two-period_model,_tietenberg

two-period_model,_tietenberg - [laminar b: WWMMMW‘#...

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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 TWO-PERIOD 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—to-pay 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 two-peri— 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 right-hand axis represents an allocation entirely to Period 1. A Two-Period 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 marginal-benefit 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 net-benefit curve for Period 2 from the right-hand 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 present-value 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 willingness-to-pay 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 right-hand 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 CONSTANT-MARGINAL-COST 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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