Increasing scarcity rent A sufficient condition

Economics Letters 0165-1765/93/$06.00

43 (1993) 235-238 0 1993 Elsevier

Increasing Darrell

B.V. All rights

reserved

condition

L. Krulce*

1000 Ehell

Received Accepted

Publishers

scarcity rent

A sufficient GTS.

235 Science

Court,

Suite 100, Palo Alto,

CA 94303,

USA

26 January 1993 24 June 1993

Abstract If the elasticity of extraction monotonically over time

cost exceeds

the elasticity

of demand.

the scarcity

rent of an exhaustible

resource

increases

1. Introduction The fundamental theorem of exhaustible resource economics, due to Hotelling (1931), is that the scarcity rent of an exhaustible resource, the difference between price and extraction cost, grows exponentially over time at the rate of interest when extraction cost is constant. When extraction cost is not constant but increases with cumulative extraction, the result is not so simple. Heal (1976) argues that in the presence of a backstop resource, available in infinite quantity at a fixed but high cost, the scarcity rent falls monotonically until vanishing when the backstop resource comes into play. Hanson (1980) shows that a rising (falling) scarcity rent implies and is implied by a convex (concave) price path and that if the time-domain extraction cost function is concave, then the scarcity rent falls monotonically. Building on this work, Chakravorty and Roumasset (1990) provide conditions, imposed on the time-domain extraction cost function, under which the scarcity rent rises and then falls. Similarly, Farzin (1992) shows that in general, depending on the time-domain extraction cost function, the time path of scarcity rent may be non-monotonic. But since the time-domain extraction cost function is a result of the optimal program, not exogenous to the model, these are not compelling conditions. The following introduces a concept of extraction cost elasticity. The larger the extraction cost elasticity, the faster extraction cost rises with cumulative extraction. It is then shown that an extraction cost elasticity greater than demand elasticity is sufficient to produce a monotonically increasing scarcity rent. The scarcity rent increases because the rapidly rising extraction cost, compared with a more slowly falling demand, causes the remaining resource to be more valuable. This suggests that resource owners who face low demand elasticities or who have a rapidly rising extraction cost will see a secular rise in scarcity rent over time. * This work

was supported

by NSF grant

SES-9122370

and the East-West

Center.

Energy

Program

236

D.L.

2. Optimal

Krulce

I Economics

Letters 43 (1993) 235-238

resource extraction

Consider a simple model of exhaustible resource extraction in the absence of technological change. Let resource demand as a differentiable function of price be D(p) 2 0, with D’(p) < 0, and resource extraction cost as a twice differentiable function of cumulative extraction be c( 4) > 0, with c’( 4) 2 0. The hypothetical planner chooses how to extract the resource over time to maximize the net present benefit, that is consumer benefit less producer cost. Given a discount rate r, this can be posed as the following optimal control problem: Choose

d, to maximize

subject

to d, 2 0 and 4, = d, with q,, given

Lm eWrr[~~‘D-‘(~)

In the above, D -’ is the inverse demand function, and qr is the cumulative extraction of the resource

du - d,c(q,)]

dt

d, is the extraction of the resource over time, up to time t. The current-value Hamiltonian is

d, H=

I0

D-‘(x)

where A, 2 0. ’ By defining an interior solution are ii,

-r/i, =g=

du - d,c(q,)

- 0,

,

price to be p, = D -‘(d,)

I

-D(p,)c'(q,)

so that d, = D(p,),

the necessary

conditions

,

(2)

$=p,-c(q,)-A,=O, I 4, = +=

t

for

D(P,) >

and lim e-“A, = 0 . I-r

(4)

3. Rising scarcity rent Rearranging (2) yields A, = p, - c( q,) and so A, is the scarcity rent, price less extraction cost. Equation (1) governs how the scarcity rent evolves over time. In general, the change in scarcity rent is complex and depends on the nature of extraction cost as well as the nature of demand. One measure of the nature of demand is the demand elasticity. The following introduces a similar concept for extraction cost. Dejinition. The elasticity of extraction cost is c”c/c’~. The

faster

the

extraction

cost

function

rises,

the

larger

the

elasticity.

For

’ By negating the state equation, the multiplier A, is positive and can be interpreted more naturally rather than the shadow cost. This changes some signs in the derivation of the necessary conditions.

example,

the

as the scarcity

rent

D.L.

Krulce

I Economics

Letters

43 (1993) 235-238

237

elasticities of the extraction cost functions c(q) = q, c(q) = q2, and c(q) = eq, are 0, l/2, and 1. Note that a positive (negative) extraction cost elasticity implies an extraction cost function that is convex (concave). To interpret the elasticity of extraction cost, consider the inverse of the extraction cost function, the cumulative extraction required for a level of extraction cost, Q = c-l. Twice differentiating the identity Q(c(q)) =q yields Q’ = l/c’ and Q”= -c”Icf3. Q’ is the marginal change in the cumulative extraction resulting from an increase in cost. The elasticity of Q ‘, the percent change in marginal cumulative extraction for a 1% change in extraction cost, is dQ’

c

dcF=Qft-&=

“I;,

_&_ C

which is the negative

C

of the extraction

cost elasticity.

Proposition. If the elasticity of extraction cost is greater than the elasticity of demand, scarcity rent rises monotonically over time. Proof. To everywhere.

show that the scarcity rent rises monotonically, First consider the behavior of A, as t-m. From

A, 2 0

e m”h,- r e-“A, = - + hrir A, +lim A, , =lim -rep” ,-a ,-@=

emr’A

!iiI A, =lim ;;f t-r e and so lim,,,

it suffices to show that (4) and L’Hospital’s rule,

then the

h, = 0. The time derivatives

A, = ri, - W,)c”(qJci,

of (1) and (2) are

(5)

- D’(P,)li,c’(q,)

and

li, = c’(s,kr + 4 Substituting

.

(6)

(3) and (6) into (5) yields:

A,= r& - (W,))2c”(q,)- ~‘(P,P(P,W(q,))2- D’(P,)c’(q,)A. NOW consider

the behavior

of A, as A,+ 0. From

hmO A, = -(wP,))2c”(q,)


3

(7)

(7),

- D’(P,)wP,)(c’(q,))2

(8)

where the third line follows from A, e 0, which implies that pI 2 c( qr), and the last line follows from the elasticity of extraction cost being greater than the elasticity of demand, -D’(p)plD(p). Inequality (8) says that whenever ii, approaches zero, A, is strictly decreasing and thus approaches from the positive side. Thus if ii, becomes negative, ii, cannot approach zero at a later

238

D.L. Krulce I Economics

Letters 43 (1993) 235-238

li, = 0, then h, can never be negative and so A, rises time. Since this contradicts that lim,,, monotonically. Q.E.D. To calculate the elasticity of a constant extraction cost function, the Hotelling case, consider the extraction cost function c(q) = cosnl(sn - q”), where s is the resource stock and 0 9 q 1s:s. As II + to, c(q) becomes a step function with c(q) = co for q
~“~cMl)=2+n-~ ~~

(c’(d)*

n

s”-q” i

4”

1.

As y1+ m, this elasticity rises to infinity for all q
References Chakravorty, U. and J. Roumasset, 1990, Competitive oil prices and scarcity rents when the extraction cost function is convex, Resources and Energy 12, no. 4, 311-320. Farzin, Y.H., 1992, The time path of scarcity rent in the theory of exhaustible resources, Economic Journal 102, no. 413, 813-830. Hanson, D.A., 1980, Increasing extraction costs and resource prices: Some further results, Bell Journal of Economics 11, no. 1, 335-342. Heal, G., 1976, The relationship between price and extraction cost for a resource with a backstop technology, Bell Journal of Economics 7, no. 2, 371-378. Hotelling, H., 1931, The economics of exhaustible resources, Journal of Political Economy 39, no. 2, 137-175.