Key words: fixed costs, resource management, technological change. There is a growing literature on the ... nomics, Iowa State University. David Zilberman is a ... versity of California at Berkeley, and a member of the Giannini. Foundation of ...
Fixed Costs, Efficient Resource Management, and Conservation Jinhua Zhao and David Zilberman This article develops a framework to analyze the impacts on resource development of efficient management practices that account for possible resource conserving technological changes. Compared with the traditional management practice ignoring these changes, efficient management may not delay or reduce resource development, depending on the level of the changes. Under efficient management, a higher probability of resource conserving technologies may not delay or reduce resource development. These ambiguities are due to the existence of fixed and increasing marginal costs in resource development. Key words: fixed costs, resource management, technological change.
There is a growing literature on the optimal management of natural resources over time under resource availability and environmental quality constraints. In particular, in assessing development plans, it is important to incorporate possible future changes in technologies. But this type of change has been ignored in many practical decision making processes. For example, there is no mention of investigation and consideration of future new technologies in many major resource management manuals, such as the Principles and Guidelines (U.S. Water Resources Council), a major document for water project evaluation in the United States, and the 1980 Operations Manual of the World Bank. Economists have long studied the effects of “incorrectly” dealing with uncertain future changes in resource management. In particular, real option theory (Arrow and Fisher; Henry; Fisher, Krutilla, and Cicchetti; Dixit and Pindyck; Kolstad; Coggins and Ramezani) investigates the impacts of the open-loop instead of the optimal closed-loop management. An open-loop approach replaces random variables with their expected values without allowing future adjustment. The literature found that, compared with the optimal management, the Jinhua Zhao is an assistant professor in the Department of Economics, Iowa State University. David Zilberman is a professor in the Department of Agricultural and Resource Economics, University of California at Berkeley, and a member of the Giannini Foundation of Agricultural Economics. We thank two anonymous referees for their helpful comments. Spiro Stefanou, the editor, provided detailed suggestions which greatly improved the article. The usual disclaimer applies.
open-loop approach leads to more or early resource development.1 However, this comparison may not be pertinent for many real world situations since policy makers often do not consider expected values of future technologies but rather ignore them. This article asks a related but different question: what is the impact of the myopic behavior of ignoring the technological changes altogether, relative to the optimal closed-loop decision? In particular, does the optimal closed-loop decision lead to early or more resource development? Many times planners use expected future prices in project assessment but do not use predictions for future technological changes. However, it is probably the oversight of the technological changes in the past that has prompted many of the current environmental restoration efforts.2 Our model is more relevant for analyzing resource management facing technological changes, and it complements other models emphasizing option values due to 1 For an exception, Miller and Lad found that the optimal closed-loop management may not reduce resource development if earlier actions affect future learning possibilities. 2 Reisner argued that there has been overdevelopment of water resources in the western United States. The design of these projects did not take into account the introduction of new water conserving technologies (drip and sprinkler irrigation), increased environmental awareness, and water trading. As an example of a large scale restoration effort, the Central Valley Project Improvement Act of 1991 requires reallocation of water from agriculture for environmental restoration (Central Valley Project Improvement Act, Public Law 1, 02-575, Washington, D.C., 1991). In another example, Fernandez and Karp discuss the restoration effort of constructing artificial wetlands, using a real option approach.
Amer. J. Agr. Econ. 83(4) (November 2001): 942–957 Copyright 2001 American Agricultural Economics Association
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price changes. Olmstead used the real option approach to show that ignoring future uncertainty in economic and institutional conditions may wrongly affect private adoption and investment decisions. Our article is developing a complimentary approach by assessing how the neglect of future technological changes may affect public investment decisions. In general, new technologies increase the production efficiency and reduce the amount of natural resources needed to produce a certain level of output. For example, high yield seed varieties will reduce the land needed to satisfy a given level of food demand. However, higher production efficiency does not necessarily lead to lower total demand for the natural resource: If the demand for food is sufficiently elastic, lower food price induced by the new seed may cause the food demand to increase sufficiently such that more land (e.g., forest land) is converted to agricultural production. In this case, the new technology is resource demanding. We formalize this argument in the article, while recognizing that there may be other reasons for a new technology to be resource demanding. For example, the resource demands for a new technology can be influenced by other production inputs (Abler and Shortle), the heterogeneity among producers (Shah, Zilberman, and Chakravorty), or a policy distortion such as a price support for the final product. However, many new technologies are likely to be resource conserving: if they do occur, there will be a smaller demand for developed resources. For example, flood zone management typically reduces the need for a flood control dam, and new irrigation technologies and water markets can reduce the necessity of an irrigation dam. Caswell surveys the empirical evidence demonstrating the resource conserving effects on a per acre basis of modern irrigation technologies, such as sprinkler and low-volume irrigation. Khanna and Zilberman (1997) provide empirical examples on the resource conserving effects of many recent technologies in agriculture and natural resources. Intuitively, it seems that ignoring these technological changes in resource management may lead to excessive development. This article shows that this conventional wisdom may not always be true. In this article, we focus on resource conserving new technologies. We show that, compared with ignoring these changes, efficient
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management may not conserve resources, depending on the level of the new technology. In particular, it may not delay resource development and may not reduce the expected development overtime. Further, under the optimal management, a higher probability of future resource conserving technologies may lead to earlier and more expected development. These results are not caused by any market distortions, but by the existence of fixed costs and increasing marginal costs in resource development. Expecting the possibility of less future demand, more resource may be developed now so that no development happens in the future and the fixed cost is saved. Thus the optimal management may lead to early development.3 Moreover, if the incentive to save the fixed cost is strong enough, current development may be so large that it cannot be fully compensated by the expected reduction in future development. In this case, the optimal management actually leads to more expected total development. The existence of fixed and increasing marginal costs is widely observed in natural resource development and restoration. For example, building an irrigation dam incurs certain design and evaluation costs that are not related to the project size. The fixed cost is also reflected through the existence of minimum economical dam capacity. Increasing marginal cost is reasonable if there is a capacity limit beyond which overtime pay is required for workers, or if the costs of conveyance facilities and maintenance are considered. Similarly, cutting forest and reforestation require significant cost of access (such as constructing roads to the forest) that is very independent of the scale of deforestation and reforestation, and marginal operation costs are typically increasing. Transaction costs, which are parts of the fixed costs, may also be significant (Rucker and Leffler, Leffler and Rucker). Essentially the existence of fixed costs is one form of nonconvex cost functions. Phillips and Zeckhauser consider a different case of nonconvexity in restoration costs, namely destination-driven costs, and find that optimal restoration is not a well-behaved function of the injury (or development) level. Clark argues that decreasing marginal fishing cost is the driving force behind pulse fishing. 3 This observation is parallel to the standard inventory management result that the incentive to save future fixed costs of placing an order prompts one to make periodic big orders that last for a number of periods of production.
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The incentive to save costs underlies all of these models: future fixed costs in our case and increasing scale economies in the other two cases. Our study indicates that the resource conservation effects of the optimal management depend on the cost structure of the problem at hand. Optimal management may cause early and more development only when fixed costs are high.4 To facilitate discussion, we call the management practice ignoring future technological changes the “traditional approach,” implying that these changes have been ignored traditionally. This approach can be viewed as a static or myopic perspective of the world that is incorporated in a long term decision procedure. We call the efficient management the “new approach,” suggesting that resource management practices will follow the optimal dynamic algorithm. Problem Definition and Model Formulation Consider a resource planning agency (the planner) developing a natural resource which provides environmental amenity in its natural state and monetary benefits if developed. For concreteness, throughout the article we use an example of cutting forest for agricultural land to illustrate our model. Let S be the total available resource, and let K0 be the current stock of developed resource, leaving S − K0 of the resource still in the natural state. Thus, of the total forest area S (measured, say, in acres), K0 acres have been cut for agricultural land. Resource in the natural state provides environmental service, whose value is given by the function V (S − K0 ) that is twice continuously differentiable, with V > 0 and V ≤ 0. Developed resource K provides monetary benefit π(K A), where A is an index or a parameter measuring the technology level. π(K A) is twice continuously differentiable in both arguments with πK > 0 and πKK < 0. In our example, V (·) measures the value of the forest, such as the benefit of water and soil conservation, providing habitat for certain species, etc. The function π(· ·) 4 Emphasizing fixed costs is consistent with the adjustment cost literature (Abel and Eberly) and the economic explanation of irreversibility (Zhao and Zilberman).
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measures the net contribution to social welfare by agricultural production, with πK measuring the agricultural demand for land. The efficiency of a new seed variety is represented by A. We assume that πKA < 0, that is, higher A reduces the society’s demand for the developed resource, so that the new technology is resource conserving. We show in the appendix that a new technology can conserve resource under two situations. One is when the agricultural output faces low demand elasticity. In this case, the demand for food does not increase much when the new technology drives down the agricultural price. But since the new technology raises substantially the agricultural productivity, less resource is needed to satisfy the demand for food, preserving the natural resource. In the United States, technological changes have led to a faster growth of food supply over demand, which has been a major stimulant of U.S. agricultural policy (Tweeten). A task force of the Council of Agricultural Sciences and Technology argued that technological development can increase productivity on intensively managed land, thereby releasing other lands from agriculture (Waggoner). In the second case, the new technology is resource-input enhancing. For example, new technologies such as drip irrigation raise the “effective input” (i.e., water reaching the plant root) for the same level of “gross input” (e.g., water diverted from a dam). Then even if the agricultural price is constant, the demand for the gross resource input will decrease if the output elasticity of the effective input is high. Khanna and Zilberman (1999) document a variety of such technologies. In the context of our example, agricultural products typically face inelastic demand and natural resources such as land are usually the limiting factors of the production process. For generations, forest lands in developed countries have been cleared and converted to agricultural production. Since the mid-twentieth century, partly due to new agricultural technologies, forestlands have not diminished and, in fact, have grown (Clawson). A natural resource can be developed and a developed resource can be restored. Development and restoration incur both fixed and variable costs. Following Abel and Eberly, we let c(I ), I > 0 be the variable cost of development and c(I ), I < 0 be that of restoration. Both variable costs are increasing and convex; i.e., c (I ) > 0 for I > 0, c (I ) < 0 for I < 0,
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and c (I ) > 0 for I = 0. The fixed cost for development is c0+ > 0 and that for restoration is c0− > 0. Figure 1 illustrates an example of the cost function. The net social benefit is additively separable in the monetary benefit of the developed resource, the cost of resource transformation, and the environmental value of the natural resource. Thus given the initial resource development K0 and the level of technology A, the social benefit associated with a resource transformation of scale I is (1)
B(K0 I A) = π(K0 + I A) − c(I ) + V (S − K0 − I )
The properties of functions π(·) c(·), and V (·) imply that B(·) is twice continuously differentiable in K0 and A, and in I except at 0. We consider a two period decision problem with two technologies: the traditional at AT and the modern (or new) at AM , with AM ≥ AT . In period one, only the traditional technology is available and used, and in period two (i.e., the future), the new technology is available and used with probability p. Without loss of generality, we assume that in the first period, it is always necessary to develop the resource further.5 Let β ∈ (0 1) be the social discount factor. Then the socially optimal resource transformation decision is (2)
W (p) = max w(I1 p) I1 ∈I 1
5 Similar methods of analysis can be applied to other situations in the first period such as restoration being necessary, both development and restoration being possible, and no transformation being optimal.
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= max B(K0 I1 AT )+βp max I1 ∈ I 1
I2 ∈I2 (I1 )
M
B(K0 +I1 I2 A )+β(1−p) max B(K0 +I1 I2 AT ) I2 ∈I2 (I1 )
where w(· ·) is the aggregate two period welfare, as a function of the first period development and the probability of the new technology. Function W (·) is the social welfare depending on the probability only. I1 = [0 S−K0 ] and I2 (I1 ) = [−K0 −I1 S−K0 −I1 ] describe sets of feasible actions. Under the traditional management approach, the planner ignores the possibility of new technologies and chooses the current development I1 by assuming p = 0. If the new technology AM does arrive in the future, the planner responds to the shock optimally, given the initial development I1 . We first analyze the deterministic version of our model, assuming p = 1 in (2). We show how the development patterns under both management schemes depend on the levels of AM . Resource Management under Certainty In this section, we fix the probability p at one and vary the new technology level. For this purpose, we denote the (varying) new technology level as A ≥ AT and the optimal transformation decisions in the two periods as {I1∗ (A) I2∗ (I1 A)}. Suppose that there is no fixed cost of transformation; i.e., c0+ = c0− = 0. Then the solution {I1∗ I2∗ } is given by the first order conditions of the optimization problem in
Figure 1. Cost function of resource transformation
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(2). Straightforward comparative statics on these conditions indicate
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technology, no agricultural land is reforested in period two. As the fixed cost c0+ becomes higher, the ∗ ∗ ∗ development jump occurs earlier so that dI1 (A) dI2 (I1 (A) A) (3) sign = sign AJ decreases. Since I1∗ is decreasing in A, dA dA I1∗ (AJ ) increases. The appendix shows that = sign(πKA ) < 0 if the fixed development cost c0+ is suffi+ +R That is, less forest is cut in both periods as ciently high (c0 ≥ c0 ), the upward jump ∗ may be so high that I1∗ (AJ ) > I1∗ (AT ). of I the new seed becomes more productive (i.e., 1 more forest conserving). We can also show That is, the first period development with the that I1∗ > I2∗ ; i.e., more forest is cut now than prospect of future new technology at AJ will in the future, since land developed in period be higher than that without this prospect. one can be used to grow crops in both peri- Without the new seed in the future, the planods (thus having a higher marginal benefit). ner cuts forests in both periods. With the Assume there are fixed costs of transfor- new seed (at AJ ), the planner cuts more in mation and I2∗ > 0 when A = AT . That is, period one and does not cut in period two without the new seed, the planner cuts the (to save fixed costs). Even though the new forest in both periods, even given the fixed seed reduces the overall demand for agricosts of doing so. Let us consider the deci- cultural land, more forest is actually cut in sion problem for A > AT . As A rises above period one with the new technology due to AT , both I1∗ and I2∗ decrease until A reaches the inaction in period two. a level AJ , at which I2∗ jumps to zero and The appendix also shows that there exists I1∗ jumps up.6 We can show that the total a unique technology level, AR > AJ , at development I1∗ +I2∗ jumps down (appendix). which I1∗ (AR ) = I1∗ (AT ). As the technolThat is, as the new seed becomes more pro- ogy becomes even more advanced so that ductive, forests cut in both periods initially A > AR , I1∗ falls below the first period develgo down. However, when the seed’s produc- opment without the new technology (i.e., tivity becomes sufficiently high, the demand I ∗ (A) < I ∗ (AT )). In this case, the new seed 1 1 for agricultural land becomes so low that the becomes so productive that even with the benefit of cutting for land in period two can- incentive to save future fixed costs, the plannot overcome the fixed cost. Then no for- ner actually cuts less forest in period one est is cut in period two (i.e., I2∗ jumps to since the overall demand for land is simply zero). In response, the planner cuts more too low. Figure 2 presents an example of the forest in period one (i.e., I1∗ jumps up) to optimal development pattern. provide some of the needed land in period Now we consider the traditional mantwo. Nevertheless, this additional cut cannot agement approach: the planner ignores the fully compensate the future inaction so that future new technology when deciding the the total forest cut jumps down at the tech- current development and optimally responds nology level AJ . to the new technology when it actually For A > AJ , I2∗ remains at zero and I1∗ occurs. Let {I1 I2 (I1 A)} be the vector of decreases in A. It is conceivable that as A solutions; then we know I1 = I1∗ (AT ) and becomes sufficiently high, resource restora- ∗ T tion becomes necessary (so that I2∗ < 0). I2 = I2 (I1 A). As A rises from A , I2 decreases and eventually jumps to zero. We For simplicity we rule out this scenario J . The
on the lev- denote this jump point of I2 as A by setting an upper bound A J R
= appendix shows that A < A whenever els of technology that we consider: A + R J supA {I2∗ (I1∗ (A) A) ≥ 0}.7 At this technol- A + ≥ A+R. Further, when c0 is not too high J < AJ . If ∗ ogy level, the initial forest cut I1 (A) is suf- (c0 < c0 + for some > 0), A the new seed is sufficiently productive, the ficiently low that even with the advanced planner, responding to the “new seed shock,” may not cut any forest in period two due 6 To derive AJ , we consider two development strategies: to the fixed cost. Further, this inaction may development in both periods (regardless of fixed costs) and become more likely than that under efficient development in period one only. The payoffs of both strateJ < AJ ) since the plangies are increasing in A, but that of the second strategy management (i.e., A increases faster. The unique crossing point of the two payner did not account for the new seed in offs (as functions of A) is at AJ . 7 deciding I1 . Empirically a socially optimal decision with certainty rarely J , no resource is develinvolves “building now and restoring later.” Thus this assumpAs A rises from A tion may not be unrealistic. Further, it does not affect the oped in period two (i.e., I2 = 0). But as A major results of the article.
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I *1 I *2
I *2
I *1
AT
~
AJ
AJ
~
AR
–
AR
A
A
~
I2
Figure 2. Resource development and regions of new technologies becomes sufficiently high, it may be optimal to restore the initial development: I2 jumps to a negative number. We denote this jump R . Since point of the new technology as A at A = AR I2 (I1 AR ) = I2∗ (I1 AR ) = 0, we R > AR . know A The optimal decisions under the two management approaches, (I1∗ I2∗ ) and (I1 I2 ), are illustrated in figure 2 for c0+ values such J < AJ . In this case, that AR exists and A we can define three regions of the new technology.8 The first region is A1 ≡ [AT , J ), in which I ∗ , I ∗ , and I2 are positive A 1 2 and decreasing in A. New technologies in this region represent modest improvement over the traditional technology. Since the resource is developed in both periods under both the traditional and new management approaches, fixed costs do not affect the project size. First order conditions then indicate that I1∗ + I2∗ < I1 + I2 . That is, the new management practice conserves resources.
Further, since I1∗ < I1 , the new practice also delays resource development. J AR ) = The second region is A2 ≡ [A J J A ) and A22 ≡ A21 ∪A22 , where A21 ≡ [A [AJ AR ). New technologies in this region represent moderate improvement over the traditional one. In this region, I1∗ + I2∗ > I1 .9 That is, the new management approach does not conserve resource; it leads to more overall resource development. When A ∈ A22 , I1∗ > I1 , and the new approach does not delay resource development.
= A31 ∪A32 , The third region is A3 ≡ [AR A] R R R A].
where A31 ≡ [A A ) and A32 ≡ [A Here the new technology represents radical improvement over the traditional one, and there is no development in period two under either management approach. Under the traditional management, restoration may even be needed. For A ∈ A31 , it is clear that I1∗ < I1 . We can show that for A ∈ A32 , I1∗ < I1 +I2 .
8 We can define similar regions for the case when AJ < J < AR . The conclusions are parallel and we thus ignore A this case.
9 When A ∈ A22 , I2∗ = I2 = 0 and I1∗ > I1 . Since I1∗ + I2∗ decreases in A and jumps down at AJ , we know I1∗ (A) + I ∗ (A) > I1 for A ∈ A21 .
2
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Thus the new approach conserves resources. We summarize the results in Remark 1. Under certainty, for modest and radical technological changes (A ∈ A1 ∪ A3 ), ignoring new technologies in resource management leads to early and more resource development. For moderate technological changes (A ∈ A2 ), ignoring new technologies leads to less, and possibly late, resource development. Rough intuition suggests that the new management approach, by accounting for modern conservation technologies, conserves resource. Thus the results for A ∈ A1 ∪ A3 are not surprising. The result for A ∈ A2 is surprising, and its intuition deserves further discussion. Essentially, under the traditional approach, the “new seed shock” in period two forces the planner to abandon the planned forest cutting (since the planner initially intended to cut in both periods), reducing the overall forest cut. If the planner had “planned well” (i.e., foreseeing the new seed), she will cut either less in period one so that she can still cut more forest in period two or cut more in period one to compensate for the inaction in period two. In both cases, overall more forest will be cut.
Resource Management under Uncertainty Remark 1 states that the efficient resource management practice recognizing the future new technology may lead to earlier or more resource development. We generalize this result to the stochastic world in this section. To do so, we first look at the efficient resource transformation patterns under uncertainty for different probabilities and technology levels. In particular, we fix the new technology AM at a certain level and investigate how the development pattern responds to the probability p of the new technology being available. I2M I2T } be the vector of soluLet { I1 (p) tions to the optimization problem under uncertainty (2), where I2M = I2∗ ( I1 A M ) and T ∗ T I2 = I2 ( I1 A ). From the first order condition of I1 and the implicit function theorem, we know β dB2M dB2T d I1 (p) (4) =− − dp D dI1 dI1
Amer. J. Agr. Econ.
when the derivative is well defined,10 where D < 0 is the second order coefficient of I1 I2∗ ( I1 AM ) AM ) is the I1 , B2M = B(K0 + second period social benefit when the new technology is available, and B2T = B(K0 + I1 AT ) AT ) is that when the new I1 I2∗ ( technology is not available, given optimal first period development. Without the fixed costs of transformation, we obtain the standard result: higher p reduces the first period development I1 and I (p) ≡ the expected total development T I1 (p) + p I2M + (1 − p) I2T . Further, I2T and I2M cannot both be negative: it is not optimal to develop the resource today, knowing that part of it will always be reversed tomorrow. This is summarized in proposition 1, the proof of which is in the appendix. Proposition 1. Without the fixed cost of transformation, a higher probability of the new technology both delays and reduces resource development. In the optimal development pattern, there is always further development in period two if the new technology does not occur (i.e., I2T > 0), and there may be development or restoration if the new technology does occur (i.e., I2M may be positive or negative), depending on the new technology level. Introducing the fixed costs of transformation greatly complicates the analysis and leads to a wide range of possible transformation patterns. To save space, we delegate the discussion to the appendix, where we show that we can obtain definitive results only when the new technology AM is in I are I1 and T regions A1 and A31 : both decreasing in p. Multiple possibilities arise for other levels of technologies. Here, we present a numerical example to illustrate these possibilities. In particular, we show I may I1 and T that when AM ∈ A2 , both be increasing in p, generalizing the results of the deterministic model. A Numerical Example of Development Patterns Again, we use our example of cutting forest for agricultural land. We assume quadratic transformation cost functions: c(I ) = c0 + vc I 2 for deforestation (I > 0) and c(I ) = 10 The appendix discusses exactly when the derivative is well defined.
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Table 1.
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Parameter Values
Parameter
Value
Meaning
S K0 AT
A β α q w au bu av c0 vc d0 vd
30 5 065 12 1/107 05 2 1 26 2 04 6 075 7 05
total land area initial agricultural land level of the traditional technology (seed) upper bound of the new technology (seed) the discount rate production function parameter price of agricultural output agriculture wage rate utility function parameter utility function parameter environmental amenity function parameter fixed cost of deforestation variable deforestation cost parameter fixed cost of reforestation variable reforestation cost parameter
d0 + vd I 2 for reforestation (I < 0). The environmental value of uncut forest is linear in the forest acreage S − K: V (S − K) = av (S − K). The gross benefit of consuming agricultural output y is assumed to be quadratic: Z(y) = au y − 21 bu y 2 (i.e., the agricultural demand function is linear). The agricultural production function is Cobb–Douglas with constant returns to scale and Hicksian neutral technological change using land and labor as inputs: g(K L A) = AK α L1−α . We assume perfect competition in both the output and labor markets, so that farmers take output price q and wage w as given. Further, the agriculture economy supported by land K is small relative to the rest of the country or the world, so that q and w are fixed and unaffected by changes in K. Finally, the amount of labor employed can be changed without any adjustment costs. Thus for any given land area K, farmers employ the optimal level of labor. The labor level is adjusted accordingly whenever deforestation or reforestation occurs. Given K, A, q, and w, we can show that the optimal labor input is 1/α q ∗ (1 − α) (5) L = A1/α K w Substituting L∗ into the production function g(·), we obtain (in reduced form) the output as a function of K and A: (6)
y = f(A K) q (1−α)/α (1 − α)(1−α)/α A1/α K = w
Since c(I ) already captures the social cost of providing agricultural land K, we only need to calculate the labor cost to account for the social cost of agricultural production. The labor cost is CL = wL∗ and from (5) and (6), we know CL (y) = (1 − α)qy. Thus, the net social benefit of the agriculture sector is the difference between the gross benefit and the cost of agricultural production: π(y) = Z(y) − CL (y) = (au − (1 − α)q)y − 21 bu y 2 . Table 1 presents the assumed parameter values. Under the traditional technology, there is a 35% yield loss, say due to pest, so that AT = 065. The new seeds have the potential of completely eliminating the pest losses and even increasing the yield, and we
= 12.11 assume A Figure 3 presents the solution of the deterministic model (with p = 1). It shows modest technologies A1 = [065 0836), moderate technologies A2 = [0836 0875), and radical technologies A3 = [0875 12]. We can also show that A21 = [0836 0845), A22 = [0845 0875), A31 = [0875 1136), and A32 = [1136 12]. Figure 4 presents the optimal transformation schedules as functions of probabilities for four technology levels: 070 ∈ A1 , 0843 ∈ A21 , 085 ∈ A22 , and 115 ∈ A32 . Both the first period development I1 and the I are continexpected total development T uously decreasing in probability p for the modest technology AM = 070, confirming our analytical results. They are continuous 11 For some crops in the United States such as wheat, corn, potatoes, and cotton, per acre yield has more than doubled between 1955 and 1985 (Antle and McGuckin).
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12
Development Levels
10
Total Development
8 I1 6
4 I2 2
0
0.7
0.8
0.9 1 Level of New Technology
1.1
Figure 3. Developments in the deterministic model but increasing in p for the moderate technology AM = 085 (and for all AM ∈ A22 ) and are discontinuous in p for the moderate technology AM = 0843 and for the radical technology AM = 115. We obtain several observations from the simulation results. First period deforestation I1 may be increasing in probability p for some technology levels (085 ∈ A22 and 0843 ∈ A21 ) and some probability levels. In these cases, resource development is not delayed as the probability of the new technology increases. Further, due to the I jumps, the expected total development T may not always be decreasing in p (e.g., for AM = 0843 at p = 08). In this case, I (p) for p ∈ (0 1) are not even I1 (p) and T I (0) and I1 (1) and T bounded by I1 (0) and I (1), respectively. In summary, T Remark 2. Under the new management approach, (i) a higher probability of the modern conservation technology may not necessarily delay resource development, and (ii) a higher probability of the modern conservation technology may not necessarily reduce the expected total resource development.
Comparison of the Two Management Approaches Now we compare the development patterns discussed above with those under the traditional resource management approach. Traditional management assumes p = 0 when choosing I1 , and optimally responds to the realized technology in the second period. Let {I1 I2M (AM ) I2T } be the vector of soluI1 (0), I2M = I2∗ (I1 AM ), tions. We know I1 = T ∗ T and I2 = I2 (I1 A ). Since the planner intends to cut the forest in both periods without the new technology, we know I2T > 0. However, facing the new technology shock, the optimal action I2M may be positive, negative, or zero, depending on the level of new technology. In our numerical example, I2M is positive (3.0952) for the modest technology AM = 070, negative (−20003) for the radical technology AM = 115, and zero for the moderate technologies AM = 0843 and 085. Figure 4(a) reveals that the efficient management does not necessarily delay resource development compared with the traditional approach (i.e., I1∗ may be higher than I1 ). In fact, it leads to earlier deforestation for all probability levels when AM = 085, and for p < 075 when AM = 0843. More
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7.5 Second Period Development With New Technology
4 0.85
First Period Development
7 0.70 6.5
0.843
6
5.5
5 1.15 4.5
4
0.70 3 0.843 2
1 0.85
0
1 1.15 2
3 0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Probability of New Technology
0.8
0.9
1
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0.1
(a) First Period Development Î1
0.2
0.3 0.4 0.5 0.6 0.7 Probability of New Technology
0.8
0.9
1
(b) Second Period Development With New Technology Î M 2
3.8
11
3.7
10
0.70
9
0.843
Expected Total Development
Second Period Development With Traditional Technology
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1.15
3.6
3.5
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3.3
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0
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(d) Expected Total Development Tˆ I
(c) Second Period Development With Traditional Technology Î T2
Figure 4. Optimal resource transformation schedules generally, since the first period development under the traditional approach equals I1 (0), efficient management delays development (i.e., leads to lower I1 ) if I1 (p) < 0 for all p and leads to earlier development (higher I1 ) if I1 (p) > 0 for all p. Consequently, efficient management delays development for new technologies in A1 and A31 and may cause earlier development for other technology levels. Next we compare the expected total development under the two management I (p) be the expected practices. Letting T total development under the traditional I (0) = T I (0). We can approach, we know T I (p) is differentiable, show that when T (7)
I (p) d T I (p) dT − dp dp
I1 d I2T d d I2M + (1 − p) = 1+p d I1 d I1 dp + ( I2M − I2T ) − (I2M − I2T )
If |d 2 I2 (I1 A)/dI1 dA| is small (i.e., the term in the square bracket of (7) is close to zero), I(p) I(p) dT < dTdp when I1 (p) < 0. That is, dp under certain conditions, a higher probability of the new technology reduces the total development if it delays the development. Then the efficient management reduces the expected development when AM is in A1 and A31 . But the result is ambiguous for other technology levels and again we resort to our numerical example to show the possibilities. Figure 5 compares the expected total development under the two management
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11
Expected Total Development
10
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9
0.843
8 0.85 7 1.15 6
5
4 0
0.1
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0.4 0.5 0.6 0.7 Probability of New Technology
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1
I and T I Figure 5. Expected total development under the two evaluation methods: T approaches for the four technology levels. The solid curves stand for total development I (p)) and under the efficient approach (T the dashed curves are for traditional management (T˜ I (p)). Total development T˜ I (p) associated with AM = 085 is extremely close to that associated with AM = 0843, and they are represented by one dashed curve in the figure. We see that the efficient management reduces the expected total development for the modest and radical technologies (0.70 and 1.15) but raises the total development for the two moderate technologies (0.843 and 0.85). Thus, Remark 3. Compared with the traditional approach to resource management, the efficient approach may not delay or reduce resource development, depending on the level of the new technology. We can interpret the probability p as a “degree of mistake” of the traditional approach: If p = 0, there is no mistake in ignoring the new technology, and if p = 1, there is a big mistake in ignoring the technology. We can interpret the difference I and T I as between total development T the “degree of mismanagement” (of the resource) of the traditional approach. There is no mismanagement of the resource (in
I = T I . Then the T I and aggregate) if T M T I curves associated with A = 0843 and AM = 115 reveal there may not be a monotonic relationship between the degree of mistake and the degree of mismanagement. In essence, fixed costs drive the results in Remark 3. As we illustrated in the deterministic model, the major characteristic of the moderate technologies in A21 is that under the traditional management, when the new technology shock occurs, the fixed cost prevents further development, resulting in less overall development. The impact of this shock increases as p rises, resulting in more overall conservation under the traditional management. For moderate technologies in A22 , in the deterministic model, more resource is developed in period one to save the fixed cost c0+ in period two. This incentive directly translates into I1 (p) > 0 in the stochastic model: as p increases, the incentive to save fixed costs becomes stronger, leading to more development in period one. Further, overall development also increases since the expected future reduction in development, (1 − p) I2T , cannot fully compensate I2M and I2M are the increase in I1 (note that both zero).
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Conclusions This article develops a framework to analyze the impacts on resource development of efficient management practices that account for possible resource conserving technological changes in the future. The impact depends on both the probability and the level of the changes. For many changes, the conventional wisdom is correct: the new management approach, by fully accounting for these possible changes, can conserve the resource by both reducing and delaying resource development. The extensive restoration efforts going on worldwide highlight the overdevelopment caused, in part, by ignoring the new technologies in resource planning. However, when there are significant fixed costs in resource development, the new (and efficient) management approach may not delay or reduce development for some moderate levels of technologies. Further, when this happens, under efficient management, a higher probability of these resource conserving changes may not conserve more resources: it may lead to more and early development. Surprisingly, these ambiguities arise without any policy or market distortions. They are due entirely to the cost structure of resource transformation, in particular the existence of fixed and increasing marginal costs. Under the efficient management, the incentive to save future fixed costs may prompt more resource development in early periods, and the expected future reduction in development may not be enough to compensate the increased earlier development. Under the traditional management, when a new technology shock occurs, fixed cost may also prevent further development. This conclusion highlights the importance of the fixed costs in stochastic dynamic investment analysis in general. Our results suggest that policy makers and the public in general should be open-minded about the possible development/restoration patterns brought forth by the new management practice. Sometimes efficient utilization of management practices that are perceived to be resource conserving may lead to increased development. Our results also call for more detailed scrutiny of the cost structure of a project, especially the differentiation between fixed and variable costs. Such differentiation is not required in some evaluation handbooks, such as the Principles and Guidelines (U.S. Water Resources Council).
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[Received January 2000; accepted October 2000.] References Abel, A.B., and J.C. Eberly. “A Unified Model of Investment under Uncertainty.” Amer. Econ. Rev. 84(December 1994):1369–84. Abler, D.G., and J.S. Shortle. “Technology as an Agricultural Pollution Control Policy.” Amer. J. Agr. Econ. 77(February 1995):20–32. Antle, J.M., and T. McGuckin. “Technological Innovation, Agricultural productivity, and Environmental Quality.” Agricultural and Environmental Resource Economics. G. Carlson, D. Zilberman, and J. Miranowski, eds., pp. 175–220. New York/Oxford: Oxford University Press, 1993. Arrow, K.J., and A.C. Fisher. “Environmental Preservation, Uncertainity, and Irreversibility.” Quart. J. Econ. 88(May 1974): 312–19. Caswell, M. “Irrigation Technology Adoption Decisions: Empirical Evidence.” The Economics and Management of Water and Drainage in Agriculture. A. Dinar and D. Zilberman, eds., pp. 295–312. Norwell MA: Kluwer Academic, 1991. Clark, C.W. Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd ed. New York: John Wiley and Sons, 1990. Clawson, M. “Forests in the Long-Sweep of American History.” Science 204(1979): 1168–74. Coggins, J.S., and C.A. Ramezani. “An ArbitrageFree Approach to Quasi-Option Value.” J. Environ. Econ. and Manage. 35(March 1998):103–25. Dixit, A.K., and R.S. Pindyck. Investment Under Uncertainity. Princeton NJ: Princeton University Press, 1994. Fernandez, L., and L. Karp. “Restoring Wetlands Through Wetlands Mitigation Banks.” Envoiron. Res. Econ. 12(October 1998):323–44. Fisher, A.C., J.V. Krutilla, and C.J. Cicchetti. “The Economics of Environmental Preservation: A Theoretical and Empericial Analysis.” Amer. Econ. Rev. 62(September 1972): 605–19. Henry, C. “Investment Decisions under Uncertainty: The Irreversibility Effect.” Amer. Econ. Rev. 64(December 1974):1006–12. Khanna, M., and D. Zilberman. “Incenties, Precision Technology and Environmental Protection.” Ecological Econ. 23(October 1997): 25–43.
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. “Barriers to Energy-Efficiency in Electricity Generation in India.” Energy J. 20(1999): 25–41. Kolstad, C.D. “Fundamental Irreversibilities in Stock Externalities.” J. Public Econ. 60(May 1996):221–23. Leffler, K.B., and R.R. Rucker, “Transactions Costs and the Efficient Organization of Production: A Study of Timber-Harvesting Contracts.” J. Polit. Econ. 99(October 1991): 1060–87. Miller, J.R., and F. Lad. “Flexibility, Learning, and Irreversibility in Environmental Decisions: A Bayesian Approach.” J. Environ. Econ. and Manage. 11(June 1984):161–72. Olmstead, J.M. “Emergying Markets in Water: Investments in Institutional and Technological Changes.” PhD dissertation, University of California at Berkeley, 1998. Phillips, C.V., and R.J. Zeckhauser. “Restoring Natural Resources with Destimation-Driven Costs.” J. Environ. Econ. and Manage. 36(November 1998):225–42. Reisner, M.P. Cadillac Desert: The American West and Its Disappearing Water. New York: Viking, 1986. Rucker, R.R., and K.B. Leffler. “To Harvest or Not to Harvest? An Analysis of Cutting Behavior on Federal Timber Sales Contracts.” Rev. Econ. and Statist. 70(May 1988): 207–13. Shah, F.A., D. Zilberman, and U. Chakravorty. “Technology Adoption in the Presence of an Exhaustible Resource: The Case of Groundwater Extraction.” Amer. J. Agr. Econ. 77(May 1995):291–99. Tweeten, L. Foundations of Farm Policy. Lincoln: University of Nebraska Press, 1974. United States Water Resources Council. “Economic and Environmental Principles and Guidelines for Water and Related Land Resources Implementation Studies.” Technical report, 1983. Waggoner, P. “How Much Land Can Ten Billion People Spare for Nature?” Task Force Rep. No. 121, Council for Agricultural Science and Technology, 1994. Zhao, J., and Zilberman. “Irreversibility and Restoration in Natural Resource Development.” Oxford Econ. Pap. 51(July 1999): 559–73.
Appendix Model Details New technologies as conservation technologies. Let f(K A) be the production function of the
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resource utilizing sector (e.g., the agriculture sector), and let U(·) be the net social benefit function of this sector’s output, with fK > 0, fKK < 0, fKA > 0, U > 0, and U < 0. Then π(K A) = U(f(K A)). Note that U (·) is the society’s (net) demand function for the agricultural output. Further, (8)
πKA = U fKA + U fA fK
Define εU = −
U U f
which represents the elasticity of the (net) demand function, f
εK =
f K df K = K dK f f
which represents the output elasticity of the resource input, and fA
εK =
dfA K f K = KA dK fA fA
which is the elasticity of the marginal output of technical change fA with respect to input K. We f fA can show that πKA = U ffKA ε/ εK − εU . Since U < 0 and fKA > 0, we know f εK (9) sign(πKA ) = sign εU − fA εK Thus πKA < 0 if the demand elasticity for the agricultural output εU is low, if land enjoys high output elasticity in agricultural production (i.e., f εK is high), and if the marginal productivity of the new technology is not too sensitive to the fA resource input (i.e., εK is low). In the numerical example in the main body of the article, we consider a Hicksian neutral technology with the production function given by g(K L A) = AK α L1−α , where L is the labor input. Then we can express (in a reduced form) the output as a function of K and A only: f(K A) = (constant)A1/α K. We can verify that in this case f fA εK /εK = 1, and the technology is resource conserving if and only if εU < 1. Even in a partial equilibrium framework where the agricultural output price is constant, a new technology can reduce the (ceteris paribus) demand for K if it is K-enhancing. From (8), we know even if U = 0, a sufficient condition or πKA < 0 is fKA < 0. Consider a production function in the form of f˜(KA), with f˜ > 0 and f˜ < 0, so that the new technology A directly enhances the effectiveness of the resource input. Examples of this kind of technologies include drip irrigation, IPM, and other precision technologies. Define the output elasf˜ ticity of the “effective input” KA as εKA = −f˜ KA/f˜ . Then we know
f˜ (10) f˜KA = f˜ + f˜ KA = f˜ 1 − εKA
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Thus the new technology reduces the resource demand if the effective resource input has a high f˜ output elasticity: εkA > 1.
and A0R . Given that AJ is continuous in c0+ , we know there exists a critical c0+ , denoted as c0+R , so that A0R ≥ AJ when c0+ ≥ c0+R .
Total development changes at AJ . We show that the total development I1∗ +I2∗ jumps down at AJ in the deterministic model. We consider two development strategies. The first one involves developing in both periods, with levels denoted as I1d (A) and I2d (A). The second strategy develops only in period one, at a level represented by I1o (A). For A < AJ , I1∗ (A) = I1d (A) and I2∗ (A) = I2d (A). For A ≥ AJ , I1∗ (A) = I1o (A). Next, we compare the first order conditions of I1d (AJ ) and I1o (AJ ). Let h(K0 + I A) = π(K0 + I A) + V (S − K0 − I ). We know hKK < 0. The first order condition for I1d (AJ ) is (11)
hK (K0 + I1d AJ ) + βhK (K0 + I1d + I2d AJ ) − c (I1d ) = 0
and that for I1o (AJ ) is (12)
hK (K0 + I1o AJ ) +βhK (K0 + I1o AJ ) − c (I1o ) = 0
Suppose I1o ≥ I1d + I2d ; then hKK < 0 and c > 0 imply that (11) and (12) conflict with each other. Thus we know I1d (AJ ) + I2d (AJ ) > I1o (AJ ). That is, total development jumps down at AJ . Nonemptiness of A2 . Formally, we define
: I 0 (A) ≥ I ∗ (AT )}, AR ≡ sup{A ∈ [AT A] 1 1 the technology level at which the strategy of developing in period one only, accounting for the new technology, would result in the same level of I1 as the traditional approach. Since at A = AT , I2∗ (I˜1 AT ) > 0, we know I10 (AT ) > I1 . Since I10 (A) decreases in A, we know AR > AT . That is, AR always exists. To show that A2 is nonempty, we need to show AR > AJ under certain conditions. Proposition 2. Given other parameters of the model, there exists c0+R > 0, such that AR > AJ if c0+ > c0+R . Proof. Consider the development strategy of developing in period one only, and let I1o (A) be the corresponding development level. Define A0R to be the technology level such that I10 (A0R ) = I1∗ (AT ). Thus, A0R = AR . Also the first order conditions for I1o (A0R ) and I1∗ (AT ) imply that A0R > AT since I2∗ (AT ) > 0. Now we only need to prove that AJ < A0R when c0+ is sufficiently high. Note that A0R is independent of c0+ . Let c¯0+ be the fixed cost level so that I2∗ jumps down to zero at A = AT . If c0+ = c¯0+ − δ1 where δ1 is an arbitrarily small number, then I2∗ jumps down at AJ (δ1 ) = AT + δ2 , and δ2 can be made arbitrarily small by reducing δ1 . Thus we can set δ1 to make sure that AJ (δ1 ) falls between AT
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For A ∈ [AJ AR ], I1∗ (A) = I10 (A). Since I10 (AR ) = I1 and I10 is decreasing in A, we know I1∗ (A) > I1 for A ∈ [AJ AR ]. J < Next, we show that whenever AR exists, A J AR ) is nonempty. AR . This means that A2 ≡ [A Further, there exist levels of the fixed cost so J < AJ . that A Proposition 3. Suppose c0+ > c0+R so that AR > J < AR . Further, there exists > 0 AJ . Then A J ≤ AJ . such that if c0+ ≤ c0+R + , A R Proof. Since at A = A , I1∗ = I1 and I2∗ = 0, the first order conditions for I1∗ and I1 imply J ≤ AR . To prove that the inequality that A J = AR . Again, let is strict, suppose A d d d (I1 (A) I1 (I1 (A) A)) be the strategy of developing in both periods (regardless of fixed costs). J indicates that the strateThe definition of A J )) and (I1 0) generate the gies (I1 I1∗ (I1 A J indisame payoff. The definition of AR = A cates that the strategy (I1 0) (at least weakly) J ) I d (I d (A˜ J ) AJ )). dominates the strategy (I1d (A 2 1 d d Since (I1 I2 ) are the optimal development levels when the strategy is to develop in both periJ )) since the ods, it strictly dominates (I1 I1∗ (I1 A latter is part of the develop-in-both-periods strategy. Thus we get the contradictory result that J < 2pt (I1 0) strictly dominates (I1 0). Thus A R A . Now we prove the second part of the propoJ < AR = AJ , sition. Since when c0+ = c0+R , A J and AJ in c + indicates that the continuity of A 0 when c0+ > c0+R but is sufficiently close to c0+R , J < AJ . Thus there exists > 0 AR > AJ and A J ≤ AJ when c + ≤ c +R + . such that A 0 0 Proof of Proposition 1. Note that without fixed costs, the first order conditions of (2) indiI2M ) and dB2T /dI1 = cate that dB2M /dI1 = c ( T c (I2 ). Since πKA < 0 and AM > AT , we know I2M for any I1 . The convexity of c(·) togeI2T > ther with (4) then implies d I1 /dp < 0. That is, higher probability of the new technology reduces first period development. Since both AT and AM are fixed, we know immediately that d I2M /dp > 0 and d I2T /dp > 0. Note that I(p)/dp dT d IM d I T d I1 = 1 + p 2 + (1 − p) 2 d I1 d I1 dp + I2M − I2T
The term in bracket is positive because d I2M / I(p)/dp < 0 I2T /d I1 < 1. Thus d T d I1 < 1 and d I2T and d I1 /dp < 0. That is, since I2M < higher probability of new technology reduces the expected total development.
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Given c0+ = 0, I2T is always positive since if T M I1 can reduce I2 ≤ 0, then I2 < 0, and reducing the overall two period cost. However, we cannot rule out the possibility that I2M is negative when the new technology is very advanced (i.e., when AM is high). Transformation patterns with fixed costs under uncertainty. We show how the transformation patterns depend on the probability and level of the new technology when there are fixed costs of transformation. Similar to the no-fixed cost case, we know I2T cannot be negative. But it can be zero due to the fixed development cost. I2M can take positive, zero, or negative values, with I2M ≤ T I2 . Then there are five possible resource transformation schemes12 in period two: (S1) develop with both traditional and modern technologies I2M > 0; (S2) develop with traditional I2T > 0, and stay-put with modern technologies I2T > 0, I2M = 0; (S3) develop with traditional and restore I2M < 0; (S4) with modern technologies I2T > 0, stay put with traditional and modern technoloI2M = 0; and (S5) stay put with gies I2T = 0, traditional and restore with modern technologies I2M < 0. Which scheme actually arises I2T = 0, depends on the fixed costs and the level of I1 , which in turn depends on the probability and level of the new technology. As p changes from 0 to 1, there may be a (finite) number of switches of development schemes. Remark 4. If higher I1 causes a change in the transformation scheme, it either reduces development or causes restoration in period two. Thus, there are two possible general trends of scheme switch if I1 rises: (S1)–(S2)–(S3)–(S5) and (S1)– (S2)–(S4)–(S5). I1 (p) jumps at a switch point but otherwise is continuous. Equation (4) is thus defined for the range of p exclusive of the jump points. As we will show later, I1 (p) is not monotonic in p. Thus Remark 4 does not lend a lot of help in characterizing the scheme switches as p changes. However, we can rule out cycles of schemes as p changes: the same scheme can appear at most once on the interval p ∈ [0 1]. Lemma 1. As p rises from 0 to 1, if a certain transformation scheme is switched out, it will not be switched back. Proof. Imagine the determination of I1 by the crossing point of the marginal cost (c (I1 )) and marginal benefit functions of I1 . For a fixed resource transformation scheme, the marginal benefit function is continuous and monotonic in p. So is the level of I1 since the marginal cost function is fixed (i.e., independent of p). 12 We use “scheme” to represent second period transformation decisions to differentiate from the overall transformation “pattern.”
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Suppose scheme (Si), i ∈ {1 2 3 4 5}, is switched out at p1 but is switched back at p2 > p1 . At p1 and p2 , the scheme is the same but I1 (p2 ). the probability is different. Thus I1 (p1 ) = I1 (p1 ) to I1 (p2 ), the Then as I1 changes from same scheme (Si) would appear twice with a different scheme in between, violating the two possible scheme switches in Remark 4. Lemma 1 suggests a simple starting point of checking the possible schemes as p changes: we can first check the schemes at the two endpoints, p = 0 and p = 1. Proposition 4. If the transformation schemes at p = 0 and p = 1 are different, there must be a finite number of switch points along p ∈ [0 1]. There is no switching if the schemes at p = 0 and p = 1 are the same. For a fixed transformation scheme, both I1 (p) I(p) are and the expected total development T continuous and monotone in p. In particular, I(p) are decrProposition 5. Both I1 (p) and T easing in p when the transformation scheme is (S1), (S3), (S4), or (S5). They may be increasing or decreasing in p when the scheme is (S2), and they are more likely to be increasing if AM is low. Proof. For scheme (S1), the fixed cost of transformation does not matter because the development is positive. From Remark 1, we I(p) are strictly decreasing know I1 (p) and T in p. Under schemes (S3) and (S5), dB2M /dI1 < 0 since I2M < 0. Since dB2T /dI1 is always positive, (4) then indicates that I1 (p) decreases in I(p) also p. It is straightforward to show that T decreases in p. Under scheme (S4), πKA < 0 and (4) indicate I1 (p) < 0. Again, it is straightforward to show I (p) < 0. that T Under scheme (S2), substituting in I2M = 0, we get (13)
dB2M = πK (K0 + I1 AM ) dI1 −V (S − K1 − I1 )
Since I2T > 0, the envelope theorem gives (14)
dB2T = πK (K0 + I1 + I2T AT ) dI1 −V (S − K1 − I1 − I2T )
The relative magnitude of dB2M /dI1 and dB2T /dI1 is thus ambiguous, depending on the difference I2T . between AM and AT , and the magnitude of If AM is low, it is more likely that dB2M /dI1 is high, or I1 (p) > 0. I(p) are continuous in While both I1 (p) and T p for a fixed transformation scheme, they jump (either up or down) at a switch point. Since in
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some cases we do not know the specific scheme associated with a certain AM and p, we do not know the corresponding properties of I1 (p) and I(p). To understand the impacts of a new techT nology AM , it is important to identify both the schemes associated with probability p and the nature of the switches as p changes. Now we try to identify the possible transformation schemes for each level of AM . If AM ∈ A1 , both I1 and I2 are positive for p = 0 and p = 1. Propositions 4 and 5 then indicate that
the scheme is (S1) for all p and both I1 and I are decreasing in p. For AM ∈ A31 , we T I2M = 0 know from figure 2 that I2T > 0 and M at p = 0 and I2 = 0 at p = 1. However, I2T > 0 at since I1∗ (AM ) < I1∗ (AT ), we know p = 1. Thus, the scheme is (S2) for all p and I1 I are decreasing in p. For other levels of and T technologies, the transformation patterns become more complicated and there might be several switches.
