Burness, H. Stuart

Lewis, Tracy R.

Matthews, Steven A.

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Monopoly and the Rate of Extraction of Exhaustible Resources: Note
Tracy R. Lewis; Steven A. Matthews; H. Stuart Burness
The American Economic Review, Vol. 69, No. 1. (Mar., 1979), pp. 227-230.
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Monopoly and the Rate of Extraction of 

Exhaustible Resources :Note 

In a recent paper appearing in this Re-
view, Joseph Stiglitz demonstrates under a 
set of familiar conditions that  a monopoly- 
owned nonreplenishable resource will tend 
to be exhausted at  a slower rate than is 
socially optimal.' This supports earlier 
views on the subject expressed by Harold 
Hotelling and Robert Solow. Stiglitz shows 
under the natural "first approximation" as-
sumptions of stationary, iso-elastic demand 
and zero extraction costs, that  monopolistic 
and socially optimal (competitive) extrac-
tion rates are identical. If demand elasticity 
increases with time or  constant unit pro-
duction costs are  positive but possibly de- 
crease with time, he shows that  competitive 
extraction rates exceed monopolistic rates 
for a t  least an  initial period of time. 
In this note we present realistic, alterna- 
tive extensions to  the iso-elastic, zero cost 
analysis which tend to  bias monopolistic ex- 
traction rates in the opposite direction, that  
is, towards excessive resource use. The first 
modification allows for costs that  d o  not 
vary with the extraction rate. Occurring in 
the form of leasing fees, capital costs, and 
*Departments of economics, University of Arizona, 
University of Illinois-Urbana, and University of New 
Mexico, respectively. This research was performed at 
the Environmental Quality Lab, California Institute of 
Technology, in part under the auspices of Ford Foun- 
dation grant 5740-0469 and Energy Research and De- 
velopment Administration grant #EX-76-G-03-1305, 
Caltech Energy Research Program and is gratefully 
acknowledged. We wish to thank Jim Quirk and 
Vernon Smith for thoughtful comments and sugges-
tions on  an earlier version of this manuscript. Thanks 
are also due to George Borts and a referee for sugges- 
tions that led to  this final form. Of course, remaining 
deficiencies are our responsibility. 
1Similar analyses comparing monopolistic and 
socially optimal extraction rates appear in John Kay 
and James Mirrlees, Tracy Lewis, James Sweeney, 
and Milton Weinstein and Richard Zeckhauser. 
Miltsn Kamien and Nancy Schwartz compare extrac- 
tion rates in a general equilibrium setting. 
maintenance fees, these quasi-fixed costs2 
are incurred only during periods of produc- 
tion and  often constitute a substantial por- 
tion of operating expense^.^ 
The second extension involves demand 
elasticities varying with consumption in-
stead of time. In particular, we consider a 
stationary demand schedule with elasticity 
increasing in consumption. A justification 
for this assumption is that  for small quanti- 
ties demand may be inelastic if certain 
amounts of the resource are  essential in the 
production of some goods. At lower prices, 
however, the resource may be used in other 
industries for which substitute inputs exist 
as well. Consequently, the elasticity of ag- 
gregate demand may increase. For example, 
the demand for natural gas by homeowners 
with gas appliances is inelastic since sub- 
stitute fuels are difficult t o  use. Since these 
homeowners are the primary users at  high 
prices, natural gas demand is inelastic a t  
high prices (low consumption). However, a t  
lower prices marginal usage occurs at  the 
extensive margin as various manufacturers 
switch to  natural gas, and aggregate de- 
mand may consequently become more 
el as ti^.^ The result in this case is that  if costs 
are quasi fixed, or  if demand elasticity in- 
creases with consumption, then a monopolist 
depletes the resource too soon. 
In general, competitive ownership of the 
resource will also result in socially non-
optimal production when fixed. operation 
costs exist. For example, if all costs are 
quasi fixed, least cost production requires 
that  only one mine operate at  a time. Yet 
with discounting there always will be an  in- 
*Fixed costs of this variety which can be avoided by 
stopping production were categorized as "avoidable 
fixed costs" by Vernon Smith, pp. 257-59. 
'For example, see James Hendry. 
4Strong income effects may also tend to cause de-
mand elasticity to increase with consumption. 
228 THE AMERICAN 
centive for competitive mining firms to  
operate simultaneously. While several al-
ternative forms of market intervention 
might limit the number of operating mines 
to the social optimum, in general the be- 
havior of an unregulated competitive in-
dustry that has quasi-fixed costs is difficult 
to assess and beyond the scope of this note.' 
Consequently, we contrast monopolistic 
with socially optimal programs of resource 
extraction. 
Letting p(q) be the inverse demand func- 
tion for the resource, Qo be the initial re-
source supply, and qs(t) and qM(t) be the 
socially optimal and monopolistic rates of 
extraction, the respective maximization 
problems for the social maximizer6 and the 
monopolist are:' 
(1) 	 maximize 
4 st').7S 
SoTS [SoYS") -p(q)dq - F] e-"dt 
subject to SoTs qS(t)dt IQo; q ~ ( t ) ,  TS 2 0 
(2) maximize 
'?M(t)>TM 
where r is the discount rate and Ts and T, 
are the terminal extraction dates. Note that 
these terminal dates are choice variables. 
Performing the indicated maximizations, 
manipulation of the necessary conditions 
for ( I )  and (2) yields, respectively; 
S T h ~ stopic IS currently being pursued by us in a 
subsequent manuscript. 
%ubject to the usual caveats, the social maximizer is 
assumed to maximize consumer's surplus, the area be- 
neath the demand curve. 
'Since cost minimization w ~ t h  zero variable and 
positive quasi-fixed costs requires that one mine 
ECONOMIC REVIEW MARCH 1979 
where R(q) = p(q)q is the revenue function, 
which we presume is concave, and e(q) is 
the demand elasticity. We assume e > 1 to 
ensure positive monopolistic output so that  
(3), (4), and el(q) 2 0 imply8 
The necessary terminal time conditions 
can be expressed asy 
where the function f(q) is defined by 
Note that f "(9) = pl(q)e(q)- '  - p(q)el(q) 
e(q)-2 < 0 since e'(q) 2 0. The concavity of 
f'together with f (0) = 0 and (6) imply l o  
Changing variables of integration from t 
to  q in the resource constraint equations 
yields 
Consistency between (5), (8), and (9) re- 
quires that qM(0) > qs(0); i.e., the mo-
nopoly initially extracts a t  a rate no slower 
than is socially optimal. Since inequality (5) 
is strict if e'(q) > 0, and inequality (8) is 
strict if F > 0, the initial monopoly extrac- 
tion rate will be excessive in either case. 
From equation (5) the time path q,(t) 
crosses q,(t) at most once, and only from 
above. Thus the monopolist either extracts 
operate at  a time, F represents the fixed operating costs 
for one mine. 
8Lest the point of this section be made vacuously we 
hasten to assert that demand functions satisfying these 
requirements exist. In particular ~f the social welfare 
function is U(q)  = In q + 2q'i2then dU/dq = q - '  + 
q - ' i 2  = p(q).  From this one easily o b t a ~ n s  e > I and 
e'(q) > 0. Moreover R"(q)  < Oeverywhere. 
9Terminal time conditions can be o b t a ~ n e d  by 
maximizing the Lagrange expression for this problem 
with respect to Ts and TM. 
I01f F = 0, we have TM = TS = z and q M ( = )  = 
q s ( z )  = 0. This follows because e ' (q)  > 0 im-
p l ~ e s  I ~ m ~ , ~ p ( q )  = s . 
VOL. 69 NO .  1 LEWIS ET AL . :  EXHA USTIBLE RESOURCES  
Frcu~kI .  NONINTERESECTING PATHSEXTRACTION 
(Iso-ELASTICD ~ M A N D  POS IT IV~AND  
FIXEDCOSTS) 
too fast for the entire extraction period be- 
fore exhaustion (see Figure l) ,  or too fast 
initially and too slowly thereafter (see 
Figure 2). Hence, the resource remaining is 
always less than socially optimal and deple- 
tion occurs too soon." This is clear for the 
case in Figure 1, and the case in Figure 2 
follows directly from q,(O) > q,(O), 
q,(x ) = q ~ ( t ~ .=) 0 and the fact that the 
paths q,(t) and qs(t) intersect only once. 
Thus we have established the following: 
PROPOSITION 1: Suppose that demand for 
a nonrenewable resource is stationary and 
everywhere elastic, and that all variable costs 
are zero. If either (a) quasi-$xed costs are 
positive and demand elasticity is nondecreas- 
ing in consumption, or (b )  quasi-$xed costs 
are nonnegative and demand elasticity is 
strictly increasing in consumption, then a 
monopolist extracts the resource faster than 
is socially optimal in the following ways: 
(i) T, 5 Ts (with T, < Ts if F > 0), (ii) 
q,(t) > qs(t) initially (and for  all t 5 T, if 
F > 0 and e' = O), and (iii) Q,(t) < Qs(t) 
f o r t  < Ts. 
At first glance the excessive extraction 
rate of the monopolist when el(q) > 0 ap- 
l 1  For F = 0, we obtain the  general result QM(t) f 
Qs(t) as e'(q) $ 0. The analysis for e'(q) < 0 is in 
Lewls. 
pears mysterious, if only because deriva-
tives of elasticities are second-order demand 
characteristics and do not affect usual 
(static) marginal analyses. But consider a 
simple two-period world, and suppose the 
monopolist is considering the socially op- 
timal schedule (q:, q i )  that is determined by 
setting the first period price equal to the dis- 
counted second period price. Marginal 
230 	 THE AMERICAN ECONOMIC REVIEW MARCH 1979 
revenue is a fraction, 1 - e - ' ,  of price, and 
that fraction increases with demand elas-
ticity. Since q4 > q i  implies e (q$)  > e ( q i ) ,  
equality of the two discounted prices im- 
plies that discounted marginal revenue is 
greater in the first period than in the second. 
The monopolist therefore adjusts ( q i ,  q;)  by 
extracting more in period one and less in 
period two, until the two discounted mar- 
ginal revenues are equal. When the number 
of periods is variable, this same reasoning 
indicates that the monopolist depletes the 
resource too soon. 
The other polar case of positive quasi- 
fixed costs and iso-elastic demand can also 
be understood more heuristically. The in- 
stantaneous net returns of the monopolist 
and social maximizer are (I - e-I) C S ( q )  -
F and C S ( q )- F, respectively, where C S ( q )  
is consumer surplus. Both the monopolist 
and social maximizer bear the same costs F 
to  operate in each period, but the mo-
nopolist captures but a fraction of the re-
turns accruing to  the social maximizer. 
There is, thus, a greater incentive for the 
monopolist to  accelerate extraction to  re-
duce total operating costs FT, (see Fig-
ure 3). 
We have shown that in two special cases 
a monopolist depletes a natural resource 
faster than is optimal. Since Stiglitz proves 
the opposite result for other special cases, 
the net effect of all these presumably realistic 
considerations is analytically indeterminate 
and must be ascertained empirically. 
REFERENCES 

J.  	B. Hendry, "The Bituminous Coal In-
dustry," in Walter Adams, ed., Structure 
ofAmerican Industry, New York 1961. 
H. Hotelling, "The Economics of Exhaustible 
Resources," J. Polit. Econ., Apr. 1931, 
39, 137-75. 
M. I. Kamien and N. L. Schwartz, "The Opti- 
mal Resource-Capital Ratio and Market 
Structure," J. Econ. Theory, forthcoming. 
J. A. Kay and J .  A.  Mirrlees, "The Desirability 
of Natural Resource Depletion," in 
David W. Pearce and James Rose, eds., 
The Economics of Natural Resource De- 
pletion, London 1975, 140-76; 2 18-20. 
T. R. Lewis, "Monopoly Exploitation of an 
Exhaustible Resource," J. Environ. Econ. 
Manage., No.  3, 1976,3, 198-204. 
Vernon L. Smith, Investment and Production, 
Cam bridge 196 1 .  
R. M. Solow, "The Economics of Resources 
or  the Resources of Economics," Amer. 
Econ. Rev. Proc., May 1974,64, 1-14. 
J. E. Stiglitz, "Monopoly and the Rate of 
Extraction of Exhaustible Resources," 
Arner. Econ. Rev., Sept. 1976,66, 655-61. 
J. Sweeney, "Economics of Depletable Re- 
sources: Market Forces and Intertem-
poral Bias," Rev. Econ. Stud., Feb. 1977, 
44, 125-42. 
M. Weinstein and R. J .  Zeckhauser, "The Opti- 
mal Consumption of Depletable Natural 
Resources," Quart. J. Econ., Aug. 1975, 
89,371-92. 
You have printed the following article:
Monopoly and the Rate of Extraction of Exhaustible Resources: Note
Tracy R. Lewis; Steven A. Matthews; H. Stuart Burness
The American Economic Review, Vol. 69, No. 1. (Mar., 1979), pp. 227-230.
Stable URL:
http://links.jstor.org/sici?sici=0002-8282%28197903%2969%3A1%3C227%3AMATROE%3E2.0.CO%3B2-%23
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References
The Economics of Exhaustible Resources
Harold Hotelling
The Journal of Political Economy, Vol. 39, No. 2. (Apr., 1931), pp. 137-175.
Stable URL:
http://links.jstor.org/sici?sici=0022-3808%28193104%2939%3A2%3C137%3ATEOER%3E2.0.CO%3B2-G
Monopoly and the Rate of Extraction of Exhaustible Resources
Joseph E. Stiglitz
The American Economic Review, Vol. 66, No. 4. (Sep., 1976), pp. 655-661.
Stable URL:
http://links.jstor.org/sici?sici=0002-8282%28197609%2966%3A4%3C655%3AMATROE%3E2.0.CO%3B2-G
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