted into recycled stock is accumulated. Finally, at time

TRx physical and economic exhaustion occurs:

(1− δ)∫ TRx

TR

u′−1[β + γδ + (1− δ)µ0e

rt]

dt = R0 −∫ TR

TS

u′−1[β + µ0e

rt]

dt, (35a)

and

pTRx= β + γδ + (1− δ)µ0e

rTRx = p. (35b)

Equation (35a) implies that net extraction over the interval [TR, TRx) equals the re-

serves remaining at time TR, since those reserves are continually sorted into recycled

stocks and reused. Equation (35b) is the no-arbitrage equation which implies that

demand is choked off at the instant supply is exhausted.

An example of this equilibrium is shown in Fig. 5. The equilibrium price path is

again the lower envelope of the four possible price paths (shown as the thicker rising

curve) and the relevant opportunity cost at which the Hotelling condition varies is

depicted as the thick step-function. This Figure shows clearly why the condition that

β + γ < p is required in order that there exist an equilibrium sequence that involves

recycling, since sorting of the waste stream does not occur until the price reaches β+γ.

5.1 Summary of Equilibria when α < β

Together, Figs. 4 and 5 show the range of possible equilibrium paths for the case

where the virgin stocks are the least-cost source and when recycling eventually becomes

feasible.

27

TS TR TRx

Α

p

Β

Α+Γ∆

Β+Γ∆

Β+Γ

Β+Γ∆+H1-∆LΜ0ert

Β+Μ0ert

Α+Λ0ertΑ+Λ0

Β+Μ0

Α+Γ∆+HΛ0-∆Μ0Lert

t

$

Figure 5: Equilibrium Sequence S → R → Rx, When Virgin Stock is Least-Cost and SortingCost Is High.

When α < β, no equilibrium can occur in which the recycled stock is utilized before

the virgin stock. This is both individually rational for the owners of the respective

stocks and is socially optimal. In every sequence, the implied price received by owners

of the recycled stock by waiting to sell their stocks in the subpaths R or Rx is higher

than the price that prevails while the virgin stock is being extracted in intervals S or

Sx. Similarly, once the virgin stock is exhausted, the prevailing equilibrium price in

intervals R or Rx is lower than what owners of the virgin stock received by selling

their stock in the intervals S or Sx. Thus, no other equilibria can occur other than

these three. These are summarized in Table 2. The critical values of γ depend upon

the size of the two initial resource stocks. When R0 = 0, for high γ, note that it is not

possible to switch to recycled stock before commencing accumulating recycled stock

from the waste stream. When γ+ β > p, no accumulation of recycled stock is feasible.

While there exist exhaustible resources for which no significant recycling occurs, I

can find no evidence of an equilibrium subpath Sx occurring. This could be because

28

Table 2: Equilibrium Outcomes when the Virgin Stock is Least-Cost

Recycled Stock SizeR0 > 0 R0 = 0

Sorting Cost / Stocks Equilibrium % Gain Equilibrium % Gainγ ‘low’ / ‘high’ Stocks Sx→ Rx 100% Sx→ Rx 100%γ ‘medium’ / ‘medium’ Stocks S → Sx→ Rx < 100% S → Sx→ Rx < 100%γ ‘high’ / ‘low’ Stocks S → R→ Rx < 100% S → Sx→ Rx < 100%γ + β > p S → R 0% S 0%Notes: The column labeled ‘% Gain’ indicates the proportion of the differencebetween (S0 +R0)/(1− δ) and S0 +R0 which is gained by recycling.

there exists a potential market failure when α < β. When sorting of the recycled

stock from the waste stream occurs in an Sx interval, the owners of the recycled stock

hold an asset which is appreciating in value at a sufficient rate to yield an equivalent

rate of return to other assets in the economy, but unlike the virgin stock, which yields

both a cash flow today and capital gains tomorrow, the promise of the recycled stock

is always far into the future. Furthermore, during the subpath Sx of recycling stock

accumulation, the cash-flow of the recycled stock firm is negative since in addition

to the costs of sorting, γ, they must also pay the costs of acquiring access to the

waste stream, δφtert. Thus, accumulating recycled stocks requires steely nerves and

an excellent credit rating. Capital market imperfections could easily undo this perfect

foresight equilibria. This may explain why such markets are not observed.

6 The Effect of LandFill and Storage Costs

Let the rental cost on land be σ per unit of land-fill and τ per unit of recycled stock.

Then the purchase price on land is σ/r per unit of waste which goes into land-fill space,

and is τ/r for each unit of recycled stock that is stored. Landfill additions are qt+yt−xtand additions to recycled stocks are xt − yt at each instant in time. Thus, one effect

of landfill and recycled stock storage costs is to raise the cost of utilizing virgin stocks

from α to α + σ/r. In addition, each unit of recycled stocks produced now has costs

β + (σ− τ)/r, and each unit of waste stream sorted into recycled stocks now has costs

γ − (σ − τ)/r. Thus, the effect of landfill and storage costs is to unambiguously raise

the cost of utilizing the virgin stock, but the effect on sorting and production from

recycled stocks depends upon the relative value of σ − τ . When σ > τ , the cost of

utilizing recycled stocks rises, while the cost of sorting into recycled stocks falls. When

29

σ < τ , the cost of utilizing recycled stocks falls, while the cost of sorting the waste

stream into recycled stocks rises. Thus, all of the analysis above holds, with virgin

stock production costs of α + σ/r, recycled stock production costs of β + (σ − τ)/r,

and with sorting costs of γ − (σ − τ)/r.

7 Conclusions

This paper considers a simple partial equilibrium model of recycling using a Herfindahl

least-cost-stock-first exhaustible resource model. Recycling is separated into two dis-

tinct activities: the accumulation of the recycled stock by sorting recyclable materials

from the waste stream, and the conversion of recycled materials into final products.

When the costs of converting recycled materials into final products is greater than

the cost of converting virgin stocks into final products, the equilibrium always consumes

the virgin stock first. In this case, when sorting recyclables from the waste stream oc-

curs, it is of a purely speculative nature, since stocks are accumulated in anticipation

of exhaustion of virgin stocks. How soon it becomes profitable for speculators to begin

accumulating recyclable stocks depends upon the costs of sorting recyclable materials

from the waste stream relative to the scarcity of the virgin stock. If the virgin stock

is relatively scarce or if sorting costs are low, speculators commence building recycled

stocks earlier. For quantities of the virgin stock sufficiently low, it may be that sort-

ing of recycled stocks occurs immediately. In any case, when the virgin stock is the

least-cost source, sorting of recyclables from the waste stream is a purely speculative

activity, and one which is fraught with difficulties, since negative current cash flows

are compensated only by capital gains based on future expectations of exhaustion of

virgin stocks. Perhaps this is why such equilibria have not been observed.

When the costs of converting the recycled stock into final goods is less than the cost

of mining virgin stocks, it always pays to use recycled stocks before virgin stocks, if such

stocks exist. Whenever costs of converting recyclables into final goods is less than the

cost of mining virgin stocks, there exists a final interval in which both recycled stocks

and virgin stocks are used simultaneously. This occurs because the accumulation of

recycled stocks is bounded by the rate of flow of the waste stream, so that as long as

100% recycling is not economically attainable, recycling cannot sustain itself without

an inflow of virgin stock. In such an equilibrium, the share of total production from

virgin and recycled stocks remains constant over time. This may explain why recycling

percentages of materials used in final goods production have remained roughly constant

30

over the last century.

Finally, I showed that when the recycled stock is the least-cost and sorting costs are

high two paradoxical equilibria may arise. The higher cost virgin stock may be utilized

because recycled stocks do not exist, or if recycled stocks exist, production from those

recycled stocks may occur, but no sorting of recyclables from the waste stream into

recyclables occurs.

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33

Appendix

Equilibrium Sequence S→ SRx Comparative Statics

In this equilibrium, α > β, but R0 = 0. The total differential of the system of equations

(20) and (21) with the endogenous variables TS , TSRx, µ0, and λ0 and exogenous

variables S0 and γ, is

rµ0e

rTS 0 erTS 0

rλ0erTS 0 0 erTS

0 r(1− δ)λ0erTSRx 0 (1− δ)erTSRx

δQTS0 0 (1− δ)2

∫ TSRx

TSertD′dt+

∫ TS

0 ertD′dt

dTS

dTSRx

dµ0

dλ0

=

1

1

−δ−(1− δ)δ

∫ TSRx

TSD′dt

dγ +

0

0

0

1

dS0,

where D′ < 0 is the derivative of output with respect to price, and QTSis production

at time TS . The determinant of the Jacobian matrix is

|J | = erTS

[r(1− δ)λ0e

rTSRx

((1− δ)2

∫ TSRx

TS

ertD′dt

+∫ TS

0ertD′dt

)− erTSrδ(1− δ)QTS

λ0erTSRx

]< 0.

Thus, by Cramer’s rule,

dTSdS0

= −|J |−1e2rTSr(1− δ)λ0erTSRx > 0,

and

dTSdγ

= |J |−1erTSr(1− δ)λ0erTSRx

((1− δ)2

∫ TSRx

TS

ertD′dt

+∫ TS

0ertD′dt+ erTS (1− δ)δ

∫ TSRx

TS

D′dt)> 0.

34

Equilibrium Sequence R→ Rx→ SRx Comparative Statics

In this equilibrium, α > β, but R0 > 0. The total differential of the system of

equations (25), (26) and (27) in the endogenous variables TR, TRx, TSRx, µ0, and

λ0, and exogenous variables S0, R0, and γ, is

rγ 0 0erTR 0

0 r(α− β) 0 erTRx −erTRx

δQTR(1− δ)QTRx

0 A 0

0 0 −r(1− δ)λ0erTSRx 0 −(1− δ)erTSRx

0 −(1− δ)QTRx0 0 (1− δ)2

∫ TSRx

TRxertD′dt

×

dTR

dTRx

dTSRx

dµ0

dλ0

=

0

0

1

0

0

dR0 +

0

0

0

0

1

dS0 +

10

0

−δ(1− δ)∫ TSRx

TRxD′dt

δ

−δ(1− δ)∫ TSRx

TRxD′dt

dγ.

whereQTRxis total production at time TRx, and A = (1−δ)2

∫ TRx

TRertD′dt+

∫ TR

0 ertD′dt.

The determinant of the Jacobian matrix is

|J | = −erTSRxr(1− δ)2λ0

×{erTRxQTRx

[rγ

(∫ TR

0ertD′dt+ (1− δ)2

∫ TRx

TR

ertD′dt)− erTRδQTR

]− (1− δ)

(∫ TSRx

TRx

ertD′dt)

×[r(α− β)

(rγ

(∫ TR

0ertD′dt+ (1− δ)2

∫ TRx

TR

ertD′dt)− erTRδQTR

)− erTRxrγ(1− δ)QTRx

]}> 0.

Thus, by Cramer’s rule,

dTRdS0

= |J |−1erTR+rTRx+rTSRxr(1− δ)2QTRxλ0 > 0,

dTRdR0

= |J |−1erTR+rTSRxr(1−δ)2(−r(α− β)(1− δ)

∫ TSRx

TRx

ertD′dt+ erTRxQTRx

)λ0 > 0,

35

and

dTRdγ

= −|J |−1erTSRxr(1−δ)2λ0

{−erTRxQTRx

[erTR(−1 + δ)δ

(∫ TRx

TR

D′dt+∫ TSRx

TRx

D′dt)

−∫ TR

0ertD′dt+ (1− δ)2

∫ TRx

TR

ertD′dt]− (1− δ)

(∫ TSRx

TRx

ertD′dt)

×[r(α− β)

(erTR(1− δ)δ

∫ TRx

TR

D′dt+∫ TR

0ertD′dt

−(1− δ)2∫ TRx

TR

ertD′dt)

+ erTRx(−1 + δ)QTRx

]}> 0.

Equilibrium Sequence S→ Sx→ Rx Comparative Statics

In this equilibrium, α < β. The total differential of the system (28), (29), and (30) in

the endogenous variables TS , TSx, TRx, µ0, and λ0, with exogenous variables S0, R0

and γ, is

rγ 0 0 erTS 0

0 r(β − γ) 0 −erTSx erTSx

0 0 −r(1− δ)µ0erTRx (1− δ)erTRx 0

0 QTSx0 −δ

∫ TSx

TSertD′dt B

−δQTSx−QTSx

0 C −δ∫ TSx

TSertD′dt

dTS

dTSx

dTRx

dµ0

dλ0

=

0

0

0

0

1

dR0 +

0

0

0

1

0

dS0 +

1

0

δ

−δ∫ TSx

TSD′dt

−(1− δ)2∫ TRx

TSxD′dt− δ2

∫ TSx

TSD′dt

dγ,

where QTSxis production at time TSx, B =

∫ TS

0 ertD′dt + δ∫ TSx

TSertD′dt, and C =

(1− δ)∫ TRx

TSxertD′dt+ δ

∫ TSx

TSertD′dt. The determinant of the Jacobian matrix is

|J | = erTRxr(1−δ)µ0

{r(α− β)

(∫ TSx

TS

ertD′dt)(

rγ(1− δ)2∫ TRx

TSx

ertD′dt− erTSδQTS

)+erTSxQTSx

[rγ(1− δ)2

(∫ TSx

TS

ertD′dt+∫ TRx

TSx

ertD′dt)− erTSδQTS

]+r(∫ TS

0ertD′dt

)×[(α− β)

(rγδ2

∫ TSx

TS

ertD′dt+ rγ(1− δ)2∫ TRx

TSx

ertD′dt− erTSδQTS

)+ erTSxγQTSx

]}< 0.

36

Thus, by Cramer’s rule,

dTSdS0

= |J |−1erTRx+rTSr(1− δ)µ0

(r(α− β)δ

∫ TSx

TS

ertD′dt+ erTSxQTSx

)> 0.

dTSdR0

= |J |−1erTRx+rTSr(1−δ)µ0

[r(β − α)

(∫ TS

0ertD′dt+

∫ TSx

TS

ertD′dt)− erTSxQTSx

]> 0.

and

dTSdγ

= |J |−1erTRxr(1− δ)µ0

{r(α− β)

[δ2(∫ TS

0ertD′dt

)∫ TSx

TS

ertD′dt

+(1− δ)2(∫ TS

0ertD′dt+

∫ TSx

TS

ertD′dt)∫ TRx

TSx

ertD′dt]

+erTSxQTSx

[∫ TS

0ertD′dt+ (1− δ)2

(∫ TSx

TS

ertD′dt+∫ TRx

TSx

ertD′dt)]

+erTS

(∫ TSx

TS

D′dt)

×[r(α− β)

((1− 2δ)δ

∫ TS

0ertD′dt+ (1− δ)δ

∫ TSx

TS

ertD′dt)

+ 2erTSx(1− δ)δQTSx

]}.

The condition δ < 1/2 is sufficient to ensure that dTSdγ is positive.

Equilibrium Sequence S→ R→ Rx Comparative Statics

In this equilibrium, α < β. The total differential of the system (33), (34), and (35)

in the endogenous variables TS , TR, TRx, µ0, and λ0, with exogenous variables S0, R0

and γ, is

r(β − α) 0 0 −erTS erTS

QTS0 0 0

∫ TS

0 ertD′dt

0 rγ 0 erTR 0

0 0 erTRxr(1− δ)µ0 erTRx(1− δ) 0

−QTSδQTR

0 (1− δ)2∫ TRx

TRertD′dt+

∫ TSx

TSertD′dt 0

dTS

dTR

dTRx

dµ0

dλ0

=

0

1

0

0

0

dS0+

0

0

0

0

1

dR0+

0

0

1

−δ−(1− δ)δ

∫ TRx

TRD′dt

dγ,

37

whereQTSis production at time TS andQTR

is production at time TR. The determinant

of the Jacobian matrix is

|J | = −erTRxr(1− δ)µ0

×{r(β − α)

(∫ TS

0ertD′dt

)[rγ

((1− δ)2

∫ TRx

TR

ertD′dt+∫ TSx

TS

ertD′dt)− erTRδQTR

]+erTSQTS

[−rγ

(∫ TS

0ertD′dt+ (1− δ)2

∫ TRx

TR

ertD′dt+∫ TSx

TS

ertD′dt)

+ erTRδQTR

]}< 0.

Thus, by Cramer’s rule,

dTSdS0

= |J |−1erTRx+rTSr(1−δ)µ0

[rγ

((1− δ)2

∫ TRx

TR

ertD′dt+∫ TSx

TS

ertD′dt)− erTRδQTR

]> 0,

dTSdR0

= −|J |−1erTRx+rTSr2γµ0(1− δ)(∫ TS

0ertD′dt

)< 0,

and

dTSdγ

= −|J |−1erTRx+rTSr(1− δ)δµ0

(∫ TS

0ertD′dt

)(−rγ(1− δ)

∫ TRx

TR

D′dt−QTR

),

which is ambiguous in sign.

38