Natural Resources Journal

21 Nat Resources J. 2 (Spring 1981)

Spring 1981

The Economics of Outer Space

Todd Sandler

William Schulze

Recommended Citation
Todd Sandler & William Schulze, The Economics of Outer Space, 21 Nat. Resources J. 371 (1981).
Available at: http://digitalrepository.unm.edu/nrj/vol21/iss2/10

This Article is brought to you for free and open access by the Law Journals at UNM Digital Repository. It has been accepted for inclusion in Natural
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 THE ECONOMICS OF OUTER SPACE*
TODD SANDLER and WILLIAM SCHULZE**

INTRODUCTION
With the advent of the space shuttle, a new natural resource-outer
space-will become increasingly available for exploitation. This article
surveys economic issues in the development of outer space. We ap-
proach this task by identifying the principal uses of this resource.
Thus, the article is organized to provide a brief economic look at
earth based activities (telecommunication satellites, military applica-
tions, and scientific exploration) as well as possible space based ex-
ploitation (space manufacturing, solar power satellites, and space col-
onization).
Three areas of economic analysis have special relevance to the
outer space resource. First, outer space is a common property re-
source, in many ways similar to the high seas. This characteristic im-
plies that allocation problems may be severe because of externalities
and public good problems. As an example we construct a formal eco-
nomic model for optimal allocation of telecommunication satellites
and examine the efficiency of the existing institutional structure.
Second, in many cases the technology for outer space exploitation
shows decreasing cost characteristics, consistent with natural monop-
oly resulting from scale economies. While exploitation to this point
has been principally by government monopoly, this may not be the
case in the future. As the private sector moves to develop further
space resources, regulation may become important. Thus we attempt
to identify those areas where natural monopolies might occur.
Third, proposals by the National Aeronautics and Space Adminis-
tration (NASA) and futurists for space manufacturing, power pro-
duction, and space colonization, if realized, could have a major im-
pact on world economic growth and distribution of wealth. However,
economic feasibility, no matter how attractive the arguments of pro-
ponents, remains a distant and open question for such applications.

*The authors appreciate helpful comments provided by Allen Kneese, Robert Ayres,
David Brookshire, Shelby Gerking, John Tschirhart, Elizabeth Knoer, and Kate Ranta. Full
responsibility rests with the authors.
**Professor and Associate Professor, respectively, of Economics, University of Wyoming,
Laramie 82071.
 NATURAL RESOURCES JOURNAL [Vol. 21

To demonstrate the economic uncertainty surrounding such pro-

posals we focus on an economic analysis of O'Neill's scheme for
space colonization presented in his book, The High Frontier(1977).
The development of a "space economy," as proposed by O'Neill, can
be simply analyzed using the traditional Harrod-Domar model of eco-
nomic growth. We then apply benefit-cost analysis to the scheme to
test the sensitivity of economic feasibility to O'Neill's underlying
assumptions.

SOME ECONOMIC ASPECTS OF TELECOMMUNICATION SATELLITES

In this section, the allocative efficiency of market and non-market
arrangements for satellite telecommunication networks are examined.
In particular, the analysis shows that a "club arrangement," charging
members (or users) a toll per signal sent, can efficiently determine
membership size, utilization rates for the network, and the number
of satellites in the system. Such an arrangement can allocate two im-
portant natural resources: frequency band width and orbital space.
The current structure of the International Telecommunications Sat-
ellite Organization's communication network (INTELSAT) is briefly
compared with the club scheme presented here.

INTELSAT and Alternative Allocative Structures

The 1969 completion of INTELSAT made telecommunications
the most important current application of satellite technology.1
INTELSAT links some 80 nations in an external communication net-
work carrying approximately two-thirds of all transoceanic messages.
Currently, the system consists of eight geostationary satellites posi-
tioned some 22,300 miles above the equator. At this altitude, the
satellites orbit the earth in the same time interval that the earth
rotates about its axis, and hence the satellites remain stationary over
a point on the earth's surface. This high altitude geosynchronous or
geostationary orbit means that only three satellites are required to

1. INTELSAT's technology, operation, and organization are discussed by Edelson,

Global Satellite Communications, 236 SCIENTIFIC AM. 58 (Feb. 1977); Fawcett, Outer
Space: New Perspectives, 49 INT'L AFF. 358 (1973); Fawcett, Satellite Broadcasting, 27
THE WORLD TODAY 76 (1971); J. GALLOWAY, THE POLITICS AND TECHNOLOGY
OF SATELLITE COMMUNICATIONS (1972); Galloway, Worldwide Corporations and
InternationalIntegration: The Case of INTELSAT, 24 INT'L ORGANIZATION 503 (1970);
Levy, INTELSAT: Technology, Politics, and the Transformation of a Regime, 29 INT'L
ORGANIZATION 655 (1975); Mickelson, Communications by Satellite, 48 FOREIGN AFF.
67 (1969); Miles, Transnationalism in Space: Inner and Outer, 25 INT'L ORGANIZATION
602 (1971); Miles, InternationalAdministration of Space Exploration and Exploitation, 8
MONOGRAPH SERIES IN WORLD AFFAIRS (1970); G. O'NEILL, THE HIGH FRON-
TIER: HUMAN COLONIES IN SPACE (1977).
April 1981] THE ECONOMICS OF OUTER SPACE

provide point-to-multipoint service almost everywhere on the earth

(except near the poles), because each satellite can communicate with
the microwave transmitters and receivers (earth stations) on one-third
of the earth. Four satellites are positioned over the Atlantic, while
two each are positioned over the Pacific and Indian Oceans. Since the
largest flow of messages transverses the Atlantic Ocean, this region
requires more communication satellites than elsewhere. Of the eight
satellites, four serve as spares and increase the system's reliability to
better than a 99.9 percent effectiveness rate. If a low orbiting system
were installed, 20 to 50 satellites (depending upon orbital altitude)
would be needed to cover the globe. INTELSAT satellites receive
weak radio signals in the mega-hertz frequency band from earth sta-
tion transmitters. After receiving these signals, the satellite amplifies
and retransmits them in the giga-hertz band to earth station receivers.2
Between earth stations and other ground points, signals travel via
microwave links and cables.
What type of allocative structure (e.g., perfect competition, regu-
lated monopoly) is most efficient for an external telecommunication
network such as INTELSAT? Since significant scale economies and
large initial investment outlays characterize these networks, perfect
competition is not a likely or desirable allocative arrangement for
these networks. Among other things, perfectly competitive markets
require a large number of sellers, none of whom have an appreciable
market share. Unfortunately, the existence of scale economies, which
lower unit costs, gives a cost advantage to whichever firm enters the
industry first. Moreover, this advantage can later be exploited to pro-
hibit others from joining the industry. In addition, large scale invest-
ment requirements can block entry into the industry for all but the
richest firms. Both scale economies and investment prerequisites
create a situation of naturalmonopoly wherein one or a few firms set
price and dominate the industry. If perfect competition and unre-
stricted entry are encouraged, losses will characterize the industry as
marginal cost pricing falls short of per unit cost owing to these scale
economies.
The scale economies of satellite communications are documented
by Snow and Edelson.3 For example, the investment cost per circuit

2. The high frequency range of the radio spectrum is assigned to satellites because high
frequency waves are able to penetrate the ionized layers (ie., the Van Allen Belts) and
clouds of the earth's atmosphere. Moreover, high frequency waves disperse less than lower
frequency ones, and hence, less ground interference results.
3. See Snow, Investment Cost Minimization for Communication Satellite Capacity:Re-
finement and Application of the Chenery-Manne.SrinivasanModel, 6 BELL J. ECON. 621
(1975) and Edelson, supra note 1.
 NATURAL RESOURCES JOURNAL [Vol. 21

year dropped from $32,500 to $800 as the size of INTELSAT satel-

lites increased from series I to V. 4 Increased INTELSAT utilization
therefore permitted larger satellites to reduce the cost per unit of
utilization. Similar scale economies characterize launch costs.'
Since elements of natural monopoly are present, a conceivable
allocative structure for satellite telecommunications is that of regu-
lated monopoly; however, there are also problems with this structure.
The extent of scale economies indicates that a global network would
minimize per unit costs, and this, in turn, means that the regulatory
agency must control an international monopoly. Thus, the agency
would require international authority to regulate price and output.
No regulatory agency has ever had these powers and, in practice, this
structure can be dismissed as an alternative.
If a good's benefits are excludable, and if the good can be simul-
taneously utilized by more than one individual, then the good is a
"club good." 6 INTELSAT qualifies as a club good, since access to
the network can, for the most part, be restricted by coding or scram-
bling signals and the network can be simultaneously used. As utiliza-
tion increases for club goods, the benefits per unit of utilization (e.g.,
per signal sent) diminishes due to congestion (e.g., interference or
noise). A club arrangement can optimally allocate utilization rates
based on the congestion phenomenon. This arrangement consists of
voluntary participants agreeing to pay either a membership charge or
a user fee (or toll) per unit of utilization. When properly formulated,
a club model can determine club size, tolls, and the optimal amount
of the shared good to provide.

A Club Model for Satellite Telecommunication Systems

When access to satellite telecommunication systems can be limited
to paying members, a club arrangement can efficiently allocate radio
frequencies and orbital space. Other forms of non-market structure,
for example, a supranational structure, may be more suitable when
access is nonexcludable. Throughout this section, efficiency refers to
Pareto optimality, which corresponds to a position in which no par-
ticipant can be made better off without harming at least one other
participant. Pareto-optimal solutions are derived by maximizing an

4. INTELSAT IV-A satellites currently contain 6,000 circuits, where each circuit con-
sists of two one-way telephone channels; hence, 6,000 simultaneous telephone conversations
can be conducted by each satellite.
5. See T. HEPPENHEIMER, COLONIES IN SPACE (1978).
6. A good's benefits are excludable if the owner can keep others from using the benefits.
For more discussion on club goods, see Buchanan, An Economic Theory of Cubs, 32 ECO-
NOMICA 1(1965).
April 19811] THE ECONOMICS OF OUTER SPACE

arbitrary participant's utility subject to the constancy of all other

participants' utilities. Moreover, all relevant constraints must be satis-
fied. Focusing on efficiency does not imply that distributional conse-
quences are ignored. Rather, we treat these separately below.
There are two aspects in allocating frequencies; frequencies must
be allocated between satellite and nonsatellite communication uses,
and frequency bands must be assigned to satellite users. The first
allocative problem can be conceptualized by treating the radio wave
portion of the electromagnetic spectrum as a joint good (b) obeying
z
the transformation b bJ, where bj is the width of the radio
j=l
spectrum allocated to the jth use. 7 That is, a joint good is purely
rival between uses; however, each use may display nonrival character-
istics so that many users can benefit from it simultaneously. In the
case of the radio spectrum, a frequency band allocated to ground
communication (like radio transmission) eliminates these frequencies
from being assigned to satellite transmission if interference between
uses is to be avoided. An optimal allocation between spectral uses re-
quires an equality between the marginal benefits of the frequency
band associated with each use. When marginal benefits are unequal,
frequencies should be reallocated until equality is reached, with wider
bands being assigned to those uses with larger marginal benefits. As
wider frequency bands are allocated to a particular use, marginal ben-
efits consist of the value of the reduction in noise and interference
experienced by the users of a particular radio spectrum allocation.
These benefits can be evaluated based both on the commercial value
per unit of time utilization associated with a radio spectrum assign-
ment, and on the time savings from not having to repeat signals as
clearer reception is achieved.
Once frequencies are distributed between uses, both orbital assign-
ments and frequency allocations for satellite users can be determined
with a club model.8 There are two congestion phenomena involved
with satellites: signal interference and satellite collision. Signal inter-
ference depends on the network's average utilization rate (k), which
equals the number of signals sent (:.x' summed over users) per unit
1
7. This analysis is similar to that found in Oakland, Joint Goods, 36 ECONOMICA 253
(1969).
8. Currently, the International Telecommunication Union (ITU) assigns orbits and fre-
quencies (except for the U.S.S.R. and the U.S. Military). However, we can find no theoreti-
cal justification for how these assignments are made. See Brown & Fabian, Toward Mutual
Accountability in the NonterrestrialRealms, 29 INT'L ORGANIZATION 877 (1975); Faw-
cett (1973), supra note 1; Galloway (1970 & 1972), supra note 1; and Miles (1970 & 1971),
supra note 1.
 NATURAL RESOURCES JOURNAL [Vol. 21

time divided by the capacity of the network for the relevant time in-
terval. Network capacity (X) is, in turn, dependent on both the fre-
quency band (bi) allocated to satellites and the number of satellites
(N) used, since satellites can use the same band provided the satellites
are sufficiently spaced. With current technology and frequency allo-
cations, each INTELSAT IV-A satellite can conduct up to 6,000 tele-
phone conversations, but as this capacity is approached through use,
background noise and interference increase. Hence, as utilization in-
creases (i. e., more signals are sent), interference congestion increases.
In contrast, an increase in either the number of satellites or the fre-
quency band lowers interference. The former increase allows each
satellite to carry fewer messages, and the latter allows a greater mes-
sage capacity per satellite. By making interference a function of k
[i.e., c = c(k),I both opposing influences on interference are cap-
tured.
The second form of congestion concerns orbital spacing. 1 0 Con-
gestion costs due to possible collision (s) increase as the number of
satellites (N) in a given orbital altitude increases; i.e., s = s(N). By
i
placing both the c and s functions in each user's utility function [u
(-)], the model can then be formulated to find the Pareto-efficient
solutions for the number of satellites for a given altitude above the
earth (say, 22,300 miles), the toll per signal sent, and the number of
users or members of the network.1 1
To find these solutions, any user's utility function must be maxi-
mized subject to the following factors: the constancy of the other
users' utilities; a private good consumption-distribution constraint; a
production possibility constraint; the frequency distribution con-
straint; and the requirement that no user utilizes the entire network
capacity. The model is briefly presented in the appendix. Equations
(1)-(3) are simplified representations for the provision, toll, and
membership conditions, respectively.
(Provision) EMBIR i = MCN + (1)
i i
(Toll) p
i MIC = MB (for allp) (2)

9. If an increase in N creates interference due to inadequate satellite spacing, then c =

c(k,N), and another marginal cost of increasing N must be balanced with the marginal bene-
fits.
10. In geostationary orbit, satellites must be separated by at least 100 miles to avoid
collision due to drift. See Fawcett (1971), supra note 1.
11. The model sketched here only concerns frequency-division multiple access (FDMA)
arrangements. Time-division multiple access (TDMA) require a more complex intergenera-
tional club model. See Sandier, A Theory of IntergenerationalClubs (1979) (unpublished
copy on file in NRJ office). See also Edelson, supra note 1, on the different frequency-
division arrangements.
April 198 11 THE ECONOMICS OF OUTER SPACE

(Membership) ATB p _ TICP (for all p) (3)

As the number of satellites in the network is increased, the bene-
fits consist of reduced interference due to increased network capac-
ity, while the costs relate to greater collision probability as well as
increased construction and launch expenses. The collision probability
(and associated costs) may be zero until a sufficient number of satel-
lites populate a given altitude. In order to determine the optimal
number of satellites at a given altitude, N should be increased until
i
the resulting marginal benefits of interference reduction (:.MBIR )
1
are equal to the sum of the marginal cost of launch and construction
(MCN) and the marginal collision costs (:.MCC' ) associated with the
1
increase in N [see equation (1)]. Both marginal collision costs and
marginal benefits of interference reduction are summed over all users,
since these benefits and costs affect all network participants. Sur-
prisingly, collision is a real problem requiring spacing of 100 miles or
more.
In equation (2), tolls are set equal to the sum of the marginal inter-
ference costs (.MIC i) imposed
1 upon the users as utilization increases,
and the pth user sends signals' 2 until his marginal benefits from utili-
zation (MBP) equal the marginal interference costs placed on the net-
work members. Since the sending of an additional signal causes the
same marginal interference costs, irrespective of user, the toll per
signal sent is identical for all members. Nevertheless, the total tolls
paid for satellite communications vary between users according to
their revealed intensity of utilization. Finally, the membership condi-
tion requires that a potential user should be admitted whenever the
total benefits from membership (ATBP) are greater than or equal to
the total interference costs imposed on the network by the potential
1
user's membership. 3

INTELSAT and Club Arrangements

The above discussion indicates that for a satellite telecommunica-
tions network, sharing arrangements can achieve allocative efficiency
with respect to utilization, membership, and provision whenever ap-
propriability is feasible.' " If costs and benefits can be monitored
accurately, an optimal sharing arrangement could, in practice, be
12. Signals can be measured in terms of the number of words or letters.
13. If the addition of a member leads to a cost or benefit in terms of private good pro-
duction, then an additional term must be included in the membership condition. See Sandier,
supra note 11.
14. The ability to force payment for giving benefits of a good to others is called appro-
priability.
 NATURAL RESOURCES JOURNAL [Vol. 21

realized. The sole difficulty concerns financing; scale economies

mean that subsidies are required to finance an optimal satellite pro-
vision.
The current structure of INTELSAT conforms closely to that of an
economic club with firms and governments as members. 1" Members
pay fees according to their utilization, and voting in the Board of
Governors is weighted according to members' utilization rates and in-
vestment shares. Although the other bodies of INTELSAT, such as
the Assembly of Parties, the Meeting of Signatories, and the Manager,
make policy recommendations, the Board of Governors is the deci-
sion-making body of INTELSAT. A weighted voting scheme based
upon utilization appears to promote optimality, since heavier users
will be serving more individuals (whose marginal benefits and costs
must be aggregated), and consequently, these users account for a
greater share of costs and benefits resulting from policy changes. As
orbital space and frequencies become scarcer, an allocative arrange-
ment similar to that proposed above should be initiated. Such a
scheme must develop proxies to measure interference costs and inter-
ference reduction benefits. Furthermore, if other types of satellites
share geostationary orbits (e.g., solar power satellites), then the deter-
mination of the optimal number of communication satellites must in-
clude other considerations.

Other Economic Consequences of INTELSAT

Allocation of orbital space and radio frequencies for INTELSAT
does not necessarily achieve full allocative efficiency, since other
kinds of externalities,' 6 not accounted for by the model, may occur.
For example, INTELSAT satellites pose a collision problem for other
satellites, for example, solar power satellites, sharing geostationary
orbit and vice versa. In determining the optimal number of powersats
(solar power satellites) or communication satellites, collision costs
must include collision externalities imposed upon all types of satel-
lites in a given altitude band. If the number of communication satel-
lites is, however, decided without including the collision externality
imposed on powersats, too many communication satellites will popu-
late the geostationary orbital band. A supranational structure may be

15. On INTELSAT organization, see Edelson, supra note 1; Galloway (1972), supra note
1; Levy, supra note 1; Miles (1970), supra note 1; and Riegel, Communications by Satellite:
The PoliticalBarriers, 11 Q. REV. ECON. & BUS. 23 (Winter 1971).
16. On transnational externalities, see D'Arge, Observations on the Economics of Trans-
national Environmental Externalities and Scott, Economic Aspects of TransnationalPollu-
tion, both in PROBLEMS IN TRANSFRONTIER POLLUTION (Organization for Economic
Cooperation and Development, eds. 1974).
April 1981] THE ECONOMICS OF OUTER SPACE

needed to allocate orbital space to include all the diverse interests

using an orbital altitude.' 7
Analogously, interference externalities may involve other types of
satellites, not part of INTELSAT. Unlike in-space collision externali-
ties, interference can result from satellites not in the same altitude
band, since radio waves transmitted at two different altitudes can still
interfere provided their paths cross. Hence, geostationary satellites'
transmissions may interfere with those of nongeostationary satellites.
A comprehensive club model must include these other interference
externalities. In so doing, network membership will be reduced and
tolls will increase, because additional interference costs are present at
the margin.
These other externalities raise a host of common property prob-
lems, owing to an absence of property rights assigned to space re-
sources (e.g., orbital space). Moreover, these resources have economic
value for two or more agents, who may want to exploit available
benefits. Essentially, outer space shares the same kinds of common
property difficulties now being confronted by the international com-
munity with regard to the oceans, especially beyond the 200-mile
limit. Without property rights assignments, the strongest and most
technologically advanced nations will claim these resources and will
exploit them as soon as the necessary technology is developed. Such
action can widen the income gap between poor and rich nations.
MILITARY RELATED USES OF OUTER SPACE
At present, military activities in outer space consist of surveillance,
navigation and communication. 1 8 Surveillance satellites are used to
gather information on missile deployment, and play an important
role in the verification of conditions such as those proposed in the
SALT II Treaty.' I Satellite based global positioning systems guide
cruise missiles to targets, while other satellite guidance systems are
used to navigate nuclear submarines. Furthermore, satellite commu-
nication networks are operated by NATO and the Warsaw Pact.2 0
Since all outer space military activities utilize satellites, the alloca-
tive issues previously discussed apply here as well. Although many

17. See Sandier and Cauley, The Design of SupranationalStructures: An Economic Per.
spective, 21 INT'L STUD. Q. 251 (1977).
18. See Brown, Reconnaissance from Space, 27 THE WORLD TODAY 68 (1971); Gallo-
way (1972), supra note 1; Riegel, supra note 15; and Miles (1971), supra note 1.
19. Surveillance satellites can discern clearly an object one foot across from an altitude
of 100 miles. See Aspin, The Verification of the SALT I Agreement, 240 SCIENTIFIC AM.
38 (Feb. 1979).
 NATURAL RESOURCES JOURNAL [Vol. 21

private corporations contract with the military for satellite related

parts, market transactions tend to exhibit non-competitive elements
with many government contracts going to large corporations. Addi-
tionally, military related externalities limit the efficiency of markets.
For example, in the early 1960s project West Ford, a Department of
Defense undertaking, placed 400 million small copper dipoles in a
belt around the earth.2 These dipoles posed an interference threat
to radio astronomy and other forms of transmitters. Even though
this project served the Department of Defense's communication
needs, it produced international externalities that were uncompen-
sated.
In 1967, members of the United Nations signed the outer space
treaty prohibiting all military activity in outer space, including the
moon and other celestial bodies.2 2 Whether this treaty will restrain
the superpowers from developing space deployed weapons is doubt-
ful, especially since these nations are already financing research on
particle beam satellites, which shoot high energy subatomic particles
at targets.2 3 Will the exploitation of outer space increase the stability
of deterrence or will conflict result due to space exploitation? Im-
provements in military surveillance as provided by satellites are prob-
ably augmenting stability, owing to the ability to verify arms limita-
tions agreements. Prior to these improvements, the stumbling block
to SALT treaties concerned verification, because neither side wanted
on-site inspections. 4 Monitoring improvements also increase stability
by reducing the chance that war will result from error when an oppo-
sing side falsely perceives an attack.
Unfortunately, space exploitation also heightens the possibility of
conflict because of common property problems, externalities, and
the vulnerability of space objects. As nations vie for space resources,
conflict can develop as two or more nations lay claim to the same re-
sources. Another potential avenue of conflict concerns satellite re-
lated externalities, for example, collision and falling debris. Even
with the "Treaty on the Liability for Damage Caused by Objects
Launched into Outer Space," in effect since 1971, the recent refusal
by the U.S.S.R. to compensate Canada for the cleanup of a Russian
surveillance satellite demonstrates that liability assignments are not
20. NATO INFORMATION SERVICE, NATO: FACTS AND FIGURES (1976).
21. Galloway (1972), supra note 1.
22. Treaty on Principles Governing the Activities of States in the Exploration and Use
of Outer Space, Including the Moon and other Celestial Bodies, opened for signature Jan.
27, 1967, 18 U.S.T. 2410, T.I.A.S. No. 6347, 610 U.N.T.S. 205.
23. Parmentola & Tsipis, Particle-BeamWeapons, 240 SCIENTIFIC AM. 54 (Apr. 1979).
24. Myrdal, The InternationalControl of Disarmament, 231 SCIENTIFIC AM. 21 (Oct.
1974).
April 1981] THE ECONOMICS OF OUTER SPACE

well-defined. 2 S Finally, conflict may be enhanced because "spy" sat-

ellites drifting in the international domain of outer space make for
tempting targets. With more military navigation being controlled by
satellites having no self-defensive capabilities, any side that destroys
the other's satellites may have a deciding first-strike advantage.

EXPLORATORY USES OF OUTER SPACE

Satellites are being used to discover new resource pools on earth,2 6
to explore the formation of stars and galaxies,2 7 and to predict long
and short term weather patterns. As satellites generate geophysical
information benefiting food and energy supplies, improvements in in-
come and hence income distributions may be made possible. Much
satellite produced data has strong publicness characteristics, since
this information is nonrival and can be distributed widely.2 8 Due to
these publicness elements, free-riding behavior can lead to an under-
supply of information. As an example, using an analogous situation,
consider the case of an ally taking a "free ride" by not buying de-
fense weapons, but rather relying on the arsenal of other allies. Conse-
quently, in the case of space exploration, nonmarket structures may
be required to cure inherent suboptimality problems. There have
been some cooperative ventures between the U.S. and the U.S.S.R.
concerning space lab experiments, and the Europeans have pooled
efforts in the European Space Research Organization; however, all
agreements have been very loose, and the efficacy of tighter struc-
tures should be explored. 2 9 Moreover, the discovery of space re-
sources through probes will further compound the common property
problem, and this may necessitate international structures to exploit
these resources.
Appropriability problems also inhibit market operation for infor-
mation producing satellites. Even though scrambling and coding de-
vices can exclude potential users from pirating satellite signals, once
the satellite owner sells satellite produced data, the buyer can sell (or
give) the information to others without the owner's permission.

25. See Canadians End Search for Debris of Soviet Satellite, N.Y. Times, April 2, 1978,
at 10, col. 1.
26. See Fawcett (1973), supra note 1, and Jastrow & Newell, The Space Program and
the National Interest, 50 FOREIGN AFF. 532 (1972).
27. Strom & Strom, The Evolution of Disk Galaxies, 240 SCIENTIFIC AM. 72 (Apr.
1979).
28. A good's benefits are nonrival when one person's consumption of a unit of the
goods does not detract from the consumption opportunities of other people, e.g., a sunset.
29. See Galloway (1972), supra note 1.
 NATURAL RESOURCES JOURNAL [Vol. 21

SPACE-BASED EXPLOITATION OF OUTER SPACE

In this section the focus is on the outward reaching, more specula-
tive, aspects of man's possible future exploitation of the outer space
resource. Since the objective is to define an "economics of outer
space," discussion must be limited to concrete proposals put forward
by NASA and others, eschewing speculation on man's possible colon-
ization beyond the solar system. Such proposals include, first, a plan
(originally conceived by Peter Glaser) to ring the earth with solar 30
power space satellites (SPSS) placed in geosynchronous orbit.
These would beam microwave energy to earth for conversion into
what is claimed to be an almost unlimited supply of environmentally
benign electric power. A second proposal, also considered, is develop-
ment of human colonies in space relatively near the earth and moon
in stable positions termed Lagrange points. L-5 is one such point
which has a number of favorable characteristics. 3 1 The second of
these proposals is closely related to the first: it appears the principal
hope for economic feasibility for space colonization depends on the
ability to manufacture such power stations in space more cheaply
than to manufacture them on earth and boost them to geosynchron-
ous orbit. Thus, Gerald O'Neill has proposed space colonies to manu-
facture SPSS. A key feature is the notion that once a first earth col-
ony is placed at L-5, raw materials from the moon can be used to
build both new colonies and solar power satellites-where growth of'
a "space economy" is self perpetuating based on sale of electric
power to earth. The plan of this section is to consider proposals for
space. manufacturing, solar power satellites, and space colonies, ad-
dressing the first issue of the economic feasibility of such proposals.
However, questions relating to the future economic growth of the
earth, the distribution of growth and income between developed and
third world nations, and finally the types of economic and political
structures which might evolve, become important if space coloniza-
tion proves economically feasible.
Space Manufacturing
Goods manufactured on earth cost about $2 per pound. 3 2 This
statement refers to ordinary commodities (such as turbogenerators,
30. See Hearings on Space Shuttle Payloads (Part 2) Before the Senate Committee on
Aeronauticaland Space Sciences, 93rd Cong., 1st Sess. 10-62 (Oct. 31, 1973) (statement of
Peter E. Glaser).
31. There exist a number of gravitationally balanced positions in space relative to the
earth, moon, and sun called Lagrange points. Some of these are dynamically unstable. One
stable position near earth is termed L-5.
32. For example, the Spring 1979 Sears Catalogue has electric stoves for $1.99/lb., table
saws for $1.90/lb., refrigerators for $1.78/lb., automobile batteries at $1.20/lb., tires at
$1.96/lb., and air conditioners for $3.63/lb.
April 19811 THE ECONOMICS OF OUTER SPACE

refrigerators, and automobiles) made principally from steel, alumi-

num, plastics, wood, glass, and other nonexotic materials. If such
goods are to be employed economically in space, they must first be
brought there at reasonable cost. Costs for placing one pound of pay-
load in low earth orbit, geosynchronous orbit or L-5, and on the
moon are now about $1,100, $4,000, and $8,000 per pound respec-
tively. It is argued that lift costs are about to fall an order of magni-
tude from the current levels, which are based on the Apollo project
that put man on the moon at a total cost of $40 billion. These lower
costs, about $110, $430, and $860 per pound respectively, could re-
sult from use of a proposed heavy lift vehicle (HLV) which is an
adaptation of the space shuttle engines to a simple payload carrying
"mule"-implying even lower costs than the more versatile shuttle.
Given the difficulties the space shuttle has had in meeting deadlines
and cost projections, it is important to note that the optimistic order
of magnitude reduction to $430 per pound lift cost to geosynchron-
ous orbit implies that ordinary manufactured goods would still cost
about $432 per pound if placed in the ideal spot for a solar power
space satellite. Clearly, only lift costs are relevant, and most ordinary
economic activities in the near term, if they rely on manufactured
goods produced on earth, are hopelessly infeasible. 3
Can one avoid lift costs from earth for space manufacturing? In
part, the answer may be yes. Certainly the asteroid belt, probably the
broken remains or unformed material of a "missing" planet between
Mars and Jupiter, contains every desirable raw material-mineral,
liquid, or gas. But again, transport costs to near-earth space would
probably be large. However, it turns out that the average soil of the
earth's moon appears to be an excellent industrial raw material. The
percentage breakdown of lunar soil by weight is about 42 percent
oxygen, 19 percent silicon, 14 percent iron, 6 percent aluminum, 6
percent titanium, and 4 percent magnesium. Thus, oxygen necessary
for life, useful both in industrial chemical reactions and as a propel-
lant; silicon, necessary for manufacture of glass; and the basic metals
constitute 91% of lunar soil. Why not then promote moon-based
manufacturing? Two problems make such an enterprise infeasible.
First, human work and life at low gravity (G) is very awkward. Sec-
ond, lift costs from earth to moon are significantly higher than those
From earth to L-5. Thus, current proposals focus on transporting

33. Of course, certain special activities including satellite communications and surveil-
[ance as outlined in the previous section are obviously feasible. Other industrial activities
,uch as growing large perfect crystals in zero-G or those requiring a high vacuum may thrive
inder the economics of the space shuttle and HLV. However, from the perspective of, say,
fhe U.S. economy, these are likely to remain unimportant even compared to satellite com-
nunications.
 NATURAL RESOURCES JOURNAL [Vol. 21

moon soils as raw materials to an industrial space colony located at

L-5. It is argued that (i) the costs of maintaining one gravity in the
living or working portions of a colony by rotation are very small; (ii)
zero gravity is available, if advantageous, for industrial processes; and
(iii) lift costs from earth are halved compared to those from the
moon. Of course, the argument also depends on an inexpensive
method of transporting lunar soil to L-5. O'Neill, for example, pro-
poses a mass-driver-a system of electromagnetically driven and levi-
tated buckets-to literally shoot lunar soil to L-5. 314 Thus, only a
small mining and transport facility is proposed for the moon itself.

Solar Power Space Satellites

Solar power space satellites as proposed by Glaser, O'Neill, and
others would consist of either orbiting concentrating mirrors heating
a working fluid to drive turbogenerators or, alternatively, panels of
solar cells for direct solar electric conversion.3" In either case, elec-
tric power would be converted to microwaves and beamed to an an-
tenna grid on earth for reconversion to electricity. Conversion to
microwaves and reconversion to electric power is estimated to occur
at 60 to 70 percent efficiency. Environmental problems are claimed
to be minimal. The microwave energy density would be relatively low
so occupants of aircraft passing through the power beam would sup-
posedly go unharmed. A 5000 megawatt power satellite would re-
quire less than 50 square kilometers of remotely placed receiver an-
tenna. The antenna array would allow most light to pass through, so
the land below could be used for grazing. However, it has been
pointed out that such facilities could make tempting military targets,
and environmental questions are far from resolved.
If a turbogenerator type power satellite were launched from earth,
even using exotic materials, the weight of components to be placed
in geosynchronous orbit would likely exceed 20 pounds per installed
kilowatt capacity. Thus, in the near term the cost would exceed
$13,600 per installed kilowatt (given lift costs of $430 per pound
and a transmission efficiency of .63) compared to about $1,000 per
installed kilowatt for a coal fired power plant, complete with emis-
sion controls, built on earth.' 6 Clearly, as noted above, the eco-
nomics of power satellites manufactured on earth are now hopeless.
Two alternative proposals for achieving feasibility for SPSS have
been put forward. First, O'Neill suggests that if space manufacturing
34. G. O'NEILL, supra note 1.
35. Id.
36. Costs for solar cells, needed for the photoelectric conversion alternative, as we point
out below, are at least for now prohibitive even without considering lifts costs.
April 19811 THE ECONOMICS OF OUTER SPACE

is possible, that is, manufactured goods can be produced in near-earth

space at costs and using techniques somewhat comparable to those
on earth, then the solar power satellite becomes an economic possi-
bility.3 7 Power satellites, in turn, may generate the revenue through
sale of power to earth to make space colonization and manufacturing
feasible.
Second, NASA has evaluated an earth-based photovoltaic SPSS
system which depends on development, in the long term, of a space
freighter capable of achieving low earth orbit at a cost of $15 per
pound. At the turn of the century the proposed space freighter would
use hydrogen and methane from coal gasification as fuel to lift SPSS
components and be completely reusable. Solar power satellite mod-
ules could then be assembled in low earth orbit and be self propelled
to geosynchronous orbit using 20 percent of the electrical output of
each module to power ion-drive electric propulsion rocket engines.3 8
Eighty percent of the solar cell array could then remain protected
during passage through the Van Allen belt from radiation which dam-
ages (irreversibly reduces efficiency of) solar cells. This "free" boost
to high orbit reduces the number of space freighter launches to low
orbit, necessary for construction of a 5,000 megawatt SPSS, by about
half. Unfortunately, near term costs of solar cells are optimistically
projected to be about $2,000 per peak kilowatt in 1982. Given lift
costs of $15 per pound, feasibility requires that solar cells cost about
$200 per peak kilowatt to manufacture. NASA thus projects feasibil-
ity for earth-based SPSS development if lift costs are reduced almost
one order of magnitude ($110 to $15 per pound) and if the costs of
solar cells is similarly reduced by one full order of magnitude ($2,000
to $200 per peak kilowatt) from near term levels.3 9 Economic feasi-
bility is only a theoretical possibility at this point, but NASA's
studies have at least defined the requirements for feasibility in a
straightforward way-one order of magnitude reductions in both lift
costs and the cost of manufacturing solar cells.

Colonization of Outer Space-A Space Economy

O'Neill's proposal for space colonization can be briefly summarized
as follows:" 0 the initial colony at L-5 would house 10,000 people in
a revolving sphere about 460 meters in diameter having a structural

37. G. O'NEILL, supra note 1.

38. Kraft, The Solar Power Satellite Concept-The Past Decade and the Next Decade
(paper prepared for the American Institute of Aeronautics and Astronautics, 1979).
39. Id.
40. G. O'NEILL, supra note 1.
 NATURAL RESOURCES JOURNAL [Vol, 21

weight of 150,000 tons.4 1 Construction of the first colony depends,

first, on construction of a moon base (about 10,000 tons transported
from earth) to supply raw materials, and placement of a construction
station located at L-5 weighing about 30,000 tons. O'Neill estimates
that "Island One" could be constructed in this way for $96 billion.
Output in manufactured goods from Island One when fully opera-
tional is estimated at 200,000 tons per year. O'Neill proposes that 60
percent of this output be used for construction of 5000 megawatt
capacity power satellites at 80,000 tons per unit which would pro-
duce revenues from power sales sufficient to allow the remaining 40
percent of total output to be "saved," that is, used in the construc-
tion of new colonies.
This proposal, given the numbers outlined above, implies an in-
credibly rapid rate of growth, both for space colonies and for power
satellites, approaching 35 percent per year. Furthermore, as is shown
below, economic feasibility depends heavily on the assumption that
this growth rate can be achieved and maintained for a period of
about 20 years. Clearly, this rate of growth needs careful examina-
tion. Fortunately, the growth process, which O'Neill describes as the
"bootstrap principle"-one colony constructed mostly from earth be-
comes the springboard for an almost entirely space based growth of
new colonies-corresponds precisely to a basic model of economic
growth which has come to be known as the "Harrod-Domar" model.4 2
This model can be explained with the following notation:
Y = total output
K = total capital
a = K/Y = fixed capital-output ratio of the economy
s = fraction of output "saved" and invested for growth
t = time (year, starting at 0).
For consistency with O'Neill, in modeling the space economy he de-
scribes, output Y will be measured in total tons of manufactured
goods, as will be capital K. Capital in the model consists of (1) lunar
bases (3 per colony at 10,000 tons each); (2) colony construction
stations (30,000 tons each); and (3) colonies themselves (170,000
tons including glass windows). Thus, to allow each colony to repli-
cate itself in about two years, as O'Neill proposes, one must satisfy a
requirement of about 230,000 tons of manufactured capital per col-
ony. In turn, each colony is supposed to be able to produce an out-

41. Additionally, 20,000 tons of glass for windows and 400,000 tons of "waste" slag fol
radiation shielding, soil, etc. would be utilized.
42. For a description, see H. WAN, ECONOMIC GROWTH (1971).
April 1981] THE ECONOMICS OF OUTER SPACE

put of 200,000 tons per year. Thus, in terms of the notation intro-
duced above, our space economy has a capital-output ratio of:
K =230000 =1.15.
To determine the rate of growth of the space economy, note that

the following relations hold:

Y = aK (the production relation) (4)

and
d K = k = sY (the investment rate, k, equals the (5)
savings ratio, s, times output).

Differentiating (4) with respect to time yields Y = K which com-

k
bined with (5) implies a percentage rate of growth of output for the
space economy of
_s
ya (6)
Since O'Neill proposes a savings ratio of s = .4 and the capital output
ratio is a = 1.15, the growth rate is '/Y = .4/1.15 = .35 or 35 percent
per year. No economic system has ever achieved such a spectacular
rate of growth. To explain this result, note first that the savings rate
proposed is very high-the U.S. economy has a savings ratio (including
business savings) of less than 15 percent while 40 percent is proposed
for the space economy. Second, the capital-output ratio is very low.
Most industrialized nations have a's of about 4, nowhere near the
1.15 proposed for the space economy. To check the Harrod-Domar
model, note that the rate of predicted growth for the U.S. economy
would be
'_-s = .15 .0375
Y_ W 4
or 334 percent per year. This is just slightly higher than the actual
long run average rate of growth for the U.S. of about three percent
per year for the last century. Thus, the Harrod-Domar model holds
up as a rough predictive tool for economic performance in spite of
the fact that much more sophisticated models are available.' 3
For purposes of argument, one might choose to agree with O'Neill's

43. Id.
 NATURAL RESOURCES JOURNAL [Vol. 21

choice of savings ratio, since very high savings rates have been ob-
tained historically through forced means-for example, the rapid rate
of growth of the Soviet Union between the world wars resulted from
a high forced savings rate. However, the choice of capital output
ratio, implying an enormously higher than expected productivity for
44
capital, seems untenable.
To explore the impact of varying the capital output ratio, the
model of the space economy can be completed as follows. Assuming
the first colony, "Island One," is in place, the number of colonies, N,
will grow from the base year, t = 0, as follows:

N(t) = l .e • (7)
Since power satellites produced by colonies consume net output
(that remaining after savings for growth are subtracted) the number
of power satellites P(t) changes over time as follows:
I(t) = (1 - s)Y(t)
80,000 tons (8)
given that each 5000 megawatt power satellite requires 80,000 tons
of manufactured goods. Note, that if one accepts O'Neill's assump-
tion that each colony can produce 200,000 tons of output per year,
then Y = 200,000 • N(t). This, in turn, implies, setting s = .4 and a =
1.15 in equations (7) and (8) above, that
35
N(t) = 1 e t
and

P(t) 4.3(e35t-1).
How many colonies and power satellites does O'Neill's scheme gener-
ate? Table I shows that if growth is allowed to proceed for only eight
years, earth will have 16 colonies and about 65 power satellites (Case
I). If growth is allowed to proceed for 20 years, earth would have
1,097 colonies and 4,711 power satellites (Case II)! Cases I and II
approximate O'Neill's assumptions, and the resultant growth rate and
numbers for colonies and "powersats" are roughly consistent with
his projections.
44. The capital-output ratio for heavy industry exceeds that predicted for the whole
space economy. For example, in the U.S. primary metals industry, the capital-value added (a
measure of final output) ratio exceeded 2 in 1971. Both for the stone, clay and glass sector
and for the chemical industry, this ratio was about 1.3. Of those industries relevant for the
proposed space economy, only metal fabricating fell below 1.15, with a ratio of.8. The en-
tire U.S. economy has a much higher overall ratio of about 4 because capital stocks in the
form of roads, hospitals, schools, private homes, etc. are added to those in basic industries
such as those noted here.
April 1981] THE ECONOMICS OF OUTER SPACE

TABLE 1
ALTERNATIVE GROWTH PATHS FOR A SPACE ECONOMY
Total Assumed
Generating Capital Benefit-
Years of # of # of Capacity Output Growth Cost Ratio
Growth Colonies Powersats MWe Ratio Rate to Earth

Case I 8 16 65 .33 x 106 1.15 .35 .4

Case II 20 1,097 4,711 23.5 x 106 1.15 .35 2.8
Case III 20 13 55 .27 x 106 3.0 .13 .68

To test economic feasibility of Cases I and II-short and long run

time frames, small and large scale colonization respectively-the
assumption is made that all power can be sold at 1.5€ per kilowatt
hour and that this figure reflects benefits to earth for power produc-
tion. Thus, with a discount rate r (a 10 percent rate will be used,
comparable to rates of return for public utilities) present value of
benefits are

B =f T ertV . P(t)dt (9)

where V, the value of annual electric power sales for each 5000 MW
powersat is taken to be $.46 x 10' (assuming a load factor of 0.7).
A 20-year lifetime for powersats is also used so that, for example, if
we allow an eight-year growth period, T in the integral above is 28
years and the time profile P (t) has the number of powersats declining
after 20 years to zero in 28 years. (In Case II the number of power-
sats would decline after 20 years to zero in 40 years).
Costs include the initial cost of setting up "Island One," Ko = $96
x 10', and an additional cost for outfitting each new colony from
earth of M = $5.5 x 109 , again following O'Neill. Discounted costs
are then

C= Ko +f t*e-rtM • fNdt (10)

where t* is the length of the growth period, eight or 20 years respec-

tively in Cases I or II.
Using these formulas, benefit-cost ratios are .4 and 2.8 for Cases I
and II respectively as shown in Table 1. Apparently, small scale col-
onization is infeasible. Thus, feasibility depends principally on two
aspects of O'Neill's arguments:
1. that a rapid expansion to large scale colonization can occur; and
 NATURAL RESOURCES JOURNAL [Vol. 21

2. that enormous quantities of electricity can be sold to finance

such growth.
In evaluating the second of these arguments, note from Table 1
that Case II electricity sales are equivalent to 23.5 x 106 megawatt
installed capacity. To put this in perspective, power sales are about
ten times the projected total installed capacity of the United States
in the year 2000! Can this much power be sold? Obviously, power
sales to the Third World must constitute the principle additional de-
mand. Further, approximately half the world's population must begin
to consume electricity at per capita rates approaching those for the
United States-a noble objective indeed, and it is argued that enor-
mous quantities of cheap power (note that a B/C ratio of 2.8 implies
one could sell power at less than half current rates and still break
even) may be just what the Third World needs to boost developing
nations into the industrial age.4 I Also, a view taken by some is that
this source of power may allow the earth to escape the limits to
growth-pollution and depletion of energy resources.4 6
Returning to the first point, the presumed 35 percent rate of
growth put forward for the space economy must be severely ques-
tioned. If a more plausible (at least to most economists) capital out-
put ratio of 3 is assumed, the rate of growth drops to 13 percent per
annum. This implies that each colony can only produce about 83,000
tons of manufactured output per year. Making this change in the,
model of the space economy results in Case III, shown in Table 1,
where again, as in Case II, a 20-year growth period is assumed. The
lowered growth rate results in a benefit-cost ratio of only .68, so
feasibility is not achieved.
This is not to argue that space colonization is necessarily infeasible.
Rather, the analysis suggests that the two key assumptions outlined
above-an extraordinarily low capital-output ratio and the ability to
sell vast quantities of electric power-need close scrutiny.

CONCLUSION
Both earth and space based exploitation of outer space have been
examined. For satellite communication networks, a club arrangement
can foster efficiency by determining the optimal number of satellites
at each altitude band, the composition of network members, and the
toll per signal sent. This arrangement will efficiently allocate the
scarce resources of orbital space and the radiowave portion of the

45. Vajk, The Impact of Space Colonization on World Dynamics, 9 TECHNOLOGICAL

FORECASTING & SOC. CHANGE 361 (1976).
46. Id,
April 19811 THE ECONOMICS OF OUTER SPACE

electromagnetic spectrum. The resulting supranational structure is

extremely loose and reflects the current INTELSAT arrangement,
where members decide their utilization rates and pay accordingly;
hence, little overall consultation of the members is required to man-
age the organization. In contrast, earth based military and explora-
tory pursuits and space based activities may generate significant
problems due to externalities and nonappropriability, and these diffi-
culties may require tighter structures.
The analysis of space based exploitation showed that existing pro-
jections for solar power space satellites lifted from earth show feasi-
bility only if order of magnitude reductions in lift costs and costs of
manufacturing solar cells occurs. O'Neill's scheme for space coloniza-
tion and SPSS growth assumes a high saving rate, a low capital-output
ratio, and an enormous projected production of electricity. When
more realistic capital output ratios and production projections are
used, the benefit-cost ratio becomes less than one, which raises ques-
tions about the feasibility of these endeavors. Other scenarios and
ratios must, of course, be analyzed before pessimistic conclusions are
justified; however, the results here raise important questions that call
for further investigation.

MATHEMATICAL APPENDIX
To illustrate the model, two uses for radio wave frequencies are as-
sumed. Superscripts on the c, k, X and b functions stand for the use,
with j = 1 corresponding to satellite communications and j = 2 corre-
sponding to a nonsatellite use. Satellite communications exhibit
interference [c' (.)] and collision [s' (.)] congestion externalities,
while the ground use experiences only interference [c2 (-)].
All values in the model are expressed in terms of a private good
serving as the numeraire. Y is the amount of the private good pro-
duced, and yi is the amount consumed by the ith individual. For the
communication uses, xij is the ith individual's utilization of the jth
use, and XJ is the provision (or capacity) of the jth use. The utility
function of the ith individual is depicted in (I-A), where kJ is the
average utilization rate for the jth use, and N is the number of satel-
lites.
ui = ui [xi I,xi2,yi,c 1 (k1),s(N),c 2(k2 )] (I-A)
where
k =. (b1,N) and k2 = xi2/X2(b2).
/ilX1
1 1
 NATURAL RESOURCES JOURNAL [Vol. 21

A number of constraints must be satisfied when finding Pareto-

optimality. These constraints are given by (2-A)-(6-A).

ui(.)> ii (Vi except i = 1) (2-A)

xJ 2 xi (Vi and j = 1,2) (3-A)

Y> . yi (4-A)

F[xl('), 2(9) 5 0 (5-A)

> b2 (6-A)

The constraints of (2-A) are those of Pareto-optimality requiring each

individual's utility level to remain no less than some beginning level
(Ui), except for one individual. In (3-A), the impurity constraint indi-
cates that no user's utilization can exceed the capacity of the particu-
lar network. The private good production-consumption constraint is
given by (4-A). This constraint requires private good consumption to
be less than or equal to the good's production. The remaining two
constraints depict the production transformation function and the
frequency band distribution constraint, respectively.
The Lagrangian is represented in (7-A), where the greek letters
stand for Lagrangian multipliers (or shadow prices).

L=ul (.) + 1Xi[ui(.) Ui] + . il [XI (N,bl)_xil] (7-A)

i=2
+ .i2 [X2(b2)-xi2] + Y(Y- . yi) - p F(')
1 1

+ .(
-0b).
J
The relevant first order conditions for this Kuhn-Tucker nonlinear
programming problem are shown in (8-A)-(14-A). By assuming Xi
(for all i), y, M*,and a > 0, and xJ (for all j), N, x ij (for all i and j), Y,
y, and b) (for all j)> 0, the conditions listed are the only relevant
ones. These assumptions remove the possibility of "corner solutions"
with respect to the utility constraints, the private good's constraint,
the transformation constraint, and the frequency constraint.
i
ayl -LXi au
aYi
u =
(Vi andX 1 =1)) 8A
(8-A)
April 1981] THE ECONOMICS OF OUTER SPACE

K-- = Y a = 0 (9-A)

OL -x OP
uP. _/p~x auOcJ1
xPi axPJ +kJi acJ x" = -0 (j = 1,2 and Vp)
axpi xpi ad ki X(10-A)

aL _xi aui ax1 k ad + jXi __ s il ax1

1
aN i1N X ak i as aN i aN

aF ax1 =0 (11-A)
ax1 aN
aL = _jXi aui axJ kJ acJ + ax ij_aF

ab i acJ abJ xJ akJ i bJ aXJ ab

= (j =1,2) (12-A)

/ijaL.. =iJ(XJ xii)=0 -X> 0

(Vi,j);XxiJ (13-A)

3ij > 0 (Vij) (14-A)

By assuming that no one utilizes the entire capacity of a radio

spectrum use [i.e., X j > x - 01j = 0, see (13-A) and (14-A)], the
conditions expressed in the paper can be derived. The elimination of
the Lagrangian multipliers, via substitution of (8-A) into (10-A),
yields the toll condition. Similarly, substitution of (8-A) and (9-A)
into (11 -A) produces the provision condition. Finally, a similar sub-
stitution into (1 2-A) gives the frequency band width allocation con-
dition. By defining the resulting "weighted" marginal rates of substi-
tution and marginal rate of transformation expressions as marginal
benefit and cost terms, respectively, equations (1) and (2) of the text
result.
The model must be reformulated to include both members and
nonmembers of the radiowave networks if the optimal membership
conditions are to be derived. 4 '

47. For this reformulation, see Sandier, supra note 11.