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Unformatted text preview: [laminar b: WWMMMW‘# N‘er {Lemme 66m mu, Gwen.
‘ eve) , l] "(m ’n seamen,” sex as Camilan M3 LT Qantas» @uwew A TWOPERIOD MODEL According to dynamic efficiency, the objective is to balance present and future uses
of this resource by maximizing the present value of the net beneﬁts derived ﬁ'orn the
use of those resources. This implies a particular allocation of the resource across
time. We can investigate the properties of this allocation and the inﬂuence of such
key parameters as the discount rate with the aid of a simple numerical example. We
beginwith the simplest of models—"deriving the dynamic efﬁcient allocation across
two time periods. In subsequent chapters we show how these conclusions general
ize to longer time periods and to more complicated situations. Assume that we have a ﬁxed supply of a depletahle resource to allocate between
two periods. Assume further that demand is constant in the two periods, the mar
ginal willingness—topay is given by the formula P = 8 — 0.49, and marginal cost is
constant at $2 per unit (see Figure 5.1). Non'ce that if the total supply were 30 or ' THE ALLOCATION OF AN ABUNDANT DEPLETABLE RESOURCE. (.3) Period 1 . (b) Period 2.
Price Price (dollars (dollars per unit) per unit)
8 8 MC MC
2 2
r O 5 10 15 20 0 5 10 15 20
Quantity Quantity (a) (units) (13) (units) 9 0 SUSTAINABLE DEVELOPMENT: DEFINING THE CONCEPT greater, and we were concerned only with these two periods, an efﬁcient allocation
would produce 15 units in each period, regardless of the discount rate. The supply is
sufﬁcient to cover the demand in both periods; the production in Period 1 does not
reduce the production in Period 2. In this case the static efﬁciency criterion is suf
ﬁcient, since time is not an important part of the problem. Examine, however, what happens when the available supply is less than 30. Sup
pose it is equal to 20. How do we determine the efﬁcient allocation? According to
the dynamic efﬁciency criterion, the efﬁcient allocation is the one that maximizes the
present value of the net beneﬁt. The present value of the net beneﬁt for both years
is simply the sum of the present values in each of the two years. To take a concrete
example, consider the present value of a particular allocation: 15 units in the ﬁrst peri
od and S in the second. How would we compute the present value of that allocation? The present value in the ﬁrst period would be that portion of the geometric area
under the demand curve which is over the supply curve—$45 .00.1 The present value
in the second period is that portion of the area under the demand cure which is over
the supply curve from the origin to the 5 units produced multiplied by 1/0 + r). If
we use it = 0.10, then the present value of the net beneﬁt received in the second peri
od is $22.73,2 and the present value of the net beneﬁts for the two years is $67.73. We now know how to ﬁnd the present value of net beneﬁts for any allocation.
How does one ﬁnd the allocation that maximizes present value? One way, with the
aid of a computer, is to try all possible combinations of q, and q2, which sum to 20.
The one yielding the maximum present value of net beneﬁts can then be selected.
That is tedious and, for those who have the requisite mathematics, unnecessary. The dynamically efﬁcient allocation of this resource has to satisfy the condition
that the present value of the marginal net beneﬁt from the last unit in Period 1 equals
the present value of the marginal net beneﬁt in Period 2 (see appendix at the end of
this chapter). Even without mathematics, this principle is easy to understand, as can
be demonstrated with the use of a simple graphical representation of the twoperi—
od allocation problem.3 Figure 5.2 depicts the present value of the marginal net beneﬁt for each of the
two periods. The net beneﬁt curve for Period 1 is to be read from left to right. The
net beneﬁt curve intersects the vertical axis at $6; demand would be zero at $8 and
the marginal cost is $2, so the difference (marginal net beneﬁt) is $6. The margin
al net beneﬁt for the ﬁrst period goes to zero at 15 units because, at that quantity,
the willingness to pay for that unit exactly equals its cost. The only tricky aspect of drawing the graph involves constructing the curve for
the present value of net beneﬁts in Period 2. Two aspects are worth noting. First,
the zero axis for the Period 2 net beneﬁts is on the right, rather than the left, side.
Therefore, increases in Period 2 are recorded from right to left. This way, any point
along the horizontal axis yields a total of 20 units allocated between the two peri—
ods. Any point on that axis picks a unique allocation between the two periods.‘i 1The height of the triangle is $6 [$8—$2] and the base is 15 units. The area is therefore (1/ Z)($6)(15) = $45.
2The undiscounted net beneﬁt is $25.00 (Why?) The discounted net beneﬁt is therefore 25/1.10 = 22.73.
3This type of analysis ﬁrst appeared in McInerney (1976). 4Note that the sum of the two allocations in Figure 5.2 is always 20. The left—hand axis represents an allo—
cation of all 20 units to Period 2, and the righthand axis represents an allocation entirely to Period 1. A TwoPeriod Model 9i THE DYNAMICALLY EFFICIENT ALLOCATION. Marginal Net Marginal Net
Benefits in Benefits in
Period 1 Period 2 d
pgflljgirts) Present Value of Margins! Net ganja?)
3 Benefits in Period 1 6p
5.45
5 . 5
Present Value of Marginal Net
Benefits in Period 2
4 4
3 3
2 2 Quantityin,
perm“ 01 2 3 4 5 e 7 e 91011121314151617181920Quantity,”
20191817161514131211109 8 7 6 5 4 3 21 O Period2 Second, the present value of the marginal beneﬁt curve for Period 2 intersects
the vertical axis at a different point than does the comparable curve in Period 1.
(Why?) This intersection is lower because the marginal beneﬁts in the second peri—
od are discounted. Thus with the 10% discount rate we are using, the marginal not
beneﬁt is $6 and the present value is 556/ 1.10 = $5.45. Notice that larger discount
rates rotate the Period 2 marginalbeneﬁt curve around the point of zero net bene~
ﬁt (q, = 5, q, = 15) toward the right—hand axis. We shall use this fact in a moment. The efﬁcient allocation is now readily identiﬁable as the point where the two
curves representing present value of marginal net beneﬁts cross. The total present
value of net beneﬁts is then the area under the marginal net—beneﬁt curve for Peri—
od 1 up to the efﬁcient allocation, plus the area under the present value of marginal
netbeneﬁt curve for Period 2 from the righthand axis up to its efﬁcient allocation.
Because we have an efﬁcient allocation, the sum of these two areas is maximized} Since we have developed our efﬁciency criteria independent of an institutional
context, these criteria are equally appropriate for evaluating resource allocations gen
erated by markets, government rationing, or even the whims of a dictator. While
any efﬁcient allocation method must take scarcity into account, the details of pre
cisely how that is done depends on the context. Intemporal scarcity imposes an opportunity cost that we henceforth refer to as
the marginal user cost. When resources are scarce, greater current use diminishes 5Demonstrate by ﬁrst allocating slightly more to Period 2 (and therefore less to Period 1) and showing
that the total area decreases. Conclude by allocating slightly less to Period 2 and showing that, in this
case as well, total area declines. ma ' 92 SUSTAINABLE DEVELOPMENT: DEFINING THE CONCEPT future opportunities. The marginal user cost is the present value of these forgone
opportunities at the margin. To be more speciﬁc, uses of those resources which would
have been appropriate in the absence of scarcity may no longer be appropriate once
scarcity is present. Using large quantities of water to keep lawns lush and green may
be wholly appropriate for an area with sufﬁciently large replenishable water supplies,
but quite inappropriate when it denies drinking water to future generations. Failure to
take the higher scarcity value of water into account in the present will lead to an inef—
ﬁciency or an additional cost to society due to the additional scarcity imposed on the
future. This additional marginal value that scarcity creates is the marginal user cost. We can illustrate how this concept is used by returning to our numerical exam
ple. With 30 or more units, each period would be allocated 15 and the resource would
not be scarce. With 30 or more units, therefore, the marginal user cost would be zero. With 20 units, however, scarcity does exist. No longer can 15 units be allocated
to each period; each period will have to be allocated less than would be the case with
out scarcity. The marginal user cost for this case is not zero. As can be seen from Fig—
ure 5 .2, the present value of the marginal user cost, the additional value created by
scarcity, is graphically represented by the vertical disrance between the quantity axis
and the intersection of the two presentvalue curves. It is identical to the present value
of the marginal net beneﬁt in each of the periods. This value can either be read off
the graph or determined more precisely from the chapter appendix to be $1.905. We can make this concept even more concrete by considering its use in a mar
ket context. An efﬁcient market would have to consider not only the marginal cost
of extraction for this resource, but the marginal user cost as well. Whereas in the
absence of scarcity, the price would equal the marginal cost of extraction; with scarci
ty, the price would equal the sum of marginal extraction cost and marginal user cost. To see this, solve for the prices that would prevail in an efﬁcient market facing
scarcity over time. Inserting the efﬁcient quantities (10.238 and 9.762, respectively)
into the willingnesstopay function (P = 8 — 0.4g) yields P, = 3.905 and P2 = 4.095.
The corresponding supply and demand diagrams are given in Figure 5.3. In an efﬁcient market the marginal user cost for each period is the difference
between the price and the marginal cost of extraction. Notice that it takes the value
$1.905 in the ﬁrst period and $2.095 in the second. In both years the present value
of the marginal user cost is $1.905. In the second year the actual marginal user cost
is $19050 + 9*). Since r = 0.10 in this example, the marginal user cost for the second
period is $2.095 .5 Thus, while the present value of marginal user cost is equal in both
periods, the actual marginal user cost rises over time. Both the size of the marginal user cost and the allocation of the resource
between the two periods is affected by the discount rate. In Figure 5 .2rbecause of
discounting, the efﬁcient allocation allocates somewhat more to Period 1 than to
Period 2. A discount rate larger than 0.10 would be incorporated in this diagram by
rotating the Period 2 curve an appropriate amount toward the righthand axis, hold—
ing the point at which it intersects the horizontal axis ﬁxed. The larger the discount
rate is, the greater the amount of rotation required. The amount allocated to the
second period would be necessarily smaller with larger discount rates. The general ‘You can verify this by taking the present value of $2 .095 and showing it to be equal to $1.905. Deﬁning lntertemporal Fairness 9 3 THE EFFICIENT MARKET ALLOCATION OF A DEPLETABLE RESOURCE:
THE CONSTANTMARGINALCOST CASE. (a) Period 1. (b) Period 2. Price Price (dollars (dollars per unit} per unit) ‘— 8" "' 8
4.095 p 2.000 MC 2.000
‘ D n D 0 10.233 20 0 9.762 0
Quantity Quantity
(a) (units) (b) (units) conclusion, which holds for all models we consider, is that higher discount rates tend
to skew resource extraction toward the present because they give the future less
weight in balancing the relative value of present and future resource use. DEFINING INTERTEMPORAL FAlRNESS While no generally accepted standards of fairness or justice exist, some have more
prominent Support than others. One such standard concerns the treatment of future
generations. What legacy should earlier generations leave to later ones? This is a
particularly difﬁcult issue because, in contrast to other groups for which we may
want to insure fair treatment, future generations cannot articulate their wishes, much ' less negotiate with current generations (“We’ll take your radioactive wastes, if you
leave us plentiful supplies of titanium”). ' One starting point for intergenerational equity is provided by philosopher John
Rawls in his monumental work/1 Theory afj’w‘tice. Rawls suggests one way to derive
general principles of justice is to place, hypothetically, eVery person in an original
position behind a “veil of ignorance.” This veil of ignorance would prevent them from
knowing their eventual position in society. Once behind this veil, people would decide
on rules to govern the society that they would, after the decision, be forced to live in. In our context this approach would suggest a hypothetical meeting of all mem
bers of present and future generations to decide on rules for allocating resources
among generations. Because these members are prevented by the veil of ignorance
from knowing the generation to which they will belong, they will not be excessive— ly conservationist (lest they turn out to be a member of an earlier generation) or
excessively exploitative (lest they become a member of a later generation). ...
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